A Case Study on the Optimization of Hydraulic Horsepower for

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A CASE STUDY ON THE OPTIMIZATION OF HYDRAULIC HORSEPOWER FOR
EFFICIENT BOTTOM HOLE CLEANING IN DRILLING
By
Orogun Humphrey Onome
Submitted in Partial fulfillment of the requirements for the degree of Masters of Engineering
Major Subject: Petroleum Engineering
At
Dalhousie University
Halifax, Nova Scotia
December, 2013
© Copyright by Orogun Humphrey Onome, 2013
i
DALHOUSIE UNIVERSITY
PETROLEUM ENGINEERING
The undersigned hereby certify that they have read and recommend to the Faculty of Graduate
Studies for acceptance a thesis entitled “A CASE STUDY ON THE OPTIMIZATION OF
HYDRAULIC HORSEPOWER FOR EFFICIENT BOTTOM HOLE CLEANING IN
DRILLING” by Orogun Humphrey Onome in partial fulfilment of the requirements for the
degree of Master of Engineering.
Dated: December 2, 2013
Supervisor:
_________________________________
Reader:
_________________________________
ii
DALHOUSIE UNIVERSITY
DATE:
AUTHOR:
TITLE:
December 2, 2013
Orogun Humphrey Onome
A CASE STUDY ON THE OPTIMIZATION OF HYDRAULIC
HORSEPOWER FOR EFFICIENT BOTTOM HOLE CLEANING IN
DRILLING
DEPARTMENT OR SCHOOL:
DEGREE:
MEng.
Petroleum Engineering
CONVOCATION:
May
YEAR:
2014
Permission is herewith granted to Dalhousie University to circulate and to have copied for noncommercial purposes, at its discretion, the above title upon the request of individuals or
institutions.
_______________________________
Signature of Author
The author reserves other publication rights. Neither the thesis nor extensive extracts from it may
be printed or otherwise reproduced without the author’s written permission. The author attests
that permission has been obtained for the use of any copyrighted material appearing in the thesis
(other than the brief excerpts requiring only proper acknowledgement in scholarly writing), and
that all such use is clearly acknowledged.
iii
DEDICATION
This project report is dedicated to my lovely family, Mr. and Mr.’s Dennis Orogun, and to my
siblings Desmond, Jane and Micheal Orogun for their support and understanding in making this
project work a huge success. Also to my roommate and course mate in the department for their
respective input.
iv
TABLE OF CONTENTS
LIST OF TABLES ...................................................................................................................... xi
LIST OF FIGURES .................................................................................................................... xii
NOMENCLATURE ................................................................................................................... xiii
ACKNOWLEDGEMENTS ...................................................................................................... xix
ABSTRACT ..................................................................................................................................xx
CHAPTER 1: INTRODUCTION .................................................................................................1
1.1 Study Objective ....................................................................................................................1
1.2 Problem Statement ...............................................................................................................1
1.3 Scope of Study .....................................................................................................................1
1.4 Limitations ...........................................................................................................................2
CHAPTER 2: LITERATURE REVIEW ....................................................................................3
2.1 Geomechanics ......................................................................................................................4
2.2 Pore Pressure ........................................................................................................................4
2.3 Causes of Over Pressure ......................................................................................................4
2.3.1 Depositional Effects ..................................................................................................5
2.3.2 Digenetic Processes ..................................................................................................5
2.3.3 Tectonic effect ..........................................................................................................5
2.3.4 Structural Causes ......................................................................................................6
2.3.5 Thermodynamic Effects ............................................................................................6
2.4 Fracture Pressure ..................................................................................................................6
2.5 Mud Weight Planning .........................................................................................................7
2.6 Factors Affecting Rate of Penetration..................................................................................7
2.6.1
Chip Hole Down Effect ...........................................................................................7
2.6.2
Effect of Bit Type ....................................................................................................8
2.6.3
Effect of Drilling Fluid Properties ..........................................................................9
2.6.4
Effect of Formation Characteristics .........................................................................9
2.6.5
Effect of Operating Conditions ................................................................................9
v
2.7 Factors Affecting Hole Cleaning ........................................................................................12
2.7.1
Drill Pipe Rotation ..........................................................................................…..13
2.7.3
Rheology ............................................................................................................…13
2.7.4
Drilling Rate.......................………………………………………………………13
2.7.5
Cutting Bed Characteristics…...…………………………………………………14
2.7.6
Hydraulics…………………………………………………………………...…...14
CHAPTER 3: PORE PRESSURE AND FRACTURE PRESSURE PREDICTION………15
3.1 Methods of Pore Pressure Predictions…………………………………………...............15
3.1.1
Resistivity Method of Pore Pressure Prediction…………………………………15
3.1.2
Sonic Methods of Pore Pressure Prediction……………………………………...17
3.1.3
Equivalent Depth Method……………………………………………………......18
3.1.4
Ratio Method.…………………………………………………………..…...…...19
3.1.5
Neutron Porosity Log…………………………………………………………….20
3.2 Methods of Fracture Pressure Predictions……………………………………………….22
3.2.1
The Hubbert and Willis Approach……………………………………………….23
3.2.2
The Mathews and Kelly Correlation…………………………………………......24
3.2.3
The Eaton’s Correlation………………………………………………………….25
3.2.4
The Macpherson and Berry Correlation……………………………………........25
3.3 Detection of Over Pressured Zone………………………………………………………..26
3.3.1
Detection of Over Pressure using Resistivity log………………………………...26
3.3.2
Detection of Over Pressure using Interval Transit Time…………………………29
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CHAPTER 4: OPTIMIZATION OF DRILL BIT HYDRAULICS.......................................31
4.1 Hydraulic Power Requirement…………………………………………………………...31
4.1.1
Surface Connection Pressure Drop……………………………………………....32
4.1.2
Drill String Pressure Drop…...……………………………………………….….33
4.1.2.1 Newtonian Fluid………………………………………………………….34
4.1.2.2 Bingham Fluid……………………………………………………….......35
4.1.2.3 Power Law Fluid………………………………………………………...37
4.1.3
Annulus Pressure Drop………………………………………………………......38
4.1.2.1 Newtonian Fluid…………………………………………………………38
4.1.2.2 Bingham Fluid……………………………………………………...........39
4.1.2.3 Power Law Fluid…………………………………………………………40
4.1.2
Drill Bit Pressure Drop………………………………………………………......40
4.2 Flow Exponent and Optimum Flow Rate…………………………………………..........42
4.3 Drill Bit Hydraulic Horsepower Criterion…..…………………………………………...42
4.4 Hydraulic (Jet) Impact Force Criterion…………………………………………………..45
4.4.1 Shallow Well Bore Formation…………………………………………………....46
4.4.2 Deep well Bore Formation ……………………………………………………….49
4.5 Bit Nozzle Selection……………………………………………………………………..52
4.6 Drill Cuttings Transport………………………………………………………………….53
4.4.1 Cutting Slip Velocity……………………………………………………………..53
4.4.2 Annular Fluid Velocity…………………………………………………………...53
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4.4.3 Flow Regime……………………………………………………………………...54
CHAPTER 5: DESIGN METHODOLOGY………………………………………………...55
5.1 Estimation of Pore Pressure……………………………………………………………...56
5.2
Estimation of Fracture Pressure……………………………………………….................57
5.3 Mud Weight Selection…………………………………………………………………...58
5.3.1 Geological Mud Specific for Pore Pressure Line………………………………...58
5.3.2 Geological Mud Specific gravity for Fracture Pressure Line…………………….59
5.3.3 Design Mud Specific Gravity for Pore Pressure Line…………………………....59
5.3.4 Design Mud Specific Gravity for Fracture Pressure Line…………….….............59
5.4 Drilling Mud Rheology………………...…………………………………………...…....61
5.4.1
Introduction ……………………………………………………………………...61
5.4.2 Objective of Experiment……………………………………………………….....61
5.4.3 Equipment Used…………………………………………………………………..62
5.4.4 Mud Mixture……………………………………………………………………...63
5.4.5 Experimental Procedure…………………………………………………………..65
5.4.6 Composition of Mud additives Used……………………………………………..66
5.5 Pressure Drop Computation……………………………………………………...............68
5.5.1 Maximum and Minimum Flow Rate Calculation………………………………...68
5.5.2 Calculating Pressure Drop in Drill String……………………...…………………70
5.5.2.1 Pressure Drop Across the Drill Pipe ……………………...……………..70
5.5.2.2 Pressure Drop Across the Drill Collar…………………………………...72
viii
5.5.3 Calculating Pressure Drop in Surface connection………………………………..74
5.5.4 Calculating Pressure Drop in Annulus…………………………………................74
5.5.4.1 Pressure Drop Across the Annulus of the Drill Pipe ……………………74
5.5.4.2 Pressure Drop Across the Annulus Drill Collar………………………….77
5.6 Optimization using the Maximum Bit Horsepower Criterion…………………………...78
5.6.1 Optimum Flow Rate to Operate the Mud Pump………………………………….79
5.6.2 Optimum Nozzle Area of the Drill Bit…………………………………………...81
5.6.3
Maximum Hydraulic Horse Power on the Drill Bit……………………………..83
CHAPTER 6: RESULTS AND DISCUSSION………………………………………….......84
6.1 Frictional Pressure Loss of Mud Samples Result……………………………..................84
6.1.1 Using Mud Sample 1 ……………………………………………………….........84
6.1.2 Using Mud Sample 2……………………………………………………………..85
6.1.3 Using Mud Sample 3……………………………………………………………..86
6.2 Result for Mud Pump Operating Conditions………………………………….................87
CHAPTER 7: CONCLUSION AND RECOMMENDATION………………………..........88
7.1 Conclusion………………………………………………………………………………..88
7.2 Recommendation…………………………………………………………………………88
REFERENCES……………………………………………………………………………….....89
APPENDIX……………………………………………………………………………………...93
APPENDIX A……………………………………………………………………………………93
APPENDIX B……………………………………………………………………………………94
APPENDIX C……………………………………………………………………………………95
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APPENDIX D……………………………………………………………………………………96
APPENDIX E……………………………………………………………………………………97
APPENDIX F………………………………………………………………………………..…..98
APPENDIX G………………………………………………………………………………..…..99
APPENDIX H………………………………………………………………………………..…100
APPENDIX I………………………………………………………………………………..….100
APPENDIX J………………………………………………………………………………..…109
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LIST OF TABLES
Table 4.1: IADC Classes of Surface Equipment…………………………………………….....32
Table 5.1: Shale Conductivity Data……..…………………………………………………..….55
Table 5.2: API Densities of Mud Additives………...…...…………………………………...…64
Table 5.3: Composition of Mud Sample 1……………………………..…………………...…..66
Table 5.4: Composition of Mud Sample 2…………………………………..………….............67
Table 5.5: Composition of Mud Sample 3……………………………………………………...67
Table 5.6: Optimum Nozzle Area and Size Across Each Depth in the Over Pressure
Zone………………………………………………………………………….……...82
Table 5.7: Optimum Hydraulic Horsepower Across the Drill Bit in the Over-Pressure
Zone…………………………………………………………………………………83
Table 6.1: Frictional Pressures Losses in the Mud Circulatory System Using Mud Sample 1...84
Table 6.2: Frictional Pressures Losses in the Mud Circulatory System Using Mud Sample 2...85
Table 6.3: Frictional Pressures Losses in the Mud Circulatory System Using Mud Sample 3...86
Table 6.4: Optimum Hydraulic Conditions for Case Study……………………………...…......87
xi
LIST OF FIGURES
Figure 2.2:
Chip Hole Down effect…………………...………………………………….......….8
Figure 2.3:
Effect of weight on the drill bit…………………………………………………....10
Figure 2.4:
Effect of rotary speed on the rate of penetration………………………..................11
Figure 3.2:
Plot of porosity with depth………………………………………………...………18
Figure 3.3:
Plot of porosity dependent parameter with depth……………………………….....20
Figure 3.4:
Plot of porosity and pressure versus depth………………………………………...21
Figure 5.1:
Semi log plot of shale resistivity versus depth………………………………….....56
Figure 5.2:
Graph of specific gravity versus depth……………………………………….……60
Figure 5.3:
Model 35 Viscometer………….………………………………………….….……60
Figure 5.4:
Model 140 Mud Balance ………………………………….……………………....63
Figure 5.5:
Critical Reynolds number for Bingham plastic fluids………………………...…...70
Figure 5.6:
Hydraulic log-log plot of parasitic pressure losses with flow rate…………...…....80
xii
NOMENCLATURE
ROP
Rate of Penetration
CHDP
Chip Hold Down Pressure
PDC
Polycrystalline Diamond Compacts
IADC
International Association of Drilling Contractors
API
American Petroleum Institute
WOB
Weight on bit
RPM
Revolution per Minute
db
Bit Diameter
W0
Threshold Bit Weight
K
Constant of Proportionality
S
Compressive Strength of the Rock
a5
Bit Weight Exponent
t
Interval Transit Time (sec)
PP
Pore Pressure (psia)
 ov
Overburden Stress or Pressure (psia)
PPN
Normal pore pressure (psia)
tN
Normal Pore Pressure Trend Line Interval Transit time value at the point of
Interest (s)
tA
Observed value of Interval Transit time at the depth of interest (sec)
Ppeq
Pore pressure at the equivalent depth (psia)
Deq
Equivalent Depth (ft)
G
Over Burden Pressure Gradient (psia/ft)
D
Depth (ft)
DCN
Normal Trend line of d Exponent
DCO
Observed d Exponent
ΦD
Porosity at a given Depth
xiii
ϕ0
Porosity in the mudline
Z
True Vertical Depth (ft)
K
Porosity Decline Constant
C
Compaction Constant
Pf
Fracture pressure (psia)
 min
Minimum principal stress (psia)

Poisson’s ratio
h
Horizontal stress (psia)
F
Stress Coefficient
(  ma )
Vertical stress (psia)
Kb
Elastic Modulus (psia)
b
Bulk Density (Ibs/gal)
RD
Resistivity at a Reference Depth (Ω)
R0
Resistivity at the Surface (Ω)
Hh
Hydraulic horse power (hp)
P
Pressure (psia)
Q
Flow rate (gpm)
Pf
Frictional Pressure Loss
( Pf ) s
Surface Pressure Loss (psi)
Ks
Surface Pressure Coefficient
m
Mud Density (lb/gal)
Pfdp
Pressure drop in the drill pipe (psia)
Lse
Surface equipment equivalent length of drill pipe (ft)
Ldp
Length of drip pipe (ft)
xiv
Ldc
Length of Drill Collar (ft)
Va
Average velocity (ft/sec)
p
Plastic Viscosity (cp)
Di
Internal diameter (in)

Pipe Roughness (in)
N RP
Reynolds number
f
Frictional factor
y
Yield Point (Ibs/100ft)
N He
Hedstrom number
N Re
Reynolds Number
Dop
Outer diameter of the drill pipe or drill collar (in)
Dh
Diameter of the hole (in)
Dp
Diameter of the Pipe (in)
De
Equivalent Diameter of the Drill Collar (in)
N
Power law index
K
Equivalent centipoise
PB
Pressure drop across the drill bit (psia)
A
Area of Nozzle (in2)
Cd
Discharge coefficient
Pdc
Pressure Drop across Drill Collar (psia)
xv
Pfadp
Pressure Drop in Annulus around Drill Pipe (psia)
Padc
Pressure Drop in Annulus around Drill Collar (psia)
( Pf ) D
Parasitic Pressure Loss (psia)
m
Flow Exponent
C
Constant that depends on mud flow properties, hole geometry and
Pipe geometry
H HB
Drill Bit Hydraulic Horse Power (hp)
Pmax
Maximum Pump Pressure (psia)
P 
Optimum Parasitic Pressure Drop (psia)
PBopt
Optimum Pressure Drop on the Drill Bit (psia)
HHPopt
Optimum hydraulic horse power at the drill bit (hp)
FJ
Jet Impact Force (Ibf)
H hp
Maximum Pump Hydraulic Horse Power (hp)
 At opt
Optimum Nozzle area (in2)
Qopt
Optimum Flow Rate (gpm)
Qmax
Maximum flow rate (gpm)
Qmin
Minimum flow rate (gpm)
 min
Minimum Annular velocity (ft/sec)

