Fluid flow in porous media

218 Pages·2018·3.534 MB·English

by  Zimmerman, Robert Wayne

The Imperial College Lectures in
PETROLEUM ENGINEERING
Fluid Flow in Porous Media
Volume
5
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Herbert Moses - Q0146 - The Imperial College Lectures in Petroleum Engineering.indd 1 01-03-18 2:44:39 PM
The Imperial College Lectures in
PETROLEUM ENGINEERING
Fluid Flow in Porous Media
Volume
5
Robert W. Zimmerman
Imperial College London, UK
World Scientific
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Published by
World Scientific Publishing Europe Ltd.
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Library of Congress Cataloging-in-Publication Data
Names: Zimmerman, Robert Wayne, author.
Title: Fluid flow in porous media / by author: Robert Zimmerman
(Imperial College London, United Kingdom).
Description: [Hackensack] New Jersey : World Scientific, [2018] |
Series: The Imperial College lectures in petroleum engineering ; volume 5 |
Includes bibliographical references.
Identifiers: LCCN 2017048948 | ISBN 9781786344991 (hc : alk. paper)
Subjects: LCSH: Oil reservoir engineering. | Petroleum--Migration. | Fluids--Migration.
Classification: LCC TN870.57 .Z56 2018 | DDC 622/.3382--dc23
LC record available at https://lccn.loc.gov/2017048948
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2018 by World Scientific Publishing Europe Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Desk Editors: Herbert Moses/Jennifer Brough/Shi Ying Koe
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Printed in Singapore
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Preface
This book is the fifth volume of a set of lecture notes based on the
Master of Science course in Petroleum Engineering that is taught
within theDepartment of EarthScience andEngineeringat Imperial
CollegeLondon.ThePetroleumEngineeringMScisaone-yearcourse
thatcomprisesthreecomponents:(a)asetoflecturesonthedifferent
topics that constitute the field of petroleum engineering, along with
associated homework assignments and examinations; (b) a group
field project in which the class is broken up into groups of about
six students, who then use data from an actual reservoir to develop
the field from the initial appraisal based on seismic and geological
data, all the way through to eventual abandonment; and (c) a 14-
week individual project, in which each student investigates a specific
problem and writes a small “thesis” in the format of an SPE paper.
The Petroleum Engineering MSc course has been taught at
Imperial College since 1976, and has trained over a thousand
petroleum engineers. The course is essentially a “conversion course”
that aims to take students who have an undergraduate degree in
some area of engineering or physical science, but not necessarily any
specific experience in petroleum engineering, and train them to the
point at which they can enter the oil and gas industry as petroleum
engineers. Although the incoming cohort has included students with
undergraduate degrees in fields as varied as physics, mathematics,
geology, and electrical engineering, the “typical” student on the
course has an undergraduate degree in chemical or mechanical
v
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vi Fluid Flow in Porous Media
engineering,andlittleifanypriorexposuretopetroleumengineering.
Althoughsomestudentsenterthecoursehavinghadsomeexperience
in the oil industry, the course is intended to be self-contained,
and prior knowledge of petroleum engineering or geology is not a
prerequisite for any of the lecture modules.
The present volume presents the equations of fluid flow in
porous media, with a focus on topics and issues that are relevant
to petroleum reservoir engineering. No prior knowledge of this topic
is assumed on the part of the reader, and particular attention is
given to a careful mathematical and conceptual development of
the governing equations, and to presenting solutions to important
reservoir flow problems. The mathematical level is intended to
be accessible to third- or fourth-year undergraduate students in
engineering. Advanced topics such as Laplace transforms and Bessel
functions, which play a key role in solving reservoir engineering
problems, are developed in a self-contained manner. Although these
notes have been written with a focus on petroleum engineering, it
is expected that they will also be useful to hydrologists and civil
engineers.
The author thanks the editorial and production staff at World
Scientific for bringing this book to completion so rapidly and
professionally. SpecialthanksgotoDr.HanlidelaPorteof PetroSim
Consultants, who read the first draft of this book, and whose
comments, criticisms, and suggestions helped to greatly improve its
readability and practical relevance.
Robert W. Zimmerman
Imperial College London
January 2018
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About the Author
Robert W. Zimmerman obtained a BS and MS in mechanical
engineering from Columbia University, and a PhD in rock mechanics
from the University of California at Berkeley. He has been a lecturer
at UC Berkeley, a staff scientist at the Lawrence Berkeley National
Laboratory, and Head of the Division of Engineering Geology
and Geophysics at the Royal Institute of Technology (KTH) in
Stockholm. He is the Editor-in-Chief of the International Journal
of Rock Mechanics and Mining Sciences, and serves on the Editorial
Boards of Transport in Porous Media and the International Journal
of EngineeringScience.HeistheauthorofthemonographCompress-
ibility of Sandstones (Elsevier, 1991), and co-author, with JC Jaeger
andNGWCook,ofFundamentals ofRock Mechanics (4thed.,Wiley-
Blackwell, 2007). He is currently Professor of Rock Mechanics at
Imperial College, where he conducts research on rock mechanics and
fracturedrockhydrology,withapplicationstopetroleumengineering,
underground mining, carbon sequestration, and radioactive waste
disposal.
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Contents
Preface v
About the Author vii
Chapter 1. Pressure Diffusion Equation for
Fluid Flow in Porous Rocks 1
1.1. Darcy’s Law and the Definition of Permeability . . . . . 1
1.2. Datum Levels and Corrected Pressure . . . . . . . . . . . 5
1.3. Concept of Representative Elementary Volume . . . . . 6
1.4. Radial, Steady-state Flow to a Well . . . . . . . . . . . . . 7
1.5. Conservation of Mass Equation . . . . . . . . . . . . . . . . 10
1.6. Diffusion Equation in Cartesian Coordinates . . . . . . . 12
1.7. Diffusion Equation in Cylindrical Coordinates . . . . . . 16
1.8. Governing Equations for Multi-phase Flow . . . . . . . . 18
Problems for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . 20
Chapter 2. Line Source Solution for a Vertical
Well in an Infinite Reservoir 23
2.1. Derivation of the Line Source Solution . . . . . . . . . . . 23
2.2. Dimensionless Pressure and Time . . . . . . . . . . . . . . 30
2.3. Range of Applicability of the Line Source Solution . . . 33
2.4. Logarithmic Approximation to the Line Source
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5. Instantaneous Pulse of Injected Fluid . . . . . . . . . . . . 37
ix