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TLFeBOOK Advanced Reservoir Engineering TLFeBOOK This page intentionally left blank TLFeBOOK Advanced Reservoir Engineering Tarek Ahmed Senior Staff Advisor Anadarko Petroleum Corporation Paul D. McKinney V.P. Reservoir Engineering Anadarko Canada Corporation AMSTERDAM • BOSTON • HEIDELBERG • LONDON...

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Advanced Reservoir Engineering

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Advanced Reservoir Engineering

Tarek Ahmed Senior Staff Advisor Anadarko Petroleum Corporation

Paul D. McKinney V.P. Reservoir Engineering Anadarko Canada Corporation

AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Gulf Professional Publishing is an imprint of Elsevier

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Gulf Professional Publishing is an imprint of Elsevier 200 Wheeler Road, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Copyright © 2005, Elsevier Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: [email protected]. You may also complete your request on-line via the Elsevier homepage (http://elsevier.com), by selecting “Customer Support” and then “Obtaining Permissions.” Recognizing the importance of preserving what has been written, Elsevier prints its books on acid-free paper whenever possible. Librar y of Congress Cataloging-in-Publication Data Application submitted British Librar y Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 0-7506-7733-3 For information on all Gulf Professional Publishing publications visit our Web site at www.books.elsevier.com 04

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Printed in the United States of America

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Dedication This book is dedicated to our wonderful and understanding wives, Shanna Ahmed and Teresa McKinney, (without whom this book would have been finished a year ago), and to our beautiful children (NINE of them, wow), Jennifer (the 16 year old nightmare), Justin, Brittany and Carsen Ahmed, and Allison, Sophie, Garretson, Noah and Isabelle McKinney.

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Preface The primary focus of this book is to present the basic physics of reservoir engineering using the simplest and most straightforward of mathematical techniques. It is only through having a complete understanding of physics of reservoir engineering that the engineer can hope to solve complex reservoir problems in a practical manner. The book is arranged so that it can be used as a textbook for senior and graduate students or as a reference book for practicing engineers. Chapter 1 describes the theory and practice of well testing and pressure analysis techniques, which is probably one of the most important subjects in reservoir engineering.

Chapter 2 discusses various water-influx models along with detailed descriptions of the computational steps involved in applying these models. Chapter 3 presents the mathematical treatment of unconventional gas reservoirs that include abnormally-pressured reservoirs, coalbed methane, tight gas, gas hydrates, and shallow gas reservoirs. Chapter 4 covers the basic principle oil recovery mechanisms and the various forms of the material balance equation. Chapter 5 focuses on illustrating the practical application of the MBE in predicting the oil reservoir performance under different scenarios of driving mechanisms. Fundamentals of oil field economics are discussed in Chapter 6. Tarek Ahmed and Paul D. McKinney

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About the Authors Tarek Ahmed, Ph.D., P.E., is a Senior Staff Advisor with Anadarko Petroleum Corporation. Before joining Anadarko in 2002, Dr. Ahmed served as a Professor and Chairman of the Petroleum Engineering Department at Montana Tech of the University of Montana. After leaving his teaching position, Dr Ahmed has been awarded the rank of Professor of Emeritus of Petroleum Engineering at Montana Tech. He has a Ph.D. from the University of Oklahoma, an M.S. from the University of Missouri-Rolla, and a B.S. from the Faculty of Petroleum (Egypt) – all degrees in Petroleum Engineering. Dr. Ahmed is also the author of 29 technical papers and two textbooks that includes “Hydrocarbon Phase Behavior” (Gulf Publishing Company, 1989) and “Reservoir Engineering Handbook” (Gulf Professional Publishing, 1st edition 2000 and 2nd edition 2002). He taught numerous industry courses and consulted in many countries including, Indonesia, Algeria, Malaysia, Brazil,

Argentina, and Kuwait. Dr. Ahmed is an active member of the SPE and serves on the SPE Natural Gas Committee and ABET. Paul McKinney is Vice President Reservoir Engineering for Anadarko Canada Corporation (a wholly owned subsidiary of Anadarko Petroleum Corporation) overseeing reservoir engineering studies and economic evaluations associated with exploration and development activities, A&D, and planning. Mr. McKinney joined Anadarko in 1983 and has served in staff and managerial positions with the company at increasing levels of responsibility. He holds a Bachelor of Science degree in Petroleum Engineering from Louisiana Tech University and co-authored SPE 75708, “Applied Reservoir Characterization for Maximizing Reserve Growth and Profitability in Tight Gas Sands: A Paradigm Shift in Development Strategies for Low-Permeability Reservoirs.”

