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Petroleum Engineering 663 — Formation Evaluation and Analysis of Reservoir Performance
Self-Study Guide (tied to Course Notes) — Analysis of Reservoir Performance [Blasingame]
Pressure Transient Analysis
Self-Study Objectives — Blasingame Notes (Pressure Transient Analysis):
Introduction:
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Be familiar with the Philosophy/Objectives of PTA [Orientation Slide] (+PTA Diagnostic Examples) ......................... 7-8
Be familiar with the Static Data required for PTA.............................................................................................................. 9
Be familiar with the production history issues as these relate to PTA ......................................................................... 10-12
Be familiar with the concept of "early" well deliverability, and the governing relation for gas wells......................... 13-15
Be familiar with the schematic of the "Reservoir/Well/Facilities System" ....................................................................... 16
Be familiar with the expected advances in Pressure Transient Analysis (PTA) and Production Analysis (PA) ............... 17
Be familiar with the "Questions to Consider" for PTA ..................................................................................................... 18
Be familiar with the Overview of Pressure Transient Analysis (PTA) section ............................................................ 19-30
— Be familiar with the reservoir properties which can be estimated using PTA ............................................................ 27
— Be familiar with the common plotting formats for PTA ............................................................................................. 28
— Be familiar with the practical challenges of PTA ....................................................................................................... 30
— Be familiar with the diagnostic methods for PTA ....................................................................................................... 30
Objective 1 — Describe the concepts of porosity and permeability and be able to relate their respective influences on fluid
flow in porous media.
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Be familiar with depositional systems for sandstone and carbonate reservoirs ........................................................... 33-34
Be familiar with the classical correlations of permeability with porosity and water saturation ................................... 35-36
Be familiar with the influence of "small-scale" heterogeneities ....................................................................................... 37
Be familiar with the scale influence permeability estimates from different sources .................................................... 38-39
Be familiar with the "Questions to Consider" for Geology/Petrophysics ......................................................................... 40
Objective 2 — Estimate oil, gas, and water properties pertinent for well test or production data analysis using industry
accepted correlations and/or laboratory data.
● Be familiar with and be able to classify Reservoir Fluids ................................................................................................. 43
● Be familiar with and be able to use the formation volume factor, viscosity, and compressibility variables ................ 44-53
● Be familiar with the "Questions to Consider" for PVT ..................................................................................................... 53
Objective 3 — Sketch pressure versus time trends and pressure versus distance trends for a reservoir system which exhibits
transient, pseudosteady-state, and steady-state flow behavior.
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Be familiar with and be able to apply the "pressure distribution" solutions for radial flow.............................................. 57
Be familiar with and be able to apply the "radius of investigation" relation for transient radial flow .............................. 57
Be familiar with the character of reservoir pressure cross-sections (or "slices") for transient radial flow ........................ 58
Be familiar with the schematic of reservoir pressure for pseudosteady-state flow conditions (radial flow) ................ 59-60
Be familiar with and be able to apply the "pseudosteady-state" flow solutions for radial flow ........................................ 61
Be familiar with the schematic of reservoir pressure for various flow conditions (radial flow) .................................. 62-66
— Constant rate, transient radial flow behavior [log(r) format] ...................................................................................... 62
— Log-linear rate decline, transient radial flow behavior [log(r) format] ....................................................................... 63
— Constant wellbore pressure, transient radial flow behavior [log(r) format] ................................................................ 64
— Constant rate, transient radial flow behavior [Cartesian r format] .............................................................................. 65
— Constant wellbore pressure, transient radial flow behavior [Cartesian r format] ........................................................ 66
● Be familiar with the "Questions to Consider" for Reservoir Pressure Trends .................................................................. 67
Objective 4 — Derive the material balance relation for a slightly compressible liquid (oil) in the presence of other phases (gas
and water), as well as the material balance relation for a dry gas.
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Be familiar with "Dake's opinion" on the roles of reservoir simulation and material balance .......................................... 71
Be familiar with and be able to apply the "Oil" material balance relations ...................................................................... 73
Be familiar with and be able to apply the "Gas" material balance relations ................................................................ 74-79
Be familiar with the "Questions to Consider" for Material Balance................................................................................. 80
(Page 2 of 2)
Petroleum Engineering 663 — Formation Evaluation and Analysis of Reservoir Performance
Self-Study Guide (tied to Course Notes) — Analysis of Reservoir Performance [Blasingame]
Pressure Transient Analysis
Self-Study Objectives — Blasingame Notes (Pressure Transient Analysis):
Objective 5 — Derive the analysis and interpretation methodologies (i.e., "conventional" plots and type curve analysis) for
pressure drawdown and pressure buildup tests, for liquid, gas, and multiphase flow systems.
● Be familiar with and be able to identify and use the appropriate relations for "wellbore storage" .............................. 83-88
— Be familiar with the base relations for wellbore storage ............................................................................................. 83
— Be familiar with the schematic Cartesian plots for wellbore storage distorted PTA data ........................................... 84
— Be familiar "exponential approximation" solution for wellbore storage ..................................................................... 85
— Be familiar with and be able to apply the "Bourdet-Gringarten" wellbore storage "type curve" ................................ 86
— Be familiar with and be able to apply the Cartesian plot for wellbore storage distorted PTA data............................. 87
— Be familiar with and be able to apply the Log-log plot for wellbore storage distorted PTA data ............................... 88
● Be familiar with and be able to identify and use the appropriate relations for "infinite-acting radial flow" (IARF) ... 89-91
— Be familiar with and be able to apply the Semilog plot for PTA data during the IARF regime ................................. 89
— Be familiar with and be able to apply the Log-log plot for PTA data during the IARF regime .................................. 90
● Be familiar with and be able to identify and use the Muskat-Arps late-time Pressure Buildup (PBU) plot ..................... 91
● Be familiar with the "Questions to Consider" for Conventional PTA Plots ...................................................................... 92
Objective 6 — Apply dimensionless solutions ("type curves") and field variable solutions ("specialized plots") for unfractured
and fractured wells in infinite and finite-acting, homogeneous and dual porosity reservoirs.
● Be familiar with PTA Model-Based Analysis Methods ............................................................................................ 95-111
— Be familiar with the Orientation Slide for PTA Model-Based Analysis ..................................................................... 95
— Be familiar with and be able to apply the models for "unfractured wells" (radial flow) (+the skin factor) ......... 99-101
— Be familiar with and be able to apply the models for "fractured wells"............................................................. 102-107
— Be familiar with and be able to apply the models for wells in "naturally fractured reservoirs" ......................... 108-111
● Be familiar with the "Questions to Consider" for Reservoir Models .............................................................................. 112
● Be familiar with and be able to apply "Type Curves" for PTA ............................................................................... 113-125
— "Bourdet-Gringarten" Type Curve: WBS and IARF ................................................................................................ 115
— "Ansah" Type Curve:
Late-Time Pressure Buildup ............................................................................ 116
— "Stewart" Type Curves:
Sealing Faults .................................................................................................. 117
— "Cinco" Type Curves:
Vertically Fractured Well (No WBS) ....................................................... 118-119
— "Economides" Type Curves:
Vertically Fractured Well (with WBS) ..................................................... 120-122
— "Onur" Type Curves:
Naturally Fractured Reservoirs (No WBS) ............................................... 123-124
— "Angel" Type Curves:
Naturally Fractured Reservoirs (with WBS) .................................................... 125
● Be familiar with the "Questions to Consider" for Type Curves ...................................................................................... 112
● Be familiar with inventory of PTA Field Case Examples ....................................................................................... 127-138
Objective 7 — Analyze production data (rate-time or pressure-rate-time data) to obtain reservoir volume and estimates of
reservoir properties for gas and liquid reservoir systems. The student should also be able to make performance
forecasts for such systems.
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Be familiar with the data issues related to Production Analysis (PA)............................................................................. 141
Be familiar with the fundamentals of PA related to the diagnosis of transient and boundary-dominated flow .............. 142
Be familiar with the applicability of PTA and PA (i.e., common characteristics and differences) ................................. 143
Be familiar with the data requirements related to Production Analysis (PA) ................................................................. 144
Be familiar with the history of PA (high-level review)............................................................................................ 146-151
Be familiar with the modern PA Methods (high-level review) ................................................................................ 152-156
Be familiar with the "Questions to Consider" for Production Analysis .......................................................................... 157
Pressure Transient Analysis
Module for PETE 663
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Pressure Transient Analysis
Objectives
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Course Objectives:
 Objectives of Pressure Transient Testing:
1. Describe the concepts of porosity and permeability and be able to
relate their respective influences on fluid flow in porous media.
2. Estimate oil, gas, and water properties pertinent for well test or
production data analysis using industry accepted correlations
and/or laboratory data.
3. Sketch pressure versus time trends and pressure versus distance
trends for a reservoir system which exhibits transient,
pseudosteady-state, and steady-state flow behavior.
4. Derive the material balance relation for a slightly compressible
liquid (oil) in the presence of other phases (gas and water), as well
as the material balance relation for a dry gas.
5. Derive the analysis and interpretation methodologies (i.e.,
"conventional" plots and type curve analysis) for pressure
drawdown and pressure buildup tests, for liquid, gas, and
multiphase flow systems.
6. Apply dimensionless solutions ("type curves") and field variable
solutions ("specialized plots") for unfractured and fractured wells in
infinite and finite-acting, homogeneous and dual porosity
reservoirs.
7. Analyze production data (rate-time or pressure-rate-time data) to
obtain reservoir volume and estimates of reservoir properties for
gas and liquid reservoir systems. The student should also be able
to make performance forecasts for such systems.
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Pressure Transient Analysis
Orientation
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Orientation: Philosophy/Objectives
 Objectives of Pressure Transient Testing:
Evaluate reservoir pressure (initial or average pressure).
Evaluate reservoir fluid (fluid samples collected for lab study).
Estimate reservoir properties (e.g., k, S, xf, λ, ω, etc.).
Estimate reservoir volumetrics (e.g., fluid-in-place, drainage area).
 Input Data:
BOTTOMHOLE pressure data (accurate to < 1 part in 10,000 or more).
SURFACE flowrate data (often poorly measured/recorded).
Fluid properties (e.g., FVF, viscosity, compressibility, ... ).
Reservoir properties (e.g., h, φ, rw, cf, ... )
 Results of PTA Interpretation:
Productive capacity of the WELL (damage/stimulation).
Productive capacity of the RESERVOIR (transmissibility).
Current average reservoir pressure.
Reservoir limits (for production to pseudosteady-state).
Well interference effects.
Well/Reservoir specific parameters (e.g., Cs, xf, λ, ω, Lfault, rcomp, kv/kh, ...).
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Orientation: PTA Diagnostic Examples
Unfractured Well
■ Field Example: (SPE 12777)
Data match for a case of radial flow —
wellbore storage signature is identified
using the pressure β-derivative. (dimensionless format — pDβd=1)
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Fractured Well
■ Field Example: (SPE 9975)
Data match for the case of a well with
an infinite conductivity vertical fracture
— formation linear flow behavior is
revealed using the β-derivative.
(dimensionless format — pDβd=1/2)
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Orientation: Static Data for PTA
 PVT Properties: (Lab report preferred, correlations acceptable)
 Black Oil: Bo, Rs, µo, co
(correlations require: T, γg,sep, pb, γSTO)
 Dry Gas: z (or Bg), µg, cg
(correlations require: T, γg,sep)
 Volatile Oil: Black oil equivalent or compositional formulation.
 Gas Condensate: Dry gas equivalent or compositional formulation.
 Water: Bw, Rsw, µw, cw
(correlations require: T, γg,sep, pbw, salinity)
 Reservoir Properties:
 Porosity (φ)
(core and/or well logs)
 Net pay thickness (h)
(core and/or well logs)
 Wellbore radius (rw)
(well completion history (bit diameter))
 Formation Compressibility (cf)
(cf=3x10-6 psia-1 or correlation)
 Well Completion History:
 Drilling records
(initial pressures, production tests)
 Well files
(well logs, core, PVT, recompletion, workover records)
 Annotated production records (records of activities — very useful)
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Orientation: Production Histories
 Allocated Rate Data:
 Common in mature producing environments (e.g., Texas).
 Common in some offshore operations (manifold rates).
 "Allocation" depends on records — and consistency checks.
 Poor/Incomplete (or Erroneous) Pressure Data:
 Virtually all production pressure measurements taken at surface.
 Completion changes often not reflected in surface pressures.
 Some pressure data are just wrong (poor gauge, poor timing, etc.).
 Well Completion Issues:
 Equipment changes, poor practices, failed equipment, etc.
 UNREPORTED activities (recompletions, workovers, treatments).
 Permanent DOWNHOLE Pressure Measurements:
 Expense is justified.
 Provides continuous evaluation of well performance.
 Data volume/sampling is an issue, but not a major problem.
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Orientation: Pressure Transient Analysis (tight gas)
 Example: Production History Plot — East TX Gas Well
 Good rate and pressure histories.
 Unique case — bottomhole and "production" pressure data.
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Orientation: Pressure Transient Analysis (tight gas)
 Example: "Log-Log" Plot (Well Test Analysis) — East TX Gas Well
 Used high frequency bottomhole pressure measurements (pws).
 Consistent match of bottomhole and "production" pressure data.
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Orientation: Early Well Deliverability
Well Deliverability:
 The first efforts to analyze well
performance were an attempt to
quantify well potential — not to
estimate reservoir properties.
 The original well deliverability
relation was completely empirical (derived from observations),
and is given as:
q = C( p2 - p2 )n
 From: Back-Pressure Data on NaturalGas Wells and Their Application to
Production Practices — Rawlins and
Schellhardt (USBM Monograph, 1935).
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wf
 This relationship is rigorous for
low pressure gas reservoirs,
(n=1 for laminar flow).
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Orientation: Derivation of Well Deliverability Relation
Q. Can the "gas deliverability" or "AOF" be derived?
A. Sort of, see steps below — assume (µgz) product is constant.
Darcy's Law:
vr =
q g Bg
Ar
=+
k  dp 
k
 dp 
A
rh
q
h
π
=
=
π
[
2
]
or
(
2
)
r
g
r dr 
µ g  dr 
µ g Bg


