Pressure Transient Testing

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Pressure Transient Testing

John Lee

Texas A&M University

John B. Rollins

IBM Corporation

John P. Spivey

Phoenix Reservoir Engineering

SPE Textbook Series, Volume 9

Henry L. Doherty Memorial Fund of AIME

Society of Petroleum Engineers

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Dedication

John Lee

To all the Aggie students and former students who have made my teaching career so much fun and so rewarding.

John Rollins

To my family—Becci, Christine, and Cathy—and to my father, J.T. Rollins, a genuine Permian Basin petroleum pioneer.

John Spivey

To my many colleagues at SoftSearch, Dwights Energy Data, S.A. Holditch and Assocs., and Schlumberger Oilfield

Technologies who have taught me, challenged me, encouraged me, and inspired me throughout my career.

Disclaimer

This book was prepared by members of the Society of Petroleum Engineers and their well-qualified colleagues from

material published in the recognized technical literature and from their own individual experience and expertise.

While the material presented is believed to be based on sound technical knowledge, neither the Society of Petroleum

Engineers nor any of the authors or editors herein provide a warranty either expressed or implied in its application.

Correspondingly, the discussion of materials, methods, or techniques that may be covered by letters patents implies

no freedom to use such materials, methods, or techniques without permission through appropriate licensing.

Nothing described within this book should be construed to lessen the need to apply sound engineering judgment

nor to carefully apply accepted engineering practices in the design, implementation, or application of the techniques

described herein.

© Copyright 2003 Society of Petroleum Engineers

All rights reserved. No portion of this book may be reproduced in any form or by any means, including electronic

storage and retrieval systems, except by explicit, prior written permission of the publisher except for brief passages

excerpted for review and critical purposes.

ISBN 978-1-55563-099-7

Society of Petroleum Engineers

222 Palisades Creek Drive

Richardson, TX 75080-2040 USA

http://store.spe.org

[email protected]

1.972.952.9393

Manufactured in the United States of America.

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SPE Textbook Series

The Textbook Series of the Society of Petroleum Engineers was established in 1972 by action of the SPE Board

of Directors. The Series is intended to ensure availability of high-quality textbooks for use in undergraduate

courses in areas clearly identified as being within the petroleum engineering field. The work is directed by the

Society’s Books Committee, one of more than 40 Society-wide standing committees. Members of the Books

Committee provide technical evaluation of the book. Below is a listing of those who have been most closely

involved in the final preparation of this book.

Book Editors

Shah Kabir, ChevronTexaco Corp., Houston

Fikri Kuchuk, Schlumberger, Dubai, UAE

Books Committee (2003)

Waldo J. Borel, Devon Energy Production Co. LP, Youngsville, Louisiana, Chairman

Bernt S. Aadnoy, Stavanger U. College, Stavanger

Jamal J. Azar, U. of Tulsa, Tulsa

Ronald A. Behrens, ChevronTexaco Corp., San Ramon, California

Ali Ghalambor, U. of Louisiana-Lafayette, Lafayette, Louisiana

Jim Johnstone, Contek Solutions LLC, Plano, Texas

Gene E. Kouba, ChevronTexaco Corp., Houston

Bill Landrum, ConocoPhillips, Houston

Eric E. Maidla, Noble Engineering & Development Ltd., Sugar Land, Texas

Erik Skaugen, Stavanger U. College, Stavanger

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Introduction

Pressure transient test analysis is a mature technology in petroleum engineering; even so, it continues to evolve.

Because of the developments in this technology since the last SPE textbook devoted to transient testing was

published, we concluded that students could benefit from a textbook approach to the subject that includes a

rep-resentative sampling of the more important fundamentals and applications. We deliberately distinguish between

a textbook approach, which stresses understanding through numerous examples and exercises dealing with

se-lected fundamentals and applications, and a monograph approach, which attempts to summarize the

state-of-the-art in the technology.

Computational methods that transient test analysts use have gone through a revolution since most existing texts

on the subject were written. Most calculations are now done with commercial software or by spreadsheets or

proprietary software developed by users to meet personal needs and objectives. These advances in software

have greatly increased productivity in this technology, but they also have contributed to a “black box” approach

to test analysis. In this text, we attempt to explain what’s in the box, and we do not include a number of the

mod-ern tools that enhance individual engineer productivity. We hope, instead, to provide understanding so that the

student can use the commercial software with greater appreciation and so that the student can read monographs

and papers on transient testing with greater appreciation for the context of the subject. Accordingly, this text is

but an introduction to the vast field of pressure transient test analysis.

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Acknowledgments

The contributions of many people were crucial in the preparation of this book. We acknowledge with heartfelt

thanks the contributions to the preparation of the subject matter by Tom Blasingame, Jay Rushing, and Jennifer

Johnston Blasingame; the contributions to the presentation of the material by Darla-Jean Weatherford; the

tech-nical audit by Shah Kabir and Fikri Kuchuk; and the SPE staff, most notably techtech-nical editors Valerie Dawe and

Jennifer Wegman. To each of you—thanks!

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1. Fundamentals of Fluid Flow in Porous Media

1 1.1 Overview

1 1.2 Derivation of the Diffusivity Equation

1 1.3 Initial and Boundary Conditions

5 1.4 Dimensionless Groups

8 1.5 Solutions to the Diffusivity Equation

10 1.6 Superposition in Space

17 1.7 Superposition in Time

19 1.8 Deconvolution

22 1.9 Chapter Summary

23 1.10 Discussion Questions

24 2. Introduction to Flow and Buildup-Test Analysis: Slightly Compressible Fluids

29 2.1 Overview

29 2.2 Analysis of Flow Tests

29 2.3 Analysis of Pressure-Buildup Tests

34 2.4 Complications in Actual Tests

41 2.5 Analysis of Late-Time Data in Flow and Buildup Tests

45 2.6 Analyzing Well Tests With Multiphase Flow

51 2.7 Chapter Summary

54 3. Introduction to Flow and Buildup-Test Analysis: Compressible Fluids

62 3.1 Overview

62 3.2 Pseudopressure and Pseudotime Analysis

62 3.3 Pressure and Pressure-Squared Analysis

63 3.4 Non-Darcy Flow

63 3.5 Analysis of Gas-Well Flow Tests

65 3.6 Analysis of Gas-Well Buildup Tests

69 3.7 Chapter Summary

73 4. Well-Test Analysis by Use of Type Curves

77 4.1 Overview

77 4.2 Development of Type Curves

77 4.3 Application of Type Curves—Homogeneous Reservoir Model, Slightly Compressible

Liquid Solution

77 4.4 Application of Type Curves—Homogeneous Reservoir Model, Compressible Fluids

91 4.5 Correcting Initial Pressure in a Well Test

93 4.6 Reservoir Identification With Type Curves

94 4.7 Systematic Analysis Procedures for Flow and Buildup Tests

95 4.8 Well-Test-Analysis Worksheets

96 4.9 Chapter Summary

96 5. Analysis of Pressure-Buildup Tests Distorted by Phase Redistribution

98 5.1 Overview

98 5.2 Description of Phase Redistribution

98 5.3 Phase-Redistribution Model

98 5.4 Analysis Procedure

101 5.5 Chapter Summary

111 6. Well-Test Interpretation in Hydraulically Fractured Wells

114 6.1 Overview

114

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6.3 Flow Geometry and Depth of Investigation of a Vertically Fractured Well

