GAS RESERVOIR ENGINEERING
John Lee
Robert A. Wattenbarger
SPE TEXTBOOK SERIES VOL. 5
Supported by the AIME James Douglas Library Fund.
SPE thanks the AIME James Douglas Library Fund for covering the cost
of converting this book to a digital format.
Gas Reservoir Engineering
John Lee
Peterson Chair and
Professor of Petroleum Engineering
Texas A&M U.
Robert A. Wattenbargar
Professor of Petroleum Engineering
Texas A&M U.
SPE Textbook Series, Volume 5
Henry L. Doherty Memorial Fund of AIME
Society of Petroleum Engineers
Richardson, TX USA
Dedication
John Lee
To the most important women in my life: Mom Phyliss, nurse Anne, minister-in-training Denise,
and renewable energy sources, Katie and Courtney.
Robert A. Wattenbargar
To my loving wife, Julie, our three sons, Mike, Chick, and Phil, and our grand twins, John and Laura.
Disclaimer
This book was prepared by members of the Society of Petroleum Engineers and their well-qualified colleagues from
material published in the recognized technical literature and from their own individual experience and expertise.
While the material presented is believed to be based on sound technical knowledge, neither the Society of Petroleum
Engineers nor any of the authors or editors herein provide a warranty either expressed or implied in its application.
Correspondingly, the discussion of materials, methods, or techniques that may be covered by letters patents implies
no freedom to use such materials, methods, or techniques without permission through appropriate licensing.
Nothing described within this book should be construed to lessen the need to apply sound engineering judgment
nor to carefully apply accepted engineering practices in the design, implementation, or application of the techniques
described herein.
© Copyright 1996 Society of Petroleum Engineers
All rights reserved. No portion of this book may be reproduced in any form or by any means, including electronic
storage and retrieval systems, except by explicit, prior written permission of the publisher except for brief passages
excerpted for review and critical purposes.
Manufactured in the United States of America.
ISBN 978-1-55563-073-7
ISBN 978-1-61399-163-3 (Digital)
Society of Petroleum Engineers
222 Palisades Creek Drive
Richardson, TX 75080-2040 USA
http://store.spe.org/
service@spe.org
1.972.952.9393
John Lee is the Peterson Chair and professor of petroleum engineering at Texas A&M U. in College
Station and executive vice president of technology at SA Holditch & Assocs. After receiving a PhD
degree from Georgia Inst. of Technology in 1963 , he worked as a senior research specialist with
Exxon Production Research Co. until 1968. He was associate professor of petroleum engineering at
Mississippi State U. from 1968 to 197 1 and technical advisor with Exxon Co. U. S.A. from 1971 to
1977 . Lee has been with Texas A&M since 1977. He received the SPE John Franklin Carll Award in
1995 and the SPE Reservoir Engineering Award in 1986. He also has been faculty advisor to the SPE
student chapter during several school years.
Robert A. Wattenbarger has been a professor of petroleum engineering at Texas A&M U. since
1983. Previously, he worked for Mobil, Mobil Research, and Sinclair Oil companies from 19 5 8 to
19 69. From 1969 to 1979 , he was vice president and director of Scientific Software-Intercomp Inc.
Since 1979 , he has consulted through Wattenbarger and Assocs. He holds BS and MS degrees from
the U. of Tulsa and a PhD degree from Stanford U. , all in petroleum engineering.
SPE Textbook Series
T he Textbook Series of the Society of Petroleum Engineers was established in 1972 by action of the
SPE Board of Directors. T he Series is intended to ensure availability of high-quality textbooks for use
in undergraduate courses in areas clearly identified as being within the petroleum engineering field.
T he w ork is directed by the Society's Books Committee, one of more than 40 Society-wide standing
committees. Members of the Books committee provide technical evaluation of the book. Below is a
listing of those who have been most closely involved in the final preparation of this book.
Book Editors
Fred Poettmann, Colorado School of Mines, Golden, CO·
Jerry Jargon, Marathon Oil Co. , Littleton, CO
Roland Horne, Stanford U. , Stanford, CA
"Deceased
Books Committee
(1996)
Dan Hill (chairman) , U. of Texas, Austin, TX
Waldo Borel, Pennzoil E&P Co. , Houston, TX
Anil Chopra, Arco E&P Technology, Plano, TX
Garry Gregory, Neotechnology Consultants Ltd. , Calgary, Alta.
T homas Hewitt, Stanford U. , Stanford, CA
John Killough, U. of Houston, Houston, TX
Susan Peterson, Halliburton Energy Svc. , Houston, TX
Rajagopal Raghavan, Phillips Petroleum Co. , Bartlesville, OK
Arlie Skov, Arlie M. Skov Inc. , Santa Barbara, CA
Allan Spivak, Intera West, Los Angeles, CA
Hans Juvkam Wold, Texas A&M U. , College Station, TX
Introduction
Natural gas production has become increasingly important in the U. S. , and the w ellhead revenue
generated from it is now greater than the wellhead revenue generated from oil production. Because
this trend eventually will be followed worldwide, we feel that it is important to emphasize gas reservoir
engineering courses at the undergraduate level and to have a textbook devoted to this purpose. T his
book also serves as an introduction to gas reservoir engineering for graduate students and practicing
petroleum engineers.
Although much of the technology for oil wells applies to gas wells, there are still many differences. It
is important to learn these differences and to have a good, fundamental background in how to
recognize and handle them. We have tried to provide practical equations and methods while
emphasizing the fundamentals on which they are based. We have not attempted to be complete in the
sense of presenting the best-known solution(s) to all problems in this area of technology. In many
cases, we didn't even present the problem, much less a solution. Instead, we concentrated on
fundamentals and hope to have made the literature in gas reservoir engineering more accessible both
now and in the future. If you don't find your favorite topic in the table of contents or in the index, it
simply didn't make our short list of fundamentals that we believed to be key parts of the literature.
We wrote this book at a time of great change in the computational methods used by petroleum
engineers. Most calculations arising frequently are done with computers and either commercial
software packages or spreadsheets written by the engineer or an associate. While clearly in the
interest of enhanced productivity, this modern trend also promotes a "black-box" approach to
engineering. We hope to have made the box a little less opaque by discussing fundamentals,
emphasizing assumptions and limitations in methods, and illustrating our recommended methods
with completely worked examples. Still, we have contributed to the computational trend on several
occasions by presenting and recommending computational techniques that would require
unreasonably complicated arithmetic if done by hand. Our intent, of course, is that these complicated
methods be implemented in a spreadsheet or other computer program. We believe that this approach
is better than providing only simple (and therefore more approximate) techniques that can be
implemented easily with a hand-held calculator.
Commercial petroleum software is changing so rapidly and, in many cases, is so specific to the
individual vender, that we cannot possibly illustrate use of the leading or most popular software for a
given application. Accordingly, we have tried to present computational methods that are generic and
that can be found in a similar form in virtually any commercial package that existed at the time of this
writing.
Acknowledgments
T his book would not exist without our students-a cliche, perhaps, but literally true in this case. Many
of the early drafts of chapters were written by students, often in preparation for lectures we gave on
gas reservoir engineering to practicing engineers in the U.S. and abroad. In many cases, their
contributions survived even the critical eye of our superb staff editor at SPE, Valerie Dawe. We would
also like to acknowledge the valuable assistance of a number of people who have contributed to this
book with word processing, proofreading, checking of technical content, and valuable suggestions.
For Chaps. 2 through 4 and 11 , we thank Bryan Maggard, James Keating, Mauricio Villegas, Liyan
Zhao, and Raj Dhir. For Chaps. 1 and 5 through 10 , special mention is due Jennifer Johnston, now
a physician-in-training, and engineers Jay Rushing and Tom Blasingame. Ede Hilton, a talented
and dedicated administrative assistant, was also a very important member of our team. To each­
thank you!
Contents
1. Properties of Natural Gases . . . .. ... ... ... .. . ... ..... .... .. .... . . ...... .... . . . . .. ... . . . .. .. .. .. . 1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.1 1
1 12
1.1 3
1.1 4
1 15
1.1 6
.
.
Introduction . ............................................................................. 1
Review of Definitions and Fundamental Principles ............................................. 1
Properties of Natural Gases ................................................................ 2
Calculation of Pseudocritical Gas Properties .................................................. 3
Dranchuk and Abou-Kassem 16 Correlation for z Factor ........................................ 1 6
Gas FVF
16
Gas Density ............................................................................. 1 7
Gas Compressibility....................................................................... 17
Gas Viscosity ............................................................................ 18
Properties of Reservoir Oils . ............................................................... 1 8
Properties of Reservoir Waters ............................................................. 23
Water Vapor Content of Gas ............................................................... 28
Gas Hydrates ............................................................................ 29
PV Compressibility Correlations ............................................................ 3 1
Gas Turbulence Factor and Non-Darcy Flow Coefficient ....................................... 3 2
Summary ................................................................................ 3 2
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2. Fundamentals of Gas Flow in Conduits.......................... .... .. ............ . . ... . . ... . .. 37
2.1
2. 2
2.3
2.4
2.5
2.6
Introduction ...............................................................................
Systems, Heat, Work, and Energy ...........................................................
First Law of T hermodynamics ...............................................................
Mechanical Energy Balance. .............................................................. .
Energy Loss Resulting From Friction .........................................................
Bernoulli's Equation .......................................................................
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37
37
39
40
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41
3. Gas Flow Measurement ....................................................................... 43
3.1
3.2
3.3
3.4
3.5
3.6
Introduction ...............................................................................
Orifice Meters .............................................................................
Orifice Meter Installation ....................................................................
Critical Flow Prover ........................................................................
Choke Nipples ............................................................................
Pitot Tube ................................................................................
4. Gas Flow in Wellbores
4.1
4.2
4. 3
4.4
4.5
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43
43
47
53
53
54
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Introduction ...............................................................................
BHP Calculations for Dry Gas Wells ........................................................ .
Effect of Liquids on BHFP Calculations .......................................................
Evaluating Gas-Well Production Performance .................................................
Forecasting Gas-Well Performance ..........................................................
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58
58
66
73
76
5. Fundamentals of Fluid Flow i n Porous Media .......... ... . ... ..... . .............. . ...... . . . .. .. 81
5.1 Introduction ............................................................................... 81
5. 2 Ideal-Reservoir Model ... ................................................................... 81
5.3 Solutions to the Diffusivity Equation .......................................................... 9 1
5.4 Radius of Investigation ..................................................................... 99
5.5 Principle of Superposition .................................................................. 1 0 1
5.6 Horner's Approximation..
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10 3
5.7 van Everdingen-Hurst Solutions to the Diffusivity Equation ..................................... 10 3
5.8 Summary ............................................................................... 1 0 6
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6 . Pressure-TransientTesting o fGas Wells .................. .... .. ... ......... . ... ... .. . . . ... . 111
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6.1
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Introduction ............................................................................. 111
Types and Purposes of Pressure-Transient Tests ............................................
Homogeneous Reservoir Model-Slightly Compressible Liquids ...............................
Complications in Actual Tests .............................................................
Fundamentals of Pressure-Transient Testing in Gas Wells .................. .................
Non-Darcy Flow ...................................................... ..................
Analysis of Gas-Well Flow Tests .................................................. ........
Analysis of Gas-Well Buildup Tests ....................... ................................
Type-Curve Analysis .......... ..........................................................
Hydraulically Fractured Gas Wells ............... .. . ........................... ........
Naturally Fractured Reservoirs ............................................................
Reservoir Model Identification by Use of Characteristic Pressure Behavior ......................
Summary ......................................................................... .. ... .
6. 2
6. 3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.1 2
6.1 3
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7. Deliverability Testing of Gas Wells
7.1
7. 2
7. 3
7.4
7.5
7.6
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168
168
1 68
1 68
171
17 2
189
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193
Introduction .......................................................... ...................
Well-Test Types and Purposes .............................................................
General Test Design Considerations .......................................................
Design of Pressure-Transient Tests ....... .. ................. .......................... .
Deliverability Test Design ..................................................................
Summary ................................................................................
19 3
19 3
194
19 6
207
210
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9. Decline-Curve Analysis for Gas Wells
9.1
9. 2
9. 3
9.4
9.5
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Introduction .............................................. ...............................
Types and Purposes of Deliverability Tests ...................................................
T heory of Deliverability Test Analysis ........................................................
Stabilization Time.........................................................................
Analysis of Deliverability Tests................................ ........................ ....
Summary ................................... ............................................
8. Design and Implementation of Gas-Well Tests
8.1
8. 2
8. 3
8.4
8.5
8.6
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111
111
11 2
1 15
1 16
117
1 25
1 31
1 39
1 50
159
1 60
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214
Introduction ..............................................................................
Introduction to Decline-Curve Analysis......... .............................................
Conventional Analysis Techniques . ........................................................
Decline Type Curves ......................................................................
Summary ................................................................................
21 4
214
214
219
225
1 0. Gas Volumes and M aterial-Balance Calculations
10.1 Introduction .................... ...................................................... .
10. 2 Volumetric Methods......................................................................
10. 3 Material-Balance Methods ............ ...................................................
10.4 Summary ...............................................................................
230
230
230
23 4
25 1
11. Reservoir Simulation
.
256
Introduction............................................................................
Finite-Difference Approach for the Diffusivity Equation (1 D) ..................................
Solution Accuracy ................ .....................................................
Gridblock Approach to Finite-Difference Equations ..........................................
A Simulator for Real-Gas Flow, x-y Coordinates ....... ....................................
Solution of Equations ...................................................................
A Single Well Simulator for Real-Gas Flow, ,-z Coordinates ...................... ..... .....
GASSIM...............................................................................
History Matching .............. ........................................................
Forecasting Performance .................................... ......................... ..
25 6
25 6
259
260
262
264
265
266
267
271
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11.1
11. 2
11. 3
11.4
11.5
11.6
1 1.7
11.8
11.9
11.10
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Appendix A Dranchuk and Abou-Kassem Equation of State for Calculating Gas z Factor
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274
Appendix B Integral Values for the Poettmann M ethod forDetermining Static BHP
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275
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Appendix C Shape Factors for Various Single-WellDrainage Areas
Appendix D Values of the Exponential Integral, -Ei(-x)
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278
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280
Appendix E van Everdingen-Hurst Solutions toDiffusivity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Appendix FDetermining PressureDerivatives
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294
Appendix G Well-Test Analysis and Reservoir IdentificationWorksheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Appendix HDeliverability Test Analysis With Pressure-Squared Techniques
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312
Appendix I Worksheets for Well TestDesign
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Appendix J Correlations for Estimating Residual Gas Saturations inGas
Reservoirs With Water Influx
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321
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326
Appendix K GASSIM Computer Program for 2D Gas Reservoir Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Author Index
SubjectIndex
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338
Chapter 1
Properties of Natural Gases
1.1 I nt roduction
This chapter presents methods for estimating reservoir fluid prop­
erties required for gas-reservoir-engineering calculations. Labora­
tory analysis is the most accurate way to determine the physical
and chemical properties of a particular fluid sample; however, in
the absence of laboratory data, correlations are viable alternatives
for estimating many of the properties . We present correlations for
estimating properties of not only natural gases but also liquid
hydrocarbons and formation waters . The correlations were chosen
for accuracy , consistency , and simplicity for manual analysis or
computer programming. Also included are correlations for estimat­
ing pore volume (PV) compressibility and the non-Darcy flow
coefficient for turbulent flow, which is common in gas wells .
1 . 2 Review o f Definitions and
Fundamental Principles
Before discussing the fluid-property calculations and correlations,
we review some definitions and fundamental principles required
to understand fluid properties and their computation with correla­
tions . This review includes the concepts of mole fraction, molar
I . The volume of the gas molecules is insignificant compared
with the total volume enclosing the gas .
2 . No attractive or repulsive forces exist among the molecules
or between the molecules and the container walls .
3 . All molecule collisions are perfectly elastic ; i . e . , there i s no
loss of internal energy upon collision .
An equation describing the relationship between the volume oc­
cupied by a gas and the pressure and temperature is called an equa­
tion of state (EOS) . 1 The form of the ideal-gas EOS was developed
from the empirical observations that, for a given mass of gas at
a constant temperature, the pressure-volume product, p V, is con­
stant (Boyle' s law) and , for a given mass of gas at a constant pres­
sure, the volumeltemperature ratio, VIT, is constant (Charles' law) .
Combining Boyle' s and Charles ' laws , we obtain the EOS for an
ideal gas :
p V=nRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 .2)
pletely and provides an excellent discussion of phase-behavior
characteristics of hydrocarbon gases and liquids.
where p =pressure , psia; V=volume, ft 3 ; n =number of pound­
moles of gas ; R=universal gas coefficient = 1 0 . 732 psia-ft 3 / ° R­
Ibm-mol; and T=absolute temperature, O R . Note that the units and
magnitude of the universal gas coefficient vary depending on the
units of the other variables in Eq. 1 .2 . Values of R in various units
are readily available. ! Eq. 1 .2 also can be developed directly from
kinetic theory . I
1 . 2 . 1 Moles a n d Mole Fraction. A pound-mole (Ibm-mol) is a
quantity of matter with a mass in pounds equal to the molecular
weight. Similar definitions apply to gram-mole, kilogram-mole, etc.
For example, 1 Ibm-mol of methane weighs 1 6 . 043 Ibm. The mole
fraction of a component in a mixture is the number of pound-moles
of that component divided by the total number of moles of all com­
ponents in that mixture . For a system with n components , the mole
fraction is
1 .2.3 Molar Volume. The concept of molar volume , Vm , is used
to convert a given mass of gas to its vapor volume at standard pres­
sure and temperature conditions . This concept implies that, for a
given set of standard conditions , the molar volume is constant and
can be used to convert mass to volume or, as some derivations re­
quire, to convert volume at standard conditions to mass.
Combining the definition of molar volume , Vm = Vln , and the
ideal-gas law given by Eq. 1 . 2 , we obtain
volume , ideal- and real-gas behavior , and the principle of corre­
sponding states. McCain 1 discusses these fundamentals more com­
Yi =n i I
nc
E
j= l
nj , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 1 )
where Yi =mole fraction of the ith component, n i =number of
pound-moles of ith component, and nc number of components in
the system.
1 .2.2 Ideal-Gas Law. To begin our discussion of the behavior of
real gases, we consider a hypothetical gas called an ideal gas . The
defining properties of an ideal gas include the following . 1
Vm =RTsclpsc, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 3 )
Assuming base o r standard conditions o f Tsc=60 ° F + 45 9 . 67 =
5 1 9 . 67 ° R and Psc= 1 4 . 65 psia, Eq. 1 . 3 becomes
Vm =
(
1 0 . 732
PSia-ft 3
Ibm-mol OR
)
(5 1 9 . 6 rR)
( 1 4 . 65 psia)
=380 . 7 scfllbm-mol .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 4)
2
GAS RESERVOI R E N G I N E E R I N G
TABLE 1 .1-PHYS ICAL PROPERTIES OF GASES AT 1 4 . 7 psla AND 60°F
Component
Hydrogen
Hel i u m
Water
Carbon monoxide
N itrogen
Oxygen
Hydrogen s u lfide
Carbon d ioxide
Ai r
Methane
Ethane
Propane
i ·B utane
n ·Butane
i ·Pentane
n ·Pentane
n ·Hexane
n ·Heptane
n ·Octane
n·Nonane
n ·Decane
C hemical
Form u l a
H2
He
H2 O
CO
N2
O2
H2 S
CO 2
-
CH4
C 2 Hs
C 3 HS
C4H10
C4H10
C SH12
C SH12
C SH14
C 7H1S
C SH1S
C 9H20
C 1O H22
Molecular
Weight
(Ibm/Ib m ·
mol)
2 . 1 09
4.003
1 8.01 5
28.01 3
28.0 1 0
3 1 .999
34.08
44. 0 1 0
28.963
1 6 .043
30.070
44. 097
58 . 1 23
58 . 1 23
72 . 1 50
72 . 1 50
86. 1 77
1 00 . 204
1 1 4.231
1 28 . 258
1 42 . 285
Critical
Tem perat u re
( O R)
Critical
Pressure
(psi a)
Liquid
Density'
(lbmIft 3)
Gas
Density
(lbm/ft 3)
Gas
Viscosity
(cp)
59.36
9 . 34
1 , 1 64.85
227. 1 6
239.26
278 . 24
672.35
547.58
238 . 36
343. 00
549.59
665 .73
734. 1 3
765. 29
828.77
845.47
9 1 3.27
972.37
1 , 023.89
1 , 070.35
1 , 1 1 1 .67
1 87.5
32.9
3,200. 1
493 . 1
507. 5
731 .4
1 ,306.0
1 ,071 . 0
546. 9
666.4
706 . 5
6 1 6.0
527.9
550. 6
490.4
488. 6
436 . 9
396 . 8
360 . 7
331 . 8
305. 2
4.432
7.802
62.336
50.479
49.231
71 . 228
49.982
51 .01 6
54.555
1 8. 7 1 0
22. 2 1 4
31 .61 9
35. 1 04
36.422
38.960
39.360
4 1 .400
42.920
44.090
45 . 020
45.790
0 .0053 1 2
0 . 0 1 055
0 . 00871
0 . 0 1 927
- 1 . 1 22
0 . 0 1 725
0 . 0 1 735
0 .02006
0 . 0 1 240
0 . 0 1 439
0 . 0 1 790
0 . 0 1 078
0 . 00901
0 .00788
0 .00732
0 .00724
'Values given are l iquid densities for those components that can exist as liquids at SooF and
naturally gases at these conditions.
The value of the molar volume depends on the standard condi­
tions of pressure and temperature , so defining these conditions is
very important. Unless otherwise noted , this textbook uses stan­
dard conditions of Psc = 14 . 65 psia and Tsc = 60 ° F . Further, to ob­
tain the molar volume in Eq. 1 . 4 , we converted the standard
temperature from degrees Fahrenheit to degrees Rankin using a con­
version constant of 459. 67 . For subsequent calculations in this chap­
ter, we use a less accurate but more common conversion constant
of 460 .
1 .2 . 4 Real-Gas Behavior. The real-gas law is simply the pres­
sure/volume relation ( i . e . , EOS) predicted by the ideal-gas law
modified by a correction factor that accounts for the nonideal be­
havior of the gas . The real-gas law is
p V=z nRT, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 5)
where z = dimensionless quantity called the z factor, the compress­
ibility factor, or the gas deviation factor. The z factor corrects the
simple EOS of Eq. 1 . 2 for an ideal gas and allows us to describe
the behavior of a real gas . Under ideal pressure and temperature
conditions , z = 1 . 0 . The z factor, which depends on pressure , tem­
perature, and gas composition, can be measured in the laboratory
on a sample of reservoir gas or, more often , obtained from corre­
lations.
1 . 2 . 5 Principle of Corresponding States. Several gas properties
have the same values for similar gases (such as paraffin hydrocar­
bons) at identical values of reduced pressure and temperature. I Re­
duced pressure and reduced temperature for pure compounds are
defined as
p , =p/Pc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 6)
and T, = T/ Tc , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 7)
respectively . Pseudoreduced pressure and pseudoreduced temper­
ature for mixtures are defined as
pp , =p/pp c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 8)
and Tp , = T/ Tp c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 9)
respectively , where Pc = critical pressure for a pure gas , psia;
Pp c = pseudocritical pressure for a gas mixture, psia; Tc = critical
temperature for a pure gas , O R ; and Tp c = pseudocritical tempera­
ture for a gas mixture , O R .
14.7
-
0 .0738 1
0 .07382
0 . 08432
0 .0898 1
0 . 1 1 60
0 .07632
0 . 04228
0 . 07924
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
psia and estimated l iquid densities for components that are
The critical point ( P c , Tc) for a pure substance is the pressure
and temperature at which the properties of the liquid and vapor
phases become identical . At pressures above P c ' liquid and gas
cannot coexist, regardless of the temperature; at temperatures above
Tc , the substance cannot be liquefied, regardless of the pressure.
For pure substances, Pc and Tc are determined experimentally . For
mixtures, Pp c and Tp c either are computed with some consistent
set of mixing rules or are estimated from correlations . These com­
puted values of Pp c and Tp c are not true criticals ; i . e . , the proper­
ties of the liquid and vapor phases do not become identical at the
point ( pp c , Tpc ) '
The observation that certain gas properties , such as the z factor,
should be approximately the same at a given reduced temperature
and pressure for pure but similar gases forms the basis for the prin­
ciple of corresponding states. This behavior also has been observed
for mixtures of chemically similar gases ; therefore, correlations
of z factors for pure gases and gas mixtures are based on this
principle.
1 . 3 P roperties of Natural Gases
Table 1 . 1 lists the physical properties of pure components that occur
in natural gases. 2 ,3 These properties , which are evaluated at stan­
dard conditions of Psc = 1 4 . 7 psia and Tsc = 60 ° F , include molecu­
lar weight, critical pressure and temperature, ideal density , and
viscosity (components lighter than pentane only) . These proper­
ties of pure components are used in calculations based on mixing
rules to develop pseudoproperties for gas mixtures, including ap­
parent molecular weight and specific gas gravity . Refs . 2 through
4 provide more complete listings of natural gas properties and con­
taminants commonly associated with natural gas production .
1 . 3 . 1 Apparent Molecular Weight of a Gas Mixture. Because
a gas mixture is composed of molecules of various sizes and molecu­
lar weights , it does not have an explicit molecular weight of its own.
However, a gas mixture behaves as if it has a definite molecular
weight . This observed molecular weight for a gas mixture with nc
components is called the apparent or molal average molecular weight
and is determined by
M=
E
i=1
Yi Mi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 1 0)
3
PROPERTIES OF NATU RAL GASES
where M=apparent molecular weight of gas mixture, lb/lbm-mol;
Mi =molecular weight of the ith gas component, lb/lbm-mol; and
Yi =mole fraction the gas phase of the ith component, fraction.
1 . 3 .2 Specific Gravity of a Gas. The specific gravity of a gas,
'Y I{' is defined as the ratio of the densities of the gas and dry air
when both are measured at the same temperature and pressure:
'Yg=Pg/Pa, .................................... ( 1 .1 1 )
where Pg=density of gas mixture, Ibm/ft3 , and Pa= density of air,
Ibm/ft 3 .
A t standard conditions (such a s 1 4.6 5 psia and 60°F), both air
and natural gas are modeled accurately by the ideal-gas law. Un­
der these conditions, if we use the definition of pound-mole
(n=m/
and density (p=mIV) and model the behavior of both
the gas and the air by the ideal-gas EOS, we can express the spe­
cific gravity of a gas mixture as
M)
g=(pM/RT)/(pMa/RT)=M/Ma' ................. ( 1 .1 2 )
where 'Y g= specific gravity of the g a s (air= 1 .0 ) ; M=apparent
molecular weight of the gas, lbmllbm-mol; and Ma=molecular
'Y
weight of air=28.962 5 Ibm/Ibm-mol.
Although Eq. 1 .1 2 is derived under the assumptions of an ideal
gas (accurate at standard conditions), its use as a definition for real
gases and real-gas mixtures is common in the natural gas industry.
1.4 Calculation of Pseudo critical Gas P roperties
This section discusses two methods for calculating the pseudocriti­
cal pressure and temperature of a hydrocarbon gas mixture. These
pseudocritical properties provide a means to correlate the physical
properties of mixtures with the principle of corresponding states.
As stated previously, the principle of corresponding states suggests
that pure, but similar, gases have the same gas deviation or z fac­
tor at the same values of reduced pressure and temperature. Other
physical properties of gases also have been correlated with the prin­
ciple of corresponding states. Mixtures of chemically similar gases
can be correlated with reduced temperature and reduced pressure.
The first method, which is a set of mixing rules developed by
Stewart
al.,5 requires the gas composition to be known.
Although their method requires more calculations than early methods
(such as Kay ' s6 procedure), Stewart
at.' s mixing rules have
been proved more accurate. The second method, developed by Sut­
7
ton, provides a method for estimating pseudocritical properties
when the gas composition is not known. Sutton' s method requires
considerably less arithmetic than Stewart at.' s mixing rules and
is the preferred method when speed is more important than the
greatest possible accuracy. Although it uses only specific gas gravity
instead of detailed hydrocarbon compositions, Sutton ' s method is
et
et
et
Kessler-Lee10 equations (Eqs. 1 .1 3 through 1 .15) are used to cal­
culate the critical properties of the heptanes-plus fraction.
Calculation Procedure-Stewart et al. Method.
1. If a significant fraction of heavy components (C7 and heavi­
er) is present in the natural gas mixture, laboratory measurements
of the molecular weight and gravity of the C7+ fraction are re­
quired to use mixing rules to calculate the mixture gravity and pseu­
docritical properties. The Whitson I I and Kessler-Lee10 equations
(Eqs. 1 .1 3 through 1 .1 5) are recommended for estimating the crit­
ical properties of the C7+ fraction.
A. First, estimate the boiling temperature of the C7+
fraction.11
TbC7+ =(4.5 579
[
B . Estimate the pseudocritical pressure of the C7+ fraction.10
1 .4 . 1 Estimating Pseudocritical Properties When Gas Compo­
sition Is Known: Stewart et al. Mixing Rules . Stewart et at.5
compared 2 1 different mixing rules and concluded that the best
method is given by Eqs. 1 .1 9 through 1 .24. Their mixing rules pro­
vide the most consistent results of the simple cubic mixing rules
when experimental data are compared with computed results. "Sim­
ple cubic" refers to the cubic EOS (e.g., van der Waals8 and
Redlich-Kwong9 EOS). Because these mixing rules give the most
accurate results, the Stewart et al. method should be used to esti­
mate pseudocritical pressures and temperatures for estimating the
z factor, gas compressibility, and gas viscosity.
A recommended procedure for using the Stewart et at. mixing
rules follows.The procedure also corrects for high-molecular-weight
7
components (Eqs. 1 .1 6 through 1 .1 8) with Sutton' s method. The
(
0.0 566
2.2898
P pcC7+ =exp 8.3634 - -- - 0.24244+-'YC7+
'YC7+
+
0.1 1 8 57
(
'Y�7+
)
TbC7+
--
1 ,000
(
+
1 .468 5+
3.648
--
) Tk
]
'YC7+
+
0.4722 7
'Y�7+
) Tlc
�
10
7
1 .6977
7+
. ...................( 1 .1 4)
- 0.420 1 9+-- __
1
1
0
�
0
'Y 7+
C. Estimate the pseudocritical temperature of the C7+
fraction.10
TpcC7+ =(3 4 1 .7 +8 1 1 'Yc7+ )+(0.4244+0.1 1 74'Yc7+
+(0.4669 - 3.2623 'YC7+ )
1 05
--
TbC7+
)TbC7+
. .................. ( 1.1 5)
2. Determine the correction factors Fj, �j' and h for high­
7
molecular-weight components using Sutton' s method.
1
Fl. =3
(T)
Y c
Pc
C7+
+
�
3
( )
y 2 Tc
Pc
�j=0.608 1 Fj+1.1 32 5F
and h=
(�)
Pc
more accurate than Kay's mixing rules.
This section also discusses correlations to correct pseudocritical
pressure and temperature for the presence of contaminants com­
monly associated with natural gas production, such as hydrogen
sulfide ( H 2 S), carbon dioxide (C0 2 ), nitrogen, and water vapor.
In addition, we illustrate a calculation technique for estimating the
specific gas gravity for wet-gas and gas-condensate fluids. This spe­
cific gas gravity then can be used with Sutton' s method to estimate
pseudocritical properties.
M27'1178'Y2711427)3 . ...............( 1 .1 3 )
C7+
, ............... ( 1.1 6)
J - 1 4.004FjYC7+ +64.434FjY�7+'
................................. ( 1 .1 7)
C7+
(0.3 1 29 Yc7+- 4.8 1 56 yE7++27.37 5 1 yt7)·
................................. (1.18)
3. Obtain the critical pressures and temperatures of the remain­
ing components from Table 1 .1 or Refs. 2 through 4.
4. Determine the pseudocritical pressure and temperature of the
gas.
A. Calculate the parameters J and K.
.........( 1 .1 9)
and K=
E
i
=1
(T)
Y c
.JP;
·
i
......................... ( 1 .20)
B. Correct the parameters J and K for the C7+ fraction.
J'=J - �j ......................................( 1 .2 1 )
and K'=K - h. .................................. ( 1 .22)
C. Calculate the pseudocritical temperature and pressure.
Tpc=K' 2 /J' ....................................( 1 .23)
and P pc =Tpc/J' ..................................( 1 .24)
·
4
(
TABLE 1 .2-COMPOSITION OF SWEET NATURAL GAS,
EXA M P L E 1 . 1
- 0.420 1 9+
Molecular
Critical
Critical
Mole
Weight
Tem perat u re Pressure
( O R)
(psia)
Co m ponent Fraction ( I b m/Ibm-mol)
0.0 1 38
28.0 1 3
227. 1 6
493. 1
N2
0.9302
1 6.043
343.00
666.4
CH4
0.0329
30.070
549.59
706.5
C2H6
0.0 1 36
44.097
665.73
6 1 6.0
C 3Ha
0.0023
58. 1 23
734.1 3
i-C4H1 o
527.9
0.0037
n-C4H10
58. 1 23
765.29
550.6
0.00 1 2
72. 1 50
i-C sH12
828.77
490.4
0.00 1 0
72. 1 50
845.47
n-C SH12
488.6
86. 1 77
9 1 3.27
0.0008
436.9
C 6 H14
1 1 4.23 1
0.0005
C 7+
1 . 15.
Tp CC 7+ = (34 1 .7 + 8 1 1 'Y C 7+) + (0.4244 +0. 1 1 74'Yc7+) TbC 7+
10 5
+ (0.4669- 3.2623 'Y C 7+) -- = 34 1 .7 + 8 1 1 (0.7070)
TbC 7+
+ [0.4244 +0. 1 1 74(0.7070)](697.6)
10 5
+ [0.4669- 3 .2623(0.7070)] -- = 1 ,005. 30oR.
697.6
2. Calculate the correction factors for the C 7 + fraction. These
factors , Fj , �j ' and � ko are defined by Eqs. 1 . 16 through 1 . 1 8 ,
-
( )
( ) [
2 0.0005 2(1 ,005) l
+ -[
=4.466x 10 -4 .
respectively.
y 2 Tc
Y Tc
� 0.0005( 1 ,005)
Fj = �
+�
=
3 Pc C 7+ 3 Pc c 7+ 3
375 . 5
x
x
TbC 7+= (4.5579 M2'/1 17 8'Y2'/1427 ) 3
= [4.5579( 1 14.2)0 .15 17 8(0.7070)0. 15427 ] 3 =697 .6°R.
B . Next , calculate the pseudocritical pressure with Eq. 1 . 14.
2.2898
0.0566
- 0.24244 +
PpcC 7+= exp 8 .3634 'Y C 7+
'Y C 7+
(
)
( � ) (0.3 129Yc 7+- 4.8 156Y�7++27.375 1yt7)·
c C 7+
)
= ( � [0.3129(0.0005) - 4 . 8 156(0.0005) 2
375.5
3 . 648 0.47227 T i c 7+
0. 1 1 857 TbC 7+
_
+ 1 .4685 + --+
'Y�7+ 1 ,000
'Y C 7+ 'Y�7+ 10 7
--
__
--
--....l.±. = exp
+ 27 .375 1 (0.0005) 3 ] =0.008054.
3.
Obtain the critical pressures and temperatures of the remain­
ing components from Table
Table 1.3 summarizes these values.
--
J= � E
3 i=1
1 .3 ,
( YPcTc )i + �3 [ iE=1 (Y�fT:-;;: )i] 2
2
1
= - (0.5405) + -(0.73 1 8) 2 =0.5372.
3
3
--
10
0.7070 0.70702
1.1.
4. Determine the pseudocritical pressure and temperature.
A . Referring to Table
calculate the parameters J and
--
--
x
h=
--
)
(
( 1 .6977 Tg c ] [8.3634 - 0.0566
- 0.4201 9+ 'Y )
0.7070
C2 7+ 10 10
( 2 .2898 + 0. 1 1 857 )-697.6
- 0.24244 +
0.7070 0.70702 1 ,000
3 . 648 0.47227 ) 697.6 2
+ ( 1 .4685 +
+
7
+
TABLE 1 . 3-PSEU DOCRITICAL PROPERTY CALCULATIONS U S I N G T H E
STEWART et a/. S MIXING RULES, EXAMPLE 1 . 1
Mole Molecular
Fract i o n , Weight,
Co m ponent
Yi
Mi
N2
C H4
C 2H 6
C 3Ha
i-C4H1 o
n -C4H10
i-C sH12
n -C sH12
C 6 H14
C 7+
0.0 1 38
0.9302
0.0329
0.0 1 36
0.0023
0.0037
0.00 1 2
0.00 1 0
0.0008
0.0005
28.0 1 3
1 6.043
30.070
44.097
58.1 23
58. 1 23
72.1 50
72.1 50
86. 1 77
1 1 4.23
E=
1 .0000
-
YiMi
0.3866
1 4.923
0.9893
0.5997
0.1 337
0.2 1 5 1
0.0866
0.072 1
0.0689
0.0571
1 7.532
]
3
375 .5
�j =0.608 1Fj + 1. 1 325FJ - 14. oo4Fj Y c 7+ +64.434Fj Y�7+
=0.608 1 (4.466 10 -4) + 1 . 1 325(4.466 x 10 -4) 2
- 14.004(4.466 10 -4)(0.0005) + 64.434(4.466 10 -4 )
X (0.0005) 2 =0.000269.
I.
---
psia.
(Ix )
Example 1 . 1-Calculation o f Pseudocritical Properties for a
Sweet Natural Gas With the Stewart et al. Mixing Rules_ Cal­
culate the apparent molecular weight , gas gravity , and pseudocrit­
ical pressure and temperature of the sweet gas 1 2 described in
Table 1 . 2 _ A sweet gas is a natural gas with no H S contamina­
2
tion. The molecular weight and g ravity of the C 7 + fraction are
1 14.2 lbmllbm-mol and 0.7070, respectively.
Solution.
1 . First , we must estimate the critical properties of the C 7 +
fraction.
A . Estimate the boiling temperature with Eq. 1 3 .
[
)
1 .6977 6 97.63
-- = 375.5
0.70702 10 10
C. Calculate the pseudocritical temperature with Eq.
---
-
]
GAS RESERVO I R E N G I N E E R I N G
Critical
Critical
Tem perature, Pressure,
pei
( O R)
Tei
(psia)
227. 1 6
343.00
549.59
665.73
734. 1 3
765.29
828.77
845.47
9 1 3.27
1 005.3
493.1
666.4
706.5
6 1 6.0
527.9
550.6
490.4
488 .6
436.9
375.5
-
-
YiTe/Pei Yi�Te/Pei YiTe/Jpei
0.0064
0.4788
0.0256
0.0 1 47
0.0032
0.005 1
0.0020
0.00 1 7
0.00 1 7
0.00 1 3
0.0094
0.6674
0.0290
0.0 1 4 1
0.0027
0.0044
0.00 1 6
0.00 1 3
0.00 1 2
0.0008
0.1 41 2
1 2.360
0.6803
0.3648
0.0735
0.1 207
0.0449
0.0382
0.0350
0.0259
0.5405
0.73 1 8
1 3.884
K.
( Y�T ) =13 . 88 .
5
PROPERTI ES OF NATU RAL GASES
nc
K= ,E
1= 1
TABLE 1 .4-CO MPOSITION OF S O U R NATURAL GAS ,
EXAMPLE 1 .2
'VPc ;
B . Correct the parameters J and K for the C7+
�j and h are calculated in Step 2 .
J' =J - �j =0.5372 - 0.000269=0.5369.
K' =K - �k =1 3 . 88 -0.008054=1 3 . 87 .
Critical
Molecular
Critical
Tem perature Pressure
Weight
Mole
( O R)
(psia)
Com ponent Fraction (Ibmll bm-mol)
fraction where
C . Calculate the pseudocritical temperature and pressure.
K' 2 ( 1 3 . 87) 2
=358.3 °R.
Tpc =- =
J'
0.5369
---
Tpc 358. 3
.
=667.4 pSla.
Ppc =-=
J' 0.5369
[
1 .3 ,
nc
M= E y ;M; =17.53
;= 1
0. 0236
0 . 0 1 64
0. 1 841
0 . 7700
0 .0042
0 .0005
0 . 0003
0 .0003
0 . 0001
0 . 0001
0 . 0001
0 . 0003
( )
28. 0 1 3
44. 0 1 0
34. 080
1 6. 043
30,070
44. 097
58 . 1 23
58. 1 23
72. 1 50
72 . 1 50
86. 1 77
1 1 4.231
( )
]
227. 1 6
547.58
672.35
343.00
549.59
665. 73
734. 1 3
765 .29
828.77
845.47
9 1 3.27
-
[
493. 1
1 ,071 . 0
1 ,306 . 0
666.4
706.5
6 1 6.0
527. 9
550. 6
490.4
488.6
436. 9
-
]
0.0003( 1 ,005)
1 Y Tc
y 2 Tc
=�
+�
Fj =375 .5
3 Pc C 7+ 3 Pc C 7+ 3
2 0.0003 2 ( 1 ,005)
=2.679x I0 -4 .
+375 .5
3
�j =0.6081Fj + 1. 1 325FJ - 14.004Fj YC 7++ 64.434Fj Y�7+
Note that these pseudocritical values are not correct (i.e., they
are incomplete) because they must be adjusted to account for the
presence of nitrogen. Correlations for these adjustments are dis­
cussed later.
We also can calculate the apparent molecular weight and spe­
cific gravity of the natural gas. From Table
the apparent
molecular weight is
5.
---
N2
CO 2
H2 S
CH4
C 2 HS
C 3Ha
i-C4H10
n-C4H10
i-C sH12
n-C SH12
C SH14
C 7+
=0.608 1 (2.679 x 10 -4) + 1 . 1 325(2.679 x 10 - 4 ) 2
- 14.004(2.679 x 10 -4)(0.0003) +64.434(2.679 x 10 - 4 )
X (0.0003) 2 =0.000162.
lbmllbm-mol.
The specific gravity of the gas mixture is
( �Pc ) 7+(0.3 129Yc 7+- 4.8 156Y�7++27.375 1yt7)
C
1 ,005 . 3 )
=(
[0.3129(0.0003) - 4.8156(0.0003) 2
'V 375 .5
M 17.53
'Y g = -=--=0.61 .
Ma 28. 96
h=
Example 1 .2-CaIculation of Pseudocritical Properties for a Sour
Natural Gas With the Stewart et aI. Mixing Rules. Calculate the
apparent molecular weight, gas gravity, and pseudocritical pres­
sure and temperature of the sour gas 12 with the composition given
in Table 1 .4 . A sour gas is a gas with H2 S contamination. The
molecular weight and gravity of the C7+ fraction are 1 14.2
Ibm/Ibm-mol and 0.7070, respectively.
Solution.
1 . The properties of the C7+ fraction, calculated in Example
1 . 1 , are
Tp CC 7+=1 ,005 .3 °R
and PpcC 7+ = 375.5 psia.
2. Calculate the correction
�
+27.375 1 (0.0003) 3 ] =0.00487.
3.
Obtain the critical pressures and temperatures of the remain­
ing components from Table
Table 1 . 5 summarizes these values.
Determine the pseudocritical pressure and temperature.
A . Referring to Table
calculate the parameters and
factors for the C7+ fraction.
1.1.
4.
nc
( ) [ (�)]
1 .5 ,
nc
1 � Y Tc
2 �
c 2
J= - i.J
+ - i.J
Y 3 ;= 1 Pc ' 3 ;= 1
PC '
1
2
= - (0.5163) + - (0.7 1 8 1) 2 =0.5 159.
3
3
-
1
1
TABLE 1 . 5-PSEU DOCRITICAL PROPERTY CALCULATIONS USING THE
STEWART et al. 5 MIXING RU LES , EXAMPLE 1 .2
Mole Molecular
Fraction , Weight,
y;
Component
N2
CO 2
H2 S
CH4
C 2 HS
C 3Ha
i-C4HlO
n-C4H10
i-C sH12
n-C SH12
C SH14
C 7+
0 . 0236
0 . 0 1 64
0 . 1 841
0 . 7700
0 .0042
0 . 0005
0 . 0003
0 . 0003
0 . 0001
0 . 0001
0 . 0001
0 . 0003
!; =
1 .0000
M;
28.01 3
44. 0 1 0
34.080
1 6. 043
30.070
44.097
58. 1 23
58. 1 23
72. 1 50
72. 1 50
86. 1 77
1 1 4.23
-
y;M;
0 . 66 1 1
0 . 72 1 8
6.2741
1 2.353
0 . 1 263
0 . 0220
0 . 0 1 74
0 . 0 1 74
0 . 0072
0 . 0072
0 .0086
0 .0343
20.250
Critical
Critical
Tem perature, Pressure,
( O R)
Te;
(psia)
227. 1 6
547.58
672.35
343.00
549.59
665.73
734. 1 3
765.29
828.77
845.47
9 1 3 . 27
1 005.3
493 . 1
1 07 1 . 0
1 306.0
666.4
706. 5
61 6.0
527.9
550 .6
490.4
488 .6
436. 9
375 .5
-
-
p c;
y/Te;IPe; y;JTe;lPe; y;Te;lJpe;
0 . 0 1 09
0 . 0084
0 .0948
0 .3963
0 . 0033
0 . 0005
0 .0004
0 .0004
0 . 0002
0 . 0002
0 . 0002
0 . 0008
0 . 0 1 60
0.01 1 7
0 . 1 32 1
0 .5524
0 . 0037
0 .0005
0 .0004
0 . 0004
0 . 0001
0 .0001
0 . 0001
0 . 0005
0 . 24 1 4
0 .2744
3.4251
1 0.231
0 . 0868
0 . 0 1 34
0 . 0096
0 . 0098
0 .0037
0 . 0038
0.0044
0 .0 1 56
0 . 5 1 63
0.71 81
1 4. 3 1 9
J
K.
6
GAS RESERVOI R E N G I N EE R I N G
�
".
iii 650
Q,.
I
IIJ
0::
:J
en
VI
IIJ
0::
Q,.
...J
-<
U
1=
."
tto
c,
II
I
c
4,
"
600
,
••
." .
'.
ii:
u
0
0
i3
VI
Q,.
550
a
�
0::
I
IIJ
0::
�
0::
IIJ
Q,.
::Ii
500
�e
.450
I:!
:i.
u
1=
ii: 4\00
U
a
a
:J
w
en
Q,.
q.
•
�
�o
,
Co�
,Co
"
0.5
0.6
0.7
0.8
D
1= 1
..
•
( y�T ) = 14.32.
LO
0.9
l'
nc
•
'\fPc ;
J and K for the C 7
�j and h are calculated in Step 2.
J' = J - �j =0.5 1 59 -0.000162 = 0.5 1 57.
K' =K - �k = 14.32 - 0.004847 = 14.32.
1.2
1.3
1..4
1.5
1.6
1.7
1. 8
CAS SPECIfiC CRAVITY
The specific gravity is
+
fraction where
C . Calculate the pseudocritica1 temperature and pressure .
K' 2 (14.32) 2
= 397.7 °R.
=
J'
0.5 157
Tpc 397.7
..
= 77 1.2 PSIa
Ppc = -=
J' 0.5 157
-
---
Note that these pseudocritical values are not correct (Le. , they
are incomplete) because they must still be adjusted for the pres­
ence of the nonhydrocarbon components (e.g . , nitrogen, CO 2 , and
H 2 S ) . Correlations for these adjustments are discussed later.
5. We also can calculate the apparent molecular weight and the
specific gravity of the natural gas . From Table 1.5 , the apparent
molecular weight is
M= E Yi M; =20.25
i= 1
1.1
F i g . 1 . 1 -Pseudocritical properties of natural gases (after Sutton 7) .
B. Correct the parameters
Tpc =
•
�.
350
300
K= E
�()
,�
(e
Ibmllbm-mol .
M 20.25
=0.70.
'Y g = -=
Ma 28.96
--
1 .4 .2 Estimating Pseudocritical Properties When Gas Compo­
sition Is Unknown: Sutton's Correlations. The method proposed
by Stewart et ai. 5 for calculating pseudocritical properties requires
information about gas composition; however, laboratory analyses
often are not available . Using data from 264 gas samples , Sutton 7
developed a correlation for estimating pseudocritical pressure and
temperature as a function of gas gravity. Sutton ' s correlation curves,
shown in Fig. 1 . 1 , are based on a larger database than that used
by Standing 1 3 and consequently differ significantly from Stand­
ing ' s curves. Sutton fit the raw data with quadratic equations and
obtained the following empirical equations relating pseudocritical
properties of the hydrocarbons to specific gas gravity :
Ppch = 756.8 - 13 1.0'Yh - 3.6 'Y � .................... ( 1.25)
and Tpch = 169.2 + 349.5 'Yh - 74.0 'Y � ' ................ ( 1.26)
where Ppch = pseudocritical pressure of the hydrocarbon compo­
nents, psia; Tpch = pseudocritical temperature of the hydrocarbon
components , OR; and 'Yh = specific gas gravity of the hydrocarbon
components (air = 1.0).
7
PROPERTIES OF NATU RAL GASES
Eqs . 1 . 25 and 1 . 26 and Fig . 1 . 1 are applicable for
0 . 57 < I'h < 1 . 6 8 . If the gas contains < 1 2 mol % CO 2 , < 3 mol %
nitrogen, and no H 2 S , then I' ll can be determined as follows.
1 . If the gas is dry (i . e . , no condensate is formed) , and if the
separator gas gravity is used , then I' h = I' g '
2 . If the gravity of the well stream fluid , I' w , is computed , then
I'h =I' w' where I'w is computed with methods presented in Sec .
1 .4 . 5 .
However, i f the gas contains > 1 2 mol % CO 2 , > 3 mol % nitro­
gen , or any H 2 S , then the hydrocarbon gas gravity should be cal­
culated by
'Yh =
I'w - 1 . 1 767y H2S - 1 . 5 1 96Y co 2
-
0 . 9672YN2 - 0 . 6220YH20
Example 1 .4- Estimating Pseudocritical Properties of a Sour
Gas With Sutton's Correlations. Using Sutton' s correlations , cal­
culate the pseudocritical pressure and temperature for the sour­
natural-gas sample 12 in Example 1 . 2 . Compare results with those
obtained with the Stewart et at. mixing rules .
Solution.
1 . Determine the gravity of the hydrocarbon components of the
mixture with Eq. 1 . 27 .
0 . 6992 - 1 . 1 767(0 . 1 84 1 ) - 1 . 5 1 96(0 . 0 1 64) - 0 . 9672(0 . 0236)
1 - 0 . 1 84 1 - 0 . 0 1 64 - 0 . 0236
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . 27)
where 1' ,,' = 'Y g if the separator gas gravity is being used .
Once the specific gas gravity of the hydrocarbon components is
estimated , the pseudocritical properties of the hydrocarbon mix­
ture are calculated with Sutton' s correlations given by Eqs . 1 . 25
and 1 . 26 or Fig . 1 . 1 . The pseudocritical properties of the entire
mixture , including contaminants , are estimated with the following
equations 1 3 :
Ppc = ( \ - Y H2S - YC02 - YN2 - YH20 )Ppch + I , 306y H2S
$ 1 ,07 1 Y co 2 + 493 , I YN2 + 3 ,200 . I Y H2 o . . . . . . . . . . . ( 1 . 28)
and Tpc = ( I -y H2S - YC02 - YN2 - YH20 ) Tpch + 672 . 35y H2S
$547 .58Y c 02 + 227 . 16Y K 2 + 1 , 1 64 . 9y H20 ' . . . . . . . ( 1 . 29)
where the coefficients of the contaminant mole fractions are the
critical pressures (Eq. 1 . 28) and temperatures (Eq. 1 .29) of the con­
taminants . Note that the forms of Eqs . 1 . 27 through 1 . 29 initially
proposed by Standing 1 3 did not have corrections for water vapor.
Note also that the pseudocritical pressure and temperature cal­
culated with Eqs . 1 . 28 and 1 . 29 are not correct if the gas mixture
is contaminated with nonhydrocarbon components . Corrections for
common natural gas contaminants , including CO 2 , H 2 S , nitrogen,
and water vapor are discussed in subsequent sections. Examples
1 . 3 and 1 .4 illustrate application of Sutton' s correlations .
= 0 . 5604 .
2 . Estimate the pseudocritical pressure and temperature of the
hydrocarbon components with Eqs . 1 . 25 and 1 . 26, respectively .
Ppch = 756 . 8 - 1 3 1 . 0'Yh - 3 . 6 1' � = 75 6 . 8 - 1 3 1 .0(0. 5604)
- 3 . 6(0 . 5 604) 2 = 682 . 3 psia.
Tpch = 1 69 . 2 + 349 . 51'h - 74 . 01' � = 1 69 . 2 + 349 . 5(0 . 5604)
- 74 . 0(0 . 5 604) 2 = 34 1 . 8 ° R.
3 . Now , calculate the pseudocritical properties of the total
mixture .
Ppc = ( l - y H2S - YC02 - YN2 - YH20 )Ppch + 1 ,306Y H2S
+ 1 ,07 1 Y co 2 +493 . I YN2 + 3 ,200 . l y H20
= ( 1 - 0 . 1 84 1 - 0 .0 1 64 - 0 .0236)(682 .3) + ( 1 , 306)(0. 1 84 1 )
+ ( 1 ,07 1 )(0 . 0 1 64) + (493 . 1 )(0 . 0236) = 799 . 0 psia.
Tpc = ( l - y H2S - YC02 - YN2 - YH z o ) Tpch + 672 . 35y H2S
+ 547.58YC02 + 227. 16YN2 + 1 , 1 64 . 9y H20
= ( 1 - 0 . 1 84 1 - 0 . 0 1 64 - 0 .0236)(34 1 . 8)
+ (672 . 35)(0 . 1 84 1 ) + (547 . 58)(0 . 0 1 64)
Example 1 .3 - Estimating Pseudocritical Properties of a Sweet
Gas With Sutton's Correlations. Using Sutton' s correlations , cal­
culate the pseudocritical pressure and temperature for the sweet­
natural-gas sample 1 2 in Example 1 . 1 . Ignore the nitrogen contami­
nation (N)- = 0 . 0 1 3 8 mol % ) for this calculation . Compare the re­
sults with those obtained using the Stewart et al. 5 mixing rules .
which usually are more accurate .
Solution. For the sweet-gas sample of Example 1 . 1 , the gas gravi­
ty of the mixture was estimated to be 0 . 6 1 . From Eqs . 1 . 25 and
1 . 26, the pseudocritical pressure and temperature for the hydrocar­
bon components are
Ppch = 75 6 . 8 - 1 3 1 .0I'h - 3 . 6 1' � = 756 . 8 - 1 3 1 .0(0 . 6 1 )
- 3 .6(0 . 6 1 ) 2 = 675 . 6 psia
and
Tpch = 1 69 . 2 + 349 . 5'Yh - 74 . 0 1' � = 1 69 . 2 + 349 . 5 (0 . 6 1 )
- 74 . 0(0 . 6 1 ) 2 = 354 . 9 ° R .
W e are ignoring the nitrogen contamination , s o the pseudocriti­
cal pressure and temperature of the gas mixture are
Ppc =Ppch = 675 . 6 psia
and Tpc = Tpch = 354.9°R.
Recall that, with the Stewart e t at. mixing rules (Example 1 . I),
Ppc = 667 . 4 psia and Tpc = 35 8 . 3 O R . Compared with the results ob­
tained using the Stewart et al. mixing rules . the errors in the pseu­
docritical pressure and temperature are 1 . 2 % and 1 . 0 % ,
respectively , with Sutton' s method. Note that the pseudocritical pres­
sure and temperature calculated with Sutton ' s method are incom­
plete because they still must be corrected for the nitrogen
contamination (Sec . 1 . 4 . 4) .
+ (227 . 1 6)(0 .0236) + ( l , 1 64 . 9)(0 . 0) = 403 . 3 O R .
Recall that ppc = 77 1 . 2 psia and Tpc = 397.rR were calculated
with the Stewart et at. mixing rules for the composition data in Ex­
ample 1 . 2 . With Sutton' s method , the errors in the pseudocritical
pressure and temperature are 3 . 6 % and 1 . 4 1 % , respectively , com­
pared with the Stewart et al. mixing rules . Note that the pseudocrit­
ical pressure and temperature calculated with Sutton ' s method are
incomplete because they still must be adjusted for H 2 S and CO 2
contamination by the correlations presented in Sec . 1 .4 . 3 .
1 .4.3. Correcting Pseudocritical Properties for H2S and CO2
Contamination. Wichert and Aziz 1 4 developed a correlation to ac­
count for the effects of CO 2 and H 2 S on the pseudocritical pres­
sure and temperature . Their correlation , which adjusts the
pseudocritical properties of the natural gas mixture to yield the cor­
rect values of estimated properties, should be applied when we use
Ppc and Tpc to estimate z factor, gas compressibility , and gas vis­
cosity .
The Wichert and Aziz correlation, shown in Fig. 1 . 2 , is
� = 1 20 (A 0 9 _A 1 6 ) + 1 5 (Bo L B 4 ) , . . . . . . . . . . . . . . . ( 1 . 30)
where the pseudocritical temperature , T;c ' and pressure , P;c ' ad­
justed for CO 2 and H 2 S contamination are
T;c = Tpc - � . . . . . . . . . . . . . . . .
and P;c =ppc T;c / [ Tpc +B( l -B)� l .
.
........
.
.
. . . . . . . (1.31)
.
. . . . . . . . . . . . . . . . . . . ( 1 . 32)
In Eqs . 1 . 30 through 1 . 3 2 , A = sum of the mole fractions of H 2 S
and CO 2 in the gas mixture and B = mole fraction of H 2 S in the
gas mixture .
8
GAS RESERVO I R E N G I N EE R I N G
70
60
(' .
'0
30
20
�
10
�
�O
��
0
0
'i'
0
...
�
10
0
20
30
040
rER CENT Hz:!
50
60
70
eo
Fig . 1 .2-Pseudocritical property corrections for H 2 S and CO 2 (after Wichert and Aziz ' 4 ) .
(Repri nted with permission from Hydrocarbon Processing, May 1 972 , pp . 1 1 9- 1 2 2 , by Gulf
Publishing Co. , all rights reserved .)
The average absolute error in the calculated z factor was 0.97 % ,
with a maximum error of 6.59% for the data set used to develop
this correlation. The correlation was developed for gases under the
following range of conditions: 154 <p(psia) < 7,026, 40 < T(OF)
< 300, 0 < C02(mol % ) < 54.56, and 0 < H2S(mol % ) < 73 . 85 .
Example 1 . 5 - Correcting Pseudocritical Properties for H2S
and CO2 Contamination. For the sour-gas sample in Example
1 .2, correct ppc and Tpc for H2S and CO2 using the Wichert and
Aziz 14 correlation. Because the composition is known, we can use
the Stewart et al. mixing rules to obtain the pseudocritical properties.
Solution.
4. The pseudocritical temperature corrected for contaminants is
T;c = Tpc - � = 397 .7 - 25.5 = 372.2°R.
The corrected pseudocritical pressure is
= 7 14.9 psia.
(77 1 .2)(372 .2)
(397 .7) + (0. 1 841)(1 -0. 1 84 1)(25.50)
1 .4.4 Correcting Pseudocritical Properties for Nitrogen and
Water Vapor Contamination. Correlations are available for cor­
recting pseudocritical properties for the presence of nitrogen and
1 . From Example 1 .2, the pseudocritical pressure and tempera­ water vapor. * These correlations are, at most, semiempirical and
ture are ppc = 77 1 .2 psia and Tpc = 397.7 °R.
should be considered accurate only in the sense that they may pro­
2. The Wichert and Aziz corrections for H2S and CO2 are
vide better results than ignoring the effects of these contaminants.
The corrections for nitrogen and water vapor are*
A = Y H 2 S + YC02 = (0. 1 84 1) + (0.0164) =0.2005
Tpc, cor = - 246. 1 YN2 + 400. Oy H20 . . . . . . . . . . . . . . . . . . ( 1 .33)
and B = y H2S =0. 1 84 1 .
Ppc,cor = - 162.0YN2 + 1270 ·Oy H 20 · . . . . . . . . . . . . . . . . . ( 1 . 34)
3 . Using the Wichert and Aziz correlation equation, we find
The corrected pseudocritical temperature and pressure are
� = 120(A o.9 - A 1 .6 ) + 15(B o. 5 -B 4 ) = 120[(0.2005)0.9
T;c - (227.2)YN2 - ( 1 , 165)YH20
- (0.2005) 1 6] + 15[(0. 1 84 1 )0.5 - (0. 1 841)4] = 25.50oR.
T;� =
+ Tpc , cor . . . . . . . ( 1 . 3 5)
Similarly, if we enter Fig. 1 .2 with the mole percent of CO2
(1 - YN2 - YH20 )
(1 . 64 %) on the vertical axis and the mole percent of H 2 S (18 Al %)
' Personal communication with J. Casey, Mobil E&P C o . , Houston (May 8, 1 990).
on the horizontal axis, we read � =25 . 5 °R.
.
9
PROPERTI ES OF NATU RAL GASES
P;� =
p;c - (4 93 . 1 )YN2 - (3 , 2 00)YH20
( 1 -YN2 -YH2 0)
+Ppc .coP
. . . . . . . ( 1 . 3 6)
where T;c and P;c are the pseudocritical temperature and pressure
corrected for H2S and CO2 with the Wichert and Aziz 14 correla­
tion. If there is no H2S or CO2 in the gas mixture, then T;c = Tpc
and P;c =PpcExample 1 . 6- Correcting Pseudocritical Properties for Nitro­
gen and Water Vapor Contamination. A gas sample was taken
from a well completed in a gas-condensate reservoir. The sample
contains significant amounts of CO2 and water vapor and a trace
of nitrogen. The uncorrected pseudocritical pressure and tempera­
ture are estimated to be ppc = 817.6 psia and Tpc =444.9°R, respec­
tively. Calculate the corrected pseudocritical properties using both
the nitrogen and water vapor corrections and the Wichert and Aziz
corrections for CO2 , The following values apply: YN2 = 0.302 % ,
Pec = 8 17.6 psia, YH2 o = 4.1 10% , Tpc = 444.9°R, and YC02 =
1 3.6 1 2 %.
Solution.
1. Correct the pseudocritical properties for the presence of H 2 S
and CO2 ,
A . For 0.0% H2S and 1 3.6 1 2 % CO2 ,
A = YH2 S + YC02 = 0.0 +0.136 12 = 0.1 3612
and B = YH2S = 0.0.
B . From the Wichert and Aziz correlation equation,
� = 120(A -A 1 .6) + 1 5(B o. 5 _ B4) = 120[(0.1 3612) 0.9
- (0.1 3612) 1 .6] + 1 5[(0.0)0.5 - (0.0)4] = 1 5.00oR.
C . The pseudocritical temperature corrected for H2S and CO2 is
0. 9
T;c = Tpc - � = 444.9 - 1 5.00 = 429.9°R.
The corrected pseudocritical pressure is
(817.6)(429.9)
(444.9) + (0.0)(1 - 0.0)( 15.00)
= 790.0 psia.
2. Correct the pseudocritical properties for nitrogen and water
vapor.
A. The pseudocritical temperature correction is
Tpc.cor = - 246.1 YN2 + 400 ·Oy H20 = - 246.1(0.00302)
+ 400.0(0.04 1 1 ) = 1 5.700R.
The pseudocritical pressure correction is
Ppc.cor = - 162.0YN2 + 1 , 270.Oy H20 = - 162.0(0.00302)
+ 1 ,270.0(0.041 1) = 5 1.7 1 psia.
B. The final corrected pseudocritical temperature is
T;� =
T;c - (227.2)YN2 - ( 1 , 165)YH20
+ Tpc .cor
( I - YN2 - YH20 )
(429.9) - (227.2)(0.00302) - ( 1 165)(0.041 1 )
+ 1 5.70
[ 1 - (0.00302) - (0.041 1)]
=414.61 °R.
1 .4.5 Estimating Pseudocritical Properties of Wet-Gas and Gas­
Condensate Reservoir Fluids: Recombination Calculations. As
discussed in Sec. 1 .4.2, Sutton developed correlations that allow
us to estimate pseudocritical properties of a natural gas without a
detailed compositional analysis of the reservoir fluid. Sutton ' s
method requires only estimates of specific gas gravity. For dry­
gas reservoirs, the specific gravity of the gas sample obtained at
the primary-separator (high-pressure) conditions equals the value
of a sample from the reservoir. However, for wet-gas and gas­
condensate reservoir fluids, which experience condensation and a
change in composition following a pressure reduction, the specific
gravities of gas samples taken from the reservoir and at the surface
differ.
The most accurate source of reservoir gas gravity, especially for
wet-gas and gas-condensate fluids, is a laboratory analysis of a reser­
voir fluid sample. Alternatively, when compositions of vapor and
liquid streams have been determined in the laboratory and when
stabilized gas and liquid flow rates have been measured at the
primary separator in the field, recombination calculations provide
useful estimates of wellstream properties. Recombination calcula­
tions of surface production data are also a rigorous way to deter­
mine the specific gravity of a reservoir fluid without requiring
bottomhole fluid sampling.
In the subsequent discussions, we first illustrate calculation of
the recombined wet-gas flow rate and cumulative production from
surface production data. Then we present correlations that allow
us to estimate reservoir specific gravity from surface production
data. These correlations are based on typical production parame"
ters, such as separator pressure, temperature, and gas gravity and
the stock-tank liquid gravity.
Example 1 . 7- Calculating Wet-Gas Flow Rate and Cumulative
Production. A gas-condensate well produced with a gas-condensate
ratio of 4,428 scf/STB at the primary separator. The liquid flow
rate at separator conditions was 50 STBID. Cumulative liquid pro­
duction was 100,000 STB. The specific gravity of the primary­
separator liquid was determined to be 0.681. The molecular weight
of the C 7 + fraction was estimated to be 143 Ibm/Ibm-mol. From
this information and the composition data in Table 1 . 6 , determine
the composition of the reservoir fluid and calculate the wet-gas flow
rate and cumulative production.
Solution.
1. First, we calculate the molar recombination ratio. For this
problem, we are assuming standard conditions of Tsc = 60°F and
Psc = 14.65 psia, so Vm = 380.7 scfllbm-mol (Eq. 1.4).
pound-moles in 1 bbi condensate =
=2.876 Ibm-mol liquid/bbi condensate.
TABLE 1 .6-GAS COMPOSITION, EXAMPLE 1 .7
p;c - (493.1)YN2 - (3 ,200)YH20
+Ppc, cor
P;� =
( I - YN2 - YH20 )
(790.0) - (493.1)(0.00302) - (3 ,200)(0.041 1 )
+ 5 1.7 1
[ 1 - (0.00302) - (0.041 1 )]
= 739.0 psia.
-------
Liquid Mole
Fraction ,
Xi
CO 2
N2
CH4
C2H 6
C 3Ha
iC4HlO
nC4H 10
iC sH 12
nC SH 12
C 6 H 14
C 7+
0 . 0778
0 . 1 002
0 . 1 508
0 . 0277
0 . 1 1 39
0 . 0352
0 . 0650
0 . 0861
0 . 3433
Gas Mole
Fraction ,
Yi
0 . 0022
0.00 1 6
0 . 7531
0 . 1 508
0 . 0668
0 . 0052
0 . 0 1 44
0 . 00 1 8
0. 0024
0.001 1
0 . 0006
1: =
1 .0000
1 . 0000
----
The final corrected pseudocritical pressure is
(O.681)(62.4)(5.615)lbm/bbl
82.88 Ibm/Ibm-mol
------
Com ponent
-
Molecular
Weight, Mi
(Ibmllbm-mol)
1 4. 5449
44. 0 1 0
28.0 1 3
1 6. 042
30. 068
1 1 . 094
58. 1 20
58. 1 20
72. 1 46
72. 1 46
86. 1 72
1 43 . 0
X i Mi
-
1 .248
3.01 3
6 . 649
1 .6 1 0
6.620
2 . 540
4.689
7.4 1 9
49.092
82.880
10
GAS RESERVO I R E N G I N E E R I N G
TABLE 1 .7-RECOMBI NATION CALCU LAT I O N , EXAMPLE 1 .7
Component _X_
i_
CO 2
N2
0 .0778
CH4
0 . 1 002
C 2 HS
0 . 1 508
C 3Ha
0 . 0277
iC4H1 o
0 . 1 1 39
nC4H1O
iC 5 H12
0 . 0352
nC 5 H12
0 . 0650
C SH14
0 . 0861
0 . 3433
C 7+
E=
1 . 0000
i _ 2.88x i + 1 1 .63YI
Y_
_
0. 0257
0 .0022
0 . 0 1 87
0 . 00 1 6
0 . 7531
9.0 1 24
0 . 1 508
2.0480
1 .2 1 33
0 .0668
0 . 1 403
0 . 0052
0 .4956
0 . 0 1 44
0 . 1 222
0 . 00 1 8
0.2 1 49
0. 0024
0.2605
0 . 00 1 1
0. 0006
0 .9943
1 . 0000
Zj
=
(2.88xj
+
1 1 . 63Yi )/
= 1 1 .63 Ibm-mol gas/bbl condensate.
Thus, we combine 2 . 876 Ibm-mol of liquid with 1 1 .63 Ibm-mol
of gas to make the recombination calculations in Table 1 . 7 to de­
termine the composition of the reservoir fluid.
2 . Now, calculate the wet-gas flow rate.
(2. 876 Ibm-moIlSTB)(50 STBID) = 1 43 . 8 Ibm-mol liquidlD.
1=1
2 .88x i
+
1 1 .63Yi
0 . 00 1 8
0 . 00 1 3
0 . 6 1 96
0 . 1 408
0. 0834
0 . 0096
0. 034 1
0 .0084
0 . 0 1 48
0 . 0 1 79
0. 0684
1 4.506
4,428 scf/STB
pound-moles in 4,428 scf gas = -----380.7 scf/lbm-mol
(E
1 .0000
The vapor equivalent of this liquid is
(143 . 8 Ibm-moIlD) (380. 7 scf/lbm-mol) = 55 MscfID.
The dry-gas surface flow rate is
qg = (50 STBID)(4,428 scf/STB) = 22 1 MscfID.
Finally, the wet-gas well stream flow rate is
q = qg + vapor equivalent = 22 1 + 55 = 276 MscfID.
3 . Similarly, we can calculate the cumulative wet-gas production.
(2 . 876 Ibm-moIlSTB)( 1 00,OOO STB) = 287,600 Ibm-mol liquid.
..
I'
.'
l
o·
l
#'
.�
I'
.!
100
"00
+"
..#
.00
• 00
)
' 00
00
...
...
t
e
i
1
2-
I
a
�
I
..
..
..
Fig. 1 .3-Additional-gas-production nomograph for three-stage separation systems (after Gold
et al. 1 5 ) .
11
PROPERTIES OF NATU RAL GASES
.t""
. .
;
�
..�
�f
.r
"
�
•
:
�
•
'000
TOI
eoo
IOO Q
••
04 0 0
,.
5
0
..
tl �; HiiItHiItHiIitt1Ht11Iti
�
uo
.
c
0
�
,.
-
1 00
t
to
Z
W
oJ
C
-
....
-
NO
t il ittifHfli'HtHtU
Fig. 1 .4-Vapor-equivalent nomograph for three-stage separation system (after Gold et al. 1 5 ) .
The vapor equivalent o f this liquid i s
(287 , 600 Ibm-mol)(380 . 7 scf/lbm-mol) = 1 09 ,489 Mscf.
Cumulative dry-gas production is
Qg , d ry = ( 1 oo,OOO STB)(4 ,428 scf/STB) = 442 , 8oo Mscf.
Cumulative wellstream production is
Qg , wet = Qg dry + vapor equivalent = 442 , 800 + 1 09 ,489
,
= 552 ,289 Mscf.
Compositions of liquid and gas streams from the primary sepa­
rator are not always available. However, reliable estimates of the
reservoir gas specific gravity can be obtained with surface produc­
tion data when the total nonhydrocarbon component is less than
20 mol % . 1 5 When gas and liquid quantities are known , the
equations 1 5 presented in Sec . 1 04 . 6 should be used to estimate the
reservoir gas specific gravity for natural gas wells that produce liq­
uid . For wells that produce only gas and no liquid hydrocarbons ,
the primary-separator gas gravity is the reservoir gas gravity .
Eq, 1 . 37 is applicable (and rigorously correct) if the gas/liquid
ratio and gas specific gravity of each separation stage and the stock­
tank liquid gravity are known . Similarly , Eq. 1 . 3 8 is the recombi­
nation expression for a two-stage separation system consisting of
a primary separator and a stock tank,
'Yw =
R J'Y I + 4 , 602'Yo + R 3 'Y 3
RI
+ 1 33 , 3 1 6'Yo /Mo +R 3
, . . . . . . . . . . . . . . . . . . . . ( 1 . 3 8)
In Eqs . 1 . 37 and 1 . 3 8 , R I = primary (high-pressure) separator
gas/stock-tank-liquid ratio, scf/STB ; R 2 = secondary (low-pressure)
separator gas/stock-tank-liquid ratio, scf/STB ; R 3 = stock-tank­
gas/stock-tank-liquid ratio , scf/STB ; 'Y I = specific gravity of
primary-separator gas (air = 1 . 0) ; 'Y 2 = specific gravity of
secondary-separator gas (air = 1 .0) ; 'Y 3 = specific gravity of stock­
tank gas (air = 1 .0) ; 'Yo = specific gravity of the liquid hydrocarbons
(water = 1 .0) ; and Mo = molecular weight of stock-tank liquid ,
lbmllbm-mol .
If Mo is unknown, then it can be approximated with either API
gravity (Eq . 1 . 39a) or specific gravity (Eq. I . 39b) 1 5 :
Mo = 5 ,954/('Y API - 8 . 8 1 1 ) . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 39a)
1 .4.6 Estimating Specific Gravity of Reservoir Gas. Eq. 1 . 37
is the recombination expression for the reservoir gas gravity , 'Y w '
for a three-stage separation system consisting of a primary (high­
pressure) separator, secondary (low-pressure) separator, and stock
tank.
or Mo = 42 A3'Yo/( l .oo8 - 'Yo) ' . . . . . . . . . . . . . . . . . . . . . ( 1 . 39b)
where 'Y API is the specific gravity (in API) of the stock-tank
hydrocarbon liquid .
Because gas rates and specific gravities from low-pressure sepa­
rators and the stock tank frequently are not measured , this gas pro­
duction must be estimated for Eq. 1 . 37 or 1 . 3 8 . These estimates
are made with correlations for additional gas production, Gpa , and
vapor equivalent, Veq , in conjunction with Eq. l AO . The correla­
tions are based on generally available production parameters, spe0
12
GAS RESERVO I R E N G I N EE R I N G
l
r
•
.I
...
•
•
ADDf1'IOIW.
� (od I STIli .ravlty
PllOOUC11OH
GAS
Fig. 1 .5-Additional-gas-production nomograph for two-stage separation systems (after Gold
et a/. 1 5 ) .
cifically the primary-separator pressure, temperature, and gas gravi­
ty ; the secondary-separator temperature ; and the stock-tank-liquid
gravity .
Gpo and Veq in Eq. 1 .40 are determined from Figs. 1.3 through
and 1 .4 apply to a three-stage separation system;
1 . 6. Fig s .
are for a two-stage separation system. Figs.
and
Fig s .
are used to estimate the additional gas production , Gpa '
and
are used to estimate the vapor equivalent,
and
and Figs .
Veq , in Eq .
In terms of Gpa and Veq , the reservoir gas specific gravity , /' w '
i s estimated by
1.1. 5 5 1. 31. 6
1. 41. 40. 1. 6
1. 3
I I +4 6 2 o +G
RI +
R /'
/' w =
, 0 /'
Veq
pa
, . . . . . . . . . . . . . . . . . . . . . . ( l . 40)
1. 4
=535 92 + 2 . 6231p so.l 79318",41 . 66 12",1.2094 rs-l O . 849 I 1 rsO2. 26987 '
. . (1. 42)
. ...
1. 5
= 1 4599(ps l - 14 ' 65) 1. 3 394 ",7.I 0943",1 .l 436rs-l O . 93446 ,
(1. 43)
. . ..
1. 6
A 0544 /, �. 0831 /, ur 12 rsl o. 79130 .
. . . . . . . . . . . . (1. 44)
Veq
I
.
·
I API
.......... .
Gpa
'
·
..... .
. .
(for a two-stage separation
I
..... .
while the correlation equation for Fig .
tion system) is
Veq = 63 5 . 5 3 + 0 . 36 1 82p
.
.
.
......
.........
The correlation equation for Fig .
system) is
·
I API
.
.............
(for a two-stage separa­
. .. . .. .....
.
.....
= 1
1. 45 1. 46:
1.3
q=qg+ 133.316(/'oqoIMo) . . . . . . . . . . . . . . . . . . . . . . . (l. 45)
q=qs l =(1 +
. . (1. 46)
.
I
O
qg
q
=2 9922(ps l _ 14 ' 65) o. 97050 ", I6 . 8049", . 0792 rs-l I .1960 rs2. 55367 '
qo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1. 4 1) qs l
where Veq = volume of secondary-separator gas and stock-tank gas
plus the volume that would be occupied by bbl of stock-tank liq­
uid if it were gas , scf/STB ; and Gpa additional gas production,
scf/STB .
(for a three-stage separa­
The correlation equation for Fig .
tion system) is
Gpa
(for a three-stage sepa­
while the correlation equation for Fig .
ration system) is
'
I
I API
The wellstream gas flow rate, representing all gas and liquid pro­
duced at the surface and including the high- and low-pressure sepa­
or
rator and stock-tank gas , can be calculated with Eq.
or
Veq IR I ) ,
.............. ..... .....
= total gas
where = total wellstream gas flow rate, MscflD;
= oil flow rate, STBID ; and
surface gas flow rate, MscflD;
= gas flow rate from primary separator, Mscf/D .
13
PROPERTIES OF NATU RAL GASES
•
",'
§
..
�
..�
i
eoo
-
400
,,0
2000
1 00 0
a ooo
VAP O R EOUlVAU H T tv"" ec l l n B
0000
Fig. 1 .6-Vapor-equivalent nomograph for two-stage separation system (after Gold et a/. 1 5 ) .
Eq. 1 .45 should b e used i f the total surface gas flow rate i s known.
Eq. 1 .46 is applicable when the secondary-separator (low-pressure)
and/or stock-tank gas production is not known.
Similarly , the cumulative wellstream gas produced at the sand­
face, Qg .wet , can be calculated . When the gaslliquid ratio and/or
the gas and liquid compositions change with time, the wet-gas cu­
mulative production can be calculated by integrating the wet-gas
rate over time . If the gas/liquid ratio and liquid and gas gravities
remain constant with time and if the cumulative surface produc­
tion of gas, Q g , dry ' and cumulative stock-tank-liquid production,
Qo ' are known, then
Qg , wet = Qg , d ry + 1 33 . 3 1 6('Y o Qo/Mo ) · . . . . . . . . . . . . . . . ( 1 . 47)
Otherwise,
Q g ,wet = Q g , d ry ( l + Veq /R , ) , . . . . . . . . . . . . . . . . . . . . . . . ( l . 48)
where Q g ,wet = cumulative wellstream gas produced at the sand­
face , Mscf; Q g , dry = cumulative surface gas production, Mscf; and
Qo = cumulative stock-tank liquid production, STB .
R 1 = 7 ,040 scf/STB .
'Y 1 = 0 . 7 1 2 .
'Yo = 0. 755 .
Qg , dry = 767 MMscf.
R 2 = 5 10 scf/STB .
'Y 2 = 1 . 02 .
M o = 125 lb/lbm-mol .
Qo = 1 00,000 STB .
R 3 = 1 20 scf/STB .
'Y 3 = 1 . 5 5 .
qo = 50 STBID .
qg = 3 84 MscflD.
Solution.
1 . Eq. 1 . 37 is applicable because every term in the recombina­
tion expression is known . The specific gravity of the reservoir gas
(air = 1 . 0) is
'Y w =
Example 1 . 8- Calculating Wellstream Gas Gravity and Flow
Rate With Known Fluid Properties. Calculate the well stream gas
gravity , the total wellstream gas flow rate, and the cumulative well­
stream production for a three-stage separation system. Measured
production data and fluid properties, including the gas and liquid
from the stock tank, are summarized below.
R , 'Y , + 4 , 602'Yo +R 2 'Y 2 + R 3 'Y 3
R , + ( l 33 , 3 1 6 'Yo/Mo ) +R 2 + R 3
(7,040)(0. 7 1 2) + (4 , 602)(0. 755) + (5 1 0)( 1 . 02) + ( 1 20)( 1 . 55)
7 ,040 + [( 1 3 3 , 3 1 6) (0 . 755)1 1 25] + 5 1 0 + 1 20
= 1 .08.
14
GAS RESERVOI R E N G I N E E R I N G
I . :'
1 .2
1.1
1 .0
N
0.9
Ii'
f:;d
ttl
.....
(I)
(I)
0.7
g: 0 . 6
W
:::E
o
u
o.
0."
0.3
0.2
o
3
PSEUDO
...
6
5
7
R E D U CED P R ES S U R F; .
n
9
,0
Pr
Fig. 1 . 7-z factors for natural gases for 0 ::5 p , ::5 1 0 (after Dranchuk and Abou·Kassem 1 6 ) .
2 . The total well stream gas rate (Eq . 1 . 45) i s
q = q g + 1 33 . 3 1 6(1' oqo IMo ) = 3 84 + 1 33 . 3 1 6[(0. 755)(50) / 1 25]
= 424 MscflD .
3 . Cumulative well stream gas production is calculated with Eq.
1 . 47:
Qg .wet = Qg . dry + 1 3 3 . 3 1 6( 'Yo QoIMo) = 767,000
+ 1 3 3 . 3 1 6 [(0. 755) ( 1 00,000) 1 1 25] = 847 ,523 Mscf
= 848 MMscf.
Example 1 .9-Estimating Wellstream Gas Gravity and Flow
Rate With Correlations. Calculate the wellstream gas gravity, total
wellstream gas flow rate, and cumulative well stream production
for a two-stage separation system.
s
R I = 7 ,040 scf/STB .
P i = 700 psia.
'YAPI = 55 . 8 ° AP!.
T1's 1l
=0.712.
= 90 ° F .
qs l = 352 MscflD.
Qg. dry = 680 MMscf.
Solution.
1 . First, determine Gpa • We can either read the value from Fig.
1 . 5 or calculate it directly with Eq. 1 .43 . For this example, we
use Eq. 1 . 43 .
Gpa = 1 . 4599( Ps l - 1 4 . 65)
= 1 . 4599(700 - 1 4 . 65)
x (90)
- 0. 93446
1. 3394 1' l'0943 'Y�M36 TsI O . 93446
1 3394 7 . 0943 1.1436
.
(0 . 7 12)
(55 . 8)
= 1 ,223 . 2 scf/STB .
2 . Similarly , we can estimate Veq from Fig. 1 . 6 or calculate it
with Eq. 1 . 44 .
I
I'
Veq = 63 5 . 53 + 0 . 3 6 1 82p
A0544 i "083 'Y �W 2 Ts1 0 . 791 30
1. 0544 (0. 7 12)5 .0831 1 . 5812
= 63 5 . 5 3 + 0 . 36 1 82(700)
X (90)
-0. 79130
= 1 , 692 . 1 scf/STB .
(55 . 8)
15
PROPERTIES OF NATU RAL GASES
2.2
2. 1
2.0
1 .9
1.8
1.3
1.2
1.1
1 .0
9
10
12
11
13
14
15
16
17
16
19
20
PSEUDO R E D UCED PRESS U R E . Pr
Fig . 1 . 8-z factors for natural gases for 9 s p
3.
'Yo = 131.5141.5+ 'YAPI 131.5141.5+ 55.8 =0. 756.
4.
'Yw= R I 'Y I + 4,602'Yo +Gpa
The specific gravity o f the stock-tank liquid.
Calculate the reservoir gas specific gravity .
l AO .
'YO '
'Y w '
is
The total well stream gas flow rate is
1+
Veq
-
---
MscflD.
S
20 (after D ranchuk and Abou.Kassem 1 6 ) .
6.
) =843
Qg,wet = Qg ,dry ( 1 + RI ) =680 ( 1 + 1,692.1
7 , 040
The cumulative well stream gas is
Veq
with Eq.
+ (4, 602)(0. 756) + 1 ,223.2 =1.112.
= (7,040)(0.712)7,040
+ 1,692.1
5.
692.1 ) =437
q=qsI ( RI ) =352 ( 1 + 1,7,040
r
MMscf.
Once we have estimated the specific gas gravity of the reservoir
gas, we then use Sutton ' s method for calculating pseudocritical pres­
sure and temperature. Sec .
outlines a systematic procedure,
including corrections for H 2 S , CO 2 , nitrogen, and water vapor
contamination, for estimating the pseudocritical properties of a nat­
ural gas .
1. 4 . 7
1 .4.7 Systematic Procedure for Calculating Pseudocritical Gas
Properties. The following procedure summarizes the techniques
outlined earlier and should be used to calculate pseudocritical pres­
sure and temperature for estimating z factor, gas compressibility ,
and gas viscosity .
Estimate pseudocritical pressure,
and temperature,
A . If laboratory analysis o f a reservoir fluid sample i s available,
and
then calculate
with the Stewart et at. mixing rules. If
1.
Ppc Tpc
Ppc,
Tpc.
16
GAS RESERVO I R E N G I N E E R I N G
the composition of both the separator liquid and gas are known, the Stewart et al. mixing rules to obtain pseudocritical properties .
then the reservoir fluid composition must first be determined from From Example 1 . 1 , ppc = 667.4 psia and Tpc = 358.3 °R. Ignore the
recombination calculations outlined in Sec. 1 .4 . 5 .
small amount of nitrogen in the sample.
B. I f no laboratory analysis o f hydrocarbon composition i s avail­
Solution.
able or if speed is more important than precision, estimate Ppc and
1 . Because the sample contains no H2S or CO2 and we are
Tpc with Sutton ' s correlation.
making no corrections for nitrogen, P;� =P;c =Ppc = 667.4 psia and
1 . Estimate the hydrocarbon gas gravity, 'Y h .
T; � = T; c = Tp c = 358 . 3 °R.
a. If the gas contains no contaminants, then:
2 . The pseudoreduced properties are
1 . If separator gas gravity, 'Y g , is used, then 'Yh = 'Y g for
a dry gas.
P 2,000
= 3 .00
Pr = - =
2 . If the gravity of the wellstream fluid, 'Y w , is used,
Pp c 667 .4
then 'Yh = 'Y w for a wet gas or a gas condensate. If the
gas/liquid ratio and separator gas gravity of each sepa­
ration stage and the stock-tank-liquid gravity are
T 200 + 460
= 1 . 84 .
known, calculate 'Y w with Eq. 1 . 37 or 1 . 38. Other­ and Tr = - =
Tpc
358 . 3
wise, calculate 'Y w with Eq. 1 .40.
b . If the gas contains more than 1 2 mol % of CO2 , more
3 . Enter Fig. 1 .7 with P r and Tr and read z =0.9 1 .
than 3 mol % of N2 , or any H2 S, then calculate the
hydrocarbon gas gravity, 'Yh , with Eq. 1 .27.
2 . Calculate Pp ch and Tp ch with Eqs. 1 .25 and 1 .26, respec­ Example 1 . 1 1-Estimating z Factors for Sour Natural Gases.
tively.
the z factor at 200°F and 2,000 psia for the gas sample
3 . Calculate Ppc and Tpc with Eqs. 1 .28 and 1 .29, respec­ Calculate
in Example 1 .2 . Because the composition is known, we can use
tively.
Stewart et at. mixing rules to obtain pseudocritical properties.
2 . Correct the pseudocritical properties for H2S and CO2 con­ the
Ignore the small amount of nitrogen contamination in the sample.
tamination.
Solution.
A . If the gas does not contain H 2 S or CO2 , then P; c =Pp c and
1 . From Example 1 .5, the pseudocritical properties corrected for
c
T/JC = Tp .
H2S
and CO2 are p;c = 7 14.9 psia and T; c = 372.2 °R.
B . If the gas contains H 2 S and/or CO2 , then calculate the cor­
2 . The pseudoreduced pressure and pseudoreduced temperature
rected pseudocritical properties, P; c and T; c, with the Wichert and are
Aziz correlation discussed in Sec. 1 .4 . 3 .
3 . Correct the pseudocritical properties for nitrogen and water
P 2,000
vapor using Casey ' s method. *
= 2 . 80
Pr = - =
A . If the gas does not contain nitrogen or water vapor, then
Ppc 714.9
p;� =p;cand T;� = T; c '
B. If the gas contains nitrogen and/or water vapor, then calcu­
T 200 + 460
and Tr = - =
= 1 . 77.
late P;� and T;; with the method outlined in Sec. 1 .4.4.
Tpc
372.2
4. P;� and T; � are the appropriate values to use in correlations
for z factor, compressibility, and viscosity.
3 . Entering Fig. 1 .7 with P r and Tr, we read z = 0. 89.
--
--
1 . 5 Dranchuk and Abou· Kassem 1 6
Correlation for z Factor
Standing and Katz 17 presented a graphical correlation of the z fac­
tor for natural gases as a function of the pseudoreduced pressure,
p " and pseudo reduced temperature, Tr . Dranchuk and Abou­
Kassem 16 fitted an l l -constant EOS to Standing and Katz's data
and extrapolated this correlation to higher reduced pressures. Figs.
1 . 7 and 1 . 8 are graphs of z factor vs. P r calculated with Dranchuk
and Abou-Kassem ' s EOS . In addition, Appendix A gives an al­
gorithm for programming this method.
Dranchuk and Abou-Kassem developed their EOS primarily to
estimate the z factor with computer routines. They fitted the EOS
to 1 ,500 data points with an average absolute error of 0.486 % when
the z factor is a function of Tr and P ro The z factor based on
the EOS is accurate within usual engineering standards for
0.2 :S P r < 30 and 1 .0 < Tr :s 3 .0, and pr < 1 .0 and 0.7 < Tr :s 1 .0.
However, the EOS gives poor results for Tr = 1 .0 and P r > 1 .0.
We recommend the following procedure for estimating the z factor
with the Dranchuk and Abou-Kassem correlation.
1 . Calculate pseudocritical properties corrected for H2S, CO2 ,
N2 , and H20, P; � and T;� . Use the procedure outlined in Sec.
1 .4 . 7 .
2. Calculate reduced properties, P r = P /P; � and Tr = TlT; � .
3 . Estimate z factor using Fig. 1 . 7 o r 1 . 8 . Alternatively , a com­
puter routine can be developed using the equations in Appendix A .
Example 1 . 10-Estimating
z
Factors for Sweet Natural Gases.
Calculate the z factor at 200°F and 2,000 psia for the gas sample
in Example 1 . 1 . Because the composition is known, we can use
•
1 . 6 G a s FYF
The FVF of a gas, B g , is defined as
B g = VRIVsc , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 .49)
where VR = volume occupied by gas at reservoir temperature and
pressure and Vsc = volume occupied by the same mass of gas at
standard conditions .
The volume of n moles of gas at reservoir conditions can be ob­
tained from the real-gas law,
VR =znRT/p, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 .50)
where T= reservoir temperature in OR and P = reservoir pressure
in psia. Similarly, the volume of n moles of gas at standard condi­
tions can be obtained from the real-gas law,
Vsc =Z scnRTsc/psc , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 5 1 )
where T= temperature at standard conditions, O R , and p =pressure
at standard conditions, psia. Substituting Eqs. 1 .50 and 1 .5 1 into
Eq. 1 .49 yields
Bg =
(znRT)/p
(z sc nRTsc)/p sc
=
zTpsc
. . . . . . . . . . . . . . . . . . . . ( 1 .52)
z sc TscP
Assuming standard conditions of Tsc = 60°F = 5 19.6rR, Psc =
14.65 psia, and z sc = l , Eq. 1 .52 becomes
Personal communication with J. Casey, Mobil E&P Co . , Houston (May 8, 1 990).
ft 3
zT( 14.65 psia)
zT
=0.0282- - . . . . . . . . . . . . . ( 1 .53)
( 1 .0)(5 19.67°R)p
P scf
-----
17
PROPERTIES OF NATU RAL GASES
.0
�
��
I
1
\
t
(J
�
\
1\\ �
1\
0.1
, \' ,\\\
\ \ \ �\\\
1---I-H++-t+lH--t-HthliITtk-�\ -! -j--j--H-+tii1
�\
100
10
0.1
REDUCED PResSuRe. Pr
Fig. 1 . 9-Variation i n c r T r for natural gases for 1 .05 � T r �
1 .4 (after Mattar et a/. 1 8 ) .
Converting the units of B g from ft 3 /scf to RB/Mscf gives
Bg =
( 0.0282ZT ft 3 ) (
scf
P
5 . 02zT RB
P Mscf
bbl
5 . 6 1 5 ft 3
)(
1 ,000 SCf
Mscf
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 54)
Substituting the definitions of mole (n = mlM) and specific volume
(v = Vim) into the real-gas law (Eq. 1 .5), we obtain
pv=zRTIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 55)
Because gas density is defined as the mass of gas per unit volume,
or simply the reciprocal of specific volume,
p = mlV= l Iv, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 56)
we can rearrange Eq. 1 .55 and solve for the gas density in terms
of pressure, temperature, and z factor:
p = l Iv =pMlzRT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 57)
In terms of specific gas gravity, 'Y g , Eq. 1 .57 becomes
( p)(28.963'Y g )
(z)(1O.732)(T)
2.70p'Y g
. . . . . . . . . . . . . . . . . . . . ( 1 . 58)
zT
where p = gas density, Ibm/ft 3 ; p =pressure, psia; "I g = specific gas
gravity (air = 1 .0) ; T= temperature, OR; and z = gas deviation fac­
tor, dimensionless.
__.c... ,
1 .8 Gas Compressibility
The definition of the isothermal coefficient of compressibility , or
simply compressibility, is
c=
-�CapV) =
V
T
i!
�
\ \ \ \\'\
\\\:\'\
o.os
0.1
10
REDUCED PRa3URE., PI'"
i
,�
100
Fig. 1 . 1 0-Variation i n c r T r for natural gases for 1 .4 � T r �
3 . 0 (after Mattar et al. 1 8 ) .
Combining Eq. 1 .57, the derivative of Eq. 1 .57 with respect to
pressure at constant temperature, and Eq. 1 .59, we can express the
gas compressibility in terms of the gas deviation factor as
Cg
=
; -� C; )
'
. . . . . , . , . . . . . . . . . . , . . , . " . . . . . ( 1 . 60)
T
which is a fundamental equation for calculating gas compressibil­
ity. Gas compressibility could be evaluated directly from p-z data;
however, we can develop an explicit relation for the gas compress­
ibility if we use the Dranchuk and Abou-Kassem correlation.
They define the pseudoreduced compressibility, C r o as
C r = CgP c - . . . . . . , . . . , . . , . , . . . . , . . , . . , . , . . . . . . . . ( 1 . 6 1 )
p
I n terms o f pseudoreduced pressures, Eq. 1 .6 1 becomes
16
1 . 7 Gas Density
p=
�
1
Bg
_ _
Cap ) = 2.p Cap ) . . . . . . . . . ( 1 . 59)
Bg
p
T
T
Cr =
� - 2.
( �)
18
. . . . . . . , . , . , . . , . . , . . , . . , . . . . ( 1 .62)
z aP r Tr
Mattar et al. used an l l-constant EOS to generate Figs. 1 . 9
and 1 . 10, where the product of the pseudoreduced compressibility
and pseudoreduced temperature, C r Tr o is plotted as a function of
pseudoreduced pressure, P r o and pseudoreduced temperature, Tr.
Recall that pseudo reduced pressure and pseudoreduced tempera­
ture for a gas mixture are defined by Eqs. 1 . 8 and 1 .9, respective­
Iy. The accuracies of the Cg and z factor calculated with the EOS
method are about the same. In addition, the c g and z-factor calcu­
lations are applicable over the range O . 2 � P r < 30 and
1 .0 < Tr ::; 3.0, and P r < 1 .0 and 0.7 < Tr < 1 .0. However, the EOS
gives poor results for Tr = l .O and P r > 1 .0.
Pr
From Figs. 1 .9
and 1 . 10, calculate the gas compressibility at 200°F and 2,000 psia
for the sweet gas described in Example 1 . 1 . Ignore the nitrogen
contamination.
Example 1 . 12-Estimating Gas Compressibility.
Solution.
1 . From Example 1 . 1 , ppc = 667 .4 psia and Tp c = 358.3°R. The
gas contains no H 2 S , CO 2 , or water vapor, and we are ignoring
the nitrogen. So, P�� =P�c =ppc = 667.4 psia and T�� = T�c = Tpc
= 35 8 . 3 °R.
18
GAS RESERVO I R E N G I N E E R I N G
2 . Pseudoreduced pressure and temperature are
p
2 ,000
= 3 . 00
p,= - =
667 . 4
Ppc
--
200 + 460
T
and T, = - =
Tpc
358 . 3
(9 . 379 + 0. 0 1 607M ) T I . 5
------
(209 . 2 + 1 9 .26M+ T)
[9 . 379 + 0 . 0 1 607(20 .25)(200 + 460) 1 . 5
------
[209 . 2 + 1 9 . 26(20 . 25) + (200 + 460)]
1 . 84 .
3 . From Fig. 1 . 1 0 , we obtain c,T, = 0 . 63 . Then, the pseudo­
reduced compressibility is
c, =
K=
c,T,
0 . 63
= -- = 0 . 342 .
1 . 84
T,
X= 3 . 448 +
986 . 4
--
T
+ 0 . 0 1 009M = 3 . 448 +
= 130.7.
986 . 4
----
(200 + 460)
+ (0 . 0 1 009)(20.25) = 5 . 1 4 7 .
--
Y= 2 . 447 - 0 . 2224X= 2 . 447 - 0 . 2224(5 . 1 47) = 1 . 302 .
The gas compressibility is
2 . Calculate viscosity with Eq. 1 . 63 .
J.t g = ( 1 x 10 - 4 ) K exp(Xp Y) = ( 1 x 1 0 - 4 ) ( 1 30.7)
exp[(5 . 1 47)(0 . 0 1 03) 1 . 3 02 ] = 0 . 0 1 7 cpo
c,
0 . 343
.
cg = - =
= 5 . 1 2 x lO - 4 pSJa - I .
667 . 4
Ppc
--
1.9 G a s Viscosity
1.10 P roperties of Reservoi r O i l s
The viscosity of a gas mixture can be estimated by interpolation
of tabulated data, 1 2 graphical interpretation, 19 semiempirical cor­
relation, 20 and a single nomograph . 3 All are accurate for sweet
natural gases , but not all are valid for gases containing H 2 S . In
this chapter, we recommend Lee et at. ' s 20 method . This semiem­
pirical method has undeservedly been considered to give poor re­
sults for sour natural gases . However, if the gas density or the z
factor has been corrected for contaminants, this viscosity correla­
tion accurately estimates gas viscosity .
The Lee et al. correlation for estimating gas viscosity is
J.t g = ( 1 x 1 O - 4 ) K exp(Xp Y) , . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 63)
where p= 1 .4935 x 1 O - 3 ( pMlz D . . . . . . . . . . . . . . . . . . . . ( 1 . 64)
Correlations of physical properties of reservoir crude oils are more
complicated and more ambiguous than those for natural gases be­
cause of the many different components that typically make up these
oils. Even though most of the components of an oil are hydrocar­
bons, the larger molecule components can be of different chemical
classes (e . g . , aromatics or paraffins) . These larger components can
strongly influence the behavior of the mixture and, because of the
different chemical structures , make a unique correlation impossible.
Aside from the oil composition, the mixing rules for liquids are
considerably different than those of gases. Most mixing rules for
multicomponent fluids (gases and liquids) are derived froin kinetic
theory . In theory , gas systems are much simpler to model than oil
systems . Also , because of the complex nature of hydrocarbon liq­
uids , simple mixing theories may not apply , and the correlations
must be completely empirical . Note that the correlations presented
here are sufficiently accurate for general reservoir engineering cal­
culations ; however, some were developed from limited databases
and should be applied judiciously .
K=
(9 . 379 + 0 . 0 1 607M ) T I . 5
(209 . 2 + 1 9 . 26M + D
. . . . . . . . . . . . . . . . . . . . . . ( 1 . 65)
986 . 4
X= 3 .448 + -- + 0 . 0 1 009M . . . . . . . . . . . . . . . . . . . . ( 1 . 66)
T
Y= 2 . 447 - 0 . 2224X, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 67)
and where J.t g = gas viscosity , cp; p = gas density , g/cm 3 ; T= tem­
perature , O R ; and M= apparent molecular weight of the gas mix­
ture, Ibm/lb-mol .
The standard deviation in the calculated gas viscosity compared
with experimental data was 2 . 69 % ; the maximum error was 9 . 0 % .
In general , Lee et at. ' s method is valid for 1 00 < p (psia) < 8 , 000
and l OO < TF ( OF) < 340. The correlation also is valid for CO 2
contamination when 0 . 90 < C0 2 (mol % ) < 3 .20.
1 . 10 . 1 Oil FVF. Standing 1 3 , 2 1 developed a correlation for the oil
FVF, B 0 ' for pressures below the original saturation or bub­
blepoint pressure, Pb ' Although other correlations for B o have
since appeared that may be more accurate for specific geological
areas or crude oil types, Standing ' s correlation frequently is ade­
quate . An additional advantage is that it is consistent with the cor­
relation presented later for adequately estimating P b .
For pressures less than or equal to Pb , Standing ' s correlation 2 1
(Fig. 1 . 1 1) is
B o = 0 . 9759 + 0 . 000 1 2F I . 2 , . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 68)
where F= Rs(-Y g l-y 0) 0 . 5 + 1 . 2 5 TF . . . . . . . . . . . . . . . . . . . . . ( 1 . 69)
Example 1 . 13-Estimating Gas Viscosity. Calculate gas viscosi­
ty at 200 ° F and 2 , 000 psia for the sour-gas sample described in
Examples 1 . 2 and 1 . 8 with the Lee et al. method . Pseudocritical
properties corrected for H 2 S and CO 2 were estimated in Example
1 . 2 , and the z factor was estimated in Example 1 . 1 1. Ignore the
nitrogen contamination.
M = 20 . 25 Ibm/lb-mol .
pp c = 7 1 4 . 9 psia.
z = 0. 89 .
Tpc = 372 . 2 ° R .
Solution.
I . First, calculate the variables defined by Eqs. 1 . 64 through 1 . 67 .
pM
(2 ,000)(20 .25)
p = 1 . 4935 x 1 0 - 3 - = ( 1 . 4935 x 1 0 - 3 )
zT
(0. 89)(200 + 460)
------
= 0 . 0 1 03 g/cm 3 .
In Eqs . 1 . 68 and 1 . 69 , B o = oil , RB/STB ; Rs = solution GaR,
scf/STB ; -y g = specific gravity of gas (air = 1 .0) ; -Yo = oil specific
gravity = 1 4 1 .5/( 1 3 1 . 5 +-y API )(water = 1 . 0) ; and TF = temperature,
OF.
The average absolute error o f the correlation given b y E q . 1 . 68
was 1 . 1 7 % with 1 05 data points covering the following ranges of
properties.
1 30 <Pb < 7 ,000 psia.
1 00 < TF < 25 8 ° F .
20 < Rs < 1 425 scf/STB .
1 6 . 5 < -y API < 63 . 8 ° API.
0 . 59 < -y g < 0. 95(air = 1 . 0) .
1 . 024 < Bo < 2 . 05 RB/STB .
The correlation in Fig . 1 . 1 1 generally is applicable for the cal­
culation of B 0 for black oils at pressures below the bubblepoint and
with properties in the above ranges of data.
19
PROPERTI ES OF NATU RAL GASES
PROPCRnES
OF" NA TURAL. HYDR O C A RIJON MIX TUR£3 or CA S ANO UQUID
/'ORMA TION VOLUME 0' 'UBBL E
POIN T UQ UIOS
txA MI't. £
flCOUIRCD :
Fon•• "." ",, 1_ • • , I()O'F" ., ..
•• I.bl. pol'" n""" ",,",'" • , • • . •"
,.,,. ., JS() CF"B• • ,... ".."" , ., 0. 75,
.A� • ,.,,/1: 011 '''' '''''' ., J() 'API.
MO CCDIJR£I
S/.,,""
.1 Me ""
oIrI. of IA. tIIerl,
,,.. . '" ".,I,.,.lan, .""., M. JJ' tFIJ
... 10 • ,.. " "''''' ., ().. 7J . F,.. N.
,"", ' tlrep ,,",Ht"" 10 M. JO 'API ""t.
p,.. .. � "" ''''' 101� (100 - ",. Io"j ""
,,*,,,,)' ...,. 10 16. 1tJ()'F" /tn.. Tj .
...... . I. fl>utW 10 ••
,...",'" _"."
t il />orr. 1 Po' ..,.,.., ., Io,,1e .11.
. -or:
FORMATION
t'OLlJMC ., 'UBItL C POINT
LIQ(I(D -
Fig. 1 . 1 1 -Standing 21 correlation for oil FVF (compliments of Chevron Research & Technology Co. , ©Chevron Research &
Technology Co . , 1 947).
For black oils at pressures above the bubblepoint, the following
relation is rigorous, not empirical , for a liquid if the liquid com­
pressibility is constant:
Bo = Bob exp[co ( Pb -p)] , . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 70)
where Bob = oil FYF at original bubblepoint pressure , RB/STB ;
CO = isothermal compressibility of oil , psi - I ; Pb = original bub­
blepoint pressure , psia; and p = reservoir pressure, psia.
1 . 10.2 Bubblepoint Pressure and Solution GOR. Standing 1 3 , 2 1
also developed the correlation shown in Fig. 1 . 12 . While the cor­
relation can be used to estimate the bubblepoint pressure when bub­
blepoint GOR, Rsb ' can be estimated accurately from early
production data, it also can be used to estimate the solution GOR,
Rs ' for pressures below the original bubblepoint pressure.
Although other correlations for Rs have since appeared, Standing ' s
correlation has proved adequate in general applications .
For pressures greater than or equal to the original bubblepoint
pressure, Standing ' s correlation for estimating Pb is 22
Pb = ( 1 8 X l OYg ) (Rsbll' g ) O . 83 , . . . . . . . . . . . . . . . . . . . . . . ( 1 . 7 1 )
where Y g = 0 . 0009 1 TF - 0 . 0 1 2SI' API ' . . . . . . . . . . . . . . . . . ( 1 . 72)
In Eqs . 1 . 7 1 and 1 . 72 , Pb = original bubblepoint pressure, psia;
I' g = gas specific gravity (air = 1 . 0) ; I' API = oil specific gravity =
( 1 4 1 . S /I' o) - 1 3 1 .S , API ; Rsb = solution GOR at Pb , scf/STB ; and
TF = formation temperature , O F .
T o determine Rs at P �Pb we use
'
Rs =1' g ( p/ 1 8 x l OYg ) 1 . 204 . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 73)
The average absolute error of this correlation was 4 . 80 % for l OS
sample points . The correlation was developed from the same data0
base as Standing ' s oil FYF correlation, so the same limits of pres­
sure , temperature, and fluid properties should apply.
Often, the reservoir discovery pressure is greater than the origi­
nal bubblepoint pressure. Under these conditions , Rs =Rsb '
Example 1 . 14-Estimating Oil FVF and Solution GOR. Use the
following data 23 to estimate Pb and FYF in a reservoir where the
discovery pressure was 4 , 000 psia and current reservoir pressure
is 2 , 1 00 psia. The solution GOR at discovery is estimated from
early production data to be 743 scf/STB . Also, estimate the solu­
tion GOR and oil FYF at current reservoir pressure . Note that a
value of oil compressibility , co ' is required to estimate FYF above
Pb ' We shall see later how to estimate this compressibility from
correlations .
TF = 220 ° F .
I' g = 0 . 786.
co = 14 . 1 3 x lO - 6 psia - I (between 2 , 609 and 4,000 psia) .
I' API = 40 . 7 ° API.
1'0 = 0 . 82 1 7 .
Solution. We first estimate P b using Rsb and Standing ' S corre­
lation given by Eqs . 1 . 7 1 and 1 . 72 . Note that Pb also can be esti­
mated with Fig . 1 . 1 2 .
Y g = 0 . 0009 1 TF - 0 . 0 1 2SI' API = (0 .0009 1 ) (220) - (0. 0 1 2S)(40 . 7)
= - 0 . 3086 .
Solving for Pb with the known value of Rsb yields
0.8 3
743 0.8 3
R
0
86
-30
)
.
lO
X
8
l
(
Pb = ( l 8 x l OYg ) �
=
0 . 786
I' g
= 2 , 609 psia.
( )
(
)
20
GAS RESERVO I R E N G I N E E R I N G
PhOPCII TIES
OF NA rUnAl.
M'DI10 CA IIION
BUBBL e
POIN T
N/rTuRCS
PRFS$UR(
or CAS AN() { lOWlJ
ElfAMP L E
'"blt', ".;,,/ "u ..,,� . 1 2OO -r
. , . Iiftv"� M ..,., . , .. . ... .7 ,.. ".. • r
JSO (Fa, " ,.. ,,.. ,,,'1, .t 0.7S, aM
" II..... ell ,,*.,,',, ., .10 -API.
IIEQUIRED:.
$1.".. , _, IA� "" .,'", .f 1M
�Ita"'. ,,-tutl ""';'."Jell, .,.", /A • .JSO
us ,;, .. ,. " .. . ,,...,.,, , .r (). TS. F,.••
II". " .,." , n.p Writ"'", ,. ".." 'YAP/
.Mil', P,.. u." lwIwri""'/Iy .+_ .. "
AI"• .,7 P"fA",''Y .t. " it IIW JOt) ar
Alit,., TA .. ,.""",. ", �"UII'" i.
rovt.tI ,. 'II' mo P$IA .
PROCEDURE:
\.
,.
. f�
�
"
I'
I·
..
�9
....
4U48l E POINT PRESSURE -
Fig. 1 . 1 2-Standing 2 1 correlation for GOR (compliments of Chevron Research & Technology Co. , © Chevron Research & Tech­
nology Co. , 1 947) .
Thus, our calculations at 2 , 1 00 psia will b e for P <Pb , while cal­
culations for 4,000 psia will be for P >Pb '
Before we can calculate the oil FVF at 2 , 1 00 psia, we need to
estimate the solution GOR at this pressure because it is required
in the correlation for estimating B o '
Solution GOR at P =2,100 psia. To determine Rs at P < Pb , we
use Eq . 1 . 73 .
1 . 204
1 . 204
P
Rs = 'Y g
= (0 . 786)
18 x 10 - 0 . 3 086
18 x I OYg
(
)
( 2,100 )
Bo = Bob exp[co( Pb -p)] = ( 1 . 45)exp [( 1 4 . 1 3 x 1 0 - 6 )
x (2 , 609 - 4 ,000)] = 1 .42 RB/STB .
1 . 10.3 Oil Compressibility . McCain et at. 24 developed correla­
tions for the isothermal coefficient of compressibility, co ' for black
oils at pressures below the bubblepoint. When Rsb at Pb is known,
oil compressibility can be estimated with
In(co) = - 7 . 633 - 1 . 497 In( p) + 1 . 1 1 5 In(T)
= 570 scf/STB .
Solution GOR at p =4,OOO psia. For P >Pb , Rs =Rsb = 743
scf/STB . Note that we have made the simplifying assumption that
'Y g and 'Y API remain relatively constant at both P b and at a later,
lower pressure . Actually , 'Y API tends to decrease, while 'Y g tends
to increase as reservoir pressure decreases below Pb '
Oil FVF at p = 2, 100 psia. To calculate Bo , we use either Stand­
ing ' s correlation given by Eqs . 1 . 68 and 1 . 69 or Fig .
F=Rs('Y g / "{ 0) ° · 5 + 1 . 2 5 TF = (5 70)(0 .786/0. 82 1 7) 0 . 5
+ ( 1 .25)(220) = 832 .
Bo = 0 . 9759 + 0 . 000 1 2F I . 2 = 0 . 9759 + (0 . 000 1 2)(832) 1 . 2
!.II.
= 1 . 36 RB/STB .
Oil FVF at P =4,000 psia. To compute the FVF at P =4,000 psia,
!.II.
we first find Bob at the bubblepoint pressure of 2 , 609 psia using
the Standing correlation (Eqs . 1 . 68 and 1 . 69), or we can use Fig .
F=Rs('Y g /'Y 0 ) ° · 5 + 1 .25 TF = (7 43)(0 . 786/0 . 82 1 7) 0 . 5
+ ( 1 .25)(220) = 1 ,002 .
Bob = 0 . 9759 + 0 . 000 1 2F I . 2 = 0 . 9759 + (0 . 000 1 2)( 1 ,002) 1 . 2
= 1 . 45 RB/STB .
We then estimate Bo at 4,000 psia using Eq. 1 . 70 .
+ 0 . 533 In('Y API) + 0 . 1 84 In(Rsb) ' . . . . . . . . . . . . . ( 1 . 74)
When Rsb is not known, oil compressibility can be estimated
with
In(co) = - 7 . 1 1 4 - 1 . 394 1n( p) + 0 . 9 8 1 1n(T)
+ 0 . 770 In('Y API ) + 0 . 446 In('Y g ) . . . . . . . . . . . . . . . ( 1 . 75)
Note that Eqs . 1 . 74 and 1 .75 calculate the apparent oil compress­
ibility , which includes the compressibility of the liquid and the gas
in solution. In Eqs . 1 . 74 and 1 .75 , Co = oil compressibility, psia - I ;
'Y API = stock-tank-oil gravity, ° API; 'Y g = weighted average of sepa­
rator and stock-tank gas specific gravities (air = 1 . 0) ; and
T= reservoir temperature, O R .
Eqs . 1 . 74 and 1 . 75 were developed with 2 , 500 samples for the
following ranges of data.
3 1 X l O - 6 < co < 6 , 600 x l O - 6 psia - I .
500 <p < 5 , 300 psia.
763 <Pb < 5 , 300 psia.
78 < TF < 330 ° F .
1 5 < Rs < 1 ,947 scf/STB .
1 8 < 'Y API < 52 ° API.
0 . 5 8 < 'Y g < 1 . 2 (air = 1 . 0) .
21
PROPERTIES OF NATU RAL GASES
1000
e o0
TO0
6 00
.
.
500
400
300
200
-
0. 1 0 0
u
80 7Oi-
•
6 o r--
o
� ��
.=
o
30
..
l"-
20
:
I
...
-
10
:
01
8
7
6
o
-
5
III
o
u
1/1
-
\
--
�
.\
_ .
I-,
'"
-
,
--
�
cs.
.
- c--
-
-
r- ,
r".
�
1-
t-
-
�
"I
'\
"\
r"-. �
"
"'"
-
3
2
:;
"
..
- .
f-
-- - -
- - - ._ --
- - _ .- ... .
_.
- 1--
-
,\�,6..�O/.-
.
.
.-
..
- - -.
.
-
.
_..
.
..
..
..
_ .. .
..
- - 1- -
�6
-- .
--
--
.. .. ..
.
.
.
1
. .
.
.. . .
-
.
.
'
_ ..
-
.
.
'
--
.
� ���
--� �I�· -"-· · " . ". -'
I
·�"'cP
:......
"'"
-
'�6
� - . -' · 0' "
-
i
-
..
" ........... "'" r:s: - . _ ' /:" .
r-..,
� �
...... ......
"b", � r::::::- "
i""-..
r---... � 6:' I"-..... ...... r---...
r---..
0.8
0.7
-
-
.
.
.
-_ .
--
_ . -
-
0.6
0.5
0• •
r--
-
. _.
t::: �,.1"" 10p .
;::::�
t....:::� [;:t'-1".......
$O i: �
.-
-
i
- ..
t-- -
0.3
�
�..
"
'O.• r--
�g:
t-
-
0. 2
0.1
10
20
30
40
S t o ck - t a n k a l l g r a v i t y .
•
50
A PI
F i g . 1 . 1 3-Dead·oi l viscosity (after William D . McCai n , Jr. ' s Properties o f Petroleum Fluids,
Second Edition, Copyright PennWell Books, 1 990 1 ) .
Oil compressibility at pressures above p b can be estimated with
the Vasquez-Beggs 2 5 correlation :
Co = (5Rsb + 1 7 . 2 TF - l , 1 801' g + 1 2 . 6 1 1' API - 1 ,433)/( 1 0 5 p) .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 76)
The average absolute error of Eq. 1 . 76 was 0 . 2 84 % for 4 ,036
sample points . Although the error analysis was not performed ex­
plicitly for the oil compressibility , this correlation is expected to
yield results as good as or better for calculating Co in black-oil
reservoirs under the following ranges of reservoir properties .
1 26 <p < 9 ,500 psig .
9 . 3 < Rs < 2 , 1 99 scf/STB.
1 .006 < Bo < 2 . 226 RB/STB .
1 5 . 3 < 'YAPI < 59 . 5 °API.
0 . 5 1 1 < 'Y g < 1 . 35 1 (air = 1 . 0) .
Example L IS-Estimating Oil Compressibility. Use the following
data 23 from Example 1 . 1 4 to estimate oil compressibility at pres­
sures of 2 , 1 00 and 4,000 psia. Recall that Pb was estimated to be
2 ,609 psia.
TF = 220 ° F .
1'0 = 0 . 82 17 .
Rs; = 743 scf/STB .
l' API
= 40 . 7 ° API .
R s = 570 scf/STB .
'Y g = 0 . 786.
Solution .
Oil Compressibility at p =4,OOO psia. In Example 1 . 1 4 , we esti­
mated Pb = 2 , 609 psia. Therefore, using the Vasquez-Beggs 25 cor­
relation (Eq. 1 . 76) for pressures above the bubblepoint gives
Co = (5Rsb + 1 7 . 2 TF - l , 1 80)' g + 1 2 . 6 1 1' API - 1 ,433)( 1 x 1 0 5 p)
= [(5)(743) + ( 1 7 . 2)(220) - ( 1 , 1 80)(0 . 786) + ( 1 2 . 6 1 )(40 . 7)
- 1 ,433] / [ 1 x 1 0 5 )(4 ,000)) = 1 4 . 1 3 x 1 0 - 6 psia - I .
Oil Compressibility at p =2, 100 psia. At p = 2 , l oo psia, we use
the McCain et at. 24 correlation (Eq. 1 . 74) for pressures below the
bubblepoint . For this problem, Rsb = 743 scf/STB .
In(co) = - 7 . 633 - 1 . 497 In( p) + 1 . 1 1 5 In( T) + 0 . 533 In('Y API )
+ 0 . 1 84 In(Rsb) = - 7 . 633 - 1 . 497 In(2 , 1 00)
+ 1 . 1 1 5 In(220 + 460) + 0 . 533 In(40 . 7) + 0 . 1 84 In(743)
or co = 1 80 . 2 x l O - 6 psi - I .
1 . 10.4 Oil Viscosity . On the basis of Beggs and Robinson' s 26
work, Ng and Egbogah 27 developed a correlation for dead-oil (gas­
free) viscosity :
22
GAS RESERVO I R E N G I N E E R I N G
....
. ..
, ...
.
.. ····
.;
mlfBm&JJmm3.M
.... k:-f'�-+-H-�-+-H-+-H�-+-+�-I-H--i-f-+-H--i-f-l
I
• •• •
� "..
of
... .
I
6 1 ",- ,, :
�� ��;�;-. ..
N... ! �� '"
.....
-r';:
! " " �m�I�
1 'Ei'm!��")...:�'g�.�j! ��
�·�:�F:·��!��l�m �
..- �-h.
! ..
•�
�H! -I·�
�.
.. .1 ! \
;
:
.s
\
r-.
"
i\
I �",
•••
a :::
.!
c oo
.
.
:
I
..
-
1
:1:....
""'_
.. �.�
:r'
. :::...
t-.-i1+'N.--I-t+'N.-+t-f"-kl-H-H
....
... t-t:h\·:-I-+-I-t-i'-r.:
-
0 1
0 ..6
0.'
0.'
0.3
0.2
0.1
./
./
V
k"
/' V �
�
0.' 0.6 0.81
2
V
3 ' 5 6 '1 , 1 0
....-
Viscosity o f gas-free
20 3 0 . 0 .6 0 801 0 0
011:/10 0 . C p
-
. _ -
200 300
:. �. .:. . ::...7 :�::g
-- :1-
.
-+-
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 77)
where I-' od = dead-oil viscosity , cp ; 'Y API = specific gravity of oil ,
° API; and TF = temperature, O F .
McCain l prepared Fig. 1 . 13 using Eq. 1 . 77 . The average ab­
solute error of the correlation given by Eq. 1 . 77 was 6 . 6 % for 394
oil systems with properties in the following ranges .
59 < TF < 1 7 6 ° F .
- 58 < Tpour < 59 ° F .
5 .0 < 'Y API < 58 . 0 ° API.
The Beggs-Robinson 26 correlation for live-oil viscosity as a
function of the solution GOR, Rs (in scf/STB) , is
l-'o =A I-' � , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 78)
where A = 10. 7 1 5(Rs + 1 00) - 0 . 5 1 5 . . . . . . . . . . . . . . . . . . . ( 1 . 79)
and B = 5 . 44(Rs + 1 50) - 0 . 33 . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 80)
McCain l prepared Fig. 1 . 1 4 using Eqs . 1 . 78 through 1 . 80 . The
average absolute error of the correlation given by Eq. 1 . 78 was
1 . 83 % for 2 ,073 data points with properties in the following ranges .
0 < p < 5 ,250 psig .
20 < Rs < 2 ,070 scf/STB .
70 < TF < 295 ° F .
1 6 < 'YAPI < 58 ° API .
At pressures above the bubblepoint, oil viscosity can be estimat­
ed with the Vasquez-Beggs 25 correlation,
1-'0 = l-'ob( p/Pb) m , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 8 1 )
where I-' ob = oil viscosity at the bubblepoint pressure , cp , and
m = 2 . 6p 1 . 1 8 7 exp( - 1 1 . 5 1 3 - 8 . 98 x 1 O - 5 p ) . . . . . . . . . . ( 1 . 82)
McCain I prepared Fig. 1 . 15 using Eqs . 1 . 8 1 and 1 . 82 . The
average absolute error of this calculation was 7 . 54 % for 3 , 143 data
points. The correlation was developed for black oils with proper­
ties in the following ranges .
1 25 < p < 9 ,500 psig .
9 . 3 < Rs < 2 , 1 99 scf/STB .
0 . 1 17 < 1-'0 < 1 4 8 . 0 cpo
,., - H-l-- �H-++"i
. I
.1.
. FF -
-
�:.t+
I· "· · "
·
.
�:f-f+J+:n:l:j�
I
- -
�
r-
-
..
.
'II.
:.: �
I i :: : . . : � ;� �. f. �. � �':I�-:I-II-HH
.,
.
. .
.
....
I I,
: :
-
_
... - - - -
- -
I lL.
i
�I- � � '4 ,;••"' ...
Fig. 1 . 1 4-Viscosity of saturated black oils (after William D .
McCain, J r. 's Properties o f Petroleum Fluids, Second Edition,
Copyright Pennwell Books, 1 990 1 ) .
log 1 0 [ log l O (l-'od + 1 )] = 1 . 8653 - 0 . 025086 'Y API - 0 . 5644 10g(TF) ,
--
-
.i
0.8
- -
-
..
-
.
.. ..
_
F . ••
.
-�H"FT-H-H .
. . � - - - :-
�:' :·::t�� ::
� � �...
"If
"�:;
.''=i
' <f:,
'. H:::Y::l
.
-
. .'
8.
1. 1
·
·
-
- l--'i-I-t::I:::I>I--t=f=t-lH-1H
't
;i
.. i
•
·
I
•
..
.
t
�
•
i
i1 f
., �
'oO ..
.
..
f"
!
f.'
.. ,
Fig. 1 . 1 S-Viscosities of undersatu rated black oils (after WiI·
liam D . McCa i n , Jr. 's Properties of Petroleum Fluids, Second
Edition, Copyright Pennwell Books, 1 990 1 ) .
1 5 . 3 < 'Y API < 59 . 5 ° API .
0 . 5 1 1 < 'Y g < 1 . 35 1 (air = 1 . 0) .
Example 1 . 1 6-Estimating Oil Viscosity. Use the following
data 23 from Example 1 . 14 to calculate the oil viscosity at pressures
of 2 , 1 00 and 4,000 psia. Recall that Pb = 2 , 609 psia.
TF = 220 ° F .
'Yo = 0 . 82 1 7 .
Rs; = 743 scf/STB .
'Y API = 40 . 7 API .
Rs = 570 scf/STB .
'Y g = 0 . 786.
0
Solution.
Oil viscosity at p = 2, lOO psia.
1 . To estimate 1-'0 at p = 2 , 1 00 psia, we first estimate the dead­
oil viscosity using the Ng and Egbogah 2 7 correlation given by Eq.
1 . 77 .
log 1 O [log l O (l-'od + 1)] = 1 . 8653 - 0 . 025086 'Y API - 0 . 5 644 10g(TF)
= 1 . 865 - 0 . 02509(40 . 7) - 0 . 5644 10g(220)
or l-' od = 1 . 1 5 cp o
Similarly, we can enter Fig . 1 . 1 3 with 'Y API = 40 . 7 ° API and
TF = 220 ° F and read l-' od = 1 . 1 5 cp o
2 . Next, estimate the live-oil viscosity at 2 , 1 00 psia using the
Beggs-Robinson 26 correlation given by Eqs . 1 . 78 through 1 . 80 .
A = 1 0 . 7 1 5 (Rs + 1 00) - 0 . 5 1 5 = 1 0 . 72(569 . 5 + 1 00) - 0 . 5 1 5 = 0 . 3756,
B = 5 . 44(Rs + 1 50) - 0 . 33 = 5 . 44(569 . 5 + 1 50) - 0 . 338 = 0 . 5887,
and 1-'0 =A I-' � = (0 . 3756)( 1 . 1 52) (0 . 588 7 ) = 0 . 4 1 cp o
Similarly , we can enter Fig . 1 . 1 4 with l-'od = 1 . 1 5 cp and
Rs = 570 scf/STB and read 1-'0 = 0 . 40 cp o
23
PROPERTIES OF NATU RAL GASES
TABLE 1 .8-UN ITS CONVERSION TABLE FOR CONCENTRATION O F
DISSOLVED SOLIDS I N WATER I
Defi nition
Symbol
U n its
Weight percent sol ids
9 solid
S
1 00 9 brine
9 solid
S ppm
Parts per m i l l ion
G rams NaCI per l iter
1 0 6 9 brine
9 solid
S'
1 . Using the Beggs-Robinson 26 correlation given by Eqs . 1 .78
through 1 . 80 , the l ive-oil viscosity at bubblepoint pressure is cal­
culated as follows .
A = 1 0 . 7 1 5 (Rs + 1 00) - 0.5 1 5 = 1 0 . 72(743 + 1 00) - 0 . 5 1 5 = 0 . 3335 ,
B = 5 .44(Rs + 1 50) - 0. 33 = 5 . 44(743 + 1 50) - 0 . 33 8 = 0 . 5472 ,
and /L ob = (0 . 3335)( 1 . 1 52) (0 . 5472 > = 0 . 36 cpo
We next estimate oil viscosity at 4 , 000 psia using the Vasquez­
Beggs 2 5 correlation given by Eqs . 1 . 8 1 and 1 . 82 .
m=2.6p 1 . 1 87 exp( - 1 1 . 5 1 3 - 8 . 9 8 x 1 0 -5p) = 2 . 6(4,000) ( 1 . 1 87)
x exp[ - 1 1 . 5 1 3 - (8 . 9 8 x 1 0 - 5 ) (4,000)] = 0 . 3424 .
The viscosity is
/L o = /L ob ( p/Pb ) m = (0 . 3603) [(4 ,000)/(2 , 620)] (0 . 3424) = 0 . 42
cpo
2 . Alternatively , we can use Fig . 1 . 1 5 . First, enter the vertical
scale on the upper left side of the graph with Pb = 2 , 609 psia. Draw
a horizontal line and intersect the isobar for the reservoir pressure
P = 4 , 000 psia. From this point of intersection, draw a vertical line
downward and intersect the line representing the viscosity at the
bubblepoint pressure , /L ob = 0 . 3 6 cp (Step 1 ) . Finally, from this
S = S ppm x 1 0 - 4
S ppm = S x 1 0 4 =
S' x 1 0 3
S' = S x P w x 1 0 =
l iter brine
Oil Viscosity a t p =4,OOO psia.
Pw
S ppm x P w
103
point, draw a horizontal line and intersect the vertical axis on the
right side of the graph. At this point, we read /L o = 0 . 40 cpo
1 . 1 1 P roperties of Reservoi r Waters
The ability to estimate properties of reservoir waters is important
for reservoir engineering calculations, especially for reservoirs with
water influx. Correlating the physical properties of reservoir water
is relatively simple because water composition generally is affect­
ed only by dissolved solids . Also, changes in the physical proper­
ties of water as functions of temperature and pressure are relatively
small and usually can be predicted .
The concentration of dissolved solids in water can be measured
with several different units, including parts per million, weight per­
cent, and grams of salt per liter. It can be confusing to convert be­
tween these unit systems, so we have included a conversion table 1
(Table 1 . 8) to simplify this process .
The correlations presented below for estimating FVF , solution
gas/water ratio, compressibility, and viscosity are the most accurate
ones available and are adequate for most reservoir engineering cal­
culations .
1 . 1 1 . 1 Water FVF. McCain 1 , 28 developed a correlation for the
water FVF , B w :
I
. q
-J -,--,-­
.
1
.
-+
.
'--1--1-J
-;.- + .j---
i
.
i
0.05
1
I
-+-H-
-0.002
000 ala
-0.004
,000
r
-;- -;
-r-I
0 .02
Conversion Equation
1
'
oJ
'0:1
-0.006
�,OOO
I
psla
-0.008
I
0.01
0.00
50
,
,000 psi.
I
-0.0 1 0
1 00
1 50
200
Temperature. o f
250
300
F i g . 1 . 1 6-.:1 V wT as a function of reservoi r temperature (after
W i l liam D. McCa i n , J r . ' s Properties of Petroleum Fluids, Sec­
ond Edition, Copyright Pennwell Books, 1 990 1 ) .
-0.0 1 2
50
1 00
1 50
200
Temperature, O F
250
300
Fig. 1 . 1 7-.:1 V wp a s a function o f reservoir pressure a n d tem­
perature (after William D. Mccain, Jr.'s Properties of Petroleum
Fluids, Second Edition, Copyright Pennwell Books, 1 990 1 ).
GAS R E S E RVOI R E N G I N E E R I N G
24
50
!Xl
I;;:;:.
u
CI)
}
rr'
40
30
20
10
1 000
2000
4000
3000
5000
6000
P ressure. psia
7000
BOOO
9000
1 00 0 0
Fig. 1 . 1 8-Solublllty of gas i n pure water (after William D . McCa i n , Jr. 's Properties of Petro­
leum Fluids, Second Edition, Copyright Pennwell Books, 1 990 1 ) .
B w = ( 1 + d Vw T)(1 + d Vwp ) ' . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 83)
where the volume corrections for temperature, d Vw T, and pres­
sure, d V"P (Figs. 1 . 16 and 1 . 17, respectively) , are estimated from
d Vw T= - 1 .000 1 O x 1 0 -2 + 1 .3339 1 x 10 -4 TF
+ 5 . 50654 x 1O - 7 T} . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 84)
and d Vwp = - 1 .9530 1 x 1O -9pTF - 1 . 72834 x 1O - p 2 TF
- 3 .58922 x 1 O - 7p - 2 . 2534 1 x 1O - l Op 2 , . . . . . ( 1 . 85)
where B w = water FVF , RB/STB ; TF = temperature , OF; and p =
13
pressure, psia.
Figs . 1 . 16 and 1 . 17 give values of d Vw T and d Vwp as functions
of pressure and temperature. This correlation is accurate to within
2 % of the limited published data. Ignoring salinity causes errors
in d Vw T and d Vwp ' However, because the errors in d Vw T and
d V"P are comparable in value but opposite in sign , the total re­
sulting error is negligible .
Example 1 . 17-Estimating Water FVF. Use the following data
to calculate the water FVF at p = 2,000 psia.
TF = 200°F.
= 2,620 psia.
S=200,000 ppm = 20
Ph
wt % solids.
Solution.
1 . First, determine the effects of temperature and pressure on
the water volume. These effects can be estimated either by reading
directly from Fig s . 1 . 16 and 1 . 1 7 or by calculating with Eqs . 1 . 84
and 1 . 85 . For this example, we use the latter approach .
d Vw T= - 1 .0001O x 10 -2 + 1 .3339 1 x 1O -4 TF + 5 .50654
x 1 O - 7 T} = - 1 .000 1 O x 10 - 2 + 1 .3339 1 x 10 -4(200)
+ 5 . 50654 x 10 - 7 (200) 2 =0.03870.
d V"P = - 1 .95301 x 1O -9pTF - 1 .72834 X 1O - 13 p 2 TF - 3 .58922
x 1 O - 7p - 2 .2534 1 x 1 O - l Op 2 = - 1 .95301 X 10 -9
x (2,000)(200) - 1 . 72834 x 10 (2,000) 2 (200)
- 3 .58922 x 10 - 7 (2 ,000) -2 .2534 1 x 10 - 10 (2 ,000)2
= -0.00254.
- 13
2. The water FVF is
B w = ( 1 + d Vw T)( 1 + d Vwp ) = ( 1 +0.03870)(1 -0.00254)
= 1 .04 RB/STB .
1 . 1 1 .2 Solution Gas/Water Ratio. McCain 1 , 28 also developed a
correlation for the solution-gas/pure-water ratio, R swp ,
Rs wp =A +Bp + Cp 2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 86)
and the partial derivative of the solution-gas/pure-water ratio with
respect to pressure,
(aR s wp laph=B+2Cp . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 87)
The coefficients of Eqs . 1 . 86 and 1 . 87 are
A = 8 . 1 5 839 - 6 . 1 2265 x 1 O - 2 TF + 1 . 9 1 663 x 1 O -4 T}
- 2 . 1 654 x 1 0 - 7 Ti , . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 88)
B= 1 .0 1 02 1 x 10 -2 - 7.4424 1 x 1O -5 TF + 3 .05553 x 1O - 7 T}
- 2 . 94883 x 10 - l O Ti , . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 89)
and C= - 1O - 7 (9.02505 -0. 1 30237TF + 8 . 53425 x 1 O - 4 T}
- 2 . 34 1 22 x 1 O - 6 Ti + 2 . 37049 x 1 O - 9 Ti ) . . . . . . . . ( 1 .90)
Values of the derivative of the solution-gas/pure-water ratio will
be needed for compressibility calculations at pressures below the
bubblepoint. In Eqs. 1 .87 through 1 .90, TF = temperature, O F , and
R s wp = solution-gas/pure-water ratio , scf/STB .
Fig. 1 . 18, which McCain generated with Eq. 1 . 86, gives values
of R S "P as a function of pressure and temperature. Similarly , Fig.
1 . 19 was generated with Eq. 1 . 87 and gives values of aR s wp lap
as a function of pressure and temperature. The accuracy of the corre­
lations given by Eqs . 1 .86 and 1 . 87 is within 5 % of published data.
In addition, the correlations were developed for 1 ,000 <p < 10,000
psia and 1 00 < TF < 340°F. Note that these correlations are ap­
plicable to formation waters , but McCain , 2 8 states that they
should never be used at pressures below 1 ,000 psia. Further, the
relations were developed strictly for pure-water systems and must
be modified for reservoir brines.
McCain proposed the following correlation to adjust pure-water
values to those for brine systems :
1
1
R s w lR s wp = l O ( - O. 0840655 S TF'°·285584) ,
.
. . . . . . . . . . . . . . (1 .91)
25
PROPERTI ES O F NATU RAL GASES
6.0X l 0
'in
';'-
:g
�
.i:
,lI-.
�
�
.,
5.5
5.0
4.5
4 .0
3.5
3.0
2.5
6.0xl 0
Pressure, ps;a
�
:0
�
..!:
5.0
4.0
ft
3.5
..
�
•
c
3.0
2.5
25
20
30
1.0 ----....---r--r---,,-,-Ir--,-
4 .5
,%
15
Tolal dissolved solids. %
Fig. 1 .20-R sw lR swp as a function of dissolved solids (after
William O. McCa i n , Jr. 's Properties of Petroleum Fluids, Sec­
ond Edition, Copyright Pennwell Books, 1 990 1 ) .
.,
5.5
�
10
1 0 x 1 03
:i
0
�
2
Pressure, psia
1 0x l 03
.
U
..
:
•
•
•
0.11
!;
Go
;;
•
U
5
10
Total dls.olved solids, wt
%
'15
20
Fig . 1 .2 1 -Effect of sal i n ity on the coefficient of isothermal
compressibility of water (after OSif 2 9 ) .
Pressure,psia
1 0x l 03
F i g . 1 . 1 9-0erivative of R swp with respect to pressure .
where S= salinity, wt% solids; TF = temperature, o f ; Rswp = solu­
tion-gas/pure-water ratio, scf/STB; and Rsw = solution-gaslbrine ra­
tio, scf/STB.
The correlation given by Eq. 1 . 9 1 and Fig. 1 .20 agreed to with­
in 3 % of published data falling within 0 < S < 30 wt % solids and
70 < TF < 250°F. The correlation is applicable to formation waters
and is used to correct the solution-gas/pure-water ratio and its
derivatives.
( )
1 OB w
= (7.033p + 54 1 .5S' - 537.0TF
B w op T
+ 403 ,300) - 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 .92)
where c w = isothermal compressibility of formation water, psi - 1 ;
cw = - -
-
p = pressure , psia ; S' = salinity , g NaCIIL ; and TF = temperature ,
of.
The correlation was developed for the following range of data.
1 ,000 <p < 20,000 psig.
0 < S' < 200 g NaCI/L.
200 < TF < 270°F.
Note that inaccurate results will be obtained if this correlation
is applied for reservoirs outside the range of data used to develop
the correlation.
Osif also developed Fig. 1 . 2 1 , a plot of the ratio of the compres­
1 . 1 1 .3 Water Compressibility. Because the reservoir water com­
sibilities of brine and pure water vs. total dissolved solids content.
pressibility is the property that is affected most by the presence of As an alternative to Eq. 1 . 92 , we can use Fig. 1 .22 to obtain the
free gas, we must use an expression for compressibility that ex­ compressibility of fresh water and then use Osifs correlation (Fig.
plicitly accounts for the presence of free gas at pressures bel ?w 1 .2 1 ) to correct for the dissolved solids content. Note that the amount
the bubblepoint. The bubblepoint pressure of a gas-saturated brme of dissolved solids has a very small effect on the water compressi­
is equal to the bubblepoint pressure of the coexisting oil. In a bility.
gas/water system, the water is considered to be at its bubblepoint
Ramey 3 l reported the following theoretically based relation for
pressure at the initial reservoir pressure. Quantitatively, the solu­ the formation water compressibility at pressures below the bub­
bility of gas in water is considerably less than that of gas in oil. blepoint:
For a pressure greater than or equal to bubblepoint pressure,
I OB w
Rsw
B
Osif29 developed a correlation to estimate the isothermal coeffi­
Cw = _ _ _
+ g
. . . . . . . . . . . . . . . ( 1 .93)
cient of reservoir water compressibility at pressures above the bub­
B w op T B w op T
blepoint pressure of the water.
( )
C )
26
GAS RESERVO I R E N G I N E E R I N G
3. Next, estimate the first term on the right side of E q . 1 .93 .
From Fig . 1 .2 1 , the ratio of the compressibilities of brine and pure
water is cwlc wp =0.7 1 . Note that we must interpolate to obtain this
ratio . From Fig . 1 .22, the compressibility of pure water is
c wp = 3 .2 X 1 0 - 6 psi - 1 . Therefore, the compressibility of brine is
C w = c wp
( :: ) = (3 . 2 x
( )
1 O - 6 psi - I )(0.7 1 ) = 2 . 3 x 1 O -6 psi - 1
l
1 OB w
= 2 . 3 x l O -6 psi - .
B w op T
or -
--
We also can estimate this value using Eq.
Temperature.
Fig. 1 .22-Coefficient of isothermal compressibility of fresh
water (after Oslf 2 9 and Dodson and Standing 30 ) .
The first term o n the right side o f E q . 1 .93 , which represents
the compressibility of the water only , is calculated with either Eq.
1 .92 or Figs. 1 .2 1 and 1 .22. The second term on the right side
represents the effects of the gas on the total water compressibility .
McCain 1 suggests the following procedure for calculating the sec­
ond term.
1 . Estimate Bg using I' g =0.63. McCain states that this value is
based on limited data with unknown accuracy, but it gives reasonable
results in further calculations .
2 . Estimate B w using Eqs . 1 . 83 through 1 . 85 and the procedure
described in Sec . 1 . 1 1 . 1 .
3 . Estimate oRswp lop from Eq. 1 . 87 or Fig . 1 . 19, and multiply
this value by the dissolved-solids-content correction factor from Fig.
1 .20.
Because this relation has not been verified with experimental data,
McCain considered the estimate of formation water compressibil­
ity with Eq . 1 .93 an approximation.
A water sam­
ple was taken from an oil reservoir having P b =2,620 psia. Using
Eq . 1 .93 and the procedure suggested by McCain, 1 estimate the
water compressibility at the current reservoir pressure , p = 2,000
psia. Include the effects of the dissolved salt content.
Example I . IS-Estimating Water Compressibility.
TF = 200°F, S' = 230 g NaCIIL = 200,000 ppm = 20 wt %
solids.
Solution.
1 . First estimate the gas FVF, Bg • For this estimate, we assume
I'g =0.63 . Using the methods described in the previous sections on
natural gas properties , we estimate the gas deviation factor to be
z =O.90. Bg at 2 ,000 psia and 200°F = 659.6rR is computed with
Eq. 1 . 54:
Bg =
2.
5.02zT (5 .02)(0.90)(660)
= 1 . 49
=
2 ,000
p
--
1 OBw
= 1 I(7 .033p + 54 1 .5S-537.0TF + 403 ,300)
B w op T
= 1 1[7 .033(2,000) + 54 1 . 5(200) - 537.0(200)
-
of
RB/Mscf.
Next, we estimate the FVF for water. From Eq.
B w = { 1 +.1VwT)(1 +.1Vwp ) ,
1 . 83 ,
where .1 Vw T and .1 V can be estimated with either Figs . 1 . 16 and
1 . 17 or Eqs . 1 . 84 and 1 . 85 . For this example , we use the graphs .
Example 1 . 17 illustrates calculation of these corrections with the
equations . From Fig . 1 . 16, .1 Vw T=0.0386. Similarly , from Fig .
1 . 1 8 , .1Vwp = -0.00255 . Note that linear interpolation may be re­
quired for pressures different from those plotted on the graph . The
water FVF is
B w = { 1 +.1VwT)(1 +.1Vwp ) = { 1 +0.0386){ 1 -0.00255)
= 1 .04 RB/STB .
( )
1 .92.
-
+ 403,300] = 2 . 39 x l O -6
psia - I ,
which is similar to the value obtained from the graphs .
4. Next, we estimate the second term on the right side of Eq.
1 .93 . We illustrate this calculation using both the graphical method
(Step 4A) and the equations (Step 4B) .
A . First, estimate R swp and (oRswp lop) T at 2 ,000 psia. From
Fig . 1 . 1 8 , the solubility of gas in pure water is R s wp = I 1 .2
scf/STB . Similarly, from Fig . 1 . 19 , the derivative is (oR s wp lop) r
=4.2 x 10 -3 scf/STB/psi.
Now we will correct the pure-water values to the appropriate brine
system using the McCain 1 correlation given by Fig . 1 .20. From
Fig . 1 .20, R s w IR s wp =0.43. Therefore,
R sw =Rswp(Rsw IRs wp) = ( 1 1 .2)(0.43) = 4 . 82 scf/STB
Rswp
R sw
ORsw
= (4.2 x lO -3 )(0.43)
= O
and
op T R s wp
op T
= 1 . 8 1 X 1 0 -3 scf/STB/psi.
( ) (
--
--
)( )
--
B . Similar results can be obtained by use of Eqs . 1 . 86 through
1 .92 . From Eqs . 1 . 88 through 1 .90,
A = 8 . 1 5839 -6. 1 2265 x 1O -2 TF + 1 .9 1 663 x 1 O -4 T;'
- 2 . 1 654 x 1 0 - 7 Tj. = 8 . 1 5839 - (6 . 1 2265 x 10 -2 )(200)
+ ( 1 .9 1 663 x 10 -4)(200) 2 - (2 . 1 654 X 1 0 - 7 )(200) 3
= 1 . 847 scf/STB .
B = 1 .0 1 02 1 x 1 0 - 2 - 7 . 4424 1 x 1 O -5 TF + 3 .05553 x 1 O - 7 T;'
- 2 . 94883 x 10 - I O Tj. = 1 .0 1 02 1 x 1 0 - 2 - (7 . 44241 x 1 0 -5 )
x (200) + (3 . 05553 x 10 - 7 )(200)2 - (2 . 94883 X 1 0 - 10 )(200) 3
= 5 . 0803 x 10 - 3 scf/STB/psia,
and C= - 10 - 7 (9.02505 -0. 1 30237TF + 8 .53425 x 1 O -4 T;'
- 2 . 34 1 22 x 1 O - 6 Tj. + 2 .37049 x 1 O - 9 Ti )
= - 10 - 7 [9.02505 - (0. 1 30237)(200) + (8 .53425 x 10 -4)
x (200) 2 - (2 . 34 1 22 x 1 0 -6)(200)3 + (2 . 37049 x 1 0 -9)
x (200)4 ] = - 2 . 1 777 X 1 0 - 7 scf/STB/psia 2 .
From Eq. 1 . 86, the solution-gas/pure-water ratio is
Rswp =A +Bp + Cp 2 = 1 . 847 + (5 . 0803 x 1 0 - 3 )(2 ,000)
+ ( - 2 . 1 777 x 10 - 7 )(2,000)2 = 1 1 . 1 37 scf/STB .
From Eq. 1 . 87, the derivative of R swp with respect to pressure is
ORs wp ) = B + 2 Cp = (5 .0803 x 1 0
( -op T
X (2 ,000) =4.209 X lO -3
- 3 ) +2( - 2 . 1 777 x
scf/STB/psi.
10 -7 )
27
PROPERTI ES OF NATU RAL GASES
2.2
2.0
1 .8
];
�
::t.
1 .6
1 .4
i-
1 .2
..
1 .0
0;
0
<J
�
>
ell
if
�
0.8
0.6
0.4
0.2
F i g . 1 .23-Water viscosity a s a function of sali nity a n d temperature (after William D . McCain ,
J r . 's Properties o f Petroleum Fluids, Second Edition, Copyright Pennwell Books, 1 990 1 • 28 ) .
1 . 1 1 .4 Water Viscosity. McCain 1 , 28 developed a correlation for
the formation water viscosity at atmospheric pressure and reser­
voir temperature, Jl- w l :
R
ST - 0 285584 )
Jl- w l =A T! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 .94)
� = I O ( _ 0.0 840655 p
R s wp
where A = 109.574 - 8.40564S+0. 3 1 33 1 4S2
+ 8 . 722 1 3 x 1 0 - 3 S 3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 .95)
285584
] = 0.4268.
= 1 0 [ - 0.0840655 (20)(200) - 0 .
and B= - 1 . 1 2 1 66 + 2 . 6395 1 x 1 0 - 2 S-6.7946 1 X 1 O -4S2
Therefore, the corrected values are
- 5 .47 1 19 x 1 0 - 5 S 3 + 1 .55586 x 10 -6S4 . . . . . . . . . ( 1 . 96)
R s w =R s wp (R ,w IR s wp ) = ( 1 1 . 1 3 6)(0 .4268) = 4.754 scf/STB
The correlation, given by Eq. 1 .94 and plotted in Fig. 1 .23, is
iJR s wp
iJR ,w
R ,w
accurate to within 5 % of published data and was developed for
= (4 .209 x 10 - 3 )(0. 4268)
and
=
1 00 < TF < 400°F and 0 < S< 26 wt % solids.
iJp T
iJp T R s wp
Once Jl- w l is estimated, the formation water viscosity at reser­
voir pressure, Jl- w ' is estimated with McCain's 1 , 28 correlation:
= 1 . 797 x 1 0 - 3 scf/STB/psi.