Separating and Integrating:
re 1
dr =
2πkh r r
w
qg
∫
∫
pe

p
T z 
1
dp  Bg ≡ sc

p µ g Bg
p
T
z
sc sc 

w
Which Reduces to: [(µgz) = constant]
qg
[ln(re /rw )] = Tsc z sc 1
2πkh
T p sc ( µ g z ) c
∫
pe
p dp
p
w
Performing the Pressure Integration:
q g = 2π
Tsc z sc
1
kh
1 2
2 → q = C p2 − p2
( pe − p w
)
( e
g
w)
ln(re /rw ) T p sc ( µ g z ) c 2
 Discussion: Derivation of Well Deliverability Relation
 Actually an empirical result (see Rawlins and Schellhardt (1935)).
 Derivation from steady-state flow (above) is useful for illustration.
 Derivation for pseudosteady-state is similar (variety of results).
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From: Energy Resources and Conservation Board, 1975, Theory
and Practice of the Testing of Gas Wells, third edition, Pub.
ERCB-75-34, ERCB, Calgary, Alberta.
Orientation: Well Deliverability (4-point test)
q = C( p2 - p2 )n
wf
a. Typical flow regimes encountered during
production (liquid system).
b. Typical "flow-after-flow" or 4-point test,
(assumes pseudosteady-state flow for each rate).
c. "Deliverability" or "Backpressure" plot used to
estimate maximum well productivity.
 Discussion: Well Deliverability (4-point test)
 Probably oldest "reservoir engineering" technique.
 Assumption of pseudosteady-state flow is the weakest link in analysis.
 Does not directly relate time, rate, and pressure performance.
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Orientation: Reservoir/Well/Facilities System
 Our focus is the reservoir ...
but, we also need to consider:
 The well completion.
 The tubulars.
 The surface facilities.
 The reservoir fluid(s).
 Overall flow system (after Fonesca). Blasingame axiom: "if there is a problem
with the analysis/interpretation of well test and/or production data — the issue
most likely stems from the well completion."
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Orientation: What Advances do We Need?
Pressure Transient Analysis: PTA
 Data processing (permanent gauges) (obvious, but...).
 Numerical modelling (advise caution in applications).
 Variable-rate analysis (deconvolution).
 Better data analysis functions (We can always hope...).
 Continuous Measurement = Continuous Assessment
Production Analysis: PA
 More consistent measurement of q and pwf.
 Pressure conversion (surface → bottomhole).
 Further implementation of semi-analytical solutions.
 Diagnostic methods for defining pressure transient
behavior in production data (model identification).
 Continuous Measurement = Productivity Optimization
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Orientation: Questions to Consider
Q1. Practical applications of Pressure Transient Analysis (PTA)?
A1. Estimate/evaluate the following:
● Reservoir properties (e.g., k, S, xf, λ, ω, etc.).
● Productivity efficiency (damage or stimulation).
● Reservoir pressure (initial or average pressure).
(rarely volume)
(direct assessment)
(pavg → long shut-in)
Q2. What are the major issues or complications in PTA?
A2. Major issues/complications in PTA:
● Planning of pressure transient test.
(always model prior to testing)
● Preparation of well for testing.
(execution failures)
● Production history is not trivial.
(can corrupt interpretation)
● Well completion — have records at hand. (leaks, tubulars, placement)
● WHY YOU ARE TESTING THE WELL — WHAT IS THE OBJECTIVE?
Q3. Comparison with of PTA with Production Analysis (PA)?
A3. Comparison of PTA and PA:
● PTA:
— HIGH resolution/HIGH frequency (pressure) data. (quality/quantity)
— Gives SNAPSHOT of the well performance at that time. (stress test)
● PA:
— LOW resolution/LOW frequency (pressure) data. (quality/quantity)
— LUMPS entire life of well into analysis.
(passive monitoring)
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Pressure Transient Analysis
Overview of
Pressure Transient Analysis
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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From: Barnum, R.S., and Vela., S.: "Testing Exploration Wells by
Objectives," paper SPE 13184 presented at the 1984 SPE Annual
Technical Conference and Exhibition, Houston, TX, 16-19 Sept. 1984.
Overview: Tubular System Schematics
a. Bottomhole equipment schematic for
surface shut-in.
b. Bottomhole equipment schematic for
bottomhole shut-in.
 Overview: Tubular System Schematics
 Bottomhole shut-in is preferred — but requires operations, time, expense.
 Surface shut-in for gas wells is common, but must take care in evaluation
and interpretation of data — data are often corrupted by wellbore effects.
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From: Barnum, R.S., and Vela., S.: "Testing Exploration Wells by
Objectives," paper SPE 13184 presented at the 1984 SPE Annual
Technical Conference and Exhibition, Houston, TX, 16-19 Sept. 1984.
Overview: Example DST — Flow/Buildup Sequences
 Overview: Example DST — Flow/Buildup Sequences
 Early flow/shut-in sequences are used to estimate initial reservoir pressure.
 Extended flow sequence used to "sample" near-well reservoir volume.
 Extended shut-in sequence used to estimate near-well reservoir properties.
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From: Schlumberger: Fundamentals of
Formation Testing (March 2006).
Overview: Example Semilog Drawdown Test Plot
 Overview: Example Semilog Drawdown Test Plot
 Schematic of the separate and combined influences of skin and WBS.
 Skin effect and wellbore storage (WBS) yield a unique combined influence.
 End of WBS effects can be difficult to distinguish (need ∆p' function).
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From: Schlumberger: Fundamentals of
Formation Testing (March 2006).
Overview: Example Log-Log Drawdown Test Plot
Distance is related
to the SKIN EFFECT.
 Overview: Example Log-Log Drawdown Test Plot
 Can see effect of separate and combined influences of skin and WBS.
 ∆p' function is horizontal for infinite-acting radial flow (IARF) flow regime.
 ∆p and ∆p' functions are "unit slope" for WBS domination.
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From: Schlumberger: Fundamentals of
Formation Testing (March 2006).
Overview: Example Semilog Buildup Test Plot
 Overview: Example Semilog Buildup Test Plot
 Similar to drawdown test, BUT pressure buildup is NOT affected by skin.
 Reversed axis is due to "superposition" to account for rate history.
 Pressure buildup data are generally better quality than drawdown data.
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From: Kamal, M.M., Freyder, D.G., and Murray, M.A.: "Use of
Transient Testing in Reservoir Management," JPT,
(November 1995), 992-999.
Overview: Summary Table — Flow Regimes
 Overview: Summary Table — Flow Regimes
 Wellbore Storage (WBS): Universal phenomena, easy to distinguish.
 Infinite-Acting Radial Flow (IARF): Traditional regime (k>0.01 md).
 Fractured Wells: Linear and Bi-Linear flow regimes possible.
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From: Kamal, M.M., Freyder, D.G., and Murray, M.A.: "Use of
Transient Testing in Reservoir Management," JPT,
(November 1995), 992-999.
Overview: Properties Obtained from PTA
 Overview: Properties Obtained from PTA
 Drawdown Tests (DD): Fine in theory, difficult in practice (rates!).
 Buildup Tests (BU): Most common PTA, several advantages (rates, skin, …).
 Falloff Tests (FO): Same as buildup tests, for injection wells.
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From: Ershagi, I. and Woodbury, J.J.: "Examples of Pitfalls in Well
Test Analysis," JPT, (February 1985),335-341.
Overview: Common Plots/Flow Regimes
Wellbore Storage
(∆p = mwbs ∆t)
a. Common Plots: Wellbore
Storage (WBS) Flow Regime.
Infinite-Acting Radial Flow
(∆p = ∆p1hr msl log[∆t])
b. Common Plots: Infinite-Acting
Radial Flow (IARF) Regime.
 Overview: Common Plots/Flow Regimes
Formation Linear Flow
(∆p = mLF Sqrt[∆t])
c. Common Plots: Formation
Linear Flow (FLF) Regime —
Fractured Wells.
 Wellbore Storage (WBS): "Unit Slope" trend (straight line on log-log plot).
 Infinite-Acting Radial Flow (IARF): Semilog plot relation.
 Formation Linear Flow (FLF): Fractured wells, high conductivity fracture.
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From: Ershagi, I. and Woodbury, J.J.: "Examples of Pitfalls in Well
Test Analysis," JPT, (February 1985),335-341.
Overview: PTA — Summary of Diagnostics
 Discussion: Pressure Transient Analysis — Summary of Diagnostics
 ∆p function (pressure drop)?
(traditional diagnostic)
 ∆pd function (Bourdet derivative)?
(primary diagnostic — IARF)
 ∆pβd function (β-derivative)?
(new diagnostic — VF, NF, Hz, Faults, etc.)
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Overview: Questions to Consider
Q1. What are the practical challenges for PTA?
A1. The practical challenges for PTA include:
● Identification of appropriate reservoir model.
● Distinguishing "artifacts" from data.
● Estimation of reservoir properties.
(interpretation)
(bad rate history, etc.)
(data quality/quantity)
Q2. What are the diagnostics for PTA?
A2. The diagnostics for PTA include:
● "Conventional" Plots: Log-log, semilog, Cartesian, root-time, etc.
● "Diagnostic" Plots: Derivative, Integral, etc. functions
● Probably the best approach is to rely mostly on the derivative for
diagnostics, and the "conventional" plots for validation.
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Pressure Transient Analysis
Objective 1
Describe the concepts of porosity and permeability and be able to relate their respective
influences on fluid flow in porous media.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Geology: Sandstone Depositional Systems
a. Various sandstone depositional
sequences — note the "transport"
system evolves basinward.
From: Reservoir Sandstones — Berg (1986).
b. These schematics illustrate similarity in
depositional processes and also give
insight into heterogeneity.
 Sandstone Reservoirs:
 Depositional sequences are well-established/accepted.
 Turbidite reservoirs are probably of most current interest.
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From: Carbonate Reservoir Characterization — Lucia (1999).
Geology: Carbonate Depositional Sys. — φ and k
a. Crossplot of permeability versus porosity
(logarithmic scales). In-cludes particle
size as a variable.
b. Permeability-porosity profiles for various
carbonate depositional sequences.
 Carbonate Reservoirs: Permeability/Porosity Character
 Porosity and permeability often weakly correlated in carbonates.
 Permeability in carbonates most often dependent on diagenetic processes.
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a. "Cartoon" of kair versus φ — illustrates k=a
exp(bφ).
b. "Cartoon" of kair versus Swi — illustrates the
influence of pore throat structure.
From: API Drilling and Prod. Prac. — Bruce and Welge (1947).
From: The Fundamentals of Core Analysis — Keelan (1972).
Petrophysics: Permeability Characterization/Correlation
 Permeability Characterization/Correlation:
 Permeability = f(φ, composition, texture, grain size, sorting, etc.).
 Simplified correlations for permeability will only be of "local" use.
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From: Archie, G.E.: "Electrical Resistivity Log as an Aid in Determining
Some Reservoir Characteristics," Trans. AIME (1942) 146, 54-62.
Petrophysics: Archie k-φ-F Relations
a. Crossplot of formation (resistivity) factor versus
permeability (F = a/φm).
Porosity Model:
R
F= o
Rw