116 6.4 Specialized Methods for Post-Fracture Well-Test Analysis

116 6.5 Post-Fracture Well-Test Analysis With Type Curves

119 6.6 Effects of Fracture and Formation Damage

130 6.7 Chapter Summary

130 7. Interpretation of Well-Test Data in Naturally Fractured Reservoirs

135 7.1 Overview

135 7.2 Naturally Fractured Reservoir Models

135 7.3 Pseudosteady-State Matrix Flow Model

136 7.4 Transient Matrix Flow Model

142 7.5 Chapter Summary

147 8. Drillstem Testing and Analysis

151 8.1 Overview

151 8.2 Conventional DST

151 8.3 Conventional DST Design

152 8.4 DST-Monitoring Procedures

154 8.5 DST Analysis Techniques

154 8.6 Closed-Chamber DST

160 8.7 Impulse Testing

164 8.8 Chapter Summary

165 9. Injection-Well Testing

168 9.1 Overview

168 9.2 Injectivity Testing in a Liquid-Filled Reservoir: Unit-Mobility-Ratio Reservoir Conditions

168 9.3 Falloff Testing in a Liquid-Filled Reservoir: Unit-Mobility-Ratio Reservoir Conditions

171 9.4 Estimating Average Drainage-Area Pressure

174 9.5 Composite-System-Test Analysis for Nonunit-Mobility-Ratio Reservoir Conditions

174 9.6 Step-Rate Testing

182 9.7 Chapter Summary

186 10. Interference and Pulse Testing

190 10.1 Overview

190 10.2 Interference Tests

190 10.3 Pulse Tests

195 10.4 Recommendations for Multiple-Well Testing

199 10.5 Chapter Summary

199 11. Design and Implementation of Well Tests

202 11.1 Overview

202 11.2 Types and Purposes of Well Tests

202 11.3 General Test-Design Considerations

203 11.4 Pressure Transient Test Design

206 11.5 Deliverability-Test Design

217 11.6 Chapter Summary

220 12. Horizontal Well Analysis

223 12.1 Overview

223 12.2 Steps in Evaluating Horizontal Well-Test Data

223 12.3 Horizontal Well Flow Regimes

223 12.4 Identifying Flow Regimes in Horizontal Wells

225 12.5 Summary of Analysis Procedures

237 12.6 Field Examples

237 12.7 Running Horizontal Well Tests

239 12.8 Estimating Horizontal Well Productivity

240 12.9 Comparison of Recent and Older Horizontal Well Models

244

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Appendix A—Dimensionless Groups

246 Constant-Rate Production—No Wellbore Storage

246 Constant-Rate Production With Wellbore Storage

247 Constant-Rate Production With Wellbore Storage and Skin

248 Linear Flow

248 Appendix B—Solutions to the Radial-Flow Diffusivity Equation

250 Introduction

250 Modified Bessel Equation and Its General Solution

250 Laplace Transformations and Their Use in Solving Partial-Differential Equations

250 Solutions to the Diffusivity Equation

251 Appendix C—Derivations of the Diffusivity Equation Multiphase Flow

(Perrine and Martin) Linear Flow of Gas

260 Introduction

260 Multiphase Flow

260 Linear Flow of Gas

263 Appendix D—Shape Factors for Various Single-Well Drainage Areas

265 Appendix E—Validation of Method of Images

267 Superposition for a No-Flow Boundary

267 Superposition for a Constant-Pressure Boundary

268 Appendix F—Determining Pressure-Data Derivatives

269 Appendix G—Reservoir-Identification Worksheets

270 Appendix H—Well-Test-Analysis Worksheets

278 Appendix I—Example Well-Test Analysis Using Worksheets, Example 4.5

287 Appendix J—Worksheets for Post-Fracture Well-Test Analysis

292 Appendix K—Worksheets for Well-Test Design

308 Appendix L—Reservoir-Fluid Properties

313 Introduction

313 Definitions

313 Correlations

317 Nomenclature

341 Author Index

349 Subject Index

351

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Chapter 1 Fundamentals of Fluid Flow

in Porous Media

1.1 Overview

In this chapter, we develop the equations to describe the flow of slightly compressible liquids and gases and the simultaneous flow of oil, water, and gas in porous media. We then define appropriate dimensionless variables that enable us to simplify the resulting par tial-differential equations. We present solutions to those differential equations subject to various inner- and outer-boundary conditions. These solutions are obtained by use of both Laplace transformations and Boltzmann's transformation. We consider radial and linear flow and superposition in space and time.

Appendix A presents a detailed method for finding dimensionless variables. Appendix B details derivations of the different solutions to the diffusivity equation. Appendix C presents the derivations of the diffusivity equation for multi phase flow and for linear flow in detail. Appendix D presents a proof of the validity of the method of images to model boundaries in a reservoir.

This chapter focuses on the mathematical basis for pressure tran sient test analysis. For those readers with little or no mathematical inclination, we note that it is not necessary to master the material in this chapter to understand the applications in the rest of the book. However, we do think that virtually all readers will derive consider able benefit from browsing through this chapter. The summary in Sec. 1.9 may be especially helpful to browsers.

1.2 Derivation of the Diffusivity Equation

1.2.1 Fundamental Physical Principles. The basic equation to de scribe the flow of fluid in porous media caused by a potential differ ence is known as the diffusivity equation. The diffusivity equation is derived from three fundamental physical principles: (1) the prin ciple of conservation of mass, (2) an equation of motion, and (3) an equation of state (EOS).

We derive the diffusivity equation in the radial coordinate system because flow in a simple, homogeneous-acting, cylindrical reser voir takes place radially from the reservoir to the well bore. We use metric units (implicitly) in derivations; later, we generalize to other systems of units used in the remainder of the text.

Principle of Conservation of Mass. The principle of conserva tion of mass states that the net rate of creation or destruction of mat ter is zero. If we consider the control volume, a fixed region in space (illustrated in Fig. 1.1), we may write

FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA

[

The mass flow

]

rate into the control volume -during time period

fl.t

[

The mass flow

]

rate out of

control volume during time period

fl.t

[

Th

�

n

�

a

�� ;

o

�

ate

1 [

The rate of mass

]

control volume accumulation in + during time period

fl.t

= _{th� co�trol vo}

_�

_ume _.

. . k dunng time penod

fl.t

owmg to source or sm

. . . .. (1.1) We now look at each part of the conservation equation, Eq. 1.1, mathematically. The mass flow rate into the system = density x ve locity x cross-sectional area of flow.

min = -

pUr Ax]

, ... (1.2) where the cross-sectional area of flow on the inflow side,

Ax],

is giv en by

Ax]

= La X

h

and the minus sign arises because the positive flow direction in the control volume has been chosen in the negative

r

direction.