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Acknowledgements As any publication reflects the author’s understanding of the subject, this textbook reflects our knowledge of reservoir engineering. This knowledge was acquired over the years by teaching, experience, reading, study, and most importantly, by discussion with our colleagues in academics and the petroleum industry. It is our hope that the information presented in this textbook will improve the understanding of the subject of reservoir engineering. Much of the material on which this book is based was drawn from the publications of the Society of Petroleum Engineers. Tribute is paid to the educators, engineers, and authors who have made numerous and significant contributions to the field of reservoir engineering. We would like to express our thanks to Anadarko Petroleum Corporation for granting us the permission to publish this book and, in particular, to Bob Daniels, Senior Vice President, Exploration and Production, Anadarko Petroleum Corporation and Mike Bridges, President,

Anadarko Canada Corporation. Of those who have offered technical advice, we would like to acknowledge the assistance of Scott Albertson, Chief Engineer, Anadarko Canada Corporation, Dr. Keith Millheim, Manager, Operations Technology and Planning, Anadarko Petroleum Corporation, Jay Rushing, Engineering Advisor, Anadarko Petroleum Corporation, P.K. Pande, Subsurface Manager, Anadarko Petroleum Corporation, Dr. Tom Blasingame with Texas A&M and Owen Thomson, Manager, Capital Planning, Anadarko Canada Corporation. Special thanks to Montana Tech professors; Dr. Gil Cady and Dr. Margaret Ziaja for their valuable suggestions and to Dr. Wenxia Zhang for her comments and suggestions on chapter 1. This book could not have been completed without the (most of the time) cheerful typing and retyping by Barbara Jeanne Thomas; her work ethic and her enthusiastic hard work are greatly appreciated. Thanks BJ.

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Contents 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Well Testing Analysis 1/1 Primary Reservoir Characteristics 1/2 Fluid Flow Equations 1/5 Transient Well Testing 1/44 Type Curves 1/64 Pressure Derivative Method 1/72 Interference and Pulse Tests 1/114 Injection Well Testing 1/133

4 4.1 4.2 4.3 4.4 4.5 5

2 2.1 2.2 2.3

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Water Influx 2/149 Classification of Aquifers 2/150 Recognition of Natural Water Influx 2/151 Water Influx Models 2/151

Unconventional Gas Reser voirs 3/187 Vertical Gas Well Performance 3/188 Horizontal Gas Well Performance 3/200 Material Balance Equation for Conventional and Unconventional Gas Reservoirs 3/201 Coalbed Methane “CBM” 3/217 Tight Gas Reservoirs 3/233 Gas Hydrates 3/271 Shallow Gas Reservoirs 3/286

5.1 5.2 5.3

6 6.1 6.2 6.3

Performance of Oil Reser voirs 4/291 Primary Recovery Mechanisms 4/292 The Material Balance Equation 4/298 Generalized MBE 4/299 The Material Balance as an Equation of a Straight Line 4/307 Tracy’s Form of the MBE 4/322 Predicting Oil Reser voir Performance 5/327 Phase 1. Reservoir Performance Prediction Methods 5/328 Phase 2. Oil Well Performance 5/342 Phase 3. Relating Reservoir Performance to Time 5/361 Introduction to Oil Field Economics Fundamentals of Economic Equivalence and Evaluation Methods 6/366 Reserves Definitions and Classifications Accounting Principles 6/375

References Index

6/365 6/372

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1

Well Testing Analysis

Contents 1.1 Primary Reservoir Characteristics 1/2 1.2 Fluid Flow Equations 1/5 1.3 Transient Well Testing 1/44 1.4 Type Curves 1/64 1.5 Pressure Derivative Method 1/72 1.6 Interference and Pulse Tests 1/114 1.7 Injection Well Testing 1/133

