Petroleum Reservoir Simulation - A Basic Approach

468 Pages·2006·15.19 MB·English

by  M. R. Islam

Petroleum
Reservoir Simulation
A Basic Approach
Jamal H. Abou-Kassem
Professor of Petroleum gnireenignE
United Arab Emirates University
AI-Ain, The United Arab setarimE
.S M. Farouq Ali
Petroleum gnireenignE Consultant
LREP adanaC Ltd.
Edmonton, Alberta, adanaC
M. Rafiq Islam
Professor dna Killam Chair ni Oil dna saG
eisuohlaD University
,xafilaH Nova ,aitocS adanaC
@
Petroleum Reservoir Simulation: A Basic Approach
Copyright © 2006 by Gulf Publishing Company, Houston, Texas. All rights
reserved. No part of this publication may be reproduced or transmitted in any
form without the prior written permission of the publisher.
HOUSTON, TX:
Gulf Publishing Company
2 Greenway Plaza, Suite 1020
Houston, TX 77046
AUSTIN, TX:
427 Sterzing St., Suite 104
Austin, TX 78704
1098765432 I
yrarbiL of ssergnoC noitacilbuP-ni-gnigolataC ataD
Petroleum Reservoir Simulation: A Basic Approach/ Jamal .H messaK-uobA ... et
al..
p. .mc
Includes bibliographical references dna index.
NBSI 0-9765113-6-3 (alk. paper)
I. Petroleum--Simulation methods, manuals, etc. 2. Petroleum--Mathematical
models, manuals, etc. 3. Hydrocarbon reservoirs--Simulation methods, ,slaunam
etc. 4. Hydrocarbon reservoirs--Mathematical models, manuals, etc. .5 Petroleum
engineering--Mathematics, manuals, etc. I. ~messaK-uobA lama3 .H (lama1 Hussein)
74R.75.078NT 6002
553.2'8015118--dc22
47692O5002
Printed in the United States of America
Printed on acid-free paper, oo
Text design and composition by TIPS Technical Publishing, Inc.
We dedicate this book to our parents.
Preface
The "Information Age" promises infinite transparency, unlimited productivity, and true
access to .egdelwonk Knowledge, quite distinct and apart from "know-how," requires a
process of thinking, or imagination--the attribute that sets human beings apart. Imagina-
tion is necessary for anyone wishing to make decisions based on science. Imagination
always begins with visualization--actually, another term for simulation.
Under normal conditions, we simulate a situation prior to making any decision, i.e., we
abstract absence and start to fill in the gaps. Reservoir simulation is no exception. The two
most important points that must not be overlooked in simulation are science and multi-
plicity of solutions. Science is the essence of knowledge, and acceptance of the multi-
plicity of solutions is the essence of science. Science, not restricted by the notion of a
single solution to every problem, must follow imagination. Multiplicity of solutions has
been promoted as an expression of uncertainty. This leads not to science or to new
authentic knowledge, but rather to creating numerous models that generate "unique" solu-
tions that fit a predetermined agenda of the decision-makers. This book reestablishes the
essential features of simulation and applies them to reservoir engineering problems. This
approach, which reconnects with the old--or in other words, time-tested--concept of
knowledge, is refreshing and novel in the Information Age.