a 
=

m
 φ 
Equating the models:
a
φm
=
A
kB
Permeability Model:
 A
= B
 k 
Ro
Rw
F=
Solving for k:
1/B
A

k =  φm
a

= αφ β
This exercise suggests that permeability and porosity
are related by a power law relation — this observation
is only true for uniform pore systems.
b. Crossplot of formation (resistivity) factor versus
permeability (F = A/kB).
 Archie k-φ-F Relations:
 Are porosity AND permeability are directly related to the formation factor?
 If this were universal, we could use well logs to estimate permeability.
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Petrophysics: Effect of Small-Scale Heterogeneities
 Weber Example Core:
 Laminated Aeolian sandstone.
 Thin beds (<1 cm) are common.
 Some laminations have zero permeability (influence on vertical flow?).
 General Considerations:
 Core-scale heterogeneities may or may
not affect overall reservoir performance
(depends on continuity).
 Attempts to correlate small-scale
heterogeneities are likely to fail, except
for isolated samples.
 Issues:
 How do such features affect:
— Pressure transient behavior (well
test time scale events)?
— Pseudosteady-state behavior
(production time scale events)?
 Solutions for increasing reservoir
exposure? (hydraulic fracturing?)
From: Weber, KJ.: "How Heterogeneity Affects Oil Recovery — from
Reservoir Characterization, Academic Press, Inc.-Harcourt Bruce
Jovanovich, Publishers, New York (1986) (Edited by: Lake, L.W.
and Carroll, H.B., Jr.).
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Petrophysics: Example —kWPA, kPTA with klog mean
15500
15500
15600
15600
15700
15700
15800
15800
15900
15900
16000
16000
16100
16100
16200
16200
16300
0.00
0.05
0.10
0.15
0.20
Porosity, fraction
0.25
0.30
From: Medina, T.: Characterization of Gas Condensate Reservoirs
Using Pressure Transient and Production Data-Santa Barbara
Field, Monagas, Venezuela, M.S. Thesis Texas A&M University
(May 2003).
Well TM-1E/Permeability Distribution with Depth
(Upper Naricual)
15400
Depth, ft
Depth, ft
Well TM-1E/Porosity Distribution with Depth
(Upper Naricual)
15400
16300
-1
10
10
0
1
2
10
10
Permeability, k, md
10
3
10
4
 Permeability Comparison: Santa Barbara Field (Venezuela)
 Major conclusion is that these data due not appear to be correlated.
 High permeability values probably "overweigh" klog mean estimate.
 kPTA values higher than kWPA, but we have only 3 (three) kPTA values.
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Reservoir Scale Issues: Halderson Schematics
Reservoir Scaling Issues
From: Simulator Parameter Assignment and the Problem of
Scaling in Reservoir Engineering — Halderson (1986).
?
NANO or ATTO
a. (Haldorsen) Four conceptual scales associated
with porous media averages.
T.A. Blasingame (2013.07.22)
b. (Haldorsen) Volume of investigation of a pressure build-up test and cross section indicating
large-scale internal heterogeneities.
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Geology/Petrophysics: Questions to Consider
Q1. Validity of correlations of petrophysical data?
A1. These will always be "local" correlations, difficult to extend or
extrapolate across depositional systems.
Q2. Role of geology in PTA?
A2. Must consider geology in general, but particularly for cases where
the following reservoir models are employed:
● Linear sealing or leaky faults.
(any geologic evidence?)
● Bounded reservoir system.
(geologic or petrophysical evidence?)
● Naturally fractured/dual porosity reservoir. (any geologic evidence?)
● Multilayered reservoirs.
(geologic or petrophysical evidence?)
Q3. Correlation of kcore with kPTA?
A3. Always a comparison of "apples and oranges" due to:
● Sample size.
● Saturation/mobility issues.
(core data are extremely localized)
(kcore evaluated using gas at low pressures)
Q4. Effect of reservoir heterogeneity on PTA?
A4. Interesting question — volume-averaging appears to dominate the
estimate of permeability obtained from PTA. Attempts to estimate
permeability "distributions" will be non-unique and/or overly simplified.
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Pressure Transient Analysis
Objective 2
Estimate oil, gas, and water properties pertinent
for well test or production data analysis using
industry accepted correlations and/or laboratory
data.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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From: Schlumberger: Fundamentals of
Formation Testing (March 2006).
PVT: Classification of Reservoir Fluids
 Overview: Classification of Reservoir Fluids
 Generic guidelines on properties of reservoir fluids.
 Useful to assess dominant component(s) and properties.
 For PTA, generally assume that a system is dry gas or non-volatile oil.
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PVT: Formation Volume Factor
Formation Volume Factor: Bo,g,w
Fluid volume at reservoir conditions
Bo,g,w =
Fluid volume at standard conditions
Bo,g,w is defined as a volume conversion for oil, gas, or
water — and is defined on a mass (or density) basis.
The Formation Volume Factor "converts" surface
volumes to downhole conditions.
Typical values: Oil:
1.2
to
Gas:
0.003 to
Water: 1.00 to
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2.4 RB/STB
0.01 rcf/scf
1.03 RB/STB
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PVT: Fluid Viscosity
Viscosity: µo,g,w
Is a measure of a fluid's internal resistance to flow
... the proportionality of shear rate to shear stress,
a sort of internal friction.
Fluid viscosity depends on pressure, temperature,
and fluid composition.
Typical values: Oil:
0.2 to 30 cp
Gas: 0.01 to 0.05 cp
Water: 0.5 to 1.05 cp
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PVT: Fluid and Formation Compressibility
Fluid Compressibility: co,g,w
1 dBo Bg dRso
co = −
+
Bo dp Bo dp
Typical values:
Oil:
5
30
Gas:
50
Water:
3
to
to
to
to
1 dBg
cg = −
Bg dp
1 dBw Bg dRsw
cw = −
+
Bw dp Bw dp
20 x10-6 psi-1 (p>pb)
200 x10-6 psi-1 (p<pb)
1000x10-6 psi-1
5
x10-6 psi-1
Formation Compressibility: cf
cf =
1 dφ
φ dp
Typical values:
Normal:
2 to 10 x10-6 psi-1
Abnormal: 10 to 100 x10-6 psi-1
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PVT: Various "Black Oil" Fluid Properties
 "Black Oil" PVT Properties: (general behavior, pb=5000 psia)
 Note the dramatic influence in properties at the bubblepoint pressure.
 The oil compressibility is the most affected variable (keep this in mind).
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PVT: 1/(µoBo) for p<pb ("Solution Gas-Drive" Case)
 "Solution-Gas Drive" PVT Properties: (1/(µoBo), p<pb, pb=5000 psia)
 Attempt to illustrate that 1/(µoBo) ≅ constant for p<pb.
 This would allow us to approximate behavior using "liquid" equations.
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PVT: z vs. ppr and ρpr (dry gas case)
a. "Standing-Katz" base plot (z vs. ppr) —
Poettmann-Carpenter Data (5960 data
points).
T.A. Blasingame (2013.07.22)
b. "Standing-Katz" plot (z vs. ρpr) —
Poettmann-Carpenter Data (5960 data
points).
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b. Original Lee, et al. correlation for hydrocarbon
gas viscosity.
a. Gas viscosity versus temperature for the Gonzalez
et al data (natural gas sample 3) compared to the
implicit correlation for gas viscosity (Londono)
and the original Lee, et al. correlation for
hydrocarbon gas viscosity.
T.A. Blasingame (2013.07.22)
From: Londono, F.E.: "Simplified Correlations for Hydrocarbon Gas
Viscosity and Gas Density: Validation and Correlation of Behavior
Using a Large-Scale Database," M.S. Thesis, Texas A&M University
(December 2001).
PVT: µg vs. T (and p) (dry gas case)
c. Londono "implicit" correlation for hydrocarbon
gas viscosity (residual viscosity type model).
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PVT: µgz vs. p (dry gas case)
 "Dry Gas" PVT Properties: (µgz vs. p)
 Basis for the "pressure-squared" approximation (i.e., use of p2 variable).
 Concept: (µgz) = constant, valid only for p<2000 psia.
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PVT: µgcg vs. p (dry gas case)
 "Dry Gas" PVT Properties: (µgcg vs. p)
 Concept: If µgcg ≅ constant, pseudotime NOT required.
 Readily observe that µgcg is NEVER constant, pseudotime required.
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PVT: Questions to Consider
Q1. Limitations of assuming a "black oil" for liquids?
A1. There are issues … but historically, the use of the constant
compressibility concept (i.e., a "black oil") has tolerated even
extreme violations of the assumption with few substantial problems.
The most obvious case where a black oil concept will not suffice is
that of a volatile oil (very high GOR).
Q2. Limitations of assuming a "dry gas" for gases?
A2. The major limitation is that of very rich gas condensate cases
(analogous to the "volatile oil" case mentioned above).
Q3. Are existing fluid properties correlations sufficient?
A3. For most cases, yes. For cases of extremely high pressure and/or
temperature, new correlations are warranted.
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Notes:
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Pressure Transient Analysis
Objective 3
Sketch pressure versus time trends and
pressure versus distance trends for a reservoir
system which exhibits transient, pseudosteadystate, and steady-state flow behavior.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Pressure Distributions: Solutions
All relations given in FIELD units.
Steady-State Solution:
q sc Bµ
pr = p w + 141.2
ln(r/rw )
kh
q sc Bµ
p r = pe − 141.2
ln(re /r )
kh
[pr — pwf form]
[pr — pe form]
Radius of Investigation:
Full Solution: (qsc=constant)
pD =
rinv = 2.434x10 -2
1
kh
( pi − pr )
141.2 qBµ
2
1  rD
≈ E1
2  4t D