Volumetric Efficiency
d N opt
Optimum Nozzle Diameter (in)
Rn
Normal Shale Resistivity (Ω)
f Dopt
xvi
R0
Observed shale Resistivity (Ω)
 pp
Density from Pore Pressure (Ibm/gal)
SG pp
Specific Gravity from Pore Pressure
w
Density of water (Ibm/gal)
 fp
Density from Fracture Pressure (Ibm/gal)
SG fp
Specific Gravity from Fracture Pressure
 dpp
Design density from pore pressure (ibm/gal)
SGdpp
Design Specific gravity from pore pressure
 dpf
Design Density from Fracture Pressure (Ibm/gal)
SGdfp
Design Specific gravity from fracture pressure
 mix
Density of the mud mixture (g/cm3)
M1
Mass of Barite (g)
M2
Mass of Bentonite (g)
M3
Mass of Water (g)
1
API Density for Barite (g/cm3
2
API Density for Bentonite (g/cm3)
3
API Density for Water (g/cm3)
PV
Plastic Viscosity (cP)
YP
Yield Point (ib/100ft)
xvii
 600
Dial reading at 600 rpm
300
Dial reading at 300 rpm
xviii
ACKNOWLEDGEMENT
I would like to express my gratitude to my project supervisor Dr. Michael Pegg, for his
contribution to this project work. I am also very thankful to my reader Dr. Steven Kuzak for
taking the time to go over my work.
I would also like to express my profound thanks and appreciation to Mr. Mumuni Amadu for his
input, advice and support during the course of this project and to Mr.Matt Kujath for his
assistance in using the laboratory equipment.
xix
ABSTRACT
Drill cuttings in the well bore cause wear and tear to the drill string and this reduces the rate of
penetration; therefore, there is need for efficient bottom hole cleaning. During drilling operation,
optimization of hydraulic horsepower at the drill bit is adopted to enhance bottom hole cleaning
and to increase the rate of penetration. Optimum drilling conditions are achieved using either the
maximum horsepower criterion or the hydraulic jet impact force criterion.
This project work focused on the application of optimization using the maximum horsepower
criterion in an over pressure zone for bottom hole cleaning and for showing the effect of mud
rheology on pressure losses in a mud circulatory system. In this work, optimum conditions for
drilling were determined by estimating pore pressure and fracture pressure from conductivity
data, selecting a suitable mud with an appropriate density based on the result of the conductivity
data analysis, studying the rheological properties of mud samples, calculating the pressure losses
in the mud circulatory system and finally applying the maximum horsepower criterion for
optimization.
Based on the results of conductivity data analysis, experimental analysis of the drilling mud
rheology and pressure loss calculation in the mud circulatory system, conditions for optimum
hydraulic horsepower across the drill bit in the problematic zone is presented in this case study.
This study shows that pressure loss in the mud circulatory system depends on the mud and the
circulating flow rate. Also, the operating conditions obtained in this study shows that the flow
rate exceeds the minimum flow rate required for drill cuttings removal. One unique aspect of this
project work is the integration of experimental work designed to generate rheological data for
theoretical computation.
xx
CHAPTER 1: INTRODUCTION
This project focuses on optimizing the hydraulic horsepower at the drill bit for the purpose of
bottom hole cleaning and to enhance the rate of drilling. This work considers the effect of mud
rheology on pressure losses in the mud circulatory system and then designing an hydraulics
system for effective drill cutting removal during drilling operation by specify the operating
conditions to maximize the power at the drill bit, using a case study where the target depth lies in
an over pressure zone.
1.1 Study Objective
The objective of this study aims at designing an hydraulic system to specify the operating
conditions to operate the mud pump for drill cutting removal and to enhance the rate of
penetration during drilling. The effect of mud rheology on the pressure losses in a mud
circulatory system will also be considered in this study.
1.2 Problem Statement
Inadequate hole cleaning can lead to a number of problems, including hole fill, packing off, stuck
pipe, and excessive hydrostatic pressure. Drill cuttings in the hole cause wear and tear of the drill
string and also reduce the rate of penetration, thereby increasing the cost and time for drilling;
hence, there is need to design a system that will efficiently remove the drill cuttings, transport
them to the surface in a cost effective manner, prepare an appropriate drilling mud and maximize
the hydraulic horse power at the drill bit.
1.3 Scope of Study
The industry has made significant progress in hole cleaning. The ability of a drilling fluid to lift
cuttings is affected by many factors, and there is no universally accepted theory which can
account for all observed phenomena, Well bore cleaning can be achieved in a number of ways
such as by increasing the drill pipe rotation, by improving the rheological properties of the mud,
the cuttings bed properties. But this study will focus on drill cutting removal by preparing a
1
suitable mud from geological data, and using the concept of hydraulic optimization to maximize
the drill bit hydraulic horsepower.
1.4 Limitations
During the course of this project work, some of the limitations of this project work were as
follows:
1. The use of conductivity data to estimate the abnormal pressure zone in this study has
some degree of inaccuracy since conductivity is affected by salinity. However, seismic
data is more accurate for detecting and quantifying abnormal pressure.
2. The Hottman and Johnson approach was used to estimate the pore pressure, which has
some degree of inaccuracy since the approach does not account for the effect of
overburden stress.
3. The Hubbert and Willis approach was used to estimate the fracture pressure, which has
some degree of inaccuracy, since the approach assumes a poisson’s ratio of 0.25 and an
over burden pressure gradient of 1psia/ft.
4. In this study, factors such as drill pipe eccentricity, drill pipe rotation and the weight on
the drill bit that affect the rate of penetration were not accounted for, this study focussed
on the mud rheology and hydraulics.
5. Due to unavailability of mud pump data, a theoretical flow exponent of 1.75 was used
for the hydraulic design (Kendall and Goin, 1960), since the flow exponent can only be
obtained by operating the mud pump on a drilling rig.
2
CHAPTER 2: LITERATURE REVIEW
The rate of penetration is considered one of the prime factors in drilling a hydrocarbon well and
it is therefore given a prime consideration when drilling an oil well. However, a lot of
extensively analyzed on ways of increasing the rate of penetration from both theoretical and
experimental standpoint has been carried out till date. Eckel (1967) was able to establish from
laboratory and field experience that the rate of drilling using mud was increased from 30 to 70
percent of those obtainable with water under the same conditions. Eckel (1967) further stipulated
that viscosity is a significant factor affecting the rate of drilling. Eckel (1967) used oil emulsion
in his experiments and he observed that the rate of drilling was improved due to their lubricative
properties. Eckel (1967) concluded that mud rheological properties have significant effect on the
rate of penetration.
Warren (1988) developed a rate of penetration (ROP) model for soft-formation bits under
conditions where cuttings removal does not impede the rate of penetration. This model relates
ROP to weight on bit (WOB), rotary speed, rock strength, and bit size. It is based on tests that
were designed to provide basic information about the interaction between the bit and rock in the
absence of complicating cuttings-removal effects. The practical application of this model to
general ROP prediction is severely limited because it does not include cuttings-removal effects.
Recently, Fear (1999) developed a method to identify the factors controlling ROP. He correlated
mud logging data, geological information and drill bit characteristics against ROP and other
drilling parameters. This statistical method suggested that factors affecting the rate of penetration
are different for different cases and Fear (1999) recommends that the method be applied for each
specific case to determine the applied drilling parameters.
Numerous factors affect the rate of penetration, and the objective of this study is to design
a hydraulic system to specifying the operating conditions to drill through the formation to
effectively remove drill cuttings and enhance the rate of penetration in an over pressured zone.
Enhancing the rate of penetration is the objective of an effective drilling program, as rate of
penetration tends to decrease with depth. Therefore a detailed study of geomechanics, factors
affecting the rate of penetration and bottom hole cleaning is discussed in this literature review.
3
2.1 Geomechanics
Geomechanic involves the geologic study of the behavior of soil and rock under mechanical
loading conditions (stresses, strain). The study of Geomechanics is paramount in predicting
important reservoir parameters such as formation porosity, permeability, pore pressure, fracture
pressure and bottom-hole pressure. This science provides us with vital knowledge of the
reservoir properties which is essential in proper well planning and in completing a successful
drilling project (Prassl, 2003).
2.2 Pore Pressure
Pore pressure is the pressure acting on the fluids in the pore space of the rock (Rabia, 2002). This
is the pressure due to the column of liquid occupying the pore space of a porous medium in a
sedimentary basin. The magnitude of pore pressure can be described as either normal pressure or
abnormal pressure. Rabia (2002) also stated that the magnitude of normal pore pressure varies
with the following: concentration of dissolved salts, type of fluid, gases present and temperature
gradient. Rabia (2002) further stated that as the concentration of dissolved salts increases the
magnitude of normal pore pressure increases.
Successfully predicting pore pressure within the formation to be drilled is one of the most
critical parameters needed in planning and drilling a well. The estimates of formation pore
pressure made before drilling are usually based on seismic data acquisition, analysis and
interpretation. To estimate formation pore pressure from seismic data, the average acoustic
velocity or the interval transit time as a function of depth must be determined.
2.3 Causes of Over Pressure
Abnormal pore pressure can be said to be any pore pressure that is greater than the hydrostatic
pressure of the formation fluid occupying the pore space. Over pressure can arise due to a
combination of geological, geochemical, geophysical and mechanical process (Rabia, 2002).
Rabia also stated some of the causes of over pressure which is show below.
4
2.3.1 Depositional Effects
The deposition of evaporites can create high abnormal pore pressures in the surrounding zones
with the pore pressure approaching the overburden gradient. When salt is deposited, the pore
fluids in the underlying formations cannot escape and therefore become trapped and abnormally
pressured (Rabia, 2002).
2.3.2 Digenetic Processes
This is a process in which sediments undergo a process of chemical and physical changes
collectively with increasing temperature and pressure (Rabia, 2002). Rabia (2002) further stated
that Diagenetic processes could be as a result of the formation of new minerals, recrystallization
and lithification. Diagenesis is the alteration of sediments and their constituent minerals during
post-depositional compaction. Diagenesis may lead to volume changes and water generation,
which if occurring in a seabed environment, may lead to both abnormal or sub-normal pore
pressure.
2.3.3 Tectonic Effects
Tectonic activity can result in the development of abnormal pore pressure as a result of a variety
of mechanisms including: folding, faulting, uplift and salt diaparism (Rabia, 2002). In folding,
abnormal pressure results when tectonic compression of a geological basin is produced. The
additional horizontal tectonic stress created by folding compacts the clays laterally. For the
formation to remain normally pressured, the increased compaction has to be balanced by pore
water expulsion, but if the formation water cannot escape, abnormal pressure will result. Also,
during faulting in sedimentary rocks, abnormal pressure is also caused by tectonic activities in
which the sedimentary beds are broken up, moved up and down or twisted. Finally, if a normally
pressured formation is uplifted to a shallower depth then the formation will appear to have an
abnormal pressure due to the fact that the formation pressure has more hydrostatic pressure than
a corresponding normally pressured zone at the same depth.
5
2.3.4 Structural Causes
Abnormal pore pressure can also exist in both horizontal and non-horizontal reservoir structures
which contain pore fluids of differing densities i.e. water, oil and gas Rabia (2002). Examples of
structures in which this may occur are lenticular reservoirs, dipping reservoirs and anticlinal
reservoirs. In dipping reservoirs, formation pressures which are normal in the deepest water zone
of the reservoir will be transmitted to the up dip part of the structure. In large structures or gas
reservoirs, the overpressure gradient contrast developed can be quite significant. Therefore,
careful drilling practices should be adopted in order to minimize the risks associated with high
overbalance as the reservoir is drilled down through the water zone.
2.3.5 Thermodynamic Effects
Thermodynamic effects such as organic matter transformation can also result in abnormal
pressure, at high temperatures and pressures associated with deep burial, complex hydrocarbon
molecules (kerogen) will break down into simpler compounds, and that Kerogen alters to
hydrocarbon at 90 0C (Rabia, 2002). Rabia (2002) stated that thermal cracking of the compound
can result in two to three fold increases in the volume of the hydrocarbon. If this occurs in a
sealed environment, high pore pressures could result. The pressures will be substantially
increased if the hydrocarbon system becomes gas generative.
2.4 Fracture Pressure
Fracture pressure is the amount of pressure it takes to permanently deform or fracture the
formation. If the formations fracture pressure is exceeded, the wellbore will fracture, which may
lead to loss of circulation as the fluids in the well are pushed into the formation through the
fractures. The fracture pressure is dependent on the formation type, overburden pressure and on
how the formation is compacted. If abnormal formation pressure is encountered, the density of
the drilling fluid must be increased to maintain the wellbore pressure above the formation pore
pressure to prevent the flow of fluids from permeable formations into the well. However, since
the wellbore pressure must be maintained below the pressure that will cause fracture in the well
6
bore. Hence, there is a maximum drilling fluid density that can be tolerated in the well bore to
maintain well bore stability. This means that there is a maximum depth into the abnormally
pressured zone to which the well can be drilled safely without cementing another casing string in
the well. Thus, the knowledge of the pressure at which formation fracture will occur at all depths
in the well is essential for well planning and in drilling an oil well.
2.5 Mud Weight Planning
Mud weight selection in a drilling program is a key factor in avoiding various borehole
problems. It is essential to select the correct mud weight for drilling the individual sections. The
following must be considered when selecting mud weight (Prassl, 2003):

A very low mud weight may result in collapse and well cleaning problems.

A very high mud weight may also result in mud losses or pipe sticking.

Excessive variation in mud weight may also lead to borehole failure; as such a more
constant mud weight must be aimed at.

A median line concept is recommended generally for mud weight planning. The midpoint is between the pore pressure and fracture pressure. Hence keeping the mud weight
within this median level causes least disturbance on the borehole wall.
2.6 Factors Affecting Rate of Penetration
The rate of penetration is considered one of the primary factors affecting drilling costs and
hence it is given consideration when planning for optimized drilling. Hence some of
numerous factors that affect the rate of penetration of a formation are Chip hold-down, bit
type, drilling fluid properties, formation characteristics, and the operating conditions (Rabia,
2002).
2.6.1 Chip Hole-Down Effect
Chip hold-down occurs when a mud filter cake or fine solids block fractures produced by the
bit. This prevents the liquid phase of the mud from invading the fractures, and results in a
7
positive pressure differential across the top surface of the chip. The chip hold-down force is
equal to the area of the chip times the differential pressure
Fig 2.1 Chip hole down effect (Rabia, 2002)
The difference between the mud hydrostatic pressure and pore pressure is called Chip Hold down
Pressure (CHDP) (Rabia, 2002). This pressure prevents formation fluids from entering the
wellbore during drilling. However, this overbalance (CHDP) also acts to keep the rock cuttings
held to the bottom of the wellbore. The effects of bit rotation and hydraulics offset this force and
ensure that cuttings are lifted from the bottom of the hole. The CHDP (differential force) has one
of the largest effects on rate of penetration especially in soft to medium strength formations
(Rabia, 2002).
2.6.2 Effect Of Bit Type
The bit type selected for drilling into a formation has a large effect on the rate of penetration. In
the case of rolling cutter bits, the initial rate of penetration of a formation is optimum when using
bits with long teeth and a large cone offset angle, but these bits are best used in soft formations
because of a rapid tooth destruction and decline in penetration rate in hard formations. While the
drag bits are designed to obtain a given penetration rate by producing a wedging type rock failure
in which the bit penetration per revolution depends on the number of blades and the bottom
cutting angle. The diamond and polycrystalline diamond compacts (PCD) bits are also designed
for a given penetration per revolution by the selection of the size and number of diamonds or
PCD blanks. The width and number of cutters can be used to compute the effective number of
8
blades. The lowest cost per foot drilled is usually obtained when using the longest tooth bit that
will give a tooth life consistent with the bearing life at optimum bit operating conditions.
2.6.3 Effect of Drilling Fluid Properties
The rate of penetration is also affected by the properties of drilling fluid used during drilling.
These properties include: rheological properties, filtration characteristics, solids content and size
distribution, and chemical composition. The rate of penetration tends to decrease with increasing
fluid density, viscosity and solids content, and tends to increase with increasing filtration rate.
The density, solid content, and filtration characteristics of the mud control the pressure
differential across the zone of crushed rock beneath the bit. The fluid viscosity controls the
system frictional losses in the drill string and thus the hydraulic energy available at the bit jets
for cleaning. The most important factor out of the drilling fluid properties is the density,
differential pressure tends to increase with increasing density, and the rate of penetration
decreases with increasing differential pressure.
2.6.4 Effecting of Formation Characteristics
The elastic limit and ultimate strength of the formation are important formation properties that
affect the rate of penetration. The permeability of the formation also has a significant effect on
the ROP. In permeable rocks, the drilling fluid filtrate can move into the rock ahead of the bit
and equalize the pressure differential acting on the chips formed beneath each tooth. This would
tend to promote the more explosive elastic mode of crater formation. The mineral composition of
the rock also affects the rate of penetration. Rocks containing hard, abrasive minerals can cause
rapid dulling of the bit teeth. Rocks containing gummy clay minerals can cause the bit to ball up
and drill in a very inefficient manner.
2.6.5 Effect of Operating Conditions
Operating conditions such as the weight on the drill bit, the rotary speed have significant effect
on the rate of penetration. The rate of penetration has been observed to increase rapidly with an
9
increase in the weight on the drill bit. In some cases, a decrease in rate of penetration is observed
at extremely high value of weight on the drill bit. This type of behaviour is often called bit
floundering. This poor response of ROP at high values of bit weight is usually attributed to less
efficient bottom hole cleaning at higher rates of cuttings.
Figure 2.2 shows the effect of the weight on the drill bit on the rate of penetration (Prassl, 2003)
Fig 2.2: Effect on weight on the drill bit (Prassl, 2003)
Figure 2.2 shows that the rate of penetration increases from point a to point d with an increase in
the weight on the drill bit, but a decrease in the rate of penetration is suddenly observed from
point d to point e with increase in the weight on the drill bit.
The rate of penetration also increases with the rotary speed while other drilling variables held
constant. The rate of penetration usually increases linearly with low rotary speed, but at higher
values of rotary speed the rate of penetration begins to decreases. The reason for decrease in the
rate of penetration is due to poor hole cleaning. Figure 2.3 shows the effect of rotary speed on the
rate of penetration (Prassl, 2003).
10
Fig 2.3: Effect of rotary speed on the rate of penetration (Prassl, 2003)
Figure 2.3 shows that the rate of penetration increases from point a to point b with an increase in
the rotary speed, but a decrease in the rate of penetration is suddenly observed from point b to
point c with futher increase in the rotary speed.
Maurer (1980) developed a theoretical equation for rolling cutter bits relating ROP to WOB,
revolution per minute (RPM), bit size, and rock strength. The equation was derived from the
following observation made in single tooth impact experiments: (1) the crater volume is
proportional to the square of the depth of cutter penetration, and (2) the depth of cutter
penetration is inversely proportional to the rock strength.
2
K
R 2
S
W  W  
     N
 db  db t 
(1.1)
Where
R= rate of penetration (ft/h)
K = constant of proportionality
S = compressive strength of the rock (ib/in2)
W = bit weight (ib)
W0 = threshold bit weight (ib)
db = bit diameter (in)
N = rotary speed(rev/min)
11
The theoretical equation of Maurer (1980) can be verified using experimental data obtained at
relatively low bit weight and rotary speeds corresponding to segment ab in Figures (2.3).
Bingham suggested the following drilling equation on the basis of considerable laboratory and
field data.
a5
W 
R  K   N
 db 
(1.2)
Where
R= rate of penetration (ft/h)
K = constant of proportionality that includes the effect of rock strength
W = bit weight (ib)
db = bit diameter (in)
a5 = bit weight exponent
N = rotary speed (rev/min)
In this equation the threshold bit weight was assumed to be negligible and the bit weight
exponent must be determined experimentally for the prevailing conditions.
2.7 Factors Affecting Hole Cleaning
To effectively remove drill cuttings during drilling, a number of factors must be put in place to
achieve optimal bottom hole cleaning. To efficiently transport cuttings out of the hole, there must
be enough energy to push the solids out of the hole and the drilling fluid must be able to suspend
the solid particles. Some of the factors that affect hole cleaning are drill pipe rotation, drill pipe
eccentricity, rheology, drilling Rate, Cutting Bed Properties, and hydraulics (Tobenna, 2010).
12
2.7.1 Drill Pipe Rotation
Pipe rotation tends to make flow turbulent and this turbulence causes an increase in shear stress
on the cutting bed surface. This increased shear stress will assist in cuttings removal. But the
impact if drill pipe rotation on hole cleaning is relatively small in vertical well but more
significant in inclined wells.
2.7.2 Rheology
Rheology refers to the study of flow properties and characteristics of a drilling fluid. These
Properties of the circulation fluid have an effect on solids transport. Bottom hole cleaning is
more effective in a vertical well bore, when a high viscosity fluid is pumped in the well in a
laminar flow regime rather than a low viscosity fluid in a turbulent flow, but for a horizontal well
bore, bottom hole cleaning is more efficient when a low viscosity fluid is pumped in a turbulent
flow regime. The rheological properties of a fluid can be described using models that provide
assistance in characterising fluid flow. These models include the Bingham plastic model, and the
Herschel-Bulkley model. The rheological properties of the mud will go a long way in
determining its flow rate and suspension characteristics. Mud rheology will be an integral part of
this project work.
2.7.3 Drilling Rate
The rate of drilling has an important effect on cuttings transport, since as the drill rate increases,
the cuttings concentration in the annulus also increases. Hence for effective removal of drilled
cuttings, as the drilling rate increases the hydraulic requirement should also increase. To ensure
good hole cleaning during high rate of penetration (ROP) drilling, the flow rate and/or pipe
rotation have to be adjusted. If the limits of these two variables are exceeded, the only alternative
is to reduce the ROP. Although a decrease in ROP may have a detrimental impact on drilling
costs, the benefit of avoiding other drilling problems, such as mechanical pipe sticking or
excessive torque and drag, can outweigh the loss in ROP.
13
2.7.4 Cutting Bed Properties
The size, distribution, shape, and specific gravity of the cuttings affect their dynamic behavior in
a flowing media. The properties of the cutting bed has a significant effect on hole cleaning, if the
bed is loose or highly porous, then it may be necessary to remove single cutting particles that are
not adhered to the bed. In which case removing the bed becomes easy. But if the cutting bed is
highly consolidated with no cutting particle free to be removed alone from the bed by the flow,
hole cleaning will be difficult.
2.7.5 Hydraulics
Mud hydraulics is a crucial aspect of effective drilling. A drilling mud hydraulics program
consist of specifying the operating conditions to operate such as the minimum mud flow rate in
the annular space required to ensure efficient drill cuttings removal .This means that for a given
flow rate an increase in mud density beyond the desired level will impose additional hydraulic
requirement on the hydraulic system, which can impact on the optimum pump power
requirement. Experimental work shows that for a given pump flow rate requirement, the density
and viscosity of the mud are important parameters that affect the overall hydraulics of the
system.
14
CHAPTER 3: PORE PRESSURE AND FRACTURE PRESSURE PREDICTION
In well planning, it is important to estimate the pore pressure and fracture pressure to be
encountered in the subsurface. These predictions are important to ensure the safety of personnel,
equipment, specify operating conditions to follow during drilling. These prediction facilities
effective well planning and in selecting the required materials required for the entire drilling
operations. Also with respect to the reservoir, the right drilling mud weight is important. If it is
too low, a blowout might occur and conversely, if it is too high, the formation might be damaged
by invasion of the drilling fluid. Pore pressure and fracture pressure prediction cannot be over
emphasised in drilling.
3.1 METHODS OF PORE PRESSURE PREDICTIONS
There are different methods of pore pressure prediction currently used in the oil and gas industry
for pore pressure prediction. Over-pressure is an important geological problem in many regions
of the world; hence the detection of pore pressure is important in drilling of well bore, in
studying hydrocarbons migrations, and also in the formation of oil and gas fields. The most
effective method of estimating geological conditions before drilling is seismic prospecting. The
zones of over-pressure are frequently characterized by an abnormal intense condition of rocks
such as abnormal porosity, resistivity and density. These factors are the physical preconditions
for overpressure prediction from seismic data. Eaton’s method is one of the most widely used
quantitative methods for pore pressure evaluation. This method applies a regionally defined
exponent to an empirical formula, which has resulted in the development of equations that may
be used for the prediction of geopressure from well logs and seismic data. Equations are given
for use with resistivity plots, conductivity plots, sonic travel-time plots, and corrected "d"
exponent plots. All equations have the same theoretical basis (Eaton, 1975). The methods of pore
pressure predictions are resistivity method, sonic method, equivalent depth method, ratio
method, and neutron porosity log.
3.1.1 Resistivity Method of Pore Pressure Prediction
Resistivity method is based on the electrical resistivity of the sample, which includes the rock
matrix and the fluid filled porosity. Shale resistivity increases with depth. The resistivity of the
sample depends on factors such as porosity and salinity (Hussain Rabia, 2002). If a zone that has
15
abnormally high porosity and high pressure is penetrated, the resistivity of the rock will be
reduced due to greater conductivity of water. This approach involves plotting shale resistivity
with depth.
Hottman and Johnson (1965) developed a technique based on empirical relationships
whereby an estimate of formation pressure could be made by noting the ratio between the
observed and normal rock resistivity. The following steps are necessary to estimate the pore
pressure