Jl- w
5. Finally, we have all the components to calculate water com­
=0.9994 + 4 . 0295 x 1 O -5p + 3 . 1 062 x 1 O -9p 2 . . . . ( 1 .97)
pressibility using Eq. 1 .93 . With the results obtained from the
Jl- w l
graphs,
This correlation, which is accurate to within 4 % for p < 10,000
psia and within 7 % for 1O,000 <p < 1 5 ,000 psia, was developed
for the limited temperature range of 86 < TF < 1 67 ° F. Correlations
for Jl- w l Jl- w l also are plotted in Fig. 1 .24.
Now we will correct the pure-water values to the appropriate brine
system using the correlation given by Eq. 1 .9 1 .
( ) (
(
)( )
)
1 .49 1 . 8 1 X IO - 3
= 4 . 89 x 1 0 -6 psia - 1 .
+ -1 ,000
1 .04
Similarly, with the results obtained from the equations,
Example 1 . 19-Estimating Water Viscosity. A water sample was
taken from an oil reservoir having Ph = 2,620 psia. Estimate the
water viscosity at p = 2 ,000 psia.
TF= 200°F.
S=200,000 ppm = 20 wt % .
Solution.
(
)
1 .49 1. 797 x 1 0 - 3
+ -= 4 . 97 x l O -6 psi - 1 ,
1 ,000
1 .04
which is comparable with results obtained from the graphs. Although
the arithmetic required for C w calculations with the equations is
quite tedious, the equations generally are more accurate than the
graphical method. In addition, the equations can be programmed
for computer calculations . The graphical method, however, is de­
sirable when speed is more important than precision.
1 . First, we estimate Jl- w l using either Fig. 1 .23 or Eq. 1 .94. For
this example, we have chosen to use the equation. The variables
A and B in Eq. 1 .94 are
A = 109 . 574 - 8.40564S+0. 3 1 33 1 4S2 + 8. 722 1 3 x 10 - 3 S3
= 1 09 .574 - 8 .40564(20) + 0. 3 1 33 1 4(20) 2 + 8. 722 1 3
x 1 0 - 3 (20) = 136.56
and B = - 1 . 1 2 1 66 + 2 . 6395 1 x 10 - 2 S-6.79461 X 1 O -4S2
- 5 .47 1 1 9 x 10 - 5 S 3 + 1 .55586 x 1 0 - 6 S4
= - 1 . 12 1 66 + 2 .6395 1 x 10 -2 (20) -6.79461 x 10 -4 (20)2
- 5 .47 1 19 x 1 0 -5 (20) 3 + 1 .55586 x 1 0 -6 (20)4 = - 1 .0543 .
28
GAS RESERVO I R E N G I N E E R I N G
2.0
e
;;
J
}
1 .8
.Q
1 .6
C:iii
1 .4
�
8
(I)
">
Q;
co
::
1 .0
2000
0
4000
6000
8000
1 0000
1 2000
Pressure, psla
Fig. 1 .24-Water viscosity ratio (after William D. McCa i n , Jr. 's Properties of Petroleum Fluids,
Second Edition, Copyright Pennwell Books, 1 990 1 , 28 ) .
TABLE 1 .9-FUNCTION F O R WATER VAPOR CONTENT
I N NATU RAL GAS
Temperature
( O F)
Vapor Pressure
(psia)
A(T)
B(T)
32
60
1 00
1 50
200
250
300
350
400
0 . 0885
0 . 256
0 . 942
3 . 72
1 1 .5
29.8
67.0
1 35 . 0
247 .0
4,21 0
1 2 ,200
45 , 1 00
1 77 , 000
547, 000
1 ,420 , 000
3 , 1 80,000
6 , 390,000
1 1 , 700 ,000
2.65
5 . 77
1 5 .3
43.2
1 04
222
430
775
1 ,360
From Eq . 1 . 94 ,
I-' w l =A TP = ( 1 36. 56)(200) ( - 1 . 0 5 43 ) = 0 . 5 1 2 1 cpo
A more rapid alternative to Eq. 1 . 94 is to use Fig . 1 . 2 3 , which
yields a similar value, I-' w l = 0 . 5 1 cp o
2 . Next, we must correct I-' w l to I-' w ' This correction is obtained
with Eq. 1 . 97 .
I-' w
- = 0 . 9994 + 4 . 0295
I-'w l
+ 4 . 0295
Therefore ,
I-' W = I-' W l
x
I-' w l
1 O - 5p + 3 . 1 062 X 1 O - 9 p 2 = 0 . 9994
1 0 - 5 (2 , 000) + 3 . 1 062
( )
I-' w
x
x
1 0 - 9 (2 ,000) 2 = 1 .0924.
= (0 . 5 1 2 1 ) ( 1 .0924) = 0 . 5 6 cpo
Similarly , from Fig . 1 . 24, I-' w l l-'w l = 1 .09 , and I-'w = I-'w l (I-'w l l-'w l )
= (0 . 5 1 ) ( 1 .09) = 0 . 56 cp, which agrees with the value calculated
with Eqs . 1 . 94 and 1 . 97 .
1 . 1 2 Water Vapor Content of Gas
The water vapor content of a gas depends on pressure , tempera­
ture , and composition of the gas . However, the gas composition
has more effect on the water vapor content at higher pressures .
Bukacek3 2 found empirically that the water vapor content of na­
tural gases can be expressed as
The function A ( T) varies directly with the vapor pressure of water.
The function B(T) falls on a straight line on a plot of log [B( T)]
vs. the reciprocal of absolute temperature . The data in Table 1 . 9
for A ( T) and B( T) are from Bukacek. 3 2 Bukacek gives the values
of A ( T) and B(T) at increments of 2 ° F from - 40 to 260 °F and
then at increments of 20 ° F up to 460 ° F . From a regression analy­
sis of the data, the variations of A ( T) and B(T) with reciprocal func­
tions of temperature in equation form are
log A ( T) = 1 O . 935 1 - 2 ,949 . 05 T- l - 3 1 8 ,045 T- 2 . . . . ( 1 . 1 02)
and log B( T) = 6 . 69449 - 3 ,083 . 87 T - l , . . . . . . . . . . . . . . ( 1 . 1 03)
where Tis in OR. These data do not include corrections for salinity
or gas gravity .
McCain 1 developed Fig. 1 .25 from the Bukacek ' s data. Figs.
1 . 26a and 1 .26b include corrections for salinity and gas gravity
proposed by McKetta and Wehe . 33 Note that Fig . 1 . 25 includes
a hydrate formation line ; i . e . , at temperatures less than values along
this line for a given pressure , hydrates may form (gas hydrates are
discussed in the next section) . The lines also end at the vapor pres­
sure of water at a given temperature .
Laboratory analyses of natural gases usually do not quantify the
amount of water vapor, so these analyses can be corrected for water
vapor content as follows.
1 . From Fig . 1 .2 5 , read the water vapor content (L e . , the solu­
bility of water in natural gas) , W. Read correction factors for sa­
linity and gas gravity from Figs. 1 . 26a and 1 . 26b . Multiply the
value of water vapor content by these factors to correct for salinity
and gas gravity .
2 . Calculate the mole fraction of water, Yw in the gas from
'
W
( WIMw) ( WI 1 8 .0 1 5)
, . . . . . . . . . . . . . ( 1 . 98)
=
Yw =
47 , 325
ng
2 , 627
where W= water vapor content, Ibm/MMscf; Mw = molecular
weight of pure water = 1 8 . 0 1 5 lbmllbm-mol ; and n g = number of
pound-moles of total gas per million scf at Psc = 1 4 . 65 psia and
Tsc = 60 ° F , or
---
--
1 06
106
� 2 , 627 Ibm-mollMMscf. . . . . . . . . . ( 1 . 99)
ng = - =
Vm
3 80 . 7
3 . Correct laboratory mole fractions o f all components o f the gas:
( Y; ) c = ( l - Yw ) ( Y; ) lab ' . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1.1 00)
W=A ( T) lp + B( T) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 1 0 1 )
where W= water vapor content, Ibm/MMscf; p = absolute pressure,
psia; and A ( T) and B( T) = functions that depend on temperature .
Example 1 .20-Calculation of Water Vapor Content. Calculate
W vs . p at 300 ° F with values of p of 2 , 000, 4 , 000, 6,000, and
29
PROPERTIES OF NATU RAL GASES
8 ,000 psia using Bukacek ' s results . Compare the results with values
from Figs. 1 . 25 and 1 . 26 .
Solution. A t 3 00 ° F , A ( T) = 3 , 1 80,000 and B(T) = 430; therefore,
the equation to calculate the water vapor content is
W= 3 , 1 80,0001p + 430.
' 0
'
-H++++H-;+++-HH-'
Table 1 . 10 gives values of W calculated from the above equa­
tion and read from Fig s . 1 .25 and 1 . 26.
1 . 1 3 Gas Hyd rates
Hydrates are physical (not chemical) combinations of water and
natural gas formed at pressures and temperatures considerably above
the freezing point of water. They are crystalline solids formed when
natural gas is in the presence of free water at or below a tempera­
ture called the " hydrate temperature . " The formation of hydrates
is not the same as the condensation of water vapor under pressure
at or below the dewpoint temperature; however, the condensed water
does provide the free water in the system necessary for hydrate for­
mation.
The primary conditions that promote hydrate formation 3 , 4 are
( 1 ) gas at or below its water dewpoint with free water present,
(2) low temperatures, and (3) high pressures. Conditions that favor
hydrate formation include ( 1 ) high velocities , (2) pressure pulsa­
tions, (3) introduction of a small hydrate crystal , (4) physical site
for hydrate formation (pipe elbows, an orifice , choke, or backpres­
sure regulators) , (5) agitation, and (6) presence of H 2 S and CO 2 ,
Operating conditions favorable to hydrate formation can be catego­
rized two way s : hydrate formation at constant pressure caused by
a sudden decrease in temperature (Fig. 1 .27) or resulting from a
sudden expansion through a flow restriction (Figs. 1 .28 through
1 .32) .
1 . Hydrate formation at constant pressure caused by a sudden
decrease in temperature in flow string or surface lines. The hy­
drate temperature, a function of pressure and gas gravity (compo­
sition) , may be estimated from Fig . 1 . 2 7 . Hydrates will form if
the temperature and pressure conditions plot to the left of the hy­
drate line for the given gas gravity . Note that Fig . 1 . 27 applies to
sweet natural gases only . The presence of contaminants (H 2 S and
CO 2 ) will increase the possibility of hydrates being formed and
must be considered .
2 . Hydrate formation resulting from a sudden expansion through
a flow restriction . A sudden expansion will be accompanied by a
temperature drop that may promote hydrate formation . Figs. 1 . 28
through 1 . 32 can be used to determine conditions for hydrate for­
mation. These figures, plotted for specific gas gravities , can be used
for gases with intermediate gravities by linear interpolation between
figures. These figures also are limited to sweet natural gases and
'0
�
::;;
' 0
'
£l
E
E
"
0
(J
8.
..
>
Q;
OJ
;:
1 0
1 0
'
,
300
200
1 00
Temperature, OF
500
400
Fig. 1 .25-Moisture content of natural gases at low pressures.
TABLE 1 . 1 0-WATER VAPOR CONTENT OF
NATU RAL GAS, EXAMPLE 1 .20
P ressu re
(psia)
2 ,000
4,000
6 , 000
8 ,000
W From Eq . 1 . 1 0 1
W From Figs. 1 . 25 and 1 .26
2 , 020
1 , 225
960
827.5
2 ,000
1 ,2 1 0
940
800
(lb m/M M scf)
(lbm/M Mscf)
Example 1 .21-Hydrate Formation Caused by Sudden Decrease
in Temperature. 3 A sudden decrease in temperature occurs in a
flowline.
A
gas with a specific gravity of 0 . 8 is at a pressure
may not predict hydrate formation accurately when applied to sour
of 1 ,000 psia. Determine the extent to which the temperature can
natural gases .
drop without hydrate formation , assuming the presence of free
water.
Molecular Weight
30
1 .0
35
1 .2
4 0
1 .4
45
1 .6
50
1 .8
Gas Gravity
Fig. 1 . 26-Water-vapor-content correction factors for (a) sal i nity and (b) gas (after McKetta and Wehe 33 ) . (Reprinted with per­
m ission from Petroleum Refiner, Aug. 1 958, pp. 1 53-1 54, by Gulf Publishing Co . , all rights reserved.)
30
GAS RESERVOI R E N G I N E E R I N G
a
...
Q.
6 00 0
/ IlL
i 3000
0
<
�
1 000
a:::
a
u..
600
w
<
a::
300
0
>:I:
/
� �� l'/
_�t".�
�"''Y
/
./
/
./
/ fJI
/, 'I
0
//J
..
Q.
�
w
a:
1 /8
/ r//
::J
V)
V)
w
0<
c..
/M
�� V/ff
7
-'
<
�
��.'//;�
�V V/#
1 00 ��
a:::
a
u..
w
a::
::>
\I)
\I)
w
,.."7
I N I T I AL
170}
TEMPERATURE OJ:...-""
5000
Z
0'/
'Y
40
so
3000
70
80
T E MPERAT U RE . · F
90
I /, 1 J
I
l3S.....
-IQS :::>
�"...
- �,...... "...
2000
�
1 000
:.;..
�.-
500
�C?'
300
60
, �o
- 1-\50
� i-" i-"
-�
'fo.�."
a::
a...
r
I
./V
V
�
V'
�,
i-'"
V
,V
�
L����-7.���������
l oo.100
2 00
400 600
1000
F I N A L PRESSU R E , P �Q
2000
4 000
Fig. 1 .27-Pressure vs. temperature curves for predicting gas
hydrate formation (after Katz et al. , Handbook of Natural Gas
Engineering, 1 95 9 , M cGraw- H i l i Book CO. , 4 courtesy of
McGraw- H i l i I n c . ) .
Fig. 1 .2S-Permlsslble expansion of a O .6-gravity natural gas
without hydrate formation (after Katz et a/. , Handbook of Na­
tural Gas Engineering, 1 959, McGraw-Hili Book Co. , 4 courte­
sy of McGraw- H i l i I n c . ) .
Solution. From Fig . 1 .27, at a specific gravity of 0 . 8 and a pres­
sure of 1 ,000 psia, the hydrate temperature is 66 °F. Thus, hydrates
may form at or below 66 ° F .
through a flow restriction occurs in a surface line. A gas with a spe­
cific gravity of 0 . 8 is flowing at an initial pressure of 1 ,000 psia and a
temperature of lOO ° F. Determine the lowest pressure to which the gas
can expand without hydrate formation if the gas is flowed through a
flow restriction in the surface line. Assume the presence of free water.
Determine the lowest pressure to which the gas can expand
without hydrate formation if the initial pressure is 800 psia. Also
Example 1 .22-Hydrate Formation Resulting From Sudden Ex­
pansion Through a Flow Restriction_ 3 A sudden expansion
10,000
8000 �188
ITIA
" 'II
"'/
t./1 I A I
6000 �
TEMP E RATURE. o F " y J
sooo
�ooo
o
J
/
�
/11
/
\19:
"",
� "..
.- �
3000
·� 2000
,/jIJh
// V/
0V
""" :\ 0
t:f��
�o_
::: 1 0 00
a..
�
70
� "..,.
so
300
200
..!2.. 1;-"
,
�v
llXloo
..
,,
200
,
,
�'
"..
,
J
I
,
/
f.0'w /
��
w
""
::>
en
en
J1/
j ,'
,
\
r
,
.
�.
//
,
,
,,
200 r--r-��;--t�-+�t++--+--+-4-+�
,/
�O , '
300 4 00 600 800 1000
F I NAL PRESSURE. psio
2000
4000
Fig. 1 .29-Permissible expansion of a D .7-gravity natural gas
without hydrate formation (after Katz et a/. , Handbook of Na­
tural Gas Engineering, 1 95 9 , McGraw-Hili Book Co. , 4 courte­
sy of McGraw- H i l i I n c . ) .
,,
1 00lOO
�-�=
2 00
��8��
��-�����OOO
��3=
07
00�-�
0 -4�
00 �
2000
F I N A L P R E S S U R E , psio
•
Fig. 1 .3D-Permissible expansion of a D .S-gravity natural gas
without hydrate formation (after Katz et al., Handbook of Na­
tural Gas Engineering, 1 95 9 , McGraw-Hili Book Co. , 4 courte­
sy of McGraw- H i l i I n c . ) .
31
PROPERT I E S OF NATU RAL GASES
I N I T IA L TEMPERATURe,
of 1\
:;90
3000
\80
. 2 2000
-166- .
Q.
'"
::>
w'
,
'"
Q.
;::
�
II /
IL L Wlt /. if,
/
3000
/
.
,bQ
Q.
�
500
60
300
50
200
/
/
/
/
/
I?
/
Q.
0::
::>
'"
on
w
0::
�-
�-
g
W
�
�. 1000
�
I
,'10
<I.
;::
/
�
210
I-t-
190 I
180
170
160
2000
1000
...
150
140
90
.....
130
500
70
/
/
-7
1 . 1 4 PV Compressibility Correlations
Newman 3 4 used 79 samples for consolidated sandstones to devel­
op a correlation between PV compressibility, Cf ' and porosity, cf>.
.�
Fig. 1 .32-Permisslble expansion of a 1 .0-gravlty natural gas
without hydrate formation (after Katz et a/. , Handbook of Na­
tural Gas Engineering, 1 959, McGraw-Hili Book Co. , 4 courte­
sy of McGraw- H i l i I n c . ) .
The data shown in Fig.
E
8
.
.
Example 1 .23-Estimation of PV Compressibility.
for a formation of unknown lithology and cf> =0.2.
Calculate cf '
'
1 0
•
1 0
'
1 0
.,
E
'0
>
·
1 0
'
•
.,
E
"
'0
:>
is
n.
were fit with the hyperbolic equation
where a = 97 . 32 x l O - 6 , b =0.6999, and c = 79 . 82 .
The average absolute error of the correlation model was 2 . 60 % .
The consolidated sandstone correlation was developed for sandstone
porosities ranging from 2 % to 23 % . Newman 34 also developed a
correlation for limestones. The data shown in Fig. 1 .34 were fit
with the hyperbolic equation
cf =a/( 1 + bccf» l ib , . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 1 05)
where a = 0 . 8535, b = 1 .075 , and c = 2 . 303 x lO6 .
The average absolute error of the correlation model was 1 1 . 8 % .
The limestone correlation was developed for a limestone porosity
of 0.02 to 0.33.
><
5.
1 .33
cf =a/( 1 +bccf» l Ib . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 1 . 1 04)
0
VI
VI
/
FINAL PRESSURE , psia
�
i
/'
1"'-
fiNAl PRESSURE, psio
determine the minimum initial temperature that will permit expan­
sion from 1 ,000 to 440 psia without hydrate formation.
Solution. From Fig. 1 .30, intersection of the 1 ,000-psia initial
pressure line and the 1 00°F initial temperature curve gives a final
pressure of 440 psia. Thus, the gas may be expanded to 440 psia
without hydrate formation.
If pressure is initially at 800 psia, the 1 00°F initial temperature
curve in Fig. 1 .28 does not intersect the 800-psia initial pressure
line. Hence, the gas may be expanded to atmospheric pressure
without hydrate formation.
For expansion from 1 ,000 to 440 psia, intersection of the 1 ,000psia initial pressure line and the 440-psia final pressure line gives
an initial temperature of l OO°F. Hence, l OO°F is the minimum in­
itial temperature to avoid hydrate formation.
/
/
���100��30�5000�--�2�000�
100�'�
100����100�--�����300��50�-�2000��30
Fig. 1 .3 1 -Permissible expansion of a O . 9-gravity natural gas
without hydrate formation (after Katz et a/. , Handbook of Na­
tural Gas Engineering, 1 95 9 , McGraw-Hili Book Co. , 4 courte­
sy of McGraw- H i l i I n c . ) .
///
./ /
.;"
60
50
V / III.
/"
II
-
80
200
!I!
f.- '"
-
300
1,'
'/
i--"
200
0
5
1 0
1 5
20
25
30
35
Initial porosity al zero net pressure
&
\
1 0
,
•
••
•
>
.,
CONSOUOATED L.t.1ES'TONES
5
•
•
1 0
•
•
•
1 5
•
20
25
30
35
Initial porosity al zero net pressure
Fig. 1 .33-PV compressibility at 75% IIthostatlc pressure vs.
I n itial sample porosity for consolidated sandstones (after
Newman 3 4 ) .
Fig. 1 .34-PV compressibility at 75% lithostatic pressure vs.
i n itial sample porosity for l imestones (after Newman 3 4 ) .
32
at
GAS RESERVO I R E N G I N E E R I N G
Solution.
1 . First, assume that the lithology is sandstone and estimate
using Newman ' s consolidated-sandstone correlation.
ct> =0.2
a
cf
97. 3200 x 1 0 -6
[1 + (0. 699993)(79. 8 1 8 1 )(0.20)] ( 1 /0.69999 3 )
( 1 +bcct» l ib
= 2 . 74 x 1 0 -6 psia - I .
2 . Next, assume that the lithology is limestone and esti­
mate Cf at 1jJ =0.2 using Newman' s limestone correlation:
cf -
a
---( 1 + bcljJ) lib
Example 1 .24-Calculation of Turbulence Factor and Non­
Darcy Flow Coefficient. A gas well is producing at a rate of 40,000
MscflD. We suspect that turbulence may be affecting the gas pro­
duction. Calculate {3 and D with the data given below . For this ex­
ample, assume that P s c = 14.65 psia and Tsc = 60° F = 520oR.
0. 85353 1
[1 + ( 1 .07538)(2 . 30304 x 106)(0.20)] ( 1 / 1 . 07538 )
= 4 . 32 x 1 0 - 6 psia - I .
1 . 1 5 Gas Turbulence Factor and
Non·Darcy Flow Coefficient
Non-Darcy flow, often called turbulent flow, may occur at high
gas flow velocity . The inertial coefficient or turbulence factor, {3,
i s defined b y the Forchheimer35 equation, which for l D linear
flow is
dp /LV
- - = - + 3 .238 x lO - 8 {3pv 2 ,
dL
.
k
.
.
.
.
.
.
.
.
.
.
.
.
•
.
.
•
( 1 . 106)
where dp/dL = flowing pressure gradient; v = fluid velocity (flow
rate divided by cross-sectional area) ; /L = fluid viscosity ;
k = formation permeability ; pv 2 = inertial flow term; and {3 = inertial
coefficient or turbulence factor.
The inertial coefficient, {3, has the dimension of reciprocal length .
Eq. 1 . 1 06 indicates that the pressure gradient required to sustain
a given flow rate through a porous medium is greater than would
be predicted by Darcy ' s flow equation when the term (3pv 2 is not
negligible. 3 6 Rewriting Eq. 1 . 1 06 gives
-
: = 7 (1 +
where
CI
:)
C 1 k PV
The non-Darcy flow coefficient is not constant but varies as a
function of pressure. After production begins, the radius of the high­
velocity flow region initially increases with time and soon stabi­
lizes . D is inversely proportional to gas viscosity evaluated at P wf '
Viscosity is directly related to pressure, which in tum is a function
of time. As pressure declines, viscosity likewise declines , causing
an increase in D. Because D is not constant, analyzing a gas-well
drawdown test with methods developed for liquid flow may give
erroneous results if non-Darcy flow affects the pressure response.
.
.
.
.
.
.
..
.
.
.
.
.
.
. .
.
.
.
.
.
. . ( 1 . 107)
.
is a constant and the right-hand term inside parentheses
represents a Reynolds number (ratio of inertial to viscous forces) .
If the Reynolds number is close to unity . then most of the flowing
pressure gradient is a result of viscous flow , and Darcy ' s equation
applies . As the Reynolds number increases, however, the inertial
forces increase significantly , and the flow can no longer be modeled
with Darcy ' s equation . We call this the non-Darcy flow effect.
Many attempts 3 6-40 have been made to relate experimental meas­
urements of {3 to rock properties. Using 355 sandstone and 29 lime­
stone samples , Jones36 experimentally determined {3 and developed
correlations describing {3 as a function of porosity and permeabil­
ity . The following correlation 3 6 is recommended for estimating {3 :
{3 = 1 . 88 x 10 1 0 k - 1 . 47 1jJ - 0 .53 . . . . . . . . . . . . . . . . . . . . . ( 1 . 108)
The non-Darcy component of the flow equation is significant only
in the area of high-velocity and high-pressure drawdown near the
wellbore, so the effect of non-Darcy flow usually is incorporated
into fluid-flow equations as an additional skin factor that is rate­
dependent. 4 1 The total skin factor, which is the value determined
from pressure-transient analysis, is an apparent value, s ' , that in­
cludes both the skin factor, s, and a term representing non-Darcy
flow effects, Dq. {3 is incorporated into the term D ,
2.715 x 1 O - 1 5 {3kMp sc
, . . . . . . . . . . . . . . . . . . . . . . ( 1 . 109)
h rw Ts c/L g ,><i
where D = non-Darcy flow coefficient, (Mscf/D) - I ; /L g ,wf = pres­
D
sure-dependent gas viscosity evaluated at the bottomhole flowing
pressure, cp ; and M = molecular weight of the gas , lbmllb-mol .
qg = 40,000 MscflD .
rw = 0.3 ft .
1jJ = 0. 1O.
P wf = 3 ,570 psia.
h =40 ft.
k = 57 md .
l' g =0.85.
/L g , wf =0.0244 cpo
Solution. The turbulence factor is computed with Eq. 1 . 108 .
(3 = 1 .88 x 1 0 1 0 k - 1 . 47 1jJ - 0. 53 = 1 .88 x 10 10 (57) - 1 . 47 (0. 10) - 0 . 53
= 1 .67 x 108 ft - I .
The non-Darcy flow coefficient is computed with Eq. 1 . 1 09 .
2 . 7 1 5 x 1 0 - 15 {3k--=-:::.
Mpsc .
D=
h r w Tsc/L g , wf
-----
(2 . 7 1 5 x 10 - 15 )( 1 .67 x 108 )(57)(0. 85 X 28 . 96)( I4.65)
(40)(0. 3)(520)(0.0244)
=6. 1 3 x 1 0 -5 D/Mscf.
1 . 1 6 Summary
Reading this chapter should prepare you to do the following tasks .
I . Calculate the pseudocritical pressure and temperature for a
sweet natural gas , given its composition, with the Stewart et al. 5
mixing rules.
2. Calculate the pseudocritical pressure and temperature for C 7 +
components in a natural gas mixture .
3 . Calculate the pseudocritical properties of a gas mixture given
its specific gravity .
4. Correct the pseudocritical properties for the presence of H 2 S ,
CO 2 , N 2 , and water vapor.
5. Estimate the pseudocritical properties of wet-gas and gas­
condensate reservoir fluids with recombination calculations.
6. Calculate the z factor for sweet and sour natural gas mixtures
given either compositional data or gas gravity .
7 . Calculate the gas FVF , gas density , gas compressibility , and
gas viscosity of a natural gas .
8. Determine the oil FVF, solution GOR, compressibility , and
viscosity for an undersaturated black oil .
9 . Determine the oil FVF , solution GOR, compressibility , and
viscosity for a saturated black oil .
10. Estimate the bubblepoint pressure of a black oil given the
solution GOR from early production data.
I I . Determine the FVF , solution gas/water ratio , compressibil­
ity , and viscosity for an oilfield brine .
12. Calculate the water vapor content of natural gases given the
pressure and temperature, and correct laboratory compositional data
for water vapor content.
1 3 . Estimate conditions of gas hydrate formation caused by either
a sudden decrease in temperature or sudden gas expansion through
a flow restriction .
14. Calculate the PV compressibility of a formation given its
porosity and lithology .
33
PROPERTIES OF NATU RAL GASES
1 5 . Calculate values of gas turbulence factor and the non-Darcy
flow coefficient.
Questions for Discussion
1 . State the three defining properties of an ideal gas . Derive the
EOS for an ideal gas in terms of pressure , volume, and tem­
perature from fundamental principles . How can this equation
be modified for a real gas?
2 . Define pseudocritical temperature and pressure . Why do we
use these values? Are the pseudocritical properties physically
realistic; i . e . , do the properties of liquid and vapor phases be­
come identical at this point?
3 . The nonhydrocarbon components in natural gases violate the
homologous series assumption of the law of corresponding states
used to develop the z-factor correlations . How do we account
for these components when calculating the z factor for a gas
mixture?
4. Define " standard conditions " and give the value of standard
temperature and pressure used in various states . Derive the nu­
merical constant for the volume occupied by 1 Ibm-mol of gas
at Texas standard conditions .
5 . What does gas viscosity measure physically? What is gas vis­
cosity a function of?
6. You are analyzing a gas reservoir that produces some conden­
sate and some water. You need to analyze gas and water flow
specifically , so you need estimates of FVF ' s , viscosities, and
compressibilities .
A . Assuming a wet-gas reservoir (not retrograde) , how d o you
take condensate production into account? What properties
are affected?
B. What data would you need (and thus request) for the most
rigorous gas property estimates possible?
C. What data would you need (and thus request) for the least
accurate methods presented in the text for gas-property es­
timates?
D. What would you do if you found that the produced gas con­
tained H 2 S , CO 2 , and N 2 ? What if it contains none of
these impurities?
E. What would you do if it contained C 7 + fractions? What
would you do if it did not?
F. Should you assume that the natural gas contains no water
vapor in the reservoir? When will it do so and when will
it not? How can you determine the amount of water vapor
and its effect on gas properties?
G. Which correlations should you use to estimate Bw , c w ' and
P- w ?
7 . What three major factors control the shrinkage of crude oil as
it is taken from reservoir to surface conditions? Sketch a graph
of the relationship between oil FVF and pressure . Why does
B o increase with a reduction of pressure from the initial con­
dition to the bubblepoint and decrease with a reduction of pres­
sure below the bubblepoint pressure?
8. Define the terms "saturated" and "undersaturated" with respect
to a black oil . Sketch a graph illustrating the behavior of solu­
tion GOR, R s ' as a function of pressure. Explain the behavior
of this function above and below the bubblepoint pressure.
9 . Sketch the typical behavior of the oil compressibility , co ' with
respect to pressure. Explain the discontinuity at the bubblepoint
pressure .
1 0 . Explain the behavior of oil viscosity, P-o ' with respect to tem­
perature, pressure, solution GOR, and size and complexity of
the hydrocarbon molecules making up the oil . Sketch a graph
of the relationship between P-o and pressure. Where does the
minimum value of P-o occur?
1 1 . You are analyzing an undersaturated oil reservoir with mobile
water saturation.
A . Which correlations should you use to estimate FVF ' s , vis­
cosities, and compressibilities?
B. What data do you need to make these estimates?
1 2 . You are analyzing a saturated oil reservoir with mobile oil,
water, and gas .
A . Which correlations should you use to estimate FVF ' s , vis­
cosities, and compressibilities?
B. What data do you need to be able to make these estimates?
C. Does the reservoir gas contain water vapor? How can you
account for the effect of water vapor on the reservoir gas
properties?
1 3 . What are the three major factors affecting the FVF of water,
Bw ? How does the magnitude of Bw compare with that of B o
and why?
1 4 . You are analyzing an aquifer. Which correlations should you
use to estimate c w ' Bw , P- w ' and c t ?
1 5 . Describe the physical state of a " gas hydrate. " At what tem­
peratures and pressures do seed crystals begin to form?
Exercises
1 . 1 . Gas was contracted at $ 1 . 1 0/Mscf at contract conditions of
1 4 . 4 psia and 80 ° F . What is the equivalent price at a legal
temperature of 60 ° F and pressure of 1 5 . 025 psia?
1 . 2 . Given the following gas composition:
mol %
0 . 00
2 . 49
0. 1 6
74 . 62
8 . 99
5. 1 1
1 . 03
1 . 86
0 . 69
0 . 76
0 . 95
3 . 34
1 00 . 00
Component
H2 S
CO 2
Nitrogen
Methane
Ethane
Propane
i-butane
n-butane
i-pentane
n-pentane
Hexanes
Heptanes plus
Total
C 7 + molecular weight
C 7 + gravity
1 26 Ibm/Ibm-mol
0 . 772
A. Calculate the average molecular weight, M, of the gas
mixture .
B . Calculate the specific gravity , 'Yg , of the gas mixture.
C . Calculate the pseudocritical pressure , Ppc ' and pseudo­
critical temperature, Tpc ' using the gas gravity .
D . Calculate the pseudocritical properties of the C 7 +
fraction .
E . Calculate the pseudocritical properties of the mixture
using the Stewart et at. 5 mixing rules, including the C 7 +
fraction .
F . Correct the pseudocritical pressure and pseudocritical tem­
perature obtained in Part E for contaminants .
G . Build a table of the following gas properties at a temper­
ature of 249 ° F and pressures of 500, 1 ,000 , 1 ,500, 2 ,000 ,
2 , 500, and 3 ,000 psia: z factor; gas density , Pg ; gas
compressibility , c /I; and gas viscosity , p-g .
1 . 3 . A cylinder i s fitted WIth leakproof piston and calibrated so
that the volume within the cylinder can be read from a scale
for any position of the piston. The cylinder is immersed in
a constant-temperature bath maintained at 1 60 ° F , which is
the reservoir temperature of the Sabine gas field. About
45 ,000 cm3 of the gas, measured at 1 4 . 7 psia and 60 ° F , is
charged into the cylinder. The volume is decreased in the steps
indicated below, and the corresponding pressures are read
with a dead-weight tester after temperature equilibrium is
reached .
V
p
(cm 3 )
(psia)
2 ,529
964
453
265
1 80
1 56 . 5
1 42 . 2
300
750
1 ,500
2 ,500
4,000
5 ,000
6 , 000
34
GAS RESERVO I R E N G I N E E R I N G
Calculate and place in tabular form the z factors and the
ideal volumes that the initial 45 ,000 cm 3 occupies at
1 60 ° F and at each pressure.
B. Calculate the gas FVF's, Bg , at each pressure in ft 3 /scf.
C . Plot the z factors and the values of Bg vs. pressure on
the same graph.
1 . 4 . A . If the Sabine field specific gas gravity is 0 . 65 , calculate
the z factors from 0 to 6 , 000 psia at 1 60 ° F in I ,OOO-psia
increments.
B . Using an appropriate mixing rule, calculate and plot the
z factors for the Sabine field at several pressures and
1 60 ° F . The gas analysis is given below.
Mole Fraction
Component
0 . 875
Methane
0 . 083
Ethane
0 . 02 1
Propane
0 . 006
i-butane
n-butane
0 . 008
0 . 003
i-Pentane
0 . 002
n-Pentane
0 . 00 1
Hexanes
0 . 00 1
Heptanes plus
1 .000
Total
C 7 + molecular .weight
1 1 4 . 2 Ibm/Ibm-mol
1 ,024 . 9 ° R
C 7 + pseudocritical temperature
C 7 + pseudocritical pressure
362 . 2 psia
C. Plot the data of Problems 3A and 4A on the same graph
for comparison.
D . Below what pressure at 1 60 ° F may the ideal-gas law be
used for the gas of the Sabine field if errors are to be
kept within 2 % ?
E. Will a reservoir contain more real or ideal gas at similar
conditions? Explain.
1 . 5 . A well is completed in an undersaturated oil reservoir with
a reservoir pressure of 3 ,500 psia ( p b = 2, 1 00 psia) and tem­
perature of 1 80 ° F . The permeability to oil, ko , is 1 30 md,
and the permeability to water, k w , is 2 1 md. The oil, water,
and gas saturations were estimated at 0 . 65 , 0 . 3 5 , and 0, re­
spectively. The porosity of the sandstone was calculated to
be 0 . 1 8 from log information.
The well produces all three phases: oil at a rate of 1 85
STBID, water at a rate of 5 STBID, and dissolved gas at a
rate of qoRs Mscf/D. A reservoir fluid sample indicated the
oil gravity was 42 ° API, and the gas gravity 0.75 . The salin­
ity of the water sample was 32,000 ppm.
Total reservoir flow rate is given by
A.
--
Total mobility is given by
ko k w k
A/ = - + - + -g ,
Jl. o Jl. w Jl.g
and total compressibility is given by
c/ = co So + cw Sw + cg Sg + cf ·
Using this information, calculate the total reservoir flow
rate in RBID, the total mobility in md/cp, and the total com­
pressibility in psia 1 .
1 . 6 . After several years, the average reservoir pressure has de­
creased to 1 ,500 psia. k o is now 1 00 md, k w is 1 .2 md, and
kg is 3 .25 md. The oil, water, and gas saturations are now
0 . 65 , 0. 3 0, and 0 . 05 , respectively.
The well still produces all three phases: oil at 85 STBID,
water at 55 STBID, and gas at 1 50 MscflD.
Using this information, calculate the total reservoir flow
rate in RB/D , the total mobility in md/cp, and the total com­
pressibility in psia I .
-
-
well is producing 4 1 . 1 ° API gravity oil and O. 762-specific­
gravity gas at 990 scf/STB. The formation temperature is
1 95 °F . The pressure is above the bubblepoint.
A . Estimate the oil FVF at the bubblepoint.
B . Estimate the bubblepoint pressure.
C . Estimate the solution GOR at pressures of 2 ,500 and 3 ,500
psia.
D. Estimate the oil compressibility at pressures of 2 , 500 and
3 ,500 psia.
E. Estimate the oil FVF at pressures of 2 ,500 and 3 ,500 psia.
F. Estimate the oil viscosity at pressures of 2 , 500, 3 ,200,
and 3 ,500 psia.
1 . 8 . Experiments were made on a bottomhole sample of the reser­
voir liquid taken from the LaSalle oil field to determine the
solution gas and the FVF as functions of pressure. The ini­
tial bottomhole pressure of the reservoir was 3 ,600 psia and
bottomhole temperature was 1 60 ° F , so all measurements in
the laboratory were made at 1 60 ° F . The following data, con­
verted to practical units, were obtained from the meas­
urements.
Solution Gas at 1 4 . 7
Pressure
psia and 60° F
FVF
(psia)
(scf/STB)
(RB/STB)
1 .7.
A
3 ,600
3 ,200
2 , 800
2 , 500
2 ,400
1 , 800
1 ,200
600
200
567
567
567
567
554
436
337
223
1 43
1 . 3 10
1 .3 1 7
1 . 325
1 . 333
1 .3 1 0
1 .263
1 .2 1 0
1 . 1 40
1 .070
A. What factors affect the solubility of gas in crude oil?
B. Plot the gas in solution vs. pressure.
C . Was the reservoir initially saturated or undersaturated?
Explain.
D . Does the reservoir have an initial gas cap?
E. In the region of 200 to 2 , 500 psia, determine the solubil­
ity of the gas from your graph in scf/STB-psi.
F. Suppose 1 ,000 scf instead of 567 scf of gas had accumu­
lated with each stock-tank barrel of oil in this reservoir.
Estimate how much gas would have been in solution at
3 , 600 psia. Would the reservoir oil then be called satu­
rated or undersaturated?
1 . 9 . From the bottomhole sample in Exercise 8 , answer the fol­
lowing.
A . Plot the FVF vs. pressure.
B. Explain the break in the curve.
C . Why is the slope above the bubblepoint pressure nega­
tive and smaller than the positive slope below the bub­
blepoint pressure?
D . If the reservoir contains 250 million RB of oil initially,
what is the initial volume of oil in place in stock-tank
barrels?
E. What is the initial volume of dissolved gas in the
reservoir?
1 . 10 . A . Estimate the viscosity of an oil at 3 ,000 psia and 1 30 ° F .
It has a stock-tank gravity of 35 ° API at 60 ° F and con­
tains an estimated 750 scf/STB of solution gas at the ini­
tial bubblepoint pressure of 3 , 000 psia.
B. Estimate the viscosity at the initial reservoir pressure of
4 ,500 psia.
C. Estimate the viscosity at 1 ,000 psia if there is an estimat­
ed 300 scf/STB of solution gas at that pressure.
I . I I . Find the compressibility and FVF for a connate water that
contains 20,000 ppm of total solids at a reservoir pressure
of 4,000 psia and temperature of 1 5 0 ° F .
1 . 12 . A container has a volume of 500 cm 3 and is full of pure
water at 1 80 ° F and 6,000 psia.
A. How much water would be expelled if the pressure were
reduced to 1 ,000 psia?
35
PROPERTI ES OF NATU RAL GASES
B. What would the volume of the expelled water be if the
salinity were 20, 000 ppm and no gas in solution?
C . Rework Part B, assuming that the water is initially satu­
rated with gas and that all the gas is evolved during the
pressure change.
D . Estimate the water viscosity .
Applied Petroleum Reservoir Engineering. 2/e. © 1 991 . p p . 49·53. reprinted by permission
Author's Note: Problems 1 . 3. 4. and B through 1 2 were taken from Ref. 42 (Craft/Hawkins.
of Prentice Hall. Upper Saddle River. NJ).
N omenclature
A(T) ,B(T) = variables used to determine water vapor content
Bg = gas FVF , RB/Mscf
B o = oil FVF, RB/STB
Bob = oil FVF at bubblepoint pressure , RB/STB
Bw = water FVF , RB/STB
cJ = PV compressibility , psia - I
Cg = gas compressibility , psia - I
Co = oil compressibility , psia - I
c, = reduced compressibility , dimensionless
cr = total compressibility , Cg Sg + co So + c w Sw + cJ '
psia - I
C w = water compressibility , psia - I
Gpa = additional gas production, scf/STB
L = gross heating value of natural gas mixture ,
Btu/scf
Lei = gross heating value of ith gas component, Btu/scf
m = mass, Ibm
M = apparent molecular weight, lbmllbm-mol
Ma = molecular weight of air = 2 8 . 9625 Ibm/Ibm-mol
M; = molecular weight of ith gas component
P = pressure , psi a
Pb = initial original bubblepoint pressure , psia
P c = critical pressure , psia
Ppc = pseudocritical pressure, psia
Pp , = pseudoreduced pressure , dimensionless
P , = reduced pressure, dimensionless
P s c = pressure at standard conditions , psia
q = total wellstream gas flow rate , MscflD
qg = total surface gas flow rate, MscflD
qo = oil flow rate, STBID
qs l = gas flow rate from primary separator, Mscf/D
R = universal gas constant
R I = primary-separator gas/stock-tank-liquid ratio,
scf/STB
Rz = secondary separator gas/stock-tank-liquid ratio,
scf/STB
R 3 = stock-tank-gas/stock-tank-liquid ratio, scf/STB
Rs = solution GOR, scf/STB
R sb = solution GOR at bubblepoint pressure, scf/STB
R s; = solution GOR at initial reservoir pressure ,
scf/STB
R sw = solution gas/water ratio, scf/STB
Rs wb = solution gas/water ratio at bubblepoint pressure ,
scf/STB
R swp = solution gas/water ratio for pure water, scf/STB
S = salinity , wt % solids
S' = salinity , g NaClIL
T = temperature, O R
Tb = boiling temperature, O F
Tc = critical temperature, O R
TF = temperature, O F
Tp c = pseudocritical temperature , O R
Tpou r = pour-point temperature , O R
Tp, = pseudoreduced temperature, dimensionless
T, = reduced temperature , dimensionless
Tsc = temperature of standard conditions, OF
V = volume, L 3 , ft 3
Vm = molar volume, ft 3 /lbm-mol
Veq
�Vw T
� Vwp
W
Y C 02
Y H20
YH2 S
Y;
YN2
z
�
'I API
1'1
'1z
'1 3
l'
g
'1 0
'1 w
P- g
P-o
P-ob
P-od
P-w
cf>
Pa
Pg
P,
Pw
= vapor equivalent of primary-separator liquid,
scf/STB
= temperature term in B w correlation
= pressure term in B w correlation
= water vapor content, lbm/MMscf
= mole fraction of CO z
= mole fraction of water
= mole fraction of HzS
= mole fraction of a particular component in gas
mixture
= mole fraction of nitrogen
= gas compressibility factor
= Wichert and Aziz contaminant correction
parameter, OR
= oil gravity , ° API
= specific gravity of primary-separator gas
(air = 1 . 0)
= specific gravity of secondary-separator gas
(air = 1 . 0)
= specific gravity of stock-tank gas (air = 1 .0)
= specific gas gravity (air = 1 . 0)
= oil gravity (water = 1 . 0 g/cm 3 )
= specific gravity of reservoir gas (air = 1 .0)
= gas viscosity , cp
= oil viscosity , cp
= oil viscosity at bubblepoint pressure , cp
= dead-oil viscosity , cp
= water viscosity , cp
= porosity , fraction
= density of dry air Ibm/ft 3
= gas density , Ibm/ft 3
= reduced density , dimensionless
= water density , g/cm 3
Subscripts
h
sc
= hydrocarbon
= standard conditions
References
1 . McCain, W . D . Jr. : The Properties of Petroleum Fluids, second edi­
tion , PennWell Publishing Co . , Tulsa, OK ( 1 990) .
2 . " Orifice Metering of Natural Gas and Other Related Hydrocarbon
Fluids , " AGA report No. 3 , American Gas Assn . , Arlington ( 1 985) .
3 . GPSA Engineering Data Book, 1 0th edition , Gas Processors Suppliers
Assn . , Tulsa, OK ( 1 987) .
4. Katz , D . L . et al. : Handbook of Natural Gas Engineering, McGraw­
Hill Book Co. Inc . , New York C ity ( 1 959) .
5 . Stewart, W . F . , Burkhardt, S . F . , and Voo, D . : " Prediction of Pseu­
docritical Parameters for Mixtures, " paper presented at the 1 959 AIChE
Meeting , Kansas C ity , May .
6. Kay , W . B . : " Density of Hydrocarbon Gases and Vapors at H igh Tem­
perature and Pressure , " Ind. & Eng. Chern. (Sept . 1 936) 1 0 1 4- 1 9 .
7 . Sutton, R . P . : " C ompressibility Factors for High-Molecular-Weight
Reservoir Gases , " paper SPE 1 4265 presented at the 1985 SPE Annu­
al Technical Meeting and Exhibition, Las Vegas, Sept . 22-25 .
8. van der Waal s , J. D . : " Over de Continuiteit van den Gas-en Vloeistof­
toestand , " dissertation , Leiden ( 1 873) .
9. Redlich , O . and Kwong , J . N . S . : " On the Thermodynamics of Solu­
tions, V-An Equation of State , Fugacities of Gaseous Solutions , "
Chern. Reviews ( 1 949) 44, 233-44 .
1 0 . Kessler, M . G . and Lee , B . I . : "Improve Prediction of Enthalpy of Frac­
tions , " Hyd. Proc. (March 1 976) 1 5 3-5 8 .
1 1 . Whitson , C . H . : " Effect o f C 7 + Properties o n Equation-of-State Pre­
dictions , " SPEI (Dec . 1 987) 685-96 .
1 2 . Theory and Practice of the Testing of Gas Wells, third edition, Energy
Resources Conservation Board , Calgary ( 1 975) .
1 3 . Standing, M . B . : Volumetric and Phase Behavior of Oil Field Hydrocar­
bon Systems, SPE, Richardson, TX ( 1 98 1 ) .
1 4 . Wichert, E . and Aziz , K . : "Calculate Z ' s for Sour Gases , " Hyd. Proc.
(May 1 972) 1 1 9-22 .
1 5 . Gold, D . K . , McCain, W . D . Jr. , and Jenning s , J . W . : "An Improved
Method for the Determination of the Reservoir-Gas Specific Gravity
for Retrograde Gases , " JPT (July 1 989) 747-52 ; Trans. , AIME, 287.
36
1 6 . Dranchuk, P . M . and Abou-Kassem, J . H . : " Calculation of Z Factors
for Natural Gases Using Equations of State, " J. Cdn. Pet. Tech. (July­
Sept. 1 975) 34-36.
1 7 . Standing , M . B . and Katz , D . L . : Density of Natural Gases, Trans. ,
AIME, 146 ( 1 942) 1 40-49 .
1 8 . Mattar, L . , Brar, G . S . , and Aziz , K . : "Compressibility of Natural
Gases , " J. Cdn. Pet. Tech . (Oct . -Dec . 1 975) 77-80.
19. Carr, N . L . , Kobayashi , R . , and Burrow s , D . B . : "Viscosity of
Hydrocarbon Gases Under Pressure , " Trans. , AIME ( 1 954) 201 ,
264-72 .
20. Lee , A . L . , Gonzalez , M . H . , and Eakin, B . E . : " The Viscosity of Na­
tural Gases , " JPT (Aug . 1 966) 997- 1 000 ; Trans. , AIM E , 237.
2 1 . Standing, M .B . : " A Pressure-Volume-Temperature Correlation for Mix­
tures of California Oils and Gases , " Drill. & Prod. Prac. , API ( 1 947)
275-87 .
22 . Beggs, H . D . : " Oil System Correlations , " Petroleum Engineering Hand­
book, H . B . Bradley (ed . ) , SPE, Richardson , TX ( 1 987) 1 , Chap . 2 2 .
2 3 . " Core Laboratory Example Reservoir Crude O i l Analysis (Good Oil
Company , Black Oil Well No. 4) , " Core Laboratories Inc . , Dallas.
24 . McCain, W . D . Jr. , Rollins, J .B . , and Villena-Lanzi, A . J . : "The Coeffi­
cient of Isothermal Compressibility of Black Oils at Pressures Below
the Bubblepoint , " SPEFE (Sept . 1 988) 659-62 .
25 . Vasquez , M. and Beggs, H . D . : " Correlations for Fluid Physical Prop­
erty Prediction , " JPT (June 1 980) 968-70 .
2 6 . Beggs , H . D . and Robinson , J . R . : "Estimating the Viscosity of Crude
Oil Systems , " JPT (Sept. 1 975) 1 1 40-4 1 .
2 7 . Ng, J . T . H . and Egbogab, E . 0 . : "An Improved Temperature-Viscosity
Correlation for C rude Oil Systems , " paper CIM 83-34-32 presented
at the 1 983 Annual Technical Meeting of the Petroleum Soc . of CIM ,
Banff, Alta . . May 1 0- 1 3 .
2 8 . McCain, W . D . Jr. : " Reservoir Water Property Correlations-State­
of-the-Art , " paper SPE 1 8573 available at SPE, Richardson, TX.
GAS RESERVO I R E N G I N E E R I N G
2 9 . Osif, T . L . : " The Effects of Salt, Gas , Temperature, and Pressure and
the Compressibility of Water, " SPERE (Feb . 1 988) 1 75-8 1 .
30. Dodson, C . R . and Standing, M . B . : " Pressure , Volume , Temperature,
and Solubility Relations for Natural Gas-Water Mixtures , " Drill. &
Prod. Prac. , API ( 1 944) 1 73-79.
3 1 . Ramey , H.J. Jr. : " Rapid Methods for Estimating Reservoir Compres­
sibilities , " Trans. , AIME ( 1 964) 23 1 , 447-54 .
3 2 . Bukacek, R . F . : " Equilibrium Moisture Content of Natural Gases , "
Bull. , Inst. of Gas Technology Bulletin ( 1 95 5 ) 8 .
3 3 . McKetta, IJ . and Wehe, A . H . : "Use This Chart for Water Vapor Con­
tent of Natural Gases , " Petroleum Refiner (Aug . 1 958) 1 53-54.
34. Newman, G . H . : " Pore-Volume Compressibility of Consolidated, Fri­
able , and Unconsolidated Reservoir Rocks Under Hydrostatic Load­
ing , " JPT (Feb . 1 973) 1 29-34.
35. Forchheimer, P . : " Wasserbewegung durch Boden, " Zeitz ver deutsch
Ing. ( 1 90 1 ) 45, 1 73 1 .
3 6 . Jones, S . C . : " Using the Inertial Coefficient, {3 , To Characterize Het­
erogeneity in Reservoir Rock, " paper SPE 1 6949 presented at the 1 987
SPE Annual Technical Conference and Exhibition, Dallas, Sept. 27-30.
37. Cornell , D. and Katz , D . L . : " Flow of Gases Through Consolidated
Porous Media, " Ind. & Eng. Chern. (Oct. 1 953) 45, 2 1 45 .
3 8 . Tek, M . R . , Coats, K . H . , and Katz, D . L . : " The Effect of Turbulence
on Flow of Natural Gas Through Porous Reservoirs , " JPT (July 1 962)
799-806 ; Trans. , AIME, 225.
39. Firoozabadi, A . and Katz , D . : "An Analysis of High-Velocity Gas Flow
Through Porous Media , " JPT (Feb . 1 979) 2 1 1 - 1 6 .
40. Geertsma, J . : " Estimating the Coefficient o f Inertial Resistance i n Fluid
Flow Through Porous Media , " SPEl (Oct. 1 974) 445-50.
4 1 . Lee , W . J . : Well Testing, Society of Petroleum Engineers , Richardson,
TX ( 1 982) 87 .
42 . Craft, B . C . et al. : Applied Petroleum Reservoir Engineering, second
edition , Prentice-Hall Inc . , Englewood Cliffs , NJ ( 1 99 1 ) .
Chapter 2
Fundamentals of Gas Flow in Conduits
2 . 1 Introduction
In this chapter, we develop the thermodynamic fundamentals of fluid
flow through conduits . Chap. 3 applies these principles to the meas­
urement of gas flow; Chap . 4 applies them to fluid flow in vertical
and inclined pipes.
Many authors
have applied the principles of thermodynam­
ics to the study of fluid flow. Thermodynamic principles are used
to generate the equations, formulas , graphs, and tables that engi­
neers use. In the development of these practical appl ications , key
assumptions are made . The student and the practicing engineer
should understand these assumptions so that exceptional cases can
be detected . If invalid assumptions are undetected , then misappli­
cations may cause significant errors in calculated results.
In this chapter, we present the development of equations that
govern fluid flow in conduits. We assume that the fluid is nonreac­
tive (not undergoing any chemical or nuclear reactions) . The re­
sult of this chapter is the development of the mechanical energy
balance, which governs steady-state fluid flow in conduits.
1 - 13
2.2 Systems, H e a t , Work, and Energy
Thermodynamics is the study of energy and the way it is transferred
in the forms of heat and work. Both heat and work may be thought
of as energy in transition. Note that heat and work have the same
units as energy .
Material properties of fluids are discontinuous on a molecular
scale. This gives rise to microscopic properties studied in statisti­
cal thermodynamics. Neither statistical thermodynamics nor
microscopic properties are considered in this chapter.
In the classic study of thermodynamics, materials are considered
to be continua. This means that the macroscopic properties of the
fluids are piecewise-continuous. Macroscopic properties, which in­
clude specific volume, pressure , and temperature, apply on a size
scale at which the properties become at least piecewise-continuous .
W e consider flowing fluids t o b e continua. A homogeneous fluid
has macroscopic properties that are a continuous function of posi­
tion. Mist is an example of a homogeneous fluid . A mist is made
up of a gas with very small, entrained liquid particles , as Fig. 2 . 1
shows. While liquid particles are present i n a mist, the properties
of the mist becomes continuous on a macroscopic scale.
A nonhomogeneous fluid has macroscopic properties that are
piecewise-continuous as a function of position. This occurs when
more than one distinct fluid phase is present. A two-phase fluid
mixture in the plug flow regime is an example of a nonhomogene­
ous fluid (Fig. 2.2) . Under conditions of plug flow, macroscopic
fluid properties are continuous within each phase but have discon­
tinuities at phase boundaries .
2 . 2 . 1 Systems. The first step i n applying thermodynamic princi­
ples to the study of fluid flow is to define the region of the universe
to be considered through specification of a system. A system can
consist of any bounded region of the universe, taken together with
its contents . The rest of the universe is considered to be the sur­
roundings (see Fig. 2.3) .
We can consider a differential element at any location in the sys­
tem. This differential element contains an amount of fluid with mass,
dm, and corresponding volume, d V. The specific volume of a fluid
is defined as the volume of fluid per unit mass at any point in the
system (vs = d VIdm) . The concepts of differential mass and differ­
ential volume are useful when considering systems that contain fluids
that do not have constant physical properties .
The boundaries of a system can be either real or imaginary . These
boundaries have two main functions : to separate the system from
the surroundings and to allow interaction between the system and
surroundings . The three system types-isolated, closed, and open
systems-are based on the interactions between the system and sur­
roundings .
Isolated systems exchange neither mass nor energy with the sur­
roundings . This means that neither crosses the system boundaries .
However, if the system and surroundings are not in thermal equi­
librium (at the same temperature) at the system boundary , then a
special type of system boundary is required to prevent energy trans­
fer in the form of heat flow . Such a system boundary is called a
perfect insulator. Perfect insulators are theoretical and can only be
approximated in practice.
Closed systems exchange only energy with the surroundings. Be­
cause a closed system does not exchange mass with the surround­
ings, it has constant mass. A closed system is not isolated from
the surroundings because energy is exchanged across the system
boundaries. This exchange of energy across system boundaries can
be heat flow or work.
Open systems exchange both mass and energy with the surround­
ings. This means that both energy and mass cross the system bound­
aries. Energy may be transferred across the system boundaries as
energy associated with mass exchange, as heat flow , and as work.
The movement of mass (fluid flow) is called convection. Energy
transfer associated with the exchange of mass between the system
and surroundings is called convective energy transfer.
38
GAS RESERVO I R E N G I N E E R I N G
Fig. 2 . 1 -Homogeneous fluid: m ist flow pattern (after Kermit
E . Brown's The Technology of Artificial Lift Methods, Volume
1 , Copyright Penwell Books, 1 977 1 3 ) .
Fig. 2 .2-Nonhomogeneous f l u i d : p l u g flow pattern (after
Kermit E . Brown's The Technology of Artificial Lift Methods,
Volume 1 , Copyright Penwell Books, 1 977 1 3 ) .
2 . 2 . 2 Heat. The flow of heat into the system, Qh , is one way that
Eq. 2 . 2 expresses the differential amount o f work, d W, done by
the system in moving a piston of constant area a differential dis­
tance under conditions of negligible friction and acceleration . The
total work done by the system in expanding the fluid from an ini­
tial system volume , V I ' to a different system volume , V2 , is ob­
tained by integrating Eq. 2 . 2 :
energy can be transferred across system boundaries . The two re­
quirements for heat flow are that physical contact is made between
the system and surroundings and that a temperature difference (or
gradient) exists at the boundary between the system and sur­
roundings .
Heat itself is not considered t o b e stored i n a system . Heat i s
energy i n transition. When energy enters a closed system a s heat
flow , the sum of the internal , potential, and kinetic energy of the
mass in the system is increased (if no work is performed) . The sign
convention used will be that energy transferred into the system as
heat is positive and energy transferred out of the system as heat
is negative .
2.2.3 Work. Work done by the system, W, is another way that
energy can be transferred across system boundaries . Work is done
whenever a force acts through a distance. The fundamental equa­
tion expressing a differential quantity of work is
d W= F/ d£. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 1 )
.
.
I n Eq. 2 . 1 , F/ is the component o f a force that acts i n the direc­
tion of a differential displacement, dE. There can be no work un­
less displacement (movement) occurs . It is not necessary for the
force to cause displacement; however, the force must act in the
direction of displacement . The sign convention used will be that
work done by the system is positive and work done on the system
is negative.
An important example of work is the expansion of fluid in a cyl­
inder by the movement of a piston (Fig. 2.4) . For this example, the
system boundaries are defined as the cylinder and the piston. This
system is closed (has constant mass) because no fluid enters or leaves
the cylinder . For this example , the properties of the fluid in the cyl­
inder are considered uniform (constant throughout the system) .
For a constant piston area, the force exerted on the face of the
piston by the fluid is equal to the product of the pressure in the
fluid and the area of the piston, PA . A differential displacement
of the piston is equal to the differential volume change of the cyl­
inder (and fluid) divided by the area of the piston (d VIA ) . Begin­
ning with our fundamental equation for work, Eq. 2 . 1 , substitution
of force and displacement for this particular example yields:
d W= PA ( d VIA ) = Pd V. . . . . . . . . . . . . . . . . . . . . .
.
.
. . . . . (2 .2)
W=
f
J
V2
VI
Pd V. . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
. . . . . (2 . 3)
Again, work done by the system (expansion) is positive, and work
done on the system (compression) is negative. While Eqs . 2 . 2 and
2 . 3 were developed for a constant piston area, they also are valid
for any system in which expansion or compression of fluids occurs.
2.2.4 Energy. Work was defined with Eq. 2 . 1 as a quantitative
and unambiguous entity . Using this definition of work, we can de­
fine energy as the ability to perform work. Total energy of a sys­
tem, E, is the sum of the internal , potential , and kinetic energy of
the system:
E = U + Ep + Ek .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 4)
.
.
.
.
Internal energy , U, is the energy contained in the molecules ,
atoms , and subparticles of mass in the system. Internal energy con­
sists of the translational, rotational, and vibration energy of these
individual molecules , atoms, and subparticles . Internal energy can
be thought of as kinetic and potential energy of mass at a microscopic
level . In classic thermodynamics, there is no way to determine ab­
solute values of internal energy . Classic thermodynamics consid­
ers only differences in internal energy, Ll U, relative to some standard
state . These changes in internal energy are determined ex­
perimentally .
If we have experimentally determined values of specific internal
energy , u, for the fluids in the system, we can integrate over the
volume of the system to obtain the total internal energy of the
system:
U=
\'
,
V, ,
y
� d V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 5)
vs
Potential energy , Ep ' is important in the study of systems con­
taining fluids. Mass at a specified elevation has the ability to per-
SURROU NDINGS
I
BOUNDARY--.
Work Done
by System
/0
Differe n t i a l
Element
(dm, d V)
Fig. 2 .3-System and surro u n d i ngs (bou ndary of the system
and a differential element also are shown).
•
(P2J � )
(P1' 'i )
Fig. 2 .4-Expansion of fluid i n cyl inder. Work, W, is done by
the system .
39
F U N DAMENTALS OF GAS FLOW I N CONDU ITS
2 . 3 . 1 Closed System. Consider a closed system made up of a cyl­
inder and piston, as shown in Fig. 2.5. No mass is allowed to cross
the system boundaries; however, both heat and work are allowed
to cross the system boundaries.
Because there is no convection for closed systems, the statement
of the energy balance reduces to
net rate of work
system ' s rate of
net rate of heat
done by system on
accumulation of = added to system
total energy
from surroundings
surroundings .
dW
dt
t
The energy balance for a closed system can be stated mathemat­
ically as
dE/dt = dQhldt - d Wldt, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 1 0)
where Qh = total heat added to the system, ft-lbf, and t = time ,
seconds.
We can consider the closed system at two specified states by in­
tegrating in time (between states) . When we do this, the energy
balance is expressed as
t::.E = Qh - W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 1 1 )
.
We can expand the term for change in total energy to reflect changes
in the total potential , total kinetic , and total internal energy of the
closed system at the two specified states. The total internal , total
potential , and total kinetic energy of the closed system would be
evaluated with Eqs . 2 . 5 , 2 . 7 , and 2 . 9 , respectively .
AU
Fig. 2 . 5 -Closed system .
form work a s i t undergoes elevation changes (e . g . , hydroelectric
power generation) . If we consider a differential fluid element with
mass dm , the potential energy associated with that differential ele­
ment is
dEp = (gZlg c )dm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 6)
The term g c relates force and mass. Multiplication by the ratio
glg c converts from mass units to force units . For an arbitrary sys­
tem with fluids, we can integrate over the system volume to obtain
the total potential energy of the system, Ep :
.
gZ
d V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 7)
Ep =
\
•
vsys
.
vsgc
Kinetic energy , Ek can be thought of as energy associated with
velocity . As expressed by Newton ' s second law { force equals mass
times acceleration, [F= m(dv/dt )] } , a force must act for the veloc­
ity of mass to change. Therefore , changing the velocity of mass
can create a force. This force performs work if there is a displace­
ment in the same direction as the force . If we consider a differen­
tial fluid element with mass dm , the associated kinetic energy is
o
dEk = (v 2 /2g c )dm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 8)
In Eq. 2 . 8 , v is the magnitude of the velocity of the differential
fluid element. Again, we can integrate over the total volume of the
system to obtain Ek :
Ek =
\.
Vsys
.
v2
--
2 g c vs
d V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 .9)
+ t::.Ep + AEk = Qh - W.
+
net rate of heat
added to system
from surroundings
.
.
. . . . . . . . . . . . . (2 . 12)
.
homogeneous fluid through an open system with a single entrance
and exit, as Fig. 2 . 6 shows.
All properties of a steady-state system are constant with respect to
time. Thus, a steady-state open system has a constant mass flow rate.
Also, the system contains a constant total mass, requiring that the mass
flow rates into and out of the system are equal. The total energy of a
steady-state system also is constant. This means that the rate of energy
entering the system must equal the rate of energy leaving the system.
The energy balance for steady-state flow through an open sys­
tem expresses the conservation of energy. The statement of the ener­
gy balance is
rate of total
energy in by
convection
rate of total
energy out by
convection
=
net rate of heat
added to system
from surroundings
net rate of work
done by system on
surroundings .
The energy balance for steady-state flow through an open sys­
tem can be stated mathematically as
( )
dE
dt
l e aving
v.,
v,
_
( )
dE
dt
e nte rin g
dt
dt
. . . . . . . . . . (2 . 1 3)
I
I
_1
net rate of work
done by system on
surrounding s .
.
2.3.2 Open System. Now consider the steady-state flow of a
2.3 First Law of Thermodynamics
The first law of thermodynamics states that the total energy of a
system, E, must be conserved . As previously stated , the units of
heat and work are the same as those of energy . This equivalence
allows us to consider the conservation of internal, kinetic, and poten­
tial energy , as well as energy in transition: heat and work.
For any system, we can express the first law of thermodynamics
by stating the energy balance as
system ' s rate of
rate of total
rate of total
accumulation of
energy in by
energy out by
total energy
convection
convection
.... ...
il Z
9.
Z,
".
Datum level
-: +Section 2 Z,
Fig. 2 . 6-0pen system (after Smith and Van Ness, Introduc­
tion to Chemical Engineering Thermodynamics, 1 987,
McGraw- H i l i Book Co. , courtesy of McGraw- H i l i Inc. 1 1 ).
40
GAS RESERVO I R E N G I N E E R I N G
Because the mass flow rate , dmldt, is constant, application of
the chain rule for differentiation to each term of Eq. 2 . 1 3 allows
us to consider the rates of convective energy transfer, heat flow ,
and work on a mass basis instead of a time basis . The equation
describing the energy balance for a steady-state open system
becomes
e l e aving - e e nte ring
= qh - w .
. . . . . . . . . . . . . . . . . . . . . . . . (2 . 1 4)
We can consider total energy to be made up of internal , poten­
tial , and kinetic energy . When we do this, the energy balance for
a steady-state open system becomes
(u + ep + ek h - (u + ep + ekh =qh - w .
. . . . . . . . . . . . . . . (2 . 1 5)
The SUbscript 1 is for entrance conditions and 2 is for exit condi­
tions of open system .
Fluid properties at the entrance and exit of the open system are
constant and described by average values . Fluid velocities at the
entrance and exit of the open system also are described by average
velocities in the direction of flow . These average velocities are cal­
culated by dividing the volumetric flow rate by the area of the flow
conduit, v= q 'IA .
Substituting our derived expressions for potential and kinetic ener­
gy per unit mass into Eq. 2 . 1 5 gives a new expression of the ener­
gy balance of a steady-state open system, as shown in Eq . 2 . 1 6 .
The unit o f all terms is energy per unit mass.
gZ
(u + gZ + �
)
- (u + + � ) =qh - w,
2
gc
gc 2
work term, w,
gc
I
2g c
. . . . . . (2 . 1 6)
.
The
includes three parts : ( 1 ) work done on the
system at the entrance, (2) work done by the system at the exit,
and (3) other work done by the system (called shaft work, ws) '
The work per unit mass done on the system at the entrance is
the work required to force a unit mass of fluid at pressure P I and
specific volume vsl into the open system . For this case , the force
exerted by the system at the system entrance is constant and equal
to P ,A , . This force is directed out of the system . The total dis­
placement per unit mass is constant and equal to vs , fA , into the
system . Thus, the work performed per unit mass of fluid flow into
the system is P I vs I ' Because the direction of the force exerted by
the system is opposite the direction of the displacement, the work
is negative. This signifies work done on the system .
The work per unit mass done by the system at the exit is the work
required to force a unit mass of fluid at pressure P 2 and specific
volume vs 2 out of the system. It is evaluated like the work at the
entrance . In this case, the direction of the force exerted by the fluid
is the same as the displacement, signifying work done by the sys­
tem. This leads to
w=P2 vs 2 -P , vs l + ws ' . . . . . . . . . . . . . . . . . . . (2 . 1 7)
where Ws = total rate of shaft work done by the system divided by
.
.
.
.
.
.
.
the constant mass flow rate , ft-Ibfllbm .
Substituting Eq. 2 . 1 7 into Eq . 2 . 1 6 gives a new expression of
the first law of thermodynamics. This expression is called the energy
balance for steady-state flow of a homogeneous fluid and is shown
in Eq. 2 . 1 8 . Differences (indicated by A) are taken between the
exit and entrance conditions of the open system :
of doing this is that the terms of the mechanical energy balance
are more directly measurable .
A differential change in internal energy of a fluid in the system
is defined as
du = Tds -Pdvs '
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 1 9)
For thermodynamically reversible processes , Tds = d qh ' How­
ever, because of the presence of friction, fluid flow is thermody­
namically irreversible . When friction is present, Eq. 2 . 1 9 becomes
du = dqh +dF-Pdvs '
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 20)
We can apply the product rule for differentiation to the last term
of Eq. 2 . 20 :
du =dqh + dF+ vsdP- d(Pvs) '
.
.
. . . . . . . . . , . . . . . . . . (2 . 2 1 )
.
Eq . 2 . 2 1 can be integrated over the open system (between the
entrance and exit conditions) to obtain
. Pz
AU =qh +F+ J vsdP- A(Pvs) "
PI
. . . . . . . . . . . . . . . . . . (2 . 22)
Eq . 2 . 22 is then substituted into Eq. 2 . 1 8 t o obtain a new equa­
tion describing the energy balance for steady-state flow of a
homogenous fluid . This new equation does not include any terms
relating specifically to internal energy or heat flow . This equation,
which considers irreversibility resulting from friction energy loss­
es, is
.\
Pz
PI
VsdP + A
) +F= - ws '
( ggcZ ) +A ( �
2gc
. . . . . . . . . . . (2 .23)
Eq. 2 .23 , is called the mechanical energy balance for steady-state
fluid flow . The advantage of this equation over Eq . 2 . 1 8 is that
the quantities are more directly measurable . Because the mechani­
cal energy balance applies for any steady-state open system (even
if it is of differential size) , we also can express the mechanical energy
balance in differential form:
v
g
v s dP+ -dZ+ - dv + dF= - dws '
gc
gc
. . . . . . . . . . . . . . . . . (2 . 24)
The mechanical energy balance was developed with a consistent
set of units to clarify the derivation . However, when applying the
mechanical energy balance (Eqs. 2 . 23 or 2 . 24) , we usually prefer
to use units of psia for pressure instead of Ibffft2 . We also usually
prefer to use fluid density , p (lbmfft 3 ) , which is the reciprocal of
specific volume . Considering these preferences for units , the in­
tegral and differential expressions of the mechanical energy bal­
ance become
.I P Z
PI
1 44
p
dP +A
) +F= - Ws ,
( ggcZ ) + A ( �
2gc
. . . . . . . . . (2 .25)
the integral form of mechanical energy balance, and
v
g
dp + -dZ+ - dv +dF= - dws '
gc
gc
p
1 44
-
. . . . . . . . . . . . . . . (2 .26)
the differential form of mechanical energy balance (note that
1 44 = in . 2 fft 2 ) .
W e will use Eqs . 2 . 25 and 2 .26 i n Chaps. 3 and 4 .
2.4 Mechanical Energy Balance
At this point, we have developed the energy balance for steady­
state flow of a homogenous fluid in an open system (Eq. 2 . 18) .
However , this form of the energy balance is undesirable for appli­
cations because it includes terms relating to internal energy and heat
flow .
In this section, we develop the mechanical energy balance. The
mechanical energy balance also expresses the first law of ther­
modynamics for steady-state flow but does not include any terms
relating specifically to internal energy or heat flow . The advantage
2 . 5 Energy Loss Resulting From Friction
The term dF in Eqs . 2 . 24 and 2 . 26 represents differential energy
loss per unit mass resulting from friction. If the fluid flow is isother­
mal (T is constant) , this energy is transferred to the surroundings
as heat qh ' If the flow is adiabatic (qh = O) , this energy increases
the internal energy of the fluid and is reflected in the temperature,
pressure , and specific volume of the flowing fluid .
The development of the empirical (experimentally developed)
methods used to evaluate energy loss per unit mass caused by fric-
41
F U N DAMENTALS OF GAS FLOW I N CONDU ITS
tion is outside the scope of thermodynamics . These empirical
methods are discussed briefly here; greater detail is provided in
Chap . 4, where these methods are applied .
One of the most general approaches to develop empirical rela­
tionships for energy loss caused by friction is to express this ener­
gy loss in terms of a friction gradient, gf, as shown in Eq. 2 . 2 7 .
The empirical relations attempt t o correlate gf with fluid proper­
ties, geometry , and fluid-flow velocity . These empirical relations
take the form of equations or graphs used to evaluate the friction
gradient :
dF= (gf /P )dL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 27)
One commonly used approach to develop empirical relationships
for energy loss caused by friction is the Moody equation for pipe
flow, Eq . 2 .2 8 . In this equation, the dimensionless friction factor,
f, is correlated as a function of fluid properties, pipe diameter, and
fluid-flow velocity . These correlations of dimensionless friction fac­
tor take the form of equations or graphs used to evaluate the value
of f:
dF= (fv 2 /2gcd')dL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 28)
2.6 Bernou lli's Equation
Bernoulli ' s equation is a famous form of the mechanical energy
balance developed well before the formal statement of the first law
of thermodynamics . However, it has limiting assumptions. If energy
loss caused by friction is neglected (F= O) and no shaft work is done
by the system (ws =0), then the differential form of the mechani­
cal energy balance, Eq. 2 . 24 , can be expressed as
g
v
vsdP + - dZ + - dv = O . . . . . . . . . . . . . . . . . . . . . . . . . . (2 . 29)
gc
gc
If the fluid also is considered to be incompressible (v is constant) ,
integration of Eq. 2 . 29 gives Eq. 2 . 30 , which is commonly called
Bernoulli ' s equation :
gZ
v2
v sP + - + - = constant . . . . . . . . . . . . . . . . . . . . . . . . (2 . 30)
gc 2g c
Bernoulli ' s equation often is applied without knowledge or con­
sideration of the inherent assumptions, which are that F=O, ws = O ,
and V s is constant. Use o f any equations , tables, o r graphs without
consideration of the inherent assumptions should be avoided be­
cause significant errors in engineering calculations can result .
Exercises
2 . 1 Beginning with the following equations , verify that the units
of work and energy are ft-lbf.
A. Work, using Eq . 2 . 3 .
B . Internal energy , using Eq. 2 . 5 .
C . Potential energy , using Eq. 2 . 7 .
D . Kinetic energy , using Eq. 2 . 9 .
2 . 2 For a rigid body , Ep = mZg/gc where m i s the total mass of
'
the body and other terms are defined in the Nomenclature. De­
rive this result from Eq. 2 . 7 .
2 . 3 For a rigid body undergoing only translational motion , Ek =
mv 2 / (2g c) , where m is the total mass of the body , v is the
translational velocity of the rigid body , and other terms are de­
fined in the Nomenclature . Derive this result from Eq . 2 . 9 .
2 . 4 A n automobile has a mass o f 2 , 500 Ibm. Assuming no fric­
tion, no drag, and 1 00 % drive-train efficiency, how much power
(in horsepower) is required to accelerate the automobile from
o to 60 miles/hr in 1 0 . 0 seconds ( 1 hp = 550 ft-lbf/sec , and 1
mile = 5 ,280 ft)? Use the result derived in Exercise 2 . 3 .
2 . 5 A gas i s confined by a cylinder and piston a s in Fig . 2 . 5 (closed
system) . The mass of the piston on a scales calibrated for stan­
dard gravitational acceleration is 20 Ibm . However, the local
gravitational acceleration, g, is 32 . 1 0 ft/sec 2 . The diameter of
the piston is 2 . 0 in.
A . Assuming no friction, no acceleration, and atmospheric
pressure (surroundings) of 1 4 . 7 psia, what is the pressure
(psia) in the gas?
B. Next, the gas in the cylinder is heated, causing gas expan­
sion. If the piston is raised 6 . 0 in. , how much work is done
by the closed system?
C . Using the result derived in Exercise 2 . 2 , what is the change
in the potential energy of the piston? Consider the piston
to be a rigid body .
2 . 6 A closed system with fixed boundaries contains 1 0 . 0 Ibm of
fluid . A sealed paddle wheel inside the system is used to trans­
fer work across the boundary of the closed system. The fluid
is initially static and in thermal equilibrium with the sur­
roundings .
A . 1 0 . 0 ft-lbf work i s added to the closed system with the pad­
dle wheel . Assuming negligible friction, what is the aver­
age velocity (absolute value , mass-weighted average) of the
fluid in the system immediately after the paddle wheel stops?
B . Assuming friction is not negligible, how much heat will flow
across the system boundaries before the fluid returns to static
conditions and thermal equilibrium with the surroundings .
2 . 7 Consider the steady-state flow of water in a pipe . The constant
mass flow rate and constant water density are 2 . 0 Ibm/sec and
62 . 4 Ibm/ft 3 , respectively . The pressure, elevation, and pipe
diameter at the entrance are 1 00 psia, 1 00 ft, and 6 . 0 in . , re­
spectively . The elevation and pipe diameter at the exit are 1 1 0
ft and 4 . 0 in. , respectively .
A . What are the assumptions implicit in Bernoulli ' s equation?
B. What is the exit pressure (psia) calculated with Bernoull i ' s
equation?
C . If the effects of friction were considered , would the exit
pressure be higher or lower?
2 . 8 Consider the steady-state, isothermal flow of gas through a pipe
(open system with a single entrance and single exit) . The con­
stant mass flow rate is 0 . 5 Ibm/sec . The pressure , elevation,
and pipe diameter at the entrance are 30 psia, 1 00 ft, and 6 . 0
in. , respectively . The elevation and pipe diameter at the exit
are 200 ft and 1 . 2 in. , respectively . Because we are at low ab­
solute pressure, it is valid to assume that the volumetric be­
havior of the gas is described by the ideal-gas law { pvM = RT,
where M= molecular weight of the gas [lbm/(lbm-mol)] , R is
the universal gas constant, [ 1 0. 73 psia-ft 3 /(lbm-mol)- O R] , and
other terms are defined in the Nomenclature} . Given that the
molecular weight of the gas is 20 Ibm/(lbm-mol) , the constant
temperature is 520 o R , friction is negligible, and no shaft work
is performed by the system, answer the following questions .
A . What is the specific volume of the gas at the entrance con­
ditions?
B. Assuming that the specific volume of the gas is constant
throughout the system and equal to the value calculated in
Exercise 2 . 8A , what is the exit pressure (psia) calculated
with Bernoull i ' s equation? Remember that the fluid veloci­
ty (Vb = q' /A) is a function of the mass flow rate, pipe di­
ameter, and gas specific volume .
C . Using the ideal-gas law to describe specific volume as a func­
tion of pressure, what is the exit pressure (psia) calculated
with the mechanical energy balance? Remember that fluid
velocity is a function of mass flow rate, pipe diameter, and
specific volume. Because this problem is nonlinear, the exit
pressure must be calculated with an iterative root-finding
technique. Continue root-finding iterations until the percent
change in exit pressure between successive iterations is less
than om % .
Nomenclature
A = area, L 2 , ft 2
d' = pipe diameter, L, ft
e = specific total energy of fluid, L 2 /t 2 , ft-Ibf/lbm
e k = specific kinetic energy of fluid , L 2 /t 2 , ft-lbf/lbm
ep = specific potential energy of fluid , L 2 /t 2 , ft-lbf/lbm
E = total energy of system , mL 2 /t 2 , ft-lbf
42
GAS RESERVO I R E N G I N E E R I N G
Ek
=
Ep =
f=
F=
F{ =
g
=
gc
=
gf =
f
=
=
m =
p=
P=
q' =
qh =
L
Qh
w
=
=
=
=
=
=
=
=
=
=
=
=
ws
=
s
S
t
T
u
U
v
Vs
V
Vsy s
total kinetic energy of system, mL 2 1t 2 , ft-lbf
total potential energy of system, mL 2 1t 2 , ft-lbf
dimensionless friction factor (Moody)
energy loss resulting from friction, L 2 It 2 , ft-lbfllbm
force component in the direction of displacement,
mLlt 2 , Ibf
local acceleration due to gravity , Lit 2 ,
usually = 32. 1 7 ft/sec 2
dimensional constant, dimensionless, 32. 17
ft-Ibm/lbf-sec 2
friction gradient, m/Ll lt 2 , Ibf/ft 2 -ft
displacement distance, L , ft
distance along flow path , L, ft
mass, m, Ibm
absolute pressure , m/Lt 2 , psia
absolute pressure, m/Lt 2 , Ibf/ft 2
volumetric fluid flow rate, L 3 It, ft 3 /sec
heat added to the system per unit mass, L 2 /t 2 ,
ft-Ibf/lbm
total heat added to the system, mL 2 It 2 , ft-lbf
specific entropy of fluid , L 2 1t 2 T , ft-lbW R-lbm
entropy , mL 2 1t 2 T , ft-lbWR
time , t, seconds
absolute temperature , T, o R
specific internal energy o f fluid, L 2 1t 2 , ft-Ibf/lbm
total internal energy of system, mLl lt 2 , ft-lbf
flow velocity of fluid = q' IA , Lit, ft/sec
specific volume of fluid , L 3 /m, ft 3 1lbm
volume , L 3 , ft 3
system volume, L 3 , ft 3
work done by the system per unit mass, L 2 /t 2 ,
ft-lbfllbm
shaft work done by the system per unit mass, Ll It 2 ,
ft-Ibfllbm
W
Ws
Z
�
p
= total work done by system, mL 2 It 2 , ft-lbf
= total shaft work done by system, mL 2 /t z , ft-lbf
= elevation relative to arbitrary horizontal datum
plane, L, ft
= difference operator
= fluid density , miO . Ibm/ft 3
References
I . Abbott, M . M . and Van Nes s , H . C . : Thermodynamics . McGraw-Hili
Book Co. Inc . , New York City ( 1 989) .
2. Petroleum Engineering Handbook, H . B . Bradley (ed . ) , SPE, Richard­
son, TX ( 1 987) Chap . 1 3 .
3 . Theory and Practice of the Testing of Gas Wells. fourth edition, Ener­
gy Resources and Conservation Board , Calgary ( 1 979) .
4. Ikoku , C . U . : Natural Gas Engineering-A Systems Approach . Penn­
Well Books, Tulsa, OK ( 1 980) .
5 . Katz , D . L . and Lee , R . L . : Natural Gas Engineering. Production and
Storage. McGraw-Hili Book Co. Inc . , New York C ity ( 1 990) .
6. Jenning s . J. et al. ,' Deliverability Testing of Natural Gas Wells, Texas
A&M U . , College Station (March 1 989) 1 1 - 1 3 .
7 . Perry , R . H . and Chilton , C . H . : Chemical Engineers ' Handbook. fifth
edition, McGraw-Hili Book Co. Inc . , New York City ( 1 973) .
8 . Manual of Petroleum Measurement Standards . AGA Report 3 ; Chap .
1 4 , API 2530; American Natl . Standards Inst . , ANSI/API 2530 1 985 ;
Gas Proces sors Assn . • GPA 8 1 85 8 5 ( 1 985) .
9 . Rawlins , E . L . and Schellhardt . M . A . : Backpressure Data on Natural
Gas Wells and Their Application to Production Practices. Monograph
Serie s , USBM ( 1 93 5 ) 7.
1 0 . Katz , D . L . et al. ,' Handbook of Natural Gas Engineering. Production
and Storage . McGraw-Hili Book C o . Inc . , New York City ( 1 959)
332-5 1 .
1 1 . Smith , I . M . and Van Ness, H . C . : Introduction to Chemical Engineer­
ing Thermodynamics. fourth edition, McGraw-Hill Book Co. Inc . , New
York City ( 1 987) 30-3 5 .
1 2 . Bird , R . B . , Stewart, W . E . , and Lightfoot, E . N . : Tramport Phenome­
non . John Wiley & Sons Inc . . New York C ity ( 1 960) 8 1 -82 .
1 3 . Brown, K . E . : The Technology ofA rtificial Lift Methods. Petroleum Pub­
lishing Co . . Tulsa ( 1 977) I, 93-96.
Chapter 3
Gas Flow Measurement
3.1 I nt roduction
Measuring the gas flow rate in some manner is necessary . Results
of flow rate measurements are used in gas sales, reservoir engi­
neering calculations, and pipeline and plant applications . Although
the volumetric flow rate (at the pressure and temperature of the
measurement point) might be important for physical calculations,
it is not as important as the flow rate in standard units . The flow
rate for gas sales is expressed in standard units , such as standard
cubic feet per day , which is equivalent to a mass flow rate.
Several common methods of gas flow measurement in flowlines
are discussed . All make use of thermodynamic principles and phys­
ical measurements. The measurements are made on a flowline,
usually at the wellhead or a testing facility .
This chapter presents the most common meters used to calculate
gas flow rates. There are two general classes of meters : volumet­
ric and dynamic . Volumetric meters usually are used to measure
gas flow rates in residential areas where flow rates are very low.
Flow rates in fields , pipelines, and plants are high, so volumetric
meters are not effective . Therefore, this chapter covers only dy­
namic meters . The focus is on the primary type of dynamic meter
used, the orifice meter. Other dynamic meters, such as critical flow
provers, choke nipples , and pitot tubes , also are covered .
3.2 Orifice M eters
The orifice meter is used most commonly in the gas production
and transportation industry because of its accuracy, simplicity, and
reasonable cost. It has an interchangeable orifice plate with a small,
circular opening, much smaller than the pipe diameter, inserted into
the flowline . Taps, either pipe or flange , are used to measure pres­
sure (Fig. 3 . 1 ) . The orifice meter has a gauge that records the pres­
sure and the pressure difference between the taps, as Fig. 3.2 shows.
The orifice meter operates on the principle that a change in the
gas-stream velocity results in a corresponding pressure change.
When the gas flow is restricted by the orifice, velocity increases
and pressure decreases . Theoretically , this can be explained by Eq.
2.23, the mechanical-energy-balance equation for steady-state flow
of a homogeneous single-phase fluid :
) + F= -ws
j' P2 VsdP + .:i ( g&Z ) + .:i ( �
2&
PI
.
. . . . . . . . . . . (2.23)
The units of this equation are often somewhat incorrectly referred
to as " feet of fluid head , " but are actually (ft-lbf/lbm) .
For the system of gas flowing through an orifice plate, Eq. 2 .23
can be simplified because the following two conditions apply :
and (2) the meter
( 1) there is no external shaft work done
is horizontal
Eq. 2 .23 becomes
(.:12=0).
j' P2VsdP+.:i ( �
2gc ) + F=0
(F=O),
3.1
I P2 vsdP + .:i ( �
2gc ) =0 . .
(ws =0)
. . . . . . . . . . . . . . . . . . . . . . (3 . 1)
.
PI
.
Our first assumption, which is that friction loss is negligible
then Eq.
becomes
.
PI
.
. . . . . . . . . . . . . . . . . . . (3 .2)
.
.
.
.
Vs
Our second assumption is a constant specific volume, to sim­
plify the integral (we later compensate for this with an expansion
factor) . Eq . 3 .2 becomes
vi -v r =2gc Vs (P I -P2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . (3.3)
vs (p\
=
v=qIA .
v,
- P2 ) is denoted h , differential head loss i n feet o f fluid
head. Note that the unit configuration is actually (ft 3 I1bm)(lbf/ft 2 )
(ft-lbf/lbm) .
Our third assumption is that
This implies that our ac­
tual velocity in our kinetic energy term, is the net flow velocity
parallel to the pipe. This net flow velocity is an average over the
flow diameter, not the actual velocity , representing the kinetic ener­
gy of the fluid.
The fourth assumption involves the location of the pressure taps.
We assume that P I is measured at A I ' which is at a point unaffect­
ed by the orifice plate , and P 2 is measured at A 2 , which is at the
orifice plate . However, Fig . 3 . 1 shows that
is not measured at
the orifice plate . We later compensate for these two important as­
sumptions with a basic orifice flow factor.
The third and fourth assumptions are the most important. With
these assumptions , Eq. 3 . 3 becomes
(q 2 /A 2 ) 2 - (q I /A 1 ) 2
. . . . . . . . . . . . . . . . . . . . . . . . . (3.4)
Now, the volumetric flow rate can be changed to a standard rate,
We
For steady-state flow,
by
continue to assume that
to give the following simpli­
fied result:
P2
=2gch
qsc ' q=(vslvssc)qsc· vsI =vs =vs qsc 1 =qsc2 =qsc·
2
qlc ? � rA� (vssc/vs ) 2 2gch . . . . . . . . . . . . . . . . . . (3 . 5)
(A -A ) =A
GAS RESERVO I R E N G I N E E R I N G
44
_--:;;� nl'no� ( throat) tops
�---J'/
/J "
( full flow) tops ----
F i g . 3 . 2 -Typical orifice meter chart a n d recording (after Ref.
2).
C' , the orifice flow constant, is
determined primarily from the basic orifice flow factor, Fb . For
routine field, pipeline, and plant operations, the first eight factors
usually are adequate to determine C' . The last three factors (Fm '
Fl , and Fa) are approximately equal to unity and do not change
C' much. Therefore, these three factors generally are used only
for gas sales and purchases . The 1 1 factors in the orifice flow con­
stant are defined as follows :
Fb is the basic orifice flow Jactor. Fb depends on the pipe and
orifice diameters and on the location of the pressure taps. Accord­
ing to the derivation of Eq . 3 . 8, Fb = 338 . 17 dli l(I -/14)O.5 . As
discussed above, the main assumptions in the derivation are our
third and fourth assumptions . Because of these two important as­
sumptions , Fb must be calibrated for Eq. 3 . 9 to match actual flow
rates . These factors are tabulated in Tables 3 . 1 and 3.2. The values
of Fb may be found in Table 3 . 1 for flange taps and Table 3 . 2 for
pipe taps. (All the orifice tables are taken from standard tables. 3 )
The tabulated values are lower than theoretical values (typically
30 % to 40% lower) .
The next five factors account for differences in our assumed con­
ditions: P sc = 14 . 73, Tsc =520, Tf =520, "{ g = l , and zf = I . To cor­
rect for the actual values of these variables , the next five factors
are used . Tables often are used in the field for these factors . How­
ever, these tables are omitted here because these factors can be cal­
culated .
Fpb = 14. 731psc (pressure base Jactor), corrects to the proper
pressure base .
F Tb = Ts/520 (temperature base Jactor), corrects to the proper
temperature base .
Fg = (11"{ g) 0. 5 (specific g ravity Jactor), corrects the proper spe­
cific gravity .
FTf = (520ITf) 0. 5 (flowing temperature Jactor), corrects to the
proper flowing temperature.
Fpv = (1 Izf) O. 5 (gas deviation Jactor), corrects to the proper Z
factor.
The next two factors are more indirect and tend to correct for
deviations from the assumptions and measurements .
FRe = 1 +bl(h w Pf) 0. 5 (Reynolds number Jactor), corrects for the
variation of the discharge coefficient with Reynolds number. The
value of b may be found in Table 3.3 for flange taps and Table
3.4 for pipe taps. The constant b is primarily a function of the lo­
cation of the pressure taps, the pipe diameter and the orifice di­
ameter.
Y (expansionJactor) corrects for the change in gas density as the
pressure changes across the orifice. The value of Y may be found
in Tables 3.5 through 3 . 9 for different taps and tap locations . The
3 . 2 . 1 Orifice Equation Factors.
Fig. 3 . 1 -0rifice meters: flange and pipe taps (after Katz,
Handbook of Natural Gas Engineering, 1 959, McGraw-Hili Book
CO . , 1 cou rtesy of McGraw- H i l i I n c . ) .
(- )
After rearranging and substituting ,
1
Vssc
,, 2 g c h , . . . . . . . . . . . . . . . . . . . (3 . 6)
" 1 _{34 Vs
where {3=dold do = orifice opening diameter, in. ; and d= pipe rD ,
qsc =A 2
r-;--;:;;t
:
�
in.
The reciprocal square-root factor is sometimes called the approach
factor. This equation can be put into a more practical form that does
not contain Vs by substituting <R TlpM for vs ' Then Eq. 3 . 6 becomes
dli
Pf Tsc �
..J -:;-::;
(3 . 7)
--hh ww,
" 1 _ {34 P sczf Tf Pf "{ g
Now , Eq. 3 . 7 is simplified further by letting P sc = 14 . 73 psia,
Tsc = Tf = 520oR, zf = l , and "{ g = I . O:
d 2 ..J
h wPf . . . . . . . . . . . . . . . . . . . . . . (3 . 8)
q gh = 338 . 17
1 - /14
E q . 3 . 8 contains the basic factors affecting the measurement pa­
qgh = 2 1 8 . 44
r-;--;:;;t
--
•
•
•
•
•
•
•
•
•
•
�
rameters but is on a very idealized basis. To modify this equation
for practical use, empirical factors are introduced to give the final
form of the basic orifice equation,
qgh = C' ..Jh wPf ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3 . 9)
where C' = FbFpbFTbFgFTfFpvFRe YFm FlFa . . . . . . . . . . . . (3 . 10)
Eqs . 3 . 9 and 3 . 10 might be considered a practical form of the
idealized flow equation , Eq . 3 . 8 . The factors in Eq. 3 . 10 are in­
troduced to match actual flow rates . The factors correct for assump­
tions made and for actual flow conditions .
LO
v
TABLE 3 . 1 -FLANGE TAPS: B A S I C O R I F I C E FACTORS, Fb (AFTER R E F . 3 )
do
(in.)
I-
z
w
:::2:
w
a:
::::l
en
«
w
:::2:
3:
0
...J
lL
en
«
(!)
0 . 250
0 . 375
0 . 50 0
0 . 62 5
0 . 750
0 . 875
1 .000
1 . 1 25
1 . 250
1 . 375
1 . 500
1 . 625
1 . 750
1 . 875
2 . 00 0
2. 1 25
2 . 250
2 . 375
2 . 500
2 . 625
2 . 750
2 . 875
3 . 000
3 . 1 25
3 . 250
3 . 375
3 . 500
3 . 625
3 . 750
3 . 875
4 . 000
4 . 250
4 . 500
1 . 689
1 2 . 695
2 8 . 474
50 . 777
8 0 . 090
1 1 7. 09
1 62 . 95
2 1 9 . 77
290 . 99
3 8 5 . 78
---
1 . 939
1 2 . 707
28.439
50.587
79.509
1 1 5.62
1 59 . 56
2 1 2 . 47
276 . 20
353. 58
448.57
d (in . )
3
2
2 . 067
1 2. 7 1 1
28.428
50.521
79. 3 1 1
1 1 5. 1 4
1 58 . 47
2 1 0.22
271 . 70
345. 1 3
433. 50
542. 26
2 . 300
1 2. 7 1 4
28.41 1
50.435
79.052
1 1 4.52
1 57 . 1 2
207.44
266. 35
335. 1 2
4 1 5 . 75
5 1 0 . 86
623 . 9 1
2 . 626
1 2. 7 1 2
28.393
50.356
78. 8 1 8
1 1 3.99
1 56.00
205. 1 8
262. 06
327. 39
402. 1 8
487. 98
586.82
701 .27
834 . 88
2 . 900
1 2. 708
28. 382
50. 3 1 3
78. 686
1 1 3. 70
1 55.41
204.04
259.95
323. 63
395. 80
477. 36
569. 65
674.44
793. 88
930 . 65
1 , 091 . 2
3. 068
1 2. 705
28. 376
50. 292
78. 625
1 1 3.56
1 55 . 1 4
203.54
259.04
322.03
393.09
472 . 96
562.58
663.42
777. 1 8
906. 0 1
1 ,052.5
1 ,223.2
3. 1 52
1 2. 703
28. 373
50.284
78 .598
1 1 3.50
1 55.03
203.33
258.65
321 . 37
39 1 . 97
471 . 1 4
559. 72
658.96
770.44
896.06
1 ,038 . 1
1 ,499. 9
4
3.438
1 2 .697
28. 364
50. 258
78. 523
1 1 3 . 33
1 54.71
202 . 75
257.63
31 9.61
389 .03
466 . 39
552 . 3 1
647. 54
753. 1 7
870.59
1 ,001 .4
1 , 1 47 . 7
1 ,31 1 .7
1 ,498 .4
6
3. 826
1 2 .687
28.353
50. 234
78.450
1 1 3. 1 5
1 54.40
202.20
256 .69
3 1 8 . 03
386.45
462.27
545 .89
637. 84
738 . 75
849. 4 1
970.95
1 , 1 04 . 7
1 ,252 . 1
1 ,4 1 5 . 0
1 ,595 . 6
1 , 797. 1
4. 026
1 2.683
28. 348
50. 224
78.421
1 1 3.08
1 54.27
201 .99
256.33
31 7.45
385. 5 1
460. 79
543.61
634.39
733. 68
842. 1 2
960.48
1 ,089.9
1 ,231 . 7
1 ,387.2
1 , 558. 2
1 , 746. 7
1 ,955.5
2, 1 94.9
4.897
5 . 1 89
5 . 76 1
6 . 065
50 . 1 97
78 . 338
1 1 2.87
1 53.88
20 1 .34
255.31
31 5.83
382 .99
456.93
537. 77
625 . 73
72 1 . 03
823. 99
934.97
1 ,054.4
1 , 1 82.9
1 , 320.9
1 ,469 .2
1 , 628.9
1 , 801 . 0
1 , 986.6
2 , 1 87.2
2,404.2
2,639.5
2,895.5
3, 1 80.8
50. 1 9 1
78.321
1 1 2 . 82
1 53 . 78
201 . 1 9
255.08
3 1 5.48
382.47
456. 1 6
536 . 64
624.09
7 1 8 . 69
820.68
930.35
1 ,048 . 1
1 , 1 74 . 2
1 ,309 . 3
1 ,453.9
1 ,608 . 7
1 , 774 . 5
1 ,952 .4
2 , 1 43.4
2 , 348 . 8
2 , 569.8
2,808 . 1
3,065.3
3 , 345.5
3,657.7
50. 1 82
78. 296
1 1 2. 75
1 53 . 63
200. 96
254 . 72
3 1 4 . 95
381 . 70
455.03
535.03
621 . 79
7 1 5.44
8 1 6. 1 3
924.07
1 ,039 .5
1 , 1 62 . 6
1 ,293. 8
1 ,433.5
1 , 582 . 1
1 , 740 . 0
1 ,907.8
2 , 086.4
2 , 276 . 5
2 ,479 . 1
2 , 695 . 1
2 , 925.7
3 , 1 72 . 1
3 , 435 . 7
3,71 8.2
4 , 354. 8
50. 1 78
78.287
1 1 2 .72
1 53.56
200 .85
254.56
3 1 4 . 72
381 . 37
454.57
534.38
620. 88
7 1 4. 1 9
8 1 4.41
921 . 7 1
1 ,036 . 3
1 , 1 58 . 3
1 ,288 . 2
1 ,426. 0
1 , 572 . 3
1 , 727 . 5
1 ,891 .9
2 , 066. 1
2 , 250.8
2 , 446. 8
2 , 654. 9
2 , 876. 0
3 , 1 1 1 .2
3 , 36 1 . 5
3 , 628 . 2
4,21 6.6
4 , 900.9
<!'J
z
a:
w
w
z
a
z
w
a:
TABLE 3 . 2- P I P E TAPS: BASI C ORI FICE FACTORS, Fb (AFTER REF. 3 )
5
>
a:
w
en
w
a:
�
<!'J
�
d (in.)
do
�
0 .250
0 . 375
0 . 50 0
0 .625
0 . 750
0 . 875
1 .0 0 0
1 . 1 25
1 .250
1 .375
1 .500
1 .625
1 . 750
1 .875
2 . 000
2 . 1 25
2 . 25 0
2 .375
2 .500
2 . 62 5
2 . 750
2 .875
3 .0 0 0
3 . 1 25
3 .250
3 . 375
3 . 50 0
3 .625
3 . 750
3 . 875
4 . 000
4 .250
4.500
4 .750
5 . 00 0
5 .250
5 . 50 0
2
1 .689
1 .939
2 . 067
2 . 300
2 . 626
1 2. 850
2 9 . 359
53.703
87. 2 1 2
1 32 . 23
1 92 . 74
275.45
391 . 93
1 2. 8 1 3
29.097
52.81 6
84.91 9
1 26.86
1 81 .02
251 . 1 0
342.98
465. 99
1 2. 800
29. 005
52.481
84.083
1 24 . 99
1 77 . 08
243.27
327.98
437.99
583.96
1 2. 782
28.882
52.0 1 9
82.922
1 22.45
1 71 .92
233.30
309 .43
404.52
524.68
679 . 1 0
1 2. 765
28.771
5 1 .591
8 1 . 795
1 20.06
1 67.23
224 . 56
293.79
377. 36
478 . 68
602.45
755 . 34
946.99
2 . 900
3. 068
3. 1 52
3 .438
1 2. 737
1 2. 745
1 2.753
1 2. 748
28.7 1 0
28.634
28. 669
28.682
5 1 . 064
5 1 . 353
5 1 . 1 96
5 1 . 243
80. 703
80.332
80.835
8 1 . 1 42
1 1 7.70
1 1 8 . 67
1 1 6.86
1 1 8. 00
1 63 . 3 1
1 61 . 1 7
1 62 . 76
1 64.58
2 1 6 . 55
2 1 7 . 52
2 1 3 . 79
2 1 9 . 76
275.42
285.48
2 8 1 .66
280.02
357. 1 2
347.03
354.45
363.41
455 .82
441 . 48
429.83
445. 74
565 . 79
543 . 3 1
525.40
549.94
635.76
697.43
672.95
662 . 8 1
763 . 5 1
856.37
8 1 9 . 05
803 . 77
1 ,050.4
9 1 1 .98
993.98
971 . 1 9
1 ,290 . 7
1 ,205 . 6
1 ,085.5
1 , 1 71 . 8
1 ,465 . 1
1 ,4 1 5 . 0
1 ,289. 7
1 ,532. 0
1 ,822 . 8
8
6
4
3
3.826
4 . 026
1 2. 727
28.598
50.936
79.974
1 1 6.05
1 59 . 57
2 1 1 .03
270.90
339.87
4 1 8 . 79
508. 76
61 1 . 1 1
727.54
860. 1 7
1 ,0 1 1 .7
1 , 1 85.3
1 ,385.4
1 ,6 1 7 . 2
1 ,887. 6
2 ,206 . 0
1 2. 722
28. 584
50.886
79.835
1 1 5. 73
1 58 . 94
209 . 9 1
269. 1 0
337.05
41 4.51
502 . 38
601 .80
7 1 4. 1 6
841 . 1 9
985 . 04
1 , 1 48 . 4
1 ,334.4
1 , 547 . 3
1 , 792. 3
2 , 075.9
2 , 407 . 0
4. 897
5. 1 89
5 . 76 1
6.065
50.739
79.436
1 1 4. 8 1
1 57. 1 1
206.62
263 . 7 1
328. 73
402.06
484.20
575.73
677.38
789.99
9 1 4.57
1 ,052 . 3
1 ,204 . 7
1 ,373.4
1 ,560.5
1 ,768. 3
1 ,999 . 8
2,258.5
2,548.6
2,875.2
3,244. 8
3,665.6
50. 705
79. 349
1 1 4.61
1 56.71
205.91
262 . 5 1
326.85
399. 30
480.23
570. 1 4
669.63
779.40
900.28
1 ,033.2
1 , 1 79.4
1 , 340 .2
1 ,51 7.2
1 ,7 1 2 . 3
1 ,927.6
2 , 1 65.9
2 ,430.2
2 , 724.4
3 , 052. 8
3,420.9
3,835.7
4,305 . 7
50.652
79.2 1 7
1 1 4. 32
1 56. 1 3
204 . 84
260 . 7 1
324.02
395.08
474.20
56 1 . 73
658.08
763. 77
879.38
1 ,005. 6
1 , 1 43 . 2
1 ,293 . 1
1 ,456.4
1 ,634 . 3
1 ,828. 3
2 , 039.9
2,271 .2
2 . 524.3
2 , 80 1 . 8
3 , 1 06.9
3 , 443.0
3 , 8 1 4.4
4 , 226.3
4 , 684.9
5 , 1 97.7
50.628
79 . 1 62
1 1 4.20
1 55.89
204.41
259.98
322.86
393.33
471 .69
558. 24
653.33
757.39
870.93
994.52
1 , 1 28 . 8
1 ,274 . 6
1 ,432. 7
1 ,604. 3
1 , 790. 3
1 ,992. 2
2 , 2 1 1 .6
2 ,450 . 1
2 ,709. 9
2 , 993.3
3,303.0
3 , 642 . 3
4,01 4.8
4,425 . 1
4,878.4
7.625
7.981
8.071
1 55. 1 0
203.00
257.62
31 9. 1 0
387.62
463.39
546.61
637 . 5 1
736 . 34
843. 34
958.78
1 ,083.0
1 ,2 1 6 . 3
1 , 359.2
1 ,51 2 . 0
1 ,675 .4
1 ,849. 9
2 , 036.0
2 , 234. 7
2 ,446. 5
2 ,672. 5
2,91 3.7
3 , 1 71 . 1
3 ,446. 0
3,739 . 9
4 ,054 . 2
4 , 75 1 .4
5 , 554. 7
6,485.3
7 , 57 1 .4
8 , 850. 3
1 54.99
202.80
257.28
3 1 8 . 56
386 . 8 1
462 . 1 9
544.92
635. 1 9
733.23
839.29
953.58
1 ,076.4
1 ,208.0
1 , 348. 8
1 ,499. 2
1 ,659 . 7
1 ,830.6
2,01 2.7
2 , 206.4
2,41 2.4
2,631 .6
2 , 864. 7
3 , 1 1 2. 7
3 ,376.6
3 ,657.6
3,957.0
4 , 6 1 6.6
5 , 369.0
6,231 . 1
7 , 224.3
8 , 376 . 3
9 , 723.8
1 54.96
202 . 75
257.20
3 1 8 .44
386.62
461 .92
544.53
634 . 65
732. 52
838.35
952.38
1 ,074 . 9
1 ,206 . 1
1 , 346 . 5
1 ,496. 3
1 ,656 . 1
1 ,826 . 3
2,007.3
2 , 1 99 . 9
2 ,404 . 7
2,622 . 3
2 ,853.7
3,099.6
3 , 36 1 . 0
3 , 639.2
3 ,935. 2
4 , 586.6
5 ,327. 9
6 , 1 75.2
7 , 1 48 . 7
8 , 274. 0
9,585 . 1
47
GAS FLOW M EAS U R E M ENT
TABLE 3 .3-FLANGE TAPS: PARAMETER USED FOR FRe , b (AFTER REF. 3)
d (in.)
do
�
0 .250
0 . 375
0 . 500
0 . 625
0 . 750
0.875
1 .000
1 . 1 25
1 .250
1 .375
1 .500
1 .625
1 .750
1 .875
2 .000
2 . 1 25
2 .250
2 .375
2 .500
2 . 625
2 . 750
1 .689
1 .939
2 . 067
2 . 300
0 . 0879
0 . 0677
0 .0562
0 . 0520
0 .0536
0 . 0595
0 . 0677
0 .0762
0 . 0824
0 . 09 1 1
0 . 0709
0 . 0576
0 . 0505
0 . 0485
0 . 0506
0 .0559
0 .0630
0 .0707
0 .0772
0 . 0926
0 . 0726
0 . 0588
0 . 0506
0 . 0471
0 . 0478
0.051 5
0 .0574
0 .0646
0 . 07 1 5
0 . 0773
0 .0950
0 .0755
0 . 061 2
0 . 05 1 6
0 .0462
0 .0445
0 . 0458
0 . 0495
0 .0550
0 . 06 1 4
0 . 0679
0 .0735
Y, Y C C ,
2 . 626
2 . 900
3 . 068
3 . 1 52
0 .0792
0 . 0648
0 . 054 1
0 .0470
0 . 0429
0 . 04 1 6
0 . 0427
0 . 0456
0 . 0501
0 .0554
0 . 06 1 3
0 . 0669
0 . 07 1 7
0 .0820
0 .0677
0 . 0566
0 . 0486
0 . 0433
0 . 0403
0. 0396
0 .0408
0 .0435
0 . 0474
0 . 0522
0. 0575
0 .0628
0 .0676
0 . 071 5
0 . 0836
0 .0695
0 .0583
0 .0498
0 .0438
0 . 0402
0 . 0386
0 .0388
0 . 0406
0 . 0436
0 .0477
0 .0524
0 .0574
0 .0624
0 .0669
0 .0706
0 .0844
0 .0703
0 . 0591
0 . 0504
0 . 0442
0 .0403
0 .0383
0 . 0381
0 .0394
0. 0420
0. 0457
0 . 0500
0 .0549
0 .0598
0 .0642
0 . 0685
1 .0.
45
120°F
Example 3 . t-Orifice Meter Calculation. Calculate the gas flow
rate through an orifice meter for the following conditions.
hwPJ ==40143
=84°F .
Psc = 14. 4
Tscd=4=60°F
.
26
0
.
g =0.. 750.
"'Ido=1
in. of water.
psig (measured upstream) .
in.
in.
Taps = flange type .
1.
C' =FbFpbF1bFgFTJFpvFReY,
3 . 1),
Fb=460 . 79
Fpb = 14. 7 3/14 . 4 = 1 . 0229,
F1b = 1 . 0
Tsc =5200R),
0
5
.
Fg =(1/0 . 7 ) = 1 . 1952,
F TJ = [520/(84+460)]°· 5 =0 . 9 777,
Fp v =(110 . 9 8)°· 5 = 1 . 0 10 (z =0 . 9 8
1),
FRe = 1 + bl[(40)(143 + 14 . 4)]°· 5 = 1 . 0004 (b=0 . 0 350
3 . 3),
3 . 5) .
Y=0 . 9971
Solution.
Determine the factors for the orifice constant. We use the ab­
breviated form here , ignoring the last three factors.
X
Calculate
scf/hr.
3.3 Orifice M eter Installation
Fig. 3.3 shows a typical orifice meter installation consisting of the
meter run and recording equipment. In this section, we discuss some
of the features and design considerations necessary for a proper
orifice meter installation.
Orifice meters most commonly are located downstream from a
gas/liquid separator to ensure that liquid has been removed from
the flow stream and to provide a lower operating pressure .
Pipe and flange taps are the standard taps used in the industry
and differ from each other in their location on the flowline . Flange
taps are located so that the centers of the taps are in. from the
respective orifice plate surfaces . Standard pipe taps are located so
pipe diameters from the orifice plate
that the upstream tap is
pipe diameters away . Fig .
surface and the downstream tap is
shows their relative locations .
Fig. 3.3 shows a permanent installation of orifice meters . Trailer­
or skid-mounted units often are used to test a new well before per­
manent equipment is available.
One major consideration in orifice meter design is sizing the
is solved for with esti­
orifice and the meter run pipe . If Eq.
mates for qgh ' h w , and PJ ' then orifice size can be estimated by
2.5
8.0
C'
3.9
do =(C' 1250) 0. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3 . 11)
000
=40
IiPJw =50,
Psc ==14143. 4
qgh
from graphical estimates of
(interpolated from Table
1
ble orifice plate diameter given the following data.
(because
and
0 .0763
0. 0653
0 . 0561
0 . 0487
0 . 0430
0 . 0388
0 . 0361
0 .0347
0 .0345
0 . 0354
0 . 0372
0 . 0398
0. 0430
0 . 0467
0 .0507
0. 0548
0. 0589
0 . 0626
0. 0659
4.026
0 .0779
0 .0670
0 . 0578
0 . 0502
0 . 0442
0 . 0396
0 . 0364
0 .0344
0 .0336
0 . 0338
0 . 0350
0 . 0370
0 . 0395
0 . 0427
0 . 0462
0 . 0501
0 . 0540
0 .0579
0 . 061 5
0 . 0647
Example 3 .2-0rifice Plate Sizing Calculation. Calculate a suita­
(from Table
Chap .
0 . 0867
0 . 0728
0 . 06 1 8
0 . 0528
0 . 0460
0 . 041 1
0 . 0380
0 . 0365
0 . 0365
0 . 0378
0 . 0402
0 . 0434
0 . 0473
0 . 05 1 7
0 .0563
0 . 0607
0 . 0648
0 .0683
3 . 826
Calculate
3. 1
psia.
where
3.438
C' .
2.
C' = FbFpbF1bFgFTJFpvFRe Y= (460 . 79)(1 . 0229)(1 . 0)(1 . 1952)
(0 . 9777)(1. 0 10)(1 . 0004)(0 . 9971) =554 . 90 .
3.
q gh '
3 4 . qgh=C'(hw
PJ) 0. 5 =554. 90 [(40)(143 + 14 . 4)] 0. 5 =44,030
specific-heat ratio, p I . is assumed to be constant in the calcu­
so is tabulated as a function of (3 and the ratio of
lation of
differential to absolute pressure.
The final three factors, Fm ' FI, and Fa ' are more likely to be
used in gas sales than in well testing. For our purposes , they can
Their values can be found in Refs . and
be set equal to
Fm ' manometer Jactor, corrects for the slight error in measure­
ment caused by having different heads of gas above the two legs
of mercury manometers .
FI, gauge location Jactor, corrects Fm for elevations other than
°.
sea level and latitudes other than
Fa ' thermal expansionJactor, corrects for expansion or contrac­
tion of the orifice opening when the operating temperature is sub­
stantially different from that at which the orifice was made . It is
or below O a F .
usually only applied for temperatures above
TJ
4
3
2
from Table
scf/hr.
in. of water.
psig (measured upstream) .
psia.
3.9.
1.
50,000 630.1.
=
C'= �
--Ih wPJ --1(40)(143 + 14 . 4)
Solution.
Determine the orifice constant from Eq.
G
Z
iI
ill
ill
Z
C3
Z
ill
a:
TABLE 3.4-PIPE TAPS: PARAMETER USED FOR F Re , b (AFTER REF. 3)
0
>
a:
ill
en
ill
a:
en
«
G
co
"<;j"
do
(in .)
--
0 . 250
0 . 375
0 . 500
0 . 625
0 . 750
0 . 875
1 .000
1 . 1 25
1 .250
1 .375
1 . 500
1 . 625
1 . 750
1 .875
2 . 000
2 . 1 25
2 . 250
2 . 375
2 . 500
2 . 625
2 . 750
2 .875
3 . 000
3 . 1 25
3 . 250
3 . 375
3 . 500
3 . 625
3 . 750
3 . 875
4 . 000
4 . 250
4 . 500
4 . 750
5 . 000
5 . 250
5 . 500
2
4
3
1 . 689
1 .939
2 . 067
2 . 300
0 . 1 1 05
0 . 0890
0 .0758
0 .0693
0 .0675
0 .0684
0 . 0702
0. 0708
0 . 1 09 1
0 .0878
0 .0734
0 .0647
0 .0608
0 .0602
0 . 06 1 4
0 .0635
0 .0650
0 . 1 087
0 .0877
0 . 0729
0 . 0635
0 . 0586
0 . 0570
0 .0576
0 . 0595
0 . 06 1 6
0 .0629
0 . 1 08 1
0 . 0879
0 .0728
0. 0624
0 . 0559
0 . 0528
0 .0522
0 . 0532
0 . 0552
0 . 0574
0 .0590
2 . 626
2. 900
3. 068
3. 1 52
3.438
0 . 0888
0 . 0737
0 .0624
0 . 0546
0 . 0497
0 . 0473
0 . 0469
0 . 0478
0 . 0496
0 . 05 1 8
0 . 0539
0 . 0553
0 .0898
0 .0750
0 . 0634
0 .0548
0. 0488
0. 0452
0 .0435
0. 0434
0 .0443
0. 0460
0. 0482
0. 0504
0 . 052 1
0 . 0532
0 .0905
0 . 0758
0 .0642
0 .0552
0 . 0488
0 .0445
0 .0422
0 . 04 1 4
0.041 8
0 . 0431
0 . 0450
0 . 0471
0 . 0492
0 . 0508
0 . 05 1 9
0 . 0908
0 . 0763
0 .0646
0 . 0555
0 . 0489
0 .0443
0 . 04 1 7
0 . 0406
0 .0408
0 . 04 1 8
0 . 0435
0 . 0456
0 . 0477
0 . 0495
0 . 0509
0 . 09 1 8
0 . 0778
0 . 0662
0 . 0568
0 . 0496
0 . 0443
0 . 0407
0 . 0387
0 . 0379
0 . 0382
0 . 0392
0 . 0408
0 . 0427
0 . 0448
0 . 0467
0 . 0483
0 . 0494
d (in .)