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WELL TESTING ANALYSIS

1.1 Primary Reservoir Characteristics Flow in porous media is a very complex phenomenon and cannot be described as explicitly as flow through pipes or conduits. It is rather easy to measure the length and diameter of a pipe and compute its flow capacity as a function of pressure; however, in porous media flow is different in that there are no clear-cut flow paths which lend themselves to measurement. The analysis of fluid flow in porous media has evolved throughout the years along two fronts: the experimental and the analytical. Physicists, engineers, hydrologists, and the like have examined experimentally the behavior of various fluids as they flow through porous media ranging from sand packs to fused Pyrex glass. On the basis of their analyses, they have attempted to formulate laws and correlations that can then be utilized to make analytical predictions for similar systems. The main objective of this chapter is to present the mathematical relationships that are designed to describe the flow behavior of the reservoir fluids. The mathematical forms of these relationships will vary depending upon the characteristics of the reservoir. These primary reservoir characteristics that must be considered include: ● ● ● ●

types of fluids in the reservoir; flow regimes; reservoir geometry; number of flowing fluids in the reservoir.

of this fluid as a function of pressure p can be mathematically described by integrating Equation 1.1.1, to give:
V
p dV dp = −c pref Vref V exp [c(pref − p)] =

V V ref

V = Vref exp [c (pref − p)]

[1.1.3]

where: p = pressure, psia V = volume at pressure p, ft3 pref = initial (reference) pressure, psia Vref = fluid volume at initial (reference) pressure, psia The exponential ex may be represented by a series expansion as: ex = 1 + x +

x2 xn x2 + + ··· + 2! 3! n!

[1.1.4]

Because the exponent x (which represents the term c (pref − p)) is very small, the ex term can be approximated by truncating Equation 1.1.4 to: ex = 1 + x

[1.1.5]

Combining Equation 1.1.5 with 1.1.3 gives: 1.1.1 Types of fluids The isothermal compressibility coefficient is essentially the controlling factor in identifying the type of the reservoir fluid. In general, reservoir fluids are classified into three groups: (1) incompressible fluids; (2) slightly compressible fluids; (3) compressible fluids.

In terms of fluid volume: −1 ∂V V ∂p In terms of fluid density: 1 ∂ρ c= ρ ∂p where

[1.1.6]

A similar derivation is applied to Equation 1.1.2, to give: ρ = ρref [1 − c(pref − p)]

[1.1.7]

where:

The isothermal compressibility coefficient c is described mathematically by the following two equivalent expressions:

c=

V = Vref [1 + c(pref − p)]

[1.1.1]

[1.1.2]

V= fluid volume ρ = fluid density p = pressure, psi−1 c = isothermal compressibility coefficient,  −1 Incompressible fluids An incompressible fluid is defined as the fluid whose volume or density does not change with pressure. That is ∂ρ ∂V = 0 and =0 ∂p ∂p Incompressible fluids do not exist; however, this behavior may be assumed in some cases to simplify the derivation and the final form of many flow equations. Slightly compressible fluids These “slightly” compressible fluids exhibit small changes in volume, or density, with changes in pressure. Knowing the volume Vref of a slightly compressible liquid at a reference (initial) pressure pref , the changes in the volumetric behavior

V = volume at pressure p ρ = density at pressure p Vref = volume at initial (reference) pressure pref ρref = density at initial (reference) pressure pref It should be pointed out that crude oil and water systems fit into this category. Compressible fluids These are fluids that experience large changes in volume as a function of pressure. All gases are considered compressible fluids. The truncation of the series expansion as given by Equation 1.1.5 is not valid in this category and the complete expansion as given by Equation 1.1.4 is used. The isothermal compressibility of any compressible fluid is described by the following expression:   1 ∂Z 1 cg = − [1.1.8] p Z ∂p T Figures 1.1 and 1.2 show schematic illustrations of the volume and density changes as a function of pressure for the three types of fluids. 1.1.2 Flow regimes There are basically three types of flow regimes that must be recognized in order to describe the fluid flow behavior and reservoir pressure distribution as a function of time. These three flow regimes are: (1) steady-state flow; (2) unsteady-state flow; (3) pseudosteady-state flow.

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Incompressible

Volume

Slightly Compressible

Compressible

Pressure Figure 1.1 Pressure–volume relationship.