 1  r2
 − E1  eD
 2  4t D


T.A. Blasingame (2013.07.22)

− r2
t
 + 2 D exp  eD
2
 4t D

r
eD


k
t
φµct
  r2
− r2

1
 +  D −  exp  eD
  2r 2
 4t D
4
  eD


Pressure Transient Analysis — PETE 663




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Pressure Distributions: Transient Flow
Radial Pressure Distribution (Lee text Fig. 1.7)
Pressure Drawdown and Buildup Cases — E1(x) Solution
2025
∆ t = 1000 hr
r 0.1 hr
p i = 2000 psia
r 1 hr
r 10 hr
r 100 hr
Legend:
r 1000 hr
2000
pD_DD(r, t_ 1Em1 hr)
∆ t = 100 hr
Pressure, psia
1950
pD_DD(r, t_ 1E0 hr)
∆ t = 10 hr
pD_DD(r, t_ 1E1 hr)
∆ t = 1 hr
pD_DD(r, t_ 1E2 hr)
pD_DD(r, t_ 1E3 hr)
1925
∆ t = 0.1 hr
1900
t = 0.1 hr
1875
1850
1825
re = 3000 ft
1975
1 hr
10 hr
pD_BU(r,tp_+_ Dt_ 1Em1 hr)
pD_BU(r,tp_+_ Dt_ 1E0 hr)
pD_BU(r,tp_+_ Dt_ 1E1 hr)
pD_BU(r,tp_+_ Dt_ 1E2 hr)
pD_BU(r,tp_+_ Dt_ 1E3 hr)
100 hr
1000 hr
1800
1775
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
Radial Distance, ft
 Pressure Distributions for Transient Radial Flow
 Note the effect of the drawdown.
 Note that the buildup pressure trends retrace last drawdown trend.
 Recall that all measurements are at the wellbore, we cannot "see" in the
reservoir — our analyses are inferred from wellbore measurements.
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Pressure Distributions: Pseudosteady-State
The physical concept of the PSEUDOSTEADY-STATE FLOW condition is defined as the condition where the pressure at all points
in the reservoir changes at the same rate. Mathematically, this
condition is given by:
d
[ p (r , t )] r = constant
dt
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Pressure Distributions: Pseudosteady-State
Concept: (pressure changes at the same rate at all
points in the reservoir)
 dp  = constant
 dr 
r
Reservoir Pressure Schematic:
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Pseudosteady-State Flow: Summary of Relations
(pr-pwf) Flow Relations: (Circular Reservoir)
 r  1 (r 2 − rw2 ) 
qBµ  re2

pr − p wf = 141 .2
+ s
ln 
−
2
2
2
2
kh  (re − rw )  rw  2 (re − rw ) 


( p -pwf) Flow Relations: (γ = 0.577216 Euler's constant)
qBµ   re  3 
p = p wf + 141 .2
(Circular Reservoir)
ln 
 − + s
kh   rw  4 
qBµ
p = p wf + 141 .2
kh
1  4 A 1  
 ln 
 + s  (General Formulatio n)
γ
2
 2  e rw C A  


Time-Dependent Pseudosteady-State Flow Relations:
qBµ   re  1 (r 2 − rw2 ) 3 
qB
ln   +
p r = pi − 141 .2
t
−  − 5.615
2
2
kh   r  2 (r − r ) 4 
Vp ct
e
w


qBµ   re  3 
qB
p wf = pi − 141 .2
s
ln
t
5
.
615
−
+
−
 


kh   rw  4 
Vp ct
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rinv = 2.434x10 -2
From: Blasingame, T.A.: Variable-Rate Analysis: Transient and PseudosteadyState Methods of Interpretation and Application, M.S. Thesis, Texas
A&M University (1986).
Pseudosteady-State Flow: Illustrative Behavior
k
t
φµ ct
Figure 2: Reservoir Pressure Distribution — Constant Rate Transient
Flow Drawdown.
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rinv = 2.434x10 -2
From: Blasingame, T.A.: Variable-Rate Analysis: Transient and PseudosteadyState Methods of Interpretation and Application, M.S. Thesis, Texas
A&M University (1986).
Pseudosteady-State Flow: Illustrative Behavior
k
t
φµ ct
Figure 4: Reservoir Pressure Distribution — Log Linear Rate Transient
Flow Drawdown.
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rinv = 2.434x10 -2
k
t
φµ ct
From: Blasingame, T.A.: Variable-Rate Analysis: Transient and PseudosteadyState Methods of Interpretation and Application, M.S. Thesis, Texas
A&M University (1986).
Pseudosteady-State Flow: Illustrative Behavior
Figure 7: Reservoir Pressure Distribution — Constant Wellbore
Pressure Transient Flow Drawdown.
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rinv = 2.434x10 -2
k
t
φµ ct
From: Blasingame, T.A.: Variable-Rate Analysis: Transient and PseudosteadyState Methods of Interpretation and Application, M.S. Thesis, Texas
A&M University (1986).
Pseudosteady-State Flow: Illustrative Behavior
Figure 52: Reservoir Pressure Distribution — Constant Rate PostTransient Flow Drawdown, Homogeneous Reservoirs.
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rinv = 2.434x10 -2
From: Blasingame, T.A.: Variable-Rate Analysis: Transient and PseudosteadyState Methods of Interpretation and Application, M.S. Thesis, Texas
A&M University (1986).
Pseudosteady-State Flow: Illustrative Behavior
k
t
φµ ct
Figure 57: Reservoir Pressure Distribution — Constant Wellbore Pressure
Post-Transient Flow Drawdown, Homogeneous Reservoirs.
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Reservoir Pressure Trends: Questions to Consider
Q1. Why study "reservoir pressure trends?"
A1. We can not measure pressure in the reservoir — only at the wellbore
(or sandface). In order to estimate the behavior in the reservoir, we
must use "model-based" pressure distributions.
Q2. Isn't the use of a simple model too limiting?
A2. Actually, no. Simple models are extremely consistent, and as such,
even when "wrong," the "trend" behavior is typically quite representative.
Q3. What is the "radius of investigation?"
A3. For the infinite-acting radial flow case, the radius of investigation is
the point in the reservoir where the logarithm of radius equation
(straight line) intersects the initial reservoir pressure. It is a fictitious
point, but it represents the "theoretical" location of the front of the
pressure distribution front.
rinv = 2.434 x10
T.A. Blasingame (2013.07.22)
−2
k
t
φµ ct
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Pressure Transient Analysis
Objective 4
Derive the material balance relation for a slightly
compressible liquid (oil) in the presence of other
phases (gas and water), as well as the material
balance relation for a dry gas.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Material Balance: Historical Perspective
"It seems no longer fashionable to apply the concept
of material balance to oilfields, the belief being that it
has now been superseded by the application of the
more modern technique of numerical simulation
modeling.
Acceptance of this idea has been a TRAGEDY
and has robbed engineers of their most powerful tool
for investigating reservoirs and understanding their
performance rather than imposing their wills upon
them, as is often the case when applying numerical
simulation directly in history matching."
L.P. Dake
The Practice of Reservoir Engineering, Elsevier (2001)
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Material Balance: Orientation
Issues:
Oil MBE (must know all data, also cf(p)).
Gas MBE (abnormal pressure, water drive).
Topics:
"Accounting" Concept of Material Balance:
 Require all inflows/outflows/generations.
 (Average) reservoir pressure profile is REQUIRED.
 Require rock, fluid, and rock-fluid properties (at some scale).
Oil Material Balance:
 Less common than gas material balance (pressure required).
Gas Material Balance:
 Volumetric dry gas reservoir (p/z versus Gp (straight-line)).
 Abnormally-pressured gas reservoirs (various techniques).
 Waterdrive/water influx cases (always problematic).
Material Balance yields RESERVOIR VOLUME!
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Material Balance: Oil
"Black Oil" Material Balance: (p>pb )
1 Bo
p = pi −
Np
Nct Boi
"Solution Gas Drive" (Oil) Material Balance: (all p)
[
]
Np Bo + ( R p − Rs ) Bg + Wp Bw =
[
N ( Bo − Boi ) + ( Rsi − Rs ) Bg
]