For angle e, the arc length is given by arc length = radius x angle, 4t =

(r

+ M) e. . ... (1.3) Therefore,

mill = -

pur(r

+

fl.r)8h

... (1.4) The mass flow rate out of the system is similarly given by

moUi =

- [pUr - fl.(pur)]Ax2,

•••••••••••••••••••• (1.5) where the term

fl.(pur)

is the change in mass flux occurring inside the control volume, and the cross-sectional area of flow on the out flow side,

Ax"

is given by

(1.6) Therefore moUi =

- [pUr - fl.(pur)]reh

. ... (l.7) We assume that there is neither a source nor a sink in the control vol ume (i.e., mass is neither being generated nor consumed). Therefore, net mass flow rate owing to source or sink = O.

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2 PRESSURE TRANSIENT TESTING

Fig. 1.1—Control volume for deriving the mass-conservation equation.

The mass in the control volume at any time is the product of the pore volume (PV) and the density of the fluid: PV+arc length width height porosity, or

Vp+rqDrhf. . . . (1.9)

Therefore m+rqDrhf ò. . . . (1.10) The rate of mass accumulation, Wa, in the control volume is given

by the change in mass in the control volume from time t to t)Dt, divided by the change in time, Dt.

Wa+

ƪ

ŤrqDrhfòŤ(t)Dt)* ŤrqDrhfòŤt

Dt

ƫ

. . . (1.11) We can now express the conservation equation, Eq. 1.1, mathe-matically by combining Eqs. 1.4, 1.7, 1.8, and 1.11:

ƪ–òur(r) Dr)qhƫ *

Ǌ

*

ƪ

òurrqh * DǒòurǓrqh

ƫǋ

+

ƪ

Dt

ƫ

. . . (1.12) Expanding Eq. 1.12 gives

* òurrqh * òurDrqh ) òurrqh * DǒòurǓrqh

+ 1_Dt

ƪ

ƫ

. . . (1.13)

Dividing Eq. 1.13 by the bulk volume of the control volume,

hrqDr, we have

*òur

r *Dǒòu

rǓ

Dr +Dt1

ƪ

ŤfòŤ(t)Dt)* ŤfòŤt

ƫ

. . . . (1.14)

Factoring out 1ńrDr on the left side and multiplying through by *1, 1 rDr

ƪ

DrǒòurǓ ) rDǒòurǓ

ƫ

+ * DǒfòǓ Dt (1.15a). . . or 1r

Ǌ

òur) r

ƪ

DǒòurǓ Dr

ƫ

ǋ

+ *DǒfòǓDt . . . (1.15b)

Taking limits of Eq. 1.15 as Dr, Dt³0, we have 1

r

ƪ

ǒòurǓ )

rēǒòurǓ

ēr

ƫ

+ *ēǒfòǓēt . (1.16). . . By the product rule,

ē

ērǒròurǓ + òurēr_{ēr )}r ē_ērǒòurǓ + òur) r ē_ērǒòurǓ. (1.17). . .

Therefore, 1r_ērēǒròurǓ + * ēērǒfòǓ. . . . (1.18)

Eq. 1.18 is known as the continuity equation, a mathematical ex-pression of the principle of conservation of mass in radial coordinates. To this point, the only assumptions we have made are that we have radial flow and that no sources or sinks are in the control volume.

Equation of Motion. An equation of motion, or flux law, relates

velocity and pressure or potential gradients within the control vol-ume. Because of the complexity of the flow paths within porous me-dia, we must use empirical relationships for the equation of motion. Liquid flow is generally governed by Darcy’s law, which states that the velocity is proportional to the negative of the gradient of the po-tential. In radial coordinates, with flow in the radial direction only, we write ur+ * kò mēFēr ,. . . (1.19) where F +

ŕ

p pb dpȀ ò ) gǒZ * Z0Ǔ , (1.20). . .

pb+pressure at a datum, and Z+Z 0.

The potential, F, consists of two terms:

ŕ

p

pb

dpȀ

ò + flow work and

g(Z*Z0)+potential head.

This form of Darcy’s law has two assumptions: (1) flow is in the laminar flow regime (low Reynolds number), and (2) the porous medium is isotropic. For single-phase flow of a slightly compress-ible liquid in a homogeneous-acting reservoir, these assumptions are generally valid.

We can now combine Eqs. 1.19 and 1.20 to express the velocity in terms of pressure, rather than potential, gradient. From Eq. 1.20,

ēF ēr +ērē

ȧ

ȱ

_Ȳ

ŕ

p pb dpȀ ò ) gǒZ * Z0Ǔ

ȧ

ȳ

ȴ

. . . . (1.21)

If we assume gravity effects are negligible, g(Z*Z0)+0.

There-fore, ēF

ēr +1òēpēr. . . (1.22) Substituting Eq. 1.22 into Eq. 1.19 gives

ur+ * kmēp_ēr. . . (1.23) EOS. An EOS relates volume, or density, to the pressure and

tem-perature of the system. We assume isothermal conditions when con-sidering the flow of a slightly compressible liquid in a reservoir be-cause the heat capacity of the fluid is generally negligible compared with the heat capacity of the rock.

The definition of fluid compressibility is

c+ * 1 V

ǒ

ēVēp

Ǔ

T+ 1ò

ǒ

ēò

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FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA 3 Treating ēòńēp as a total derivative, dòńdp, for an isothermal

sys-tem and rearranging Eq. 1.24 gives

cdp+ 1ò dò. . . (1.25) For a fluid of small and constant compressibility, we integrate Eq. 1.25 to obtain c

ŕ

p pb dpȀ +

ŕ

ò òb 1 ò dò, (1.26). . . where òb+density at base pressure, pb. Integrating, we obtain

cǒ p* pbǓ + ln ò * ln òb (1.27a). . .

and cǒ p* pbǓ + ln

ǒ

ò

òb

Ǔ

. (1.27b). . .

Exponentiating both sides gives

ò + òbexp

ƪ

cǒ p* pbǓ

ƫ

. . . (1.28)

This is the EOS that we use when we assume that the fluid is slightly compressible and the compressibility is constant.