6
3.826
4. 026
4.897
0 .0799
0. 0685
0 . 0590
0 . 05 1 3
0 .0453
0 . 0408
0 . 0376
0 .0358
0 . 0350
0 . 0351
0 . 0358
0.0371
0. 0388
0 . 0407
0 . 0427
0 . 0445
0 .0460
0 . 0472
0.081 0
0. 0697
0. 0602
0. 0524
0 . 0461
0 . 04 1 2
0 . 0377
0 . 0353
0 .0340
0. 0336
0. 0340
0. 0349
0. 0363
0 .0380
0 . 0398
0 . 04 1 7
0 . 0435
0 . 0450
0. 0462
0 . 0850
0 .0747
0 .0655
0 .0575
0 . 0506
0 .0448
0 . 0401
0 .0363
0 .0334
0 . 031 3
0 .0300
0 . 0293
0 . 0292
0. 0297
0 .0305
0 . 031 6
0 .0330
0. 0345
0. 0362
0. 0379
0.0395
0 . 041 0
0 . 0422
0 . 0432
5. 1 89
5 . 76 1
6. 065
0. 0762
0.0672
0.0592
0. 0523
0. 0464
0 . 041 3
0.0373
0.0340
0 . 031 5
0 .0298
0. 0287
0 . 0281
0 . 0281
0.0285
0 . 0293
0 .0304
0 . 031 7
0 . 0331
0. 0347
0. 0364
0. 0380
0. 0394
0 . 0408
0 . 041 9
0 .0428
0. 0789
0. 0703
0. 0625
0 . 0556
0. 0495
0 . 0442
0 . 0397
0. 0360
0. 0329
0. 0304
0. 0285
0. 0273
0. 0265
0 . 0261
0. 0262
0. 0267
0 . 0274
0 . 0284
0 . 0295
0. 0308
0. 0323
0 .0338
0 . 0353
0. 0367
0 . 0381
0. 0393
0 . 0404
0 . 041 3
0 .0802
0.071 8
0 .0642
0 . 0573
0 . 05 1 2
0 . 0458
0 . 04 1 2
0 .0372
0 .0339
0.031 1
0 .0290
0 . 0273
0 .0262
0 .0258
0 . 0253
0 . 0254
0 . 0258
0 . 0265
0 . 0274
0 .0285
0 . 0297
0 . 03 1 1
0 .0325
0 .0339
0 .0354
0 . 0367
0 . 0380
0 . 0391
7. 625
0 . 07 1 6
0 . 0652
0 . 0592
0 .0538
0 . 0489
0 . 0445
0 .0404
0 . 0369
0 . 0338
0.031 1
0 . 0288
0 . 0268
0 . 0252
0 . 0239
0 . 0230
0 . 0224
0 .0220
0 . 02 1 9
0 . 0220
0. 0223
0 .0228
0. 0235
0 .0243
0 . 0252
0 . 0262
0 . 0273
0. 0296
0 . 0321
0 . 0 344
0 . 0364
0.0381
8
7.98 1
8.071
0 . 0730
0 . 0668
0 . 0609
0 . 0555
0 . 0506
0 . 0462
0 . 042 1
0 . 0384
0 . 0352
0 . 0323
0 . 0298
0 . 0277
0 . 0259
0 . 0244
0 . 0232
0 .0224
0 . 02 1 8
0 . 02 1 4
0 . 02 1 3
0 . 02 1 4
0 . 02 1 6
0 . 0221
0 . 0227
0 . 0234
0 . 0243
0 . 0252
0 . 0273
0 . 0296
0 . 0320
0 . 0342
0 . 0361
0 . 0377
0 .0733
0 . 0662
0.061 3
0 . 0560
0 . 05 1 0
0. 0466
0 . 0425
0 . 0388
0 . 0355
0. 0327
0.0301
0 .0280
0 . 0261
0 .0246
0 . 0233
0. 0224
0 . 02 1 8
0 . 02 1 3
0.021 1
0 . 02 1 2
0 . 02 1 4
0.021 8
0 . 0224
0 . 0230
0 . 0238
0 . 0246
0 . 0268
0 . 0290
0.031 4
0 .0336
0 .0356
0 .0372
49
GAS FLOW M EAS U R E M ENT
TABLE 3 . 5-FLANGE T A P S : EXPANSION FACTORS, Y, STATIC PRESSURE
TAKEN FROM U PSTREAM TAPS (AFTER REF. 3)
(3 do ld
=
h w lp f
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
h w lpf
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
0.1
--
1 .0000
0 . 9989
0 . 9977
0 . 9966
0 . 9954
0. 9943
0 .9932
0 . 9920
0 . 9909
0. 9898
0 .9886
0 . 9875
0. 9863
0 . 9852
0 . 9841
0. 9829
0 . 98 1 8
0 . 9806
0 .9795
0 .9784
0 . 9772
0 . 9761
0 .9750
0 . 9738
0 . 9727
0.971 5
0. 9704
0 . 9693
0 . 9681
0 .9670
0 .9658
0 . 9647
0 . 9635
0 .9624
0 . 96 1 3
0 . 9602
0 .9590
0 . 9579
0. 9567
0 . 9556
0. 9545
0 . 63
--
1 .0000
0. 9987
0 .9974
0 . 9961
0 .9948
0 . 9935
0 . 9923
0 . 991 0
0. 9897
0 . 9884
0 . 9871
0 . 9858
0 . 9845
0 . 9832
0.98 1 9
0 . 9806
0 . 9793
0 . 9780
0 . 9768
0 . 9755
0 .9742
0 .9729
0.971 6
0 . 9703
0 .9690
0 . 9677
0 .9664
0 . 9651
0 . 9638
0. 9625
0 . 96 1 3
0 . 9600
0.2
--
1 . 0000
0 . 9989
0 . 9977
0 .9966
0 . 9954
0 .9943
0 . 9932
0 . 9920
0 .9909
0. 9897
0 .9886
0 . 9875
0 . 9863
0 .9852
0 .9840
0 . 9829
0.981 8
0. 9806
0 .9795
0 .9783
0. 9772
0 . 9761
0 .9749
0 .9738
0 . 9726
0.971 5
0 .9704
0 . 9692
0 . 9681
0 .9669
0 .9658
0 .9647
0 . 9635
0. 9624
0 . 96 1 2
0 . 9601
0 . 9590
0 . 9578
0 . 9567
0 . 9555
0 . 9544
0 . 64
1 . 0000
0. 9987
0 .9974
0 . 9961
0 .9948
--
0 . 9935
0 .9922
0 .9909
0. 9896
0 . 9883
0 . 9870
0 . 9857
0 . 9844
0 .9831
0 . 98 1 8
0 . 9805
0 .9792
0 . 9779
0 .9766
0 . 9753
0 .9740
0 . 9727
0 . 97 1 4
0 . 970 1
0 .9688
0 . 9675
0 .9662
0 .9649
0 . 9636
0 . 9623
0.961 0
0 .9597
0.3
--
1 .0000
0 . 9989
0. 9977
0 . 9966
0 . 9954
0 . 9943
0 . 9931
0 . 9920
0 . 9908
0 . 9897
0 . 9885
0. 9874
0 . 9862
0 . 9851
0 . 9840
0 . 9828
0 . 981 7
0 . 9805
0. 9794
0 . 9782
0 . 9771
0 . 9759
0 . 9748
0 . 9736
0 . 9725
0 . 971 3
0 . 9702
0 . 9691
0 . 9679
0 .9668
0 . 9656
0 . 9645
0 . 9633
0. 9622
0 . 96 1 0
0 . 9599
0 .9587
0 . 9576
0 .9564
0 . 9553
0. 9542
0 . 65
1 . 0000
0. 9987
0 . 9974
0 . 9961
0 . 9948
--
0 . 9934
0 . 9921
0 . 9908
0. 9895
0 . 9882
0 . 9869
0 . 9856
0 .9843
0 . 9829
0 . 98 1 6
0. 9803
0 . 9790
0 . 9777
0 . 9764
0 . 9751
0 . 9738
0 . 9725
0 . 97 1 1
0 . 9698
0 . 9685
0 . 9672
0 . 9659
0 . 9646
0 . 9633
0 . 9620
0 . 9606
0 . 9593
0.4
--
1 .0000
0 . 9988
0. 9977
0 . 9965
0 . 9953
0. 9942
0 . 9930
0.991 9
0 .9907
0. 9895
0 .9884
0 . 9872
0 . 9860
0 . 9849
0. 9837
0 . 9826
0.981 4
0 . 9802
0 . 979 1
0 .9779
0 . 9767
0 . 9756
0 . 9744
0 .9732
0 . 972 1
0 . 9709
0 . 9698
0 .9686
0 . 9674
0 . 9663
0 . 9651
0 . 9639
0 . 9628
0 . 96 1 6
0. 9604
0 .9593
0 . 9581
0 . 9570
0 .9558
0 . 9546
0. 9535
0 . 66
1 .0000
0 . 9987
0 .9974
0 . 9960
0 . 9947
--
0 . 9934
0 . 9921
0 . 9907
0. 9894
0 . 9881
0 . 9868
0 .9854
0 .9841
0. 9828
0.981 5
0. 9802
0 .9788
0 . 9775
0 . 9762
0 . 9749
0 . 9735
0. 9722
0. 9709
0 . 9696
0 . 9683
0 . 9669
0. 9656
0 .9643
0 . 9630
0.961 6
0. 9603
0 . 9590
0 . 45
--
1 .0000
0 . 9988
0. 9976
0 . 9965
0 . 9953
0 . 9941
0 . 9929
0 . 99 1 8
0 . 9906
0. 9894
0 .9882
0 . 9870
0 . 9859
0. 9847
0 . 9835
0 . 9823
0.98 1 1
0 .9800
0 . 9788
0 .9776
0 .9764
0 . 9753
0 . 974 1
0 .9729
0.971 7
0 .9705
0 .9694
0 . 9682
0 . 9670
0 . 9658
0 . 9647
0 .9635
0 . 9623
0 . 96 1 1
0 . 9599
0 .9588
0 . 9576
0 .9564
0 . 9552
0 .9540
0 . 9529
0 . 67
1 .0000
0 . 9987
0 . 9973
0 . 9960
0. 9947
--
0 . 9933
0 . 9920
0 . 9907
0. 9893
0 .9880
0 .9867
0 .9853
0 .9840
0 . 9827
0 . 98 1 3
0. 9800
0 .9787
0 . 9773
0 . 9760
0 . 9747
0 . 9733
0 .9720
0. 9706
0 .9693
0 . 9680
0 .9666
0. 9653
0 .9640
0 . 9626
0 . 961 3
0 . 9600
0 . 9586
0 . 50
--
1 .0000
0. 9988
0 . 9976
0. 9964
0 . 9952
0 . 9940
0. 9928
0 . 99 1 6
0 . 9904
0. 9892
0 . 9880
0 . 9868
0 . 9856
0 . 9844
0 . 9832
0 .9820
0. 9808
0 . 9796
0 .9784
0. 9772
0 . 9760
0 . 9748
0 .9736
0 . 9724
0 . 971 2
0 . 9700
0 . 9688
0 . 9676
0 .9664
0 . 9652
0 .9640
0 . 9628
0 . 96 1 6
0 . 9604
0 . 9592
0 . 9580
0. 9568
0 .9556
0 . 9544
0 . 9532
0 . 9520
0 . 68
1 . 0000
0 . 9987
0 . 9973
0 .9960
0 .9946
--
0 . 9933
0 . 99 1 9
0 . 9906
0. 9892
0 . 9879
0 . 9865
0 . 9852
0 . 9838
0 . 9825
0 . 98 1 2
0 . 9798
0 . 9785
0 . 9771
0 . 9758
0 . 9744
0 . 9731
0 . 971 7
0. 9704
0 .9690
0 .9677
0 . 9663
0 . 9650
0 . 9637
0 .9623
0 . 96 1 0
0 . 9596
0 . 9583
0 . 52
--
1 . 0000
0. 9988
0. 9976
0. 9964
0 .9952
0. 9940
0 . 9927
0.991 5
0 .9903
0 . 9891
0. 9879
0. 9867
0 .9855
0. 9843
0 . 9831
0 . 98 1 9
0 . 9806
0. 9794
0 .9782
0 . 9770
0 . 9758
0 .9746
0 .9734
0. 9722
0 . 971 0
0 .9698
0. 9686
0 . 9673
0 . 9661
0. 9649
0 .9637
0. 9625
0.961 3
0 . 9601
0 . 9589
0 . 9577
0 . 9565
0 . 9553
0. 9540
0 . 9528
0 . 95 1 6
0 . 69
1 . 0000
0. 9986
0 .9973
0 . 9959
0 .9946
0 . 54
--
1 .0000
0 . 9988
0. 9976
0 .9963
0 . 9951
0 .9939
0 .9927
0 . 99 1 5
0 . 9902
0 . 9890
0 . 9878
0 . 9866
0 . 9853
0 .9841
0 . 9829
0 . 98 1 7
0 .9805
0 . 9792
0 . 9780
0 .9768
0 . 9756
0 .9744
0 . 9731
0.971 9
0 . 9707
0 . 9695
0 .9683
0 . 9670
0.9658
0 .9646
0 . 9634
0 .9622
0 .9609
0 . 9597
0 . 9585
0 . 9573
0 .9560
0 .9548
0 . 9536
0 .9524
0 . 95 1 2
0 . 70
1 . 0000
0 . 9986
0 . 9973
0 . 9959
0 .9945
0 . 56
--
1 .0000
0 . 9988
0. 9975
0 .9963
0 . 9951
0 . 9938
0 . 9926
0 . 99 1 4
0 . 9901
0 . 9889
0 . 9877
0 . 9864
0 . 9852
0. 9840
0. 9827
0 . 98 1 5
0. 9803
0 . 9790
0 . 9778
0 . 9766
0 . 9753
0 . 9741
0 .9729
0 . 971 6
0 . 9704
0 .9692
0 . 9679
0 . 9667
0. 9654
0 .9642
0 . 9630
0 . 96 1 7
0 . 9605
0 . 9593
0 . 9580
0 . 9568
0 . 9556
0 .9543
0 . 9531
0 . 95 1 9
0 . 9506
0.71
1 . 0000
0. 9986
0 . 9972
0 . 9958
0. 9945
0.58
--
1 .0000
0. 9988
0. 9975
0.9963
0 .9950
0 .9938
0. 9925
0.99 1 3
0 .9900
0 . 9888
0. 9875
0 . 9863
0 .9850
0. 9838
0. 9825
0 . 98 1 3
0 . 9800
0 . 9788
0 .9775
0 . 9763
0 . 9750
0. 9738
0. 9725
0 . 971 3
0 . 9700
0 .9688
0. 9675
0 .9663
0 . 9650
0. 9638
0. 9626
0 . 96 1 3
0 . 960 1
0 . 9588
0. 9576
0. 9563
0 . 9551
0 .9538
0 . 9526
0 . 95 1 3
0 . 9501
0 . 72
0 . 60
--
1 .0000
0 . 9987
0. 9975
0 .9962
0 .9949
0 . 9937
0 . 9924
0 . 99 1 2
0 .9899
0 .9886
0. 9874
0 . 9861
0 .9848
0. 9836
0. 9823
0.981 0
0 .9798
0 .9785
0 . 9772
0 . 9760
0 .9747
0 .9734
0 . 9722
0 . 9709
0 . 9697
0 .9684
0 . 9671
0 . 9659
0 .9646
0 .9633
0 . 9621
0 .9608
0 . 9595
0 .9583
0 . 9570
0 . 9558
0 .9545
0. 9532
0 .9520
0.9507
0 . 9494
0 . 73
1 . 0000
0. 9986
0 . 9972
0 . 9958
0 .9943
0.61
--
1 . 0000
0 . 9987
0 . 9975
0 .9962
0 .9949
0 . 9936
0 . 9924
0 . 99 1 1
0 . 9898
0 . 9885
0 . 9873
0 . 9860
0 . 9847
0 . 9835
0 .9822
0 .9809
0 . 9796
0 . 9784
0 . 9771
0 . 9758
0 . 9745
0 .9733
0 .9720
0 .9707
0 . 9694
0 . 9682
0 . 9669
0 . 9656
0 .9644
0 . 9631
0 . 961 8
0 . 9605
0 .9593
0 . 9580
0 . 9567
0. 9554
0 .9542
0 . 9529
0 . 95 1 6
0 . 9504
0 . 9491
0 . 74
1 .0000
0. 9986
0 . 9971
0 . 9957
0 . 9943
0 . 62
--
1 .0000
0 . 9987
0. 9974
0 . 9962
0. 9949
0 .9936
0 .9923
0.991 0
0 . 9897
0.9885
0. 9872
0 .9859
0 . 9846
0 . 9833
0 . 9821
0 . 9808
0 . 9795
0 .9782
0 . 9769
0. 9756
0 .9744
0 . 9731
0 . 97 1 8
0. 9705
0. 9692
0 . 9680
0 . 9667
0. 9655
0 . 9641
0 .9628
0 . 961 5
0 . 9603
0 .9590
0 . 9577
0 . 9564
0 . 9551
0 . 9538
0 . 9526
0 . 95 1 3
0 . 9500
0 . 9487
0 . 75
1 . 0000
0 . 9986
0 . 9971
0. 9957
0. 9942
--
--
--
--
--
--
--
0 . 9932
0 . 9931
0 . 9931
0 . 9930
0 . 9929
0 . 9929
0 . 9928
0 . 99 1 8
0. 9905
0 . 9891
0 . 9878
0 . 9864
0 . 9851
0 .9837
0 .9823
0 . 98 1 0
0. 9796
0 . 9783
0. 9769
0 . 9755
0 . 9742
0 . 9728
0.971 5
0. 9701
0 . 9688
0 . 9674
0 . 9660
0. 9647
0 .9633
0 . 9620
0. 9606
0. 9592
0 .9579
0 . 99 1 8
0 . 9904
0. 9890
0 . 9877
0 .9863
0 . 9849
0 . 9835
0 . 9822
0 .9808
0. 9794
0 . 9781
0 . 9767
0 . 9753
0 .9739
0 . 9726
0.971 2
0. 9698
0 . 9685
0 . 9671
0 . 9657
0 .9643
0 .9630
0 . 96 1 6
0 . 9602
0 . 9588
0 .9575
0 . 99 1 7
0 . 9903
0. 9889
0 . 9875
0 . 9861
0 . 9848
0 .9834
0 . 9820
0 . 9806
0. 9792
0 . 9778
0 . 9764
0 . 9751
0 . 9737
0 . 9723
0 . 9709
0. 9695
0 . 9681
0 . 9668
0 . 9654
0 . 9640
0 . 9626
0 . 96 1 2
0. 9598
0 . 9584
0 . 9571
1 .0000
0. 9986
0. 9972
0 . 9958
0 .9944
0 . 99 1 6
0 . 9902
0. 9888
0 .9874
0. 9860
0. 9846
0. 9832
0 . 98 1 8
0. 9804
0. 9790
0. 9776
0. 9762
0. 9748
0. 9734
0. 9720
0. 9706
0. 9692
0. 9678
0 . 9664
0 . 9650
0. 9636
0 .9622
0. 9608
0. 9594
0 . 9580
0 .9566
0.99 1 5
0.9901
0. 9887
0 .9873
0 . 9859
0. 9844
0. 9830
0 . 98 1 6
0 . 9802
0. 9788
0 .9774
0 .9760
0 . 9745
0 . 9731
0.971 7
0. 9703
0. 9689
0 . 9675
0 . 9661
0 .9646
0 .9632
0.961 8
0 . 9604
0. 9590
0 . 9576
0 . 9562
0 . 99 1 4
0 . 9900
0 . 9886
0 . 9871
0 . 9857
0 . 9843
0 . 9828
0 . 98 1 4
0 . 9800
0. 9786
0 . 9771
0 . 9757
0. 9743
0. 9728
0 . 971 4
0 . 9700
0. 9685
0 . 9671
0 . 9657
0 . 9643
0 . 9628
0 . 961 4
0. 9600
0. 9585
0 . 9571
0 . 9557
0 . 99 1 3
0 . 9899
0. 9884
0 . 9870
0 . 9855
0 . 9841
0 .9826
0.981 2
0 . 9798
0. 9783
0 .9769
0 . 9754
0 . 9740
0 . 9725
0.971 1
0 . 9696
0. 9682
0 . 9667
0 .9653
0 .9639
0 . 9624
0 . 961 0
0 .9595
0 . 9581
0. 9566
0 . 9552
50
GAS RESERVO I R E N G I N E E R I N G
TABLE 3 . 5-FLANGE TAPS: EXPANSION FACTORS, Y, STATIC PRESSURE
TAKEN FROM U PSTREAM TAPS (AFTER REF. 3) (Continued)
h w1p t
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
(3 = do ld
0 . 63
0 .9587
0 . 9574
0 . 9561
0 . 9548
0 .9535
0 . 9522
0. 9509
0 . 9496
0 . 9483
--
0 . 64
0 . 9584
0 . 9571
0 . 9558
0 . 9545
0 . 9532
0.951 8
0 . 9505
0 . 9492
0 . 9479
--
0 . 65
0 . 9580
0 . 9567
0 . 9554
0 . 954 1
0 . 9528
0 . 95 1 5
0 . 9502
0 . 9488
0 . 9475
--
0 . 66
0 . 9577
0 .9564
0 . 9550
0 . 9537
0 .9524
0.951 1
0 . 9497
0 . 9484
0 . 9471
--
0 . 67
0 .9573
0 .9560
0 . 9546
0 . 9533
0 .9520
0 .9506
0 . 9493
0 . 9480
0 . 9466
0 . 68
0 .9569
0 . 9556
0 .9542
0 . 9529
0 . 95 1 5
0 . 9502
0 . 9488
0 . 9475
0 . 9462
--
--
0 . 69
0 .9565
0 .9552
0 . 9538
0 . 9524
0.951 1
0 . 9597
0 .9484
0 . 9470
0 . 9457
0 . 70
0 . 9561
0 .9547
0 .9534
0 .9520
0 .9506
0 . 9492
0 . 9479
0 . 9465
0 . 9451
--
--
0.71
0 . 9557
0 .9543
0 . 9529
0 . 95 1 5
0 .950 1
0 . 9487
0 . 9474
0 . 9460
0 . 9446
--
0 . 72
0 .9552
0 . 9538
0 .9524
0 . 95 1 0
0 . 9496
0 . 9482
0 . 9468
0 .9454
0 .9440
--
0 . 73
0 .9547
0 . 9533
0.951 9
0 . 9505
0 . 949 1
0 . 9477
0 .9463
0 . 9448
0 .9434
--
0 . 74
0 .9542
0 . 9528
0 . 95 1 4
0 . 9500
0 .9485
0 . 9471
0 . 9457
0 . 9442
0 . 9428
--
0 . 75
0 . 9537
0 .9523
0 .9508
0 . 9494
0 . 9480
0 .9465
0 . 945 1
0 .9436
0 . 9422
--
TABLE 3 . 6- P I P E TAPS: EXPANSION FACTORS, Y, STATIC PRESSURE
TAKEN FROM U PSTREAM TAPS (AFTER REF. 3)
h w1p t
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
h w1p t
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
{3 = dold
0.1
1 .0000
0 .9990
0 . 9981
0 .9971
0 . 9962
0 .9952
0 .9943
0 . 9933
0 . 9923
0 . 99 1 4
0 . 9904
0 .9895
0 . 9885
0 .9876
0 . 9866
0 .9857
0 .9847
0 . 9837
0 . 9828
0 . 98 1 8
0 . 9809
0 .9799
0 .9790
0 . 9780
0 . 9770
0 . 9761
0 . 975 1
0 .9742
0 .9732
0 . 9723
0.971 3
0 .9704
0 . 9694
0 . 9684
0 .9675
0 .9665
0 . 9656
0 .9646
0 .9637
0 . 9627
0.961 7
--
0.2
1 .0000
0 . 9989
0 .9979
0 . 9968
0 .9958
0 .9947
0 . 9937
0 .9926
0 . 99 1 6
0 .9905
0 . 9895
0 .9884
0 .9874
0 .9863
0 . 9853
0 . 9842
0 . 9832
0.9821
0.981 1
0 . 9800
0 . 9790
0 . 9779
0 . 9768
0 . 9758
0 .9747
0 . 9737
0 . 9726
0 . 97 1 6
0 . 9705
0 .9695
0 .9684
0 .9674
0 . 9663
0 . 9653
0. 9642
0 . 9632
0 . 9621
0.961 1
0 .9600
0 . 9590
0 .9579
--
0.3
1 .0000
0 . 9988
0 .9976
0 .9964
0 .9951
0 . 9939
0 . 9927
0.991 5
0 .9903
0 . 9891
0 . 9878
0 . 9866
0 . 9854
0 .9842
0 .9830
0.981 8
0 . 9805
0 .9793
0.9781
0 .9769
0 .9757
0 .9745
0 .9732
0 . 9720
0 . 9708
0 .9696
0.9684
0 .9672
0 .9659
0 .9647
0 . 9635
0 . 9623
0.961 1
0 . 9599
0 . 9587
0 .9574
0 . 9562
0 . 9550
0 .9538
0 . 9526
0 . 95 1 4
--
0.4
1 .0000
0 . 9985
0 . 9971
0 . 9956
0 . 9942
0 . 9927
0 . 99 1 3
0 . 9898
0 . 9883
0 .9869
0. 9854
0 .9840
0 . 9825
0.981 1
0 . 9796
0 . 9782
0 .9767
0 . 9752
0 . 9738
0 .9723
0 . 9709
0 . 9694
0 . 9680
0 .9665
0 .9650
0 . 9636
0 . 9621
0 . 9607
0 . 9592
0 . 9578
0. 9563
0 .9549
0 .9534
0.95 1 9
0 . 9505
0 . 9490
0 . 9476
0 . 9461
0 .9447
0 . 9432
0 . 941 7
--
0.45
1 .0000
0 .9984
0 . 9968
0 . 9952
0 . 9936
0 . 99 1 9
0 . 9903
0 . 9887
0 .9871
0 . 9855
0 . 9839
0 . 9823
0 . 9807
0 .979 1
0 .9775
0 .9758
0 .9742
0 . 9726
0.971 0
0 .9694
0 .9678
0 .9662
0 .9646
0 . 9630
0 . 96 1 3
0 . 9597
0.9581
0 .9565
0 .9549
0 . 9533
0 . 95 1 7
0 .950 1
0 . 9485
0 . 9469
0 . 9452
0 . 9436
0 . 9420
0 . 9404
0 . 9388
0 . 9372
0 .9356
--
0.50
1 .0000
0 . 9982
0 .9964
0 .9946
0 . 9928
0 . 99 1 0
0 . 9892
0 .9874
0 . 9857
0 .9839
0 . 9821
0 .9803
0 . 9785
0 .9767
0 .9749
0 . 9731
0 . 97 1 3
0 .9695
0 .9677
0 . 9659
0 . 9641
0 . 9623
0 .9605
0 . 9587
0 . 9570
0 . 9552
0 .9534
0 . 95 1 6
0 . 9498
0 . 9480
0 .9462
0 .9444
0 . 9426
0 . 9408
0 . 9390
0 . 9372
0 . 9354
0 .9336
0.931 8
0.9301
0 .9283
--
0.52
1 . 0000
0 . 9981
0 . 9962
0 .9944
0 .9925
0 .9906
0 .9887
0 .9869
0 .9850
0 . 9831
0.981 2
0 . 9794
0 .9775
0 .9756
0 .9737
0.971 9
0 . 9700
0 . 9681
0 .9662
0 .9643
0 .9625
0 .9606
0.9587
0 .9568
0 .9550
0 . 9531
0 . 95 1 2
0 . 9493
0 . 9475
0 . 9456
0 . 9437
0.941 8
0.9400
0.9381
0 . 9362
0 .9343
0.9324
0 . 9306
0 .9287
0 .9268
0 .9249
--
0 . 54
1 .0000
0 . 9980
0 . 9961
0 . 9941
0.9921
0 . 9902
0 .9882
0. 9862
0 .9843
0 .9823
0 .9803
0 .9784
0 .9764
0 .9744
0 .9725
0 .9705
0.9685
0 . 9666
0 .9646
0 .9626
0 .9607
0 . 9587
0 .9567
0 .9548
0 . 9528
0 . 9508
0 . 9489
0 . 9469
0 . 9449
0 . 9430
0 .941 0
0 . 9390
0 . 9371
0 . 9351
0 . 9331
0 . 93 1 2
0 . 9292
0 . 9272
0 . 9253
0 .9233
0 . 92 1 3
--
0 . 56
1 .0000
0 .9979
0 .9959
0 . 9938
0 . 99 1 7
0 .9897
0 . 9876
0 .9856
0 . 9835
0 . 98 1 4
0 .9794
0 . 9773
0 . 9752
0 . 9732
0.971 1
0 . 9690
0 .9670
0 .9649
0 . 9628
0 . 9608
0 . 9587
0 . 9566
0 .9546
0 . 9525
0 .9505
0 . 9484
0. 9463
0 .9443
0 . 9422
0 . 9401
0 . 9381
0 . 9360
0 . 9339
0.931 9
0 .9298
0 .9277
0 .9257
0 .9236
0 . 92 1 6
0 . 9 1 95
0 . 9 1 74
--
0.58
1 .0000
0 .9978
0 . 9957
0 . 9935
0 . 99 1 3
0 . 9891
0 . 9870
0 . 9848
0 . 9826
0 . 9805
0 .9783
0 . 9761
0 .9739
0.971 8
0 .9696
0 .9674
0 . 9652
0 . 963 1
0 .9609
0 . 9587
0 .9566
0 .9544
0 . 9522
0 . 9500
0 .9479
0 . 9457
0 . 9435
0 . 941 4
0 . 9392
0 . 9370
0 .9348
0 . 9327
0 . 9305
0 . 9283
0 . 9261
0 . 9240
0.921 8
0 . 9 1 96
0 . 9 1 75
0 . 9 1 53
0.91 31
--
0 . 60
1 .0000
0 . 9977
0 .9954
0 . 9931
0 . 9908
0 . 9885
0 .9862
0 .9840
0.981 7
0 .9794
0 . 9771
0 .9748
0 .9725
0 . 9702
0 .9679
0 . 9656
0 .9633
0 . 96 1 0
0 . 9587
0 .9565
0 .9542
0 . 95 1 9
0 .9496
0 . 9473
0 . 9450
0 .9427
0.9404
0 . 938 1
0 .9358
0 .9335
0 . 93 1 2
0 . 9290
0 . 9267
0 .9244
0 .922 1
0 . 9 1 98
0 . 9 1 75
0 . 9 1 52
0 . 9 1 29
0 . 9 1 06
0 . 9083
--
{3 = do ld
0.61
1 . 0000
0 .9976
0 . 9953
0 . 9929
0 .9906
0 . 9882
0 . 9859
0 . 9835
0 . 98 1 1
0 .9788
0 . 9764
0 .9741
0.971 7
--
0 . 62
1 . 0000
0 . 9976
0 . 9951
0 . 9927
0 . 9903
0 . 9879
0. 9854
0 . 9830
0 . 9806
0 . 9782
0 . 9757
0 . 9733
0 . 9709
--
0 . 63
--
1 . 0000
0 .9975
0 . 9950
0 . 9925
0. 9900
0 . 9875
0 . 9850
0 . 9825
0 .9800
0 . 9775
0 . 9750
0 . 9725
0 .9700
0.64
1 .0000
0 .9974
0 .9948
0 . 9923
0.9897
0 . 9871
0 .9845
0 . 98 1 9
0 . 9794
0 .9768
0 .9742
0.97 1 6
0 . 9690
--
0 . 65
1 . 0000
0 . 9973
0 .9947
0 . 9920
0 . 9893
0 . 9867
0 . 9840
0.981 3
0 . 9787
0 .9760
0. 9733
0 . 9707
0 .9680
--
0 . 66
1 .0000
0 .9972
0 .9945
0 . 99 1 7
0 . 9890
0 .9862
0 .9834
0 . 9807
0 . 9779
0 .9752
0 .9724
0 . 9696
0 .9669
--
0 . 67
1 .0000
0 .9971
0 . 9943
0 . 99 1 4
0 . 9886
0 . 9857
0 . 9828
0 . 9800
0 . 9771
0 . 9742
0 . 97 1 4
0 . 9685
0 .9657
--
0 . 68
1 . 0000
0 .9970
0 . 9941
0.991 1
0 . 9881
0 . 9851
0 . 9822
0. 9792
0. 9762
0 . 9733
0 . 9703
0. 9673
0 .9643
--
0 . 69
1 . 0000
0 .9969
0 . 9938
0 . 9907
0 .9876
0 .9845
0 . 98 1 4
0 .9784
0 .9753
0 .9722
0 .969 1
0 .9660
0 .9629
--
0 . 70
1 . 0000
0 .9968
0 .9935
0 . 9903
0 . 9871
0 . 9839
0 . 9806
0 .9774
0 .9742
0.971 0
0 . 9677
0 . 9645
0 . 961 3
--
51
GAS FLOW M EAS U R E M ENT
TABLE 3 .6-PIPE TAPS : EXPANSION FACTORS, Y, STATIC PRESSURE
TAKEN FROM U PSTREAM TAPS (AFTER REF. 3) (Continued)
(3 d o ld
=
h w1p t
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3. 1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
0. 6 1
--
0 .9694
0 . 9670
0 . 9646
0 .9623
0 .9599
0 .9576
0 .9552
0 . 9529
0 . 9505
0 . 9481
0 . 9458
0 .9434
0 . 941 1
0 .9387
0 . 9364
0 .9340
0 . 93 1 6
0 .9293
0 .9269
0 .9246
0 . 9222
0 . 9 1 99
0 . 9 1 75
0.91 51
0 . 9 1 28
0 .9 1 04
0 . 9081
0 . 9057
0 . 62
--
0 . 9685
0 . 9660
0 . 9636
0 . 961 2
0 . 9587
0 . 9563
0 . 9539
0 . 95 1 5
0 . 9490
0 . 9466
0 . 9442
0 . 94 1 8
0 . 9393
0 . 9369
0 . 9345
0 . 9321
0 . 9296
0 . 9272
0 . 9248
0 . 9223
0 . 9 1 99
0 . 9 1 75
0.91 51
0 . 9 1 26
0 . 9 1 02
0 . 9078
0 .9054
0 . 9029
0 . 63
--
0 .9675
0 .9650
0 .9625
0 .9600
0 .9575
0 . 9550
0 .9525
0 .9500
0 .9475
0 .9450
0 . 9425
0 .9400
0 .9375
0 . 9350
0 . 9325
0 .9300
0 .9275
0 . 9250
0 . 9225
0 . 9200
0 . 9 1 75
0 . 9 1 50
0 . 9 1 25
0 . 9 1 00
0 .9075
0 .9050
0 . 9025
0 . 9000
0 . 64
--
0 . 9664
0 . 9639
0 . 96 1 3
0 . 9587
0 . 9561
0 . 9535
0.951 0
0 . 9484
0 . 9458
0 . 9432
0 . 9406
0 . 9381
0 . 9355
0 . 9329
0 .9303
0 . 9277
0 . 9252
0 . 9226
0 . 9200
0 . 9 1 74
0 . 9 1 48
0 . 9 1 22
0 . 9097
0 .9071
0 . 9045
0 . 90 1 9
0 . 8993
0 . 8968
0 . 65
--
0 .9653
0 . 9627
0 . 9600
0 .9573
0 .9547
0 . 9520
0 . 9493
0 .9467
0 . 9440
0 . 941 3
0 . 9387
0 .9360
0 . 9333
0 .9307
0 . 9280
0 .9253
0 . 9227
0. 9200
0 . 9 1 73
0 . 9 1 47
0 . 9 1 20
0 .9093
0 .9067
0 . 9040
0.901 3
0. 8987
0 .8960
0 . 8933
0 . 66
--
0 . 9641
0 . 96 1 4
0 .9586
0 . 9558
0 . 9531
0 . 9503
0 .9476
0 .9448
0 . 9420
0 . 9393
0 .9365
0 .9338
0 . 93 1 0
0 . 9282
0 . 9255
0. 9227
0 . 9200
0 . 9 1 72
0 . 9 1 44
0.91 1 7
0 .9089
0 .9062
0 .9034
0 . 9006
0 . 8979
0 . 8951
0 . 8924
0 .8896
0 . 67
--
0 . 9628
0 . 9599
0 . 9571
0 . 9542
0 . 95 1 4
0 . 9485
0 . 9456
0 . 9428
0 . 9399
0 . 9371
0 . 9342
0 . 93 1 3
0 . 9285
0 . 9256
0 . 9227
0 . 9 1 99
0 . 9 1 70
0 . 9 1 42
0.91 1 3
0 . 9084
0 . 9056
0 . 9027
0 . 8999
0 . 8970
0 . 8941
0 . 89 1 3
0 .8884
0 . 8856
0.69
0 . 68
--
--
0 .9598
0 . 9567
0 .9536
0 . 9505
0 .9474
0 .9443
0 . 94 1 2
0 . 9381
0 .935 1
0 . 9320
0 .9289
0 . 9258
0 . 9227
0 . 9 1 96
0 . 9 1 65
0 . 9 1 34
0 . 9 1 03
0 .9072
0 . 9041
0.901 0
0 .8979
0 .8948
0 . 89 1 8
0 . 8887
0 .8856
0 . 8825
0 . 8794
0 . 8763
0.961 4
0 .9584
0.9554
0. 9525
0 . 9495
0 .9465
0 .9435
0 .9406
0 .9376
0 .9346
0.931 7
0. 9287
0 . 9257
0. 9227
0 . 9 1 98
0 .9 1 68
0 . 9 1 38
0 . 9 1 08
0 . 9079
0 .9049
0 . 90 1 9
0 .8990
0 . 8960
0 . 8930
0. 8900
0. 8871
0 . 8841
0.881 1
0 . 70
--
0 . 9581
0 .9548
0.951 6
0 .9484
0 . 9452
0 .941 9
0 . 9387
0 .9355
0 .9323
0 . 9290
0 . 9258
0 .9226
0 . 9 1 94
0 . 9 1 61
0 . 9 1 29
0 . 9097
0 .9064
0 . 9032
0 .9000
0 .8968
0 .8935
0 . 8903
0 . 8871
0 . 8839
0 . 8806
0. 8774
0 . 8742
0.871 0
TABLE 3 . 7-FLANGE TAPS: EXPANSION FACTORS , Y, STATIC PRESSU R E
TAKEN F R O M DOWNSTREAM TAPS (AFTER REF. 3 )
h w1p t
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
0.1
1 . 0000
1 . 0007
1 .00 1 3
1 . 0020
1 . 0027
1 .0033
1 .0040
1 .0047
1 .0053
1 .0060
1 . 0067
1 . 0074
1 .0080
1 .0087
1 .0094
1 .0 1 00
1 . 0 1 08
1 .0 1 1 4
1 .0 1 2 1
1 .0 1 28
1 .0 1 34
1 .0 1 40
1 . 0 1 47
1 .0 1 54
1 .0 1 60
1 .0 1 67
1 .0 1 74
1 .0 1 83
1 .0 1 87
1 . 0 1 94
1 . 0200
1 .0207
1 . 021 3
1 .0220
1 .0227
1 .0233
1 .0240
1 . 0246
1 .0252
--
0.2
1 .0000
1 .0007
1 .00 1 3
1 .0020
1 .0027
1 .0033
1 .0040
1 .0047
1 .0053
1 .0060
1 .0066
1 . 0073
1 .0080
1 .0087
1 .0093
1 .0 1 00
1 .0 1 07
1 .0 1 1 4
1 .0 1 20
1 .0 1 27
1 .0 1 33
1 .0 1 40
1 .0 1 47
1 .0 1 54
1 .0 1 60
1 .0 1 67
1 .0 1 73
1 .0 1 82
1 .0 1 86
1 .0 1 94
1 .0200
1 .0206
1 .021 3
1 .0220
1 .0227
1 .0233
1 . 0239
1 .0246
1 .0252
--
0.3
1 . 0000
1 .0007
1 .00 1 3
1 .0020
1 .0027
1 .0033
1 .0040
1 . 0047
1 . 0053
1 . 0060
1 . 0066
1 . 0073
1 .0079
1 .0086
1 . 0093
1 . 0099
1 . 0 1 06
1 .01 1 3
1 . 0 1 20
1 .0 1 26
1 .0 1 32
1 .0 1 39
1 .0 1 46
1 . 01 53
1 .0 1 59
1 .0 1 66
1 .0 1 72
1 .0 1 8 1
1 .0 1 85
1 . 0 1 92
1 .0 1 98
1 . 0205
1 . 021 1
1 . 02 1 8
1 . 0225
1 .0231
1 .0237
1 . 0244
1 .0250
--
0.4
1 .0000
1 .0006
1 .001 3
1 . 0020
1 . 0026
1 .0032
1 .0039
1 .0045
1 .005 1
1 .0058
1 .0064
1 . 0071
1 .0077
1 .0084
1 . 0090
1 . 0097
1 .0 1 04
1 .01 1 0
1 .0 1 1 7
1 .0 1 23
1 .0 1 29
1 .0 1 36
1 . 0 1 42
1 .0 1 49
1 . 0 1 54
1 .0 1 62
1 . 0 1 68
1 . 0 1 76
1 .0 1 80
1 .0 1 87
1 . 0 1 93
1 . 0200
1 . 0206
1 . 021 3
1 .021 9
1 .0225
1 .0232
1 .0238
1 . 0244
--
0 . 45
1 .0000
1 .0006
1 .001 3
1 . 001 9
1 .0026
1 .0031
1 .0038
1 .0044
1 .0050
1 .0057
1 .0063
1 .0069
1 .0075
1 . 0082
1 .0088
1 .0094
1 .0 1 0 1
1 .0 1 08
1 .0 1 1 4
1 .0 1 20
1 .0 1 26
1 .0 1 33
1 .0 1 39
1 .0 1 46
1 .0 1 51
1 .0 1 58
1 .0 1 64
1 . 0 1 72
1 .0 1 76
1 .0 1 83
1 .0 1 89
1 . 0 1 95
1 . 0201
1 .0208
1 .021 4
1 .0220
1 .0227
1 .0232
1 . 0238
--
0 . 50
1 .0000
1 . 0006
1 .001 2
1 .001 8
1 . 0025
1 .0030
1 .0036
1 .0043
1 .0049
1 .0055
1 .0061
1 . 0067
1 .0073
1 .0080
1 . 0086
1 .009 1
1 .0097
1 .0 1 04
1 .0 1 1 0
1 .0 1 1 6
1 .0 1 22
1 .01 29
1 .0 1 35
1 .0141
1 . 0 1 46
1 .0 1 53
1 .0 1 59
1 . 0 1 67
1 .0 1 71
1 .0 1 77
1 . 0 1 83
1 . 0 1 89
1 .0 1 95
1 . 0202
1 .0208
1 . 021 4
1 .0220
1 .0225
1 .0231
--
(3
=
d old
0 . 52
1 .0000
1 . 0006
1 . 001 2
1 . 001 8
1 .0024
1 . 0029
1 .0036
1 . 0042
1 .0048
1 .0054
1 .0060
1 .0066
1 .0072
1 . 0078
1 .0084
1 .0090
1 .0097
1 .0 1 03
1 .0 1 08
1 . 01 1 5
1 .0 1 2 1
1 . 0 1 27
1 . 0 1 33
1 . 0 1 39
1 .0 1 44
1 .0 1 50
1 .0 1 56
1 .0 1 64
1 .0 1 68
1 .0 1 75
1 .0 1 80
1 .0 1 86
1 .0 1 92
1 .0 1 99
1 . 0205
1 .021 0
1 .021 6
1 .0221
1 .0227
--
0 . 54
1 .0000
1 .0006
1 .001 2
1 .001 8
1 .0024
1 .0029
1 .0035
1 .0041
1 .0047
1 .0053
1 .0059
1 .0065
1 .0071
1 .0077
1 .0083
1 .0088
1 .0095
1 .0 1 0 1
1 .0 1 06
1 .0 1 1 2
1 .0 1 1 8
1 .0 1 24
1 .0 1 30
1 . 0 1 36
1 .0141
1 .0 1 48
1 .0 1 54
1 .0 1 6 1
1 .0 1 65
1 .0 1 72
1 .0 1 77
1 .0 1 83
1 .0 1 89
1 .0 1 95
1 .020 1
1 .0207
1 .021 2
1 . 021 7
1 .0223
--
0 . 56
1 .0000
1 .0006
1 .00 1 2
1 .001 7
1 .0023
1 . 0028
1 .0034
1 .0040
1 . 0046
1 . 0052
1 .0058
1 .0063
1 .0069
1 .0075
1 .0081
1 . 0086
1 . 0093
1 .0099
1 .0 1 04
1 .0 1 1 0
1 .0 1 1 6
1 .0 1 22
1 . 0 1 28
1 . 0 1 33
1 .01 38
1 . 0 1 45
1 .01 51
1 . 0 1 58
1 .0 1 62
1 .0 1 68
1 .0 1 73
1 .0 1 79
1 . 01 85
1 .0191
1 . 0 1 97
1 .0202
1 . 0208
1 . 021 3
1 .021 9
--
0 . 58
1 . 0000
1 . 0006
1 . 001 1
1 .00 1 7
1 .0023
1 .0028
1 .0033
1 .0039
1 .0045
1 .0050
1 .0056
1 . 0061
1 . 0067
1 . 0073
1 . 0079
1 . 0084
1 . 0090
1 .0096
1 . 0 1 02
1 . 0 1 07
1 .0 1 1 3
1 .0 1 1 9
1 . 01 25
1 . 01 30
1 . 01 35
1 .0 1 4 1
1 . 0 1 47
1 .0 1 54
1 .01 58
1 .01 64
1 .0 1 69
1 .0 1 75
1 . 01 80
1 . 0 1 86
1 .0 1 92
1 .0 1 98
1 .0203
1 .0208
1 .02 1 4
--
0 . 60
1 .0000
1 .0005
1 .001 1
1 .00 1 6
1 .0022
1 .0027
1 .0032
1 . 0038
1 .0044
1 .0049
1 .0055
1 .0060
1 . 0066
1 .0071
1 .0077
1 .0082
1 .0088
1 .0094
1 .0 1 00
1 .0 1 04
1 .0 1 1 0
1 .01 1 5
1 .0 1 2 1
1 . 0 1 27
1 .0 1 33
1 .0 1 37
1 .0 1 43
1 .0 1 50
1 .0 1 54
1 .0 1 60
1 .0 1 65
1 .0 1 71
1 .0 1 76
1 .0 1 81
1 .0 1 87
1 .0 1 92
1 .0 1 98
1 .0203
1 .0209
--
0.61
1 . 0000
1 .0005
1 . 001 1
1 .001 6
1 . 0022
1 . 0027
1 .0032
1 .0038
1 . 0043
1 .0048
1 . 0054
1 . 0059
1 .0065
1 .0070
1 .0076
1 .0081
1 .0086
1 .0092
1 . 0097
1 . 0 1 03
1 .0 1 08
1 .0 1 1 4
1 .0 1 20
1 .0 1 25
1 . 01 30
1 . 0 1 35
1 .0141
1 .0 1 48
1 . 0 1 52
1 . 0 1 57
1 .01 62
1 .0 1 68
1 . 01 73
1 .0 1 78
1 .01 84
1 .0 1 90
1 .01 95
1 . 0200
1 .0206
--
0 . 62
1 . 0000
1 .0005
1 .001 1
1 . 00 1 6
1 . 0021
1 .0027
1 .0031
1 .0037
1 . 0043
1 .0048
1 .0053
1 . 0058
1 .0064
1 .0069
1 .0074
1 . 0080
1 . 0085
1 . 0091
1 . 0096
1 .0 1 0 1
1 .0 1 06
1 .01 1 1
1 .0 1 1 8
1 .0 1 23
1 .0 1 28
1 .0 1 33
1 . 0 1 39
1 .0 1 46
1 .0 1 49
1 . 0 1 55
1 .0 1 60
1 . 01 66
1 . 0 1 71
1 . 0 1 76
1 .0 1 82
1 .0 1 87
1 .0 1 92
1 .0 1 97
1 .0203
--
GAS RESERVOI R E N G I N E E R I N G
52
TABLE 3 . 7-FLANGE TAPS: EXPANSION FACTORS, Y, STATIC PRESSURE
TAKEN FROM DOWNSTREAM TAPS (AFTER REF. 3) (Conti nued)
{3 = do ld
h w lP r
--
h w lP r
--
3.9
4.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
0.1
1 .0259
1 . 0265
0 . 63
1 . 0000
1 .0005
1 . 001 0
1 . 001 5
1 .002 1
1 . 0026
1 . 003 1
1 . 0036
1 . 0042
1 . 0047
1 . 0052
1 . 0057
1 . 0062
1 .0067
1 . 0073
1 .0078
1 .0083
1 . 0089
1 .0094
1 .0099
1 .0 1 04
1 .01 1 0
1 .01 1 6
1 .0 1 2 1
1 . 0 1 26
1 .01 31
1 . 0 1 36
1 . 0 1 43
1 . 0 1 47
1 . 0 1 52
1 .0 1 57
1 . 0 1 63
1 . 0 1 68
1 . 0 1 73
1 . 0 1 79
1 . 0 1 83
1 . 0 1 88
1 .0 1 93
1 . 0 1 99
1 . 0205
1 . 021 0
0.2
1 .0259
1 .0265
--
0 . 64
1 .0000
1 .0005
1 .00 1 0
1 . 00 1 5
1 .0021
1 .0026
1 .0030
1 . 0036
1 .0041
1 .0046
1 .0051
1 .0056
1 .0061
1 .0066
1 .0072
1 . 0077
1 .0081
1 .0087
1 .0092
1 .0097
1 . 0 1 02
1 .0 1 08
1 .01 1 4
1 .01 1 9
1 . 0 1 24
1 . 0 1 28
1 .0 1 34
1 .0 1 41
1 .0 1 44
1 .0 1 50
1 . 0 1 55
1 . 0 1 60
1 . 0 1 65
1 .0 1 70
1 .0 1 75
1 .0 1 80
1 .0 1 85
1 .0 1 90
1 .0 1 96
1 .020 1
1 .0206
--
0.3
1 . 0257
1 .0263
--
0 .65
1 . 0000
1 . 0005
1 .00 1 0
1 . 00 1 5
1 . 0020
1 . 0025
1 .0030
1 . 0035
1 . 0040
1 .0045
1 . 0050
1 . 0055
1 . 0060
1 . 0065
1 . 0070
1 .0075
1 . 0080
1 . 0086
1 .009 1
1 .0095
1 .0 1 00
1 . 0 1 05
1 .01 1 1
1 .01 1 6
1 .0 1 2 1
1 . 0 1 26
1 .01 31
1 .0 1 38
1 . 0 1 41
1 . 0 1 47
1 . 0 1 52
1 . 0 1 57
1 . 0 1 62
1 . 0 1 66
1 . 0 1 72
1 . 0 1 77
1 . 0 1 82
1 .0 1 87
1 . 0 1 92
1 . 0 1 97
1 . 0202
--
0.4
--
1 . 0250
1 . 0256
0 . 66
1 . 0000
1 . 0005
1 . 001 0
1 . 001 5
1 . 0020
1 . 0025
1 .0029
1 .0034
1 . 0039
1 . 0043
1 . 0049
1 . 0054
1 . 0059
1 .0064
1 . 0069
1 . 0074
1 . 0078
1 . 0084
1 . 0089
1 . 0093
1 . 0098
1 . 0 1 03
1 . 0 1 09
1 .01 1 4
1 .01 1 9
1 .0 1 23
1 .0 1 28
1 . 0 1 35
1 . 0 1 39
1 . 0 1 44
1 . 0 1 49
1 .0 1 53
1 . 0 1 58
1 . 0 1 63
1 . 0 1 69
1 .0 1 73
1 . 0 1 78
1 . 0 1 83
1 .0 1 89
1 . 0 1 93
1 . 0 1 98
--
0.45
--
1 .0245
1 .0251
0 . 67
1 .0000
1 .0005
1 .00 1 0
1 .00 1 4
1 .001 9
1 . 0024
1 .0028
1 .0033
1 .0038
1 .0042
1 . 0048
1 .0053
1 .0057
1 .0062
1 .0067
1 .0072
1 .0076
1 .0082
1 .0087
1 .0091
1 .0096
1 .0 1 0 1
1 . 0 1 07
1 . 01 1 1
1 .0 1 1 6
1 .0 1 20
1 .0 1 25
1 . 0 1 32
1 .0 1 36
1 .0 1 41
1 .0 1 46
1 .0 1 50
1 . 0 1 55
1 . 0 1 59
1 .0 1 65
1 .0 1 69
1 .0 1 74
1 .0 1 79
1 .0 1 85
1 . 0 1 89
1 . 0 1 94
--
0.50
--
1 .0237
1 . 0243
0 . 68
1 . 0000
1 . 0005
1 . 0009
1 . 00 1 4
1 . 00 1 9
1 . 0024
1 . 0027
1 . 0032
1 . 0037
1 . 004 1
1 . 0046
1 . 0051
1 . 0056
1 . 0061
1 . 0065
1 . 0070
1 . 0075
1 . 0080
1 . 0084
1 . 0089
1 . 0094
1 . 0099
1 . 0 1 04
1 .0 1 09
1 .01 1 4
1 .0 1 1 8
1 . 0 1 23
1 . 0 1 29
1 . 0 1 32
1 . 0 1 37
1 .0 1 42
1 . 0 1 47
1 . 01 52
1 . 0 1 56
1 .0161
1 . 0 1 65
1 . 0 1 70
1 . 0 1 75
1 . 0 1 80
1 . 0 1 85
1 . 0 1 90
--
0 . 52
1 . 0234
1 . 0240
--
0 . 69
1 . 0000
1 . 0004
1 . 0009
1 . 00 1 4
1 . 001 8
1 .0023
1 . 0027
1 . 003 1
1 . 0036
1 .0040
1 . 0045
1 . 0050
1 . 0054
1 . 0059
1 . 0063
1 .0068
1 .0073
1 . 0078
1 . 0082
1 . 0087
1 . 0092
1 . 0096
1 .01 01
1 . 0 1 06
1 .01 1 1
1 .01 1 5
1 . 0 1 20
1 . 0 1 26
1 . 0 1 29
1 . 0 1 34
1 . 0 1 39
1 . 0 1 43
1 . 0 1 48
1 . 0 1 52
1 . 0 1 57
1 .01 61
1 .0 1 66
1 .0 1 71
1 . 0 1 76
1 . 0 1 80
1 . 0 1 85
--
0 . 54
--
1 .0229
1 .0235
0 . 70
1 . 0000
1 .0004
1 .0009
1 .00 1 3
1 .00 1 8
1 .0022
1 .0026
1 . 003 1
1 . 0035
1 . 0039
1 .0044
1 .0049
1 .0053
1 .0058
1 .0062
1 .0067
1 . 0071
1 .0076
1 .0080
1 .0084
1 .0089
1 .0094
1 .0098
1 .0 1 03
1 .0 1 08
1 .0 1 1 2
1 .0 1 1 6
1 .0 1 23
1 .0 1 26
1 .0 1 30
1 . 0 1 35
1 . 0 1 39
1 . 0 1 44
1 .0 1 48
1 . 0 1 53
1 .0 1 57
1 .0 1 62
1 . 0 1 66
1 . 0 1 71
1 .0 1 76
1 . 0 1 80
--
0.56
1 . 0225
1 . 0231
--
0.71
1 . 0000
1 . 0004
1 .0008
1 .001 3
1 . 00 1 7
1 . 0022
1 . 0025
1 .0030
1 .0034
1 . 0039
1 .0043
1 . 0047
1 .005 1
1 .0056
1 . 0060
1 .0065
1 .0069
1 .0074
1 .0078
1 .0082
1 .0087
1 .009 1
1 .0095
1 .0 1 00
1 .0 1 05
1 . 0 1 09
1 .01 1 3
1 .0 1 1 9
1 . 0 1 22
1 . 0 1 27
1 .0 1 3 1
1 . 0 1 35
1 . 0 1 40
1 .0 1 44
1 .0 1 49
1 .0 1 53
1 . 0 1 58
1 . 0 1 62
1 . 0 1 67
1 . 0 1 71
1 . 0 1 75
--
0.58
--
1 . 0220
1 . 0225
0 . 72
1 . 0000
1 . 0004
1 .0008
1 . 00 1 2
1 . 00 1 7
1 . 0021
1 . 0024
1 . 0029
1 . 0033
1 . 0037
1 . 0041
1 .0046
1 .0050
1 .0054
1 .0058
1 . 0063
1 .0066
1 .0071
1 . 0075
1 . 0079
1 .0084
1 .0088
1 . 0094
1 . 0097
1 . 0 1 02
1 . 0 1 06
1 .01 1 0
1 .0 1 1 5
1 .01 1 8
1 .0 1 23
1 . 0 1 27
1 .0 1 3 1
1 . 0 1 36
1 . 0 1 39
1 . 0 1 44
1 .0 1 48
1 . 0 1 53
1 . 0 1 57
1 . 0 1 62
1 . 0 1 66
1 . 0 1 70
--
0 . 60
1 .021 4
1 .0220
--
0 . 73
1 .0000
1 . 0004
1 . 0008
1 . 001 2
1 . 001 6
1 .0020
1 .0023
1 .0028
1 .0032
1 .0036
1 .0040
1 .0044
1 .0048
1 .0052
1 .0056
1 .0060
1 .0064
1 .0069
1 .0073
1 .0077
1 .008 1
1 .0085
1 .0089
1 .0093
1 .0098
1 .0 1 02
1 .0 1 06
1 .0 1 1 1
1 .0 1 1 4
1 .01 1 9
1 . 0 1 23
1 .0 1 27
1 .0 1 31
1 . 0 1 35
1 . 0 1 39
1 .0 1 43
1 . 0 1 48
1 . 0 1 52
1 .0 1 56
1 .0 1 60
1 .0 1 64
--
0.61
1 .021 1
1 . 02 1 7
--
0 . 74
1 . 0000
1 . 0004
1 . 0008
1 . 001 1
1 . 00 1 5
1 .00 1 9
1 .0022
1 . 0026
1 .003 1
1 .0035
1 .0039
1 . 0042
1 .0046
1 .0050
1 . 0054
1 . 0058
1 .0062
1 . 0066
1 .0070
1 .0074
1 .0078
1 . 0082
1 . 0086
1 . 0090
1 . 0095
1 . 0098
1 . 0 1 02
1 .0 1 07
1 .0 1 1 0
1 .01 1 4
1 .01 1 9
1 . 0 1 23
1 . 0 1 27
1 . 0 1 30
1 . 0 1 35
1 .0 1 39
1 . 0 1 43
1 .0 1 46
1 .0 1 5 1
1 .0 1 55
1 . 0 1 59
--
0 . 62
1 . 0208
1 . 021 3
--
0 . 75
1 . 0000
1 . 0004
1 .0007
1 . 001 1
1 . 00 1 5
1 .001 9
1 . 0022
1 . 0025
1 .0029
1 . 0033
1 . 0037
1 . 0040
1 . 0044
1 . 0048
1 . 0052
1 . 0056
1 . 0059
1 .0064
1 . 0067
1 . 0071
1 . 0075
1 . 0079
1 . 0083
1 .0086
1 .009 1
1 .0094
1 . 0099
1 . 0 1 03
1 .0 1 06
1 .0 1 1 0
1 .01 1 4
1 .0 1 1 8
1 . 0 1 22
1 . 0 1 25
1 . 0 1 29
1 .0 1 33
1 .0 1 38
1 . 0 1 41
1 .0 1 45
1 .0 1 49
1 . 0 1 53
--
TABLE 3 .S-PIPE TAPS: EXPANSION FACTORS, Y, STATIC PRESS U R E
TAKEN F R O M DOWNSTREAM TAPS (AFTER REF. 3 )
h w lP r
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
{3 = do ld
0.1
1 . 0000
1 .0008
1 .001 7
1 . 0025
1 .0033
1 .0042
1 .0051
1 .0059
1 .0068
1 .0076
1 .0084
1 .0093
1 .0 1 0 1
1 .01 1 0
1 .01 1 9
1 . 0 1 27
1 . 0 1 36
1 . 0 1 43
1 . 0 1 52
1 .0 1 6 1
1 . 0 1 69
1 . 0 1 77
--
0.2
1 .0000
1 .0007
1 . 001 5
1 . 0023
1 . 0031
1 . 0037
1 .0045
1 .0053
1 .0060
1 .0068
1 .0075
1 .0082
1 .0090
1 . 0098
1 .0 1 06
1 .01 1 3
1 .0 1 2 1
1 . 0 1 28
1 .0 1 36
1 .0 1 43
1 .0 1 50
1 . 0 1 58
--
0.3
1 .0000
1 .0006
1 . 001 2
1 .001 8
1 .0024
1 .0029
1 .0035
1 .0042
1 .0048
1 .0053
1 .0059
1 .0065
1 . 0071
1 . 0077
1 . 0083
1 .0089
1 .0095
1 .0 1 0 1
1 . 0 1 07
1 .01 1 3
1 .0 1 1 9
1 . 0 1 25
--
0.4
1 .0000
1 .0003
1 . 0007
1 .00 1 0
1 .00 1 4
1 .001 8
1 .002 1
1 .0024
1 .0028
1 .0032
1 .0036
1 .0039
1 .0043
1 . 0047
1 .0050
1 .0054
1 .0058
1 . 0062
1 .0065
1 .0069
1 .0073
1 .0077
--
0.45
1 .0000
1 .0002
1 .0004
1 .0006
1 .0008
1 . 001 0
1 .00 1 2
1 . 001 4
1 .001 6
1 .00 1 8
1 .0021
1 .0023
1 .0025
1 . 0027
1 .0030
1 . 0032
1 .0034
1 . 0036
1 . 0038
1 .0041
1 . 0044
1 . 0046
--
0 . 50
1 .0000
1 .0000
1 . 0000
1 .0000
1 .0000
1 .0001
1 . 000 1
1 .000 1
1 .0002
1 .0003
1 . 0003
1 .0004
1 .0004
1 .0004
1 .0005
1 .0005
1 .0006
1 .0006
1 .0007
1 .0008
1 . 0008
1 . 0008
--
0.52
1 .0000
0 . 9999
0 . 9998
0 . 9998
0 . 9997
0 . 9997
0 . 9996
0 . 9996
0 . 9995
0 . 9995
0 . 9994
0 . 9994
0 . 9994
0. 9994
0 . 9993
0 . 9993
0 . 9993
0 . 9993
0 . 9992
0 . 9992
0 . 9992
0 . 9992
--
0 . 54
1 . 0000
0 . 9998
0 . 9997
0 . 9995
0 . 9994
0 . 9993
0 . 9991
0 . 9989
0 . 9988
0 . 9987
0 . 9986
0 .9984
0 . 9984
0 . 9982
0 . 9981
0 . 9980
0 .9979
0 .9978
0 . 9977
0 . 9976
0 . 9975
0 .9974
--
0 . 56
1 .0000
0 . 9997
0 . 9995
0 . 9992
0 . 9989
0 . 9987
0 . 9985
0 . 9983
0.9981
0 . 9978
0. 9976
0 .9974
0 . 9972
0 .9969
0 . 9967
0 . 9965
0 .9964
0 .9962
0 .9960
0 .9958
0 . 9956
0 .9954
--
0 . 58
1 .0000
0 . 9996
0 . 9993
0 . 9989
0 . 9985
0 . 9982
0 . 9979
0 . 9976
0 .9972
0. 9969
0 . 9965
0 . 9962
0 . 9959
0. 9956
0 . 9953
0 . 9950
0 . 9947
0 . 9944
0 . 9942
0 . 9938
0 . 9935
0 .9933
--
0.60
1 .0000
0 . 9995
0 . 9990
0 . 9985
0 . 9980
0 . 9975
0. 9972
0 .9968
0 .9963
0. 9958
0 .9954
0 .9949
0 .9945
0 . 9941
0 . 9937
0 . 9932
0 . 9928
0 . 9924
0 . 9920
0 . 99 1 6
0 . 99 1 2
0 . 9908
--
53
GAS FLOW M EASU R E M ENT
T A B L E 3 . S-PIPE TAPS: EXPANSION FACTORS, Y, STATIC PRESSURE
TAKEN FROM DOWNSTREAM TAPS (AFTER REF. 3) (Continued)
h w1p t
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
(3= do ld
0.1
--
1 .0 1 85
1 .0 1 94
1 .0202
1 .021 0
1 . 02 1 9
1 . 0230
1 . 0236
1 .0244
1 .025 1
1 .0259
1 .0267
1 .0276
1 .0284
1 .0291
1 .0301
1 .0309
1 . 031 6
1 .0324
1 .0332
0.2
--
1 .0 1 65
1 .0 1 73
1 .0 1 80
1 .0 1 88
1 .0 1 95
1 .0205
1 .021 0
1 .021 7
1 .0224
1 .0232
1 .0239
1 .0247
1 .0253
1 .0261
1 .0268
1 .0276
1 .0287
1 .0290
1 .0297
0.3
--
1 .0 1 3 1
1 .0 1 37
1 .0 1 42
1 . 0 1 48
1 .0 1 54
1 .0 1 62
1 .0 1 66
1 .0 1 73
1 .0 1 79
1 .0 1 85
1 .0 1 90
1 .0 1 96
1 . 0202
1 .0208
1 .02 1 4
1 . 021 9
1 .0225
1 .0231
1 .0237
0.4
--
1 .0081
1 .0084
1 .0089
1 .0092
1 .0096
1 .0 1 0 1
1 .0 1 04
1 .0 1 07
1 .0 1 1 1
1 .01 1 5
1 .0 1 1 9
1 .0 1 22
1 .0 1 26
1 .0 1 30
1 .0 1 34
1 .0 1 37
1 . 0 1 41
1 . 0 1 45
1 .0 1 49
0.45
--
1 .0048
1 .0050
1 .0053
1 .0056
1 .0058
1 .0061
1 .0063
1 .0065
1 .0067
1 .0070
1 .0072
1 .0075
1 .0077
1 .0080
1 .0083
1 .0085
1 .0088
1 .0091
1 . 0093
2.
do
3.11.
do =(C' 1250) 0. 5 =(630.11250) 0 . 5 = 1. 5 88
1. 5
3
Pf
Calculate
from Eq.
in.
The minimum meter-run ID should be about
times the orifice
size but rarely is smaller than in. in nominal diameter for pro­
duction measurement. Orifice plates can then be changed as flow
in the desired range .
varies to keep h w and
Some other rules o f thumb used in sizing installations are that
( I ) the numerical value of h w should not exceed pf , the meter­
run ID should be at least one-third larger than the orifice, and
the
recording chart pens should operate in the middle % of the chart
range .
Recording gauges (Pig .
record h w and
The recording
charts have either linear or square-root scales , as Fig. 3.4 shows.
Modern meters use transducers for pressure recordings and can
produce digitized values for cumulative flow rates with tempera­
ture corrections . However, the recording charts with mechanical­
ly actuated pens are still most common in production operations .
In some cases, the swirling motion of the gas flow may cause
unreliable readings . Straightening vanes l (Fig. 3.5) can be in­
stalled upstream of the orifice but should be used only if necessary
because they are susceptible to erosion, introduce additional pres­
sure loss , and clog easily .
3.3)
(2) (3)
60
Pf'
3.4 Critical Flow Prover
A critical flow prover is a special pipe nipple with an orifice flange
on the end . Unlike that for an orifice meter, the orifice plate is thick­
er and has a rounded edge facing upstream (see Fig. 3.6) .
A critical flow prover can b e used if the gas is vented t o the at­
mosphere or if the pressure drop across the device is large. The
critical flow prover is not as accurate as an orifice meter but some­
times is convenient when reasonable accuracy is sufficient. The crit­
ical flow prover is based on the assumption that critical flow
velocity , the velocity of sound at the existing conditions, has been
reached . The criterion for reaching critical flow velocity is that the
ratio between the downstream and upstream pressures must be below
a critical ratio. Por ideal gases, this critical ratio is determined by
0 . 54
0 . 52
0.50
--
--
--
0 . 9992
0 .9992
0 . 9992
0 . 9992
0 . 9992
0 . 9992
0 .9992
0 . 9992
0 . 9992
0 . 9993
0 . 9993
0 . 9993
0 .9994
0 . 9994
0 .9994
0 . 9995
0 . 9995
0 . 9995
0 . 9995
1 .0009
1 .00 1 0
1 .001 1
1 .00 1 2
1 .00 1 3
1 .00 1 4
1 .00 1 4
1 .001 5
1 .001 7
1 .001 8
1 .001 8
1 .001 9
1 .0020
1 .0021
1 .0022
1 .0023
1 .0024
1 .0025
1 . 0025
0 .9973
0 . 9972
0 . 9971
0 . 9971
0 . 9970
0 .9969
0 .9969
0 . 9968
0 . 9967
0 . 9966
0 . 9966
0 .9966
0 . 9965
0 .9964
0 .9964
0 .9964
0 .9963
0 . 9963
0 . 9963
0. 9953
0 . 9951
0 .9949
0 .9948
0 .9946
0 .9944
0 .9943
0 . 9941
0 . 9939
0 .9938
0 .9936
0 . 9935
0 .9934
0 . 9933
0 . 9931
0 . 9930
0 . 9929
0 . 9928
0 . 9927
0. 58
--
0 .9930
0 . 9928
0 .9924
0 . 9922
0 . 99 1 9
0.991 6
0 . 99 1 4
0 . 99 1 2
0.991 0
0 . 9907
0 . 9905
0 . 9903
0 . 9901
0 .9898
0 . 9896
0 . 9894
0 . 9892
0 . 9889
0 . 9887
0 . 60
--
0 . 9905
0 . 9902
0 . 9897
0 .9894
0 .9890
0 . 9885
0 . 9882
0 .9879
0 . 9875
0 .9872
0 .9869
0 .9866
0 . 9863
0.9860
0 . 9857
0 .9854
0 . 9850
0 .9847
0. 9844
upstream temperature, gas gravity , and orifice diameter.
The equation for determining gas flow rate with a critical flow
prover is
qgh=CPt'-J'Yg Tf ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3. 1 3)
3.13 1 . 0 .
14. 4
60oP.
where C= coefficient, listed in Table 3 . 10.
Eq .
is for gas flow rate at
psia and
It also as­
sumes that z =
Corrections can be made for the same factors
as in the orifice meter equation. However, this is often not done
if the flow rate is considered a rough estimate .
Example 3.3-Critical Flow Prover Calculation. Calculate the
gas flow rate through a critical flow prover.
dofd=2=0.150875
PTf
in.
in.
=
psia.
= 70 o P .
'Y g = 0 . 7 .
1.
qgh
C=309.3
3.10) .
gh
qgh= (0.(309.7)(703 )(150)
+ 460) =2,408. 7
Solution.
Determine the factors to calculate
.
(from Table
2.
Calculate
q
.
scf/hr.
vi
3.5 Choke N ipples
Choke nipples sometimes are used to control gas flow rates. Fig.
3.7 shows a sketch of a choke nipple . If flow through the choke
nipple reaches critical (downstream pressure less than about one­
half the upstream pressure) , then Eq .
can be used to calculate
flow rate . The coefficients for Eq .
also are given in Table
but are identified as being for choke nipples .
A more common choke is the bean choke in which a cylinder,
similar to a choke nipple , is inserted into a choke assembly . Por
estimating flow rate , bean chokes can be treated the same as choke
nipples .
3.13
3.13
P cdIPf=[2/(k + l)]kI(k - l ), . . . . . . . . . . . . . . . . . . . . . . . . (3.12) 3 .1 0
Pf=
k=CpICv;
0. 5
where p cd = critical downstream flowing pressure , psia;
flowing pressure (upstream) , psia;
Cp = constant­
pressure specific heat, Btullbm- Op; and Cv= constant-volume spe­
cific heat , Btullbm o p .
This critical ratio is about
for most gases. This means that
the downstream pressure should be lower than one-half the upstream
flowing pressure .
The gas flow rate is directly proportional to the absolute upstream
pressure , so a reduction in downstream pressure does not affect
flow rate. The flow rate depends only on the upstream pressure ,
0 . 56
--
Example 3 . 4-Choke Nipple Calculation. Calculate the gas flow
rate through a choke nipple .
do=0.250
Pf =305
in.
psia.
GAS RESERVOI R E N G I N E E R I N G
54
TABLE 3 .9-FLANGE TAPS: EXPANSION FACTORS, Y STATI C PRESSURE
M EAN O F U PSTREAM AND DOWNSTREAM (AFTER REF. 3)
h w lp f
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
0.1
1 .0000
0 . 9998
0 .9995
0 . 9993
0 . 9991
0 . 9988
0 .9986
0. 9984
0.9981
0. 9979
0. 9977
0 . 9975
0 . 9973
0 . 9971
0 . 9968
0. 9966
0 . 9964
0 . 9962
0 . 9959
0 . 9957
0 .9955
0 . 9953
0 . 9951
0 . 9949
0 . 9947
0 . 9945
0 . 9943
0. 9941
0 . 9939
0 . 9937
0 . 9935
0 .9932
0 . 9931
0. 9929
0 . 9927
0 . 9925
0 . 9923
0 . 9921
0 . 99 1 9
0.991 8
0 . 99 1 6
--
0.2
1 .0000
0. 9998
0 . 9995
0 . 9993
0 . 9991
0 . 9988
0 . 9986
0 . 9984
0 . 9981
0 . 9979
0 . 9977
0 . 9974
0 . 9973
0 . 9970
0 . 9968
0 . 9966
0 .9964
0 . 9962
0 .9959
0 . 9957
0 .9954
0 .9953
0 . 9951
0 .9949
0 .9947
0. 9944
0 .9942
0 . 9941
0 . 9938
0 . 9936
0 . 9934
0 .9932
0 .9931
0 . 9928
0 . 9926
0. 9924
0 . 9923
0 .9920
0 . 99 1 8
0 . 99 1 7
0 . 99 1 5
--
0.3
1 .0000
0 . 9998
0 . 9995
0 . 9993
0 . 9990
0 .9988
0 . 9985
0 .9983
0 . 9981
0 . 9978
0 . 9976
0 .9974
0 . 9972
0 . 9970
0 . 9967
0 . 9965
0 . 9962
0 . 9960
0 . 9958
0 .9956
0 .9953
0 . 9952
0 .9950
0 .9947
0 . 9945
0 .9943
0 . 9941
0 . 9939
0 . 9937
0 . 9934
0 . 9932
0 .9930
0 . 9929
0 .9926
0 . 9924
0 . 9922
0 . 9920
0 . 99 1 8
0 . 99 1 6
0 . 99 1 5
0 . 99 1 3
--
0.4
1 . 0000
0 . 9997
0 . 9995
0. 9992
0 . 9990
0 . 9987
0 . 9984
0 . 9982
0 . 9979
0 .9977
0 .9975
0 .9972
0 .9970
0 . 9967
0 . 9965
0 . 9962
0 . 9960
0 . 9958
0 . 9955
0 .9953
0 . 9950
0 .9948
0 .9947
0 .9944
0 . 9941
0 .9938
0 .9936
0 .9934
0 .9932
0 . 9929
0. 9927
0 . 9925
0 .9923
0 . 9921
0.991 8
0 . 99 1 6
0 . 99 1 4
0 . 99 1 2
0.991 0
0 . 9908
0 . 9906
--
(3
0.45
1 .0000
0 . 9997
0 . 9994
0 . 9992
0 . 9989
0 . 9986
0 .9984
0 . 9981
0 .9978
0 .9975
0 . 9973
0 . 9971
0 . 9968
0 .9965
0 .9963
0 .9960
0 . 9957
0 . 9955
0. 9953
0 .9950
0 .9946
0 .9945
0 .9943
0. 9940
0 .9938
0 .9935
0 .9933
0 .9930
0 . 9928
0 . 9925
0 . 9923
0 . 9920
0 . 99 1 9
0 . 99 1 6
0 . 99 1 3
0 . 99 1 1
0 . 9909
0 . 9907
0 . 9905
0 . 9902
0 . 9900
--
0 . 50
1 .0000
0 . 9997
0 .9994
0 . 9991
0 . 9988
0 . 9985
0 . 9982
0 . 9979
0. 9976
0 . 9973
0 . 9971
0 . 9968
0 . 9966
0 . 9963
0 .9960
0 . 9957
0 .9954
0 . 9952
0 .9949
0 .9948
0 .9942
0 .9940
0 . 9938
0 . 9936
0 . 9933
0 . 9930
0 . 9928
0 . 9925
0 . 9922
0 . 99 1 9
0 . 99 1 7
0 . 99 1 4
0 . 99 1 2
0 . 9909
0 . 9907
0 . 9905
0 . 9902
0 . 9899
0 . 9897
0 .9894
0 . 9892
--
TJ = 80°F.
"Ig =O.77.
=
do ld
0 .52
1 . 0000
0 . 9997
0 . 9994
0 . 9991
0 . 9988
0 . 9985
0 . 9982
0 . 9979
0. 9976
0 .9973
0 .9970
0 . 9967
0 .9964
0 . 9961
0 . 9958
0 . 9955
0 .9952
0 . 9950
0 .9947
0 . 9944
0 . 9940
0 . 9938
0 . 9936
0 . 9933
0 .9930
0 . 9927
0 . 9925
0 . 9922
0.991 9
0 . 99 1 6
0 . 99 1 4
0.991 1
0 . 9909
0 . 9906
0 . 9903
0 . 9901
0 . 9898
0 . 9895
0 .9893
0 . 9890
0 . 9888
--
0 . 54
1 .0000
0 . 9997
0 .9994
0 . 9990
0 . 9987
0 .9984
0.9981
0 .9978
0 . 9975
0 .9972
0 .9969
0 .9966
0 .9963
0 .9960
0 . 9957
0 .9954
0 . 9950
0 .9948
0 .9945
0 . 9942
0 .9938
0 .9936
0 .9934
0 . 9931
0 . 9927
0 .9924
0. 9922
0.991 9
0 . 99 1 6
0 . 99 1 3
0 . 99 1 1
0 . 9908
0 . 9905
0 . 9902
0 . 9899
0 . 9897
0 .9894
0.9891
0 . 9889
0 .9886
0 .9884
--
0 . 56
1 .0000
0 . 9997
0 . 9993
0 . 9990
0 . 9987
0 . 9983
0 . 9980
0 . 9977
0 .9974
0 . 9971
0 . 9968
0 .9965
0 . 9961
0 . 9958
0 . 9955
0 .9952
0 . 9948
0 . 9946
0 .9943
0 . 9940
0 . 9935
0 .9933
0 . 9931
0 . 9928
0 . 9924
0 . 9921
0.991 9
0.991 6
0.991 3
0 . 9909
0 . 9907
0 . 9904
0 . 9901
0 . 9898
0 . 9895
0 . 9893
0 . 9890
0 . 9887
0 .9884
0 . 9881
0 . 9879
--
0.58
1 . 0000
0 . 9997
0 . 9993
0 . 9990
0 . 9986
0 .9983
0 .9979
0 .9976
0 . 9972
0 .9969
0 . 9967
0 . 9963
0 . 9960
0 . 9956
0 . 9953
0 . 9950
0 . 9946
0 . 9943
0 . 9940
0 . 9937
0. 9932
0 .9930
0 . 9927
0. 9924
0 . 9921
0.991 8
0.991 5
0 . 99 1 2
0 .9909
0 . 9905
0 .9903
0 . 9900
0 . 9897
0 . 9894
0 . 9890
0 . 9888
0 . 9885
0 . 9882
0. 9879
0 . 9876
0. 9873
--
0 . 60
1 .0000
0 . 9996
0 . 9993
0 . 9989
0 .9986
0 . 9982
0 .9978
0 . 9975
0 . 9971
0 .9968
0 .9965
0.9961
0 . 9958
0 .9954
0 . 9951
0 .9948
0 .9944
0 . 9941
0 .9937
0 .9934
0 . 9929
0 . 9927
0. 9924
0 .9921
0.991 7
0 . 99 1 4
0.991 1
0 . 9907
0 .9904
0 . 9901
0 . 9898
0 .9895
0 .9892
0 . 9889
0 .9885
0 .9883
0 . 9878
0. 9876
0 . 9873
0 . 9870
0 . 9867
--
0.61
1 .0000
0 . 9996
0 . 9993
0 . 9989
0 . 9985
0 . 9981
0 . 9978
0 . 9974
0 . 9970
0 . 9967
0 .9964
0 .9960
0 . 9957
0 . 9953
0 .9949
0 .9946
0 . 9943
0 . 9939
0 .9936
0 .9933
0 . 9927
0 . 9925
0 . 9922
0 . 99 1 9
0 . 99 1 5
0 . 99 1 2
0 . 9909
0 . 9905
0 . 9902
0 . 9899
0 . 9895
0 . 9892
0 . 9889
0 . 9886
0 .9882
0 . 9880
0 . 9876
0 . 9873
0 . 9870
0 . 9867
0 . 9864
--
0 . 62
1 .0000
0 . 9996
0 . 9992
0 . 9989
0. 9985
0 . 9981
0 . 9977
0 . 9973
0 .9970
0 .9967
0 .9963
0 .9959
0 . 9955
0 .9952
0. 9948
0. 9945
0 . 9941
0 . 9938
0 .9934
0 . 9931
0 . 9925
0 . 9923
0 .9920
0 . 99 1 7
0 . 99 1 3
0 . 9909
0 . 9907
0 .9903
0 . 9900
0 . 9896
0 .9892
0 .9890
0 . 9886
0 . 9883
0 . 9879
0 . 9877
0 . 9873
0 . 9870
0 . 9866
0 . 9863
0 . 9861
--
where P imp = impact pressure on pitot tube, psia; h w = pressure dif­
ferential , in. of water.
Solution.
1 . Determine the factors to calculate qgh '
C=26.5 1 (from Table 3 . 10) .
2. Calculate qgh .
(26.51)(305)
= 396.5 scf/hr .
qgh = ..J
(0.77)(80 +460)
3.6 Pltot Tube
A pitot tube also is used to measure gas flow rate by indirectly meas­
uring the velocity head of the gas flow rate. Pitot tubes are used
commonly on airplanes to determine velocity . Their use in the gas
industry is uncommon . They usually are limited to certain labora­
tory purposes . Fig. 3.8 is a sketch of a pitot tube .
The pressure measured at the end of the pitot tube is called the
impact pressure, Pimp ' At this point, the kinetic energy of the gas
is transformed into pressure energy . In other words, gas velocity
equals zero. However, the pressure in the moving gas stream is
commonly called a static pressure . The difference in these pres­
sures is measured as h w ' The flow rate can be determined with the
impact pressure and Eq. 3 . 14. To obtain an accurate rate, a steady­
state flow rate should be obtained .
qgh =29 1 .67d2 Fg FrJ..Jh wPimp , . . . . . . . . . . . . . . . . . . . . (3 . 14)
Example 3.5-Pitot Tube Calculation. Calculate the gas flow rate
through a pitot tube .
TJ = 78°F.
Pimp = 24.4 psia.
h w = 7 . 3 in .
d = 3 . 826
'Y g =0.73.
in.
Solution.
1 . Determine Fg and FrJ .
Fg = ( 1 .010.73)0.5 = 1 . 1704.
FrJ = [520/(78 +460)]°·5 = 0.983 1 .
2 . Calculate qgh '
qgh =29 1 .67(3 . 826) 2 ( 1 . 1704)(0.983 1)..J (7 . 3)(24 . 4)
=65 ,564.7 scf/hr.
Exercises
3. 1
Calculate the gas flow rate through an orifice meter.
h w = 54 in . of water.
(measured upstream) .
PJ = 194 psig
TJ = 67°F.
55
GAS FLOW M EAS U R E M ENT
TABLE 3 .9-FLANGE TAPS: EXPANSION FACTORS , Y, STATIC PRESSURE,
MEAN O F U PSTREAM AND DOWNSTREAM (AFTER REF. 3) (Continued)
(3 = do ld
h w lP I
--
--
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 .0
1 .1
1 .2
1 .3
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
1 . 0000
0 . 9996
0 . 9992
0.9988
0 . 9984
0.9981
0 . 9977
0 . 9973
0 . 9969
0 .9966
0 . 9962
0 . 9958
0 . 9954
0.9951
0 .9947
0 . 9944
0 . 9940
0 . 9936
0 .9932
0 . 9929
0 . 9923
0.992 1
0.99 1 8
0 . 99 1 5
0 . 99 1 1
0 . 9907
0. 9904
0 . 9900
0 . 9897
0 . 9893
0 .9890
0 . 9887
0 . 9883
0 . 9880
0.9876
0 .9873
0 . 9869
0.9866
0 .9863
0 . 9860
0 . 9857
1 .0000
0 .9996
0. 9992
0 . 9988
0 . 9984
0 . 9980
0 .9976
0.9972
0 . 9968
0 . 9965
0 . 9961
0 . 9957
0 . 9953
0 . 9949
0 . 9945
0 . 9942
0 . 9938
0 . 9934
0.9930
0 . 9927
0 . 992 1
0 . 99 1 9
0.991 6
0.99 1 3
0 . 9909
0 . 9905
0 .9902
0.9898
0 . 9895
0 . 9891
0 .9887
0 . 9884
0 . 9880
0 . 9877
0 .9873
0.9870
0 . 9866
0 . 9863
0 .9859
0 . 9856
0 . 9853
0 . 63
0 . 64
0.65
0 . 66
0.67
0.68
0 . 69
0.70
0.71
0 . 72
0 . 73
0 . 74
0 . 75
--
--
--
--
--
--
--
--
--
--
1 .0000
0 . 9996
0 . 9992
0 .9987
0 . 9983
0 . 9979
0 . 9975
0 . 9971
0 .9967
0 . 9963
0 . 9959
0 . 9955
0.9951
0 .9946
0 . 9942
0 . 9939
0 . 9935
0 . 9931
0 . 9927
0 . 9923
0.991 7
0.99 1 5
0.99 1 1
0 . 9908
0. 9904
0 . 9900
0 . 9896
0 . 9892
0.9889
0. 9885
0 .9881
0 . 9877
0. 9873
0 .9870
0 .9866
0 . 9863
0 .9859
0 . 9855
0 . 9851
0 .9848
0 . 9845
1 .0000
0 .9996
0 . 9991
0 . 9987
0 . 9983
0 . 9978
0 . 9974
0 . 9970
0 . 9965
0 . 9962
0 . 9958
0 . 9953
0 . 9949
0.9945
0 . 9941
0.9937
0 . 9933
0 . 9929
0 . 9925
0 . 9921
0 . 99 1 5
0 . 99 1 3
0.9908
0 . 9905
0.9901
0 . 9897
0 .9893
0 . 9889
0 . 9886
0 .9882
0.9877
0 . 9874
0 .9870
0 . 9866
0 . 9862
0 . 9859
0 . 9855
0.9851
0 . 9847
0 .9844
0. 9840
1 . 0000
0 . 9996
0 . 9991
0 . 9987
0 . 9982
0 . 9978
0.9973
0 . 9969
0 . 9965
0 . 9961
0 . 9957
0 . 9952
0 . 9948
0 . 9943
0 .9939
0 . 9935
0 . 9931
0.9927
0 . 9923
0 . 99 1 9
0 . 991 2
0 . 99 1 0
0 . 9906
0 . 9902
0 . 9898
0. 9894
0 . 9890
0 . 9886
0 . 9882
0.9878
0 . 9874
0 . 9870
0 . 9866
0 . 9863
0 . 9858
0 . 9855
0 . 9851
0. 9847
0 . 9843
0 . 9839
0 .9836
1 . 0000
0. 9995
0 . 999 1
0 . 9986
0.9982
0.9977
0.9973
0 .9968
0 . 9964
0 . 9960
0.9955
0.995 1
0.9946
0. 9942
0.9938
0. 9933
0 .9929
0.9925
0.9920
0.99 1 6
0.99 1 0
0. 9908
0 . 9903
0 . 9900
0.9895
0.989 1
0.9887
0.9883
0.9879
0. 9874
0 .9870
0 . 9867
0 . 9862
0. 9859
0 . 9854
0 .9850
0 . 9846
0 .9842
0. 9838
0.9835
0 . 9831
1 .0000
0 . 9995
0.9991
0.9986
0 .998 1
0.9977
0 . 9972
0.9967
0.9963
0 . 9959
0.9954
0 .9949
0 .9945
0 .9940
0.9936
0 . 9932
0 . 9927
0.9923
0.99 1 8
0 . 99 1 4
0 . 9907
0 . 9905
0.9900
0 .9897
0.9892
0.9888
0 . 9884
0 .9880
0 .9875
0 . 9871
0 .9867
0 . 9863
0 . 9858
0.9854
0.9850
0 .9846
0 . 9842
0 .9838
0.9834
0 . 9830
0.9826
1 . 0000
0 . 9995
0 . 9990
0 . 9986
0 . 998 1
0 . 9976
0 . 9971
0 . 9966
0 . 9962
0 . 9957
0 . 9953
0 . 9948
0 . 9943
0 . 9938
0 . 9934
0 . 9929
0 . 9925
0 . 9921
0 . 99 1 6
0 . 99 1 1
0 . 9904
0 . 9902
0 . 9897
0 . 9894
0 . 9889
0 . 9885
0 . 9880
0 . 9876
0 . 9871
0 . 9867
0 . 9863
0 . 9859
0 . 9854
0 . 9850
0 . 9845
0 . 9841
0 . 9837
0 . 9833
0 . 9829
0 . 9825
0 . 9820
--
1 . 0000
0 . 9996
0 . 9992
0 . 9988
0 . 9984
0 . 9980
0 . 9975
0 . 9971
0 .9967
0 . 9964
0 . 9960
0 . 9956
0 . 9952
0 . 9948
0 . 9944
0 . 9941
0 . 9937
0 . 9933
0 . 9928
0 . 9925
0.99 1 9
0 . 99 1 7
0 . 99 1 3
0 . 99 1 0
0 . 9906
0 . 9902
0 . 9899
0 . 9895
0 . 9892
0 . 9888
0 . 9884
0 .988 1
0 . 9877
0 . 9873
0 . 9869
0 . 9866
0 . 9862
0 . 9859
0 . 9855
0 . 9852
0 . 9849
1 .0000
0 . 9995
0 .9990
0.9985
0. 9980
0. 9975
0 . 9970
0 . 9965
0.9961
0.9956
0 . 9951
0 .9946
0 . 9941
0.9936
0 .9932
0 . 9927
0.9922
0.99 1 8
0.99 1 3
0.9908
0 . 9901
0 .9899
0 .9894
0 . 9891
0.9886
0.9881
0.9876
0 . 9872
0 . 9867
0. 9863
0 . 9859
0 .9854
0. 9850
0 . 9845
0.9841
0.9837
0 . 9833
0 .9828
0. 9824
0.9820
0.98 1 5
1 . 0000
0 . 9995
0 .9990
0 . 9985
0 . 9980
0 .9974
0 .9969
0 .9964
0 . 9960
0.9955
0.9950
0 . 9945
0 .9940
0 .9934
0.9930
0.9925
0.9920
0.99 1 6
0 . 99 1 1
0.9906
0. 9899
0.9896
0 .9892
0 . 9887
0 . 9882
0.9878
0.9873
0 .9869
0 .9864
0.9859
0 .9854
0 . 9850
0 .9845
0 . 9841
0 .9836
0 . 9832
0.9827
0.9822
0.981 8
0.98 1 4
0.9809
1 . 0000
0 . 9995
0 . 9989
0 . 9984
0 . 9979
0 . 9974
0 . 9968
0 . 9963
0 . 9959
0 . 9953
0 . 9948
0 . 9943
0 . 9938
0 . 9933
0 . 9928
0 . 9923
0 . 99 1 8
0 . 99 1 3
0 . 9908
0 . 9903
0 . 9896
0 . 9893
0 . 9888
0.9884
0 . 9878
0 . 9874
0 . 9869
0 . 9864
0.9859
0 . 9855
0.9850
0. 9845
0 . 9840
0 . 9836
0 . 9831
0.9826
0 . 9822
0.981 7
0 . 98 1 3
0 . 9808
0 . 9803
1 .0000
0.9995
0 . 9989
0 .9984
0 . 9978
0.9973
0 . 9967
0 . 9962
0 . 9957
0 .9952
0 .9947
0.9941
0 . 9936
0.9931
0 . 9926
0.9920
0 . 99 1 5
0 . 99 1 0
0 . 9905
0.9900
0 . 9892
0.9890
0.9884
0.9880
0 . 9875
0 . 9870
0 . 9865
0.9860
0 . 9855
0 . 9850
0 .9845
0 . 9840
0 . 9835
0.9831
0 . 9825
0.9821
0.98 1 6
0.981 1
0.9807
0.9802
0 . 9797
Psc = 14.0 psia.
Tsc = 70°F.
d=4.897 in.
do =2.75 in.
/' g =O.60.
Taps = pipe type .
3.2 Calculate the gas flow rate through an orifice meter.
h w = 14 in. of water.
PI = 1 19 psig (measured upstream) .
TI = 87°F.
Psc = 14.6 psia.
Tsc =65 °F.
d = 3 . 826 in.
do =2.00 in.
i'g =0.68.
Taps = pipe type .
3 . 3 Describe each orifice factor and discuss why they are needed.
3 .4 Which orifice factors are commonly neglected? Why?
3 . 5 What is the most common location for an orifice meter? Why
is it the most common location?
Fig . 3 . 3-An orifice meter with flange taps and three pens (for
temperature, pressure, and pressu re differential).
3.6 Calculate an appropriate orifice plate diameter.
PI = 3 15 psig (measured downstream) .
h w = 35 in. of water.
qgh =216,000 scf/hr.
56
GAS RESERVOI R E N G I N E E R I N G
F i g . 3 .4-(a) D i rect-read ing chart; (b) square-root chart (after Katz, Handbook of Natural Gas Engineering, 1 959, McGraw-Hili
Book Co. , 1 cou rtesy of McGraw- H i l i I n c . ) .
3.7 Discuss three rules of thumb for sizing orifice installations .
3.8 What are the advantages and disadvantages o f straightening
vanes?
3 . 9 Calculate the gas flow rate through a critical flow prover.
d=4 in.
do =2.00 in.
Pj =75 psia.
Tj =75 °F.
I'g =0.73 .
3 . 10 When can a critical flow prover be used?
3 . 1 1 When is critical flow obtained?
3 . 12 Calculate the critical downstream flowing pressure .
Cp =0.6 BtulIbm- o F .
Cv=O.4 BtulIbm- o F .
Pj = 75 psia.
3. 13
Pj = 19.4 psig .
d=5. 1 89 in .
I' g =0.63.
h w = 5 . 8 in.
3 . 15 Describe how velocity is determined with a pitot tube .
3 . 16 Derive Eq. 3.7 from Eq. 3.6.
3 . 17 Derive Eq. 3 . 8 from Eq. 3 . 7 .
Nomenclature
Al =
area of flowline, L 2 , ft 2
Calculate the gas flow rate through a choke nipple .
do =0.250 in.
Pj = 305 psia .
Tj = 80°F.
I' g =O.77.
3 . 14 Calculate the
Tj = 87°F.
gas flow rate through a pitot tube.
Fig. 3 . 5-Straightening vanes for orifice i nstal lation (after
Katz, Handbook of Natural Gas Engineering, 1 959, McGraw­
H i l l Book Co. , 1 courtesy of McGraw-H i l i I n c . ) .
.Hole drive fit for //00
thermometer well
1�h
foce
Std pipe
Section A -A
Fig. 3 . 6-Design of 2-in . critical flow prover (after Katz, Handbook of Natural Gas Engineering, 1 959, McGraw- H i l i Book Co. , 1
courtesy of McGraw- H i l i I n c . ) .
57
GAS FLOW M EAS U R E M E N T
TABLE 3 . 1 0-COEFFICIENTS F O R CRITICAL FLOW
PROVER AND CHOKE N I PPLE (AFTER REF. 1 )
Val ue of C
Orifice size
(in.)
1j1 6
3/3 2
1ja
3/1 6
7/32
1j4
0/1 6
o/a
7/1 6
1j2
5/a
0/4
'l'a
0 . 063
0 . 094
0. 1 25
0. 1 88
0.21 8
0. 250
0.31 3
0 . 375
0 .438
0 . 500
0 . 625
0 . 750
0. 875
1 . 000
1 . 1 25
1 .250
1 .375
1 . 500
1 . 750
2 . 000
2. 250
2. 500
2. 750
3.000
1
1 1ja
1 114
1 3/a
1 1j2
1 3/4
2
2 1j4
21j2
23/4
3
=
b=
A2
C
C'
Cp
Cv
=
=
=
=
d =
do =
F=
Fa =
Fb =
Fg =
F[ =
F =
Fpb =
Fp v =
F
=
F =
F Tf =
g=
m
Re
Th
gc =
h
hw
k
M
Pcd
Pf
Pimp
Psc
P
q
q gh
qsc
Critical Flow P rover
2-i n . p i pe
1 .524
3 . 355
6.301
1 4.47
1 9.97
25.86
39. 77
56.58
8 1 .09
1 01 .8
1 54 . 0
224.9
309. 3
406 . 7
520. 8
657. 5
807 .8
1 ,002 . 0
-
4-i n . pipe
-
24.92
-
56.01
-
1 00 . 2
1 56 . 1
223. 7
304 . 2
396. 3
499 . 2
6 1 6.4
742 . 1
884.3
1 ,208
1 ,596
2 , 046
2 , 566
3 , 1 77
3 , 904
C hoke N i pple
6.25
1 4.44
26. 5 1
43.64
61 .21
85. 1 3
1 1 2 . 72
1 79 . 74
260.99
area of orifice opening , L 2 , ft 2
parameter used in calculating FRe ' Reynolds number
factor
coefficient listed in Table 3 . 10 (Eq. 3 . 13)
orifice flow constant
constant-pressure specific heat, L 2 /t 2 T , Btullbm- o F
constant-volume specific heat, L 2 /t 2 T , Btu/lbm- o F
pipe diameter, L, in.
orifice opening diameter, L , in.
energy loss resulting from friction, L 2 It 2 , ft-Ibf/lbm
thermal expansion factor
basic orifice flow factor
specific gravity factor
gauge location factor
manometer factor
pressure base factor
gas deviation factor
Reynolds number factor
temperature base factor
flowing temperature factor
local acceleration owing to gravity , Lit 2 ,
usually = 32 . 1 7 ft/sec 2
dimensional constant, dimensionless, 32. 17
ft-Ibm/lbf-sec 2
pressure differential across orifice , L 2 It 2 , ft-Ibf/lbm
pressure differential across orifice , L, in. of water
=
=
= Cp lCv
= molecular weight, m, Ibmllbm-mol
= critical downstream flowing pressure , m/t 2 L , psia
= flowing pressure, m/t 2 L , psia
= impact pressure on pitot tube, m/t 2 L , psia
= pressure at standard conditions, m/t 2 L , psia
= absolute pressure, m/t 2 L , Ibf/ft 2
= volumetric fluid flow rate, L 3 It, ft 3 /sec
= gas flow rate measured at Psc and Tsc , L 3 /t, scf/hr
= volumetric flow rate at standard conditions , L 3 It,
Tf =
Tsc =
scf/sec
flowing temperature, T, o R
temperature a t standard conditions, T , o R
F i g . 3 . 7-Choke nipple (after Katz, Handbook of Natural Gas
Engineering, 1 95 9 , M cGraw- H i l i Book CO. , l courtesy of
McGraw- H i l i Inc.).
t1+-- 0 --+VI
Oifferen tial
p/�at tube
Static
pressure
Velocity plus
static pressure
Water
Manom eter
Velacity af
flowinq qas
Fig. 3 . S-Pitot tube (after Katz, Handbook of Natural Gas
Engineering, 1 959, McGraw-Hi l i Book CO. , l courtesy of
McGraw-H i l i I nc . ) .
v=
v =
vssc =
s
w
s=
Y=
zf =
Z=
(3 =
=
'Y g =
Ll
net flow velocity of fluid, Lit, ft/sec
specific volume of fluid , L 3 /m, ft 3 /1bm
specific volume of fluid at standard conditions,
L 3 /m, scf/lbm
shaft work done by the system per unit mass, L 2 It 2 ,
ft-Ibf/lbm
expansion factor
gas deviation factor of flowing gas
elevation relative to arbitrary horizontal datum
plane, L, ft
dold
difference operator
gas specific gravity (air = 1 . 0)
References
1 . Katz , D . L . et al. : Haruibaok ofNatural Gas Engineering, McGraw-Hili
Book Co. Inc . , New York C ity ( 1 959) Chap . 8, 332 .
2 . Ikoku, C . U . : Natural Gas Engineering: A Systems Approach, PennWell
Books , Tulsa, OK ( 1 980) Chap . 6, 224 .
3 . Petroleum Engineering Handbook, H . B . Bradley (ed . ) , SPE Richard­
son , TX ( 1 987) Chap . 1 3 .
4 . AGA report No . 3 ( 1 985) ; API, Manual ofPetroleum Measurement Stan­
dards , Chap . 1 4 , API 2530; American National Standards Inst . ,
ANSI/API 25 30- 1 9 8 5 ; Gas Processors Assn . , GPA 8 1 85-85 ( 1 98 5 ) .
Chapter 4
Gas Flow in Wellbores
4.1 I ntroduction
This chapter presents the theoretical development and practical uses
of methods for predicting the flowing pressure behavior in gas wells.
Specific applications of interest to gas reservoir engineers include
estimating bottomhole pressures (BHP ' s) from surface measure­
ments in gas wells and evaluating gas-well production performance.
The chapter begins with methods for calculating static and flowing
BHP ' s (BHSP and BHFP) from surface measurements in wells
producing dry gas . We discuss the l imiting assumptions associated
with each method and present example problems illustrating cal­
culation techniques. Next, we present several correlations for pre­
dicting the pressure behavior when both liquid and gas phases are
flowing . We discuss factors affecting gas-well production perform­
ance, including pressure losses in the wellbore and surface equip­
ment and liquid loading problems . Finally , we show how to forecast
production rates for gas wells .
4.2 B H P Calculations for Dry G a s Wells
Accurate BHP values are essential for gas reservoir engineering
calculations . Ideally , these pressures are measured directly with
pressure gauges placed at the well bottom. However, bottomhole
measurements are often impractical . Therefore, several methods
have been developed for estimating BHP ' s from surface meas­
urements.
All the BHP calculation techniques presented in this chapter are
based on an energy balance in the wellbore. These methods differ
only in the degree of computational complexity and in the simplify­
ing assumptions made to develop the technique . Recall from Chap.
2 that Eq. 2.26 is the differential form of the mechanical energy
balance describing steady-state flow in a conduit :
v
144
g
- dp + - dZ+ - dv + dF= - dws '
p
gc
gc
4 . 1 describes the relationships among the var­
ious forms of energy for a flowing fluid in a conduit. However,
for a static gas column, the kinetic energy and friction effects are
zero and can be eliminated from the equation.
4.2 . 1 BHSP's. Eq.
. . . . . . . . . . . . . . . (2.26)
In this chapter, we will consider only cases with no shaft work,
so dws =0. Then our mechanical energy balance simplifies to
144
v
g
- dp + -dZ+ - dv + dF=O . . . . . . . . . . . . . . . . . . . . . (4. 1)
p
gc
gc
Eq. 4. 1 is used for both BHSP and BHFP methods . Some methods
were intended to be used for hand calculations , others for comput­
er calculations .
144
g
dp +
dZ=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4 . 2 )
p
gc
-
-
Assuming that the local gravitational acceleration equals the
gravitational acceleration constant ( g = gc ) ' Eq. 4 . 2 is rearranged
to yield
dp =
P
- -
1 44
dZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.3)
Referring to Fig. 4. 1 , Z is the true vertical depth of the wel l .
For a slanted well with a total length L and depth Z,
cos
() =ZIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.4)
or, in differential form,
()
dZ= cos
dL,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 5)
where dZ is the change in the elevation in the upward direction and
dL is positive upward.
Substituting Eq. 4.5 into Eq. 4.3 yields
P
dp = - - cos () dL .
144
. . . . . . . . . . . . . . . . . . . . . . . . . . . (4.6)
.
.
Assuming a single-phase fluid that obeys the real-gas equation
of state (EOS ) , we can express the gas density as a function of
pressure,
28. 97'Y gP
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.7)
zRT
Combining Eqs . 4.7 and 4 . 6 yields
0.01875"1 gP
cos () dL . . . . . . . . . . . . . . . . . . . . . . . (4. 8)
dp =
zT
Eq. 4.8 forms the basis for all the methods developed t o esti­
pM
pg = - =
zRT
-
.
mate BHSP ' s from surface measurements. As we illustrate later,
the methods differ only in the limiting assumptions and degree of
computational complexity .
59
GAS FLOW I N WELLBORES
4. Calculate Pws with Eq.
Pws converges.
4. 10.
Iterate on Steps 2 through 4 until
Example 4.1-Calculating BHSP With the Average Tempera­
ture and z-Factor Method. With the fluid property and well data 3
given below, calculate the BHSP.
'Y g = 0.75.
f!.p c :: 667 l sia.
lpc 408 R.
P ts = 2,500 psia.
Tts = 35 °F.
Tws = 245 °F.
L = 1 0,000 ft.
() = 0° (vertical well) .
-
z
Solution.
1 . Compute an initial estimate of Pws ' A good initial guess is
2 ,500 10,000
Pws =2,500 + 0.25 100 ---wo- = 3 , 125 psia.
)(
(
Fig. 4 . 1 -Schematic of wel l geometry.
Average Temperature and z-Factor Method. Eq. 4 . 8 relates the
change in wellbore pressure as a function of depth and gas den�ity .
Recall from Chap. 1 that both the gas density, P g ' and z factor (or
gas deviation factor) are functions of pressure and temperature. In
addition, wellbore temperature changes with depth. Therefore, solv­
ing the differential equation given by Eq. 4 . 8 is difficult. To sim­
plify the solution, the average temperature and z factor method 1
(sometimes called the exponential method) makes the assumption
that the z factor and temperature are constant and can be repre­
sented by average values. Typically, these average values are deter­
mined at the arithmetic average of the surface and bottomhole
temperature (BHT) and pressure; however, other averaging tech­
niques have also been proposed. 2
Substituting an average temperature, T, and an average z factor,
z, into Eq. 4 . 8 and separating variables, we can derive an equation
for computing BHP as follows:
r ts dp = _ 0.01 87�� cos () r
dL , . . . . . . . . . . . . . . . . . (4.9)
z
T
P
0
P ws
which integrates from bottom to top of the wellbore. Following in­
tegration, the solution is
Pws =p ts e s I2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 10)
where s =
0.0375 1' gL cos ()
. . . . . . . . . . . . . . . . . . . . . . . . . (4. 1 1)
ZT
_
Because z depends on Pws ' the solution to Eq. 4.9 involves an
iterative process. In the calculation procedure described below and
illustrated with Example 4 . 1 , we apply this iterative scheme in a
one-step computation from the surface to the bottom of the well.
However, better accuracy can be achieved by dividing the well­
bore into a finite number of increments and successively applying
the calculation procedure to each increment.
Calculation Procedure.
1 . Assume a value of BHSP, Pws ' A good guess for the initial
calculations is
( )( L cos () ) .
P ts
Pws "' P ts +0.25 -1 00
---
1 00
2. Compute the arithmetic average pressure and temperature.
3 . Calculate average z factor using the average pressure and tem­
perature from Step 2 .
)
2. Compute the arithmetic average wellbore pressure and tem­
perature: p = 2 , 8 1 2 . 5 psia and T=6000R.
3. Calculate average z factor using the average pressure and tem­
perature from Step 2 . We find that Pp r =4.22, Tp r = 1 .47, and
z=0.77.
4. Calculate Pws ' Calculate first s, defined by Eq. 4. 1 1 , then Pws
with Eq. 4. 10.
0.0375 1' gL cos () (0.0375)(0.75)( IO,OOO)(cos 0°)
s=
=0. 6 1 .
zT
(0.77)(600)
Pws =p ts e sl2 =2,500e O .6 1 12 = 3 ,392 psia.
Second Iteration. Compute a new value of P using the most recent
value of Pws ' p = (3 , 392 + 2 ,500)/2 = 2,946 psia. This now gives
z= 0.775 and s =0.60. Then,
Pws =p ts e s l2 =2,5OOe O .60/2 = 3 ,375 psia.
Third Iteration. Now p = (3 ,375 + 2,500)/2 = 2 ,937.5 psia. However,
the value of s is still 0.60, so we have converged on Pws = 3 ,375
psia.
Because of the simplifying assumptions made in its development,
this method is not sufficiently accurate for all applications. In gener­
al, this method is applicable to shallow gas wells. For deeper gas
wells, we suggest using alternative methods that are more rigorously
correct, such as the method developed by Poettmann. 4
PoeUlnann 's Method. Like the average temperature and z-factor
method, the method proposed by Poettmann4 assumes an average,
but constant, value for temperature. Unlike the previous method,
however, Poettmann allows the z factor to vary with pressure and
therefore provides a more rigorous calculation technique. Substitut­
ing an average temperature, T, into Eq. 4 . 8 and separating varia­
bles yields
ts zdp
_ 0.01 875 1' g cos
dL . . . . . . . . . . . . . . . . (4. 12)
=
T
P
°
P ws
So, after integrating the right side, we have
0.0 1 875 L cos ()
ts zdp
=
. . . . . . . . . . . . . . . . . . . . (4. 1 3)
T
P
PM
Eq. 4. 1 3 is the form proposed originally by Fowler. 5 However,
to allow this method to be applied to a wide range of pressures .and
temperatures, Poettmann rewrote the integral in Eq. 4. 1 3 in terms
of pseudoreduced pressure:
0.0 1 875 1' gL cos ()
p r,ts zdPp r
=
. . . . . . . . . . . . . . . . (4. 14)
T
Pp r
Ppr,ws
() r
r
r
r
�
_
_
60
GAS RESERVO I R E N G I N E E R I N G
Further, Poettmann broke the integral into parts , choosing an ar­
bitrary integration limit of 0.2:
I
zdPpr f Ppr.ts zdPpr _ 0.01 8751' gL cos 0
=
+
, . . (4. 15)
T
Ppr 0.J 2 Ppr
O. 2
Ppr, ws
5 . Return to the tables in Appendix B with
r Ppr.ws z dPp r
-- and
J
Ppr
is P ws = Pp r,wsPpc =
0. 2
Tpr = 1 .47 and read Ppr ,ws = 5 .0.
(5 .0)(667) = 3 , 335 psia.
The BHSP
j' Ppr.ws zdPpr = \' Ppr.ts zdPpr + 0.01875'Y_gL cos 0 . . . . . . (4. 16)
The Poettmann method accounts for the variation of z factor with
pressure but not with temperature . If we eliminate the constant­
temperature assumption, we can achieve even more accuracy .
The integrals in Eq.
are tabulated in Appendix B as a function of the pseudoreduced
temperature, Tp" and pseudoreduced pressure, Ppr' Thus, the right
side of Eq. 4. 1 6 can be evaluated explicitly .
Note that, unlike the iterative process required with the average
temperature and z-factor method, the Poettmann method does not
require iteration. In addition, because of the inclusion of the z fac­
tor in the integrals , this calculation technique is potentially more
accurate and applicable over a wider range of pressure and tem­
perature conditions than the exponential method .
Cullender and Smith Method. The Cullender and Smith 6
method makes no simplifying assumptions for the variation of either
temperature or z factor in the wellbore. As a result, this method
is more rigorous than the previous calculation techniques and is
applicable over a much wider range of gas-well pressures and tem­
peratures . Again , we begin with Eq. 4 . 8 . After separating varia­
bles and rearranging, we have
which he then put in his final equation form:
Ppr
0. 2
T
Ppr
4. 16 have been evaluated numerically and
0.2
Calculation Procedure.
1 . Compute the value of the parameter K defined by
0.01 8751' gL cos-0
K= ----"--T
2. Compute the pseudoreduced wellh�ad pressure, Ppr,ts ' and the
average pseudoreduced temperature, Tpr .
_
3 . Enter the tables in Appendix B with Ppr , ts and Tpr from Step
2 and read .
Ppr.ts zdppr
I0. Ppr .
2
4.
\' Ppr.ws -zdPpr
using Eq. 4. 16.
Calculate
0. 2
5.
With
.
\ Ppr.ws zdPpr
Ppr
4 and 2,
return to the ta-
bles in Appendix B and read Ppr , ws ' From the definition of pseu­
doreduced pressure , compute P ws = Ppr. wsPpc '
Example 4.2-Calculating BHSP With the Poettmann Method.
U sing the well data given for Example 4. 1 , calculate the BHSP.
Solution.
1.
K
2.
The first step is to calculate the parameter K .
0.01 875 'Y gL cos 0
T
=0.23.
(0.01 875)(0.75)( 1O,000)(cos 0°)
(600)
Calculate the pseu�oreduced wellhead pressure and tempera­
Ppr.ts = 3. 75 and Tpr = 1 .47.
_ 3. Enter the tables in Appendix B with Ppr , ts = 3 .75 and
Tpr = 1 .47, and read
ture:
\' Ppr.ts zdppr -_ 2.6.
Ppr
0. 2
Note that interpolation is required to read the value from the table.
4.
Calculate
\' Ppr.ws zdPpr
-- using Eq.
0. 2
'
Ppr
4. 16.
( Ppr.ws zdPpr \ Ppr,ts zdPpr
-- =
-- +K=2.6 +0.23 = 2 . 83 .
J
0. 2
Ppr
0. 2
Ppr
P ws P
g
'L
\
cos O dL,
0
. . . . . . . . . . . . . . . (4. 17)
which reduces to
Tz
\' Pws -dp
=0.01875 'Y L cos O .
. . . . . . . . . . . . . . . . . . . . (4. 1 8)
g
.
Pts P
We observe that the integral in Eq. 4. 1 8 contains both pressure­
and temperature-dependent variables, thereby making an exact eval­
uation of the integral difficult. Cullender and Smith used trapezoi­
dal numerical integration to evaluate the integral on the left side
of Eq. 4. 1 8 . Their numerical method incorporates a two-step cal­
culation procedure that uses an intermediate value of pressure at
the midpoint of the production string , or
\' Pws -Tz dp
P ts
Ppr
-- from Steps
0. 2
Tz
\' Pts -dp
= - 0.01875'Y
P
. . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . (4. 19)
where I is the integrand evaluated at either the surface , midpoint,
or bottomhole conditions (denoted by the subscripts ts, mp, and
ws, respectively) and is defined as
1= Tz/p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.20)
While this two-step procedure is reasonable for hand calcula­
tion, more than two steps can, and should, be used to improve the
accuracy in computerized versions of this procedure. Alternative­
Iy, Cullender and Smith suggested a method for improving the ac­
curacy of the BHP obtained from a two-step calculation . The
two-step calculation procedure is discussed below and illustrated
with Example 4.3. Note that this same procedure is valid for any
number of steps .
Calculation Procedure.
I . Calculate the quantity on the right side of Eq.
4. 18. Call this a.
a =0.0 1 875 'Y gL cos O .
2. Evaluate I i n Eq. 4 . 1 8 at the wellhead temperature and pres­
sure . Call this Its '
3 . Compute the midpoint pressure, P mp ' of the production string .
Pmp =Pts + --­
Imp + Its
For the initial estimate, assume Imp = Its ' Then iterate with new
values Imp for each new estimate of Pmp ' Stop when Pmp con­
verges.
4. Compute the BHSP of the production string .
P ws =Pmp + --Iws + lmp
For the initial estimate , assume Iws = lmp . Then iterate with new
values of Iws for each new estimate of P WS ' Stop when Pmp con­
verges .
61
GAS FLOW I N WELLBORES
01 "
r
LAMINAR Fl!JN
rCRITlCAL ZONE
\ ,TRANSITION ZONE
. l n l NI
009 It
-+�
"
..,
�'-+008U.
1 ....�"""'"
]\
I
� �:� �I
....•
�
�
.gu
J:
�
0 5."
0.
0.
04
003
0.02 5
:\$� : .
CO.2
\
�
�::
� ....
o.�
0.004
mrt-+9=f!1:wF=�::#�==R��O'OO2
��
I� ....
0.001
\
���
"7,Q :O:::
I
I
8
�
2 3 4 56 8 1
�
2 3 4 56 8 1
�
::::- ",, I:::
-:,
�s _
C.o.l H-++I-+-H-f+fttt--H-t-I-ttl'lt--; (
000
'd
r
=
r--
til
�
::l
�
�
o.W �
.
III '
�
NRc
6
=
.....
I
0 000 00 1
2 3 4 56 8 1
Reynolds Number.
vi
0.0000 .;::
�
00. 15 H-++I-+-H-ttlfttt--H-t-I-ttlrr'
0..00
0..05
004
0.0.3
.02
-+-+++I+I++-++i+tlH1t-H+:Httt1 o.0.0.15
:"
?=t-i�!-#I++�!4£E::
I
..R ., ......
1 1 1 11111
III
J l
1 1 1 111
COMPLETE TUR8lA..E NCE,ROlXiH PIPES
=
0 . 000 , 005
�
2 3 4 56 8 1
�
7
1 488 p v d'
•
0000 2 �
I
0.000
.
0.00,0 0.5
0000,0.1
2 3 4 56 8 l
, a
�
I.l
Fig. 4 .2-Moody friction factor for fluid flow in pipe (after Moody 1 2 ) .
5 . U s e Simpson ' s rule t o obtain a more accurate value of P ws '
6a
P ws = Pts +
Its + 4Imp + lws
----­
Cullender and Smith suggested using Simpson ' s rule, given in
Step 5, to achieve better accuracy than trapezoidal integration. How­
ever, according to a study performed by Young 7 , applying Simp­
son ' s rule does not always improve the accuracy of the calculation.
Second Iteration . Using P ws = 3 ,490 psia, we calculate
Iws =0. 1 8 . Then,
140.625
a
= 3,02 1 +
= 3 ,447 psia.
P ws =Pmp +
0. 1 8 +0. 1 5
Iws + lmp
Third Iteration. Using P ws = 3,447 psia, we calculate Iws =0. 18.
Because this value did not change, w e have converged to
3,447 psia.
Therefore, ifmore accuracy is desired, we suggest using more incre­ P w5.s =Use
Simpson ' s rule to obtain a more accurate value of P ws '
'
ments, especially when computer versions of Cullender and Smith s
method are used.
6(140.625)
6a
=2,500 + ------P ws =Pts +
0. 1 2 + 4(0. 1 5) +0. 1 8
Its +4Imp + lws
Example 4.3-Calculating BHSP With the Cullender and Smith
= 3,437 psia.
Method. With the well data 3 given for Examples 4. 1 and 4.2, cal­
culate the BHSP using the Cullender and Smith method with a two­
step calculation.
'YgL
Solution.
1 . Calculate a =0.01875
cos 8 =0.01 875(0.75)(10,000)(1)
= 140.625 .
2. Evaluate Its using the psuedoreduced wellh�ad pressure , the
temperature, and the
factor. Pp r,ts = 3.75, Tpr , ts = 1 .2 1 , and
Z ts =0.585 . Using Eq. 4.20 gives Its =0. 12.
3 . Compute the midpoint pressure, P me , of the production string.
For this initial estimate , assume Imp = lts .
140.625
a
3,086 psia.
=2,500+
Pmp =Pts +
0. 12 +0. 12
Imp + Its
Second Iteration. Using Pmp = 3 ,086 psia, we calculate
Imp =0. 15. Then ,
140.625
= 2,500 +
= 3,02 1 psia.
P mp =Pts +
0. 15 +0. 12
Imp + Its
Third Iteration. Using P mp = 3,02 1 psia, w e calculate Imp =0. 15.
T � is value did not change, s o w e have converged t o Pmp = 3 ,02 1
z
pSla.
4.
Compute P ws ' For this initial estimate , assume
P ws =Pmp +
Iws = Imp .
140.625
a
3 ,02 1 +
= 3,490 psia.
0. 15 +0. 15
Iws + lmp
4.1.2 BHFP's. The methods developed for computing BHFP ' s from
surface measurements consider the flowing wellhead pressure , the
pressure exerted by the weight of the gas column in the production
string , and the energy losses resulting from gas flowing through
pipe . Changes in kinetic energy typically are small for gas flow ,
so these effects are not included in the analysis. Therefore, the ener­
gy balance given by Eq. 4. 1 simplifies to
144
p
g
gc
fv 2
- dp + -dZ+ -- dL = O .
2gcd
'
. . . . . . . . . . . . . . . . . . . . (4.21)
Using the real-gas EOS , we can write this in terms of pressure ,
53.34Tz dp g
fv 2
+ -dZ+ -- dL = O . . . . . . . . . . . . . . . . . (4.22)
2gc d'
P gc
'Yg
For a real gas flowing through a conduit with a circular cross
section, the average gas velocity at any point is
v=
q' = ( I , Oooqg ) ( T ) ( Psc ) ( z ) ( 4 )
-::t 8.64 x I04 Tsc -; z sc d '
4. 152 x 1O - 4 Tzqg
. . . . . . . . . . . . . . . . . . . . . . . . . . (4.23)
1f
pd ' 2
2
62
GAS RESERVOI R E N G I N E E R I N G
.
.0 5
. 04
1
. 0 3 '"
. 02
"'
. 0 I t\..
.ooa
.00
6 . �"
.00 5
. 00 3
VJ"
VJ
. 002
G)
I:
�
::a
�
�
''iii
v
c.:::
.00 I
.4
.
\.
"-
Internal Pipe Diameter d'. ft
� �
II �
�.
I'\..
�
'\
'-.
i"
" r-.
''-
R V
T
" ""
" "
" "
, � i"
"
N
1'\
r-.
�
1'\
R
\,
"
. 000 4
o�"
. 0002
. 000 1
. 0 0008
-
. 0000 2
,
'\
"
�
�.'� ��
��
""� I
�
<.
� '>i>
0
0"
R-I-;..
"
('�
��
�'"
o�
'",0",
,
'\
f'\.
,
1\
1'\
1\. '
1\ 1\
"
. 000008
"
��o,-
"
",, �
5 6
'0
�o' 0
U �.t
8 10
20
30
4 0 50 60
- -
8 0 100
,.....
"
· oo?..... �
'" , ,,
,"
�
:
Internal Pipe Diameter d, inches
025
'-'
r-. 02
I'\.
'"
v
0.
i:l:
..c
01)
::l
0 18
['I..
t'-..
�
"
"
��.. �
i"
�
'Vi'
03
�" f-I��
1-. 0 1 4
X .,.
"
.......
I�;r-I-
" 1&"),
'"
t\
. 05
0 16
'-
i"
1'*�,
I
,
�"
,
i"
ol";..'�o "
1-
y�
r--e"
"
I"
�o
i'
1'\ ",
1"'\
. 0000 1
. 000006
"
,
. 00003
.0000 0 5
I,
\.",I�D� r\..
""9(,,
. 06
035
"-
':'''1''
07
1- . 04
I'-
" ol";..
. 0003
. 00004
"
,
�'"
5
-
r\..
'-
'"
1'\
"
ED
L
E
WOOD
TAVE
I"
"
.0006
0005
. 00005
I �
-
I'\..
.0008
. 00006
}O
•
1'-
� �
:004
�
.�
I
Oil
�
<1)
v
0..
E
(;
....
<2
�
o
U
'"
t.t..
c:
.gu
·c
t.t..
01
009
0 08
'"
200 300
Fig. 4 . 3-Relative pipe roughness as a function of pipe d iameter (after Brown 2 6 ) . (f is i n feet
in this figure only.) Cou rtesy of ASME I nti .
For these calculations, we have that assumed standard conditions
are Psc = 1 4 . 65 psia, Tsc = 520 oR, and z sc = 1 .0 . Substituting Eq.
4 . 23 into 4 . 22 yields
5 3 . 34 Tz dp
'Y g
P
g
gc
+ -dZ+
(- )
2 . 667 x 10 - 9f TZ 2
d' 5
P
(qg ) 2 dL = 0 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4 . 24)
If we convert the dimensions of the pipe diameter from feet to
inches (d' to d) and substitute dZ= cos (JdL, Eq. 4 . 24 becomes
( )
5 3 . 34 Tz dp
6 . 67 X 1 O - 4f TZ 2
g
- (q g ) 2 dL = 0 .
--- - + - cos (J dL +
d5
'Y g
P
P gc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4 .25)
Eq. 4 . 25 forms the basis for all methods developed to estimate
BHFP ' s from surface measurements in gas wells . The methods differ
only in the limiting assumptions made to simplify the calculations.
Frictional Pressure Losses. Before using Eq. 4.25, we must evalu­
ate the Moody friction factor, f Generally, for single-phase gas flow
in pipes , the friction factor is a function of the gas properties, gas
flow rate, type of flow regime (laminar, turbulent, or a transition
regime) , and the internal pipe roughness . If the gas flow is charac­
terized as laminar, the pressure losses can be predicted exactly from
theoretical fluid flow considerations. However, for transition or tur­
bulent flow, the friction losses cannot be calculated directly, and we
must rely on correlations developed from laboratory data.
The Reynolds number , NRe (which is the dimensionless ratio of
the fluid inertial forces to the viscous forces) , often is used to iden­
tify the nature of the flow regime and is defined by
NRe =
1 ,488p vd'
P-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4 . 26)
In terms of field units for gas flow ,
20'Y g qg
NRe = -- .
p- g d
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 27)
63
GAS FLOW IN WELLBORES
Laminar flow is defined as flow in which the fluid moves in im­
aginary layers . Each layer glides smoothly over an adjacent layer
with only a molecular interchange of momentum. Any tendencies
toward instability and turbulence are damped out by viscous shear
forces that resist the relative motion of adj acent fluid layers. Con­
sequently , during laminar flow, frictional pressure losses are caused
primarily by the shear forces . 8 .9 For most engineering calcula­
tions, we characterize the flow in pipes as laminar when
NRe ::; 2,000. Under these conditions , the energy losses are directly
proportional to the average velocity , and the Moody friction factor
is inversely proportional to NRe :
j=64/NRe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.28)
For 2,000 <NRe < 4,000, both viscous and inertial forces be­
come important, and the flow regime is classified as unstable. As
NRe exceeds 4,000, the flow regime reaches a transition region be­
tween laminar and fully turbulent flow conditions . Generally , the
flow is considered fully or completely turbulent when NRe � 4,000.
Turbulent flow is characterized by very erratic motion of fluid par­
ticles with interchanges of momentum in the transverse direction. 9
Because of the erratic nature of the particle motion during the
unstable and turbulent flow regimes, we cannot predict pressure
losses from theoretical fluid flow considerations . As a result, we
must rely on correlations derived from laboratory experiments .
Colebrook empirically developed an equation for estimating the
friction factor for the unstable region in which 2,000 <NRe <
4,000:
10
I 17
.JJ = . 4 -2 log
(--;;2E + NRe18.7.JJ ) . . . . . . . . . . . . . . . . (4.29)
In the fully turbulent flow region (NRe � 4,000), the friction fac­
tor is independent of the value of NRe and depends only on the rela­
tive roughness . Under these conditions , Nikuradse I I developed an
empirical relationship between j and the relative roughness, E/d:
� = 1.74 2
-
lOg
e: )
'
. . . . . . . . . . . . . . . . . . . . . . . . (4.30)
which is the same as Eq . 4.29 but with the final term set equal to
zero. Fig. 4.2 illustrates the relationship between NRc and j for var­
ious flow regimes.
For unstable and turbulent flowing conditions (NRe � 2,000) and
assuming E =0.0023 in . , the correlation developed by Jain and
Swamee l 3 (Eq . 4.31) provides another method for calculating j
directly :
12
(
I
0.0023 2 1 .25
f= 4 l2 . 2 8 - 4 lo g -- +
d
NR:
)l - 2
. . . . . . . . . . . (4 . 3 1 )
The relative roughness of the production string depends on the
manner in which the pipe is constructed . Fig. 4.3, a plot of the
relative roughness as a function of pipe diameter,
is recom­
mended for estimating the relative roughness if the manufacturer
does not provide values. Typically , we use an absolute roughness
equal to 0.0006 in. for tubing in gas wells when measured values
are not available.
Eqs . 4.28 through 4.3 1 can be used to calculate j as a function
of the flow regime , gas properties , gas flow rate, and internal pipe
roughness. We can also use these equations to calculate the Fan­
ning friction factor, fF, which is related to the Moody friction fac­
tor by j=4fF. Unless otherwise noted , we will use the Moody
friction factor in this chapter.
Annular Flow. For flow through tubing , d is the tubing !D . How­
ever, for flow through the tubing/casing annulus , we must modify
the diameter. Following the derivation presented by Bertuzzi et
aZ. , 8 we define the hydraulic radius, rh , as the area of flow divid­
ed by the wetted perimeter. For flow inside a circular pipe,
14.15
rh =
'lrd
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.32)
d
['Ir(d? -di )]l4 d l - d2 , . . . . . . . . . . . . . . . . . . . . (4.33)
rh =
'Ir(d +d2 ) 4
d l 4.33 ,
d2
4.32
d eq ,
where
is tubing
!D .
For the tubing/casing annulus,
--
I
where
is the casing !D and
is the tubing aD . Combining
Eqs .
and
we can develop an equivalent diameter,
for flow in the annulus ,
deq =dl -d2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.34)
For flow through the tubing/casing annulus , we substitute deq
for d, where d relates to the hydraulic radius concept in Eqs . 4.21
and 4.26. However, d 2 i n Eq . 4.23 becomes dr - d i for convert­
d5
ing velocities to flow rates. Thus, for annular flow, the
in Eq.
then becomes (d I
(d I - d 3 . The d in Eq. 4.27 be­
comes (d I
Other relationships derived from these equations
can be modified equivalently when applied to annular flow .
Average Temperature and z-Factor Method. Like the BHSP cal­
culation technique, the average temperature and z factor method
for BHFP calculations assumes that the z factor and the tempera­
ture can be represented with an average value calculated at the aver­
age of surface temperature and BHT and surface pressure and BHP.
With this assumption , Eq. 4.25 is rearranged to yield
4.25
+d2 ) 2
+d2 ).
53.34 IT
P
-- - dp =
19
l
- cos
2)
2
0+ 6.67xd51O-4j (-ITP ) (qg) 2 J dL,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.35)
where we also assume g = gc - After separating variables and rear­
ranging , we have
l
53 . 34 IT
-dp
P
-19
cos
2 = - dL
0+ 6. 67x d51O - 4jql (-ITP ) J
0,
After mUltiplying both sides by cos
53 .34 IT
--dp
P
19
--6-.6-7-X-':I 0-:"' --4-jq-1-(---=-IT---"2
1+ d5 0 P )
=
.
. . . . . . . . . . (4.36)
we can rearrange to yield
0
- cos dL. . . . . . . . . . (4.37)
cos
Further, if we multiply the numerator and denominator on the
left side by p 2 and apply the limits of integration , we have
.\. PIJ
1' "1
pdp
------
p 2 + 6. 67 Xd51O - 4jT02 z2 qg2
cos
=-
19
\. L
cos
53.34IT ·
0
0 dL.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . �.3�
Following integration , w e can express Eq. 4.38 i n terms o f the
BHFP , Pwj :
+ 6.67 xd51 0 -4q1JT 2 Z 2 (e' - I), . . . . . .
=0 03 7 5 gL OnT,
4. 1 1 .
4.39
p aj =piJe s
cos
0
.
.
. .
(4.39)
where s
.
cos
as given in Eq.
1
The calculation procedure involves use of Eq.
either in a
one-step calculation from the wellhead to the bottom or as a mul­
tistep calculation with intermediate pressures at several points in
the production string . The average z factor and viscosity are ob­
tained by estimation or iteration at each step . We illustrate the cal­
culation procedure with a one-step calculation .
64
GAS RESERVO I R E N G I N E E R I N G
Calculation Procedure.
1 . Assume a BHFP. A good initial guess is
cos 8
P tf
.
P wf "" P tf +0.25 100
100
2. Compute the arithmetic average wellbore pressure and tem­
L )
) (--
(
perature .
3 . Calculate the average z factor and viscosity using the average
pressure and temperature from Step 2.
4. Calculate the friction factor, f
A . Calculate NRe using Eq. 4.27.
B . Depending o n the value of NRe , compute f using Eqs . 4.28
through 4.3 1 .
5 . Calculate P wf using Eq. 4.39. Iterate on Steps 2 through 5 un­
til P wf converges .
Example 4.4-Calculating BHFP With the Average Tempera­
ture and z-Factor Method. For the data 16 given below, calcu­
late the BHFP using the average temperature and z-factor method .
Assume that a one-step calculation scheme is sufficiently accurate .
'Y g =
Ppc =
Tpc =
qg =
P tf =
Ttf =
=
8=
wJ =
d=
T
L
0.65 . .
667 pSJa.
366°R.
6,300 MscflD.
2 , 1 75 psia.
1 1 8 °F.
6,818 ft .
0° (vertical well) .
2 16°F.
2 .44 1 i n . (flow through tubing) .
Second Iteration. First, calculate the new average pressure.
p = (2,656 +2, 175)/2 =2,415 psia.
This now gives z=0. 865 and I-' g =0.017 cpo These values are
not different from the initial values calculated in Step 3 , so we have
converged on P wf = 2,656 psia.
Sukkar and Cornell Method. Similar to the Poettmann method
for calculating BHSP ' s , Sukkar and Cornell ' s 1 7 method assumes
that the wellbore temperature can be represented by an average
value . Originally , Sukkar and Cornell proposed using a log mean
temperature when the wellbore temperature profile is linear. How­
ever, subsequent calculations suggested that the averaging method
has little effect on the result. Arithmetic average temperature could
be used . Unlike the average temperature and z-factor method, the
z factor is incorporated into the integral as a pressure-dependent
variable. In terms of average temperature,
Eq. 4.25 becomes
l
53 .34 Tz
-'Y g P
- dp = - cos
·
( 2, 175 )( 6,8 1 8 )
P wf = 2 , 1 75 +0.25 Wo Wo = 2,546
' Pt/
J
P »j
psia.
Compute the arithmetic �verage wellbore pressure and tem­
perature : p=2,361 psia and T= 627°R.
3 . Determine a n average z factor and viscosity using the aver­
age pressure and temperature from Step 2. We find that ppr = 3 .54
and Tp r = 1 .7 1 . Then , z =0.865 and I-' g =0.017 cp o
4. Calculate the friction factor.
A. Calculate the Reynolds number using Eq. 4.27.
---
20'Y q (20)(0.65)(6,300)
NRe = g g =
= 1 ,974,000.
(0.017)(2.441)
I-' g d
NRe � 4,000 indicates turbulent flowing conditions ..
B . Calculate f with Eqs . 4.28 through 4.3 1 . For thiS problem ,
we have elected to use the Iain-Swamee correlation given by Eq.
[
O.0023
l -+
]J -2
2 1 .25
=0.0196.
2.44 1 ( 1 ,974,000)0.9
5. Calculate P wf using Eq. 4.39. We must first calculate the pa­
rameter
log
s
0.0375 'Y gL cos 8 (0.0375)(0.65)(6,8 1 8)
0.3 1 .
(0. 865)(627)
zT
From Eq. 4.39, the BHFP is
s
l
P wf = (2, 175) 2 eO. 3 1
+
x
(6.67 1 0 -4)(0.0196)(6,300) 2 (627) 2 (0.865) 2
(2 .44 1 ) 5 COS 0°
]112
(eO.3 1 - I) =2,656
psia.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.40)
P
=-�
53 . 34T °
6. 67xlO-4fq2g (-TZ) 2
1+
d5
8
P
......................
cos
·
Compute an initial estimate of BHFP.
f=4 2 .28 -4
P
zdp
2.
4.3 1 .
d5
Rearranging and integrating from bottom to top of the wellbore,
we have
Solution.
1.
T,
6.67X lO-4f (-TZ) 2 (qg ) 2 ] dL .
8+
r
.. ..
.
dL .
...
. . (4.41)
Similar t o the Poettmann method for calculating BHSP ' s , Suk­
kar and Cornell ' s method used pseudoreduced pressure, Pp n
zdp
' Pt/
,\
PwJ
0.01875'Y gL cos 8
P
1+
-4fq2 ( Tz ) 2
T
. . . . . . . . . . . . . . . . . . . . (4.42)
6.67 x 10
g
d5p'/;c cos 8 Ppr
.. .. .. .....
·
.
We can simplify this to
zdp
' Prj
0.01 875 'Y gL
P
zy
L I +B (_
Ppr
cos
T
8
. . . . . . . . . (4.43)
where a friction factor term is defined as
B
x -4fqJ T2
6.67 10
d5 pfoc cos 8
.
.
. . . . . . . . . . . . . . . . . . . . . . . . (4.44)
Like Poettmann, Sukkar and Cornell expressed the integral as
z
I' Ppr, »j
dppr
Ppr
\' Ppr,t/
zdpp
r
Ppr
0. 2 l +B( _z_ y 0. 2 l +B (� Y
Ppr
Ppr
+
0.01 875 'Y gL cos 8
, . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.45)
-
T
where 0.2 is an arbitrary constant. The integrals in Eq. 4.45 have
been evaluated numerically and are tabulated in various
GAS FLOW I N WELLBORES
65
18.
references-for example, Ref.
The solution procedure is simi­
lar to the Poettmann method for BHSP.
Cullender and Smith Method. Unlike the two previous methods ,
the Cullender and Smith 6 method makes n o simplifying assump­
tions for the variation of temperature and factor in the wellbore.
To achieve accuracy , the wellbore is divided into two (or more)
segments .
Beginning with Eq.
and separating variables, we can write
z
4. 25
Tz p
p
=-�dL . . . . . . (4. 46)
6.
6
7
x 10 -4f ( TZ ) 2
0 + d5 -P (qg) 2 53.34
4. 46 (Tz/p) 2
�dp
"{ d . . . . . . . . (4. 47)
Tz
=
2
4
i
O
1
x
6.67
( p ) 0 + ------"- 53.�4 L
d5
Tz
2 . Calculate the friction factor term O .
A . Calculate the gas viscosity ,
Evaluate
at flowing well­
head pressure and temperature.
B . Calculate
using Eq. 4 . 2 7 .
C . Depending o n the value o f
calculate f using Eqs . 4 . 2 8
through 4 . 3 1 .
D . Calculate O .
3 . Evaluate the integrand given b y Eq.
a t the wellhead
temperature and pressure conditions . Call this
Compute the midpoint pressure,
of the production string.
JJ.g '
NRe,
NRe
JJ.g
4. 50, I,Itf.
4.
Pmp '
a ·
Pmp =P tf + --Imp+ ltf
Imp = Iptf'mp '
Imp
5.
P wf'
P wf=P mp+ --­
IWf + lmp
IWf= lmpP wf'.
Iwf
6.
6a
P dp
tf
=
wf
+
P
P
----­
Itf + 4lmp+ lwf
Tz
Pwf
=0.01875
L.
.
.
.
(4.
4
8)
g
"{
)
2 0 + -----"6. 67 X 1 O-4 i
6wf'
Ptf ( p
IW
f
P
d5
Tz
4. 48
-d
cos
Dividing the numerator and denominator of the left side of Eq.
by
and rearranging yields
-
-
cos
fq
After integrating the right side and reversing the limits of integra­
tion, we have
f
J
-
fq
cos
Pp
Tz
)
2
6.67 X 1O-4fq2g
Ptf ( p
0
+
d5
Tz
+ (lwf+ lmp ) ( P wf-Pmp ) ' . . . . . . . . . . . . . . . . . . . . . . . . (4. 49)
I
mp, wf,
P
1= --:-:--:)""'Tz2'�--- ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.50)
( Z +0
0,
0= 6. 67 xd510 -4fqi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.51)
improveusingthemore
accuracy
ofespeci
the aCullender
and
Tosuggest
Smi
t
h
method,
we
steps,
lly
wi
t
h
com­
puteri
ons of the calculation technique.
zed versiProcedure.
Calculation
1.
4. 4 8
a.
a=0. 0 1875"{g L
-d
J
-
cos
2
where is the integrand evaluated at the surface, midpoint , or bot­
tomhole conditions (denoted by the subscripts if,
and
re­
spectively) of pressure and temperature and is defined as
The friction factor term,
is
In the following calculation procedure and example, we use a
two-step scheme.
Calculate the quantity on the right side of Eq.
defined as
pmp
P wf
For the initial estimate, assume
Then, iterate with new
Stop when
con­
values of
for each new estimate of
verges.
Use Simpson' s rule to obtain a more accurate value of
It may improve accuracy to iterate on Step
ation values of
from the new values of
not done in the following example .
As in their BHSP calculation technique, the integral on the left
side of Eq.
is approximated with a numerical integration
scheme. Assuming a two-step calculation procedure that considers
an intermediate value of pressure at the midpoint of the production
string , the integral is approximated as
i' P wf
For the initial estimate, assume
Then , iterate with new
values of
for each new estimate of
Stop when
con­
verges.
Compute the BHFP,
of the production string .
P wf'
with improved iter­
However, this is
4. 4 ,
Example 4.5-Calculating BHFP's With the Cullender and
Smith Method. With the same data given in Example
calcu­
late the BHFP using the Cullender and Smith method and assum­
ing a two-step calculation scheme is sufficiently accurate.
1.
4. 4 8.
a =0. 0 1875,,{gL =(0.01875)(0. 65)(6,818) =83. 09.
2.
1, JJ.g =0.NRe,
0 162
20,,{gqg = (20)(0. 65)(6,300) =2,071,100.
NRe=--J.l. g d (0. 0 162)(2,441)
NRe 4,000
4. 2 8 4. 3 1.
4. 3 1.
21. 25 ) ] - 2 =0. 0 195.
f =4 [2. 2 8-4 ( 0.2.0402341 + 2,071,100
0.9
Solution.
Calculated defined by Eq.
Calculate O .
A . First, w e must evaluate the gas viscosity . From correlations
presented in Chap .
cp o
is
B . The Reynolds number,
�
indicates turbulent flowing conditions .
C . Calculate f with Eqs .
through
For this example ,
w e have chosen the Jain-Swamee correlation given b y E q .
log
--
D . The friction factor term is
0= x d510 -4fqi = (6.67 x 10 (2. 441)5
5. 95.
3.
Itf z Ppr.tf=:.3 .26, Tpr.tf= 1. 5 8,
�
ztf4.- 0.815.
4. 50, PItfmp'- 0.169.
Imp =Itf.
a = 2,175 +
P mp =P tf + --Imp + Itf
0.169 + 0.169 =2,421
6 . 67
- 4 ) (0 . 0 1 95)6 , 3002
Evaluate
using the pseudoreduced wellhead pressure and
erature a?d the
factor.
and
Usmg Eq.
we get
Compute the midpoint pressure,
o f the production string.
For this initial estimate , assume
te
83 .09
psia.
66
GAS RESERVOI R E N G I N E E R I N G
Second Iteration.
Using Pmp = 2 ,42 1 psia and 0 = 5 . 99 , we cal­
culate Imp = 0 . 172. Then,
.
� . 09
a
= 2 , 4 1 9 psia.
PmI' = Ptf + --- = 2 , 1 75 +
0 . 1 72 + 0. 1 69
Imp + Itf
Second
I"f + Imp
= 2 ,4 1 9 +
83 . 09
0 . 1 72 + 0 . 1 72
a
l"f + Imp
= 2 ,4 1 9 +
P wf = P tf +
Itf + 4Imp + I wf
6(83 .09)
------
0 . 1 69 + 4(0 . 1 72) + 0 . 175
I
o
o
.
. . . .. .
' . .
>.
. .
�
;::
:.
g
o
. .. . . . .
z
�
o
�
,
'
,"
I ·
Slug
.
) . . '.
g . ' ', '
AnnularSlug
.
.
.' .
Annular-Mist
Transition
Fig. 4 .4-Classification of flow regi mes for two-phase flow
i n vertical production stri ngs (after Orkiszewski 2 0 ) .
uid . Although the liquid slugs always move upward i n the direc­
tion of flow , the liquid film may move upward but possibly at a
lower rate, or it may even move downward . 20 Because of the dif­
ferential liquid movement, the frictional losses vary, and under some
conditions, accumulations of liquid film or liquid
occurs
along the pipe wall . The liquid may also be entrained in the gas
bubbles at high velocities . In slug flow , both the liquid and gas
phases contribute to the nowing pressure gradient.
In annular/slug transition now, the liquid phase becomes discon­
tinuous as the liquid slugs between the gas bubbles disappear. Even­
tually, the gas phase becomes the continuous phase, and a significant
amount of liquid becomes entrained in the gas phase. Although the
liquid phase still affects the flowing pressure behavior, the flow­
ing pressure losses during transition flow are affected primarily by
the gas phase. 8
In annular mist flow, the gas phase is continuous , with liquid
occurring both as entrained droplets in the gas stream and as a liq­
uid film that wets the pipe wall . Generally , the gas velocity moves
the liquid film up the tubing wall. However, the liquid film has
little effect on the flowing pressure losses.
A number of two-phase correlations have been presented . Each
correlation, however, is based on the mechanical energy balance
equation (with w, = 0) in terms of the nuid mixture:
holdup
= 2 , 65 8 psia.
4.3 Effect of Liquids on BHFP Calculations
The methods presented in the previous sections for computing
BHSP ' s and BHFP ' s were developed assuming a single-phase gas
in the wellbore. In practice, liquids (i. e . , water and/or condensate)
often are produced with the gas , and the single-phase calculation
techniques are not adequate for predicting BHP ' s , especially when
significant quantities of liquids are produced . A common technique
for including the effects of liquid production is modification of the
gas gravity term to account for the additional fluid density caused
by the presence of liquids . Generally, this technique is valid only
for producing gas/liquid ratios (GLR ' s) in excess of 1 0,000
scf/STB . 1 9 For lower producing GLR ' s (i . e . , larger volumes of
liquids) , two-phase correlations must be used. In the following sec­
tions , we present several two-phase methods for calculating BHP ' s
in wells producing liquids .
4.3 . 1 Two-Phase Vertical Flow Correlations. In practice, signif­
icant quantities of liquids may be produced with the gas in the flow
string , especially from wells completed in either deep , high­
pressure, gas-condensate reservoirs or dry gas reservoirs under­
lain by water. We cannot use the simple techniques presented in
the previous sections . In these cases, we must use correlations de­
veloped empirically from laboratory or field data to predict the var­
iation of pressure with elevation along the length of the flow string .
According to a study conducted by Orkiszewski, 20 multiphase flow
in vertical pipes is characterized by four different regimes. These
flow regimes (Fig. 4.4) consist of bubble, slug , annular/slug tran­
sition, and annular mist now.
In bubble flow, the liquid phase is continuous with the gas phase,
which exists as bubbles randomly distributed in the fluid mixture .
Under these conditions , the pipe is almost completely filled with
liquid, and the percentage of the gas phase is small. Depending on
their size, the gas bubbles may move at different velocities than
the liquid. Generally , the liquid moves at a uniform velocity . The
difference between liquid and gas velocities often is called the s ip
Except for its effect on fluid density , the gas contributes
little to the flowing pressure gradient of the fluid mixture .
In slug flow . the liquid is still the continuous phase; however,
the gas bubbles coalesce and form stable bubbles that almost fill
the production string but are separated by slugs of liquid . Typical­
ly , the gas bubbles are rounded on their leading edge, relatively
flat on their trailing edge, and surrounded on their sides by a thin ,
liquid film. 8 The gas bubble velocity is greater than that of the liq-
velocity.
.
Bubble
83 . 09
= 2 ,658 psia.
.
0 . 1 75 + 0 . 1 72
= 2 , 1 75 +
10
•
= 2 ,66 1 psia.
Because the latest value of P wf agrees with the initial value, we
have converged to P Wf = 2 , 65 8 psia.
6 . Use Simpson ' s rule to obtain a more accurate estimate of
BHFP.
6a
.
o
�
...
Iteration . Using p wf = 2 ,66 1 psia, and 0 = 5 . 99 , we cal­
culate Iwf = 0 . 1 75 . Then,
P wf = Pmp +
•
.
The latest value of Pmp agrees with the initial value, so we have
converged to PmI' = 2 ,4 1 9 psia.
5. Compute the BHFP, P wf , of the production string . For this
initial estimate , assume IWf = Imp .
P Wf = Pmp +
-
l
vm
g
�
dp + - dZ+ - dv m + -dL = O, . . . . . . . . . . . . . . (4 . 52)
Pm
Pm
1 44
-
which is equivalent to Eq. 4 . 1 except the subscript m denotes mix­
ture properties . The term with gf is used here for the friction loss
term, dF. The definitions of Pm and v m differ slightly , depending
on the correlation. Substituting dZ= cos e dL , we can rewrite Eq.
4 . 52 in terms of a total pressure gradient,
dp
dL
=
(
)
_ !.!!!... � cos & + gf + vm dVm ' . . . . . . . . . . . (4 .53)
1 44
gc
Pm
gc
dL
We have added cos (} to generalize for deviated wells. Note, how­
ever, that the following methods were developed only for vertical
wells , except for the Beggs and Brill method.
Numerical Integration . For two-phase now, we must calculate
BHFP with numerical integration rather than attempt to have aver­
age flow properties over a long distance. Therefore, the methods
discussed here will relate to calculating pressure gradients rather
than pressures. These methods are used repeatedly at different ele­
vations until the desired elevation is reached. This integration usually
goes from top to bottom of the wellbore. Orkiszewski 20 present­
ed a step-wise procedure for the numerical integration that can be
applied to any of the gradient methods. His procedure is as follows.
67
GAS FLOW I N WELLBORES
��I�EII�II
100
\
,
10
�
..:
-
0
U
<IS
I \t
�
!�
1 .0
�
r:::
-
g
.9
u
' r::
�
0.0
r:::
0. 1 0
l"
"
�Lii:
0.0 1
1=
ffff-
I< 0
•
" r\
Flowing wells
- G a s l i f t w ell s '
- Bureau of
M i n e s d a t a +-+++1-1'1:1
1I
1I I UL
1 1 ..u..
I I I ----I.1 �u..u.u
.
.L-.
1 .,L
1 J..l.
-I I�
...L I 1U,.i.
1 1 ll....J L.
o . a a I '---.I'-
0.1
�
1 .0
p,.v.,d'
10
100
Fig. 4.5-Correlation for Fanning friction factor for vertical, two­
phase flow (after Kermit E. Brown's The Technology of Artifi­
cial Lift Methods, Volume 1 , Copyright Pennwell Books, 1 977).
1.
Pick a point in the flow string (e. g . , wellhead or bottomhole)
where the flow rates , fluid properties , temperature, and pressure
are known.
2. Estimate the temperature gradient of the well.
3 . Fix the tJ.p at about
of the measured or previously cal­
culated pressure, which may be at either the top or bottom of the
increment. Find the average pressure of the increment.
Assume a depth increment, tJ. Z, and find average depth of
increment. Calculate tJ. L = tJ. Z/cos (J .
From the temperature gradient, determine average tempera­
ture of increment.
Calculate pressure gradient, tJ.p/tJ.L, from one of the two-phase
flow methods .
7 . Calculate tJ.L from the fixed tJ.p and the calculated pressure
gradient.
8 . Iterate if necessary , starting with Step until the calculated
tJ.L equals the assumed tJ. L .
9. Determine the value of p and L for that increment from tJ. p
and tJ. L .
Repeat the procedure from Step 3 until the sum of tJ.L equals
the total length of the flow string .
These steps are easy to program and fast to compute . If the ac­
curacy of integration is in question , take smaller increments to im­
prove accuracy . With this incremental approach, it is certainly
possible that the deviation angle, (J, and flow diameter may vary
with depth. Thus, this incremental approach allows a very general
approach for a complicated wellbore.
We will now turn our attention to the different methods for cal­
culating pressure gradients. In his review , Orkiszewski 20 identi­
fied three categories of correlations based on similarities in
assumptions and theoretical concepts used to develop the correla­
tions and evaluate the effects of each pressure gradient term. In
the following sections, we present an example of each category .
Fluid Properties. The two important phases are considered to
be liquid and gas . The oil and water phases are combined for the
10%
4.
5.
6.
4
10.
60 1,5,500000
56 11 ,000
o
'c
a
�
liquid phase. We assume that the oil and water phases travel at the
same velocity and are lumped together into liquid flow parame­
ters . Oil and water properties are determined at a particular pres­
sure and temperature. These fluid properties include Rs ' Bo ' Po '
IL o ' (lo , Bg , P g , IL g ' B w ' Pw' P w ' and (lw · Liquid properties are
then taken to De volumetric averages of oil and water properties .
Poettmann-Carpenter Method. The Poettmann-Carpenter 2 1 cor­
relation is a semiempirical approach based on actual field data taken
from many flowing and gas-lift oil wells . This method typifies corre­
lations that do not require recognition of flow regimes. Other corre­
lations in this category include work by Baxendell and Thomas , 22
Tek, 23 and Fancher and Brown. 24 The correlation was developed
with a wide range of data that included nominal tubing diameters
of 2 , 2 V2 , and 3 in . ; producing GLR ' s up to
scf/bbl of total
liquid; total liquid production rates from to
STB/D; produc­
ing WaR' s up to
STB/STB ; oil gravities between 30 and
API ; and well depths to
ft .
With these field data, they correlated the combined energy loss­
es resulting from liquid holdup, frictional effects , and other ener­
gy losses in terms of a friction factor. Rather than evaluating the
various components making up the total energy loss , the flowing
fluid was considered to be a single, homogeneous mass. Mixture
density was assumed to be a composite density of the fluids pro­
duced at the surface but corrected for BHP and temperature condi­
tions . The correlation does , however, consider mass transfer
between phases as fluid flows up the production string . The liquid
holdup and frictional losses are not specifically evaluated separately
but are correlated with an empirical friction factor. Finally, the cor­
relation assumes that flow is completely turbulent. Because viscosity
effects often are negligible during turbulent flow , viscosity is not
used as a correlating factor.
Poettmann and Carpenter presented the flowing pressure gradient
in terms of the Fanning friction factor.
tJ.p
tJ.L
=
54
_ _144I_ [p m (J + 39.4(f7F2P) mmWr' 5 ] . . .
d
.
cos
.
.
.
.
.
.
.
. (4.54)
.
We have taken g = gc and have included (J to generalize for deviated
wells. However, this method was developed only for vertical wells .
(Step of the following calculation proce­
dure was adapted from Orkiszewski ' s 20 work . )
Calculate the values o f flow rates and densities a t flow condi­
tions in the desired units from field units. This accounts for dis­
solved gas .
Calculation Procedure. 1
1.
qi =6.49 x 1O- 5 (qoBo +qwBw)' . . . . . . . . . . . . . . . . . . (4. 5 5)
zT
x 1O- 7 ( I , OOOqg -qo Rs )-' . . . . . . . . . . . . . . (4. 5 6)
�
q;=q{ +q� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4 5 )
x 10 - 7 Rs qo 'Y g '
wL =4.05 x 10 - 3 (qo 'Yo
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 5 8)
Wg x 1O- 7 (1,000qg- Rs qohg
(4. 59)
Wr=WL + wg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 60)
pm =w/qr' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.61)
q = 3 .27
p
. 7
+ qw 'Yw ) + 8 . 85
= 8 . 85
•
.
.
.
.
.
.
.
•
•
.
.
•
.
.
•
2 . Calculate the group o f terms P m vm d' , for use i n Fig. 4 . 5 (as
the x axis) .
from Fig.
3 . Find the Fanning friction factor,
Calculate the pressure gradient from Eq.
( fF) m , 4. 54. 4.5.
4.
Example 4.6-Estimating Two-Phase B HFP Gradient Using the
Poettmann-Carpenter Correlation. The following data were taken
from a well producing both water and gas . Estimate the flowing
pressure gradient (for the first increment) using the Poettmann­
Carpenter correlation .
'Ygqg 0.200
1006 5.
=
=
qw =
MscflD .
BID .
68
GAS RESERVO I R E N G I N E E R I N G
0.05
,
-l----/--;�V
----l
0.6 f--�---!-/
�-l------l
1 .0
0.8 I--
CNL
0.01
0.4
Correta1ion based on:
Tubing Sizes: 1 in
in.
Viscosities: 0.86 cp . 1 1 0 cp
..2
1-----+------4��---I-----1
--:::;
0.2 1-___-+��/l7
=--+-
0.01
0.001
0. 1
NL
Fig . 4 .6-Correlation for viscosity n u m ber coefficient, C NL
for Hagedorn-Brown method (after Economides, H i l l , EhligEconomides, Petroleum Production Systems, © 1 99 3 , pp.
1 57, 1 58 . 32 Reprinted by permission of Prentice Hall, Upper
Saddle River, New Jersey).
PwL == 5,63.00002
Pdrj == 1,1. 099500
Trj8 == 0°.70°F.
TwJ = 150°F.
Ibm/ft 3 .
ft .
psia.
in. (flow through tubing) .
L..���...L�__...._
..l.
�_�.L-_�_...J
10.6
1 0.5
10"
1 0.3
10"
Fig. 4 . 7-Holdup factor correlation (after Economides, H i l i ,
Ehlig-Economides, Petroleum Production Systems, © 1 993,
pp. 1 57, 1 58 . 32 Reprinted by permission of Prentice Hal l , U p­
per Saddle River, New Jersey).
-0.15
liquid
Hagedorn-Brown Method. The second category of vertical , two­
I1p,
I1p=
100
I1L= l1p/gJ = 100/( -0.15) = -666. 6 -0.15
5,000]( -666.{p=6)1,=81000 + 100/2= 1,050 z=0.84.T=70 +[(70-50)/
1.
(qo =0, Bw = 1. 0).
qi =6. 49x 1O -5(100x 1)=6.50x 10 -3
q;=3. 27X 1O- 7 (0.84)(1,000x200-0) --1, 050
=27. 99xlO -3
wt=4. 05 x 10 -3 [0 + (100) ( 6:��:) ] + 8.85 X 10- 7
x( I ,OOO x 200)(0. 65) =0. 5241
0. 5241 --- =15. 20 Ibm/ft -3 .
---P m = -qL,--,
+qg (6. 5 0 + 27. 99) x 10- 3
pmvmd'
2.
4.5.
4 (1. 995/12)
Pm vm d'= � d'=(0. 5241) 11'( 1. 9 95112)
2
=4. 0 14
(fF) m ,
4.5.
(fF)4.3. m =0. 60.
4. 54.
4(0. 60)(0. 5241) 2 -- l
I1p = --1 [(15.20) + ---I1L 144
39. 72(15.20)(1. 995112) 2
= -0.165
ft 3 /sec .
(75 + 460)
ft 3 /sec .
Ibm/sec .
WI
for use in Fig .
W
A
Ibm/ft-sec .
Read the Fanning friction factor,
Calculate the pressure gradient using Eq.
psi/ft .
--1
--
Because this pressure gradient is slightly different than the
assumed gradient of
psi/ft, we may choose to iterate on this
pressure increment.
Solution. W e begin b y assuming a pressure increment,
and
a corresponding length increment, 11L. Assume
psia.
We will assume a pressure gradient of
psi/ft . Then,
ft.
Then the average pressure and temperature of the average depth
increment
psia and
OF} .
Determine the gas deviation factor,
Now , determine the pressure gradient .
Calculate the values of rates and densities at flow conditions
in the desired units from field units . This accounts for dissolved
gas
assume
Calculate the group
0.0
1
0.001
-+-
--
from Fig .
holdup
phase flow correlation is characterized by the inclusion of
resulting from differences between gas and liquid veloci­
ties in the production string . In general, liquid holdup is either cor­
related separately or combined in some manner with the friction
losses. However, similar to the Poettmann-Carpenter method, this
category of correlation does not distinguish individual flow regimes .
Unlike the Poettmann-Carpenter method , the Hagedorn­
Brown 2 5 correlation was developed specifically to include liquid
viscosity effects . A
experimental well was constructed to
study the pressure gradients in tubing diameters ranging from
to
in. The tests were conducted for wide ranges of liquid flow
rates, producing GLR ' s , and liquid viscosities . For this study , the
liquid holdup was not measured directly but was used as a correlating
factor to make the experimental and calculated results match .
Hagedorn and Brown ' s pressure gradient equation is similar to
the Poettmann-Carpenter method , except the inertial term is re­
tained . Again , we have taken
and have included () to gener­
alize for deviated wells . However, this method was developed only
for vertical wells.
1. 5
1 ,500-ft
1
g = gc
I1p = --1 [Pm 8 + fm wr + pm l1(vm/2gc) 2 ] .
39. 72Pmd' 5
I1L
I1L 144
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 62)
(fp)m
fm , f=4(1) fF), (2)
(3)
Pm
cos
However,
they use the Moody friction factor of the mixture,
rather than the Fanning friction factor of the mixture
(remember
they include an inertial term, and
their
method of determining friction factor and
is completely differ­
ent than the Poettmann-Carpenter method. They used the concept
of liquid holdup to account for liquid accumulation in the pipe. The
holdup factor, Hr o is defined as the volumetric fraction of liquid
at a point in the flow string .
They defined four dimensionless variables in their correlation
method and based their determination of liquid holdup factor on
these four dimensionless variables . The liquid velocity number is
NLv= 1. 938vSL �PL/UL ' . . . . . . . . . . . . . . . . . . . . . . . . . (4. 63)
Ngv=1. 938vSg �PL/UL ' . . . . . . . . . . . . . . . . . . . . . . . . . (4. 64)
Nd,=120.872d'-JPL/UL , . . . . . . . . . . . . . . . . . . . . . . . . . (4. 65)
The gas velocity number is
The pipe diameter number is
69
GAS FLOW I N WELLBORES
2 . 0 ,-----,
1 .8
4. Read the value of CNL from Fig. 4.6.
5.
Find the value of HL N.
A . Calculate the value o f the correlating function (the x axis of
Fig. 4.7) as follows:
(NL vfN2v5 7 5 )( pIPsc)O. 1 O ( CNL INd' ) '
4.
B . Read the value o f HL N from Fig. 7 .
Find the liquid holdup factor, HL .
A . Calculate a value of the correlating function (the x axis of Fig.
6.
4.8) :
1 .2
'''_-'-_...._..J...
..
_-'--_.1.....
____'-----'_....
1 .0 '-----'''''---'0.00
0.0 1
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.C9 0.10
NrJiL/J.JIfJIN/"
j\J . 3 8 0
INd2". 1 4
NgV" LO
4.8.
HL = (HLN) x if; . . . . . . . . . . . . . . . . . . . . . . .
B . Read the value of if; from Fig .
C . Multiply the value of HLN obtained in Step 5B by
tain the value for HL .
.
F i g . 4 .8-Correlatlon for secondary correction factor for
Hagedorn-Brown method (after Economides, H i l l , Ehlig­
Economldes, Petroleum Production Systems, © 1 993, pp.
1 57 , 1 58 . 32 Reprinted by permission of Prentice Hall, Upper
Saddle River, New Jersey) .
.
.
.
.
..
.
if; to ob.
.
(4. 70)
7. Find the value of the friction factor , 1m .
A . Calculate mixture density and vicosity of the mixture as
follows:
. . . . . . . . . . . . . . . . . . (4.71)
. . . . . . . . . . . . . . . . . . . . (4. 72)
P m =PLHL + p g ( l -HL)
and p.m = p.J!L p. j - HL
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
B. Calculate the two-phase Reynolds number.
1 , 4 88Pm vm d'
=0. 15726p.L � l IpLC1l ' . . . . . . . . . . . . . . . . . . . . . . . (4. 66)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 73)
Calculation
Procedure.
P. m
4. 55
e/d,
1m
4.60.2.1 .
8.
!J.p/!J.L
4. 62.
v SL = qiJA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 6 7)
VSg =q; /A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 6 8)
v m = vSL + vsg , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 6 9)
AL ' HL < AL ,
HL
AL '
3. 4. 66.
4. 63
AL
and the liquid viscosity number is
NL
NRe =
Calculate the values of in-situ rates from Eqs .
through
Calculate superficial velocities.
Calculate the four dimensionless variables defined in Eqs .
through
C . Use NRe and roughness ratio,
from Fig, 4.9.
Calculate
from Eq.
to obtain a value for
Modified Hagedorn and Brown Method. The Hagedorn and
Brown method has been modified to improve its accuracy . Accord­
ing to Brown, 26 these suggested changes were made by Brill and
Hagedorn. The first modification involves the calculation of a " no­
slip" holdup factor,
If
then
is set equal to
The value of
is given by
�
..:
0
....
0
Cd
�
�
c:
.....
2
.
vi'
0
;:
�
\
\
\
�
-t::
�
I
�
E-c
00 4
\
(1)
.02
.01 5
\
\
\
\
\
\
Vl
(1)
s::
-t::
OJ)
::l
0
,:x:
(1)
.001
.�
.00 0 8 C<l
.006
. 0004 Q3
,:x:
.0002
\
SM OOTH PI P E S
. 000 1
.000,05
Two-Phase Reynolds Number,
NRe
=
1 ,4 8 8 Pm Vm d'
Ilm
Fig . 4.9-Frictlon factor correlation (after Kermit E. Brow n ' s The Technology of Artificial Lift Methods, Volume 1 , Copyright
Pennwe l l Books, 1 977).
70
GAS RESERVOI R E N G I N E E R I N G
SEGREGATED
100
Ll
--- - --
"
"
-- Or i g i nal up
up
-- -Revised
Stra t i fi ed
-::-��:
I
"
-:=
Wa vy
"
"
,
,
"
L
F l o. "s1o,.
�
- ---- - '.-.,. -.,.,.. - -..... _ - .,------.... �.. - � - ... -
:.:r--::- �
,
,
,
2 ',
Segregated
I n t. ... l tt.nt
1�1
��:!�l�:d
0 . 1b';===:;;:=::=::;.':;;0.000 1
0.001
0.01
I nput l iquid tont.nt .
,
,
',
,
,
'
IV �
,
L3
,
--;;'0 �
. 1 �'�...\.�'�1 .0
"
___
A
L
Fig. 4 . 1 1 -Horizontal flow-pattern map for Beggs-Bril l method
(after Kermit E. Brow n ' s The Technology of Artificial Lift
Methods, Volume 1 , Copyright Pennwell Books, 1 977) .
[
I NTERM I TTENT
:.,/
]
ity and liquid velocity. Griffith recommends using the following:
Vs =0.8 ft/sec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.77)
2. The mixture density is calculated with Eg. 4.7 1 , using the new
value of HL .
3 . Because the continuous phase in bubble flow is the liquid, the
friction loss gradient, gf, is based on the flow of a single-phase
liquid . The liquid velocity, vL , is corrected for gas voidage:
Pl ug
.
1---- ..-
J( )
1
vm
vm 2 4vs
HL = I - - 1 + - - 1 + - - -g , . . . . . . . . (4.76)
2
Vs
Vs
Vs
where Vs is the "slip velocity, " the difference between gas veloc­
Annu l a r
. . �"
"' . ' v . '"
. .
.
•
. ..... .
t·
- -
Sl ug
D I ST R I BUT E D
f'\... a .. ,...----., .. ,...,.... . .. C:J " ..
��. �-!.!
",:--,
, .. .. --- -= - -- -- - - - - - ---- - -
-
Bubb l e
Fig. 4 . 1 0-Horizontal flow patterns for Beggs-Bril l method (af­
ter Kermit E. Brow n ' s The Technology of Artificial Lift
Methods, Volume 1 , Copyright Pennwe l l Books, 1 977).
AL = qi/q; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.74)
The second modification is to use the Griffith 27 method for bub­
ble flow. Griffith and Wallis 28 defined the conditions for bubble
flow. First, the bubble flow parameter, LB, is calculated accord­
ing to
vL = vsLIHL · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.78)
4. 1m is obtained from a Moody friction factor chart (Fig. 4.2)
as a function of NRe , defined by
1 ,488PLd'vL
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.79)
NRe =
P-L
5 . The pressure gradient is then calculated with Eq. 4.62, using
wL for and PL for P m in the friction gradient term.
Wt
To illustrate the third category of two-phase
correlations, we selected the method proposed by Orkiszewski20 ;
however, other correlations2 7-3 l are available. The distinguishing
characteristics of the third category of vertical, two-phase correla­
tions are ( 1 ) the inclusion of correlations for identifying the flow
regime and (2) separate correlations for estimating the liquid holdup
and friction factors as a function of the flow regime. Generally,
the friction factor is determined from the properties of the continu­
ous phase. The continuous phase varies with the flow regime. For
example, during bubble flow the liquid phase is the continuous
phase; during mist flow, the gaseous phase is continuous.
In a detailed study of two-phase correlations, Orkiszewski com­
pared pressure behavior predicted from various correlations with
actual field data and concluded that no correlation was sufficiently
accurate for all flow regimes. He did, however, identify several
correlations that were accurate for a specific flow regime. In sum­
mary, Orkiszewski suggested using ( 1 ) the Griffith 27 method for
bubble flow, (2) a modification of the Griffith-Wallis28 method for
slug flow, (3) a combination of the modified Griffith-Wallis and
the Duns-Ros 3 l methods for annular/slug flow, and (4) a modi­
fied Duns-Ros method for mist flow. The form of the equation pro­
posed by Orkiszewski for calculating flowing pressure gradient is
Orkiszewski Method.
(
)
1
P m cos O + gf
f:.. p
= _
, . . . . . . . . . . . . . . . . . (4. 80)
q;
144
f:.. L
wt
1 - ---"-4,637A 2 p
If it is determined that bubble flow exists, then the following steps
where P m ' the density of the fluid mixture, and gf , the friction loss
are used.
gradient, depend on the flow regime. Again, we have taken g = gc
1 . The holdup factor is now calculated as
LB = 1 .07 1 -0.22 1 8v�/d' , . . . . . . . . . . . . . . . . . . . . . . . . (4.75)
with a lower limit of 0. 1 3 for LB' Bubble flow exists if q; /q; < LB'
_ _
71
GAS FLOW I N WELLBORES
TABLE 4 . 1 -CONSTANTS FOR BEGGS AND BRILL M ETHOD
a
b
c
Segregated
I nterm ittent
Distributed
0 .98
0 .845
1 . 065
0.4846
0 .535 1
0 . 5824
0.0868
0 . 01 73
0 . 0609
Segregated u p h i l l
I nterm ittent u p h i l l
Distributed u p h i l l
All regimes down h i l l
0.01 1
2 . 96
Flow Regime
d
g
e
- 3. 768
3 . 539
- 1 .614
0 .0978
- 0 . 4473
0 . 305
No Correction C 0
0 . 1 244 - 0. 5056
- 0. 3692
=
4 . 70
and have included 0 to generalize for deviated wells. However, this
method was only developed for vertical wells.
By using variations of existing methods, Orkiszewski covered
the spectrum of the four flow regimes he used. His method was
validated by comparison with 148 measured pressure drops in ac­
tual wells, including 22 new measured pressure drops in actual wells.
However, his method was limited to vertical wells. Today, highly
deviated and horizontal wells have become common. The most com­
mon method for these wells, and perhaps the most versatile method '
is the Beggs-Brill method.
Beggs and Brill Method. The Beggs-BrilP O method may have
become the most widely used method because it is not limited to
vertical wells. It can be used not only for highly deviated wells but
also for surface flowlines. The Beggs-Brill method was developed
from experimental data obtained in a small scale experimental test
facility. The facility consisted of two 90-ft. sections of acrylic pipe
that had 1 .0- and l .5-in. diameters, respectively. The pipe sections
could be positioned at any inclination angle. The parameters studied
and their ranges of variation were ( 1 ) gas flow rate, 0 to 300
MscflD; (2) liquid flow rate, 0 to 1 ,000 STBID; (3) average sys­
tem pressure, 35 to 95 psia; (4) pipe diameter, 1 .0 and 1 .5 in. ;
(5) liquid holdup, 0 to 0.87; (6) pressure gradient, 0 to 0.8 psia/ft;
(7) inclination angle (from horizontal) , - 90 through + 90 ° ; and
(8) horizontal flow regime (segregated, intermittent, and distrib­
uted) . The fluids used in the empirical study were air and water.
For each pipe size, liquid and gas rates were varied so that all
flow patterns were observed when the pipe was horizontal. After
a particular set of flow rates was specified, the inclination was varied
throughout the complete range of angles so that the effect of incli­
nation angle on liquid holdup and pressure gradient could be ob­
served. Liquid holdup and pressure gradient were measured at angles
of 0, ±5, ± 10, ± 15, ± 20, ± 35, ±55, ± 75 and ± 90 ° . The corre­
lations were developed from a total of 584 measured flow tests.
Different correlations for liquid holdup are presented for each
of three horizontal flow regimes. The liquid holdup for horizontal
flow is calculated; then a correction is made for the pipe inclina­
tion angle. The horizontal flow regimes are illustrated in Fig.4.10.
Their original flow-pattern map was later modified to include a tran­
sition zone between the segregated and intermittent flow re­
gimes. 26 The modified flow pattern map is shown with the original
flow pattern map in Fig.4. 1 1 . A two-phase friction factor is calcu­
lated with equations that are independent of flow regime but de­
pend on liquid holdup. Pressure gradient is calculated with Eq. 4.80.
The two-phase density of the mixture, P m ' and two-phase fric­
tion gradient, gf' depend on flow regime. The definition of the
term wt differs from that used by Poettman-Carpenter, Hagedorn
and Brown, and Orkiszewski . Those authors used wt as the total
mass flow rate in units of Ibm/sec. For the Beggs-Brill correlation,
wt = pm v rnA , where, P m i� the volume-in-place weighted average
densIty. But Beggs and Bnll use the same definition for superficial
mixture flow velocity, Vm (Eq. 4.69) .
Calculation Procedure.
1 . Calculate in-situ flow rates with Eqs . 4 .55 through 4.57.
2 . Calculate dimensionless flow regime parameters. Superficial
velocities are calculated from Eqs. 4.67 through 4.69 . A no-slip
holdup factor, f..L is calculated with Eq. 4.74. Then the Froude
number, NFR , is calculated according to
NFR = vJ, /gd' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 8 1 )
I
3 . Determine the flow regime. The values o f the dimensionless
parameters, f..L and NFR are used for the modified determination
of flow regime, as seen in Fig. 4. 1 1 . Following are the steps to
determine flow regime: The first step is to determine flow regime
limits, L l o L2 , L 3 , and L4 ·
L ] = 3 16. f..2 - 302 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 82)
L 2 =0.0009252f..L- 2 4684 , . . . . . . . . . . . . . . . . . . . . . . . . . (4. 83)
L 3 =O. l Of..L- 1 . 45 ] 6 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 84)
and L4 =0.5f..L-6. 73 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 85)
Then the flow regime is determined by the following rules (this
is equivalent to the regions separated by straight lines in Fig. 4. 1 1 ) .
Segregated: f..L < 0.01 and NFR < L ] ,
or f..L � 0.01 and NFR < L 2 ·
Transition:
f..L � 0.01 and L 2 < NFR ::; L 3 .
Intermittent: 0.01 ::; f..L < 0.4 and L 3 < NFR ::; L ] ,
or f..L � O.4 and L 3 <NFR ::; L4 .
Distributed: f..L < 0.4 and NFR �L I ,
or f..L � 0.4 and NFR > L4 '
When the transition region is encountered, the liquid holdup for
the segregated and intermittent regimes (both corrected for incli­
nation angle) are averaged with the equation below. The actual
values for liquid holdup of the segregated and intermittent regimes
are calculated as described in the next two subsections.
Transition averating :
HLO, transition = (1 -B) HLO, segregated + BHL O, intermittent ' . , . (4. 86)
where, B= 1 - (L 3 - NFR )/(L 3 -L2) . . . . . . . . . . . . . . . . . . . . (4. 87)
4. Determine horizontal liquid holdup. Eq. 4.88 is used to cal­
culate liquid holdup for all flow regimes under horizontal flow con­
ditions, HLO O ' Values of a, b, and c are determined from Table
4.1.
HLoo =
a (f..L ) b
-; HLO o � f.. L ' . . . . . . . . . . . . . . . . . . . . . . . (4. 88)
(NFR) C
5 . Correct the liquid holdup for inclination angle. The horizon­
tal liquid holdup is corrected for the actual inclination angle. The
following method is used. Note that 0 is the inclination angle from
vertical, while 0: is the inclination angle from horizontal.
HLO = t/;HLO ° , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 89)
where the inclination correction factor is
t/; = 1 + C[sin( 1 . 80:) -0.333 sin 3 ( 1 . 80:)] , 0: = 90 ° - 0 . . , . (4.90)
C is a coefficient determined by
C= ( 1 - f..L )ln(df..[.N{vN�0 , C � O , . . . . . . . . . . . . . . . . . (4.91)
and NLV is the liquid velocity number,
PL 0 . 25
. . . . . . . . . . . . . . . . . . . . . . . . . (4.92)
NL V= 1 .938v,L
uL
(d, e, i, and g are also from Table 4. 1 .)
6. Calculate the two-phase density. The two-phase mixture den­
sity is calculated from
P m = PL HLO + P g ( 1 - HLO ) ' . . . . . . . . . . . . . . . . . . . . . . . . (4. 93)
7. Determine the two-phase friction factor. A two-phase friction
f�ctor, ftp , is u �ed in calculating the friction pressure-loss gradient.
FlfSt, the no-slip Reynolds number, NRenp is used to calculate the
no-slip friction factor (Moody), fns ' The two-phase friction factor
ratio is then used to correct for the two-phase friction factor value.
( P L v sL + p g v sg ) d'
. . . . . . . . . . . . . . . . . . . . . . . . ( 4.94)
NRens =
( )
Jl m
and in , = { 2 log [NRens /(4.5223 log NRen;' - 3 . 82 1 5)] } - 2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.95)
72
GAS RESERVO I R E N G I N E E R I N G
PRESS U R E .
1 60 P S I
100
PSI
VERTICAL
56
FLOWING PRESSURE GRADIENTS
(OIL PERCENT - 50)
TUBING SIZE . 2.44'-IN. 10
PRODUCTION RATE . ' .000 BLiD
• 0.65
FLOWING TEMPERATURE
OI L AP1 GRAVITY - 35.0 AP1
GAS SPECIFIC GRAVITY
AVERAGE
. '5O"F
WATER SPECIFIC GRAVITY • • • 07
GIVEN DATA:
211o-tN. 00 TUBING (2.44'·IN. 10)
LIOUID FLOW RATE . ' .000 BID (50% WATER)
DEPTH · '2.000 FT
PRODUCING GOR
•
800 SCFIBBL
•
PROOUCING GLR • 800 /2
400 SCF/BBL
W E L L H EAO PRESSURE • • 60 PSI
8
FINO THE FLOWING BOTTOM·HOLE PRESSURE.
o
m
-0
-i
::r:
- 10
o
,.
is at a depth of 1 .400 '1. Note the pressure scale
u n t i l i n tersecting the 4OD-scflbbl GLR fine. This
IS I n eO-psi increments and the depth scale is in
o
o
"T1
-i
Find Ihe equivalen, deplh correspondino 10 .60
psi wellhead pressure. To do this. proceed
vertically downward from 1 60 psi at zero depth
2. Add I h e equivalenl deplh 01 1 .400 11 10 Ihe well
deplh 01 '2.000 II and oblain 1 3.400 II.
l00-fl i n c rements.
