Quantitative Methods in Reservoir Engineering

489 Pages·2002·2.385 MB·English

by  Wilson C. Chin PhD

Contents
Preface, xi
1. Motivating Ideas and Governing Equations, 1
Examples of incorrect formulations, 3
Darcy’s equations for flow in porous media, 7
Logarithmic solutions and beyond, 11
Fundamental aerodynamic analogies, 12
Problems and exercises, 18
2. Fracture Flow Analysis, 19
Example 2-1. Single straight-line fracture in an isotropic circular
reservoir containing incompressible fluid, 19
Example 2-2. Line fracture in an anisotropic reservoir with
incompressible liquids and compressible gases, 27
Example 2-3. Effect of nonzero fracture thickness, 32
Example 2-4. Flow rate boundary conditions, 34
Example 2-5. Uniform vertical velocity along the fracture, 35
Example 2-6. Uniform pressure along the fracture, 37
Example 2-7. More general fracture pressure distributions, 38
Example 2-8. Velocity conditions for gas flows, 39
Example 2-9. Determining velocity fields, 40
Problems and exercises, 41
3. Flows Past Shaly Bodies, 43
Example 3-1. Straight-line shale segment in uniform flow, 43
Example 3-2. Curved shale segment in uniform flow, 49
Example 3-3. Mineralized faults, anisotropy, and gas flow, 49
Problems and exercises, 50
4. Streamline Tracing and Complex Variables, 52
Discussion 4-1. The classical streamfunction, 52
Discussion 4-2. Streamfunction for general fluids in heterogeneous and
anisotropic formations, 55
Discussion 4-3.Subtle differences between pressure and streamfunction
formulations, 57
Discussion 4-4. Streamline tracing in the presence of multiple wells, 60
v
Discussion 4-5. Streamfunction expressions for distributed line sources
and vortexes, 63
Discussion 4-6. Streamfunction solution using complex variables
techniques, 65
Discussion 4-7. Circle Theorem: Exact solutions to Laplace’s equation, 66
Discussion 4-8. Generalized streamline tracing and volume flow rate
computations, 68
Discussion 4-9. Streamline tracing in 3D flows, 70
Discussion 4-10. Tracer movement in 3D reservoirs, 73
Fluid flow instabilities, 76
Problems and exercises, 78
5. Flows in Complicated Geometries, 79
What is conformal mapping? 80
Using “simple” complex variables, 82
Example 5-1. The classic radial flow solution, 84
Example 5-2. Circular borehole with two symmetric radial fractures, 86
Example 5-3. Circular borehole with two uneven, opposite, radial
fractures; or a single radial fracture, 88
Example 5-4. Circular borehole with multiple radial fractures, 89
Example 5-5. Straight shale segment at arbitrary angle, 91
Example 5-6. Infinite array of straight-line shales, 94
Example 5-7. Pattern wells under aquifer drive, 95
Three-dimensional flows, 96
Example 5-8. Point spherical flow, 97
Example 5-9. Finite line source with prescribed pressure, 97
Example 5-10. Finite line source with prescribed flow rate, 99
Example 5-11. Finite conductivity producing fracture having
limited areal extent, 100
Example 5-12. Finite conductivity nonproducing fracture having
limited areal extent, 101
Borehole interactions, 101
Example 5-13. Producing fracture near multiple wells under aquifer
drive, 102
Example 5-14. Producing fractures near multiple wells in solid
wall reservoirs, 103
Example 5-15. Straight-line shale segment near multiple wells in
uniform flow, 104
Examples 5-16 and 5-17. Nonproducing faults in solid wall and
aquifer-driven reservoirs, 105
Example 5-18. Highly curved fractures and shales, 106
Problems and exercises, 107
6. Radial Flow Analysis, 108
Example 6-1. Steady liquids in homogeneous media, 108
Example 6-2. Simple front tracking for liquids in homogeneous,
isotropic media, 109
vi
Example 6-3. Steady-state gas flows in homogeneous, isotropic media, 111
Transient compressible flows, 113
Example 6-4. Numerical solution for steady flow, 114
Example 6-5. Explicit and implicit schemes for transient compressible
liquids, 116
Example 6-6. Transient compressible gas flows, 118
Problems and exercises, 121
7. Finite Difference Methods for Planar Flows, 122
Finite differences: basic concepts, 122
Formulating steady flow problems, 126
Steady flow problems: seven case studies, 128
Isotropy and anisotropy: fluid invasion in cross-bedded sands, 153
Problems and exercises, 158
8. Curvilinear Coordinates and Numerical Grid Generation, 160
General coordinate transformations, 162
Thompson’s mapping, 163
Some reciprocity relations, 164
Conformal mapping revisited, 165
Solution of mesh generation equations, 167
Problems and exercises, 172
9. Steady-State Reservoir Applications, 174
Governing equations, 176
Steady areal flow: generalized log r solution, 177
Streamline tracing in curvilinear coordinates, 181
Calculated steady flow examples, 183
Example 9-1. Well in Houston, 184
Example 9-2. Well in Dallas, 189
Example 9-3. Well in center of Texas, 190
Example 9-4. Fracture across Texas, 192
Example 9-5. Isothermal and adiabatic gas flows, 194
Mesh generation: several remarks, 197
Problems and exercises, 201
10. Transient Compressible Flows: Numerical Well Test
Simulation, 202
Example 10-1. Transient pressure drawdown, 203
Example 10-2. Transient pressure buildup, 207
Problems and exercises, 211
11. Effective Properties in Single and Multiphase Flows, 212
Example 11-1. Constant density liquid in steady linear flow, 212