The normal trend is established by plotting the logarithm of shale resistivity with depth.

The top of the pressure interval is found by noting the depth at which the plotted points
diverge from the trend.

The pressure gradient at any depth is found by taking the ratio of the extrapolated normal
shale resistivity to the observed shale resistivity, and then the formation pressure
corresponding to the calculated ratio is found.
Eaton (1975) also developed an equation using resistivity data for pore pressure predictions. This
equation is given as
R
Pp   ov   ov  PPN  o
 RN
1.2



(3.1)
Where
PP = pore pressure (psia)
 ov = overburden pressure (psia)
PPN = normal pore pressure (psia)
RN= normal resistivity (Ω)
RO = observed resistivity (Ω)
16
3.1.2 Sonic Method of Pore Pressure Prediction
To estimate formation pore pressure from seismic data, the average acoustic velocity as a
function of depth must be determined. The sonic log is provides the most reliable
estimate of pore pressure. Sonic log are usually more accurate because they are relatively
unaffected by borehole size, formation temperature and pore water salinity (Rabia, 2002).
Sonic logs measure the transit time for a compressional sonic wave to travel through the
formation from a transmitter to a receiver. The time to travel one foot is termed the
interval transit time (IIT). In a shale sequence showing a normal compact profile the
transit time should decrease with depth due to decreased porosity and increased density.
Abnormal pressure shales tend to have higher porosity and lower density than normally
pressured shales at the same depth; Hence, ITT values will be higher. By plotting ITT
against linear depth, a normal pressure trend line can be established through clean shales.
Abnormally pressured shales will therefore show an increased ITT above normal trend
line values. Rabia (2002) developed the following procedure to estimate pore pressures
knowing the acoustic travel time for shale formations:

The normal compaction trend for the area of interest is established by plotting
the logarithm of interval transit time versus depth.

The top of the over pressure formation is found by noting the depth at which
the plotted Points diverge from the trend line.

The fluid pressure gradient of a reservoir at any depth is found as follows:
 The divergence of adjacent shales from the extrapolated normal line is
measured.
 The pore pressure gradient corresponding to the interval transit time
value is found from a figure showing the relationship between the shale
acoustic parameter and the reservoir pressure gradient.

The pore pressure is then obtained by multiplying the pore pressure gradient
by the depth to obtain the reservoir pore pressure.
Eaton (1975) also developed a similar equation to that used in resistivity, to be used with interval
transit time data this equation can be used for both sonic and seismic data.
17
t 
Pp   ov   ov  PPN  N 
 tA 
3
(3.2)
Where
PP = pore pressure (psia)
 ov = overburden pressure (psia)
PPN = normal pore pressure (psia)
t N = normal pore pressure trend line interval transit time value at the point of interest(s)
t A = observed value interval transit time at the depth of interest (s)
3.1.3 Equivalent Depth Method
The equivalent depth method (Ham, 1966) is based on the reasonable assumption that formations
with the same physical properties such as resistivity, interval velocity, density or porosity would
have the same effective vertical stress irrespective of the depth. Figure 3.1 shows that every point
A in an under compacted clay is associated with a normally compacted point B. The compaction
at point A is assumed to be identical to that at point B. The depth of point B, ZB, is called the
equivalent depth, which is also called the isolation depth (Soufi, 2009).
Fig 3.2: Plot of porosity with depth (Soufi, 2009)
18
The pore pressure is given as
Pp  Ppeq  G( D  Deq )
(3.3)
Ppeq  0.465Deq
(3.4)
Where
Pp = pore pressure (psia)
Ppeq = pore pressure at the equivalent depth (psia)
Deq= equivalent depth (ft)
G = over burden pressure gradient (psia/ft)
D = depth (ft)
3.1.4
Ratio Method
The ratio method is based on the concept that the difference between the observed and normal
values of formation parameters is proportional to the increase in pressure (Soufi, 2009). Thus the
ratio of the observed (dco) to the normal (dcn) value is Proportional to the formation pressure.
D 
P p  PPN *  CN 
 DCO 
(3.5)
Where
PP = pore pressure gradient (Psia/ft)
PPN = normal pore pressure gradient (psia/ft),
DCN = normal trend line of d exponent
DCO = observed d exponent.
19
Fig 3.2: Plot of porosity dependent parameter with depth (Soufi, 2009)
The ratio method is considered unsuitable for use in most shale sequences. However, it has been
found to give accurate results in interpreting pore pressures from elastic limestone data in the
Middle East.
3.1.5 Neutron Porosity Log
Pore pressure can also be predicted from the neutron porosity log. The under-compaction of
sediments is the primary cause of formation overpressure, which occurs primarily in rapidly
subsiding basins and in rocks with low permeability. Pore pressure and formation porosity are
higher in under-compacted sediments than those in the normal compaction condition. It is
commonly accepted that porosity decreases exponentially as depth increases in normally
compacted formations. One commonly used relationship between porosity and depth is given by
the equation below (Athy, 1930)
D  0e KD
(3.6)
20
Where
 D = porosity at a given depth
ϕ0 = porosity in the mudline
D = true vertical depth (ft)
K = porosity decline constant.
Therefore porosity is a function of effective stress and pore pressure, particularly for the
overpressures generated from under-compaction and hydrocarbon cracking. Therefore, pore
pressure can be estimated from formation porosity. In a formation with under-compaction,
porosity and pore pressure are higher than those in a normally compacted one.
Fig 3.4: Plot of porosity and pressure with depth (Athy, 1930)
Figure 3.3 shows the schematic porosity (a) and corresponding pore pressure (b) in a
sedimentary basin. The dashed porosity profile in (a) represents a normally compacted
formation. In the over-pressured section in (a), a porosity reversal occurs. In the over -pressured
section, there is a deviation from the trend line (ϕn) which indicates an abnormal pressure.
Heppard (1998) used an empirical porosity equation similar to Eaton's sonic method to predict
pore pressure using shale porosity data. Heppard (1998) derived a theoretical equation for pore
pressure prediction shown in the relationship below:
21
Pp   ov   ov  PPN ln o  ln cz
(3.7)
Where
PP = pore pressure (psia)
 ov = over burden pressure (psia)
ϕ = porosity in shale
ϕ0 = porosity in the mudline
Z = true vertical depth below the mudline (ft)
C = compaction constant (1/ft)
Porosity tends to decrease with depth in the wellbore, but when there is a sudden increase in
porosity with depth, under compaction is said to have occurred which indicates an abnormal
pressure zone. Compaction occurs in a normally pressured zone.
3.2 METHOD OF FRACTURE PRESSURE PREDICTIONS
Formation fracture pressure predictions are made based on empirical correlations. Formation
fracture pressure is deduced from formation pore pressure. The commonly used fracture pressure
prediction equations are Hubbert and Willis equation, Mathews and Kelly equation and Eaton’s
correlation.
3.2.1 The Hubbert and Willis Approach
The Hubbert and Willis (1957) approach states that the minimum wellbore pressure required to
fracture a formation is the sum of the pore pressure and the minimum principal stress (  min ). The
Hubbert and Willis equation is given below (Hubbert and Willis, 1957):
Pf   min  Pp
(3.8)
Where
Pf = fracture pressure (psia)
 min = minimum principal stress (psia)
22
Pp = pore pressure (psia)
The minimum principal stress occurs in the horizontal plane and if this horizontal stresses (  h )
are equal to the local stress concentration at the borehole wall. Thus, the pressure required to
initiate fracture in a homogeneous, isotropic Formation is given in the relationship below:
Pf  2 h  PP
(3.9)
Where
Pf = fracture pressure (psia)
 h = horizontal stress (psia)
Pp = pore pressure (psia)
The horizontal stress is given as
 1  2
 1 
h  

 ov

(3.10)
Where
 h = horizontal stress (psia)
Pp = pore pressure (psia)
 ov = over burden pressure (psia)
 = passion ratio
Since the earth is so inhomogeneous and anisotropic, with many existing joints and bedding
planes, fracture pressure is generally used for well planning and casing design.
Furthermore Hubbert and Willis (1965) also concluded that the minimum stress in the shallow
sediments is approximately one-third the vertical matrix stress resulting from the weight of the
overburden. Therefore the formation fracture pressure is shown as (Hubbert and Willis, 1957)
23
Pf 

ov
 2 Pp 
3
(3.11)
Where
Pf = fracture pressure (psia)
Pp = pore pressure (psia)
 ov = over burden pressure (psia)
 h = horizontal stress (psia)
3.2.2 The Mathews and Kelly correlation
Mathews and Kelly in 1976 established that the assumption made by Hubbert and Willis that the
minimum stress was one-third the matrix stress is not valid for deeper formation, as formation
fracture gradient tends to increase with depth. They provided an equation for calculating the
minimum stress (  min ).
 min  F  ma
(3.12)
Where
 min = minimum stress (psia)
F = Stress coefficient
 ma = vertical stress (psia)
Where the stress coefficient ( F ) is determined empirically from field data in a normally
pressured formation. The vertical stress (  ma ) at the normally pressured zone is calculated with
the relationship below
 ma   ov  Ppn
(3.13)
Where
 ma = vertical stress (psia)
24
 ov = over burden pressure (psia)
Ppn = normal pressure gradient (psi/ft)
Matthews and Kelly assumed that the average overburden stress (  ob ) is 1psi/ft and the average
normal pressure gradient ( Ppn ) is 0.465psi/ft. Then the fracture pressure of the formation is
calculated with the Hubbert and Willis equation given in equation (3.7).
3.2.3 The Eaton Correlation
Eaton’s (1957) fracture prediction approach is the most widely used strategy. Eaton’s correlation
can be used anywhere in the world as long as the area specific over-burden stress gradient, the
pore pressure of the well and the area-specific Poisson’s ratio is known. Eaton’s equation for
fracture pressure is given below (Eaton, 1975):
Pf
D

   ov PP  PP
 

1   D
D D
(3.14)
Where
Pf = fracture pressure (psia)
 = poisson’s ratio
D = depth (ft)
Pp = formation pore pressure (psia)
 OV = overburden stress (psia)
3.2.4 The MacPherson and Berry correlation
Macpherson and Berry (1972) were able to estimate formation fracture pressure by developing a
correlation between elastic modulus for a compressional wave and formation fracture pressure.
They made the following conclusion from studies on the prediction of fracture pressure. They
25
made measurements of the interval transit time by means of a sonic log. The elastic modulus is
computed using the relationship below:
Kb  1.345 *1010
b
t2
(3.15)
The fracture pressure is then obtained from the Macpherson and Berry empirical correlation
between
Kb
 ov
and the fracture pressure.
Where
Kb = elastic modulus (psia)
 b = the bulk density (ib/gal)
t = the interval transit time (s)
 ov = the overburden stress (psia)
3.3 DETECTION OF OVER PRESSURED ZONE
The over-pressure zone can be detected using a number of approaches; each approach relates
formation properties such as porosity, resistivity, interval transit time, and density with depth.
These empirical relationships are obtained using Athy’s equation, which relates porosity and
depth.
3.3.1 Detection of Over-Pressure Using Resistivity Log
An over pressured zone can be detected from resistivity data by relating resistivity to porosity. A
basis for most rock resistivity studies was provided by Archie (1942) who examined the
relationship between resistivity and porosity in sandstone cores from the U.S. Gulf Coast region.
He empirically established that the resistivity is inversely proportional to porosity. Archie
26
established an exponential empirical relationship between resistivity and the porosity which can
be described by the equation below
R
K
(3.16)

Where
R = resistivity at a reference depth (Ω)
K = compaction factor which depends on the formation
 = porosity
At the surface of the well bore
R0 
K
0
(3.17)
This can be written as
0 
K
R0
(3.18)
At a given depth in the well bore
RD 
K
D
(3.19)
This can be written as
D 
K
RD
On substituting both equation (3.18) and equation (3.19) into equation (3.5)
Therefore,
27
K
K
 0e  KD
RD R0
1
1
 0e  KD
RD R0
Taking the logarithm of both side of the equation
 1 
 1 
  Log    KD
Log 
 RD 
 R0 
This can also be written as
Log1  LogRD   Log1  LogR0   KD
Therefore
LogRD   LogR0   KD
(3.20)
Where
RD = resistivity at a reference depth (Ω)
R0 = resistivity at the surface (Ω)
D = depth (ft)
K = compaction factor which depends on the formation
Equation (3.20) shows that the resistivity is expected to increase linearly with depth; hence overpressured zone is detected when there is an abnormal deviation from the normal trend line in a
semi-log plot of resistivity with depth.
28
3.3.2 Detection of Over Pressure Using Interval Transit Time
The interval transit time depends on the elastic properties of the rock matrix, the properties of the
fluid in the rock, and the porosity of the rock. Wyllie (1958) proposed that the interval transit
time can be represented as the sum of the transit time in the matrix fraction and the transit time in
the liquid fraction and that this interval transit time is directly proportional to porosity. The
Wyllie relationship between interval transit time and porosity can be written as:
t  K
(3.21)
Where
t = interval transit time (sec)
K = compaction factor or porosity decline factor
 = porosity
At the surface of the well bore
t0  K0
This can be written as
0 
t0
K
(3.22)
At a given depth in the well bore
t D  KD
This can be written as
D 
t D
K
(3.23)
29
On substituting equation (3.22) and equation (3.23) into equation (3.16)
Therefore,
t D t0  KD