3, From 13.400 II on Ihe vertical sc.'e. proceed
12
horizontally to the "00 scf/bbl line and read a
flowing pressure of 3 .360 psi.
o
14
50
1 00
16
18
F i g . 4 . 1 2-Example of pressure traverse curves for a flowing well (after Kermit E. Brow n ' s
The Technology of Artificial Lift Methods, Vo lume 4 , Copyright Pennwe l l Books, 1 984) .
Now calculate the parameter S .
S=
In(y)
-------
-0.0523 + 3 . 1 82 In(y) -0. 8725[ln(y)] 2 +0.01853 [ln(y)] 4
y ::5 1 .0,y ;:d .2, . . . . . . . . . . . . . . . . . . . . . . . . (4.96)
where y = ALlHro , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4.97)
or S= ln(2 .2y - 1 .2), 1 .0 < y < 1 .2 . . . . . . . . . . . . . . . . . . . . (4. 98)
Finally, fm =fns e s . . . . . . . . . . . . . . . . . . . . . (4. 99)
8. All the parameters for the pressure gradient equation, Eq. 4.80,
and the pressure gradient are calculated.
The Beggs-Brill method was determined to be the best empirical
correlation for use with directional wells 3 2 because ( 1 ) it was de­
veloped from extensive laboratory experiments, including varia­
tion in inclination angle (584 flow tests) ; (2) it applies for any
inclination angle; and (3) it is well accepted (in terms of accuracy)
by the industry for use with directional wells.
.
.
.
. .
Pres­
sure traverse or gradient curves are an alternative to the tedious
calculations required for most two-phase correlations. These curves
are graphical solutions or results from the two-phase flow correla­
tions and usually include potential and kinetic energy effects, fric­
tion losses, and variations in wellhead pressure. However, a
particular traverse curve represents only a given flowing situation
and calculation procedure.
Fig. 4. 12 shows an example 33 of a traverse curve. The
Hagedorn-Brown method was used to compute this set of curves.
This set was prepared for a well with tubing ID of 2 . 44 1 in. and
a liquid production rate of 1 ,000 STBID. The liquid is one-half
water (/' w = 1 .07) and one-half oil (0 API = 35 .0) . Various GOR's
are shown from 0 to 3 ,000 scf/STB with /' g =0.65 . The average
flowing temperature is 1500P. If this particular set of traverse curves
approximates a well of interest, then they provide a very simple
calculation of BHFP. Example 4.7, taken from Ref. 33, shows how
to use these curves.
4.3.2 Pressure Traverse Curves for Vertical Gas Flow.
.
.
.
.
. .
.
. . .
.
.
.
GAS FLOW I N WELLBORES
73
=--- SALES
LI N E
GAS
LIQU I D
lI P I
!:>P7
°
Pwf - Pw h
A PZ
A P3
IN
MEDIUM
o
Pr - Pwfs
•
LOSS
•
Pwfs Pwf
-
•
LOSS ACROSS COMP LET ION
lI P4
•
AP
S
•
A Ps
POROUS
PUR - POR
P
P
USy - OSy
REST R I C T I O N
SAFETY VALVE
•
Pwh- Pose '
P
P
ose- sep
=
A P7
•
Pw f - Pw h
•
A Ps
•
Pw h - Psep
=
S U RFACE C H O K E
IN
FLOW L I NE
TOTAL LOSS IN T U B I N G
"
FLOW L I N E
F i g . 4 . 1 3-Schematic showing pressure losses I n a gas-well production system (after Ker­
mit E. Brown's The Technology of Artificial Lift Methods, Volume 4, Copyright Pennwell Books,
1 984) .
Example 4.7-Calculating BHFP With Pressure Traverse
Curves.
'Y g =
P t! =
L ==
qo
qw
qg
()
dT
=
=
=
=
=
0 . 65 .
1 60 psia.
12,000 ft.
500 STBID .
500 STB/D .
400 MscflD .
0 ° (vertical well) .
2 . 44 1 inches. (flow through tubing) .
1 50 ° F .
Solution (see Fig.
4. 12) .
1 . Find the equivalent depth corresponding to 1 60-psia wellhead
pressure. To do this, proceed vertically downward from 1 60 psia
at zero depth until intersecting the 400 scflbbl line. This is at a depth
of 1 ,400 ft .
2 . Add the equivalent depth o f 1 ,400 ft to the well depth o f 12,000
ft and obtain 1 3 ,400 ft .
3 . From 1 3 ,400 ft o n the vertical scale, proceed horizontally to
the 400 scf/bbl line and read a BHFP of 3 ,360 psia.
4.4 Evaluating Gas-Well P roduction Performance
In this section, we discuss factors affecting the production perform­
ance of gas wells . The performance characteristics of a gas well
can be analyzed as two components, inflow and outflow perform­
ance. The inflow performance, or deliverability , is a measure of
the reservoir ' s ability to produce gas to the wellbore. The inflow
performance relation (IPR) is used to describe the relationship be­
tween gas production rate and BHFP. The IPR is controlled by reser­
voir rock and fluid properties, including near-wellbore effects and
heterogeneities in the drainage area of the well, average reservoir
pressure, and field development practices . Chap. 7 gives methods
for determining the IPR of gas wells from field tests . In this chap­
ter, only stabilized , or pseudo-steady-state, flow is considered, not
transient flow .
The outflow performance involves flow from bottomhole condi­
tions to the surface, usually through tubing . As shown in Fig. 4.13,
a schematic of a gas-well production system, outflow performance
may include various wellbore restrictions and surface equipment.
The presence of liquids in the wellbore increases not only the fric­
tion losses but also the flowing pressure gradient required to lift
the denser liquids. In this chapter, only outflow performance through
the tubing is considered .
4 . 4 . 1 Four-Point Deliverability Test. A number of testing tech­
niques have been developed to assess gas-well deliverability or in­
flow performance characteristics . Although detailed discussions of
these techniques are presented in Chap. 7, here we briefly sum­
marize deliverability test analysis, especially for developing inflow
performance curves . The most common well test, the convention­
al backpressure test, was proposed by Rawlins and Schellhardt 34
in 1 935. They observed that, when gas production rates are plot­
ted v s . the square of the difference between the average reservoir
pressure and the BHFP on log-log coordinates, the relationship is
represented by a straight line given by
qg = ( 1 ,000C)( p2 -p aj) n , . . . . . . . . . . . . . . . . . . . . . . . (4. 100)
where C is the stabilized performance constant and n is the reciprocal
of the slope of the line when log( p2 -P t.t) is plotted vs. log qg . Ex­
trapolation of the straight line to the square of the pressure difference
evaluated at PI1j = 14.7 psia defines the absolute open-flow (AOF)
potential of the well . Fig. 4.14 shows an example of a four-point
deliverability test. Theoretically, the AOF is the rate at which the
well could produce if the BHFP were maintained at atmospheric pres­
sure. In practice, the well cannot produce at this pressure. However,
the AOF is a common measure of well deliverability and often is
used by U.S. regulatory agencies to establish field proration schedules
and to set maximum allowable production rates for individual wells.
Rawlins and Schellhardt 3 4 dealt with relatively shallow and sim­
ple wells in 1935 . Their simplified theory indicated that slope , n,
was directly related to turbulence, or non-Darcy flow . The inverse
of the slope, n, may range in value from 0 . 5 , indicating fully tur- .
bulent flow, to 1 . 0, indicating laminar flow. However, current the­
ory and field experience show that Eq. 4. 1 00 is not adequate for
many actual wells . Corbett and Wattenbarger 3 5 showed that the
log-log plot is not a straight line and the values of C and n depend ,
in part, on gas properties that change with time as the reservoir
74
GAS RESERVOI R E N G I N E E R I N G
..
..0.,
..
-
.
...
H
......
c..
P.L-:':'�
\'
)
.� "
'.0
...
V
1£
e 'i .S!
..!!
E
�
.!!
III
x
I e.
-I""
' - �I
�
";
Q,
..
/
V
V
0.,
�
I!
'"
0
.c
0
:::
t::
'.0
0.0
Gas Flow Rate. Mscf7D
x
1 0J
I�
il
�I l
t'-.. V
.......
III
os
PlIO
c
II
ID
:: �."'
...... .
f'.- t-....
=
II
I
1/
't'
0.'
F::
lY
1/
t--...
:aiii
"-
c
i'-..
�
"" .... .
Gas Flow Rate. MscflO
x
f\.
r-
�
r�
1 0J
Fig. 4 . 1 5-Example of gas i nflow performance curves in a
tight gas formation (after Greene 3 6 ) .
TABLE 4 . 2-RESULTS F O R EXAMPLE 4 . 8
Fig . 4 . 1 4-Example plot of a gas four-point test (after
Greene 3 6 ) .
P wf
(psia)
1 4.65
500
1 , 000
1 , 500
2 ,000
2 ,500
3 , 000
3 , 360
depletes . Although the four-point deliverability test i s common and
can be used for interpolation in the range of rates used in a recent test,
this method should not be used for detailed analysis or for forecasting
future performance. Rather, the following methods are more accurate.
4.4.2 Gas-Well IPR Curves. As mentioned, Eq. 4. 100 is not suffi­
ciently accurate for most wells . It is necessary to account for vari­
ation of fluid properties and non-Darcy flow. This must be done
by going to the fundamentals of gas flow . Theoretical develop­
ments , 36-39 described in Chap . 7, have shown that a better analy­
sis can be made with the following improved equation,
pp ( p) - Pp ( P "f) =aq g + bqj , . . . . . . . . . . . . . . . . . . . . (4. 101)
where pp ( p) is the real-gas pseudopressure at pressure p. This i s
a n integral function defined by
. p
P
pp ( p) =2 J -dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 102)
Po P- g Z
The coefficients a and b are given by
1 ,422T
1O.06A 3
a=
1 . 15 1 log
- - +s . . . . . . . . . . (4. 103)
4
kg h
CA r a
and b = (1 ,422Tlkg h)D, . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 104)
2 . 7 1 5 10 - 1 5 (3kg Mp sc
. . . . . . . . . . . . . . . . . . (4. 105)
where D =
h p-g ( p wf )rw Tsc
Note that the viscosity in the denominator of D is taken at the
BHFP and given as P- / Pwf ) ' The value o f {3 can b e empirically
--
[
(
--
) l
x
x
estimated from
{3 = 1 . 88
10 1 0 k
-
1
.
47 <p - 0 .5 3 .
. . . . . . . . . . . . . . . . . . . . (4. 106)
The parameter CA is the Dietz shape factor. A value of 3 1 . 62
should be used for a circular drainage area with the well in the
center. See Appendix C for other shapes.
Essentially the same result can be obtained with a simpler equa­
tion involving p 2 rather than pseudopressure . By making an ap­
proximation of the integral equation, we can write
[
(
) l
p 2 -p� =aqg +bqj , . . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 107)
1O.06A 3
1 .422ZiI T
where a =
1 . 15 1 log
- - + s . . . (4. 108)
4
kg h
CA r a
---
and
qg
(Mscf/O)
1 3 , 1 99
1 3, 024
1 2 , 220
1 0, 828
8 , 848
6 , 1 99
2 , 858
0
b = ( 1 ,422ziI Tlkg h)D . . . . . . . . . . . . . . . . . . . . . . . . . . (4. 109)
Typically , the inflow reservoir performance is presented as a plot
of gas production rate vs. BHFP. This is called the inflow perform­
ance curve (sometimes called the IPR curve) . Fig. 4 . 1 5 gives an
example . Curve C is calculated from Eqs . 4. 105 through 4. 109.
The gas flow rate at P wf=P atm is the AOF potential o f the well .
Note that the AOF is 5.5 MMscflD compared with 5 . 3 MMscflD
for Curve B , which is calculated by Eq. 4. 100. (Actually ,
Greene 36 did not include the non-Darcy flow term, Dqg , in his ex­
ample but this does not detract from the generality of his paper. )
The IPR curve applies to only one reservoir pressure. The average
reservoir pressure is read at q g = O. As the reservoir depletes, an
IPR curve must be developed for each new reservoir pressure . Re­
member that the equations used here apply only to single-phase gas
flowing in the reservoir. If significant liquids do flow in the reser­
voir, a more complicated method must be used to develop IPR
curves .
Example 4.8-Calculating Gas-Well Inflow Performance. Using
the following data, develop the gas-well inflow performance curve.
P = 3 ,360 psia .
"/g = 0.65.
T = 150°F.
k$ : 1 .0 md .
cp - 0. 1 3 .
h = 1 70 ft .
r w = 0.3 ft .
s = 1.1.
A = 3.4848 X 10 6 ft (80 acres) .
Solution. Eqs . 4. 105 through 4. 109 were used to calculate P wf
at various rates. With these equations, the coefficient a is constant
but the coefficient b must be updated for the changing wellbore
viscosity . For P wf = 14.65 psia, a =793.0589 and b = 0.0047 16.
75
GAS FLOW I N WELLBORES
. 01
;;;
Co
e
::s
::!
e
"'-
co
c
.�0
ii:
.!t
0
.e
e
0
;::
S
.. ......
- .......
.......i'-... ; ....
kiIoM ":' -lou
'"
"
"
�
.-. '..
....
..
r.:::
_S: ; aMW '"
"-.
/'
'"
�
-
'"
I
"0
Iii
"0
01
..
:5 u
.
�
I
0
\
'"
•
Gas Flow Rate, Mscf/D
\
\
x
. ..;;; - .......
- I'......
e
�c
i'...i'-...
.... "' � .. ::
e
"'K
'"
.�
.......
.ii:
.!t
�
e
� ��
�
;::
"
S
,,' . ' � ��
�
." Iii
'\ "" <, �...... \
.".. ...
c.
"VI
::s
OJ)
c
0
0
.e
0
,
:5
U
�
/
/
,
.
1 0)
�
�
JOG
'\
•
'\
Gas Flow Rate, MscflD
�� \
x
\ \\
1 0)
F i g . 4 . 1 6-Example of gas-well outflow performance curve
(after Greene 3 6 ) .
Fig. 4 . 1 7-Effect of tubing size on gas-wel l outflow perform·
ance (after G reene 3 6 ) .
Table 4.2 gives the rates and pressures. The resulting IPR curve
4.4.4 Tubing Performance Curves. Another approach to the prob­
will be used later in Example
4.9.
4.4.3 Gas-Well Outflow Performance Curves. W e now have the
tools to calculate the pressure drop in the formation and flow string
simultaneously . For a given flow rate, we can calculate P wf with
Eqs . 4. 101 through 4. 1 09. Then we can calculate the surface pres­
sure, P tf, with one of the pipe flow methods discussed previously .
The pressure in the pipe is calculated from the bottom up .
Greene 36 described the " outflow performance " of a gas well.
This relationship describes the variation of flowing wellhead pres­
sure rate with flow rate . Fig. 4 . 1 6 shows a typical outflow per­
formance curve for a well producing liquids . He called the maximum
P tf the flow point. This corresponds to the minimum sustainable
flow rate possible . Lower rates are unstable (the well will die be­
cause the gas velocity will not remove sufficient liquids from the
well) . A dry gas well does not have a flow point and can produce
at any rate on the curve.
The gas outflow performance depends on tubing size . Fig. 4. 17
illustrates the variation of well outflow performance for four tub­
ing sizes. Note that the smaller tubing strings have better flow ef­
ficiencies at lower gas production rates, while the larger tubing
strings are more efficient at higher gas production rates .
..
(;;
. Co
i
'"
- i"'--_
\
r---....
I>co
""
.e
0
.
......
�
- t-.....
...;;;
I'
/
1'0....
- h\ \
,V
41 /""
e - �\ \
V
"""
/
....... V
e - \\ l\ '"
,
......
k
1"-
-. .
,
'"
.... "
-e
I'--. ';* ' ,/
- ....... \
... ��
.S
--- �
� ...
.Sl
"'... \. -..
.. '"'"* '"
'0
...... ....
u;.n I_L' ..::" - �
e ......
;::
...
\
S
\
lem is to develop tubing performance curves. These curves show
the variation of BHFP with rate while the surface pressure is held
constant. These curves are plotted with IPR curves to determine
the BHFP and rate for the given wellhead pressure. Fig. 4. 18 shows
an example 36 of a tubing performance curve . With the IPR curve
for a reservoir pressure of 2,000 psia, Point P shows the intersec­
tion. This is sometimes called the natural flow point. The flow rate
is 3 . 6 MMscflD and the BHFP is 1 ,060 psia. However, when the
reservoir pressure depletes to 1 ,300 psia. the well is barely flow­
ing at 1 . 1 MMscflD . At reservoir pressures below 1 ,300 psia, the
well is dead. Artificial lift or a change in tubing size is required
to continue production.
The tubing performance curve represents the BHFP performance
for a given wellhead pressure and tubing size . When the wellhead
pressure or tubing size is changed , the tubing performance curve
also changes .
When liquids are being produced, these curves typically are J-shaped
and may cross the IPR curve at two points . In Fig. 4. 18, Point P rep­
resents a stable rate when the reservoir pressure is 2,000 psia. Al­
though the two lines may also intersect at about 0.3 MMscflD, this
intersection does not represent a stable condition and is meaningless.
,
..liN
,
,
,
\
\
\
Gas Flow Rate, Mscf7D
x
1 0)
Fig. 4 . 1 8-Example of a tubing performance curve (after
Greene 3 6 ) .
II
Co
::s
WI
WI
I>co
c
.�0
..
'0 e
;:: -
ii:
.e
0
�� <-
V
",
'"
r---
-. .....
'-
S
Gas Flow Rate, MscflO
x
'"
.-S2 -II)
1\
V�
\
1 0J
Fig. 4 . 1 9-Effect of tubing size on tubing performance curves
(after Greene 3 6 ) .
76
G A S RESERVO I R E N G I N E E R I N G
Example 4.9-Calculating Tubing Performance Curve and Na­
tural Flow Point for a Gas Well. For the following data, develop
TABLE 4 .3-RES U L TS F O R EXAMPLE 4 . 9
a tubing performance curve using Eq. 4 . 39 . In addition, plot the
tubing performance curve with the IPR curve developed in Exam­
ple 4 . 8 and determine the natural flow point.
PIt =
TIl =
L=
Twl
=
() =
d=
1 ,000 psia.
75 ° F .
6 , 000 ft.
1 50 ° F .
0 ° (vertical well).
1 .995 in.
Solution. Because this is a dry gas well and is not very deep,
the average temperature and z-factor method was used to compute
Pwf at various rates. Remember, the surface pressure is fixed at
1 ,000 psia. Table 4.3 gives the results. This tubing performance
curve is plotted in Fig. 4.20, along with the IPR curve from Ex­
ample 4 . 8 . The intersection is the natural flow point that gives values
of q g =9,333 MscflD (9 . 333 MMscflD) and P Hj = 1 , 896 psia.
4.5 Forecasting Gas-Well Performance
The most common reason for making forecasts is to estimate the
reserves (future production) of wells . It is also necessary to fore­
cast cash flow in many cases and to compare cash flows for alter­
native producing decisions. Here, we discuss a method for predicting
well performance for a single well producing from a volumetric
reservoir. This combines the well flow methods we have just dis­
cussed with the material-balance methods (plz curves) from Chap .
1 0 . This can easily be generalized to mUltiple wells producing from
a volumetric reservoir.
The preceding sections discussed \yell performance at a particu­
lar time . The flowing tubing pressure and the average reservoir
.1;1c:l.
3000
�
2500
Q::
.�Ii:
O +-+-+-+-+--l--l-++++-+-+-I
3
4
5
6
7
8
9
0
1 ,889
3,61 0
5 , 877
1 0 , 1 79
Calculation Procedure.
�
2
1 , 1 62
1 ,200
1 ,300
1 ,500
2 , 000
1 . Select a sequence of average reservoir pressures . For each
average reservoir pressure, develop an IPR curve in a manner simi­
lar to Example 4 . 8 .
oJ
1
(MsctIO)
pressure were specified, along with the GLR for the well . These
values change with time for any particular well. The reservoir pres­
sure declines as the reservoir is depleted . The GLR may change
with time. The flow string (tubing) and the flowing tubing pres­
sure may change owing to operating conditions. We often need a
method of making (on a time scale) forecasts that account for these
changing conditions .
In addition to forecasting rates, we also forecast the average reser­
voir pressure and BHFP. The forecast of flow rates and abandon­
ment time are then used to calculate reserves (remaining economical
production) .
For the following procedure, the usual case is assumed . That is,
the IPR curve shifts downward as the reservoir depletes, while the
tubing performance curve remains the same throughout time. This
is true only if the tubing diameter, flowing tubing pressure, and
GLR are constant throughout the producing life . If these change,
then the procedure must be modified to reflect the changing flow
conditions .
3 500
o
qg
P wf
(psia)
2000
1 500
Gas Flow Rate, MMscf/D
F i g . 4 . 20-Determ i n i ng a gas-wel l natu ral flow point,
Examples 4.8 and 4 . 9 .
I'-t--,
.......
r---
"'-.
I'-
1"-
f'-..
,......
f'-..
1"-
f'-..
l"--
�
l"-
o
1
2
"\
l\.
3
b-.
"\
4
1\
5
�
f'...
K f"...
e>V
f',
l'--b:" l"-� .......
""
�f- "'"
""
o
10 1 1 12 13 14
t--..
\
6
i"-, � [\
\
1\
7
8
\
�
\
9 10 1 1 12 13 14
Gas Flow Rate, MMscfID
Fig. 4 . 2 1 -IPR and tubing performance curves, Example 4 . 1 0.
TABLE 4 .4-RESULTS OF STEP-WISE CALCU LATIONS FOR EXAMPLE 4 . 1 0
(psia)
P
p lz
3 , 360
3,895 . 7
Gp
(Bct)
Il G p
(Bct)
qg
(MsctIO)
3,529.4
2 ,976.2
2 , 366. 9
4. 7092
1 , 734 . 1
6. 6584
6,91 6
246. 4
4,883
384.4
2 ,566 . 5
759. 5
3, 733
1 .9492
1 , 500
131 .2
6,033
2. 8324
1 .8768
2 , 000
8 , 566
7,800
1 . 1 283
1 .7041
2 , 500
M
(days)
9 , 333
0
1 . 1 283
3 , 000
q
(MscflO)
1 ,400
t
(days)
P wf
(psia)
0
1 ,896
1 31 .2
1 , 729
377.6
1 ,542
762 . 0
1 , 333
1 , 521 . 2
1 ,208
GAS FLOW IN WELLBORES
4000
77
10
_---.---..---"""'T'"--,...---,----,
8
1\
� �
a
�
�
t3
o
2.S
S
7.S
10
12.S
Cumulative Gas Production, BCF
IS
Fi g . 4 . 22-plz plot, Example 4 . 1 0 .
2 . Develop a tubing performance curve for the tubing conditions
specified . The intersections with the IPR curves represent the na­
tural flow points at the various average reservoir pressures.
3. Find the cumulative production, Gp • that corresponds to each
average reservoir pressure . This can easily be done with a plz plot
for a closed reservoir .
4 . Now put the results on a time basis. Find an average rate be­
tween each pair of average reservoir pressures . Divide the change in
cumulative production between reservoir pressures by the corre­
sponding average rates. This gives the elapsed time between average
reservoir pressures. All the results can now be put on a time basis .
This procedure is general . It can be modified to handle multiple
wells producing from the same reservoir. The plz plot can be modified
to account for formation and water compressibility (see Chap. 1 0) .
Example 4. 10-Forecasting Flow Rates, Abandonment, and
Reserves . We are going to forecast the performance of the dry gas
well described in Examples 4 . 8 and 4 . 9 . We will assume that the
well flows at a constant P if and that the tubing is not changed dur­
ing the production life. The well is producing from a reservoir origi­
nally containing 12 Bscf at an initial reservoir pressure of 3 , 360
psia . All other data are the same as in Examples 4 . 8 and 4 . 9 . Stop
the forecast when production rate drops below an economic limit
of 400 Mscf/D or at 4 years , whichever occurs first .
Solution. The procedures of Examples 4 . 8 and 4 . 9 were repeat­
ed for average reservoir pressures of 3 ,000, 2,500, 2,000, and 1 ,500
psia. The resulting plots of the IPR curves and the tubing perform­
ance curve are shown in Fig. 4.2 1 . The BHFP and flow rate are
taken at each intersection . These are tabulated in Table 4.4.
The cumulative production at each average reservoir pressure is
found from material balance, as Fig. 4.22 shows. Incremental cu­
mulative productions are then calculated between average reser­
voir pressures . Average production rates are found for each interval .
Then the elapsed time during each interval is simply found by divid­
ing the incremental production by the average rate. Table 4 . 4 shows
the complete spreadsheet results .
The last line shows a cumulative production of 6 . 6584 Bscf at
a time of 1 ,52 1 .2 days. This is 6 1 .2 days past the 4-year limit ( 1 ,460
days) . The reserves for this well would then be 6 . 6584 Bscf
- (2 . 566 . 5 MscfID x 6 1 . 2 days X 1 0 - 6 Bscf/Mscf) = 6 . 50 1 Bscf.
Fig. 4.23 shows the time schedule for the natural flow rate .
4.6 Summary
This chapter covered various aspects of the wellbore effects on a
gas well. There were several hand methods given for static pres­
sure and flowing pressure of a dry gas well . For a shallow well,
6
"
4
""
2
o
500
,
�
1000
Time, Days
...... 11500
2000
Fig. 4 . 23-Forecast of natural flow rate with time, Example
4.1 0.
these methods may be adequate . For a deep well, these methods
should be adapted to a computer program or spreadsheet that breaks
the wellbore down into small increments .
The problem becomes more complicated when liquids are pres­
ent. We have shown various types of correlations for the two-phase
cases. The Beggs-Brill method is the most common , mainly be­
cause it is appropriate for nonhorizontal segments . Many directional
wells have deviation angles that change with depth. It is then re­
quired to break the wellbore down into smaller segments with differ­
ent angles for these segments . The Hagedorn-Brown method should
be adequate for nearly vertical wells .
The final section o f this chapter tied reservoir performance to
tubing performance . This technology is a combination of reservoir
engineering and production engineering . The IPR curve represents
the reservoir performance at a particular time but must be tied to
the tubing performance to determine actual flowing conditions . The
IPR curve changes with time owing to changes in average reser­
voir pressure . In practice, the relative permeability to gas may also
change with time . It is typical for a well to produce more water
as the average reservoir pressure declines.
A complete understanding of the relationship between reservoir
flow and tubing flow is often lacking in practice . It is common for
inefficient operations to occur simply because the production engi­
neer or the reservoir engineer does not have a complete understand­
ing or does not look at the complete situation . A common problem
is to have the wrong tubing size, which either has too much fric­
tion loss or does not adequately flow liquids out of the well .
Exercises
4 . 1 Calculate P ws for the following data using :
A . The average temperature and z-factor method .
B . The Poettmann method .
e . The Cullender-Smith method .
Data:
L
= 1 2 ,000 ft .
= 0.6.
Tws = 1 90 ° F .
TIS = l O a F .
(j = 1 0 0 .
P ws = 2 , 000 psia.
4.2 Example 4 . 1 shows the calculation of P ws using the average
temperature and z-factor method .
A . Modify this method to divide the wellbore into two seg­
ments [first calculate the midpoint pressure , then the bot­
tomhole pressure (BHP») . Remember to take the average
temperature for each segment, using linear interpolation.
'Yg
78
4.3
GAS RESERVO I R E N G I N E E R I N G
B . Repeat with four segments , then eight segments , etc .
C . Tabulate each calculated BHP and corresponding number
of segments . Comment on the number of segments required
to give good accuracy .
We can modify the average temperature and z-factor method
so that no iteration is required for z. Repeat Exercise 4.2 but
use values of T, and z , at top of each interval. Use Eqs . 4. 10
and 4. 1 1 but replace P ws and P IS by P 2 and p " respectively ,
as follows :
4. 1 1
4. 12
T
qg [1 ,000/24] = 18.062 �
P 2 = p , e sI2 ,
. s=
wIth
4.4
4.5
0.03751'g (Z2 -Z, )
z i Ti
Psc
.
[
p, g
- (Z -Z, ) + for (L 2 -L , )
2g cd'
144 g c 2
4.6
4.7
4.8
--
l
.
The right side is evaluated entirely at Point I - i . e . , the top
of the current segment. Repeat Exercise 4.4 using the above
equation to find P 2 for each segment .
Derive Eq. 4.76, which is the holdup fraction, HL ( the
volume fraction of the liquid in the flow stream) . Hint: Begin
with q ; = qi + q� . You will use substitutions such as
qL = vL AHL ·
Example 4.9 shows the estimation of two-phase BHFP gra­
dient using the Poettmann-Carpenter correlation. Repeat the
example problem using :
A . Hagedorn-Brown method.
B . Beggs-Brill method .
Calculate and plot the IPR curve for the following data .
if
= 4, 100 psia.
k = 5 md .
re = 1 ,000 ft .
TWf = 2 10°F.
h = 42 ft .
rw = 0.25 .
c/> = 0. 14.
I' g = 0.7 1 .
s = 1 .5 .
4.9
Calculate and plot the tubing performance curve for the well
in Exercise 4.8, given the following .
() =
=
L=
Ttf =
d =
P tf
[
( P 2 _ p 2 )d S .333
'_ 2
l
°.5
g TzLl5,280
(The numbers 1 ,000, 24, and 5 ,280 are included here to
convert to our usual units where q g is in Mscf/D and L is in
The temperature profile is still linear. All values will be
known from the previous segment, so you can work down
the flow string without iteration. This method may be slight­
ly less accurate than in Exercise 4.2 for a given number of
segments but requires less work. It should converge to an ex­
act solution . Hint: For two segments , (Z2 -Z, ) = 5 ,000 ft,
p , =2,500 psia, T, = (35 + 460) OR, z , is taken at p " T, .
Example 4.4 shows the calculation of BHFP using the aver­
age temperature and z-factor method .
A . Modify this method to divide the wellbore into two seg­
ments (first calculate the midpoint pressure, then the BHP) .
Remember to take the average temperature for each seg­
ment using linear interpolation .
B . Repeat with four segments , then eight segments, etc.
C . Tabulate each calculated BHP and corresponding number
of segments . Comment on the number of segments required
to give good accuracy .
Modify Eq . 4.21 to solve for P wf' using a finite-difference
approach without iteration. Replace dp by ( P 2 -P I ) ' dZ by
(Z2 -Z, ), and dL by (L 2 -L , ) in Eq . 4.21 as follows:
( P 2 -P , ) = - -
Make a production rate forecast o f the well described in Ex­
ercises 4.8 through 4. 10. Follow the procedure of Example
4. 10. Assume that P tf remains constant at 750 psia.
Suppose the well in Exercises 4.8 and 4.9 had a I -mile (5,280ft) horizontal flowline with d= 1 .50 in. Rather than the well­
head pressure being held constant at 750 psia, the downstream
end of the pipeline is held constant at 750 psia. A simple equa­
tion for horizontal pipeline flow is Wehmouth ' s equation:
0° (vertical well ) .
750 psia.
1 1 ,000 ft.
80°F.
1 .995 in.
4. 10 Plot the IPR curve and tubing performance curve from Exer­
cises 4.8 and 4.9. Determine the natural flow point .
4. 1 3
l'
ft. The average values, of course, are averages for the horizon­
tal pipeline flow) . Now , make a production rate forecast of
this well. Note that the IPR curves do not have to be redone.
The "tubing performance curve" will now include the effect
of the horizontal flowline.
Repeat Exercise 4. 12 with the same conditions , except that
the well is producing at a gas/water ratio of 5,000 scf/STB .
Assume that the IPR curves are the same as above and that
the water is not removed from the flow string until it reaches
a separator at the end of the horizontal flowline. (Use the Beggs
and Brill method . )
Nomenclature
a
=
A =
b=
B=
Bg =
stabilized deliverability coefficient defined by
Eq. 4. 103 or 4. 108; mIL 4 t 2 ,
(psia 2 /cp)/(MMscf/D) for Eq. 4. 103 ;
m 2 /L 5 t 3 , psia 2 /(MMscf/D) for Eq. 4. 108
area, L 2 , ft 2
deliverability equation coefficient defined by
Eq. 4. 104 or 4. 109; m/Ut ,
(psia 2 /cp)/(MMscf/D) 2 for Eq. 4. 104;
m 2 /L 8 t 2 , psia 2 /(MMscf/D) 2 for Eq. 4. 109
Beggs and Brill parameter defined by Eq.
4.87
gas FVF at flowing conditions , dimensionless,
RB/STB
Bo = oil FVF at flowing conditions , dimensionless ,
RB/STB
B w = water FVF at flowing conditions ,
dimensionless , RB/STB
C = Beggs and Brill parameter defined by Eq. 4.91
C = stabilized performance coefficient used i n Eq .
4. 100, L S t 2 /m, (MMscf/D)/psia 2
CNL = Hagedorn and Brown viscosity number
coefficient (Fig . 4.6)
d = pipe diameter, L, in.
d l = casing !D, L , in.
d2 = tubing OD , L , in.
d' = pipe diameter, L , ft
D = non-Darcy flow coefficient defined by Eq.
4. 105 , t/L 3 , (Mscf/D) - '
deq = equivalent diameter, L , in.
e = natural logarithmic base
I = Moody friction factor, dimensionless
(iF )m = Fanning friction factor of fluid mixture (Fig .
4.5), dimensionless
1m = Moody friction factor of fluid mixture ,
dimensionless
Ins = no-slip friction factor, dimensionless
irp = two-phase friction factor, dimensionless
F = energy loss resulting from friction, L 2 /t 2 ,
ft-Ibf/lbm
g = local gravitational acceleration , Llt 2 , ft/sec 2
g c = gravitational acceleration constant,
dimensionless, 32.2 ft-Ibm/lbf-sec 2
gf = friction-loss gradient, m/L 2 t 2 , Ibf/ftLft
GAS FLOW I N WELLBORES
h
H
HLO, intennit entL
HLO, segregated
HLO,transition
Imp
= net formation thickness, L , ft
= liquid holdup factor, dimensionless
= liquid holdup for intermittent flow regime cor­
rected for inclination angle, dimensionless
= liquid holdup for segregated flow regime cor­
rected for inclination angle, dimensionless
= liquid holdup for transition flow regime cor­
rected for inclination angle, dimensionless
= modified Cullender and Smith integral evaluat­
ed at midpoint of production string and used
in static (Eq. 4.33) and flowing (Eq. 4.90)
BHP calculations , ° R/psia
= modified Cullender and Smith integral evaluat­
ed at top of production string and used in
BHFP calculations (Eq. 4 . 85) , ° R/psia
= modified Cullender and Smith integral evaluat­
ed at top of production string and used in
static pressure calculations (Eq. 4.28),
° R/psia
IWf = modified Cullender and Smith integral evaluat­
ed at bottom of production string and used
in BHFP calculations (Eq. 4.94) , ° R/psia
= modified Cullender and Smith integral evaluat­
ed at bottom of production string and used
in BHSP calculations (Eq. 4 . 37) , ° R/psia
= effective permeability to gas , L 2 , md
K = Poettmann gravity term
L = vertical direction along flow string , positive
upward, L, ft
L = flow-string length (absolute value) , L, ft
LB = parameter defining boundary between bubble
and slug flow regimes, dimensionless
L = equivalent depth corresponding to the well­
head flowing pressure, L , ft
L Wf = equivalent depth corresponding to the BHFP,
L , ft
L 1 L 4 = Beggs and Brill boundary parameters for flow
regime determination defined by Eqs . 4 . 82
through 4 . 85
M = molecular weight of gas mixture, m ,
Ibmllbm-mol
n = inverse slope (exponent) of deliverability
curve used in Eq. 4 . 1 00, dimensionless
= pipe diameter number defined by Eq. 4.65 ,
dimensionless
= Froude number, dimensionless
g = gas velocity number defined by Eq. 4.64,
dimensionless
= liquid viscosity number defined by Eq. 4.66,
dimensionless
= liquid velocity number defined by Eq. 4 .63 ,
dimensionless
= Reynolds number, dimensionless
= Reynolds number of the mixture, dimen­
sionless
= no-slip Reynolds number, dimensionless
= absolute pressure, m/Lt 2 , psia
if = average reservoir pressure, m/Lt 2 , psia
= pressure at production-string midpoint, m/Lt 2 ,
psia
p ( jJ) = average reservoir pseudopressure, m/Lt 3 ,
psia 2 /cp
pp ( P Hi) = flowing sandface pseudopressure, m/Lt 3 .
psia 2 /cp
= pseudocritical pressure of gas mixture, miLt 2 ,
psia
= pseudoreduced pressure , p/p ' dimensionless
Itf
Its
Iws
kg
tf
-
Nd,
NFR
Nv
NL
NLv
NReNRem
NRens
P
Pmp
p
Ppc
Ppr
pc
P =Ptf'
P =Pts '
Ppr,tf
Ppr,ts
Ppr,v.i
Ppr, ws
PPsctf
PPPwswtfs
79
= pseudoreduced pressure evaluated at
dimensionless
= pseudoreduced pressure evaluated at
dimensionless
= pseudoreduced pressure evaluated at p =p wf'
dimensionless
= pseudoreduced pressure evaluated at p =p ws '
dimensionless
= standard pressure, m/Lt 2 , psia
= flowing tubing-head pressure , m/Lt 2 , psia
= static tubing-head pressure , m/Lt 2 , psia
= BHFP, miLt 2 , psia
= BHSP , m/Lt 2 , psia
f1plf1L = pressure gradient, m/L 2 t 2 , psi/ft
q = gas flow rate, L 3 It, ft 3 /sec
q' = volumetric fluid flow rate, L 3 It, ft 3 /sec
= gas flow rate, L 3 It, MscflD
q ,� = gas flow rate, L 3 It, ft 3 /sec
qi = liquid flow rate, L 3 It, ft 3 /sec
= oil flow rate, L 3 It, STBID
= total flow rate of well stream, L 3 It, Mscf/D
q ; = total flow rate of well stream, L 3 It, ft 3 1D
qw = water flow rate, L 3 It, BID
rh = hydraulic radius defined by Eqs . 4.53 and
4 . 54 , L, ft
R = universal gas constant = 10.73 ft 3 -psia/lbm­
mol-O R , L 2 /t 2 mT
R s = solution GOR at flowing conditions , dimen­
sionless, scf/STB
s = average temperature and z-factor method pa­
rameter defined by Eq. 4 . 1 1 , m/L 2 T
s = skin factor, dimensionless
S = Beggs and Brill parameter defined by Eq.
4.97 or 4.98, dimensionless
= absolute temperature, T, O R
f = average temperature, T , O R
= pseudocritical temperature, T , O R
= average pseudoreduced temperature,
dimensionless
= pseudoreduced tubing-head static temperature,
dimensionless
= standard temperature, T, O R
= tubing-head flowing temperature, T , O R
= tubing-head static temperature, T , O R
TWf = BHFT, T , O R
v = bulk flow velocity o f fluid, Lit, ftlsec
vL = liquid velocity , Lit, ft/sec
v m = bulk flow velocity of fluid mixture, Lit, ft/sec
v = slip velosity , Lit, ft/sec
vSL = superficial liquid velocity , Lit, ft/sec
VSg = superficial gas velocity, Lit, ft/sec
WIi = gas mass flow rate, mit, Ibm/sec
wL = liquid mass flow rate, mit, Ibm/sec
w , = shaft work done by the system per unit mass,
L 2 It 2 , ft- Ibf/lbm
wt = total mass flow rate, mit, lbm/sec
y = Beggs and Brill parameter defined by Eq.
qg
qoqt
T
Tp'ipcr
Tpr,ts
TscTtf
Tts
s
4.96
z = gas deviation factor, dimensionless
z = gas deviation factor evaluated at if, dimen­
sionless
zsc = gas deviation factor at standard conditions,
dimensionless
Z = true vertical depth of flow string (absolute
value) , L, ft
Z = vertical direction (elevation) positive upward,
L , ft
80
G A S RESERVO I R · E N G I N E E R I N G
a =
a =
{3
'Y
=
g=
'Y 0
=
'Y w
=
() =
f
=
=
AL =
f
/d
Ii
iJ. g
iJ.L
iJ. o
iJ. w
P
Pg
PL
Pm
Po
=
=
=
=
=
=
=
=
=
=
Pw =
Uo =
UL =
Uw =
if; =
if; =
{} =
flow angle measured from horizontal = (90-0) ,
degrees
modified Cullender and Smith term defined by
either Eq. 4 .27 or 4 . 84, ° R-Ibm/lbf
non-Darcy flow coefficient defined by Eq.
4. 1 06, I /L , ft - I
specific gravity of gas mixture (air = 1 . 0) ,
dimensionless
specific gravity of oil (water = 1 . 0),
dimensionless
specific gravity of water (pure water = 1 .0) ,
dimensionless
well deviation angle measured from vertical ,
degrees
absolute pipe-wall roughness, L, in.
relative roughness ratio, dimensionless
liquid holdup factor for no-slip conditions
(Eq. 4 . 74) , dimensionless
gas viscosity evaluated at p, mILt, cp
gas viscosity , mILt, cp
liquid-phase viscosity , mILt, cp
oil or condensate viscosity , mILt, cp
water viscosity , mILt, cp
fluid density , m/L 3 , Ibm/ft 3
gas density , m/O , Ibm/ft 3
liquid density , m/L 3 , Ibm/ft 3
fluid mixture density , m/L 3 , Ibm/ft 3
oil or condensate density , m/L 3 , Ibm/ft 3
water density , m/L 3 , Ibm/ft 3
oil surface tension, m/t 2 , dyne/cm
surface tension of liquid/gas interface, m/t 2 ,
dyne/cm
water surface tension, m/t 2 , dyne/cm
Hagedorn and Brown secondary correction
factor (Fig . 4 . 8)
Beggs and Brill parameter defined by Eq.
4.90
modified Cullender and Smith friction factor
term defined by Eq. 4 . 83 , Llt 2
References
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Natural Gas Through Pipe , " Trans. , AIME ( 1 95 1 ) 192, 3 1 7-24 .
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( 1 954) 201 , 279.
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Through Experimental Pipe Lines and Transmission Lines , Monograph
9, U . S . Bureau of Mines .
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road Commission, Texas A&M U . , College Station, TX ( 1 989) .
1 7 . Sukkar, Y . K . and Cornell, D . : "Direct Calculation of Bottomhole Pres­
sures in Natural Gas Wells , " Trans. , AIME, ( 1 95 5 ) 204 .
1 8 . Messer, P . H . , Raghaven, R . , and Ramey , H . J . , Jr. : "Calculation of
Bottom-Hole Pressures for Deep , Hot, Sour Gas Wells , " IPT (Jan .
1 974) 85-94 .
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OK ( 1 984) .
20. Orkiszewski , J . : " Predicting Two-Phase Pressure Drops in Vertical
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2 1 . Poettmann, F . H . and Carpenter, P . G . : "The Multiphase How o f Gas ,
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dients in High-Rate Howing Wells , " IPT (Oct. 1 96 1 ) 1 023-2 8 ; Trans. ,
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Vertical Flow Strings , " IPT (Oct. 1 96 1 ) 1 029-36; Trans. , AIME, 222.
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for M1,Iltiphase How in Tubing , " SPEJ (March 1 963) 59-69 ; Trans. ,
AIM E , 228.
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Gradients Occuring During Continuous Two-Phase Flow in Small­
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234.
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1 9-26 , 1 963) Sec . II, Paper 22-PD6 .
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Production Systems, Prentice-Hall, Inc . , Englewood Cliffs, NJ ( 1 993) .
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Publishing Co. , Tulsa , OK. ( 1 984) 4.
34. Rawlins , E . L . and Schellhardt, M . A . : Backpressure Data on Natural
Gas Wells and Their Application to Production Practices, Monograph
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3 5 . Corbett , T . G . and Wattenbarger, R . A . : " A n Analysis o f and Cor­
rection Method for Gas Deliverability Curves , " SPE 1 4208 , 60th An­
nual Fall Meeting of the SPE held in Las Vega s , Nevada , Sept . 22-2 5 ,
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3 6 . Greene, W . R . : "Analyzing the Perfonnance of Gas Wells , " Proc. ,
25 A nnual Southwestern Petroleum Short Course, Lubbock, TX ( 1 978)
1 29-3 5 .
3 7 . AI-Hussainy , R. , Ramey , H . J . , Jr. , and Crawford, P . B . : " The Flow
of Real Gases Through Porous Media , " IPT (May 1 966) 624-3 6 .
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Chapter 5
Fundamentals of Fluid Flow
in Porous Media
5. 1 Introduction
This chapter discusses the differential equations most often used
to model unsteady-state fluid flow in porous media. We begin with
the derivation of the diffusivity equation for slightly compressible
fluid flow in homogeneous-acting reservoirs . We also present forms
of the equation applicable to single-phase gas and multiphase flow.
We then present some of the more useful solutions to the fluid-flow
equations . In particular, we emphasize the exponential-integral so­
lution, which forms the basis of most pressure-transient test analy­
sis techniques . In addition , we discuss the use of dimensionless
variables , the radius-of-investigation concept, and the principle of
superposition. Finally , we conclude with a discussion of the van
Everdingen-Hurst solution to the diffusivity equation .
The mass flow rate out of the system,
by
wout = [ - p U r - ll( - pu r )]A r = [ - p U r + Il ( PU r)] A r , . . . . . . (5 . 4)
where A r = h X r O at radius r and ll(p U r) = change in mass flux in­
side the control volume . The negative sign accounts for flow in
the - r direction . Therefore ,
Wout = [ -pur + ll( pur)] r Oh . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 5)
The mass in the control volume at any time is the product of pore
volume (PV) and the fluid density . The PV is
Vp = r Ollrhc/> .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .6)
.
To develop analysis and design techniques for well testing , we must
make several simplifying assumptions about the well and reservoir
being modeled. These assumptions are introduced as needed to com­
bine the principle of mass conservation , an equation of fluid mo­
tion, and an equation of state (EOS) . From these fundamental
relations , we derive the diffusivity equation, which is used to model
fluid flow in a porous medium . In a homogeneous-acting , cylin­
drical reservoir, fluid flows radially from the reservoir to the well­
bore . Therefore, our diffusivity equation will be derived in terms
of a radial coordinate system.
5.2.1 Diffusivity Equation for Flow of Slightly Compressible
Fluids . We begin with the derivation of the diffusivity equation
for the flow of a slightly compressible fluid (i . e . , a liquid) . Con­
sider the small control volume illustrated in Fig. 5 . 1 . We assume
that this element is a porous medium fixed in space and that flow
is radial in the - r direction (from reservoir to wellbore) . Assum­
ing that the net generation of matter is zero , we can write a mass
balance on the element as
Wi n - Wout = q m ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .7)
m = rMrhc/>p .
The rate of mass accumulation in the control volume during the
time intervai llt (from time t to time t + llt) is equal to the change
in mass in the control volume during this time interval divided by Ilt:
qm =
r OllrhfjJp l t + ll t - r OllrhfjJp I t
Ilt
Combining Eqs .
pur A "
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .2)
The negative sign accounts for flow in the - r direction . The arc
length is equal to the radius times the angle , or (r + llr)O. Hence,
Win = - pur(r + llr)Oh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 3)
.
5.3, 5.5,
and
. . . . . . . . . . . . . . . . . . . (5 . 8)
.
5.8,
we can express Eq.
5.1
as
- pur(r + llr)Oh - [ -pur + ll(pur)] r Oh
r OllrhfjJp l t + A t - r OllrhfjJp l t
------
Ilt
Expanding Eq.
5.9
. . . . . . . . . . . . . . . . . . . . . . (5. 9)
gives
- pu r r Oh - pu r llr Oh + PU r r Oh - 1l( pur)rOh
=
r OllrhfjJp l t + ll t - rMrhfjJp l t
III
The mass flow rate into the system at radius r + llr is given by
-
.
Hence, the mass in the control volume at any time is
5.2 Ideal·Reservolr Model
win =
W out' at radius r is given
Dividing Eq.
yields
5 . 10 by
PU r
Il ( pu r)
r
Ilr
.
.....
..
. . . . . . . . . . . . (5 . 10)
.
the bulk volume of the element, rMrh ,
fjJp l t + ll t - fjJp l t
. . . . . . . . . . . . . . . . (5 . 1 1)
82
- (p u , )
/
h
,
"
I
,
,
(J
'-\
. . ...
I
••
-
•.••
-
r
r
+ L1 r
••-
•• �
I
4--
Factoring out ( - I /,l1r) on the left side of Eq. S . 1 1 and simplifying
the right side gives
1
11 ( �p)
- - [l1rpu r + rl1( pur)] = -- . . . . . . . . . . . . . . . . . . (S . 1 2)
rl1r
I1t
[
Rearranging Eq. S . 1 2 , we obtain
r
pu r + r
I1( P U r)
I1r
l
=
_ 11 ( �p)
. . . . . . . . . . . . . . . . . . . . . (S . 1 3 )
I1t
Taking limits in Eq . S . 1 3 as I1r and I1t--+O gives
�
r
[
P U r +r
a( pur)
ar
l
a( � p)
=-
at
.
. . . . . . . . . . . . . . . . . . . . (S . 1 4)
By the chain rule,
ar
a( pur)
--- = p u r - + r -- . . . . . . . . . . . . . . . . . . . . . . . (S . I S)
or
ar
Substituting from Eq . S. IS into Eq . S. 1 4 yields
�
r
[
a(rp U r)
ar
l
=
dp
- + 0 . 00694(z - z o ) , . . . . . . . . . . . . . . . . . . . . . . (S . 1 8)
Ph P
a <l>
a
-=ar
ar
[ ].
P dp
- + 0 . 0069 4 (z - z o )
Ph p
_ a( � p)
at
. . . . . . . . . . . . . . . . . . . . . . . . . . (S . 1 6)
Eq . S . 1 6 , the continuity equation , is a mathematical expression
of the principle of conservation of mass in radial coordinates.
An equation of motion , or flux law, relates velocity and pres­
sure (or potential) gradients within the control volume . Because
of the complexity of the flow paths within porous media, empiri­
cal relationships must be used for the equation of motion . Liquid
flow generally is modeled by Darcy ' s law , which states that veloc­
ity is proportional to the negative of the gradient of the potential .
In radial coordinates, with flow in the radial direction only , we write
kp a<l>
u r = - 0 . 00 1 1 27 - - , . . . . . . . . . . . . . . . . . . . . . . . . . (S . 1 7a)
p. ar
where u r = volumetric fluid velocity , ft 3 /hr-ft 2 , or for velocity
measured in RB/D-ft 2 ,
kp a<l>
u r = - 0 . 0002637 - - . . . . . . . . . . .
p. ar
l
l· p dP
a
a
=- + - [0 . 00694(z -zo)] . . . . . . . . . . . . . . . . . . (S . 19)
ar P p
ar
h
a<l>
1 ap
- = - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (S . 2 0 )
ar
p ar
Substituting Eq . S. 20 into Eq . S. 1 7 gives
k ap
u r = - 0 . 0002637 - - . . . . . . . . . . . . . . . . . . . . . . . . . . . (S . 2 1 )
p. ar
An EOS relates volume (or density) to pressure and temperature .
We can assume isothermal conditions for flow of a slightly com­
pressible liquid in a reservoir because the heat capacity of the fluid
generally is negligible relative to the heat capacity of the rock. Fluid
compressibility is defined as
C= -
(� :; ) (; :; )
T
. . . . . . . . . . . . . . (S . 1 7b)
In Eqs . S . 1 7a and S . 1 7b , k = permeability in the radial direction ,
p. = fluid viscosity , and <I> = potential ,
=
T
'
.
. . . . . . . . . . . . . . . . . . . . . (S . 22)
Treating the partial derivative as a total derivative for an isother­
mal system and rearranging Eq . S . 22 , we obtain
1
a(rpu r)
ar
·
Assuming that gravity effects are negligible so that a[0 . 00694(z Zo)]lar is very small compared with the integral, Eq. S . 1 9 becomes
Fig . S . 1 -Control vol u m e , radial coord i nate system .
�
]p
where P b = pressure at a datum z o o Darcy ' s law assumes that flow
is in the laminar flow regime . For single-phase flow of a slightly
compressible liquid , these assumptions generally are valid .
We can express ur in terms of a pressure gradient rather than
a potential gradient by combining Eqs . S . 1 7 and S . 1 8 . From
Eq . S . 1 8 ,
- (p U ,) + L1 (P U , )
,
<1> =
GAS RESERVO I R E N G I N E E R I N G
c dp = - dp .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (S . 23)
p
For a fluid of small and constant compressibility and with p b de­
fined as density at a low base pressure ,
to obtain
Pb, we integrate Eq. S . 23
P = P b exp[c( p - P b )] ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . (S . 24)
We shall use Eq . S . 24 as an EOS , assuming a slightly compressi­
ble fluid of constant compressibility .
To derive the diffusivity equation, we combine the continuity
equation (Eq . S . 1 6 ) , the equation of motion (Eq. S . 2 1 ) , and the
EOS (Eq . S . 24) . Combining Eqs . S . 1 6 and S . 2 1 , we obtain
��
r ar
(
rp
� ap
p. ar
)
=
1
�
0 . 0002637 at
( p � ) . . . . . . . . . . . . . . . (S . 2S)
Assuming constant permeability and viscosity and applying the
chain rule gives
1 a
� ar
( )
rp
ap
ar
=
1
p.
(
ap
0 . 0002637 k � at
+p
a�
a;
)
. . . . . . . . . (S . 26)
Expanding Eq . S . 26 by the chain rule yields
( )
�
-; :; : )
p a
ar
+
r
ap
a;
+
ap ap ap
a; ap a;
=
1
p.
0 . 0002637 k
�p
(-;;
1 ap ap
ap
a;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (S . 27)
83
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
5. 24 ,
ap b exp[c(p-Pb»), . . . . . . . . . . . . . . . . . . . . . . . . . (5. 28)
-=cP
ap
C
aplap=cp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 29)
Cf,
cf=(lIr/» (ar/>lap) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.30)
cp
cf =c + cf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.31)
5. 22, 5.30, 5.31,
5.27
r/>p,C, - p-.
ap . . . . . . . . (5.32)
� !... (/p ) ( ap ) 2
cp
---'--+
r ar ar ar 0. 0002637k at
5.32
�!... (r ap ) + c ( ap ) 2 r/>p,C, ap . . . . . . . . . . (5.33)
r ar ar ar 0. 0002637k at
5. 3 3,c(aplar) 2
From Eq.
where
is assumed to be small and constant. Then,
We define PV compressibility ,
and total compressiblity ,
as
Applying Eqs .
Dividing Eq.
and
by
as
p
we rewrite Eq.
as
gives
p ap f+ c )
=r/>--(c
z at
p ap . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.37)
=r/>c,--,
z at
cf
5.30
5. 22,
Cg = � :; = ; � (� ). . . . . . . . . . . . . . . . . . . . . . . . . (5.38)
5.36 5.37
�!... (r� ap ) = r/>p,cf � ap . . . . . . . . . . . . . . (5.39)
r ar p,z ar 0. OOO2637k p,z at
5.39 5.34),
g
where
is defined by Eq.
and, from Eq.
.
Combining Eqs .
and
.
and using field units, gives
Diffusivity Equation in Terms of Pseudopressure and Pseudo­
time Variables. To express Eq.
in a form similar to the diffu­
sivity equation for liquid flow (Eq.
pseudopressure 1 function as
!, P
we define a
P
pp (p)=2 J -p,z dp, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 40)
Pb
5.39
app = app ap = 2p ap . . . . . . . . . . . . . . . . . . . . . . . . . (5. 4 1)
at ap at p,z at
1 ar (r ap ) = r/>p,cf ap . . . . . . . . . . . . . . . . . . . (5.34) app 2p ap
-=-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 42)
� a a; 0. 0002637k at
ar
ar
5. 4 1 5.42 5. 39,
0.5.34
0002637klr/>p,c,
�!... (r app ) = r/>p,c, app . . . . . . . . . . . . . . . . . . (5. 43)
r ar ar 0. 0002637k at
5. 43
tap(t)= \ �
p,cf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 44)
p =(MIR D(plz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 35)
5.25
atap =- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 45)
at p,c,
1 a (r k ap ) = 1 a (r/> p )
� ar
� --; a; 0. 0002637 at
�
app app atap 1 app . . . . . . . . . . . . . . . . . . . (5. 46)
-=--=-�!... (r� ap ) = 1 1 !... (r/> � ) . . . . . . . . . . . (5.36) at atap at p,c, atap
r ar ar 0. 0002637 k at z
5. 45 5. 46 5. 43,
5. 3 6,
aPp . . . . . . . . . . . . . . . . . (5. 47)
�!... (r app ) =
r ar ar 0. 0002637k atap
Assuming that, for radial flow of a fluid of small and constant
compressibility ,
is negligible compared with the other
terms in Eq.
we obtain (in field units) the partial-differential
equation
Ph
where
is a low base pressure. Now , i n terms o f pseudopres­
sures, the differentials in Eq .
are
and
Eq.
is called the diffusivity equation for liquid flow . The
term
is called the hydraulic diffusivity and fre­
quently is denoted by Tf (with units of square feet per hour) .
p,Z
Substituting Eqs.
and
into Eq.
we obtain the diffu­
sivity equation for gas flow in terms of pseudopressure and time:
5.2.2 DifTusivity Equation for Flow of Compressible Fluids. We
can develop a diffusivity equation for compressible fluid (L e . , gas)
flow using various forms of pressure and time variables . In the fol­
lowing discussion, we derive the diffusivity equation for gas flow
in terms of pseudopressure and time, pseudopressure and pseudo­
time, pressure and time , and pressure-squared and time. From the
real-gas law ,
Assuming constant k and negligible gravity effects , we can re­
write Eq.
as
M P
M
RT
RT
We can modify Eq.
to account for variations in gas proper­
ties with time by defining a pseudotime, 2
o
'
.
..
Then, as a nonrigorous result, 3
and
or
p,Z
Now , expanding the differential on the right side of Eq.
we have
Substituting Eqs.
and
into Eq.
we obtain the diffu­
sivity equation for gas flow in terms of pseudopressure and pseu­
dotime:
r/>
under the assumptions that Lee and Holditch 3 discussed .
pip,z
r/>
5. 3 6
� !... (r� ap ) = r/> a (� ) . . . . . . . . . . . . (5. 48)
r ar p,z ar 0. 0002637k at z
Diffusivity Equation in Terms of Pressure and Time Variables.
If we assume that
is constant, we can write the diffusivity equa­
tion for gas flow in terms of pressure and time. With assumed
constant, Eq.
becomes
.
.
84
GAS RESERVO I R E N G I N E E R I N G
2SOOOO
2SOOOO
200000
t
.u;
Il.
..r
1
,
lSOOOO -
-:r 1()()()()() "'-
SOOOO
0
f
!
.'
.. . �
.. ... . ...... ... ...
t:.. .. ... . �
. ...
---.
J.
•
-
e . _ . e - ... -
.....
.
.
___
�
0
••
. . .. ......
� . ... .
. ... .
. .
. .... . .. . .
.. . •
. ..
SG = 0.6
.
.. .. ...
�
. u;
Il.
..r
.
i
Pressure. psia
1 50000
.......
.. . . .. ... . . . ......
.,.....
.. .. ..
.,... .. - . ... ....
... ...
... . .... .
"'-
.. .• ..
�
/
....
or I ()()()()()
�----II-'�"
'"
SG · l.2
�
I
4000
c.
SG • 1 0
.
.
I..... . . ..
SG = 0.8
.
--
2()()()()()
;... ..... . -.. ..i... - ..
... .. .. .. . .. . .. .. .. · . · . .
--. - ---
.... .
-.......
.
...... --- .
.. .. .. · .. .. .. ..
. .. ... ..... ..
- - . . ... . .
... . .
0
T
.
·
J
.. 1. .. . .. . .
SG =
- . -.. ,- L.
... .
-.
�=M
.. ..... .. .
1 .0
-...
. ... . . .. . . .. . . . .
...
,
SOOOO
10000
S O = 0.6
- /-.
SG = 1 .2
0
4000
2000
8000
6000
10000
Pressure. psia
Fig. 5.2-Range of applicability of pressure methods at 1 00 ° F .
Fig. 5.3-Range of applicability of pressure methods at 200° F .
a(p 2 )
p a;ap = ZI a:We note that
Then,
........ . .
.
.
.
.
.
. .
.
.
.
.
.
.
. . . . . . (5 . 56)
.
.
.
and that
=� : ( ; - � :; )
.
. . . . . . . . . . . . . . . . . . . . . . (5 . 49)
From the real-gas law (Eq . 5 . 35 ) ,
� + ;p [ �
=
(�) t,
C;) R Z
T
.... .... .
.
.
.
.
.
.
.
(5 . 50)
.
Cg = ; + z [ a: (�) t = ; -�G;)
T
'
.
.
.
.
.
.
.
•
.
.
.
.
.
(5
.
51
)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 52)
For conditions of single-phase gas flow and constant <p, we can
replace c with c 1 in Eq . 5 . 52 to obtain
. . . . . . . . . . . . . . . . . . . . . (5 . 53)
Combining Eqs . 5 . 48 and 5 . 5 3 , we obtain the diffusivity equation
for gas flow in terms of pressure and time , under the assumption
that � is constant :
pi
I a (r ap ) = <Pw/ ap
at
� ar a;
. . . . . . . . . . . . . . . . . . . (5 . 54)
�CI
This equation can be linearized if we assume that
can be treat­
ed as a constant and evaluated at average drainage area pressure , p.
Diffusivity Equation in Terms of Pressure-Squared and Time
Variables. If we assume that � is constant, then we can write the
diffusivity equation for gas flow in terms of pressure squared and
time. Rewriting Eq . 5 . 48, we obtain
�� (rp ap ) =
r ar ar
<p �
0 . 0002637k
.
.
.
. ... ..
.
...... ..
.
.
.
.
.
.
.
a (!!.)
at Z
. .
<p�
.
.
.
.
.
.
.
. . . . . . . . (5 . 57a)
.
. . . . . . . . . . . . . (5 .57b)
.
.
.
a(p 2 ) l
[-;- ZI &
0 . 0002637k
c1
.
.
. . . .
.
.
.
.
.
. . (5 . 55)
.
.
.
. . . (5 .58)
.
Simplifying , we obtain the diffusivity equation for gas flow in
terms of pressure squared and time , under the assumption of con­
stant
�Z:
�� [r a(p 2 ) ] = <P�CI a(p 2 )
r ar ar
at
0 . 0002637k
Substituting Eq. 5 . 5 1 into Eq . 5 . 49 gives
:t (�)=Cg � :
g
:t (� ) =C/ � : =�CI : :
or
. . ..
Substituting Eqs . 5 . 5 3 , 5 . 56 , and 5 . 57 into Eq . 5 . 55 gives
assuming isothermal conditions . Combining Eqs . 5 . 22, 5 . 35 , and
5 . 50, we can write gas compressibility as
0 . 0002637k
a(p 2 ) =2p a;ap
&
a(p 2 )
p a;ap = ZI &'
I a [r I a(p2 ) l =
� ar Z a:-
.
.
.
.
.
.
.
.
.
.
.
. . . (5 . 59)
�CI
Again , this equation can be linearized if we assume that
can
be treated as a constant and evaluated at average drainage area pres­
sure , p.
Effect of Pressure-Dependent Gas Properties. Because of the
pressure dependence of the gas properties , Eq. 5 . 39 is a nonlinear,
partial-differential equation . If we assume that the quantity �
is constant with respect to pressure and that
can be evaluated
at p and treated as constant, we can solve Eq. 5 . 39 in terms of pres­
sure (Eq . 5 . 54) . Generally , these assumptions are valid only for
very high pressures and temperatures. Figs. 5.2 through 5.4, which
show the relationship between pressure and � at different tem­
peratures for various gas gravities , illustrate that
is essential­
ly constant with pressure at pressures exceeding 3 ,000 psia for
1 00 ° F , 5 ,000 psia for 200 ° F , and 6 , 500 psia for 300 ° F . In gener­
al , we note that the higher the gas gravity and pressure , the more
the
product varies with pressure. Figs. 5 . 2 through 5 . 4 imply
that the solutions to the real-gas diffusivity equation in terms of
pressure, which assume a constant � product, should be used
only for gases at very high pressures.
Assuming that
is constant with pressure and that
can be
evaluated at p and treated as constant, we can solve Eq . 5 . 39 in
terms of pressure-squared (Eq . 5 . 59) . The assumption of constant
� is valid only for very low pressures and gas gravities at high
temperatures . Figs. 5.5 through 5.7 illustrate the variation of the
product � with pressure for different gas gravities at temperatures
of 1 00 , 200 , and 300 ° F , respectively . Note that � is essentially
pi
�Cg
pi pi�Z
p/�z
�z
pi
�Cg
85
F U N DAM ENTALS OF FLU I D FLOW I N POROUS M EDIA
2SOOOO
200000
0.
.�
;;;
0.
..
:r
c:..
....
....
... .
. . ..... .
.
... ....
.,��.*. "''' ..... . -
fl.'
�
-,.
.
-
.
.
-
,
0. 1 5
. ....... ..
SO = 0.8
. ... . .. . . . .
..
. ..·7· ·... · • ·
·
�...
1 00000
so =
50000
0
.
... . ....
SO = 0.6
. .... ..
..!... . .. .. .... .. .... .
1 50000
.
. ... .
.
0.20 -,------,
�
. ;;;
0.
..
0. 1 0
..."
:r
1 .2
0.05
..
0
2000
4000
6000
8000
2000
0
t. . ·· :
;f�.
·.
0.00
1 0000
constant with pressure at pressures less than 1 ,200 psia for 1 00 ° F ,
1 ,750 psia for 200 ° F , and 2 , 200 psia for 300 ° F . The higher the
gas gravity , the more the p.z product varies with pressure, so solu­
tions to the real-gas diffusivity equation should be used in terms
of pressure-squared only at very low pressures and gas gravities
at high temperatures.
A more rigorous method of linearizing Eq. 5 . 39 is with the real­
gas pseudopressure transformation introduced by Al-Hussainy et
al. 1 The pseudopressure transformation allows Eq. 5 . 39 to be
solved without the limiting assumptions that certain gas properties
are constant with pressure. Note that the diffusivity equation given
by Eq . 5 . 43 is not completely linear because the p. c/ term depends
on pressure and pseudopressure . As an acceptable approximation,
we assume the quantity p. c / is constant and evaluate it at current
drainage area pressure, p, and denote the product as iig c, . Under
most conditions , the use of pseudo time more completely linearizes
Eq. 5 . 39 .
5.2.3 Diffusivity Equation for Muitiphase Flow. T o develop the
diffusivity equation for multiphase flow , we begin with principles
and then develop continuity equations for each phase flowing in
the reservoir. Consider a unit control volume that contains oil, water,
and gas with saturations So , S w , and Sg ' respectively . The mass
of oil in the reservoir in the unit reservoir volume is
c/JSo
mo = -- p os ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 60)
Bo
Similarly , the mass of water in the reservoir in the unit reservoir
volume is
0.
�
.
SO = 0.8
-- '
.
.. .
"", ."",. "
.. .. ..
.
.,
,
..... '<41('
"-..
. ... . ..• · • .. .. . .... .
....
..
·. 4· ... ·
SO = 0 .6
1 0000
8000
6000
4000
c/JS w
m w = --p ws ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 6 1 )
Bw
Gas exists i n two phases: a s free gas and a s gas dissolved i n the
oil and water. The mass of the free gas is
m g .F =
c/J Sg
B
g
PgS ;
the mass of the dissolved gas is
Therefore, the total mass of gas in the unit control volume is
c/J Sg
c/JRs So
c/JR sw S w
m g = -- Pgs + -- Pgs +
Pgs ' . . . . . . . . . . . . (5 . 62)
Bg
Bo
Bw
Eq. 5 . 62 contains terms representing both free gas (terms with
the subscript g) and gas dissolved in oil and water (terms with the
subscripts 0 and w , respectively) .
We can use Darcy ' s law for an equation o f motion into the con­
trol volume for each phase. For the oil , water, and gas phases,
op
ko
P ouro = - 0 . 0002637 -- pos - , . . . . . . . . . . . . . . . . . (5 . 63)
or
p. o B o
0 10
.
0. 1 2
0.08
.•
••
Fig. 5 . 5-Range of applicability of pressure-squared methods
at 1 00 ° F.
0. 1 4
�'"
.
. .•
. ..
.
.«
1 .0
Pressure, psia
Fig. 5 .4-Range of applicability of pressure methods at 300 ° F.
0.
,.It
..
.. ..A:
....• ·..
.. •
: .a ·:.. ·
Pressure, psia
0.10
�· ·
It··t·
/" .A1"
SO .
0.
0.
�
. ;;;
0.
.;
0.06
SO = 1 .0..
0.08
0.06
:r 0.04
0.04
0.02
-.,..
0,02
0.00
0.00
0
2000
4000
6000
8000
1 0000
Pressure, psia
Fig . 5 .6-Range of applicability of pressure-squared methods
at 200 ° F.
/ ..•
.
0
£
2000
•
.A:
�
··
.
SO = 0.8
/ .ff'
. ,. .
,
.. ...
..
.
•
• • It
.A/' K
• • • . . ...
,�
.
. .
.
.. ..
. ... . .... . .... .. . � .
·
SO = 0.6
.•.4
�:�...-:: :
. 1O�
.. . � ... It
..
4000
"
:
;
6000
.
.
8000
.4
\
1 0000
Pressure, psia
Fig. 5 . 7-Range of applicability of pressure-squared methods
at 300 ° F.
86
GAS RESERVO I R E N G I N E E R I N G
ap
kw
P w urw = -0.0002637--p ws - , . . . . . . . . . . . . . . . . (5 .64)
/l w B w
ar
and P g Urg -0.0002637
=
kg
ap Rs ko ap Rsw k w ap
x -- P gs - + --P gs - +
P gs - . . . . . (5 .65)
ar
/l g Bg ar /lo Bo ar /lwBw
(
--
)
A continuity equation can now be written for each phase because
the rate of mass into the control volume less the rate of mass out
is ( lIr)(a/ar)(rpur), and the rate of change of mass in the control
volume is am/at. Therefore, the continuity equation for any Phase
x is
1 a
amx
- - ( - rpx urx ) = -- '
r ar
at
For oil, we combine Eqs .
ty equation,
)
(
(
(
)
( )
(
)
( )
so the oil equation becomes
Similarly , for water,
cP
� � � ap
� Sw , . . . . . . . . . (5 .67)
r
=
0.0002637 at B w
r ar /l w B w ar
) ]
kg ap
1 a
Rs ko Rsw k w
r
+
+
� ar /lo Bo /lw B w /l g Bg a;
(
)
X � Rs So + Rsw Sw + � . . . . . . . . . . . . . . . . . . . . . (5. 7 1)
at Bo
Bw
Bg
Now, we expand the partial derivatives with respect to time for
Eqs . 5.66 through 5 . 68. Because saturations, FVF ' s , and solubili­
ties are functions of pressure, the oil equation becomes
We can expand the oil equation, Eq.
to obtain
For convenience, we let all partial derivatives with respect to pres­
sure be denoted by B� , S� , etc . , and we use the fact that
( )
1
� _
_I_ aBo
�
.
= (B o- I ) = _
ap Bo
ap
BJ ap
( )
1 a ap
/loBo S� SoB� ap
cP
r
a; '
=
� ar a; 0.0002637 ko Bo B3
( )
)
5 .66, using the product rule
( )
( )
a So
1 ko
a ap ap a
ko
- -- - r- + - - -- = 0.0002637- - .
at Bo
r /lo Bo ar ar
ar ar /lo Bo
If we assume that ko is constant with respect to pressure , we
know that ko is a function of both oil and water saturations, So and
Sw , whereas /lo and Bo are functions of pressure but not satura­
tion. We can then use the chain rule to expand the second term
to obtain
)
(
Therefore,
and the oil equation becomes
a Rs So Rsw Sw Sg
cP
= 0.0002637 a; --;;: + ----;;:- + Bg . . . . . . . . . . . . (5 .68)
( ) ( )
(
. . . . . . . . . (5.70)
)
a So
1 a
ko ap
cP
. . . . . . . . . . . (5 .66)
r
� ar /lo Bo ar = 0.0002637 at Bo
[(
( )
1 a ap
/l w B w a Sw
cP
r
=
-;::- at B w
-� ar a; 0.0002637
5.60 and 5 . 63 to form the oil continui­
1
1 a
a cP So
ap
ko
r
P as
=
Ps .
a;
Bo a
� ar /lo Bo ar 0.0002637
Because P os is the density at standard conditions, it is a constant,
and for gas,
( )
Similarly , for water and gas, we can write
)
(
1 a ap
/lo
SoB� ap
cP
. . . . . . (5 . 72)
r
S� =
;: a; '
--;
0.0002637 ko
� ar ik
( )
1 a ap
- - r- =
r ar
ar
cP
0 . 0002637
)
(
Similarly , for water,
/l w
SwB �' ap
S,:, - -- - . . . . . . (5 .73)
kw
B w at
The gas equation is slightly more difficult because it contains solu­
bility terms that also are functions of pressure . Expanding the right
side of Eq. 5 .68 with the chain rule gives
( )
(
(
)
-I
1 a ap
k
cP
Rs ko Rsw k w
r
+
+ g
=
� ar a; 0.0002637 /lo Bo /l w B w /lg Bg
)
X � � Rs So + Rsw Sw + � ap .
Bw
Bg at
r ap Bo
Or we can write
( )
a So
cP
= 0.0002637 at B o .
If we assume that (ap/ ar) 2 and the products of the pressure and
saturation gradients , [(ap/ar)/(a so/ar)] and [(ap/ar)(aSw /ar)] , are
( )
( )
negligible compared with these gradients , the oil equation becomes
l a ap
/lo B o a So
cP
. . . . . . . . . . . (5 .69)
r
=
� ar a; 0.0002637 ko at Bo
+
)
Rs:v Sw Rsw Sw B .(; S� Sg B�
+ - - -- . . , . . . . . . . . . . . (5.74)
Bg B�
Bw
Ba,
--
We now define mobility as the ratio of the effective permeabil­
ity of the porous medium to fluid viscosity. Eqs. 5.75 through 5.77
define the oil , water, and gas mob ilities, respectively :
87
F U N DAM ENTALS OF FLU I D FLOW I N POROUS M E DIA
A o = ko /p. o ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .75)
A w = kw /p. w , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .76)
and A g = kg /p. g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .77)
After we substitute Eq. 5 . 75 into Eq . 5 .72, the oil equation
)
(
( )
becomes
Grouping terms on the right side gives
1 a ap
cf>
SoB� ap
. . . . . . . . (5 .78)
r
S� - ; ar a;. = 0.0002637)... 0
--;: ;;;
Similarly, combining Eqs . 5.76 and 5 .73 for water and Eqs . 5.77
and 5 . 74 for gas gives
(
( )
)
1
So B�
_
s� Bo
Ao
)
(
)
(
)
(
)(
=
So B�
Sw B�
AW
. . . . . . . . . . . . . . . . . (5 . 81)
s� _
= s� _
Bo
Bw
Ao
Similarly , we equate the right sides of the oil and gas equations
(Eqs .
and
again realizing that the left sides are
equivalent:
5 .78
5 . 80),
�(s�- So B� ) ap = <I> ( Rs Ao
Ao
+
Bo
at
Bo
+
Rsw Aw
Bw
+
)
Rs'w Sw Rsw Sw B� S� Sg B�
.
+ _
_
Bw
B a,
Bg Bi
Canceling like terms and rearranging gives
)
(
(
(
)
Ag - l
Bg
[S� - (So B�)lBo] '
�)
)( ) [
(
1
So Bo �
Sg B�
R;So Rsw Sw _
+
+
=
s� Bo
Bo
Bw
Ao Bg
Bg
Bg
or multiplying through by Bg , we obtain
S� -
(
, So B�
So - -Bo
)( ) (
)] ,
)
Ag
, Sg B�
R; So Bg Rs'w Sw Bg
+ S g - -- .
+
-- -_
Bg
Bw
Bo
Ao
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 83)
---
We know that, for a three-phase system ,
Sg + Sw + So = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .84)
Differentiating Eq. 5 . 84 with respect to pressure gives
S� +S� + S� = O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 85)
Also, the total mobility of a three-phase system is the sum of
the individual mobilities :
A t = A O + A W + Ag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .86)
Substituting Eqs . 5 . 85 and 5 . 86 into Eq . 5 .83 for Sg and Ag ,
(
, So B�
So - -Bo
Rs S� R; So Rs So B� Rsw S�
+
+
_
Bo
Bo
Bt
Bw
]
R;SO Rs'w Sw S� Sg �
,
+ _
+
Bo
Bw
Bg Bg
)(
)
respectively , gives
=
)
which simplifies to
)
Simplifying gives
Rs Ao Rsw Aw Ag
+
+
Bo
Bw
Bg
Factoring all terms on the left side containing
we obtain
5 . 79) ,
� S� - So B� ap = !...- s� - SwB� ap .
Ao
Bo at Aw
B w at
]
S� Sg B�
Rsw Aw , So B�
+ - - So - -- + - - -- .
B w Ao
Bo
Bg Bi
respectively .
Next, we derive a single equation to describe multiphase flow
through a porous medium. First, we equate the right sides of the
realizing that the left
oil and water equations (Eqs .
and
5 . 78
)(
(
Rs'w Sw Rsw SwB� S� Sg B�
, . . . . . . . . . . . . . (5 . 80)
+ --z;:Bg Bi
B a,
sides of both are equivalent:
)
(
1 a ap
SwB� ap
cf>
. . . . . . (5 .79)
r
S� - =
; ar ar
0.0002637 Aw
-;:: ;;;
+
(
S� Sg B�
Rsw , Sw B�
. . . . . . . . . . . . . . . (5. 82)
+ Sw B a,
Bw
Bg Bi
Substituting Eq. 5 . 8 1 into Eq. 5 . 82 gives
+
A t - Ao - Aw R; So Bg Rs'w SwBg
+ ----"=
Bo
Bw
Ao
, S B'
- So' - Sw _ --L..!..
Bg
(So B�)/Bo
If we add the terms
the equation, we can
[S� - (So B�)/Bo] :
again
(Sw B�)/B w to each side
and
group
terms
that
of
contain
88
GAS RESERVO I R E N G I N E E R I N G
W e can now substitute Eq. 5 . 94 into the original oil , gas, o r water
equation to obtain the diffusivity equation for rilUltiphase flow. For
the oil phase,
Sw B";' Sg B�
So B�
R; So B R 'w Sw Bg
----"-g + s
.
- S� + -- - S";' +
Bo
Bw
Bo
B w -Bg
--
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 87)
Recall Eq. 5 . 8 1 :
)
) (
(
Sw B";'
So B�
Aw
.
= S";' S� Ao
Bw
Bo
(
So B�
So - -Bo
)(
)
)
becomes
( ) (
)
1 a ap
ap
tPc t
r a; =
'
ih
0.0002637At
� ar
. . . . . . . . . . . . . . . . (5 . 95)
Eq. 5 . 95 is similar to the single-phase diffusivity equation for
slightly compressible liquids:
( ) (
)
ap
tPP, o ct
� � r ap =
,
r ar ar
0.OOO2637ko at
Substituting Eq. 5 . 8 1 into Eq. 5 . 87 yields
,
(
( )
So B� ap
tP
� � r ap =
. . . . . . . (5 . 78)
S� r ar ar 0.0002637Ao
Bo at
A t - A o - A w So B� Sw B";'
+ -- +
Ao
Bo
Bw
--
. . . . . . . . . . . . . . . . (5 . 96)
which implies that the solutions to the single-phase diffusivity equa­
tion presented later in this chapter apply to multiphase flow as long
as c t is defined by Eq. 5 . 93 and we use A t instead of k o / P, o .
5.2.4 Dimensionless Forms of the Diffusivity Equation. The
_
Sg B�
Bg
Again, factoring all terms containing
side gives
[S� - (So B�)/Bo] on the left
So B� Sw B";' Sg B�
R;So B R 'w Sw Bg ----"-g + s
- -Bo
Bw
Bo
Bw
Bg
--
or finally ,
(S� - SBo B� )(�A ) = R; BSooBg
o
_
o
Sw B";' _ Sg B�
Bg
Bw
_
So B� R s'w S w Bg
+
Bo
Bw
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 88)
For a system with multiple phases,
C t = So c o +Sw cw + Sg Cg ,
. . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 89)
where , below the bubblepoint,
1
c o = - - (B� -R; Bg ) . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 .90)
Bo
Similarly , water compressibility , c w ' below the bubblepoint is
1
c w = - - (B";' -Rs'wBg ) , . . . . . . . . . . . . . . . . . . . . . . . . (5 . 9 1 )
Bw
while gas compressibility , C g ' is
cg = -B�/Bg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 92)
.
Substituting Eqs . 5 . 90 through 5 . 92 into Eq . 5 . 89 gives the ex­
pression for calculating total compressibility :
Sw B";' _ Sg B�
Bg
Bw
. . (5 . 93)
( )
1 a ap
tPp,ct ap
r a; =
� ar
0.0002637k at
. . . . . . . . . . . . . . . . . . . (5 . 34)
The initial condition is
p(r,t=O) =Pi '
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 97)
.
.
The inner boundary condition (constant-rate production,
r=rw ,t > O is
q=
:
0.001 l 2 (27rrw h) k
p,
(:)
qBp,
.
0.OO7082kh
------
r = rw
. . . . . . . . . . . . . . . (5 .98a)
. . . . . . . . . . . . . . . . . . . . . (5 . 98b)
The outer boundary condition (no-flow outer boundary,
r=re ,t > O) is
q =O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 99a)
=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 99b)
or (ap/ar)
For this problem, we choose rw as a convenient reference length.
.
r - re
_
Substituting Eq . 5 .93 into Eq . 5 . 88 yields
(s� - SBoo B� )(�A ) = ct . . . . . . . . . . . . . . . . . . . . . . . . . . 5
o
diffusivity equation has several parameters ( tP , p" cp and k) that
describe a wide variety of specific situations . By defining some
dimensionless variables , we can write the diffusivity equation in
a convenient, general form. These dimensionless variables are not
unique but are defined (rather than derived) quantities selected for
convenience in a particular situation. To write the diffusivity equa­
tion with dimensionless variables , we find logical groupings of vari­
abIes that appear in the differential equations and initial and
boundary conditions .
First, w e consider radial flow o f slightly compressible liquids
being produced at constant rate and at constant bottomhole pres­
sure (BHP) . We then derive the dimensionless diffusivity equation
for gas in terms of pseudopressure, pressure, and pressure-squared
variables .
Constant-Rate Production, Slightly Compressible Liquids. For
the case of constant-rate liquid production , we assume ( 1 ) uniform
pressure , Pi ' throughout the reservoir before production,
(2) constant-rate production from a single well of radius r w cen­
tered in a cylindrical reservoir, and (3) no flow across the reser­
voir outer boundary at radius re o The diffusivity equation and the
initial and boundary conditions are expressed mathematically as
follows.
The diffusivity equation is
( . 94 )
Using the reference length , we define dimensionless radius as
rD = r/rw' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 100)
We need to define a dimensionless pressure group, so we work
with the time-independent inner boundary condition. Substituting
Eq. 5 . 100 into Eq. 5 . 98b (with r = rw and hence rD = 1) gives the
inner boundary condition as
qBp,
( )
ap
or
arD
qBp,
--
. . . . . . . . . . . . . . . . . . . . . (5 . 10 1 )
0.007082kh
If we tentatively let dimensionless pressure be PD =plp c , where
Pc is some characteristic pressure (a constant) , and substitute it into
Eq. 5 . 10 1 , we obtain
----
rD = 1
(Pc aarPDD )
qBp.
. . . . . . . . . . . . . . . . . . . . (5 . 102)
0.007082kh
With the initial condition (Eq. 5 . 97) in mind, we would like
PD = O when p =p i ,PD to increase with time, and pD to be posi­
tive. These preferences suggest that we rewrite P D in the form
P D = ( Pi -p)lp c ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 103)
Differentiating Eq. 5. 103 gives
---
rD = i
1 ap
apD
-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 104)
arD
Pc arD
-
( aarPDD )
( aarDPD )
Substituting Eq. 5 . 1 04 into Eq. 5 . 10 1 yields
- Pc
or
qBp,
qBp.
rD = i
0.OO7082khp c
. . . . . . . . . . . . . . . . . (5 . 105)
We now choose
Pc =
qBp,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 106)
0.OO7082kh
so that the inner boundary condition simplifies to
( :D )
D
= - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 107)
rD = i
0.0070 82kh( Pi - P )
and p D =
. . . . . . . . . . . . . . . . . . . . . (5 . 108)
qBp.
( )
Substituting from Eq. 5 . 1 00 into Eq. 5 . 34 gives
ap
cPp.c t r � ap
� _a_
=
rD
. . . . . . . . . . . . . . (5 . 1 09)
rD arD
arD 0.0002637k at
Substituting from Eq. 5 . 1 03 into Eq. 5 . 109 yields
[
]
1 a
a
cPp,c t r � a
- - rD - ( p i - PDPc ) =
( Pi -P DPc ) '
rD arD
arD
0.OO02637k at
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 0)
Noting that ap;larD = 0 and ap;lat=o and dividing by - Pc ' we
have
( )
a_ aP D
cPp.c t r � aP D
rD
=
. . . . . . . . . . . . . (5 . 1 1 1)
rD arD
arD 0.0002637k at
_l_
_
cPp, c t ra
0.0002637k
We now determine a dimensionless time group. Letting dimensionless time be tD = tltc , where tc is some characteristic time (a
constant) , Eq. 5 . 1 1 1 becomes
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 3)
then Eq. 5 . 1 12 simplifies to
� : (rD :; ) :;
=
r aD
where tD =
0.0002637kt
cPp. c t r�
, . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 4)
. . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 5)
Finally, we express the outer boundary condition in dimension­
less terms. Using Eq. 5 . 1 00 with rw * O, Eq. 5 . 99b becomes
)
c:
( :D )
a
D
=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 16)
rD = re 1rw
Using Eq. 5 . 1 04 in Eq. 5 . 1 16 with nonzero Pc ' we get
D
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 7)
rD = re 1rw
Defining a dimensionless outer radius as reD = re 1r w yields
( :D )
D
0.007082kh
rD = i
aP D
1 a
cPp, c t ra apD
-- . . . . . . . . . . (5 . 1 12)
- -- rD - =
rD arD
arD 0.0002637k a(tDtc )
If tc =
0 .OO7082kh
89
( )
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 8)
rD = reD
In summary, the diffusivity equation and initial and boundary con­
ditions for constant-rate liquid production with a no-flow outer
boundary are as follows.
The diffusivity equation is
�r a:D (rD :; ) ::
=
. . . . . . . . . . . . . . . . . . . . . . . (5 . 1 1 9)
The initial condition is
PD ( rD ,tD =O) = O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 120)
The inner and outer boundary conditions are
( :D )
( aa: )
D
and
rD = i
D
= - 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 2 1)
=0, . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 122)
rD = reD
respectively.
Constant-BHP Production, Slightly Compressible Liquids. For
liquid production at constant BHP, we assume ( 1 ) uniform pres­
sure, Pi , throughout the reservoir before production, (2) constant
BHP during production from a single well of radius r w centered
in a cylindrical reservoir, and (3) no flow across the reservoir out­
er boundary at radius re ' The diffusivity equation and the initial
and boundary conditions can be expressed mathematically as
follows.
The diffusivity equation is
a
(
r )=
ar ar
1 a
�
p
cPp.ct
ap
0.0002637k at
. . . . . . . . . . . . . . . . . . . (5 . 34)
and the initial condition is
p(r,t=O) =Pi ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 123)
90
GAS RESERVO I R E N G I N E E R I N G
The inner boundary condition (constant-BHP production,
r = rw , t > O) is
p ( rw , t = O) =pwj ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 124)
the outer boundary condition (no-flow outer boundary, r = re ,t > O)
is
( : ) r = re
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 125)
We now define the dimensionless variables for this case. Dimen­
sionless radius, rD ' is defined by Eq. 5 . 1 00; while dimensionless
time, tD ' is defined by Eq. 5 . 1 15 . The development of these vari­
ables and the dimensionless diffusivity equation (Eq. 5 . 1 14) are
described by Eqs. 5 . 1 10 through 5 . 1 15 . The dimensionless form
of the outer boundary condition is developed in Eqs. 5 . 1 17 and
5 . 1 1 8 . The initial and inner boundary conditions suggest that we
want P D = 0 when P = Pj , P D = 1 when P = P wj' P D to increase with
time, and P D to be positive. Thus,
PD =
Pj - P
' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 126)
Pj -P w[
--
Substituting Eq. 5 . 126 into Eq. 5 . 124 simplifies the inner bound­
ary condition to
( P D) rD = l = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 127)
In summary, the diffusivity equation and initial and boundary con­
ditions for liquid production at constant BHP with a no-flow outer
boundary are as follows.
The diffusivity equation is
� � (rD ::) = ::
r aD
. . . . . . . . . . . . . . . . . . . . . . . (5 . 128)
The initial, inner boundary, and outer boundary conditions are
P D(rD ,tD =0) =0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 129)
( P D) rD = l = l , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 30)
and
( :DD ) rD =reD
.
=0, . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 13 1)
respectively.
Constant-Roie Gas Production. Pseudopressure and Pseudotime
Formulation. For constant-rate gas production, we assume
( 1 ) uniform pressure, Pj , throughout the reservoir before produc­
tion, (2) constant-rate production from a single well of radius r w
centered in a cylindrical reservoir, and (3) no flow across the reser­
voir outer boundary at radius re ' The diffusivity equation and the
initial and boundary conditions in terms of pseudopressure and pseu­
dotime can be expressed mathematically as follows.
The diffusivity equation is
( )
r ar ar
� � ,app
app
. . . . . . . . . . . . . . . . . (5 . 47)
0.OOO2637k atap
cf>
and the initial condition is
pp (r,t=O) =ppj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 132)
The inner boundary condition (constant-rate production, r=rW '
t > O) is derived beginning with Eq. 5 . 98b,
qBp.
0.007082kh
where q = gas production in STBID. For q in ft 3 1D, Eq. 5 . 98b is
qBp.
. . . . . . . . . . . . . . . . . . . . . . (5 . 133a)
0.03977kh
In addition, the
B=
z Tp sc
FVF ,
B , for gas is
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 133b)
--
z sc TscP
From Eq. 5 .42, we can write the pressure derivative in Eq. 5.98b:
ap p.Z app
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 33c)
=
ar 2p ar
Substituting Eqs. 5 . 1 33b and 5 . 1 33c into Eq. 5 . 1 33a and solv­
ing for the derivative of pseudopressure with respect to radius yields
an equation for the inner boundary condition:
50,300P sc Tq
--...:..::.... . . . . . . . . . . . . . . . . . . . . (5 . 1 33d)
khTsc
The outer boundary condition (no-flow outer boundary, r = re ,
t > O) is
( a:; ) r= re
=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 134)
rD is defined by Eq. 5 . 100. The inner boundary condition (Eq.
5 . 1 33) suggests that we define dimensionless pseudopressure as
kh Tsc ( ppj -pp )
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 1 35)
PpD =
50,300qTp sc
Eq. 5 .47 suggests that we define dimensionless pseudotime as
0.0002637ktap
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 136)
-----'cf> r a
With substitutions from Eqs. 5 . 1 00 , 5 . 135, and 5 . 1 36, Eq. 5 . 47
simplifies to the dimensionless diffusivity equation in terms of pseu­
dopressure and pseudotime. Substituting Eq. 5 . 1 35 into Eq. 5 . 1 32
reduces the initial condition to dimensionless form. The inner and
outer boundary conditions (Eqs. 5 . 133 and 5 . 1 34) are reduced to
dimensionless form by substitution of Eqs. 5 . 1 00 and 5 . 1 35 . The
resulting diffusivity equation and initial and boundary conditions
for constant-rate gas production in terms of pseudopressure and
pseudotime are as follows.
The diffusivity equation is
� � (rD a::: ) = ;:;
r aD
, . . . . . . . . . . . . . . . . . . . . (5 . 137)
and the initial condition is
ppD(rD ,taD = 0) =0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 138)
The inner boundary condition (constant-rate production, r=r W '
t > O) is
( aaPrPDD ) rD = l =
- 1 ', . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 139)
the outer boundary condition (no-flow outer boundary, r=re , t > O)
is
( a::DD ) rD =reD =
0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 140)
Pressure and Time Formulation. The diffusivity equation and the
initial and boundary conditions in terms of pressure and time can
be expressed as follows.
The diffusivity equation is
( )
� � , ap =
r ar
ar
ap
. . . . . . . . . . . . . . . . . . . . . (5.54)
0.OOO2637k at
cf>p. c t
91
F U N DAM ENTALS OF FLU I D FLOW I N POROUS M EDIA
The initial condition is
p(r ,t=O) = Pi ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.141 )
the inner boundary condition (constant-rate production, r =r w ,t
> 0) is
25 ,1 50P sc Tqp.Z
. . . . . . . . . . . . . . . . . . . (5 .142)
pk hTsc
and the outer boundary condition (no-flow outer boundary , r =re'
t > O) is
(�r )
u
r=re
=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 143)
5 . 142)
The inner boundary condition (Eq .
is similar to that for
the case of pseudo pressure and pseudotime in Eq.
Eq.
can readily be obtained from Eq .
by assuming constant pip.Z
in Eq .
and substituting the resulting expression , p( 2plp. z) , for
in Eq .
i s defined i n E q .
The inner boundary condition (Eq.
suggests that we define dimensionless pressure as
Pp
rD
5 . 142)
5 .40
5 . 133.
5 . 133
5 . 100.
pk h Tsc( P i -p)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 144)
PD =
25, 150qTji z p sc
Note that we chose to evaluate pip.Z , which is assumed to be con­
stant, at average drainage area pressure,
Eq.
suggests that
p.
5 .54
0.0002637k t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 145)
tD =
tPjic/�
Here , we simply assume that we can evaluate p.c( at p with satis­
----
factory results .
With substitutions from Eqs .
and
Eq.
simplifies to the dimensionless diffusivity equation in terms of pres­
sure and time , assuming constantplp.Z . Substituting Eq.
into
Eq .
reduces the initial condition to dimensionless form . The
inner and outer boundary conditions (Eqs .
and
are
reduced to dimensionless form by substitution of Eqs .
and
The resulting diffusivity equation and initial and boundary
conditions for constant-rate gas production in terms of pressure and
time , assuming constant plp.Z , are as follow s .
The diffusivity equation is
5 . 100, 5 . 144,
5 . 142
5 . 144.
5 . 145,
5.54
5 . 144
5 . 142
r =re ,t> O)
50,300p sc Tqjiz
and
(:)
5 . 143)
5 . 100
r=re
. . . . . . . . . . . . . . . . . (5 .1 51 )
=0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 152)
respectively .
The inner boundary condition (Eq .
is similar to that for
the case of pseudo pressure and pseudotime in Eq.
Eq.
can readily be obtained from Eq.
by assuming constant p.z
in Eq.
and substituting the resulting expression , p 2 1p.Z, for
in Eq .
defining gas properties at average drainage area
pressure,
i s defined by Eq.
Eq .
suggests that we define
as in Eq .
The inner boundary condition (Eq .
sug­
gests that
5 . 151)
rD
5 . 133.
5 . 133
5.40
5 . 133,
p.
5 . 142 Pp
5 . 133.
r =r w ,t> O)
The inner (constant-rate production ,
and outer (no­
flow outer boundary,
boundary conditions are
5 .1 00.
5 . 145 .
5.59
5 . 15 1
5 . 151)
tD
k hTsc ( p f _p 2 )
. . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 153)
50,300qTji z p sc
With substitutions from Eqs . 5 . 100, 5 . 145, and 5 . 153, Eq . 5.59
PD =
simplifies to the dimensionless diffusivity equation in terms of pres­
sure squared and time , assuming that p.Z is constant. Substituting
Eq.
into Eq .
reduces the initial condition to dimen­
sionless form . The inner and outer boundary conditions (Eqs .
and
are reduced to dimensionless form by substitution of
Eqs .
and
The resulting diffusivity equation and ini­
tial and boundary conditions for constant-rate gas production in
terms of pressure and time , assuming constant p.Z , are as follows .
The diffusivity equation is
5 . 153
5 . 152)
5 . 100
5 . 150
5 . 151
5 . 153 .
� : (rD :;) = ::
r OD
.
. . . . . . . . . . . . . . . . . . . . . . (5 . 154)
The initial condition is
P D(rD ,tD =0) =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 155)
The inner boundary condition (constant -rate production , r = r w '
t > O) is
OP D
=
; . . . . . . . . . . . . . . . . . . . . . . (5 . 146)
rD
= - 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 156)
r OD
o rD ,v=1
the initial condition is
and the outer boundary condition (no-flow outer boundary , r = r e'
P D(rD ,tD) =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 147) t > O) is
The inner boundary condition (constant-rate production , r = rw ,t
PD
> 0) is
=0. . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 157)
rD 'V='eV
OP D
= - 1 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 148)
o rD ,v=1
5.3 Solutions to the DiHusivity Equation
We now state some useful solutions to the diffusivity equation
and the outer boundary condition (no-flow outer boundary , r =re'
describing the flow of a slightly compressible liquid in a porous
t > O) is
medium . Four solutions to Eq . 5.34 are particularly useful in well
D
testing: (1) the solution for a bounded cylindrical reservoir, (2) the
=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 149) solution
for an infinite reservoir with a well treated as a line source
r D 'V='eV
with zero wellbore radius , (3) the pseudosteady-state solution, and
Pressure-Squared and Time Formulation. The diffusivity equa­ (4) the solution that includes wellbore storage for a well in an in­
� : ( :;) ::
( )
( )
(: )
.
u
.
.
.
(� )
u
tion and the initial and boundary conditions in terms of pressure
squared and time can be expressed as follow s .
The diffusivity equation i s
[ ]
tPp.c( 0 ( p 2 )
�� 0(p 2 )
r
=
. . . . . . . . . . . . . . (5 . 59)
r or
or
0.0002637k ot
The initial condition is
p(r ,t=O) =P i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 150)
finite reservoir .
Before discussing these solutions, we should summarize the as­
sumptions used in developing Eq.
Specifically , these assump­
tions include
a homogeneous and isotropic porous medium of
uniform thickness ,
pressure-independent rock and fluid prop­
erties ,
small pressure gradients ,
radial flow,
applicability
of Darcy ' s law (Le . , laminar flow) , and
negligible gravity ef­
fects . Further assumptions will be introduced as needed to obtain
solutions to Eq.
(1)
(3)
5.34.
(2)
5.34.
(4)
(6)
(5)
GAS RESERVOI R E N G I N E E R I N G
92
kh( Pi -P)
Cente r of
Wellbo re
where PD=
...........................(5.162)
14 1.2 qBt-t
The exponential integral or Ei function is defined as
J
-Ei(-x)=
, co e-U
x
r
w
5 . 3 . 1 Bounded Cylindrical Reservoir. Solution of Eq. 5.34 re­
quires that we specify two boundary conditions and an initial con­
dition. A realistic and practical solution is obtained if we assume
that ( 1) a well produces at constant rate, qB, into the wellbore (where
q=flow rate in STBID at surface conditions and B=FVF in
RB/STB) ; ( 2) the well , with wellbore radius r w is centered in a
cylindrical reservoir of radius re , with no flow across this outer
boundary ; and (3) the reservoir is at uniform pressure , P i' before
production.
The most useful form of the solution relates flowing pressure ,
P wj, at the sandface to time and to reservoir rock and fluid prop­
erties. In dimensionless form , the solution is4
................................ (5.158)
Before we examine the properties and implications of Eq. 5.161 ,
we must answer a logical question: because Eq. 5.158 is an exact
solution and Eq. 5.161 clearly is based on idealized boundary con­
ditions , when (if ever) are pressures calculated from Eq. 5.161 at
satisfactory approximations to pressures calculated from
radius
Eq. 5.158? Analysis of these solutions6 shows that the Ei-function
solution is an accurate approximation to the more exact solution
for times in the range
the assumption of rw=O
For times less than
(i.e., assuming the well to be a line source or sink) limits the ac­
the
curacy of the equation. At times greater than
reservoir boundaries begin to affect the pressure distribution in the
reservoir, so the reservoir is no longer infinite-acting.
A further simplification of the solution to the flow equation is
possible. For x<0.0 2 , Ei( -x) can be approximated with a
error by
kh( P i -P wj )
4 . t-t
The terms an are the roots of
J \ (anr eD) Y\ (a n ) -J \ (an ) Y\ ( anr eD) =0 , ............(5.160)
where J \ and Y\ are Bessel functions. ct is used in all equations
---'----''-. .............................(5.159)
1 1 2 qB
in this chapter because even formations that produce a single-phase
oil contain an immobile water phase and have formation compress­
ibility.
Although Eq. 5.158 may appear formidable , using this equation
is un­
in its complete form to calculate numerical values of
necessary. Instead , we shall use limiting forms of the solution in
most of our computations. The most important fact about Eq. 5.158
is that , under the assumptions made in its development, it is an ex­
and is sometimes called the van Everdingen­
solution to Eq.
Hurst5 constant-terminal-rate solution , discussed in Sec. 5.7. Be­
cause the solution is exact, it serves as a standard with which we
may compare more useful, albeit approximate, solutions. Two such
approximate solutions are the solution for an infinite cylindrical
reservoir with a line-source well and the pseudosteady-state solution.
P wj
5.34
act
5.3.2 Infinite Cylindrical Reservoir With Line-Source Well. We
assume that ( 1) a well produces at constant rate , qB; ( 2) the well
has zero radius ; (3) the reservoir is at uniform pressure, P i ' be­
fore production, and (4) the well drains an infinite area (i.e., P -+P i
as r -+oo). Under these conditions , the solution to Eq. 5.34 is
�( : )
Ei
- 948 t-t Ctr 2 .
(3.79XI05 c/>t-t ctr a lk ) < t«948c/>t-t ctr;lk ) .
3.79 x 105 c/>t-t ctr a lk ,
948c/>t-t ctr;lk ,
< 0.6%
Ei( -x)=ln( 1.78 1 x) . ............................(5.164)
To evaluate the Ei function, we can use the table in Appendix
D for O.0 2<x51 O.9. For x50.0 2, we use Eq.5.164. For x> 10.9 ,
E i (-x ) can b e considered zero for reservoir engineering appli­
cations.
Near-Wellbore Damage or Stimulation Effects. In practice, most
wells have reduced permeability (damage) near the wellbore. This
damage results from drilling or completion operations. Many other
wells are stimulated by acidization or hydraulic fracturing. Eq.5.161
fails to model such wells properly ; its derivation includes the ex­
plicit assumption of uniform permeability throughout the well
drainage area up to the wellbore. Hawkins 7 pointed out that, if the
damaged or stimulated zone is considered equivalent to an altered
then the
and outer radius ,
zone of uniform permeability ,
across this zone can be modeled by
additional pressure drop ,
the steady-state radial flow equation (Fig. 5.8) .
Therefore , the additional pressure drop resulting from alteration
of the formation near the wellbore is
t:.p s ,
where tD is defined by Eq. 5.1 15 and
P D=-
......................... (5.163)
u
rw
Fig. 5.8-Two-region reservoir model of altered zone near the
wellbore.
PD
- duo
.................... (5.161)
=14 1.2
qBt-t
kh
rs'
k s,
(�ks } ( r w )
-I
n
!.!...
. ............... (5.165)
In dimensionless form ,
(�k s } ( r w )
t:.P SD =
-l
r t:.p sD =
whee
n
!'!"'
kht:.p s
14 1.2 qBt-t
, .......................(5.166)
. .........................(5.167)
Eq. 5.166 states that the pressure drop in the altered zone de­
than on and that a correction to the pressure
pends more on
drop in this region must be made.
Combining Eqs. 5.161, 5.162, and 5.165 , we find that the total
pressure drop at the wellbore is
k
ks
( - 948c/>t-tk t Ctr a ) +t:.ps
[ ( - 948c/>t-tCtr a ) ( k ) ( r s
kh
ks
r w )l
kt
qBt-t .
P i -P wj = - 70.6--El
kh
qBt-t
.
=-70.6- El
-2 --1 In -
.
................................ (5.168)
FU N DA M ENTALS OF FLU I D FLOW I N POROUS M EDIA
(
Or in dimensionless form,
93
0
---- =2.35 hours.
)
(3. 79x 1 5 )(0 .23)(0 .72)(1 .5 X 10 -5 )(0 .5) 2
1 . -9481/>p.C/r�
0 .1
+ /lP sD' . . . . . . . . . . . . . . (5.1 69)
PD = - -El
kt
2
This value is less than t = 3 hours, as required. Thus, we can use
Eq. 5 .1 61 with satisfactory accuracy if the reservoir is still infinite­
For r =r w' the argument o f the E i function i s sufficiently small
after a short time that we can use the logarithmic approximation.
Thus , substituting for
from Eq.
the drawdown is
1
P D = - - ln
2
(
/lP sD
1 ,6881/>p. c/r�
b
5 .1 66,
) + (-k -1) (-r s ) . . . . . (5.1 70)
�
In
�
We define a skin factor, s, in terms of the properties of the equiva­
lent altered zone as 7
s
=
(� - l) (::) . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.1 71 )
ln
(
Thus, the drawdown is
)
1 ,6881/>P.Cr '�
+s . . . . . . . . . . . . . . . . . . . (5 .1 72)
P D = -!.. In
kt
2
Eq. 5 .1 71 provides some insight into the physical significance
of the sign of the skin factor. If a well is damaged (k s < k ) , then
s will be positive. The greater the contrast between k s and k and
the deeper into the formation that the damage extends, the larger
the numerical value of s will be. There is no upper limit for s. Some
newly drilled wells will not flow at all before stimulation; for these
wells,
and s-+oo. Conversely , if a well is stimulated
then s will b e negative. The deeper the stimulation, the greater the
numerical value of s is. Rarely does the stimulated well have a skin
factor less than
or
such skin factors arise only for wells
with deeply penetrating , highly conductive hydraulic fractures. If
a well is neither damaged nor stimulated
then s=O. Bear
in mind that Eq.
is best applied qualitatively ; actual wells
rarely can be characterized
by such a simplified model.
An altered zone near a particular well affects only the pressure
near that wel l ; i.e. , the pressure in the unaltered formation away
from the well is not affected by the existence of the altered zone.
Put another way, we use Eq.
to calculate pressure at the sand­
face of a well with an altered zone, but we use Eq.
to calcu­
late pressures beyond the altered zone in the formation surrounding
the well. We have presented no simple equations that can be used
to calculate pressures for
but this presents no difficul­
ties in reservoir engineering analysis.
k s:::: O
(k s >k),
-7
-8;
(k s =k ) ,
5 .1 71
exactly
5 .1 72
5 .1 61
rw <r <r S '
Example S.l-Calculating Pressures Beyond the Wellbore With
the Ei-Function Solution. A well is producing only oil at a con­
stant rate of 20 STBID. Data describing the well and formation are
summarized below. Calculate the reservoir pressure at radii of 1 ,
1 0, and 1 00 ft after 3 hours of production.
rw =0.5 ft.
p. =0.72 cpo
k =O. l md.
1/> =0.23.
r e = 3 ,000 ft.
Bo = 1 .475 RBISTB.
Pi = 3,000 psia.
h =1 50 ft.
c/ =1 .5 x lO -5 psi-1
s =O.
Solution.
1 . First , we must determine whether the Ei-function solution is
valid for the desired times. The Ei function is not an accurate solu­
tion to flow equations until t > 3.79 X1 051/>p. ctrMk . In this case,
3 .79 X1 05 I/>p.c tr �
t > ---- ----'----'
k
acting at this time. The reservoir will act as an infinite reservoir
as long as
In this case ,
t <9481/>p. ctr;lk .
9481/>p. c tr;
t<------'�
k
------- =21 1 ,900 hours.
(948)(0.23)(0 .72)(1 . 5 x 1 0 -5)(3,000)2
0.1
Thus, for times less than 21 1 ,900 hours , we can use Eq. 5 .1 61 .
2. Combining Eqs. 5 .1 61 and 5 .1 62, we can develop an expres­
sion to calculate pressure in terms of the well and reservoir prop­
erties :
P =P i +
(
70.6qBp. . -9481/>p. ctr 2
El
kh
kt
)
.
Or , in terms of the well and reservoir properties,
(70.6)(20)(1 .475)(0.72)
p = 3 ,000 + ------(0.1 )(1 50)
.[ ( - 948)(0.23)(0.72)(1 5 x 1 O -6)r2 ]
(O. l)t
-2.35 x lO -2r 2
).
= 3,000+ 99.97 (
t
XEl
Ei
3.
At a radius of
x=
1
2.35 x lO -2r 2
ft , the argument of the Ei function is
2.35 x lO -2 (1 ) 2
= 7 . 85 x lO -3
3
x < 0.02,
Because
we can use the logarithmic approximation to
the Ei function given by Eq.
The reservoir pressure at
ft is
p = 3 ,000 +99.97 Ei
5 . 1 64.
( -2.35 xt lO -2r 2 )
r =1
= 3 ,000 +99.97 1n(1 .78lx)
= 3,000+ 99.97 In[(1 . 781 )(7. 85 x 1 0 -3)]
=2,573 psia.
4. At a radius of 1 0 ft, the argument of the Ei function is
2.35 X1 O -2r 2 2.35 X1 0 -2 (1 0) 2
= 7 . 85 x lO - 1
x=
3
0.02 < x < 1 0.9,
Since
we can evaluate the Ei function using the
table in Appendix D. Therefore, the reservoir pressure at
ft is
p = 3 ,000 +99.97 Ei
( -2.35 xt lO -2r 2 )
r =1 0
= 3,000 +99.97 Ei( - 7 . 85 x lO - 1 )
= 3 ,000 + (99.97)( -0.31 9)
=2,968 psia.
5 . At a radius of 1 00 ft, the argument of the Ei function is
2.35 x lO -2r 2 2.35 x lO -2 (1 00) 2
x=
=78.5.
3
94
GAS RESERVO I R E N G I N E E R I N G
Since x> 1 0.9, the Ei function equals zero, and the reservoir pres­
sure at r 1 0 ft is
=
( -2.35 1O - 2 2 )
t
=3, 000+(99.97)(0.0)
=3, 000
r
X
p = 3 , 000 + 99 . 97 Ei
or P P f
-w=
5.158,
t 48 p. k.
PD=-2t-2D + 1n reD- -43 ' ..........................(5.173)
tD
5.115,
PD= kh(141P.i-2qBp.P wf) . .............................(5.174)
5.173tD' PD:
Pwf=Pi- 141.2qBp. [0.OO05274kt (!i.w ) _�4 ].
................................(5.175)
5.175
ap wf =- 0.07447qB . ..........................(5.176)
at
... , ... , ....................... , .. (5.177)
apwf 0, 234qB . ........... ,."., .......... (5.178)
so--=at
reD
Eq.
nitions of
reD = re1rw and
'
can be written in dimensional form by use of the defi­
reD and
'
2
kh
</!p.ctre
Differentiating Eq.
-
+ In
</!c thr;
The liquid-filled PV of the reservoir (in cubic feet) is
Vp ='u;hc/),
ct Vp
Thus, during this time period , the rate of pressure decline is in­
versely proportional to Vp . Thi� re�ul� leads !o a for� of well test­
ing sometimes called reserv Oir-lImits testmg , which seeks to
determine reservoir size from the rate of pressure decline in a well­
bore with time .
Another form of Eq .
is useful for some applications . In
this form, the original reservoir pressure ,
is replaced with the
average pressure ,
within the drainage volume of the well . The
volumetric average pressure within the drainage volume of the well
can be found from material balance. The pressure decrease resulting
from the removal of a fluid volume equal to
(RBID) x (hours),
for a total volume removed equal to
ft 3 ) , is
p,
Pi'
�V
ct(71'r;h</!)
</!cthr;
Substituting into Eq .
</!c thr;
re
In -
we obtain
</!cthr;
,
(5. 1 8 0)
5.180
�
(!i.rw )- 4 , ............................. (5.181)
p-Pwf) ........ .. ...... . ......(5.182)
PD= kh(141.2qBp.
5.173 5.175) 5.181 5.180)
becomes
where
.
. .
Eqs .
(or
and
(or
become more useful
in practice if they include a skin factor to account for the fact that
most wells are either damaged or stimulated . For example , in Eq.
( ) 3
rw 4
( - )--3 +s. ........(5.183)
rw 4
ka'
5.180
qBP.[ ( )--3 ]
P-Pwf=141.2kah rw 4
[ln (!i. )-� +s] , ............ (5.184)
=141.2 qBP.
kh rw 4
5.181,
re
PD = ln - - - + �p sD = ln
re
Further, we can define an average permeability ,
becomes
so that Eq.
re
In -
from which
ka
s=O
proves to be of value in well-test analysis , as we will see later.
Note that , for a damaged well ,
is lower than the true bulk for­
mation permeability , in fact, these quantities are equal only when
( i . e . , no formation damage or improvement) . Because we
sometimes estimate the permeability of a well from PI measure­
ments, and because the PI, J (STB/D-psi) , of an oil well is defined as
k;
ka
q
kh
J=--=
P-Pwf 141.2Bp.[ln ( :: ) - +s] ..........(5.186)
�
does not ko '
,
this method
necessarily provide a good estimate of bulk
formation permeability ,
Thus, a more complete means of
characterizing a producing well than using PI information only is
needed .
1, 500 100
10
0.25
0.5
Example S.2-Analysis of a Well PI Test. A well produces
STBID of oil at a measured flowing BHP (BHFP) of
psia.
A recent pressure survey showed that the average reservoir pres­
sure is
psia, while logs indicate a net sand thickness of
ft. Geological data suggest the well drains a n area with a drainage
radius , re, of
ft. The borehole radius is
ft . Fluid sam­
ples indicate that, at current reservoir pressure , oil viscosity is
cp and the FVF is
RB/STB . Core data from the well indicate
an effective permeability to oil of
md . Estimate the PI for the
tested well and formation permeability from these data. Does tbis
suggest that the well is either damaged or stimulated? Determine
the apparent skin factor .
Solution.
To estimate PI, we use Eq.
2,000
qB
t
5.615qBt124(res
1,000
5.615qBt
1
24
0.07447qBt
. ........(5.179)
Pi- P=-1.5
50
5.175,
Pwf= P + 0.07447qBt 0.07447qBt
1.
5.186.
100 -=0.2
q ---qBP.[ ( )--3 ]
J
-141.2kh rw 4
P-Pwf (2, 000 - 1,5 00)
c t Vp
.
r
gives
5.175
..
P D = ln
5.3.3 Bounded Cylindrical Reservoir, Pseudosteady-State Flow.
We now discuss the next solution to the radial diffusivity equation
that will be used extensively in this introduction to gas reservoir
engineering . The pseudosteady-state solution is simply a limiting
form of Eq.
which describes pressure behavior with time
for a well centered in a cylindrical reservoir of radius reo The limit­
ing form of interest is the one that is valid for long times, so the
summation involving exponentials and Bessel functions is negligi­
ble . Here , long times are > 9 </! c t r'; l For long times, the
pressure response at the wellbore is given by
is defined by Eq.
.
In dimensionless form, Eq.
psia.
where
[ ( w ) -�]4 . ... .......
1 41 . 2 qBp. In !i.
kh
r
= --
STBID-psi.
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
95
Iogt
Fig. 5.9-Characteristic curve shapes for various flow regimes
on a semllog plot.
2.
We do not have sufficient information to estimate formation
permeability . We can calculate only
which is not necessarily
a good approximation of
From Eq .
ka,5.186,
Fig . 5. 1 0-Characteristic curve shapes for various flow re­
gi mes on a Cartesian plot.
tAD<"
tAD=0. 0002637ktlcpJ-tc(A,
C(AtAD . ...... ........ ........... ...... (5.189)
t< 0.CPOJ-tOO2637k
"< 1
tAD>" 1
CtA tAD . . ... ................. ........ . . (5.190)
t> 0.CP0J-t002637k
tAD>'"
in Appendix C. Because
time in hours is calculated from
the
k.
3]
ka-_ 141.2h JBJ-t [ (Te)
rw 4
(141. 2)(0. 2)(1. 5)(0. 5) [ (-1,000 )-0. 75]
10
0. 2 5
=16
k
50
P wf ,
3.
5.185
s:
5.172,
s c: - 1) [ ( ::)-�] C: -1) [ C��;50) - 0. 7 5]
t. 5.187
=16.
5.180 5. 175
In -
The time required for the pseudosteady-state equation to be ac­
curate within % can be found in the column titled
% Error
for
in Appendix C and from
In
=
md .
Core data often can provide a better estimate of than that de­
rived from the PI, particularly for a well that is badly damaged .
Cores indicate a permeability of
md , so we conclude that this
well is damaged .
We can use Eq .
to estimate
=
ln
=
ln
A large positive skin factor indicates damage o r reduced perme­
ability in the formation adj acent to the wellbore .
5.3.4 Flow Equations for Generalized Reservoir Geometries. Eq.
is limited to a well centered in a circular drainage area. A
similar equation9 that models pseudosteady-state flow in more
general reservoir shapes is
5.183
The time required for the pseudosteady-state equation to be ex­
act is found from the entry in the column titled "Exact for
Figs. 5.9 and 5 . 1 0 show the flow regimes during different time
in a well producing at a constant rate , q, is
ranges. BHFP ,
plotted as a function of time on both semilogarithmic and linear
scales. In the transient region, the reservoir is infinite-acting and
is modeled by Eq.
which implies that P wf is a linear func­
tion of log
In the pseudosteady-state region , the reservoir is
modeled by Eq.
in the general case or by Eqs .
and
for the special case of a well centered in a cylindrical reser­
voir . Eq .
shows the linear relationship between P wf and
during pseudosteady -state flow . This linear relationship also ex­
ists in generalized reservoir geometries .
At times between the end o f the transient region and the begin­
ning of the pseudosteady-state region is the transition region, some­
times called the late-transient region. No simple equation is available
to predict the relationship between BHP and time in this region .
ln
where
is defined by Eq .
= drainage area (ft2 ) , and
= dimensionless shape factor for a specific drainage area shape
and well location . Values of
are given in Appendix C . 1O The
PI,
can be expressed for general drainage area geometry as
q
=
Other numerical constants tabulated in Appendix C allow us to
calculate
the maximum elapsed time during which the reser­
voir is infinite-acting so that the Ei-function solution can be used ,
the time required for the pseudosteady-state solution to predict
pressure drawdown within % accuracy , and
the time required
for the pseudo steady-state solution to be exact .
For a given reservoir geometry , the maximum time that a reser­
voir is infinite-acting can be determined with the entry in the column
entitled " Use Infinite-System Solution With
% Error for
<1
t
This region is of short duration (or, for practical purposes, nonex­
2. ( 1O. 06A )-� +s, ..................... (5.187)
PD
2 CA r� 4
5.182, A
CA PD
CA
J,
5.161
.
........
(5.188)
J= P-P wf [ 2.l (0.IO.00708kh
�
BJ-t 2 n CAr�06A ) _ 4 +s]
Ik.
5.161 5.158).5.175)
(1)
(2)
1
(3)
=
5.175
istent) for a well centered in a circular , square , or hexagonal
drainage area, as Appendix C indicates . However, for a well off­
center in its drainage area, the late-transient region can span a sig­
nificant time period .
Note that determination of when the transient region ends or when
the pseudo steady-state region begins is somewhat subjective . For
example , our stated limits on applicability of Eqs .
and
are not the same as given in the table of shape factors , although
the difference is slight. Other authors 4 consider the deviation from
Eq.
to be sufficiently great for
that a late­
transient region exists even for a well centered in a cylindrical reser­
voir between this lower limit and an upper limit of
These apparently contradictory opinions are nothing more than
different judgments about when the slightly approximate solutions
(Eq .
and
can be considered identical to the exact so­
lution (Eq .
Example
illustrates these concepts.
5. 161 5.175
t>379cpJ-tc(r�/k
t< I ,136cpJ-tctr�
5. 3
Example 5.3-Flow Analysis for Generalized Reservoir Geom­
etries. For each of the following reservoir geometries, calculate
the time in hours for which
the reservoir is infinite-acting, ( 2) the
pseudosteady-state solution is exact, and
the pseudosteady-state
equation is accurate to within % . The geometries are
a well
(I)
1
(3)
(1)
GAS RESERVO I R E N G I N E E R I N G
96
TABLE 5 . 1 -DATA CALCULATED F O R EXAM PLE 5. 3 , PART 1
Pseudosteady State
I nf i n ite Solution
Geometry
C i rcular
Square,
centered
Square,
quadrant
t
t
0.06
�
0.1
(hou rs)
1 3.2
1 1 .9
0.05
6.60
0.1
1 3 .2
3.3
0.3
0.6
79 .2
(hours)
1 3.2
0.09
0.025
tAD
39.6
--- -
TABLE 5.2-DATA CALCULATED FOR EXAMPLE 5.3, PART 2
J
�
Geometry
Circular
Square, centered
Square , quadrant
q
(STB/O-psi)
0 . 586
0. 586
0 .535
3 1 .62
30.88
4.51 3
(STB/O)
293
293
267
(2)
p-Pw/=500
centered in a circular drainage area,
a well centered in a square
drainage area, and
a well centered in one quadrant of a square
drainage area . In addition , estimate the PI and the stabilized pro­
duction rate for each well, given
psi . For the well
centered in one quadrant of a square drainage area, write equations
relating constant flow rate and wellbore pressure drops at elapsed
times of
and
hours .
(3)
3, 20, 40
B=
1.
23 RB/STB.
rc/Jw=0.
J.!h=lO
==0.I 2 .
s=3.
0 . O-5
=
x
A=1.
k=IOO742x106 (40
t
(hou rs)
7.92
�
0.1
Exact
Approximate
SURFACE RATE, q
'.
"
.
"
..
.....
,
....-_
.
SANDFACE RATE, q
TIME ---+
sf
8IIDOF WEUBORE STORAGE
Fig. S . 1 1 -Effect of well bore storage on sand face flow rate .
ft .
3.
We now can write equations relating constant flow rate and
wellbore pressure drops for the well centered in one quadrant of
a square drainage area .
A . For
hours , the reservoir is infinite-acting . Thus, the draw­
down is given by Eq .
where
is defined by Eq.
with a pressure drawdown of
cpo
ft.
ct
l
l
md .
psi - I .
ft 2
acres) .
Solution.
We first calculate the group
1.
(0. 2)(1)(1x 10-5)(1. 742 X 106) = 132.
0. 0002637k (0. 0002637)(100)
------
Using values from the table of shape factors in Appendix C, we
calculate the data in Table 5 . 1 .
For the off-center well , the reduction i n time for which the infinite­
system solution is accurate and the increase in time required to
achieve pseudosteady-state flow , relative to the centered wells, are
noteworthy .
Next, calculate the PI and stabilized production rate. The PI is
2.
J=
kh
�
[
1O.
141. 2BJ.! 2ln ( 06A ) _2.4 l
(100)(10)
742X106)l 3. 0
(141)(1. 2)(1)[2I [(1O. 06)(1.
CA(0. 3) 2 4 J
5. 9
CAr�
- In
+s
3 +
--
The stabilized production rate i n terms o f the P I is
q=J(p-Pw/)=500J .
Using values of CA from the table of shape factors , we prepare
the data in Table 5.2.
t=3 5.172, PD
Pi -P w/ :
PD= 2I ( I kt )
t=20
t=40
(t>
39.
6
5.182 :
PD
PD= �2 ( 1O. 06A ) _2.4
- - In
,688c/JJ.!Ctr�
+s
5.162
.
B. For
hours , the reservoir is no longer infinite-acting and
the pseudosteady-state equation is not yet accurate
hours
(3. 3
5.187,
<t<39.6 hours). Accordingly , no simple equation can be written .
C . For
hours , the pseudosteady-state equation is accurate
hours) . Thus, the drawdown is given by Eq.
where
is defined by Eq .
ln
CAr�
+s.
5.3_5 Radial Flow in an Infinite Reservoir With Wellbore
Storage. The next solution to the radial diffusivity equation incor­
porates wellbore storage . Consider a shut-in oil well in a reservoir
with uniform and unchanging pressure . Reservoir pressure will sup­
port a liquid column to some equilibrium height in the wellbore .
If we open a valve at the surface and initiate flow, the first fluid
produced will be that stored in the wellbore , and the initial flow
rate from the formation (sandface) to the well will be zero . With
increasing flow time and at constant surface producing rate ,
the
flow rate at the sandface,
will approach the surface rate and
the amount of liquid stored in the wellbore will approach a con­
stant value . Fig. 5 . 1 1 shows the effect of wellbore storage on the
sandface flow rate .
The propensity of the wellbore to store or unload fluids per unit
change in pressure is called the wellbore storage coefficient, C,
which has field units of bbllpsi . The definition of C depends on
wellbore conditions . We consider two cases : a liquid/gas interface
in the wellbore (as in a pumping or gas-lift well) and a single-phase
fluid in the wellbore .
qsf,
q,
97
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
...-- --.... q
P
t
AREA=Awb
....
P
wi
Fig. 5. 1 2-Wellbore with moving liquid/gas i nterface.
Wellbore Storage With a Liquid/Gas Interface. Consider a well
with a liquid/gas interface in the wellbore , as shown in Fig. 5 . 12 ,
with a mechanism (a pump o r gas l ift) t o deliver l iquid to the sur­
face. The surface rate, q , i s constant. In terms of a mass balance
on the liquid in the wellbore , the rate of liquid entering the well­
bore i s 5.615 qs j Bp (lbmlD) , the rate of liquid leaving the well­
bore is 5.615 qBp (lbmlD) , and the rate of liquid mass accumulation
in the well bore is
d
d
- ( 24 PwbVwb ) =24- ( PwbZ4wb )' ................ (5.19 1)
d
d
t
t
Then , assuming constant wellbore area, Awb , constant oil den­
sity , P =pwb=constant , and constant FVF , B=constant , the bal­
ance becomes
--
24Awb dZ
-=( qs j - q)B. ......................... (5.19 2)
5.615 dt
For a well with surface pressure Pr and neglecting frictional pres­
sure losses in the flow string , the FBHP i s
PwbZ g
Pwj =Pr+---' ........................... (5.193)
144 gc
where Pwb is the density (lbm/ft3) of the l iquid in the wellbore and
g /gc=1.0 lbf/lbm. Differentiating with respect to time gives
Pwb g dZ
---. ...................... (5.194)
144 gc dt
Combining Eqs. 5.19 2 and 5.194 yields
---
(24)( 144) gc
d ( Pwj -Pr )
=(qs j -q)B. ..........(5.195)
-Awb
5.615 Pwb g
dt
We now define C as the multiplier of the pressure derivative :
C=
144Awb gc
5.615 Pwb g
. ............................. (5.196)
Rewriting Eq. 5.195 and rearranging , we find that the sandface
rate is
qSj=q+
-
24C d
- ( Pwj -Pr )' ...................... (5.19 7)
B dt
To help us formulate the solution to flow problems that include
wellbore storage , we need to introduce dimensionless variables.
We let qi be the surface rate at t=O. PD is defined by Eq. 5.108
...
q sl --------'
p
wi
Fig. 5 . 1 3-Wellbore with single-phase fluid in the wellbore .
tD
with qi replacing q.
is defined by Eq. 5.1 15. Substituting these
definitions for Pw and
we obtain
-dpw
dt
-
t,
0.037 23 qiB
0.OOO2637k dPD
qiBP,
c/>p,crr3;,
0.00708 2kh
c/>crhr3;,
dtD
dpD
dtD
................................ (5.198)
Applying Eq. 5.198 and assuming zero or unchanging surface
pressure, Pr (a major and not necessarily valid assumption) , we can
write Eq. 5.197 as
qs j=q-
0.8936qi C
c/>crhr w2
dpD
dtD
.
.
.
.
.
.
.
.
.
•
.
.
.
.
.
.
.
.
.
.
.
•
•
(5.199)
If we define a dimensionless wellbore storage coefficient as
CD=
0.8936C
c/>crhr3;,
, ............................... (5.200)
we can rewrite Eq. 5.199 as
qSj=qi
(:i.
qi
-CD
D
dtD
dP
)
. ........................ (5.20 1)
For constant-rate production, q(t )=qi and Eq. 5.20 1 becomes
qs j
q
=1-CD
dPD
dtD
. ............................. (5.20 2)
Eq. 5.20 2 is the inner boundary condition for the problem of
constant-rate flow of a slightly compressible liquid with wellbore
storage. Note that , for small CD or for small
qs [lq= 1 ;
i.e., the effect of wellbore storage o n flow rates i s neglIgible.
Wellbore Storage With a Single-Phase Fluid. Now consider a
well with a single-phase fluid in the wellbore that is produced at
a surface rate, q (see Fig. 5. 13) . The surface rate, q , is constant.
If we let Vwb be the volume (in barrels) of wellbore open to the
formation and Pwb be the fluid density in the wellbore (evaluated
at wellbore conditions) , the mass-balance components are ( 1) the
rate of fluid entering the wellbore , qs j Bp ; (2) the rate of fluid leav­
ing the wellbore, qBp ; and (3) the rate of fluid accumulation in the
wellbore , d( 24 PwbVwb )/dt. The mass balance becomes
dpDldtD ,
qsjJ p-qBp=24Vwb
-d Pwb
dt
' ..................... (5.203a)
98
GAS RESERVO I R E N G I N E E R I N G
l� �----,
10'
Fig. 5. 1 4-Ramey's 1 8 type curve.
From the definition of compressibility ,
sure , P p' and pseudotime , tap ' Defining PpD by Eq. 5.135 , tap D
by Eq. 5.136, and
by Eq. 5. 100 and substituting into Eq. 5.206,
we obtain
rD
( apP D) - - 1+
arD
1 dp
C=-- ,
p dp
--
we can write
d Pwb
=
--
dt
d p"b dPwb
=Pwbc"b
----
dt
dPwb
dPwb
--
dt
·
as
Substituting this expression into Eq. 5.203a yields
Because the pB product is constant and thus the same at surface
and reservoir conditions , the mass balance becomes
qsf=q+
24cwbVwb
B
p
( --;; ) dr'
PWb
d w
.
. ..... .... .. .... (5.204)
In this case , we define =cwbV wb' If we assume pll'h P R (not
necessarily a valid assumption for a gas) , then the mass balance
becomes
C
qsf=q+
S dr
24C dpll'
==
' ............................ (5.205)
C
Eq. 5.205 is identical to Eq. 5.19 7 , but has a different defini­
tion in this case. Thu s , Eqs. 5.20 1 and 5.20 2 also apply to the case
of a single-phase fluid in the wellbore.
From Eq. 5.200 ,
for this case is
CD=
0.S936C
<pcthr�.
CD
=
0.S936cwbVwb
<pcthra
. ................ (5.205a)
For a gas well , cwb is the gas compressibility in the wellbore
and is strongly dependent on pressure (as an approximation,
cll'b=lIPll'b )' From a mass balance on a wellbore of volume Vwb
and constant temperature Twb , Lee and Holditch 3 derived the fol­
lowing inner boundary condition (r=r
0) for a gas well with
constant-rate production and well bore storage:
( aPP ) ar
r"
-
w ,t >
50 ,300Psc T q
khrll' Tsc
+
3,390TVwb
khrll' Tll'b
( P )
dP Wb
--
dlap
. ...(5.206)
Eq. 5.206 replaces Eq. 5.133 as the inner boundary condition
for the gas diffusivity equation (Eq. 5.47) in terms of pseudopres-
ro=1
0.S936TVwh
<phr\�,Twb
( apP D )
aD
tap
rD=1
. ... (5.207)
We define an effective dimensionless wellbore storage coefficient
CeD
=
dPwb
qsfBp-qBp=24 PwbVwbcwh- -- . ............... (5.203b)
dt
_
0.S936TVwb
<phraTll'b
............................(5.20 S)
The inner boundary condition can now be written in dimension­
less form as
( apP D) = - 1+CeD( apP D )
a r£)
aD
'D=
tap
I
rD=
.... .. ..... (5.209 )
I
For the well bore-storage case , Eq. 5.209 replaces Eq. 5.139 as
the inner boundary condition for the gas diffusivity equation in terms
of pseudopressure and pseudotime , in dimensionless form. Because
Eq. 5.209 has the same form and conditions as the liquid solution
presented by Agarwal et a l.
the l iquid solution should model
a gas-well test when the gas temperature in the wellbore is con­
stant , as assumed in deriving Eq. 5.209.
Radial Diffusivity Equation With Wellbore-Storage Effects. The
radial diffusivity equation for liquid flow-with the wellbore-storage
equation (Eq. 5.20 2) as an inner boundary condition, infinite
drainage radius , uniform initial formation pressure , and formation
damage or stimulation (characterized by s)-has been solved both
analytically and numerically. 8.11 Fig. 5. 14 gives the analytical so­
lution. From this figure , values of PD (and thus Pw) can be deter­
mined for a well in a formation with given values of
and
s. Two properties of this log-log graph warrant special mention.
First , at earliest times for a given value of
and for most
values of s, a unit-slop e line (i.e. , a line with 45° slope) appears
on the graph. This line appears and remains as long as all produc­
tion comes from the wellbore and none comes from the formation.
Eq. 5.20 2 leads us to expect this line. For qstiq=O and integrat­
ing from t =O (where PD=O) , Eq. 5.20 2 becomes
, II
CD
D
dtD
D CD pD
CDPD=tD'
tD' CD'
d PD
l-CD-=O.
Thu s , d t =
and
d
................................. (5.2 10)
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
99
TABLE 5.3-PRESSURE DISTRIBUTION RESULTS
FROM APPLICATION OF EO. 5.220
5000
!?
.9:
'
iii
�
::>
(J)
(J)
CD
a::
t
(hours)
'j
(tt)
0.0001
0.01
1
1 00
1
10
1 00
1 ,000
4900
4800
4700
4600
10
Radius (ft)
0.1
4500
100
1000
Fig. 5.l5-Pressure d istribution in the formation near a
producing wel l .
Taking logarithms of both sides, we obtain
log CD +log PD = log tD ' ........................ (5.2 1 1)
Thus, a s long a s q sf =0, theory leads us to expect that a graph
of log PD vs. log tD will have a slope of unity and that any point
( P D ,tD) on this unit-slope line must satisfy the relation
CDP D
= 1. ................................... (5.2 12)
tD
Second. when wellbore storage has cea sed (when q sf =q), we
should expect the solution to the flow equations to be the same a s
i f there had never been any wellbore storage: i.e., we expect it to
be the same a s for CD =0. In Fig. 5.14, we see that the solutions
for finite CD and for CD = 0 do in fact become identical after suffi­
cient elapsed time. One useful empirical observation is that this time ,
t wbs (the end of wellbore-storage distortion). occurs approximate­
ly 11/2 to 2 log cycles a fter the disappearance of the unit-slope line.
Another empirical observation that is very useful in well-test anal­
ysis is that the 'D at which wellbore-storage distortion cea ses is ap­
proximated by
tD = (60 +3.5s)CD. ............................. (5.2 13)
--
vestigation, ri ' we mean the distance over which a pressure tran­
sie nt, moving into a formation following a rate change in a well.
has a significant influence on reservoir pressure at that distance. Be­
fore developing a quantitative means of calculating ri, we shall ex­
amine pressure distributions at ever-increasing times to develop a
feel for the movement of a transient into a formation. Fig. 5.15 shows
pressure as a function of radius for 0.0001. 0.01. I, and 100 hours
after a well begins to produce from a formation originally at 5,000
psi. These pressure distributions were calculated with the Ei-function
solution to the diffusivity equation given by Eq. 5.161 and for a well
and a formation with the following characteristics.
h = 150 ft.
rw =O.1 ft.
q = l77 STBID.
¢ =0.15.
s =O.
k = lO md.
c, =7.03x lO -6 psi -I.
B= 1.2 RB/STB.
!J. = 1 cpo
Two observations about Fig. 5.15 are particularly im portant.
First, the pressure in the wellbore at r = rw decreases steadily with
increa sing flow time. Likewise , pressures at other fixed values of
r also decrease with increasing flow time. Second , the pressure dis­
turbance (or pressure transie nt) caused by producing the well moves
further into the reservoir as flow time increases. For the range of
flow times shown, a point always exists beyond which the draw­
down in pressure from the original pressure is negligible.
Now consider a well in which we instantaneously inject or produce
a volume of liquid. This impulse introduces a pressure disturbance
into the formation; the disturbance at radius ri will reach its max­
imum at time 'max after introduction of the impulse. We seek the
relationship between ri and tmax' From the solution to the diffu­
sivity equation for an instantaneous line source in an infinite
medium , 1 3
Cl
p-Pi = - e - r-/4*, . ............... .......... (5.2 17)
Linear flow
occurs in some petroleum reservoirs with long, highly conductive
vertical fractures. Thus, we revie w one of the fundamental equa­
tions describing linear flow in a reservoir. Consider a situation with
linear flow (in the direction, for convenie nce) of a slightly com­
pressible fluid in an infinite and homogeneous reservoir initially
t
at P i ' Fluid is produced at constant rate , qB. over an area Af . If where CI =constant
related to impulse strength. We find (max at
Af represents a vertical fracture with two equal-length wings of
by
differentiating
Eq.
5.217 with respect to time and setting the
r
i
length Lf (feet) and height h (feet), then Af = 4hLf with flow en­
tering both sides of each fracture wing. This situation is modeled result equal to zero:
by the diffusivity equation in the form
Clr 2
dp - c l
-=
e - r2/4ryt +
e - r-/4ry' =0. ............ (5.2 18)
¢Jl.C,
ap
2
dt
,
41'/( 3
......................... (5.214)
ax 2 0.0002637k at
Thus, the maximum effect of the impulse on reservoir pressure
at
radius ri will be felt at time tmax given by
2
For the conditions stated, the solution 1 to this equation at =0
rf 948¢!J.c, r 2
is
tmax= =
. ........................ (5.2 19)
41'/
k
qB !J.t \
P i - P"f = 16.26) .................... (5.2 15)
Stated another way, at time t, a pressure disturbance reache s a
Af k¢c"
distance ri, given by
For linear flow into vertical fractures, Af = 4hLf and
kt
ri=
............................. (5.220)
qB
!J.t
948¢!J.C ,
P i -Pwf =4.064- -- . .................. (5.216)
hLf k¢c,
The radius of investigation given by Eq. 5.220 also proves to
be
the distance that a significant pressure disturbance is propagated
5.4 Radius of Investigation
(by time t) by constant-rate production or injection. For example ,
The radius-of-inve stigation concept is of both quantitative and for the formation with pressure distributions shown in Fig. 5.15,
qualitative value in well-test design and analysis. By radius of in- application of Eq. 5.220 yields the results given in Table 5.3.
5.3.6 Linear Flow in Infinite-Acting Reservoirs.
x
-
�
__
'---
.
--
-
(
x
--
(
-
v,
) 1/2
(
)1/2
I
.
�
1 00
GAS RESERVO I R E N G I N E E R I N G
Comparison of these results with the pressure distributions plot­
ted in Fig.
shows that calculated from Eq.
is near
the point at which the drawdown in reservoir pressure caused by
producing the well becomes negligible .
We also use Eq.
to calculate the achieved at any time
after any rate change in a well. This fact is significant because the
distance a transient ha s moved into a formation is approximately
the distance from the well at which formation propertie s are being
investigated at a particular time in a well test.
Our understanding of can be enhanced through the following
alternative derivation of Eq.
We want to find the length of
time to achieve stabilized flow, which is the time required for
a pressure transient to reach the boundaries of a tested reservoir.
This time occurs when the pressure change with time is the same
for pseudosteady-state flow and unsteady-state (transie nt or infinite­
acting) flow-i.e . , the time at which a bounded reservoir of radius
ceases to behave like an infinite reservoir. Rewriting Eq.
in dimensional form , the unsteady-state flow equation is
5.15
5. 220
ri
5. 220
ri
ri 5. 220.
tS'
a hexagon
and, approximately, a square
with a centered well
for pseudosteady-state .flow and
for unsteady-state flow).
We solve Eq.
for when the first boundary is encoun­
tered by substituting the appropriate definition of for each reser­
voir shape . For a circular reservoir,
(tAD=O.l),
(tADAD=O.l),
=O.l
(t
0. 09
5. 226 ri
A='1l'r2 ,
A
I
r .= ( 948tPktsf-tcr )
For a hexagonal reservoir with sides of length L and perpendic­
ular distance ri from center to side ,
A=3Lri=2-J"3 rf
rf) ,
ts= 948tPf-tc('1l't(2-J"3
)k
( kts )
and r·=
1, 045tPJLcr
For a square reservoir with sides of length 2ri,
A=4r2 ,
ts= 948tPJLc('1l')rkC4rf)
( kts )
and r.=
1,207tPJLcr
for applicable reservoir shapes, these modifi cations of Eq.
5.acting
2Thus,
20 give
rio As a practical matter, ri is the same when infinite­
flow ceases regardless of drainage area shape for a well cen­
and
'h
I
5. 172
Pwf=Pi+ 70.kh6qBJLlIn ( 1 ,688tPktJLcrra ) - 2sl . . . . . . . . (5. 221)
Rewriting Eq. 5.175 to include s, the pseudosteady-state flow
equation is
3 l.
(-re )--+s
+In
Pwf=Pi- 141.kh2qBJLlO. OtPJLOO5274kt
Crr� r 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 222)
Rewriting Eq. 5. 222 in terms of drainage area , A, and shape fac­
tor, CA, the pseudo steady-state equation becomes
3 l.
kt +-In1 (--1O. 06A ) - -+s
P wf=Pi- 141.kh2qBJLlO. 0005274'1l'
tPJLcrA 2 CAra 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 223)
In Eq. 5. 2 23, A is a function of r� and the constant '1l'maintains
equivalence of the first term in brackets when we replace r� with
A.We find ts at which the pressure change with time is the same tered in its drainage area .
for pseudosteady- and unsteady-state flow. Derivatives of Eqs. 5. 223
The radius of investigation ha s several uses in pre ssure-transient
and 5. 221, respectively, are
test analysis and design. A qualitative use is to help explain the
shape of a pressure-buildup or pressure-drawdown curve . For ex­
ample , a buildup curve may have a shape or slope that is difficult
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(5.
2
24)
to interpret at earliest times when ri is in the zone of altered per­
(d:;)pss
meability, kS' nearest the wellbore . More commonly, a pressure­
curve may change shape at long times when ri reaches the
70. 6qBf-t . . . (5. 225) buildup
general yicinity of a reservoir boundary (such as a sealing fault)
or some massive reservoir heterogeneity. In practice , we find that
khts
a heterogeneity or boundary has a clear, unambiguous influence
Equating derivatives and solving for ts yields
on the pressure re sponse in a well when the calculated ri is about
twTheice theradius-of-investigation
distance to the heterogeneity.
( d:7) = (d:;f)
concept provides a guide for well­
test
design.
For
example
,
we
may
want to sample reservoir prop­
pss
uss
erties at lea st 500 ft from a tested well. Thus, we must determine
how long to run the test. Using the radius-of-investigation concept,
0. 07447'1l' qBJL 70. 6qBJL
we can estim ate (rather than guess) the time required to test to the
desired distance into the formation.
khts
tPf-tcthA
Although useful, the radius-of-investigation concept has limita­
tions. First, Eq. 5. 220 is exactly correct only for a homogeneous,
948tPJLcrA . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5. 226) isotropic reservoir of an applicable shape ; reservoir heterogenei­
Thus, ts=
tie s will decrease the accuracy of the estimate of ri from Eq. 5. 220.
'1l'k
Further, Eq. 5. 220 is exact only for describing the time at which
Replacing A by its definition in terms of re , we can solve Eq. the maximum pressure disturbance reaches radius ri following an
5. 226 for that value of re (which we shall denote by ri) reached instantaneous burst of injection into or production from a well. The
by a pressure transient at time ts ' The derivation of Eq. 5. 226 ap­ exact location of ri becomes less well-defined for continuous in­
plies only for reservoir shapes where there is no transition period jection or production at a constant rate following a rate change .
between unsteady- and pseudosteady-state flow. Applicable shape s With these limitations kept in mind , though, the radius-of­
(from Appendix C) are those shapes for which tAD is the same for investigation concept can be very useful in pre ssure-transient
the exact pseudo steady-state and infinite-reservoir ca ses. We find analysis.
re
w
'/2
I
I
I
O.07447'1l'qBf-t
that the only reservoir shape s meeting this criterion are a circle
'h
1 01
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
Example 5.4-Calculation of Radius of Investigation. A flow
test is to be run on an exploratory well. The flow test should be
long enough to ensure that the well will drain a cylindrical reser­
voir with more than a 1 ,000-ft radius. A preliminary well- and fluid­
data analysis suggests that k=l0 0 md, c/>=0.2, ct=2xlO-5
psi -1, and p, =0.5 cpo Determine the appropriate flow test dura­
tion. What flow rate should be used?
Well A
Solution.
1. The minimum
length of time for the flow test would propa­
gate a pressure transient approxim ately 2,000 ft from the well (twice
the required radius of investigation, just to be certain of attaining
a 1 , OOO-ft radius) . From Eq. 5.2 2 0 , the required flow time is
t=
948c/>p,ct
k
r7
-------
=
(948)(0.2)(0.5)( 2 x 10-5) ( 2 , 000) 2
100
hours.
2. In principle, any flow rate would suffice because the time re­
quired to achieve a particular ri is independent of flow rate. In
practice, we require a flow rate large enough that the pressure
change with tim e can be recorded with sufficient precision to be
useful for analysis. What constitutes sufficient precision depends
on the particular pressure gauge used .
=75.8
5.5 Pri nciple of Superposition
At this point, the most useful solution to the flow equation, the Ei­
function solution, appears to be applicable only for describing the
pressure distribution in an infinite reservoir, with the pressure dis­
tribution caused by production from a single well in the reservoir,
and , most restrictive of all, well production at a constant rate be­
ginning at time zero. In this section, we demonstrate how applying
the principle of superposition can remove some of these restrictions.
For our purposes, we state the principle of superposition in the
following way. The total pressure drop at any point in a reservoir
is the sum of the pressure drops at that point caused by flow in
each well in the reservoir. The simplest illustration of this princi­
ple is the case of more than one well in an infinite reservoir. As
an example, consid er three wells (Wells A, B, and C) that start
to produce at the same time from an infinite reservoir (Fig. 5 . 1 6) .
Applying the principle of superposition shows that the total pres­
sure drop observed at Well A is
Well C
Fig. S . 1 6-Multiple-wel l system in an i nfinite reservoir.
No-Flow Boundary
q
Image Well
C indicate the wells
contributing
the pressure drop.
Eq. 5.2 27 can be written in terms of Ei functions and logarith­
mic approximations:
)
-- (
where
and = the rates at which Wells A, B, and
spectively, produce.
- 70.6
q A' q B,
q cB p,
qc
kh
L
o
+( Pi- Pwf ) B
where the subscripts A, B, and
L
f
q
�1q
.
t
kt
C,
re­
Eq. 5.2 28 includes a skin factor for Well A but not for Wells
B and C. Because most wells have a nonzero skin factor and be­
cause we are modeling pressure inside the zone of altered permea­
bility near Well A, we must include its skin factor. However, a
nonzero skin factor at either Well B or C affects pressure only in­
side that well ' s own zone of altered permeability and has no influ-
Actual Well
_
�
Well 1
Well
. -948c/>p,ctr1 c
El
, ......(5.2 28)
q
Fig. S . 1 7-lmage-well technique for a wel l near a no-flow
boundary .
( Pi- Pwj ) t,A =( Pi- Pwf ) A
+( Pi -Pwj>C, ........ .............. (5.2 27)
Well B
_
2
lWell
3
Fig . S . 1 8-Production schedule for variable-rate wel l .