1.2.2 Diffusivity Equation for Radial, Single-Phase Flow of a Liquid With Small, Constant Compressibility. To derive the

dif-fusivity equation, we must combine the continuity equation, 1

r_ērēǒròurǓ + * ēērǒfòǓ, (1.18). . .

the equation of motion,

ur+ * kmēp_ēr , . . . (1.23)

and the EOS for the appropriate fluid,

ò + òbexp

ƪ

cǒp* pbǓ

ƫ

. . . (1.28)

Combining Eqs. 1.18 and Eq. 1.23, we obtain 1

r_ērē

ǒ

rò kmēp_ēr

Ǔ

+ ē_ērǒfòǓ . . . (1.29)

If we assume constant permeability and viscosity, using the product rule gives

1

r_ērē

ǒ

ròēp_ēr

Ǔ

+m_k

ǒ

fēò_{ēt ) ò}ēf_ēt

Ǔ

. . . (1.30)

We can now expand Eq. 1.30 by use of the chain rule: ò r_ērē

ǒ

rēp_ēr

Ǔ

) 1r rēp_ērēò_{ēr +}m_k

ǒ

fēò_ēpēp_{ēt ) ò}ēf_ēpēp_ēt

Ǔ

ò r_ērē

ǒ

rēp_ēr

Ǔ

)ēp_ērēò_ēpēp_{ēr +}m_{k fò}

ǒ

1òēò_ēpēp_{ēt )}_f1ēf_ēpēp_ēt

Ǔ

. (1.31) . . . From Eq. 1.28, ēò ēp +còbexp

ƪ

cǒp* pbǓ

ƫ

, (1.32). . .

where compressibility, c, is small and constant. Therefore, ēò

ēp +cò. (1.33). . . By remembering the definition of compressibility,

c+ 1òēò_ēp , . . . (1.34)

and defining a formation compressibility,

c_f+ 1fēf

ēp , . . . (1.35) we can define a total compressibility, ct, as

ct+c)cf , (1.36). . .

and write Eq. 1.31 as ò r_ērē

ǒ

rēp_ēr

Ǔ

) cò

ǒ

ēp_ēr

Ǔ

2 + fmct k ò ēp ēt. (1.37). . . We know that ò00; therefore, we can divide the equation through by density. 1 r_ērē

ǒ

rēp_ēr

Ǔ

) c

ǒ

ēp_ēr

Ǔ

2 + fmct k ēp ēt. (1.38). . . We now assume for radial flow of a fluid of small, constant com-pressibility that cǒēpńērǓ2 is negligible compared to ēńērǒrēpńērǓ and ēpńēr, so the final partial differential equation is

1

r_ērē

ǒ

rēp_ēr

Ǔ

+ fmc_ktēp_ēt. . . (1.39)

Summary of Assumptions for Eq. 1.39.

1. Radial flow.

2. Laminar (or Darcy) flow.

3. Porous medium has constant permeability and compressibility. 4. Negligible gravity effects.

5. Isothermal conditions.

6. Fluid has small, constant compressibility.

7. Compressibility/pressure-gradient-squared product, cǒēpńērǓ2, is negligible.

1.2.3 Diffusivity Equation for Radial, Single-Phase Flow of a Gas. The continuity equation and equation of motion for radial

single-phase gas flow through porous media are the same as those equations used for slightly compressible liquid flow.

Continuity Equation.

1

r_ērēǒròurǓ + * ēētǒfòǓ. . . . (1.18) Equation of Motion (Darcy’s Law).

ur+ * kmēp_ēr. . . (1.23) EOS. The EOS used for slightly compressible liquids does not,

however, model gas behavior. The equation most commonly used to model real-gas pressure/volume/temperature (PVT)behavior is the real-gas law given by

ò +_zRTpM . . . . (1.40) We can now combine the continuity equation, Eq. 1.18, and the equation of motion, Eq. 1.23, to obtain

* 1r ēēr

ǒ

ròk_m ēp

ēr

Ǔ

+ * ēētǒfòǓ (1.41a). . . or 1r_ērē

ǒ

ròk_m ēp

ēr

Ǔ

+ ēētǒfòǓ. . . . (1.41b) Now, substituting the real-gas law, Eq. 1.40, into Eq. 1.41b, we ob-tain 1 r_ērē

ǒ

r kpM mzRTēpēt

Ǔ

+ ēēt

ǒ

f pM zRT

Ǔ

. . . . (1.42)

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4 PRESSURE TRANSIENT TESTING Because R, T, and M are constant and considering the special case

with kconstant, we find that 1 r_ērē

ǒ

r p mzēpēr

Ǔ

+ 1kētē

ǒ

f p z

Ǔ

. . . (1.43)

We can expand the right side of Eq. 1.43 using the product rule as follows: 1 r_ērē

ǒ

r p mzēpēr

Ǔ

+ 1k

ƪ

p zēf_{ēt )}fē_ēt

ǒ

p z

Ǔ

ƫ

. . . (1.44)

We can use the chain rule to obtain another expansion of the right side of Eq. 1.44: 1 r_ērē

ǒ

r p mzēpēr

Ǔ

+ 1k

ƪ

p zēf_ēpēp_{ēt ) f}_ēpē

ǒ

p z

Ǔ

ēp_ēt

ƫ

(1.45a). . . or 1r_ērē

ǒ

r_mzp ēp ēr

Ǔ

+ pf zk ēp ēt

ƪ

f1ēfēp )zp_ēpē

ǒ

p z

Ǔ

ƫ

. . . (1.45b)

The compressibility of gas is defined similarly to the compressibili-ty of a liquid in terms of the densicompressibili-ty:

cg+ 1òēò_ēp. . . (1.46)

Substituting density from the real-gas law, Eq. 1.40, into this defini-tion gives cg+ zRT pM ēpē

ǒ

pM zRT

Ǔ

+ z p_ēpē

ǒ

p z

Ǔ

. . . (1.47)

We define formation compressibility as

cf+ 1f

ēf

ēp. . . (1.48) We can now substitute Eqs. 1.47 and 1.48 into Eq. 1.45b, which gives 1 r_ērē

ǒ

r p mzēpēr

Ǔ

+ pf zk ēp ēt

ǒ

cf) cg

Ǔ

. . . (1.49)

If we define total compressibility for this case as

c_t+c_g)c_f, . . . (1.50) we have 1 r_ērē

ǒ

r p mzēpēr

Ǔ

+ pfct zk ēp ēt. . . (1.51) Eq. 1.51 is a nonlinear partial-differential equation and cannot be solved directly. We generally consider three limiting assumptions,

p/mz is constant, mct is constant, and the real-gas pseudopressure

transformation.

Diffusivity Equation for Gas in Terms of Pressure. If we assume

that the term p/mz is constant with respect to pressure, and therefore radius, Eq. 1.51 can be written as

1 r p mz_ērē

ǒ

rēp ēr

Ǔ

+ pfct zk ēp ēt . . . (1.52) or, cancelling terms,

1 r_ērē

ǒ

rēp_ēr

Ǔ

+fmc t k ēp ēt. . . (1.53) Eq. 1.53 is the same as the diffusivity equation for slightly compress-ible liquids, Eq. 1.39, and can be solved similarly (when mc_t can be con-sidered to be constant). Eq. 1.53 has the following assumptions.