The petroleum industry is known as the biggest user of computer models. Even though
space research and weather prediction models are robust and are often tagged as "the
mother of all simulation," the fact that a space probe device or a weather balloon can be
launched--while a vehicle capable of moving around in a petroleum reservoir cannot--
makes modeling more vital for tackling problems in the petroleum reservoir than in any
other discipline. Indeed, from the advent of computer technology, the petroleum industry
pioneered the use of computer simulations in virtually all aspects of decision-making. This
revolutionary approach required significant investment in long-term research and
advancement of science. That time, when the petroleum industry was the energy provider
of the world, was synonymous with its reputation as the most aggressive investor in engi-
neering and science. More recently, however, as the petroleum industry transited into its
"middle age" in a business sense, the industry could not keep up its reputation as the big-
gest sponsor of engineering and long-term research. A recent survey by the United States
Department of Energy showed that none of the top ten breakthrough petroleum technolo-
gies in the last decade could be attributed to operating companies. If this trend continues,
major breakthroughs in the petroleum industry over the next two decades are expected to
be in the areas of information technology and materials science. When it comes to reser-
voir simulators, this latest trend in the petroleum industry has produced an excessive
emphasis on the tangible aspects of modeling, namely, the number of blocks used in a
simulator, graphics, computer speed, etc. For instance, the number of blocks used in a res-
ervoir model has gone from thousands to millions in just a few years. Other examples can
xi
xii Preface
be cited, including graphics in which flow visualization has leapt from 2D to 3D to 4D,
and computer processing speeds that make it practically possible to simulate reservoir
activities in real time. While these developments outwardly appear very impressive, the
lack of science and, in essence, true engineering render the computer revolution irrelevant
and quite possibly dangerous. In the last decade, most investments have been made in
software dedicated to visualization and computer graphics with little being invested in
physics or mathematics. Engineers today have little appreciation of what physics and
mathematics provide for the very framework of all the fascinating graphics that are gener-
ated by commercial reservoir simulators. As companies struggle to deal with scandals trig-
gered by Enron's collapse, few have paid attention to the lack of any discussion in
engineering education regarding what could be characterized as scientific fundamentals.
Because of this lack, little has been done to promote innovation in reservoir simulation,
particularly in the areas of physics and mathematics, the central topical content of reser-
voir engineering.
This book provides a means of understanding the underlying principles of petroleum res-
ervoir simulation. The focus is on basic principles because understanding these principles
is a prerequisite to developing more accurate advanced models. Once the fundamentals are
understood, further development of more useful simulators is only a matter of time. The
book takes a truly engineering approach and elucidates the principles behind formulating
the governing equations. In contrast to cookbook-type recipes of step-by-step procedures
for manipulating a black box, this approach is full of insights. To paraphrase the caveat
about computing proposed by R. W. Hamming, the inventor of the Hamming Code: the
purpose of simulation must be insight, not just numbers. The conventional approach is
more focused on packaging than on insight, making the simulation process more opaque
than transparent. The formulation of governing equations is followed by elaborate treat-
ment of boundary conditions. This is one aspect that is usually left to the engineers to
"figure out" by themselves, unfortunately creating an expanding niche for the select few
who own existing commercial simulators. As anyone who has ever engaged in developing
a reservoir simulator well knows, this process of figuring out by oneself is utterly con-
fusing. In keeping up with the same rigor of treatment, this book presents the discretiza-
tion scheme for both block-centered and point-distributed grids. The difference between a
well and a boundary condition is elucidated. In the same breadth, we present an elaborate
treatment of radial grid for single-well simulation. This particular application has become
very important due to the increased usage of reservoir simulators to analyze well test
results and the use of well pseudo-functions. This aspect is extremely important for any
reservoir engineering study. The book continues to give insight into other areas of reser-
voir simulation. For instance, we discuss the effect of boundary conditions on material-
balance-check equations and other topics with unparalleled lucidity.
Even though the book is written principally for reservoir simulation developers, it takes an
engineering approach that has not been taken before. Topics are discussed in terms of sci-
ence and mathematics, rather than with graphical representation in the backdrop. This
makes the book suitable and in fact essential for every engineer and scientist engaged in
modeling and simulation. Even those engineers and scientists who wish to limit their
Preface xiii
activities to field applications will benefit greatly from this book, which is bound to pre-
pare them better for the Information Age.
stnemgdelwonkcA
We, the authors, are grateful to many colleagues, friends, and students who have contrib-
uted to this book. We thank Dr. M. S. Osman, of the Kuwait Oil Company, for his contri-
butions, fruitful discussions, and critiques at the various stages of writing this book, during
his tenure at the UAE University. We are indebted to Dr. T. Ertekin, of The Pennsylvania
State University, and Dr. R. Almehaideb, of UAEU, for their reviews and comments on
chapters of the book. We thank the many students at UAEU who took the undergraduate
reservoir simulation course during writing the book. We thank Mr. Othman N. Matahen,
of the UAEU Center of Teaching and Learning Technology, for his most skillful computer
drafting of all figures in this book.