1 02
ence on the pressure at Well A, if Well A is not within the altered
zone of either Well B or C.
Using the superposition method, we can treat any number of wells
flowing at constant rate in an infinite-acting reservoir. Thus, we
can model so-called interference tests, which basically are designed
to determine reservoir properties from the observed response in
one well (e.g., Well A) to production from one or more other wells
(e.g., Well B or C) in a reservoir. A relatively modern method
of conducting interference tests, called pulse testing, is based on
these ideas. 10
Our next application of the principle of superposition is to simu­
late pressure behavior in a bounded reservoir. Consider a well lo­
cated a distance L from a single no-flow boundary (such as a sealing
fault). Mathematically, this problem is identical to the problem of
a well located a distance 2L from an "image" well (i.e., a well
with the same production history as the actual well). Fig. 5.17 shows
the producing well and an image well. This two-well system simu­
lates the behavior of a well near a boundary because a line equidis­
tant between the two wells can be shown to be a no-flow boundary;
i.e., along this line, the pressure gradient is zero, so there can be
no flow. Thus, this is a simple problem of two wells in an infinite
reservoir modeled as
1 ,688cf>p.ctra
qBp.
__
_
---In
Pi Pwj - 70. 6
kh
kt
(
. --;- Ei [
- 70 6
qBp.
- 948cf>P.Ct (2L) 2
kt
l
1ln
L
[
l
1 ,6 88 cf>p.C t r a _
2
k(t - tl )
1
) l
j
. . . . . . . . . . . . . . . . . . . . . (5 . 2 3 1 )
l
I
[ 1,688cf>p.Ctra - 2sI . . . . . . . . . . . . . . . . . . . . . (5 . 232)
n
J
Li k(t - t2 )
The total drawdown for the well with two rate changes is
Pi -P wf = (Llp) I + (Llp h + (Llp h
1 ,688cf>p.Ctra
ql Bp. [
= - 70 . 6 -- In
- 2s
kh
kt
x
(
. . . . . . . . . . . (5 . 229)
ciple i s to model variable-rate produc ing wel l s . To illu strate this
)
Similarly, the contribution of a third well, Well 3, is
- 70 . 6
Again, whether or not the image well has a nonzero skin factor
is immaterial. Its influence on pressure outside its zone of altered
permeability is independent of the existence of this zone.
Extensions of the imaging technique also can be used, for exam­
ple, to model ( I) the pressure distribution for a well between two
boundaries intersecting at 90° , (2) the pressure behavior of a well
between two parallel boundaries, and (3) pressure behavior for wells
in various locations completely surrounded by no-flow boundaries
in rectangular reservoirs. The last case has been studied in depth;
the technique developed by Matthews et al. 14 is used frequently
to estimate average drainage area pressure from pressure-buildup
tests.
Our final and most important application of the superposition prin­
application, consider the case in Fig. 5 . 1 8 where a well produces
at rate q I from time t = 0 to I ' At t I ' the rate is changed to q2 ,
and at time 1 , the rate is changed to q ' We wish to determine
the pressure 2at the sand face of the well3 at some time t > t2 ' To
solve this problem, we shall use superposition as before, but in this
case, each well that contributes to the total pressure drawdown will
be at the same position in the reservoir. The wells simply will be
"turned on" at different times.
The first contribution to the total drawdown in reservoir pres­
sure is from a well producing at rate ql starting at t = O . This well,
in general, will be inside a zone of altered permeability; thus, its
contribution to reservoir pressure drawdown is
1,688cf>Jl.Ctra
qI BP.
- 2s .
(Llp) I = ( Pi - Pwj ) 1 = - 70 . 6 ---;;- In
kt
[(
x
GAS RESERVO I R E N G I N E E R I N G
(q 3 - q 2 )Bp.
kh
[In[
) l
I,688cf>p.ctra
k(t - t2 )
l }
- 2s .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 233)
Proceeding similarly, we can model a well with dozens of rate
changes. We also can model the rate history for a well with a con­
tinuously changing rate, with a sequence of constant-rate periods
at the average rate during the period. But in many cases, such ap­
plications of superposition yield lengthy equations that are tedious
to evaluate by hand. Note, however, that such an equation is valid
only if Eq. 5 . 1 72 is valid for the total time elapsed since the well
began to flow at its initial rate; i.e., for time t, ri must be r
:$
Example 5.5-Using Superposition To Model the Pressure Be­
havior in a Well.
A flowing well is completed in a reservoir that
has the properties listed below. Determine the pressure drop in a
ing well has been shut in for I day following a flow period of 5
days at 300 STBID. The initial reservoir pressure, Pi' was 2,500
psia.
cf> = 0 . 1 6 .
p. = 0 . 44 cpo
h = 43 ft.
B= 1 . 32 RB/STB.
k = 25 md.
ct = 1 8 x l O - 6 psi - I .
Solution. Because of the rate change, we must superimpose the
contributions of two wells.
shut-in well located 500 ft from the flowing well , when the flow­
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 230)
Starting at I ' the new total rate is q . We introduce a second
well, Well 2, producing at rate (q2 - q l )2 starting at tl , so the total
rate after t I is the required q 2 ' Total elapsed time since this well
started producing is (t - tl ) ' Furthermore,
this well is still inside
a zone of altered permeability. Thus, the contribution of Well 2
to reservoir pressure drawdown is
e'
1
We calculate the term
- 948cf>p.ctr 2
[(948)(0. 1 6)(0 . 44)( 1 . 8 x 10 - 5 ) (500) 2 ]
k
(25)
= 1 2 . 01 .
1 03
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
Then,
- 12.01 l
P i -P= - [ (70.6)(0.44)(1 .32) l [(3OO)Ei [ --(6)(24)
(25)(43)
+(0-3OO)Ei [ - 12.01 ]]
(1)(24)
= 1 1 .4[ -Ei( -0.0834) + Ei( -0.5)]
= 1 1 .44(1 .989 -0.560)
= 16.35 psi.
5.6 Horner's Approximation
Horner 15 presented an approximation that can be used in many
cases to avoid using superposition to model the production history
of a variable-rate well. With this approximation, we can replace
the sequence of Ei functions, reflecting rate changes, with a single
Ei function that contains a single producing time and a single produc­
ing rate. The producing rate, q ' is the most recent nonzero rate
at which the well was produced.lastThe single producing time is found
by dividing cumulative production from the well by the most re­
cent rate. This producing time, which we call the pseudoproduc­
ing time, is defined by
tp = 24Np lq last ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.234)
where tp is in hours; Np =cumulative production, STB; and q last
is the most recent flow rate, STB/D. To model pressure behavior
at any point in a reservoir, we can use the simple equation
r2 ) . . . . . . . . . . . (5.235)
P i -P= - 70. 6qhlastBP. El. ( -948cpp.c(
k
ktp
Two questions arise at this point. First, what is the basis for this
approximation? Second, under what conditions is it applicable? The
basis for the approximation is intuitive, not rigorous, and is found­
ed on two criteria. First, if we use a single rate in the approxima­
tion, the clear choice is the most recent rate. Such a rate, maintained
for any significant period, determines the pressure distribution
nearest the wellbore and approximately out to the radius of inves­
tigation achieved with that rate. Second, given the single rate, in­
tuition suggests that we choose an effective production time so that
the product of the rate and the production time results in the cor­
rect cumulative production. Thus, material balances will be main­
tained accurately.
The approximation is adequate if the most recent flow rate is main­
tained long enough. If we maintain the most recent rate too brief­
Iy, then previous rates will play a more important role in determining
the pressure distribution in a tested reservoir. We can offer two
helpful guidelines for how long the most recent rate should be in
effect. First, if the most recent rate is maintained long enough for
the radius of investigation achieved at this rate to reach the drainage
radius of the tested well, then Horner's approximation always is
sufficiently accurate. This rule is quite conservative, however. Sec­
ond, for a new well that undergoes a series of rapid rate changes,
it is usually sufficient to establish the last constant rate for at least
twice as long as the previous rate. When any doubt exists about
whether these guidelines are satisfied, the best approach is to use
superposition to model the production history.
Follow­
ing completion, a well is produced for a short time and then shut
in for a buildup test. Table 5.4 gives the production history. Cal­
culate tp and determine whether Horner's approximation is ade­
quate for this case. If it is not, how should the production history
for this well be simulated?
Example 5.6-Application of Horner's Approximation.
TABLE 5. 4-PRODUCTION DATA , EXAMPLE 5 . 6
Production Time
(hou rs)
Total Production
(STB)
25
12
26
72
52
o
46
68
Solution.
1 . First, we calculate the most recent flow rate,
68 STB X 24 hours =22.7 STBID.
q last =
72 hours day
2. The pseudoproduction time is
24Np = (24)(52+0+46+68) = 175.5 hours.
tp = -22.7
q last
3. Next, determine whether Horner's approximation is valid. In
this case,
72 = 2.77 2.
--Llt---"last-'-'-- = Lltnext-to -Iast 26
Thus, Horner's approximation probably is adequate for this case.
Therefore, it is not necessary to use superposition, which is required
when Horner's approximation is not adequate.
>
5.7 van Everdlngen·Hurst Solutions
to the Diffusivity Equation
Having introduced the concept of superposition, we can now dis­
cuss the van Everdingen-Hurst5 solutions to the diffusivity equa­
tion. van Everdingen and Hurst developed their solutions for two
cases important in reservoir engineering applications: production
at a constant rate at the inner boundary and production at constant
BHP at the inner boundary. Both cases are discussed in subsequent
chapters.
For the constant-rate case, we consider both an infinite-acting
and a bounded reservoir. An infinite-acting reservoir is a special
case of a bounded reservoir in that a bounded reservoir is infinite­
acting until a boundary is encountered by a pressure transient. The
bounded reservoir may be characterized either by no flow across
the outer boundary or by constant pressure at the outer boundary.
For the constant-BHP case, we consider both an infinite-acting reser­
voir and a bounded reservoir. The bounded reservoir is character­
ized by no flow across the outer boundary.
5 . 7 . 1 Constant-Rate Production. No-Flow Outer Reservoir
Boundaries.
The first case of the van Everdingen-Hurst solution
to the diffusivity equation is for production at a constant rate at
the inner boundary. The first subcase of this solution models radi­
al flow of a slightly compressible liquid for the following condi­
tions: (1) a homogeneous reservoir of uniform thickness, with the
reservoir at uniform pressure, P i ' before production; (2) no flow
across the outer boundary at r = re; and (3) production at a con­
stant rate, q, from a single well of radius rw centered in the reser­
voir. The reservoir is infinite-acting until a boundary is encountered.
The solution (pressure as a function of time and radius for fixed
values of re, rw' and rock and fluid properties) for this subcase
is expressed most conveniently in terms of dimensionless variables
and parameters as
PD =/(tD,rD,reD)' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.236)
where PD, tD, and rD are defined by Eqs. 5. 108, 5. 1 15, and
5. 100, respectively, and reD =relrw' Eq. 5.236 states that PD is
a function of the variables tD and rD for a fixed value of the pa-
1 04
GAS RESERVO I R E N G I N E E R I N G
TABLE 5 .5-DATA FOR EXAMPLE 5 . 7
t
(hour)
0.001
0.01
0.1
to
Po
4
40
400
1 .275
2.401
9.675 1
Source of P o
Table E . 1 (infinite-acting reservoi r)
Tab le E . 2 (reO 1 0)
Eq. 5 . 240
=
rameter reD ' The most important solution is for pressure at the
wellbore radius (r = rw or rD = I ) :
pi = 25 psia.
ct = 0 . l 1 X I O - 3 psi - I .
q = 1 . 0 STBID.
B= 1 . 0 RB/STB .
cf> = 0 . 3 .
Solution.
1 . We first calculate tD and reD ' The dimensionless radius is
reD = re1rw = 10/1 = 1 0 .
Dimensionless time i s
0 . 0002637kt
( P D ) 'D = l =/(tD , reD) ' . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 237)
When expressed in terms of dimensionless pressure evaluated at
�D = I. ' Eq . 5 . 1 5.8 shows the functional form of /(tD , reD) ' which
IS an mfimte senes of exponentials and Bessel functions. This se­
ries has been evaluated 5 for several values of reD over a wide
range of values of tD ' A modified version of Chatas ' 1 6 tabulation
of these solutions is presented in Appendix E. Some important char­
acteristics of these data include the following .
1 . Table E- I presents values of P D in the range tD < 1 ,000 for
an infinite-acting reservoir. For tD < 0 . 0 1 , P D can be approximated
by the relation
P D = 2--J tD I7r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 23 8)
2 . Table E- l also is valid for finite reservoirs with tD < 0 . 25 r�D
because a bounded reservoir is infinite-acting until a boundary is
encountered by the pressure transient .
3 . For 1 00 < tD < 0 . 2 5 r�D ' Table E- I can be extended by
P D = 0 . 5 (ln tD + 0 . 80907) . . . . '. . . . . . . . . . . . . . . . . . . . (5 . 239)
4. Table E-2 presents P D as a function of tD for 1 . 5 < reD < 1 0
for a bounded reservoir with a no-flow outer boundary .
A . For values of tD smaller than the value listed in the table for
a given reD , the reservoir is infinite-acting and Table E- l should
be used to determine P D '
B . For values of tD larger than the largest value listed for a given
reD (or , more correctly , for 25 < tD and 0 . 25 r�D < tD ) ' P D can
be calculated from 1 6
PD =
2(tD + 0 .25)
(3r�D - 4r�D In reD - 2r�D - I )
r�D - I
4(r�D _ 1 ) 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 240)
C. A special case of Eq. 5 . 240 arises when r�D � 1 . In that case,
2tD
3
P D = - + ln reD - - ' . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 24 1 )
4
r�D
5 . The P D solutions in Tables E- l and E-2 also apply to a reser­
oir
of radius r w surround� by an aqUifer of radius re when there
�
IS a constant rate of water mflux from the aquifer into the reser­
voir. For this situation , values of reD between 1 . 5 and 1 0 . 0 (Ta­
ble E-2) are of practical importance . For most well problems,
reD > 1 0 . 0 , and the approximations given by Eq. 5 . 239 for
l OO < tD < 0 . 2 5 r�D and Eq . 5 . 240 for 0 . 2 5 r�D < tD are the real­
ly useful results of the van Everdingen-Hurst analysis .
6. When analyzing a variable rate problem with the PD solution,
we use superposition .
Example 5.7-Using pD Solutions To Model Pressure Behavior
in a Reservoir With a No-Flow Outer Boundary. In a labora­
tory flow experiment , fluid was produced into a perforated cylin­
der with a I -ft radius from a sand-packed model with a lO-ft radius .
N o fluid flowed across the external radius of the model . Sandpack
and produced-fluid properties are given below . Estimate the pres­
sure on the inner boundary of the sandpack at times of 0 . 00 I , 0 . 0 I ,
and 0 . 1 hours .
/-, = 2 cp o
k = 1 darcy .
h = 0 . 5 ft .
(0 . 0002637) ( 1 , 000)t
(0 . 3)(2)(0 . 1 1 X 1 0 - 3 )( 1 )
= 4 ,ooot.
2 . Next , estimate values o f P D from Tables E- I and E-2 using
various values of tD (see Table 5.5) . Note that for tD = 400,
tD > 0 . 25 r�D = (0 . 25)( 1 0) 2 = 25 . Thus, Eq . 5 . 240 is used to cal­
culate P D '
PD =
2 (tD + 0 . 25)
(3r�D - 4r�D In reD - 2 r�D - I )
r�D - I
4(r�D _ 1 ) 2
2 (400 + 0 . 25)
[3 ( 1 0) 4 - 4 ( 1 0) 4 In( l O) - 2 ( 1 0) 2 - I ]
( 1 0) 2 - I
4[( 1 0) 2 - 1 ] 2
= 9 . 675 1 .
3 . Calculate values of real pressure using the dimensionless pres­
sures from Step 2 . From the definition of P D '
qB/-,
( 1 4 1 . 2 ) ( 1 . 0) ( 1 . 0) (2)
p =pi - 1 4 1 . 2 -PD = 25 PD
kh
( 1 ,000)(0 . 5 )
= 2 5 - 0 . 565 P D '
With this equation , we calculate the following estimates of the
� ressure on the inner boundary of the sandpack at the indicated
times .
t
(hours)
P
(psia)
0 . 00 1
0.01
0. 1
24 . 2 8
23 . 64
1 9 .53
Constant-Pressure Outer Reservoir Boundaries. The second sub­
case of the van Ev� rdingen-Hurst solution for constant-rate pro­
.
ductIo ? model � �adlal flow of a slightly compressible liquid for the
followmg conditIOns: ( 1 ) a homogeneous , bounded reservoir of uni­
form th !ckness , with the reservoir at uniform pressure , P i ' before
productIOn; (2) unchanging pressure equal to P i at the outer bound­
ary (r = re ) ; and (3) production at a constant rate , q, from a single
well of radius r w centered in the reservoir.
The solution � , PD evaluated at rD = I as a function of tD for fixed
values of reD m the range 1 . 5 < reD < 3 , 000 , are given in Table
E-3 for a bounded reservoir with constant pressure at the outer
boundary . The �imensionless variables P D ' tD ' rD , and reD have
.
the saI?e defimtlOns as �or the no-flow boundary . Following are
some Important properties of these tabulated solutions .
I . For values of tD smaller than the smallest value listed in Ta­
ble E-3 for a given reD , the reservoir is infinite-acting , and thus
Table E- I should be used to determine P D '
2 . For values of tD larger than the largest value listed in Table
E-3 for a given reD (or for tD > r�D ) '
PD = ln reD ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 242)
Example 5.8-Using pD Solutions To Model Pressure Behavior
i� a Res�rvoir ":ith a Constant-Pressure Outer Boundary. An
011 well IS producmg 300 STBID from a waterdrive reservoir that
maintains pressure at the original oil/water contact (OWC) at a con-
1 05
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
stant 3 ,000 psia. The well is 900 ft from the OWC . Well and reser­
voir properties are summarized below . Calculate pressure in the
wellbore for production after 0 . 1 , 1 . 0 , and 1 00 . 0 days.
c/ = 20 x 1 0 - 6 psi -I .
k = 80 md .
rw = 0 .45 ft .
p, = 0 . 8 c p o
h = 1 2 ft .
B = 1 . 25 RB/STB .
q, = 0 . 2 1 .
= 7 . 45 x 1 0 5 t.
The dimensionless radius is
re
900
reD = - = -- = 2 ,000 .
rw
0 . 45
2 . Calculate the dimensionless pressures for the dimensionless
times in Table 5.6. For reD = 2 ,000 , tD = 7 .45 X 1 04 is smaller than
the smallest value listed in Table E-3 . This means that the reser­
voir is infinite-acting at tD = 7 . 45 x 1 0 4 , so Table E- l should be
used to obtain P D . For tD = 7 . 45 X 1 0 5 , P D is found in Table E-3 .
For tD = 7 .45 X l O ? , which is larger than the largest value listed
for reD = 2 ,000 in Table E-3 , P D is calculated from Eq. 5 . 242 .
3 . Calculate dimensionless pressures corresponding to each
dimensionless pressure .
qBp,
0 . 00708 1kh
= 3 ,000 -
PD
(300)( 1 .25)(0 . 8)
(0 . 00708)(80) ( 1 2)
PD
= 3 ,000 - 44 . 14 P D .
1L
6.015
7. 161
7 . 60 1
Pwf
(psia)
2 ,735
2 ,684
2 , 665
5.7.2 Production at Constant BHP. The second case of the van
Everdingen-Hurst solution is for production at constant BHP. This
solution of the diffusivity equation models radial flow of a slightly
compressible liquid for the following conditions: ( 1 ) a homogene­
ous reservoir of uniform thickness, with the reservoir at uniform
pressure , P i ' before production ; (2) no flow across the outer
boundary (r = re ) ; and (3) production at constant BHP , P wf , from
a single well of radius r w centered in the reservoir. The reservoir
is infinite-acting until a boundary is encountered .
The solution is expressed most conveniently in terms of dimen­
sionless variables in the form given by Eq. 5 . 236, where tD and
rD are defined by Eqs . 5 . 1 1 5 and 5 . 1 00 , respectively ; reD = re1rw ;
and P D is defined for a pressure P at radius r as
PD =
P; -P
--
P; -P wf
6.01 5
7.1 61
7.601
qBp,
qD =
0 . 00708kh( p ; -P wf )
and Q D =
P
(0 . 2 1 )(0. 8)(20 x 1 0 - 6 ) (0 . 45) 2
q,p,c/ra
�
to
7.45 x 1 0 4
7.45 x 1 0 5
7.45 x 1 0 7
Sou rce
0.5 ( I n t o + 0.80907)
Table E-3
I n reO
--
Dimensionless production rate, qD , and dimensionless cumulative
production , QpD , are defined as
(0 . OO02637)(80)(24)t(days)
0 . 0002637kt
-----
P wf =P ; -
t
(days)
0.1
1 .0
1 00.0
Solution.
1 . First , we calculate dimensionless time and radius. Rearrang­
ing the definition of tD (t in hours) ,
tD =
TABLE 5. 6-DIMENSIONLESS TI MES FOR EXAMPLE 5.8
For this problem , the instantaneous rate , q, and cumulative pro­
duction , Qp , are of more practical importance than P D . These
quantities can be derived from the fundamental solution , P D .
B
qDdtD =
1 . 1 1 9q,c/hra( p ; -pwf )
Qp . . . . . . (5 . 245)
QpD is tabulated in Table E-4 for an infinite-acting reservoir and
in Table E-5 for a finite reservoir with a no-flow outer boundary ,
with 1 . 5 :5 reD :5 1 0 6 . For r eD � 20, values for both qD and QpD
are given in Table E-5 . In Table E-5 , for values of tD smaller than
the smallest value for a given reD , the reservoir is infinite-acting
and Table E-4 should be used . For values of tD larger than those
for a given value of reD in Table E-5 , the flow rate has diminished
to essentially zero , and
r�D - l
QpD = -- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 246)
2
For reD = 00 and tD � 200 , QpD can be approximated as l ?
QpD =
( - 4 . 2988 1 + 2 . 02566tD)
In tD
. . . . . . . . . . . . . . . . . . (5 . 247)
Since both qD and QpD are based on P D solutions that, for rD = 1
(L e . , r = rw) , depend only on tD and reD , qD and QpD also should
depend only on tD and reD . Table E-5 confirms thIS expectation .
For given reD and tD , QpD is determined uniquely . Although the
QpD solutions can be used to model individual well problems, they
are used more often to model water influx from an aquifer of radius
re into a petroleum reservoir of radius r w for a fixed pressure , P wf .
Superposition is used to model a variable pressure history .
Example 5.9-Using QpD Solutions To Model a Reservoir With
a No-Flow Outer Boundary . An oil well is produced with a con­
stant BHP of 2,000 psia for 1 hour from a reservoir initially at 2,500
psia. The reservoir is finite with no flow across the outer bounda­
ry . Calculate Qp in barrels .
c, = 20 X I O - 6 psi - I .
q, = 0 . 1 5 .
rw = 0 . 5 ft .
B = 1 . 2 RB/STB .
h = 1 5 ft .
re = I ,OOO ft .
p, = l cp o
k = 0 . 294 md .
p; = 2 , 500 psia.
Solution. We calculate tD and reD and then read QpD from Ta­
ble E-4 or E-5 . Dimensionless time is
tD =
· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5 . 243)
' /D
\
iJ
. . . . . . . . . . . . . . . . . . . . . . . (5 . 244)
0 . 0002637kt
----­
q, w /ra
(0 . 0002637)(0 . 294)( 1 )
-----
(0 . 15)( 1 )(20 x 1 0 - 6 )(0 . 5) 2
The dimensionless radius is
reD =
1 ,000
--
0.5
= 2 , 000 .
= 1 03 .
1 06
GAS RESERVOI R E N G I N E E R I N G
There is no entry in Table E-5 at this value of fD ' Thus , the
reservoir is infinite-acting , and from Table E-4 QpD = 44. 3 . The
cumulative production (Eq . 5 .245) at I hour is
,
QpD
2 p ; -pwj ) -Qp = 1 . 1 1 9cpc1 hrw(
B
( )
44 . 3
= ( 1 . 1 19)(0 . 1 5)(20 x 1 0 - 6 )( 1 5)(0 . 5) 2 (2,500 - 2,000) 1 .2
= 0 .233 STB .
From Table E-5 , w e read QpD = 1 . 89 X 10 7 . From E q . 5 .245,
the cumulative production for Well 2 is
QpD
Qp 2 = 1 . 1 1 9cpct hr� ( p; -P lif ) -B
= ( 1 . 1 19)(0 . 2 1 )(2 x 1 0 - 5 )(20)(0 . 3 3) 2 ( 1 ,000)
= 1 . 55 X 105 STB .
(
1 .89 X 107
1 .25
)
5 . For imaginary Well 3,
tD = ( 1 . 664 x 107 )(6) = 1 .0 x 1 0 8 .
Example S . IO-Analysis of Variable Pressure History With
QpD Solution. A well is completed in a reservoir with an initial
pressure of 6,000 psia. The well is centered in the cylindrical reser­
voir with no flow across the outer boundary . The well produced
for 6 months with a FBHP of P Wj = 5, 500 psia, then for 6 more
months with P wj = 4,500 psia, and finally for 6 more months with
P lif = 5,OOO pSla. Calculate the cumulative production after 1 8
months o f production .
ct = 20 x 10 -6 psi -I .
cp = 0 .2 1 .
r w = 0 . 3 3 ft .
1-'0 = 0 . 8 cp o
h = 20 ft .
re = 3,300 ft .
Bo = 1 .25 RB/STB .
k = 3 1 . 6 md .
reD = 3,300/0 . 33 = 1 0,000
0. 0002637kt
-----
cpl-'ctr�
(0.0002637)(3 1 .6)(730 hours/month)(t months)
(0 . 2 1 )(0.8)(2 x 10 - 5 )(0 . 3 3) 2
= 1 . 664 X 107 ( t months) .
3 . For the first imaginary well , Well 1 , we calculate
tD = ( 1 . 664 x 10 7)( 18) = 3 . 0 x 1 0 8 .
From Table E-5 , we read QpD = 2 . 54 x 107 . From Eq . 5 . 245,
the cumulative production for Well 1 is
tD = ( 1 . 664 x 10 7)( 12) = 2 . 0 x 1 0 8 .
1 .25
)
= 2 . 1 6 x 105 STB .
Well 3 produces for ( 18 - 12) = 6 months
with ( Pwj 2 -Pwj 3 ) = 4, 500 - 5,000 = - 500 psi .
2. For the real well , calculate the dimensionless radius and time:
4. For imaginary Well 2,
1 .06 X 107
Qp = Qp l + Qp 2 + Qp 3
= 1 . 04 x 105 + 1 . 55 X 1 0 5 - 0 . 434 X 105
Well 2 produces for ( 1 8 - 6) = 12 months
with ( P lifl - P lif2 )= 5,500 - 4,500 = 1 ,000 psi .
= 1 .04 X 1 0 5 STB .
(
6 . The cumulative production for the real well is
Well I produces for ( 1 8 - 0) = 1 8 months ,
with ( p ; - Pwjl ) = 6,000 - 5,500 = 500 psi .
= ( 1 . 1 1 9)(0.2 1 )(2 x 1 0 - 5 )(20)(0 . 3 3) 2 (500)
= ( 1 . 1 1 9)(0 . 2 1 )(2 x 1 0 - 5 )(20)(0 . 3 3) 2 ( - 500)
= - 0. 434 x 105 STB .
Solution.
1 . Because of the variable pressure history , superposition is re­
quired to solve this problem . We can calculate the cumulative pro­
duction from this well by adding the cumulative production from
three imaginary wells , each beginning to produce when P lif is
changed and each producing with a pressure drawdown equa1 to
the difference in pressures before and after the change :
and tD =
From Table E-5 , we read QpD = 1 .06 X 107 . The cumulative pro­
duction for Well 3 is
(
2 . 54 X I07
1 . 25
)
5.8 Summary
This chapter should prepare you to do the following .
1 . Derive the continuity equation for 1 D linear and radial flow .
2 . Derive the diffusivity equation from first principles (Darcy ' s
law , an EOS , and the continuity equation) for either a slightly com­
pressible liquid or a gas .
3 . Summarize all the simplifying assumptions on which the var­
ious forms of the diffusivity equation are based .
4 . Transform the diffusivity equation for a gas into forms writ­
ten in terms of pressure , pressure squared , and pseudopressure ,
and be able to state the assumptions required to derive each form
of the equation .
5 . Reduce the diffusivity equation, for either liquid or gas , and
its initial and boundary conditions to dimensionless forms with ap­
propriate and convenient definitions of dimensionless variables .
6 . State the assumptions on which each of the following solu­
tions of the diffusivity equation is based and the conditions of ap­
plicability of each solution : constant-rate production, bounded
cylindrical reservoir, no-flow outer boundary ; constant-rate pro­
duction , infinite reservoir, line-source well; bounded cylindrical
reservoir, pseudosteady-state flow ; radial flow in an infinite reser­
voir with wellbore storage; linear flow , constant-rate production,
semi-infinite reservoir.
7 . Use the shape-factor table to write a pseudosteady-state flow
equation for generalized reservoir geometry , determine PI, deter­
mine the duration of transient flow , the time region spanned by the
"transition" region, determine the start of pseudosteady-state flow ,
and calculate the PV in communication with a well in pseudosteady­
state flow .
8. Derive equations for wellbore-storage coefficient for wells with
a moving liquid/gas interface and wells filled with a single-phase
liquid or gas . State the assumptions on which the final results are
based . Reduce the final equations to dimensionless form .
9 . Derive an equation that models the unit-slope line on Ramey ' s
type curve . Calculate the wellbore-storage coefficient from well
characteristics or from a unit-slope line .
10. Derive the standard equation for calculating radius of inves­
tigation in two ways: considering an impulse (instantaneous injec­
tion of fluid into the wellbore) and considering both transient and
pseudosteady-state equations for situations in which we have
constant-rate production or injection .
1 07
F U N DAMENTALS OF FLU I D FLOW I N POROUS M EDIA
1 1 . Calculate radius of investigation.
12. Calculate the pressure at any point in an infinite-acting reser­
voir with any number of wells present, a no-flow boundary pres­
ent , and after any number of rate changes .
1 3 . Calculate effective producing time using Horner ' s approxi­
mation and state the conditions under which the approximation is
accurate .
1 4 . Calculate pressure in a well following a sequence of rate
changes in a finite , cylindrical reservoir using the van Everdingen­
Hurst constant-rate solutions .
1 5 . Calculate rate and cumulative production (or water influx)
in a well (or reservoir) following a sequence of discrete pressure
drops in a finite , cylindrical reservoir using the van Everdingen­
Hurst constant-pressure solutions .
Questions for Discussion
1 . Your boss has asked you to identify a good method to predict
well performance as a function of time so that she can provide up­
per management with financial forecasts. However, because you
are inexperienced , she wants considerable backup on your com­
putational method so that all assumptions are clearly identified and
the applicability of the method can be checked. Accordingly , please
provide the following information for the method you propose for
each of the field cases specified later .
• The equation or computational method you will use to provide
the boss with the desired information .
• The most important, possibly limiting , assumptions inherent
in your chosen method .
• The basis for your belief that these are the key assumptions .
This information should include , but should not be limited to , the
form of the diffusivity equation and the appropriate initial and bound­
ary conditions that provide the basis for your method .
• Additional information you will look for or request to check
the applicability of the method .
Field Cases.
A . A 20-year production forecast for a gas well in a very low­
permeability formation and with a 640-acre drainage area and a
1 2 , OOO-psi discovery pressure . The well will be produced at con­
stant surface pressure .
B . A forecast of oil production for the next year for a well in
a high-permeability formation . The well produces significant quan­
tities of water and free gas . The well will be produced at constant
BHP.
C. A l O-year production forecast for an oil well currently capa­
ble of producing at rates far exceeding its allowable from a high­
permeability formation . The discovery pressure was bubblepoint
pressure in the reservoir .
D . An estimate of formation permeability for a gas well in a very
low-pressure , high-permeability formation that , unfortunately , is
in a field developed on 40-acre spacing .
E. A PI for an oil well in a highly undersaturated , low­
permeability reservoir.
2. What are the boundary conditions for steady-state flow?
3. Are steady-state flow solutions included in the van Everdingen­
Hurst tables? Where?
4. What are the boundary conditions for pseudosteady-state flow?
5 . Are pseudo steady-state flow solutions included in the van
Everdingen-Hurst tables? Where?
6. What are the boundary conditions for unsteady-state flow?
7 . Are unsteady-state flow solutions included in the van
Everdingen-Hurst tables? Where?
8 . How can we calculate P wf v s . t for a well produced at con­
stant rate in an infinite-acting reservoir? In a finite reservoir? What
geometry is assumed? What are the practical limits on the reser­
voir size if we use the van Everdingen-Hurst tables? Other methods?
9 . How can we calculate q and Q vs. t for a well produced at
constant BHP in an infinite-acting reservoir? In a finite reservoir?
What geometry is assumed? What are the practical limits on the
reservoir size if we use the van Everdingen-Hurst tables? Other
methods?
10. How can we estimate rate and cumulative production vs. time
for a well produced at variable BHP?
1 1 . How can we estimate BHP v s . time for a well produced at
rate?
1 2 . Define steady-state flow , unsteady-state flow , pseudosteady­
state flow , radial flow , and linear flow .
1 3 . Write equations in field units , expressing oil flow rate
(STBID) as a function of well and reservoir properties , for the fol­
lowing situations :
A . Steady-state radial flow .
B . Pseudosteady-state radial flow .
C . Unsteady-state radial flow .
D . Steady-state linear flow .
E . Pseudosteady-state linear flow .
F. Unsteady-state linear flow .
1 4 . Write the differential form of Darcy ' s law with each quanti­
ty in field units and include standard condition flow rate , q, and
area, A. Then write special forms for radial flow and for linear
flow . Write these equations for both slightly compressible liquid
flow and gas flow (with rate in MscflD for gas) .
1 5 . On what assumptions are the following equations from Question 13 based?
A. Eq . 1 3a .
B . E q . 1 3b .
C . Eq . 1 3 c .
D . Eq . 1 3d .
1 6 . Write the steady-state radial flow equation for gas , express­
ing gas flow rate in MscflD as a function of well and reservoir prop­
erties , and state the assumptions on which your equation is based .
1 7 . Write equations relating pressure and volume at constant temperature for oil , gas , and water. Identify important assumptions im­
plied in these equations .
1 8 . Define formation compressibility , cf ' How is it related to
rock volume change? PV change?
1 9 . Wellbore radius , rw ' appears in several solutions of flow
equations . What is rw? Casing ID? Tubing ID? What?
20. A. Dimensionless time , tD is sometimes defined as
'
variable
0 . 0002637kt
cp p. c t r a
-----
. Prove that tD is truly dimensionless .
B . Dimensionless pressure , P D ' is sometimes defined as
khl1p
----
1 4 1 . 2qBp.
. Prove that P D is dimensionles s .
2 1 . A . Explain the difference between the expressions dp/dr and
aplar.
B. Is the equation d 2 pldr 2 = dpldt in proper form? Explain.
22 . Consider a 1 x 1 x 1-ft cube of reservoir rock, with fluid en­
tering and leaving the block. Express the " law of conservation of
mass" in words for this system .
Exercises
5 . 1 A. Derive the diffusivity equation in a form that models gas
flow in 1D linear flow . Use x for the space variable . Sum­
marize all assumptions made to derive this equation . Ex­
press all variables in conventional field units in all equations
used in the derivation .
B . The situation to be modeled with the equation derived in
Problem 1 is constant-rate production from the reservoir
face at x = O , no flow across the outer boundary at x =L ,
and initial pressure uniform throughout the reservoir at the
value P i ' Express the initial and boundary conditions as
equations .
C . Define dimensionless variables in an appropriate way (so
that no variables or parameters remain in the differential
equation or conditions) , and reduce the differential equa­
tion , initial condition , and boundary conditions to dimen­
sionless form .
5 . 2 A well has a shut-in BHP of 2 , 300 psia and flows 2 1 5 STBID
of oil under a 500-psi drawdown . The well produces from
a formation of 36-ft net productive thickness . Use r w = 6 in . ,
re = 660 ft , p. = 0 . 88 cp , Bo = 1 . 32 RB/STB .
1 08
GAS RESERVO I R E N G I N E E R I N G
A . What is the well ' s PI?
B. What is the average formation permeability?
C. What is the formation capacity?
5 . 3 Given a reservoir with these properties , perform the following calculations .
q = 250 STBID .
J.I. = 0 . 75 cp o
B = 1 . 25 RB/STB .
k = 25 md .
h = 60 ft .
4> = 20 % .
c, = 6 . 0 x l O - 6 psia - I .
re = 3 ,000 ft .
rw = 0 .25 ft .
s=5.
p; = 3 ,500 psia .
A . Calculate the time at which the Ei-function solution first
becomes valid .
B . Calculate the time after which the Ei-function solution is
no longer valid .
C . Calculate and plot pressure as a function of position on
semilog graph paper at t = 1 00 hours using the Ei-function
solution (or the logarithmic approximation) for the follow­
ing radii: 0 . 2 5 , 1 , 2 , 1 0 , 20, 50, 1 00 , 200 , 500 , 1 ,000,
2 , 000, and 3 ,000 ft .
D . Calculate the additional pressure drop resulting from the
skin factor.
5 . 4 Consider a well centered in a 2 X 1 rectangular drainage area,
producing single-phase oil and solution gas at constant rate .
The well and reservoir have the following characteristics .
q = 40 STBID.
cg = 4 .26 X 1 0 - 4 psia - I .
Cf= 2 . 58 X I O -6 psia - I .
Bo = 1 . 234 RB/STB .
Bg = 0 . 659 RB/STB .
Sg = 0 . 05 .
'Yo = 40 o API .
Rp = 1 ,226 scf/STB .
A = 1 20 acres.
k = 8 md .
rs = 1 2 ft .
4> = 28 % .
p = 4 ,999 psia at 20 day s .
p = 4 , 998 . 3 psia a t pseudosteady state .
c o = 7 . 1 89 x l O - 5 psia - I .
c w = 3 . 99 x l O - 6 psia - I .
J.l.o = 0 . 459 cp o
B w = 1 . 0 1 4 RB/STB .
T= 200 o P .
Sw = 0 . 3 .
'Y g = 0 . 8 .
R sw = I 1 . 2 scf/STB .
rw = 0 . 3 3 ft .
ks = O . 1 md .
h = 45 ft .
p; = 5 ,000 psia .
p = 4 ,952 . 1 psia at I year .
A . Calculate the wellbore pressure 3 hours after production
began. What is the flow regime in the reservoir at this time?
B . Calculate the wellbore pressure 20 days after production
began. What is the flow regime in the reservoir at this time?
C . Estimate the well ' s skin factor .
D . Calculate tpss ' the time a t which pseudosteady state i s
reached . How much sooner (or later) would pseudosteady
state be reached if the well were centered in a circular reser­
voir of the same area?
E. Calculate the pressure in the wellbore at tpss (for the rec­
tangular drainage area) .
P . Calculate the well ' s PI .
G. Calculate the rate at which pressure is declining at a time
of 2tpss at the wellbore and at 1 00 ft from the well .
H . I f the well depth is 8 ,452 ft and the tubing ID i s 2 . 44 1
in. (packed off) , calculate the wellbore-storage coefficient.
I. What would be a good estimate of the dimensionless time ,
t D , when wellbore-storage effects become negligible?
J. What would the radius of investigation be if the well could
continue to produce oil (and solution gas) at a constant rate
of 40 STBID for 1 year? Can the well produce at this rate
for 1 year? Explain your answer .
5 . 5 Consider a well producing single-phase oil at constant rate .
The well and reservoir have the following characteristics.
q = 300 STBID .
Bo= 1 . 32 RB/STB .
k = 25 md .
4> = 1 6 % .
c ,= 1 8 x l O - 6 psia - I .
J.l.o = 0 . 44 cpo
h = 43 ft .
p; = 2 , 500 psia .
A . Plot pressure vs. radius on both linear and semilog paper
at times of 0 . 1 , 1 . 0 , 1 0 , and 1 00 day s .
B . Assuming that a 5-psi pressure drop can b e detected easi­
ly with a pressure gauge, how long must the well be flowed
to produce this drop in a well located 1 ,200 ft away?
C . Suppose the flowing well is located 200 ft due east of a
north/south fault . What pressure drop will occur after 1 0
days o f flow i n a shut-in well located 600 ft due north of
the flowing well?
D. What will the pressure drop be in a shut-in well 500 ft from
the flowing well when the flowing well has been shut in
for 1 day following a flow period of 5 days at 300 STB/D?
5 . 6 Calculate the time required to reach pseudosteady state for
a well in the center of a circular , 640-acre reservoir in each
of the following situations (h = 75 md , cf = 3 X 10 -6 psia - I ,
rw = 0 . 2 5 ft) .
A . Undersaturated oil reservoir : p; = 4 , 000 psia, T= 200 o P ,
'Y o = 3 8 ° API , J.l.o = 0 . 46 cp , c o = 1 . 28 x l O - 5 psia - I ,
4> = 25 % , k = 350 md .
B . High-pressure gas reservoir : p; = 1 2 ,000 psia , T= 350 o P ,
'Y g = 0 . 67 , J.l. g = 0 . 03636 cp , cg = 3 . 74 X I O - 5 psia - I ,
4> = 7 % , k = O . O I md .
C . Low-pressure gas reservoir: p; = 600 psia, T= l l O o P ,
'Y g = O . 7 2 , J.l. g = 0 . 0 1 l 67 cp , cg = 1 . 83 X IO - 3 psia - I ,
4> = 1 5 % , k = 1 O md .
D . Saturated oil reservoir: p ; = 2500 psia, T= 225 ° P ,
'Yo = 45 ° P , J.l.o = 0 . 36 c p , S o = 6 0 % , co= 1 . 8 X I O - 4
psia - I , S = 25 % , cg = 3 . 92 x l O - 4 psia - I , Sw = 1 5 % ,
c
c w = 4 x 10. - 6 psia - 1 , 4> = 2 1 % , k = 1 00 md . Calculate c,
using c,= Soco + Sg C g + S w c w + cf . Assume that oil is the
only mobile phase .
5 . 7 Consider a well centered in a circular reservoir producing
single-phase oil at constant rate . The well and reservoir have
the following characteristics .
t = 1 ,000 STBID.
Bo= I . 475 RB/STB .
re = 3 ,000 ft .
k = 1 O md .
4> = 23 % .
s=5.
c,= 1 . 5 x l O - 5 psia - I .
J.l.o = O . 72 cp o
rw = 0 . 5 ft .
h = 1 50 ft .
p; = 3500 psia .
The well produces for 1 ,000 hours at 1 ,000 STBID . There
is a linear, no-flow boundary 75 ft away from the well . Cal­
culate and plot the pressure at the wellbore as a function of
time on semilog graph paper (put time on the logarithmic axis)
using the method of images . Use times of 0 . 1 , 0 . 2 , 0 . 5 , 1 ,
2 , 5 , 1 0 , 20, 50, 1 00 , 200 , 500 , and 1 ,000 hours .
5 . 8 Consider a well centered in a circular reservoir producing
single-phase oil at constant rate . The well and reservoir have
the following characteristics .
F U N DA M ENTALS OF FLU I D FLOW I N PORO U S M EDIA
s=5.
Bo = I . 475 RB/STB .
re = 3,000 ft .
k = l O md .
c/> = 2 3 % .
q l = 1 000 STBID .
Q2 = 0 STBID .
ct = 1 . 5 x l O - 5 psia - I .
/-to = 0 . 72 cp o
r w = 0 . 5 ft .
h = 1 50 ft .
Pi = 3, 500 psia .
tl = 1 ,000 hours .
t2 = 1 , 0 1 O hours .
The well produces for 1 ,000 hours at 1 ,000 STBID and
then is shut in. Calculate and plot the pressure at
t2 = 1 ,0 1 0 hours as a function of radius on semilog graph
paper (put radius on the logarithm axis) using superposi­
tion in time . Use radii of 0 . 5 , 1 , 2, 5, 1 0 , 20, 50, 1 00 ,
200 , 500, 1 , 000, and 2,000 ft .
5 . 9 A sandstone reservoir has the following properties .
q = 1 000 STBID .
Bo = 1 . 56 RB/STB .
k = l O md .
c/> = 25 % .
r w= 0 . 5 ft .
A = 20 acres .
ct = 1 8 . 5 X l O - 6 psia - I
/-to = 0 . 33 cp o
h = 40 ft .
Pi = 4 ,000 psia .
s= O .
Pb = 3, 200 psia .
A . Estimate the time at which the pressure at the wellbore
will fall below the bubblepoint .
B . Estimate the average reservoir pressure at the time the pres­
sure at the wellbore falls below the bubblepoint pressure .
5 . 1 0 A new reservoir has three wells . Well A produces at 1 00
STBID , Well B at 500 STBID , and Well C at 1 ,500 STBID .
Well B is 660 ft from Well A and Well C is 1 , 320 ft from
Well A . Well A has a skin factor of + 2 , Well B has a skin
factor of + 3, and Well C has a skin factor of - 5 . The fluid
and rock properties are given blow . All three wells begin
producing at the same time .
1 09
i.e. , there is no replenishment of water from the surface. Some
other important aquifer properties are given below .
cj = 4 x l O - 6 psia - I .
B w = 1 .0 1 3 RB/STB .
k = 1 1 5 md .
c/> = 1 5 % .
c w = 3 x l O - 6 psia - I .
/-tw = 0 . 86 cp o
h = 30 ft .
Pi= 3,526 psia .
The pressure at the aquifer/reservoir interface was instan­
taneously dropped to 3 , 462 psia and held constant for 1 month.
It was then lowered to 3 , 364 psia and held constant for 6
months . Then the pressure was dropped again to 3 , 2 1 2 psia
and held constant for 8 months.
A . Estimate the cumulative water influx into the reservoir after
1 , 7 , and 1 5 months .
B . Estimate the water influx rate at 1 , 7 , and 1 5 months.
5 . 1 2 The Barboza No. 2-A well has produced a 48 ° API oil at a
constant rate of 85 STBID for 30 minutes . Before that , the
well had produced at 1 00 STBID for 3 hours , and before that
it had produced at 1 45 STB/D for 1 5 hours . Calculate the cur­
rent BHP given that the well is centered in a closed , circular
drainage area with the following properties .
cj = 3 X l O - 6 psia - I .
c w = 3 . 2 x l O - 6 psia - I .
Bo= 1 . 25 RB/STB .
S w = 25 % .
rw = 0 . 25 ft .
k = 0 . 32 md .
c/>= 12 % .
co= 1 . 25 x 1 O - 5 psia - 1 .
/-to= 1 . 1 cp o
Pi= 2,320 psia.
Pb = 4 , 1 20 psia.
re = 6 , 000 ft.
h = 60 ft.
s= O.
5 . 1 3 A well is located in the center of the northwest quadrant of
a 640-acre square drainage area , as shown below .
q = 1 000 STBID .
Bo= 1 . 56 RB/STB .
k = l O md .
c/>= 25 % .
rw = 0 . 5 ft .
A= 20 acres .
ct = 1 8.5 x l O -6 psia - I .
/-to = 0 . 33 cp o
h = 40 ft .
Pi = 4 , 000 psia.
s = O.
Pb = 3,200 psia .
A . Estimate the time at which Well A exhibits a pressure
response caused by production from Well B .
B . Estimate the time at which Well A exhibits a pressure
response caused by production from Well C .
C . Calculate the pressure drop at Well A resulting from pro­
duction from Well A after 250 hours .
D . Calculate the pressure drop at Well A resulting from pro­
duction from Well B after 250 hours .
E. Calculate the pressure drop at Well A resulting from pro­
duction from Well C after 250 hours .
F . Calculate the pressure at Well A caused by combined pro­
duction from Wells A through C after 250 hours .
5 . 1 1 A roughly cylindrical oil reservoir is surrounded by and es­
sentially centered in an aquifer. The reservoir area is 360 acres,
and the aquifer area is 1 2 ,960 acres . The aquifer is closed ;
The well , formation, and fluid properties are given below .
q = 606 STBID .
A = 640 acres .
Bo = 1 . 593 RB/STB .
s= 1 2 .
k = 1 7 . 5 md .
ct>= 1 2 % .
ct= 1 8 x l O - 6 psia - 1 •
/-to = 0 . 306 cpo
Pi= 4,250 psia.
rw = 0 . 25 ft.
h= 1 03 ft .
T= 275 ° F .
A . Calculate the time at which the error i n using the Ei­
function solution becomes greater than l % as a result of
boundary effects .
B . Calculate the time after which the pseudosteady-state equa­
tion can be used with less than I % error.
C. Calculate the additional pressure drop owing to the skin
factor, il ps '
Author's note: Probl e m s 5 . 2 and 5 . 5 were taken from Craft/Haw k i n s , Applied Petroleum
Reservoir Engineering, second edition , © 1 99 1 , p p . 266-267. 1 8 Reprinted by permission
of Prentice Hall, Upper Saddle River, NJ .
GAS RESERVO I R E N G I N E E R I N G
1 10
Win
=
=
Y1 =
z =
!J. P s =
c/> =
'Y g =
'A =
p. =
p =
D . Calculate the pressure at the wellbore , P wf , when the
average reservoir pressure , p , is 4 , 1 00 psia.
E . Calculate the flow rate that would be obtained for a draw­
down pressure , P -P wf , .of 500 psia if an acid treatment
were performed that changed the skin factor from 12 to
-2.
wout
Nomenclature
A
Ar
Bg
B0
Bw
=
=
=
=
=
=
=
=
=
=
=
=
=
drainage area , L, acres
cross-sectional area, L2 , ft2
gas FVF , RB/Mscf
oil FVF , RB/STB
water FVF , RB/STB
Cf formation compressibility , Lt2 /m, psia - I
Cg
gas compressibility , Lt2 /m, psia - I
Co
oil compressibility , Lt 2 /m , psia - I
c,
total compressibility , Lt2 /m, psia - I
Cw
water compressibility , Lt 2 /m, psia - I
C well bore-storage coefficient, bbl/psi
CA shape factor
h
net pay thickness, L, ft
J = PI , L 4 t/m, STBID-psi
J 1 = first-order Bessel function
ka = average permeability , L2 , md
kg = gas permeaiblity , L2 , md
ks = altered-zone permeability , L2 , md
Lf = fracture half-length, L, ft
m g , dissolved = mass of dissolved gas in the reservoir per unit
reservoir volume , m
m g Jree = mass of free gas in reservoir per unit reservoir
volume , m
mo = mass of oil in the reservoir per unit reservoir
volume , m
m w = mass of water in the reservoir per unit reservoir
volume , m
M = number of moles
q = flow rate , L3 It, MscflD for gas or STB/D for
oil
q m = rate of mass accumulation , Ibm/sec
Qp = cumulative production, L 3 , Mscf for gas or
STB for oil
P = average drainage area pressure , m/Lt 2 , psia
P i = initial pressure , m/Lt 2 , psia
Pp = pseudopressure function, psia2 /cp
Pwf = flowing wellbore pressure , m/Lt 2 , psia
re = outer radius, L , ft
rs = altered-zone radius , L, ft
r w = wellbore radius , L, ft
s = skin factor
Sg = gas saturation , fraction
So = oil saturation , fraction
S w = water saturation , fraction
t = time , t, hours
tap = pseudotime function , hr-psia/cp
T = temperature, T, O R
ur = fluid velocity , Lit, ftlsec
VP = PV " L3 ft 3
W = mass flow rate , Ibm/sec
mass flow rate into system, Ibm/sec
mass flow rate out of system , Ibm/sec
first-order Bessel function
gas compressibility factor
pressure drop owing to skin factor , m/Lt2 , psia
porosity , fraction
gas specific gravity
mobility , L 3 t/m, md/cp
viscosity , miLt, cp
fluid density , m/L 3 , Ibm/ft3
Subscripts
A ,B, C =
t =
wells
total
References
I . AI-Hussainy , R . , Ramey , H . J . Jr. , and Crawford, P . B . : " The Flow
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
of Real Gases Through Porous Media, " JPT (May 1 966) 624-36 ;
Trans. , AIM E , 237.
Agarwal , R . G . : " Real Gas Pseudo-Time-A New Function for Pres­
sure Buildup Analysis of MHF Gas Wells , " paper SPE 8279 present­
ed at the 1 979 SPE Annual Technical Conference and Exhibition, Las
Vegas , Sept . 23-26 .
Lee , W . J . and Holditch, S . A . : "Application of Pseudo time to Buildup
Test Analysis of Low-Permeability Gas Wells With Long-Duration Well­
bore Storage Distortion , " JPT (Dec. 1 9 82) 2877-87 .
Matthews, C . S . and Russel l , D . G . : Pressure Buildup and Flow Tests
in Wells, Monograph Series, SPE, Richardson, TX ( 1 977) 1 .
van Everdingen , A . F . and Hurst , W . : " Application o f the Laplace
Transformation To Row Problems in Reservoirs, " Trans. , AIME ( 1949)
186, 305-24.
Slider, H . C . : Practical Petroleum Reservoir Engineering Methods, Pe­
troleum Publishing Co. , Tulsa, OK ( 1 976) 70.
Hawkins, M . F . Jr. : "A Note on the Skin Effect, " Trans. , AIME ( 1 956)
207, 356-57 .
Theory and Practice of the Testing of Gas Wells, third edition, pub .
ERCB-74-34, Energy Resources and Conservation Board , Calgary
( 1 975) .
Odeh , A . S . : " Pseudosteady-State Flow Equation and Productivity In­
dex for a Well With Noncircular Drainage Area, " JPT (Nov. 1 978)
1 630-32.
Earlougher , R.C. Jr. : Advances i n Well Test Analysis, Monograph Se­
ries, SPE, Richardson, TX ( 1 977) 5.
Agarwal , R . G . , AI-Hussainy , R . , and Ramey , H . J . Jr. : "An Investi­
gation of Wellbore Storage and Skin Effect in Unsteady Liquid Row-I .
Analytical Treatment, " SPEI (Sept. 1 970) 279-90; Trans. , AIME, 249.
Katz , D . L . et al. : Handbook of Natural Gas Engineering, McGraw­
Hill Book Co. Inc . , New York City ( 1 959) 4 1 1 .
Carslaw , H . S . and Jaeger , J . C . : Conduction ofHeat in Solids, second
edition, Oxford at the Clarendon Press ( 1 959) 2 5 8 .
Matthews, C . S . , Brons, F . , and Hazebroek, P . : " A Method for De­
termination of Average Pressure in a Bounded Reservoir, " Trans. ,
AIME ( 1 954) 201 , 1 82-9 1 .
Horner, D . R . : " Pressure Buildup i n Well s , " Proc. , Third World Pet .
Cong . , The Hague ( 1 95 1 ) Sec . II, 503-23 ; Pressure A nalysis Methods,
Reprint Series, SPE , Richardson, TX ( 1 967) 9, 25-43 .
Chatas, A. T . : "A Practical Treatment of Nonsteady-State Flow Prob­
lems in Reservoir Systems , " Pet. Eng. (Aug. 1 953) B-44-56 .
Edwardson, M . J . et al. : "Calculation of Formation Temperature Dis­
turbances Caused by Mud Circulation , " JPT (April 1 962) 4 1 6-26 ;
Trans. , AIM E , 225.
Craft, B . C . et al. : Applied Petroleum Reservoir Engineering, second
edition, Prentice-Hall , Englewood Cliffs , NJ ( 1 99 1 ) .
Chapter 6
Pressure·Transient Testing
6.1 Introduction
This chapter presents the underlying theory and practical applica­
tions of pressure-transient testing in gas wells . Beginning with the
line-source (Ei-function) solution to the diffusivity equation, we de­
velop analysis techniques for flow and buildup tests in homogeneous­
acting reservoirs . Both semilog and log-log plotting analysis tech­
niques are discussed and illustrated with examples . We also dis­
cuss non-Darcy flow effects that are more pronounced in gas-well
testing . We introduce special analysis techniques for well tests from
hydraulically fractured gas wells and from wells completed in natur­
ally fractured gas reservoirs .
6.2 Types and Purposes of
P ressure· Transient Tests
The term pressure-transient test refers to a test in which we gener­
ate and measure pressure changes in a well as a function of time .
From this measured pressure response , we can determine impor­
tant formation properties of potential value in optimizing either an
individual completion or a depletion plan for a reservoir. Pressure­
transient tests can be grouped into two broad categories-single­
PWf= P; +
6.3 Homogeneous Reservoir Model­
Slightly Compressible Liquids
The basis of well-test analysis techniques for homogeneous-acting
reservoirs is the line-source (Ei-function) solution to the diffusivi­
ty equation. 1 - 3 As shown in Chap. 5, the relationship between bot­
tornhole flowing pressure (BHFP) , P wf ' and the formation and well
characteristics for a well producing a slightly compressible liquid
at a constant rate is
kh
[(
In
Gas Wells
1 ,688c/>P, c t ra
kt
) ]
- 2s . . . . . . . . . . (6 . 1 )
If we change from natural logarithms to base 1 0 logarithms and
simplify , we can rewrite Eq. 6 . 1 in a more familiar form,
Pwf =P; -
1 62 . 6 qBp,
kh
[ (
log
kt
--
c/>p, c t rw2
)
]
- 3 .23 + 0 . 869s , . . (6 . 2)
where the skin factor, s, is used to quantify either formation damage
or stimulation . Skin effects are discussed later.
6 . 3 . 1 Analysis of Constant-Rate Flow Tests. Eq. 6.2 describes
the variation of the wellbore pressure with time when a well is pro­
duced at constant rate . 1 - 3 Production at a constant rate can be con­
sidered a pressure-drawdown or -flow test. Comparing Eq. 6 . 2 with
the equation of a straight line, y= mx +b, suggests an analysis tech­
nique in which the following terms are analogous :
Y - P wf ,
x - log t,
well and multiwell tests .
Single-well tests measure pressure buildup , drawdown, and fall­
off, as well as injectivity . In these tests, we use the measured pres­
sure response to determine average properties in part or all of the
drainage area of the tested well . Multiwell tests, which include in­
terference and pulse tests , are used to estimate properties in a region
centered along a line connecting pairs of wells. Therefore , they
are more sensitive to directional variations in properties . In a mul­
tiwell test, the approach is to produce from (or inject into) one well ,
called the active well , and observe the pressure response in one
or more offset wells, or observation wells. Although we concen­
trate in this chapter on analysis techniques for single-well tests , the
fundamental principles presented can be extended to develop
methods for analyzing multi well tests .
70. 6qBp,
of
m- -
1 62 . 6qBp,
kh
and b -p ; -
1 62 . 6qBp,
kh
[ (
log
k
---
c/>p, c t rw2
)
]
- 3 . 23 + 0 . 869s .
These analogies indicate that a plot of P wf vs. log t will exhibit
a straight line from which the slope , m, allows us to estimate k and
s. Fig. 6 . 1 is an example semilog graph of constant-rate flow test
data. The slope of the line , m, is the difference between two pres­
sures, P wf , 1 and P wf, 2 , one log cycle apart, or m=pwf, 2 -P wf, l .
For single-phase flow, the formation permeability in the drainage
area of the well is computed from
k=
1 62 . 6qBp,
mh
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 3 )
where the absolute value of m is used . Rearranging Eq. 6 . 2 and
combining with Eq. 6 . 3 gives
s= 1 . 1 5 1
[
( P; -P wf )
m
- IOg
(� ) ]
c/>p, ct rw
+ 3 .23 . . . . . . . . . . (6 . 4)
112
GAS RESERVO I R E N G I N E E R I N G
Rate
41 00
4000
q l-------.
390 0
co
'"
.8;
3800
tp
3700
•
Q.
3600
t1 t
.
3500
3400
0.'
o
Time (hours)
Fig. 6 . 1 -Analysis technique for constant-rate flow test data.
P I hr
P I hr
For convenience, we set the flow time , t, equal to 1 hour, and
use the symbol
nec­
for the BHFP at this time . Note that
essarily lies on the semi log straight line . Substituting these into Eq .
6 . 4 yields
s= 1 . 151
)
[ ( Pi-P l hr) (-c/>p.c,rw2 l
k
- log
m
+ 3 .23 . . . . . . . . . (6 . 5 )
6.3.2 Analysis o f Pressure-Buildup Tests. A n equation model­
ing a pressure-buildup test can be developed by use of superposi­
tion in time . 1 -3 In terms of the line-source solution given by Eq.
6 . 2 , the bottomhole pressure (BHP) for the rate history shown in
Fig. 6.2 is
Pws=Pi
-
1 62 . 6qBP.
kh
[ ( c/>p.c,r 2 )
w
log
k(tp + .:lt»
[ (--2 )
c/>p.c,rw
Pws
=
Pws=P i- �:q [ [ :; lJ
1 62 . 6( - q)Bp.
-
kh
where
k.:lt
log
- 3 . 23 + 0 . 869s
l
l
- 3 .23 + 0 . 869s , . . . . . . (6 . 6)
= shut-in BHP, tp = duration of the constant-rate produc­
tion period before shut- i n , and .:It
duration of the shut- in period .
If we combine terms and simplify , Eq . 6 . 6 can be rewritten as
BP.
1 62
IOg
(tp
t)
. . .
m- -
and
x
=
1 62 . 6qBp.
(---;;:;- )
tp + .:lt
.
This suggests that a plot of shut-in BHP,
from a buildup
test as a function of the log of the Horner 4 time ratio function ,
(tp + .:It)/.:lt, will exhibit a straight line with slope m . The slope is
one
the difference between two values of pressure,
and
log cycle apart. To calculate permeability , we use the absolute value
of the slope , or
Pws '
Pws.l Pws.2 '
k=
1 62 . 6qBp.
mh
2050
Cl.
'"
1 90 0
.8;
'"
5
'"
'"
'iii
2000
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 8)
m
1 95 0
Q)
Time
0
a
I
Pws, 2 - PWS, '
Pi
,
i
1 85 0
0
1 800
E
g
1 75 0
0
aJ
1 70 0
1 0
4
3
2
Horner Time Ratio, ( t + ;\t)
p
1 0
1 0
1 0
/.<11
'
1 0
°
Fig. 6.3-Graphical analysis technique for pressure-buildup
test data.
Pi '
From the semilog graph , the original reservoir pressure ,
is
estimated by extrapolating the straight line to infinite shut-in time
where (tp + .:It )/.:lt = 1 . Fig . 6.3 illustrates calculation of the slope
and original reservoir pressure .
We also can solve for the skin factor, s , from a pressure-buildup
test . At the instant a well is shut-in , the BHFP is
P f =Pi....
1 62 . 6qBp.
kh
[ ( c/> )
w,rw
l
k tp
log -2
- 3 . 2 3 + 0 . 869s . . . (6.9)
Combining Eqs . 6 . 7 , 6 . 8 , and 6 . 9 , we can derive an expression
for the skin factor:
s= 1 . 1 5 1
[( Pws -Pwf ) ( )
c/>p.c,rw2
- I Og
m
k.:lt
+ 3 .23 + I O g
( )l
tp + .:It
tp
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 1 0)
P I h r Pws
) l
[ ( P l hr-Pwf) (�
c/>p.c,rw
Pwf
P ws
Pi ,
where m = slope of the semilog straight line . Setting .:It = 1 hour,
introducing the symbol
at .:It = 1 hour on the semilog
for
straight line , and neglecting the term log [(tp + 1 )/tp l gives
s = 1 . 15 1
kh
- log
.,
. . . . . . . . . . . . (6 . 7)
Comparing Eq . 6 . 7 to the equation of a straight line , y mx + b ,
gives
=
F i g . 6 .2-Model i ng a pressure-buildup test i n terms of
variable-rate production.
.r;
In summary , from the straight line predicted by theory for a plot
of constant-rate flow test data on semilog graph paper, we can es­
timate k and s .
t1 t
m
- IOg
+ 3 . 23 , . . . . . . (6. 1 1 )
where
= BHFP at the instant of shut-in. In summary , using in­
formation obtained from a plot of
vs . log (tp + .:It )/.:It, we can
estimate k,
and s .
6.4 Complications in Actual Tests
The analysis techniques presented in the previous section were de­
rived assuming a homogeneous reservoir model and therefore rep­
resent ideal conditions . In reality , reservoirs are not homogeneous ,
and the actual pressure response during a flow o r buildup test devi­
ates from the ideal behavior (i . e . , the semilog straight line predicted
by theory may not be present) . These deviations usually are caused
by conditions in the wellbore and drainage radius of the reservoir
113
PRESS U R E-TRA N S I E N T TESTI N G OF GAS WELLS
5000
5000
<is
·iii
.e:
5
'"
'"
�
Q)
Q.
<is
·iii
.e:
4900
4800
5
'"
'"
Q)
4900
4800
470 0
c:
Q)
470 0
4600
10
Radius (It)
0.1
4500
1 00
1000
4600
0.1
4500
10
Radius (It)
100
1000
Fig. 6.4-Radlus of i nvestigation as a fu nction of flow time
d u ring a pressu re-drawdown test.
Fig. 6 . S-Radius of Investigation as a function of shut-in time
d u ring a pressure-buildup test.
that are not considered in the simple model described by Eq. 6 . 2 .
W e use the concept o f radius o f investigation, introduced i n Chap .
5 , to understand the causes of the nonideal behavior .
test is different from that encountered later (away from the well) ,
we should not be surprised that the slope of the curve of pressure
v s . the appropriate time function on a semilog graph is different
at early and late times . Similarly , because the Ei function solution
assumes an infinite-acting reservoir, we should expect the slope
of a buildup or flow test plot to change shapes at late times when
the radius of investigation reaches the reservoir drainage boundaries .
6.4. 1 Radius of Investigation Concept. Consider a graph (Fig.
6.4) of pressure as a function of radius for constant-rate flow at
various times since the beginning of flow . The pressure in the well­
bore continues to decrease as flow time increases . Simultaneously ,
the area from which fluid is drained increases , and the pressure
transient moves further out into the reservoir. The radius of inves­
tigation, defined as the point in the formation beyond which the
pressure drawdown is negligible , is a measure of how far a tran­
sient has moved into a formation following any rate change in a
well and physically represents the depth to which formation prop­
erties are being investigated at any time in a test . The approximate
position of the radius of investigation at any time is estimated with
the relation 3
ri =
(
:) ... .. . ... .
948 I-'C
�
. . . . . . . . . . . . . . . . . . . . (6 . 1 2)
Similarly, for a buildup test, pressure distributions following shut­
in have the profiles illustrated in Fig. 6.5. The radius at which the
rate of pressure change becomes negligible by a particular shut-in
time moves farther into the reservoir with time , and the radius
reached by this pressure level is given by
ri =
C ::: )
4
C
�.
. . .. .
. .... ..
.. .........
.. . ..
. .
(6 . 1 3)
As an example , if the permeability encountered by the radius of
investigation near the wellbore at earliest times in a buildup or flow
M IDDLE
TIMES
EARLY
TI M ES
Q"i
0.1
1 0
( h o u rs )
LATE
T I M ES
1 00
1 000
Fig. 6 . 6-Characteristic curve shapes exhibited during a flow
test.
6.4.2 Time Regions on Test Plots. On an actual flow or buildup
test plot , the straight line predicted by ideal theory rarely occurs
over the entire range of test times . Instead, the curve is shaped more
as illustrated in Fig. 6.6 or 6.7. To help understand the causes of
the nonlinear portions of the curve , we subdivide the flow test data
into three time regions-early , middle, and late time-based on the
radius-of-investigation concept .
1 . Early time . The pressure transient is near the wellbore in a
damaged or stimulated zone . Wellbore unloading or afterflow of
fluid (defined in Sec . 6 . 4 . 3 ) stored in the wellbore also distorts the
test data during this period .
2 . Middle time . The pressure transient has moved into the un­
damaged formation . A straight line , with a slope related to the ef­
fective permeability of the flowing phase , usually occurs during
this period . This flow period , called the radial flow or middle-time
region, is the basis of conventional well-test analysis techniques .
3 . Late time . The pressure transient encounters reservoir bound­
aries, interference effects from other producing wells, or massive
changes in reservoir properties . The flow test curve deviates from
the straight line established during the middle-time region .
6.4.3 Wellbore-Storage Effects. Only in rare cases is the time re­
quired for the radius of investigation to move through the altered
zone near the wellbore of significant duration. In most cases , the
L
EARLY
TIMES
MIDDLE
TIM ES
----- .
P
�
LATE
TIM ES
Q. 1t
t
+ L1 t
P
l o g --.:...
M
__
1
Fig. 6 . 7-Characteristic curve shapes exhibited du ring a
pressure-buildup test .
1 14
GAS RESERVO I R E N G I N E E R I N G
i
SU RF.'/C,E RI\JE
I�
�
- - - - - __----------
��
o
OOTTa.4HOlE RATE
(ASSUMED CONSTANT
IN IDEAL THEORy)
((
q
�-- t
o
TIM E ----4��
P
BOlTOMHOlE RATE
(ASSUMED TO BE ZERO
IN IDEAL THEORy)
--��--� A t
L1 t
=
0
TIM E ----.�
Fig. 6 .8-Su rface production rate schedule d u ring wellbore­
storage period .
Fig. 6.9-Bottomhole flow rate or afterflow following well
shut-in at the surface.
length of the early-ti
me region
isdata.
determi
ned tests,
by thea speci
duratialoncaseof
wellbore-storage
distortion
of
test
In
flow
ofwhich
the wellbore-storage
phenomenon
isproducti
called wellbore
unloading,
occurs
because
the
i
n
i
t
i
a
l
flui
d
o
n
measured
atfromthe
surface
ori
g
i
n
ates
from
flui
d
s
stored
i
n
the
wellbore
rather
than
the formatie oflow
n. Only
after what
mayy equal
be atheprolonged
time does the
bottornhol
rate
approxi
m
atel
surface
rate
l then,equatitheoassumpti
on ofngconstant
bottornhole
rate,notonsatiwhisfied.ch
theUntiWellbore
flow
n
and
graphi
techni
q
ue
are
based,
is
storageatalsotheaffects
thefluid
earlyconti
buildup
pressure
response.
Following
shut-in
surface,
n
ues
to
flow
from
the
reservoi
r
i
n
to
the
wellbore,
compressing
the
gas
and
li
q
ui
d
already
iwhin thech wellbore
and storitypeng more
fluid. Thistorage,
s contiins uedcalledproduction,
also Unti
is a special
of
wellbore
afterflow
l
the
rate
of
afterflow
diminishes
to
less
than
about
1theoryof thefor rate
before
shut-in,
the
strai
g
ht
line
predi
c
ted
by
ideal
a
Homer
plot
of
bui
l
dup
test
data
does
not
appear.
Following a mass
wellbore-storage
coefficibalance
ent, C,inasthe wellbore,3 we define a
C=!:J.V/!:J.p , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.14)
whereand!:J.V=change
in wel
lbore fluipsi.dThe
voluform
me at ofwellbore
condiontiothens,
bbl,
!:J.
p
=change
in
BHP,
C
depends
fluid iphases
the wellbore.
that
s eitherinrising
or falling,For a well with a liquid/gas interface
C= 25.P6wb4Awb , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.15)
whereAwb=wellbore
the wellbore
phase
fluid (either liquidarea.orIfgas),
then contains only a single­
C=Vwbcwb, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.16)
where Vwbat=wellbore
wellboreconditions.
volume and wb = fluid compressibility
evaluated
Many welresulting
ls either from
have
adrilling
zone ofor reduced
permeabi
l
i
t
y
near
the
wellbore
completic fracturi
on operating.oAnscommon
or have been
stiquemulated
by acidi­
zation
or
hydrauli
techni
for
i
n
corporat­
ing the effects of altered conditions near the wellbore is with a skin
factor.
Historically,
effects permeabi
have beenlimodeled
asformati
an infinitesimal­
lyAnother
small modeli
zone ofskin
reduced
t
y
on
the
oben aface.two­
n
g
technique
considers
the
formati
o
n
to
region
1-3ered equivalentintowhian caltered
h the damaged
or stimulat­
edmeabizonelireservoir
ity,s consi
d
zone
of
uniform
per­
ks , extending out to a radiu s, rs ; outside this zone of
altered permeability, the formation has a permeability, k, unaffected
byto quanti
drillinfyg oreicompl
etion operati
ons. Wiorthstithimsulatimodel,
theterms
skin offactorthe
t
her
formati
o
n
damage
o
n
in
properties of the altered zone is
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.17)
s=(�ks -I}n( �).
rw
Accordi
ningditocatesthe damage
sign conventi
on usedlitiny reducti
this book,
a posi
tive
skin
factor
or
a
permeabi
o
n,
and
a
nega­
tive skin orfactorhydrauli
indicates
an improvement
in permeabi
ltheity well­
from
acidizing
c
fracturing.
If
the
formati
o
n
near
is neial tdepth
her damaged
noronstidamage,
mulated, rthes , canskinbefactor
is zero.
If
thebore
radi
of
formati
determi
n
ed
or
assumed,
we canzone,
estimateks ' theAlternati
correspondi
nifgthevaluepermeabi
of perme­
abireducti
lity oinn then
theratio,altered
v
ely,
lity
ks /k, is available from laboratory measurements,
theAnother
depth ofusedamage,
rs ' can be calculated directly with Eq. 6.17.
of
the
bore radiu s, rwa. 2, 5 skin factor is in terms of an effective well­
rwa=rwe-s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.18)
Noteve wellbore
that, for posi
tivemodels
skin factors
(i.wie .,thdamaged
zones),but thewithef­a
fecti
radius
a
well
no
damage
smaller
radius and
a larger
pressurewellbore
drop at theradiuswell.models
Conversely,
forstimstiulated
mulated
wells,
the
effective
anof un­the
well
wi
t
h
a
very
l
a
rge
wellbore.
An
appli
c
ati
o
n
concept
the observati
on that,
foreffectivertive-well
cally bore-radius
fractured wells
withisinbased
finitelyon conducti
ve fractures
(Fig. 6.8) .
(Fig. 6.9) .
%
C
Center of
We l lbore
6.4.4 Damage and Stimulation Analysis.
5
(Fig. 6. 10)
r
w
Fig. 6 . 1 0-Two-region reservoir model of altered zone near
the well bore.
115
PRESS U R E-TRANSIENT TESTI N G OF GAS WELLS
having two wings each of equal length , Lf , the relationship be­
tween rwa and Lf is
6
Lf = 2rwa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 9)
Thus , if the skin factor for a fractured well can be estimated from
a well test, and if the fracture is assumed to be infinitely conduc­
tive, then Lf can be estimated . Well-test analysis of hydraulically
fractured wells is presented in Sec . 6. 1 0 .
W e also can quantify the skin factor in terms of the additional
pressure drop 1 associated with the damaged zone , or
1 4 1 . 2qBfJo
t::.. P s
--- s o
kh
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 20)
For example, if the total drawdown is 1 ,500 psi and t::.. P s = 1 ,000
psi , then the skin factor has provided some useful information re­
garding the incentive for well stimulation . Either the pressure draw­
down can be reduced by about 1 ,000 psi and the same flow rate
maintained , or the same pressure drawdown can be maintained and
the rate increased by a factor of about three .
6.5 Fundamentals of P ressure· Transient
Testing i n Gas Wells
The basis of flow-test analysis techniques for gas wells is the line­
source (Ei function) solution to the diffusivity equation given by
Eq. 6 . 2 . Eq. 6 . 2 is valid for a slightly compressible liquid with
relatively constant properties . However , for flow of compressible
gases in which the properties are strong functions of pressure , Eq.
6 . 2 often is not sufficiently accurate for analyzing gas-well tests .
In this section, we introduce pressure and time transformation vari­
ables that account for the variation of gas properties with pressure .
6.5_1 Pseudopressure and Pseudotime Variables. The equations
developed for slightly compressible fluids (i . e . , liquids) can be al­
tered by replacing pressure and time with real-gas pseudopressure
and pseudotime variables , respectively . 7 - 1 O These transformations
account for variations in gas properties with pressure . Accuracy
is improved for both semilog and type-curve analysis of gas-well
tests by replacing pressure with the real-gas pseudopressure func­
tion , P ' 7
P
pp ( p) = 2
I
o
·p
P
fJo g ( p)z( p)
dp . . . . . . . . . . . . . . . . . . . . . . . . (6 . 2 1 )
For type-curve analysis (discussed in Sec . 6 . 9) , particularly of
wellbore-storage-distorted data from both flow and buildup tests ,
accuracy also is improved by replacing time with pseudotime ,
tap ( p) , 8
tap ( p) =
\
t
dt
o fJo g ( p) Ct ( p)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 22)
For convenience, although not by necessity , pp and tap can be
normalized 1 1 to have units of psia and hours, respectively, like the
original variables , P and t. Normalization also gives the pseudopres­
sure and pseudotime variables magnitudes comparable with those
of the untransformed pressure and time ; the unnormalized varia­
bles Pp and tap typically have values of 1 0 5 to 1 0 8 . Reference
values of pressure used for normalization are arbitrary . In this chap­
ter, we define normalized variables as
Pn =
l
2
( )
fJo g Z
P
r
pp =
( )
fJo g Z
P
and tn = ( fJo g Ct ) rtap = ( fJog Ct ) r
I,
\' P P d P
r iJ
t dt
--
o fJo g Ct
fJo g Z
. . . . . . . . . . . . . (6 . 23)
, . . . . . . . . . . . . . . . . . . (6 . 24)
where the subscript n refers to the normalized variables and r refers
to the reference values of properties used in the normalization
process .
Some engineers prefer properties evaluated at original reservoir
pressure, P i ' Because BHFP can be measured directly , others pre­
fer at the end of a flow period . In this chapter, we use the current
static drainage area pressure, p. Although p may not be available
at the start of an analysis, using the pressure p* (the pressure on
the semilog straight line extrapolated to a Horner time ratio of uni­
ty) as an estimate of p is quite satisfactory for buildup tests be­
cause the choice of a reference pressure is completely arbitrary (i .e. ,
the value of Pr has no effect on results) . For a flow test, of course ,
the pressure at the start of the test is p . We shall call our normal­
ized variables adjusted pressure , Pa ' and adjusted time , ta ' and we
define them as
( )
\,'
_ I jig Z
_ jig Z p p dp
. . . . . . . . . . . . . . . . . . . (6.25)
Pa - - -- Pp - -2
p 0 fJo g Z
P
and ta = ( jig ct )tap = ( jig ct )
--
j,
t dt
-- , . . . . . . . . . . . . . . . . . . . (6 . 26)
o fJogCt
where p =- p* and p = Pi for a new well . In terms of adjusted vari­
ables , the unsteady-state equation for slightly compressible liquids
(Eq . 6 . 2 ) becomes
Pa . ; -Pa , wf =
1 62 . 6q g Bg jig
kh
l
log
(
kt
_ _
tP fJo g Ct r ",2
)
l
- 3 . 23 + 0 . 869s ' ,
. . . . , . . . . . . . . . . . . . . , . . . . . . . . . . . , . (6. 27)
s ' = total
where
skin factor that includes the skin resulting from true
formation damage or stimulation , s, and a non-Darcy effect (dis­
cussed in Sec . 6 . 6) .
For semilog analysis o f buildup tests , adjusted pressure and ad­
juste� time should �e used , but the adjusted producing time , tpa '
used III the Homer tune ratIO IS evaluated at current average drainage
area pressure ,
fJo g Ct
tpa = � tp = tp ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 28)
fJo g Ct
Consequently , tpa = tp ' Adjusted shut-in time , t::.. ta , is evaluated
from the integral
t::.. ta = (jig ct )
d(t::.. t )
\, �t -,
o
fJo gCt
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 29)
with fJog and c[ evaluated at shut-in BHP, P ws • at values of t::.. t dur­
ing the test. For semi log analysis of a flow test , adjusted pressure
should be used , but adjusted flowing time , ta , is evaluated at p .
Consequently , Eq . 6 . 2 9 , which involves no numerical integration,
can be used to calculate adjusted flowing time . This is equivalent
to using actual rather than adjusted flowing time , and we will write
all working equations using this result .
The logic behind these rules is that, for semilog analysis (data
not distorted by wellbore storage) , gas properties should be evalu­
ated at the pressure at the radius of investigation reached at the time
under consideration . The pressure at the radius of investigation is
p for flow tests and P ws (the current shut-in pressure in the well­
bore) for buildup tests . Additional discussion about the pressure
at which the gas properties are evaluated in the pseudotime and pseu­
dopressure transformations is given in Sec . 6 . 9 , which addresses
type-curve analysis of gas-well tests .
6.5.2 Pressure and Time Variables. Use of adjusted time and ad­
justed pressure in formulating equations for analysis of transient
tests in gas wells is not always necessary . For some (although cer­
tainly not all) gases at high pressure (e . g . , above 3 , 000 psia) , an
adequate approximation is p/ fJog z = constant =p/ jig Z. When this approximation is valid , Eq . 6 . 25 becomes
Pa =
\
jig Z p p dp
P 0
fJo g Z
=p . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 30)
116
GAS RESERVOI R E N G I N E E R I N G
TABLE 6 . 1 -SU M MARY OF WORKING EQUATIONS F O R PRESSURE-TRANSIENT TEST ANALYSIS I N G A S WELLS
Flow Test
Semi log graph
variables
Permeabi l ity from m
of sem ilog straight
line
Oil
Gas, Using Adj usted Variables
P wt vs. t
P a. wt vs. t
[
2)
(
k
S = 1 . 1 5 1 P i - P 1 h' - IO g
m
¢fJ. c C t 'w
Skin factor calculation
B u i l d u p Test
Semi log graph
variables
Permeabil ity from m
of semi log straight
l i ne
P ws vs. (t p
k=
[
+
1 62 . 6q
giig lig
mh
2)
(
kh
·
log
kt
cPl-'g C t 'w2
_ _
)
]
]5
r
0
p dp =
I-' g Z
�,
2]5
a
f
5 7 , 9 1 Oq gP sc T z jig
kh Tsc
·
[
log
(
kt
cPl-'gC t ' w2
_ _
)
f
a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 3 3)
1 , 637qg TZ lig
kh
·
]
- 3 . 23 + 0 . 869s ' .
[ (
For P sc = 1 4 . 7 psia and Tsc = 520 o R , Eq . 6 . 3 3 becomes
P -p [ =
log
kt
_ _
cPl-'gC t 'w2
)
]
- 3 .23 + 0 . 869s ' .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 34)
.
[
(
_
�
2)
�
2)
+
3 .23
]
]
S = 1 . 1 51 P a. 1 h, - P a. wt - IO g
m
¢fJ. g C t 'w
kh( p ; - P a l
70.6Q g B g fJ. g
_
+
3 .23
]
monly used unsteady-state flow equation for pressure-drawdown
analysis with constant-rate gas production is based on the solution
for slightly compressible liquid flow with pressure replaced by pseu­
dopressure :
- 3 . 23 + 0 . 869S
}
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 35)
}
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 36)
In terms of normalized pseudopressure (adjusted pressure) , P o '
the unsteady-state flow equation becomes
. . . . . . . . . . . . . . . . . . . . . . . . . (6. 32)
and the unsteady-state flow equation can be written in terms of
pressure-squared ,
P -p [ =
3 .23
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 3 1 )
6 . 5 . 3 Pressure-Squared and Time Variables. An adequate ap­
proximate for some gases at low pressures (e . g . , below 2 , 000 psia)
is I-' g Z = constant = lig Z. When this approximation is valid, Eq . 6 . 25
becomes
jig Z
(
- 3 . 23 + 0 . 869s ' ,
where the average gas FVF , jjg ' and viscosity , li ' are evaluated
s
at the average pressure in the drainage area of the wel l .
Po =
+
70.6Q o B o fJ. o
[ (
[
S = 1 . 1 51 P a.I - P a. 1 h' - IO g
m
¢fJ. g C t 'w
1 62.6Q o B o fJ. o
Thus, i n the unsteady-state flow equation given b y Eq. 6 . 27 in
terms of adjusted pressure , Po can be replaced by ordinary pres­
sure , p . The unsteady-state flow equation then becomes
p ; -P w[ =
]
----­
kh( p ' - p )
Definitio n of P MBH.D
3 . 23
M)/!:l.t
k
S = 1 . 1 51 P 1 h' - P wf - IOg
m
¢fJ. o C t 'w
Skin factor calcu lation
+
6.5.4 Summary of Working Equations for Gas-WeU-Test Analy­
sis. The modified unsteady-state flow equations for gas wells (Eqs.
6 . 2 7 , 6 . 3 1 , and 6 . 34) serve as the basis for buildup- and flow-test
analysis techniques for gas well s . Table 6 . 1 summarizes the in­
terpretation and analysis equations for the variables frequently used
in gas-well-test analysis . We included variables for well-test analy­
sis of slightly compressible liquids for comparison .
6.6 Non-Darcy Flow
The transient pressure response of a gas well may be affected by
high-velocity or non-Darcy flow near the wellbore . 1 2 The com-
- 3 .23 + 0 . 869S
where s' = s +Dqg is an effective skin factor that includes true for­
mation damage (or stimulation) and the effects of non-Darcy flow .
Eq . 6 . 35 can be written in dimensionless form:
ppD ( l ,tD) = 0 . 5 [ ln(tD) + 0 . 80907] + s +Dqg , . . . . . . . . . . (6. 37)
where PpD ( l ,tD ) is the dimensionless (or normalized) pseudopres­
sure at the wellbore and dimensionless time
tD =
0 . 0002637kt
-----
cPligc/a
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 38)
The usual assumption in conventional calculations is that the non­
Darcy flow effect can be represented as a rate-dependent pseudoskin
defined as Dqg , where D is a constant known as the non-Darcy
flow coefficient (in D/Mscf) and q g is the flow rate (in Mscf/D) .
The true skin factor, s , reflecting formation damage o r stimula­
tion near the wellbore cannot be determined from a single draw­
down or buildup test . Rather , the apparent or total skin factor,
s ' = s + Dqg , is obtained . If s and D are to be determined separate­
ly, then two flow tests can be run at different rates so that two equa­
tions (given by Eq . 6 . 37) can be solved simultaneously I3 for the
117
PRESSURE-TRAN SI ENT TESTI NG OF GAS WELLS
TABLE 6 . 1 -SUMMARY OF WORKING EQUATIONS FOR PRESSU RE-TRANSIENT TEST ANALYSIS IN GAS WELLS (Continued)
Gas, Using Pressure and Time
Gas , Using Pressu re-Sq uared and Time
P wr vs. t
P ;f vs. t
Flow Test
Semi log graph
variables
Permeabil ity from m
of semilog straight
line
Skin factor calculation
1 63 7q g T Z Jl g
k = ____-".:-_,--".
mh
8 = 1 . 1 51
[
B u i l d u p Test
Semilog graph
variables
Permeab i l ity from m
of semilog straight
line
Skin factor calculation
(
�
15 - P '
1 h - IO g
m
cf>ll g C t fw2
_
[
(
�
S = 1 . 1 51 P 1 h, - P Wf - IO g
m
cf>ll g C t fw2
2 . 7 1 5 x 1 O - 15 (3kMp sc
------ ,
hrw Ts cJl- g , wf
_
. . . . . . . . . . . . . . . . . . . . . . . (6.39)
10 1 0k - 1 .47 ct> - 0 .5 3 . . . . . . . . . . . . . . . . . . . . . . (6.40)
6 . 7 Analysis of G as-Well Flow Tests
Gas wells are produced at conditions approximating constant well­
head pressure or variable bottomhole rates , rather than at constant
bottomhole rates , in most gas field operations . In addition, many
gas-well tests , especially deliverability tests , are conducted under
variable-rate conditions . In this section, we begin with a brief discus­
sion of constant-rate gas flow tests , but we concentrate on analysis
techniques for variable gas rates , including gas-well tests with dis­
crete rate changes and tests in which the rates are smoothly changing .
We also address non-Darcy flow effects in flow tests. Finally , we
briefly discuss the effects of reservoir boundaries on gas-well testing.
6.7. 1 Constant-Rate Gas Flow Tests. In terms of adjusted varia­
bles , Eq. 6.27 describes the pressure drop at the wellbore as a func­
tion of time when a well is produced at a constant rate . Similar
to the analysis of slightly compressible liquids, the form of Eq. 6.27
suggests that a plot of P a , wf vs. log t will form a straight line from
which the slope , m, allows estimation of k and s .
For single-phase gas flow , the formation permeability i n the
drainage area of the well is computed from
162.6qgi3giig
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.41)
m
Combining Eqs . 6.27 and 6.41 , we also can develop an expres­
sion for
s:
) l
+ 3.23
) ]
+ 3.23
kh( p ' 2 15 2 )
7 1 1 qg TZ Jl g
kh( p ' - 15 )
70.6q g B g ll g
where M= gas molecular weight and Jl-g , wf = pressure-dependent
gas viscosity evaluated at P wj . (3, which is a turbulence parameter
inversely proportional to permeability , can be determined ex­
perimentally or from 1 4
k=
_
_
1 63 7q g T Z Jl g
k = . ____�___�
mh
1 62.6q g B!.... g Jl.2.
..e..:... g .
k = --:.!..
mh
two unknowns, s and D . If only one test is available, then D can
be estimated by
(3 = 1 . 88 X
(
[
+ 3.23
P ws vs. (t p + llt)/llt
Defin ition o f P MBH,D
D
2
k
8 = 1 . 1 5 1 15 - P th' - log
m
g
cf>ll C t f w2
) l
_
For convenience , w e set the flow time , t, equal to 1 hour and
use Pa , l hr for the adjusted BHFP, Pa , wj' at this time . Substituting
these into Eq . 6.42 yields
s= 1 . 151
[ ( Pa,; -mPa, l hr) (
- IOg
k
_ _ rw2
ct>Jl-gC
t
) + 3.23] . . . . . (6.43)
Note that P a , Ihr necessarily lies on either the semilog straight line
or its extrapolation . Table 6 . 1 summarizes the working equations
for semilog analysis of gas flow tests in terms of pressure and
pressure-squared .
6.7.2 Gas Flow Tests With .Discrete Rate Changes. We model
a variable-rate gas flow test using superposition in time . 1 -3 First,
we consider the pressure drawdown in terms of adjusted pressures
in an infinite-acting gas reservoir resulting from a single produc­
tion rate . We assume, for now , negligible non-Darcy flow effects .
Eq . 6.27 can be rewritten as
-
3 2 3 + 0 869S
.
.
l
·
·
·
·
or Pa ,; - Pa , wj =m ' qg(log
where
m' =
and S = IOg
162.6iigiig
kh
(
�
ct> Jl-_g Ct rw2
·
. . . . . . . . . . . . . . . . . . . . . . . . . . . (6.44)
t+S), . . . . . . . . . . . . . . . . . . . . . (6.45)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.46)
) - 3.23 +0.869S . . . . . . . . . . . . . . . . (6.47)
Now , for the variable-rate production history in Fig. 6 . 1 1 , the
pressure drawdown resulting from n discrete rate changes and for
time t> tn I is 1 5
Pa ,; -Pa , wj =m 'ql (log t+ S ) +m ' (q -ql )[log(t-t l ) + S]
2
+m'(q 3 - q2 ) [log(t-t2 ) +s ]
+ . . . +m ' (qn -qn - I )[log(t-tn _ l ) +S ] . . . . . (6.48)
118
GAS RESERVO i R E N G I N E E R I N G
1 .9
Rate
.�'"
c
'"J:'"
�
I::!
o
Time
Fig. 6 . 1 1 -Variable-rate production history.
i
1 .8
1 .7
1.6
1 .5
!
.
_
.
. ..
_ ... . ......
:;:
:
"1 '1'
. . . . . . . .:.:,.
1
.
·
. ..
· ··_·······_·· · ·····_····· ·
·· M .
. ... . . M
..
. .
·· ·
·
. . .
· · r··· · ··
-r! .. . .
.
. ..
.
..
.
.
·
.. .
··· ······ .
1,
.
.t
··
.· ·
·· ·
••••
1
.
.
· . .
.. .
- ········ ·· . · ·· · ·····
. .. .
.
.
. ... .... ... .. l:;
j
···
· .. ·m· .
·· ··· ··
T
.
·
·· .
·· ········ ·-
1.
.. .. . .
=. 0.. 2. 3. .6...-.
.
.. .
.
.
�:J+++=1=
I
I
f !
I
1 .4 +--t--+--+-t--+--+---1
0.0
0.2
0.4
0.6
0.8
1 .0
1 .2
1.4
TIJlle-Rate Function
Eq. 6.48 can be rewritten as
Fig. 6 . 1 2-Cartesian plot of m u ltirate test data, Example 6 . 1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.49)
The form of Eq. 6.49 suggests that we plot
I
1 n�P a. i -P a. wf vs. i.J f:.qj I og(tn -tj )
qn
qn j = o
on Cartesian coordinate paper, where P a i = adjusted initial reser­
voir pressure, psia; Pa.wf =adjusted FBHP at time tn ' psia; qn = last
of n different flow rates, MscflD; f:.qj = qj + 1 -qj ' (qo = 0); tn =
total (cumulative) flowing time for n constant-rate flow periods,
hours; and tj =time at which rate was changed, hours. A straight
line with slope m' proportional to k should result from this plot.
Specifically,
162.6Bgiig
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.50)
k=
--"-"'---....:.:.:.
:!.
-. ..
m 'h
where m ' is defined by Eq. 6.46. The skin factor also can be deter­
mined from this plot. If b is the intercept of the plot (the value of
f:.p/q at which the time plotting function is zero), then
S = 1 . 15 1
_ 2 ) +3. 2 3] . . . . . . . . . . . . . . (6.51)
[ � - IOg ( c/>IJ._ kCtTw
m
g
The plotting method assumes that the reservoir is infinite-acting
at all times up to tn - Once boundary effects are felt by the pres­
sure transient, the method is invalid. This analysis method can be
used with four-point deliverability or backpressure tests (discussed
in Chap. 7), which are simply variable-rate tests with discrete
changes in flow rate.
Non-Darcy flow effects in a gas-well test can be included by
plotting
1 n;.,1
( P a. i -P a.wf - D'q� )
-------"
--- vs. - i.J f:.qj log(tn - tj ) '
qn j = o
qn
where D' = (141 .2Bg iigD)/(kh) is a constant, and D is the non­
Darcy flow coefficient defined by Eq. 6.39. With the non-Darcy
flow term, D' q� , we are forced either to assume D ' =0 or to find,
by trial and error, the value of D' that results in the best straight-line
plot. The slope of the straight line, m ' , provides an estimate of per­
meability and skin factor by use of Eqs. 6.50 and 6.5 1 , respectively.
Although this method is potentially quite useful because back­
pressure data are available for virtually all gas wells, it has at least
two limitations. First, a well may not be cleaned up at the time
of testing, especially when the well is tested following a workover
operation or a hydraulic fracture treatment. Second, the method
assumes negligible wellbore-storage distortion; however, wellbore
storage almost certainly distorts some of the test data, particularly
for short-duration flow periods and in low-permeability formations.
Example 6 . 1-Determining Permeability and Skin Factor From
a Multirate Test. A gas well was tested with a conventional flow­
after-flow backpressure test. Table 6.2 summarizes the measured
pressures and rates. Assuming that non-Darcy flow effects (D' qD
are negligible, estimate effective gas permeability and skin factor
using the following well, rock, and gas properties.
'Yg = 0.7.
Tw = 0.255 ft.
h = 19 ft.
T = 200°F.
psia - I .
ct =
Bg = 0.58778 RB/Mscf.
c/> = 0. 176.
ii = 0.0286 cpo
73. 3 xlO-6
Solution.
1 . Our objective is to prepare a plot of
1 n
P a. i -P a. wf - D'q� vs. ----"''--'
- -'''- -'- -'-E f:.qj 10g(tn -tj _ I ) '
qn
qn j = 1
Thus, the first step is to calculate the pressure/rate and time/rate
plotting functions for each of the four flow periods. For example,
at t=6 hours, n = 1 , t l =6 hours, and ql =2,7 1 1 MscflD,
Pa .i -Pa .wf 4,709.4 -66.4
= 1 .7 1 3 psi/MscflD.
2,7 1 1
n
1
E f:.qj 10g(tn -tj _ I ) = - [f:.ql log(t l -to ) ]
qn j = 1
ql
-----
=
1
-[(2,71 1 -0)log(6 -0) ]
2,7 1 1
=0.7782.
At t=9.5 hours, n = 3 , t3 =9.5 hours, and q 3 =2,504 MscflD,
P a .i -P a. wf 4,709.4 -258.9
1 .777 psi/MscflD.
2,504
1
1 n
q3
qn j = 1
log(t
+ f:.q2 log(t3 -t l ) +f:.q 3
3 -t2 ) ]
1
= -- [(2,7 1 1 -0)log(9.5 -0)
2,504
+ (2,607 -2,7 1 1)log(9.5 -6)
+ (2,504 -2,607)log(9.5 - 8)]
= 1 .0287.
- E f:.qj 10g(tn -tj _ I ) = - [f:.ql log(t3 - tO )
1 19
PRESSURE-TRAN SIENT TEST I N G OF GAS WELLS
TABLE 6 . 1 -SUMMARY OF WORKING EQUATIONS FOR PRESSURE-TRANSIENT TEST
ANALYSIS I N GAS WELLS (Continued)
Gas, Using Pseudopressure and Time
Flow Test
Semilog graph
variables
Permeabi lity from m
of semi log straight
line
Pp
vs. t
k = _1 637...:qo!.g T
__ _
mh
Skin factor calculation
B u i l d u p Test
Semi log graph
variables
Permeabi l ity from m
of semilog straight
l i ne
Skin factor calculation
P p vs. ( tp + t:..t)/A t
k = 1 63 7...:qo!.g T
s
= 1 . 1 51
[
___ _
mh
P p, 1 hr - P p, wf - IO g
m
(162.6)(0.5878)(0.0286)
=0.61 md .
m 'h
(0.236)(19)
3 . Next, calculate the skin factor using Eq. 6.5 1 .
k=
162.6Bgiig
=
s = 1 . 15 1 � - 10g
[
) + 3 .23]
(�
cpp. Ctr�
1 .532
0.61
[
[ 0.236
(0. 176)(0.0286)(7.33 x 10 - 5 )(0.255) 2 ]
- 3 .23]
= 1 . 15 1
m'
--
- log
= 2.6 .
A positive skin factor indicates damage resulting from a reduc­
tion in permeability in the formation adj acent to the wellbore .
6.7.3 Variable-Rate Gas Flow Tests With Smoothly Changing
Rates. In many testing situations , a strictly constant producing rate
is impractical or impossible to maintain . A more probable mode
of operation is production at a constant surface pressure , and if tub­
ing friction effects are negligible , the BHP also is constant . At ear­
ly times , however, both BHP and bottornhole rate may be changing
rapidly . Data obtained under these nonideal test conditions can be
analyzed accurately with a simple modification of the transient flow
equation for constant-rate production .
For slightly compressible liquids , Winestock and Colpitts 1 6
showed that, even when both P w and q vary with time , Eq. 6.52
f
can be used as long as the rate IS changing slowly and smoothly
rather than abruptly :
Pi -Pwf 162.6Bp.
=
kh
q
--
[ ( ) - 3 .23 +0. 869s . . . (6.52)
]
cpp.ctr w
log
-2
kt
_
�
2)
CP/L g C t ' w
kh(p; - p )
p
Defi nition of P M B H , D
Calculated values of the pressure- and time-rate plotting functions
(Table 6.3) are plotted in Fig_ 6. 12.
Using least-squares regression analysis, calculate the best-fit
straight line through the data . From this line , we determine
m ' =0.236 and b = 1 .532.
2 . The effective g a s permeability is calculated with Eq . 6.50.
(
+ 3.23
]
71 1 qg T
To analyze transient data, we prepare a semilog graph of
( Pi -P wf)lq as a function of t. To analyze variable-rate tests. in gas
wells in which non-Darcy effects are important, we can reWrIte Eq.
6.52 in terms of adjusted pressures :
P a ,i -P a , wf - D'qJ 162.6iigBg
=
kh
qg
]
[ ( CPiigC/� )
IOg
kt
-3 .23 +0. 869S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.53)
where
D'
Either the non-Darcy flow effects (the D' qJ term) must be ne­
glected or the value of D' that leads to the best straight line in the
middle-time region must be found iteratively . Once we have iden­
tified the semilog straight line indicative of the middle-time region,
k is estimated from the m ' of this line :
k
and
162.6Bgiig
--� ,
m 'h
S = 1 . 15 1
. .
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.54)
[ � ( Pa'i -qPa , wf )
m
- IOg
( cpp.gctr� w ) + 3 .23] ,
_
2
1hr
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.55)
where [( P a ,i -Pa , wf )lqh hr must lie on the semilog straight line or
its extrapolation to 1 hour. Similar to constant-rate production data,
the variable-rate data may be affected by wellbore storage at early
times and reservoir boundaries at late times, thus distorting the pres­
sure response and possibly masking the correct sernilog straight line
indicative of the radial flow or middle-time region .
Example 6.2-Analysis of a Variable-Rate Gas Flow Test. A gas
well was produced at a constant BHP of 1 ,000 psia. Known data
are summarized below and in Table 6.4. Average gas and forma­
tion properties are evaluated at the arithmetic average of the initial
and BHFPs , ji =2,825 psia. Assuming non-Darcy flow effects
1 20
GAS RESERVO I R E N G I N EE R I N G
TABLE 6.2-FLOW-AFTER-FLOW TEST DATA, EXAMPLE 6 . 1
t
(hou rs)
0
6
8
9.5
1 2. 5
P wt
q
P a,wt
(psia)
6 , 1 80
566
832
1 , 1 30
1 ,646
TABLE 6 . 4-TI M E AND FLOW DATA F O R EXAMPLE 6.2
(psia)
4 , 709.4
66.4
1 42 . 2
258, 9
535 . 2
t
(MscflD)
0
2,71 1
2,607
2,504
2,309
TABLE 6 . 3-PRESS U R E AND TIME FUNCTIONS,
EXAMPLE 6 . 1
j
Pressure
Function
(psi/Mscf·D)
Time/Rate
Function
1 .7 1 3
1 . 752
1 . 777
1 .808
0.7782
0.9271
1 .0287
1 .1819
(D ' q ) are negligible , estimate formation permeability and skin
factor using the semilog analysis technique for variable-rate gas
flow tests .
rw =
jig =
Pi =
Pwf =
h =
ct =
P a ,i =
P a , wt =
t/> =
Jig =
0 . 365 ft .
0 . 0 1 87 cp o
4 , 650 psia .
1 ,000 psia.
23 ft.
2 1 . 5 5 x 1 0 - 5 psia - 1 .
3 ,6 1 6 . 9 1 psia.
23 1 . 7 1 psia.
0. 14.
1 . 0485 RB/Mscf.
3 ,6 1 6 . 9 1 - 23 1 . 7 1
405 . 9
k
j
'
10
.
10
'
TImc. hr
1 62 . 6Jig ji
m'h
( 1 62 . 6) ( 1 . 05)(0 . 0 1 87)
( 1 . 09)(23)
= 0 . 1 3 md ,
4 . Next, calculate the skin factor . From Fig . 6 . 1 3 , we see that
the value of the pressure function at the extrapolation of the straight
line to t = 1 hour is
= 8 . 34 .
Similarly , plotting functions for each measured rate are summar­
ized in Table 6.5.
2 . The data from Table 6.5 also were plotted in Fig. 6.13. From
type-curve analysis (discussed in Sec . 6 . 9 ) , we have identified the
.. . , .. -.. � . . .. � . . , .........�.. :
...... , .... , . ... .. , . :,.j
(Mscf/D)
432. 0
405 . 9
392.9
385.4
380 .3
376 . 3
372 . 9
370 . 0
367. 5
365. 2
361 .4
358.3
355. 7
353. 5
35 1 . 7
348 . 1
345 . 1
342. 8
340 . 8
339 . 6
337.4
335.9
334. 5
332 . 1
330.4
328 . 1
beginning o f the middle-time region a t approximately 3 0 hours .
Therefore, the slope of the line drawn through data points after this
time is m' = 1 .09 psi/cycle .
3 . The formation permeability is
Solution.
1 . The first step is to prepare a plot of ( P a , i -Pa , wt - D'q )/qg
vs. log t. Using the adjusted initial and BHP ' s in Table 6 . 4 , calcu­
late the plotting functions assuming negligible non-Darcy flow ef­
fects (i.e. , D =O) . For example, at t = 4 . 8 hours , the plotting function
is
------
q
(hou rs)
2.4
4.8
7.2
9.6
1 2. 0
1 4.4
1 6. 8
1 9. 2
21 .6
24. 0
28.8
33.6
38.4
43 . 2
48. 0
60. 0
72 . 0
84. 0
96.0
1 08 . 0
1 20 . 0
1 32 . 0
1 44.0
1 68 . 0
1 92 . 0
2 1 6.0
'
10
10
'
Fig. 6 . 1 3-Semilog plot of variable-rate data for Example 6 . 2.
( Pa,i -Pa, wt )
qg
O
I hr
= 7 . 8 psi/Mscf/D,
�
L
1 --1�---'�
�'---1�
---1��
�
�--1��
6 --�0�
1�
O4
1 6-O
O
10
Fig. 6 . 1 4-Effects of non-Darcy flow and skin factor on
transient-pressure response (after Fligelman e t al. 1 2 ) .
1 21
PRESS U R E-TRANSIENT TESTI N G OF GAS WELLS
30
TABLE 6 . 5-PLOTT I N G FUNCTIONS FOR EXAMPLE 6 . 2
t
( P a,i - P a,w, )/q g
(psi/Mscf-D)
7.84
8.34
8.62
8 . 78
8.90
9 . 00
9. 1 5
9.21
9.27
9 . 37
9.45
9.52
9.58
9 . 63
9 . 72
9.81
9.88
9 . 93
1 0.03
1 0.08
1 0. 1 2
1 0. 1 9
1 0.25
1 0.32
(hou rs)
2.4
4.8
7.2
9.6
1 2.0
1 4.4
1 9. 2
21 .6
24.0
28.8
33.6
38.4
43.2
48.0
60.0
72.0
84.0
96.0
1 20 . 0
1 32 . 0
1 44 . 0
1 68 . 0
1 92 . 0
2 1 6.0
[ ( )
l
[ 1
0. 13
= 1 . 15 1 7.8-- - 10g
1 .09
(0. 14)(0.0187)(2 1 .55 x 10 -5 )(0.365) 2
+ 3.23J
6.55,
1 a - a
k
s = 1 . 15 1 -, P , j P , wf - log _ _ + 3 .23
m
q
cf>JL gCtTw2
1hr
and from Eq.
25
20
.0'"
' 5
' 0
5
0
We assumed negligible non-Darcy effects , s o the skin factor rep­
resents true formation damage .
6.7.4 Gas Flow Tests With Non-Darcy Flow. As noted previously,
the pressure drawdown in a gas well can be affected by high­
velocity ,
non-Darcy
flow near the wellbore .
Unsteady-state ,
8
' 0
Fig. 6 . 1 5-Correlation of b D and 8q D (after Fligelman et
al. ' 2 ) .
bulent flow and a positive skin factor , indicating that the non-Darcy
term is not additive and thereby confirming that D is not constant.
Fligelman et at. 1 2 presented a method to estimate k and true s
from pressure-drawdown data for a gas well . Their equations , writ­
ten in terms of pseudopressures, are modified here for use with
adjusted pressures . They define the dimensionless group
2.715 1O - 1 5 {3kMP sc qg
, . . . . . . . . . . . . . . . . . . . (6.57)
Tsch Tw iig
where ii = gas viscosity evaluated at ji . For a constant skin fac­
x
BqD
tor, the &mensionless intercept of the semilog straight line in a plot
of dimensionless pseudopressure as a function of the log of dimen­
sionless time is defined as
------
=4.8.
6
2
0
P a ,j - m log
bD =
( 0.OOO2637k ) -Pa, l hr
_ _
cf> JLgCt Tw2
qDP a ,j
, . . . . . . . . . . (6.58)
where m and P a , 1hr are the slope and intercept, respectively , of the
semilog straight line in a plot of P a vs. log t. The dimensionless
form of the slope is
mD =mlqDP a ,j , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.59)
and the dimensionless flow rate is
qD =
0. 138qgPsc T
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.60)
kh Tsc P a ,j
The intercept b D i s related t o BqD b y the correlation
bD = c 1 (BqD) + c2 ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.61)
Table 6.6 summarizes selected values o f the constants c 1 and
ppD( 1 ,tD) =0.5[ln(tD) +0.80907] +s +Dqg ' . . . . . . . . . . (6.56) c2 ' The correlation of bD and BqD is shown in Fig. 6 . 1 5 for a
range of skin factors .
where D frequently is treated as a constant. In reality , D is not con­
A second correlation relates the pseudopressure response with
stant. Fig. 6.14 show the effects of non-Darcy flow and skin damage non-Darcy flow to the group BqD ' The dimensionless wellbore
on the pressure response . The dimensionless pseudopressure pressure response over dimensionless time is given by
Responses A, B, and C are simulated with Eq . 6.56 for the condi­
P aD = (mD - 1 . 15 1 )logtb +bD - 0.4045 - s, . . . . . . . . . . . (6.62)
tions shown . The lowest solid line is the liquid-flow response with
a slope of 1 . 15 1 . Response B, with a slope of 1 . 163, shows only where mD = slope of the straight line in a plot of PpD vs. log tD '
the effect of non-Darcy flow . Response C is a translation of The time tb is defined as
Response A with a positive skin factor, assuming that the non-Darcy
0.OO02637k
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.63)
tb =
pseudoskin effect is an additive term to the pressure response.
iigi:t Ta
Response D , with a slope of 1 . 1 83, is the actual response with tur-
constant-rate gas flow can be modeled in terms of dimensionless
pseudopressure :
TABLE 6 .6-CONSTANTS c , A N D c FOR NON-DARCY
2
FLOW CORRELATIONS
s
o
5
10
C,
___
1 .001 3
3. 96273
7.4733
C2
0.3205
5.241 1
1 0. 1 993
TABLE 6 .7-CONSTANTS c • C 4 . AND C S FOR NON-DARCY
3
FLOW CORRELATIONS
s
o
5
10
C3
0.002
4 . 1 68
7.624
C4
___
- 0.002
- 0.202
- 0.366
...EL
0.002
0.049
0. 1 34
1 22
GAS RESERVOI R E N G I N EERING
S.O
10.0
IS.O
:m.0
..
25.0
:10.0
ic���.-���-.���--��.-.-����-.-.��---��
: : : : : : : : : :
:
:
:
:
:
:
:
:
:
:
:
Fig. 6 . 1 6-Correlation of P :D and Bq D (after Fligelman et al. 1 2 ) .
The pressure response with turbulent flow ,
BqD by the correlation
P �D '
is related to
2
P:V = C 3 (BqD ) + c 4 (BqD) + C S (BqD) 3 , . . . . . . . . . . . . . . (6 . 64)
where selected values of the constants c 3 , c 4 , and C s are summar­
ized in Table 6.7. The correlation of p:V and BqD also is shown
in Fig. 6.16 for a range of skin factors.
When non-Darcy flow effects are significant , we suggest the fol­
lowing procedure 1 2 for pressure-drawdown analysis in terms of
adjusted pressures .
1 . Convert the sandface pressures , measured at a constant flow
rate , to adjusted pressures, Pa ( p) . Construct a log-log plot of ad­
justed pressure change as a function of time . Use conventional type­
curve matching (discussed in Sec . 6 . 9) with a wellbore-storage/skin
type curve to estimate the beginning of the semilog straight line .
2 . Plot Pa as a function of log t. Draw a straight line through
the semilog-straight-line data on the semilog plot. Find the slope ,
m, and the intercept, P a. 1hr ' Calculate a first approximation of for­
mation permeability by
k
3.
1 62 . 6q g iig iig
---"---'
'-''- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 65)
mh
Calculate the total skin factor, s ' .
(
a. -Pa
S' = S +Dqg = 1 . 15 1 P i . l hr - IOg _ k_
m
ct> /JogCtrw
[
2
)
]
+ 3 . 23 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 66)
4. Calculate (3 , BqD ' bD , and qD using Eqs . 6 . 40 , 6 . 5 7 , 6 . 5 8 ,
and 6 . 60 , respectively . Find the true skin factor, s, from Fig . 6 . 1 5 .
5. Using BqD and s , find p :V from Fig . 6 . 1 6 . Calculate a n ap­
proximation of mD by rearranging Eq . 6 . 62 to
mD = 1 . 1 5 1 + ( p:V - bD + 0 . 4045 + s)/log(t6) · . . . . . . . . (6 . 67)
6. Using mD , calculate a second approximation of k by
0. 138 TP scqg
k=
mD ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 68)
mh Tsc
7 . Repeat Steps 4 through 6 until the k estimate from Eq. 6 . 65
converges . The convergent value is the correct estimate of k, and
the associated value of s is the correct skin factor .
Note that this procedure has certain limitations . First, only non­
Darcy flow effects are considered . The method does not consider
the effects of D on a phase change (L e . , condensation of the gase­
ous phase) as the pressure is reduced near the wellbore . Second ,
because the correlations are based on positive skin factors, the proce­
dure applies only when s > O . Third , the correlations were devel­
oped for a damaged-region/wellbore radius ratio greater than 1 0 .
For a higher ratio and a given value o f skin factor, the slope of
the semilog straight line is unaffected , but the non-Darcy effect is
smaller . Finally , the accuracy of this method depends on the ap­
plicability of the empirical correlation for {3 (Eq. 6 . 40) , which is
subject to considerable uncertainty . 1 4
Example 6.3-Constant-Rate Gas-Well Drawdown Analysis
With Non-Darcy Flow. Table 6.8 summarizes pressure and time
data from a constant-rate drawdown gas-well test 1 3 ; other known
data are summarized below . Determine k and s. In addition , esti­
mate the skin factors caused by non-Darcy flow and by true for­
mation damage or stimulation .
qg =
ct> =
T=
ct =
Pi =j5 =
h=
Sw =
Z=
40 , 000 MscflD.
0. 10.
200 ° F .
l 1 2 x l O - 6 psia - 1 .
4,290 psia.
40 ft.
0 . 25 .
0 . 92 cp o
TABLE 6 . S-PRESSURE DRAW DOWN DATA, EXAMPLE 6 . 3
Time
(hours)
0.00
0 . 75
1 .00
1 . 25
1 .50
1 . 75
2.00
2.25
2.50
3.00
Pressure
(psia)
4,290
3,602
3 , 596
3,591
3,587
3,583
3,580
3,577
3,575
3,570
Adj usted Pressure
(psia)
2 , 370 .4
2 , 364.5
2 , 359.5
2 , 355.5
2,351 . 6
2 , 348. 6
2 , 345. 6
2,343. 6
2 , 338.7
1 23
PRESS U RE·TRANSIENT TESTI N G OF GAS WELLS
cu
'iii
...
S:
CD
:::l
<Il
<Il
�
2370
2350
:::l
2340
iii
'0
<{
5
\..
2360
c..
"0
CD
6
1\
2380
2330
'\
Po FOR lIQUID
Ppo FOR NATURAl GAS (qo = 0.05)
" Ppo FOR IDEAL GAS (qo = 0.05)
o Ppo FOR NATURAL GAS (qo = 0.1 0)
'" PpD FO R IDEAL GAS (qD = 0. 1 0) .... ..a"
o
Time (hou rs)
1 0
=
=
=
=
=
=
0.7163 RB/Mscf.
3 ,056 psia .
0.3 ft.
0.85 .
2,364 psia.
0.0254 cp o
0.1
1 62.6qg Bg iig
:.
....> ....o:..
k = --.:e...
mh
(
=8.3.
= 1 . 88 x 10 - 10 (57.0) - 1 . 47 (0. 10) - 0 . 53
= 1 .67 x 108 ft - I .
The group Bqv is
2.715 x 1O - 1 5{3kMP scqg
Bqv = --------------�
Tsch Tw iig
(2.715 x 10 - 1 5 )(1 .67 x 108)(57.0)(0.85)(29)(14.7)(40,000)
(520)(40)(0.3)(0.02544)
=2.36.
The dimensionless rate i s
(0. 138)(40,000)(14.7)(660)
(57.0)(40)(520)(3,056)
=0.0148.
The term b v is
0.0002637k
-P a . l hr
P a ,i - m log
/
cf>/L
C
T
w
g
-'- ----bv = ------------"-'-----
(
(162.6)(40,000)(0.7163)(0.0254)
(52)(40)
= 57 md .
3 . The total skin factor is estimated to be
- log
10
Fig. 6 . 1 8-Comparison of dimensionless pressure responses
for liquid and gas solutions (after AI-Hussainy 1 8 ) .
Solution.
With the high flow rate of 40,000 MscflD, the skin factor from
conventional semilog analysis can be expected to reflect both a true
skin factor (formation condition near the wellbore) and a pseudoskin
resulting from non-Darcy flow near the wellbore . The procedure
adapted from Fligelman et al. 12 can be used to find k and true s
when non-Darcy flow has a significant effect on the pressure
response . For convenience in analyzing a gas-well drawdown test,
we convert pseudopressures to adjusted pressures using viscosity ,
iig ' compressibility factor , z, and total compressibility , c/' evalu­
ated at average pressure , p . Table 6.8 gives plotting variables .
1 . Qualitative type-curve analysis (discussed in Sec . 6.8) using
adjusted pressure indicates that all data points are in the region of
the semilog straight line .
2 . Adjusted pressure is plotted as a function of real time in Fig.
6.17, and the semilog straight line drawn through the data points
has a slope m = 52.0 psi/cycle . Adjusted pressure at t= 1 hour is
P a , l hr =2,364 psia. The initial estimate of formation permeability is
= 1 . 15 1
-
eginning of Pseudosteady State
Fig. 6 . 1 7-Semilog plot, Example 6 . 3 .
Bg
P a ,i =Pa
Tw
'Yg
Pa I hr
' iig
-0-
\.
1
.1
-
3,056 - (52.0)log
[ 3 ,056 - 2,364
[
_ _
2
)
]
0.0002637(57.0)
- 2,364
(0. 10)(0.02544)(0.0001 12)(0.3) 2
(0.0148)(3 ,056)
= 8.68 .
From Fig . 6. 15 with Bqv =2.36 and bv = 8.68, estimate s =2. 7 .
From Fig . 6. 16 with Bqv =2.36 and s =2.7, we find p � =
The term tjj is
5.
5.8.
52
) ]
57
+ 3 .23
(0. 10)(0.0254)(1 12 x 10 -6)(0.3) 2
4. The turbulence factor i s estimated with Eq . 6.40.
{3= 1 . 88 x IO l O k - 1 .47 cf> - 0 . 53
tjj =
0.0002637k
iig C/ T�
0.0002637(57.0)
(0.0254)(0.0001 12)(0.3) 2
=58,615.
1 24
GAS RESERVO I R E N G I N E E R I N G
And mD is
mD = 1 . 1 5 1 + ( P:D - bD + 0 . 4045 + s)/log(tLj )
= 1 . 1 5 1 + (5 . 8 - 8 . 68 + 0 . 4045 + 2 . 7)/log(5 8 , 6 1 5)
= 1 . 12 .
6 . Using mD ' compute a second approximation o f permeability .
7A. The permeability estimate of 6 1 .9 m d from Step 5A still does
not agree with the previously calculated value of 55.5 md. For better
accuracy , another iteration will be made .
Second Iteration.
4B . The turbulence factor is
(3 = 1 . 88 X 10 1Ok - 1 . 47 c/> - 0 . 53
= 1 . 88 x 1 0 - 1 0 (6 1 . 9) - 1 . 47 (0 . 10) - 0 . 53
= 1 . 48 x 1 0 8 ft - I .
The group BqD is
BqD =
(0 . 1 3 8)(660) ( 1 4 . 7)(40 ,000) ( 1 . 1 2)
(2 . 7 1 5 x 1 0 - 1 5 ) ( 1 .48 x l O 8 ) (6 1 . 9)(0 . 85)(29) ( 1 4 . 7)(40,000)
(52 . 0)(40)(520)
(520)(40)(0 . 3) (0 . 0254)
= 5 5 . 5 md .
7 . For illustration purposes, consider the permeability estimate
of 55 . 5 md from Step 5 sufficiently different from the initial esti­
mate of 5 7 . 0 md that iteration is required .
First Iteration.
4A . The new value for the turbulence factor is
= 2.27.
The dimensionless rate i s
qD =
(3 = 1 . 88 x lO I Ok - 1 . 47 c/> - 0 . 53
(0 . 1 3 8) (40,000) ( 1 4 . 7)(660)
(6 1 . 9)(40)(520)(3 ,056)
= 0 . 0 1 36 .
= 1 . 88 x 1 0 10 (55 . 5 ) - 1 . 47 (0 . 1 0) - 0 . 53
The term b D is
= 1 . 74 x 1 0 8 ft - l .
bD =
The group BqD is
BqD =
(2 . 7 1 5 x 1 0 - 1 5 ) ( 1 . 74 x 1 0 8 )(55 . 5) (0 . 85)(29) ( 1 4 . 7) (40 ,000)
-----
3 ,056 - (52 . 0)log
(520)(40)(0 . 3)(0 . 0254)
(55 .5)(40)(520)(3 ,056)
tLj =
= 0 . 0 1 52 .
The term b D is
(0 . 0254)(0 . 000 1 1 2)(0 . 3) 2
[
0 . 0002637(55 . 5 )
(0. l O)(0 . 0254)(0 . 000 1 1 2)(0 . 3) 2
(0 . 0 1 420)(3 , 056)
]
The term mD is
mD = 1 . 1 5 1 + (6 . 3 - 9 . 39 + 0 . 4045 + 3 . 0)llog(6 3 , 754)
= 1 . 22 .
- 2 , 3 64
6B . Using mD ' calculate another value of permeability .
k=
From Fig . 6 . 1 5 with BqD = 2 . 3 9 and bD = 8 . 45 , s = 2 . 5 .
5A. From Fig . 6 . 1 6 with BqD = 2 . 39 and s = 2 . 5 , p � = 6 . 0 . The
term tLj is
0 . 0002637(55 . 5)
-----­
(0 . 0254)(0 . 000 1 1 2)(0 . 3) 2
= 5 7 , 1 62 .
The term m D is
mD = 1 . 1 5 1 + (6 . 0 - 8 . 45 + 0 . 4045 + 2 . 5 )/log(57 , 1 62)
6A . Using mD ' calculate a second approximation of permea­
bility .
(0 . 1 3 8) (660) ( 1 . 47)(40 ,000) ( 1 . 22)
------
(52 . 0)(40)(520)
= 60 . 4 md .
7B. The permeability estimate of 60 . 4 md from Step 5B differs
from the previous estimate of 6 1 . 9 md , so iterations could continue;
however, we will accept this result here because our main objec­
tive is to illustrate the computational procedure . Thus, formation
permeability is k = 60 . 4 md . From Step 3 B , the corresponding
"true " skin factor is s = 3 . 0 .
8 . The error from the exclusion o f non-Darcy flow effects in the
drawdown analysis can be assessed by comparing these results with
results from conventional calculations. The non-Darcy coefficient
is estimated to be
= 1 . 25 .
k=
0 . 0002637(6 1 . 9)
= 63 , 754 .
= 8 . 45 .
tLj =
- 2 , 364
From Fig . 6 . 1 5 with BqD = 2 . 27 and bD = 9 . 39 , s = 3 . 0 .
5 B . From Fig . 6 . 1 6 with BqD = 2 .27 and s = 3 . 0 , p � = 6 . 3 . The
term tLj is
(0 . 1 3 8)(40,000)( 1 4 . 7) (660)
3 ,056 - (52 . 0)log
(0 . 1 0) (0 . 0254)(0 . 000 1 1 2) (0 . 3 ) 2
]
= 9.39.
The dimensionless rate is
bD =
0 . 0002637(6 1 . 9)
(0 . 0 1 36)(3 ,056)
= 2 . 40 .
qD =
[
D=
2 . 7 1 5 x 1 O - 1 5 {3kMp sc
-------
hrwTsc JJ.g.wf
(0 . 1 3 8) (660) ( 1 4 . 7)(40, 000) ( 1 .25)
(2 . 7 1 5 x 10 - 1 5 )( 1 . 67 x 1 0 8 ) (57 . 0) (0 . 85)(29)( 1 4 . 7)
(52 . 0) (40)(520)
(40) (0 . 3) (520)(0 . 0244)
-------
= 6 1 . 9 md .
= 6 . 1 5 x l O - 5 D/Mscf.
PRESS U R E-TRAN SI ENT TESTI N G OF GAS WELLS
As an estimate , Jl. g, w! is evaluated at the final flowing pressure
during the test, Pw! = 3 ,570 psia, and is equal to 0.0244 cp o The
pseudoskin , Dq g , resulting from high-velocity flow near the well­
bore is
Dqg = (6. 15 x 10 - 5 )(40,000) =2.5.
9 . The true skin factor, s , resulting from formation damage near
the wellbore is estimated to be
s=s
'
6.7.5 Gas Flow Tests in Bounded Reservoirs. When a pressure
transient encounters reservoir boundaries during a gas-well test,
the liquid solution no longer describes the pressure behavior . 17 , 1 8
With boundary effects , variations in pressure-dependent gas prop­
erties affect the pressure response . In the semilog plot in Fig. 6.18, "
dimensionless pseudopressure solutions are compared with the liquid
solution .
Dimensionless time is
0 . 0002637ktp
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.69)
tAD =
cfJiig Ct A
with fluid properties evaluated at ji . After about one log cycle of
dimensionless time , the gas solutions deviate substantially from the
liquid solution. Because the gas physical properties are significantly
affected by boundary effects, PV cannot be found with techniques
developed for slightly compressible fluids (i . e . , liquids) . However,
Fraim and Wattenbarger 1 9 developed an iterative procedure that
accounts for variations in fluid properties during pseudosteady-state
flow in a gas well producing at constant BHP.
&.8 Analysis of G as-Well Buildup Tests
This section discusses analysis techniques for pressure-buildup tests
in wells completed in gas reservoirs . We begin with buildup tests
with constant-rate production before shut-in but then discuss the
more probable testing scenarios-discrete rate changes or constant­
pressure production before shut-in . Finally , we illustrate how aver­
age drainage area pressure is determined from a gas-well buildup
test .
l
162.6qiig iig
ktp
log
- 3 .23 +0.869s' .
_ _
kh
cfJJl.gCt rw2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 72)
Combining Eqs . 6.70 and 6.72 yields an expression for the ap­
)
parent skin factor ,
- Dqg = 8 . 3 - 2 . 5 = 5 . 8 .
Note that the estimate o f s = 5 . 8 obtained with conventional cal­
culations is almost twice that obtained when non-Darcy flow is in­
cluded (s = 3 . 0) .
(
l
1 25
+ IOg
( tp :pllta )l,
. ... ... ..
. . . . . . . . . . . . . . . . . . . . . (6.73)
where m = slope of the semilog straight line . Setting Ilta = 1 hour,
introducing the symbol P a , I hr for P a , ws at Ilta = 1 hour on the semi­
log straight line , and neglecting the term log [(tp +llta )/lltp ] , we
can rewrite Eq. 6.73:
( a - a )
S' = 1 . l 5 1 P , l hr P ,wf - IOg
[
m
( cfJiJ.g_ kc_trw2 ) + 3 23l
.
. . (6.74)
,
where P a ,wf = adjusted BHFP at the instant of shut-in .
Similar to the technique for slightly compressible liquids, we can
estimate k, Pi ' and s ' by using information obtained from a graph
of P a , ws vs. log (tp +Ilta )/Ilt a . Although presented in terms of ad­
justed variables, similar analysis techniques are possible with either
pressure or pressure-squared variables . Table 6. 1 gives the appro­
priate working equations .
6.8.2 Buildup Tests With Discrete Changes in Rate Before Shut­
In. Superposition Method. We can develop an analysis technique
for buildup tests with discrete rate changes before shut-in using su­
perposition in time 3 for (n - I) rates preceding the pressure-buildup
test (Fig. 6.19) .
For the general case where q n =0 and for (n - 1 ) different rates
before shut-in,
. _
p,
x log
P ws =
162.6qn _ IJl.B
kh
q2
[ ( qnq-l I ) ( t -tt l ) + (-)
qn I
--
log
--
"
-
( tt -- tt2l ) + . . . + ( qqnn - 2I ) ( tt -- ttnn - 2 ) ( tt -- ttnn _- 21 ) l '
3
log
+ log
6.8. 1 Buildup Tests With Constant-Rate Production Before
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.75)
Shut-In. An equation modeling a pressure-buildup test 1 -3 in a gas
well also can be developed by use of superposition in time. In terms where t - tn _ 1 =Ilt (time elapsed since shut-in) and qn - I is the gas
of adjusted variables , Eq. 6.7 for a slightly compressible liquid production rate just before shut-in .
Note the fundamental assumption on which Eq. 6.75 is based
becomes
(that , for t=tpl +tp 2 + . . . +tp , n - I +Ilt, the reservoir is infinite­
acting) rarely will be valid for large values of t. Nevertheless , when
162 . 6qiig iig I (tp +Ilta )
eog Ilta , . . . . . . . (6.70) Eq . 6.75 is used to model a buildup test , the following analysis
kh
P a , ws =P a ,i procedure in terms of adjusted variables is recommended.
1 . Calculate the plotting function .
where P a , ws = adjusted shut-in BHP , tp = duration of the constant­
rate production period before shut-in, and Ilta = adjusted shut-in
t t
� I Og
time . Comparison of Eq. 6 . 70 to the equation of a straight line sug­
X=
+ . . . + IOg = n - 2
t tl
t tn - I
qn - I
gests that a plot of adjusted shut-in BHP, P a , ws ' from a buildup
test as a function of the log of the adjusted Homer 4 time ratio func­
2. Plot P a,ws vs. X on Cartesian coordinate graph paper.
tion , (tp +llta )/llta , will exhibit a straight line with slope m. To
3. Determine the slope m of the straight line .
calculate permeability , we use the absolute value of the slope , or
4. Calculate permeability to gas using the slope from Step 2.
162.6qgiig iig
162 . 6qn l iig iig
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.71)
k=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.76)
k=
mh
[
lJ
[ ( ) (� )
)l.
(
_
mh
From the semilog graph , the original reservoir pressure , Pi ' is
estimated by extrapolating the straight line to infinite shut-in time
where (tp +Ilta )/Ilta = 1 .
We also can solve for the apparent skin factor, s ' , from a pressure­
buildup test . At the instant that a well is shut in,
5.
Calculate the skin factor .
a
S = 1 . l 5 1 P , l hr -P a , W! - l Og
[
m
( cfJiJ.g_ �ctrw2 ) + 3 23l . . . . . (6.77)
.
GAS RESERVO I R E N G I N E E R I N G
t
.oM
pn-l
-
-
-
t n-1
Fig. 6 . 1 9-Rate history for buildup test following n
6. The initial adjusted formation pressure , P a ,; '
P a , ws on the straight line extrapolated to X = O .
(
)
162.6q*Bg iig
t; + llta
log
, . . . . . . . . . . (6.78)
Ilta
kh
where the modified production time , t;, and flow rate , q*, are
[
1
- 3.23 +0.869S ,
---
qj (tJ - t]- I )
j E= 1
t; = 2 tn - --n
. . . . . . . . . . . . . . . . . . (6.79)
2 E qj (tj - tj _ l )
j=1
n
I
and q* = - E qj (tj - tj _ I ) ' . . . . . . . . . . . . . . . . . . . . . . (6. 80)
t; j = 1
------
The Odeh-Selig method , approximate but accurate, is applica­
ble only for pressures at Ilta values greater than actual producing
time . This condition is likely only in a drillstem test or short pro­
duction test .
Homer's4 Approximation. Horner reported an approximation
that can be used in many cases to avoid the use of superposition
in modeling the production history of a variable-rate well . He de­
fined a producing time, tp ' for production from a gas well as
tp = Gp /q last ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 8 1 )
where Gp = cumulative production from the well, Mscf, and q last =
the most recent production rate, MscflD. For tp in hours , Eq . 6.81
then becomes
-
[ (
Horner proposed to model the effect of the entire rate history by
)
]
ktp
162.6q lastBg iig
log
- 3.23 +0,869s .
_ _
kh
cPJ.tg c[rW2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 83)
An equation modeling a pressure-buildup test 1 can be written
,
by noting that pressure buildup is a special case of variable-rate
production . Assuming that Horner ' s approximation adequately
models the production history before shut-in , the entire production
history can be modeled as production at q last for tp ' If Ilta denotes
time elapsed since shut-in, then superposition in time with Eq . 6.27
yields Eq. 6.84, which describes P a , ws :
J
[ [
]
J
162.6( - q last )Bg iig
k(llta )
log
- 3 .23 +0.869s' .
kh
cPiig c[r�
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.84)
----
[ [ (tp +llta ) ]J .
Ilta
Combining terms and simplifying yields
_
P a , ws -Pa ; '
162.6qlastBg iig
kh
log
.
. . . . . (6. 85)
As in the analysis technique for pressure-buildup tests preceded
by a constant-rate production period , we simply plot P a , ws as a
function of the log of the Horner time ratio function based on the
pseudoproducing time, (tp + Ilta )/ Ilta . To calculate permeability ,
we use the absolute value of m of the semilog straight line,
k=
162.6q lastBg iig
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.86)
mh
From the semilog graph, the original adjusted reservoir pressure,
P a ,; ' is estimated by extrapolating the straight line to infinite shut­
in time , where (t +Ilta )/Ilt a = l .
The apparent ski n factor is estimated from
[
(
)
S' = 1 . 15 1 P a , l hr -P a ,wf
m
- IOg
( cPJ.tgkc[rW ) +3 .23] , . . (6.87)
_ _
2
where m = slope of the semilog straight line , P a , l hr = P a , ws at
hour on the semilog straight line , and Pa , wf = adjusted
BHFP at the instant of shut-in.
tp =24Gp /q last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 82) Ilta = l
P a , wf =P a ,; -
1 different flow rates.
is the value of
Odeh-Selig20 Method. A s a n alternative t o superposition, Odeh
and Selig suggested that a buildup test following n different rates
could be analyzed by a method similar to the Horner method . The
shut-in pressure response is
P a " ; -P a ws =
-
6.S.3 Buildup Tests With Constant-Pressure Production Before
Shut-In. Conventional buildup test analysis techniques have been
developed primarily for wells producing at a constant rate before
shut-in. However, some common situations involve production at
constant BHP rather than at constant rate . Examples include
declining-rate production during reservoir depletion, fluid flow into
a constant-pressure separator, or open wells flowing at atmospher­
ic pressure. With slight modification, conventional Horner and type­
curve analyses can be used to analyze buildup test data following
production at constant BHP . 2 1 ,22 The Horner method gives the
correct semilog straight line from which k, s, and average drainage
area pressure can be found .
1 27
PRESS U R E-TRANSIENT TESTI N G OF GAS WELLS
.. .. .. .. .. ..
MTR
ETR
P
ETR
M1R
l o g -..!...P-­
ii t
The usual Homer time plotting function uses a producing time
defined by Eq. 6.82.
�
�
�
p.
I
1
l o g --,-P-­
itt
Fig. 6 . 2 1 -Buildup test plot for a well near a boundary in a
reservoir with negligible pressure depletion (L TR = late-time
region).
=
tp 24Gp /q la st ,
where q l a st is the
last established flow rate before shut-in . For
Homer analysis of a buildup test following constant-pressure pro­
duction , the actual producing time , t actual , rather than tp defined
by Eq. 6 . 82 is used . As an example, suppose that a gas well has
been produced at constant BHP for 1 year (8,760 hours) . At the
time of shut-in for a buildup test , Gp 100,000 Mscf and q l a st =
100 MscflD . From Eq . 6 . 82, the conventional Homer producing
time is
=
(24)(100,000)
- = 24,000
---100
hours ,
Ilta 10 hours, the Homer plotting function is
tp + llta 24,000 + 10
= 2,401 .
10
Ilta
Using t actu al = 8,760 hours rather than tp =24,000 hours , w e ob­
=
�
�
t + ii t
Fig. 6 . 20-Bu lldup test plot for a well in an infinite-acting
reservoir with negligible pressure depletion (ETR = early-time
region; MTR = midd le-time region).
=
�
�
1
t + ii t
and at
LTR
----
.. .. .. ..
1
P
.�
LTR
ws
...
t + ii t
log P
L1 t
Fig. 6.22-Extrapolated pressure, MBH 23 method .
tain the Homer plotting function of
log analysis must be used for reliable formation evaluation . Fur­
thermore, when non-Darcy flow effects are significant, a buildup
test gives more reliable results than a drawdown test. 22
vs. the adjusted Homer time ratio based on t actu­
A plot of Pa
al gives the c�rrect semilog straight line with slope m. k is com­
puted with
6.8.4 Determining Average Drainage Area Pressure for Gas
Wells. The average pressure in the drainage area of a well repre­
sents the driving force for fluid flow and is useful in material-balance
calculations . For a well in a new reservoir with negligible pres­
sure depletion , extrapolation of buildup test data to infinite shut-in
time on a Homer semilog plot provides an estimate of original (and
current) drainage area pressure, Pi ' For a well in a reservoir where
the average pressure has declined from its original value because
of fluid production, the pressure extrapolated to infinite shut-in time,
p *, is related but not equal to the current average pressure in the
drainage area of the well .
We consider two possibilities for a well in a reservoir with negligi­
ble pressure depletion . 3 First, if the pressure-transient data are not
influenced by boundaries during the production period before the
buildup test, a typical buildup test will have the shape shown in
Fig. 6.20. The original reservoir pressure is obtained by extrapolat­
ing the middle-time semilog straight line to (tp + Ilt)/ Ilt = 1 .
Second , for a well with one or more boundaries relatively near
the well (and encountered by the radius of investigation during the
production period) , a buildup test will exhibit the shape shown in
Fig. 6.21, and the late-time semilog straight line is extrapolated .
For a reservoir where the pressure has been depleted , the
Matthews-Brons-Hazebroek23 (MBH) method can be used to es­
timate the average drainage-area pressure . The MBH method is
based on theoretical correlations between the extrapolated pressure,
ws
k
=
8,760 + 10
--= 877.
10
162.6q lastBg iig
mh
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 86)
where for the constant-pressure case q l a st is the last established
producing rate, not the average rate over the producing period . The
skin factor also is calculated in the usual way ,
[
(
)
S' = 1 . 15 1 P a , l hr -Pa , ws - IOg
m
(
_
k
_
</>/lgCt rw2
) +3.23J . . . (6. 87)
Extrapolation of the semilog straight line to infinite shut-in time
(adjusted Homer time equal to one) gives P a , i for an infinite-acting
reservoir.
For buildup tests following constant-pressure production, the ef­
fects of wellbore storage , skin damage, and non-Darcy (high­
velocity) gas flow usually are short-lived and do not affect the slope
of the semilog straight line . However, in gas reservoirs with k > O . l
md , non-Darcy flow effects may cause substantial errors in esti­
mates of the apparent skin factor, s ' , from match-point data in type­
curve analysis (discussed in Sec . 6.9). Under these conditions , serni-
1 28
GAS RESERVO I R E N G I N E E R I N G
F i g . 6.23-MBH 2 3 dimensionless pressures f o r various w e l l locations i n a square drainage
area (after Earlougher 2 ).
6
5
4
Fig. 6.24-MBH 2 3 dimension less pressu res for various well locations in a 2 : 1 rectangular
drainage area (after Earlougher 1 ).
p *,
shown in Fig . 6.22 and current average drainage area pres­
sure , p , for various drainage area configurations .
Figs. 6.23 through 6.24 show two of the numerous correlation
charts available . Fig . 6 . 23 applies to wells in various locations in
square drainage areas. The most common approximation of drainage
area is that the well is centered in a square drainage area with an
area equal to the acreage assigned to the well. Fig . 6 . 24 applies
to
rectangular drainage areas . Here , (tAD) pss indicates the be­
ginning of pseudo steady-state flow , where tAD is defined by Eq.
6 . 89 . Similar charts for other drainage area shapes are available
elsewhere . 1 -3 , 23
2:I
A dimensionless pressure , P MBHD , is plotted as a function of a
dimensionless time , tAD ' in Figs . 6 . 23 and 6 . 24 . The dimension­
less variables are defined as
P MBHD
and tAD =
kh(p* -p )
70. 6qBp,
O. 0002637ktp
c/Jp,ctA
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 88)
, . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 89)
where A = drainage area of the tested well , ft2 .
1 29
PRESS U R E-TRANS I ENT TESTI N G OF GAS WELLS
TABLE 6.9-GAS-WELL PRESSURE-BUILDUP TEST DATA,
EXAMPLE 6.4
ilt
P w.
(hou rs)
(psia)
0
0 . 0 1 00
0. 0 1 49
0 . 0221
0 . 0329
0 .0489
0 .0728
0 . 1 08
0.1 61
0.240
0.356
0.530
0 . 788
1 .17
1 . 74
2.59
3.86
5.74
8.53
1 2 .7
1 8 .9
28. 1
41 .8
62. 1
92.4
1 37
204
304
452
672
1 ,000
6,287 . 1
6 , 296. 6
6,301 . 1
6,307.8
6 , 3 1 7.7
6 , 332 . 1
6 , 353 . 1
6,383. 5
6 ,427. 1
6,488.6
6,573.6
6,687.9
6 , 834. 7
7 , 0 1 1 .8
7,208.3
7,405.9
7,586.0
7,738. 7
7,864. 9
7,971 .4
8,065 .6
8 , 1 53 . 2
8,234.4
8 , 3 1 3.4
8,389 . 8
8,463. 7
8 , 534.9
8 , 602. 9
8,666. 6
8 , 725.3
8 , 777.6
Adj usted
Horner
Time Ratio
P ;.
Adj usted
Pressure
(psia)
Horner
Time
Ratio
39.528
39. 647
39. 704
39.788
39.9 1 3
40.095
40.362
40.749
4 1 .308
42. 1 02
43. 2 1 2
44.728
46.7 1 3
49 . 1 65
5 1 .960
54.847
57.547
59.887
6 1 .857
63.543
65.054
66.475
67. 805
69 . 1 1 3
70.389
7 1 .634
72.845
74.0 1 0
75. 1 1 0
76. 1 31
77.046
4,804 . 1
4,81 3.9
4,8 1 8.5
4,825.4
4,835 .5
4,850.3
4,871 . 9
4 , 903.0
4,947.8
5 , 0 1 0.8
5,098.0
5,21 5.1
5 , 365 .5
5,546.9
5 , 748 .0
5 ,950 . 1
6, 1 34 . 1
6,289.8
6,41 8.3
6,526.6
6.622.3
6,71 0.3
6,793.5
6,873.5
6,950.7
7,025.4
7,097.2
7, 1 65.7
7,229.8
7,288.8
7,34 1 .3
-
-
200,000
1 34,230
90,499
60,791
40,901
27,474
1 8 ,520
1 2 ,423
8 , 334. 3
5,61 9.0
3,774. 6
2 , 539 . 1
1 , 7 1 0.4
1 , 1 50.4
773.20
51 9. 1 3
349.43
235.47
1 58.48
1 06.82
72 . 1 74
48.847
33.206
22.645
1 5. 599
1 0 . 804
7.5789
5.4248
3.9762
3.0000
286,370
1 92 , 1 20
1 29,460
86,887
58,386
39 , 1 48
26,230
1 7,589
1 1 , 737
7,853.9
5,22 1 . 7
3 ,464.2
2,292.4
1 ,509 . 1
990. 1 6
648.42
426. 1 1
280.88
1 85.36
1 22.80
8 1 .709
54.543
36. 6 1 5
24.677
1 6.81 1
1 1 .5 1 9
7.9970
5 .6678
4 . 1 1 60
3.0794
(psia 2
x
1 0 -6)
The advantages of the MBH method are that it does not require
data beyond the middle-time region and that it is applicable to a
wide variety of drainage area shapes. The disadvantages are that
the drainage area size and shape must be known and that reliable
estimates of rock and fluid properties , such as ct and r/J, must be
available . In addition, the method is limited to well tests in single­
layer formations and cannot be applied accurately to multilayer for­
mations .
Although developed for slightly compressible liquids , the aver­
age drainage area pressure , p , for a gas well can be found with
the MBH p* method. To find the correct p , pressure-dependent
fluid properties must be evaluated at p . This suggests an iterative
procedure . For gas , we use P a ' The procedure (outlined below)
involves making an initial estimate of Pa (the adjusted pressure cor­
responding to p) , evaluating fluid properties at the pressure associat­
ed with p, calculating the MBH pressure function pa . MBHD on the
basis of those properties , and obtaining another estimate of Pa from
7.
Calculate the next estimate of Pa :
mP a ,MBHD
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.91)
2.303
8. If the Pa estimate from Eq. 6.91 is different from the previ­
ous estimate, evaluate the gas properties at the pressure associated
with the new estimate of Pa ' Repeat Steps 5 through 7, recalculat­
ing tAD at the new values of the fluid properties, until the estimate
of Pa from Eq . 6.91 converges . The correct P is the pressure cor­
responding to the final Pa value .
Kazemi 1 7 and Reynolds et al. 24 presented other methods to es­
timate P for gas reservoirs . Kazemi ' s method requires an iterative
procedure ; Reynolds et al. 's technique requires a Homer plot in
_
P a =P a* -
terms of real time and pseudopressure rather than adjusted time
and adjusted pressure .
Example 6.4-Analysis of a Gas-Well Pressure-Buildup Test
P a ,MBHD '
1 . Assume a value ofp , calculate the adjusted pressure and time, With Constant-Rate Production Before Shut-In. A pressure­
buildup test was run on a gas well in a newly discovered reservoir.
and make a Homer plot of the buildup data, using values of P a , ws
Known and calculated data are summarized below and in Table
and (tp + l1ta )/l1ta .
6.9.
Because the reservoir pressure has not been depleted, the semi­
2 . Extrapolate pa , ws to (tp + l1ta )/l1ta = l . The extrapolated ad­
log straight line should extrapolate to original reservoir pressure
justed pressure, P �, may be an improved estimate of Pa ; however,
at a Homer time ratio of unity ; i . e . , P * = Pi ' Prepare Homer plots
if p� is clearly larger than the apparent level value of Pa ' continue
of (l) p ws vs. (tp +l1t)/l1t; . (2) pJs vs. (tp +l1t)/l1t, and (3) P a , ws
to use the initial estimate of Pa in subsequent steps.
vs. (tp +l1ta )/l1ta . Determme k and from each plot .
3. Estimate the drainage area shape. In practice, the drainage area
h = 2 1 ft .
usually is reasonably symmetrical .
tp = 2,000 hours .
4 . Select the appropriate MBH chart 1 -3 ,23 for the drainage area.
iig = 0.03403 cp o
5. Calculate the tAD with Eq. 6.89 with JL and ct at the pressure
Pi = 9,000 psia.
corresponding to the most recent estimate of p.
'Y
g = 0.659.
6. From the MBH chart at the calculated value of tAD ' read the
r w = 0.365 ft .
value of the MBH pressure function:
q,g = 100 Mscf/D .
ct = 35 .5 x lO -6 psia - I .
kh ( P � -Pa )
. . . . . . . . . . (6.90)
P a ,i = 7,560 psia .
P a ,MBHD
70.6qiig iig
m
Sw = 0.36.
s
1 30
GAS RESERVO I R E N G I N E E R I N G
�
'iii
Co
9 0 0 0 ,-------��
.. 8 5 0 0
,f
! 8000
�
�
0..
G>
:g
E
!
7500
7000
6500
•
•
•
•
•
•
•
..
.. .. . . .
6 0 0 0 4-----�--�r_--�--_.--�
•
Horner Time Ratio.
(I + M) 1M
p
Fig. 6 .2S-Horner plot using pressure, Example 6.4.
=
T=
Bg =
Z=
0. 10.
6700R (210°F).
0.497 RB/Mscf.
1 .325 .
</>
Solution. Figs. 6.25 through 6.27 give Horner plots in terms
of pressure , pressure-squared, and adjusted pressures, respective­
ly . Note that semilog straight lines passing through the later data
and through Pi (equal to p* in this case) appear in each case .
Analysis Using Pressure Variables (Fig . 6.25).