e
K
K
t D  t0e KD
Taking the logarithm of both side of the equation
LogtD   Logt0   KD
Therefore
LogtD   Logt0   KD
(3.20)
Where
t D = interval transit time at a reference depth (s)
t0 = interval transit time at the surface (s)
D = depth (ft)
K = compaction factor which depends on the formation
The equation above shows that the interval transit time is expected to increase linearly with
depth. An over pressured zone is detected when there is an abnormal deviation from the normal
trend line in a semi-log plot of interval transit time with depth.
30
CHAPTER 4: OPTIMIZATION OF DRILL BIT HYDRAULICS
The hydraulic power from mud pump must be efficiently utilized for efficient drilling operation
to be achieved. Hence, the hydraulic power across the drill bit needs to be maximized at the point
of contact between the drill bit and the formation so as to provide enough jet impact force to
transport the cuttings as the formation is been drilled. Therefore, to efficiently remove drill
cuttings and transport cuttings up the annulus, it is important to minimize power loss in the mud
circulatory system so has to have adequate hydraulic horsepower across the drill bit.
4.1 Hydraulic Power Requirement
The power involved in the mud circulating system is made up of the mechanical horse power
which is the power needed to drive the mud pump, the fluid hydraulic horsepower, which is the
fluid power which will provide a jet impact force and the bit hydraulic horse power, which is the
power at the drill bit. The fluid hydraulic horsepower and the bit hydraulic horse power are the
main design parameter for an effective hydraulic program design needed for effective bottom
hole cleaning, and improved rate penetration. The main component of a hydraulic system is the
mud pump at the surface, the surface connection, the drill pipe, the drill collars, the drill bit, and
mud tank at the surface. However, the mud pump is the main source of circulating drilling fluid
in the mud circulating system.
Hydraulic power is define as the product of pressure and the corresponding flow rate (Azar and
Robello 2007)
Hh  P * Q
(4.1)
In field unit this relationship is given
Hh 
P *Q
1714
(4.2)
Where
Hh = hydraulic horse power (hp)
31
P = pressure (psia)
Q =flow rate (gpm)
4.1.1 Surface Connection Pressure Drop
In the mud circulation system the first pressure drop is experienced in the surface equipment.
The surface equipment of a drilling rig includes the standpipe, rotary hose, swivel wash pipe,
along with the gooseneck and Kelly bushing. The pressure drop in the surface connection is
substantial during drilling fluid circulation and this loss depends on the type of surface
connection. Surface equipment has been grouped into four classes by the International
association of drilling contractors (IADC).Table 4.1 shows IADC classes of surface equipment
of a drilling rig.
Table 4.1 IADC Classes of Surface Equipment (Baker-Huges Drilling Engineering Workbook,
1995)
Class #1(Coefficient 19)
Class #2 (Coefficient 7)
Class #3 (Coefficient 4)
Class #4 (Coefficient 3)
40 ft & 3 in. I.D. standpipe
40 ft &3.5 in. I.D. Standpipe
5 ft & 2.5 in. I.D. Swivel
6 ft & 3 in. I.D. Swivel
45 ft & 2 in. I.D. Hose
55 ft & 2.5 in. I.D. Hose
45 ft & 4 in. I.D. Standpipe
45 ft & 4 in. I.D.Standpipe
4 ft & 2 in. I.D. Swivel
5 ft & 2.5 in. I.D. Swivel
40 ft & 3.25 in. I.D. Kelly
40 ft & 4 in. I.D. Kelly
40 ft & 2.25 in I.D. Kelly
40 ft & 3.25 in. I.D. Kelly
55 ft & 3 in. I.D. Hose
55 ft & 3 in. I.D.Hose
Therefore, when calculating surface pressure losses, the surface pressure coefficient that
corresponds to the surface equipment on the rig is chosen and the following relationship below is
used (Baker Huges drilling engineering workbook 1995).
Pfs  105 * K s * m * Q1.86
(4.3)
32
Where
Pfs = surface pressure loss (psia)
Ks = surface pressure coefficient
 m = mud density (lb/gal)
Q = flow rate (gal/min)
However, the surface connection pressure loss can also be obtained from the equivalent length of
surface equipment of drill pipe. The surface connection pressure loss can also be calculated by
Pfs  Pfdp *
Lse
Ldp
(4.4)
Where
Pfs = surface pressure loss (psi)
Pfdp = pressure drop in the drill pipe (psia)
Lse = surface equipment equivalent length of drill pipe (ft)
Ldp = length of drip pipe (ft)
4.1.2 Drill String Pressure Drop
After the drilling fluid passes through the surface connection it flows through the drill string. As
the drilling fluids flow through the drill pipe and the drill collar, the walls of the drill strings
create a resistance against fluid flow. This drag force from the drill string walls and irregularities
caused by the drill string joints and sudden contractions of the internal diameter from the drill
pipe to the drill collar produce eddies in the drill string. These eddies cause cross-flow and
countercurrents, which create frictional resistance. This frictional resistance results in pressure
loss in the drill string. The pressure drop in the drill pipe and the drill collar can be calculated for
33
laminar and turbulent flow criteria depending on the drilling fluid type used. The correlations for
pressure drop in the drill string for both laminar and turbulent flow regimes are given below (
Bourgoyne, 1991) :
4.1.2.1 Newtonian Fluid
For a Newtonian fluid in a laminar flow the pressure drop in the drill pipe and the drill
collar is given below
Pf 
L *  p * Va
2
1500 * Didp
Va 
Q
2
2.45 * Didp
(4.5)
(4.6)
Where
Pf = frictional pressure loss in the drill pipe or the drill collar (psia)
Di = internal diameter of the drill pipe or drill collar (in)
Va = average velocity of the drill pipe or drill collar (ft/min)
Q = flow rate (gpm)
L = length of the drill pipe or the drill collar (ft)
 p = plastic viscosity (cP)
The equation for pressure drop using a Newtonian fluid in a turbulent flow regime, in the drill
pipe and the drill collar is given by the equation below (Bourgoyne, 1991):
34
Pf 
f *  m * Va2
25.8 * Di
(4.7)
Where the friction factor as given by Colebrook’s equation
  21.25 
1
 2.28  4 Log   0.9 
N RP 
f
 Di
(4.8)
Where
Pf = friction pressure loss in the drill pipe or the drill collar (psia)
f = friction factor
 m = mud density (ib/gal)
Di
Va
= internal diameter of the drill pipe or drill collar (in)
= average velocity of the drill pipe or drill collar (ft/min)
 = pipe roughness (in)
N RP = reynolds number
4.1.2.2 Bingham Fluid
For a Bingham fluid in a laminar flow regime the pressure drop in the drill pipe and the
drill collar is given by the relationship below (Bourgoyne, 1991):
Pf 
L *  p * Va
2
idp
1500 * D

y *L
225 * Di
(4.9)
Where
P f = frictional pressure loss in the drill pipe or the drill collar (psia)
Di = internal diameter of the drill pipe or drill collar (in)
35
Va = average velocity of the drill pipe or drill collar (ft/min)
Q = flow rate (gpm)
L = length of the drill pipe or the drill collar (ft)
 y = yield point (ib/100ft)
 p = mud plastic viscosity (cP)
For turbulent flow, a turbulence criterion for fluids that follows the Bingham plastic model was
presented by Hanks (1967) which he calls the Hedstrom number.
The Hedstrom number equation is given as
N He 
37100 *  m * y * d 2
 p2
(4.10)
Where
N He = hedstrom number
 m = density of mud (ib/gal)
 p = plastic viscosity (cP)
d = diameter of drill pipe or drill collar (in)
Hanks (1967) also found that the Hedstrom number could be correlated to the critical Reynolds
number from a chart. A turbulent flow is said to exist when this critical number is less than the
Reynolds number .The pressure drop across the drill pipe for a turbulent flow can be calculated
using the relationship given below (Bourgoyne, 1991):
36
Pf 
m0.75 * V 1.75 *  0p.25 * L
(4.11)
1.25
1800 * Di
Where
P f = frictional pressure loss in the drill pipe or the drill collar (psia)
Di = internal diameter of the drill pipe or drill collar (in)
Va = average velocity of the drill pipe or drill collar (ft/min)
L = length of the drill pipe or the drill collar (ft)
 m = density of mud (ib/gal)
 p = plastic viscosity (cP)
4.1.2.3 Power Law Fluid
For a Power law fluid in a laminar flow the pressure drop in the drill pipe and the drill collar is
given by the relationship below (Bourgoyne, 1991):
Pf 
L * K * V n  3n  1 


143700 * Dipn 1  0.0416 
n
(4.13)
Where
K = equivalent centipoise
n = power law index
L = length of the drill pipe of drill collar (ft)
V = velocity (ft/min)
Dip = Internal diameter (in) of drill pipe or drill collar
37
4.1.3 Annulus Pressure Drop
The pressure drop in the annulus of the drill pipe and the drill collar mainly depend on the
external diameters of the drill collar and the drill pipe, the bore hole size, the internal diameter of
the casing and the drilling fluid flow rate. The cross-sectional fluid flow area in the annulus is
larger compared to inside the drill string. The flow in the annulus is usually assumed to be
laminar due to low fluid pressure and velocity. The frictional pressure loss in the annulus of the
drill pipe and the drill collar can be calculated for both laminar and turbulent flow criteria
depending on the type of drilling fluid used. The correlations for pressure drop in the annulus of
the drill pipe and drill collar for both laminar and turbulent flow regimes are given below
(Bourgoyne, 1991):
4.1.3.1 Newtonian Fluid
For a Newtonian fluid in a laminar flow the pressure drop in the annulus of the drill pipe and the
drill collar is given by the relationship below (Bourgoyne, 1991):
 p * Va * L
Pa 
1500
 2
D 2  Do2 
 Dh  Do2  h

in ( Dh / Do ) 

(4.14)
Where
Va 
Q
2.45 * ( Dh2  D02 )
(4.15)
Where
Pa= frictional pressure loss in the annulus of the drill pipe or the drill collar (psia)
Di = internal diameter of the drill pipe or drill collar (in)
38
Va = average velocity of the drill pipe or drill collar (ft/gal)
Q = flow rate (gpm)
L = length of the drill pipe or the drill collar (ft)
Dh = diameter of the hole (in)
Do = outer diameter of the drill collar or drill pipe (in)
 p = Plastic viscosity (cP)
For a Newtonian fluid in a turbulent flow regime the pressure drop in the drill pipe and the drill
collar is calculated using equation (4.7) where the internal diameter ( Di ) is replaced with the
equivalent diameter ( De )of the drill pipe or drill collar.
4.1.3.2 Bingham Fluid
For a Bingham fluid in a laminar flow the pressure drop in the annulus of the drill pipe and the drill collar
is given below (Melrose, 1985).
Pf 
L *  p * Va
1000 * Dh  Dod 
2

y *L
200 * Dh  Dod 
(4.16)
Where
P f = frictional pressure loss in the drill pipe or the drill collar (psia)
Di = internal diameter of the drill pipe or drill collar (in)
Va = average velocity of the drill pipe or drill collar (ft/min)
Q = flow rate (gpm)
L = length of the drill pipe or the drill collar (ft)
 y = yield value (ib/100ft)
 p = plastic viscosity (cP)
39
4.1.3.3 Power Law Fluid
For a power law fluid in a laminar flow the pressure drop in the annulus of the drill pipe and the
drill collar is given below (Dodge and Metzner, 1957):
L * K *V n
 2n  1 
Pf 

n 1 
143700 * Dh  Do   0.0208 
n
(4.17)
Where
P f = frictional pressure loss in the annulus of the drill pipe or the drill collar (psia)
K = equivalent centipoise
n = power law index
L = length of the drill pipe of drill collar (ft)
V = velocity (ft/s)
Do = outer diameter of the drill pipe or drill collar (in)
Dh = diameter of the hole (in)
4.1.4 Drill Bit Pressure Drop
The pressure drop across the drill bit is the most important element in a hydraulics equation and
is mainly due to the change of fluid velocities in the nozzles and the flow rate of the drilling
fluid. The amount of hydraulic horsepower available at the drill bit is influenced by the size of
nozzles used, the mud density and the flow rate. The pressure drop across the bit is given by the
relationship below (Azar and Samuel, 2007):
8.3 *105 *  m * Q
Pb 
A2 * Cd2
2
(4.18)
40
Where
Pb = pressure drop across the drill bit (psia)
Q = flow rate (gpm)
A = area of nozzle (in2)
Cd = discharge coefficient
 m = density of mud (ib/gal)
Thus, the total pressure coming from the mud pump system consists of the pressure drop across
surface connections (Pfs), the pressure drop across the drill bit (PB), the pressure drop across drill
pipe (Pdp), the pressure drop across drill collar (Padc), the pressure drop in the annulus of the drill
pipe (Padp) and the pressure drop in the annulus of the drill collar (Padc).
The total circulating pressure (Pmax)
Pmax  Pfs  Pdp  Pdc  padp  Padc  PB
(4.19)
The sum of all pressure drops except the pressure drop across the drill bit is called the parasitic
pressure drop ( Pf ) D
( Pf ) D  Pfs  Pdp  padp  Padc
(4.20)
Hence the Total circulating standpipe pressure ( Pmax )
Pmax  ( Pf ) D  PB
(4.21)
41
4.2 Flow Exponent and Optimum Flow Rate
The flow exponent (m) between two points is deduced from the relationship between frictional
pressure loss and flow rate. The flow exponent has a theoretical value of 1.75 (Bourgoyne,
1991):
( Pf ) D  CQ m
(4.22)
Where
( Pf ) D = Parasitic pressure loss (psia)
Q = Flow rate (gpm)
m = Flow exponent
C = constant that depends on mud flow properties, hole geometry and pipe geometry
On taking the logarithm of both side of the equation
log( Pf ) D  log C  m log Q
(4.23)
The plot of equation (4.23) is a straight-line with a slope of m and an intercept of log C .
Therefore, if the mud pump is operated at two different flow rates, the flow exponent (m) can be
obtained.
There are two basic criteria that are used in analyzing bit hydraulics for hole cleaning either the
drill bit hydraulics horsepower or the hydraulic jet impact force
4.3 Drill Bit Hydraulic Horsepower Criterion
Drill bit hydraulic horsepower criterion is based on the fact that cuttings are best removed from
beneath the bit by delivering the most power to the bottom of the hole. The amount of pressure
lost at the bit, or bit pressure drop, is essential in determining the hydraulic horsepower. This
criterion states that the optimum hole cleaning is achieved if the hydraulic horsepower across the
bit is maximized with respect to the flow rate (Azar and Samuel, 2007).
42
H HB  PB * Q
.
(4.24)
Where
H HB = drill bit hydraulic horse power (hp)
PB = pressure across the drill bit (psia)
Q = flow rate (gpm)
Kendall and Goins (1960) derived an equation for calculating optimum parasitic pressure loss
this is given below.
On substituting equation (4.21) into equation (4.22) and making P B subject of the formula
PB  Pmax  CQ m
(4.25)
On substitution equation (4.26) into equation (4.25)
H HB  Pmax Q  CQ m 1
For maximum condition to occur
dH HB
0
dQ
Hence
dH HB
 Pmax  (m  1)CQ m  0
dQ
(4.26)
43
But recall that
( Pf ) D  CQ m
Therefore
dH HB
 Pmax  (m  1)( Pf ) D  0
dQ
Therefore the optimum parasitic pressure loss is given by the equation below (Kendall and
Goins, 1960)
( Pf ) Dopt 
Pmax
m 1
On the basis of the maximum bit hydraulic horsepower criterion, the optimum bit hydraulic will
be achieved if frictional pressure loss in the circulating system is maintained at an optimum
value given below (Kendall and Goins, 1960).
( Pf ) Dopt 
Pmax
m 1
(4.27)
Where
Pmax = maximum pump pressure (psia)
m = flow exponent
( Pf ) Dopt = optimum parasitic pressure drop (psia)
The resulting optimum pressure drop across the drill bit is derived below (Kendall and Goins,
1960):
But recall from equation (4.21)
( Pf ) Bopt  Pmax  ( Pf ) Dopt
On substituting equation (4.28)
44
( Pf ) Bopt 
m
Pmax
m 1
(4.28)
Where
( Pf ) Bopt = optimum pressure drop on the drill bit (psia)
Pmax = maximum pump pressure (psia)
m = flow exponent
Using the optimum hydraulic horsepower criteria, the hydraulic horse power at the bit can be
determined from the relation below (Kendall and Goins, 1960):
HHPopt 
PBoptQopt
1714
(4.29)
Where
HHPopt = optimum hydraulic horse power at the drill bit (hp)
PBopt = optimum pressure drop across the drill bit (psia)
Qopt = optimum flow rate (gpm)
4.4 Hydraulic (Jet) Impact Force Criterion
Hydraulic (jet) impact force criterion is based on the fact that drill cuttings are best removed
from beneath the bit when the force of the fluid leaving the jet nozzles and striking the bottom of
the hole is very high. The maximum jet impact force criterion states that the bottom-hole
cleaning is achieved by maximizing the jet impact force with respect to the flow rate. The jet
impact force at the bottom of a wellbore can be derived from Newton’s second law of motion
and is given by the equation below (Azar and Samuel, 2007):
45
FJ  0.01823Cd Q PB m
(4.30)
Where
FJ = jet impact force (ibf)
Cd = discharge coefficient
 m = mud density (ib/gal)
Q = flow rate (gpm)
PB= pressure on the drill bit (psia)
But recall from equation (4.26)
PB  Pmax  CQ m
On substituting equation (4.37)
FJ  0.01823Cd Q m ( Pmax  CQ m )
(4.31)
The hydraulic jet impact force criteria has two basic limitations which are due to the maximum
available pump hydraulic horsepower and the maximum allowable surface operating pressure
4.4.1 Shallow Well Bore Formation
When drilling a shallower portion of a wellbore formation, the frictional pressure loss is usually
low and the flow rate requirement is large. Therefore, the hydraulic jet impact force is limited
only by the limited pump hydraulic horse power. This relationship is relationship is derived
below (Azar and Samuel, 2007):
Pmax 
H H max
Q
(4.32)
46
Where
Pmax = maximum pump pressure (psia)
Q = flow rate (gpm)
H H max = maximum pump hydraulic horsepower (hp)
On substituting equation (4.33) into equation (4.32)
FJ  0.01823Cd m ( H P max Q  CQ m  2 )
(4.33)
For maximum condition to occur
dFJ
0
dQ
On differentiating the equation (4.34)
dFJ 0.009115Cd m ( H H max  (m  2)CQ m 1 )

0
dQ
m ( H H max Q  CQ m  2 )
To obtain a valid solution of the differential equation above, the numerator must be equal to zero.
Hence
0.009115Cd m ( H H max  (m  2)CQm 1 )  0
Therefore
( H H max  (m  2)CQm 1 )  0
(4.34)
But recall
H H max  Pmax Q
On substituting into equation (4.35)
47
Pmax Q  (m  2)CQmQ  0
Also recall that
( Pf ) D  CQ m
Hence,
Pmax Q  (m  2)( Pf ) D Q  0
This gives
( Pf ) Dopt 
Pmax
m2
Therefore the optimum frictional pressure loss that can be obtained using the jet impact force
criterion for a shallow portion of the well is given below (kendall and Goins, 1960):
( Pf ) Dopt 
Pmax
m2
(4.35)
Where
Pmax = maximum pump pressure (psia)
m = flow exponent
( Pf ) Dopt = optimum parasitic pressure drop (psia)
The resulting optimum flow drop across the drill bit for a shallow depth based on the jet impact
force criterion of optimization is given below (kendall and Goins, 1960):
But recall,
( Pf ) Bopt  Pmax  ( Pf ) Dopt
(4.36)
Where
48
( Pf ) Bopt = optimum pressure drop on the drill bit (psia)
Pmax = maximum pump pressure (psia)
( Pf ) Dopt = optimum parasitic pressure drop (psia)
On substituting equation (4.35) into equation (4.36)
( Pf Bopt ) 
m 1
Pmax
m2
(4.37)
Where
( Pf Bopt ) = optimum pressure drop on the drill bit (psia)
Pmax = maximum pump pressure (psia)
m = flow exponent
4.4.2 Deep Well Bore Formation
When drilling a deeper portion of the wellbore, the frictional pressure loss increases while the
flow rate requirement decreases. Therefore, the hydraulic jet impact force will be limited by the
limited maximum allowed pump pressure Pmax. This relationship is derived as shown below
(Azar and Samuel, 2007):
From equation (3.32)
FJ  0.01823Cd Q m ( Pmax  CQ m )
This can also be written as
FJ  0.01823Cd m ( Pmax Q2  CQ m  2 )
(4.38)
For the maximum condition to occur,
49
dFJ
0
dQ
On differentiating equation (4.39)
dFJ 0.009115Cd  m (2 Pmax Q  (m  2)CQ m 1 ) _