 Bg
+ mNBoi 
− 1
 Bgi 
(cw S wi + c f )
+ (1 + m) NBoi
( pi − p )
(1 − S wi )
+ We Bw
T.A. Blasingame (2013.07.22)
(Withdrawal (RB))
(Oil Expansion (RB))
(Gas Cap Expansion (RB))
(Water Exp./Pore Vol. Comp. (RB))
(Water Influx (RB))
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Material Balance: Gas
General Gas Material Balance:
p
[1 − ce ( p )( pi − p )] =
z

pi pi 1 
1
(Wp − Winj ) Bw − We 
−
Gp − Ginj + Wp Rsw + 5.615
zi zi G 
Bg

[
]
"Dry Gas" Material Balance: (no reservoir liquids)
p pi
=
z zi
 −1G 
1 G p 
"Abnormal Pressure" Material Balance: (cf=f(p))
 Gp 
1
p pi
1−
=
z zi [1 − ce ( p )( pi − p )] 
G 

 VpNNP
1
ce ( p ) =
 S wi cw + c f +  
(1 − S wi ) 
  VpR
T.A. Blasingame (2013.07.22)
 VpAQ
+
  VpR

  (cw + c f
 
Pressure Transient Analysis — PETE 663

)

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Material Balance: Gas — Abnormal Pressure
 Gas Material Balance: Abnormally Pressured Reservoir Schematic
 Normal pressure production sequence (volumetric depletion, G).
 Abnormal pressure production sequence (Gapp).
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Material Balance: Gas — Abn. Pressure (US GOM)
Should this extrapolation
be a straight-line?
 Gas Material Balance: Abnormally Pressured Reservoir Schematic
 Normal pressure production sequence (volumetric depletion, G).
 Abnormal pressure production sequence (Gapp).
 Note the position of the "pivot point" is at hydrostatic pressure.
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Material Balance: Gas — Normal Pressure
 "Dry Gas" Material Balance: Normal Pressured Example
 Volumetric reservoir — no external energy (gas expansion only).
 p/z versus Gp yields unique straight-line trend.
 Linear extrapolation yield gas-in-place (G).
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Material Balance: Gas — Abnormal Pressure
 "Dry Gas" Material Balance: Abnormally Pressured Reservoir
 Volumetric reservoir — no water influx or leakage.
 p/z versus Gp yields unique quadratic trend (approximate MBE).
 Quadratic extrapolation yield gas-in-place (G).
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From: Unsteady-State Performance of Water Drive Gas Reservoirs,
Agarwal (Texas A&M Ph.D., 1967).
Material Balance: Gas — Water Influx
a. Gas Material Balance Plot: p/z vs. Gp —
simulated performance. Note effect of
aquifer permeability on field performance.
b. Gas Material Balance Plot: p/z vs. Gp —
simulated performance. Note effect of
displacement efficiency (Ep).
 Gas Material Balance: Water Drive Gas Reservoir
 Pressure (hence p/z) is maintained during production via water influx.
 Agarwal used an unsteady-state aquifer to generate these cases.
 Numerous other aquifer models (analyst must choose).
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Material Balance: Questions to Consider
Q1. What is the "weakest link" in material balance?
A1. Need for very accurate production records — particularly average
reservoir pressure (which is rarely available).
Q2. What is the strength of material balance?
A2. It is an accounting method, essentially independent of the reservoir
model. It provides an estimate of initial reservoir volume being
sampled by the wells under production.
Q3. Future of material balance?
A3. Difficult to say, material balance essentially being (or has been)
replaced by numerical reservoir simulation.
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Pressure Transient Analysis
Objective 5
Derive the analysis and interpretation methodologies (i.e., "conventional" plots and type curve
analysis) for pressure drawdown and pressure
buildup tests, for liquid, gas, and multiphase
flow systems.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
T.A. Blasingame (2013.07.22)
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Wellbore Storage: Base Relations/Concepts
 dpwf dptf 
(q sf − q ) B = 24 C s 
−

dt 
 dt
(field units form)
(Definition of Cs for a fluid filled wellbore)
C s = c wVwb
(Definition of Cs for a well with a rising
or falling liquid level)
Awb
144
Cs =
5.615 ρ ( g / g c )
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Wellbore Storage: Base (Cartesian) Plots
(Drawdown Case)
p wf = pi −
T.A. Blasingame (2013.07.22)
qB
t
24 C s
(Buildup Case)
qB
p ws = p wf (∆t = 0) +
t
24 C s
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Wellbore Storage: Approximate Solution
Dimensionless Pressure Relation: Constant Approximation for pSD(tD)



− tD
p wCD (t D ) = p sD (t D ) 1 − exp 


 p sD (t D )C D  
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Wellbore Storage: "Bourdet-Gringarten" Type Curve
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Wellbore Storage: Cartesian Plot
pws = pwf (∆t = 0) + mwbs∆t
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Wellbore Storage: Log-Log Plot
∆p = pws − pwf (∆t = 0) = + mwbs∆t
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Infinite-Acting Radial Flow (IARF): Semilog Plot
pws = pwf ,1hr + msl log( ∆t )
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Infinite-Acting Radial Flow (IARF): Log-Log Plot
dpws
1
∆p ' ≡ ∆t
and for radial flow only, (∆p' ) rf =
msl
ln(10 )
d∆t
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Average Reservoir Pressure: Cartesian Plot
1 d
pws = p − a exp[ −b∆t ] and pws = p −
[ pws ]
b d∆t
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Conventional PTA Plots: Questions to Consider
Q1. What are the "conventional" PTA plots?
A1. Listing of plots:
● Log-log (diagnostic) plot (∆pw and ∆pw') (reservoir boundaries, WBS, k)
● Semilog plot (pw vs. log[t])
(k,s)
● Early-time Cartesian plot (pw vs. t)
(WBS)
● Late-time Cartesian plot (∆pw vs. d∆pw/dt) (average reservoir pressure)
Q2. Strengths of "conventional" PTA plots?
A2. Sampling:
● Observation of straight-line or constant behavior.
● Simplified "flow regime" relations that can be used to estimate reservoir
and/or well properties.
Q3. Weaknesses of "conventional" PTA plots?
A3. Sampling:
● Can observe "artifact" regimes (e.g., linear flow) that are not real.
● Observe only part of the overall behavior in time, can be limiting in a
diagnostic sense.
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Pressure Transient Analysis
Objective 6
Apply dimensionless solutions ("type curves")
and field variable solutions ("specialized plots")
for unfractured and fractured wells in infinite and
finite-acting, homogeneous and dual porosity
reservoirs.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
T.A. Blasingame (2013.07.22)
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Orientation: PTA Model-based Analysis
 Reservoir Models:
 Unfractured Well
 Fractured Well
 Naturally Fractured Reservoir
 Type Curve Library:
 Unfractured Well: WBS + IARF
("Bourdet-Gringarten")
 Pressure Buildup in a Rectangle (Unfractured Well)
("Ansah")
 Linear (Sealing) Reservoir Boundaries
("Stewart")
 Fractured Well: no WBS
("Cinco and Samaniego")
 Fractured Well: WBS
("Economides")
 Naturally Fractured Reservoir: Unfractured well
("Angel")
 Field Examples: (from SPE 103204)
 Unfractured oil wells.
 Hydraulically fractured gas wells.
 Hydraulically fractured water injection wells.
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Pressure Transient Analysis
Objective 6
Reservoir Models
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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From: Matthews, C. S. and Russell, D. G.: Pressure Buildup and
Flow Tests in Wells. Monograph Series, Society of
Petroleum Engineers of AIME, Dallas (1967) 1.
Unfractured Well: Flow Regimes
 Discussion: Flow Regimes (Unfractured Wells)
 INFINITE-ACTING RADIAL FLOW (IARF) is the most "popular" regime.
 PSEUDOSTEADY-STATE (PSS) flow → CLOSED BOUNDARIES.
 STEADY-STATE (SS) flow → CONSTANT PRESSURE (not realistic).
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From: Earlougher, R.C. Jr.: Advances in Well Test Analysis,
Monograph Series, SPE, Dallas (1977) 5.
Unfractured Well: Orientation and Solutions
 Discussion: Orientation and Solutions (Unfractured Wells)
 Pressure profile propagates radially away from well (homogeneous).
 Cylindrical source solution → finite wellbore.
 Line source solution → infinitesimal wellbore (i.e., a line).
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From: Earlougher, R.C. Jr.: Advances in Well Test Analysis,
Monograph Series, SPE, Dallas (1977) 5.
Unfractured Well: Skin Factor Concept
 Discussion: Skin Factor Concept (Unfractured Wells)
 Finite skin concept → zone of "altered" permeability near the well.
 Infinitesimal skin concept → mathematical convenience.
 Negative skin has mathematical (and physical) limitations.
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From: Cinco-Ley, H., Samaniego-V., F.: "Transient Pressure
Analysis for Fractured Wells," JPT (September 1981)
1749-1766.
Fractured Well: Flow Regimes
 Discussion: Flow Regimes
 FORMATION LINEAR flow DOES NOT EXIST (a few seconds at most).
 FORMATION linear flow → High fracture conductivity.
 BILINEAR flow → Low fracture conductivity.
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From: Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.: "Transient
Pressure Behavior for a Well with a Finite-Conductivity Vertical
Fracture," SPEJ (August 1978) 253-264.)
Fractured Well: Fracture Flux Distributions
 Discussion: Fracture Flux Distributions
 Discretized fracture must be solved numerically.
 High-conductivity fracture → flux distribution IS NOT significant at well.
 Finite-conductivity fracture → flux distribution IS significant at well.
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From: Cinco-Ley, H., Samaniego-V., F.: "Transient Pressure Analysis:
Finite Conductivity Fracture Case Versus Damaged Fracture
Case," paper SPE 10179 presented at the 1981 SPE Annual
Technical Conference and Exhibition, San Antonio, TX, 5-7 Oct.
Fractured Well: Fracture Damage Comparison
 Discussion: Fracture Damage Comparison
 Argument: Finite conductivity can be modeled as damage...
(false!)
 "Fluid loss" damage is no referred to as "fracture face" skin.
 "Choked fracture" damage is just a constant skin factor.
(not correct)
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Fractured Well: Analytical Solution (Uniform Flux)
General (Uniform Flux) Solution: (Infinite Conductivity Solution; xD≈0.732)
Short-Time Solution: Linear Flow
Long-Time Solution: Pseudoradial Flow (Infinite Conductivity Fracture)
Identities:
2
erf ( z ) =
π
z
∫0
E1 ( z ) =
exp(−t 2 ) dt
[erf (0) = 0; erf (∞) = 1; erf (−∞) = − 1]
T.A. Blasingame (2013.07.22)
∞ e −t
∫z
t
dt
 1 
[ E1 ( z < 0.01) ≅ ln 
; E1 (∞) = 0]
γ
 ze 
(γ = 0.577216... Euler' s constant )
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From: Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.: "Transient
Pressure Behavior for a Well with a Finite-Conductivity Vertical
Fracture," SPEJ (August 1978) 253-264.
Fractured Well: Finite-Conductivity Type Curve
Slope = 1/4
Slope = 1/2
 Discussion: Finite-Conductivity Type Curve
 FORMATION LINEAR flow → High fracture conductivity.
 BILINEAR flow → Low fracture conductivity.
 Linear and bilinear flow end due to "tip effect" — flow around fracture tip.
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From: Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.: "Transient
Pressure Behavior for a Well with a Finite-Conductivity Vertical
Fracture," SPEJ (August 1978) 253-264.
Fractured Well: Skin Factor Correlation
 Discussion: Skin Factor Correlation
 Developed to relate PSEUDORADIAL flow skin factor and fracture cases.
 Useful in 1980's to generate a skin factor that managers could understand for cases of fractured wells (still used for that purpose …).
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a. Natural fracture dependence on stress state
and orientation.
b. Schematic of compressional and tensional
fracturing in-situ.
From: Leroy, G.: "Cours de Geologie de Production, Inst. Francais du
Petrole. Ref. 24,429 (1976).
From: Stearns, D.W. and Friedman, M.: "Reservoirs in Fractured Rock
in Stratigraphic Oil and Gas Fields Classification, Exploration
Methods and Case Histories" AAPG Mem. 16. (1972) 82-106.
Naturally Fractured Reservoirs: Fracture Patterns
 Discussion: Fracture Patterns
 Fracture patterns are due to stress orientation.
 Large-scale fractures can yield tremendous productivity.
 Stress state changes during production (depletion) — re-fracture?
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From: Najurieta, H.L.: "A Theory for Pressure Transient Analysis in
Naturally Fractured Reservoirs," JPT (July 1980) 1241-1250.
From: de Swaan, O.A.: "Analytic Solutions for Determining Naturally
Fractured Reservoir Properties by Well Testing," SPEJ, (June 1976)
117-122; Trans., AIME, 261.
Naturally Fractured Reservoirs: Fracture Models
 Discussion: Fracture Models
 Kazemi initially produced "slab" model using numerical simulator.
 De Swaan developed the solution for transient interporosity flow.
 Najurieta developed Laplace domain form of De Swaan result.
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(1/2)
From: Warren, J.E., and Root, P.J.: "The Behavior of Naturally Fractured
Reservoirs," SPEJ (Sept. 1963) 245-55; Trans. AIME, 228.
Naturally Fractured Reservoirs: W&R Model
 Discussion: Warren and Root Model
 "Borrowed" (i.e., stolen) from Barenblatt and Zheltov.
 By far the most popular "heterogeneous" reservoir model.
 Some physical limitations, but its simplicity provides unique flexibility.
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Naturally Fractured Reservoirs: W&R Model
(2/2)
Laplace Domain Solution:
s
1
(Line Source Solution)
K 0 ( uf (u ) rD ) +
u
u
1  4 1 1  s
ln
≈
+ (" Log" Approximation)
2u  eγ r 2 uf (u )  u
D