1 . Calculate th.e slope of the line drawn through the semilog
straight line .
m = 8,989.7 - 8,545 . 1 = 444.6 psi/cycle .
2. The permeability to gas is
162.6qg Bg Jig
k=
mh
(162.6)(100)(0.497)(0.3403)
(444.6)(2 1)
=0.029 md .
3 . Calculate the skin factor . The pressure P l hr at (tp +.1t)/.1t=
(2,000 + 1)11 =2,001 is 7,522 psia . Thus ,
S' = 1 . 15 1 P : hr -P W! - IOg _ k_
+3.23
m
</>p-g c t rw2
--"-�"-
--
( ) ]
[
(7,522 - 6,287. 1)
= 1 . 15 1 [
444.6
0.029
] + 3 .23{j
[ (0. 1)(0.03403)(0.0000355)
(0.365) 2
'f'
�
..
til
'iii 7 0
�
....
it
Q.
ti
�::>
Analysis Using Pressure-Squared Variables (Fig . 6.26).
1 . Calculate the slope of the line drawn through the semilog
straight line .
m = 80.67 x 106 - 73 .07 x 106 =7.60 x 1 0 6 psia2 /cycle .
2. The permeability to gas is
1 ,637qg TZJig
k = ---"---"­
mh
( 1 ,637)(100)(670)( 1 .325)(0.03403 )
(0.760 x 10 7 )(21)
=0.03 1 md .
60
m
_
7.6
x 1 0·
II 5 0
l!!
::>
fI)
fI)
l!!
40
0..
• • •• • •
6
5
psla2/cycle
•••
•
•
•
•
•
4
•
•
o
2
3
Homer Time Ratio. (t + dt) Idt
p
Fig. 6 .26-Horner plot using pressure squared , Example 6.4.
�
.e:
'0;
8 0 0 0 ,-------,
7500
p; - Pa. ' - 7560 psla-----�
5 7000
Q.
e
�
6500
� 6000
*
0..
5500
% 5000
•• • • • •
0(
••
•
•
•
•
•
•
•
•
•
Adjusted Ratio. (I +
ps
M� /dl.
1 0'
Fig. 6. 27-Horner plot using adjusted pressure and adjust­
ed Horner time ratio, Example 6 . 4 .
3. Calculate the skin factor.
2,001 is 55.6 x 106 psia2 , so
s, =
[ P ihrm-P !
�
= 1 . 15 1
x
- IOg
= - 0.292.
80
><
- IOg
(
k
_ _
The term
</>p-g ctrw2
p ihr
at
(tp + .1t)/.1t=
) + 3.23]
[ 55.6 x 106 - 39.53 x 106
7.60 x 10 6
- log
0.03 1
] + 3 .23]
[ (0. 1)(0.03403)(0.0000355)(0.365)
2
= - 1 .08.
A nalysis Using Adjusted Pressure Variables (Fig . 6.27).
1 . Calculate the slope of the line drawn through the semilog
straight line .
m = 7,553 . 8 - 7 , 1 17.5 =436.3 psi/cycle .
2. The permeability to gas is
k
162.6qgBg Jig
=mh
-"--"-"-
-
(162.6)( 1 00)(0 .497)(0.03403)
(436.3)(21)
=0.030 md .
PRESS U RE·TRANSIENT TESTI N G OF GAS WELLS
3. Calculate the skin factor. At an adjusted Homer time ratio of
(tp + �ta )/�ta = (2,000 + 1)/1 =2,00 1 , Pa . ws =6, 1 14 psia.
Pa, l hr -Pa , wf 1
0g
+ 3 .23
S I = 1 . 15 1
c/>p.gctrw2
m
[
_
[
(6, 1 14 -4,804 . 1 )
= 1 . 15 1
436.3
- IOg
( � ) ]
] + 3.23(j
[ (0. 1)(0.03403)(30.030
.55 10 - 5)(0.365) 2
x
= -0.04.
I n this example, there i s little difference i n results when the pres·
sure , pressure·squared , or adjusted-pressure analysis is used on a
Homer semilog graph . However, when wellbore-storage-distorted
data are analyzed on a type curve , the differences can be signifi­
cant . We discuss type-curve analysis in next section .
PD = -
� ( ::: )
Ei
At the well , where
PD =P wD = where P wD
1 31
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.97)
rD = 1 ,
± {:� )
E
.
the solution simplifies to
. . . . . . . . . . . . . . . . . . . . . . . . (6.98)
kh ( p ; -P wj ) . . . . . . . . . . . . . . . . . . . . . . . . . (6.99
)
141 .2qBp.
Eq . 6.98 implies that we can develop a type curve from a plot
of P wD as a function of the single variable tD ' Generating a single
graph in terms of P wD is much simpler than attempting to plot P wf
as a function of t for all reasonable values of the variables that ap­
pear in the dimensional form of the line-source solution (Eq. 6.92).
Thus, with this type curve, we can analyze any pressure-transient
test conducted under conditions satisfying the assumptions made
in deriving the Ei-function solution . Again , before using any type
curve , we must fully understand the assumptions inherent in the
model being used .
6.9 Type·Curve Analysis
Type curves, which are plots of theoretical solutions to flow equa­
tions , are very useful in well-test analysis, particularly when used
with semilog analysis techniques . Type curves can help identify
the appropriate reservoir model , identify the appropriate flow re­
gimes for analysis , and estimate reservoir properties . They are es­
pecially helpful for analyzing gas-well tests when the data are
distorted by wellbore storage . In this section, we explain what type
curves are , identify some of their more useful properties , and il­
lustrate how they can be used to improve pressure-transient test
analysis. We begin with type curves developed for slightly com­
pressible liquids and then present modifications for application to
gas-well test analysis .
6.9.1 Development of Type Curves. Type curves can be generat­
ed for virtually any kind of reservoir model for which a general
solution describing the flow behavior is available. For a type curve
to be applied correctly, the assumptions underlying the solution must
be understood . Furthermore, those assumptions must accurately
model the well or reservoir conditions being analyzed . As a matter
of convenience, type curves usually are presented in terms of dimen­
sionless rather than real variables . The definition of the dimension­
less variables varies according to the reservoir model. For example,
recall the line-source or Ei-function solution for slightly compres­
sible liquids,
(
70.6qBp. . -948c/>p.ctr2
El
kh
kt
If we rearrange Eq . 6.92 as
P; -P =
[
)
.
. . . . . . . . . . . . . . . (6.92)
]
kh( p ; -p)
1 .
- (rlr w ) 2
El
, . . . . . . . . . . . (6.93)
0.00026 7kt
141 .2qBp. - '2
4
c/>p.ctrw
then the form of Eq . 6.93 suggests the following convenient dimen­
(
� )
sionless variables:
kh ( p ; -p)
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.94)
141 .2qBp.
rD = rlrw ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.95)
0.0002637kt
and tD =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6.96)
c/>p.ctr;
6.9.2 Application of Type Curves-Homogeneous-Reservoir
Model, Slightly-Compressible-Liquid Solution. Although many
type curves have been developed in the petroleum industry , we use
the Gringarten et al. 2 5 type curves to illustrate the important char­
acteristics of type curves and their applications to well-test analy­
sis . Gringarten et al. ' s type curves are based on solutions to the
radial-diffusivity equation modeling the flow of a slightly compres­
sible liquid in a homogeneous formation . The initial condition is
uniform pressure throughout the drainage area of the well . The outer
boundary condition specifies an infinite-acting or unbounded reser­
voir; the inner boundary condition is constant-rate flow with well­
bore storage and skin effects .
As Fig. 6.28 shows, Gringarten et al. plotted PD vs . tDICD on
log-log coordinate s , where the dimensionless pressu re ,
P D =/ (tD , s , CD ) , is a function of tD (Eq . 6.96), s, and dimension­
less wellbore-storage coefficient , CD :
0. 8936C
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 100)
c/>ct hr;
C= wellbore-storage coefficient defined by Eqs . 6. 15 and
CD =
where
6. 16.
The family o f type curves is shown i n Fig . 6.28 a s a function
of the correlating parameter CDe 2s . The value of CDe 2s indicates
whether a well has skin damage, has zero skin , is acidized , or is
hydraulically fractured .
In using the Gringarten et al. and other type curves , we com­
pare a plot of test data with the type curve . The test data are plot­
ted on either tracing paper or log-log graph paper (the same size
used to plot the type curve) . The data plot is tlIen matched to tlIe
type curve having the same shape as the test data plot (Fig. 6.29) .
The theory underlying type-curve matching is that the type curve
and test data plots differ only by displacement of both coordinates
by constants . This concept is illustrated as follows . Taking
logarithms of the dimensionless time group ,
c/>cthr; ) ]
)
(
( CDtD ) [( 0.0002637kt
0. 8936C
c/> p.ctr;
log - = log
PD =
In terms of these dimensionless variables , the line-source solu­
tion becomes
= log t + log
1kh )
( 0.OOO295
,
p.
C
suggests that the horizontal axes of the type curve and test data plots
are displaced by the constant
log
( 0.000295 1kh )
p. C
.
1 32
GAS RESERVO I R E N G I N E E R I N G
10 '
•
lllIlllml
' :wmt.ilmIlllll
tfp
HlllH
llDlIllll
mmHl
SPf nos
" COM;NPlISOH elTwtOl tK',[A(Hl
I
IKIN
... ,.., 0 Wl1.l80fll (
1T00A.Ol
f'rP1:oO.1PM.t ,on !ARt! T I M !
TIUJfSI(HT ANAUSII by . A C Qll\ IHOA�T l H . D . BOU"Of: l . P. U.A"'
•
DH.
Y. KHIAI [ "
.
..
'0' .
i
10 "
�
•
..!£ ,
11
Co
c
c
,
•
f
• ,
�
. C_
E_
P_
_
TY
WQL WITH W_
_
A
R_
___________________
_
O
_
E_F
__________
_
RV
U_
K I_
S_
O_
N_
A_
E_
E_
G_
_A_
R (_�O R
0_
8_
L L_
N__
____
__
__
__
____
__
____
__
__
__
____
__
__
Fig. 6.28-Gringarten et al. 25 type curve. Courtesy of Schlumberger
( 141 kh.2qB/L )
Similarly , the logarithm of dimensionless pressure,
log P D = I Og( P; -Pwj ) + IOg
( 141 kh.2qB/L )
,
indicates that the vertical axes are displaced by the constant
1
0
g
.
Therefore, the plot of constant-rate flow test data should be iden­
tical to a plot of log P D vs. log tD ICD but with the horizontal and
vertical axes displaced , as Fig . 6.29 shows. Before illustrating the
type-curve matching procedure , consider the following important
properties of the Gringarten et al. type curve .
1 . As long as wellbore unloading accounts for 100% of the flow
during a draw down test or, in a buildup test , afterflow accounts
for 100% of the flow rate before shut-in, a unit-slope line will form
at early times . The unit-slope line has the property
. . . . . . " . . , . . . . . . . . . . ' , . , . . . . . . . (6, 101)
The implication o f E q . 6. 101 i s that C is constant and can be
tD ICDP D = l .
.
determined from any point on the unit-slope line . Substituting the
definitions Of P D , tD , and CD (Eqs . 6.99, 6.96, and 6. 100, respec­
tively) for a slightly compressible liquid into Eq . 6. 101 , we find
that, for a point on the unit-slope line ,
C=
�: C : )
f:.t
p
SL ' . .
U
, ' , . . . . . . . . . , . . , . , . . . . . . (6. 1 02)
Note that the unit-slope lines converge at earliest times on the
Gringarten et al, type curve. The tick marks indicate the end of
each unit-slope line for each value of CDe2 s .
2. The " original " curve marks rigorously the approximate start
of the semilog straight line (i , e . , the end of wellbore-storage ef­
fects for all but fractured wells) ; the "preferred " curve indicates,
less rigorously but adequately , the end of wellbore-storage distor­
tion of test data . Accordingly , the point at which the test data cross
the preferred' curve in a type-curve match represents the approxi­
mate beginning of the semilog straight line indicative of radial flow .
This straight line, commonly called the middle-time region , is the
basis of the semilog analysis techniques presented previously . Note
that, the more pronounced the effect of wellbore storage is on pres­
sure response during a test , the longer the delay in the onset of this
middle-time region i s .
A nearly horizontal line crossing the type curves for fractional
values of CDe2 s lies in the " fractured-well" region on the Grin­
garten et al. curve. This line also indicates the beginning of a straight
line on a semilog graph of test data . For the fractured-well solu­
tions plotted on the Gringarten et al. type curves , however, well­
bore storage does not delay the onset of the middle-time region.
Rather, the nearly horizontal line marks the transition from a flow
regime characterized by linear flow from the reservoir into a verti­
cal , high-conductivity , hydraulic fracture to essentially radial flow
(called pseudoradial flow) . The transition from linear to pseudoradial
flow is not complete , nor will a semilog straight line appear, until
the plot of the test data crosses this line . Sec . 6. 10 gives additional
details on the analysis of pressure-transient data from hydraulical­
ly fractured wells.
3. The Gringarten et al, type curves are based on solutions to
equations mQdeling constant-rate flow ; however, they also can be
used to analyze buildup tests if the appropriate change of variables
is made to account for the differences in flow and buildup tests .
For a drawdown test , we plot ( p; -Pwj ) as a function of t, For a
buildup test, we plot ( P ws - P wj ) as a function of the Agarwal26
equivalent time , f:.te :
l
l ( �; )J . , . . . . . . , . . , . . . . . . . . . . . . . . , (6. 103)
f:.te = f:.t l +
A given pressure change, f:.p , that occurred at shut-in time , f:.t,
during a buildup test would have occurred at an equivalent time,
f:.t e ' during a constant -rate flow test. This definition of f:. t e is
rigorous only for radial flow in homogeneous , infinite-acting reser­
voirs , with test data undistorted by wellbore storage. Within limits ,
1 33
PRESS U RE-TRAN SI ENT TESTI N G OF GAS WE LLS
TABLE 6 . 1 0-EQUIVALENT VARIABLES IN O I L AND GAS FLOW
Gas, Using Pseudopressure
and Pseudotime
Oil
p
kMp
-'--PD = 1 41.2qB J-t
p dp
pp (p) = 2 r p J
P o p.z
j. t dt
h
( )
PD
IIp MP
O.0002637k
cf>p.c rra,
(
s' = 5 + D l q l
s' = s + D l q l
C . = C(T cd/(T wb c Wb )
= V wb T c t I T wb
C . = C( T cd/(T wb C wb )
= V wb T c t I T wb
C .D = O . 8936 V wb Tlc/>hr � T wb
C eD = O .8936 V wb Tlc/>hr � T wb
. . . . . . . . . . . . . . . . . . . . . . . . . (6. 104 )
t or ilte
tD I CD
)
MP
. . . . . . . . . . . . . . . . . . (6. 105 )
If a unit-slope line , which is indicative of a constant wellbore­
storage coefficient during the test , is present in the data plot, CD
also can be determined from a point on the line .
nPe CI.RIE
j
w
1 00
DATA
r
s
From the time match point (tD I CD ,t)MP for drawdown tests or
Ilte for buildup tests and the definition of tD given by Eq . 6.96,
we can calculate
J-t c t
r
however, it may be used to analyze radial-flow data distorted by
wellbore storage and test data affected by boundaries . For linear
flow , which occurs at early test times in many hydraulically frac­
tured wells, another expression for Ilte is more appropriate .
4. Once a match between the plotted data and the type curve is
found , a match point can be used to estimate k, s, and C. A match
point , chosen while the test plot is in the matched position , is any
convenient value of PD and tDICD from the type curve and the cor­
responding values of IIp and Ilt from the test data plot . When a
fit of the data to the type curve is found , the ratios of PD to IIp
and tDICD to Ilt will be constant .
We can estimate k from the pressure match point , ( P D , llp)MP ,
and the definition of P D '
141 .2 qBp.
to
r
C D = O .8936CIc/>c t hr �
k=
kMP a
P aD = 1 41 .2qB g jl.
khT se !1p p
P pD = 50,300q e T
ps
t ap ( P) =
Gas, Using Adj usted Pressure
and Adj usted Time
LOG cI B.APSB> 11ME
�
'II
§
w
�
'l5
§
s =O.5 In
LOG cI DlMEHSlCH.ESS TNE GRCXF
Fig. 6 .29-Example type-curve m atch of well-test data using
Gringarten et al. 2 5 type curve.
( C�:2S) .
.
. . . . . . . . . . . . . . . . . . . . . . . . (6. 106)
.
.
.
6.9.3 Gas-Well-Test Analysis Using Type Curves. Spivey and
Lee 27 showed that , for both buildup and flow test analysis in gas
wells , the use of pseudotime and pseudopressure (or adjusted time
and adjusted pressure) is essential for analyzing early-time data dis­
torted by wellbore storage. Without this transformation of varia­
bles , conventional type curves cannot model the early pressure
behavior . For a gas well , the wellbore-storage coefficient is
c wb Vwb , where c wb is a strong function of pressure . Thus, in a test
in which the pressure changes by one to two orders of magnitude ,
the wellbore-storage coefficient also will change drastically . Lee
and Holditch9 showed that, with the change in variables to pseu­
dopressure and pseudotime, the modified CD becomes independent
of pressure and can be treated as a constant. Therefore, type curves
derived on the assumption of a constant wellbore-storage coeffi­
cient can be used for gas-well-test analysis .
Table 6 . 10 summarizes the analogies between variables used in
type-curve analysis for slightly compressible liquids and gases using
the pseudopressure and pseudotime transformations . When adjust­
ed time and adjusted pressure are used for gas-well-test analysis,
the same modified wellbore-storage coefficients arise . The defini�
�
><
i
�
I
The skin factor can be estimated with the correlating parameter,
CDe 2 s , and the calculated CD :
c:
�
.!
1 000
'"
.s
..
co
c:
.,
c3
i!!
�
a
0
1 00
1 0
'0
5!
"
Cl.
0.01
0.1
1 0
Equivalent Time (hours)
1 00
1 000
Fig. 6. 30-Effect o f pseudopressure and equ ivalent time
variables o n curve shape .
1 34
GAS RESERVO I R E N G I N E E R I N G
TABLE 6 . 1 1 -S U M MARY OF ANALYSIS TECH N I Q U ES F O R TYPE-CURVE PLO,TS
Case
I nterpretation
of Pressure
M atch Point
Defi nition
of CD
I nterpretation
of U n it S lope Line
Slightly Compressible Fluids (Oil)
t¥J vs. tlt
t¥J vs. tlt
CD
CD =
O.0372q g B g
cf> h C t '�
CD =
O.0372q g B g
cf>h C t '�
CD =
CD =
(
( )
( )
= 22.92A wb
------'=-
P Wb cf>Ct h' �
Compressible Fluids (Gas)
= O.8936V wb C wb
-
tl t
tl p USL
CD
tl t
tl p USL
C D = --:-�=­
k=
O.8936V wb C wb
C D = ---;;.;:,--=-
k=
1 422qg Tiil �
h
(tl p 2 )
k=
1 422q g T �
h
(tl pp )
-
O.8936V wb T
cf> h '� Twb
)
tlt
O.375q g Ti
2
tl( p-2-) USL
cf> hc t , w
( )
C t ' W2
cf> C t h'�
O.375q g T -tl t
h
tl
p p USL
cf> C t '�
Notes:
1 41 .2q g B g il
h
(!!..E- )
tl p
(
( )
)
MP
MP
MP
1 . For drawdown test analysis. the p l ott i n g functions are defined as follows.
L;fJ = P - P wi (or equivale nt) .
.:1.1 = I = flow t i m e (or equivalent).
2 . For buildup test analys i s . the p l otting functions are defined as follows:
L;fJ = P w. - p wd.:l.l = 0) (or equ ivalent) .
.:1.1 = .:l.t . = .:1.1/(1 + .:1.1/1 p) (or equivalent).
TABLE 6 . 1 1 -S U M MARY OF ANALYSIS TECH N IQ U ES FOR TYPE-CU RVE PLOTS (continued)
Case
I nterpretation
of Time M atch Point
i n Fractured Wel l
Type Cu rves
I nterpretation
Time M atch Point i n
Gringarten et a/. Type C u rve
Slightly Compressible Fluids (Oil)
t¥J vs. tlt
CD =
(
tl t
O.0002637k -t
D /C D
cf>p. o C t '�
)
MP
Lf =
Com pressible Fluids (G as)
t¥J vs. tlt
CD -
( )
(�)
(�)
(�)
tl t
O.0002637k -t D /C D
cf>J1C t '�
CD
= O.0002637k
CD
= O.0002637k
CD
= O.0002637k
cf>J1C t '�
cf>J1C t '�
cf>pi5t '�
O.0002637k
cf>p. o C t
)( )
�
'·
tL I D
MP
] '12
MP
t D /C D
MP
t D /C D
MP
t D /C D
MP
tions of dimensionless pressure and time variables have the same
form as for slightly compressible liquids , except that the gas prop­
erties are evaluated at ji .
Figs . 6.30 and 6.31 show the effect of replacing time with pseu­
dotime on a logarithmic plot of a gas-well buildup test . Fig . 6.30
[(
plots pseudopressure change as a function o f equivalent time for
a simulated gas-well test. The early-time data fall on a line with
a slope greater than unity and are followed by a sharp flattening
at later times, indicating a changing wellbore-storage coefficient
during the test. Because the Gringarten et at. type curves were de-
1 35
PRESSU RE-TRANSIENT TESTI N G OF GAS WELLS
...
'0
)(
c:
�
TABLE 6 . 1 2-GAS-WELL BUILDU P-TEST DATA,
EXAMPLE 6 . 5
Adj usted
Pressure
C h ange
(psia)
Adj usted
Equ ivalent
Time
(hours)
Adj usted
Pressure
Change
(psia)
Adj usted
Equ ivalent
Time
(hours)
9.7476
1 4.365
21 .239
31 .397
46. 1 72
67.71 9
98.906
1 43.63
206.70
293.84
41 0.95
561 .33
742.73
943.90
1 , 1 46.0
0.0069839
0.0 1 04 1 0
0.01 5449
0.0230 1 8
0.034255
0.051 088
0.075987
0. 1 1 370
0. 1 7041
0.25465
0.38302
0.57734
0.87244
1 .3253
2.01 99
1 ,329.9
1 ,485.7
1 ,61 4.2
1 ,722.5
1 ,81 8.2
1 ,906.2
1 ,989.4
2,069.3
2 , 1 46.6
2,221 .2
2,293.0
2,361 .5
2,425.7
2,484.7
2 ,537.2
3.0844
4.6937
7 . 1 204
1 0.790
1 6.287
24.477
36.668
54.622
81 .047
1 1 8.97
1 73.63
250.09
352.87
485.91
649.47
1 000
NO!
�
'"
go
'"
.t:.
U
I
1 00
1 0
0.01
1 0
0.1
Equivalent Pseudo time (hr-psia/cp)
1 00
Fig. 6 . 3 1 -Effect of pseudopressure and equivalent pseudo­
time variables on curve shape.
veloped assuming a constant wellbore-storage coefficient, no sin­
gle curve fits all the test data,
When the same data are plotted as a function of adjusted equiva­
lent time, the earliest data fall on a unit-slope line, indicating a con­
stant wellbore-storage coefficient. In addition , all test data are fit
by a single type curve, Table 6 . 1 1 summarizes the interpretation
of the unit-slope line and the pressure and time match points for
type-curve analysis of gas-well tests using the common plotting func­
tions , including adjusted pressure and time , For comparison, we
also include interpretation equations for type-curve analysis for oil­
well tests,
Type-Curve Analysis Procedure. To analyze gas-well buildup
or flow tests using type curves, we recommend the following proce­
dure , Although presented in terms of adj usted pressure and time,
other variables , such as pressure or pressure-squared, also can be
used .
Plot ( P a , j -P a , wj) vs, t for a drawdown test or [ Pa, ws ­
P a ,w! (.:1tae =0)] vs, [.:1tae =.:1ta l(1 + .:1ta ltp )] for a buildup test,
Make the plot on either tracing paper or log-log paper the same
size as the Gringarten et al. type curve,
If a unit-slope line is present , calculate CD ' from Eq,
1.
2,
0. OOO2637k ( ) " " " " " " " " (6.110)
cp/ig ctra
7.
s
6.
s=0. 5 ( C�:2S). . . . . . . . . " . . . . . . . " . . . . " . (6.111)
6.
CD from the time match point .
t or .:1tae
.
tDICD
using CDe 2 s and CD from Step
Calculate or confirm
CD
Calculate
MP
.
In
.
Example 6.5-Gas-Well Buildup Test Analysis Using Type
Curves. Using the gas-well buildup test data presented in Exam­
ple
estimate the formation permeability and skin factor using
the Gringarten type curves . Use adjusted pressure and time varia­
bles as the plotting functions.
Solution.
Because we are analyzing a gas-well buildup test, the first step
in our analysis procedure is to prepare a log-log plot of
[ Pa, ws -Pa ,w! (.:1t=O)] vs . .:1tae (Fig. 6.32) . Table 6 . 12 gives the
plotting functions .
From the log-log plot, we observe a unit-slope at very early
times . With the point .:1Pa
psia and .:1tae
hours
on the unit-slope line , we estimate CD '
6. 4 ,
1.
6,107, .
/ig
2
O,
0
372q
t .:1tae )
.
(
=0
=21.
2
39
0 15449
.
CD =
"
_
"
"
"
"
,
,
,
(6.107)
CP/ig ctra .:1Pa
/ig ( t .:1tae ) . . . . . . . . . . . . . . . . . . . . . . .
q
(6.108)
C=
24 .:1Pa
(0 . 0372)(100)(0 . 497) ( 0. 0 15449 )
CD
(0.10)(21 )(0. 0000355)(0 . 365) 2 21.239
6.
3.
=135.
CDe 2s
3.
CDe 2s = 100 .
4.(.:1Pa ,PD) (t,tDICD)
.:1p a = 385
P D = 1,
(.:1tae ,tDICD)
.:1tae =2. 7
tDICD = 10.
4. B
k = 141. 2qg g/ig ( .:1PPaD )
5.
k = 141. 2qgBg /ig ( .:1PPaD ) ' " . " " " " " " " " . (6 . 109) (141. 2)(100)(0 . 497)(0 . 034) (_1_ )
21
385
( Pa ,ws -Pa , wj)
.:1Pa = ( Pa, j -Pa,wj)
=0. 03
or
,
USL
If desired , C can be calculated from
or
USL
If a unit-slope line is not present,
can be calculated from the
time match point as described in Step
Find the type curve that most nearly fits all plotted test data.
Record the value of the parameter
corresponding to the
matched type curve .
With the test data plot fitted to the type curve, record values
of
and
for a drawdown test or
for a buildup test a t the match point . The match-point values are
corresponding values of the pressure and time variables on the test
data plot and the type curve.
Calculate permeability from the pressure match point.
h
where
for a buildup test .
The best fi t o f all the data is found with the curve f()r
With the test data in the matched position , we select
a pressure match point of
psia,
and a time match
point of
hours ,
Permeability is estimated from the pressure match point .
h
MP
for a drawdown test or
md .
MP
1 36
GAS RESERVO I R E N G I N E E R I N G
Equivalent Time, Lite (hours)
Fig. 6.32-Type-curve match for Example 6 . 5 .
5 . Using the time match point, we confirm our estimate of CD
from the unit-slope line .
0 . 0002637k
cPjig Ctr�
(
t:..tae
tDI CD
)
MP
0 . 0002637(0 . 03)
(0 . 1 0) (0 . 034)(0 . 0000355)(0. 365) 2
= 133.
( )
2.7
10
This value agrees well with CD = 1 35 from Step 2 .
6 . The skin factor i s estimated from CDe2s = 1 00 chosen from
the type-curve match and CD = 1 33 from Step 5 .
s =0.5
In ( C�:2S)
= 0 . 5 In( 1 00/ 1 33)
= -0. 14.
pressure/time plots , the pressure derivative is used widely , specif­
ically to identify reservoir types . In this section , we introduce the
basic concepts of well-test analysis using pressure derivatives and
illustrate how pressure derivatives can aid in reservoir identification.
Bourdet et al. 28 Derivative Type Curve. Although a number of
pressure-derivative type curves have been presented, we have select­
ed the derivative type curve developed by Bourdet et at. 28 for il­
lustration purposes. Their type curve is based on the analytical
solution for a slightly compressible liquid derived by Agarwal et
at. 29 and plotted on the Gringarten el at. type curve . The Bourdet
et at. type curve also includes a pressure-derivative function. The
variable Pv (tDI CD ) is plotted on the derivative type curve as a
function of IDI CD for various values of CDe2s , where
dP D
--- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 1 12)
d(tDICD )
The derivative type curve also has the following useful properties.
1 . For test data on the unit-slope line, P D = tD I CD . Therefore,
dPD
---
d(tD I CD )
One inherent problem i n type-curve analysis is the difficulty in
finding a unique match of the data. Because of the similar shapes
of the type curves for a wide range of the type-curve correlating
parameters , the data can be matched on more than one curve; con­
sequently , more than one solution is possible. In contrast, if a semi­
log straight line indicative of the middle-time or radial flow region
is present and can be identified correctly , semilog analysis can pro­
vide a correct estimate of formation permeability . These observa­
tions suggest that type curves should not be the primary analysis
technique but should be used to complement, rather than replace,
semilog analysis . We illustrate these concepts in a later section .
6 . 9 . 4 Type-Curve Analysis With Pressure Derivatives­
Homogeneous-Reservoir Model, Slightly Compressible Liquids.
As discussed , type curves must accurately model the reservoir or
well conditions being analyzed . Therefore , the key to using type
curves is identifying the correct reservoir type (homogeneous , dual­
porosity , hydraulically fractured , etc . ) so that we may select the
appropriate type curves . Unfortunately , conventional log-log pres­
sure/time plots are relatively insensitive to the changes in pressure
data characteristic of various reservoir types . As a supplement to
,
= PD = 1
and pv(tD I CD ) = tD I CD . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 1 1 3)
Then, log [ Pv(tDICD)] = 10g(IDICD ) , and thus the slope of a plot
of [ P v (tDI CD)) vs. IDI CD on a log-log graph is unity . Therefore,
at early times, for the same times during which a unit-slope line
appears on a plot of P D v s . IDI CD , a unit-slope line also will ap­
pear on a plot of [ P v (tDICD)] vs. IDI CD .
2 . For test data on the semilog straight line, P D can be modeled
with the logarithmic approximation to the line-source solution:
P D = 0 . 5(1n tD + 0 . 80907 + 2s) . . . . . . . . . . . . . . . . . . . . . (6. 1 1 4)
Adding and subtracting In CD inside the parentheses in Eq.
6 . 1 1 4 gives
P D = 0 . 5 [ln tD - ln CD + 0 . 80907 + 1n CD + ln(e2s )]
= 0 . 5 [ln(tD I CD ) + 0 . 80907 + In(CDe2 s )] .
Thus,
dPD
d(IDI CD )
,
= PD =
0.5
(tDI CD )
o r Pv(tD I CD ) = 0 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 1 5)
1 37
PRESS U R E-TRANSIENT TESTI N G OF GAS WELLS
10'
.�
10'
� 10'
Cl
·c
�
�
�
�
10"
10"
10°
1 0"
Dimensionless nme. lo I Co
10'
10'
10'
6.9.5 Gas-WeU-Test Analysis With Derivative Type Curves. The
following procedure, adapted from slightly-compressible-liquid anal­
ysis techniques using derivative type curves , can be applied to gas­
well-test data analysis . We present the procedure in terms of ad­
justed pressure and time variables; however, it also can be used
with either pressure or pressure-squared variables (with possible
loss of accuracy) .
1 . From test data, calculate the following derivatives . For a draw­
down test ,
-:;.:.:� = { :;;)wf ] =t£\p� , . . . . . . . . .
and for a buildup test,
]
...
. . . . . (6. 1 1 6a)
dPa , ws
dPa. ws
=£\tae
=£\tae£\P � ' . . . . . . . . . . (6. 1 l6b)
£\tae )
d (£\tae )
d(ln
[
10°
1 0'
Data that lie on a straight line on a semilog graph will lie on a
horizontal line at P v (tDI CD ) =0.5 on the derivative type curve.
Fig. 6.33 illustrates derivative curves with unit-slope lines at early
times , a horizontal line at late times, and more complex shapes at
intermediate times .
3 . If a test data plot has both a unit-slope line on a logarithmic
type-curve plot and a straight line on a semilog plot, then a match
of the early and late test data is clearly and uniquely defined on
the derivative type curve . A unique value of CDe2s can then be
determined, unlike Gringarten et at. ' s type curve of P D vs. tDI CD ,
which has uniqueness problems for large values of CDe2s .
4. The type curves of PD and P v vs. tDI CD can be plotted on
a single graph, thus permitting simultaneous and less ambiguous
type-curve analysis . Fig. 6.34 shows these curves .
d
10"
10"
1 0'
Fig. 6 .33-Pressure-derlvatlve type curve.
_
10°
We present a recommended pressure-derivative calculation tech­
nique in Appendix F .
2. Plot t£\p � (or £\tae £\P�) and £\ Pa as functions of t (or £\tae )
on either tracing paper or log-log graph paper the same size as the
type-curve graph .
3 . If possible , force a match of the data to the type curve in the
vertical direction by aligning the flat region of the test data with
the P v (tD I CD ) =0.5 line on the type curve . The flat region is the
middle-time or radial flow region . If this flat region is not present,
semilog analysis is not possible .
4. If possible, force a match in the horizontal direction by align­
ing the unit-slope regions of the test data derivative plot and the
derivative type curve .
5 . Determine CDe 2s from the matching parameter of the deriva­
tive type-curve match . This same matching parameter character­
izes the fit on the pressure-change type curve ( P D vs. tDI CD ) .
6. With the test data plot fitted t o the type curve , record values
of (£\ Pa 'P D ) and (t,tD I CD ) for a drawdown test or (£\ ta e,tDICD )
for a buildup test at the match point . The match-point values are
DimaIsionle•• 1ime, 10 I Co
10'
10'
10'
1 0'
1 0'
Fig. 6 .34-Pressure-change and pressure-derivative type
curves.
corresponding values of the pressure and time variables on the test
data plot and the type curve.
7. Calculate permeability from the pressure match point.
k=
141 .2qgBgjig
h
( £\Pa )
PD
,
MP
. . . . . . . . . . . . . . . . . . . . (6. 109)
£\Pa = ( Pa. i - Pa. wj ) for a drawdown test or
£\Pa = ( Pa. ws -Pa.wf ) for a buildup test .
8. Calculate or confirm CD from the time match point .
0.0002637k t or £\ tae
. . . . . . . . . . . . . . . . . (6. 1 10)
CD tDI CD
cf>jig Ctr�
9. Calculate s using CDe 2s and CD from Step 8.
where
S
=0 . 5 In
)
(
_
MP
( C�:2S ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 1 1 )
Example 6.6-Gas-Well Buildup Test Analysis With Pressure­
Derivative Type Curves. With the gas-well buildup test data
presented in Example 6.4, estimate k and s using the Bourdet et
at. derivative type curves. Known and calculated data are summar­
ized in Table 6 . 1 3 . Use adjusted pressure and time variables as
the plotting functions .
Solution.
1 . Plot tlPa and tltaetlp � v s . tltae on log-log paper (Fig. 6.35) .
2 . Using the Bourdet et al. type curve , try to match both tl Pa
and tltaetl p � . Note that the early-time data form a unit-slope line,
indicating wellbore-storage effects. Similarly , several adjusted
pressure-derivative data points at the end of the test begin to flat­
ten and appear to form a middle-time region . As Fig . 6.35 shows,
we could match both adjusted pressure change and pressure deriva­
tive with the type curves characterized by CDe 2 s = 100. In addi­
tion, we can match several of the last few pressure-derivative points
on the type curve for P v (tDI CD) =0.5, indicating the presence of
the middle-time region .
3. From the match in Fig . 6.35, we obtain a pressure match point
of £\ Pa = 3 80 psi , PD = I , and a time match point of £\tae = 2 . 7
hours , tD ICD = 10.
4. Determine k using the pressure match point .
k=
14 1 .2qgBgjig
h
( £\Pa )
PD
MP
( )
(141 .2 ) (100) (0.497) (0.03403 ) 1
__
21
380
=0.03 md .
=
1 38
GAS RESERVO I R E N G I N E E R I N G
TABLE 6 . 1 3-PRESSURE-BUILDUP-TEST DATA, EXAMPLE 6 . 6
Adjusted
Pressure
Change
(psi)
9 . 7476
1 4.365
2 1 .239
31 .397
46. 1 72
67.71 9
98.906
1 43.63
206.70
293.84
4 1 0.95
561 .33
742.73
943.90
1 1 46.0
Adjusted
Equ ivalent
Time
(hou rs)
0 .0069839
0 . 0 1 041 0
0 . 0 1 5449
0 . 0230 1 8
0 .034255
0.051 088
0 .075987
0 . 1 1 370
0 . 1 704 1
0 .25465
0 .38302
0 .57734
0 .87244
1 . 3253
2 . 0 1 99
5.
)
(
Adj usted
Pressure
Derivative
1 1 .350
1 4. 233
2 1 .021
30.741
44.657
65.028
92.859
1 30.88
1 83.02
246.92
320 .44
395 .20
451 .41
471 .23
448 .37
CD from the time match point .
0.OO02637k Atae
CD =
cpjig ctra tDICD MP
Calculate
( )
2.7
0.0002637(0.03)
(0. 10)(0.034)(0.0000355)(0.365) 2 10
= 133.
=
5
6. Now, estimate the skin factor using the value of CD from Step
and CDe 2s = 100.
s =0.5 In
( C;:2S )
=0.5 I n(1001133)
= -0. 14.
Note that simultaneously matching both the adjusted pressure­
change and the adjusted pressure-derivative curves eliminates much
Match Point:
dPa = 385
Lit«
1 04
.�
=
CD e2s = 1 02
psi, PD
=
1
2.7 hr, tD I CD = 10
Adj usted
Equ ivalent
Time
(hours)
3.0844
4. 6937
7. 1 204
1 0. 790
1 6. 287
24.477
36. 668
54.622
8 1 . 047
1 1 8.97
1 73.63
250 . 09
352.87
485 . 9 1
649.47
Adj usted
Pressure
Change
(psi)
1 ,329 . 9
1 ,485 . 7
1 ,6 1 4.2
1 , 722. 5
1 ,81 8.2
1 ,906.2
1 ,989.4
2,069.3
2, 1 46.6
2,221 . 2
2,293.0
2,361 .5
2,425.7
2,484. 7
2,537.2
Adj usted
Pressure
Derivative
394.88
333.00
278 .95
241 .63
220.83
205.85
1 97.60
1 93.95
1 91 .64
1 88.41
1 85 . 1 2
1 83.29
1 8 1 .69
1 79 . 08
1 79 . 24
of the ambiguity associated with conventional type-curve match­
ing with only the pressure-change data. As an additional verifica­
tion of our analysis , we could precalculate a pressure match point
using results from a semilog analysis. Further, we should not rely
only on one well-test analysis technique but should use the charac­
teristics of each method . From these observations , we suggest the
following general analysis procedure, which integrates conventional
semilog analyses with type curves . In addition , we have provided
a worksheet (Appendix G) , which summarizes the procedure .
1 . Perform a preliminary or qualitative analysis of the data using
both the pressure-change and pressure-derivative data. The objec­
tive of this preliminary analysis is to identify the time regions (early-,
middle- , and late-time regions) .
2. Next, try a semilog analysis if the middle-time or radial flow
region is present and is identified correctly from the preliminary
type-curve analysis.
3. Finally , perform a quantitative type-curve analysis using both
pressure and pressure-derivative data. Precalculate a pressure match
point using k from the semilog analysis . If a reasonable match is
obtained , we have confidence in our analysis . If we cannot obtain
a reasonable match with the precalculated match point , we should
iterate between the semilog and type-curve analyses until we ob­
tain consistent results .
I--------�
ca
;.
. t:
0 103
�
J:
'"
'"
"0 102
Iii
0)
t>I)
c
ca
0
e lOI
::I
e
'"
'"
p...
1 0°
1 0- 3
Equivalent Time, dte (hours)
Fig. 6.35-Type-curve match using Bourdet et a/. 28 pressure type curve, Example 6 . 6 .
1 39
PRESS U R E-TRANSIENT TESTI NG OF GAS WELLS
Wellbore
(a) Fracture Linear Flo\\
Fracture
(b) Bilinear Flow
Fracture
a
a
-
Wellbore
-b
(d) Elliptical Flow
(c) Fonnation Linear Flow
/
/
f
(e) Pseudoradial Flow
Fig. 6.36-Flow periods in a vertically fractured wel l (after Cinco-Ley and Samaniego-V. 30 ) .
6. 1 0 Hydraulically Fractured G a s Wells
Many gas wells , particularly those in low-permeability formations ,
require hydraulic fracturing to be economically viable producers .
Interpretation of pressure-transient data in hydraulically fractured
wells is important for evaluating the success of fracture treatments
and predicting the performance of fractured wells. This section
describes graphical techniques , including semilog , log-log , and
Cartesian coordinate , for analyzing postfracture pressure-transient
tests . Before beginning our discussion of these analysis techniques ,
we identify several flow patterns characteristic of hydraulically frac­
tured wells . Often, identification of specific flow patterns can aid
in well-test analysis .
6.10.1 Flow Patterns in Hydraulically Fractured Wells. Five dis­
tinct flow patterns (Fig. 6.36) occur in the fracture and formation
around a hydraulically fractured well. 30-32 Successive flow pat­
terns, often separated by transition periods, include fracture-linear,
bilinear , formation-linear , elliptical , and pseudoradial flow .
Fracture-linear flow (Fig . 6 . 36a) is very short-lived and may be
masked by wellbore-storage effects . During this flow period , most
of the fluid entering the wellbore comes from fluid expansion in
the fracture , and the flow pattern is essentially linear . Because of
its extremely short duration, the fracture-linear flow period often
is of no practical use in well-test analysis . The duration of the frac­
ture linear flow period is estimated by 30-32
tL
jD ==
where
tLjD
0. 1 CJ
--- '
'rIf2D
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 1 7)
tLjD is dimensionless time in terms of fracture half-length,
==
0 . 0002637kt
rjJp.ct LJ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 1 8)
The dimensionless fracture conductivity is
Cr = wkfl7rkLf ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 1 1 9)
.
.
and the dimensionless hydraulic diffusivity is
'rIfD =
kf rjJct
'
krjJf Cft
--
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 20)
.
.
.
Bilinear flow (Fig . 6 . 36b) evolves in finite-conductivity fractures
as fluid in the surrounding formation flows linearly into the frac­
ture and before fracture-tip effects begin to influence well behavior.
Fractures are considered to have finite conductivity when Cr < 1 00 .
GAS RESERVO I R E N G I N E E R I N G
1 40
I O'-:r:::=""
'''
...... I..U.W'--'''''.u.=J
lL
---L-.L-U.J.I.ILL-....L.J...u..u.",..I
1 0-2 �...LJ..JJ.l
1 0-5
1 0-4
1 0 -2
1 0-1
1 0-3
Dimensionless Time
Fig. 6 .37-Bili near flow region for a finite-conductivity verti­
cal fracture [after Economides, M . J . : " Post-Treatment Eval­
uation and Fractu red We l l Performance , " Reservoir
Stimulation, M . J . Econom ides and K.G. Nolte (eds.), Schlum­
berger Educational Services, Houston (1 987) 1 1 -1 -1 1 -1 7. 31
Cou rtesy of Schlumberger] .
Most of the fluid entering the wellbore during this flow period comes
from the formation. During the bilinear flow period , P wf is a linear
function of t v. on Cartesian coordinate paper. A log-lot plot of
( Pi -Pwf) as a function of time exhibits a slope of one-quarter; the
pressure derivative also has a slope of one-quarter during this time
period . The bilinear flow period may last for a significant period ,
as Fig. 6.37 shows .
The duration of bilinear flow depends on dimensionless fracture
conductivity and is given by Eqs . 6 . 1 2 1 a through 6 . 1 2 1 c for a range
of dimensionless times and fracture conductivities :
tL D =
f
C2'
0.01
,
C, > 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 2 1 a)
tL D = 0 . 0205 (C, - 1 . 5) - 1 . 5 3 , 1 . 6 s C, s 3 , . . . . . . . . (6. 1 2 1 b)
f
-4
and tL D =
2.5
, C, < 1 . 6 . . . . . . . . . . . . . . (6. l 2 I c)
_
f
[ :;, ]
Formation-linear flow (Fig . 6 . 36c) occurs only in high­
conductivity (C, 2! 100) fractures . This period continues to tLiD =
0 . 0 1 6 . The transition from fracture-linear to formation-linear flow
is complete by tL D = 1 x 10 - 4 . On Cartesian coordinate paper,
f
P wf is a linear function of t 'l2 , and log-log plots of both ( Pi -P wj )
and the pressure derivative as a function of time exhibit a slope
of one-half.
Elliptical flow (Fig . 6 . 36d) , a transitional flow period , occurs
between a linear or nearly linear flow pattern at early times and
a radial or nearly radial flow pattern at late times .
Pseudoradial flow (Fig . 6 . 3 6e) occurs with fractures of all con­
ductivities . After a sufficiently long flow period , the fracture ap­
pears to the reservoir as an expanded wellbore (consistent with the
effective wellbore radius concept Prats et al. 6 suggested) . At this
time , the drainage pattern can be considered a circle for practical
purposes . The larger the fracture conductivity is , the later the de­
velopment of an essentially radial drainage pattern is. If the frac­
ture length is large relative to the drainage area , then boundary
effects distort or entirely mask the pseudoradial flow regime . Pseu­
doradial flow begins at tLfD = 3 for high-conductivity fractures
(C, 2! 1 00) and at slightly smaller values of tLiD for lower values
of C, . Fig. 6.38 shows the beginning of pseudo radial flow, indi­
cated by the start of a straight line on a graph of Pwj vs. log t, for
several values of C, . The pressure derivative flattens when pseu­
doradial flow begins . This characteristic often is used to identify
pseudo radial flow .
Dimensionless Time, 'IP
Fig. 6 .38-Di mensionless pressure/time plot showing start
of pseudo radial flow period for finite-conductivity vertical frac­
tures (after Ci nco-Ley and Samaniego-V. 30 ) .
These flow patterns also appear i n pressure-buildup tests and occur
at about the same dimensionless times as in flow tests . The physical
interpretation is that the pressure has built up to an essentially uni­
form value throughout a particular region at a given time during a
buildup test . For example , at a given time during bilinear or forma­
tion-linear flow, pressure has built up to a uniform level through­
out an approximately rectangular region around the fracture . At a
later time during elliptical flow , pressure has built up to a uniform
level throughout an approximately elliptical region centered at the
wellbore . At a given time during pseudoradial flow, pressure has
built up to a uniform level throughout an approximately circular
region centered at the wellbore . The area of the region and the pres­
sure level within that area increase with increasing shut-in time .
6. 10.2 Specialized Methods for Postfracture Well Test Analy­
sis. In general , the objectives of postfracture pressure-transient test
analysis are to assess the success of the fracture treatment and to
estimate fracture length , fracture conductivity , and formation per­
meability . In this section, we discuss three specialized methods for
analyzing postfracture pressure-transient tests-pseudo radial flow
analysis , bilinear flow analysis, and linear flow analysis.
Pseudoradial Flow Method. The pseudoradial flow method 32
applies when a short, highly conductive fracture is created in a high­
permeability formation and pseudoradial flow develops in a short
time . The time required to achieve pseudoradial flow for an infinitely
conductive fracture (C, > 1 00) in either a flow test or a pressure­
buildup test is defined by tL D = 3 , or
f
tL D =
f
0 . 0002637kt
ifJp.c t LJ
=3.
The beginning of pseudoradial flow is characterized by the start
of a semilog straight line . Hence , when the pseudoradial flow re­
gime is reached , conventional semilog analysis can be used to cal­
culate k and s. As discussed in Sec . 6 . 4 . 4 , s is related to fracture
half-length , Lj , for a highly conductive fracture by 6
Lj = 2r w e - s .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 1 22)
Table 6 . 1 summarizes working equations for pseudoradial flow
analysis for gas wells. The procedure for analyzing test data from
the pseudoradial flow regime is presented in terms of adjusted pres­
sure and time plotting functions ; however, the procedure can be
modified to incorporate other suitable plotting functions .
1 . For a drawdown test , make a semilog plot of P a , wj vs. log
t. For a buildup test, make a semilog plot of P a ' ws vs. adjusted
Horner time ratio .
2. Determine the position and slope m of the sernilog straight line.
3 . Using m, calculate values of k and s (or s ' ) with the appropri­
ate equations in Table 6 . 1 .
4 . Calculate Lj using Eq. 6 . 1 2 2 .
The pseudoradial flow method has limitations that make i t sel­
dom applicable in practice. 32 First, a short, highly conductive frac-
141
PRESS U R E-TRAN SIENT TESTI N G OF GAS WELLS
TABLE 6 . 1 4-S U MMARY OF WORKING EQUATIONS, LINEAR FLOW ANALYSIS I N HYDRAULICALLY FRACTURED WELLS
Pressu re Squared and Time
Pressure and Time
Cartesian-coord i n ate
graph variables
.jk L , from slope, m L
o f straight l i n e
Cartesian-coord i n ate
graph variables
.jk L, from slope, m L
o f straight l i n e
Flow Test
Ji
P wi vs.
P � vs.
.jkk L , = 4.064q g B g
mLh
P ws vs .
Jllte
l1te = l1t/(l + l1tit p )'
ji g
---
fj>ct
.jkk L , = 40.93q g TZ
'h
mLh
B u i l d u p Test
P �s vs .
.jkk L , = 4.064q g B g
mLh
( )
( ) '/2
fj>ct
mLh
ture in a high-permeability formation, generally required to achieve
pseudoradial flow, rarely occurs . The most common application
of hydraulic fractures (wells with long fractures in low-permeability
formations) requires impractically long test times to reach pseu­
doradial flow . Second , an incompletely perforated interval causes
distortion of test analysis results . Third , for gas wells , s' calculat­
ed from test data is distorted by non-Darcy flow . Fourth, the method
applies only to highly conductive (Cr �
fractures . For lower­
conductivity fractures , fracture lengths calculated with s (Eq. 6 . 122)
will be too low . Fig_ 6.39 relates effective wellbore radius, r wa '
to Cr and thus to
This chart can be useful in estimating
if
both s and Cr can be estimated , because r wa = rw e - s . Then , r wa
can be substituted for r w e - s in Eq. 6 . 1 2 2 .
Bilinear Flow Method. The bilinear flow method 32 applies to
test data obtained during the bilinear flow regime in wells with finite­
conductivity vertical fractures . For a constant-rate flow test from
a gas well , bilinear flow is indicated by a quarter-slope line on a
log-log graph of ( P a . i -P a . wj ) vs. t or, for a buildup test ,
( P a , ws -P a , wj ) vs. !J.tae ' A recommended procedure for analyzing
gas-well-test data obtained in the bilinear flow regime follows .
For a constant-rate flow test, plot P a , wj vs. t IA on Cartesian
coordinate paper. For a buildup test , plot Pa .ws v s . !J.ta� '
2 . Determine the slope of the linear region of the plot, mB '
3 . From independent knowledge of (e . g . , from a prefracture
well test) , estimate the fracture conductivity ,
using mB and
the relationship
100)
Lf
1.
k
0. 1
IrC,
-
1 0
wkl L,k
wf kf,
1 00
Jflte
rk L , = 40.93q g TZ
ji g
---
Lf .
Ji
P •. wI vs.
( )'/2
fj>ct
( )
ji g
-
fj>ct
Ji
.jkk L , = 4.064q g B g
ji g
-
mLh
P •. ws vs .
Jilt .e
.jkk L , = 4. 064q g B g
'h
mLh
0. 5
B
(
1
g
)
44.1Qg
g
)
(
iL
_
-wf kf4>iLgl:tk .
2
hmB
( ) '/2
ji g
---
fj>ct
( ) '/2
ji g
---
fj>ct
. . 1
. . . . . . . . . (6. 23 )
The bilinear flow analysis method has important limitations. First,
the method does not provide an estimate of
Second , in wells
with low-conductivity fractures , wellbore storage frequently dis­
torts early test data for a sufficient time that the unit-slope line char­
acteristic of wellbore storage completely masks the quarter-slope
line of bilinear flow on a log-log plot of test data. Finally , the
greatest limitation is that an estimate of is required .
Linear Flow Method. The linear flow method 32 applies to test
data obtained during formation linear flow in wells with high­
conductivity , vertical fractures . Formation linear flow occurs up
to tLtD
for production into a highly conductive (Cr �
vertical fracture and after wellbore-storage effects have ceased to
distort the pressure data. During formation linear flow , the dimen­
sionless pressure response is modeled by
Lf .
k
=0. 0 16
100)
PD = ( 7rtLtD) 'h .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 2 )
14
According to Economides, 3 1 for a buildup test Agarwal ' s 26
equivalent time function, !J.te =!J.tl(l + !J.tltp ) , is used in place of
t in the definition of dimensionless time in Eq . 6 . 1 1 8 . However,
this definition of
ilte
applies rigorously only for radial flow in an
infinite-acting reservoir . For purely linear flow , the definition of
equivalent time is .,j!J.tenL =;;t; +$1 -.,jtp +!J.t .
Table 6 . 14 gives working equations for linear flow analysis of
gas-well-test data for the following cases: pressure and time,
1 000
Fig. 6 . 39-Effective well bore radius vs. d imensionless
fracture conductivity for a vertical fracture (after Ci nCO-Ley
and Samaniego-V . 30,33 ) .
Adj usted Pressure and Time
10
1
°
10
10
Dimensionless Time, IflO
'
10
10
'
'
10
Fig. 6.40-Gringarten et al. 34 type cu rve for an infinite­
conductivity, vertically fractured well in the center of a closed
square .
1 42
GAS RESERVO I R ENG I N E E R I N G
TABLE 6 . 1 4-SU M MARY OF WORKING EQUATIONS,
LINEAR FLOW ANALYSIS I N HYDRAULICALLY
FRACTURED WELLS (continued)
Pseudopressure a n d Time
Flow Test
Cartesian-coord inate
graph variables
Jk L f from slope,
o f straight l i n e
P p vs.
Ji
mL
B u i l d u p Test
Jk L f from slope,
o f straight l i n e
'
10
mL
pressure-squared and time , adjusted pressure and time , and pseu­
dopressure and time as plotting functions .
The following procedure is recommended for analyzing gas-well­
test data in the linear flow regime, using adjusted pressure and time
variables.
1 . For a constant-rate flow test, plot Pa. Mf v s . t 'h on Cartesian
coordinate paper. For a buildup test , plot Pa. ws vs . dt�1 .
2 . Determine the slope of the linear region of the plot, mL '
3 . From independent knowledge of k, estimate LJ , using the ap­
propriate equation for .../k LJ from Table 6. 14.
The linear flow analysis method also has limitations . First, the
method applies only for fractures with high conductivities . Strictly
speaking , linear flow occurs for uniform flux into a fracture (same
flow rate from the formation per unit cross-sectional area of the
fracture at all points along the fracture) rather than for infinite frac­
ture conductivity . Therefore, only very early test data (tLtD :::;;
0.016) exhibit linear flow data available for analysis. Third , the
estimation of LJ requires an independent estimate of k.
6. 10.3 Postfracture Well-Test Analysis Using Type Curves. Be­
cause they span the entire range of flow regimes and include the
intervening transition region s , type-curve methods 3 1 • 32 are more
general than the specialized pseudoradial , bilinear, or linear flow
analysis methods . We illustrate type-curve analysis methodologies
with the following type curves : (1) the Gringarten et al. , 34 (2) the
Cinco-Ley et al. , 35 (3) the Agarwal et al. , 36 and (4) the Barker­
Ramey 37 type curves . Table 6. 1 1 summarizes the interpretation
of match points and parameters for type-curve plots. The follow­
ing sections and example problems also offer explanation.
Gringarten et al. 34 Type Curve. The Gringarten et al. type
curve (Fig. 6.40) , is useful for postfracture analysis of data from
a constant-rate flow test or a pressure-buildup test . The type curve
is a graph of solutions to flow equations modeling a vertical hy­
draulic fracture in a finite reservoir under the following assump­
tions : (1) the fracture is infinitely conductive; (2) the well is centered
in a square drainage area with no-flow boundaries; (3) the fracture
has two equal-length wings; and (4) wellbore-storage effects are
negligible .
We suggest the following procedure for analyzing test data with
the Gringarten et al. type curve. Although presented in terms of
adjusted variables, the procedure also is applicable with pressure
or pressure-squared variables. In addition , the worksheet in Ap­
pendix G summarizes the procedure .
1 . Plot the adjusted pressure change since the beginning of the
test, d Pa , vs. t for a constant-rate flow test or dta e for a buildup
test on tracing paper or log-log paper with the same grid size as
the type curve .
2 . Perform a qualitative type-curve analysis to obtain an initial
match and to identify any flow regimes characteristic of infinite­
conductivity vertical fractures .
'
10
10
Cartesian-coordinate
graph variables
'
Time, he
10
10
'
10
'
10
'
Fig. 6 . 4 1 -Type-curve match using the Grlngarten et a/. 34
model , Example 6 . 7 .
A . If the type-curve match indicates early data points with
tLtD < 0.016 and a slope equal to one-half, then the linear flow pat­
tern may be present, and we can analyze the data using the tech­
nique presented previously .
B . If the type-curve match indicates several data points with
tLtD > 3, then pseudoradial flow has occurred, and we can analyze
the data using the technique presented previously .
C . If boundary effects occur during the test, then the test data
will deviate upward from the infinite-acting curve defined by
LelLJ = 00 and will match one of the curves for a finite reservoir
characterized by the parameter LeILJ < 00 .
3 . I f any flow regimes characteristic o f infinite-conductivity ver­
tical fractures are identified , perform a postfracture analysis using
the specialized techniques discussed previously .
A . If the preliminary type-curve match indicates data points for
tLtD < 0.016, then plot Pa. Mf vs . t 'h on Cartesian coordinate paper
for a constant-rate flow test or Pa. ws vs. dta� for a buildup test.
A straight line should appear with a slope inversely proportional
to .../k LJ . If permeability can be estimated from a pseudoradial
flow analysis or if a prefracture value is available , then we can es­
timate LJ .
B . If the type-curve match indicates several data points with
tLtD > 3 , then plot Pa. Mf vs. log t for a constant-rate flow test or
Pa. ws vs. log (tp +dta)/dta for a buildup test. If a semilog straight
line appears, then a unique estimate of k can be made.
C. If boundary effects occur during the test and if we have an
estimate of the drainage area shape and relative location of the well
in this area, we can estimate drainage area using the parameter
LelLJ and the estimate of LJ from the linear flow analysis .
4. Next, perform a quantitative type-curve analysis . The purpose
is either to confirm the results from the specialized analysis tech­
niques or to obtain estimates when these analysis techniques are
not possible .
A . Using the k estimate from the pseudoradial flow analysis or
the prefracture k estimate, precalculate a pressure match point. Use
a convenient and arbitrary value of P D to precalculate d Pa '
(dPa ) MP =
141 .2qgiig iig
( P D) MP '
kh
. . ..... ...
. . .
..
.
(6. 125)
Compare the type-curve match with the preliminary match ob­
tained in Step 2. If the matches are greatly different, then repeat
Steps 2 and 3 until a consistent match is obtained .
B . If a k value is not available , then choose a pressure match
point ( P D ,dPa ) from the qUalitative type-curve match and estimate
k:
k=
( ) ..
141 .2qg Bg iig P D
.
h
dPa MP
. ..
. ... . . .
.
. ... . ..
(6. 126)
1 43
PRESS U R E·TRAN SIENT TESTI N G OF GAS WELLS
41000
.
TABLE 6 . 1 5-CONSTANT·RATE DRAWDOWN DATA,
EXAMPLE 6 . 7
sooo
!
is. 4000
If
j
I
3000
�
:zooo
1000
Fig. 6.42-Pseudorad ial flow analysis, Example 6 . 7 .
. . . ..
----. . - . .
4OOO �===_=
· ··=
_r=_..===· ·=
··.. 4======+====�
�--·· _· _ .
.
3900 - ==- -
0.0
0.5
.
1 .0
Sq..... ROCK of TIme, hrl/Z
.
1 .5
.
2.0
Fig . 6 . 43-Linear flow analysis, Example 6 . 7 .
[ 0.0002637k ( t
)
C . Choose a time match point
Lf =
t:.tae
tLtD
or
(t or t:.tae .tLtD) and estimate Lf :
]
�.
. . . . . . . . . . . . . (6. 127)
ef>p.g ct
MP
The value of Lf computed with the time match point should be
_
_
consistent with the value obtained from the linear flow analysis.
If the values do not compare , repeat Steps 2 and 3.
Example 6 .7-Analysis o f a Postfracture Constant-Rate Flow
Test With Boundary Effects. A constant-rate drawdown test was
run in a gas well following a fracture treatment. Table 6 . 1 5 sum­
marizes pressure and time data; other known data are given be­
low. Assume that wellbore·storage effects are negligible. Determine
k, s, Lf , and wfkf using the Gringarten et al. type curve, and con­
firm the analysis with specialized analyses .
qg =
rw =
ct =
/i =
Pi =P =
h =
ef> =
Bg =
Pa =
Pa , l hr =
3,000 MscflD.
0.25 ft .
2.084 x lO -4 psia - 1 .
0.01961 cp o
5,000 psia.
60 ft.
0. 10.
0.7085 RB/Mscf.
3 ,974 psia.
5 ,239 psia.
Solution. Preliminary Type-Curve Analysis.
1 . Plot t:.P a vs. t on log-log paper with the same grid size as the
type curve (Fig. 6.41) .
2. From a preliminary type-curve analysis using the curve for
Le /Lf "" 10, we observe the following : formation linear flow , which
Adjusted
Time
Pressure Pressu re
(hours)
(psia)
(psi a)
0.001 4 1 0 4,993 . 1 3 , 966.6
0 . 0028 1 3 4 , 990. 2 3,963.6
0 . 0056 1 3 4 , 986.2 3,959.5
0 . 0 1 1 1 99 4 , 980.5 3 , 953.6
0.028 1 3 1 4,969 . 1 3 , 941 .8
0. 070662 4,95 1 . 0 3 , 923.2
0 . 1 7750
4,922.4 3,893.7
4 , 877.0 3,846.7
0.44586
4,806. 2 3 , 773.5
1 . 1 200
4,758. 3 3,723.8
1 . 7751
4,668. 5 3,630.6
3 . 5421
5 . 6 1 44
4 , 594. 6 3,553.8
8 .8995
4 , 507.3 3,462. 8
4,458.4 3,41 1 .8
1 1 .205
4 , 347.7 3,296. 3
1 7. 764
4 , 284. 9 3,230. 7
22.368
4 , 1 42 . 8 3,082 . 1
35.473
3 , 977. 9 2,909. 8
56.275
3 , 792.9 2,71 6.4
89.236
1 41 . 9 1
3 , 588.9 2,503.3
3,369. 2 2,274.9
225. 78
360.02
3 , 1 35.3 2,033.9
3 , 0 1 6.8 1 ,9 1 3.0
455.26
2 , 892.6 1 , 787.5
576.26
730.43
2 , 766. 7 1 ,661 .8
2 , 637. 6 1 ,534.7
927.60
2 , 502.5 1 ,404 . 1
1 , 1 80.7
2 , 355.7 1 ,265 . 5
1 ,507.4
2 , 1 88.4 1 , 1 1 2. 2
1 ,932.0
1 ,984.4
933. 36
2,488.9
71 8.29
3,227.3
1 , 720. 3
4,22 1 . 8
1 ,347. 9
453. 1 4
683. 8
1 1 9.69
5 , 588.5
Adj usted Square Root
Time
Pressure
Function
Change
(sq rt hour)
(psia)
0 .037550
7. 1 1 1 3
0.053038
1 0. 1 02
0 .074920
1 4.226
0 . 1 0583
20. 1 04
0 . 1 6772
3 1 . 859
0 .26582
50.522
0.42 1 3 1
80.050
0.66773
1 26 . 96
1 .0583
200 . 23
1 .3323
249. 88
1 .8820
343. 1 0
2 . 3695
4 1 9 . 95
2 .9832
5 1 0 . 88
3 . 3474
561 .87
4.21 47
677.42
4.7295
743.02
5 . 9559
891 .57
7.50 1 7
1 ,063 . 9
9 . 45 1 2
1 ,257.4
1 1 .913
1 ,470.4
1 5.026
1 ,698.8
1 8. 974
1 ,939.8
2 1 .337
2,060. 7
24.005
2 , 1 86.2
2 , 3 1 1 .9
27.026
30.457
2,439.0
34.361
2 , 569.6
2 , 708.2
38.826
2,86 1 . 5
43.955
49.888
3,040. 3
3,255.4
56.809
3 , 520.6
64.975
3,854.0
74. 756
ends at tL D =0.016, appears to exist for 0.0014 s t s 1 .775 1
t
hours ; pseudoradial flow , which begins at tL D = 3 , appears to be
present for 141 .91 S t S 1 ,932.0 hours ; and oundary effects are
exhibited for t > 1 ,932.0 hours, indicating a finite reservoir with
Le /Lf "" 10. Because the linear and pseudoradial flow regimes have
been tentatively identified, we should try to analyze the data in these
regimes and obtain an estimate of k.
Pseudoradial Flow Analysis.
1 . First, plot P a vs . the log of flow time (Fig. 6.42) .
2 . The slope of the straight line through the points during the
time period 141.91 S tS 1 ,932.0 hours is m = 1 ,284.5 psi/cycle, and
b
162.6qg Bg /ig
k= ---"--"-"'mh
162.6(3,000)(0.7085)(0.01961)
( 1 ,284.5)(60)
=0.088 md .
3 . From Table 6. 1 , the appropriate equation for s'
= 1 . 15 1
[ 3 ,974 - 5,239
1 ,284.5
log
0.088
] + 3.23J
[ (0. 10)(0.01961)(0.0002084)(0.25)
2
= -4.94.
is
1 44
GAS RESERVO I R E N G I N E E R I N G
4. For an infinite-conductivity fracture,
we -s
Lf = 2
= (2)(0 . 25)e - ( - 4.94)
r
= 70 ft .
Linear Flow A nalysis.
1 . First, plot Pa vs . t lA (Fig. 6.43) .
2 . The slope of the straight line through the data points during
the time period 0 . OO 1 4 s t s 1 . 775 1 hours is mL = 1 93 . 9 psi/
..J hours . With this slope, we estimate
�
4 064q g B g fig
Lf =
..Jk
( )
.
rjJ ct
mLh
=
4 . 064(3000)(0 . 7085)
( 1 93 . 9) (60) ..J0 . 088
[
0 . 0 1 96 1
(0. 1 0) (0 . 0002084)
]
�
= 77 ft .
This value agrees well with the estimate o f 7 0 ft from the pseu­
doradial flow analysis. Because this estimate is obtained from a
straight-line slope of data in a particular flow regime, we have more
confidence in its accuracy than in an estimate from type-curve anal­
ysis or from the pseudoradial flow analysis .
Quantitative Type-Curve Analysis.
1 . Because the pseudoradial flow analysis provided an estimate
of k, we may now confirm our analysis using type curves . First,
calculate a pressure match point using an arbitrary value of PD = 1 .
( LiP a) MP =
1 4 1 . 2q
giig fig
kh
( P D ) MP
( 1 4 1 . 2)(3 ,000)(0 . 7085)(0 . 0 1 96)
-----
(0 . 088)(60)
(1)
= 1 , 1 1 5 psi.
2. Maintaining the pressure match point from Step 1 , slide the
log-log plot horizontally until a match is obtained . As Fig . 6 . 4 1
shows, the best match is obtained at a n interpolated value o f the
parameter Le/Lf = 8 . 5 . This value is approximately equal to that
obtained in the preliminary type-curve analysis. Now , choose a time
match point, Lita = 1 ,000 hours and tLtD = 1 1 , and calculate a frac­
ture half-length,
Lf =
=
[
[
0.
�2:37k
rjJp.g ct
( Lita ) ]
tLtD
�
MP
(0 . 0002637)(0 . 088)
(0. 1 0) (0 . 0 1 96 1 )(2 . 084 x 1 0 -4 )
( )]
1 ,000
-11
y,
assumptions : ( 1 ) the fracture has finite conductivity that is uniform
throughout the fracture; (2) the fracture has two equal-length wings;
and (3) wellbore-storage effects are ignored .
We recommend the following procedure presented in terms of
adjusted pressure and time variables for analyzing test data with
the Cinco-Ley et al. type curve . This procedure is also applicable
to analyses with either pressure or pressure-squared plotting
functions .
1 . Plot LiPa vs. t for a constant-rate flow test or L1tae for a build­
up test on tracing paper or log-log paper with the same grid size
as the type curve .
2 . Perform a preliminary or qualitative type-curve analysis to
obtain an initial match and to identify any flow regimes character­
istic of finite-conductivity vertical fractures. Because of the simi­
lar curve shapes, a unique match is difficult to obtain, especially
for low-conductivity fractures .
A . For infinite-conductivity fractures , pseudoradial flow begins
at tLt D = 3 . Although not theoretically correct for finite­
conductivity fractures, we also can apply this criterion for a prelimi­
nary analysis . If the type-curve match indicates several data points
with tLtD > 3 , then pseudoradial flow regime may be present .
B . Bilinear flow , which may evolve with a finite-conductivity
fracture , may be evident in the time ranges summarized in Eqs .
6 . 1 2 1 a through 6 . 1 2 1 c if wellbore-storage effects do not distort the
pressure response. If the type-curve match indicates early data points
with a one-quarter slope and for the data are in the proper time
range , then the bilinear flow pattern may be present.
C . If boundary effects occur during the test, then the test data
will deviate from the type curves developed assuming an infinite­
acting reservoir.
3 . If any flow regimes characteristic of finite-conductivity verti­
cal fractures are identified , perform a postfracture analysis using
the specialized techniques discussed previously .
A . If the type-curve match indicates several data points with
tLtD > 3, then plot Pa . w.f v s . log t for a constant-rate flow test or
Pa .ws vs . log (tp + Lita )/Lita for a buildup test . If a semilog straight
line appears , then a unique estimate of k can be made.
B . If the type-curve match indicates data points for tLiD values
in the appropriate time ranges given by Eqs. 6 . 1 2 1 a through 6 . 1 2 1 c ,
then plot Pa . w.f vs. t 14 on Cartesian coordinate paper for a constant­
rate flow test or Pa .ws vs . Lita� for a buildup test. We can estimate
fracture conductivity from the slope of the straight line indicative
of bilinear flow , but we must have an estimate of k, from either
a prefracture well test or the pseudoradial flow analysis .
4. Next, perform a quantitative type-curve analysis either to con­
firm the results from the specialized analysis techniques or to ob­
tain estimates when these analysis techniques are not possible.
A . Using the permeability estimate from the pseudoradial flow
analysis or the prefracture estimate of k if available , precalculate
a pressure match point. Use an arbitrary value of P D to precalcu­
late Li Pa '
= 72 ft .
This value is consistent with the values obtained from the linear
and pseudoradial flow analyses .
3 . If we have an estimate of the drainage area shape and relative
location of the well in this area, we may estimate the drainage area.
For example , assuming a square drainage area, we can estimate
Le =Lf (Le/Lf ) = (72) ( 8 . 5 ) = 6 1 2 ft .
W e also can estimate the size of the drainage area:
A = (2Le) 2 = [2 (6 1 2)] 2 = 1 . 50 X 1 06 ft2 = 34 . 4 acres.
Cinco-Ley, et a1. 3 5 Type Curve. The Cinco-Ley et at. type
curve (Fig. 6.44) can be used for postfracture analysis of data from
a constant-rate flow test or a pressure-buildup test . The type curve
is a graph of solutions to flow equations modeling a vertical hy­
draulic fracture in an infinite-acting reservoir under the following
Compare the type-curve match with the preliminary match ob­
tained in Step 2 . If the matches are significantly different, then repeat
Steps 2 and 3 until a consistent match is obtained .
B . If a value of k is not available, then choose a pressure match
point ( P D ,LiPa ) from the qualitative type-curve match and estimate
k=
1 4 1 . 2 q g Bgfig
( PD )
LiPa
h
MP
. . .. ... .. ... . ..
.
.
.
.
. . (6 . 1 26)
Note that , without an independent estimate of k, unambiguous
type-curve matches are difficult to obtain, especially for low­
conductivity fractures .
C . Choose a time match point (t or Lita e h D) and estimate
Lf =
[
0 . OO02637k
_
_
rjJ p.g ct
(
Litae
tLtD
t or
)]
MP
�
t
. . . . . . . . . . . (6. 1 27)
.
.
.
1 45
PRESSURE-TRANSIENT TESTI N G OF GAS WELLS
1� '-----r---�--'---;--,
r�10·'
J
110..
10' .,,---,----;--"[--,..--,---,
� 100
�
.. ::1--
Ie
..· . -r':;:: . ·· ··.· ····· ···· ·· i· +··- · =�r'T
..... -+O. I =
� 10- ' .J:±'"' I �-=··++····· ······ ·· :::.;Ir+-!,....
. o ···· ····· · · + ·-t·· · ·:;�..,,·
·
, ,,
· · ··
-- �=l£=t - ---
c
..J,..��+----l--+--�f---t
10'
10"
Fig. 6 .44-Cinco-Ley et al. 35 type curve for a vertically frac­
tured well with a finite-conductivity fractu re .
The value of Lf computed with the time match point should b e
consistent with that obtained from the linear flow analysis. If the
values do not compare , repeat Steps 2 and 3 .
D . Once k and Lf have been determined , find the fracture con­
ductivity , wf Lf ' by using the value of Cr from the type-curve
match .
wf kf = Cr 7rkLf · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 28)
The value of fracture conductivity computed with Eq . 6 . 1 28
should be comparable with the value estimated from the bilinear
flow analysis; however, the value obtained from the bilinear flow
analysis should be more accurate because that method uses the pres­
sure behavior from a specific flow pattern .
Example 6.8-Postfracture Pressure-Buildup Test Analysis in
a Gas Well With a Finite-Conductivity Vertical Fracture. A
pressure-buildup test was run following a fracture treatment in a
gas well . Table 6 . 1 6 summarizes the pressure and time data; other
known data are given below . A prefracture pressure-buildup test
estimated k = O . 1 md . Determine Lf and wf kf using the Cinco-Ley
et al. type curves .
IP =
=
qg =
Pi =J5 =
ct
0 . 20 .
1 . 057 x 1 0 - 4 psia - I .
1 ,000 MscflD.
5 , 000 psia .
h = 1 00 ft .
ii = 0 . 6992 RB/Mscf.
g
J
1 0. 2
1 0·'
;
Pressure Ocrivativc I
.. ..j
...j ........
.. l+
... f+........
., ...j-+'....�I
.j ..,..,...
... r\-r
........
-Hi-r-r...,."
H'
.�
.•
.•
Time
(hou rs)
0.0480
0 . 1 440
0.3360
0 .7200
1 .4880
3 .0240
6.0960
1 2. 240
23. 1 29
39.257
62.537
86. 537
1 1 0 . 54
1 34 . 54
1 58.54
1 68 . 00
Press u re
(psia)
4,673. 8
4 , 677. 1
4,680.7
4,685 . 1
4,69 1 . 1
4,698.9
4 , 709.8
4 , 725 . 1
4 , 742. 6
4 , 76 1 .6
4,78 1 . 0
4 , 796.0
4 , 808 . 7
4,818.6
4 , 827.9
4,831 . 1
I
i
'
·'
10 ·me, 'lID 10
10
10
Adjusted Equivalent Tunc, hr
'
10
10'
Fig. 6 .45-Type-curve match with Cinco-Ley et al. 35 model
for a finite-conductivity fracture, Example 6 . 8 .
tp
P wf
rw
Ji
=
=
=
=
1 68 hours .
4 , 664 psia.
0 . 25 ft.
0 . 02442 cp o
Solution.
Qualitative Type-Curve Analysis.
1 . Plot il P a v s . iltae for a buildup test on tracing paper or log­
log paper with the same grid size as the type curve (Fig. 6.45) .
2 . We observe the following from a preliminary type-curve
analysis.
A . For infinite-conductivity fractures , pseudoradial flow begins
at tL D = 3 . Although not theoretically correct for finite­
f
conductivity fractures , we can apply this criterion for a prelimi­
nary analysis of this well test. Based on the match shown in Fig .
6 . 45 , the pseudoradial flow regime probably is not present .
B . Bilinear flow , which may evolve with a finite-conductivity
fracture, may last for a significant time. If wellbore-storage effects
do not distort the pressure response, bilinear analysis of the early
data may be possible . The preliminary type-curve match suggests
that the first several points may exhibit bilinear flow .
C . No boundary effects are exhibited at the end of the test, in­
dicating an infinite-acting reservoir.
Because the bilinear flow regime may be present, we should at­
tempt to analyze data in this regime .
Bilinear Flow Analysis.
1 . First, we plot Pa vs. the fourth root of time function (Fig.
6.46) .
2 . The first several data points appear to be linear . The slope
of the straight line through the data for (iltae) \4 ::5 (0 . 07) \4 = 0 . 5 1
TABLE 6 . 1 6-POSTFRACTURE B U I LDU P-TEST DATA, EXAMPLE 6 . 8
Eq u ivalent
Adj usted
Time
(hours)
0 .044595
0 . 1 3390
0 . 3 1 259
0 .67002
1 .3847
2.81 20
5 . 6557
1 1 .293
2 1 . 1 03
35.202
54.660
73.666
9 1 .671
1 08.74
1 24.94
1 31 . 1 1
1
Adj usted
P ressure
(psia)
3,098.2
3, 1 01 .4
3, 1 05.0
3 , 1 09.4
3 , 1 1 5. 3
3, 1 23. 1
3 , 1 33.9
3, 1 49 . 1
3 , 1 66.5
3 , 1 85.4
3,204. 7
3 , 2 1 9.6
3,232.3
3,242. 2
3,25 1 .4
3,254.6
Adj usted
Press u re
Change
(psi)
9.4277
1 2. 703
1 6.275
20.642
26.596
34.346
45. 1 79
60.384
77.776
96.683
1 1 5.99
1 30.92
1 43.57
1 53.44
1 62 . 70
1 65.89
Fou rthRoot-of-Time
Function
(hours v. )
0.45954
0.60492
0.74773
0.90474
1 .0848
1 . 2949
1 .5421
1 .8332
2 . 1 433
2.4358
2.71 91
2 .9297
3.0943
3. 2292
3.3433
3 .3838
1 46
GAS RESERVO I R E N G I N E E R I N G
TABLE 6 . 1 7-POSTFRACTURE PRODUCTION TEST DATA , EXAMPLE 6 . 9
Time
(hou rs)
qg
(Mscf/D)
1 .5 1 02
1 .90 1 3
2 . 3935
3 . 0 1 33
3 . 7935
4.7757
6 . 0 1 23
7.5690
9 .5289
1 1 . 996
1 5. 1 02
1 9. 0 1 3
23.935
30. 1 33
37.935
47. 757
60. 1 23
75.690
95.289
6,693 . 7
6,071 .2
5 ,493 . 1
4,958.9
4 ,467.8
4 ,0 1 8.2
3,608.3
3,235.9
2 ,898.7
2 , 594.3
2 , 320.0
2 , 073. 3
1 ,85 1 .9
1 ,653.5
1 ,475 . 7
1 ,3 1 6.8
1 , 1 74.7
1 ,048.0
935 . 1
1 lq g ( x 1 0 6 )
1 49.4
1 64.7
1 82.0
20 1 .7
223.8
248.9
277. 1
309.0
345.0
385.5
43 1 .0
482.3
540 .0
604.8
677.6
759.4
85 1 .3
954.2
1 ,069.4
(hour) 'A is mB =22.36 psi/hr 'A . Using this slope with Eq.
we estimate
wfkf -_
=
6. 123 ,
( 44. 1qhmBiig/ig ) 2 ( c/J/ig1ctk ) 0.5
--
[ (44. 1)(I ,OOO)(0.6�92)(0.0244) ] 2
(100)(22.36)
1
[ (0.20)(0.0244)(1 .057
X 10 - 4 )(0. 1)
]Q 5
=499 md-ft.
Quantitative Type-Curve Analysis.
1 . Find the type curve (Fig . 6.45) that best matches the test data.
For low-conductivity fractures , the data may fit more than one
curve . For this test , we have an estimate of k before the fracture
treatment, so we precalculate the pressure match point using an ar­
bitrary value of P D = 1 .
(ilP a ) MP =
141 .2Qiig/ig
( PD ) MP
kh
(141 .2)(1 ,000)(0.6992)(0.0244)
(1)
(0. 1)(100)
=24 1 psi .
----
2 . Now , simply find the best match by sliding the log-log plot
horizontally while maintaining the pressure match point . The time
match point, found for Cr = lO, is iltae = 8.2 hours and tL D =O.OI ,
t
and
�
Lf =
[ 0.0c/J�P-gC2�t37k ( ilttL aDe ) MP]
[
t
8.2 �
(0.0002637)(0. 1)
=
(0.20)(0.0244)(1 .057 x 10 - 4 ) 0.01
=205 ft .
3 . Find wf kf using Cr = 10 from the type-curve match ,
wf kf = 7rkC,Lf
= 7r(0. 1O)(10)(205)
= 644 md-ft.
( )]
Time
(hours)
1 1 9.96
1 5 1 . 02
1 90 . 1 3
239. 35
30 1 .33
379.35
477.57
601 . 23
756 . 90
952.89
1 , 1 99 . 6
1 ,5 1 0.2
1 , 90 1 . 3
2 , 393.5
3,01 3.3
3 , 793.5
4,775.7
6 ,0 1 2 . 3
7,569. 1
qg
(Mscf/D)
---
834. 7
745. 5
666.4
596 . 1
533 . 7
478.0
428. 5
384.3
344.8
309.6
278.4
250.6
225 . 7
203.6
1 83 . 9
1 66.4
1 50 . 8
1 36 . 9
1 24.5
1 Iq g ( x 1 0 6 )
1 , 1 98 . 1
1 ,341 .4
1 ,500.7
1 ,677.6
1 ,873. 9
2,091 . 9
2 , 333. 8
2 , 602.2
2 ,899. 8
3 , 229.5
3 , 592 . 1
3 , 990.3
4,43 1 .6
4 , 9 1 2.7
5 ,438 . 5
6 ,0 1 0 . 1
6,631 .4
7,303.7
8,029.5
This value is consistent with the bilinear analysis, thus confirm­
ing our identification of the bilinear flow regime .
Agarwal et al. 36 Type Curve. This type curve (Fig. 6.47) is use­
ful for analyzing flow tests or long-term production data in wells
produced at essentially constant BHP. The type curve is a graph
of solutions to flow equations modeling a vertical hydraulic frac­
ture in an infinite-acting reservoir under the following assumptions:
(1) the fracture has finite conductivity that is uniform throughout
the fracture and (2) the fracture has two equal-length wings. When
a well produces at constant BHP , wellbore-storage effects (other
than wellbore unloading immediately after production is begun from
a previously shut-in well) are not present, so wellbore storage is
not of concern in this test data analysis . Following is a recommended
procedure for analyzing test data with the Agarwal et al. type curve.
Although presented in terms of adjusted variables , the procedure
also is applicable when pressure or pressure-squared variables are
used .
1 . Plot the reciprocal of flow rate , l Iq, vs. t on tracing paper
or log-log paper with the same grid size as the type curve .
2 . Find the type curve that best matches the test data . The data
may fit more than one curve, particularly for lower Cr values. An
unambiguous match cannot be obtained without an independent es­
timate of k (e . g . , from a prefracture test analysis) .
3 . If a type-curve match is attempted without an independent es­
timate of permeability , determine k using a rate match point ( 1 / q ,
lIqD ) MP from the type-curve match :
k=
141 .2Bg /ig
h( P a ,i -P a , wf )
( llIqIqD )
MP
. . . . . . . . . . . . . . . . . . . (6. 129)
In addition, record a time match point (t,tL D) MP '
4. If an independent estimate of k is available,tthen a unique type­
curve match can be found . First, a rate match point is precalculat­
ed by selecting an arbitrary value for lI qD and calculating l Iq.
141 .2Bg /ig
(1/qD) MP' . . . . . . . . . . . . . . (6. 130)
kh( P a ,i -P a , wf )
Align the matching values of l Iq and l IqD ' and move the test
(lIq) MP =
data horizontally to obtain a unique match . Record a time match
point (t,tL D ) MP from the match .
t Lf from a time match point and k:
5 . Estimate
Lf =
6.
Eq .
[ 0.OOCPP-gC�2�t37k ( tLtD ) MP]
t
__
;'
'
.
. . . . . . . . . . . . . . . . . (6. 127)
Once k and Lf have been determined , calculate wf kf using
6. 128 and the Cr value from the type-curve match .
1 47
PRESSURE-TRANSIENT TESTI N G OF GAS WELLS
T
!
-=='::
-=
::--=
--==
===::;::=
-'====
"==='-"=
='=
-""
" -'-=
3 20 0 �
'-::::
" '=
: "
==
-::::
'
i
i
,
,
: ""-11---+-1--+ .
�
i
3 1 00
....
3000
j --- -- -.-
I
'
•
.
•
I
I·
•
i
I
----i----L-]
-11-_1_1- I I I
L
_____
0.0
I.
.J
IO' ...----,---,--.,---r---.
.
__ _______
0.5
..
I'
J
.
______ . __._.
1 .0
L
1 .5
..
..
_ _._ __
Fourth Root o f Time Function
i
__
I
J._
2.0
....._._ ..._ ..___.
I'
J
2.5
Fig. 6 . 46-Bilinear flow analysis, Example 6 . 8 .
1O.2+-....,.
....- nt.,...,. ....,.,.
....- nt..,. .,...,...,.
.. .j.,.., .,.......
.,.., .. .J---,,-,-,..,.,,,r---.,....,...., ....,,j
Fig. 6 .47-Agarwal et a/. 36 type curve for constant-BHP pro­
duction from a finite-conductivity, vertical fracture.
Estimate
Lf and wf kf using the Agarwal et al.
Tw = 0 . 25 ft .
fig
Example 6.9-Analysis of a Postfracture Production Test Un­
der Constant BHFP Conditions. A hydraulically fractured gas well
produced at constant BHFP with the rate history given in Table
6.17. Analysis of a prefracture buildup test suggests k = O . O I md .
10 '
10"
=
pj =ji =
P wf =
h =
ct =
1 . 3639 RB/Mscf.
2 , 400 psia.
2 , 000 psia.
50 ft.
4 . 5 x l O - 4 psia - 1 .
TABLE 6 . 1 8-PRESSURE-BU ILDUP DATA, EXAMPLE 6 . 1 0
Time
(hours)
0 . 0041 04
0 .008242
0 . 0 1 6445
0 .020703
0 .026063
0.0328 1 2
0 . 04 1 307
0 .052003
0 .065468
0 . 0824 1 9
0 . 1 0376
0 . 1 3063
0 . 1 6445
0 .20703
0.26064
0 . 328 1 2
0.41 309
0 . 52003
0 .65470
0 . 82420
1 .0376
1 . 3063
1 .6446
2 . 0705
2.6066
3 . 28 1 5
4. 1 3 1 2
5.201 1
6.5478
8 . 2436
1 0. 379
1 3. 067
1 6.45 1
20.71 4
26.080
32.834
4 1 . 350
52.071
65.576
82.591
1 04.03
1 3 1 . 06
1 65 . 1 3
Pressure
(psia)
3,249 . 1
3,249.3
3,249.5
3,249.7
3,249.8
3,250.0
3 , 250.3
3,250.6
3 ,250 . 9
3,25 1 .4
3,252.0
3,252.7
3,253.5
3,254.5
3,255.7
3,257.2
3,259.0
3,261 . 1
3,263.6
3,266.6
3,270 . 1
3,274. 2
3,279.0
3,284.5
3,290.9
3,298.3
3,306.6
3,31 6.1
3,326.6
3 , 338.3
3,35 1 . 3
3 , 365.5
3,381 . 0
3,397.9
3,41 6.2
3,436 . 1
3,457.6
3,480.9
3 , 506.0
3 , 530.0
3 , 556.0
3,582 . 0
3 , 609.0
Equ ivalent
Adj usted
Time
(hou rs)
0.00332 1
0.006671
0 . 0 1 331 0
0 . 0 1 6757
0.021 096
0.026560
0 .033437
0.042097
0 .053000
0. 066727
0.0840 1 1
0 . 1 0578
0. 1 33 1 8
0 . 1 6769
0.21 1 1 4
0 .26586
0 .33480
0 . 42 1 59
0.53096
0.66871
0 . 84228
1 .061 0
1 .3367
1 .6841
2 . 1 220
2.674 1
3.3704
4.2485
5.3558
6. 7525
8 . 5 1 44
1 0 .736
1 3. 536
1 7. 066
21 .51 2
27. 1 03
34. 1 41
42.972
54.035
67. 854
85.049
1 06.36
1 32.59
Adj usted
Pressure
(psia)
1 ,822 .0
1 ,822. 2
1 ,822.4
1 ,822.6
1 ,822.7
1 ,822 . 9
1 ,823. 2
1 ,823.5
1 , 823.7
1 ,824.2
1 ,824. 8
1 ,825.4
1 ,826. 2
1 , 827. 1
1 , 828. 2
1 ,829 .7
1 ,831 . 3
1 , 833 . 3
1 ,835 .7
1 , 838 .5
1 ,841 . 8
1 ,845.6
1 ,850. 2
1 ,855. 3
1 ,861 .4
1 ,868.4
1 ,876. 2
1 , 885 .2
1 ,895 . 1
1 , 906. 2
1 ,9 1 8.6
1 ,932 . 1
1 ,946. 8
1 ,962 .9
1 ,980.4
1 ,999.5
2,020 . 1
2,042.5
2,066.7
2,089.8
2 , 1 1 5. 0
2 , 1 40 . 2
2, 1 66.5
Adj usted
Pressure
Change
(psia)
0 .094238
0.28235
0 .47021
0.65820
0 .75220
0.9403 1
1 .2224
1 .5046
1 .7864
2.2565
2.8208
3.4788
4.23 1 2
5. 1 71 3
6.2996
7.71 00
9.4023
1 1 .377
1 3.727
1 6. 548
1 9.839
23.694
28.207
33.398
39.443
46.433
54.273
63.247
73. 1 88
84.290
96.624
1 10.1 1
1 24.88
1 40.99
1 58 .48
1 77.52
1 98 . 1 5
220.53
244.72
267.88
293.05
3 1 8.27
344.50
Sq uareRoot-ot-Time
Fu nction
(hours lh )
0.057631
0.081 674
0 . 1 1 537
0 . 1 2945
0 . 1 4524
0 . 1 6297
0 . 1 8286
0.205 1 8
0.23022
0. 25832
0.28985
0.32523
0.36494
0.40949
0 .45950
0 . 5 1 562
0.57862
0.64930
0.72867
0 . 8 1 775
0.91 776
1 .030 1
1 . 1 56 1
1 .2977
1 .4567
1 .6353
1 . 8359
2.06 1 2
2.31 43
2.5986
2.91 79
3.2765
3.679 1
4. 1 3 1 2
4.638 1
5.2061
5.843 1
6.5553
7.3508
8 .2373
9.2222
1 0.31 3
1 1 .5 1 5
type curve.
1 48
GAS RESERVO I R E N G I N E E R I N G
Match Poim (C, = 1 (0)
l/qg = 1 .37 x 10-3 psi, I/qD = 1 x 1 0- 1
-4
, = 100 hr, 'lID = 9.75 x 1 0
1 0 ' .."....---,----,---...---,,--,
Dimensionless Time, tLjD
1 0'
1 0"
1 0'
Fig. 6 .48-Type-cu rve match , Example 6.9.
P a ,; =
P a , wf =
cf> =
/i =
1 ,476 psia.
1 ,077 psia.
0. 10.
0.0142 cp o
Solution.
1 . First, plot (Fig. 6.48) 11q vs . t on log-log paper with the same
grid size as the type curve. Use real time rather than adjusted time .
2. We have a prefracture estimate of k, so precalculate a rate
match point using an arbitrary value of (lIqD) MP = 1 X 10 - 1 and
obtain a unique match.
(llq) MP =
141 .2Bg /ig
( l IqD) MP
kh( P a ,; -P a , wf )
(141 .2)( 1 . 364)(0.0142)
(10 - 1 )
(0.01)(50)(1 ,476 - 1 ,077)
=0.00137 D/Mscf.
------
Maintaining the rate match point, the best match again is obtained
with Cr = 100. In addition, the time match point is t= loo hours
and tLtD =9.75 x 10 - 4 .
3 . The hydraulic fracture half-length is
Lf =
[ 0.00�2�37k ( ) ]
[
cf>p.g c t
'h
t
__
tL D MP
t
(
100
(0.0002637)(0.01)
(0. 10)(0.0142)(4.5 x 10 - 4 ) 9.75 x lO - 4
=65 1 ft .
=
4. The fracture conductivity i s
wf kf = 7rCrkLf = 7r(1oo)(0.01)(65 1 ) =2,045
)]
y,
md-ft.
Barker-Ramey 3 7 Type Curve. The Barker-Ramey type curve
(Fig. 6.49) includes wellbore-storage effects on a postfracture
constant-rate flow test or pressure-buildup test . The type curve is
a graph of solutions to flow equations modeling a vertical hydrau­
lic fracture in an infinite-acting reservoir under the following as­
sumptions: (1) the fracture is infinitely conductive and (2) the
fracture has two equal-length wing s . A recommended procedure
for analyzing test data with the Barker-Ramey type curve follows .
Although presented in terms of adjusted variables , the procedure
also is valid when pressure or pressure-squared variables are used .
1 . Plot t.P a vs . t for a constant-rate flow test or t.tae for a build­
up test on tracing paper or log-log paper with the same grid size
as the type curve .
2. Perform a qualitative type-curve analysis to obtain an initial
match and to identify any flow regimes characteristic of infinite­
conductivity vertical fractures .
1 0'
10'
Fig. 6.49-Barker-Ramey 37 type curve for a vertically frac­
tured well with well bore storage (after Ramey, J . J . and Grln­
garten , A . C . : " Effect of High Volume Vertical Fractures on
Geothermal Steam Well Behavior, " Proc., Second United Na­
tions Symposium on the Use and Development of Geother­
mal Energy, San Francisco [May 20-29, 1 975]). Cou rtesy of
Lawrence Berkeley Laboratory.
A . Wellbore-storage effects are indicated for data matched on
type curves for CfD > O. A plot of early-time data with a slope sig­
nificantly steeper than that of any type curve usually indicates
wellbore-storage distortion of the test data .
B . If the type-curve match indicates several data points with
tLtD > 3, then pseudoradial flow has occurred and we can perform
a pseudoradial flow analysis .
C . If the type-curve match indicates early data points with a slope
of one-half and for tL D < 0.016, then the linear flow pattern may
t
be present .
D . If boundary effects occur during the test, then the test data
will deviate from the type curves developed assuming an infinite­
acting reservoir.
3. If any flow regimes characteristic of infinite-conductivity ver­
tical fractures are identified , perform a postfracture analysis using
the specialized techniques discussed previously .
A . If the type-curve match indicates several data points with
tL D > 3 , then plot P a , wf vs. log t for a constant-rate flow test or
t
P a , ws vs. log (tp +t.ta )lt.ta for a buildup test . If a semilog straight
line appears , then a unique estimate of formation permeability can
be made .
B . If the type-curve match indicates data points for tL D < 0.016,
t
then plot P a. wf vs. t V, on Cartesian coordinate paper for a constant­
rate flow test or P a . ws vs. t.ta� for a buildup test . A straight line
should appear with a slope inversely proportional to .JkLf ' If k
can be estimated from a pseudoradial flow analysis or if a prefrac­
ture value is available, then estimate Lf .
4. Next, perform a quantitative type-curve analysis either to con­
firm the results from the specialized analysis techniques or to ob­
tain estimates when these analysis techniques are not possible.
A . Using the estimate of k from the pseudoradial flow analysis
or a prefracture estimate if available , precalculate a pressure match
point. Use a convenient and arbitrary value of P D to precalculate
t.P a ·
Compare the type-curve match with the preliminary match ob­
tained in Step 2. If the matches are significantly different, then repeat
Steps 2 and 3 until a consistent match is obtained .
B . If a value of k is not available, then choose a pressure match
point ( P D ,t. P a ) from the qualitative type-curve match and estimate
k=
( )
14 1 .2qg Bg /ig P D
t.P a
h
MP
. . . . . . . . . . . . . . . . . . . . . (6. 126)
1 49
PRESSURE-TRANSIENT TESTIN G OF GAS WELLS
10'
...---..,---"T'""--"T'"--'
1900
..
's.
e
�
'
10
10 '
°
10
Adjus1ed Equivalent Tunc.At". (hours)
10"
10
'
11
II
.880
.860
'6'
0(
1820
'
10
Fig. 6 . 50-Type-curve matc h , Example 6 . 1 0 .
[ 0.OO�2�37k ( t
)]
C . Choose a time match point
(t or Il.tae .tLfD)
Il.tae
tLfD MP
or
and estimate
�
. . . . . . . . . . . . . . (6 . 127)
c/>""g ct
The value of Lf computed with the time match point should be
consistent with that obtained from the linear flow analysis . If the
values do not compare , repeat Steps 2 and 3 .
Lf =
D . The matching parameter i s a dimensionless fracture storage
coefficient (analogous to the dimensionless wellbore storage coeffi­
cient) , CfD , defined by
0.8936C
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 3 1)
CfD =
c/>hcrLt
where C can be estimated from the estimated Lf and the value of
CfD from the type-curve match.
E . We also can estimate wf kf . For an infinite-conductivity frac­
ture , Cr � 100 and
wf kf � Cr ll'kLf � 100 ll'kLf ·
=
0.25 ft.
1 .506 x 10 - 4 psia - 1 •
1 ,400 MscflD .
3 ,249 psia .
100 ft.
0.78549 RB/Mscf.
1 ,644 hours .
4,000 psia .
0. 12.
0.02162 cp o
Solution.
Type-Curve Analysis.
1 . First, plot Il.P a vs. Il.tae on log-log paper with the same grid
size as the type curve (Fig. 6 . 50) .
2 . From the preliminary type-curve match with the curve for
CfD = 0. 1 , observe the following .
A . Pseudoradial flow , which begins at tL D = 3 , does not appear
t
to be present for this test .
B . Linear flow , which continues until tLf D = 0.016, appears to
be present for Il. tae ::s; 2.0 hours .
C . No boundary effects are present, indicating an infinite-acting
reservoir .
3.0
2.S
Because we have tentatively identified a linear flow period, attempt a linear flow analysis .
Linear Flow Analysis.
1 . Plot P a vs . t � (Fig. 6.51) .
2. As Fig . 6.5 1 shows, a straight line with slope mL = 38.6
psi/hr � can b e drawn through most of the data i n this period. The
-:..rk Lf product is estimated from
�
r.
_
'I k Lf -
4.
( )
4.064qgBg Jig
mLh
c/> ct
[
(4.064)(1 ,400)(0.7855)
0.0216
=
(0. 12)(1 .506 x 10 -4 )
(38.6)(100)
= 4� -ft.
] 'h
Quantitative Type-Curve A nalysis.
1 . We have an estimate of k, so precalculate a pressure match
point . Choose P D = 3.0 and
141 .2qgfigJig
( P D) MP
(Il.P a ) MP =
kh
Example 6 . 10-Analysis of a Constant-Rate, Postfracture Pres­
sure-Buildup Test With Wellbore-Storage Distortion. A pressure­
buildup test was run following a fracture treatment in a gas well.
Table 6. 18 gives pressure and time data. Permeability is estimat­
ed to be 0. 1 md from a pre-fracture test. Determine Lf . In addi­
tion , verify the type-curve analysis using any specialized analyses .
=
ct =
qg =
P wf =
h =
fig =
tp =
Pi =p =
c/> =
Ji
2.0
1.5
SqIUR Root of Time, hrl/2
Fig. 6 . 5 1 -Ll near flow analYSis, Example 6 . 1 0 .
'
rw
1 .0
O.S
(141 .2)(1 ,400)(0.7855)(0.0216)
(3 .0)
(0. 1)(100)
= 1 ,000 psi .
-------
With this precalculated match point, the type curve for CfD =
In addition , we obtain the following time
match point :
0.05 matches the data.
Il.tae = l ,OOO hours , tLfD =6.35.
2. Computed Lf using the time match point.
�
0 . �2�37k l:itae
Lf =
tLtD MP
c/>""g ct
[
[
=
( )]
( )]
(0.0002637)(0. 10)
1 ,000
(0. 12)(0.02 16)(1 . 506 X 10 -4 ) 6.35
= 103 ft.
�
The linear flow analylsis provided a n estimate o f -Jk L f
-v'iiid - ft; using k=O. l md . from the pre-fracture test,
Lf = -Jk Lf / -Jk
=4� -ft/-../D.l
= 126 ft.
=
40
1 50
GAS RESERVO I R E N G I N E E R I N G
Wel lbore
fractures , respectively . Economides 3 1 stated that, in terms o f its
impact on the effective wellbore radius , 6 defined as wa = w e - s =
Lf12, the fracture skin effect is negligible for practical purposes .
The second type of damage, often described as a choked frac­
ture, is thought to be caused by proppant crushing and embedding
in the formation, which reduces the near-wellbore fracture perme­
ability (Fig. 6.55) . The skin factor representing a choked fracture
is given by
r r
Fract u re
sfs . ch =
Fig. 6.S2-lnfinite-conductivity vertical fracture with fracture­
face damage caused by fluid losses.
This value is consistent with the value of Lf = 103 ft from the
quantitative type-curve analysis .
5 . Compute the wellbore-storage coefficient , C, using the type
curve correlating parameter , CfD .
C--
c/>hcrL}
-CD
0. 8936 t
(0. 12)(100)( 1 .506 10 -4)(126)2
---------- (0.05)
0.8936
x
= 1 .6 1 bbl/psi .
6. For an infinite-conductivity fracture (i.e. ,
ture conductivity is estimated to be
Cr � 100), the frac­
wf kf � 1rCrkLf = 1r(100)(O. 1)(127) = 3,990 md-ft .
6. 10.4 Effects o f Fracture and Formation Damage. Cinco-Ley
and Samaniego-V . 33 , 38 suggested that two types of fracture
damage may occur during hydraulic fracturing : in the formation
around the fracture and within the fracture adj acent to the well­
bore . The first type , quantified as a fracture-face skin, sfs ' proba­
bly is caused by fluid losses into the formation during the fracture
treatment (Fig. 6.52) . The fracture-face skin is defined by
sfs =
:; (� - 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 132)
where Ws = depth of fluid-loss damage into the formation and nor­
mal to the fracture face , ft; Lf = length of the hydraulic fracture ,
ft; k s = permeability of the damaged zone, md; and kf = fracture
permeability , md .
Figs. 6.53 and 6.54 show the effect of a positive skin factor on
the pressure response during a test for low- and high-conductivity
1rLs k
-wf kfs
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 133)
where sfs ch = skin factor for choked fracture , dimensionless ;
.
L s = length of choked portion of fracture, ft; and kfs = permeability
of the choked portion, md .
The pressure response in a well with a choked fracture is similar
to that for a fracture with skin damage (see Fig. 6.56) .
6. 1 1 Naturally Fractured Reservoirs
Many important natural gas reservoirs are naturally fractured . In
this section, we describe two models for these naturally fractured
reservoirs . Further, we introduce special semilog and type-curve
analysis techniques for gas-well tests in these reservoirs .
6. 1 1 . 1 Naturally-Fractured-Reservoir Models. A characteristic
of naturally fractured reservoirs is the presence of two distinct
porosity types, matrix and fracture porosities. Fig. 6.57 illustrates
an actual dual-porosity reservoir composed of a rock matrix sur­
rounded by an irregular system of vugs and natural fractures .
Fortunately , it has been observed that a real heterogeneous, dual­
porosity reservoir can be modeled with an equivalent homogene­
ous , dual-porosity model like that shown in the idealized sketch.
With this idealized reservoir, the dual-porosity characteristics are
modeled as an array of regular but discontinuous matrix blocks sur­
rounded by continuous channels representing the natural fracture
network. The conduits for fluid flow in dual-porosity reservoirs
are primarily the natural fractures , which have high permeabilities
but low porosities. The unfractured matrix blocks contain the majori­
ty of the storage capacity of the reservoir but contribute little to
the overall reservoir conductivity .
Warren and Root 39 introduced two dual-porosity parameters for
describing the characteristics of naturally fractured reservoirs . The
first parameter, interporosity flow coefficient, measures how easi­
ly fluid flows from the matrix to the fractures . Warren and Root
defined the interporosity flow coefficient , A, as
A = arMkma lkf), . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 34)
where kma = matrix permeability and kf = fracture permeability .
The parameter a , which characterizes system geometry , is defined
by 40
a =4j (j+2)IU , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 135)
�t r-----�
�I<
, ...
...
..
Q.
CI)
- N•
0..
:; i
Q
10 '
s-,
100
ICr4
0..1
�
5
D-
•
D-4
I
10-3
,
D-t
I
eo-!
I
0°
I
I
0.000263711'
l.o . ---��
." C t Lfl
Fig. 6 . S3-Effect of skin damage on the pressure response in a vertically fractu red well ,
C r = 0 . 2 (after Ci nco-Ley and Samaniego-V. 38 ) .
1 51
PRESS U R E-TRANSIENT TEST I N G OF GAS WELLS
----,
.10 1 --
J
-
..
CD
..
•
CL fit
I:.
#I
i
..
100
10"1
•••
o.z
wkf
er " "'- ( II. " 10 0
co
Fig. 6 . 54-Effect of skin damage on the pressure response In a vertically fractu red well,
C, = 1 00 (after Cinco-Ley and Samaniego-V. 38 ) .
k
Fig. 6 .55-Schematlc of a choked fracture.
where L = characteristic dimension of a matrix block and j = number
of normal sets of planes limiting the less permeable medium (j = 1 ,
2, 3 ) . For example, j = 3 in the idealized reservoir model in Fig .
6 . 5 7 . Similarly , for a multilayered or " slab " model , j = 1 . In this
case, by letting L =hma (the thickness of an individual matrix
block) , A becomes
A = 12r�
l!!
kma
.
kfh tna
--
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 36)
10 r-----�--�--�
:::I
'"
'"
l!!
The second dual-porosity parameter, the storativity ratio , is de­
fined by 39
(cpVCt )f
( cpVCt )f + (cpVct ) ma
-----'--- , . . . . . . . . . . . . (6. 1 37)
where V= ratio of the total volume of one medium to the bulk
volume of the total system and cp = ratio of the PV of one medium
to the total volume of that medium. The subscripts/and/ +ma refer
to the fracture and the total system (fractures plus matrix) , respec­
tively . w is a measure of the relative fracture storage in the reservoir.
Two models of interporosity flow from the less permeable medium
(matrix) to the more permeable medium (fractures) currently are
used . Barenblatt et al. 4 1 and subsequent authors 39,42-44 assumed
pseudosteady-state flow; others , 45-5 1 notably deSwaan, 45 assumed
Q.
'"
'"
CD
'2
.Q
�
'"
c:
CD
E
Ci
A
A
---�--�
I� �0l
10
I�
10
I�
I d""
I o-S
Dimensionless TIme
Fig. 6.56-Effect of choked fracture on the pressure response
In a vert i c a l l y fractu red w e l l (after C i nco-Ley a n d
Samaniego-V. 33 ) .
.�
....T"IX
,
"IIACTUIIE
AOTUAL RESERVOIR
....TIUX
.
\
i:::'
�
?
?
!?
��
:::::" �
/
....
"ItACTU " E S
MOOEL "ESERVOI "
Fig. 6.57-Warren and Root 39 dual-porosity reservoir model.
1 52
GAS RESERVO I R E N G I N E E R I N G
1 800
I
1 60 0
<II
.;;
a.
t
C4
1400
,
.
1200
.
.
-�..,
,�
:
.
r
,
8 .l
I
"
:.I'
�
v-:
10"
�
/
:,..-""
L
//
'/
Ea1�apolGlc 10 PUll
4V
l/
I
p.
end
-
V
V
1 01
10"
Fig . 6 . 58-Characteristic pressu re-buildup response for the pseudosteady-state matrix flow
mode l .
transient flow in the matrix . Gringarten 40 pointed out that ,
although many pressure-transient tests exhibit pseudosteady-state
interporosity flow behavior, other tests suggest transient interporosi­
ty flow . A possible explanation of this seeming inconsistency is
that interporosity flow always occurs under transient conditions but
can exhibit a pseudosteady-state-like behavior if a significant im­
pediment to flow exists between the less permeable medium and
the more permeable one .
6 . 1 1 . 2 Pseudosteady-State-Matrix-Flow Model. The pseudo­
steady-state-flow model assumes that, at a given time, the pressure
at all points in the matrix is decreasing at the same rate and thus
flow from the matrix to the fracture is proportional to the differ­
ence between matrix pressure and pressure in the adj acent frac­
ture. Specifically, this model does not allow unsteady-state pressure
gradients within the matrix . Further, pseudosteady-state flow con­
ditions are assumed to be present from the beginning of fluid
movement.
Because it assumes a pressure distribution in the matrix that would
be reached only after what could be a considerable flow period ,
the pseudosteady-state-flow model is oversimplified. However, this
model seems to match field data in a number of cases . One possi­
ble reason is that damage to the matrix face, similar to the fracture­
face skin effects discussed in Sec . 6. 10.4, acts like a choke or flow
restriction, thereby causing the flow from the matrix to the frac­
ture to be proportional to pressure differences upstream and down­
stream of the matrix face .
Semilog Analysis Technique. The pseudosteady-state interporosi­
ty flow solution developed by Warren and Root predicts that , un­
der ideal conditions , two parallel straight lines will develop on a
semilog graph of test data. Curve A in Fig. 6.58 shows this pres­
sure response.
The initial straight line reflects flow in the fracture system only .
At this time , the formation is behaving as a homogeneous forma­
tion with no contribution from the matrix. The slope of the initial
semilog straight line is proportional to the kh product of the frac­
ture system , just as for any homogeneous system. Later, the matrix
begins to produce fluid to the fracture , and a rather flat transition
region appears . Finally , the matrix and fracture reach equilibrium,
and a second straight line appears .
The slope of the second straight line is almost identical to that
of the initial straight line . At this time , the reservoir again is be­
having as a homogeneous system ; however, now the system con-
sists of both the matrix and the fractures . The slope of the second
semilog straight line is proportional to the total kh of the
matrix/fracture system . Because the fracture system permeability
usually greatly exceeds the matrix permeability , the slope of this
line is almost identical to that of the initial straight line , which
reflects contribution from the fractures only .
The shape of a semilog plot of test data from a naturally frac­
tured reservoir is almost never the same as predicted by Warren
and Root ' s model . Wellbore storage usually distorts the initial
straight line and often obscures part of the transition region between
the straight lines . Curve B in Fig . 6.58 shows a commonly observed
pressure response .
The reservoir kh [actually the kh of the fractures , or (kh)f ' be­
cause (kh)ma usually is negligible] can be estimated from the slope
of the two semilog straight lines . The storativity ratio can be deter­
mined from their vertical displacement, op, while the interporosi­
ty flow coefficient, A, can be obtained from the time at which a
horizontal line , drawn through the middle of the transition curve,
intersects with either the first or second semilog straight line . 40
When semilog analysis is possible (L e . , when both of the correct
semilog straight lines can be identified) , the following procedure,
in terms of adjusted pressure and time variables , can be used for
semilog analysis of buildup or drawdown test data .
1 . From the slope of the initial straight line (if present) or final
straight line (more likely to be present) , determine kh. In either
case, m is related to the total system kh which is essentially all in
the fractures.
_
(kh)f =kh =
162.6qglig iig
m
. . . . . . . . . . . . . . . . . . . . . . (6. 138)
,
where k = (kh)f /h. Strictly speaking, the slope of the second straight
line is related to [(kh)f + (kh)ma ] ' but (kh)ma typically is much
smaller than (kh)f.
2 . If both initial and final straight lines can be identified (or the
position of the initial line can at least be approximated) and the pres­
sure difference, op, established, then
w = lO -oplm
•
.
.
.
.
.
.
.
.
.
.
.
•
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(6 . 1 3 9)
3 . Draw a horizontal line through the middle of the transition
data . The times of intersection of this line with the first and second
1 53
PRESSURE-TRANSIENT TESTI N G OF GAS WELLS
lOS
�
�
::J
0
ell
CII
a..
0.-
'"
O'l
.S!!
.2
c:
<n
c
CIl
�
0
102
__
-_
Cae2s
;.,e2s
%0
10
1010
10'
1
S
43
10
3 X 10....:1
100
IO� I[L--L.:...--L__L-.l...__
.-L___L-__-L__---.:L__..J
101:
10'
10""
10�
DimensionlQss lima Group,
10<4
IO�
10:1
tr:!=o
Fig . 6 .59-Bo u rdet and Gringarten 5 2 type curves for pseudosteady-state matrix flow.
semilog straight lines are ta, 1 and ta , respectively , The inter­
,2
porosity flow coefficient, A, is calculated 52 for a drawdown test by
) jigra ( cPVCt �f+majigra
(
A = cPVCt f
=
, . . . . . . . . . . . . . (6. 140)
,1ita, 1
'Ykta , 2
C
or for a buildup test,
)
(
)
With the estimate of w from Step 2, ( cP VCt )f can then be calcu­
lated with Eq. 6. 142 .
4. The second semilog straight line should be extrapolated to p *
(Fig . 6.58). From p * , f5 can be found using conventional methods
(e . g . , the MBH23 p * method) .
5 . The second semilog straight line should be extrapolated to
P a, l hr ' The skin factor is then calculated from
[
(
)
l
k
a l hr
(cPVCt�fjigra p + Ala, I
(cPVCt )f� majigra tp + A ta , 2
s = 1 . 15 1 AP - log
+ 3.23 , . . . . . . . . (6. 143)
_ _
,
=
2
g
m
cP
ctr
p'Ykt
'Yktp
t
Ala, 2
p
A a, 1
APa, l hr = P a ,i - P a, l hr for a drawdown test o r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 141) where
a, l hr -P a ,wj(Atae =O) for a buildup test .
P
Type-Curve Analysis. When wellbore storage distorts the data,
where "I = exponential o f Euler ' s constant ("1 = 1 .78 1). Because ta 1
and ta can only be approximated the value of A obtained by thi s type curves are quite useful in identifying and analyzing dual­
,2
A=
method may not be very accurate but usually is of the same order
of magnitude as the correct value. The terms ( cP II) ma and (ct ) ma
in Eq. 6. 141 are obtained by conventional methods . A porosity log
usually reads only the matrix porosity (not the fracture porosity)
and thus gives cPma , while (ct ) ma is calculated as the sum of co So '
CgSg , cw Sw , and cf ' Vma , the fraction of the total system (frac­
ture plus matrix) that is matrix, is essentially unity . From the defi­
nition of w in Eq. 6. 137,
( )
l
( cPVCt )f = (cP VCt ) ma __ . . . . . . . . . . . . . . . . . . . . . (6. 142)
l -w
102
--
cr/"
--
.1.Cd(1-ot
3 x 10..;!
w
porosity systems . Fig. 6.59 shows an example of the Bourdet and
Gringarten 52 type curves for pseudosteady-state matrix flow . At
first, test data will follow a curve for some value of CD e 2s . The
data then will deviate from the early fit and follow a transition curve
characterized by the parameter Ae - 2 s . Finally , the data will fol­
low another CD e 2 s curve .
In the example in Fig . 6.59, the earliest data for Well A follow
the curve for CD e 2s = 1 , while the transitional data follow the curve
for Ae -2s =3 x lO -4 . The late data then follow the CD e 2 s =0. 1
curve . At earliest times, the reservoir behaves as a homogeneous
reservoir with all flow in the fracture system . During intermediate
times, there is a transition region as the matrix begins to produce
10"
- -- ACdtd,,1-tD)
'
--
3 X 10-;l
10'
3 x 10'"
10'
10'
3 X 10-4
Fig. 6 . 60-Bourdet et al. 5 3 derivative type cu rves for pseudosteady-state matrix flow.
1 54
GAS RESERVO I R E N G i N E ER i N G
Recimc t
Flow
t
Recime 2
Flow
Regime 3
Flow
tl
'i tO
d 9
2
Q.
0
!!!
=>
0)
'"
Wellbote
Sroraae
o 0
S tape - m
0 I
0 I
S
I \opo m l
Time. hr
Fig . 6 . 6 1 -Characteristic flow regimes in a dual-porosity sys­
tem with transient matrix flow.
into the fractures . At late times, the system again behaves as a
homogeneous system with matrix and fractures contributing and
with an equilibrium established between the matrix and fractures .
Fig. 6.60 shows derivative type curves for a formation with
pseudosteady-state matrix flow 53 . The most notable feature , char­
acteristic of naturally fractured reservoirs , is the dip below the
curves that characterizes a homogeneous reservoir (the dotted line
in Fig . 6.60).
The downward-dipping curves are characterized by the parame­
ter ACDlw(l - w). The curves returning to the homogeneous reser­
voir curves are characterized by the parameter ACDI(l -w). Test
data that follow this pattern on the derivative type curve can be
interpreted reasonably as identifying a dual-porosity reservoir with
pseudosteady-state matrix flow (a theory that needs to be confirmed
with geological information and reservoir performance) .
Pressure and pressure-derivative type curves can be used together
to analyze dual-porosity reservoirs . The following procedure, given
in terms of adjusted pressure and time variables, is for analyzing
buildup or drawdown data with type curves .
1. Plot adjusted pressure and adjusted pressure-derivative o n log­
log graph or tracing paper. Also make a semilog plot of test data.
2. If there is a horizontal line at middle times on the derivative
plot of the test data, lay this line over the (tDICD)piJ = 0 . 5 line on
the type curve . A straight line on the semilog graph should span
the same time range as the horizontal line on the derivative type
curve . If this semilog straight line is present , then determine sys­
tem permeability from its slope and use this calculated permeabil­
ity to force the pressure match point . The result should be the same
vertical match that resulted from the derivative type-curve match,
but the forced pressure match point should remove any ambiguity
in the vertical match .
3. If there is a unit-slope line at early times, then fix the horizontal
match of the data by overlaying the unit-slope line on the type curve.
If no unit-slope line is present , then try to fix the horizontal match
by finding the best match possible for the early data on the deriva­
tive type curve . Choose pressure and time match points .
4. With the type curve in fitted position , determine the value of
CDe 2s characterizing the fit of the earliest data on the pressure type
curve . The fit on the derivative type curve should confirm this pa­
rameter value . The earliest data are those that appear before the
transition region. This value of CD e 2s , which we call (CDe 2 s )f'
characterizes the fracture system .
5 . With the type curve remaining in the same fitted position, de­
termine the value of CDe 2s , which we call (CDe 2s )f + m£1' that char­
acterizes the test data after the transition to total system flow has
been completed . The parameter will be read from the pressure type
curve.
6. Read the value of Ae -2s that characterizes the horizontal tran­
sition curve crossed by the test data on the pressure type curve at
an intermediate time in the transition region . There is a significant
!!!
Q.
<II
'"
CD
"2
.Q
en
c
m
c3
6
7
),:01
6
0 10"7
:;
Co to·'
0 10· &
"
t�
fat
t�
toG
loT
toa
Dimensionless Time, tcj
104
t()ll
tdO
toll
Fig. 6.62-Estlmating w' and A' using data from Flow Regimes
1 and 2 (after Serra et al. 50 ) .
10'
r------,--.,--r--,
10....
10� ��--��----�------�----�----�
10'
10'
10"
10�
Dimensionless llme, 'clCD
Fig. 6 . 63-Bourdet et al. 5 4 type curves for transient matrix
flow.
uniqueness problem in determining the best-fitting transition curve.
For Test B in Fig . 6.59, the transition curve chosen was for
Ae -2s = 1 X 10 -7 .
7. Calculate w from the ratio of the two CD e2s values .
(CDe 2s )f + ma
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 144)
(CDe 2S )f
8. Calculate k from the pressure match point (unless a forced pres­
sure match point was used in Step 2, in which case permeability
w=
is already known from semilog analysis) .
9. Compute the wellbore-storage coefficient from the time match
point.
CD _-
(
)
0.OOO2637k t or tJ.tae
. . . . . . . . . . . . . . . . (6. 145)
tD ICD MP
cf>jig ctr;
10. From the value of CD determined in Step 9 and the value
of (CDe 2s )f + ma determined in Step 4, determine s .
(CD
ma
. . . . . . . . . . . . . . . . . . . . . . . (6. 146)
s = 0 . 5 In
.
[ e;�f+ ].
1 1 . Using the estimate of s from Step 10 and the matching pa­
rameter Ae -2s from Step 6, calculate A. This value should be con­
sistent with the estimate of A from semilog analysis . If it is not,
then refine the type-curve match .
6 . 1 1 .3 Transient-Matrix-Flow Model. The more probable flow
regime in the matrix is unsteady-state or transient flow, during which
1 55
PRESS U RE-TRAN S I E N T TESTI N G OF GAS WELLS
an increasing pressure drawdown starts at the matrix/fracture in­
terface and moves further into the matrix with increasing time .
Pseudosteady-state flow should be achieved only at late times. In
some cases , however, a matrix undergoing transient flow but hav­
ing a thin , low-permeability damaged zone at the fracture face may
behave as predicted by the model of pseudosteady-state matrix flow .
A semilog graph of test data for a formation with transient matrix
flow has a characteristic shape different from that for pseudosteady­
state . Three distinct flow regimes are identified in Fig. 6.61 . Flow
Regime 1 occurs at early times, during which a ll production comes
from the fracture system . Flow Regime 2 occurs when production
from the matrix into the fracture begins and continues until the
matrix/fracture transfer reaches equilibrium. This eqUilibrium point
marks the beginning of Flow Regime 3 , during which total system
flow , from matrix to fracture to wellbore , is dominant. The same
three flow regimes appear when there is pseudosteady-state matrix
flow . However, the durations and shapes of the transition flow re­
gimes are considerably different for the two matrix-flow models .
Serra e t a l.
observed that pressures from each o f these flow
regimes will plot as straight lines on conventional semilog graphs .
Flow Regimes 1 and 3 correspond to the classic early- and late­
time semilog straight-line periods, respectively .
The semilog
straight lines of these flow regimes have essentially the same slope.
Flow Regime 2 is an intermediate period between Flow Regimes
1 and 3 . In practice , if all or any two of these regimes can be iden­
tified , then a complete analysis is possible with semilog methods
alone . Unfortunately, certain nonideal conditions may make this
analysis difficult to apply . Flow Regime 1 often is distorted or to­
tally obscured by wellbore storage , which makes this flow regime
difficult, if not impossible, to identify . Flow Regime 2 , the transi­
tion region , also may be obscured by wellbore storage. Flow Re­
gime 3 sometimes requires a long flow period followed by a long
shut-in time to be observed and consequently may be obscured by
boundary effects.
Semilog Analysis Techniques. Serra et a t . presented a semi­
log method for analyzing well-test data in dual-porosity reservoirs .
They found that the existence of the transition region , Flow Re­
gime 2, and either Flow Regime 1 or 3 is sufficient to obtain a com­
plete analysis of drawdown or buildup test data . They assumed
unsteady-state flow in the matrix, no wellbore storage, and rectan­
gular matrix-block geometry in their work . From our experience,
the rectangular matrix-block geometry is adequate, although differ­
ent assumed geometries can lead to slightly different interpretation
results . The weakness of the Serra et at. method is the assumption
of no wellbore storage . In many cases , Flow Regimes 1 and 2 are
partially or totally obscured by wellbore storage , making analysis
by this method impossible or difficult. Despite this limitation , the
Serra et a l. method has great practical value when used in conjunc­
tion with type-curve methods .
Semilog A nalysis Based o n Flow Regimes 1 and 2. The first cal­
culation procedure, which uses Fig. 6.62, is applicable when Flow
Regimes 1 and 2 are present . The procedure is presented in terms
of adjusted time and pressure for which gas properties are evaluat­
ed at p .
1 . Estimate
= k from the slope of the semilog straight line
(m or m * , where m* = mI2) .
50
compressibility , (psi - I ) ; and = fracture permeability , md . Sec­
ond , assuming a value of ePf h c ,
s = 1 . 15 1
kftf f
[ -APa , l hr
[ (kfhft) ]
ePfhft cf �g rw2
- log
m
+ 3 .23
_
}
.
. . . . (6 . 1 49)
3 . Plot AP a vs . t (or Atae for buildup) on tracing paper or semi­
log graph paper with the same scale as Fig'. 6.62.
4. Choose an arbitrary value o f AP a and calculate PwD '
f f
k h tAPa
-'---"-'--_ '---
PwD
141 .2qgBgiLg
-so
. . . . . . . . . . . . . . . . . . . . . . . . (6. 150)
5. Using the chosen value of AP a and the value of PwD from Eq.
6. 150 for a vertical match point, match the data in Fig . 6.62. Record
the values of the parameters A'W' ,t ( ,t I D ' and (tD , t or Atae) at the
match point, where t ( is the adjusted time at the point of intersec­
tion of the lines representing Flow Regimes 1 and 2, and t I D is
the dimensionless adjusted time at this point of intersection .
0.OOO2637kt(
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 5 1 )
ePiLg Ctr�
39,45
ePf cf hft = 2.637 �g10rw2-4 kfhft ( t
6.
Estimate
ePf cf hft
x
_
532.3 iLg ( C
t(
8. Assume a value of ePmacma
n
2kmaePmacma
)
from the time match point .
Atae
tD
or
ePf f hft) 2
=
.
MP
. . . . . . (6. 152)
. . . . . . . . . . . . . (6. 153)
and estimate
kma hfna .
/
50
kfhft h
m*
m
2k2 .maePmacma
Estimate
n
in one of two ways. First , assume a value of
and calculate s :
s
s =0.5756 \ AP ��l hr
l
. . . . . . . . . (6. 147)
m
- IOg
[ k (kf hft) 2 _ 4 ] + 3 .7291 ,
n
2 maePmacma �g rw
Assume that
ePmacma h
hmat h
=
J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 148)
ht
hf) ;
and estimate
A'
and w ' .
ePf cfhft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 155)
kma h
A' = 12- -- r� . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 156)
h'lna kfhft
w' =
10.
.
Estimate w and 'A .
w = 1/( 1 +w') .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 157)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 158)
1 1 . If the values of
assumed in Step 2 and calculated
Use
in Step 6 do not agree , then assume a new value of
the value that was calculated in Step 6 and return to Step 2.
12. I f the values of n ma ma cma assumed in Step 2 and calcu­
lated in Step 7 do not agree , then assume a new value of n 2 kma
ePmacma ' Use the value that was calculated in Step 7 and return to
Step 2 .
'A == 'A ' .
ePf Cf hft
2k eP
ePf Cfhft.
Semi log A nalysis Based on Flow Regimes 2 and 3. Because of
wellbore-storage distortion of Flow Regime 1 , the case with Flow
Regimes 2 and 3 is observed more commonly in practice . The fol­
lowing procedure , presented in terms of adjusted time and pres­
sure , can be used to analyze test data under these conditions .
1 . Estimate
= from the slope of the semilog straight line
(m or m * , where m* = m/2) by Eq. 6. 147.
2. Assume a value o f ePmacma �g and calculate m / fna .
kf hft kh
where n = number of fractures (equal to net matrix thickness, ma '
divided by individual matrix-block thickness , h ma or to net frac­
ture thickness,
divided by individual fracture thickness ,
kma = matrix permeability , md; ePma = matrix porosity ; cma = matrix
hfP
9.
kma
hfna
- =
532.3 ePmacma iLg
t*
k ah
, . . . . . . . . . . . . . . . . . . . . . . . . (6. 159)
where t* = adjusted time at which lines representing Flow Regimes
intersect . (Use At� for buildup tests . )
2 and 3
1 56
GAS RESERVOI R E N G I N E E R I N G
TABLE 6 . 1 9-PRESSU RE·BUILDUP TEST DATA, EXAMPLE 6 . 1 1
t
(hou rs)
0 . 0 1 54
0.0239
0.0369
0 . 0569
0 .0880
0 . 1 360
0 . 2 1 00
0 . 3240
0.501 0
0 . 7740
1 .2000
1 .8500
2.8500
4.41 00
6.81 00
1 0 .500
1 6. 200
25. 1 00
38.800
59.900
92.500
1 43.00
221 . 00
341 .00
527.00
8 1 4.00
1 ,260 .0
1 ,940.0
3 , 000.0
3.
Estimate
P wf
(psia)
1 1 0.35
1 1 0.54
1 1 0.82
1 1 1 .26
1 1 1 .92
1 1 2. 9 1
1 1 4.39
1 1 6.57
1 1 9.71
1 24. 1 0
1 29 . 98
1 37.41
1 46.05
1 55 . 1 5
1 63.85
1 71 .59
1 77.30
1 84. 1 6
1 9 1 .53
1 95.04
203.37
2 1 0.70
220 . 1 0
229.40
238 .50
247.30
255. 70
262. 70
269 .62
Adj usted
Pressure
(psia)
1 8 .605
1 8 .670
1 8 .766
1 8. 9 1 7
1 9 . 1 43
1 9.493
20.0 1 8
20.799
2 1 .953
23.61 8
25.951
29.046
32.864
37. 1 35
41 .461
45.5 1 4
48 .626
52.493
56. 8 1 9
58.941
64. 1 25
68 .871
75 . 1 97
8 1 .739
88.399
95.085
1 0 1 .70
1 07.38
1 1 3. 1 5
Equ ivalent
Adj usted
Time
0. 00572
0. 00888
0 . 0 1 372
0.02 1 20
0. 03289
0.051 06
0. 07936
0 . 1 2366
0 . 1 9400
0 .30586
0.48759
0. 77665
1 . 2435
2 . 0 1 55
3 .2704
5.2946
8. 5406
1 3 .782
22. 1 49
35.383
56.406
90. 1 39
1 44. 1 1
230.26
368.22
586.93
933.62
1 ,466.6
2 , 292. 0
'A ' .
kma h mat rw2
k
h
;;: 12 '; -- r�, . . . . . . . . . . . (6. 160)
kf hft h�
h ma kf hft
where h ;;: h mat .
4. Estimate the adjusted time, tb2 (or tltb2 for buildup) , at which
Flow Regime 2 begins, and calculate
kf hft q,r:..a "; h'A'tb2
. . . . . (6 . 161 )
q,f cf hft = 8 . 33 x 10 - 4
p-g rw
5 . Estimate
and 'A.
h
w' = q, macma mat , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 162)
q,f cf hft
where h ;;: h mat .
w = lI(1 + w') . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 163)
'A ;;: 'A' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6 . 164)
6. Estimate
kf hf
tl a
�
S = 1 . l5 1 P , l hr - IOg
+ 3 . 23 . . . . . (6 . 165 )
m
q, macm h p-g rw2
For a flow test, tl P a, l hr = P a ,i -P a , l hr - For a buildup test,
tlP a , l hr =P a , l hr -P a ,wf (tltae =0) .
'A' = 12
'
w , w,
[
s.
(
) !h
C
(
)
]
Both o f these semilog methods require some knowledge o f or
assumptions about matrix and fracture properties . Like the semi­
log analysis techniques presented for homogeneous reservoirs , the
advantage of semilog analysis is the unique interpretation possible
when the correct semilog straight lines are identified . The disad­
vantages are that ( 1 ) wellbore storage can distort data to the extent
that application of semilog analysis is impossible and (2) the test
duration is sometimes insufficient to reach Flow Regime 3 . When
much of the test data is distorted by wellbore-storage effects , type
curves become a viable analysis technique .
Adj usted
Pressu re
C h ange
0 . 1 1 98 1
0 . 1 8485
0. 28070
0.43 1 33
0. 65726
1 . 0076
1 . 5327
2 . 3 1 35
3.4671
5. 1 326
7.4652
1 0.560
1 4. 378
1 8 .650
22. 975
27. 028
30. 1 4 1
34.007
38. 333
40.455
45. 639
50. 385
56.71 1
63.253
69. 9 1 3
76.600
83.2 1 0
88.895
94.663
Pressure
Derivative
0. 1 21 1 1
0 . 1 8406
0.283 1 7
0.43004
0.65543
0.993 1 8
1 .4746
2 . 1 581
3. 1 071
4.32 1 9
5 .8259
7.3757
8.4734
8.89 1 0
8. 6738
7.4573
7.2951
8.60 1 7
6.8089
7.8303
1 0.622
1 1 .801
1 3.721
1 4.074
1 4.265
1 4. 292
1 3.403
1 2. 753
1 2. 9 1 8
Adj usted
Horner
Time Ratio
7,971 ,300
5 , 1 3 1 , 900
3 , 3 1 9 , 700
2 , 1 48 ,600
1 ,385,200
892,260
573,990
368,370
234,820
1 48 , 940
93,426
58 ,654
36, 634
22 ,602
1 3 ,929
8,603.9
5 ,333.8
3,305.4
2,056.7
1 ,287.5
807.61
505.38
316.1
1 97.84
1 23 . 72
77. 6 1 4
48.793
3 1 .062
1 9 .875
Type-Curve Analysis. Bourdet et al. 54 presented type curves for
analyzing well tests in dual-porosity reservoirs . These type curves,
which include the effects of wellbore storage and transient matrix
flow , are useful complements to the Serra et a l. semilog analysis.
Fig. 6.63 gives an example of the pressure and pressure-derivative
type curves for transient matrix flow .
Early-time data, dominated by fracture properties , are fit by a
CDe 2s pressure curve for a homogeneous reservoir . Data in the
transition region are fit by curves characterized by a parameter {3 ' .
Finally , data in the homogeneous-acting , fracture-plus-matrix flow
regime are fit by another CDe 2s curve . Early data also are fit by
a derivative curve for a homogeneous reservoir. If wellbore-storage
distortion ceases before the transition region begins , the derivative
data will be horizontal and should be aligned with the P D =0.5
curve . I n the transition region , m e derivative will decrease and the
data will fall below the earlier-fitting curve . Because the slope of
the semilog straight line during Flow Regime 2 equals one-half of
those during Flow Regimes 1 and 3 , the derivative curve during
the transition region will flatten and should be aligned with the
PD =0.25 curve . The homogeneous (fracture-plus-matrix) data
should , after wellbore distortion has ceased and before boundary
effects have appeared , be horizontal on the derivative type curve
and should be aligned with the P D =0.5 curve. We recommend the
following analysis procedure for using the Bourdet et at. type curves .
Although it is presented in terms of adjusted pressure and time vari­
ables, the procedure also is applicable to other plotting variables
with some modifications .
1 . Plot data as tlP a vs. t or tltae on tracing paper or on log-log
graph paper with the same scale as the type curves .
2. Overlay the log-log graph on the type curves and move the
graph horizontally and vertically until a good match is obtained .
A unit-slope line on the test data should overlay that line on the
derivative type curve . Horizontal lines should overlay either the
PD =0.25 line (transition region) or the PD =0.5 line (homoge­
neous-acting region) on the derivative type curve . Record values
of ( P D ,tl P a ), (tDICD ,t) or (tDICD ,tltae), (CDe 2s )f ,(CDe 2s )f +ma ,
and {3 ' . (CD e 2 s )f is the matching curve parameter of the fracture­
dominated region , while (CDe 2 s )f +ma is the parameter for the
total-system response region . A pressure match point ( P D ,tl P a )
1 57
PRESS U R E-TRAN SIENT TESTI N G OF GAS WELLS
TABLE 6 . 20-FLU I D PROPERTIES, EXERCISE 6 . 1
Pressu re
(psia)
1 4.6
1 75
350
525
700
875
1 ,050
1 ,225
1 ,400
1 ,575
1 , 750
1 ,925
2 , 1 00
2 , 275
2 , 450
2,625
2,800
2 , 975
3 , 1 50
3,325
3,500
3 , 675
3,850
4 , 025
4 , 200
4,375
4,550
4 , 725
Bg
(scflft 3 )
Viscosity
(cp)
0 .77438
9 . 4 1 76
1 9. 1 29
29. 1 25
39.391
49.904
60.632
7 1 .533
82.558
93.649
1 04 . 74
1 1 5 .78
1 26.69
1 37.43
1 47.93
1 58. 1 6
1 68.09
1 77.69
1 86.95
1 95 . 85
204.41
21 2.61
220.48
228. 02
235 . 23
242 . 1 4
248. 76
255 . 1
0.01 3 1 2
0 . 0 1 325
0.01 345
0 . Q 1 370
0 . 0 1 399
0 . 0 1 432
0 . 0 1 468
0.01 509
0 . 0 1 553
0 . 0 1 600
0.01 651
0.01 705
0 . 0 1 762
0 . 0 1 821
0 . 0 1 883
0 . 0 1 946
0.020 1 0
0.02076
0.021 42
0 .02209
0.02276
0.02343
0.0241 0
0.02477
0 .02543
0 .02609
0 .02674
0. 02738
can be forced if the correct semilog straight line can be identified ,
and kh can be calculated from its slope .
3 . Calculate kh from the pressure match point ( PD ,ilP a ) ,
( )
z
factor
--0 . 9987
0 .9843
0 . 9692
0 . 9548
0 . 941 3
0 .9288
0.91 73
0.9071
0.8983
0 . 8909
0 . 8850
0.8807
0 .8780
0 .8769
0 .8773
0.879 1
0.8824
0 .8869
0. 8925
0 .8993
0.9070
0 . 9 1 56
0.9250
0 .9350
0.9458
0.9571
0 .9689
0 . 98 1 1
cg
(psia 2 /cp)
6.849 x 1 0 - 2
5 . 804 x 1 0 - 3
2 . 944 x 1 0 - 3
1 .988 x 1 0 - 3
1 .508 x 1 0 - 3
1 .2 1 7 x 1 0 - 3
1 .020 x 1 0 - 3
8.765 x 1 0 - 4
7.661 x 1 0 - 4
6.776 x 1 0 - 4
6 .043 x 1 0 - 4
5.422 x 1 0 - 4
4.886 x 1 0 - 4
4.41 9 x 1 0 - 4
4.007 x 1 0 - 4
3.643 x 1 0 - 4
3.320 x 1 0 - 4
3.032 x 1 0 - 4
2 . 776 x 1 0 - 4
2 . 547 x 1 0 - 4
2.343 x 1 0 - 4
2 . 1 60 x 1 0 - 4
1 .995 x 1 0 - 4
1 .848 x 1 0 - 4
1 .7 1 5 x 1 0 - 4
1 .596 x 1 0 - 4
1 .488 x 1 0 - 4
1 .390 x 1 0 - 4
1 .627 x 1 0 4
2.560 x 1 0 6
9.608 x 1 0 6
2 . 1 33 x 1 0 7
3.766 x 1 0 7
5.848 x 1 0 7
8.365 x 1 0 7
1 . 1 30 x 1 0 8
1 .462 x 1 0 8
1 .831 x 1 0 8
2 .234 x 1 0 8
2.668 x 1 0 8
3. 1 30 x 1 0 8
3.61 6 x 1 0 8
4. 1 25 X 1 0 8
4.654 x 1 0 8
5 . 1 98 x 1 0 8
5.758 x 1 0 8
6.329 x 1 0 8
6.91 0 x 1 0 8
7.500 x 1 0 8
8.096 x 1 0 8
8.698 x 1 0 8
9.305 x 1 0 8
9.91 4 x 1 0 8
1 .053 x 1 0 9
1.114x 109
1 . 1 76 x 1 0 9
(psia - 1 )
Pp
a naturally fractured reservoir known to exhibit transient matrix
flow behavior . Well and reservoir data are summarized below ;
Table 6 . 1 9 gives the buildup test data. Determine permeability ,
skin factor, storativity ratio , and interporosity flow coefficient .
- = 14 1 .2q B- il -PD
. . . . . . . . . . . . . . . . . . . . . (6. 1 66)
kh
g g g
qg = 126.7 MscflD .
il P a MP
= 0.26 ft .
rw
4. Calculate (CDe 2s )f +ma from the time match point (tD ICD ,
= 0.06.
�
ilta ) by
P i =P = 319. 1 psia.
P a . i =Pa = 158 psia .
0.OO02637k t or iltae
. . . . . . . . . . (6. 167)
(CD)f +ma ct = 1 .649 x 10 -3 psia - 1 .
iig � macmara tDICD MP
h = 1 10 ft .
tp = 45,554 hours .
5 . Calculate the skin factor .
ilg = 0.Q1 1 1 cp o
CDe 2S
a
P
l hr = 2,364 psia .
.
s = 0 . 5 I n -. . . . . . . . . . . . . . . . . . . . . . . . . (6. 168)
Bg = 8.422 RB/Mscf.
CD f +ma
Solution. Qualitative Type-Curve Analysis. For this example, we
6. Calculate A.
begin with a qualitative type-curve analysis . The objective is to de­
(CDe 2S )f +ma
termine whether Flow Regime 1 , 2, or 3 can be identified .
A= 1 . 8914
. . . . . . . . . . . . . . . . . . . . . . . . (6. 169)
1 . Plot adjusted pressure change and adjusted pressure deriva­
s
({3') Mpe -2
tive vs . adjusted time on a log-log plot (Fig. 6.64) .
2. Using the Bourdet et al. type curves, match both adjusted pres­
7. Calculate w.
sure
and pressure derivative data (Fig . 6.64). Dual-porosity behavior
(CDe 2s )f +ma
is
evident
from the match of the test data with the type curve for
w=
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6. 1 70)
transient interporosity flow . Specifically , the derivative data from
(CDe 2s )f
the longest shut-in times (iltae � 230 hours) can be aligned on the
If (CD e 2s )f cannot be determined uniquely , then
(tDICD)PD =0.5 line, whereas earlier derivative data (5 :5 iltae :5 35
hours) match the (tDICD)P D =0 . 25 line, characteristic of a dual­
(CDe 2s )f +ma
w :5
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(6.
171)
porosity system. With the earliest data trending toward a unit-slope
(CDe 2s )f
line, we also can establish a horizontal match. The type-curve match,
_
(
)
(
)
Although type curves are particularly advantageous for analyz­
ing tests that have wellbore-storage distortion, they do require the
existence of Flow Regime 3 for good interpretation . For best re­
sults, the type curves and the semilog analysis methods should be
applied simultaneously until a consistent interpretation can be found
with both approaches . Often, this may require an iterative process .
Example 6 . 1 1 -Buildup Test Analysis in a Naturally Fractured
Gas Reservoir. A pressure-buildup test was run on a gas well in
in particular the derivative type-curve match , suggests that Flow
Regimes 2 and 3 may be present .
Semilog Analysis Based on Flow Regimes 2 and 3. We have ten­
tatively identified Flow Regimes 2 and 3, so we now attempt a semi­
log analysis.
1 . Plot adjusted pressure vs . adjusted Homer time on a semilog
plot. On the Homer plot in Fig. 6.65, a final straight line with slope
m = 9 1 . 2 - 59.5 = 3 1 . 7 psi/cycle is a reasonable fit of the later data
(Flow Regime 3). We can force a line with slope m * = m /2 =76 . 6 60.7 = 15.9 psi/cycle though earlier data (Flow Regime 2), which
1 58
GAS RESERVO I R E N G I N E ER I N G
1 0 1 .r--------r-------.---��_.--��--,_------_.-p' = 10
__.
----
1 � �;::::;�41;;::::�'IT,----�\lr__t----Jt::.;1����:::_l
�[)ata
I
AdjllSled J'ressurc Change !
Pressure {)crivauvc
. .. ._
. . --.-11
!I
..._. . ....._
.
:i
.....--_..
C"':2s
1 0,1 --a---,."""T'"T"T'T, .�, = O.1
10'3
,
•
•
!
ii
_._.
II
1 0'I.
Ii
II
I!
l
10'
�djusted Equiv�ent Thne,
.1 "' ''T'
-t---""""T'"T"'rn ...r""''
'' rnr
'T' r!
101
i:
i
!
I
10'
10
hr
--r-r'T'T'rT'ft'1'i--r-T"T'TTr'm'i--r-T"T'TTr'....i---r-r'T'T',.,.".j
10
tIfCD
Fig . 6 . 64-Type-cu rve match , Example 6 . 1 1 .
fall on the (tDICD)P D =0.25 line on the derivative type curve.
These lines intersect at (�t: +tp )/�t: = 830, or �t: = tp I829 =
45,554/829 = 55.0 hours .
2. The beginning of Flow Regime 2, I::..ta. b2 , is unclear because
of wellbore-storage distortion of the test data. From the semilog
graph , we assume this to be at (�ta. b2 +tp )ll::.. ta. b2 = 5 ,333.8, or
�ta . b2 =45,554/5,332 . 8 = 8.54 hours .
3. To calculate skin factor , we need P a. l hr on the semilog
straight line of Flow Regime 3 . At &a = 1 hour, the adjusted Hom­
er time ratio is (tp + �ta )/�ta = (45,554 + 1)/1 =45,555. At this
value in Fig . 6.65 , we �btain P a. l hr =6.92 psia.
4. Determine kf hft =. kh.
_ 162.6qiig iig
kf hft =. kh = ----"-"m
162.6(126.7)(8.422)(0.01 1 1)
3 1 .7
= 60.8 md-ft,
or for h = 1 10 ft , k = 6O.8/l 1O=0.55 md .
5 . Calculate kma 1h?na . From the available data, we have tPma =
0.06, Cma = C t = 1 .649 x 10 -3 psia - 1 , and iig =O.Ol l 1 cp o
kma 532.3 tPmacma iig
532.3(0.06)(1 .649 x 10 -3)(0.01 1 1)
55.0
= 1 .063 x 10 -5 md/ft2 .
6. Estimate A' .
�
(
k/ h/ttP ma C mahA 'l::..ta. b2 )
tPf cf hft =8.33 x 10 -4
_ 2
7.
From the time,
mate
tPf cf hft .
/log rw
[
= (8.33 x 10 -4»
(60.8)(0.06)(1 .649 x 10 -3 )(1 10)(1 .56 x 10 -5)(8.54)
x
(0.Q1 1 1)(0.26) 2
=2.86 x l O -4 .
8 . Estimate w' , w , and A , assuming h =. h mat .
a
a
]�
a
w' = tPm cm hm t
tPf cf hft
-----
(0.06)(1 .649 x 10 -3 )(1 10)
2.86 x 10 -4
= 38. 1 .
w = I /(1 +w ' ) = 1I(1 + 38. 1) =0.026. A is A =. A ' = 1 .56 x lO -5 .
9. Estimate s .
[( )
6.92
= 1 . 15 1 3 1 .7
- log
(12)(1 .063 X 10 -5 )(1 10)(0.26) 2
60. 8
= 1 .56 x 10 -5 .
�ta. b2 ' at which Flow Regime 2 begins , esti­
[ (0.06)(1 .649 x 10 -360.)(1810)(0.01 1 1)(0.26) 2 ] + 3.23]
= - 3 .94.
Quantitative Type-Curve Analysis.
1 . Using the log-log plot in Fig . 6.64, find the match points and
matching parameters . In this case, we have an estimate of kh from
semilog analysis, so we precalculate a pressure match point. We
1 59
PRESS U RE-TRAN SIENT TESTI N G OF GAS WELLS
ro
'iii
.B:
1 00
�
80
Q.
60
=>
<Il
<Il
Q)
"0
Q)
iii
40
'0
«
=>
20
Adjusted Horner Time R atio
Fig. 6. 65-Horner plot, Example 6 . 1 1 .
arbitrarily choose P D = 1 and calculate the value of lip a that must
overlay P D = 1 on the type-curve match.
IiP a
1 4 1 . 2qg BgiLg
PD
kh
(141 .2)(126.7)(8.422)(0.01 1 1)(1)
(60.8)
=27.5 psi .
This value is in only fair agreement with A ' = 1 .56 x 10 - 5 . The
lack of good agreement is at least partially a result of our inability
to determine {3' accurately .
6. Calculate w .
(CDe 2s )!+ma
(CDe 2s )!