1. Radial flow.

5. Isothermal conditions. 6. Fluid obeys the real-gas law.

7. The term p/mz is constant with respect to pressure.

Diffusivity Equation for Gas in Terms of Pressure Squared. We can

write Eq. 1.51 in terms of pressure squared, p2_{, by use of the fact that}

pēp ēr +12 ēp2 ēr ,. . . (1.54) pēp_{ēt +}1 2 ēp2 ēt, (1.55). . . and 1r_ērē

ǒ

_mzr ēp 2 ēr

Ǔ

+fckzt ēp2 ēt . . . (1.56) If we assume that the term mz is constant with respect to pressure and therefore radius, Eq. 1.56 can be written as

1 rmz1 _ērē

ǒ

rēp 2 ēr

Ǔ

+fckzt ēp2 ēt . . . (1.57) or, multiplying through by the term mz, as

1 r_ērē

ǒ

rēp 2 ēr

Ǔ

+fmck t ēp2 ēt. . . (1.58) Eq. 1.58 is also similar to the diffusivity equation for slightly compressible liquids, Eq. 1.39, but the dependent variable is pres-sure squared. Therefore, Eq. 1.58 has solutions similar to those of Eq. 1.39 except these solutions are in terms of pressure-squared. These equations also require that mc_t be constant. Eq. 1.58 has the following assumptions.

1. Radial flow.

7. The term mz is constant with respect to pressure.

Diffusivity Equation for Gas in Terms of Pseudopressure. The

assumptions we have discussed so far to obtain the linear diffusivity equation for gas are applicable only under certain conditions. Figs.

1.2 and 1.3 illustrate the range of applicability of Eqs. 1.53 and 1.58,

respectively. Fig. 1.2 shows for gases of different specific gravity when the term p/mz is constant with pressure for a constant tempera-ture. The figure shows that we could use Eq. 1.53 for very high pres-sures. Fig. 1.3 shows for gases of different specific gravity when the term mz is constant with pressure for a constant temperature. This figure shows that we could use Eq. 1.58 for very low pressures.

We prefer to have an accurate solution for all pressure ranges. A more rigorous method of linearizing Eq. 1.51 (at least partially) is by use of the real-gas pseudopressure transformation introduced by Al-Hussainy et al.1_{The pseudopressure transformation allows the}

general gas diffusivity equation, Eq. 1.51, to be solved without the limiting assumptions that certain gas properties are constant with pressure. We define a pseudopressure, pp, by

pp+ 2

ŕ

p

p0

p

mz dp. . . . (1.59) Using Liebnitz’s Rule for differentiating an integral,2

ē ēx

ŕ

h (x) f ǒ xǓ g (u)du+

Ǌ

g [h(x)]ēƪh(x)ƫ ēx *g

ƪ

f ǒ xǓ

ƫ

ē

ƪ

f ǒ xǓ

ƫ

ēx

ǋ

, (1.60) . . . the derivative of pseudopressure is

ēpp

ēr +2

p

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250000 200000 0. � 150000 .� " '" =- ₁₀₀₀₀₀ Co 50000 2000 4000 6000 Pressure, psia " SG = 1.0 8000 10000

Fig. 1.2-Range of applicability of pressure methods (200°F). with respect to radius and

(1. 62) with respect to time.

Rearranging Eqs. 1. 61 and 1. 62, we can substitute for ap/ ar and ap/atinto Eq. 1.52 to obtain

t

:r [r

:z

(i;

?:

)

]

=

p�t

(i; a::)

... (1. 6 3)

or, simplifying,

t:r(ra::)

=

¢Jt

t

a:t

... (1. 64)

Eq. 1. 64 is not completely linear because the flCt term depends on pressure and therefore on pseudopressure, but we can approximate this quantity as constant) and evaluate it at current drainage area pressure, p.

Eq. 1. 64 is also similar to the diffusivity equation for slightly compressible liquids, Eq. 1. 39, but in terms of pseudopressure. Therefore, Eq. 1. 64 has solutions similar to those ofEq. 1. 39, except these solutions are in terms of pseudopressure. Eq. 1. 64 has the fol lowing assumptions.

Summary of Assumptions for Eq. 1.64.

1. Radial flow.

1.2.4 Diffusivity Equation for Radial, Multipbase Flow. Martin3

developed a diffusivity equation for multiphase flow, Eq. 1. 65, which looks very similar to the diffusivity equation for single-phase flow, Eq. 1. 39.

t:r(r��)

=

(i�

t

)

?r

. . . (1. 65)

dl �

(

ap

)

= ,f, flCtap an

_{r ar r ar}

't' k at' . . . (1. 39)

Appendix C presents the derivation in detail. Note that the only dif ference between these equations is the use of total mobility, At, and a more general definition of the total compressibility, Ct. We define the total mobility of a three-phase system as the sum of the individu al mobilities,

, ko kw kg , , ,

/l,t = _{("" 0}'iJ + u + 'iJ _f"'lV _rg= /1,0 + /l,w + /l,g.

FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA

(1. 6 6) 0.14 -,---_ 0.12 0.10 e- 0.08 N "" :::1. 0.06 0.04 2000 4000 6000 8000 10000 Pressure, psia

Fig. 1.3-Range of applicability of pressure-squared methods (200°F).

The total compressibility, Ct, for a system with pressure-dependent porosity is defined as

Ct = Soco + Swcw + SgCg + cJ' ... (1. 67)

The similarities between the multiphase flow and single-phase flow equations imply that the solutions to the single-phase diffusivity equation presented later in this chapter also apply to multiphase flow as long as Ct is defined by Eq. 1. 67 and we use At instead of ko1flo.

Summary of Assumptions for Eq. 1.65.

1. Radial flow.

2. Laminar (or Darcy) flow. 3. Uniform porous medium. 4. Negligible gravity effects. 5. Isothermal conditions.

6. Effective permeability varies with saturation, but not pressure. 7. Small pressure- and saturation-gradient terms.

8. Negligible capillary pressure.

1.3 Initial and Boundary Conditions

The general diffusivity equation for fluid flow in porous media is a partial-differential equation for pressure with respect to both space (radius) and time.

t:r(r��)

=

(

¢

�

Ct

)

?r

. . . (1. 68)

To solve Eq. 1. 68, we must know how the pressure behaves at spe cific distances and time; that is, we must specify conditions to solve the equation. Conditions specified at different extremes of distance are known as boundary conditions , whereas the condition specified at initial time, t = 0, is known as the initial condition.

We note that the partial-differential equation is "second order" with respect to space; in other words, we have taken the partial de rivative of pressure with respect to radius twice as indicated on the left side of Eq. 1. 68. Likewise, the diffusivity equation is "first or der" with respect to time. A second-order equation requires two conditions to obtain a solution. Therefore, we must have two bound ary conditions. In radial flow, we usually specify a condition on pressure at the wellbore (the inner-boundary condition) and at the edge of the drainage area of the reservoir ( the outer-boundary condi tion). Similarly, a first-order differential equation requires only one condition; therefore, we need only a single condition for time (i.e., the initial condition).

In this section, we will discuss possible initial and boundary conditions for different reservoir models and production schemes.

1.3.1 Initial Condition. We always assume that the reservoir is ini tially at a uniform, constant pressure throughout the reservoir at a time t=O.

p(r,O) = Pi- ( 1. 69)

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6 PRESSURE TRANSIENT TESTING

Fig. 1.4—Surface and sandface rates during wellbore storage.