Fluid Density

Compressible

Slightly Compressible

Incompressible

0

Pressure Figure 1.2 Fluid density versus pressure for different fluid types.

Steady-state flow The flow regime is identified as a steady-state flow if the pressure at every location in the reservoir remains constant, i.e., does not change with time. Mathematically, this condition is expressed as:   ∂p =0 [1.1.9] ∂t i This equation states that the rate of change of pressure p with respect to time t at any location i is zero. In reservoirs, the steady-state flow condition can only occur when the reservoir is completely recharged and supported by strong aquifer or pressure maintenance operations. Unsteady-state flow Unsteady-state flow (frequently called transient flow) is defined as the fluid flowing condition at which the rate of change of pressure with respect to time at any position in the reservoir is not zero or constant. This definition suggests that the pressure derivative with respect to time is essentially

a function of both position i and time t, thus:     ∂p = f i, t ∂t

[1.1.10]

Pseudosteady-state flow When the pressure at different locations in the reservoir is declining linearly as a function of time, i.e., at a constant declining rate, the flowing condition is characterized as pseudosteady-state flow. Mathematically, this definition states that the rate of change of pressure with respect to time at every position is constant, or:   ∂p = constant [1.1.11] ∂t i It should be pointed out that pseudosteady-state flow is commonly referred to as semisteady-state flow and quasisteadystate flow. Figure 1.3 shows a schematic comparison of the pressure declines as a function of time of the three flow regimes.

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WELL TESTING ANALYSIS

Location i Steady-State Flow

Pressure

Semisteady-State Flow

Unsteady-State Flow

Time Figure 1.3 Flow regimes.

Plan View

Wellbore

pwf

Side View

Flow Lines

Figure 1.4 Ideal radial flow into a wellbore.

1.1.3 Reservoir geometry The shape of a reservoir has a significant effect on its flow behavior. Most reservoirs have irregular boundaries and a rigorous mathematical description of their geometry is often possible only with the use of numerical simulators. However, for many engineering purposes, the actual flow geometry may be represented by one of the following flow geometries: ● ● ●

radial flow; linear flow; spherical and hemispherical flow.

Radial flow In the absence of severe reservoir heterogeneities, flow into or away from a wellbore will follow radial flow lines a substantial distance from the wellbore. Because fluids move toward the well from all directions and coverage at the wellbore, the term radial flow is used to characterize the flow of fluid into the wellbore. Figure 1.4 shows idealized flow lines and isopotential lines for a radial flow system. Linear flow Linear flow occurs when flow paths are parallel and the fluid flows in a single direction. In addition, the cross-sectional

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WELL TESTING ANALYSIS

p1

p2

1/5

area to flow must be constant. Figure 1.5 shows an idealized linear flow system. A common application of linear flow equations is the fluid flow into vertical hydraulic fractures as illustrated in Figure 1.6.

A Spherical and hemispherical flow Depending upon the type of wellbore completion configuration, it is possible to have spherical or hemispherical flow near the wellbore. A well with a limited perforated interval could result in spherical flow in the vicinity of the perforations as illustrated in Figure 1.7. A well which only partially penetrates the pay zone, as shown in Figure 1.8, could result in hemispherical flow. The condition could arise where coning of bottom water is important.

Figure 1.5 Linear flow. Well Fracture Isometric View

h

Plan View Wellbore

1.1.4 Number of flowing fluids in the reservoir The mathematical expressions that are used to predict the volumetric performance and pressure behavior of a reservoir vary in form and complexity depending upon the number of mobile fluids in the reservoir. There are generally three cases of flowing system: (1) single-phase flow (oil, water, or gas); (2) two-phase flow (oil–water, oil–gas, or gas–water); (3) three-phase flow (oil, water, and gas).

Fracture

Figure 1.6 Ideal linear flow into vertical fracture.

The description of fluid flow and subsequent analysis of pressure data becomes more difficult as the number of mobile fluids increases.

Wellbore

Side View

Flow Lines

pwf

Figure 1.7 Spherical flow due to limited entry.

Wellbore

Side View

Flow Lines

Figure 1.8 Hemispherical flow in a partially penetrating well.

1.2 Fluid Flow Equations The fluid flow equations that are used to describe the flow behavior in a reservoir can take many forms depending upon the combination of variables presen...