0
dQ
m ( Pmax Q 2  CQ m  2 )
To obtain a valid solution of the differential equation above, the numerator must be equal to zero.
Hence


0.009115Cd m 2Pmax Q  (m  2)CQm 1  0
2Pmax Q  (m  2)CQ m 1  0
Recall that
( Pf ) D  CQ m
Therefore
2Pmax Q  (m  2)( Pf ) D Q  0
This gives
( Pf ) Dopt 
2 Pmax
m2
Therefore the optimum frictional pressure loss that can be obtained using the jet impact force
criterion for a deeper portion of the well is given below (kendall and Goins, 1960):
( Pf ) Dopt 
2 Pmax
m2
(4.39)
Where
( Pf ) Dopt = optimum parasitic pressure drop (psia)
50
Pmax = maximum pump pressure (psia)
m = flow exponent
The resulting optimum flow drop across the drill bit based on the jet impact force criterion of
optimization is given below
Recall,
(Pf ) Bopt  Pmax  (Pf ) Dopt
(4.40)
On substituting equation (4.39) into equation (4.40)
( Pf ) Bopt 
m
Pmax
m2
(4.41)
Where
( Pf ) Bopt = optimum pressure drop on the drill bit (psia)
Pmax = maximum pump pressure (psia)
m = flow exponent
Using the optimum jet impact force criteria for both cases, the hydraulic jet impact force is given
below:
m
FJ  0.01823Cd * Qopt m ( Pmax  C * Qopt
)
(4.42)
Where
FJ = jet impact force (ibf)
Cd = discharge coefficient
 m = mud density (ib/gal)
51
Qopt = optimum flow rate (gpm)
PB= pressure on the drill bit (psia)
4.5 Bit Nozzle Selection
The process of bit nozzle selection involves running a circulating pressure test at the rig site,
while keeping the rotary speed and weight-on-bit constant. This usually involves varying the
mud pump speed and recording the pump pressure and circulating rate at each speed. The
objective is to determine the optimum pressure drop across the bit nozzles and the optimum flow
rate as discussed above and from thence determine the nozzle sizes to support these optimum
conditions. The necessary conditions for attaining optimal bottom hole cleaning below a drill bit
is usually approximated via the optimization of the two design criteria. The optimum nozzle area
and optimum nozzle diameter is given by the relationship below (Azar and Samuel, 2007):
( At )opt 
2
8.311 *105 *  m * Qopt
2
Cd * Pbopt
(4.43)
Where
( At ) opt = optimum nozzle area (in2)
Qopt = optimum flow rate (gpm)
m = flow exponent
Pbopt = optimum pressure on the drill bit (psia)
Cd = discharge coefficient
 m = mud density (ib/gal)
If there are three nozzles and of equal diameter the optimum nozzle diameter is given below
(d N )opt 
4( At )opt
3
(4.44)
52
Where
(d N ) opt = optimum nozzle diameter (in)
( At ) opt = optimum nozzle area (in2)
4.6
Drill Cutting Transport
Drill cuttings in the annular space are subjected to numerous forces such as gravitational forces,
buoyancy, drag inertia, friction and interparticle contact. The flow of cuttings in the annulus is
dictated by these forces. Some of the factors that affect the capacity of drilling fluids to transport
drilled cuttings through the annular space are cutting slip velocity, annular fluid velocity and
flow regime.
4.6.1 Cutting Slip Velocity
The cutting slip velocity is the rate at which drill cuttings fall. For the fluid to lift the drill
cuttings to the surface, the fluid annular average velocity must be in excess of the cuttings
average slip velocity. To maintain good hole cleaning, the velocity of the drilling fluid
has to be greater than the slip velocity of the cuttings. The slip velocity depends on the
difference in densities, viscosity of the fluid and the size of the cuttings. Several
empirical correlations such as the Chien correlation, the Moore correlation and theWalker
and Mayes correlation have been developed to predict the slip velocity.
4.6.2 Annular Fluid Velocity
The annular fluid velocity when drilling a vertical well has to be sufficiently high to
avoid cuttings from settling and to transport these cuttings to the surface. The increasing
radial component of a particle slip velocity pushes the particles towards the lower wall of
the annulus, causing cuttings bed to form. Therefore the annular velocity has to be
sufficiently high to avoid bed formation.
53
4.6.3 Flow Regime
Flow regime describes the manner in which a drilling fluid behaves when flowing. The
flow regime could be laminar or turbulent. Fluid flow may also be predominantly laminar
at very low pump rates, but can become turbulent either at high pump rate or during pipe
rotation. The characteristics of laminar flow that is useful to the drilling engineer are the
low frictional pressures and minimum hole erosion. Laminar flow can be described as
individual layers moving through the pipe or annulus. The center layers moves at rates
greater than the layers near the well bore or pipe. The variations in velocity of this layer
are controlled by the shear-resistant capability of the mud. A high yield point for the mud
tends to make the layers move at more uniform rates. Cuttings removal is often discussed
as being more difficult with laminar flow. Turbulence occurs when increased velocities
between the layers create shear stresses exceeding the capacity of the mud to remain in
laminar flow. Turbulence occurs commonly in the drill string and occasionally around the
drill collars. Reynolds number can be used to determine flow regime.
54
CHAPTER 5: DESIGN METHODOLOGY
The aim of a drill bit hydraulic design is to provide sufficient hydraulic horsepower to the drill
bit for efficient bottom hole cleaning and to ensure an effective rate of penetration during
drilling. Hence for a drill bit hydraulic design, the pump operating requirements, appropriate
drilling mud, optimum flow rate and the corresponding optimum drill bit nozzle size are
necessary to ensure optimum drilling conditions. In this project work a case study will be
considered in the design of a hydraulic system to enhance the rate of penetration in an overpressured zone and to also enhance bottom hole cleaning by providing the optimum operating
conditions. In this case study the reservoir interval lies in an over-pressured zone; therefore, it is
critical to drill efficiently and safely in this zone.
This project work will involve estimation of pore pressure and fracture pressure using
geological data, mud weight selection, laboratory work on drilling fluid rheology as well as
calculations of pressure drop across the hydraulic system using the maximum horse power
criterion for optimization purposes. The geological data available for this study is the Frio shale
conductivity data (Mian,1991) acquired from an offshore well drilled in Nueces County in
Texas. Table 5.1 below shows of shale conductivity data obtained from a vertical well.
Table 5.1 Shale Conductivity Data from Nueces County Texas (Mian, 1991)
Conductivity (m/Ωm2)
Depth(ft)
7400
7500
7550
8300
8350
8400
8500
9200
9300
9550
9600
9700
9750
9900
9950
710
780
790
710
690
680
690
600
590
570
590
610
621
700
830
Conductivity (m/Ωm2)
Depth(ft)
10000
10050
10150
10200
10300
10500
10600
10650
10850
11000
11050
11200
11300
11500
950
1100
1200
1240
1310
1250
1350
1370
1500
1280
1400
1650
1840
1920
55
5.1 Estimation of Pore Pressure
The pore pressure from geological data (well log) is obtained using the Hottman and Johnson
procedure (Mian, 1991), by making a semi-log plot of shale resistivity versus depth using the
well data given. The plot of resistivity versus depth for the well is shown in Fig 5.1.
In(R)
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
7000
7500
8000
Normal Presssure
Zone
8500
Depth(ft)
9000
9500
Top of Over
Pressure
10000
10500
Over Pressure
Zone
Normal Shale
Resistivity
Trend Line(Rn)
11000
11500
Observed shale
Resistivity
Trend Line (R0)
12000
Fig 5.1 – Semi log plot of shale resistivity versus depth
56
From Figure 5.1, it shows that the point of entry into the overpressure zone occurs at a depth of
9550 ft. Therefore the over pressure zone of interest extends with depth from this point. Figure
5.1 also shows that the normal pore pressure trend is defined by the traditional semi log plot
characterised by a straight line that extends from 7400ft to 9550ft. This has a pore pressure
gradient 0.465psia/ft. The pore pressure at each depth is obtained using the relationship below
Pp  GD
(5.1)
Where
G = pore pressure gradient (psia/ft)
D = depth (ft)
PP = pore pressure (psia)
The pore pressure values in the normally pressured zone is shown in Appendix A
The pore pressure in the over pressure zone that lies from 96000 ft to 11500 ft is obtained from
the Hottman and Johnson procedure using equation 5.1 by taking the ratio of the observed shale
resistivity ( R0 ) and the normal shale resistivity ( Rn ) from Figure 5.1 at each depth. The pore
pressure gradient at each depth in the overpressure zone is obtained from the chart of shale
resistivity ratio versus pore pressure gradient given in Appendix E.
The estimates of the pore pressure at each depth in the well are shown in appendix A.
5.2 Estimation of Fracture Pressure
The fracture pressure of the formation in this case study is estimated using the Hubbert and
Willis equation. The Hubbert and Willis approach assumes that overburden pressure gradient
G= 1psia/ft and Poisson’s ratio = 0.25 (Bourgoyne, 1991).
57
The semi-log plot of resistivity versus depth shows that the over pressure zone starts from a
depth of 9600 ft.
Fracture pressure at the normal pressure zone which lies from 7400 ft to 9600 ft is and the over
pressure zone which lies between 9600 ft to 11500 ft is obtained using the Hubbert and Willis
equation (Hubbert and Willis, 1957) given in equation (3.9) and equation (3.10).The estimates of
the fracture pressure at each depth in the well are shown in Appendix A.
5.3 Mud Weight Selection
The mud weight to be selected for this design work is obtained by plotting a graph of specific
gravity versus depth. The calculation approach of specific gravity from pore pressure and
fracture pressure data for each depth is outlined in the following section.
5.3.1 Geological Mud Specific Gravity from Pore Pressure
Geological mud specific gravity from pore pressure is the specific gravity calculated from well
log data. The relationship below is given below ( Bourgoyne, 1991):
p 
PP
0.052 * D
SG p 
p

 p

8.33
(5.2)
(5.3)
w
Where
 p = density from pore pressure (ib/gal)
D = depth (ft.)
SG p = specific gravity from pore pressure
 w = density of water (ib/gal)
PP = pore pressure (psia)
58
5.3.2 Geological Mud Specific Gravity for Fracture Pressure
Geological mud specific gravity from fracture pressure is the specific gravity calculated from
fracture pressure data obtained using well log data. This can be calculated using equation (5.2)
and (5.3) where the pore pressure is replaced with fracture pressure and the specific gravity for
pore pressure is replaced with specific gravity for fracture pressure.
5.3.3 Design Mud Specific Gravity from Pore Pressure Line
In designing for mud specific gravity a safety factor called the trip margin is considered. The trip
margin is necessary to compensate for swab pressure. Redmann (1991) suggested a trip margin
of 0.5 ib/gal for pore pressure. The relationship below was used:
dp   p  0.5
(5.4)
Where
 dp = design mud density from pore pressure (ib/gal)
 p = mud density from pore pressure (ib/gal)
The design mud specific gravity can then be calculated using equation (5.3) where the mud
density from pore pressure is replaced with the design mud density from pore pressure.
5.3.4 Design Mud Specific Gravity from Fracture Pressure
In designing for the specific gravity, a kick tolerance of 0.5 ib/gal is used as kick margin for
fracture pressure (Redmann, 1991). To avoid fracturing the formation Redmann (1991)
recommended a kick tolerance of 0.5 ib/gal. The design mud specific gravity for fracture
pressure can be calculated using equation (5.3) where the specific gravity from pore pressure is
replaced with design specific gravity from fracture pressure and mud density from pore pressure
is replaced with design mud density from fracture pressure.
59
The design mud density from pore pressure and design mud density from fracture pressure used
for this work are shown in Appendix A. Figure 5.2 shows a plot from Appendix B that gives the
design mud specific gravity from pore pressure and fracture pressure.
Specific gravity
0.5
1
1.5
2
2.5
3
3.5
4
4.5
7000
7500
8000
8500
Depth(ft)
9000
9500
10000
10500
11000
11500
Geological
specific
gravity for
pore
pressure
line
Design
specific
gravity for
fracture
pressure
line
Geological
specific
gravity for
fracture
preesure
line
Design
specific
gravity for
pore
pressure
line
12000
Fig 5.2: Graph of specific gravity versus depth
60
Figure 5.2 shows that the mud density required to drill through the formation lies between the
pore pressure and fracture pressure line. The density of the drilling mud affects the rate of
penetration. Therefore based on fig 5.2, a mud specific gravity ( SGm ) of 1.5 will be chosen to
drill effectively and efficiently through the formation at a faster rate and to avoid the chip hole
down problem. Hence the mud density is given below:
Mud density (  m ) = mud specific gravity ( SGm )*density of water (  w )
1.5*8.33= 12.495 ib/gal
5.4 DRILLING MUD RHEOLOGY
5.4.1 Introduction
Based on the mud density (12.495 ib/gal) obtained from the geological data of this case study as
given above, the drill mud rheology properties were studied. It is important to study the
rheological properties of the mud to be used. This study will enable the determination of the
rheological properties of the mud, such as the plastic viscosity and the yield point of the drilling
mud to be used in the mud circulatory system. These rheological properties are vital in
determining the flow regime in the mud circulatory system. These rheological properties are also
essential in calculating the entire pressure drop in the mud circulatory system.
5.4.2 Objective of Experiment
The aim of the experimental part of this project work is to prepare a suitable drilling mud based
on the result of the mud density obtained from the geological data. This requires the
determination of the Plastic viscosity and the yield point of the drilling mud. A rotational
viscometer was used to acquire the plastic viscosity and yield strength of the drilling mud. Mud
samples with varying concentration of mud additives were prepared to achieve a suitable mud
with suitable rheological properties, since pressure losses strongly depend on the rheological
properties of the drilling mud. Based on the geological data of my case study a water-based mud
with a density of 12.5 ib/gal was prepared in the laboratory using a mixture of barite, bentonite
and water. A water-based mud was chosen for this work because the rate of penetration with
water-based mud is generally slightly faster than with oil-base mud for both roller cutting bits
61
and diamond bit (Cheatham, 1985). Three mud samples were prepared for this purpose in order
to determine the mud sample with the smallest parasitic pressure losses.
5.4.3 Equipment Used
Viscometer: A model 35 Fann viscometer manufacture by Fann Instrument Company was used
for this experiment, this viscometer was used to determine the viscosity and yield point of the
mud sample that was prepared for this project work. The diagrammatic representation of the
model 35 viscometer can be seen in the Figure 5.3.
Fig 5.3: Model 35 Viscometer
Mud Balance: A model 140 Mud balance manufactured by NL Bariod was used for mud density
measurement which has an accuracy of ±0.05ib/gal.The mud density test for each mud sample
was conducted using a mud balance, which consists of a base and a balance arm with cup, lid,
knife edge, rider, level glass, and counterweight. The cup is attached to one end of the balance
62
arm and the counterweight is at the opposite end. The diagrammatic representation of the mud
balance can be seen in the Figure5.4.
Fig 5.4: Model 140 Mud Balance
5.4.4 Mud Mixture: The mud composition used in preparing a water based mud for this
experimental work is barite, bentonite and water. The equation below was used in preparing the
mud samples to obtain a mud density of 12.5 ib/gal (Mian, 1991):
 mix 
M1  M 2  M 3
M1 M 2 M 3


1
2
(5.5)
3
Where
 mix = density of the mud mixture (g/cm3)
M 1 = mass of barite (g)
M 2 = mass of bentonite (g)
M 3 = mass of water (g)
1 = API density for barite (g/cm3
 2 = API density for bentonite (g/cm3)
3 = API density for water (g/cm3)
63
Table 5.2 shows the American Petroleum Institute (API) density for the additives used for this
work.
Table 5.2 API Densities of Mud Additives (Bourgoyne, 1991)
Mud Additive
Density ( ib/gal)
Barite
35
Bentonite
2.7
Water
8.33
5.4.5 Experimental procedure
The following test procedures were carried out in the laboratory at a temperature of 220c

The masses of barite, bentonite and water were each measured using the mass balance

The barite, bentonite and water were mixed and poured in the mud cup of the mud
balance and the mud density was measured by adjusting the rider in the balance arm until
a point of equilibrium was achieved. The mud density was then read from the level glass
indicator of the mud balance.

More additives was added to the mixture because the laboratory mud density of the
mixture was less than the theoretical mud density, until the mud density of 12.5 ib/gal
was obtained using the mud balance.

The mud mixture with a density of 12.5 (ib/gal) was filled in the stainless steel sample
test cup of the viscometer to the scribed line and placed on the instrument stage.

The lock nut of the viscometer was loosened and the instrument stage with the stainless
steel in it was raised until the rotor was immersed to the proper immersion depth of the
stainless steel cup and the lock nut was tightened.

The rotor of the viscometer was operated in a high speed position of 600 rpm with the
gear shifted down. The dial reading on the viscometer is recorded when the indicator
became steady.