1.3.2 Outer-Boundary Conditions. We consider three cases for the

outer boundary of the reservoir. It may be infinite-acting (i.e., it is so large that the outer boundary effects are never felt at points in the reservoir at practical distances from the source or sink). The reser-voir may be bounded by a no-flow boundary (i.e., a volumetric res-ervoir). The reservoir could be bounded by a constant-pressure boundary, such as a reservoir/aquifer system.

Infinite-Acting Reservoir. As the radius becomes very large,

ap-proaching infinity, the pressure approaches the initial pressure, pi,

for all times.

p(r³ R, t) + pi (1.70). . .

or Dp(r ³ R, t) + pi* p(r ³ R, t) + 0.. . . (1.71) No-Flow Boundary. For a cylindrical reservoir with a no-flow

boundary a distance re from the well the flow rate at r+re will be

q+0 for all times greater than zero.

Darcy’s law states that qTēp_ēr . . . . (1.72)

q+ * C kA_m ēp

ēr , . . . (1.73) where C+constant00, k+permeability00, A+area (cross-sec-tional)00, and m+viscosity00. Therefore,

ǒēpńērǓre+ 0. . . (1.74)

Constant-Pressure Outer Boundary. For a cylindrical reservoir

with a constant-pressure boundary at distance r_e from the well, the pressure at the outer boundary will be equal to the initial pressure,

p_i, for all times.

p(r+ re, t)+ pi , . . . (1.75)

or Dp(r + re, t)+ 0. (1.76). . . 1.3.3 Inner-Boundary Conditions. A well may be produced at

constant rate or constant pressure and have wellbore storage effects.

Constant-Rate Production. If a well is produced at a constant

sandface rate, this rate of flow from the formation into the wellbore of radius rw may be described by Darcy’s law. At r+rw ,

qB+

ǒ

_akA

1m

ēp ēr

Ǔ

(r+rw)

, (1.77). . . where a1+conversion constant [e.g., in “field units” a1+141.2

(2p), and A+cross-sectional area+ 2 r_wh (in square feet)].

If we substitute the definition of area into the inner-boundary condition, we have

Fig. 1.5—Wellbore diagram for a well with a liquid-gas interface.

qB+

ǒ

2_ap 1 kh m rwēp_ēr

Ǔ

(r+rw) . . . . (1.78) Rearranging Eq. 1.78, the constant-rate inner boundary condition becomes

ǒ

rēp

ēr

Ǔ

(r+rw)

+a1qBm

2pkh. . . (1.79)

Constant-Pressure Production. This inner-boundary condition

is valid when the reservoir is initially at uniform pressure through-out the reservoir and is produced by simply lowering the wellbore pressure to a constant value, p_wf, and producing at a variable sand-face rate.

p(rw, t)+ pwf+ constant. . . (1.80) Wellbore Storage. Wellbore storage may occur if a well is set to

produce at constant surface rate after a shut-in period. Initially, fluid will unload from the wellbore with no flow from the formation to the wellbore. As time passes, the sandface rate will equal the surface rate, with the amount of liquid stored being constant; see Fig. 1.4.

We call the ability of the wellbore to store or unload fluids per unit change in pressure the wellbore-storage coefficient, C(bbl/psi). The definition of the wellbore-storage coefficient depends on the situa-tion in the wellbore. We consider the following two cases: a liquid/ gas interface in the wellbore and a single, compressible fluid in the wellbore.

Liquid/Gas Interface. For a pumping well or a well produced by

gas lift, the wellbore will have a column of liquid with a column of gas at the top of the wellbore.

If we let the surface rate, q, be constant, a mass balance for the wellbore shown in Fig. 1.5 would be

ǒ

Rate of flow of mass into wellbore

Ǔ

*

ǒ

Rate of flow of mass out of wellbore

Ǔ

+

ǒ

Rate of accumulationof mass in wellbore

Ǔ

, which is

ǒ

qsfBòsf

Ǔ

* ǒqBòscǓ + d

dt

ǒ

24òwbVwb

5.615

Ǔ

, . . . (1.81) where time is in hours and the volume of the wellbore,Vwb , is

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FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA 7

Fig. 1.6—Wellbore diagram for a well producing a single-phase fluid.

Vwb+ AwbZ. (1.82). . .

If we assume a constant wellbore area and constant density (ò_sf+ ò_sc+ ò_wb), we can write the mass balance as

ǒ

qsf* q

Ǔ

B+ 24_5.615Awb

dZ

dt. (1.83). . . The surface pressure, pt, is related to the bottomhole pressure, pw,

at any time by

pw+ pt) òZ

144 , . . . (1.84) where ò+density of liquid.

Differentiating with respect to time gives d

dt(pw* pt)+₁₄₄ò dZ_dt. (1.85). . .

Substituting for dZńdt from the mass balance gives d dtǒpw* ptǓ

ǒ

144 ò

Ǔ

+

ǒ

24A5.615_wb

Ǔ

ǒ

qsf* q

Ǔ

B

ǒ

qsf* q

Ǔ

B+ (24)(144) 5.615ò Awb_dtd(pw* pt). (1.86). . .

We now define a wellbore-storage coefficient, C, as

C+144Awb

5.615ò bbl

psi , . . . (1.87) where Awb is in square feet, the constant 144 in.2/ft2 converts square

feet to square inches, ò is in lbm/ft3_{, and the constant 5.615 ft}3_/bbl

converts barrels to cubic feet.

If we substitute C into the equation relating sandface and surface rate, we obtain

qsf+ q ) 24C_B

d

dt(pw* pt). (1.88). . . The sandface rate is given by Darcy’s law as

qsf+ 2pkh_a

1Bm

ǒ

rēp

ēr

Ǔ

(r+rw)

, (1.89). . . where a₁+141.2(2p) in field units, thus

q+ 2pkh

a1Bm

ǒ

rēp

ēr

Ǔ

(r_+rw)

* 24C_B _dtd(pw* pt). . . (1.90)

This is the inner-boundary condition for wellbore storage for a well with a gas-liquid interface. In some cases, we can assume that

dpt

dt[0, . . . (1.91) and the boundary condition becomes

q+ 2pkh a1Bm

ǒ

rēp ēr

Ǔ

(r+rw) * 24C_B dpw dt . . . (1.92)

Single Phase in Wellbore. In this case, we consider a well that is

producing a single-phase fluid, either liquid or gas, at a constant sur-face rate, as illustrated in Fig. 1.6. The mass balance for this system would be

ǒ

Rate of mass flow into wellbore at

sandface

Ǔ

*

ǒ

Rate of mass flow out of wellbore at

surface

Ǔ

+

ǒ

Rate of accumulation_{of mass in wellbore}

Ǔ

qsf Bòsf* qBscòsc+

ǒ

24Vwb dò_wb dt

Ǔ

+

ǒ

24V_wbò_wbc_wbd pwb dt

Ǔ

, (1.93). . . becausedòwb dt + òwbcwb dpwb dt . (1.94)

The density/volume factor product is constant and thus the same at both surface and reservoir conditions. Thus, if we define C for the single-phase case as

C+ Vwbcwb , . . . (1.95)

the mass balance becomes

qsf+ q ) 24C_B

dpw

dt . . . (1.96) The wellbore-storage boundary condition is the same, despite the different definition of C. q+ 2pkh a1Bm

ǒ

rēp ēr

Ǔ

(r+rw) * 24C_B dpw dt. . . (1.97)

Skin Factor. To account for the additional pressure drop near the

wellbore caused by reduction in permeability owing to adverse drill-ing and completion conditions, Hawkins4 developed the idea of a fi-nite skin zone around the wellbore. This skin zone can cause the measured pressure drop to be much greater than the pressure drop calculated from solutions to the diffusivity equation. We assume that the shaded zone in Fig. 1.7 has a constant permeability, k_s, and extends only a short distance, r_s, from the center of the wellbore into the reservoir. Fig. 1.8 shows the effect that this altered zone would have on the pressure drop at the wellbore.