λ + ω (1 − ω )u
f (u ) =
λ + (1 − ω )u
p D (u , rD , ω , λ , s ) =
(No Wellbore Storage)
Real Domain Solution: (Derived from the Log Approximation Solution)
1  4
p D (tD , rD , ω , λ , s ) ≈ ln
2  eγ


 1  λ
tD  1  λ
tD  + E1 
tD  + s
− E1 
rD2  2  ω (1 − ω )  2  (1 − ω ) 

 −λ
 1
 −λ
' ( , , , , )≈ 1 + 1
p D tD rD ω λ s
tD  − exp 
tD 
exp 
2 2
 (1 − ω ) 
 ω (1 − ω )  2
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Reservoir Models: Questions to Consider
Q1. What are the "traditional" reservoir models?
A1. Listing:
● Infinite-Acting Radial Flow (IARF) model (unfractured well)
● Vertically Fractured Wells:
— Infinite-Conductivity Vertical Fracture
— Finite-Conductivity Vertical Fracture
● Naturally-Fractured/Dual Porosity Reservoirs:
— Pseudosteady-State Interporosity Flow (Warren and Root)
— Transient Interporosity Flow (Kazemi-De Swaan-Najurieta)
● Horizontal Wells (problematic, requires interactive model (computer))
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Pressure Transient Analysis
Objective 6
Type Curves
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Type Curves: WBS + IARF ("Bourdet-Gringarten")
● Type Curve: WBS + IARF ("Bourdet-Gringarten") (unfractured
■ "Starting point" for virtually all pressure transient test analysis.
well)
■ ∆p': WBS domination = "unit slope;" Infinite-acting radial flow (IARF) = 1/2.
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From: Ansah, J., Likitsupin, P., and Blasingame, T.A.: "Determination of
Reservoir Pore Volume From Pressure Buildup Tests Using Type Curves,"
paper SPE 29584 presented at the 1995 Joint Rocky Mountain
Regional/Low Permeability Reservoirs Symposium, Denver, CO, 20-22
March, 1995.
Type Curves: Late-Time Buildup ("Ansah")
● Type Curve: Late-Time Buildup ("Ansah") (unfractured
well)
■ A "correlation" of late-time cases of pressure buildup in a rectangle.
■ Helps to distinguish the Muskat (late-time) pressure buildup behavior.
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From: Stewart, G., Gupta, A., and Westaway, P.: "The Interpretation of Interference
Tests in a Reservoir With Sealing and Partially Communicating Faults,"
paper SPE 12967 presented at the 1984 European Petroleum Conference
held in London, England 25-28 Oct. 1984.
Type Curves: Sealing Faults ("Stewart")
● Type Curve: Sealing Faults ("Stewart") (unfractured well)
■ Solutions for sealing faults have a distinct (and unique) behavior.
■ The radial composite model can be virtually indistinguishable (check the geology!).
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From: Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.: "Transient Pressure
Behavior for a Well with a Finite-Conductivity Vertical Fracture," SPEJ
(August 1978) 253-264.
Type Curves: Fractured Well (No WBS) ("Cinco")
● Type Curve: Fractured Well (No WBS) ("Cinco")
■ Distinctive flow regimes — pressure and derivative functions very strong diagnostics.
■ Somewhat impractical (no wellbore storage effects).
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From: Cinco-Ley, H., Samaniego-V., F., and Dominguez, N.: "Transient Pressure
Behavior for a Well with a Finite-Conductivity Vertical Fracture," SPEJ
(August 1978) 253-264.
Type Curves: Fractured Well (No WBS) ("Cinco")
● Type Curve: Fractured Well (No WBS) ("Cinco") (β-Derivative Formulation)
■ pDβd is a very strong diagnostic (linear and bi-linear flow).
■ pDβd is less distinctive for infinite-acting radial flow, but still useful.
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From: Hosseinpour-Zonoozi, N., *Ilk, D., and Blasingame, T.A.: "The Pressure
Derivative Revisited — Improved Formulations and Applications,"
paper SPE 103204 presented at the 2006 Annual SPE Technical
Conference and Exhibition, Dallas, TX, 23-27 September 2006.
Type Curves: Frac-Well (WBS) ("Economides") (CfD=1)
● Type Curve: Fractured Well (WBS) ("Economides") (CfD=1)
■ CfD=1: VERY LOW fracture conductivity (similar to damage).
■ Very strong bi-linear flow signature.
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[CfD=FcD]
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From: Hosseinpour-Zonoozi, N., *Ilk, D., and Blasingame, T.A.: "The Pressure
Derivative Revisited — Improved Formulations and Applications,"
paper SPE 103204 presented at the 2006 Annual SPE Technical
Conference and Exhibition, Dallas, TX, 23-27 September 2006.
Type Curves: Frac-Well (WBS) ("Economides") (CfD=10)
● Type Curve: Fractured Well (WBS) ("Economides") (CfD=10)
[CfD=FcD]
■ CfD=10: MEDIUM fracture conductivity.
■ Pressure drop and pressure derivative signatures vary (linear and bi-linear flow).
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From: Hosseinpour-Zonoozi, N., *Ilk, D., and Blasingame, T.A.: "The Pressure
Derivative Revisited — Improved Formulations and Applications,"
paper SPE 103204 presented at the 2006 Annual SPE Technical
Conference and Exhibition, Dallas, TX, 23-27 September 2006.
Type Curves: Frac-Well (WBS) ("Economides") (CfD=103)
● Type Curve: Fractured Well (WBS) ("Economides") (CfD=1x103)
[CfD=FcD]
■ CfD=1x103: VERY HIGH fracture conductivity (infinite fracture conductivity).
■ Very strong (formation) linear flow signature.
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T.A. Blasingame (2013.07.22)
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(No Wellbore Storage)
From: Onur, M., Satman, A., and Reynolds, A.C.: "New Type Curves for
Analyzing the Transition Time Data From Naturally Fractured Reservoirs,"
paper SPE 25873 presented at the SPE Rocky Mountain Regional/Low
Permeability Reservoirs Symposium, Denver, CO, 12-14 April, 1993.
Type Curves: Naturally Fractured Res. (No WBS)
● Type Curve: Naturally Fractured Reservoir (No Wellbore Storage)
■ Pseudosteady-state "interporosity" flow case.
■ This is the "cubes" or "Warren and Root" model.
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(No Wellbore Storage)
From: Onur, M., Satman, A., and Reynolds, A.C.: "New Type Curves for
Analyzing the Transition Time Data From Naturally Fractured Reservoirs,"
paper SPE 25873 presented at the SPE Rocky Mountain Regional/Low
Permeability Reservoirs Symposium, Denver, CO, 12-14 April, 1993.
Type Curves: Naturally Fractured Res. (No WBS)
● Type Curve: Naturally Fractured Reservoir (No Wellbore Storage)
■ Transient "interporosity" flow case.
■ This is the "slabs" or "Kazemi" model.
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(Includes Wellbore Storage and Skin Effects)
From: Angel, J.A.: Type Curve Analysis for Naturally Fractures Reservoir
(Infinite-Acting Reservoir Case) ─ A New Approach, M.S. Thesis,
Texas A&M U., College Station, Texas (2000).
Type Curves: Naturally Fractured Reservoir (WBS)
● Type Curve: Naturally Fractured Reservoir (WITH Wellbore Storage)
■ Pseudosteady-state "interporosity" flow case shown for emphasis.
■ This is the "Angel" type curve format.
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Type Curves: Questions to Consider
Q1. Any advantage of using type curves instead of interactive models?
A1. Type curves are static representations of solutions … but, this helps
the analyst to develop a visual diagnostic for each reservoir model.
Both type curves and interactive models are useful for pressure
transient analysis (PTA) and production analysis (PA).
Q2. Best diagnostics on type curves?
A2. Sampling:
● Pressure derivative function changed diagnostics (1980's).
● Pressure integral function(s) never caught on (1990's).
● β-pressure derivative identifies "power law" regimes (2006).
Q3. Advice/cautions?
A3. Sampling:
● Fractured wells can be extremely difficult to analyze, should have an
estimate of permeability to "lock" that aspect of the analysis.
● The dual porosity model is often abused — i.e., used where it is not
warranted.
● Be VERY careful with "flexible" models such as the radial composite
(reservoir) model and the changing wellbore storage (wellbore) model.
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Pressure Transient Analysis
Objective 6
Field Case Examples
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Infinite-Acting Radial Flow (IARF)
●Unfractured oil well: (SPE 11463)
■ Strong wellbore storage signature (pDβd =1).
■ Transition region from wellbore storage to infinite-acting radial flow.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Infinite-Acting Radial Flow (IARF)
●Unfractured oil well: (SPE 12777)
■ This result is an excellent match of all functions.
■ β-derivative function is an excellent diagnostic for the well-bore storage and
transition flow regimes.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Dual Porosity, Infinite-Acting Radial Flow
●Unfractured oil well in dual porosity system: (SPE 13054)
■ Derivative functions indicate dual porosity signature — good match.
■ "Less-than-perfect" late time data match may be due to rate history effects.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Dual Porosity, Infinite-Acting Radial Flow
●Unfractured oil well in dual porosity system: (SPE 18160)
■ Strong performance of the β-derivative function — particularly in the region
defined by transition from wellbore storage to transient interporosity flow.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Hydraulically Fractured Wells
●Fractured gas well: buildup test (SPE 9975 — Well 5)
■Wellbore storage effects (pDβd=1).
■Linear flow regime could be diagnosed clearly (pDβd=1/2) — very good match.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Hydraulically Fractured Wells
●Fractured gas well: buildup test (SPE 9975 — Well 10)
■ pDβd=1 indicates wellbore storage effect.
■ The well is either poorly fracture-stimulated, or a "skin effect" has obscured
any evidence of a fracture treatment.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Hydraulically Fractured Wells
●Fractured gas well: buildup test (SPE 9975 — Well 12)
■ Wellbore storage domination regime (pDβd=1).
■ The pDd and pDβd signatures during mid-to-late times confirm the well is
highly stimulated.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Hydraulically Fractured Wells
●Fractured water injection well: fall off test (Samad thesis — Well 207)
■ β-derivative function confirms the existence of an infinite conductivity
vertical fracture for this case (pDβd=1/2).
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Hydraulically Fractured Wells
●Fractured water injection well: fall off test (Samad thesis — Well 5408)
■ Wellbore storage domination (pDβd = 1) and infinite-acting radial flow (pDd
=1/2) — good match with infinite conductivity fracture type curve.
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Pressure Derivative Revisited — Improved Formulations and
Applications," paper SPE 103204 presented at the 2006 Annual SPE
Technical Conference and Exhibition, Dallas, TX, 23-27 September 2006.
 From: Hosseinpour-Zonoozi, N., Ilk, D., and Blasingame, T.A.: "The
Field Cases: Hydraulically Fractured Wells
●Fractured water injection well: fall off test (Samad thesis — Well 2403)
■ From these data we can ob-serve the flow regimes for wellbore storage
domination (pDβd = 1), and the infinite-acting radial (pDd =1/2).
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Pressure Transient Analysis
Objective 7
Analyze production data (rate-time or pressurerate-time data) to obtain reservoir volume and
estimates of reservoir properties for gas and
liquid reservoir systems. The student should
also be able to make performance forecasts for
such systems.
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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Notes:
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Orientation: Production Analysis (PA) Data
Q. What is Production Analysis (PA)?
A. Combined analysis of rate and flowing bottomhole pressure data.
Rate Data:
Flowrates are measured
on a per-well basis for
most gas wells — oil
flowrates are often allocated (this is a major
issue).
Pressure Data:
Measured bottomhole
pressure data are essentially non-existent —
surface pressure data
are often available for
gas wells, flowing (surface) pressure data for
oil wells are rare at
best.
 Discussion: Orientation — Production Analysis (PA) Data
 PA is a "passive technology" — can it yield high resolution results?
(yes)
 What is/are the key data issue(s)? (q and pwf data accurate and correlated)
 Erratic q and pwf data — how to analyze? (boundary-dominated flow theory)
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Orientation: PA Data Analysis Example (Gas)
Q. Does material balance time work? (prove with an example)
A. Typical low productivity gas well example (mid-Continent US).
Transient Behavior:
Appears to reflect behavior of a vertical well
with a finite-conductivity vertical fracture.
 ∆m( p)