=0. 113.6
=0.028.
w e;
The match also forces the late-time derivative data to overlay
(tD /CD)PV =0.5 and the early derivative data to overlay (tD/
CD)pv =0.25 for (CDe 2s )!+ma =0. 1 . A time match point is chos­
en as lita e = 123.4 and tD/CD = 100. {3' is estimated to be 10. The
parameter (CD!!.?S )! is 3 .6.
2. Calculate kh from the pressure match point. Because the pres­
sure match was forced , kh = 60.8 md-ft.
3. Calculate (CD)! +ma using the time match point .
0.OOO2637k litae
(CD)!+ma =
cf>macmaiLg rw2 tD/CD MP
(
--
)
(
)
(0.0002637)(0.608)
123 .4
(0.06)(1 . 649 X 10 - 3)(0.01 1 1)(0.26) 2 100
=2,665.
4. Compute s.
CDe 2s
s =0.5 In
CD !+ma
=0.5 In(0. 1I2665)
= -5. 10.
This value is i n reasonably good agreement with s = - 3.94 from
=
)
(
semilog analysis .
5 . Calculate A.
-----' --"-.c.:.::.::...
1 . 8914(CDe 2s )!+ma
({3') Mpe -2s
( 1 . 8914)(0. 1)
( l O)e -2 ( - 5 . 10)
=7.03 x lO -7 .
A=
__
_
6. 1 2 Reservoir Model Identification by U s e o f
Characteristic P ressure Behavior
In the previous sections , we presented analysis techniques for
pressure-transient tests in gas wells . For each technique, we have
made the implicit assumption that we know a priori the correct reser­
voir model to use for the analysis. In some cases , however, the
engineer may not have sufficient information from which to choose
a reservoir model, especially in newly discovered fields . Fortunate­
ly , wells exhibit characteristic pressure responses that vary depend­
ing on the near-wellbore conditions and heterogeneities in the
drainage area of the well . In fact, the basis of each analysis tech­
nique presented in this chapter is recognition of a curve shape rep­
resentative of some reservoir model . For example , a flow pattern
characteristic of fmite-conductivity fractures is bilinear flow . During
bilinear flow , a log-log plot of Pi -P wf vs. flowing time, t, exhibits
a straight line having a slope of one-quarter .
As another example , consider the u s e o f type curves for well­
test analysis . The underlying principle is that, if a plot of test data
exhibits the same shape as a type curve in all time regions , then
the reservoir is of the same type as that characterized by the type
curves. Unfortunately , this principle is not infallible. Different reser­
voir types sometimes exhibit essentially the same shape on a type­
curve plot . In addition, both semilog and log-log plots of pres­
sure/time data often are insensitive to pressure changes character­
istic of a specific reservoir model. As an alternative to pressure/time
plots , the pressure derivative often is used specifically to identify
reservoir types.
In fact , the derivative type curve is the most definitive of the type
curves for identifying reservoir type. It can identify subtle but char­
acteristic changes in slope that may be masked or not apparent on
a pressure/time type curve . However, both the derivative and the
pressure/time type curves are better than a semilog graph for iden­
tifying reservoir type . A type curve spans all time regions, while
on the semilog plot we usually examine only the semilog straight
line (middle-time region) . Furthermore, semilog analysis general-
GAS RESERVO I R E N G I N E E R I N G
1 60
TABLE 6 . 2 1 -PRESSURE DRAW DOWN DATA, EXERCISE 6 . 1
Time
(hou rs)
1
1 .5
2
2.5
3
4
5
6
7
8
9
10
11
12
15
18
21
24
48
72
96
1 20
1 44
1 68
1 92
216
240
264
288
31 2
336
360
384
408
432
456
480
504
528
552
576
600
624
648
672
Pressure
(psia)
3 , 1 1 8 .4
2,921 . 8
2,786.4
2,689.3
2 , 6 1 6.0
2,522.6
2,455 . 7
2 , 406.0
2 , 367. 1
2 , 335.4
2 ,308.8
2,285.9
2,265.9
2,247.9
2,207.7
2 , 1 73 . 1
2 , 1 43.8
2 , 1 1 8.8
2 , 0 1 7.8
1 ,940.0
1 ,882. 3
1 , 837.4
1 ,800. 3
1 , 768 .7
1 , 74 1 . 1
1 , 7 1 6.7
1 , 694. 7
1 ,674.7
1 ,656.4
1 ,639 . 5
1 ,642 .0
1 ,609 . 7
1 ,596.4
1 ,583. 9
1 , 572 . 0
1 , 560.7
1 ,549.8
1 , 539. 5
1 ,529. 6
1 ,520 . 1
1 ,5 1 1 .0
1 ,502 . 3
1 ,493.9
1 ,485 .7
1 , 477.8
Pp
(psia 2 /cp)
6. 1 8 1 x 1 0 8
5 . 546
5. 1 1 8
4.8 1 6
4.592
4.31 0
4.1 1 2
3.966
3.853
3 . 762
3.686
3.621
3.564
3.51 4
3.402
3.306
3 .226
3 . 1 58
2.889
2.687
2.541
2.430
2 . 340
2.263
2 . 1 98
2 . 1 40
2.089
2.043
2.001
1 . 963
1 .928
1 .896
1 .866
1 .839
1 .8 1 3
1 .788
1 . 764
1 . 742
1 .721
1 .700
1 .68 1
1 .663
1 .645
1 .628
1 .6 1 1
P.
(psia)
1 ,762 .4
1 , 58 1 .3
1 ,459 . 3
1 , 373.4
1 , 309 .4
1 , 229 . 1
1 , 1 72.5
1 , 1 3 1 .0
1 , 098.8
1 ,072 . 8
1 ,051 . 1
1 ,032.5
1 ,0 1 6.4
1 , 002 .0
970 . 0
942 . 8
91 9.9
900.4
823. 8
766.3
724.7
692.9
667 . 1
645 .4
626.7
61 0.3
595.7
582 .6
570.5
559.7
549.7
540 . 5
532.2
524. 3
5 1 6.8
509. 7
503 . 1
496 .7
490 .7
484.8
479 .4
474. 1
469. 1
464.2
459 . 5
l y assumes a homogeneous reservoir, while a type curve reflects
a particular reservoir type .
The best approach for identifying the correct reservoir model in­
corporates three major plotting techniques: the ordinary type curve,
the derivative type curve, and a " specialized graph" for a test. Prop­
erties can be deduced from a specialized graph when a straight line
develops during a certain time region . These graphs include the
Homer plot for a homogeneous reservoir, a square-root-of-time plot
for a well with a high-conductivity fracture , and a fourth-root-of­
time plot for a well with a low-conductivity fracture. When the reser­
voir type is correctly identified , all three plots will confirm or at
least be consistent with the hypothesized reservoir type . We now
consider specific characteristics of the derivative type curve that
are useful for identifying reservoir type from pressure-transient tests
in gas wells .
1 . A maximum in the curve at early times indicates wellbore
storage and skin . The greater the maximum , the more severely the
well is damaged . Conversely , the absence of a maximum suggests
a stimulated well (i . e . , acidized or fractured) .
2 . A minimum in the curve at intermediate times indicates a devi­
ation from homogeneous reservoir behavior (i . e . , a reservoir het­
erogeneity) . Examples include dual-porosity (naturally fractured)
or layered reservoirs .
3 . Stabilization or flattening at later times indicates radial flow
and corresponds to the semilog straight line on a semilog graph .
TABLE 6 .22-PRESSURE B U I L D U P DATA, EXERCISE 6.2
Time
(hours)
0.01
0.01 5
0.0225
0. 0338
0.0507
0.076
0.1 1 4
0.1 71
0.257
0 . 385
0.578
0.867
1 .3
1 . 95
2.93
4.39
6.59
9.88
1 4.8
22.2
33.4
50.0
75. 1
1 1 3. 0
1 69.0
P ressure
(psia)
2 , 1 04.5
2 , 1 04.8
2 , 1 05.3
2 , 1 06.0
2, 1 07. 1
2 , 1 08 . 9
2 , 1 1 1 .8
2 , 1 1 6. 3
2 , 1 23 . 1
2 , 1 33. 2
2 , 1 47.3
2 , 1 65.4
2 , 1 86.0
2,205.4
2,21 9.6
2,227.8
2,232.2
2 , 235.3
2,238.0
2,240.6
2,243 . 1
2,245.6
2,248 . 0
2,250.3
2,252.6
Adj usted
Time
(hou rs)
0.001 67
0 .00254
0. 00394
0.006 1 9
0. 00983
0 . 0 1 57
0.0252
0 . 0406
0 . 0655
0. 1 06
0. 1 72
0.278
0.452
0. 734
1 .19
1 .94
3. 1 5
5. 1 2
8.31
1 3.5
21 .8
35.2
56.7
91.1
1 46.0
Adj usted
Horner
Time Ratio
0.430 1 9 x 1 0 7
0.28331 x 1 0 7
0 . 1 8277 x 1 0 7
0 . 1 1 626 x 1 0 7
0. 73245 x 1 0 6
0.45849 x 1 0 6
0.2857 x 1 0 6
0 . 1 7745 x 1 0 6
0 . 1 0994 x 1 0 6
67,983.0
4 1 ,972.0
25,880.0
1 5 , 942.0
9 , 8 1 3.5
6.038.3
3 , 7 1 4.9
2,286.0
1 ,407.6
867.68
535.68
331 .42
205 . 6 1
1 28.02
80.07
50.376
Adj usted
Pressure
(psia)
782.54
782.73
783.04
783.53
784. 32
785.58
787.56
790.66
795.39
802.40
8 1 2 .25
825 . 0 1
839.58
853.39
863.57
869.39
872 .56
874.80
876. 79
878.67
880.48
882.24
883.96
885.66
887.33
Once we have identified this region on a derivative plot, we can
estimate permeability and skin factor using the semilog analysis .
4 . An upward or downward trend of the data at the end of the
test indicates the presence of a reservoir boundary . An upward trend
is characteristic of one or more boundaries having been encoun­
tered with the reservoir still open in at least one direction . An ex­
ample of this situation is a single well centered in a rectangular
reservoir . Similarly , a downward trend in a buildup test indicates
reservoir closure ; all boundaries , either no flow or constant pres­
sure , affect the pressure transient.
6. 12. 1 Procedure for Identifying Reservoir Type. The follow­
ing procedure provides a systematic approach for identifying reser­
voir type . To facilitate application of this procedure , we have
developed the reservoir identification worksheets in Appendix G .
1 . Using the methods described previously , plot t(MPa /d<1ta )
vs. t a and <1 Pa vs. t a for drawdown tests . For buildup tests ,
equivalent time , <1tae , should be calculated and used in the plot­
ting functions. In addition , prepare a specialized graph depending
on the reservoir model .
2 . As illustrated in Appendix G, perform a preliminary screen­
ing with the derivative type curve. This initial analysis is divided
into early-, middle- , and late-time regions . From the characteristic
derivative curve shapes discussed previously , the early-time analy­
sis indicates the presence of wellbore-storage and skin effects . Simi­
larly, the middle-time region analysis indicates whether the reservoir
is homogeneous-acting or has heterogeneities . Finally, if the test
duration is long enough , a late-time region analysis will provide
information on reservoir boundaries .
3 . Next, using all three plotting techniques (specialized graph,
log-log plot of the pressure/time data, and log-log plot of pressure­
derivative data) , confirm the preliminary results from Step 2. Note
that each plotting technique exhibits a unique pressure response .
However, rather than relying on one plot, use the information from
all three plots before selecting a reservoir model .
6 . 1 3 Summary
Reading this chapter should prepare you to do the following .
1 . Define the term " pressure-transient test, " list the two broad
categories of transient tests , and list and describe the maj or types
of tests in each broad category .
2 . State the purpose of pressure-transient testing .
1 61
PRESS U R E-TRAN SIENT TEST I N G OF GAS WELLS
TABLE 6.23-FL U I D PROPERTIES, EXERCISE 6 . 3
Pressure
(psia)
(scf/ft 3 )
8g
Viscosity
(cp)
1 4.6
1 00
200
300
400
500
600
700
800
900
1 ,000
1 , 1 00
1 ,200
1 ,300
1 ,400
1 ,500
1 ,600
1 ,700
1 ,800
1 ,900
2 , 000
2 , 1 00
2 , 200
2 , 300
2 ,400
2 , 500
2 , 600
2 , 700
2,800
2 , 900
0 .83037
5 . 7402
1 1 . 605
1 7.595
23.71
29.95
36. 3 1 2
42. 794
49.39
56. 096
62.904
69. 804
76. 785
83.835
90.941
98.085
1 05.25
1 1 2,42
1 1 9 .58
1 26 . 7 1
1 33.79
1 40.8
1 47.74
1 54.58
1 61 .31
1 67.92
1 74.41
1 80 . 76
1 86 . 98
1 93.04
0.01 238
0.01 245
0.01 255
0 .0 1 268
0 . 0 1 282
0 . 0 1 298
0.01 31 5
0 . 0 1 334
0 . 0 1 354
0 . 0 1 376
0 . 0 1 399
0 . 0 1 424
0 . 0 1 450
0 . 0 1 478
0 . 0 1 507
0 . 0 1 537
0.01 569
0 . 0 1 602
0 . 0 1 637
0 . 0 1 673
0 . 0 1 709
0 . 0 1 747
0 . 0 1 786
0 . 0 1 826
0 . 0 1 866
0.01 907
0 . 0 1 949
0 . 0 1 991
0 . 02033
0 . 02076
3. List the major assumptions on which the line-source solution
is based .
4. Define skin factor, write the line-source solution at a well in­
cluding the skin factor , and state four methods of translating the
skin factor into more concrete reservoir characterizations .
5 . Determine effective penneability and skin factor from graphical
analysis of an idealized constant-rate flow test .
6. Derive an equation modeling an ideal pressure-buildup test
and explain a graphical method to analyze data from this test.
7. Estimate effective penneability , skin factor, and initial drainage
area pressure from graphical analysis of an idealized pressure­
buildup test .
8. Sketch pressure vs. log (radius) diagrams for constant-rate pro­
duction and shut-in of ideal reservoirs , and use these diagrams to
explain the radius of investigation concept qualitatively .
9. Sketch and explain the time regions on flow and buildup test
plots .
10. Explain why wellbore storage distorts the shapes of flow and
buildup test data plots .
1 1 . Determine current drainage pressure for a new well and for
a well with pressure depletion , in the latter case using the MBH
method .
1 2 . Make the following determinations for a buildup test in a well
with single-phase flow of a gas .
A . End of wellbore-storage distortion .
B . Start of boundary effects .
C . Effective permeability to the produced fluid .
D . Skin factor, flow efficiency , and related measures of damage
and stimulation .
E. Current average reservoir pressure .
1 3 . Use semilog plots , pressure change , and pressure-derivative
log-log plots in a systematic , coordinated manner.
14. From a drawdown test conducted at either constant or smooth­
ly changing rate , calculate permeability and skin factor from in­
formation in the transient region .
1 5 . Identify and achieve the obj ectives of postfracture transient
test analysis using the methods below .
A . " Pseudoradial " method .
B . Analysis of linear or bilinear flow regime .
z
factor
---
0.9984
0 .9893
0 .9786
0.9682
0 .9580
0.9480
0 .9383
0 .9289
0 . 9 1 98
0.91 1 1
0.9027
0.8949
0.8875
0.8806
0 .8742
0.8684
0.8632
0.8587
0 .8548
0.85 1 5
0.8489
0.8469
0.8456
0 .8449
0.8449
0.8454
0.8465
0.8482
0.8504
0.8531
cg
(psia 2 /cp)
6.849 x 1 0 - 2
1 .001 x 1 0 - 3
5 . 1 08 x 1 0 - 3
3,440 x 1 0 - 3
2.606 x 1 0 - 3
2. 1 04 x 1 0 - 3
1 . 769 x 1 0 - 3
1 . 528 x 1 0 - 3
1 .347 x 1 0 - 3
1 .205 x 1 0 - 3
1 .090 x 1 0 - 3
9.946 x 1 0 - 4
9 . 1 40 x 1 0 - 4
8.445 x 1 0 - 4
7.837 x 1 0 - 4
7.298 x 1 0 - 4
6.81 4 x 1 0 - 4
6. 376 x 1 0 - 4
5.976 x 1 0 - 4
5.609 x 1 0 - 4
5.269 x 1 0 - 4
4.955 x 1 0 - 4
4.663 x 1 0 - 4
4.391 x 1 0 - 4
4. 1 38 x 1 0 - 4
3.902 x 1 0 - 4
3.682 x 1 0 - 4
3.476 x 1 0 - 4
3.284 x 1 0 - 4
3. 1 05 x 1 0 - 4
1 .7243 X 1 0 4
9,4755 x 1 0 5
3.3882 x 1 0 6
7.46 1 2 x 1 0 6
1 .3 1 64 x 1 0 7
2.0486 x 1 0 7
2.94 1 5 x 1 0 7
3. 9929 x 1 0 7
5. 2004 x 1 0 7
6. 5608 x 1 0 7
8.0707 x 1 0 7
9.7260 x 1 0 7
1 . 1 522 x 1 0 8
1 .3454 x 1 0 8
1 .551 6 x 1 0 8
1 . 7702 x 1 0 8
2 . 0007 x 1 0 8
2. 2424 x 1 0 8
2.4946 x 1 0 8
2.7567 x 1 0 8
3.0279 x 1 0 8
3.3077 x 1 0 8
3.5953 x 1 0 8
3.8901 X 1 0 8
4, 1 9 1 4 x 1 0 8
4,4987 x 1 0 8
4.81 1 4 x 1 0 8
5 . 1 290 x 1 0 8
5,4509 x 1 0 8
5. 7767 x 1 0 8
(psia - 1 )
Pp
C . Type-curve analysis using the Gringarten et al. , Cinco Ley ,
Agarwal , and Barker-Ramey type curves .
1 6 . Analyze buildup o r drawdown data from naturally fractured
formations using both semilog and type-curve techniques .
Questions for Discussion
1 . We can estimate a number of important formation properties
from flow and buildup tests . When we do, we follow certain proce­
dures (make plots and use methods to interpret these plots) ; we also
need certain data. The interpretation procedures are based on ideal­
ized models of reservoirs , which include a number of important,
and possibly limiting , assumptions . For each of the tests listed be­
low , state the properties obtainable from the analysis, the analysis
procedure (what to plot and how to interpret the plot) , the data re­
quired for interpretation and possible real-world sources of the data,
and assumptions stated or implied in the analysis technique.
A. Constant-rate tests .
B . Flow tests with smoothly changing rates.
C. Flow tests with discrete rate changes .
D . Two-rate flow tests .
E. n-rate flow tests .
F . Buildup tests with constant-rate production before shut-in .
G . Buildup tests preceded by two different rates.
H . Buildup tests preceded by (n - l ) different rates .
2. What typical " complications" might one encounter i n the
analysis of test data?
3 . Describe the characteristics of the "two-region" reservoir
model used as the basis for most transient test interpretation
methods . How is the skin factor related to altered zone properties?
Describe four methods that can be used to translate the abstract con­
cept " skin effect" into a more concrete characterization of a well ' s
condition .
4. Define pseudopressure, adjusted pressure , pseudotime , and
adjusted time . What is the purpose and value of these transformed
variables? How is adjusted time calculated for flowing times? Shut-in
times? What is the logic behind these methods of calculating pseu­
dotime? When are pressure and pressure-squared analyses adequate
for gas-well test analysis?
1 62
G A S RESERVO I R E N G I N E E R I N G
TABLE 6 .24-BUILDUP TEST DATA, EXERCISE 6 . 3
Time
(hours)
0.25
0.5
0.75
1
2
3
4
5
6
9
12
15
18
21
24
48
96
1 44
1 92
240
288
336
384
432
480
504
Pseudotime
(hr-psia/cp)
2.6492 x 1 0 4
5.6859 x 1 0 4
8.8275 x 1 0 4
1 .20 1 9 x 1 0 5
2.4967 x 1 0 5
3.8052 x 1 0 5
5. 1 225 x 1 0 5
6.4459 x 1 0 5
7.7731 x 1 0 5
1 .7777 x 1 0 6
1 .5803 x 1 0 6
1 .9845 x 1 0 6
2.3901 x 1 0 6
2.7967 x 1 0 6
3.2043 x 1 0 6
6.4837 x 1 0 6
1 .3 1 05 x 1 0 7
1 .9779 x 1 0 7
2.6488 x 1 0 7
3.3221 x 1 0 7
3. 9974 x 1 0 7
4.6741 x 1 0 7
5.3522 x 1 0 7
6.031 3 x 1 0 7
6.71 1 4 x 1 0 7
7.051 7 x 1 0 7
Adj usted
Time
(hours)
0. 1 82 1 8
0.39 1 0 1
0.60704
0.82651
1 . 71 69
2.61 67
3.5226
4.4327
5.3453
8.0986
1 0.867
1 3. 647
1 6.436
1 9.232
22. 035
44.586
90. 1 21
1 36 . 02
1 82 . 1 5
228.45
274.89
321 .43
368.05
4 1 4.76
461 .52
484.93
pp
(psia 2 /cp)
3.5632 x 1 0 8
3.9380 x 1 0 8
4 . 1 063 x 1 0 8
4. 1 950 x 1 0 8
4. 3229 x 1 0 8
4.391 1 x 1 0 8
4.4367 x 1 0 8
4.4708 x 1 0 8
4.4982 x 1 0 8
4.5529 x 1 0 8
4.5929 x 1 0 8
4.6241 x 1 0 8
4.6493 x 1 0 8
4.6707 x 1 0 8
4.6890 x 1 0 8
4. 7709 x 1 0 8
4.8568 x 1 0 8
4.91 07 x 1 0 8
4. 9484 x 1 0 8
4.9768 x 1 0 8
4.9991 x 1 0 8
5.01 75 x 1 0 8
5.0329 x 1 0 8
5.0461 x 1 0 8
5. 0577 x 1 0 8
5.0631 x 1 0 8
P.
(psia)
1 , 1 06.7
1 ,223 . 1
1 ,275.4
1 ,303.0
1 , 342 . 7
1 , 363. 9
1 ,378.0
1 ,388. 6
1 ,397. 1
1 ,4 1 4 . 1
1 ,426.6
1 ,436.2
1 ,444 . 1
1 ,450. 7
1 ,456.4
1 ,481 .8
1 ,508.5
1 , 525.2
1 ,537.0
1 ,545 .8
1 , 552 . 7
1 ,558.4
1 ,563.2
1 ,567. 3
1 ,570 . 9
1 ,572 . 6
5 . How can w e estimate average drainage area pressure for gas
wells? Why can ' t we use the same methods (identical in all detail)
as for oil wells?
6. In constructing type curves, what are the major differences
in the following categories that lead to different type curves : reser­
voir type , initial condition , outer boundary condition, inner bound­
ary condition , test type modeled? Why do these conditions matter
for a type curve that we might plan to use for test analysis?
7. How can we correct for an incorrect initial time or pressure
in a transient test? Why should a correction be necessary? When
should the correction be made? How could we determine whether
to correct time or pressure?
Exercises
6 . 1 . A 2 8-day drawdown test has been run on a new gas well that
was producing at a constant rate of 2 , 500 Mscf/ D . The reser­
voir and fluid properties are given below .
rw =
Pi =
h=
¢ =
T=
'Y g =
0 . 25 ft .
4 ,500 psia.
1 75 ft .
0 . 08 .
210°F.
0 . 726 (with 2 . 5 mol % H 2 S) .
The fluid properties are given in Table 6.20 as a function of
pressure .
Analyze the drawdown test data in Table 6.21 to obtain gas
permeability , skin factor, and radius of investigation at the end
of the flow test.
6 . 2 . A buildup test was run on a gas well that produced gas at a
constant rate of 3 1 2 MscflD for 7 , 200 hours (tp = 7 ,200
hours) . The BHP just before the well was shut in for the buildup
test was 2 , 1 00 psia ( P w[a = 782 . 1 6 psia) . The reservoir prop­
erties are given below.
rw =
Pi =
T=
h=
¢ =
Sw =
0.28 ft.
3 , 500 psia.
2 12°F.
8 8 ft.
0 . 1 8 (sandstone reservoir) .
0.22.
TABLE 6 . 25-BU I LD U P TEST DATA, EXERCISE 6.4
Time
(hours)
0 . 0 1 667
0.03333
0.08333
0 . 1 667
0.5
1
2
3
4
6
8
10
12
16
20
24
36
48
72
96
1 44
1 92
240
288
336
384
432
480
528
576
624
672
Pressure
(psia)
1 ,325 . 7
1 ,327. 1
1 ,329.4
1 ,331 .8
1 ,337.5
1 ,343.0
1 ,350.6
1 ,356.9
1 ,361 .2
1 ,369.6
1 ,375.4
1 ,38 1 . 1
1 ,386.7
1 ,394.4
1 ,402.2
1 ,409.6
1 ,424.5
1 ,438. 9
1 ,457.2
1 ,475.0
1 ,500. 9
1 ,52 1 .0
1 ,537.3
1 ,550.9
1 ,562.5
1 ,572.5
1 ,58 1 .2
1 ,589.0
1 ,595.9
1 ,602.2
1 ,608.0
1 ,61 3.2
TABLE 6 .26-PREFRACTURE BUILDUP TEST DATA,
EXERCISE 6 . 5
Time
(hours)
0.01 667
0.03333
0.08333
0 . 1 667
0.333
0.6667
1
2
3
4
6
8
10
12
16
20
24
36
48
72
96
1 44
1 92
240
288
336
Pressure
(psia)
1 ,992.6
1 ,995 . 1
2,002.8
2 , 0 1 5.3
2,039.8
2,087.2
2 , 1 33.3
2,262.5
2 , 380 . 1
2,487.7
2,674.5
2,829.9
2 ,958.5
3,064.8
3,226.8
3,340.6
3 ,422. 6
3,568.0
3,645. 7
3 ,733.5
3,786 . 1
3,849.7
3,889.2
3 , 9 1 7.8
3,939.4
3,956.5
Estimate gas properties at initial pressure , given that 'Y g =
0 . 6 with 8 % H 2 S . Also, use engineering judgment and assume
appropriate values for any properties you cannot calculate .
Analyze the test data in Table 6.22 as completely as possible .
6 . 3 . A new gas well has been produced for 9 1 days at a constant
BHFP of 450 psia. Cumulative production is 76,272 Mscf.
1 63
PRESS U R E-TRAN SIENT TEST I N G OF GAS WELLS
TABLE 6 .27-POSTFRACTURE BUILDUP TEST DATA,
EXERCISE 6 . 5
Time
(hou rs)
Pressure
(psi a)
0.01 667
0 . 03333
0 .08333
0 . 1 667
0.5
1
2
3
4
6
8
10
12
16
20
24
36
48
72
96
1 44
1 92
240
288
336
384
432
480
528
576
624
672
2 , 007.4
2 , 020.3
2 , 05 1 .3
2 , 086.5
2 , 1 56.9
2 , 203.2
2,253.4
2,285.5
2 ,309.3
2 ,344.5
2 , 370.2
2 , 39 1 . 1
2 ,409 .9
2 ,440 .5
2 , 466 . 1
2 ,487. 8
2 , 538.4
2 , 576.8
2 ,633.5
2,678 . 1
2 , 747.6
2 , 802.7
2,847.3
2,885 . 3
2 , 9 1 8.7
2 , 948.4
2,975.2
2 , 999.7
3 , 022.4
3,Q43.7
3 , 063.6
3 , 082.6
TABLE 6 . 28-PRESSU REITIME DATA, EXERCISE 6 . 6
The flow rate at the end of the 9 1 -day flow period is 8 1 8
MscflD . The reservoir and fluid properties are given below .
r w = 0 . 475 ft .
h
=
I/> =
T=
"I g =
37 ft .
0. 12.
1 65 ° F .
0.7.
The fluid properties are given i n Table 6_23 a s a function
of pressure .
Analyze the buildup test data in Table 6.24 obtained from
a 504-hour test using a bottornhole shut-in to minimize the ef­
fects of wellbore storage . Estimate formation permeability ,
skin factor , initial reservoir pressure, and radius of investiga­
tion at the end of both the flow period and the buildup test.
6 . 4 . A 28-day pressure-buildup test has been run on a newly fractured well . The reservoir and fluid properties are given below.
r w = 0 . 3 ft .
Pi = unknown .
cf
h
=
=
I/> =
T=
"I g =
6 . 0 x lO - 6 psia - 1 .
77 ft .
0 . 09 .
0 1 60 ° F .
0 . 7 1 0 (no impurities) .
The well produced 482 . 5 MMscf during the 1 80-day flow
period immediately before the buildup . The flow rate was es­
sentially constant during the flow period. At the instant of shut­
in, the BHFP was 1 , 3 1 8 psia and the flow rate was 2 ,43 1 . 6
MscflD . The shut-in BHPs in Table 6.25 were measured. Ana­
lyze the test as thoroughly as possible . Try to determine for­
mation permeability , skin factor and an estimate of initial
pressure (from analysis of the pseudoradial flow period) , and
fracture half-length (from type-curve matching and/or analy­
sis of the linear or bilinear flow period) . If you cannot deter­
mine all these properties , explain why not .
t
(days)
(psia)
0.041 67
0. 08333
0. 1 25
0 . 1 667
0.2083
0.25
0 . 29 1 67
0. 3333
0.375
0.41 67
0 .8333
1 .25
1 .6667
2 .0833
2.5
2 .9 1 67
3.3333
3.75
4. 1 667
8.3333
1 2. 5
1 6. 667
20.833
25.0
29. 1 67
33.333
37.5
4 1 .667
83.333
1 25.0
1 66.67
208.33
250.0
291 .67
333.33
365. 0
4,492.388
4,489.76
4,487.901
4,486.401
4,485 . 1 24
4,484.001
4,482.992
4,482.074
4 ,481 .229
4,480.446
4,474.676
4 ,470. 863
4,468.008
4,465.721
4,463.803
4 ,462 . 1 94
4,460.737
4,459.447
4,458.344
4,450 .635
4,446.061
4,442. 793
4,440.248
4 ,438 . 1 52
4,436.353
4,434. 752
4,433.289
4,431 . 9 1 9
4,41 9 .905
4,408.372
4,396.849
4,385 .323
4,373.785
4,362.254
4,350.72
4,34 1 .954
P wt
6 . 5 . A new gas well produced for 28 days and was then shut-in
for a 1 4-day prefracture pressure-buildup test. The well was
then hydraulically fractured and allowed to produce for 1 80
day s , and a 28-day postfracture buildup test was conducted.
In both buildup tests , a bottornhole shut-in was used to
minimize the effects of wellbore storage . The reservoir and
fluid properties are given below .
rw = 0 . 25 ft .
Pi = unknown .
5 . 5 x l O - 6 psia - 1 .
5 1 ft .
0 . 04 .
2 1 5 °F .
0 . 6 1 (0 . 5 % H 2 S , 2 . 5 % CO 2 , 1 % N 2 ) .
The well produced 3 . 1 42 MMscf during the 2 8-day flow
period immediately before the prefracture buildup test. The
flow rate was essentially constant during the flow period . At
the instant of shut-in, the BHFP was 1 ,990 psia; the flow rate
was 1 04 MscflD. The shut-in BHPs in Table 6.26 were meas­
ured in the prefracture test .
The well produced 1 3 7 . 7 MMscf during the 1 80-day flow
period immediately before the postfracture buildup test. The
flow rate was essentially constant during the flow period . At
the instant of shut-in, the BHFP was 1 ,990 psia; the flow rate
was 4 1 5 MscflD . The shut-in BHP ' s in Table 6.27 were meas­
ured in the postfracture test .
Analyze both pre- and postfracture pressure-buildup tests
as completely as possible . Try to determine formation perme­
ability, skin factor, an estimate of initial pressure, fracture half­
length, and fracture conductivity . If you cannot determine all
these properties , explain why not.
6 . 6 . The data in Table 6.28 are for 1 year of pressure history
(BHFP) from an Austin Chalk well . Unfortunately , the opercf
=
=
I/> =
T=
"I g =
h
1 64
GAS RESERVOI R E N G I N E E R I N G
TABLE 6 . 29-DRAWDOWN TEST DATA, EXERCISE 6 . 8
Time
(hou rs)
0. 024
0.048
0.072
0. 1 2
0. 1 92
0.24
0.48
0.72
1 .2
1 .44
2.04
2. 1 6
2.4
26.4
50.4
1 94.4
242 .4
362.4
482 .4
602.4
722.4
842. 4
962.4
1 ,226.4
1 ,466.4
1 ,706.4
1 ,946.4
2 , 1 86.4
2,546.4
4 ,946.4
7 ,346.4
Pressure
(psia)
4 , 566
4 , 559
4,554
4,547
4,540
4,536
4,521
4,51 1
4,495
4,489
4,475
4,473
4,468
4 , 278
4, 1 79
3,856
3,783
3,632
3,501
3,405
3,31 3
3,230
3 , 1 54
3,006
2 , 889
2 , 783
2 , 686
2,596
2,472
1 , 826
1 ,298
P ressu re
Derivative
1 0. 092
1 1 . 500
1 2. 922
1 4. 294
1 6. 839
1 8.71 6
23.499
27.536
34.327
37.235
44.398
44.61 5
47. 577
1 35.61
1 77.64
309.31
335.46
385 .52
425 . 1 7
456. 08
483 . 8 1
506.25
525 . 86
562.71
590.00
61 2. 1 1
628.90
646.95
667.99
74 1 .65
760. 79
Adj usted
Pressure
(psia)
3,036.4
3,029.4
3,024.4
3 , 0 1 7.4
3 , 0 1 0.4
3,006.4
2 , 99 1 .5
2,981 .5
2 , 965.5
2 ,959.6
2 , 945 .6
2 , 943.6
2 , 938 .6
2 , 750 . 1
2 , 652.5
2 , 337.5
2 , 267.2
2 , 1 23.0
2 , 007.9
1 , 909. 8
1 , 824.8
1 , 748 . 8
1 , 679 . 9
1 ,547.7
1 ,445 . 1
1 ,353. 9
1 , 271 .9
1 , 1 97 . 1
1 , 096.5
625 .43
324.50
Adj usted
Pressure
Change
(psi)
31 .980
38.975
43.972
50.958
57.943
61 .935
76.904
86.883
1 02.83
1 08.81
1 22.76
1 24.75
1 29 . 73
31 8.27
41 5.89
730.84
80 1 . 1 3
945 .33
1 ,060.5
1 , 1 58.6
1 ,243.6
1 ,3 1 9.6
1 ,388.5
1 ,520.7
1 ,623.2
1 ,71 4.5
1 , 796.5
1 ,871 .2
1 ,971 .8
2 ,442 .9
2 , 743 .9
ators did not perform a prefracture buildup test on this well ,
so we must analyze the drawdown data and hope that there
is enough data to determine formation permeability . The well
has produced at a constant rate of 300 Mscf/D for 1 year. Well
and reservoir data include the following .
=
=
=
=
=
=
=
=
=
=
=
ct =
Pi
"( g
T
h
fig
jig
r/>
Sw
rw
Cw
cf
4 , 500 psia .
0.7.
1 80 ° F .
6 0 ft .
0 . 68220 RB/Mscf.
0 . 02683 cp o
0. 1 17.
0 . 46 .
0 . 365 ft .
3 . 5 x 1 O - 6 psia - I .
4 . 0 x l O - 6 psia - I .
8 . 4939 x 1 0 - 5 psia - I .
Required.
A . Estimate gas permeability .
B . Estimate fracture half-length .
C . Estimate the distance to the boundary , Le , assuming that
the well is centered in a square drainage area .
D . List the three "types" of analysis possible with this test
and state the results that can be obtained with each method .
E . Why does this well exhibit l inear but not bilinear flow?
How can you tell?
F. Explain, in your own words , how a prefracture buildup
test could have increased the economic feasibility of run­
ning a postfracture buildup test .
G. How can we calculate depth of investigation for hydrauli­
cally fractured wells?
6 . 7 . A prefracture buildup test indicated a gas permeability of 0 . 079
md for this well . After a massive hydraulic fracture treatment,
the well produced at a constant BHP with the following rate
profile .
TABLE 6 . 30-BU ILDUP TEST DATA, EXERCISE 6 . 9
Time
(hou rs)
0.01 67
0.0333
0.05
0 .0833
0.1 1 7
0 . 1 67
0.333
0.5
0.75
1
1 .5
2
3
5
7
10
12
15
18
24
36
48
72
96
1 44
1 92
240
288
336
384
432
480
528
576
672
744
Pressu re
(psia)
557.3
6 1 0.5
662.4
761 .5
854 . 1
981 . 1
1 ,345 .9
1 ,632 .8
1 ,953 .8
2, 1 64.2
2,439 .7
2,59 1 . 5
2 , 740.5
2,844.8
2,887.6
2 , 923.6
2 , 940 .2
2 , 959.4
2 , 975 . 1
3,000 . 1
3,039.4
3,072 . 1
3, 1 24.7
3 , 1 67.8
3,232.6
3,28 1 .7
3,320.4
3,352.3
3,378 .6
3,400 .8
3,41 9 . 9
3 ,436.6
3,45 1 .5
3,464.8
3,488 . 1
3,503.5
t
(days)
q
(McflD)
1
4
13
40
121
365
7 , 747 . 5
4,719.2
3,419.4
2 ,678 . 2
2 , 1 89 . 9
1 , 844 . 2
Well and reservoir data include the following .
Pi =
Pp ( Pi ) =
P wf =
Pp ( P wf ) =
"{ g =
T=
h=
fig =
jig =
r/> =
Sw =
rw =
cw =
cf =
cg =
3 ,450 psia.
7 . 36 1 6 x 108 psia2 /cp .
400 psia.
1 . 2405 X 1 07 psia2 /cp .
0 . 675 .
220 ° F .
1 03 ft .
0 . 92327 RB/Mscf.
0 . 02089 cp o
0.08.
0.4.
0 . 33 ft .
3 . 6 x l O - 6 psia - I .
4 . 0 x l O - 6 psia - I .
2 . 484 x 1 0 - 4 psia - I .
Required.
A. Estimate fracture length and fracture conductivity .
B . Comment on the analysis that would have been possible
without the prefracture test data.
6 . 8 . A hydraulically fractured well produced at a constant rate of
1 50 MscflD for 300 days, and drawdown data were recorded
for this period . Well and reservoir data include the following .
1 65
PRESSU R E-TRANSIENT TESTI N G OF GAS WELLS
Pi =
'Yg =
T=
h=
cP=
Sw =
'w =
Pa ( P i) =
Bg =
ilg =
Cw =
cf =
cg =
4598 psia.
0 . 744 .
250 ° F .
30 ft .
0 . 07 6 .
0.33.
0 . 3 3 ft .
3068 . 36 psia.
0 . 7696 1 RB/Mscf.
0 . 02364 cp o
.
l
3 . 6 x l O - 6 pSla - .
I
4 . 0 x l O - 6 psia - .
1 . 5 3 x 1 0 - 4 psia - I .
Drawdown test data are recorded in Table 6.29. Analyze
the test to obtain estimates of formation permeability , frac­
ture length, and fracture conductivity .
6 . 9 . A gas well in a suspected dual-porosity reservoir produced for
366 days and then was shut in for a 3 1 -day pressure-buildup
test. The well produced a total of 603 MMscf of gas during
the flow period . The production rate was 1 ,537 MscflD; the
FBHP at the moment of shut-in was 500 psia . Well and reser­
voir data include the following .
Pi =
Cf=
h=
cP=
'w =
unknown.
4 x 1 0 - 6 psia - I .
1 1 0 ft .
0 . 055 .
0 . 3 ft .
T= 120°F.
'Y g = 0 . 6 (no impurities) .
Buildup test data are recorded in Table 6.30. Analyze the
test as thoroughly as possible . Try to determine estimates of
initial pressure, gas permeability , skin factor, interporosity flow
coefficient, and storativity ratio . If you are unable to deter­
mine all these properties , explain why not .
Nomenclature
a =
A =
A Wb =
b =
b' =
bD =
B=
Bg =
Bg =
C=
C =
Cma =
Ct =
ct =
cwb =
c I , c2 , c 3 ,
c4 , c5 =
C =
C =
CD =
A
CfD
=
(CD)f +ma =
Cr
D
=
=
Ei( -x)
=
major axis of elliptical flow regime
drainage area, L 2 , ft 2
wellbore area, L 2 , ft 2
intercept of ( Pi - P wf)lq n plot, psi/STB-D
intercept on Odeh-Jones plot
dimensionless intercept in plot of PpD vs. log tD
FVF , L 3 IL 3 , res vollsurface vol
5 . 04 Tzlp , FVF gas , 0 /L 3 , RB/Mscf
gas FVF evaluated at p, 0 10 , RB/Mscf
1
compressibility , mILt, psi compressibility evaluated at p , mILt, psi - I
matrix compressibility , mILt, psi - 1
1
total compressibility , mILt , psi total compressibility evaluated at p, mILt, psi - I
compressibility evaluated at average pressure and
temperature in wellbore , mILt, psi - I
constants in non-Darcy flow correlations
wellbore-storage coefficient, bbllpsi
shape constant or factor
0 . 8936 Cl cf> ct h'�, dimensionless wellbore-storage
coefficient (slightly compressible liquid)
0 . 8936 Cl cpcthL} , dimensionless wellbore-storage
coefficient for fractured wells (slightly
compressible liquid)
0 . 8936 Cl cpcth'�, dimensionless wellbore-storage
coefficient (slightly compressible liquid) based
on matrix and fracture system properties in
naturally fractured reservoir
wkf l7rLfk, dimensionless fracture conductivity
non-Darcy flow coefficient , tlO , D/Mscf
oo
(e - u lu)du
-j
x
Gp =
h=
hf =
hft
hma
h mat
hp
ht
k
=
=
=
=
=
=
k=
kf =
kft =
kg =
kma =
ks =
Le =
Lf =
Ls =
m =
m* =
m' =
mB
mL
=
=
=
=
P=
p =
p' =
P* =
M
n
Pa =
=
p� =
pa
P aD
p�
=
=
P a ,i =
=
P a , ws =
P a ,wf =
Pa, l =
P a , MBHD
hr
PD =
PD
P MBHD
Pi
Pn
Po
Pp
=
=
=
=
=
=
PpD =
=
Pp MBHD
Ppi =
Ppwf =
P sc =
PwD =
Pwf =
cumulative gas production , L 3 , Mscf
net pay thickness , L , ft
individual fracture thickness in naturally
fractured reservoir , L, ft
total or net fracture thickness , L , ft
thickness of individual matrix block, L, ft
total or net matrix thickness, L, ft
height of net perforated interval , L, ft
total net formation thickness , L, ft
effective formation permeability , L 2 , md
(kf hft +kma h mat )lh, average permeability of
naturally fractured reservoir , L2 , md
fracture permeability , L2 , md
permeability of choked portion, L 2 , md
permeability to gas , L2 , md
matrix permeability , L2 , md
permeability of damaged fracture face, L2 , md
perpendicular distance from well to reservoir
boundary , L, ft
length of one wing of vertical fracture with two
equal-length wings, L, ft
length of choked portion of fracture , L, ft
slope of semilog straight line, psi/log 10 cycle
m12 , psi/log l O cycle
1 62 . 6BpJkh , slope of drawdown curve with
( Pi -P wf )lq as abscissa; 1 62 . 6 BlJ.lkh, slope of
Odeh-Jones plot, psi/STBID-cycle
slope of linear region of plot, psi/hr 'A
slope of square-root-of-time-plot straight line ,
psi/hr �
molecular weight of gas , lbm/lbm-mol
h ma/h ma =hftlhf , number of fractures
pressure , m/Lt2 , psi
static drainage area pressure , m/Lt2 , psi
derivative of pressure, m/Lt2 , psi/hr
MTR pressure trend extrapolated to infinite shutin time , m/Lt2 , psi
(jic/2j5)pp , adjusted pressure , m/Lt2 , psi
adjusted pressure evaluated at p, mILt2 , psia
adjusted pressure extrapolated to Horner time
ratio of unity , m/Lt2 , psia
dimensionless wellbore pressure response
dimensionless wellbore pressure response with
turbulent flow
adjusted pressure evaluated at Pi ' m/Lt2 , psia
adjusted P MBHD , dimensionless
adjusted pressure evaluated at Pws ' m/Lt2 , psia
adjusted pressure evaluated at P wf' m/Lt2 , psia
adjusted pressure at 1 hour test time and on
semi-log straight line , m/Lt2 , psia
khApI141 .2qBIJ. , dimensionless pressure (slightly
compressible liquid)
dpD /d(tD I CD ) , dimensionless pressure derivative
2.303 ( p* -p)lm , dimensionless
original reservoir pressure , m/Lt2 , psi
normalized pressure , m/Lt2 , psia
base pressure , m/Lt2 , psi
2
rpl [lJ.( p)z( P)]dp, gas pseudopressure , psia2 /cp
o
dimensionless pseudopressure , dimensionless
dimensionless pseudopressure P MBHD ,
dimensionless
pseudopressure evaluated at Pi ' psia2 /cp
pseudopressure evaluated at P wf, psia2 /cp
pressure at standard conditions, m/Lt2 , psia
dimensionless pressure evaluated at P wf or Pws
FBHP at shut-in in buildup and drawdown tests ,
m/Lt2 , psi
1 66
GAS RESERVO I R E N G I N E E R I N G
P ws = shut-in BHP, m/Lt2 , psi.
.
.
= pressure on semilog straIght lme at test time of 1
hour, m/Lt2 , psia
P !hr = pressure on half-slope semilog line at test time of
1 hour, mIlt 2 , psia
IIp = pressure change from start of transient test,
m/Lt2 , psi
IIp* = P * -Pw , m/Lt2 , psi
IIp a = change in adj usted pressure from start of
transient test , m/Lt2 , psi
Il PD = khllpI 1 4 1 . 2qBp. , dimensionless pressure response
amplitude
Ilps = 1 4 1 . 2qBp.(s)lkh = O. 869ms, additional pressure
drop across altered zone , m/Lt2 , psi
IIp = pseudopressure change, m/Lt 2 , psi
= liquid flow rate at surface, L 3 It , STBID
q* = modified flow rate (Odeh-Selig method) , L 3 It,
STBID
qD = 1 4 1 . 2qBp.lkhllp, dimensionless flow rate (slightly
compressible liquid)
qg = gas flow rate at surface, L 3 It, MscflD
ijg = average gas flow rate , L 3 It , MscflD
q l ast = most recent flow rate (Horner approximation) ,
L 3 It, MscflD
q = last flow rate of n different rates, L 3 It, STBID
q = sandface flow rate after shut-in, L 3 It, STBID
r = distance from wellbore center, L, ft
rD = rlr W ' dimensionless radius
re = drainage radiu s , L, ft
reD = dimensionless drainage radius , reI r w
r i = radius of investigation, L, ft
rs = radius of altered zone near the well, L, ft
r w = wellbore radius , L, ft
r wa = r w e - s , effective wellbore radius , L, ft
s = skin factor , dimensionless
f = log(kl q, p. ct r�) - 3 .23 + 0 . 869s
s' = s +Dq g ' effective skin factor for gas well ,
dimensionless
sfs = (1['/2)(ws ILf )[(klks) - l ] , fracture-face skin factor,
dimensionless
sJ�S ' Ch = skin factor representing choked fracture at
sandface
t = time , t, hours
t* = time at intersection of Flow Regimes 2 and 3 for
naturally fractured reservoirs , t, hours
ta = jiCtap ' adjusted time , t, hours
tAD = O .OO02637ktlq,p.ctA , dimensionless time based on
drainage area
(tAD)pss = dimensionless time at beginning of pseudo­
steady-state flow
P l hr
�
;
tap =
I
o
t
dtlp.ct , pseudotime , hours-psia/cp
tb2 ,lltb = time at start of Flow Regime 2 for naturally
2
fractured reservoirs , t, hours
tD = O. OOO2637ktlq,p.ctr� , dimensionless time (slightly
compressible liquid)
tD* = dimensionless time at intersection of Flow
Regimes 2 and 3 for naturally fractured
reservoirs , t, hours
tLofD = O. OOO2637ktlq,p.ctL , dimensionless time referred
to fracture half-length (slightly compressible
liquid)
tLtD ,max = dimensionless time achievable in the longest flow
and shut-in periods possible
tn = normalized time , t, hours ; final (nth) time in
variable-rate sequence , t , hours
tp = effective producing time , hours ; pseudoproducing
time , 24 Gp lq last
}
t;
tp ,aetua l
tp ,mm
.
tPa
tpa ,actuaI
tpa ,nun
.
tpss
ts
t wbs
t *I
tID
Ilt
Ilta
Iltae
Ilte
Iltend
IltenL
IltenR
Iltx
T
Vma
Vwb
wf
wf kf
W
�
y
z
z
IX
{3
{3'
l'
�
A'
p.
p. g
jig
p. g , w1
P wb
q,
q,f
q, ma
w
w
'
= modified production time i n Odeh-SeJig analysis,
t, hours
= effective producing time in actual test, t, hours
.
= minimum producing time reqUIred to move
.
beyond wellbore-storage distortion (Odeh-SelIg
method) , t, hours
= producing time expressed as adjusted time , t, hours
= adjusted producing time in actual test, t, hours
.
.
= minimum adJ' usted producing tIme requlfed to
.
.
move beyond wellbore-storage dIstortI On , t ,
hours
= time required to achieve pseudosteady state, t,
hours
= time for well to stabilize , t, hours
= duration of wellbore-storage distortion, t, hours
= time at intersection of Flow Regimes 1 and 2
(naturally fractured reservoirs) , t, hours
= dimensionless time at intersection of Flow
Regimes 1 and 2 for naturally fractured
reservoirs , t, hours
= time elapsed from start of transient test , t, hours
= adjusted time elapsed from start of transient test,
t, hours
= Iltal( l + Iltaltpa) ' adjusted equivalent time , t ,
hours
= Iltl( l + Iltltp ) , equivalent time , t, hours
= time MTR ends , t, hours
= Agarwal ' s equivalent time for linear flow , t ,
hours
= Agarwal ' s equivalent time for radial flow , t ,
hours
= time at which middle- and late-time straight lines
intersect , t, hours
= reservoir temperature, T, O R
= fraction of total system (fracture + matrix) made
up of matrix
= wellbore volume, L 3 , bbl
= fracture width , L, ft
= fracture conductivity , md-ft
= depth of fluid-loss damage into formation, L, ft
= independent variable in equation of straight line
= dependent variable in equation of straight line
= gas-law deviation factor, dimensionless
= gas-law deviation factor evaluated at j5,
dimensionless
= interporosity-flow shape factor
= 1 . 88 x 1 0 1 0 k - 1 .47 q - 0 .53 , turbulence parameter
a
= { 1 . 89 1 4[(CDe 2 s )f + maIAe - 2s ] } m , Bourdet et at.
type-curve parameter (for slab-matrix blocks)
= gas specific gravity , air = 1 .0
= IXkm hr�/kf hfth;' , interporosity-flow coefficient,
dimensionless
= (IXkmhmr1kf hft)(r�/h;')
= viscosity , mILt, cp
= gas viscosity , mIlt, cp
= gas viscosity evaluated at j5 , cp
= pressure-dependent gas viscosity evaluated at Pwf'
cp
= density of liquid in wellbore , Ibm/ft 3
= total porosity , fraction
= fracture porosity , fraction
= matrix porosity , fraction
= (q,cV)f l(q,cV)t , storativity ratio, dimensionless
= ( l - w)lw
Subscripts
f = fracture
rna = matrix
1 67
PRESSU R E-TRAN SIENT TESTI N G OF GAS WELLS
MP
=
r =
USL
=
match point
reference
unit-slope line
References
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Chapter 7
Deliverability Testing
of
Gas Wells
7.1 I ntroduction
This chapter discusses the implementation and analysis of the four
most common types of gas-well deliverability tests : flow-after-flow,
single-point , isochronal , and modified isochronal tests . We begin
with a summary of the fundamental gas-flow equations , both theo­
retical and empirical , used to analyze deliverability tests . These
equations are presented in terms of either pressure squared or the
real-gas pseudopressure function . Next , we focus on specific tests
and discuss testing procedures, advantages and disadvantages of
each testing method , and common analysis techniques . We also pro­
vide examples illustrating the analysis of deliverability tests .
7.2 Types and Pu rposes of Deliverab llity Tests
Deliverability testing refers to the testing of a gas well to measure
its production capabilities under specific conditions of reservoir and
bottomhole flowing pressures (BHFP ' s) . A common productivity
indicator obtained from these tests is the absolute open-flow (AOF)
potential . The AOF is the maximum rate at which a well could flow
against a theoretical atmospheric backpressure at the sandface.
Although in practice the well cannot produce at this rate, regulato­
ry agencies often use the AOF to establish field proration sched­
ules or to set maximum allowable production rates for individual
wells .
Another, and possibly more important, application o f delivera­
bility testing is to generate a reservoir inflow performance rela­
tionship (IPR) or gas backpressure curve . The IPR curve describes
the relationship between surface production rate and BHFP for a
specific value of reservoir pressure ( i . e . , either the original pres­
sure or the current average value) . The IPR curve can be used to
evaluate gas-well current deliverability potential under a variety
of surface conditions , such as production against a fixed backpres­
sure . In addition, the IPR can be used to forecast future production
at any stage in the reservoir ' s life .
Several deliverability testing methods have been developed for
gas wells . Flow-after-flow tests are conducted by producing the
well at a series of different stabilized flow rates and measuring the
stabilized bottomhole pressure (BHP) . Each flow rate is established
in succession without an intermediate shut-in period . A single-point
test is conducted by flowing the well at a single rate until the BHFP
is stabilized . This type of test was developed to overcome the limi­
tation of long testing times required to reach stabilization in the
flow-after-flow test .
The isochronal and modified isochronal tests also were devel­
oped to shorten test times for wells that need long periods to stabi-
lize. An isochronal test consists of a series of single-point tests usual­
ly conducted by alternately producing at a stabilized (or slowly
declining) sand face rate and then shutting in and allowing the well
to build to the average reservoir pressure before the next flow peri­
od . The modified isochronal test is conducted similarly , except the
flow periods are of equal duration and the shut-in periods are of
equal duration (but not necessarily the same as the flow periods) .
7.3 Theory of Deliverability Test Analysis
This section summarizes the theoretical and empirical gas-flow
equations used to analyze deliverability tests . The theoretical equa­
tions , developed by Houpeurt, l are exact solutions to the gener­
alized radial-flow diffusivity equation , while the Rawlins and
Schellhardt2 equation is derived empirically . All basic equations
presented here were developed with radial flow in a homogene­
ous , isotropic reservoir assumed and therefore are not applicable
to the analysis of deliverability tests from reservoirs with heter­
ogeneities , such as natural fractures or layered pay zones. These
equations also cannot be used to analyze tests from hydraulically
fractured wells , especially during the initial , fracture-dominated,
linear flow period . Finally , these equations assume that wellbore­
storage effects have ceased . Unfortunately , wellbore-storage dis­
tortion may affect the entire test period in short tests , especially
those conducted in low-permeability reservoirs .
7.3.1 Theoretical Deliverability Equations. The generalized diffu­
sivity equation for radial flow of a real gas through a homogene­
ous , isotropic porous medium is
0
1 ( P -op ) 1 = -t/JctP -op . . . . . . . . . . . . . (7 . 1)
Il-g Z or 0 . 0002637 kgz ot
7. 1
7. 1
p/ll-g Z
Ilg
Ct
7. 1 7. 1
��(/p ) = . t/Jll-g Ct k oP . . . . . . . . . . . . . . . . . . . . (7 . 2)
r o r o r 0 0002637 g ot
-- r
r or
--
Because o f the pressure dependence o f the gas properties , Eq .
is a nonlinear partial-differential equation ; however, Eq .
can be linearized partially with several limiting assumptions . If we
assume that the quantity
is constant with respect to pressure
can be evaluated at p and treated as constant, Eq.
and that
can be solved in terms of pressure . With these assumptions,
Eq .
becomes
,
1 69
D ELIVERABI LITY TESTI N G O F GAS WELLS
2SOOOO ,-----,
sa = 0.6
2SOOOO
200000 -
c..
1SOOOO
.�
;;;
c..
..
:r 100000 Q.
SOOOO -
,..'
= 0.6
. ... . . _ ... . . ..- . . - .. -.. .. .. .. SO
•...
- ,. .. . .
_
'.,': .. ..... ..- . -. _ - .
I. ...
"
1/�
.
.
.
-. -
2000
-.
0
------
.
0.8
.. - +
��,.
*
I
SOOO
Pressure. psia
6000
..
..
which is the same linear differential equation that we solve for slight­
ly compressible liquid flow . The solution is the familiar exponen­
tial integral (Ei) solution .
The assumptions made to obtain Eq. 7 . 2 generally are valid only
for very high pressures and temperatures . Figs. 7 . 1 through 7.3,
which show the relationship between pressure and
g Z at differ­
ent temperatures for various gas gravities , illustrate that P- g Z is
essentially constant with pressure at pressures exceeding 3 ,000 psia
for 1 00 ° F , 5 ,000 psia for 200 ° F , and 6 ,500 psia for 300 ° F . Note
that l g z varies with pressure more at high gas gravity . Figs . 7 . 1
through 7 . 3 imply that the solutions to the real-gas diffusivity equa­
tion in terms of pressure should be used only for gases at very high
pressures .
As an alternative solution, we can rewrite Eq. 7 . 1 in terms of
pressure-squared variables using the following relationships :
pip- pi
p p-
1
.
2
1
and
2
.
.
. . . .
. . ..
.
.
. .
.
.
.
.
.
.
. .
. . . . . . . . . . (7 . 3)
.
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 4)
Substituting Eqs . 7 . 3 and 7 . 4 into Eq. 7 . 1 yields the resulting
equation :
( r op 2 ) =
a;
� or
1 a
P- g Z
cPC t
op 2
0 .0002637kg Z at
.
.
..
.
. . . . . . (7 . 5)
.
.
Jf
If we assume that P- g Z is constant with pressure and that P- C
can be evaluated at ji and treated as constant, Eq . 7 . 5 beco e
2SOOOO ,-------------________________�
c..
�
200000 -
.;;;
l SOOOO -
Q.
1 00000
�
SOOOO -
/.
"
..
.
..
, . ,II"
. .
� ..
-
. .. .
-..
:r
Q.
50000
:
...... . . . .. . . .. . .. . ..
.
so
..... .- .
= 0.6
- .. � .. - . _
so
,
= o.S
���:i�.�(
.
.
. � *.
SO =
1.2
--­
O �---,r----.----r---�----�
o
2000
4�
�
8000
I�
Pressure. psi a
Fig. 7.3-Range of applicability of pressure methods at 300 o F .
�..
..... -..--... -._- .
.
.... .. ... .. .... .....
... - Il _
. _.
..
.. .
, .. . .. . ..
.
J'/
,
!
.
,;.'/
.
.
..
..
i.
SO = 0.8
..
.. . .........
.
.../-..-..­
_
... .. . ... . .. .
.
.
.. ... .
.
so = 1 .0
-L
- . .....
.... - · -.. ....
,
. .. . . .._
.
:. '
... . . -
SO = 1 .2
O �----,I ----.----,�
I ----.I ----�
o
4000
I�
8�
Pressure. psia
2000
6000
Fig. 7.2-Range of applicability of pressure methods at 200 ° F .
�!.. (r op 2 ) cPP-g Ct op 2
at
r or or
, . . . . . . . . . . . . . . . . . (7 . 6)
0 . 0002637kg
p2
which is a linear differential equation in the variable
and thus
has the familiar Ei-function solution. The assumption of constant
P- g Z is valid only for very low pressures and gas gravities at high
temperatures . Figs. 7.4 through 7.6 illustrate the variation of the
product P- g Z with pressure for different gas gravities at 1 00, 200,
and 300°F, respectively . Note that P- g Z is essentially constant with
pressure at pressures < 1 ,200 psia for 100 ° F , < 1 ,750 psia for
200 ° F , and < 2 ,200 psia for 300 ° F . The P- g Z product varies with
pressure more at higher gas gravities. Therefore, solutions to the
real-gas diffusivity equation should be used in terms of pressure
squared only at very low pressures and gas gravities at high tem­
peratures .
A more rigorous method of linearizing Eq . 7 . 1 is with the real­
gas pseudopressure transformation introduced by Al-Hussainy et
at. , 3
!, P
P
pp =2 J -dp
P- g Z
Po
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
. .
.
.
. .
.
.
(7 . 7)
With the pseudopressure transformation, Eq. 7 . 1 can be solved
without the limiting assumptions that certain gas properties are con­
stant with pressure . Eq. 7 . 1 can be written in terms of the pseu­
dopressure function as
�!.. (r OPp ) =
r or or
cPP-g ( p) Ct( p)
oPP
0 . OOO2637kg at
.
.
.
.
.
•
•
.
.
.
.
.
•
•
.
•
•
.
(7 . 8)
0.20 ,----,
.. · .... · .. - · · ... · · 7· - · · .... ·
-.. � . - .
1 00000
,........
b./
....... .. •· .. · .. • · .... .. ..
·
1SOOOO
�
Fig. 7 . 1 -Range of applicability of pressure methods at 1 00oF.
op op 2
p-=-or or
op op 2
p-=-at
at
;;;
c..
..
SO - I .2
�
4000
=
4:
'�
-L.
..
·
�
· -· · · · · · · · - · · · ·· . . . . .
·
·
0
SO
.�
200000
0. 1 5
so = 1.0
c..
.�
;;;
c..
..
:r
0. 1 0
.
.
�
.-: . ...��,� .. :::: . . .. . ...
0.05
1{
.
............
-
-
·
.. �
.
;.
.It
·$
.'
so = 0.8
/4.. '*""
,. �
. .
,
.
.
..
. ... ..
. . � ..
• .. ..
. ...
.. ... .
$
..
....
... ....
.
..
.
...
..
.
so = 0.6
.
. ...
" . . "' . , .. . ...
.
-t----,-----.--.,-.J
.
0.00
.,
...
1t ·
·
It ·
o
2000
Pressure, psia
4000
6000
8000
1 0000
Fig. 7.4-Range of applicability of pressure-squared methods
at 1 00 o F .
1 70
GAS RESERVO I R E N G I N E E R I N G .
0. 14
0. 12
0.10
�
0.08
'Vl
Po.
.;
0 06
,r '
0.04
.
....
. ..
. •. .
°
4000
2000
. �.,.
j.
SG . l �..
.'
. ...
.. .. ··SG "' 0.8
/ .,tII'
. ...
• • ••
."
6000
••
., ••
.
"
SG = 0.6
Fig. 7.S-Range of applicability of pressure-squared methods
at 200 ° F .
P-g(p)ct( p)
Eq. 7 . 8 is not completely linear because
depends on
pressure and pseudopressure . An acceptable approximation is the
assumption that
is constant and can be evaluated at some p.
We denote this product
Thu s , familiar solutions, such as the
Ei-function solution, are reasonably accurate for gas wells when
we use the pseudopressure transformation .
The early-time or transient solution to Eq . 7 . 8 for constant-rate
production from a well in a reservoir with closed outer boundaries is
p-gCt /i.gCt .
[
X 1 . 1 5 1 10g
( 1, 688kgt4>/i.gctra )
]
+ S +Dq , . . . . . . . . . . . . . (7 .9)
where p s = the stabilized shut-in BHP measured before the deliver­
ability test . In new reservoirs with little or no pressure depletion,
this shut-in pressure equals the initial reservoir pressure
while i n developed reservoirs ,
The late-time or pseudosteady-state solution to Eq. 7 . 8 is
( Ps =Pi )'
Ps < Pi '
[
X 1 . 1 5 1 10g
( CAra ) �4
1 O · 06A
_
]
+ S + Dq , . . . . . . . . . . . . . (7 . 1 0)
where p = current drainage area pressure . Gas wells cannot reach
true pseudosteady state because
changes as p decreases .
Note that, unlike p (which decreases during a pseudosteady-state
flow test) ,
remains constant.
Eqs . 7 . 9 and 7 . 1 0 are quadratic in terms of the gas flow rate ,
q . For convenience, Houpeurt 1 wrote the transient flow equation
as
Ps
Po.
�
'Vl
Po.
.;
,r
P-g (p)Ct(p)
. . . . . . . . . . . . . . . . (7 . 1 1 )
0.08
0.06
0.04
0.Q2
0.00
10000
8000
Pressure, psia
t/' •
.
. . ., .... . "'\:.
_K
... . ....
. ...
. . . .. ..
... 1 _ .. . .. ... . .
..
. ...;.. .� .. .
0 . 02
0.00
.
. ...
.. .
0.10
0
2000
4000
6000
8000
Pressure, psia
10000
Fig. 7.6-Range of applicability of pressure-squared methods
at 300 ° F .
and b =
1 . 422 106 TD
kgh
q(at
X
. . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 1 5)
a
The coefficients of
for transient flow and for pseudosteady
state flow) include the Darcy flow and skin effects and are meas­
ured in (psia2 .cp)/(MMscf-D) when q is in MMscf/D . The coeffi­
cient of the q2 term represents the inertial and turbulent flow
effects and is measured in (psia2 -cp)/(MMscf-D) 2 when q is in
MMscflD. Commonly called non-Darcy effects , the inertial and
turbulent flow effects result from high gas velocities near the well­
bore and consequently cannot be modeled with Darcy ' s law . The
non-Darcy flow coefficient, D, is defined in terms of a turbulence
factor , {3 , which has been correlated with rock properties , permea­
bility , and porosity , 4
D=
-12 (3kgMpsc
hP-i p wj)rw Tsc
lk
P-g
pwf.
2.715 x 10
and {3 = 1 . 88 x 1 O O
1 4 4>
- . 7
-
. . . . . . . . . . . . . . . . . . . . . . (7 . 1 6)
. . . . . . . . . . . . . . . . . . . (7 . 1 7)
0 . 53 ,
P-gb
where
is evaluated at
Note that, because
is not constant
with respect to pressure , D and the coefficient are not strictly
constant at different BHFP ' s ; however, the assumption of a con­
stant D is adequate for most practical applications .
The Houpeurt equations also can b e written i n terms o f pressure
squared and are derived directly from the solutions to the gas diffu­
sivity equation , assuming that
is constant over the pressure
range considered . For transient flow ,
P-g Z
Ilp 2 =p� -p aj=atq+bq 2 ,
Ilp 2 -p aj=aq+bq2
. .
and for pseudosteady-state flow ,
.
. . . .
. . . . . . . . . . . . . . . (7 . 1 8)
.
. .
. .. ... . .. . . ....... ..
=p2
. .
.
(7 . 1 9)
The flow coefficients are
.
gZT[ ( kgt ) ]
IlPp =Pp (ps) -pp( pwj)=atq+bq2
at= 1 . 422X106/i.
kgh
/i.gctra
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
.( 2
IlPp =Pp ( p ) -pp (pwj)=aq+bq , .
( kgt ) ]
at = 1 . 422X106T[
a= 1 . 422 kgh106/i.gZT [ ( CAra ) 4 ]
kgh
/i.gctra
..
. . (
1 . 422 X 106/i.g zm . . . . . . . . . . . . . . . . . . . . . . . . ( 22
( ) ]
b=
a= 1 . 422X106T[
kgh
kgh
CAra 4
1 . 1 5 1 log
and the pseudo steady state flow equation as
7. 1 )
where
1 . 1 5 1 log
.
. . . . . . . . . . . . . . . .
1 . 1 5 1 log
1 O . 06A
--
. . . . .
3
.
x
. . . . . . 7 . 1 3)
- - + S , . . . . . (7 . 1 4)
+S ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7. 20)
+S ,
1 , 688 4>
1 ,688 4>
and
3
1 O . 06A
1 . 1 5 1 10g -- - - + s , . . (7 . 2 1 )
7.
)
1 71
D ELIVERABI LITY TESTI N G OF GAS WELLS
TABLE 7 . 1 -EFFECTS OF PERMEABILITY AND DRAINAGE
AREA ON STABI LIZATION TIME
SOOO
'2
!
�
::s
�
k
4900
4800
4700
4600
4500
0.1
10
Radius (fl)
1 00
1000
Fig. 7 . 7-Radius of i nvestigation as a function of flow time.
When the Houpeurt equation is presented in terms of pressures
squared , the coefficients of q are measured in psia2 /(MMscf-D)
when q is in MMscflD, while the coefficient of q2 is measured
in units of psia2 /(MMscf-D) 2 when q is in MMscflD . The units
of the flow coefficients given by Eqs . 7 . 1 1 through 7 . 1 3 and 7 . 1 8
through 7 . 20 vary depending o n the units o f flow rate and whether
the pseudopressure or pressure-squared formulation is used . For
consistency , all equations and example problems in this chapter are
presented with q measured in MMscflD .
Recall from previous discussions that, because of the nonlinear
behavior of the gas properties at high pressures, the pressure-squared
form of the equation should be used only for gas reservoirs at low
pressures and high temperatures . To eliminate doubt about which
equations to choose, we recommend using the pseudopressure equa­
tions , which are applicable at all pressures and temperatures . Con­
sequently , all the analysis procedures in this chapter are presented
in terms of pseudopressure. The same procedures and example prob­
lems , however, are presented in Appendix H in terms of pressure
squared .
An advantage of the pseudopressure form of the theoretical
deliverability equation is that the flow coefficients are independent
of the average reservoir pressure and therefore do not change as
P decreases during a flow test conducted under pseudosteady-state
flow unless s, k, or A changes . Because of the dependency of the
non-Darcy flow coefficient on p, g C P Mj), only the coefficient b will
change slightly if the BHFP is changed . Conversely , because of
the pressure dependency of the gas properties on average reser­
voir pressure , the flow coefficients for the pressure-squared form
of the deliverability equation must be recalculated for every new
p value. When s , k, or A changes with time, the only way to up­
date the deliverability curve is to retest the well .
7.3.2 Empirical Deliverability Equations. In 1 935 , Rawlins and
Schellhardt2 presented an empirical relationship that is used fre­
quently in deliverability test analysis. The original form of their
relation given by Eq. 7.23 in terms of pressure squared, is applicable
only at low pressures:
q = C(p 2 _p ?it ) n .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 2 3 )
In terms of pseudopressure , Eq . 7 . 23 becomes
q = C[ Pp ( p) -pp (p wj) ] n , . . . . . . . . . . . . . . . . . . . . . . . .
(acres)
(hours)
0. 0 1
0.01
0.1
0.1
1 .0
1 .0
1 0.0
1 0.0
1 00.0
1 00.0
1 ,000.0
1 ,000.0
40
640
40
640
40
640
40
640
40
640
40
640
25 ,953 (3 years)
41 5 ,242 (47 years)
2,595 ( 1 08 years)
41 , 524 (4.7 years)
259.5 ( 1 0 . 8 years)
4 , 1 52.4 ( 1 73 days)
25.95 ( 1 . 1 days)
41 5.2 ( 1 7.3 days)
2 . 59 (0 . 1 1 days)
41 .52 ( 1 .73 days)
0.259 (0.01 1 days)
4. 1 5 (0. 1 73 days)
real gas through porous media . Although the Rawlins and Schell­
hardt equation is not theoretically rigorous , it is still used widely
in deliverability analysis and has worked well over the years , es­
pecially when the test rates approach the AOF potential of the well
and the extrapolation is minimal .
7.4 Stabilization Time
Unlike pressure-transient tests , the analysis techniques for conven­
tional flow-after-flow and single-point tests require data obtained
under stabilized flowing conditions . Although isochronal and modi­
fied isochronal tests were developed to circumvent the requirement
of stabilized flow , they still may require a single, stabilized flow
period at the end of the test. Consequently , we must understand
the meaning of stabilization time and have a method to estimate
its value .
Stabilization time is defined as the time when the flowing pres­
sure is no longer changing or is no longer changing significantly .
Physically , stabilized flow can be interpreted as the time when the
pressure transient is affected by a no-flow boundary , either a natu­
ral reservoir boundary or an artificial boundary created by active
wells surrounding the tested well . Consider a graph of pressure as
a function of radius for constant-rate flow at various times since
the beginning of flow . As Fig. 7.7 shows, the pressure in the well­
bore continues to decrease as flow time increases . Simultaneously ,
the area from which fluid is drained increases , and the pressure
transient moves further out into the reservoir.
The radius of investigation, the point in the formation beyond
which the pressure drawdown is negligible, is a measure of how
far a transient has moved into a formation following any rate change
in a well . The approximate position of the radius of investigation
at any time is estimated by 5
,; =
J :;�
94
.
.
H
.
.
.
.
H
.
.
.
.
H
.
H
.
H
.
H
.
H
.
H
" ,
( 72 5 )
Stabilized flowing conditions occur when the radius of investi­
gation equals or exceeds the distance to the no-flow boundary of
the well , i . e . , 'i ?; 'e ' Substituting ,e and rearranging Eq. 7 . 2 5 , we
can develop an equation for estimating the stabilization time , ts '
for a well centered i n a circular drainage area:
(7 . 24 )
which is applicable over all pressure ranges . In Eqs . 7 . 23 and 7 . 24 ,
C = stabilized performance coefficient and n = inverse slope of the
line on a log-log plot of the change in pressure squared or pseu­
dopressure vs. gas flow rate . Depending on the flowing conditions ,
the theoretical value of n ranges from 0 . 5 , indicating turbulent or
non-Darcy flow, to 1 . 0 , indicating flow behavior described by Dar­
cy ' s equation . Note that the value of C changes depending on the
units of flow rate and whether Eq. 7 . 23 or 7 . 24 is used . Again,
all equations and example problems in this chapter are presented
with q measured in MMscflD.
Houpeurt proved that neither Eq. 7 . 23 nor Eq. 7 . 24 can be de­
rived from the generalized diffusivity equation for radial flow of
ts
A
(md)
ts =
948C/>jii:t'�
kg
.
. .. .
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
(7 . 26)
As long as the radius of investigation is less than the distance
to the no-flow boundary , stabilization has not been attained and the
pressure behavior is transient. To illustrate the importance of stabili­
zation times in deliverability testing, we calculated stabilization times
as a function of permeability and drainage area for a well produc­
ing a gas with a specific gravity of 0 . 6 from a formation at 2 1 0 ° F
and a n average pressure o f 3 , 500 psia ( jig = 0 . 02 c p and
ct =2.468 x 1O - 4 psia - 1 ) , with a porosity of 1 0 % . Table 7 . 1
shows that, for wells completed i n low-permeability reservoirs ,
several days, if not years, are required to reach stabilized flow ,
1 72
GAS RESERVO I R E N G I N E ER I N G
while wells completed in high-permeability reservoirs stabilize in
a short time .
A more general equation for calculating stabilization time is
TABLE 7.2-EFFECTS OF PERMEABILITY AND HYDRAULIC
FRACTURE HALF-LENGTH O N TIME TO REACH
PSEUDO RADIAL FLOW
cjJji/:rA tDA
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.27)
0.OO02637kg
where tDA = dimensionless time for the beginning of pseudo steady­
state flow . Values for tDA are given in Appendix C for various
ts =
reservoir shapes and well locations . 5,6 The time required for the
pseudosteady-state equation to be exact is found from the entry in
the column " Exact for tDA > . "
We should emphasize that the Rawlins-Schellhardt and Houpeurt
deliverability equations assume radial flow . If pseudoradial flow
has been achieved, however , these analysis techniques can be used
for hydraulically fractured wells. The time to reach the pseudoradial
flow regime , tp rJ , occurs 7 , 8 at tL D = 3 and is estimated with
j
I l ,400cjJjigctL}
. . . . . . . . . . . . . . . . . . . . . . . . . . . (7.28)
tprJ s
kg
To illustrate the importance of achieving pseudoradial flow dur­
ing a deliverability test , we calculated values of tp rJ for a hydrau­
lically fractured well completed in a reservoir with cjJ =0. 15,
jig =0.03 cp , and ct = 1 x 10 4 psia - I and with the range o f per­
meabilities and hydraulic fracture half-lengths in Table 7.2. The
results illustrate that a well with a long fracture in a low-permeability
formation will take far too long to stabilize for conventional deliver­
ability testing .
-
1
2
3
t Prf
Lf
k
(md)
1
0.01
0.01
Case
--
.J!!L
(hours)
5 1 .2
5, 1 20 (21 3 days)
5 1 2,000 (58 years)
1 00
1 00
1 ,000
g(.)'"
6
q3
0-
i
..
c.:
�
£'"
0
..
Time, t (hours)
7.5 Analysis of DeliverabUlly Tesls
This section discusses the implementation and analysis of the flow­
after-flow , single-point , isochronal , and modified isochronal tests.
We present both the Rawlins and Schellhardt and Houpeurt analy­
sis techniques in terms of pseudopressures. The same analysis tech­
niques are presented in Appendix H in terms of pressure squared .
7 . 5 . 1 Flow-After-Flow Tests. Flow-after-flow tests, sometimes
called gas backpressure or four-point tests, are conducted by produc­
ing the well at a series of different stabilized flow rates and meas­
uring the stabilized BHFP at the sandface . Each different flow rate
is established in succession either with or without a very short in­
termediate shut-in period . Conventional flow-after-flow tests often
are conducted with a sequence of increasing flow rates; however,
if stabilized flow rates are attained , the rate sequence does not af­
fect the test . 9 The requirement that the shut-in and flowing peri­
ods be continued until stabilization is a major limitation of the
flow-after-flow test, especially in low-permeability formations that
take long times to reach stabilized flowing conditions . Fig. 7.8 il­
lustrates a flow-after-flow test .
Rawlins-Schellhardt Analysis Technique. Recall the empirical
equation introduced by Rawlins and Schellhardt :
q = q Pp ( p ) -Pp (p wf)] n . . . . . . . . . . . . . . . . . . . . . . . . . (7.24)
Taking the logarithm of both sides of Eq . 7.24, we obtain the
equation that forms the basis for the Rawlins-Schellhardt analysis
technique :
log(q) = log( e) + n { log [ pp ( p ) -Pp ( P wf)] } ' . . . . . . . .
. . (7.29)
if we plot log ( q) vs. log
The form of Eq . 7.29 suggests that,
(App ) , we will obtain a straight line with a slope n and an inter­
Time, t (hours)
Fig. 7.S-Pressure and flow rate history of a typical flow-after­
flow test.
points at the lower rates may not fit on the same straight line as
the later points . In this case, the points at the lower rates should
be ignored for all subsequent calculations .
2. Determine the deliverability exponent, n , by simply calculat­
ing the slope of the best-fit straight line through the data points or
with least-squares regression analysis. For a regression analysis, use
N
N
N
log qj E (log App )j
'=l
l �__�j_
j�
=
=_l__�_____
_
__
��
________
�
�
J
�
n=
N E
(log
N
j
q log App )j - E
�l
(log
App)j -
[ j�l
(log
App )j
r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.30)
Alternatively , lin is the slope of the graph of log App vs. log
cept of log (e) . The conventional graphical method for analyzing
flow-after-flow test data, however, is to plot log (A pp) vs. log (q) ,
q . If (ql ' APpI) and (q z , App2 ) are two points on the perceived
giving a straight line o f slope l in and an intercept of { - lin
"best " straight line fitting the test data , then
[loge e)] } . The AOF potential is estimated from the extrapolation
log(App2 ) -!og(Appl )
of the straight line to App evaluated at a P wf equal to atmospheric
pressure (sometimes called base pressure) , P h ' This analysis tech­
l og (qz ) - log (ql )
n
nique is described in the next section and is illustrated with Exam­
ple 7. 1 . In addition , we present the same example in Appendix H or the reciprocal of the slope is
(Example H . I ) in terms of pressure squared .
A nalysis Procedure.
1 . Plot A Pp =Pp ( p ) -Pp( P wf) vs. q on log-log graph paper.
Construct the best-fit line through the data points . Some of the data
n = ------
10g(AppZ ) -log(Appl )
. . . . . . . (7.31)
1 73
DELIVERABI LITY TESTI N G OF GAS WELLS
TABLE 7 . 3-FlOW-AFTER-FlOW TEST DATA ,
EXAMPLE 7 . 1
T" t!
p p ( P wt l
P wl
P t!
q
( M Mscf/D)
( O F)
(psia)
(psia)
(psia 2 /cp)
0
4,288
9,265
1 5.552
20 . 1 77
75
70
73
77
77
375,2
371 , 2
361 .3
343. 8
327 . 1
p = 407,60
403, 1 3
393.03
375.79
359 .87
1 ,6 1 73 x 1 0 7
1 ,58 1 7 x 1 0 7
1 . 5032 x 1 0 7
1 .3736 x 1 0 7
1 .259 1 x 1 0 7
The value of n should be between 0 . 5 and 1 .0. A value of 0 . 5 indi­
cates turbulent flow throughout the entire drainage area (improba­
ble) , while a value of 1 .0 indicates completely laminar flow .
If n does not lie in this range , we recommend conducting the test
again at flow rates higher than the highest rate achieved in the first
test . If absolutely necessary , the original test can be analyzed with
approximate techniques outlined in Step 4.
3 . If 0 . 5 S n S 1 .0, calculate the AOF using either Step 3 A o r 3B.
A . Estimate the stabilized AOF potential of the well from the
extrapolated straight line on the deliverability plot to the flow rate
that corresponds to l1Pp =Pp ( p) -Pp ( P b ) ' where Pp ( P b ) is the
pseudopressure evaluated at P b '
B . Alternatively, determine the stabilized performance coefficient,
C:
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.32)
c= 1 0 "' ,
where a is determined from least-squares regression analysis of
the points on the best-fit straight line ,
a=
N
N
N
N
E log qj E (log t..pp )J -E (log l1pp )j E (log q log l1pp )j
j=1
j=1
j=1
j=1
]
N
N E
j=1
(log l1pp ) -
[ .f
J
=1
(log l1pp )j
r
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.33)
Alternatively (and less accurately) , we can determine C from a
perceived "best " straight line using the value of n determined in
Step 2, one point (q l ,11pp l ) on the straight line , and
C= q l /(l1pp l ) n .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.34)
.
q,
MMscfID
Fig. 7. 9-Rawli ns-Schellhardt analysis, Example 7. 1 .
C=
q
[ pp ( p) _pp ( p wj-» ) n
, . . . . . . . . . . . . . . . . . . . . . . . . (7.36)
where pp ( Pwj ) is the point through which the line was construct­
ed . Calculate the AOF potential with Eq . 7.35 using C from Eq.
7.36 and n = 0 . 5 or 1 .0.
Houpeurt Analysis Technique. Flow-after-flow tests require
stabilized data or data measured during pseudo steady-state flow .
Recall that Houpeurt gives the theoretical equation for pseudosteady­
state flow, which was derived directly from the gas diffusivity equa­
tion , as
l1pp =pp ( p) -pp ( pwj ) = aq + bq 2 . . . . . . . . . . . . . . . . . . (7. 12)
In addition , recall that coefficients a and b have theoretical bases
and can be estimated if reservoir properties are known . We can
also determine these stabilized flow coefficients from flow-after­
flow test data . Dividing both sides of Eq . 7. 12 by the flow rate ,
q, and rearranging , we obtain the equation that is the basis for the
Houpeurt analysis technique:
l1pp
q
pp ( p) -pp ( Pwj )
�----"=---=-
q
= a + bq .
. . . . . . . . . . . . . . . . . . (7.37)
where pp ( P b) = pseudopressure evaluated at P b '
4. I f n < 0 . 5 , set n = 0 . 5 and construct a line with a slope of 2 . 0
through the data point corresponding t o the highest flow rate . Simi­
larly , if n > 1 .0, set n = 1 .0 and construct a line with slope of 1 .0
through the data point corresponding to the highest flow rate . De­
termine the stabilized AOF potential of the well from the extrapo­
lated straight line on the deliverability plot to the flow rate
corresponding to l1 Pp =Pp ( p) -Pp ( P b ) ' Alternatively, calculate C
with n = 0 . 5 or 1 .0:
The form of Eq. 7 . 37 suggests that , if we plot tJ.. p lq v s . q, we
p
will obtain a straight line with a slope b and an intercept a. The
AOF is estimated in the Houpeurt deliverability analysis by solv­
ing Eq. 7. 12 for q = q AO F at Pwj =P b '
Analysis Procedure.
1 . Plot l1pp lq = [ pp ( p) -pp ( Pwj » ) /q v s . q on Cartesian graph
paper. Construct the best-fit line through the data points . Some of
the data points at the lower rates may not fit on the same straight
line as the points corresponding to the higher rates . In this case ,
ignore the points at the lower rates and use only the points at the
higher rates for all subsequent calculations .
2. Calculate a and b by least-squares regression analysis (Eqs .
7.38 and 7.39, respectively) of the points on the best-fit line .
TABLE 7.4-PlOTTING FUNCTIONS F O R
RAWlINS-SCH EllHARDT ANALYSIS, EXAMPLE 7 . 1
TABLE 7 . S-lEAST-SQUARES REGRESSION,
RAWlINS-SCHEllHARDT ANALYSIS, EXAMPLE 7.1
We can then calculate the AOF potential with
q
( M MscflD)
4.288
9.265
1 5 .552
20 . 1 77
3.560 x 1 0 5
1 . 1 41 x 1 0 6
2 .437 x 1 0 6
3.582 x 1 0 6
Point
1
2
3
4
E=
log q
0. 6323
0.9668
1 . 1 91 8
1 . 3049
4. 0958
log Il p p
5.55 1 4
6.0573
6.3879
6.5541
24.5507
(log Il p p ) 2
30. 8 1 86
36.6909
40.8053
42. 9562
1 5 1 .27 1 0
log q log II p p
3 . 5 1 02
5. 8562
7. 6 1 31
8.5524
25 .531 9
GAS RESERVO I R E N G I N E ER I N G
1 74
20c000 ,---�--�--�
a=
g
. . (7.38)
o
i
�
�
. ;;;
"' ",
�
p..
. . . . . . (7.39)
-<:]
An alternative but less accurate method is to use two points,
[q 1 ,(� pp /q) d and [qz ,(� pp /qh ] , on the perceived "best" line
through the test data and the following equations:
( �pp ) - ( �Pp )
q
q
b = ___-=z___-=-I . . . . . . . . . . . . . . . . . . . . . . . . (7.40)
and a =
( �pp ) -bql . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.41)
q I
3 . Calculate the AOF with Eq. 7.42, where Pp ( Pb ) is the pseu­
dopressure evaluated at Pb and a and b are from Step 2:
-a + -Ja Z +4b[ pp (p ) -Pp ( Pb ) ]
. . . . . . . . . . (7.42)
and qAOF =
2b
Example 7. I-Analysis or a Flow-After-Flow Test. Estimate the
initial stabilized AOF potential of a well 1 0 having the well and
reservoir properties listed below. Use both the Rawlins-Schellhardt
and the Houpeurt analysis techniques. In addition, estimate the AOF
potential 10 years later when the static drainage area pressure has
decreased to 350 psia. Evaluate the AOF potential at Pb = 14.65
psia. Table 7.3 summarizes the flow-after-flow test data.
'Y g = 0.715.
L
=
3 , 050 ft .
= 0.5 ft.
= 20.71 Ibm/Ibm-mol.
A = 640 Ac.
e/> = 0.25.
Tj, wj = 90°F.
CA = 30.8828.
h = 200 ft.
Current p =407.6 psia, pp ( p =407.6) = 1 .6173 x 107 psia Z /cp.
p after 10 years =350 psia, pp (p =350) = 1 .2239 x 107 psia Z /cp.
Pb = 14.65 psia, Pp ( Pb ) =2,674. 8 psia z /cp.
rw
Ma
Solution.
Rawlins-Schellhardt Analysis.
1 . Plot � Pp vs. q on log-log graph paper (Fig. 7.9) . Table 7.4
gives the plotting functions. Construct the best-fit line through the
data points. Note that all data points lie on the best-fit line and will
be used for all subsequent calculations.
2. Determine the deliverability exponent using least-squares
regression analysis (Eq. 7.30). Table 7.5 summarizes the calcu­
lations.
n=
N
N
N
N E (log q log �pp )j
E log qj E (log �pp )j
_I_ _____�
�J�
' =
j_
=I
_�
j_
=_
I _ __
�
��
�
N
j
-
� (log �pp)J [ � (log �pp)jr
I
-
j
I
_
l SOOOO
I
I OCOOO
, ····· .... ······ .. ···· .... ··· · ··_ .. ······· ....··· __ ···· ····-..... _
t
. . .... ... ./. . ... . . . . . . . . . . .
_
... .... .. i!'�P.!,7.-''. ······················ ..·········· .. . . ... .....�... ... . .. . . .-.-.....!............................ .
I
50000
0 4----+---�--4---_+--�
o
10
q,
15
20
MMscf/D
25
Fig. 7 . 1 O-Houpeurt analysis, Example 7 . 1 .
4(25 .53 19) - (4.0958)(24.5507)
4(151 .2710) - (24.5507) Z
=0.67.
Alternatively, because Points 1 and 4 both lie on the perceived
"best" straight line through the test data, the reciprocal slope is
estimated to be
log(20. 177/4.288)
=0.671 .
log(3 .582 x 106/3 .560 x 105)
3 . Now, calculate the AOF of the well. Because 0.5 s n s 1 .0,
we can calculate C using either regression analysis or a point from
the best-fit straight line through the test data.
A. We can estimate C with regression analysis using Eq. 7.33
where the exponent a is
a=
-------
N
N
N
N
j=1
j=1
j=1
j=1
E log qj E (log �pp)J - E (log �pp)j E (log q log �pp )j
Nj
�I (log �pp)J - [ �1 (log �pp)jr
j
(4.0958)(15 1 .2710) - (24.5507)(25 .5319)
4(151 .27 10) - (24.5507) 2
= - 3 .09.
The stabilized performance coefficient is
C= 10 " = 10 ( - 3 .09) = 8 . 1 3 x 10 -4 .
B . We also can estimate C using Point 4 from the best-fit line
and Eq. 7.34.
20. 177
= 8.07 x lO -4 .
(3 .582 X 106) 0.671
C. Therefore, the AOF potential of this well is
qAOF = C [ Pp ( p) -pp (14.65)] n
= 8 . 13 x 10 -4(1 .6173 X 107 -2.6738 X 10 3 ) 0. 67
=55.0 MMscflD at p =407.6 psia.
4. To update the AOF to a new average reservoir pressure, recall
that, for pseudopressure analysis, both C and n do not change as
drainage area pressure decreases. Therefore, recalculate AOF for
the new drainage area pressure using Eq. 7.35.
q AOF = C [ Pp ( p ) -pp (14.65) ] n
= 8 . 13 X 10 - 4(1 .2239 X 107 -2.6748 X 10 3 ) 0. 67
=45.6 MMscflD at p = 350 psia.
------
1 75
DELIVERABI LITY TESTI N G OF GAS WELLS
TABLE 7 .7-LEAST-SQUARES REGRESSION,
HOUPEURT ANALYSIS, EXAMPLE 7 . 1
TABLE 7 . 6-PLOTTING FUNCTIONS FOR THE
HOUPEURT ANALYSIS, EXAMPLE 7 . 1
t:.P p
-- =
p p ( p ) - p p ( P wt )
q
q
(psia 2 /cp)/(M M scflD)
q
(MMscf/D)
0.8302 x 1 0 5
1 .232 x 1 0 5
1 .567 x 1 0 5
1 .775 X 1 0 5
4.288
9.265
1 5. 552
20. 1 77
Point
2
3
4
E
=
q2
q
9.265
1 5. 552
20. 1 77
44.994
85.84
241 .87
407. 1 1
734.82
t:.p p lq
t:. Pp
1 . 1 41 x
2.437 x
3.582 x
7. 1 60 x
106
1 06
106
1 06
1 .232 x 1 0 5
1 .567 x 1 0 5
1 . 775 x 1 0 5
4.574 x 1 0 5
Alternatively, we can use Points 2 and 4 from the line drawn
Houpeurt Analysis.
1 . Plot ilpp lq vs. q on Cartesian graph paper (Fig. 7 . 1 0) . Ta­ through the test data to calculate a and b using Eqs. 7.40 and 7.4 1 ,
respectively.
ble 7.6 gives the plotting functions. Construct the best-fit line
through the last three data points. The first point, corresponding
ilPp
ilpp
to the lowest flow rate, does not follow the trend and will be ig­
nored in subsequent analyses.
q 4
q 2
b=
2. Determine the deliverability coefficients, a and b, from a least­
q4 -q 2
squares regression analysis (Eqs. 7.38 and 7.39, respectively) of
the points used to develop the best-fit straight line. Table 7.7 sum­
1 . 775 X 105 - 1 .232 X 105
marizes the calculations. Note that Point I is not included in the
calculations.
20. 177 -9.265
=4.976 x 10 3 psia 2 /cp/(MMscf-D) 2 .
ilPp
(qj ) 2 E qj E (ilpp )j
E
ilPp
j=1 q j j=1
j=1 j=1
a=
-bq(
a=
ql I
N � (qj ) 2 - � qj
= 1 . 775 X 105 - (4,976)(20. 177)
j (
j 1
=7.71 x 104 =psia 2 /cp/(MMscf-D) 2 .
(4.574 x 105 )(734. 82) - (44.994)(7. 160 x 106)
3 . From Eq. 7.42,
3(734.82) - (44.994) 2
- a + ..Ja 2 +4b[ pp(p ) -pp (14.65) ]
=
q
2
AOP
= 7.75 x 104 psia /cp/(MMscf-D) .
2b
= - (7.75 x 104)
+ ..J (7. 75 x 104) 2 +4(5.00 x 10 3 ) [ ( 1 . 62 x 107) -2,674.8] I
2(5.00 x 10 3 )
=49.7 MMscflD at p = 407.6 psia.
4. To update the AOF, recall for pseudopressure analysis that
3(7. 160 x 106 ) - (44.994)(4.574 x 105)
both a and b do not change as drainage area pressure changes.
Therefore, the AOF for the new drainage area pressure is
3(734. 82) - (44.994) 2
-a + ..Ja 2 +4b[pp ( p ) -pp (14.65)]
=5.00 x 10 3 psia 2 /cp/(MMscf-D) 2 .
q AOP =
( ) ( )
_
N( ) E
N
N
(N
_
r
N
( )
2b
= - (7.75 X 104)
+ ..J (7. 75 x 104) 2 +4(5.00 x 10 3 ) [(1 .2239 x 107) -2,674.8] I
2(5 .00 x 10 3 )
=42.2 MMscflD at p = 350 psia.
. . . . . Rawlins-ScheUhardl Equation
- Houpeun Quadlatic Equation
• Aow·Aflet·Flow Data
i
• • • _ •••••• H ••••••••••••• •• •••••••• _ •• _
•
I
• •••••••• _••••••••••••_• •_••••••••••••••••••••• ••••••••••••••••••••••_••••••••••••••••••••••••
!:
!
!
q,
A comparison (Fig. 7 . 1 1 ) of the results from Example 7. 1 shows
that the Rawlins-Schellhardt equation appears to be valid for this
range of test data; however, the line representing the Houpeurt equa­
tion deviates from the Rawlins-Schellhardt equation as BHFP
decreases. Although the Rawlins-Schellhardt method is valid un­
der many testing conditions, this deviation suggests that, when we
extrapolate the empirical equation over large variations in pressure,
we may not predict well behavior correctly.
A single-point test is an attempt to over­
come the limitation of long test times. A single-point test is con­
ducted by flowing the well at a single rate until the sandface pressure
is stabilized. One limitation of this test is that it requires prior knowl­
edge of the well ' s deliverability behavior, either from previous well
tests or possibly from correlations with other wells producing in
7.5.2 Single-Point Tests.
MMscf/D
Fig. 7 . 1 1 -Comparlson of results from the Rawlins-Schellhardt
and Houpeurt methods, Example 7 . 1 .
1 76
GAS RESERVOI R E N G I N E E R I N G
the same field under similar conditions. We also must ensure that
the well has flowed long enough to be out of wellbore storage and
in the pseudosteady-state flow regime. Similarly, for hydraulical­
ly fractured wells, we must ensure that the well has flowed long
enough be in the pseudoradial flow regime.
To analyze a single-point test with the Rawlins-Schellhardt
method, n must be known or estimated. An estimate of n can be
obtained either from a previous deliverability test on the well or
from correlations with similar wells producing from the same for­
mation under similar conditions. The calculation procedure is similar
to that presented for flow-after-flow tests. The AOF can be esti­
mated graphically by drawing a straight line through the single flow
point with a slope of lin and extrapolating it to the flow rate at
/lpp = Pp ( p) -PP ( Pb ) ' Or the AOF can be calculated with Eq.
7.3S, where C is estimated with Eq. 7 . 36 .
To use the Houpeurt analysis technique, the slope, b, of the line
on a plot of /lPp /q = [ pp ( p) -pp ( p w[)]/q vs. q must be known.
We have only one point from a single-point test, so we must esti­
mate b using Eq. 7 . IS . Note that we must have estimates of the
formation properties to use Eq. 7. 1S. The remaining analysis proce­
dure is similar to that for flow-after-flow tests. The AOF can be
calculated with Eq. 7.42 where the intercept, a, is estimated with
Eq. 7 .4 1 .
Extended
Flow RalC
Time, r
Time, I
(hours)
Fig. 7 . 1 2-Pressure and flow rate history of a typical
7.5.3 Isochronal Tests. The isochronal test is a series of single
point tests developed to estimate stabilized deliverability charac­ isochronal test.
teristics without actually flowing the well for the time required to
Further, we define an effective drainage radius as
achieve stabilized conditions. The isochronal test is conducted by
alternately producing the well, then shutting in the well and allow­
ing it to build to the average reservoir pressure before the begin­
ning of the next production period. Pressures are measured at several
; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 44)
'd �
,
37
time increments during each flow period. The times at which the
pressures are measured should be the same relative to the begin­
If we substitute Eq. 7.44 into Eq. 7.43 and rearrange, the tran­
ning of each flow period. Because less time is .required to build
to essentially initial pressure after short flow periods than to reach sient solution becomes
stabilized flow in a flow-after-flow test, the isochronal test is more
practical for low-permeability formations. Although not required
for analyzing the test, a final stabilized flow point often is obtained
at the end of the test. Fig. 7 . 12 illustrates an isochronal test.
The isochronal test is based on the principle that the radius of
investigation established during each flow period is not a function
x In
- + s +Dq . . . . . . . . . . . . . . . . . . . . . . . . (7 .4S)
of the flow rate but depends only on the length of time for which
the well is flowed. Consequently, the pressures measured at the
which is valid at any fixed time because r is a function of time
same time periods during each different rate correspond to the same
drainage radius. Under these conditions, isochronal test data can only, not flow rate. Note that rd has no physical significance but
be analyzed using the same theory as a flow-after-flow test, even is simply the radius that forces the pseudosteady-state equation to
though stabilized flow is not attained. In theory, we should obtain resemble the transient equation. 1 0 In addition, do not confuse rd
a stabilized deliverability curve from transient data if a single stabi­ with ri ' which is the transient radius of investigation given by Eq.
lized rate and the corresponding BHP have been measured and are 7.2S .
available.
Similar to Houpeurt ' s equations, we can rewrite Eq. 7.4S as
Recall that the transient solution to the real-gas diffusivity equa­
pp ( ps) -pp ( p w[) = a , q + bq 2 , . . . . . . . . . . . . . . . . . . . . . (7 . 46)
tion in terms of pseudopressures is
1 .422 x 106 q T
1 .422 X 106 T
rd
3
pp ( Ps ) -Pp ( Pwf ) =
In
- - + S . . . . . . . . . . . (7 .47)
where a, =
rw
4
kg h
J 7;;,
[ C: )
�
J
d
[( ) l
-
[
X 1 . 1S 1 10g
(1
kg t
,688 <1>jig ctr�
)
l
+ S +Dq , . . . . . . . . . . . . . (7 . 9)
and b =
1 .422 X 106DT
kg h
. . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 1 S)
where p s = stabilized BHP measured before the test. We can re­
Note that b is not a function of time and will remain constant.
write the transient equation in a form similar to the late-time solu­ Similarly, the intercept at is constant for each fixed time line or
tion (Eq. 7 . 1 0) :
isochron.
The theory of isochronal test analysis implies that the transient
pressures corresponding to the same elapsed time during each differ­
ent flow period will plot as a straight line with a slope b. In addi­
tion, the slopes of each line or isochron representing a fixed flow
time will have the same value. The intercept at will increase with
increasing time. Therefore, we can draw a line with the same slope,
kg�
- � + S +Dq . . . . . . . . (7 . 43) b, through the final, stabilized data point, and use the coordinates
X ln � +ln
4
377 <1>p- g ct
rw
of the stabilized point and the slope to calculate a stabilized inter-
[( ) (
_)
112
l
1 77
DELIVERABI L ITY TESTI N G OF GAS WELLS
TABLE 7.S-ISOCHRONAL TEST DATA, EXAMPLE 7.2
Time
(hou rs)
q
( M MscflD)
P wt
(psia)
p p ( P wd
(psia 2 /cp)
0.5
1 .0
2.0
3.0
0.5
1 .0
2.0
3.0
0.5
1 .0
2.0
3.0
0.5
1 .0
2.0
3.0
214
(Extended
flow poi nt)
0.983
0. 977
0.970
0.965
2.631
2 . 588
2.533
2.500
3. 654
3.565
3.453
3.390
4. 782
4.625
4.438
4.31 8
1 . 1 56
344.7
342.4
339. 5
337.6
329.5
322.9
3 1 5.4
31 0.5
3 1 8.7
309. 5
298.6
291 .9
305. 5
293. 6
279.6
270.5
291 .6
9.6386 x 1 0 6
9. 5406 x 1 0 6
9.41 79 x 1 0 6
9.3381 x 1 0 6
9.0027 x 1 0 6
8.7351 x 1 0 6
8.4371 x 1 0 6
8.2458 x 1 0 6
8.5674 x 1 0 6
8.2071 x 1 0 6
7.7922 x 1 0 6
7.5435 x 1 0 6
8.0534 x 1 0 6
7.61 36 x 1 0 6
7.0990 x 1 0 6
6 .7797 x 1 0 6
7.5285 x 1 0 6
B . Alternatively, determine C using the coordinates of the stabi­
lized, extended flow point and n = n.
C=
qs
[ pp ( P s ) -pp ( P w/ W
[( ) J
Analysis Procedure.
. . . . . . . . . . . . . . . . . . . . . . . . (7 . 36)
Calculate the AOF potential using calculated values of n = n from
Step 3 and C from Eq. 7.36.
q AOP = C [ Pp ( ps) -Pp ( Pb)] n . . . . . . . . . . . . . . . . . . . . . . (7.35)
5 . If n< 0.5, set n =0.5 and construct a line with a slope of 2.0
through the extended flow data point. Similarly, if n > 1 .0, set
n = 1 .0 and construct a line with slope of 1 .0 through the extended
flow data point. Extrapolate this line to the flow rate at App =Pp
( Ps ) -Pp ( Pb ) and read the AOF directly from the graph. Alterna­
tively, calculate C using Eq. 7.36 with either n =0.5 or 1 .0 where
pp ( Pl\js) is the pseudopressure evaluated at the extended flow
point. The AOF potential of the well is computed with Eq. 7.35.
Houpeurt Analysis. Recall that the Houpeurt equation for analyz­
ing flow-after-flow tests is
App
cept, a, independent of time, where the stabilized flow coefficient
is defined by
1 .422 X 106 T !!... _ �
ln
a=
+ s . . . . . . . . . . . . . . . . (7.48)
kg h
4
'w
Rawlins-Schellhardt Analysis. Recall the empirical equation in­
troduced by Rawlins and Schellhardt for analysis of flow-after-flow
test data:
log(q) = log(C) + n{log [ pp( j) -pp ( P l\j)] } ' . . . . . . . . . . (7.29)
For isochronal tests, we plot transient data measured at different
flow rates but taken at the same time increments relative to the be­
ginning of each flow period. Therefore, the lines drawn through
data points corresponding to the same fixed flow time are parallel,
so the value of n remains constant and is independent of time. How­
ever, the intercept, log (C), is a function of time, so we must cal­
culate a different intercept for each isochronal line. We call this
the "transient" intercept, log (Ct ) . In terms of this transient in­
tercept, Eq. 7 .29 becomes
10g (q) = log ( Ct ) + n { log [ pp ( ps) -pp ( Pl\j )] } ' . . . . . . . . . (7,49)
The conventional Rawlins-Schellhardt method of isochronal test
analysis is to plot 10g[App =pp ( Ps) -pp ( Pl\j)] vs. log (q) for each
time, giving a straight line of slope lin and an intercept of
{ - l /n [ l og( Q ] } .
.
q
=
pp ( p) -pp ( Pwj )
q
a + bq .
. . . . . . . . . . . . . . . . . . (7.37)
Remember that Eq. 7.37 assumes stabilized flow conditions; how­
ever, in isochronal testing, we are measuring transient data. Con­
sequently, for each isochronal (or fixed time) line, the equation for
transient flow conditions is
A pp
q
=
pp ( Ps ) -pp ( Pl\j )
where at =
q
at + bq,
. . . . . . . . . . . . . . . . (7.50)
[( ) ]
1 .422 X 106 T
3
rd
In - - - +s . . . . . . . . . . . (7,47)
4
kg h
'w
..
1 ,422 X 106DT
. . . . . . . . . . . . . . . . . . . . . . . . . (7. 15)
kg h
The form of Eq. 7.50 suggests that a plot of App lq = [ pp ( Ps) ­
pp ( Pwj )]lq vs. q will yield a straight line with slope b and inter­
cept at . We can then extend this theory to the stabilized point and
calculate a stabilized intercept, a, using the coordinates of the stabi­
lized point. The slope b remains the same.
and b=
Analysis Procedure.
1 . Plot App lq = [ pp ( Ps) -pp C p l\j)]lq vs. q on Cartesian graph
paper. Also plot Apps lqs = [ pp ( Ps) -pp( pl\js )]lqs vs. qs ' which
( P wj,s ,qs) '
corresponds to the stabilized extended flow point
Con­
struct best-fit lines through the data points for the same flow times,
ignoring the lower flow rates that do not follow the same trend as
the higher flow rates.
2. Calculate the slopes b for each time line by least-squares regres­
sion analysis (Eq. 7.39), or use two points, [q l ,(App lq) d and
[q 2 , (Applqh] , on the perceived "best" line through the test data
and Eq. 7,40. Calculate the arithmetic average value of the slopes.
3 . Calculate the stabilized isochronal deliverability line intercept
using the average b from Step 2 .
1 . Plot App =pp ( Ps) -pp ( Pl\j) vs. q on log-log graph paper.
Also plot Apps =pp ( Ps) -pp ( P wf, s) vs. qs ' which corresponds to
the stabilized, extended flow point ( P wf, s ,qs) ' Construct the best­
fit line through the data points for the same flow times, i.e. , draw
the best-fit line through the isochrons. Ignore the data points at the
lower flow rates if they do not follow the same trend as the higher
flow rates.
2 . Determine the deliverability exponent, n , for each line or
isochron either by drawing the perceived best-fit line through the
[ pp ( Ps ) -pp ( P wf, s)]
- bq s ' . . . . . . . . . . . . . . . . . (7. 5 1)
a=
data and calculating the slope (Eq. 7.31) or by least-squares regres­
qs
sion analysis (Eq. 7.30).
The theoretical value of n should be between 0.5 and 1 .0. If n
The AOF is calculated by Eq. 7,42.
is not in this range for any of the times chosen, we recommend
conducting the test again at flow rates higher than the highest rates
- a + .,ja 2 + 4b[ pp ( Ps ) -Pp ( Pb)]
q AOP =
. . . . . . . . . . (7,42)
achieved in the first test. If absolutely necessary, this test can be
2b
analyzed using approximate techniques outlined in Step 4.
3 . Calculate an arithmetic average of the n values, n , from Step 2.
4. If 0.5 :5 n :5 1 .0, estimate the AOF using either Step 4A or 4B.
A . Calculate the pseudopressure at p b ' Draw a line of slope l in Example 7.2-Analysis of Isochronal Tests. Estimate the AOF
through the stabilized, extended flow point, extrapolate the line to of this well I I using both the Rawlins-Schellhardt and the Houpeurt
the flow rate at App =Pp ( Ps ) -Pp ( Pb ) ' and read the AOF directly analyses. Table 7.8 summarizes the isochronal test data. Assume
Pb = 14.65 psia.
from the graph.
. .
.
1 78
GAS RESERVO I R E N G I N E E R I N G
TABLE 7 .9-PLOTTING FU NCTIONS F O R T H E
RAWLINS-SCHELLHARDT ANALYSIS, EXAMPLE 7 . 2
t
!J. P p
t
q
q
(hours) (MMscf/O) (psia 2 /cp) (hours) ( M Mscf/O)
0.5
0.983
0.3329 x 1 0 6 2 . 0
3.654
0.977 0.4309 x 1 0 6
3.565
0.970
0 .5536 x 1 0 6
3.453
0.965
0.6334 x 1 0 6
3.390
1 .0
2.631
0 .9688 x 1 0 6 3.0
4. 782
2 .588
1 .2364 x 1 0 6
4. 625
2.533
1 .5344 x 1 0 6
4.438
2 . 500
1 . 7257 x 1 0 6
4.31 8
214
1 . 1 56
--
3 ..
1 O'-+---!---.-4--;---r-i"-;-;.+---1-+--r---;-.;....;...�
3
4 5 6 7 8 9
0. 1
10
q . MMscf/D
Fig. 7 . 1 3-Rawli ns-Schellhardt analysis, Example 7 . 2 .
!J. P p
(psia 2 /Cp)
1 .4041 x 1 0 6
1 . 7644 x 1 0 6
2 . 1 793 x 1 0 6
2.4280 x 1 0 6
1 .9 1 8 1 x 1 0 6
2.3579 x 1 0 6
2 . 8725 x 1 0 6
3. 1 9 1 8 x 1 0 6
2.443 x 1 0 6
TABLE 7 . 1 O-LEAST-SQUARES REGRESSI O N ,
RAWLINS-SCHELLHARDT ANALYSIS, EXAMPLE 7.2
Point
2
3
4
E=
log q
0.4201
0.5628
0.6796
1 .6625
log !J. P p
5 .9862
6 . 1 474
6.2829
1 8 .41 65
(log !J.p p ) 2
35.8346
37. 7905
39.4748
1 1 3.0999
log q log !J. P p
2 .5 1 48
3.4598
4.2699
1 0 .2445
To determine the AOF graphically, first calculate the pseudopres­
sure at P b and compute
tJ.Pp =Pp ( Ps ) -pp ( Pb) = 9 . 9 7 l 5 x 106 - 2,098.7 =9.969 X 106 .
Solution.
Then, draw a line of slope l in through the stabilized flow point,
Rawlins-Schellhardt Analysis Technique.
1 . First, plot tJ.Pp =Pp ( Ps) -pp ( P wj ) vs. q on log-log graph extrapolate the line to the flow rate at tJ.Pp =Pp ( P s) -Pp ( Pb ) ' and
paper (Fig. 7 . 13) and include the single stabilized, extended flow read the AOF directly from the graph.
Houpeurt Analysis Technique.
point. Table 7.9 gives the plotting functions.
1 . Plot tJ.Pp /q = [ pp ( Ps ) -pp ( Pwj )]/q vs. q on Cartesian graph
2 . Calculate the deliverability exponent, n , for each line or
isochron using least-squares regression analysis (Eq. 7 . 30) . Note paper (Fig. 7 . 15) . Table 7 . 12 gives the plotting functions. Con­
that, because the first data point for each isochron does not align struct best-fit lines through the isochronal data points for each time.
with the data points at the last three flow rates (Fig. 7 . 1 3) , the first Note that, for each flow time, the point corresponding to the lowest
rate does fit on the same straight line, so all four data points will
data point is ignored in all subsequent calculations.
For example, at t = 0 . 5 hours, the deliverability exponent is cal­ be used for the analysis of each isochron.
2 . Determine the slope b of each line or isochron. For this ex­
culated as follows (see Table 7. 10) :
ample, use least-squares regression analysis (Eq. 7 . 39) . For ex­
ample, at t = 0 . 5 hour (see Table 7. 13) :
N E (log q log tJ.Pp )j - E log q E (log tJ.Pp )j
psia, pp ( p) ","pp ( pJ = 9 . 97 I 5 x 106 psia 2 /cp.
Pb = 1 4 . 65 psia, Pp ( Pb) = 2, 09 8 . 7 psia 2 /cp.
p "'"Ps = 352 . 4
N
j =1
j�1 (log
N
N
j =1 j j =1
T
J - [ j �1 (log
n l = �����----���--��--------
N
tJ. pp )j
tJ.Pp )
(3)( 1 0 .2445) - (1 .6625)(18 . 4 1 65)
(3)( 1 1 3 .0999) - (18 .4 165) 2
=0.88.
Table 7 . 1 1 summarizes the deliverability exponents determined
with a least-squares regression analysis for each isochron.
3 . The arithmetic average of the n values in Table 7. 1 1 is
_ n l +n 2 +n 3 +n
4 0.88 +0.91 +0.89 +0.88 0.89.
n=
4
4
Because 0.5 :s; n :s; 1 .0, estimate the AOF either using Eq. 7 . 35
or graphically using Fig. 7. 14. For this example, we use Eq. 7 . 3 5 .
First, determine the stabilized performance coefficient using the
coordinates of the stabilized, extended flow point and n=n:
1 . 156
4.
------
(2 . 443 X 106 ) 0 89
Calculate the AOF potential using Eq.
q AoF = C [ Pp ( Ps ) -Pp ( Pb)] n
7.35.
= 2 . 39 x 10 - 6 (9 . 97 1 5 X 106 - 2 ,09 8 . 7 ) 0 89
= 4 . 04
MMscflD.
= 2 . 39 x l O - 6 .
_
b, -
N
j�1
qj j�1 ( tJ.;p)j
j
�1
�'j N (qj )L ( j�'N jr
( tJ.Pp )r
N
q
(4)(4 . 6239 x 106 ) - ( 1 2 . 050)( 1 4 . 923 X 105)
(4)(44 . 108) - ( 1 2 . 050) 2
psia 2 /cp/(MMscf-D) 2 .
Other values of b are summarized in Table 7. 14.
3. The arithmetic average value of the slopes in Table 7 . 1 4 is
_ bl + b + b + b
2 3 4
b=
= 1 . 644 x 14
-------
4
( 1 . 644 + 1 .904 + 2 .255 + 2 . 492) x 104
4
= 2 . 074 X 104 psia 2 /cp/(MMscf-D) 2 .
4 . Calculate the stabilized isochronal deliverability line intercept
using tJ.pp /q = 2 . I 1 3 x 106 psia 2 /cp/(MMscf-D) at the extended,
stabilized point.
a = ( tJ. pp /q) - bq
= 2 . 1 1 3 X 106 - (2 . 074 x 104)( 1 . 156)
= 2 . 1 09 x 106 psia 2 /cp/(MMscf-D) .
1 79
DELIVERABI LITY TESTI N G OF GAS WELLS
5 . Calculate the AOF potential using the average value of b from
Step 3 and the stabilized value of a calculated from Step 4.
TABLE 7 . 1 1 -DELIVERABILITY EXPONENTS FOR
VARIOUS FLOWING TIMES, EXAMPLE 7.2
t
q AOP =
n
(hours)
0.88
0.91
0 .89
0.88
0.5
1 .0
2.0
3.0
7������
,
+
l i fl l,: :: :T ' �' TTml
I Os-+---r---;-..<;--;..+1-H+-----if--+--+-+-+1+H
3
4 5 6 7 1 9
0.1
,
10
F i g . 7 . 1 4-Rawlins·Schelihardt analysis resu lt, Example 7 . 2 .
l .° Tr==:;:;::;:�----;--"""""T -T--l
o
•
0.8 - ..
0. 6 -
OO
. xl
O'
._
..
•
O.S hr
,
I
r
1 .0 ht
2.0 ht
,
.
_
·,
3 .0 ht ·························· ···_······················· ············· ··_···· r-···_·_· __·······_···
. . ....._.... .. ..
I
:
_-+..__._ -_._._-_. -
:
!
" 'T"
.
_ _--t ·_·· _ ....
·· ····
··
�
I
!
-
'l--'
3
q , MMscf/D
!
_ .._..! .••••.
4
- (2. 109 X 106) +-!(2. 109 X 106) 2 +4(2.074 X 104)(9.97 X 106)
2(2.074 X 104)
=4.53 MMscflD.
Fig. 7 . 1 6 illustrates the results.
7.5.4 Modified Isochronal Tests. The time to build up to the aver­
age reservoir pressure before flowing for a certain period of time
still may be impractical, even after short flow periods. Consequent­
ly, a modification 1 2 of the isochronal test was developed to short­
en test times further. The objective of the modified isochronal test
is to obtain the same data as in an isochronal test without using
the sometimes lengthy shut-in periods required to reach the aver­
age reservoir pressure in the drainage area of the well.
The modified isochronal test (Fig. 7. 17) is conducted like an
isochronal test, except the shut-in periods are of equal duration and
the flow periods are of equal duration. The shut-in periods, how­
ever, should equal or exceed the length of the flow periods. Be­
cause the well does not build up to average reservoir pressure after
each flow period, the shut-in sandface pressures recorded immedi­
ately before each flow period rather than the average reservoir pres­
sure are used in the test analysis. As a result, the modified isochronal
test is less accurate than the isochronal test. Note that, as the dura­
tion of the shut-in periods increases, the accuracy of the modified
isochronal test also increases. Again, a final stabilized flow point
usually is obtained at the end of the test but is not required for analyz­
ing the test data.
The well does not build up to the average reservoir pressure during
shut-in; the analysis techniques for the modified isochronal tests
are derived empirically. Recall the transient flow equation, ex­
pressed in terms of the reservoir pressure at the start of flow, on
which isochronal testing is based:
1 .422 X 106 q T
3
In
- - +s +Dq .
pp( P s) -Pp( Pwf) =
kg h
rw
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.45)
In new reservoirs with little or no pressure depletion, P s equals
the initial reservoir pressure ( P s =Pi ) ; in developed reservoirs,
in Eq. 7.45
P s < Pi ' In addition, the transient drainage radius,
is defined as
[ ( rd )
j
j
S
Fig . 7 . 1 S-Houpeurt analysis of isochronal test data, Example
7.2.
'd� J 37;;;, <,'
.
.
.
.
!:J.. p p lq
(psia 2 Icp/M Mscf.D)
3.387 x 1 0 5
3.682 x 1 0 5
3.843 x 1 0 5
4.01 1 x 1 0 5
4.41 0 x 1 0 5
4.777 x 1 0 5
4.949 x 1 0 5
5.098 x 1 0 5
t
q
(hou rs) (MMscflD)
2.0
0. 970
2.533
3.453
4.438
3.0
0.965
2.500
3.390
4.31 8
214
1 . 1 56
.
rd,
. . . . . . . . . . . . . . . . . . . . . . . . . (7.44)
TABLE 7 . 1 2-PLOTTING FUNCTIONS FOR
HOUPEURT ANALYSIS, EXAMPLE 7.2
t
q
(hou rs) ( M M scf/D)
0.5
0 .983
2.631
3.654
4.782
0.977
1 .0
2.588
3.565
4.625
]
-
+----ir---+----t---+---�
o
- a + -!a2 +4b [ pp( Ps) -Pp( Pb)]
2b
!:J.. p Iq
(psia 2 /cp/M M SCf-D)
5.707 x 1 0 5
6.058 x 1 0 5
6.31 1 x 1 0 5
6.473 x 1 0 5
6. 564 x 1 0 5
6.903 x 1 0 5
7. 1 62 x 1 0 5
7.392 x 1 0 5
2.1 1 3 x 1 06
1 80
GAS RESERVOI R E N G I N E E R I N G
TABLE 7 . 1 3-LEAST ·SQUARES REGRESSION,
HOUPEURT ANALYSIS, EXAMPLE 7.2
-
Point
1
2
3
4=
q0.983
2.631
3.654
4.782
1 2.050
E
rd
�
0.966
6. 922
1 3. 352
22.868
44. 1 08
0.3320 x
0. 9688 x
1 .4041 x
1 .9 1 8 1 X
4.6239 x
106
106
106
106
106
3.387 x
3. 682 x
3.843 x
4.01 1 x
1 4. 923 x
TABLE 7 . 1 4-SLOPES, b , OF ISOCHRONS, EXAMPLE 7 . 2
105
105
105
105
105
Because i s a function only of time and not flow rate, Eq. 7.45
is valid at any fixed time. For modified isochronal tests, Eq. 7.45
is approximated by Eq. 7.52 where the stabilized shut-in BHP, P s '
is replaced with the shut-in BHP, P ws ' measured before each flow
period, where P ws s P s '
1 .422 X 106 q T
3
In
- - + s +Dq .
pp( P ws ) -pp( Pwj )
kg h
w 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7.52)
We can rewrite Eq. 7.52 as
pp( p ws ) -pp ( p wj) =atq+bq 2 , . . . . . . . . . . . . . . . . . . . . (7.53)
1 .422 X 106 T
3
In
where at =
- - +s . . . . . . . . . . . (7.47)
kg h
w 4
[ (-rd )
1
r
and b =
[ (-rd ) 1
r
1 .422 X 106 DT
. . . . . . . . . . . . . . . . . . . . . . . . . . . (7. 15)
kg h
Eq. 7. 15 indicates that b is independent of time and will remain
constant during the test. Similarly, Eq. 7.47 illustrates that a t is
constant for a fixed time. The similarity of Eqs. 7.46 and 7.53 for
the isochronal and modified isochronal tests, respectively, suggests
that the modified isochronal test data can be analyzed like those
from an isochronal test.
The theory developed for the modified isochronal test implies
that, if the empirical approximation of using P ws instead of P s is
valid, the transient data will plot as straight line for each time hav­
ing the same slope, b. The intercept, at, will increase with increas­
ing time. Therefore, we can draw a line with slope b through the
stabil ized data point and use the coordinates of the stabilized point
and the slope to calculate a stabilized intercept, a, that is indepen­
dent of time, where
1 .422 X 106 T � �
_ + s . . . . . . . . . . . . . . . . (7.48)
a=
ln
kg h
w 4
[ (r ) ]
b
(hours)
t
[psia 2 /(M Mscf-O) 2 )
0.5
1 .0
2.0
3.0
1 . 644 x 1 0 4
1 . 904 x 1 0 4
2.255 x 1 0 4
2 .492 x 1 0 4
To calculate the AOF of the well, we use the average reservoir
pressure, P S I measured before the test instead o f the P ws value, or
- a + ..Ja 2 +4b[pp ( P s ) -Pp ( P b )]
. . . . . . . . . . (7.44)
2b
We now consider two variations of the modified isochronal test:
tests with a stabilized flow point obtained at the end of the test and
tests run without that final point.
� ---!.
-----
-
--
Modified Isochrotud Tests With a Stabilized Flow Point. Rawlins­
Recall the empirical Rawlins and Schellhardt
equation in terms of transient isochronal test data:
log(q) = log(Ct ) + n { log [ pp( p s ) -pp ( Pwj)] } ' . . . . . . . . . (7.49)
As in the graphical analysis techniques for isochronal tests, we
plot several trends of data taken at different times during a modi­
fied isochronal test. The slope n of each line through points at equal
time values will be constant. However, the intercept, log (Ct), is
a function of time but not flow rate. Therefore, we must calculate
a different intercept for each isochronal line. In addition, for modi­
fied isochronal tests, we use pp( P ws ) instead of pp( P s ) in Eq. 7.49,
which gives
log(q) = log(Ct ) +n{log[ pp( p ws ) -pp( P w/)] } ' . . . . . . . . (7.54)
The conventional analysis technique for modified isochronal test
data is to plot log [ pp( P ws ) -pp( P w/)] vs. log (q) for each time,
giving a straight line of slope l in and an intercept of { - l In [ log
(Ct)] ) . The Rawlins-Schellhardt analysis procedure for modified
isochronal tests with a stabilized flow point is similar to that present­
ed for isochronal tests, except the plotting functions are developed
in terms of the shut-in pressure measured immediately before the
next flow period. Only the stabilized, extended flow point is plot­
ted in terms of the average reservoir pressure measured before the
Schellhardt Analysis.
test . Example 7 . 3 illustrates the procedure .
Extended
Flow Rate
�======�-r--�----1
4 ��
o O.S hr
1 .0 hr
2.0 hr
3.0 hr
•
3 - ...
2
1
.
..
•
"
.
. :.: ..� . .T m
..:..�
Point
-
S labilized
- ................ ..........·� .
.. .
. .
•
!= 2 1 4
hr
..
"
,
t .:. : - - - :. t=.:.: . :. :..�!: .
:.�.:,.:..
- ...... -.... -. . · - ·
.
t
.
·
. .· .. · · ·
· . . · ·· ·· .. · · ·· .. ·· · · . . . . r · · _ ·· · ·· · .. ··· · . . · ·· .. · · ·
·T·
·
. . . . . . . . . . . . . . . . . . . . . . . . ·. . ·
.
�
.�
..
.
.
:..:..:.. �.. :
t ... . . . . . . . . r. . . . . . . . . . . . . .
..
.
. ..
.
. .
. .
.. . .
.
.
Ox lO -+----;----;----+----+-----...1
o
:
�
q,
�
MMscf/D
Time. t (hours)
4
Fig. 7 . 1 6-Houpeu rt analysis of Isochronal test data result,
Example 7.2.
Time, (hours)
t
Fig. 7 . 1 7 -Pressure and flow rate history of a typical modified
isochronal test.
1 81
D ELIVERABI LITY TESTI N G OF GAS WELLS
TABLE 7 . 1 S-MODIFIED ISOCHRONAL TEST DATA, EXAMPLE 7 . 3
P wI (psia)
Time
(hours)
o ( P w. )
0.5
1 .0
1 .5
2.0
q = 1 . 520 M M scflD q = 2 . 04 1 M M scflD q = 2 .688 M MscflD q = 3. 1 22 M MscflD
70 1 . 2
703.5
706.6
706. 6
541 . 7
578. 5
624.5
655.6
537.8
573.9
620.7
653.6
572 .3
536.3
61 9.9
652 . 1
570.8
534.7
619.1
651 . 3
5.093 x 1 0 7
3.403 x 1 0 7
3.348 x 1 0 7
3.330 x 1 0 7
3.31 2 x 1 0 7
5.093 x 1 0 7
3.970 x 1 0 7
3.922 x 1 0 7
3.91 1 x 1 0 7
3.901 x 1 0 7
5.093 x 1 0 7
4 .379 x 1 0 7
4.352 x 1 0 7
4 .332 x 1 0 7
4.321 x 1 0 7
O[ p p ( P w. » )
0.5
1 .0
1 .5
2.0
5.01 5 x 1 0 7
2.979 x 1 0 7
2.936 x 1 0 7
2.9 1 9 x 1 0 7
2.902 x 1 0 7
Extended Flow Point
t = 24 hours
p wI = 567.7 psia
p = 706.6 psia
q = 2 . 665 M Mscf/D
pp ( p wI ) = 3.276 x 1 0 7 psi a 2 /cp
pp ( 1 4.65) = 2 ,766.6 psia 2 /cp
Houpeurt Analysis. As shown previously, the Houpeurt deliver­ point and calculate a stabilized intercept, a, using the coordinates
of the stabilized point, or
ability equation in terms of transient isochronal test data is
Ilpp
q
=
Pp ( Ps ) -pp ( P wj )
q
=at + bq . . . . . . . . . . . . . . . . . (7 . 50)
a=
[ pp ( Ps ) -p/ P wj )]
q
Ilpp
bq = - -bq . . . . . . . . . . . . . (7 . 5 1 )
q
The slope b of the line through the stabilized point should re­
For modified isochronal test data, we must modify Eq. 7 . 50 with
the assumption that we can use pp ( Pws) instead of pp ( Ps) ' With main the same. In addition, we must use the average reservoir pres­
sure, which is measured before the test, to evaluate the
this assumption, Eq. 7 . 50 becomes
pseudopressure, pp ( Ps) , in Eq. 7 . 5 1 . Example 7 . 3 illustrates the
Ilpp pp ( P ws) -pp ( P wj )
Houpeurt analysis procedure for modified isochronal tests with a
=
at + b q, . . . . . . . . . . . . . . . (7 . 55) stabilized flow point, which is similar to that presented for isochronal
q
q
tests.
1 .422 X 1 0 6 T
where at =
and b =
kg h
1 . 422 x 106DT
[In (- )
3
- - +s
rw
4
rd
]
. . . . . . . . . . . (7 .47)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 1 5)
kg h
The form of Eq. 7 . 55 suggests that, if we plot IlPp /q = [ pp ( Pws)
vs. q, we will obtain a straight line with a slope b
and intercept a t . We also can extend this theory to the stabilized
-pp ( P wj )]/q
1
�
C'l.�Ul
c:-
�
_
--:-�·s
�:-T-"
-" '-'--" -" " --" --'-1" '----" " '-" -1'-" -'- ·······1
A 1 .0 hr
I
,
i
i
I .S hr ............. ............................ ................,. .............................. ,. ..................... 1
o 2.0 hr
i
! i
I
lIE
.j.- . . t . . . I . LJ
. . . . . . . . . . . . . . . . .-..t. . . . . . . . . ,J . _. . . . . /
I
I i I
t
!
I
I'
I
i i i
II
:1
;
i
I
...
!
i
f
.. . . .. ....
'
i
i
!
!
i
I
.. . . . .. .
.
.. . . . . .. . . .
,
!
i!.
i
. . . . .. .. .
.
..
,
. ..
.
i
!
j,.
.
� "': mf��
I
•
9
q, MMscf/D
Fig. 7 . 1 8-Rawllns-Schellhardt plot of the modified Isochronal
test data, Example 7 . 3 .
Example 7.3-Analysis of a Modified Isochronal Test With a
Stabilized Flow Point. Using the following data taken from Well
4 of Ref. 1 3 , calculate the AOF using both Rawlins and Schell­
hardt and Houpeurt analysis techniques. Assume Pb = 1 4 . 65 psia,
where pp ( Pb) = 5 . 093 x 107 psia 2 /cp. Table 7 . 1 5 gives the test
data.
h = 6 ft .
r w = 0 . 1 875 ft .
!/> = 0 . 27 1 4 .
T = 5400R (80°F) .
P "" Ps = 706 . 6 psia.
jig = 0 . 0 1 5 cpo
Z = 0 . 97 .
cg = 1 . 5 x l O - 3 psia - 1 .
'Y g = 0 . 75 .
Sw = 0 . 30 .
cf = 3 x l O - 6 psia - 1 .
A = 640 acres (assume that
square drainage area).
the well is centered in a
Solution.
Rawlins-Schellhardt Analysis.
Plot Ilpp =pp ( Pws) -pp ( P wj ) vs. q on log-log graph paper
gives the plotting functions. In addition,
plot on the same graph the value of Ilpp that corresponds to the
stabilized, extended flow point evaluated at Ps '
1.
(Fig. 7. 18) . Table 7 . 1 6
Il Pp =Pp ( Ps ) -Pp ( P wf ) = 5 . 093 x 1 0 7 - 3 . 276 x 1 0 7
psia 2 /cp.
For each time, construct the best ..fit line through the data points.
Because the first data point for each isochron does not follow the
trend of the higher rate points, it will be ignored for all subsequent
calculations.
= 1 . 817 x 107
GAS RESERVO I R EN G I N EERING
1 82
TABLE 7 . 1 6-PLOTTING FUNCTIONS F O R RAWLINS-SCHELLHARDT ANALYSIS, EXAMPLE 7 . 3
Time
(hours)
0.5
1 .0
1 .5
2.0
t. P p (psia 2 /cp)
q = 1 .520 M MscflD q = 2.041 M MscflD q = 2.688 M M scf/D q = 3. 1 22 M MscflD
2.039 x 1 0 7
1 .645 x 1 0 7
1 . 1 23 x 1 0 7
0.71 4 x 1 0 7
2 . 082 x 1 0 7
1 . 700 x 1 0 7
1 . 1 71 x 1 0 7
0 . 74 1 x 1 0 7
2 . 099 x 1 0 7
1 .71 8 X 1 0 7
0 . 76 1 X 1 0 7
1 . 1 82 x 1 0 7
2. 1 1 3 x 1 0 7
1 . 736 x 1 0 7
0.772 x 1 0 7
1 . 1 92 x 1 0 7
2 . Calculate the deliverability exponent, n , for each line or stabilized C value is evaluated at P s measured at the beginning of
isochron. For this example, we use least-squares regression analy­ the test, rather than P ws ' From Eq. 7 . 36 ,
sis (Eq. 7 . 30) . For example, at t = 0 . 5 hour (see Table 7 . 1 7) ,
N
N
N
C=
q
2 . 665
[ pp ( ps) -pp( pw/ )] n
(5 . 093 X 1 07 - 3 . 276 x 1 07 ) 0.74
-------
= 1 . 1 32 x lO - 5 .
From Eq.
7.35,
q AOF = C [ Pp ( Ps ) -Pp ( Pb)] n
= 1 . 132 x 10 - 5 (5 . 0935 x 1 0 7 - 2 ,766 . 6) 0.74
(3)(8 . 8963) - ( 1 .2336)(2 1 . 5753)
= 0 . 72 .
summarizes deliverability exponents.
The arithmetic average of the values in Table 7 . 1 8 is
Table 7 . 18
3.
n=
MMscflD.
We can also determine the AOF graphically by drawing a line
of slope lin through the extended flow point, extrapolating the line
to the flow rate at I1Pp =Pp( Ps ) -Pp ( Pb ) , and reading the AOF
directly from the graph (Fig. 7 . 1 9) .
=5.7
(3) ( 1 5 5 . 1988) - (2 1 . 5753) 2
n l + n2 + n 3 + n4
0 . 72 + 0 . 74 + 0 . 74 + 0 . 7 8
4
4
0 . 74 .
Houpeun A nalysis.
1 . Plot I1pplq = [ pp ( Pws ) -Pp ( P wf )]/q vs. q on Cartesian graph
paper (Fig. 7.20) . In addition, plot the I1pp lq value that corre­
sponds to the stabilized, extended flow point. Table 7 . 1 9 gives the
plotting functions. Construct best-fit lines through the modified
4 . Because 0 . 5 $ n $ 1 .0 , determine the stabilized performance isochronal data points for each time. The first data point at the lowest
coefficient, C, using the coordinates of the stabilized, extended flow rate for each isochron does not fit on the same straight line as the
point and n =n. Note that the pseudopressure used to calculate the last three rate points and is ignored in subsequent calculations.
TABLE 7 . 1 7-LEAST-SQUARES REGRESSION,
RAWLI NS-SCHELLHARDT ANALYSIS, EXAMPLE 7.3
Point
2
3
4
E=
log q
0 .3098
0.4294
0 .4944
1 .2336
log t. P p
7.0503
7.21 62
7. 3088
2 1 . 5753
(log t. p p ) 2
49.7067
52.0735
53.4 1 86
1 55 . 1 988
TABLE 7 . 1 8-SLOPES, n , O F ISOCHRONS, EXAMPLE 7 . 3
t
(hou rs)
0.5
1 .0
1 .5
2.0
log q log t. P p
2 . 1 842
3.0986
3.61 35
8.8963
n
0.72
0 . 74
0.74
0 . 78
7.5
§
u
�
�
6..
u
'"
::::::'
' C;;;
<'I ",
p.
�
ct
�
7.0
6.5
6.0
5 .5
5.0
4 .5 , \ 0
'
1.0
q , MMscf/D
Fig. 7 . 1 9-Rawllns-Schellhardt analysis of a modified
isochronal test, Example 7 . 3 .
1 .5
2 .0
2.5
3.0
3.5
q , MMscf/D
Fig. 7.20-Houpeurt plot of the modified isochronal test data,
Example 7 . 3 .
1 83
DELIVERABILITY TEST I N G OF GAS WELLS
TABLE 7 . 1 9-PLOTTING FUNCTIONS FOR HOUPEURT ANALYSI S , EXAMPLE 7 . 3
t. p p lq [psia 2 /(c p 1 M Mscf/D»)
Time
(hou rs)
q = 1 .520 M MscflD q = 2 . 04 1 M MscflD q = 2 . 688 M Mscf/D q = 3. 1 22 M MscflD
0.653 x 1 0 7
0.61 2 x 1 0 7
0.550 x 1 0 7
0 .470 x 1 0 7
0.632 x 1 0 7
0.667 x 1 0 7
0.488 x 1 0 7
0 .574 x 1 0 7
0.672 x 1 0 7
0.579 x 1 0 7
0.501 X 1 0 7
0 . 639 x 1 0 7
0.584 x 1 0 7
0.678 x 1 0 7
0.646 x 1 0 7
0 . 508 x 1 0 7
0.5
1 .0
1 .5
2.0
2 . Detennine the slopes of the lines, b , for each isochron by least­
squares regression analysis of the best-fit lines through the data
points . For example, at t = 0 . 5 hour (see Table 7.20) ,
bl =
Nj
�
1
N
( ilpp )j -
N j �1
qAOP =
� N� ( ; )
(� )
j 1
( qj ) 2 -
qj
il p
j 1
2(8 . 723 x 1 0 5 )
= 5 . 5 MMscflD .
Fig. 7.21 shows the data for this example .
(3)(2 1 . 1 38) - (7 . 85 1 ) 2
= 9 . 654 x 10 5 psia2 /cp/(MMscf-D) 2 .
Table 7.21 summarizes the slopes of the isochrons .
The arithmetic average value o f the slopes i n Table 7 . 2 1 i s
b2 + b3 + b4
b = ---3
_
( 8 . 678 + 8 . 7 1 1 + 8 . 780) x 1 0 5
3
= 8 . 723 x 1 0 5 psia2 /cp/(MMscf-D) 2 .
3 . Calculate the stabilized isochronal deliverability line intercept,
a.
2 . 665
( 8 . 723 x 1 0 5 )(2 . 665)
TABLE 7 .20-LEAST-SQUARES REGRESSION,
HOUPEURT ANALYSI S , EXAMPLE 7 . 3
Point
2
3
4
E=
q
2.041
2 . 688
3 . 1 22
7.85 1
�
4 . 1 66
7.225
9 . 747
2 1 . 1 38
Modified Isochronal Tests Without a Stabilized How Point. Be­
cause the well is not required to build up to the average reservoir
pressure between the flow periods, the modified isochronal approx­
imation shortens test times considerably . However , the test analy­
sis relies on obtaining one stabilized flow point . Under some
conditions , environmental or economic concerns prohibit flaring
produced gas to the atmosphere during a long production period,
thus preventing measurement of a stabilized flow point. These con­
ditions often occur when new wells are tested before being con­
nected to a pipeline .
Two methods have been developed to analyze modified isochronal
tests without a stabilized flow point. The Brar and Aziz 13 method
was developed for the Houpeurt analysis , while the stabilized C
method 1 4 was developed for the Rawlins and Schellhardt analy­
sis . The stabilized C method requires that we have prior knowl­
edge of permeability and skin factor or that we detennine these
properties using the methods Brar and Aziz proposed for analyz­
ing modified isochronal tests . Both methods require knowledge of
the drainage area shape and size .
Brar and Aziz Method-Houpeurl Analysis. The Brar and Aziz
method is based on the transient Houpeurt deliverability equation,
ilpp =pp ( ps) -pp ( pwj ) = a( q + bq 2 , . . . . . . . . . . . . . . . . (7. 1 1 )
= 4 . 493 x 106 psia2 /cp/(MMscf-D) .
--
tip p
1 . 1 23 x 1 0 7
1 .645 x 1 0 7
2 . 039 x 1 0 7
4 . 807 x 1 0 7
ti p p 'q
0.550 x
0.61 2 x
0.653 x
1 .81 5 x
107
107
107
107
g'"
<.)
�
�
6.
<.)
"=:::
"'
TABLE 7 . 2 1 -SLOPES , b, O F THE ISOCHRONS, EXAMPLE 7 . 3
t
(hours)
0.5
1 .0
1 .5
2.0
2b
...; (4 . 493 X 1 0 6 ) 2 + 4(8 . 723 x 1 0 5 ) [ (5 . 093 x 1 0 7 ) - 2 , 766. 6] I
qj
(3)(4 . 807 X 1 07 ) - (7 . 85 1 ) ( 1 . 85 1 X 1 07 )
1 . 8 1 7 x 10 7
- a + "';a 2 + 4b[ pp ( p ) -pp ( l 4 . 65) ]
= - (4 . 493 x 1 06 ) +
j
2
j 1
Calculate the AOF potential using b from Step 2 and the stabi­
lized a value:
b
[psia 2 /cp/(M Mscf-D) 2 )
9. 654 x 1 0 5
8 .678 x 1 0 5
8.71 1 x 1 0 5
8 . 780 x 1 0 5
",
'r;;
p..
:i
�
"<:]
7.0
6. 5
6 .0
5.5
5.0
4.5x 1 0
6
1 .0
1 .5
2.0
2.5
3.0
3.5
q, MMscf/D
Fig. 7.21 -Houpeurt analysis of modified Isochronal test data,
Example 7 . 3 .
1 84
1 . 422 X 10 6 T
where a t =
kgh
[
1 . 1 5 1 log
(
GAS RESERVO I R E N G I N EE R I N G
) ]
kg t
1 ,688 cJ>jrgc",�
can u se the gas permeability calculated from E q . 7 . 62 and the skin
factor from Eq. 7.63 to calculate
+s ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 1 3)
b=
1 .422 x 10 6 D T
kgh
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 1 5)
and P s = stabilized BHP measured before the deliverability test.
The transient-gas-flow equation derived from the diffusivity equa­
tion for a homogeneous-acting , isotropic reservoir producing at a
constant sandface rate is
[
X 1 . 15 1 10g
]
kgt
)
( ,688cJ>jrgc
tr�
+ S +Dq . . . . . . . . . . . . . . (7 . 9)
1
The form of Eq . 7 . 9 suggests that, for a single and constant flow
rate, a plot of tJ.Pp =Pp ( Ps) -pp ( P wj ) v s . log t will be a straight
line with slope m, where
m=
1 . 422 x 1 0 6 ( 1 . 1 5 1 )qT
1 . 632 x 1 0 6 qT
kgh
kgh
a=
[
---kgh
kgf
)
( ,688cJ>jrg
ct r�
� (� ) �
j l
kgh
1 . 422 X 106 T
kgh
[
+ S + Dq . . . . . . . . . . . . . (7 . 57)
.
. . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 59)
1 . 1 5 1 log
kg
( ,688cJ>jrgc
t r� )
1
]
+ s +Dq
Substituting the definition of b (Eq. 7. 15) into Eq. 7 . 60 gives
an equation for the intercept, c' :
c' =
kgh
[
1 . 1 5 1 log
(
kg
1 ,688cJ>jrgct r�
) ]
kg =
m 'h
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 62)
Combining Eqs . 7 . 59 and 7 . 6 1 yields an equation for the skin
factor,
S = 1 . 15 1
[ (� ) ( cJ>p,_gkct� rw ) ]
m
- IOg
2
CA
�
j l
� �
(� )
(qj ) 2 _
N
qj
j l
N
(qj ) 2 _
N
N E
j l
(tJ.Pp )j
2
qj
j l
+ 3 . 23 . . . . . . . . . . . (7 . 63)
Estimating the AOF potential of the well requires that we have
a stabilized value of a . If we know the drainage area shape , we
N
N
(at)j E (log t )j
j=l
j=l
j=l
--��----��--�-m' =
. . . . . (7 . 65)
N
and
(at log t )r
�
E
c' =
(� Y
j l
N
N
j=l
E
(log tj ) 2 _
j l
(a t )j
E
j=l
N
+ s . . . . (7 . 6 1 )
m' and c' can be calculated using regression analysis of Eq. 7 . 5 8 .
Alternatively , these variables can b e computed from a plot o f a t
vs. log t. We then can calculate the permeability from the slope,
1 . 632 x 1 0 6 T
3
- - + S . . . . . . (7 . 14)
4
4. Plot a t vs. log t and draw the best-fit l ine through the data.
If the earliest-time data do not lie on the straight line, these data
should be omitted from subsequent calculations. Calculate the slope,
m ' , and intercept, c' , of the l ine either directly from the graph or
using Eqs . 7 . 65 and 7 . 66 , respectively .
- bq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 60)
1 .422 X 1 0 6 T
]
)
r�
1 O . 0M
--
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 64)
at = m ' l og(t) + c ' , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 5 8 )
and c' =
j j l
N
Dividing Eq. 7 . 5 7 by rate , q, gives
1 . 632 x 1 0 6 T
(
. . . . . . . . . . (7 . 42)
2b
A nalysis Procedure-Brar and Aziz Method.
1 . Plot tJ.pp lq = [ pp ( Pws) -pp ( p wj) ]lq vs. q on Cartesian graph
paper. Construct best-fit lines through the modified isochronal data
points for each time , ignoring the data at the lower rates that do
not follow the trend of the data at higher rates .
2 . Determine the slopes, b , of the lines for each time by 1east­
squares regression analysis (Eq . 7 . 39) or simply by drawing the
best-fit line through the data. Calculate the arithmetic average of
the slopes, b.
3 . Calculate the transient deliverability line intercepts , a t , for
each isochronal line either directly from the graph or with Eq. 7 . 64 .
. . . . . . . . . (7 . 56)
]
1
1 . 1 5 1 log
- a + -Ja 2 + 4b[ pp ( Ps ) -Pp ( P b )]
q AOP =
1 . 422 x 1 0 6 qT
X 1 . 1 5 1 10g
where m' =
kgh
[
Appendix C gives shape factors , CA ' for various reservoir
shapes and well locations . This stabilized value of a then is used
in Eq. 7 . 42 to calculate the AOF of the well :
Equating Eqs . 7 . 9 and 7 . 1 1 yields
atq + bq 2 =
1 . 422 x 1 0 6 T
log tj
N
N
(log tj ) 2 -
�
j l
E (a t
j=l
(log tj ) 2 -
log t )j
(� Y
j l
E
j=l
(log t )j
log tj
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 66)
5 . Calculate the formation permeability to gas using the slope
of the semilog straight line , m ' , from the plot of a t vs. log t.
kg =
1 . 632 x 1 0 6 T
m'h
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7. 62)
6 . Calculate the skin factor using c ' .
S = 1 . 15 1
[ (� ) ( cJ>p,_gkct� rw ) ]
]
)
(
[
r�
kgh
m
- IOg
2
+ 3 .23 . . . . . . . . . . . (7 . 63)
7 . Calculate the stabilized line intercept, a .
a=
1 .422 X 1 0 6 T
1 . 15 1 log
1 0 . 0M
--
CA
3
- - + S . . . . . . (7 . 14)
4
1 85
DELIVERAB I LITY TESTI N G OF GAS WELLS
8 . Calculate the AOF potential using b from Step 2 and the stabi­
lized a value from Step 7 .
q AOF =
- a + -J a 2 + 4b[ pp ( p ) -Pp ( P b )]
2b
' . . . . . . . . . . . (7 . 42)
a( l - n)
qe =
where P C P b ) = pseudopressure evaluated at P b '
p
StabilIZed C Method-Rawlins-Schellhardt Analysis. Although the
Houpeurt equation has a theoretical basis and is rigorously correct,
the more familiar but empirically based Rawlins and Schellhardt
equation continues to be used and is indeed favored by the natural
gas industry. Consequently , in this section we combine the Houpeurt
and Rawlins-Schellhardt analysis techniques and develop a more
accurate version of the Rawlins-Schellhardt method for analyzing
modified isochronal tests. This analysis technique, called the stabi­
lized C method, 1 4 is derived by equating the stabilized Rawlins
and Schellhardt empirical backpressure equation with the stabilized
theoretical Houpeurt equation to obtain equations for the delivera­
bility exponent, n, and the stabilized flow coefficient , C, in terms
of the Houpeurt flow coefficients , a and b .