Dp1 represents the pressure drop from a radius rs to the wellbore

radius, rw, that would normally occur because of flow through the

altered zone. Dp₂ represents the pressure drop from a radius r_s to the wellbore radius, r_w, that would have occurred had there been no change in permeability in the altered zone (i.e., if the permeability in this zone remained the average formation permeability, k). The additional pressure drop that results across the skin zone is therefore equal to Dp_s, where

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r s

Fig. 1.7-lIIustration of the zone of altered permeability around the well bore.

r

w r s

Fig. 1.8-The effect of the skin zone on the well bore pressure drop.

Because rs is small, we can assume steady-state flow in the altered zone and write the steady-state radial-flow equations for the pres sure drops in this region.

I-.p, =

inq:,�

In

(;,:)

... (1.9 9) and I-.P2 =

a;:::

In

(;,:)

, ... (1.100) Combining Eqs. 1.98 through 1.100,

I-. = a,qB .,u 1

(�)

_ a,qB,u1

(�)

P s _{2Jrk ,h} n r w 2Jrkh n rw = alqB,uln

(

rS

)

(

� _ 1

)

2Jrh rw k,. k = a [qB,u In

(

rs

)

[

(

klks

)

- 1

]

2nh rw k a,qB,u

(

k

)

(

rs

)

= 2Jrkh

k,

- 1 In rw' ... (LlOl) We define a skin factor, 5, on the basis of the properties of the altered

zone.

5 =

(t

- 1

)

In

(;,:)

, ... (1.102) 8

From Eqs. 1.10 1 and 1.102, the definition of skin factor becomes 2JrkMps

5 = a ,qB,u . ... (1.103) We note that Eq. 1.102 provides some insight into the physical significance of the sign of the skin factor. If a well is damaged (ks < k), 5 _{will be positive; and the greater the contrast between}ks

and k and the deeper into the formation the damage extends, the larg er the numerical value of 5. _{There is no upper limit for}5. _{Some newly}

drilled wells will not flow at all before stimulation; for these wells, ks = 0 and 5-> 00 • If a well is stimulated (ks > k), 5 will be negative; and the deeper the stimulation, the greater the numerical value of 5.

Rarely does a stimulated well have a skin factor less than -7 or -8, and such skin factors arise only for wells with deeply penetrating, highly conductive hydraulic fractures. We should note finally that, if a well is neither damaged nor stimulated (ks = k), 5 = O. We caution that Eq. 1.102 is best applied qualitatively; actual wells rarely can be characterized exactly by such a simplified model. We also note that an altered zone near a particular well affects only the pressure near that well; i.e., the pressure in the unaltered formation away from the well is not affected by the presence of the altered zone.

1.4 Dimensionless Groups

We use dimensionless groups to express our equations more simply. Many well-test-analysis techniques use dimensionless variables to depict general trends rather than working with specific parameters (e.g., k and h). To define appropriate dimensionless variables, we find logical groupings of variables that appear naturally in differen tial equations and initial and boundary conditions.

In this section, we present dimensionless groups used for radial flow of slightly compressible liquids that are being produced at ei ther constant rate, with and without wellbore storage, and constant bottomhole pressure. Appendix A provides a complete explanation of how these dimensionless groups are derived.

1.4.1 Radial Flow-Constant-Rate Production. For this case, we define the following dimensionless variables and use conventional field units. Dimensionless Pressure. kh PD = 141.2qB

/

Pi - p) . ... (1.104) Dimensionless Radius. r D - rlV - � and reD =

;:

. . ... . Dimensionless Time. tD = 0.00026

:;

7 kt . ¢,uc/ rw

Dimensionless Wellbore-Storage Coefficient.

CD = 0.8 936C . ... . ¢c/h r�v Skin Factor. (1.105) (1.106) (1.107) (1.108) khl-.ps 5 = 141.2qB,u' ... (1.10 9) The diffusivity equation and various initial and boundary condi-tions can be rewritten in terms of these dimensionless variables.

Partial-Differential Equation.

r

�

a

�

D

(

rD

��)

=

?r:

. ... (1.1 10) Initial Condition.

PD(rD,tD = 0) = O . ... (1.1 1 1) PRESSURE TRANSIENT TESTING

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FUNDAMENTALS OF FLUID FLOW IN POROUS MEDIA 9

Outer Boundary Condition. Infinite-Acting Reservoir.

pD(rD³ R, tD)+ 0. . . (1.112) No-Flow Boundary.

ǒ

ēpD ērD

Ǔ

_r eD + 0. . . (1.113) Constant-Pressure Boundary. pD

ǒ

rD+ reD,tD

Ǔ

+ 0. . . (1.114) Inner-Boundary Condition. Constant-Rate Production.

ǒ

ēpD

ērD

Ǔ

_ǒ_r

D+1Ǔ

+ * 1. (1.115). . .

Constant Rate Production With Wellbore Storage. CD dpwD dtD *

ǒ

rDēp D ērD

Ǔ

_ǒ_r D+1Ǔ + 1. (1.116). . . Skin Factor. pwD(tD)+ pD(1, tD)) s. . . . (1.117) 1.4.2 Radial Flow—Constant-Pressure Production. This case

re-quires a different definition of dimensionless pressure. Dimension-less time and length are defined the same as for the constant rate case. In addition, we must define dimensionless rate and cumulative production. Dimensionless Pressure. pD+ pi* p pi* pwf . . . (1.118) Dimensionless Rate. qD+ qBm 0.00708kh

ǒ

pi* pwf

Ǔ

. . . (1.119)

Dimensionless Cumulative Production.

QpD+

ŕ

tD 0 qDdtȀ + B 1.119fcthr2w

ǒ

pi* pwf

Ǔ

Qp. . . (1.120)

The diffusivity equation and various initial and boundary condi-tions can be rewritten in terms of these dimensionless variables.

Partial-Differential Equation. 1 rD ē ērD

ǒ

rD ēpD ērD

Ǔ

+ ēpD ētD . . . (1.121) Initial Condition. pD(rD, tD+ 0) + 0. . . (1.122) Outer-Boundary Condition. Infinite-Acting Reservoir.

pD(rD³ R, tD)+ 0. (1.123). . . No-Flow Boundary.