The rotor of the viscometer was then switched to 300 rpm speed with the gear still shifted
down. The dial reading on the viscometer is also recorded when the indicator became
steady.
64

The plastic viscosity and the yield of the mud sample was obtained using the relationship
below (Model 35 viscometer instruction manual) :
PV  600  300
YP  300  PV
(5.6)
(5.7)
Where
PV = plastic viscosity (cP)
YP = yield point ib/100ft
 600 = dial reading of viscometer at 600 rpm
300 = dial reading of viscometer at 300 rpm
5.4.6 Composition of Mud Additives used
Table 5.3 shows the composition of mud additives used in the laboratory for preparing the three
mud samples.
Table 5.3: Composition of Mud Sample 1
Mud Additives
Barite
Bentonite
Water
Mass (g)
250
25
400
Taking measurements from the viscometer while running the viscometer at 600 rpm and 300 rpm
600  43
300  24
The plastic viscosity was obtained using equation (5.6)
PV  600  300
65
PV  19cP
The Yield point is obtained using equation (5.7)
YP  300  PV
YP  5ib / 100 ft 2
Table 5.4: Composition of Mud Sample 2
Mud Additives
Barite
Bentonite
Water
Mass (g)
350
40
550
Taking measure from the viscometer running the viscometer at 600 rpm and 300 rpm
600  80
300  46
The Plastic viscosity for this mud sample is obtained using equation (5.6)
PV  600  300
PV  34cP
The Yield point for this mud sample is obtained using equation ( 5.7)
YP  300  PV
YP  12ib / 100 ft 2
Table 5.5: Composition of mud sample 3
Mud Additives
Barite
Bentonite
Water
Mass (g )
560
90
795
66
Taking measure from the viscometer running the viscometer at 600rpm and 300rpm
600  85
300  72
The Plastic viscosity for this mud sample is obtained using equation (5.6)
PV  600  300
PV  15cP
The Yield point for this mud sample is obtained using equation (5.7)
YP  300  PV
YP  57 Ibs / 100 ft 2
5.5 PRESSURE DROP COMPUTATIONS
The pressure drop computation in the mud circulatory system for this work was done based on
the result obtained from the experimental work. The pressure drop across the mud circulatory
system comprises the pressure drop across the surface connection, the pressure drop in the drill
pipe and drill collar, the pressure drop in the annulus of the drill pipe and the drill collar and the
pressure drop across the drill bit. The pressure losses in the mud circulatory system will be
calculated using the properties of the three mud samples prepared in the laboratory. The mud
pump data, drill string data, drill bit and hole data used in the minimum flow rate, maximum
flow rate and pressure losses computation are given in appendix C.
5.5.1 Maximum and Minimum Flow Rate Calculation
The optimum flow rate to be used in drilling must be between the maximum and minimum flow
rates. The maximum flow rate in the well can be achieved at maximum mud pump pressure with
maximum pump horse power and pump efficiency. The minimum flow rate is a critical
parameter that should be high enough to carry the cuttings from the bottom hole. The minimum
flow are depends on the minimum annular velocity. Laboratory and field work carried out by
Williams and Bruce (1951) show that the minimum annular velocity necessary to remove drill
cuttings from a hole ranges from 100 to 125 ft per min (Williams and Bruce, 1951).
67
Maximum flow rate can be calculated with the relationship given as (Bourgoyne, 1991):
Qmax  1714 * *
H hp
Pp max
(5.8)
Where
Qmax = maximum flow rate (gpm)
H hp = hydraulic horse power (hp)
Pp max = maximum allowed operating pressure of the pump (psia)
 = volumetric efficiency
Therefore the maximum flow rate in the well can be calculated using the given pump data as
shown below:
Qmax  1714 * 0.8 *
1765
4000
Qmax = 605.04 gpm
Minimum flow rate can be calculated with the relationship given as (Bourgoyne, 1991):
Qmin  2.448 * ( Dh2  Dp2 ) * vmin
(5.9)
Where
Qmin = minimum flow rate (gpm)
Dh = diameter of the hole (in)
Dp = diameter of the pipe (in)
vmin = minimum annular velocity (ft/s)
A minimum annular velocity of 120 ft/min is used for this design in according with Williams and
bruce (1951).
68
Therefore the minimum flow rate in the well can be calculated as shown below:
 120 
Qmin  2.448 * (7.8752  42 ) * 

 60 
Qmin = 378.35 gpm
Therefore, the operating flow rate ranges from the minimum (378.35 gpm) to maximum (667.4
gpm). Hence the parasitic pressure losses will be computed at a flow rate between the maximum
and minimum flow rate (Bourgoyne, 1991). A flow rate of 400 gpm (Bourgoyne, 1991) will be
chosen for the parasitic pressure losses computation in this work.
5.5.2 Calculating Pressure Drop in the Drill String
In calculating the pressure drop in the drill string it is very important to know the flow pattern in
the drill pipe and in the drill collar. The pressure drop across the drill pipe and drill collar was
calculated for each depth starting from the point of entry into the over pressured zone which is at
a depth of 9600 ft.
5.5.2.1 Pressure Drop Across the Drill Pipe
The flow regime drilling at a depth of 9600ft was determined using the Hedstrom number
criterion using a Mud Sample 1 which has a viscosity of 19 cP and yield point of 5 ib/100ft.
The Hedstrom number given in equation (4.10)
N He 
37100 *12.495 * 5 * 42
 102728.97
192
69
The critical Reynolds number is obtained using the graph below :
Fig 5.5 Critical Reynolds number Bingham plastic fluids (Bourgoyne, 1991)
The Critical Reynolds number = 7000 as seen in Figure 5.5, using the Hedstrom Number (
102728.97 )
The Reynolds number is given by (Bourgoyne, 1991):
N Re 
928 *  m * * d
p
(5.10)
Where
N Re = reynolds number
 m = density of mud (ib/gal)
 p = mud plastic viscosity (cP)
 = mean fluid velocity (ft/s)
d = diameter of drill pipe (in)
The mean fluid velocity in the drill pipe is given as (Bourgoyne, 1991):
70
V
Q
2
2.45 * Didp
(5.11)
Where
V = mean fluid velocity in the drill pipe (ft/s)
Q = flow rate (gpm)
Didp = internal diameter of the drill pipe or drill collar (in)
V
Q
400

 10.2 ft / s
2
2.45 * Didp 2.45 * 42
Hence,
N Re 
928 *12.495 *13.28 * 4
 24899.487
19
The critical Reynolds number is less than the Reynolds number; therefore the flow regime in the
drill pipe is turbulent. The frictional pressure loss in the drill pipe for turbulent flow is given in
equation (4.11).
Pdp 
Pdp 
m0.75 * 1.75 *  0p.25 * Ldp
1.25
1800 Didp
12.4950.75 *10.21.75 *190.25 * (9600  1000)
1800 * 41.25
Therefore the pressure drop in the drill pipe at a depth of 9600 ft is given as
Pdp  682.33 psia
71
The same procedure is repeated for the pressure drop across the drill pipe at subsequent depth in
the over pressure zone. This calculation is also repeated using Mud Samples 2 and 3 which is
given in Appendix G and H.
5.5.2.2 Pressure Drop across the Drill Collar
The flow regime will be determined using the Hedstrom number criteria using a Mud Sample 1
Hedstrom number equation is given by equation (4.10) as shown below:
N He 
N He 
37100 * m * y * de2
 p2
37100 *12.495 * 5 * 4.52
 130016.36
192
From Figure 5.5, The Critical Reynolds number =22000 is obtained using the Hedstrom number
(130016.36).
The mean velocity in the drill pipe is obtained using equation (5.11), replacing the diameter of
the drill pipe with the diameter of the drill collar. The fluid velocity in the drill collar is given
below.
V
Q
2
2.45 * Didc
V
Q
400

 8.06 ft / s
2
2.45 * Didc 2.45 * 4.52
Therefore the Reynolds number is given below:
72
N Re 
928 *12.495 * 8.06 * 4.5
 22134.93
19
The Critical Reynolds number is less than the Reynolds number therefore the flow regime in the
drill collar is turbulent. The frictional pressure loss in the drill collar for a turbulent flow is given
in equation (4.11)
Pdc 
 m0.75 * 1.75 *  0p.25 * Ldc
1.25
1800 Didc
12.4950.75 * 8.061.75 *190.25 *1000
Pdc 
 48.54 psia
1800 * 4.51.25
Therefore the pressure drop across the drill collar is:
Pdc  48.54 psia
This calculation is repeated using mud sample 2 and 3 which is given in Appendix G and H.
5.5.3 Calculating Pressure Drop in Surface Connection
The surface connection pressure drop will be calculated using the relationship below since the
equivalent surface equipment length of drill pipe is known. The equation for calculating pressure
drop in surface connection in equation (4.4) is given below.
Pf S  Pdp *
Lse
Ldp
Pf S  682.33 *
340
(9600  1000)
Pf S  26.98 psia
73
5.5.4 Calculating Pressure Drop in Annulus
In calculating the pressure drop in the annulus it is very important to know the flow regime in the
annulus of the drill pipe and in the drill collar. The pressure drop across the annulus of the drill
pipe and drill collar will then be calculated for each depth starting from the point of entry into the
over pressured zone which is at a depth of 9600ft.
5.5.4.1 Pressure Drop across the Annulus of Drill Pipe
The flow regime will be determined using the Hedstrom number criteria using Mud Sample
1.The Hedstrom number equation which is given by equation (4.10) where the internal diameter
is replaced with the equivalent diameter.
N He 
37100 * m * y * de2
 p2
The equivalent diameter de is given as (Bourgoyne, 1991):
de  0.816 * (dh  dodp)
(5.12)
Where
d e = equivalent diameter (in)
d h = diameter of the hole (in)
d odp = outer diameter of the drill pipe (in)
de  0.816 * (7.875  5.25)
74
de  2.142in
N He 
37100 *12.495 * 5 * 2.1422
 29458.59
192
Figure 5.5, the critical Reynolds number =10000 is obtained using the Hedstrom number
(29458.59)
The mean velocity in the drill pipe is given as (Bourgoyne, 1991):
V
Q
2
2.45 * ( Dh2  Dodp
)
(5.13)
Where
Va = average velocity of fluid in the drill pipe (ft/s)
Q = flow rate (gpm)
Dodp = outer diameter of the drill collar (in)
Dh = diameter of the hole (in)
Va 
Q
400

 4.74 ft / s
2
2
2.45 * ( Dh  Dodp ) 2.45 * (7.8752  5.252 )
N Re 
928 *12.495 * 4.74 * 2.142
 6196.24
19
The critical Reynolds number is greater than the Reynolds number, therefore the flow regime in
the drill collar is laminar. The frictional pressure loss in the annulus of the drill pipe for a laminar
flow regime starting at a depth of 9600 ft is calculated by the relationship given in equation
(4.16).
75
Pfadp 
Pfadp 
Ldp *  p * Va
1000( Dh  Dodp)
2

 y * LdP
200( Dh  Dodp)
(9600  1000) *19 * 4.74
5 * (9600  1000)

2
1000 * (7.875  5.25)
200 * (7.875  5.25)
Pfadp  194.30 psia
The same procedure is repeated for the pressure drop across the annulus of the drill pipe at
subsequent depths in the over pressure zone. This calculation is also repeated using Mud
Samples 2 and 3 which is given in Appendix G and H.
5.5.4.2 Pressure Drop across the Annulus of the Drill Collar
The flow regime will be determined using the Hedstrom number criteria using mud sample 1.
Hedstrom number equation is given by equation (4.10).
N He 
37100 * m * y * de2
 p2
The equivalent diameter de is calculated using equation (5.12) where the outer diameter of the
drill pipe is replaced with the other diameter of the drill collar.
de  0.816 * (dh  dodc)
de  0.816 * (7.875  6.5)
de  1.122in
76
N He 
37100 *12.495 * 5 *1.1222
 8082.74
192
From Figure 5.5, the Critical Reynolds number =3200 is obtained using the Hedstrom number
(8082.72).The Reynolds number is given by equation (5.10).
The mean velocity in the annulus of the drill collar is calculated using equation (5.13) where the
outer diameter of the drill pipe is replaced with outer diameter of the drill collar given below.
Va 
Q
2
2.45 * ( Dh2  Dodc
)
Va 
Q
400

 8.26 ft / s
2
2
2.45 * ( Dh  Dod ) 2.45 * (7.8752  6.52 )
N Re 
928 *12.495 * 8.26 *1.122
 5655.92
19
The critical Reynolds number is less than the Reynolds number, therefore the flow regime in the
drill collar is turbulent. Since the flow is turbulent the frictional pressure loss in the annulus of
the drill collar is calculated using equation (4.11) where the internal diameter is replaced with
equivalent diameter as shown below.
Padc 
Padc 
 m0.75 * 1.75 *  0p.25 * Ldc
1800 * De1.25
12.4950.75 * 8.261.75 *190.25 *1000
 268.63 psia
1800 *1.1221.25
Therefore the pressure drop across the drill collar is given below:
77
Padc  268.63 psia
This calculation is also repeated using Mud Samples 2 and 3 which are given in Appendix G and
H.
5.6 Optimization using the Maximum Drill Bit Horsepower Criterion
The condition for maximum drill bit horse power derived by Kendal and Goins (1960) states that
bit hydraulic horse power is maximum when the parasitic pressure loss is given by the equation
(4.28) shown below.
( Pf ) Dopt 
Pmax
m 1
Kendal and Goins (1960) stated that the theoretical value for the flow exponent (m) is 1.75
Therefore the optimum parasitic pressure loss using the maximum bit horsepower criteria for this
case study is given below:
( Pf ) Dopt 
Pmax
4000