 q g

 Gp
≈ m̂ g , pss 

 trans
 q g



(1/4)
BDF/PSS Behavior:
Clear indication of
(approximate) material
balance behavior.
 ∆m( p)

 q g

 Gp
≈ m̂ g , pss 

 bdf
 q g



(1)
Material balance behavior is independent of
reservoir shape!
 Discussion: Orientation — PA Data Analysis Example (Gas)
 What are the limitations of this approach? (poor rate data, poor sampling)
 What is the "best" data frequency? (minimum preferred frequency = daily)
 What is the effect of pressure? (pressure data critical — but in practice ...)
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Orientation: PA Objectives
Q. What are the objectives of Production Analysis (PA)?
A. Estimate reservoir properties and volume, and predict performance.
Transient Radial Behavior: (PTA/PA)
r(t)
(infinite-acting
radial flow behavior)
Common Characteristics:
 Constant rate (q).
 Constant reservoir properties.
 Constant fluid properties.
Differences:
 Volume ONLY from pseudosteady-state data.
 Reservoir properties ONLY
from transient flow data.
∆p = ( pi − p wf ) = btrns ,cr + mtrns ,cr ln(t )
d
d
[∆p] = mtrns ,cr
∆p' = t [∆p ] =
dt
dln(t )
mtrns ,cr = f (k , ...)
Pseudosteady-State Behavior: (PA)
btrns ,cr = f (k , s, ...)
re
 dp 
 dr  = 0
 r
e
(closed outer
boundary)
∆p = ( pi − p wf ) = bpss ,cr + m pss ,crt
d
d
[∆p] = m pss ,crt
∆p' = t [∆p ] =
dt
dln(t )
m pss ,cr = f ( N , ...) bpss ,cr = f (k , s, ...)
 Discussion: Orientation — PA Objectives
 In THEORY — any differences in PA and PTA? (none, could use same tools)
 In PRACTICE — differences in PA and PTA? (PTA data is higher resolution)
 In FUTURE — what will happen with PA and PTA? (applications will merge)
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Orientation: PA Data Requirements
Q. What are the data requirements for Production Analysis (PA)?
A. Rate and pressure data, reservoir and fluid properties, well history.
 Discussion: Orientation — PA Data Requirements
 Issues with pressure?
(measured infrequently at surface)
 Issues with rate?
(gas → usually good, oil → problematic, water → poor)
 Issues with well completion?
(review history, keep completion simple)
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History: Production Analysis (PA) — q(t)=qiexp(-Dit)
Q. Can the "exponential" rate-time relation (q(t)=qiexp(-Dit)) be derived?
A. Yes, see steps below — slightly compressible liquid, pwf=constant.
Oil Material Balance Eq. (MBE): (p>pb)
The q(t)=qiexp(-Dit) form
is correct for boundary1 Bo
dominated flow behavior
p = pi − mo , pss Np where mo , pss =
— slightly compressible
Nct Boi
liquid, pwf=constant.
Oil Pseudosteady-State Flow Eq. (PFE): (p>pb)
µ o Bo  1  4 1 A  
 ln 
 + s
p = p wf + bo , pss qo where bo , pss = 141.2
2
γ
kh  2  e CA rw  

 

Steps:
1. Differentiate oil MBE and oil PFE with respect to time.
2. Assume: pwf = constant [i.e., d(pwf)/dt = 0].
3. Equate results → 1st order o.d.e.
4. Separate/integrate.