T o obtain a n equation for the exponent n , w e begin b y taking
the logarithm of both sides of the stabilized Rawlins and Schell­
hardt empirical backpressure equation (Eq. 7 . 24) ,
10g(q) = l og(C) + n 10g[ pp ( p) -p ( Pwj )] ' . . . . . . . . . . . (7 . 67)
p
Rearranging Eq. 7 . 67 and solving for n, we see that n is the slope
of a log-log plot of q v s . tlp - Alternatively , n can be expressed
p
as the derivative of log q With respect to log tlpp ,
d log(q)
n = --------��----d 10g[ pp ( p) -pp ( P wj ) ]
dq
q d 10 g[ pp ( p) -pp ( Pwj ) ]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 68)
Similarly , if we take logarithms of both sides of the stabilized
Houpeurt equation given by Eq. 7 . 1 2 , we obtain
10g [ pp ( p) - Pp ( P lif ) ] = log [aq + bq 2 ] . . . . . . . . . . . . . . . (7 . 69)
Differentiating log tlpp with respect to q gives
d 10g[ p ( p) -P ( P ) ]
p
p lif
aq + bq 2
dq
where qe = the un ique rate at which the pseudopressure drops from
Eqs . 7 . 72 and 7 . 73 are equal . In terms of the Houpeurt coefficients
and the deliverability exponent, from Eq. 7 . 7 1 ,
d(aq + bq 2 )
a + 2bq
dq
aq + b q 2
b(2n - I )
Equating Eqs . 7 . 68 and 7 . 70 shows that n can be expressed in
Note that n is different for different rates , but we will clear up
this difficulty shortly .
To develop an expression for the performance coefficient, C, we
first take the logarithm of the original Rawlins and Schellhardt equa­
tion in terms of pseudopressures, or
10g
( �)
Similarly , taking the logarithm of the Houpeurt equation in terms
of pseudopressures gives
log [ pp ( p) -Pp ( P lif )] = log(aq + bq 2 ) . . . . . . . . . . . . . . . (7 . 73)
Equating Eqs . 7 . 72 and 7 . 73 and solving for C yields
C=
qe
(aqe + b q � )
n '
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 74)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 75)
The stabilized C method is limited by the need for values of reser­
voir properties determined before the deliverability test. However,
these properties can be estimated either from drawdown or build­
up test analysis or from the Brar and Aziz method .
Analysis Procedure-Stabilized C Method.
1 . Plot tl Pp =Pp ( Pws) -pp ( Pwj ) vs. q on log-log graph paper.
Construct the best-fit line through the data points for each isochron.
Some of the data points at lower rates may not agree with the general
trend of the data at higher rates , so the data points at the lower
rates should be ignored in all subsequent calculations .
2 . Determine the deliverability exponent, n , with least-squares
regression analysis (Eq . 7 . 30) or simply by taking the slope of the
perceived "best-fit " line through the test data . In addition , calcu­
late the arithmetic average of the exponents , n.
3. Determine the theoretical values of the Houpeurt coefficients,
a and b, using permeability , skin, and non-Darcy flow coefficient
values obtained from drawdown and buildup tests on the well , or
alternatively from the modified isochronal test data using the Brar
and Aziz method.
a=
1 . 422 X 1 0 6 T
kg h
[
1 . 1 5 1 10g
1 . 422 x 106DT
and b = -----
( a) l
1 O . 06A
--
CA r
3
- - +s
4
. . . . . . (7 . 1 4)
. . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 1 5)
4 . Calculate the rate at which the change in pseudopressure ,
tlpp , determined with the Rawlins-Schellhardt equation is equal
to the change in pseudopressure determined using the Houpeurt
equation . Use the average slope , n, of the deliverability plot and
theoretical stabilized a and b values from Eqs . 7 . 14 and 7. 1 5 , re­
spectively .
qe =
= n 10g[ pp ( p) -pp ( Pwj ) ] ' . . . . . . . . . . . . . . . . (7 . 72)
.
To apply the stabilized C method , we must assume that the slope,
n , of the empirical deliverability plot remains constant with time .
This assumption implies that , if we can calculate values of a and
b for given reservoir properties , we can calculate a flow rate from
Eq. 7 . 75 where the change in pseudopressure calculated by the
Rawlins-Schellhardt equation is equal to the change in pseudopres­
sure calculated by the theoretical Houpeurt equation. We then sub­
stitute this flow rate into Eq. 7 . 74 and calculate a stabilized C value.
We use the constant n and calculated stabilized C to calculate a value
of AOF :
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 70)
terms of the gas flow rate and the Houpeurt flow coefficient s ,
.
a( 1 - n)
b(2n - l )
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 75)
5 . Calculate the stabilized C value using n from Step 2 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7 . 74)
6. Substitute the stabilized C value from Step 5 into Eq . 7 . 35
to calculate the AOF potential of the well . Use n from Step 2 .
1 86
GAS RESERVOI R E N G I N E E R I N G
Example 7.4-Analysis o f a Modified Isochronal Test Without
a Stabilized Data Point. The purpose of this example is to com­
pare results obtained from the analysis of a modified isochronal
test (see Table 7.22) with and without an extended , stabilized data
point. Calculate the AOF for the following modified isochronal test
data without the extended flow point. Use both the Brar and Aziz
and the stabilized C methods . Compare these results with the re­
sults obtained by using the extended flow point. This example is
Well 8 taken from Ref. 1 3 . Only the last four flow points from
the test are used in the analysis . Reservoir data are summarized
below . In addition, the results from a drawdown test in this well
indicate kg = 4 .23 md and s = - 5 . 2 .
h
rw
rP
T
ji
Ii
Z
cg
'Y g
Sw
A
CA
=
=
=
=
=
=
=
=
=
=
=
=
454 ft .
0 . 2 6 1 5 ft .
0 . 0675 .
7 1 8 ° R (25 8 ° F) .
4,372 . 6 psia.
0 . 023 cp o
0 . 87 .
1 . 69 x 1 0 -4 psia - I .
0 . 65 .
0.3.
640 acres .
30. 8828 (assume that the well is centered in a
square drainage area) .
The Rawlins and Schellhardt analysis with extended flow point
gave C = 2 . 426 x l O - 3 , n = 0 . 54 , and q Aop = 1 80 . 1 MMscflD.
The Houpeurt analysis with extended flow point gave a =
1 . 4 5 5 x 1 0 6 psia 2 /cp/MMscf- D , b = 1 . 774 x l 0 4 psia 2 /cp/
(MMscf-D) 2 , and q A op = 205 . 6 MMscflD.
Solution.
Brar and Aziz Method.
1 . Plot t:.pp /q = [ pp ( Pws) -Pp ( P wf»)/q vs. q on Cartesian graph
paper (Fig . 7 . 22) . Table 7.23 gives the plotting functions. Con­
struct best-fit lines through the modified isochronal data points for
each time . Although the data are scattered , we used all flow rates
for each isochron.
2 . Determine the slopes of the line s , b, for each time by least­
squares regression analysis. For example , at t = 3 . 0 hours (see Ta­
ble 7.24) ,
2.4
2.2
g
,-..
2.0
u
�
�
fr
'"
1 .8
,
'-'
No"""
co:S
' r;;
1 .6
1 .4
Po
�
�
'<:I
1 .2
'
I.Ox I O
20
40
30
q,
50
MMscf/D
60
bI =
Nj
�
I
N
N�
( t:. Pp )r
j I
� N� ( ; )
(� )
j I
(qj ) 2 _
qj
t:.
p
j I
j
2
j I
qj
(4)(3 . 1 67 X 1 0 8 ) - (202 . 477)(5 .961 X 1 0 6 )
(4)( 1 . 1 07 X 1 04 ) - (202 . 477) 2
= 1 . 823 X 1 0 4 psia2 /cp/(MMscf-D) 2 .
Table 7.25 summarizes the slopes for all isochrons .
The arithmetic average value of the b values in Table 7 . 25 is
( 1 . 823 + 1 . 870 + 1 . 8 8 1 + 1 . 939) x 1 04
4
= 1 . 878 X 104 psia2 /cp/(MMscf-D) 2 .
P wt (psia)
o
( P ws )
3.0
4.0
5.0
6.0
O [ pp ( P ws »)
3.0
4.0
5.0
6.0
q = 3 1 . 6 1 2 M MscflD q = 44. 3 1 3 M M scf/D q = 56.287 M M scf/D q = 70.265 M MscflD
4,343.6
4,344.3
4,356.0
4,372.6
4 , 1 1 1 .8
3,994.6
4 , 206.5
4,274 . 1
3,973.3
4 , 1 03.3
4 , 1 99 . 3
4,266.8
3 , 959.6
4,097.0
4 , 1 92 . 0
4,262.5
3,951 . 9
4,093.5
4 , 1 90.4
4 , 258.3
1 .049 x
1 .0 1 1 x
1 .008 x
1 . 006 x
1 .005 x
109
109
109
109
109
1 . 042 X 1 0 9
9.848 x 1 0 8
9.821 x 1 0 8
9 . 793 x 1 0 8
9.787 x 1 0 8
1 .038 x 1 0 9
9.488 x 1 0 8
9.456 x 1 0 8
9.433 x 1 0 8
9.41 9 x 1 0 8
1 .037 X 1 0 9
9 . 047 x 1 0 8
8 . 968 x 1 0 8
8.91 6 x 1 0 8
8.888 x 1 0 8
Extended Flow Point
P wI = 3,794. 0 psi a
p = 4,372. 6 psia
pp ( P w, ) = 8. 303 x 1 0 8 psi a 2 /cp
80
Fig. 7.22-Brar-Aziz analysis of modified isochronal test data,
Example 7.4.
TABLE 7.22-MODIFIED ISOCHRONAL TEST DATA , EXAMPLE 7.4
Time
(hours)
70
t = 72 hours
q = 77.346 M Mscf/D
pp ( 1 4. 65) = 2 , 003.8 psi a 2 /cp
1 87
DELIVERABI L ITY TESTI N G OF GAS WELLS
TABLE 7 .23-PLOTTING FU NCTIONS FOR BRAR AND AZIZ ANALYSIS, EXAMPLE 7 . 4
b.p p lq [psia 2 /(cp-M MscflD)]
Time
(hours)
q = 3 1 .6 1 2 M M scflD q = 44.31 3 M Mscf/D q = 56.287 M Mscf/D q = 70.265 M MscflD
1 .585 X 1 0 6
1 .291 X 1 0 6
1 . 883 X 1 0 6
1 .202 x 1 0 6
1 .642 X 1 0 6
1 .352 X 1 0 6
1 .995 X 1 0 6
1 .297 X 1 0 6
1 .41 5 x 1 0 6
2.069 x 1 0 6
1 .360 x 1 0 6
1 .682 x 1 0 6
1 . 707 X 1 0 6
2 . 1 21 X 1 0 6
1 .428 X 1 0 6
1 .392 X 1 0 6
3.0
4.0
5.0
6.0
TABLE 7 .24-LEAST-SQUARES REGRESSION FOR
SLOPES, EXAMPLE 7.4
-
Point
1
2
3
4
E=
q3 1 .61 2
44.31 3
56.287
70. 265
202.477
0 . 999 x
1 .965 x
3 . 1 68 x
4.937 x
1 . 1 07 x
103
103
103
103
104
b.P p
3.800 x 1 0 7
5 . 724 x 1 0 7
8 . 920 x 1 0 7
1 .323 x 1 0 7
3. 1 67 x 1 0 8
TABLE 7.25-SLOPES, b, OF THE ISOCHRONS, EXAMPLE 7.4
N
E
j=!
at ! =
C� )
pp
N
N�
q
E
N
(N
�
c' =
( qj ) 2 _
j !
qj
r
�
!
(log tj ) 2 -
� r
log tj
j !
(2 . 553 x 1 0 6 )( 1 . 684) - ( 1 .65 1 x 106 )(2 . 556)
5 . Calculate the formation permeability to gas using the slope
of the semilog straight line .
kg =
= 5 . 677 x 1 0 5 psia2 /cp/MMscf-D .
Table 7.27 gives the intercepts for each isochron .
4 . Prepare a graph of a/ v s . log t (Fig. 7.23) and draw the best­
fit line through the data. Using all four data points, calculate m '
and c' o f the best-fit line of the plot o f a/ vs. log t using least­
squares regression analysis (Eqs . 7 . 65 and 7 . 66 , respectively) (see
Table 7.28) .
N
---­
1 . 632 x 1 06 T
1 . 632 x 1 0 6 (7 1 8)
m'h
(3 . 87 1 x 1 0 5 ) (454)
S = 1 . 15 1
= 1 . 15 1
[
[ (� ) ( _k� ) ]
[(
)
- IOg
m
CP/J- g c/ ,w2
+ 3 .23
3 . 909 X l O S
3 . 87 1 x 1 0 5
6.6
(0 . 0675)(0 . 023)(0 . 000 169)(0 . 26 1 5 ) 2
= - 5.0.
(4)( 1 . 65 1 X 1 0 6 ) - (2 . 553 x 106 )(2 . 556)
= 3 . 87 1 X 1 0 5 psia2 /(cp-MMscflD)/cycle .
TABLE 7 .26-LEAST -SQUARES REGRESSION FOR
I NTERCEPTS, EXAMPLE 7 . 4
q31 .61 2
44.31 3
56.287
70.265
202. 477
0 . 999 x 1 0 3
1 . 965 x 1 0 3
3. 1 68 x 1 0 3
4.937 x 1 0 3
1 . 1 07 x 1 0 4
3.800 x 1 0 7
5 . 724 x 1 0 7
8.920 x 1 0 7
1 .323 x 1 0 7
3. 1 67 x 1 0 8
] ]
+ 3 .23
This value agrees with S = - 5 . 2 estimated from the drawdown test
analysis .
7 . Calculate the stabilized flow coefficient, a. Assume that the
well is centered in a square drainage area with CA = 30 . 8828 (Ap­
pendix C) .
(4) ( 1 . 684) - (2 . 556) 2
-
6 . 6 md ,
which compares with kg = 4 .23 md estimated from the drawdown
test analysis.
6 . Calculate the skin factor with Eq . 7 . 63 .
- log
Point
1
2
3
4
E=
N
= 3 .909 x 1 0 5 psia2 /(cp-MMscflD) .
(4)( 1 . 107 X 1 04 ) - (202 . 477) 2
N
N
(
(4) ( 1 . 684) - (2 . 556) 2
(5 .96 1 x 1 0 6 )( 1 . 1 07 x 1 04 ) - (202 . 477)(3 . 1 67 x 1 0 8 )
N
N
N
E (a/ )j E (log tj ) 2 - E (a/ log t )j E (log t )j
j=!
j=!
j=!
j=!
Nj
( qj ) 2 _ E qj E ( t:..Pp )j
j=!
j=!
j j=!
j !
[psia 2 /(cp-M MscflD)]
1 . 823 x 1 0 4
1 . 870 x 1 0 4
1 .881 x 1 0 4
1 . 939 x 1 0 4
(hours)
3.0
4.0
5.0
6.0
3 . Using Eq. 7 . 64 , calculate the transient deliverability line in­
tercepts for each isochronal line. For example, at t = 3 . 0 hours (see
Table 7.26) ,
N
b
t
b. p p lq
1 .202 x 1 0 6
1 .291 x 1 0 6
1 .585 X 1 0 6
1 .883 x 1 0 6
5.961 x 1 0 6
fl. p p lq
1 .202 x 1 0 6
1 .291 x 1 0 6
1 .585 X 1 0 6
1 .883 X 1 0 6
5.961 X 1 0 6
TABLE 7.27-TRANSIENT DELIVERABILITY LINE I NTERCEPTS,
8 " OF THE ISOCHRONS, EXAMPLE 7.4
t
(hou rs)
3.0
4.0
5.0
6.0
b
(psia 2 /cp/M Mscf-D)
5 .677 x 1 0 5
6.247 x 1 0 5
6. 794 x 1 0 5
6.807 x 1 0 5
GAS RESERVOI R E N G I N E E R I N G
1 88
8x lOS
�
7
�
6
fr
:::::'r;;
<'I",
10'
5
0..
c::r
2
6
Time, hr
7
I
9
3x 10
[109(t)) 2
0.228
0. 362
0.489
0.605
1 .684
1 .422 x 106 T
kg h
[
1 . 1 5 1 log
( 1 . 422 x 1 0 6 ) (7 1 8)
(6. 6)(454)
-�
+ ( - 5 .0)
]
[
at
5 . 677 x 1 0 5
6.247 x 1 0 5
6 . 794 x 1 0 5
6.807 x 1 0 5
2.553 x 1 0 6
(-- )
1 O . 06A
CA r�
1 . 1 5 1 log
3
- - +s
4
]
. .. .
.
·········· .. ···· ..·.. ···
..
1
. .. . . . .
...
q.
N
[ ------- ]
( 1 0 .06)(640)(43 ,560)
...
. . _ . _.
j
.. .
..
__
. �.....
I
.
. ....._
·······
··················
i
.. .+--_........
·····················
··_········_·······
!
· · .. ······ . . · · · · .
··············· . . . .
MMscfID
N
N
(30. 8828)(0.2615) 2
(4)(53 . 056 1 ) - (6. 7435)(3 1 . 409 1 )
(4)(246 . 7983) - (3 1 . 409 1 ) 2
8 . Now , calculate the AOF potential using b from Step 2 and
the stabilized a value calculated in Step 7 .
[
.. .
. . + . . .. . . .
1 .. .
. .. _ . .
Stabilized C Method.
1 . Plot tl.Pp =Pp ( P ws) -pp ( P w/ ) vs. q on log-log graph paper
(Fig. 7.24) . Table 7.29 gives the plotting functions . Construct best­
fit lines through the data.
2 . Calculate the deliverability exponent, n , for each line. For this
example, use least-squares regression analysis of all points for each
isochron . For example, for t = 3 . 0 hours (see Table 7.30) ,
a t 109(t)
2.708 x 1 0 5
3.761 x 1 0 5
4. 749 x 1 0 5
5.296 x 1 0 5
1 . 651 x 1 0 6
= 1 .227 x 1 0 6 psia2 /cp/(MMscf-D) .
q AOF =
.
.
Fig. 7 .24-Stabi lized C analysis of modified Isochronal test
data, Example 7 . 4.
TABLE 7 .28-LEAST-SQUARES REGRESSION ON
TRANSIENT I NTERCEPTS, a " EXAMPLE 7.4
109(t)
0.477
0.602
0. 699
0. 778
2 . 556
.. ...
. ...
' �--w
10
Fig. 7 .23-Brar-Azlz plot of a , vs. t.
a
.... . . . . .... �
-_._
4
Point
1
2
3
4
E=
. . . . . . . . . . . . f . . . . . . .. . .. . . . . .. r . l .
. . . . . . . . 1" . . . . . . . . . , t
. .._·····.._····..···············
_ _
= 0 . 63 .
Table 7.31 summarizes values of the deliverability exponent for
each isochron . The arithmetic average slope of the values in Table
7 . 3 1 is
- a + -Ja 2 + 4b[ pp ( p ) -pp ( 1 4 . 65)]
2b
- ( 1 . 277 x 1 0 6 ) +
-J ( 1 . 277 x 1 0 6 ) 2 + 4( 1 . 878 x 1 04 ) [ ( 1 . 049 x 1 0 9 ) - 2 ,003 . 8]
2( 1 . 878 x 1 04 )
= 205 . 9 MMscflD.
]1
n=
------0 . 63 + 0 . 64 + 0 . 65 + 0 . 65
4
0 . 64 .
3 . Calculate the theoretical value of the Houpeurt coefficient, a ,
using permeability and skin factor values calculated previously with
the Brar and Aziz analysis (L e . , kg = 6 . 6 md , s = - 5 . 0) .
TABLE 7 .29-PLOTTING FUNCTIONS F O R STABILIZED C METHOD, EXAMPLE 7.4
t. P p (psia 2 /cp)
Time
(hours)
3.0
4.0
5.0
6.0
q = 31 .6 1 2 M M scf/D q = 44.31 3 M Mscf/D q = 56.287 M MscflD q = 70.265 M MscflD
5.720 x 1 0 7
3. 800 x 1 0 7
8.920 x 1 0 7
1 .323 x 1 0 8
5 . 990 x 1 0 7
4. 1 00 x 1 0 7
9.240 x 1 0 7
1 .402 x 1 0 8
6.270 x 1 0 7
4. 300 x 1 0 7
9.470 x 1 0 7
1 .454 x 1 0 8
6. 330 x 1 0 7
4.400 x 1 0 7
9.61 0 x 1 0 7
1 .490 x 1 0 8
DE LIVERABI LITY TESTI N G OF GAS WELLS
1 89
TABLE 7 . 3 1 -DELIVERABILITY EXPONENTS, n, OF THE
ISOCHRONS, EXAMPLE 7 . 4
TABLE 7 . 30-LEAST-SQUARES REGRESSION, STABILIZED
C METHOD, EXAMPLE 7.4
Point
1
2
3
4
E=
a=
=
-
log q
1 .4999
1 . 6465
1 . 7504
1 . 8467
6.7435
1 . 422 x 1 0 6 T
kgh
7.5789
7. 7574
7. 9504
8. 1 2 1 6
31 .4091
[
1 . 1 5 1 log
1 . 422 x 1 0 6 (7 1 8)
(6. 6)(454)
l ]
(log � p p ) 2
log � P p
[
( --- ) - - + sl
1 O . 06A
3
CA r a
4
1 . 1 5 1 log
[
t
log q log � P p
1 1 .3686
1 2 .7726
1 3. 9 1 64
1 4.9982
53.0561
57.4531
60. 1 773
63.2089
65.9604
246 . 7983
1 0 . 06(640) (43 ,560)
(30. 8828)(0 . 26 1 5) 2
7 . 6 Sum mary
l
-5.0
Use the average value for the coefficient , b = 1 . 873 x 1 04
psia 2 /(cp-MMscfID 2 ) , obtained from the Brar and Aziz analysis .
4. Calculate the rate at which the change in pseudopressure deter­
mined with the Rawlins-Schellhardt equation equals the change in
pseudopressure determined with the Houpeurt equation . Use the
average value for the coefficient , b = 1 . 873 x 1 04 psia 2 /(cp­
MMscf/D 2 ) , obtained from the Brar and Aziz analysis and the a
coefficient from Step 3 .
a( l - n)
b(2n - l )
=
1 . 227 x 1 0 6 ( 1 - 0 . 64)
1 . 873 x 1 0 4 [2(0 . 64) - 1 ]
0.63
0.64
0.65
0.65
3.0
4.0
5.0
6.0
= 1 . 227 x 1 0 6 psia 2 /(cp-MMscflD) .
qe =
n
(hou rs)
= 84 . 2 MMscflD .
Reading this chapter should prepare you to do the following :
• Define the term " deliverability test" and describe the major
types of deliverability tests .
• State the purpose of deliverability testing .
• List the major assumptions on which the theory of deliverabil­
ity test analysis is based and the reservoir models for which this
theory is not applicable .
• Derive the Houpeurt deliverability equation from equations for
gas flow in porous media, and state the theoretical significance of
the Houpeurt equation coefficients .
• State the physical significance of the Rawlins-Schellhardt
deliverability equation coefficients .
• Calculate stabilization time and give the significance of this
value .
• Interpret flow-after-flow , isochronal , modified isochronal, and
single-point gas-well deliverability tests , with or without a stabi­
lized flow point , and from this interpretation , construct stabilized
deliverability curves for present and future conditions .
• Determine effective permeability and skin factor from the Brar
and Aziz analysis .
Questions for Discussion
5 . Calculate the stabilized C value .
84 .2
[( 1 . 227 x 1 0 6 )(84 .2) + ( l . 873 x 1 04 )(84 .2) 2 ] 0. 64
= 3 . 69 x l O -4 .
7 . Calculate the AOF potential of the well using n from Step 2 .
q AOF = C [ Pp C D ) - pp ( 1 4 . 65)] n
= 3 . 69 x 1 O - 4 ( l . 049 x 1 0 9 - 2 ,003 . 8 ) 0. 64
= 2 1 8 . 7 MMscfl D .
Table 7.32 compares the results of the analyses with and without
the extended , stabilized flow point . In general , the results are com­
parable and illustrate the validity of the Brar and Aziz and the stabi­
lized C methods for modified isochronal tests with no extended ,
stabilized flow point .
1 . We can establish important reservoir characteristics from
deliverability tests . When we do , we should follow certain proce­
dures (plots to make and methods of interpreting these plots) ; we
also need certain data . The interpretation procedures are based on
idealized models of reservoirs , which include a number of impor­
tant , and possibly limiting , assumptions . For each of the delivera­
bility tests listed below , state the information obtainable from the
analysis and how to use that information ; the analysis procedure
(what to plot and how to interpret the plot) using both the Rawlins­
Schellhardt and the Houpeurt analyses ; the data required for in­
terpretation and possible real-world sources of the data ; and as­
sumptions stated or implied in the analysis technique.
Deliverability Tests.
A. Flow-after-flow tests .
B . Single-point tests .
C . Isochronal tests .
D . Modified isochronal tests with stabilized data .
E. Modified isochronal tests without stabilized data .
2 . Derive Houpeurt ' s deliverability equation from pressure­
transient test theory . What are the limitations and assumptions for
this equation?
TABLE 7 . 32-COMPARISON OF RESU LTS FROM EXAMPLE 7.4
Parameter
n
C, (MMscflD)/
(psia 2 /Cp)
n
a, psia 2 /(cp-
MMscf/D)
b, psia 2 /(cpM M scflD 2 )
AOF, M Mscf/D
Without Stab il ized Data
Stab i l ized C
Brar and Aziz
Method
Method
0.64
3.60 x l 0 - 4
With Stab il ized Data
Rawl i nsSche l l hardt
Houpeurt
0.54
2.426 x 1 0 - 3
1 . 227 x l O B
1 . 227 x l 0 B
1 .455 x l o B
1 .873 x 1 0 4
1 . 873 x 1 0 4
1 . 774 x 1 0 4
2 1 8.7
205.9
1 80 . 1
205 .6
GAS RESERVO I R E N G I N E E R I N G
1 90
TABLE 7.33-ISOCHRONAL TEST DATA, PROBLEM 7.2
t
(hou rs)
1
2
3
q
( M Mscf/D)
1 .224
1 .2 1 5
1 .200
P wt
(psia)
425 .75
422.37
41 6.32
p p ( P wt )
(psia 2 /cp)
1 .4692 x 1 0 7
1 .4483 x 1 0 7
1 .41 1 6 x 1 0 7
c.P p
(psia 2 /cp)
0.538 x 1 0 6
0 . 747 x 1 0 6
1 . 1 1 4 x 106
c.P p /q
(psi a 2 /cp-M M scflD)
4.395 x 1 0 5
6. 1 48 x 1 0 5
9 . 283 x 1 0 5
1
2
3
4.262
4.1 1 4
4.022
371 .08
355. 77
345. 89
1 . 1 577 x 1 0 7
1 . 0797 x 1 0 7
1 .031 3 x 1 0 7
3.653 x 1 0 6
4.433 x 1 0 6
4.91 7 x 1 0 6
8.571 x 1 0 5
1 .078 x 1 0 6
1 . 223 x 1 0 6
1
2
3
1 .7 1 0
1 .691
1 . 680
41 4.47
409 .59
406.33
1 .4004 x 1 0 7
1 .371 5 x 1 0 7
1 . 3523 x 1 0 7
1 . 226 x 1 0 6
1 .51 5 x 1 06
1 . 707 x 1 0 6
7. 1 70 x 1 0 5
8 . 959 x 1 0 5
1 .0 1 6 x 1 0 6
1
2
3
2 . 1 07
2.073
2 . 054
407. 35
400.45
396 . 0 1
1 . 3583 x 1 0 7
1 .3 1 83 x 1 0 7
1 . 293 1 x 1 0 7
1 . 647 x 1 0 6
2.047 x 1 0 6
2.299 x 1 0 6
7.8 1 7 x 1 0 5
9 .875 x 1 0 5
1 .1 1 9 x 106
1
2
3
3.057
2. 986
2.942
389. 9 1
379. 66
373.04
1 .2590 x 1 0 7
1 . 2031 x 1 0 7
1 . 1 679 x 1 0 7
2.640 x 1 0 6
3. 1 99 x 1 0 6
3.551 X 1 0 6
8.636 x 1 0 5
1 . 071 x 1 0 6
1 .207 X 1 0 6
3.238
72
(Extended flow poi nt)
259 .79
0 . 6883 x 1 0 7
8.347 x 1 0 6
2 . 579 x 1 0 6
3. What would you do at the well site if you were asked to con­
duct a flow-after-flow test? An isochronal test? A modified
isochronal test?
4. What are the three different states of flow observed when a
well is producing at a constant rate from a closed reservoir? List
them in the chronological order they appear . What is characteris­
tic about the last state of flow? 1 5
5 . How could you determine whether the deliverability test you
were asked to perform would yield useful results for your particu­
lar well?
6. Why might the first point on a flow-after-flow test analysis
plot be considered a "bad data point " ?
7 . What are the differences between gas-well transient (build­
up/drawdown) tests and deliverability tests? How are they simi­
lar? List and compare the information that can be obtained from
transient and deliverability tests .
Exercises
1 . Obtain the stabilized AOF of this well using both the Rawlins­
Schellhardt and the Houpeurt analysis techniques . The data for this
exercise are taken from a 1 989 draft manual prepared for the Texas
Railroad Commission . 1 0
Flow-After-Flow Data
q
(MMscflD)
o
1 .012
2 . 248
3 . 832
5 . 480
Pwj
(psia)
Pp ( P wf )
(psia2 /cp)
7 . 5980 x 1 0 8
3 ,360
3,317
7 . 440 1 x 1 0 8
3 ,2 1 5
7 . 0679 x 1 0 8
3 ,020
6 . 3675 x 1 0 8
2 ,724
5 . 3373 x 1 0 8
pp ( 1 4 . 65) = 1 4 , 066 psia 2 /cp
2. Obtain the stabilized AOF potential of this well using both the
Rawlins-Schellhardt and the Houpeurt analysis techniques . The data
for this exercise are taken from Ref. 1 5 . A gas specific gravity of
0 . 6 was assumed to calculate pseudopressures at 1 80 ° F .
Flow-After-Flow Data
q
(MMscflD)
o
1.8
2.7
3.6
4.5
P wj
(psia)
pp ( Pwj )
(psia2 /cp)
5 . 3373 x 1 0 8
2 , 800
4 . 93 6 1 x 1 0 8
2 ,680
4 . 6420 x 1 0 8
2 ,590
4 . 3539 x 1 0 8
2 , 500
4 . 1 1 87 x 1 0 8
2 ,425
pp ( l 4 . 65) = 2 , 1 92 . 6 psia2 fcp
TABLE 7.34-MODIFIED ISOCHRONAL TEST DATA ,
PROBLEM 7.4
t
q
( M M scflD)
(hou rs)
1 .9
2
2.7
2
3.6
2
4.5
2
4.5
16
(Extended flow point)
P wf
(psia)
2 ,9 1 0
2,860
2,804
2 , 750
2 , 630
p p ( P wt )
(psia 2 fcp)
5.71 35 x 1 0 8
5.541 5 x 1 0 8
5.3509 x 1 0 8
5 . 1 690 x 1 0 8
4. 7720 x 1 0 8
3 . Table 7.33 gives the data for this example, Well 2 from Ref.
1 1 . Calculate the AOF potential of this well using the Rawlins­
Schellhardt and the Houpeurt analyses . The isochronal test data are
p =436.0 psia and pp ( p) = 1 . 5230 X 1 0 7 psia2 fcp; and pp ( l 4 . 65) =
2 , 326.6 psia2 fcp .
4. Table 7.34 gives the data for this example, which is from Ref.
I S . Calculate the AOF potential of this well using Rawlins­
Schellhardt and the Houpeurt analyses . A gas specific gravity of
0 . 6 was assumed to calculate pseudopressures at a temperature of
1 80 ° F . The isochronal test data are p = 3 ,000 psia and
pp ( p) = 6 . 0268 x 1 0 8 psia2 fcp; and pp ( 1 4 . 65) = 2 , 1 92 . 6 psia2 fcp .
5 . This example (see Table 7.35) is Well 5 from Ref. 1 3 . It is
a modified isochronal test with a stabilized flow point . Calculate
the AOF potential of this well using the Rawlins-Schellhardt and
the Houpeurt analyses .
6. Use the data from Table 7.35 without the stabilized flow point,
and calculate permeability and the AOF potential using the
stabilized C and Brar and Aziz methods . Compare the AOF
potential obtained from these methods to the results from
Problem 5 , which were calculated with knowledge of the extended
flow point.
7 . This example is Well 7 from Ref. 1 3 (see Table 7.36) . It is
a modified isochronal test with a stabilized flow point . Calculate
the AOF potential of this well using the Rawlins-Schellhardt and
the Houpeurt analyses .
8 . Use the data from Table 7 . 36 without the stabilized flow point,
and calculate permeability and the AOF potential using the
stabilized C and Brar and Aziz methods . Compare the AOF
potential obtained from these methods to the results from
Problem 7 , which were calculated with knowledge of the extended
flow point .
DELIVERABILITY TESTING
OF
191
GAS WELLS
TABLE 7.35-MODIFIED ISOCHRONAL TEST DATA, PROBLEM 7.5
Reservoir Data
h = 4 ft
' w = 0.12 ft
t/l = 0 .19
T = 602°R (142°F)
p = 1,188.5 psia
jI = 0.015 cp
Z = 0.902
Cg = 8. 8x10 -4 psia-1
A = 640
acres
(assume well is
centered in a square
drainage area)
Modified Isochronal Test Data
Pwf (psia)
Time
(hours)
o
q = 2. 1 04 MMscf/D q = 3.653 MMscflD q = 4.026 MMscflD q = 5.079 MMscflD
1 ,1 87 . 1
1 ,1 86.4
1 ,1 86.0
1 ,1 88 . 5
954. 0
721 . 1
1 ,072. 9
887.2
946. 9
71 5.5
1 ,072 . 6
883. 5
944.4
71 3.5
1 ,071 .4
882. 5
944.2
7 1 3.2
1 ,070 . 6
882 .3
(Pws)
0.5
1 .0
1 .5
2.0
O[App(Pws»)
0.5
1 .0
1 .5
2.0
1 1 .230x 1 07
9 . 1 846x 1 07
9 . 1 795x 1 07
9 . 1 593x 1 07
9 . 1 458x 1 07
Pwf = 738. 0
1 1 . 204x 1 07
7 .2776x 1 07
7. 1 700x 1 07
7 . 1 323x 1 07
7 . 1 293x 1 07
1 1 . 1 91 x 1 07
6 .2940x 1 07
6.24 1 4x 1 07
6. 2272x 1 07
6 .2243x 1 07
1 1 . 1 84x 1 07
4. 1 355x 1 07
4.0704x 1 07
4.0473x 1 07
4.0438x 1 07
Extended Flow Point
psia
psia
p = 1,1 88 . 5
Pp(Pwf) = 4.3353 X 1 07
App(P) = 6 .8947x 1 07
psia2/cp
psia2/cp
t = 7 hours
q = 4.964 MMscflD
AP p/q = 1 .3889x 1 07 psia2/cp/MMscf-D
Pp ( 1 4. 65) = 2,326.6 psia2/Cp
TABLE 7.36-MODI FIED ISOCHRONAL TEST DATA, PROBLEM 7.7
Reservoir Data
h = 35 ft
' w = 0 . 1 875
t/l = 0.08
T = 7 1 5°R
p = 7,1 2 1 psia
jI = 0.0286 cp
Z = 1 . 1 45
Cg = 7.4x1 0 - s
ft
A = 1 60
acres
(assume well is
centered in a square
drainage area)
psia-1
Modified Isochronal Test Data
Time
(hours)
o
(Pws)
6.0
7.0
8.0
9.0
Pwf (psia)
q = 8.584 MMscflD
7,1 2 1
6,250
6,238
6,226
6,2 1 6
q = 9 .879 MMscf/D
7,1 0 1
6,006
5,989
5,975
5,965
2 . 1 545x 1 0 9
1 .8297x 1 0 9
1 .8233 X 1 0 9
1 .8 1 88x 1 0 9
1 .8 1 50x 1 0 9
2 . 1 470x 1 0 9
1 .7358x 1 0 9
1 . 7293x 1 0 9
1 .7240x 1 0 9
1 .7203x 1 0 9
O[App ( P ws»)
6.0
7.0
8.0
9.0
Pp(Pwf)
psia2/cp
t = 1 20 hours
q = 9.225 MMscf/D
p p ( 1 4 . 65) = 1 3,788 psia2/cp
psia
psia
psia2/cp
N omenclature
a
= stabilized deliverability coefficient, (psia 2 _
at
calculations in terms of pressure squared
= transient deliverability coefficient, (psia 2 -
cp)/(MMscf-D) for calculations in terms of
pseudopressure or psia 2 /(MMscf-D) for
cp)/(MMscf-D) for calculations in terms of
pseudopressure or psia 2 /(MMscf-D) for
A
b
2 . 1 41 1 x 1 0 9
1 .501 8x 1 0 9
1 .491 2x 1 0 9
1 .4828x 1 0 9
1 .4756x 1 0 9
Extended Flow Point
Pwf = 5845
P = 71 2 1
pp ( Pwf) = 1 . 5054 X 1 0 9
q = 1 2.867 MMscf/D
7,085
5,388
5,360
5,338
5,31 9
calculations in terms of pressure squared
= drainage area of well , ft 2
= deliverability equation coefficient, (psia 2 cp)/(MMscf-D) 2 for calculations in terms of
pseudopressure or psia 2 /(MMscf-D) 2 for
calculations in terms of pressure squared
c'
= constant in Eq.
C
= gas compressibility , psia - 1
ct
= total system compressibility , psia - 1
Cf
g
Cg
Cw
C
7. 58
= formation compressibility , psia - 1
= gas compressibility at average reservoir pressure ,
psia
= water compressibility , psia - 1
= stabilized performance coefficient, (MMscf­
D)/(psia 2 -cp) n for calculations in terms of
pseudopressure or (MMscf-D)/psia 2n for
CA
Ct
calculations in terms of pressure squared
= shape constant or factor for well drainage area
= transient performance coefficient, (MMscf­
D)/(psia 2 -cp)n for calculations in terms of
GAS RESE RVOI R E N G I N E E R I N G
192
pseudopressure or (MMscf-D)/psia 2n for
calculations in terms of pressured squared
D
h
= net formation thickness , ft
kg
= reservoir effective permeability to gas , md
= non-Darcy flow constant, D/MMscf
j = summation parameter
L
m
m
'
n
N
P
Pa
Ph
Pi
= flow-string length from surface to middle of
perforations , ft
= slope of at or (at +bq) v s . log (t) plot, (psiaL
cp)/(MMscf-D) per cycle for calculations in terms
of pseudopressure or psia 2 /(MMscf-D) per cycle
for calculations in terms of pressure squared
constant in Eq. 7. 58
inverse slope (exponent) of deliverability curve
number of data points used in regression analysis
pressure , psia
atmospheric pressure , psia
base pressure , psia
initial reservoir pressure , psia
= gas pseudopressure , psia 2 /cp
= average reservoir pseudopressure , psia 2 /cp
=
=
=
=
=
=
=
Pp
pp (ji)
Pp ( P wf)
= flowing sandface pseudopressure, psia 2 /cp
pp ( Pws)
= static sandface pseudopressure , psia 2 /cp
ji = average reservoir pressure , psia
P s = stabilized shut-in BHP measured before the
deliverability test, psia
/lp 2 = difference of squared static and flowing pressures,
psia 2
/lpp = difference of static and flowing sandface
pseudopressure s , psia 2 /cp
P tf = flowing wellhead pressure , psia
P wf = BHFP, psia
Pws = shut-in BHP, psia
q = total wellstream gas flow rate , MMscflD
qAOF = AOF potential , MMscflD
q e = rate at which pressure drops from Houpeurt and
r
rd
re
rj
rw
s
Sg
So
Sw
t
tD
=
=
=
=
=
=
=
=
=
=
=
Rawlins-Schellhardt equations are equal , MMscflD
radial distance from center of wellbore , ft
effective (transient) radius of drainage, ft
external radius of drainage , ft
radius of investigation , ft
wellbore radius , ft
skin factor, dimensionless
gas saturation, fraction of PV
oil saturation, fraction of PV
water saturation, fraction of PV
elapsed time , hours
dimensionless time
{LtD = dimensionless time calculated with fracture half-
length
t s = well stabilization time , hours
T = temperature , OR
Tf = temperature , OF
z = gas-law deviation factor , dimensionless
z = gas-law deviation factor at average reservoir
pressure and temperature , dimensionless
a = exponent in Eq.
(3 = turbulence factor
7. 3 2
'Y g = gas specific gravity (air =
/Jog
iig
cP
1 . 0)
= gas viscosity , cp
= gas viscosity at average reservoir pressure and
temperature, cp
= porosity of reservoir rock, fraction
References
I. Houpeurt, A.: "On the Flow of Gases in Porous Media," Revue de
L'/nstitut Franfais du Ntrole ( 1 959) XIV (11) , 1 468- 1 684.
2 . Rawlins, E.L. and Schellhardt, M . A . : Backpressure Data on Natural
Gas Wells and Their Application to Production Practices, Monograph
Series, USBM ( 1 93 5 ) 7 ,
3 . AI-Hussainy, R . , Ramey, H . J . J r . , and Crawford, P . B . : " The Flow
of Real Gases Through Porous Media," JPT (May 1 966) 624-36 ;
Trans., AIME, 237.
4. Jones, S . c . : " U sing the Inertial Coefficient, (3. To Characterize Het­
erogeneity in Reservoir Rock," paper SPE 1 6949 presented at the 1 987
SPE Annual Technical Conference and Exhibition, Dallas, Sept. 27-30.
5. Lee, W . J . : Well Testing, Textbook Series, SPE, Richardson, TX ( 1 977)
1.
6. Earlougher, R . C . Jr.: Advances in Well Test Analysis, Monograph Se­
ries, SPE, Richardson, TX ( 1 977) 5.
7 . Gringarten, A . C ., Ramey, H . J . Jr., and Raghavan, R.: " Unsteady­
State Pressure Distributions Created by a Well With a Single Infinite­
Conductivity Vertical Fracture," SPEl (Aug . 1 974) 347-60; Trans.,
AIME, 257.
8. Gringarten, A . C . , Ramey, H . J . Jr., and Raghavan, R.: "Applied Pres­
sure Analysis for Fractured Wells," JPT (July 1 975) 887-92 ; Trans.,
AIME, 259.
9. Theory and Practice o/the Testing o/Gas Wells, third edition, Energy
Resources and Conservation Board, Calgary ( 1 978) .
1 0 . Jennings, J . W . et al.: " Deliverability Testing of Natural Gas Wells,"
prepared for the Texas Railroad Commission, Petroleum Engineering
Dept . , Texas A&M U . , College Station, TX (Aug . 1 989) .
II. Cullender, M . H .: "The Isochronal Performance Method of Determining
the Flow Characteristics of Wells, " Trans., AIME ( 1 955) 2 04, 1 37-42 .
1 2 . Katz, D . L . et al.: Handbook 0/Natural Gas Engineering, McGraw­
Hill Publishing C o . , New York City ( 1 959) .
1 3 . Brar, G . S . and Aziz, K . : "Analysis of Modified Isochronal Tests To
Predict the Stabilized Deliverability Potential of Gas Wells Without Using
Stabilized Flow Data," JPT (Feb . 1 978) 297-304 ; Trans., AIME, 265.
1 4 . Johnston, J . L . , Lee, W.J., and Blasingame, T . A . : "Estimating the Stabi­
lized Deliverability of a Gas Well Using the Rawlins and Schellhardt
Method: An Analytical Approach," paper SPE 23440 presented at the
1 99 1 SPE Eastern Regional Meeting, Lexington, Oct . 22-2 5 .
1 5 . Donohue, D . A . T . and Ertekin, T . : Gaswell Testing, IntI . Human
Resources Development Corp . , Boston, MA ( 1 982) .
Chapter 8
Design and Implementation
of Gas-Well Tests
8.1 Introduction
This chapter presents fundamental concepts for designing and im­
plementing gas-well tests , including pressure-transient and well­
deliverability tests . These design concepts include recommenda­
tions for selecting the appropriate well test to achieve the desired
test obj ective s , estimating pretest formation properties , selecting
the appropriate flow-rate schedule and sequence for the test , and
choosing the test duration required to sample a desired reservoir
volume and/or to reach stabilized flow conditions . Examples illus­
trate the design concepts .
8.2 Wel l Test Types and Purposes
Gas-well tests can be grouped into two general categories based
on their primary function . ! The first category , pressure-transient
tests, includes tests designed to quantify important reservoir rock
and fluid properties (e. g . , permeability , porosity , and average reser­
voir pressure) and to locate and identify reservoir heterogeneities
(e. g . , sealing faults , natural fracture s , and layers) . The second
category , deliverability tests , includes tests designed to evaluate
a well's production potential .
8 . 2 . 1 Pressure-Transient Tests. Pressure-transient tests refer to
well tests in which we generate and measure pressure changes with
time . These tests allow us to evaluate not only near-wellbore con­
ditions but also the in-situ reservoir properties beyond the region
affected by drilling and completion operations . In addition , these
well tests can characterize important formation features needed to
design the optimal depletion plan for the reservoir, including pres­
sure in the drainage area of tested wells , possible presence and ap­
proximate location of flow barriers (e . g . , sealing faults) , and the
dominant properties of reservoir heterogeneities (e . g . , fracture and
matrix properties in a naturally fractured reservoir or individual
layer properties) . Pressure-transient tests can be subdivided fur­
ther into single-well and multiwell tests .
Single-Well Tests. As the name implies , single-well tests involve
only one well in which the pressure response is measured follow­
ing a rate change . From the measured pressure response , we can
characterize average properties in a portion or all of the drainage
area of the tested well . A common single-well test, a pressure­
buildup test , is conducted by first stabilizing a producing well at
some fixed rate , placing a bottomhole pressure (BHP) measuring
device in the well , and shutting in the well . Following shut-in, the
BHP builds up as a function of time , and the rate of pressure build­
up is used to estimate well and formation properties . From a
pressure-buildup test , we can estimate average reservoir pressure
and permeability in the well's drainage area and the properties of
the region immediately adj acent to the wellbore .
Another common pressure-transient test, a drawdown or flow test,
is conducted by producing a well at a known and constant rate while
measuring BHP changes as a function of time . Drawdown tests are
designed primarily to quantify the reservoir flow characteristic s ,
including permeability and skin factor . I n addition, when the pres­
sure transient is affected by outer reservoir boundaries , drawdown
tests can be used to establish the outer limits of the reservoir and
to estimate the hydrocarbon volume in the well's drainage area.
These specific drawdown tests are called reservoir-limit tests . When
economic considerations require a minimum loss of production time,
pressure-drawdown tests also can be used to estimate well deliver­
ability and , if conducted and analyzed properly , are viable alterna­
tives to deliverability tests . !
A pressure-falloff test is similar to a pressure-buildup test, ex­
cept that it is conducted on an inj ection well. Following stabiliza­
tion at some inj ection rate , the well is shut in. BHP , which then
begins to decline , is measured as a function of time . An alternative
test for an injection well is an injectivity test, in which we inject
into a well at a measured rate and measure pressure as it increases
with time . Inj ectivity testing is analogous to pressure-drawdown
testing .
Multiwell Tests. When the flow rate is changed in one well and
the pressure response is measured in one or more other well s , the
test is called a multi well test . Multiwell tests are designed to deter­
mine properties in a region centered along a line connecting pairs
of test wells and therefore are more sensitive to directional varia­
tions of reservoir properties , such as permeability . In addition, these
tests can determine the presence or lack of communication between
two points in the reservoir . The basic concept in a multiwell test
is either to produce from or to inject into one well , the active well,
and observe the pressure response in one or more offset wells , or
observation well s . From these data, we can estimate both permea­
bility and porosity in the drainage area of the wells and can quanti­
fy some reservoir anisotropies . For example , multiwell tests can
be used to determine the orientation of natural fractures and to quan­
tify the ratio of the porosity-compressibility products of the matrix
and fracture systems . 2
Interference and pulse tests are two common multiwell tests. In
interference testing , the active well is produced at a measured, con­
stant rate throughout the test . Other wells in the field must be shut
in so that any observed pressure response can be attributed to the
194
GAS RESERVOI R E N G I N E E R I N G
active well only . In pulse tests , the active well produces and then
is shut in, returned to production , and shut in again . This produc­
tion/shut-in sequence, which is repeated with production and shut­
in periods rarely exceeding more than a few hours , produces a pres­
sure response in the observation wells that usually can be inter­
preted unambiguously even when other wells in the field continue
to produce.
8.2.2 Deliverability Tests. A s discussed in Chap . 7, deliverabili­
ty tests are flow tests designed to measure the production capabili­
ties of a well under specific reservoir conditions . Although these
tests are used primarily for gas wells , deliverability testing also is
applicable to oil well s . Unlike most pressure-transient tests , some
deliverability tests require stabilized flowing conditions for proper
analysis . A common productivity indicator obtained from gas-well
deliverability tests is the absolute open-flow (AOF) potential .
Another application of deliverability testing is for generating a reser­
voir inflow-performance relationship (IPR) or gas backpressure
curve. The IPR, which describes the relationship between surface
production rates and bottomhole flowing pressures (BHFP's ) , is
used to design surface production facilities .
Common deliverability tests include flow-after-flow, single-point,
isochronal , and modified isochronal tests . Described in detail in
Chap . 7, these tests are briefly reviewed here to complete our dis­
cussion of well-test design and implementation.
Flow-After-Flow Tests. Flow-after-flow tests, sometimes called
gas backpressure or four-point tests, are conducted by producing
the well at a series of different stabilized flow rates and measuring
the stabilized BHFP at the sandface. Each different flow rate is es­
tablished in succession either with or without a very short inter­
mediate shut-in period. Typically, conventional flow-after-flow tests
are conducted with a sequence of increasing flow rates; however,
if stabilized flow rates are attained , the rate sequence does not af­
fect the test . 1 The requirement that the flowing periods be con­
tinued until stabilization is sometimes a maj or limitation of the
flow-after-flow test , especially in low-permeability formations that
take a long time to reach stabilized flowing conditions .
Single-Point Tests. A single-point test is an attempt to overcome
the limitation of long test times required for the flow-after-flow tests
in low-permeability formations . Single-point tests are conducted
by flowing the well at a single rate until the BHFP is stabilized .
Single-point tests are particularly appropriate when the well's
deliverability characteristics are being updated, as required by many
regulatory agencies . One limitation of this type of test , however,
is that analysis of the results requires prior knowledge of the well's
deliverability behavior , either from previous testing on the well or
from correlations with other wells producing in the same field un­
der similar conditions. In these tests , we also must ensure that the
well has flowed long enough to be out of wellbore storage and in
the pseudosteady-state flow regime .
Isochronal Tests. The isochronal test also was developed to short­
en test times in wells that take long periods of time to stabilize.
Specifically , the isochronal test is a series of single-point tests de­
veloped to estimate stabilized deliverability characteristics without
actually flowing the well for the time required to achieve stabilized
conditions . The isochronal test is conducted by alternately produc­
ing and then shutting in the well and allowing it to build up to the
average reservoir pressure before the next production period . Pres­
sures are measured at several time increments during each flow
period . The times at which the pressures are measured should be
the same relative to the beginning of
flow period . For exam­
ple , we may choose to measure the BHFP's at 0. 5, 1 . 0, 1 . 5, and
2. 0 hours after the beginning of each flow period. Because less time
is required to build up to essentially initial pressure after short flow
periods than to reach stabilized flow in a flow-after-flow test , the
isochronal test is more practical for low-permeability formations .
Although not required for analyzing the test, a final , stabilized flow
point usually is obtained at the end of the test.
Modified Isochronal Tests. The time to build up to the average
reservoir pressure before flowing for a certain period of time may
still be impractical , even after short flow periods in an isochronal
test. Consequently , a modification of the isochronal test was de-
each
veloped t o shorten test times further. The obj ective of the modi­
fied isochronal test is to obtain the same data as in an isochronal
test without the sometimes lengthy shut-in periods required to reach
the average reservoir pressure in the well's drainage area. The modi­
fied isochronal test is conducted like an isochronal test , except that
the shut-in periods are of equal duration but should equal or ex­
ceed the length of the flow period . Because the well often does not
build up to average reservoir pressure after each flow period , the
shut-in sandface pressures recorded immediately before each flow
period are used in the test analysis instead of the average reservoir
pressure. Consequently , the modified isochronal test is less accurate
than the isochronal test . Note that , as the duration of the shut-in
periods increases , the accuracy of the modified isochronal test also
increases . Again , a final stabilized flow point often is obtained at
the end of the test but is not required for test data analysis.
8.3 General Test Design Considerations
If properly designed and implemented, a well test can provide much
useful information about individual wells and the reservoir . Com­
mon applications of oil- and gas-well tests include identifying and
locating important reservoir heterogeneities, such as sealing faults ,
reservoir layering , or natural fractures; calculating the resource in
place and estimating the reserves for various stages of reservoir
depletion; selecting optimal field development and production strat­
egies; and estimating future deliverability for design of surface pro­
duction and processing equipment. Other applications include
determining the nature of the formation fluids and obtaining fluid
samples for laboratory analysis .
Over the years , a variety of well tests have been developed. Many,
however, are appropriate only under specific testing conditions ,
and each test yields different information about the well and/or the
reservoir . Therefore , one of the first steps in well-test design is
to identify the test obj ectives clearly . Once these obj ectives have
been identified, we can proceed with the test design, including select­
ing a particular well test to achieve the desired objectives, developing
procedures for safely and economically implementing the test, select­
ing test equipment required to obtain the proper data , and collect­
ing and analyzing the data.
In general , the goals of a well test are not only to obtain suffi­
cient data to meet the stated obj ectives, but also to accomplish these
tasks in a timely and inexpensive manner. To satisfy these general
goal s , the design engineer must recognize the various well-testing
environments and understand how these environments affect well­
test design and implementation . In this section. we summarize some
of the testing environments commonly encountered in well-test de­
sign and discuss how these various situations affect test design .
8 . 3 . 1 Well Type and Status. When deciding the type of well test
to use, the foremost design considerations are the well type and
status . 3 The type of well refers to its maj or function , such as de­
velopment/exploration or producer/inj ector; the status indicates
whether the well is active or shut-in. Further , depending on the
type and status of the well and the information required , we may
select either a pressure-transient or a deliverability test.
Development vs. Exploration Wells. The maj or difference be­
tween testing development and exploration wells is the type of in­
formation needed . In a developed field where the general geology
and reservoir behavior are known , development wells are drilled
either to increase reserves or to accelerate production. Consequently,
testing of development wells concentrates on obtaining specific prop­
erties of the individual well rather than of the reservoir . For exam­
ple , we may choose to run a pressure-buildup test to estimate the
current average reservoir pressure in the well's drainage area , the
amount of skin damage from a recent workover , or the properties
required for stimulation design and evaluation .
Conversely , for an exploration well drilled in an area with few
or no wells , the tests must yield more information about the reser­
voir in a large region around the well rather than near the individual
well . More emphasis is placed on evaluating deliverability, estimat­
ing reserves , and obtaining geologic information. For example, in
a wildcat well in a new reservoir for which little or no data are
DESIGN AND I M P L E M ENTATION OF GAS-WELL TESTS
available , we may choose to run a drillstem test (DST) to obtain
fluid samples for laboratory analysis, estimate well productivity ,
and calculate the formation permeability and initial static reservoir
pressure.
Producing vs. Injection Wells. The major differences in testing
these wells are the type of test and the equipment needed. To esti­
mate reservoir properties , we typically conduct a pressure­
drawdown or -buildup test , while a four-point or single-point test
is used to estimate future well deliverability in a producing well.
Conversely , reservoir properties can be estimated from an inj ec­
tion or falloff test in an inj ection well. In general , because of the
equipment required to perform an inj ection test, testing inj ection
wells is more expensive than testing producers.
Gas storage fields are one of the few examples of reservoirs de­
veloped by wells that alternate between injection and production,
depending on the time of the year. During late spring to early fall ,
gas is inj ected into the wells; during late fall through the winter
and early spring , the wells are produced to satisfy the short-term
demand for gas associated with cold weather. Other applications
of gas inj ection wells include pressure maintenance projects in gas­
condensate reservoirs and CO 2 displacement programs in oil
reservoirs.
Shallow vs. Deep Wells. The maj or difference between these test­
ing environments is the type of equipment required to implement
the test. The volume of fluids stored in the wellbore distorts the
early-time pressure response and controls the duration of wellbore
storage , especially in deep wells with large wellbore volumes. If
wellbore-storage effects are not minimized or if the test is not con­
tinued beyond the end of the wellbore-storage period , the test data
will be difficult, if not impossible , to analyze with conventional
methods. To minimize wellbore-storage distortion and to keep well
tests within reasonable and economic lengths of time , it may be
necessary to run tubing , packers , and bottomhole shut-in devices.
Although this additional equipment greatly reduces or even elimi­
nates wellbore-storage effects , it also increases the cost and time
required to perform the test.
Yet another consideration is the higher reservoir pressure and
temperature encountered in deep wells. In deep , high-pressure gas
well s , we must specify equipment that can safely accommodate the
highest pressure and flow rate expected during the test. For exam­
ple , threaded pipe connections on flow pipe and lubricators should
be used only at lower reservoir pressure s , while at higher pres­
sures , welded connections , integral connections , and/or flanges
should be used to connect lubricators to the wellhead. In general ,
equipment designed to sustain higher pressures also is more costly.
In addition , because temperature increases with depth ,
temperature-related tool failures occur more frequently in deeper
wells and must be incorporated into the design process. A very com­
mon temperature-related problem is failure of lubricants in mechan­
ical tools. Care should be taken to lubricate all moving tool parts
with high-temperature oils and greases. Another common
temperature-related problem is with battery packs in electric gauges
with bottomhole memories. Generally , to prevent tool failures, high­
temperature batteries must be used when the bottomhole tempera­
ture (BHT) exceeds 300°F.
Stimulated vs. Unstimulated Wells. The pressure-transient data
used to calculate permeability and skin factor must be taken from
the middle-time region or data representing radial flow in an infinite­
acting reservoir. These data are no longer affected by wellbore
storage and reflect the effective reservoir permeability that is un­
altered by stimulation or damage. Many well s , particularly those
in low-permeability gas-bearing formations , require hydraulic frac­
turing to be economically viable producers. In these stimulated
wells , the pressure transient often is affected by the hydraulic frac­
ture long after wellbore storage ceases. In low-permeability wells
that have been hydraulically fractured , pressure-transient data
representing the radial flow period may not appear for many months
or even years.
Because of these long testing times , we generally recommend
an initial well test before the fracture treatment. A reasonably short
prefracture test will yield data that can be used to determine for­
mation permeability. Then , a postfracture test can be used to as-
195
sess the success of the fracture treatment and to evaluate the fractured
well's performance. If the formation permeability was estimated
previously , the postfracture test can be terminated at a reasonable
time after wellbore-storage effects end , thus considerably shorten­
ing the postfracture testing time and reducing costs.
Another consideration in testing hydraulically fractured wells is
the proper selection of surface equipment. Often, sand used as prop­
pant will be displaced and produced during the flowback period
of the test. Because sand is very abrasive, it can destroy chokes ,
valve s , and flowlines. If sand production is expected to be a prob­
lem , sand catchers and knock-outs should be placed upstream of
the production measuring equipment.
8.3.2 Effects of Reservoir Properties. Low-- vs. High-Permeability
Formations. Formation properties also affect test design. Principal
among these properties is permeability , which dictates the test flow
rate and duration. When selecting the test flow rate and flow times ,
we must satisfy several criteria. First, the test must be maintained
long enough to obtain data beyond near-wellbore effects , such as
the period of wellbore-storage distortion, formation damage, or
stimulation. In addition, the test must be run long enough to reach
the desired radius of investigation and to evaluate a representative
volume of the formation. In low-permeability reservoirs, the flowing
time required to satisfy both these criteria often is prohibitive , es­
pecially when gas is flared to the atmosphere.
The duration of the wellbore-storage period depends on several
wellbore characteristics (e.g., wellbore volume and fluid compress­
ibility) and reservoir properties (e.g., porosity , permeability , net
pay thickness, and fluid properties). Although permeability affects
the duration of wellbore-storage distortion , it is not the only prop­
erty that must be considered in well-test design. For example , de­
pending on well depth and reservoir pressure , low-permeability
wells can exhibit wellbore-storage effects for several weeks. How­
ever, we may encounter a reservoir with equally low permeability
and the same reservoir pressure but at a shallower depth and, be­
cause of the smaller wellbore volume , the duration of wellbore
storage could be much shorter. The engineer should estimate the
duration and severity of wellbore-storage effects of each well-test
design.
Although we cannot change reservoir properties to eliminate or
reduce wellbore-storage effects , we can alter the wellbore mechan­
ically to change its properties. For example, by running a bottom­
hole packer on a string of tubing , we can reduce the effective
wellbore volume by a factor of three or four and consequently reduce
the duration of wellbore-storage effects. The wellbore-storage period
can be reduced further by running bottomhole valves in the tubing
string , which allows us to shut in the well just above the sandface
rather than at the surface. Note that all these special tools are ex­
pensive. Therefore, the design engineer must compare the cost of
longer test times with the cost of running tubing packers and bot­
tomhole shut-in valves.
Like the duration of wellbore-storage effects , the time to reach
a desired radius of investigation in a reservoir increases with
decreasing permeability. In this case, however, we cannot shorten
the flowing time by changing the wellbore configuration. Additional
test time will result in increased costs , especially when production
losses become significant during the test period.
When selecting the test flow rate , we must recognize that it is
necessary to create a measurable drawdown at the sandface and in
the reservoir during the test. The magnitude of the pressure draw­
down depends on the reservoir properties and the fluid flow rate
in the reservoir. Because reservoir properties are fixed, we can con­
trol the pressure drawdown only by varying the flow rate. The differ­
ence between the sandface and reservoir pressures varies directly
with the change in flow rate and inversely with permeability. For
higher-permeability well s , a given flow rate will yield a smaller
pressure drawdown than the same rate in lower-permeability wells.
Therefore, a higher rate must be used to increase the pressure
drawdown.
Note that higher rates require larger separators and meter runs
at the surface. In addition, more gas will be wasted through vent­
ing and flaring unless the well is connected to a pipeline. There-
196
GAS RESERVOI R E N G I N E E R I N G
fore , the test cost per day generally increases in higher-permeability
wells. The increased cost, however, is offset somewhat by the sig­
nificantly reduced test time in high-permeability wells. Finally ,
higher rates can create larger pressure drawdowns , which may re­
sult in undue retrograde condensation in gas-condensate reservoirs
or formation sloughing in unconsolidated sandstones. Both condi­
tions may damage the reservoir immediately adjacent to the
wellbore.
Single vs. Multiple Zones. Again, the primary difference under
these conditions is the type of equipment used to conduct the test.
The test obj ectives and techniques are generally the same. In a sin­
gle zone , drawdown and buildup tests typically are run to deter­
mine such reservoir properties as permeability , skin factor, or, in
hydraulically fractured wells, the fracture half-length. During the
test , the producing zone communicates with the wellbore through
an openhole interval or perforated casing.
In wells completed in multiple zones, the same drawdown and
buildup tests used in the single zone are used. The testing proce­
dure, however, must be modified to isolate the tested zone from
the other producing zones surrounding it. The equipment used to
isolate individual zones varies depending on the isolation method.
For example, when isolating the zone from below, we may use either
a wireline or a mechanical bridge plug. After the plug is placed
between zones, the wireline or tubing is pulled out of the hole. The
test is conducted with bottomhole gauges placed in the well on a
wireline.
When isolating the zone from above , we normally use a packer
and tubing. The packer is set between the test zone and the producing
zones above it. The test is conducted with gauges on a wireline
run through the tubing. Alternatively , we could place the gauges
in a bomb carrier mounted below a bridge plug set between the
test zone and the zones above it. To isolate the test zones above
and below, we set a bridge plug between the test zone and the zones
below. In addition , we use a packer on tubing to isolate the test
zone from the zones above it. Pressure gauges are run on a wire­
line through the tubing.
8.3.3 Safety and Environmental Considerations. Sweet vs. Sour
and Corrosive Gases. The type of gas in the reservoir is a major
design consideration , especially when implementing safety proce­
dures in the actual test and incorporating corrosion protection of
the equipment. Sweet gases refer to hydrocarbon mixtures with little
or no impurities , while sour and corrosive gases contain large quan­
tities of nonhydrocarbon gases , such as hydrogen sulfide (H 2 S) and
carbon dioxide (C0 2 ),
When testing in environments containing H 2 S , all industry­
standard safety procedures must be followed. Generally , H 2 S is
recognized by its characteristic foul odor. Prolonged exposure to
low concentrations tends to affect the olfactory nerves , thereby dull­
ing the sense of smell. When high concentrations are present, how­
ever, death may occur before the odor is detected. Recommended
operating and testing procedures are provided in API RP 55. Met­
al embrittlement and fatigue also can be problems when testing wells
with H 2 S. New surface equipment has failed during tests in H 2 S
wells. Therefore , it is very important to use equipment specifically
designed to accommodate H 2 S. In addition, all personnel at the
testing site must be trained in H 2 S safety procedures.
Wire pitting and wireline failure can occur in wells where other
corrosive gase s , such as CO 2 , are present. Wire metallurgies are
designed to accommodate various environments, so to prevent costly
fishing j obs related to wireline failures and lost tools , corrosion­
resistant materials should be used in environments with corrosive
gases. Again , this special equipment is more costly but must be
included in the design process.
Environmental Concerns. Finally , the design engineer must be
aware of environmental hazards, especially when flaring large quan­
tities of natural gas to the atmosphere. Ideally , the test is designed
to minimize the volume of gas flared. Fortunately , the volume of
gases released to the atmosphere can be minimized when the well
is connected to a pipeline. When designing any well test, however,
the engineer should be aware of and adhere to all environmental
regulations.
4
8.4 Design
of
Pressure· Transient Tests
In this section, we address specific design procedures for gas-well
pressure-transient tests. Although our discussions are limited to
pressure-buildup and -drawdown tests , these concepts generally can
be extended to other pressure-transient tests. Included are pretest
estimates of reservoir properties , appropriate test flow rate s , and
test duration requirements.
8 . 4 . 1 Pretest Estimates of Reservoir Properties. Many aspects
of pressure-transient test design depend on accurate estimates of
permeability and skin factor. Although estimates of other rock and
fluid properties are required, permeability and skin factor estimates
usually are the most difficult because these estimations are frequently
a test objective. Possible means of obtaining permeability estimates
include laboratory analysis of core data, results from other well
tests in the formation , productivity tests (interpreted with the
pseudosteady-state radial flow equation with an estimated skin fac­
tor) , or, for a well in an infinite-acting reservoir, the unsteady-state
flow equation (also with an estimated skin factor).
In this section , we offer suggestions for obtaining pretest esti­
mate� of both skin factor and permeability. As shown previously ,
transient flow of gas is modeled most rigorously in terms of either
pseudopressure and pseudotime or adjusted pressure and adj usted
time variables. However, given the numerous approximations of
properties that must be made in well-test design, we think that the
design equations formulated in terms of ordinary pressure and time
are not only simpler to use but also sufficiently accurate.
Skin Factor. The skin factor is a dimensionless quantity used
to quantify the additional pressure difference from a zone of al­
tered permeability in the formation immediately adjacent to the well­
bore. Because of drilling and/or completion operations , this
�ear-wellbore zone can be damaged, resulting in a permeability that
IS lower than the unaltered, in-situ formation permeability. Under
these conditions, the skin factor is a positive quantity. Larger posi­
tive values of skin factors indicate greater reductions in near­
wellbore permeability. Conversely, if the formation around the well­
bore is stimulated , such as by acidizing or fracturing , the skin fac­
tor is negative.
The skin factor can be estimated in various ways. An estimate
can be obtained from other pressure-transient tests in the same for­
mation and in wells with similar completions, or it can be approxi­
mated from the well's completion and/or stimulation technique. For
low-permeability wells that do not flow at measurable rates before
eithe� a breakdown or an acid treatment, the skin factor can be ap­
proximated by
for a breakdown treatment (e.g., with nitro­
- 2 for acidization. Similarly , Table 8 . 1
gen or KCI water) or
provides estimates o f skin factors for various completion and stimu­
lation strategies (based on a modification and update of Ref. 5) .
Because of non-Darcy effects (Le. , turbulence and inertial ef­
fects that cannot be modeled by Darcy's equation) characteristic
of gas flow , an additional pressure drop similar to the skin effect
in the formation may occur immediately adj acent to the wellbore.
Unlike the skin effect caused by alterations in the near-wellbore
p�rmeability , the non-Darcy effect is not constant but varies directly
With flow rate. Consequently , the skin factor estimated from a gas­
well test is an apparent skin factor, s/, defined as
s= - 1 s=
s'=s+Dqg ' .....................................(8.1)
s=
Dq g =
8.1,
D,
1O - 1 5 (3kgM psc . ....................... (8.2)
D= 2. 7l 5xhrwTs
c P-g, wt
(3,
7
.
5
(3 = 1.88 x 10 10k 1 .47 - 0 3. ...................... (8.3)
D
where
true skin factor , sometimes called mechanical skin , re­
sulting from the zone of altered permeability around the wellbore ,
and
the additional , rate-dependent skin factor. In Eq.
the turbulent or non-Darcy flow coefficient,
is given by6
The turbulence coefficient,
is estimated with the correlation
c/>
Although
is not really constant, the estimate given by Eq. 8.2
is adequate for well-test design purposes.
197
DESIGN A N D IM P L E M E N TATION OF GAS-WELL TESTS
Formation Permeability. A s stated, possible sources of permea­
bility estimates include laboratory analysis of core samples from
the well to be tested or adjacent wells in the same formation. Well
tests in adjacent wells are other sources. Alternatively , if produc­
tivity tests have been conducted on the well to be tested , the
pseudosteady-state flow equation (written here for a slightly com­
pressible liquid) can be used to estimate permeability :
141.2qBJL [( � - � +s
k= h(pP wf) rw ) 4 ]
In
.
.
.
.
TABLE S.1-SKIN FACTOR ESTIMATES FOR VARIOUS
WELL STIMULATION AND COMPLETION STRATEGIES 5
N atural completion
Small acid treatment
I ntermediate acid treatment
Large acid or small fracture treatment
I n termediate fracture treatment
Large fracture treatment i n low-permeab i l i ty
reservoi r
Very large fracture treatment i n
low-permeab i l i ty reservoi r
. . . . . . . (8.4)
.
.
.
.
. .
.
.
.
When a flowing gas well is stabilized, permeability can be esti­
mated by
q/igiig [ ( re ) - 43 +s ,] , . . . . . . . . . . . . . . . (8.5)
kg = 141.2
h(p-Pwf) rw
8.1
Bg
fig =5.04 Tzlp .
In
where s' = skin!actor defined by Eq.
and qg = gas flow rate.
The gas FVF
is evaluated at average reservoir pressure:
Note that the length o f time to stabilize varies depending on the
formation permeability and the stage of field development. In new
wells in undeveloped fields and most low-permeability gas well s ,
the well may not reach stabilized flowing conditions for several
weeks or even months. In this case , the permeability is estimated
with the unsteady-state flow equation. Eq.
is the unsteady-state
flow equation in a form applicable when the gas flow rates change
smoothly :
8.6
Ps - Pwf 70.6figiig [( kgt ) - 2s ,] , . . . . . (8.6)
kgh 1,6884>iigCt r�
qg
Ps
Ps = Pi'
Ps <Pi '
8.6 kg ,
qg'
t,
--
In
=
s
Type of Sti m ulation or Completion
2.
3.
5.4.
1,
-
-
-
0
1.0
2.0
3.0
4.0
6.0
8.0
8. 9 kg .
Using the value o f rd from Step
solve Eq.
for
Com­
pare this calculated value of
with the initial guess. If the values
agree , stop; otherwise, continue with the iterative process .
Using the estimate of
from Step
calculate an improved
estimate of rd'
With this improved estimate of d, solve Eq.
again for
Repeat Steps
and
until the value of
converges . Con­
vergence generally occurs in two or three calculations.
kg
kg
2,
r
3 4
kg 8. 9
kg .
Example S.l-Pretest Permeability Estimate. In preliminary test­
ing , 9 a gas well in a low-permeability formation was produced for
hours at a final BHP,
of
psia. At the end of the pro­
duction period ,
=
MscflD and Gp =
M scf. Before test­
ing , the formation was broken down with KCI water, so assume
s' =
Using the well and formation data below , estimate the
permeability for designing a well test. Because the well is in a low­
permeability formation, estimate assuming unsteady-state flow­
ing conditions .
20
qg 110 Pwf' 400 110
- 1. 0 .
k
4> 0.118.
Pwf
Vwfib 400
15
gqg 1.1105
h 6
Pi 3,200
2. 9 x lO -4
iig 0.015
110
p 0.365
r
w
t=24Gp lqg , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.7) s' - 1. 0 .
ct
rd,
1.
t.
24Gp (24)(110
24
rd C77;;g )
(8.8)
C
qg 110
2. rd'
kg =0.1
8.10,
iig
8.8
8.6
( kgt ) [ (0. 1)(24) -4 ]
rd
3774>iigCt (377)(0. 118)(0.015)(2. 0 x 10 )
141.2
fi
ii
rd
]
q
[
,
g
g
g
kg = h(ps -Pwf) ( rw ) - 0. 75+s . . . . . . . . . . . . . . (8.9) =134
3. 2. 8.12,
kg
rd
8.9
kg
where
= stabilized, bottomhole shut-in pressure measured before
the beginning of the test. In new reservoirs with little pressure deple­
tion, this shut-in pressure equals the initial reservoir pressure , i . e. ,
while i n developed reservoirs ,
Again, gas proper­
ties are evaluated at the average drainage area pressure for wells
in developed fields or at the initial reservoir pressure in new fields.
To solve Eq.
for
we use the flow rate from the longest­
duration flow period during which a smoothly changing rate ,
was maintained . Effective producing time ,
is equal to cumula­
tive production divided by the chosen flow rate , or
where Gp = cumulative gas production and qg = gas production
rate . Further, we define an effective transient radius of drainage, 8
as
=
y"
•
.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
.
•
•
•
•
•
•
•
•
•
•
•
where
and ct = average viscosity and compressibility values , re­
spectively , evaluated at current average pressure in the well's
drainage area. Incorporating Eq.
into Eq.
and rearranging
yields
In
First, calculate the effective producing time,
t = -- =
mate
Msct)
------
M scflD
hours.
As a first guess, assume
Y,
=
md . Using Eq.
=
esti­
y,
ft.
Iterative Procedure for Estil1UJting Permeability.
kg ,
First, make an initial estimate of
and using this value, cal­
culate rd from Eq.
The initial estimate of
may be no more
than a guess.
8.8.
Solution.
-
Eq.
can be solved iteratively for
with the following rec­
ommended calculation procedure . Note that, if the flow data are
distorted by wellbore storage, this procedure overestimates perme­
ability. During wellbore storage , most of the surface production
is simply fluid being unloaded from the well , so the sandface rate
is actually less than the flow rate measured at the surface.
1.
=
=
psia.
=
bbl.
=
RB/Mscf.
=
MscflD.
=
ft.
=
psia.
cwb =
psi - I .
=
cpo
G =
Mscf.
=
ft.
=
= 2 . 0xlO - 4 psi - I .
kg
Using Eq.
from Step
solve for
with the value of
obtained
(141.2)(110)(1. 5)(0.015) [ ( 134 -0.7 5-1. 0]
(6)(3,200-400)
0.365 )
=0. 0864
=
In
md.
--
198
GAS RESERVOI R E N G I N E E R I N G
4.
3
The permeability calculated in Step is different from our in­
itial estimate, so we must calculate a new value of d using the
from Step
most recent value of
t )
r 377rjJkg/igl:t
125
kg 5..
d=
(
=
!hkg
=
ft .
r
3.
[ (0.0864)(24)
(377)(0.118)(0.015)(2. 0 x 10
rd
4,
With the new value o f
from Step
-4)
]!h
In
--
Although the permeability calculated in Step is closer to the
value calculated in Step
iterate once more. With the new value
of
the new value of
is
Ih
=
-4)
ft.
Finally , the latest value of
In
is
-
In
=
md .
This value is very close to that calculated in Step
have converged to the correct value .
Thu s , we
8.4.3 Estimating Test Duration. Homogeneous-Acting Reser­
voirs. As an absolute minimum, the flow period must be sufficiently
long so that the pressure transient moves beyond the wellbore­
storage distortion period . In terms of dimensionless variables , the
duration of wellbore storage during a drawdown or injection test
in a well with s�
is estimated by the empirical fit to the
Agarwal et al. 10 type curve ,
- 3.5
tD =(60+3.5s)CD , ........... .... ... . . .......... (8.13)
t
tD = 0. 0rjJ002637kt
p.ct r� . . ...... ..... . ................. (8.14)
8.13 8.14
where, for
8.13
0.8936C ..... ........ . . . ................ (8.15)
rjJc hr�
t
'
................................... (8.17)
In terms of dimensional variables , the duration of wellbore
for a flow test, the flow period before shut-in for a
storage,
buildup test, or an injection test is estimated by 3
,
where C = wellbore-storage coefficient. When both gas and liquid
phases exist in the wellbore , for a changing gas/liquid interface ,
.
kh /
Eq .
often is modified I I to model the duration of wellbore­
storage distortion for pressure-buildup tests; however, for simplicity ,
we will use the same equation for the design of both drawdown
and buildup tests .
If a flow time equal to or greater than the duration of the wellbore­
storage period is not attained, then the flow period data and data
from a buildup test following the flow period cannot be analyzed
correctly with semilog plotting techniques . Further, conventional
type-curve analysis of these data also is impossible. Under these
conditions , the well test should be run again with a longer flow
period before shut-in .
If a feasible test length for a drawdown test, t max ' or for a build­
up test, �tmax ' is greater than four times the duration of wellbore
for a drawdown test or �tmax >
storage (Le . , tmax >
for a buildup test) and occurs before boundary effects appear, then
surface shut-in is possible. For a buildup test with a low probabili­
ty of reaching the semilog straight line , a bottomhole shut-in device
should be considered. With this device , wellbore-storage distor­
tion of the test data should be negligible . For a flow test, there is
no simple way to limit the wellbore volume to be unloaded and
thus to reduce the duration of wellbore-storage distortion.
Semilog analysis techniques are inherently more accurate than
type-curve methods and consequently , are preferred for analyzing
pressure-transient test data. Semilog analysis techniques , however,
require at least one-half a log cycle of middle-time data. This criteri­
on suggests that the time to reach the end of the middle-time or
radial flow region during either a drawdown or a buildup test should
be at least three to four times greater than the time to reach the
end of wellbore storage . The end of the middle-time region in a
homogeneous-acting reservoir can be estimated for either a pressure­
drawdown or -buildup test , respectively , as
4�twbs
4twbs
rjJ
tend �tend 948 kp.ct ....... ....... ........ (8.20)
8.20
or
=
L2
,
where L = distance from the well to one or more no-flow bounda­
ries . Eq.
also is valid for gas-well tests when average proper­
ties are used . The no-flow boundaries may be either natural reservoir
boundaries (sealing faults, low-permeability barriers , etc . ) or ar­
tificial boundaries created by adjacent producing well s . For a well
centered in a drainage area having a radius re, Eq.
can be
rewritten for a buildup test as
8.20
in hours ,
For gas wells , the gas properties in Eq .
are evaluated at
the average reservoir pressure . Eq.
predicts the duration of
wellbore unloading for a well produced at a constant sqrlace flow
rate, but the relationship also is accurate for well-test de sign when
the flow rates vary smoothly . 9 The dimensionless wellbore-storage
coefficient, CD' in Eq.
is defined by
CD =
V
wb wb
twbs
twbs (200,000+ 12,000s)C
. ............ . . ..... . (8.18)
p.
8.18
=
md .
=
C= c
refine the estimate of
(141.2)(110)(1.5)(0.015) [ ( 125 )- 0.7 5 - 1. 0]
0.365
(6)(3,200 - 400)
=0.0850
6.
5
3,
kg ,
rd
( kg! ) [ (0.0850)(24) ]!h
rd
377rjJ/i g ct (377)(0.118)(0.015)(2. 0 x 10
=124
kg
7.
qglig /i [ ( rd ) - 0.7 5+s']
kg = 141.2
h( Pi - Pwf) r w
( 14 1.2)(110)(1. 5 )(0. 0 15), [ (� ) - 0.7 5 - 1.0]
0.365
(6)(3,200 - 400)
=0.0848
5,
=
When the wellbore is filled with a single-phase fluid ,
� tend =
237rjJp.ct r� . .. . .... ................. ....(8.21)
k
8.20
8.20 8.21
.
For a drawdown test , however, Eq.
should be used with
replacing L. Similar expressions can be written for gas wells
by replacing the fluid properties in Eqs .
and
with aver­
age values .
Hydraulically Fractured Wells. A s when designing well tests for
homogeneous-acting reservoir s , the engineer requires guidelines
for identifying both the end of wellbore-storage distortion and the
onset of boundary effects in a hydraulically fractured wel l . In this
section, we provide suggestions for estimating these flow periods
for both infinite- and finite-conductivity vertical fractures .
Infinite-Conductivity Vertical Fractures. Although limited in the­
ory to infinite-conductivity (i. e . , Cr�
fractures in infinite­
acting reservoirs, the Barker-Ramey 1 2 type curves (Chap .
Fig .
provide a convenient method for predicting the end of
wellbore-storage effects . Analysis of their type curves indicates that,
re
C=25.65(A wb/Pwb)' .. . ... . .... ... ..... ...... .... (8.16) 6.49)
100)
6,
1 99
D E S I G N A N D I M PL E M ENTATION OF GAS-WELL TESTS
Lt ,
regardless of fracture half-length,
the dimensionless time at the
end of wellbore-storage distortion of the test data is
tLfD = 15 CtD ,
tD = 15 CD , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.22)
tLfD = 0.0002637kt
l/>p.ct L} . . . . . . . . . . . . . . . . . . . . . . . . . (8.23)
CfD = 0.8936C
I/>ct hL} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.24)
Cr
wf kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.25)
Cr=--.
7rkLf
or more simply ,
where
and
is defined by
The start of the middle-time region in a test of a hydraulically
fractured well usually is the beginning of pseudoradial flow, which
is the point when the transient moves beyond the influence of the
fracture and is no longer affected by fracture properties. The larg­
er the
value, the later the development of an essentially radial
drainage pattern. For infinitely conductive fractures in a well cen­
tered in a square drainage area, 1 3 pseudoradial flow begins ap­
proximately when
Cr
Depending on the value of the dimensionless fracture conduc­
tivity and the duration of wellbore storage, several intermediate flow
periods l6 -fracture l inear, bilinear, and formation linear-can
occur before the onset of pseudoradial flow (Chap. 6, Fig. 6 . 36) .
Of these flow regimes , the fracture linear flow occurs first , but it
is very short-lived and usually is masked by wellbore-storage ef­
fects. Because of its extremely short duration , this flow period is
of no practical use for most well-test analyses.
The next possible flow period is bilinear flow , which occurs in
finite-conductivity fractures as fluid in the surrounding formation
flows linearly into the fracture. Depending on the duration of
wellbore-storage effects, the bilinear flow period may last for a sig­
nificant time period .
Formation linear flow occurs only in infinite-conductivity frac­
tures. Between the end of formation linear flow and the beginning
of pseudoradial flow patterns , a transitional flow regime occurs.
This flow regime is characterized by an elliptical flow pattern.
For a well with a fmite-conductivity vertical fracture in an infinite­
acting reservoir, C inco-Ley and Samaniego-V.'s I 7 type curves
(Chap.
Fig.
provide estimates of the time at which pseu­
doradial flow begins. Note that the type curves in Fig.
do not
include wellbore-storage effects and thus may underestimate the
beginning of the semilog straight line . Again, these curves are suffi­
ciently accurate for designing well tests.
Naturally Fractured Reservoirs. In well tests in naturally frac­
tured reservoirs, early-time data are necessary for estimating frac­
ture and matrix system properties. Often, these early-time data are
masked by the effects of wellbore storage. Mavor and Cinco-Ley 1 8
suggested that , if
6, 6.44)
6.44
tLfD �3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.26)
w(1 - w) , . . . . . . . . . . . . . . . . . . . . . . . . . . . (S.2S)
tprJ'
CD > 36A(6O
+3.5s)
1
l,4001/>
p.
c
L}
t
. . . . . . . . . (S.27)
tprJ
k
In dimensional form, the onset of the pseudoradial flow period,
is
_
=
. .
. .
.... . .
.. .
.
. .. .
In tests of hydraulically fractured wells , having one-third to one­
half log cycle of data in pseudoradial flow is desirable so that more
precise semilog rather than type-curve analysis can be used. This
objective is idealistic , however, and rarely is achieved , especially
in low-permeability formations with long fractures. As a general
rule , the pseudoradial flow period has not begun in a fractured­
well test at the time when wellbore-storage distortion ends.
If the fracture half-length is large relative to the drainage area,
then boundary effects may quickly mask the infinite-acting pseu­
doradial flow regime. For infinite-conductivity fractures in wells
centered in square reservoirs , Gringarten
's 14 type curves
(Chap.
Fig .
provide a method to estimate the time at which
boundary effects will appear. Examination of these curves indicates
that
6, 6. 40)
et al.
tend = 94SI/>kp.ct L� , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (S.20)
Le
Finite-Conductivity Vertical Fractures.
where
is the distance from the well to the side of the square
reservoir .
Finite-conductivity ver­
tical fractures are characterized by measurable pressure drops in
the fracture . Because of the number of flow regimes exhibited by
finite-conductivity vertical fractures , simple equations for estimat­
ing the duration of the wellbore storage period are difficult to ob­
tain. Type curves 15 for finite-conductivity vertical fractures and
including wellbore-storage effects have been developed; however ,
we recommend using the Barker-Ramey 12 type curves (Fig.
for estimating the time when wellbore-storage effects end. Although
their type curves are not theoretically correct for analyzing well­
test data from finite-conductivity vertical fractures , we have found
them to be sufficiently accurate for well-test design purposes, es­
pecially for estimating the duration of wellbore-storage effects. Anal­
ysis of these type curves indicates that, regardless of
the
duration,
of wellbore storage is
6. 49)
Lf ,
tD '
tD = 15CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (S.22)
then wellbore-storage effects will distort the early-time data. Un­
der these conditions , the design engineer should consider using a
bottomhole shut-in device. For a pressure-buildup test, this device
should reduce wellbore-storage effects . For a flow test , there is
no simple way to limit the wellbore volume to be unloaded and
thus reduce the duration of wellbore-storage distortion.
Mavor and Cinco-Ley also suggested that the duration of
wellbore-storage effects in a naturally fractured reservoir can be
estimated with the same equation developed for a homogeneous­
acting reservoir,
tD =(6O+3.5s)CD , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8.29)
tD (I/>ma0.c0ma002637kt
. . . . . . . . . . . . . . . . . . . . . (S.30)
+I/> f c f ) /J.'a
et al.
6. 63 6.59 6.60
where
------
1 9 ,20 developed type curves (Figs.
and
Bourdet
for pseudosteady-state matrix flow and Fig.
for transient matrix
flow) for estimating the beginning of the semilog straight line un­
der a variety of conditions of wellbore storage and skin factor.
8.4.4 Radius of Investigation. The radius-of-investigation concept
is of both quantitative and qualitative value in well-test design.
Radius of investigation,
, is the distance that a pressure transient
has moved into the formation following a rate change in a well.
The distance traveled by the pressure transient is a function of not
only the formation rock and fluid properties but also the time elapsed
since the rate change. To evaluate a representative reservoir volume
or to identify a suspected reservoir heterogeneity , we must design
a well test that achieves the desired radius of investigation .
For well tests designed to analyze productivity problems, a radius
of investigation of
s S
ft during a flow test or during the
flow period before shut-in for a buildup test should be achieved.
In a buildup test , a similar radius of investigation during the shut­
in period is desirable. In well tests designed for sampling condi­
tions at the reservoir boundaries ,
==
should be achieved dur­
ing both the flow period and the shut-in period . For identifying an
important heterogeneity , such as a sealing fault , an
at least four
times the expected distance to the heterogeneity should be attained .
ri
50 ri 200
ri re
ri
200
GAS RESERVOI R E N G I N E E R I N G
il
D
�.
..:
k
�
.
I
" '1
:- f - ·
, .. 1-
,
h
�
�
�
/
v
v,
:ts
/. / /
V
v
V
�
V / .v
!. V
f- ' · ·
/)
/ /
;.
��r' I - · -· ·· ' � ��
:,;/!.�/
- ��
/
�! . - .
. •.
-
·l· i -
'- " 2 �r;
I
�.-. I" ' ;
�
_.
y .__
./ � v
./
:....::
ij(�
��
v
�
/.v /
�� :a� ./v V
r-v
...... � v A �
,,,..,
V
J. V
�
�V- A �
......
'/.
�
./-� ��
r...
.- I� ���
�4
�
�3
:::I
i
w2
� � � r....--i-" � .- ��� ...... � -�....... - - f-� ��t:=-� .-��l.-I-- -
i
�I
�
�o
�
I
....
5,
«
••
•
I
--
�� -
-�
-
--
-
.�J)�
./ � �
'}
r....--
.-
::J
i
\l1o v
r-
..} f-"
.-
i..!�1===
- --
- l-
20
V
......
./ ,.....
.•
V ,�
/,% �
� �� 'j;
� ��'
-
--
5
10
FlOWING W E W£AD PRESSU RE · 1 0 0 PS IA
l/ v
Vv /
V
V
k1L
l� !Z�
v
:!�
.-
I
2
30
Fig. B . 1 -Minimum gas flow rate required to unload liquids from a gas well (after Duggan 25 ) .
The radius o f investigation2 1 for a well i n a homogeneous-acting
formation and during pseudoradial flow in a hydraulically fractured
well is
ri =
C:)
48 /l-C
'h
. . . .
. . .
. . .
. .
.
. . .
. .
. .
. . .
. . . . . . .
(8 . 3 1 )
Comparing Eqs . 8. 21 and 8. 3 1 , we observe that the time at which
boundary effects begin to be seen in the pressure response at the
wellbore is one-fourth the time required for ri to reach re during
a pressure-buildup test .
8.4.5 Flow-Rate Requirements. Constant-Rate Testing. Most of
the well·test analysis techniques discussed here were developed as­
suming constant production rates during the entire test. In a draw­
down test or during the flow period before a pressure-buildup test,
a constant rate should be maintained throughout the test whenever
possible . Often , constant or nearly constant rates can be achieved
with an adjustable choke at the wellhead .
For flow before a pressure-buildup test and for a well with es­
tablished production characteristics , the historical producing rate
should be maintained essentially constant for at least the expected
duration of the shut-in period . Unfortunately , maintaining constant
rates during a drawdown test or during the flow period before a
pressure-buildup test is difficult , especially early in the life of a
gas well . When a constant rate cannot be maintained , the instanta­
neous rates should be measured continuously throughout a flow test,
and variable-rate analysis techniques should be used.
Whenever possible, the pressure gauge should be placed into the
hole with the well flowing . If this cannot be done , then the same
constant rate established before the well was shut in should be re­
established after the gauge is placed in the hole and maintained for
at least eight times the duration of the shut-in period .
Initiating Flow in Low-Permeability Wells. Wells in low­
permeability formations, especially natural gas reservoirs, are tested
infrequently before fracturing because they often will not flow at
measurable rates, if at all , without stimulation . Experience indi­
cates that a prefracture breakdown with acid , KCI water, or nitro­
gen usually will enable a low-permeability well to flow at rates
adequate for prefracture testing . 9
Given the importance of pre fracture tests , these breakdown treat­
ments are a good investment in formation evaluation . Bostic and
Graham22 reported success in achieving flow in the Cotton Val­
ley formation (permeabilities ranging from 0. 001 to 0. 01 md) with
2% KCl water breakdown. Similar success has been experienced
in the tight gas sands in the Travis Peak formation . 23 Holgate et
at. 24 suggested nitrogen breakdown treatments , which have proved
successful in initiating flow in wells completed in the lower-pressure
Eastern Devonian gas shale reservoirs .
Liquid Loading. During either a drawdown test or the flow period
before a buildup test in a gas well that produces significant quanti­
ties of liquids , the minimum flow rate should equal that required
to lift the liquids continuously from the wellbore . Liquid loading
or accumulation occurs when the gas phase cannot provide ade­
quate energy for the continuous removal of liquids in the wellbore .
When liquids are produced with gas , the pressure drops in the tub­
ing increase because of not only the additional weight of the liquid
in the gas stream but also the accumulation of the liquids in the
wellbore bottom . As a result, these liquids impose an additional
backpressure on the formation and can significantly affect the test
results .
Ideally , we like to maintain gas velocity at some minimum value
sufficient to eliminate or greatly reduce liquid holdup in the well­
bore , thereby continuously removing liquid s . When the gas rate
falls below a rate that is inadequate to remove the liquids continu­
ously , however, the liquid accumulates in the wellbore . We pre­
sent two simple methods for estimating the minimum gas flow rate
for continuous liquid removal in the wellbore : Duggan ' s 25 and
Turner et at. ' s 26 methods .
Duggan 's Method. I n 196 1 , Duggan developed a method for es­
timating the gas flow rate required to keep a gas well continuously
unloaded . The method is based on the assumption that a minimum
wellhead velocity of 5 ft/sec is required to lift liquids continuously
in the wellbore . The method is empirical because the velocity of
5 ft/sec was selected after observing the flowing performance of
a number of wells under a wide range of operating conditions of
liquid loading . Fig. 8 . 1 shows the results of Duggan ' s study .
Note that Duggan ' s method is empirical and that several field
studies subsequent to his work have suggested that higher gas ve­
locities may be required . For example, Libson and Henry 27 indi­
cated that gas velocities exceeding 1 5 ft/sec are required in many
deep gas wells in southwest Texas . In low-pressure gas wells in
the Hugoton field , Smith 2 8 found that velocities of 5 to 1 0 ft/sec
were necessary to remove condensates, while water removal re-
201
DESIGN AN D IM PLEMENTATION OF GAS-WELL TESTS
quired velocities of 10 to 20 ft/sec . These additional results sug­
gest that Duggan ' s method should be applied judiciousl y .
Turner et al. Method.
Turner
et al.
proposed two physical models
for predicting when a gas well will experience liquid loadup:
(1) liquid fIlm movement along the pipe walls and (2) liquid droplets
entrained in a high-velocity gas . Using theoretical particle and drop­
breakup fluid mechanics , they calculated the minimum gas veloci­
ty required to remove either water or condensates continuously from
the wellbore . From comparisons between theoretical predictions
and field results , they concluded that the controlling mechanism
for continuous liquid removal is the movement of the entrained liquid
droplets .
Turner et al. defined the minimum gas velocity that will move
the largest liquid drops possible in a gas stream as
vt = 20 . 4
u 'A ( P L - P g ) 'A
P'f
.
. . . . . . . . . . . . . . . . . . . . . . . . . (8 . 32)
We usually do not have fluid properties , however , especially in­
terfacial tensions . Therefore, by assuming typical values for the
fluid properties , Turner et al. developed the following equations
for the minimum velocities of the gas phase required to lift water
and condensate , respectively :
vg , w =
5 . 62 (67 - 0 . 003 1ptf) 'A
(0 . 003 1P tf) 'h
and vg , c =
4 . 02 (45 - 0 . 003 1ptf) 'A
(0. 003 1P tf) 'h
. . . . . . . . . . . . . . . . . . . . . (8 . 33)
a
8.4.6 General Design Procedures-Single-Well Tests. I n this sec­
tion, we offer general procedures for designing single-well pressure­
buildup tests in unstimulated and stimulated well s . Note that these
procedures are suggestions and that each test design is unique and
will require different considerations , depending on the testing ob­
j ectives and environment .
Prefracture Pressure-Drawdown and -Buildup Tests. The fol­
lowing procedure , based on the theoretical and operational princi­
ples described previously , should prove adequate for designing
drawdown and buildup tests in a well that has not been stimulated
by either hydraulical fracturing or acidizing. If a fracture treatment
is planned, the procedure applies to the design of a prefracture build­
up test. In all equations in the following procedure , the gas prop­
erties are evaluated at the current average reservoir pressure ,
p,
Test Design Procedure.
1 . Estimate the well and reservoir properties required for the test
design .
A . Select the flow rate for the drawdown period or for the flow
time before shut-in . Using the methods given in Sec . 8 . 4 . 5 , ensure
that this rate is sufficient to lift liquids continuously from the well­
bore for gas wells producing liquid s .
B . Estimate P j ( o r
and P wj (t:.. t = O) . Possible sources inciude
data from a previous DST or pressure surveys from active wells
completed in the same formation .
p)
C . Estimate fluid properties (i . e . , Jig , Jig , and ct for gas wells)
at
In the absence of laboratory data, correlations generally are
adequate for estimating fluid properties .
D . Estimate k, h , and c/!. k can be estimated with the suggestions
in Sec . 8 . 4 . 1 . The best sources for estimating h are well log s ; c/!
estimates can be obtained from cores or well log s .
E . Estimate the PI ,
p.
J=
q
(p
C
only a single fluid phase is produced , estimate
C
with Eq. 8 . 1 7 .
temperature and BRT . For single-phase gas in the wellbore , we
can estimate Cwb = cg at p and average wellbore temperature .
G. Estimate the drainage radius , r e , as the well spacing from
known field development patterns or the distance L to a nearby reser­
voir no-flow boundary. Distances to natural no-flow boundaries may
be estimated with geologic data .
2 . Estimate the duration of wellbore-storage distortion, t WbS '
during the production and/or shut-in period .
A . If the PI is unknown and kh/p, and s estimates are available, use
twbs =
(200 ,000 + 1 2 ,000s)C
kh/p,
,s � - 3 . 5 .
. . . . . . . . . . . . . ( 8 . 1 8)
B . If the PI is known , use
twbs = 200CI1B .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8 . 1 9)
Note that if the well is stimulated (negative skin factor) , using
s = O in Eq. 8 . 1 8 will lead to a more conservative test design.
3. Estimate the flowing or shut-in time required to achieve the
design
rj ,
=
948 c/! p, c t r
r
. . . . . . . . . . . . . . . . . . . . . . . ( 8 . 20)
k
Guidelines for selecting the design
rj
include the following .
A . For evaluating near-wellbore condition s , achieve rj = 200 ft
to sample a representative portion of the reservoir .
B . If rj = 200 ft cannot be attained in a reasonable time , then a
smaller rj can be selected. This smaller rj , however, must be sever­
al times the estimated depth of damage or stimulation , rs (e . g . ,
rj � 5rs ' with rs calculated with estimated properties in the altered
zone near the wellbore rather than in the formation) .
C . To sample the entire drainage area, rj = re ' where r e is esti­
mated from Step 1 G .
D . T o confirm the presence o f a flow barrier a n estimated dis­
tance L from the well , use rj > 4L .
4. Estimate the flowing or shut-in time ,
at which bounda­
ry effects may appear.
A. For a well a distance L from the nearest boundary ,
tend ,
tend t:..tend
or
=
948 c/! p, c t L 2
k
.
. . . . . . . . . . . . . . . . . . . . . . (8 . 20)
For a drawdown test, Eq . 8 . 20 should be used with
re
replacing L .
B . For a well centered i n a circular drainage area, the shut-in
time is
t:..tend
==
237 c/! p, c t r
k
;
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8 . 2 1 )
5 . Select a flow time for the drawdown test or shut-in time for
the buildup test that is the greater of 4twbs and
6 . An important component in designing a well test is selecting
a pressure gauge with the proper sensitivity for our test. A recom­
mended procedure for estimating the gauge sensitivity is as follows.
t mi n '
A . Estimate the slope , m , of the semilog straight line expected
in the flow or buildup test :
m=
1 62 . 6qBp,
kh
The pressure change between the time at the end of the straight
line and 90 % of that time is
t:.. p = m log( 1 /0 . 90) = 0 . 0458m .
-P wj )
or alternatively ,
with
Average wellbore temperature i s the arithmetic average o f surface
or
. . . . . . . . . . . . . . . . . . . (8 . 34)
C. For wells
for a liquid/gas interface in the wellbore is given by Eq . 8 . 1 6 ; if
tmin t:..tmin
Eqs . 8 . 33 and 8 . 34 were developed with the following assumed
fluid properties : 'Y = 0 . 6 ; T= 1 2 0 ° F ; P w = 67 Ibm/ft 3 ; P L = 45
Ibm/ft3 ; ug, w = 60 ynes/cm; and ug,L = 20 dynes/c m .
for gas-well desig n .
F. Estimate the wellbore-storage coeffIcient ,
both gas and liquid (L e . , condensate and/or water) production ,
s.
B . Choose a pressure gauge sensitive enough to respond to the
expected pressure change calculated in Step A .
G A S RESERVOI R E N G I N E E R I N G
202
C. Estimate the maximum pressure to be encountered in the test
(e . g . , in a new reservoir ,
P max =p j ) .
D . Choose a pressure-gauge range s o that the maximum test pres­
sure falls between 60 % and 80 % of the upper gauge limit .
E. If possible , choose the clock on a self-contained gauge so that
most of the chart is used but the gauge is run only once during the
test. Tandem gauges are essential . An excellent alternative to the
conventional mechanical gauge is the memory-type gauge .
To facilitate well-test design, we have developed worksheets that
incorporate many of the suggestions listed in the design procedure.
These worksheets, useful for oil and gas wells, are given in Ap­
pendix I .
Example S.2-Design o f a Prefracture Pressure-Buildup Test
for a Gas Well. The following example well-test design is repro­
duced from Ref. 9. We want to run a pressure-buildup test in a
gas well before fracturing . Our obj ective is to design a test that
will provide estimates of formation permeability and initial reser­
voir pressure . To minimize flaring gas , the operator would prefer
to produce the well for 1 day or less and would prefer to limit the
shut-in period to less than 3 day s . Based on existing development
patterns in the field , the drainage area of the well is estimated to
be 640 acres. The well is currently producing 100 Mscf/D at a BHFP
of 400 psia.
I/> = 0 . 1 1 8 .
r w = 0 . 365 ft .
Vwb = 15 bbl .
Bg = 1 . 5 RB/Mscf.
P i = 3 , 200 psia (preliminary estimate) .
h = 6 ft .
cwb = 2 . 9 x l O - 4 psi - I .
jig = 0 . Q 1 5 cpo
P wf = 400 psia .
= 0 . 0848 md .
c, = 2 . 0 x l O - 4 psi - I .
A = 640 acres .
".B
Solution .
1 . First , estimate gas and formation properties .
A . Estimates of pressures and gas properties are provided above.
Note that Bg , C" and ji are evaluated at ji. We choose to set s' = 0
for conservative test desig n .
B . Assuming only g a s in the wellbore ,
C = cwb Vwb = (2 . 9 x 1 0 - 4 ) ( 1 5) = 0 . 00435 bbl/psi .
C . Produce the well at P Mif = 400 psia and assume that the cur­
rent well production rate , q g = 1 00 Mscf/ D , can be maintained at
the end of the flow period .
D. For 640-acre spacing and a circular drainage area ,
re =
.JA l 7r = .J (640)(43 , 5 6O)11r = 2 , 979
ft .
2 . From Eq . 8 . 1 8 , the duration o f wellbore-storage distortion is
twbs =
(200 , 000 + 1 2 , 000s ' ) C
[200 ,000 + ( 1 2 , 000)(0) ] (0 . 00435)
(0 . 0848)(6)/(0 . 0 1 5 )
= 2 6 hours .
3 . The shut-in time required to investigate a minimum distance
of 200 ft into the reservoir is
� t mi n
948l/>jigc,r
r
== ---"--
kg
(948)(0 . 1 1 8) (0 . 0 1 5) (2 . 0 x 1 0 -4 ) (200) 2
0 . 0848
= 1 5 8 hours .
4 . The shut-in time , �tend ' at which boundary effects appear for
a 640-acre drainage area is
�tend ==
237I/>jic,r�
-----'--'­
kg
(237)(0 . 1 1 8) (0 . 0 1 5) (2 . 0 x 10 -4 ) (2 , 979) 2
0 . 0848
= 8 , 7 80 hours .
5 . The flow time should be the greater of 4twbs = (4)(26 hours)
= 1 04 hours and �tmi n = 1 5 8 hours . Therefore, the well should be
flowed for 7 days because �tmi n = 1 5 8 hour s . With this flow peri­
od, about 200 ft of formation will be investigated . A l OO-MscflD
rate will be maintained at the end of this period .
6 . Similarly , a shut-in time of 7 days should be used .
7 . Estimate the pressure-gauge sensitivity required for the test .
A . Estimate the slope , m, of the semilog straight line expected
in the flow or buildup test :
( 1 62 . 6) ( 1 00)( 1 . 5) (0 . 0 1 5)
(0 . 0848)(6)
= 7 1 9 psi/cycle.
The pressure change between the time at the end of the straight
line and 90 % of that time is
�p = m log ( 1 I0 . 90) = 0 . 0458m
or �p = 0 . 0458m = (0 . 0458)(7 1 9) = 33 psi .
B . Choose a pressure gauge sensitive enough to respond to the
expected pressure change calculated in Step A . An ordinary Bour­
don tube gauge is sufficiently sensitive for this test. However, any
gauge with greater sensitivity is acceptable.
C . Estimate the maximum pressure to be encountered in the test
(e . g . , in a new reservoir, P max = P i ) ' For this test, we anticipate
P i = 3 , 2oo psia.
D. Choose a pressure-gauge range so that the maximum test pres­
sure falls between 60 % and 80 % of the upper limit of the gauge ,
or 4 , 000 and 5 , 300 psi .
E . Choose a 1 20-hour clock for the pressure gauge .
In summary , the operator cannot test as he originally proposed
(flow period of 1 day or less and buildup period of < 3 days) and
obtain a proper test . Instead , he should flow the well for 7 days
at P Mif = 4oo psia and shut in the well for 7 day s . During both the
flow period before shut-in and the pressure-buildup test, wellbore­
storage distortion should not be a problem. In addition , boundary
effects will not be encountered .
Postfracture Pressure-Drawdown and -Buildup Testing. The
procedure outlined below provides recommendations for design­
ing postfracture buildup tests . A s with the prefracture test design,
a pretest estimate of effective permeability to the produced phase
must be available. For gas-well test design, use gas properties evalu­
ated at ji for all equations .
Test Design Procedure.
1 . Estimate the well and reservoir properties required in the test
design.
A . Select the flow rate for the drawdown period or for the flow
time before shut-in. Using the methods given in Sec . 8 . 4 . 5 , ensure
that this rate is sufficient to lift liquids continuously from the
wellbore .
B . Estimate Pi (or ji) and P Mif (�t = O) . Possible sources include
data from a previous DST or pressure surveys from active wells
completed in the same formation.
203
DESIGN AND I M PLEMENTATION OF GAS-WELL TESTS
C . Estimate fluid properties (i . e . , ii , Ji , and ct for gas wells)
g g
at j5. In the absence of laboratory data, correlations generally are
adequate for these estimations .
D . Estimate k, h , and r/J . k can be estimated using the sugges­
tions given in Sec . 8.4. 1 . The best sources for estimating h are well
logs , and r/J estimates can be obtained from either cores or well logs.
E . Estimate C. For wells with both gas and liquid (i . e . , conden­
sate and/or water) production, use Eq. 8 . 1 6 to obtain C for a liq­
uid/ gas interface in the wellbore; estimate C with Eq . 8 . 1 7 if only
a single fluid phase is produced . Average wellbore temperature is
the arithmetic average of surface temperature and BHT . For single­
phase gas in the wellbore, estimate C wb == CIi at p and average well­
bore temperature .
F. Estimate Lf and the perpendicular distance to the side of a
square drainage area, L e , in which the well is assumed to be com­
pleted . In addition, estimate the distance L to any single boundary
nearby .
2. Estimate twbs '
A . Calculate CD :
CD =
0. 8936C
r/Jct hr�
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8 . 1 5)
B. Determine the time tD at which wellbore-storage effects end.
For infinite-conductivity fractures (Cr 2! 1 00) ,
buildup test design is reproduced from Ref. 9. The gas well cur­
rently is producing 1 , 500 MscflD at a BHFP of 400 psia. The long­
est possible flow period before the buildup test is 1 month , and the
longest possible buildup test is 2 weeks . Design a pressure-buildup
test to evaluate the success of the fracture treatment.
¢ = 0. 1 1 8 .
rw
Vwb
iig
Le
Pi
h
cwb
Jig
Lf
P wf
kg
ct
qg
twbs =
8 . 23 ,
calculate
1 . The wellbore-storage coefficient (single phase flowing in the
wellbore) is
C = cwb Vwb = ( 2. 9 x 1 0- 4 )( 1 5) = 0. 0043 5 bbl/psi .
2. Estimate the duration of wellbore-storage distortion .
A . Calculate CD :
CD =
twbs :
¢p.. c t r�tD
tp if =
( 0. 1 1 8) ( 2. 0x 1 0-4 )(6)(0.365) 2
¢ct hr�
= 206.
tD at which wellbore-storage effects end .
}.
C . Using the definition of dimensionless time , calculate
t Wbs =
. . . . . . . . . . . . . . . . . . . . . . . . . . . (8. 27)
Estimate tend ' the time at which boundary effects will appear.
A. For a well centered in a symmetrical drainage area with a dis­
tance from well to boundary . L e ,
948¢p..c tL�
k
twbs :
¢Jig Ct r�tD
0. 000263 7kg
( 0. 1 1 8) ( 0. 01 5) ( 2. 0 x 1 0 -4)(0.365) 2 (3 , 09 0)
4.
tend ==
------
tD == 1 5CD = ( 1 5)( 206) = 3 , 09 0.
1 1 ,4 00¢p.. ct L
k
( 0. 8936) ( 0. 0043 5)
0. 8936C
---
B . Determine the time
0. 000263 7k
3 . Estimate the time required to reach pseudoradial flow , tpif .
For infinite- and finite-conductivity fractures , the beginning of the
semilog straight line in dimensional form is approximated by
_
0. 365 ft .
1 5 bbl .
1 . 5 RB/Mscf.
2,64 0 ft .
3 , 200 psia .
6 ft.
2.9 x lO -4 psi - I .
0.D 1 5 cp o
200 ft .
4 00 psia .
0. 0848 md .
2. 0x 1 0-4 psi - I .
1 , 500 Mscf/ D .
Solution .
tD == 1 5CD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8. 22)
For well-test design , Eq. 8 . 22 is an adequate approximation for
finite-conductivity fractures .
C . Using the definition of tD given by Eq.
=
=
=
=
=
=
=
=
=
=
=
=
=
( 0. 000263 7)(0. 0848)
= 6 . 5 hours .
3 . Estimate tpif . For infinite-conductivity fractures , the onset of
pseudoradial flow occurs at
B . For a well much closer to one boundary than others, estimate
_
tpif =
te nd from
1 1 ,400¢/ig ctL}
kg
1 1 ,4 00( 0. 1 1 8) ( 0. 01 5) ( 2. 0 x 1 0 -4 )( 200) 2
5. Select a flow time for the test . The flow time must exceed
4twbs ' In addition , the flow time should at least equal the lesser
of 4tp if or the maximum flow time possible for the test . Finally ,
the well should flow at least long enough to clean up .
6. Select the shut-in time for the buildup test. The shut-in time
' must exceed 4twbs ' If impractical , then consider using a bottom­
hole shut-in device . In addition, the shut-in time should equal the
lesser of 4tp if or the maximum shut-in time , Ll.tmax ' possible for
the test (which should be less than or equal to the maximum flow
time possible) .
7. An important component in well-test design is selecting a pres­
sure gauge with the proper sensitivity for the test . A recommended
procedure for estimating the gauge sensitivity is given in the proce­
dure for designing prefracture well tests .
Example 8.3-Design of a Postfracture Pressure-Buildup Test
for a Gas Well. The following example postfracture pressure-
0. 0848
= 1 ,9 04
hours or
79
day s .
This flow time is longer than the longest fl o w period possible
month) and longest shut-in period possible ( 2 weeks) . There­
fore , pseudoradial flow will not be reached during this test, and
semilog analysis of the pressure-transient data will not be possible .
4 . The time to boundary effects , either during the flow period
or the shut-in period , is
(1
948¢Jig ctL�
kg
tend or dtend == ---"'---­
(948) ( 0. 1 1 8)(0. 01 5) ( 2. 0x 1 0-4)(2,64 0) 2
0. 0848
= 27,6 00 hours
or
1 , 1 50 day s .
204
G A S RESERVO I R ENG I N E ER I N G
Thu s , boundary effects are not likely to distort data in this test .
5 . The flow time for this test will be the maximum possible, or
1 month .
6. The shut-in time for this test will be the maximum possible,
or 2 weeks .
7 . Estimate the pressure-gauge sensitivity required for the test .
A . Estimate the slope , m , of the semilog straight line expected
in the flow or buildup test .
1 62 . 6qg1igiig
m=
ri o
is the distance from the active well to
the observation wel l , i . e . , ri = r :
t mi n =
948r/>p.ct r
J
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8 . 20)
k
3 . Using the estimated well and reservoir properties , calculate
the following dimensionless variables :
0 . OOO2637kt
and tD =
( 1 62 . 6) ( 1 , 5 00)( 1 . 5) (0 . 0 1 5)
,
r/>p.ct r�
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (8 . 36)
where t = desired flowing time . This flowing time should equal or
(0 . 0848)(6)
= 1 0 , 800 psi/cycle .
The pressure change between the time at the end of the straight
line and 90 % of that time i s
exceed t m in calculated in Step 2 .
4 . Using either Eq . 8 . 37 or a graph 3 of the exponential-integral
solution, calculate the dimensionles s pressure , P D .
PD = -
ilp = m log ( 1 I0 . 90) = 0 . 045 8m
or ilp = 0 . 0458m = (0 . 045 8 ) ( 1 O , 800) = 495 psi .
B . Choose a pressure gauge sensitive enough to respond to the
expected pressure change during the test . An ordinary Bourdon tube
gauge is sufficiently sensitive for this test , but any gauge with greater
sensitivity is acceptable .
C . Estimate the maximum pressure to be encountered in the test
(e. g . , in a new reservoir, P m ax =P i ) . For this test , we anticipate
P i = 3 ,200 psia.
D. Choose a pressure-gauge range so that the maximum test pres­
sure falls between 60 % and 80 % of the upper limit of the gauge ,
or 4 , 000 and 5 , 300 psi .
E. Choose a 1 80-hour clock for the pressure gauge .
In summary , before shut-in the operator should produce the well
for 1 month at P wj = 400 psia and qg = 1 , 500 Mscf/ D . In addition,
the pressure-buildup test should last 2 weeks . During the flow and
buildup test s , wellbore-storage effects will last about 6 hours .
Neither boundary effects nor pseudoradial flow will be achieved
during the test . Therefore , semilog analysis of the data is impossible.
8.4.7 General Design Procedures-Multiwell Tests. Interference
Test Design . In designing an interference test , our obj ective is to
obtain a sufficiently large pressure drawdown at the observation
well within an acceptable period of time . As a general guideline ,
we require a pressure drawdown , ilp, of several pounds per square
inch with a test duration of no more than about 1 day . Although
pressure gauges are sensitive enough to detect a pressure drawdown
of a fraction of 1 psi , we cannot be certain whether such a small
pressure response results from production or injection at the active
well or simply from reservoir " noise" from other wells in the
reservoir.
Test Design Procedure.
1 . Estimate the well and reservoir properties required for the test
design .
A . Select the flow rate for the interference test. Using the methods
given in Sec . 8 . 4 . 5 , ensure that thi s rate is sufficient to lift liquids
continuously from the wellbore for gas wells producing liquid s .
B . Estimate P i (or ji) and P wj (ilt = O) . Pos s ible sources include
data from a previous DST or pressure surveys from active wells
completed in the same formation .
C . Estimate fluid properties (i . e . , fig , iig , and ct for gas wells)
at ji. In the absence of laboratory data , correlations generally are
adequate for estimating fluid propertie s .
D . Estimate k, h , and r/> . k can be estimated with the suggestions
given in Sec . 8 . 4 . 1 . The best sources for estimating h are well log s ;
r/> estimates c a n b e obtained from either cores or well log s .
active well to the observation well s .
ri
rD = r/rw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8 . 35)
kh
E . F o r interference testing , estimate the distances ,
2 . Estimate the flowing time required t o achieve the design
For interference testing ,
r,
from the
� ( ::: )
Ei
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8 . 37)
5. U sing the definition of dimensionless pressure given by Eq .
8.38,
PD =
kh( P i -P r )
1 4 1 . 2qBp.
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8 . 3 8)
calculate the pressure drawdown , ilp :
P i -P r = ilp =
1 4 1 . 2qBp.P D
kh
.
. . . . . . . . . . . . . . . . . . . . . . ( 8 . 39)
If ilp is less than several pounds per square inch for a 24-hour
production period , then an interference test may not be effective
in this reservoi r . In this case , we may consider running a pulse test
in which the allowable pressure drawdown can be much smaller
for a given test duration .
6. An important component in well-test design is selecting a pres­
sure gauge with the proper sensitivity for the test . Use the recom­
mended procedure g iven for single-well tests to estimate gauge
sensitivity .
Example 8.4-Interference Test Design. We want to conduct an
interference test using one active well and one observation well 10cated 600 ft from the active well . Estimate the expected pressure
drawdown at the observation well after 24 hours of production . De­
termine whether an interference test is appropriate for this reservoir.
k = 50 md .
iig = 0 . 0 1 cp o
= 0 . 3 ft .
q = 10 MMscflD .
rw
r
fig =
=
h =
ct =
r/> =
1 . 0 RB/Mscf.
600 ft .
2 5 ft .
3 x l O - 4 psi - I .
0.20.
Solution. W e have estimates o f the well and reservoir proper­
ties , so we begin our design by calculating the minimum flowing
time required for
to reach the observation well 600 ft from the
active well .
1 . Estimate the flowing time required to achieve the design ri o
For interference testing , ri is the distance from the active well to
ri
the observation well .
tm in =
948r/>iig Ct
k
= 4 . 1 hours .
r[
(948)(0 . 20)(0 . 0 1 ) (3 . 0 x 1 0 - 4 ) (600) 2
50
205
DESIGN A N D I M PLEM ENTATION OF GAS-WELL TESTS
..
.--,
u
�
.J
W
o
0.
<l
OJ
a
::>
..
.J
Q.
:2
<!
OJ
a
::>
>­
J
Q.
�
OJ
'"
Z
OJ
'"
z
a
Q.
'"
OJ
a:
�
'"
OJ
a:
OJ
OJ
'"
.J
::>
Q.
j
2
�
10'"
Fig. 8.2-Relationship between time lag and amplitude for the
first odd pulse.
Therefore , if the test is conducted for 24 hours , the pressure tran­
sient will be detected at the observation well .
2 . Using Eqs . 8 . 35 and 8 . 36 and the estimated well and reser­
voir parameters , calculate the dimensionless radius and time :
600
rD = - = - = 2 , OOO
0.3
and
0 .0002637kt
tD = -----
¢J/i g ct ra
tD
"2"
rD
"
<l
w
1
P D = - - Ei
2
1 .46.
�
�'"
0.0020
OJ
'"
Z
OJ
a:
OJ
'"
..J
::>
Q.
O.CO I S
0.0010
Our estimate of A p = 7 . 7 psi at t = 24 hours meets the criterion
that A p be no less than several pounds per square inch during a
test less than 1 day long . Therefore, we can run an interference
test in this reservoir and be somewhat confident that the pressure
data we obtain can be distinguished from reservoir noise . If some
of the parameters had been different (e. g . , larger k or larger
then A p might have been too small by t = 24 hours for the interfer-
r),
=0. 17,
�
�.J
W
o
"
<l
0.0030
:::;:
<!
0.002�
�
OJ
�
�
'"
.. . . .
. .
\oJ
a:
OJ
�
'
1 "
10-I
( TIME I..A Gl/cCYCI.. E L E NGTH I.
5
I L /lIle
1
7 . ,
Fig . S.4-Relationship between time lag and amplitude for the
first even pulse.
�
ii'
••
O.OO3S
�;J
>-
,
Q.OO4II ���-
:1 0.0040
o.003�
0.OO2�
J
Q.
4tD
1
1
= - - Ei ( - 0 . 1 7) = - ( l . 35 8) = 0 . 68 .
2
2
(50)(25)
(2 , 000) 2
4(5 . 86 x 1 0 6 )
0.0030
--
(0.20)(0 . 0 1 )(3 x 1 0 - 4 )(0 . 3) 2
5 . 86 x 1 0 6
----
0
::>
>-
( -rl )
from which we calculate
1 4 1 . 2 ( 1 0 , 000)( 1 ) (0 . 0 1 )(0. 68)
<J 0.0040 . ' :
.J
II-Idle
0 . 0002637(50)(24)
N
�
LAG I /(CYCI.. E LENGTH ) .
• •
= 7 . 7 psi.
(2 , 000) 2
�
10-1
4 . Using Eq. 8 . 39 with P D = 1 .68 , the pressure drawdown at the
observation well after t = 24 hours is
3 . We can use tables of Ei functions in Appendix C with a value of
4tD
I ,
Fig. 8.3-Relationship between time lag and amplitude for all
odd pulses except the first.
= 5 . 86 x 1 0 6 .
Thu s ,
7
( TIME
(TIME I.. AGI/CCYCl..E I..ENGTH). II-Idle
r
rw
,
0.0 0 1 !!)
0.00 1 0
; '
10-'
( TIME LAGI/(CYCLE LENGTH I .
IL/dle
Fig. 8 .S-Relationship between time lag and amplitude for all
even pulses except the first.
206
N
GAS RESERVOI R E N G I N E E R I N G
N
0
O
--::..
0
oJ
'-
�
oJ
<5
<t
...J
W
::!i
i=
'"
'"
W
.J
Z
-
<t
...J
.;
O. 1 2 !5
w
::!i
i=
0,100
'"
'"
W
...J
z
Q
Z
Q
Z
'"
'"
w
::!i
a
w
�
0.0 5 0
0
10- 1
( TI M E L.AGl/(CYCL.! L.ENGTH ) .
.0- 1
tJlote
(TIME L.AGI/(CYCl.! L.!NGTH I . tL/tote
Fig. 8 . 6-Relationship between time lag and cycle length for
the first odd pulse.
Fig. 8.7-Relationship between time lag and cycle length for
all odd pulses except the first.
ence test data to be useful . In that case, we would have recommended
running a pulse test instead .
2. Select Fig . 8 . 2 for the first odd pulse and Fig . 8 . 3 for all other
odd pulses , or Fig . 8 . 4 for the first even pulse and Fig . 8 . 5 for
all other even pulses . On the appropriate graph, read the maximum
value of Il P D (tLIllte) 2 and the corresponding value of tLIllte from
the chosen F' curve .
3 . Select Fig . 8 . 6 for the first odd pulse and Fig . 8 . 7 for all other
odd pulses , or Fig . 8 . 8 for the first even pulse and Fig . 8 . 9 for
all other even pulses . On the appropriate graph using the chosen
F' curve , read the value of
corresponding to the value of
tLIllte obtained from Step 2 .
4 . Using the best estimates available for k , cp , jig , and ct ' along
with the value of twlr obtained from Step 3 , calculate the lag
time , tL :
Pulse- Test Design . Designing a pulse test 2 9 requires the pulse
times and the expected pressure response to be determined . Esti­
mation of the expected pressure response before a pulse test is run
is important for the range and sensitivity of the pressure gauge and
the test duration to be specified properly . The test design proce­
dure is based on maximizing the amplitude response for a pulse
while minimizing the cycle length . Figs. 8.2 through 8.9 are used
for pulse-test design.
Test Design Procedure.
I . Select the pulse ratio , F' . If a specific ratio is most convenient
for a particular application, then use that value. Otherwise, F' should
be based on whether the odd or even pulses will be analyzed . The
choice of odd or even is arbitrary . To maximize the amplitude of
the pressure response, II p , to the odd pulses, F' should be 0 . 7 (Figs .
8 . 2 and 8 . 3 ) . To maximize the amplitude of the even pulses , F'
should be 0 . 3 (Fig s . 8 . 4 and 8 . 5) . In no case should F' be less than
0 . 2 or greater than 0 . 8 .
N
twlr8
8
tL =
(twlr8) Pig .
2
8 . 5 r cp jig Ct
0 .0002637k
. . . . . . . . . . . . . . . . . . . . .
(8 . 40)
tL is defined as the time between the end of the pulse and the
pressure peak caused by that pulse .
NO
O
'-
�
"
.!2
-..;'
..;'
<t
<I
...J
.;
...J
w
::!i
i=
.;
W
::!i
i=
'"
'"
w
.J
'"
'"
W
.J
Z
Z
0
Q
'"
z
w
::!i
<ii
z
w
::!i
a
a
10 .1
( TI M E L.AG)j(CYCL.E L.ENGTH ) .
O.1 2 �
0. 1 00
0.075
o.o � o
.
��T'�' ::::::'::k3 t · ·
Fig. 8 . 8-Relationship between time lag and cycle length for
the first even pulse.
..
.
­
.
'
: ::' � ?: :O.6 : -::: ? ::C: ::' ·
.
..
:: :::::: 0.7 1 :: ?IS ..:: :
. . . . . : :::: : 0.8 I : 0t':: .. ' :
: : ::+::::: 0:'9 T
: :: ;;;.: :. " ,
. •
ELm.l2tb2tJiUI
�
o,o2 � 3 : · : h$
t L /tote
.•..
� : ·0.4 i · ';;;; ·
:� :: ' 0'.5 : : : ;;;;� :� :
6
1
1 '
1 0 ··
' : : r
! :::�: ·1 : ·
( TIME L.AGI/( CYCL.E L.ENGTH I .
· . . . ..
"
·
1 1 1
t L /to t e
Fig. 8.9-Relationship between time lag and cycle length for
all even pulses except the first.
207
DESIGN AND I M PLEME NTATION OF GAS-WELL TESTS
Using the value of lag time calculated in Step 4 and the value
from Step
calculate the cycle length :
5.
tL ID.te
2,
tL
D.te = (tL ID.te)
fig
6. F'
F' =D.tp ID.te ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 42)
F'
5
D.tp .
D.tp =F' D.te. . . . . . . . . . . . . . . . . . . . ( 43)
7.
D.p.
Ji
,
k,
h,
,
Ji
g
g
D.pv (tL ID.te)2 tL ID.te
2.
(t. r /D.te) 2 ] Fig . 8 . 5 . . . . . . ( . 44)
D.p= 141. 2qkhBg[JL(tgL[Pv
ID.te)2hig . 8 .4
D.p
D.tp .
D.te,
t=D.te -D.tp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 45)
D.p
of
TABLE B.2-EFFECTS OF PERMEABILITY AND
DRAINAGE AREA ON STABILIZATION TIME
k
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8 . 4 1 )
.
is defined as the pulse/cycle length ratio :
8.
Rearranging this equation , we can use the chosen value of
and
the cycle length calculated from Step to estimate the pulse length
(producing time) ,
.
.
. . .
. . .
. . .
.
. .
.
.
8.
Estimate the amplitude of the pressure pulse,
We need
good estimates of q,
and
as well as the dimension­
less groups
and
from Step
.
.
.
. .
. .
8.
The choice of pressure-gauge sensitivity should be based on the
calculated
value .
Example 8.5-Pulse-Test Design for Analysis of First Even
Pulse. Design a pulse test , based on analyzing the first even pulse ,
for the wells and reservoir with the estimated properties given be­
low . Determine the pulse length and the gauge sensitivity required
for a pulse test that will provide better estimates of and
k cpc,.
Jigkr === 0.206600 1
251.1
Jihg === 30,000
= 03 1 4
q
ct
Solution .
1.
2.
D.te.
[D.PV (tLID.te)2 hig . 8.4 =0. 0042
(tLID.te) Fig . 8. 4 =0.33.
3.tLDlrb
tLID.te =0.33 F' =0.3,
(tLDlrb ) Fig . 8 . 5 =0.122 tLID.te=0. 3 3.
4.
g . 8. 5 r2cpJig Ct
tL (tLDlr0.b )0Fi002637k
(0.122)(660)2 (0.18)(0.01)(3 10 -4 )
0. 0002637(20)
=5.4
=0.3
and
of
and
at
From Eq . 8 . 40 ,
x
hours .
40
640
40
640
40
640
40
640
40
640
40
640
25 ,953 (3 years)
41 5 ,242 (47 years)
2 , 595 (1 08 days)
4 1 , 524 (4. 7 years)
259. 5 (1 0.8 days)
4 , 1 52.4 ( 1 73 days)
25.95 (1 . 1 days)
41 5.2 (1 7.3 days)
2.59 (0. 1 1 days)
41 .52 (1 . 73 days)
0.259 (0. 0 1 1 days)
4. 1 5 (0. 1 73 days)
From Eq. 8 . 4 , we estimate
(tL ID.te) Fig . 8.4
=5.4/0. 3 3 = 16.4
6.
D.tp =F'D.te =(0. 3 )(16. 4)=4.9
D.p 14 1. 2qJigJig [Pv (tL ID.te)2 ] Fig. 8. 5
hours .
From Eq. 8 . 43 , the pulsing period is estimated to be
hours .
7 . From Eq. 8 . 44 ,
141.2(30,000)(1.1)(0. 0 1)(0. 0042)
(20)(25)(0.33)2
=3. 6
D.tt=D.te
p =4.9- D.tp =11.5
D.p
psi .
8.5 Dellverability Test Design
F' =0.3
8 . 5 at
0.01
0.01
0.1
0.1
1 .0
1 .0
1 0. 0
1 0 .0
1 00.0
1 00.0
1 ,000.0
1 ,000.0
A . Shut i n the well for
hours .
B. Produce the well for
hours .
C . Use a pressure gauge with at least a O . l -psi resolution (to per­
mit detection of
to one decimal place) .
Select
to maximize the pulse response amplitude in
even pulse s . (We could just as well have chosen to analyze the odd
pulse s . )
W e want t o maximize the amplitude response for the first even
pulse , while minimizing the cycle length ,
From Fig . 8 . 4 , for
the first even pulse , we choose the maximum point on the F'
curve. The coordinates o f this point are
From Fig .
of
(hours)
8 . In summary , our test design is as follows .
ft .
cp o
md .
ft .
RB/Mscf.
Mscf/D .
. 8.
x l O - psi - I .
=
ts
(acres)
8
Note that
is directly proportional to q and can be increased
by increasing q .
8 . In summary , w e shut in the well for a time equal t o the pulse
length ,
The total cycle length is
so we produce for a time
cp
5.
A
(md)
we read a value
Many of the design considerations presented for pressure-transient
tests in Sec . 8 . 4 are also applicable to deliverability testing . Un­
like pressure-transient tests, most analysis techniques for delivera­
bility tests require data obtained under stabilized flowing conditions .
Therefore, a critical component in designing well-deliverability tests
is estimating the time required to reach such flowing conditions .
I n this section , w e discuss well stabilization time and introduce the
concept of drainage radius for designing deliverability tests . In ad­
dition , we discuss selection of the type of test and the flow require­
ments for each test .
8.5. 1 Stabilization Time. The analysis techniques for conventional
flow-after-flow tests require stabilized flowing conditions. Although
isochronal and modified isochronal tests were developed to circum­
vent the long stabilized flow times required in low-permeability
reservoirs , they may require a single , stabilized flow period at the
end of the test . Stabilization time is defined as the time when the
flowing pressure is no longer changing or is no longer changing
significantly . Physically , stabilized flow can be interpreted as the
time when the pressure transient is affected by a no-flow bounda­
ry , either a natural reservoir boundary or an artificial boundary
created by active wells surrounding the tested wel l .
I n high-permeability reservoirs , the time t o reach stabilized flow­
ing conditions is reasonably short ; however , in low-permeability
formations, the stabilization time can be months or even years , de­
pending on the formation permeability . For example, consider a
well producing a gas with a specific gravity of
from a forma-
0. 6
208
GAS RESERVO I R E N G I N E E R I N G
tion at 2 10°F and an average pressure of 3 ,500 psia (cg =2.468 x
1 0 4 psia - 1 and itg = 0.02 cp) , with a porosity of 10 % . Table
8.2 clearly shows that several day s , if not years , may be required
to reach stabilized flow in low-permeability formations, while high­
permeability reservoirs stabilize very quickly .
Therefore , the design engineer must estimate the time to reach
stabilized flow before designing the test . The duration of the stabi­
lized time can affect not only the validity of the data gathered dur­
ing the test but also the type of deliverability test selected . The
stabilization time , ts , for a well centered in a circular drainage area
is estimated with
-
ts =
948 cp jig c/�
kg
,
t min =
where ts = stabilization time , hours ; cp = total porosity , fraction ;
viscosity evaluated at p , cp; c1 = total compressibility
evaluated at p , psi - 1 ; Te = distance to no-flow boundary , ft; and
kg = effective permeability to gas , md .
The average gas properties are evaluated at the average pressure
in the drainage area of the well .
Similar to the design process for pressure-transient tests , estimat­
ing the stabilization time for deliverability tests requires pretest es­
timates of formation and fluid propertie s . If the average reservoir
pressure at the time of the test is known or can be estimated , gas
properties usually can be obtained with correlations . In addition ,
the distance to the no-flow boundary can be estimated from geo­
logic data or known locations of adjacent producing wells . Final­
ly , the formation permeability can be estimated with the methods
discussed in Sec . 8.4.
The criterion for identifying the beginning o f stabilized flowing
conditions varies within the natural gas industry . For example , the
natural gas regulatory agency in the state of Texas , the Railroad
Commission of Texas , 3 0 defines stabilized flow when two con­
secutive pressure readings over a I S -minute period agree within
0. 1 psi. Similarly , the Interstate Oil Compact Commission 3 1 de­
fines stabilized flow as a constant wellhead flowing pressure or static
column wellhead pressure and rate of flow for at least 15 minutes.
Rather than basing a definition on a prescribed rate of pressure
decline , a better criterion for establishing the beginning of stabi­
lized flow is based on the radius of investigation concept . 1 The
radius of investigation , the point beyond which the pressure draw­
down is negligible, is a measure of the distance a pressure tran­
sient has traveled into a formation following a rate change in a well .
It represents the depth to which the formation properties are being
investigated at any time during a test. The approximate position
of the radius of investigation is estimated with the relation
J 8;�g
94
CI
' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8.47 )
where t = time since last rate change , hours; kg = formation perme­
ability , md; and p = current average drainage area pressure , psia.
Stabilized flowing conditions occur when Ti ?:. T e' As long as Ti
is less than the distance to the no-flow boundary ( Ti < T e ) ' stabili­
zation has not been attained and the pressure behavior is transient.
Therefore, knowing the distance to a no-flow boundary allows us
to estimate the time required to reach stabilized flowing conditions
and to select the appropriate deliverability test .
8.5.2 Estimating Deliverability Test Duration. Analysis tech­
niques for deliverability tests may require data obtained under stabi­
lized flowing conditions . Therefore , the flow periods for the
conventional four-point and single-point tests and the extended flow
points for the isochronal and modified isochronal tests must equal
or exceed the stabilization time estimated by Eq . 8 .46. The dura­
tion of the isochronal periods for the isochronal and modified
isochronal tests , however, is determined by different criteria. Like
the flow periods in pressure-transient tests , the isochronal flow peri­
ods must exceed the time period when wellbore storage distorts the
data. The duration of wellbore storage can be estimated with
(200,000 + 12,000s)C
h
k
/ jig
,
. . . . . . . . . . . . . . . . . . . . . (8. 1 8)
where C is defined by Eq . 8 . 16 or 8 . 17. Ref. 1 suggests a flow
time equal to four times the duration of wellbore-storage effects.
Yet another consideration in selecting the duration of the
isochronal test periods is that the pressure transient must move be­
yond the altered zone near the wellbore . This altered zone can be
either damage from drilling and/or completion operations , or stimu­
lation resulting from acidizing or hydraulic fracturing . For test de­
sign purposes , we recommend using 50 :5 Ti :5 200 ft . The flowing
time required to reach the design T; is
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8.46)
jig = gas
T; =
t wbs =
948 cp jig C1 T r
kg
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8.48 )
If the well is hydraulically fractured , however, then the test must
be designed so that the pressure transient moves beyond the effects
of the fracture and pseudoradial flow is achieved . In other words ,
the dimensionless time based on fracture half-length must equal or
exceed three . If pseudoradial flow is not attained , then the analysis
techniques discussed previously are not valid for hydraulically frac­
tured wells. In summary , the flow time for the isochronal periods
should be the greater value of 4 t wbs and t min ' Flow time in con­
ventional tests also must meet these criteria, of course .
8.5.3 Flow-Rate Requirements. Constant-Rate Production . Con­
ventional deliverability test analysis techniques are based on con­
stant production rates during each flow period . Therefore , the test
should be designed to maintain constant or near-constant rates . In
practice, constant production rates are difficult, if not impossible ,
to attain during a flow test , especially in low-permeability reser­
voirs which take long times to reach stabilized flow . Work by
Winestock and Colpitts 3 2 indicates, however , that the analysis
techniques based on constant-rate production are valid provided the
actual rates are declining slightly and vary smoothly and continu­
ously with time .
Sequence of Flow Rates. Deliverability tests typically are con­
ducted with a sequence of increasing flow rates ; i . e . , each succeed­
ing rate is higher than the previous rate. 1 If truly stabilized flowing
conditions are attained following each rate change during a con­
ventional four-point test , the rate sequence does not affect test in­
terpretation . Either an increasing or decreasing rate sequence will
provide the correct deliverability relationship. To maintain accuracy
during an isochronal or modified isochronal test, however , an in­
creasing rate sequence is recommended .
In addition , the timing of the extended flow period for the
isochronal and modified isochronal tests is immaterial . If a well
is already producing , then the extended flow period can be run at
the beginning of the test. Otherwise , this flow period can be placed
at the end of the test . Note , however, that if the extended test is
conducted at the beginning , the well must be shut in long enough
to reach stabilized conditions before the beginning of the isochronal
flow period s . This shut-in period may be prohibitively long in low­
permeability reservoirs .
A final consideration in the selection of the rate sequence is the
presence of liquids in the wellbore and the formation of hydrates.
If liquid holdup in the production string is a problem , a decreasing
rate sequence is preferred . Methods for predicting the minimum
flow rate required to lift liquids continuously in the wellbore are
discussed in Sec . 8.4.5. Similarly , if hydrate formation is proba­
ble , a decreasing rate sequence is recommended for a convention­
al four-point test.
Minimum and Maximum Flow Rates, Several factors must be
considered when selecting the flow rates for testing the well . 1 The
minimum flow rate should equal or exceed the minimum value re­
quired to lift liquids continuously from the wellbore, especially when
liquid slugging problems are possible . Methods for estimating the
minimum liquid unloading rate are given in Sec . 8 . 4 . 5 . Another
consideration is that the minimum flow rate must be adequate to
maintain a wellhead temperature above the point at which hydrate
formation becomes a problem .
209
DESIGN AND I M PLEM ENTATION OF GAS-WELL TESTS
If hydrate formation and liquid accumulations in the wellbore do
not affect flow-rate selection, Ref. 1 suggests choosing minimum
and maximum rates that produce pressure drops at the well that
equal S % and 25 % , respectively , of the shut-in pressure . Alterna­
tively , if the well ' s absolute open-flow (AOF) potential is known
or can be estimated from correlation with similar wells, another
criterion is minimum and maximum rates of 1 0 % and 75 % , re­
spectively , of the AOF . If reservoir rock and fluid properties can
be estimated , the AOF can be calculated from
q AOF =
_
[ (- )
1 4 1 .2j1g B g ln In
re
rw
3
- - +s'
4
]
. . . . . . . . . . . ( 8 . 49)
The maximum rate should not create excessive pressure draw­
downs so that the well is damaged from water coning or formation
sloughing. Further, the maximum flow rate should not create a pres­
sure drawdown that would cause retrograde condensation in the
reservoir , especially in the area adj acent to the wellbore . Finally ,
if the well is new and has not been connected to a pipeline or pro­
duction facilities, the maximum flow rate should be chosen to
minimize the quantity of gas flared to the atmosphere .
2.
Estimate
ts =
ts :
948cf>jlg ct r;
k
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 8 . 4 6)
g
3 . Using the length of time required to reach stabilized flowing
conditions, select the type of deliverability test. In general , if the
stabilization time is a few hours , a conventional four-point test is
selected . If the stabilization time greatly exceeds a few hours, then
an isochronal or modified isochronal test should be considered . Fi­
nally , when a well' s deliverability characteristics are known or can
be estimated from previous well tests, choose a single-point test.
4. Select the duration of the flow periods for the deliverability
test. For conventional four-point tests, single-point tests , and the
extended flow points in isochronal and modified isochronal tests ,
the flow time must equal or exceed the stabilization time calculat­
ed in Step 2, or t 2! ts and also must exceed the duration of well­
bore storage , twbs '
For the flow periods during isochronal or modified isochronal
tests , select the greater value of 4twbs and tmin '
A . The duration of wellbore-storage effect s , twbs ' is estimated
with
8.5.4 Selection of Deliverability Test. The most important con­
sideration when selecting the deliverability test is the time required
for stabilization . I If only a few hours are needed for stabilization,
a conventional four-point or backpressure test is appropriate. If the
stabilization time greatly exceeds a few hours , either an isochronal
or a modified isochronal test should be considered . The isochronal
test is more accurate than the modified isochronal test and there­
fore should be used when more accuracy is required.
When a well ' s deliverability characteristics are known or can be
estimated from previous well tests , a single-point test is suitable.
Single-point tests are especially appropriate when these delivera­
bility characteristics are being updated , as required by many regula­
tory agencies . A convenient time to conduct a single-point test is
before shut-in for a pressure survey of the field and when the well
to be tested is stabilized . Under these conditions , only measure­
ments of the flow rate and flowing pressure are required .
8.5.5 General Deliverability Test Design Procedures. In this sec­
tion, we offer general procedures for designing deliverability tests
in both unstimulated and stimulated gas wells . Note that these proce­
dures are suggestions and that each test design is unique and will
require different considerations , depending on the testing objec­
tives and environment . In all equations presented in the following
procedure, the gas properties are evaluated at the current average
reservoir pressure , p .
Test Design Procedure.
1 . Estimate well and reservoir properties required for the test
design .
A . Estimate the initial reservoir pressure, P i ' or the current aver­
age reservoir value, p . Possible sources include data from a previ­
ous DST or pressure surveys from active wells completed in the
same formation .
B . Estimate gas properties ( L e . , ii , jig , and ct ) at p . Correla­
g
tions generally are adequate for estimating these properties .
C . Estimate k, h , and cf> . k can b e estimated with the suggestions
given in Sec . 8 . 4 . 1 . The best sources for estimating h are well logs,
while cf> estimates can be obtained from either cores or well logs.
D . Estimate C. For gas , use Eq. 8 . 1 7 . Average wellbore tem­
perature is the arithmetic average of surface temperature and BHT .
For single-phase gas in the wellbore , we may estimate cwb =- Cg at
p and average wellbore temperature .
E . Estimate the drainage radius , re , as the well spacing from
known field development patterns or the distance L to a nearby reser­
voir no-flow boundary . Distances to natural no-flow boundaries may
be estimated with geologic data. In addition , if the well is hydrau­
lically fractured , estimate Lt .
Example 8.6-Designing a Gas-Well Deliverability Test. Using
the following data, design a suitable deliverability to assess the pro­
duction potential of a new gas well . Note that the well is not yet
connected to a pipeline .
cf> =
=
=
=
Pi =
h =
ct =
iig =
=
kg =
jig =
rw
cwb
Vwb
A
0. 1 18.
0 . 365 ft .
2 . 9 x lO - 4 psi - I .
1 5 bbl .
p = 3 ,200 psia.
6 ft.
2 . 0 x l O - 4 psi - I .
1 . 5 RB/Mscf.
640 acres .
1 0 md .
0 . 0 1 5 cp o
21 0
GAS RESERVO I R E N G I N E E R I N G
Solution .
1 . First, estimate the gas and formation properties .
A . Estimates of pressures and gas properties are provided above.
Note that the gas properties (Jig , ct and iI) are evaluated at p . We
'
choose to set s / = 0 for conservative test design .
B . The wellbore-storage coefficient is
C = c wb Vwb = (2 . 9
x
(640 acres)(43 , 560 ft 2 /acre)
2 , 979 ft .
-------
+s/
1 4 1 .2iIg Jig ln
]
In summary , test the gas well using a modified isochronal test
with the duration of the flow periods equalling 1 . 5 hours . A suit­
able flow schedule is 366 , 854 , 1 , 342 , and 1 , 830 MscflD . During
each flow period , we recommend measuring the flowing pressure
at 0 . 2 5 , 0 . 5 , 1 . 0 , and 1 . 5 hours .
x
10 -4 )(2 , 979) 2
10
8.6 Summary
= 298 hours o r 1 3 day s .
3 . Because of the long stabilization time , we choose to conduct
a modified isochronal test .
4 . Select the duration of the flow periods for the deliverability
test . For the flow periods during modified isochronal tests , we
should select the greater value of 4t wbs and t
A . Assuming s / = 0 , estimate t wbs :
min .
(200,000 + 1 2 ,000s / ) C
[200 ,000 + ( 1 2 , 000)(0)] (0. 00435)
( 1 0)(6)/(0 . 0 1 5)
= 0 . 22 hour.
B. Estimate the flowing time required to achieve the design
50 :s; ri :S; 200 ft. For this design, use r; = 200 ft.
r; ,
948cJ>iIg ct r T
(948)(0 . 1 1 8)(0 . 0 1 5)(2 . 0
x
10 -4 )(200) 2
10
= 1 . 34 hours .
tmin ;
Now , select the greater value of 4t wbs and
i . e . , choose a
flow time for the isochronal periods equal to t = 1 . 34 rounded up
to t= 1 . 5 hours .
5 . Calculate the minimum and maximum flow rates :
=
Reading this chapter should prepare you to do the following tasks :
• Be able to determine which well test (pressure-transient or
deliverability) will achieve your objectives based on the type and
status of the well to be tested , reservoir properties, and safety and
environmental considerations .
• Be able to estimate pretest values of skin factor and formation
permeability .
• Be able to estimate test duration for any reservoir model
(homogeneous , hydraulically fractured , and/or naturally fractured) .
• Be able to calculate the radius of investigation .
• Be able to determine the appropriate flow rate for the well test
considering the cases of low permeability and liquid loading .
• Be able to design prefracture pressure-drawdown and -buildup
tests .
• Be able to design postfracture pressure-drawdown and -buildup
tests .
• Be able to design interference tests.
• Be able to design pulse tests .
• Be able to design deliverability tests.
Questions f o r Discussion
tmin = kg
[ ( )-� ]
1 4 1 .2iIg Jig ln
where
-"::-
= 1 , 830 MscflD .
(948)(0 . 1 1 8)(0 . 0 1 5)(2 . 0
�P min
and �P max
[ ( ::) -� ]
kgh(�Pmax)
----
[( )
kg
or
=
2 , 979
3
( 1 4 1 .2)(0 . 0 1 5)( 1 . 5) In -- - - + 0
0. 365
4
948 cJ> iIg Ct r�
qgmin qgmax
qgmax
( 1 0)(6)(800)
2. Estimate ts :
t wbs =
and
10 - 4 ) ( 1 5) = 0 .00435 bbllpsi .
C . For 640-acre spacing and a circular drainage area,
ts =
= 3 6 6 MscflD,
�
rw
4
,
+s/
= 0 . 05p = (0 . 05)(3 ,200) = 1 60 psi
= 0 .25p = (0 . 25)(3 ,2oo) = 800 psi.
[( )
( 1 0)(6)( 1 60)
( 1 4 1 .2)(0 . 0 1 5)( 1 . 5)ln
3
2 ,979
-- - - + 0
0. 365
4
]
1 . For the following well conditions and test objectives, select
the most appropriate well test (pressure-transient , multiwell , or
deliverability test) . In each case , list the data you would require
for the test design , the possible sources of these data , and any test
limitations .
A . Determine permeability and skin factor for a low-permeability
gas well that has been acidized .
B . Determine whether a fault exists near a gas well drilled in
the Devonian Shale (naturally fractured) formation .
C . Determine the average pressure and permeability of a deep
gas well that has been hydraulically fractured .
D . Determine wheth