m  1 1.75  1
( Pf ) Dopt  1454.55 psia
Kendal and Goins further stated that for bit hydraulic horse power to be maximum the pressure
across the drill bit is given by the relationship below.
( Pf ) Bopt  Pmax  ( Pf ) Dopt
(5.14)
Where
( Pf ) Bopt = optimum pressure drop on the drill bit (psia)
( Pf ) Dopt = optimum Parasitic pressure drop (psia)
Pmax = maximum pump pressure (psia)
78
Therefore for optimum condition to be achieved the pressure across the drill bit must be
maintained, in this case study the pressure across the drill bit is given below :
( Pf ) Bopt  4000  1454.55  2545.45 psia
In other to maintain the optimum pressure across the drill bit in this case study, the pump must
be operated at the optimum flow rate to the target depth of 11500 ft.
5.6.1 Optimum Flow Rate to Operate the Mud Pump
Pressure drop increases with depth, hence in order to drill at the optimum condition the pump
must be operated at the optimum flow rate at each depth as the well is being drilled to the target
depth. In other to achieve this objective in this case study a graphical approach is used. The
optimum flow rate across each depth in the overpressure zone is obtained from the hydraulic
plot. The hydraulic plot is a log-log plot of parasitic pressure loss against flow rate. The data for
the parasitic pressure drop and flow rate used for the hydraulic plot is given in Appendix D.
79
3.185
Line extends to maximum
mud pressure Log(Pmax)=
3.6
Optimum
Parasitic
pressure
loss
log(Pdopt)
=3.163
3.165
11500ft
3.145 y = 1.75x - 1.295
Log Pd
11300ft
11200ft
10850ft
3.125
Optimum
hydraulic
plot
11050ft
11000ft
m= 1.75
10650ft
10600ft
10500ft
Over
pressure zone
line of
interest
10300ft
10200ft
10150ft
3.105
10050ft
10000ft
9950ft
9900ft
9750ft
9700ft
3.085
2.45
9600ft
Qmin
2.55
2.65
Log Q
Qmax
2.75
Fig 5.6: Hydraulic log-log plot of parasitic pressure losses versus flow rate using Appendix D
80
The optimum flow at each depth in the over pressured zone is obtained by taking a straight line
with a slope =1.75 across each depth on the over pressure line, and recording the point where the
line hits the optimum hydraulic path line, which gives the optimum flow rate at each depth. This
is given in Appendix F.
5.6.2 Optimum Nozzle Area of Drill Bit
The optimum nozzle area and diameter is obtained at optimum conditions to drill to the target
depth of 11500ft. It is obtained by using the equation below (Azar and Samuel, 2007):
( At )opt 
2
8.311 *105 *  m * Qopt
2
Cd * Pbopt
(5.15)
Where
( At ) opt = optimum nozzle area (in2)
Qopt = optimum flow rate (gpm)
m = flow exponent
Pbopt = optimum pressure on the drill bit (psia)
Cd = discharge coefficient
 m = mud density (ib/gal)
The optimum nozzle area is calculated using the equation below assuming that the three nozzles
of the drill bit are of equal area.
d N opt 
4( At )opt
3
(5.16)
81
Where
d N opt
= optimum nozzle diameter (in)
( At ) opt = optimum nozzle area (in2)
Table 5.6 shows the optimum nozzle area and optimum nozzle size of the drill bit using equation
5.15 and 5.16.
Table 5.6: Optimum Nozzle Area and Size Across Each Depth in the Overpressure zone.
Depth(ft) Qopt(gpm) Aopt(sq in) Dopt(in)
9600
439.54
0.364
0.393
9700
438.53
0.363
0.392
9750
438.53
0.363
0.393
9900
434.51
0.36
0.39
9950
433.51
0.36
0.39
10000
432.51
0.358
0.389
10050
431.52
0.358
0.389
10150
429.54
0.356
0.389
10200
428.55
0.355
0.388
10300
426.58
0.354
0.388
10500
421.69
0.349
0.385
10600
420.73
0.349
0.385
10650
419.75
0.348
0.384
10850
416.87
0.345
0.383
11000
413.99
0.343
0.382
11050
412.09
0.342
0.381
11200
410.21
0.34
0.38
11300
408.32
0.338
0.379
11500
403.65
0.335
0.377
In other to drill at optimum conditions in the over pressured zone to the target depth of 11500 ft
using the maximum hydraulic horse power criterion, the flow rates at which the pump must be
operated are given in the Table 5.6 as obtained from the hydraulic plot of Figure 5.6.
82
5.6.3 Maximum Hydraulic Horse Power on the Drill Bit
Using the optimum hydraulic horsepower criteria, the hydraulic horse power at the bit can be
determined from the relationship given in equation (4.29) as shown below (Kendall and Goins,
1960):
HHPopt 
PBopt * Qopt
1714
Table 5.7: Optimum Hydraulic Horsepower at Each Depth in the Overpressure zone.
Depth(ft) Qopt(gpm) Aopt(sq in) Dopt(in)
HHPbopt(hp)
9600
439.54
0.364
0.393
652.76
9700
438.53
0.363
0.392
651.26
9750
438.53
0.363
0.393
651.26
9900
434.51
0.36
0.39
645.29
9950
433.51
0.36
0.39
643.8
10000
432.51
0.358
0.389
642.32
10050
431.52
0.358
0.389
640.85
10150
429.54
0.356
0.389
637.91
10200
428.55
0.355
0.388
636.44
10300
426.58
0.354
0.388
633.51
10500
421.69
0.349
0.385
626.25
10600
420.73
0.349
0.385
624.82
10650
419.75
0.348
0.384
623.37
10850
416.87
0.345
0.383
619.09
11000
413.99
0.343
0.382
614.81
11050
412.09
0.342
0.381
611.99
11200
410.21
0.34
0.38
609.2
11300
408.32
0.338
0.379
606.39
11500
403.65
0.335
0.377
599.46
83
CHAPTER 6: RESULT AND DISCUSSION
The results of this project work are based on experimental work carried out and pressure loss
calculation.
6.1 Frictional Pressure Loss of Mud Samples Result
Frictional pressure losses are observed to increase with depth across the drill pipe, and the
annulus of the drill pipe, but the frictional pressure drop across the surface connection, drill
collar and the annulus of the drill collar remain the same.
6.1.1 Using mud sample 1
Mud Sample 1 has a viscosity of 19 cP and a yield point of 5 lb/100ft2. Table 6.1 shows the
pressure losses in the mud circulatory system when Mud Sample 1 was used in the design work
with an assumed flow rate of 400 gpm. This results shows that a larger percentage of the pressure
loss in the mud circulatory system was found to occur in the drill pipe which is as a result of
turbulent flow in the drill pipe. Table 6.1 shows all the frictional pressure losses observed when
using this mud.
Table 6.1 Frictional Pressures Losses in the Mud Circulatory System Using Mud Sample 1
Depth (ft) Ps (psia) Pdp (psia) Pdc (psia) Padp (psia) Padc (psia) Pd (psia)
9600
9700
9750
9900
9950
10000
10050
10150
10200
10300
10500
10600
10650
10850
11000
11050
11200
11300
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
26.98
682.33
690.26
694.23
706.13
710.09
714.06
718.03
725.96
729.93
737.86
753.73
761.66
765.63
781.5
793.4
797.37
809.27
817.2
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
48.54
194.3
196.56
197.69
201.08
202.21
203.34
204.47
206.73
207.86
210.11
214.63
216.89
218.02
222.54
225.93
227.06
230.45
232.71
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
268.63
1220.78
1230.96
1236.06
1251.35
1256.45
1261.54
1266.64
1276.83
1281.93
1292.12
1312.51
1322.7
1327.8
1348.19
1363.48
1368.57
1383.86
1394.06
Ps = pressure losses in surface connection (psia)
Pdp = pressure losses in the drill pipe (psia)
Pdc = pressure losses in the drill collar (psia)
84
Padp = pressure losses across the annulus of the drill pipe (psia)
Padc = pressure losses across the annulus of the drill collar (psia)
Pd = parasitic pressure loss (psia) is the summation of the above losses.
6.1.2 Using mud sample 2
Mud Sample 2 has a viscosity of 34cP and a yield point of 12 ib/100ft2. Table 6.2 shows the
pressure losses in the mud circulatory system when Mud Sample 2 was used in the design work
with an assumed flow rate of 400 gpm. This result also shows that a larger percentage of the
pressure loss in the drill pipe. However Mud Sample 2 has a higher parasitic pressure loss
compare to Mud sample 1 which is due to the difference in viscosity and yield point. Table 6.2
shows all the frictional pressure losses observed when using this mud.
Table 6.2 Frictional Pressures Losses in the Mud Circulatory System using Mud Sample 2
Depth(ft) Ps (psia) Pdp (psia) Pdc (psia) Padp (psia) Padc (psia) Pd (psia)
9600
31.19
789.1
56.14
558.35
310.66 1745.44
9700
31.19
798.27
56.14
564.84
310.66
1761.1
9750
31.19
802.86
56.14
568.08
310.66 1768.94
9900
31.19
816.63
56.14
577.82
310.66 1792.44
9950
31.19
821.21
56.14
581.07
310.66 1800.27
10000
31.19
825.8
56.14
584.31
310.66 1808.11
10050
31.19
830.39
56.14
587.56
310.66 1815.94
10150
31.19
839.56
56.14
594.05
310.66 1831.61
10200
31.19
844.15
56.14
597.3
310.66 1839.44
10300
31.19
853.33
56.14
603.79
310.66 1855.11
10500
31.19
871.68
56.14
616.78
310.66 1886.45
10600
31.19
880.86
56.14
623.27
310.66 1902.11
10650
31.19
885.44
56.14
626.52
310.66 1909.95
10850
31.19
903.79
56.14
639.5
310.66 1941.28
11000
31.19
917.56
56.14
649.24
310.66 1964.79
11050
31.19
922.14
56.14
652.48
310.66 1972.62
11200
31.19
935.91
56.14
662.22
310.66 1996.12
11300
31.19
945.08
56.14
668.71
310.66 2011.79
11500
31.19
963.43
56.14
681.7
310.66 2043.13
Ps = pressure losses in surface connection (psia)
Pdp = pressure losses in the drill pipe (psia)
Pdc = pressure losses in the drill collar (psia)
Padp = pressure losses across the annulus of the drill pipe (psia)
85
Padc = pressure losses across the annulus of the drill collar (psia)
Pd = parasitic pressure loss (psia) is the summation of the above losses.
6.1.3 Using mud sample 3
Mud Sample 3 has a viscosity of 15 cP and a yield point of 57 ib/100ft2. Table 6.3 shows the
pressure losses in the mud circulatory system when Mud Sample 3 was used in the design work
with an assumed flow rate of 400gpm. A larger percentage of the pressure loss was found to
occur in the annulus of the drill pipe using this mud sample which is due to a high yield point of
the mud. Table 6.3 shows all the frictional pressure losses observed when using this mud.
Table 6.3 Frictional Pressures Losses in the Mud Circulatory System Using Mud Sample 3
Depth (ft) Ps (psia)
Pdp (psia) Pdc (psia) Padp (psia) Padc (psia) Pd (psia)
9600
25.32
640.53
45.57
1020.96
271.71 2004.09
9700
25.32
647.98
45.57
1032.83
271.71 2023.41
9750
25.32
651.7
45.57
1038.77
271.71 2033.07
9900
25.32
662.87
45.57
1056.57
271.71 2062.05
9950
25.32
666.6
45.57
1062.51
271.71 2071.71
10000
25.32
670.32
45.57
1068.45
271.71 2081.37
10050
25.32
674.04
45.57
1074.38
271.71 2091.03
10150
25.32
681.49
45.57
1086.25
271.71 2110.35
10200
25.32
685.22
45.57
1092.19
271.71 2120.01
10300
25.32
692.66
45.57
1104.06
271.71 2139.32
10500
25.32
707.56
45.57
1127.8
271.71 2177.96
10600
25.32
715.01
45.57
1139.67
271.71 2197.28
10650
25.32
718.73
45.57
1145.61
271.71 2206.94
10850
25.32
733.62
45.57
1169.35
271.71 2245.58
11000
25.32
744.8
45.57
1187.16
271.71 2274.56
11050
25.32
748.52
45.57
1193.09
271.71 2284.22
11200
25.32
759.7
45.57
1210.91
271.71
2313.2
11300
25.32
767.14
45.57
1222.78
271.71 2332.52
11500
25.32
782.04
45.57
1246.52
271.71 2371.16
Ps = pressure losses in surface connection (psia)
Pdp = pressure losses in the drill pipe (psia)
Pdc = pressure losses in the drill collar (psia)
Padp = pressure losses across the annulus of the drill pipe (psia)
Padc = pressure losses across the annulus of the drill collar (psia)
86
Pd = parasitic pressure loss (psia) is the summation of the above losses.
6.2 Results for Pump Operating Conditions
Based on the analysis from the three mud samples, mud sample 1 was selected for this design
work, because it has the lowest parasitic pressure losses and it produces the least strain on the
mud pump. To optimize the hydraulic power across the drill bit to the target depth of 11500 ft, in
order to maintain an optimum parasitic pressure loss of 1454.55psia and an optimum pressure
drop of 2545.55psia across the drill bit, mud pump must be operated at the drilling conditions
given in Table 6.4.
Table 6.4 Optimum Hydraulic Conditions for This Case Study
Depth(ft) Qopt(gpm) Aopt(sq in) Dopt(in)
HHPbopt(hp)
9600
439.54
0.364
0.393
652.76
9700
438.53
0.363
0.392
651.26
9750
438.53
0.363
0.393
651.26
9900
434.51
0.36
0.39
645.29
9950
433.51
0.36
0.39
643.8
10000
432.51
0.358
0.389
642.32
10050
431.52
0.358
0.389
640.85
10150
429.54
0.356
0.389
637.91
10200
428.55
0.355
0.388
636.44
10300
426.58
0.354
0.388
633.51
10500
421.69
0.349
0.385
626.25
10600
420.73
0.349
0.385
624.82
10650
419.75
0.348
0.384
623.37
10850
416.87
0.345
0.383
619.09
11000
413.99
0.343
0.382
614.81
11050
412.09
0.342
0.381
611.99
11200
410.21
0.34
0.38
609.2
11300
408.32
0.338
0.379
606.39
11500
403.65
0.335
0.377
599.46
Qopt = optimum flow rate (gpm)
Aopt = optimum nozzle area (in2)
Dopt = optimum nozzle diameter (in)
HHPbopt= optimum hydraulic horsepower across the drill bit (hp)
87
CHAPTER 7: CONCLUSIONS AND RECOMMENDATION
7.1 Conclusions
Drilling is the most capital intensive stage of hydrocarbon exploration projects and as such
requires geological, technical and economic evaluations. The basis for optimization studies in
this project work was geological data based on conductivity measurement in a potential
hydrocarbon basin. In this project, a laboratory based approach was used for mud rheology
determination for pressure loss computations. In the petroleum industry, optimization techniques
are based on the maximum horse power criterion and the jet impact force criterion, however this
project work considers the maximum bit horse power criterion. The following are the principal
conclusions of this study.
1. The results for pore pressure and fracture pressure predictions obtained are in accordance
with correlations used in the industry and in other published works.
2. The mud sample with a density of 12.5 ib/gal, a viscosity of 19 cP and a yield point of 5
ib/100ft2 gave the least parasitic pressure loss.
3. The optimum flow (403.65 gpm) rate required to operate the mud pump to the target
depth lies within the minimum flow rate (378.35 gpm) and maximum flow rate (605.04
gpm) which is in accordance with industry standard for bottom hole cleaning.
7.2 Recommendation
Pore pressure and fracture pressure prediction is the basis for well planning, hence it is not
sufficient to make accurate prediction of pore pressure and fracture pressure using only
conductivity data as carried out in this work. Hence, an integrated approach using other well
logging data should be incorporated to deduce pore pressure and fracture pressure; this would
help in understanding the uncertainty in each method. Also, the flow exponent should be
obtained directly by operating the mud pump on a drilling rig instead of using a theoretical flow
exponent as carried out in this study.
88
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Prassl, W. F., & Dipl, L. (2003). Drilling Engineering. Curtin University of Technology.
Rabia, H. (2002). Well Engineeering and Construction. Elsevier Science Publishers.
Redmann, K. P. (1991). Understanding Kick Tolerance and Its Significance in Drilling Planning
and Execution. SPE Drilling Engineering, 245-249.
90
Reynolds, o. (1895). on the dynamic theory of incompressible viscous fluids and the
determination of the criterion. London: Royal Society of London.
Soufi, N. (2009). Shale Pressure Measurements methods. Institute of Petroleum Engineering
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Tobenna, C. U. (2010). Hole Cleaning and Hydraulics.
Warren, T. M. (1987). Penetration rate performance of roller cone bit . SPE Drilling
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Petroleum Engineering.
Williams, C. E., & Bruce, G. H. (1951). Carrying Capacity of Drilling Muds. Journal of
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Wyllie, M. R., Gregory, A. R., & Gardner, G. H. (2008). An experimental investigation of factors
affecting elastic wave velocities in porous media. Standford: American Geological
Institute.
91
APPENDIX
Appendix A
Depth (ft) C
7400
710
7500
780
7550
790
8300
710
8350
690
8400
680
8500
690
9200
600
9300
590
9550
570
9600
590
9700
610
9750
621
9900
700
9950
830
10000
950
10050 1100
10150 1200
10200 1240
10300 1310
10500 1250
10600 1350
10650 1370
10850 1500
11000 1280
11050 1400
11200 1650
11300 1840
11500 1920
R
0.00141
0.00128
0.00127
0.00141
0.00145
0.00147
0.00145
0.00167
0.00169
0.00175
0.00169
0.00164
0.00161
0.00143
0.0012
0.00105
0.00091
0.00083
0.00081
0.00076
0.0008
0.00074
0.00073
0.00067
0.00078
0.00071
0.00061
0.00054
0.00052
ln R
PP
FP
MDPP DMDPP MDFP
6.56526
3441 13312.6 8.94231 9.44231 34.5962
6.65929 3487.5 13492.5 8.94231 9.44231 34.5962
6.67203 3510.75 13582.5 8.94231 9.44231 34.5962
6.56526 3859.5 14931.7 8.94231 9.44231 34.5962
6.53669 3882.75 15021.7 8.94231 9.44231 34.5962
6.52209
3906 15111.6 8.94231 9.44231 34.5962
6.53669 3952.5 15291.5 8.94231 9.44231 34.5962
6.39693
4278 16550.8 8.94231 9.44231 34.5962
6.38012 4324.5 16730.7 8.94231 9.44231 34.5962
6.34564 4440.75 17180.5 8.94231 9.44231 34.5962
6.38012
4464 17270.4 8.94231 9.44231 34.5962
6.41346 4510.5 17450.3 8.94231 9.44231 34.5962
6.43133 4533.75 17540.3 8.94231 9.44231 34.5962
6.55108 4603.5 17810.1 8.94231 9.44231 34.5962
6.72143
4975 18248.3 9.61538 10.1154 35.2692
6.85646
5100
18440 9.80769 10.3077 35.4615
7.00307 5326.5 18733.2 10.1923 10.6923 35.8462
7.09008 5379.5 18919.6 10.1923 10.6923 35.8462
7.12287
5508 19114.8 10.3846 10.8846 36.0385
7.17778
5562 19302.2 10.3846 10.8846 36.0385
7.1309
5775
19782 10.5769 11.0769 36.2308
7.20786
6042 20182.4 10.9615 11.4615 36.6154
7.22257 6070.5 20277.6 10.9615 11.4615 36.6154
7.31322 6184.5 20658.4 10.9615 11.4615 36.6154
7.15462
6270
20944 10.9615 11.4615 36.6154
7.24423
6409 21149.7 11.1538 11.6538 36.8077
7.40853
6608 21548.8 11.3462 11.8462
37
7.51752
6667 21741.2 11.3462 11.8462
37
7.56008
6785
22126 11.3462 11.8462
37
MDPP= Mud Density from Pore Pressure (psia)
MDPF= Mud Density from Fracture Pressure (psia)
DMDPP= Design Mud Density from Fracture Pressure (psia)
R= Resistivity (Ω)
C=Conductivity (1/Ω)
PP = Pore Pressure (psia)
FP =Fracture Pressure (psia)
DMDFP
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.0962
34.7692
34.9615
35.3462
35.3462
35.5385
35.5385
35.7308
36.1154
36.1154
36.1154
36.1154
36.3077
36.5
36.5
36.5
92
Appendix B
Depth (ft) MDPP
7400 8.942308
7500 8.942308
7550 8.942308
8300 8.942308
8350 8.942308
8400 8.942308
8500 8.942308
9200 8.942308
9300 8.942308
9550 8.942308
9600 8.942308
9700 8.942308
9750 8.942308
9900 8.942308
9950 9.615385
10000 9.807692
10050 10.19231
10150 10.19231
10200 10.38462
10300 10.38462
10500 10.57692
10600 10.96154
10650 10.96154
10850 10.96154
11000 10.96154
11050 11.15385
11200 11.34615
11300 11.34615
11500 11.34615
DMDPP
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
9.442308
10.11538
10.30769
10.69231
10.69231
10.88462
10.88462
11.07692
11.46154
11.46154
11.46154
11.46154
11.65385
11.84615
11.84615
11.84615
MDFP
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
34.59615
35.26923
35.46154
35.84615
35.84615
36.03846
36.03846
36.23077
36.61538
36.61538
36.61538
36.61538
36.80769
37
37
37
DMDFP
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.09615
34.76923
34.96154
35.34615
35.34615
35.53846
35.53846
35.73077
36.11538
36.11538
36.11538
36.11538
36.30769
36.5
36.5
36.5
SG-PP
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.073506
1.154308
1.177394
1.223566
1.223566
1.246653
1.246653
1.269739
1.315911
1.315911
1.315911
1.315911
1.338997
1.362083
1.362083
1.362083
DSG-PP
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.13353
1.214332
1.237418
1.28359
1.28359
1.306677
1.306677
1.329763
1.375935
1.375935
1.375935
1.375935
1.399021
1.422107
1.422107
1.422107
SG-FP
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.1532
4.234001
4.257087
4.30326
4.30326
4.326346
4.326346
4.349432
4.395604
4.395604
4.395604
4.395604
4.418691
4.441777
4.441777
4.441777
DSG-FP
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.093176
4.173977
4.197063
4.243236
4.243236
4.266322
4.266322
4.289408
4.33558
4.33558
4.33558
4.33558
4.358667
4.381753
4.381753
4.381753
MDPP= Mud Density from Pore Pressure (psia)
MDPF= Mud Density from Fracture Pressure (psia)
DMDPP= Design Mud Density from Fracture Pressure (psia)
SG-PP= Specific gravity from Pore pressure
DSG-PP=Design Specific gravity from Pore pressure
SG-FP= Specific gravity from Fracture pressure
DSG-FP=Design Specific gravity from Fracture pressure
93
Appendix C
In this case study, the following data’s were used for (Azar and Samuel, 2007).
Given the following well data:
TARGET DEPTH (TM): 11500 ft
Hole size to TD = 7 7/8 in
DRILL PIPE