mo , pss 

5. Exponentiate result — final form: q = qi exp( − Di t )  Di =
b

o , pss 
 Discussion: History — Production Analysis (PA) — q(t)=qiexp(-Dit)
 Is the q(t)=qiexp(-Dit) relation rigorous?
(oil (p>pb) — yes, gas — ?)
 Other formulations?
(rate-cumulative: q(t)=qi – DiNp)
 Applications?
(estimate reserves (Np,max = Np(q→0)), q(t) predicton)
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History: Production Analysis (PA) — q(t)
Q. "Rate-Time" Plot: qo versus t (or qg versus t)?
(original purpose?)
A. Base "performance" plot for PA.
(originally used for taxation...)
— From: Fetkovich, M.J.: "Decline Curve Analysis Using Type
Curves," JPT (June 1980) 1065-1077.
Theory: (exp model)
Assume slightly compressible liquid, pwf=
constant — use oil MBE
and oil PSS FE to yield:
(p<pb)
q = qi exp( − Di t )
Theory: (hyp model)
APPROXIMATE derivation using oil MBE
(p<pb), and oil PSS FE
(also p<pb):
q=
qi
(1 + bDi t ) (1 / b)
 Discussion: History — Production Analysis (PA) — q(t)
 Is the q(t)=qiexp(-Dit) model supported by theory?
(yes — liquid; pwf=con)
 Is the q(t)=qi/[(1+bDit)^(1/b)] model supported by theory?
("not exactly")
 Issues?
(exponential = conservative; hyperbolic = liberal → use caution)
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History: Production Analysis (PA) — (EUR)exp
Q. What is an Estimated Ultimate Recovery (or EUR) plot?
A. Plot q(t) vs. Np, extrapolate to zero rate using a straight line ((EUR)exp).
— From: Fetkovich, M.J.: "Decline Curve Analysis Using Type
Curves," JPT (June 1980) 1065-1077.
Theory: (exp model)
The q(t)=qiexp(-Dit)
model (i.e., slightly compressible liquid, pwf=
constant) is integrated
to yield:
q = qi − Di Np
Application: (EUR)exp
Plot q(t) versus Np, extrapolate trend using a
straight-line model to
Np(q=0) — this gives
estimated ultimate recovery (EUR)exp.
 Discussion: History — Production Analysis (PA) — (EUR)exp
 Is the estimated ultimate recovery (EUR)exp supported by theory?
(yes)
 Is the (EUR)exp conservative or liberal?
((EUR)exp is always conservative)
 Other issues? (use of other EUR models (e.g., hyperbolic) requires caution)
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History: Production Analysis (PA) — EUR vs. qo,1yr
Q. Origin and purpose of the EUR versus qo,1yr correlation?
A. Potential value as a correlation, but must quantify theory (N, k, s, etc).
— From: Manual for The Oil and Gas Industry Under The
Revenue Act of 1918, Treasury Department — United States
Internal Revenue Service (1919).
"Ancient" Technique:
The proposed correlation of EUR vs. qo,1yr was
used to estimate oil reserves from initial production performance
data.
Modern Application:
Approach is based in
theory — EUR = f[k, s, xf,
... and contacted fluids
in-place (i.e., N or G)].
Could be used as a "reservoir characterization"
tool to classify well performance.
 Discussion: History — Production Analysis (PA) — EUR vs. qo,1yr
 Origin?
 Theory?
 Rationale?
(1919 — production data correlation (>85 years old)!)
(Constant pwf (liquid) boundary-dominated flow conditions)
(Correlate reserves versus production (or reservoir properties))
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History: Production Analysis (PA) — Arps
— From: Arps, J.J: "Analysis of Decline Curves," Trans., AIME
(1945) 160, 228-247.
Q. Theory for Arps' relations?
A. Arps derived the exponential and hyperbolic relations from loss ratio.
Case
Rate Relation
Cumulative Relation
qi
Exponential: (b=0) q = qi exp(− Di t )
Np =
[1 − exp(− Di t )]
Di
qi
qi
Np =
1 − (1 + bDi t )1−(1 / b)
Hyperbolic: (0<b<1) q =
(1 − b) Di
(1 + bDi t ) (1 / b)
qi
qi
q
=
N
ln(1 + Di t )
=
Harmonic: (b=1)
p
(1 + Di t )
Di
[
Arps' observations:
b=0
—
b=0
—
b=0.5 —
b=0.667 —
b=0.333 —
Reservoir is highly undersaturated (p>pb).
Gravity drainage and no free surface.
Gravity drainage with free surface.
Soln. gas-drive reservoir ( p 2 vs. Np → linear).
Soln. gas-drive reservoir ( p vs. Np → linear).
Theory???
]
Loss Ratio:
a≡
1
q
≡−
D
dq/dt
Theory???
Loss Ratio Derivative:
b≡
d
[a] ≡ d  1  ≡ − d
dt
dt  D 
dt
 q 


 dq/dt 
 Discussion: History — Production Analysis (PA) — Arps
 "Theory" for the Arps' relations?
(loss ratio (exp) and its derivative (hyp))
 Validity of the Arps' observations? (only qualitative (except for p>pb case))
 Graphical analysis using the hyperbolic relation? (only using Fetkovich TC)
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History: Production Analysis (PA) — Fetkovich
— From: Fetkovich, M.J.: "Decline Curve Analysis Using Type
Curves," JPT (June 1980) 1065-1077.
Q. What is the "Fetkovich" Decline Type Curve, and how is it used?
A. A composite of analytical (pwf=con) and empirical (Arps) solutions —
used as a "type curve" (data overlay) to estimate reservoir properties.
Variables for the Fetkovich Decline Type Curve
t Dd =
q Dd =
0.00633 kt
1
φµct rw 2 1   r  2    r  1 
  e  − 1 ln  e  − 


2  rw 

q(t )
kh ( pi − p wf )
 r
   wa 
2 
rwa = rwe − s
Transient Stems: (left)
 Infinite-acting radial
flow model (pwf = con).
 q(t) is concave up.
Depletion Stems: (right)
 Bounded circular reservoir (pwf = con).
 q(t) is concave down.
 b=0: pwf = con.
 b=1: qo = con. (qo/∆p).
 b>1: transient flow or
external drive energy.
Reservoir Properties:
 k — y-axis match.
 N — x&y-axis matches.
 s — reD match.
  r  1
141.2 µB ln  e  − 
  rwa  2 
 Discussion: History — Production Analysis (PA) — Fetkovich
 An original purpose of the Fetkovich TC? (graphical solution of Arps Eqs.)
 Use of "transient" stems?
(estimate reservoir properties — k and s)
 Use of "depletion" stems?
(estimate reservoir volume, predict rate)
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History: Production Analysis (PA) — Carter
— From: Carter, R.D.: "Type Curves for Finite Radial and
linear Gas Flow Systems: Constant Terminal Pressure
Case," SPEJ (October 1985) 719-728.
Q. What is the "Carter" Decline Type Curve, and how is it used?
A. A numerically-generated gas rate solution (pwf=con) — used as a "type
curve" (data overlay) to estimate reservoir properties.
Transient Stems: (left)
 Numerical flow model
(pwf = con).
 q(t) is concave up.
Depletion Stems: (right)
 q(t) is concave down.
 b=0: pwf = con.
 b=1: qo = con. (qo/∆p).
 b>1: transient flow or
external drive energy.
 λ: numerical gas flow
cases (λ =f(pwf/(pi)).
Reservoir Properties:
 k — y-axis match.
 G — x&y-axis matches.
 s — reD match.
Variables for the Carter Decline Type Curve
t Dd =
0.00633 kt
1
φµ gi cti rw 2 1   r  2    r  1 
  e  − 1 ln  e  − 
   rwa  2 
2   rw 


rwa = rwe − s
q (t )
kh ( pi − p wf )
q Dd =
  r  1
141.2 µ gi B gi ln  e  − 
  rwa  2 
 Discussion: History — Production Analysis (PA) — Carter
 Genesis of the Carter TC?
 Use of "transient" stems?
 Use of "depletion" stems?
T.A. Blasingame (2013.07.22)
("correction" of Fetkovich gas flow solutions)
(estimate reservoir properties — k and s)
(estimate reservoir volume, predict rate)
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Modern PA: Doublet Type Curve (Constant Rate Eq.)
Q. Type curve solution for variable-rate/variable pressure case?
A. "Doublet" type curve uses material balance time function.
 Auxiliary functions:
 Material Balance
Time:
t
1
t=
qo dt
qo 0
 Rate integral:
t q 
 qo 
1
o
  =
  dt
 ∆p  i t 0  ∆p 
 Rate integralderivative:
 qo 
d   qo  
t
=
  
 
p
d
t
∆
  ∆p  i 
  id
∫
∫
 Discussion: Modern PA — Doublet Type Curve (constant rate eq.)
 Material Balance Time: tmb = (1/q) Int(qo, 0, t). (material bal. "deconvolution")
 Auxiliary Functions: Defined using qo and ∆p. (accounts for qo and ∆p = f(t))
 Validity: Variable-rate/variable pressure drop cases. (rigorous for pss flow)
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Modern PA: Analysis-by-Modelling (Gas Case)
(1/4)
Q. Analysis-by-modelling for gas cases?
A. Same general procedure, must use pseudopressure/pseudotime, or
numerical (or semi-analytical) gas solutions.
Points to Consider:
 Using both PA and PTA
can yield complimentary analyses.
 Accurate rate and pressure data are required.
 Discussion: Modern PA — Gas Case
 Origin of data?
(DAILY data: rate → surface, pressure → surface)
 Likelihood of a successful analysis? (high to very high, data well-correlated)
 Integration of PTA data? (designed for production, modelling for shut-ins)
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Modern PA: Analysis-by-Modelling (Gas Case)
(2/4)
Q. Behavior of "normalized PI" and "Blasingame" plots for this case?
A. Both plots perform EXTREMELY well → driven by data quality.
a. "Normalized PI" Plot: (∆pp/qg) functions
versus Gp/qg — excellent agreement in
data and model functions.
b. "Blasingame" Plot: (qg/∆pp) functions versus
Gp/qg — excellent agreement — note that the
qDd functions converge (confirms BDF).
 Discussion: Modern PA — "Normalized PI" and "Blasingame" plots
 Transient flow data?
(confirms fractured well behavior, moderate FcD)
 Boundary-dominated flow?("convergence" (Blasingame plot) confirms BDF)
 What is required to achieve similar results?
(accurate q and pwf data)
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Modern PA: Analysis-by-Modelling (Gas Case)
(3/4)
Q. How well can a gas case be modeled?
A. Depends on the data — can be excellent...
 Discussion: Modern PA — PA model/data match
 Rate match?
(EXCELLENT — near perfect match until late times)
 Pressure match?
(virtually perfect — including shut-ins)
 Accounting for "damage?" (use PTA, or "cheat" → use variable skin effect)
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Modern PA: Analysis-by-Modelling (Gas Case)
(4/4)
Q. Analysis of individual pressure transient tests?
A. Should be straightforward — vigilance in data acquisition is required.
Comment:
 Excellent match on log-log
plot (pressure drop functions) and very good match
of the entire production
pressure history.
 Daily rate and pressure
data are sufficient for this
low permeability reservoir
case.
 Discussion: Modern PA — PTA model/data match
 PTA match?
(VERY GOOD — note that production pwf data also matched)
 Comparison of PA and PTA results? (minor differences, due to data quality)
 etc.?
(note that pressure history is matched for PTA (entire history))
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Production Analysis: Questions to Consider
Q1. Accuracy of PA relative to PTA?
A1. Pressure transient analysis (PTA) is an "event" analysis approach
that requires very controlled input conditions (i.e., the flowrate).
Production analysis (PA) is a "holistic" (or complete) analysis
approach — in theory, PTA would be a part of PA, but in practice the
production data are not carefully acquired (compared to PTA). In
simple terms, if the production history "high frequency/high
resolution" then the results of PA and PTA should be the same.
Q2. Why the recent emphasis on production analysis (PA)?
A2. For low/ultra-low permeability reservoirs, it is virtually impossible to
conduct a successful pressure transient test — production analysis
becomes the only viable method of evaluation.
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Notes:
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Pressure Transient Analysis
Thomas A. Blasingame, Ph.D., P.E.
Department of Petroleum Engineering — Texas A&M University
College Station, TX 77843-3116 (USA)
+1.979.845.2292
t-blasingame@tamu.edu
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