We are most deeply thankful to all our "teachers," from whom we have learned all that we
know, and to members of our families, for their encouragement, their support, and most
importantly their patience and tolerance during the writing of this book.
J. H. Abou-Kassem
S. M. Farouq Ali
M. R. Islam
Introduction
In this book, the basics of reservoir simulation are presented through the modeling of
single-phase fluid flow and multi-phase flow in petroleum reservoirs using the engi-
neering approach. This text is written for senior-level B.S. students and first-year M.S.
students studying petroleum engineering and aims to restore engineering and physics
sense to the subject. In this way, it challenges the misleading impact of excess mathemat-
ical glitter that has dominated reservoir simulation books in the past. The engineering
approach employed in this book uses mathematics extensively but injects engineering
meaning to differential equations and to boundary conditions used in reservoir simulation.
It does not need to deal with differential equations as a means for modeling, and it inter-
prets boundary conditions as fictitious wells that transfer fluids across reservoir bound-
aries. The contents of the book can be taught in two consecutive courses. The first, an
undergraduate senior-level course, includes the use of a block-centered grid in rectangular
coordinates in single-phase flow simulation. This material is mainly included in Chapters
2, 3, 4, 6, 7, and 9. The second, a graduate-level course, deals with a block-centered grid in
radial-cylindrical coordinates, a point-distributed grid in both rectangular and radial-cylin-
drical coordinates, and the simulation of multiphase flow in petroleum reservoirs. This
material is covered in Chapters 5, 8, and 10 in addition to specific sections in Chapters 2,
4, 6, and 7 (Secs. 2.7, 4.5, 6.2.2).
Chapter 1 provides an overview of reservoir simulation and the relationship between the
mathematical approach presented in simulation books and the engineering approach pre-
sented in this book. In Chapter 2, we present the derivation of single-phase, multidimen-
sional flow equations in rectangular and radial-cylindrical coordinate systems. In Chapter
3, we introduce the Control Volume Finite Difference (CVFD) terminology as a means to
writing the flow equations in multidimensions in compact form. Then we write the general
flow equation that incorporates both (real) wells and boundary conditions, using the block-
centered grid (in Chapter 4) and the point-distributed grid (in Chapter 5), and present the
corresponding treatments of boundary conditions as fictitious wells and the exploitation of
symmetry in practical reservoir simulation. Chapter 6 deals with wells completed in both
single and multiple layers and presents fluid flow rate equations for different well operating
conditions. Chapter 7 presents the explicit, implicit, and Crank-Nicolson formulations of
single-phase, slightly compressible, and compressible flow equations and introduces the
incremental and cumulative material balance equations as internal checks to monitor the
accuracy of generated solutions. In Chapter 8, we introduce the space and time treatments
of nonlinear terms encountered in single-phase flow problems. Chapter 9 presents the basic
direct and iterative solution methods of linear algebraic equations used in reservoir simula-
tion. Chapter 10 is entirely devoted to multiphase flow in petroleum reservoirs and its sim-
ulation. The book concludes with Appendix A, which presents a user's manual for a single-
phase simulator. The CD that accompanies the book includes a single-phase simulator
VX
xvi Introduction
written in FORTRAN 95, a compiled version, a users' manual, and data and output files for
four solved problems. The single-phase simulator provides users with intermediate results
as well as a solution to single-phase flow problems so that a user's solution can be checked
and errors are identified and corrected. Educators may use the simulator to make up new
problems and obtain their solutions.