OD = 5 ¼ in


ID = 4 in
Air weight = 26.66 ib/ft
DRILL COLLAR



Length = 1000 ft
OD = 6 ½ in
ID = 4.5 in
MUD PROGRAM


Bingham plastic model
Mud density= 12.459 ppg
PUMP



Maximum allowed Operating Pressure = 4000psia
Hydraulic Horsepower = 1765hp
Volumetric Efficiency = 80%
Drill bit
12 7/8 in tricon with 3-14 nozzles to 9600ft
Minimum required annular fluid velocity: 120ft/min
Surface Equipment: Equivalent to 340ft of drill pipe
Field data: using the 8 7/8 tricone 3-14 Nozzle
94
Appendix D
Using Mud Sample 1
D(ft) Ps(psia) Pdp(psia) Pdc(psia)
9600 26.976
682.33
48.54
9700 26.976 690.258
48.54
9750 26.976 694.225
48.54
9900 26.976 706.126
48.54
9950 26.976 710.093
48.54
10000 26.976
714.06
48.54
10050 26.976 718.027
48.54
10150 26.976 725.961
48.54
10200 26.976 729.928
48.54
10300 26.976 737.862
48.54
10500 26.976
753.73
48.54
10600 26.976 761.664
48.54
10650 26.976 765.631
48.54
10850 26.976 781.499
48.54
11000 26.976
793.4
48.54
11050 26.976 797.367
48.54
11200 26.976 809.268
48.54
11300 26.976 817.202
48.54
11500 26.976
833.07
48.54
Padp(psia) Padc(psia) Pd(psia) LOG Pd Q (gpm) LOG Q
194.3
268.63 1220.78 3.0866
400
196.5591
268.63 1230.96 3.0902
400
197.6888
268.63 1236.06
3.092
400
201.0777
268.63 1251.35 3.0974
400
202.2074
268.63 1256.45 3.0991
400
203.337
268.63 1261.54 3.1009
400
204.4667
268.63 1266.64 3.1027
400
206.726
268.63 1276.83 3.1061
400
207.8556
268.63 1281.93 3.1079
400
210.1149
268.63 1292.12 3.1113
400
214.6335
268.63 1312.51 3.1181
400
216.8928
268.63 1322.7 3.1215
400
218.0225
268.63 1327.8 3.1231
400
222.5411
268.63 1348.19 3.1297
400
225.93
268.63 1363.48 3.1346
400
227.0597
268.63 1368.57 3.1363
400
230.4486
268.63 1383.86 3.1411
400
232.7079
268.63 1394.06 3.1443
400
237.2265
268.63 1414.44 3.1506
400
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
Ps loss = Surface connection pressure loss (psia)
Pdopt =Optimium parasitic loss =1454.55 psia
Pdp loss =Drill Pipe pressure loss
Pbopt =Optimum drill bit pressure =2545.45psia
Pdc loss =Drill Collar Pressure loss (psia)
Padp loss = Annulus of the Drill Pipe pressure loss (psia)
Qmin =Mimimum flow rate =378.35gpm
Padc loss = Annulus of the Drill Collar pressure loss (psia)
Qmax =Maximum flow rate= 605.04gpm
Pd Loss = Parasitic Pressure loss (psia)
Viscosity = 19CP
Q= Assumed Flow Rate (gpm)
Yield Point = 5
D =Depth (ft)
P max= Maximum pump pressure =4000psia
LOG Qmin = 2.576
LOG Qmax = 2.782
LOG Pdopt = 3.163
LOG Pmax = 3.6
95
Appendix E
Hottman and Johnson relationship between formation pore pressure gradient with shale
resistivity ratio (Bourgoyne, 1991).
Plot of formation pore pressure gradient with shale resistivity ratio (Bourgoyne, 1991)
96
Appendix F
Depth(ft)
Log Qopt
Qopt(gpm)
2.643
439.54
2.642
438.53
2.642
438.53
2.638
434.51
2.637
433.51
2.636
432.51
2.635
431.52
2.633
429.54
2.632
428.55
2.63
426.58
2.625
421.69
2.624
420.73
2.623
419.75
2.62
416.87
2.617
413.99
2.615
412.09
2.613
410.21
2.611
408.32
2.606
403.65
Qopt = Optimum flow rate obtained by taking the antilog
Log Qopt= obtained from the hydrualic plot
9600
9700
9750
9900
9950
10000
10050
10150
10200
10300
10500
10600
10650
10850
11000
11050
11200
11300
11500
97
Appendix G
Mud Sample 2
Depth(ft) Ps loss(psia) Pdp loss (psia) Pdc loss (psia) Padp loss (psia) Padc loss (psia) Pd loss(psia)
9600
31.19
789.1
56.14
558.35
310.66
1745.44
9700
31.19
798.28
56.14
564.84
310.66
1761.1
9750
31.19
802.86
56.14
568.08
310.66
1768.94
9900
31.19
816.63
56.14
577.82
310.66
1792.44
9950
31.19
821.21
56.14
581.07
310.66
1800.27
10000
31.19
825.8
56.14
584.31
310.66
1808.11
10050
31.19
830.39
56.14
587.56
310.66
1815.94
10150
31.19
839.56
56.14
594.05
310.66
1831.61
10200
31.19
844.15
56.14
597.3
310.66
1839.44
10300
31.19
853.33
56.14
603.79
310.66
1855.11
10500
31.19
871.68
56.14
616.78
310.66
1886.44
10600
31.19
880.85
56.14
623.27
310.66
1902.12
10650
31.19
885.44
56.14
626.52
310.66
1909.95
10850
31.19
903.79
56.14
639.5
310.66
1941.29
11000
31.19
917.56
56.14
649.24
310.66
1964.79
11050
31.19
922.15
56.14
652.49
310.66
1972.62
11200
31.19
935.91
56.14
662.22
310.66
1996.12
11300
31.19
945.08
56.14
668.72
310.66
2011.79
11500
31.19
963.43
56.14
681.7
310.66
2043.13
Ps loss = Surface connection pressure loss (psia)
Pdp loss =Drill Pipe pressure loss
Pdc loss =Drill Collar Pressure loss (psia)
Padp loss = Annulus of the Drill Pipe pressure loss (psia)
Padc loss = Annulus of the Drill Collar pressure loss (psia)
Pd Loss = Parasitic Pressure loss (psia)
98
Appendix H
Mud Sample 3
Depth(ft) Ps loss(psia) Pdp loss (psia) Pdc loss (psia) Padp loss (psia) Padc loss (psia) Pd loss(psia)
9600
25.32
640.53
45.57
1020.96
271.71
2004.09
9700
25.32
647.98
45.57
1032.83
271.71
2023.4
9750
25.32
651.7
45.57
1038.77
271.71
2033.07
9900
25.32
662.87
45.57
1056.57
271.71
2062.05
9950
25.32
666.6
45.57
1062.52
271.71
2071.71
10000
25.32
670.32
45.57
1068.45
271.71
2081.37
10050
25.32
674.04
45.57
1074.38
271.71
2091.03
10150
25.32
681.49
45.57
1086.25
271.71
2110.35
10200
25.32
685.22
45.57
1092.19
271.71
2120.01
10300
25.32
692.66
45.57
1104.06
271.71
2139.33
10500
25.32
707.56
45.57
1127.8
271.71
2177.97
10600
25.32
715.01
45.57
1139.68
271.71
2197.28
10650
25.32
718.73
45.57
1145.61
271.71
2206.94
10850
25.32
733.63
45.57
1169.35
271.71
2245.58
11000
25.32
744.8
45.57
1187.16
271.71
2274.56
11050
25.32
748.52
45.57
1193.1
271.71
2284.22
11200
25.32
759.7
45.57
1210.91
271.71
2313.2
11300
25.32
767.14
45.57
1222.78
271.71
2332.52
11500
25.32
782.04
45.57
1246.52
271.71
2371.16
Ps loss = Surface connection pressure loss (psia)
Pdp loss =Drill Pipe pressure loss
Pdc loss =Drill Collar Pressure loss (psia)
Padp loss = Annulus of the Drill Pipe pressure loss (psia)
Padc loss = Annulus of the Drill Collar pressure loss (psia)
Pd Loss = Parasitic Pressure loss (psia)
99
Appendix I
Calculating the Pore Pressure in the Over-Pressure Zone

Over pressure at 9600 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=6.34
Rn=6.34
R0
Rn

6.34
1
6.34
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
D
 0.465 psi /
ft
Hence Pore pressure = 4464 psia

Over pressure at 9700 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=6.42
Rn=6.32
R0
Rn

6.42
6.32
 1.01
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.465 psi /
D
ft
100
Hence Pore pressure = 4510.5 psia

Over pressure at 9750 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=6.44
Rn=6.3
R0
Rn

6.44
 1.02
6.3
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
D
 0.465 psi /
ft
Hence Pore pressure = 4533.75psia

Over pressure at 9900 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=6.56
Rn=6.28
R0
Rn

6.56
 1.04
6.28
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.465 psi /
D
ft
101
Hence Pore pressure = 4603.5psia

Over pressure at 9950 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=6.72
Rn=6.27
R0
Rn

6.72
6.27
 1.07
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.5 psi /
D
ft
Hence Pore pressure = 4975 psia

Over pressure at 10000 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=6.86
Rn=6.26
R0
Rn

6.86
 1.09
6.26
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.51 psi /
D
ft
102
Hence Pore pressure = 5100psia

Over pressure at 10050ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7
Rn=6.25
R0
Rn

7
 1.12
6.25
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.53 psi /
D
ft
Hence Pore pressure = 5326.5psia

Over pressure at 10150ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.08
Rn=6.24
R0
Rn

7.08
 1.13
6.24
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.53 psi / ft
D
Hence Pore pressure = 5379.5 psia
103

Over pressure at 10200 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.12
Rn=6.22
R0
Rn

7.12
 1.14
6.22
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.54 psi /
D
ft
Hence Pore pressure = 5508 psia

Over pressure at 10300 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.18
Rn=6.21
R0
Rn

7.18
 1.16
6.21
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship chart
Pore pressure gradient
P
 0.55 psi /
D
ft
Hence Pore pressure = 5562 psia
104

Over pressure at 10500 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.13
Rn=6.21
R0
Rn

7.13
 1.15
6.21
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.55 psi /
D
ft
Hence Pore pressure = 5775 psia

Over pressure at 10600 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.2
Rn=6.16
R0
Rn

7.2
6.16
 1.17
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.57 psi /
D
ft
Hence Pore pressure = 6042 psia

Over pressure at 10650 ft
105
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.22
Rn=6.14
R0
Rn

7.22
 1.18
6.14
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.57 psi /
D
ft
Hence Pore pressure = 6070.5 psia

Over pressure at 10850 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0 = 7.31
Rn = 6.2
R0
Rn

7.31
 1.18
6.2
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.57 psi / ft
D
Hence Pore pressure = 6184.5 psia

Over pressure at 11000 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.16
Rn=6.08
106
R0
Rn

7.16
 1.18
6.08
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.57 psi /
D
ft
Hence Pore pressure = 6270 psia

Over pressure at 11050 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.24
Rn=6.07
R0
Rn

7.24
 1.19
6.07
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
D
 0.58 psi /
ft
Hence Pore pressure = 6409 psia

Over pressure at 11200 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.41
Rn=6.04
R0
Rn

7.41
 1.23
6.04
107
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
 0.59 psi / ft
D
Hence Pore pressure = 6608 psia

Over pressure at 11300 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0 =7.52
Rn = 6.03
R0
Rn

7.52
 1.25
6.03
From Hottman and Johnson Pore pressure gradient and shale resistivity relationship
Pore pressure gradient
P
D
 0.59 psi /
ft
Hence Pore pressure = 6667 psia

Over pressure at 11500 ft
From Figure 5.1 the normal shale resistivity and observed shale resistivity is given below
R0=7.56
Rn=6
R0
Rn

7.56
 1.26
6
Pore pressure gradient
P
D
 0.59 psi /
ft
Hence Pore pressure = 6785 psia
APPENDIX J
Calculating the Fracture Pressure in the Over Pressured Zone

Fracture pressure at the normal pressure zone which lies from 9600ft to11500 is obtained
using the Hubbert and Willis equation.
108
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where pore pressure at 9600 ft is given as 0.465 psia/ft
PP 
PP
*D
D
PP= 0.465*9600= 4464 psia
Also
 ov  G * D
 ov =1*9600=9600 psia
 1  2(0.25) 
= 6403.2 psia
h  
 1  (0.25) 
9600


Hence
Pf
= 2(6403.2) +4464 = 17270.4 psia
Therefore the Fracture pressure at a depth of 9600 ft = 17270.4 psia

Fracture pressure at 9700 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 9700 ft = 4510.5 psia
109
Also
 ov  G * D
 ov =1*9700=9700 psia
 1  2(0.25) 
h  
 1  (0.25) 
9700  6469.9 psia


Hence
Pf
= 2(6469.9) +4510.5= 17450.3 psia
Therefore the Fracture pressure at a depth of 9700 ft = 17450.3 psia

Fracture pressure at 9750 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 9750 ft =4533.75 psia
Also
 ov  G * D
 ov =1*9750=9750 psia
 1  2(0.25) 
h  
 1  (0.25) 
9750  6503.25


Hence
Pf
= 2(6503.25) +4533.75 = 17540.25 psia
Therefore the Fracture pressure at a depth of 9750 ft = 17540.25 psia
110

Fracture pressure at 9900ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 9900ft =4603.5psia
Also
 ov  G * D
 ov =1*9900=9900psia
 1  2(0.25) 
h  
 1  (0.25) 
9900  6603.3 psia


Hence
Pf
= 2(6603.3) +4603.5= 17810.1psia
Therefore the Fracture pressure at a depth of 9900ft = 17810.1psia

Fracture pressure at 9950 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 9950 ft =4975 psia
Also
111
 ov  G * D
 ov =1*9950=9950psia
 1  2(0.25) 
h  
 1  (0.25) 
9950  6636.65 psia


Hence
Pf
= 2(6636.65) +4975= 18248.3 psia
Therefore the Fracture pressure at a depth of 9950 ft = 18248.3 psia

Fracture pressure at 10000 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 10000 ft =5100 psia
Also
 ov  G * D
 ov =1*10000=10000 psia
 1  2(0.25) 
h  
 1  (0.25) 
10000  6670 psia


Hence
Pf
= 2(6670) +5100= 18540 psia
Therefore the Fracture pressure at a depth of 10000 ft = 18440 psia

Fracture pressure at 10050 ft
112
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 10050 ft =5326.5 psia
Also
 ov  G * D
 ov =1*11000=10050 psia
 1  2(0.25) 
h  
 1  (0.25) 
10050  6703.35 psia


Hence
Pf
= 2(6703.35) +5326.5= 18733.2 psia
Therefore the Fracture pressure at a depth of 10050 ft = 18733.2 psia

Fracture pressure at 10150 ft
Pf  2 h  PP
Recall that
1  2 
 o v
 1  
h  

Where Pore pressure at 10150 ft =5379.5 psia
Also
 ov  G * D
 ov =1*10150=10150 psia
113
 1  2(0.25) 

10150  6770.05 psia
 1  (0.25) 
h  

Hence
Pf
= 2(6770.05) +5379.5= 18919.6 psia
Therefore the Fracture pressure at a depth of 10150 ft = 18919.6psia

Fracture pressure at 10200 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 10200 ft =5508 psia
Also
 ov  G * D
 ov =1*10200=10200 psia
 1  2(0.25) 
10200  6803.4 psia
h  
 1  (0.25) 


Hence
Pf
= 2(6803.4) +5508= 19114.8 psia
Therefore the Fracture pressure at a depth of 10200 ft = 19114.8 psia

Fracture pressure at 10300ft
Pf  2 h  PP
114
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 10300 ft =5562 psia
Also
 ov  G * D
 ov =1*10300=10300psia
 1  2(0.25) 
h  
 1  (0.25) 
10300  6870.1 psia


Hence
Pf
= 2(6870.1) +5562= 19302.2 psia
Therefore the Fracture pressure at a depth of 10300 ft = 19302.2 psia

Fracture pressure at 10500 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 10500ft =5775psia
Also
 ov  G * D
 ov =1*10500=10500psia
115
 1  2(0.25) 

 h  
10500  7003.5 psia
 1  (0.25) 
Hence
Pf
= 2(7005.5) +5775= 1978.2 psia
Therefore the Fracture pressure at a depth of 10500 ft = 1978.2 psia

Fracture pressure at 10600 ft
Pf  2 h  PP
Recall that
1  2 
 o v
 1  
h  

Where Pore pressure at 10600 ft =6042 psia
Also
 ov  G * D
 ov =1*10600=10600 psia
 1  2(0.25) 
h  
 1  (0.25) 
10600 = 7070.2 psia


Hence
Pf
= 2(7070.2) +6042=20182.4
Therefore the Fracture pressure at a depth of 10600 ft = 20182.4 psia

Fracture pressure at 10650 ft
Pf  2 h  PP
116
Recall that
1  2 
h  

 o v
 1 
Where Pore pressure at 10650 ft =6070.5 psia
Also
 ov  G * D
 ov =1*10650=10650 psia
 1  2(0.25) 
h  
 1  (0.25) 
10650  7103.55 psia


Hence
Pf
= 2(7103.55) +6070.5= 20277.6 psia
Therefore the Fracture pressure at a depth of 10650 ft = 20277.6 psia

Fracture pressure at 10850ft
Pf  2 h  PP
Recall that
1  2
 1 
h  


 ov

Where Pore pressure at 10850 ft =6184.5 psia
Also
 ov  G * D
 ov =1*10850=10850psia
117
 1  2(0.25) 
10850  7236.95 psia
 h  
 1  (0.25) 
Hence
Pf
= 2(7236.95) +6184.5= 20658.4 psia
Therefore the Fracture pressure at a depth of 12100 ft = 20658.4 psia

Fracture pressure at 11000 ft
Pf  2 h  PP
Recall that
1  2 
 ov
 1  
h  

Where Pore pressure at 11000ft =6270 psia
Also
 ov  G * D
 ov =1*11000=11000 psia
 1  2(0.25) 
h  
 1  (0.25) 
11000 = 7337 psia


Hence
Pf
= 2(7337) +6270= 20944 psia
Therefore the Fracture pressure at a depth of 11000 ft = 20944 psia

Fracture pressure at 11050 ft
118
Pf  2 h  PP
Recall that
1  2 
 ov
 1  
h  

Where Pore pressure at 11050 ft =6409 psia
Also
 ov  G * D
 ov =1*11050=11050 psia
 1  2(0.25) 
h  
 1  (0.25) 
11050 = 7370.35 psia


Hence
Pf
= 2(7370.35) +6409= 21149.7 psia
Therefore the Fracture pressure at a depth of 11050 ft = 21149.7 psia

Fracture pressure at 11200 ft
Pf  2 h  PP
Recall that
1  2 
 ov
 1  
h  

Where Pore pressure at 11200 ft =6608 psia
Also
 ov  G * D
119
 ov =1*11200=11200 psia
 1  2(0.25) 
= 7470.4 psia
h  
 1  (0.25) 
11200


Hence
Pf
= 2(7470.4) +6608= 21548.8 psia
Therefore the Fracture pressure at a depth of 11200 ft = 21548.8 psia

Fracture pressure at 11300 ft
Pf  2 h  PP
Recall that
1  2 
h  

 o v
 1  
Where Pore pressure at 11300 ft =6667 psia
Also
 ov  G * D
 ov =1*11300=11300psia
 1  2(0.25) 
h  
 1  (0.25) 
11300 = 7537.1 psia


Hence
Pf
= 2(7537.1) +6667= 21741.2 psia
Therefore the Fracture pressure at a depth of 11300 ft = 21741.2 psia

Fracture pressure at 11500 ft
120
Pf  2 h  PP
Recall that
1  2 
 ov
 1 
h  

Where Pore pressure at 11500 ft =6785 psia
Also
 ov  G * D
 ov =1*11500=11500 psia
 1  2(0.25) 

 h  
11500  7670.5 psia
 1  (0.25) 
Hence
Pf
= 2(7670.5) +6785= 22126 psia
Therefore the Fracture pressure at a depth of 13300 ft = 22126 psia
121
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