ǒ

ēpD ērD

Ǔ

_r eD + 0. . . . (1.124) Constant-Pressure Boundary. pD

ǒ

rD+ reD,tD

Ǔ

+ 0. . . (1.125) Inner-Boundary Condition for Constant-Pressure Production.

p_D(r_D+1,t_D)+1. . . (1.126)

Fig. 1.9—Linear flow to a fractured well system.

1.4.3 Linear Flow—Constant-Rate Production—General Case.

For the general linear-flow case, we define the following dimen-sionless variables on the basis of a cross-sectional area. In Sec. 1.4.4, we will present the specialized case for hydraulically frac-tured wells. Note that the diffusivity equation that models linear flow may be derived from a shell balance exactly as in the radial-flow diffusivity equation, but with rectangular coordinates. Appen-dix C presents this derivation.

Dimensionless Pressure. pD+ k AǸ 141.2qBm(pi* p). . . . (1.127) Dimensionless Length. xD+ x A Ǹ . . . (1.128) Dimensionless Time. tAD+ 0.0002637kt fmctA . . . (1.129) The diffusivity equation and various initial and boundary condi-tions can be rewritten in terms of these dimensionless variables.

Partial-Differential Equation. ē2_p D ēx2 D +ēpD ētD . . . (1.130) Initial Condition. pD(xD, tD+ 0) + 0 . . . (1.131) Outer-Boundary Condition. Infinite-Acting Reservoir.

pD(xD³ R, tD)+ 0. (1.132). . . No-Flow Boundary.

ǒ

ēpD ēxD

Ǔ

_x eD + 0. (1.133). . . Constant-Pressure Boundary. pD

ǒ

xD+ xeD,tD

Ǔ

+ 0. . . (1.134) Inner-Boundary Condition for Constant-Rate Production.

ǒ

ēpD

ēxD

Ǔ

_x

D+1

+ * 1. . . . (1.135)

1.4.4 Linear Flow—Constant-Rate Production—Hydraulically Fractured Wells. Linear flow occurs in hydraulically fractured well

systems because the fracture behaves as a “plane source” with the fluid flowing linearly to the fracture. Fig. 1.9 illustrates this system. For this case, the cross-sectional area denoted in the general case represents a vertical fracture with two equal-length wings of length

L_f and height h. Therefore, A+4hL_f, with flow entering both sides of each wing of the fracture.

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The dimensionless pressure for fractured wells is defined the same as in the radial-flow constant-rate cases, but dimension less length and time are defined on the basis of the fracture half length, 4. Dimensionless Pressure. kh PD = 14 1.2qB,u (Pi - p). . ... (1.136) Dimensionless Length. LD = � Lf Dimensionless Time. (1.137) tL D = 0.00026327kt . . . .. (1.138) I _¢,uctLf

The diffusivity equation and various initial and boundary condi tions can be rewritten in terms of these dimensionless variables.

Partial-Differential Equation.

(1.139) Initial Condition.

PD(LD,tD = 0) = O. (1.140)

Outer-Boundary Condition. Infinite-Acting Reservoir.

Po(Lo -+ 00, to) = O. ... (1.14 1)

No-Flow Boundary.

(�f:)

= O. ... (1.142)

L,D

Constant Pressure Boundary.

PD

(

LD = LeD/D

)

= O. . ... (1.143)

Inner-Boundary Condition for Constant-Rate Production.

(�f:)

= - 1. ... (1.144) LD�I

1.5 Solutions to the Diffusivity Equation

There are several different solutions to the diffusivity equation, de pending on the initial and boundary conditions used to solve the equation. In this chapter, we present the solutions for the following reservoir models.

1. Transient radial flow, constant-rate production from a line source well, both without skin factor and with skin factor and well bore storage.

2. Pseudosteady-state radial flow, constant rate production from a cylindrical-source well in a closed reservoir.

3. Steady-state radial flow, constant-rate production from a cylin drical-source well in a reservoir with constant pressure outer bound aries.

4. Transient linear flow, constant rate production from a hydrauli cally fractured well.

There are numerous possible reservoir models with different boundary conditions, but the solution techniques for all models are similar. Appendix B gives a full explanation of these solution tech niques. We also give examples of how to implement these solutions in solving flow problems in reservoirs.

1.5.1 Transient Radial Flow, Constant-Rate Production From a Line-Source Well. In this case, we assume that the well can be rep resented as a "line source;" in other words, the wellbore is infinitesi mally small (rw-+O). This well produces at a constant rate with no wellbore storage or skin from an infinitely large reservoir. This does not describe a real situation; all reservoirs have a finite areal extent, 10

and all wells that are drilled have a certain wellbore radius. Howev er, the well bore radius is small compared with the radius of the reser voir, so a line-source assumption is not unreasonable. Also, at early producing times, the effects of the outer boundaries of the reservoir are not seen and the reservoir acts as if there were no boundaries (i.e., the reservoir is infinite-acting).

The partial-differential equation for this case is given by . ... (1.1 10) Initially, pressure in the reservoir is uniform throughout the reser voir, so the initial condition is given by

po(ro,to = 0) = O. .. ... (1.Ul)

The reservoir is infinite-acting; therefore, as the dimensionless radius tends toward infinity, the pressure at that radius will be the same as initial pressure and the dimensionless pressure function will be zero. The outer boundary condition is therefore written as

PD(rD -+ oo,tD) = o . ... (1.1 12) The reservoir is producing at constant sandface rate with no well bore storage or skin from a line-source well (i.e., the wellbore radius approaches zero). The inner-boundary condition for this case is

= - 1. ... (1.145)

Remember, the inner-boundary condition is for a "line-source" well. This is a limiting condition as rw-+O of the constant-rate boundary condition for a finite wellbore.

Using either Laplace transforms or the Boltzmann transforma tion, as explained in Appendix B, we can derive the line-source solution in dimensionless variables, given here by

PD = -

�

Ei

(

�

t

:

�

)

. ... (1.146)

where Ei is the exponential integral defined as

00

Ei( -x) = -

J

e

;

Y dy. .. ... (1.147)

x

Substituting in the appropriate definitions for dimensionless vari ables as given in Sec. 1.4, we can write the line-source (or Ei-func tion) solution in terms of field variables

... (1.148) The line-source solution is an approximation of the more general cylindrical-source solution, so we must define limits of its applica bility. It has been shown to be accurate for the range

(3.975 X 105)¢,uctr�v 948¢,uctr�

k < t < k . . . . . .. (1.149) At times less than the lower limit, the assumption of zero well size limits the accuracy of the equation. At times greater than the upper limit (for a well centered in a circular drainage area), the reservoir boundaries will affect the pressure distribution in the reservoir so that the reservoir is no longer infinite-acting.

When the argument of the Ei function, x , is greater than 0.0 1, we use Table 1.1 to estimate the Ei-function value for a given x value. We then use that value in Eq. 1.148 to calculate the pressure.

For values of x less than 0.0 1, this solution can be further simpli fied by making an approximation to the exponential integral func tion, Ei( -x). This approximation is given by Eq. 1.150.

Ei( -x) = In(1.78 1x). (1.150)