Nomenclature
a n = coefficient of unknown Xn,n.,,, , B i = fluid formation volume factor in
defined by Eq. 9.46f block i , RB/STB m3/std m 3
A = parameter, defined by Eq. 9.28 in B o = oil formation volume factor,
Tang' s algorithm RB/STB m3/std m 3
IAI = square coefficient matrix Bob = oil formation volume factor at
A x = cross-sectional area normal to x- bubble-point pressure, RB/STB
direction, ft 2 m 2 m3/std m 3
xlxA = cross-sectional area normal to x- Bei = formation volume factor of phase p
direction at x, ft 2 me1 in block i
x Axlx+
= cross-sectional area normal to B w = water formation volume factor,
x-direction at x + Ax, ft 2 m 2 RB/B m3/std m 3
Ax 2,,_,xI cross-sectional area normal to B ° = fluid formation volume factor at
. =
x-direction at block boundary reference pressure p° and
2/l+~x , ft 2 m 2 reservoir temperature, RB/STB
b = reservoir boundary m3/std m 3
b ,L = reservoir east boundary c = fluid compressibility, psi -1 lkPa-ll
b L = reservoir lower boundary c e = coefficient of unknown of block i
b N = reservoir north boundary in Thomas' algorithm
b s = reservoir south boundary c n = coefficient of unknown x,, defined
b U = reservoir upper boundary by Eq. 9.46g
b w = reservoir west boundary c u = coefficient of unknown x N in
b = coefficient of unknown x,_,~,, r , Thomas' or Tang's algorithm
defined by Eq. 9.46a c o = oil-phase compressibility, psi -1
kPa -1
B = parameter, defined by Eq. 9.29 in
Tang' s algorithm c¢ = porosity compressibility, psi -1 kPa -1
B = fluid formation volume factor, c a = rate of fractional viscosity change
RB/STB m3/std m 3 with pressure change, psi -1 kPa -1
= average fluid formation volume factor C = parameter, defined by Eq. 9.30 in
in wellbore, RB/STB m3/std m 3 Tang' s algorithm
gB = gas formation volume factor, CMe = cumulative material balance
RB/scf m3/std m 3 check, dimensionless
xvii
xviii Nomenclature
poC = coefficient of pressure change over ~F = ratio of wellblock i area to
time step in expansion of oil theoretical area from which well
accumulation term, STB/D-psi withdraws its fluid (in Chapter 6),
std m3/(d.kPa) fraction
woC = coefficient of water saturation F m = argument of an integral evaluated
change over time step in expansion at time t m
of oil accumulation term, STB/D F(t m) = argument of an integral
std m3/d evaluated at time t"
p~,C = coefficient of pressure change over F ° = argument of an integral evaluated
time step in expansion of water at time t"
accumulation term, B/D-psi F(t") = argument of an integral
std m3/(d.kPa) evaluated at time t"
wwC = coefficient of water saturation F l+n = argument of an integral evaluated
change over time step in expansion at time t l+n
of water accumulation term, B/D F(t )l+n = argument of an integral
std m3/d evaluated at time t ~+"
¢~ = vector of known values F 2/1+n = argument of an integral
D = parameter, defined by Eq. 9.31 in evaluated at time t "~ln
Tang' s algorithm F(t )z/l+n = argument of an integral
di = known RHS of equation for evaluated at time t 2/1+"
block i in Thomas' algorithm g = gravitational acceleration, ft/sec 2
dm~ = maximum absolute difference ms 2
between two successive iterations gi = element i of a temporary vector
d = RHS of equation for gridblock n, (~) generated in Thomas'
defined by Eq. 9.46h algorithm
~e = coefficient of unknown of G = geometric factor
block i + 1 in Thomas' algorithm G. = well geometric factor, RB-cp/D-psi
,,e = coefficient of unknown +nX 1 m3.mPa.s/(d.kPa)
,
defined by Eq. 9.46d ,wG = well geometric factor for
e u coefficient of unknown ~x in wellblock i , defined by Eq. 6.32,
=
Tang' s algorithm RB-cp/D-psi m3.mPa.s/(d.kPa)
f ( ) = function of G* = well geometric factor of the
fp = the pressure dependent term in theoretical well for wellblock i ,
transmissibility RB-cp/D-psi m3.mPa.s/(d.kPa)
f,.l ~,,,p = nonlinearity, defined by Eq. 8.17 Gy i .... = interblock geometric factor
F(t) = argument of an integral at time t between block i and block i -T- 1