Example 11-2. Lineal multiphase flow in two serial cores, 215
Example 11-3. Effective properties in steady cylindrical flows, 219
Example 11-4. Steady, single-phase, heterogeneous flows, 219
Example 11-5. Time scale for compressible transients, 219
Problems and exercises, 221
vii
12. Modeling Stochastic Heterogeneities, 222
Observations on existing models, 222
A mathematical strategy, 224
Example 12-1. Contractional fractures, 226
Problems and exercises, 228
13. Real and Artificial Viscosity, 229
Real viscosity and shockwaves, 229
Artificial viscosity and fictitious jumps, 232
Problems and exercises, 234
14. Borehole Flow Invasion, Lost Circulation, and Time
Lapse Logging, 235
Borehole invasion modeling, 235
Example 14-1. Thin lossy muds (that is, water), 236
Example 14-2. Time-dependent pressure differentials, 237
Example 14-3. Invasion with mudcake effects, 237
Time lapse logging, 238
Lost circulation, 243
Problems and exercises, 244
15. Horizontal, Deviated, and Modern Multilateral Well
Analysis, 245
Fundamental issues and problems, 246
Governing equations and numerical formulation, 252
Example calculations, 266
Example 15-1. Convergence acceleration, two deviated horizontal
gas wells in a channel sand, 267
Example 15-2. Dual-lateral horizontal completion in a fractured,
dipping, heterogeneous, layered formation, 270
Example 15-3. Stratigraphic grids, drilling dome-shaped structures, 273
Example 15-4. Simulating-while-drilling horizontal gas wells
through a dome-shaped reservoir, 275
Example 15-5. Modeling wellbore storage effects and compressible
borehole flow transients, 281
Problems and exercises, 287
16. Fluid Mechanics of Invasion, 288
Qualitative ideas on formation invasion, 290
Background literature, 294
Darcy reservoir flow equations, 297
Moving fronts and interfaces, 303
Problems and exercises, 305
17. Static and Dynamic Filtration, 306
Simple flows without mudcake, 306
Flows with moving boundaries, 312
Coupled dynamical problems: mudcake and formation interaction, 316
Dynamic filtration and borehole flow rheology, 325
viii
Concentric power law flows without pipe rotation, 334
Concentric power law flows with pipe rotation, 336
Formation invasion at equilibrium mudcake thickness, 337
Dynamic filtration in eccentric boreholes, 338
Problems and exercises, 340
18. Formation Tester Applications, 341
Problems and exercises, 351
19. Analytical Methods for Time Lapse Well Logging Analysis, 352
Experimental model validation, 352
Characterizing mudcake properties, 356
Porosity, permeability, oil viscosity, and pore pressure determination, 360
Examples of time lapse analysis, 367
Problems and exercises, 372
20. Complex Invasion Problems: Numerical Modeling, 373
Finite difference modeling, 373
Example 20-1. Lineal liquid displacement without mudcake, 381
Example 20-2. Cylindrical radial liquid displacement without cake, 386
Example 20-3. Spherical radial liquid displacement without cake, 389
Example 20-4. Lineal liquid displacement without mudcake, including
compressible flow transients, 391
Example 20-5. Von Neumann stability of implicit time schemes, 393
Example 20-6. Gas displacement by liquid in lineal core without
mudcake, including compressible flow transients, 395
Example 20-7. Simultaneous mudcake buildup and displacement front
motion for incompressible liquid flows, 399
Problems and exercises, 407
21. Forward and Inverse Multiphase Flow Modeling, 408
Immiscible Buckley-Leverett lineal flows without capillary pressure, 409
Molecular diffusion in fluid flows, 416
Immiscible radial flows with capillary pressure and prescribed mudcake
growth, 424
Immiscible flows with capillary pressure and dynamically coupled
mudcake growth, 438
Problems and exercises, 452
Cumulative References, 453
Index, 462
About the Author, 472
ix
Preface
Most reservoir flow analysis books introduce the basic equations, such as
Darcy’s law, single-phase radial flow solutions, simple well test models, and the
usual descriptions of relative permeability and capillary pressure and explain
elementary concepts in finite difference methods and modeling before referring
readers to commercial simulators and industry case studies. These books, and
the courses that promote them, are useful in introducing students to fundamental
methodologies and company practices. However, few develop the physical and
mathematical insight needed to create the next generation of models or to
evaluate the limitations behind existing simulation tools. Many analysis
techniques and computational approaches employed, in fact, are incorrect,
despite their common use in reservoir evaluation.
I earned my Ph.D. at MIT and earlier degrees from Caltech and NYU. My
major areas were high-speed aerodynamics and wave propagation, which are
synonymous with applied math and nonlinear differential equations – specialties
that focus on rigorous solutions to practical problems. From MIT, I joined
Boeing’s prestigious computational fluid dynamics group in Seattle and, three
years later, headed up engine flow analysis at United Technologies’ Pratt and
Whitney, the company that develops the world’s most powerful jet engines.
But the thrill of the hunt lost its allure, despite the thrill of being published
in journals and attending high-tech conferences. Like all of you, I was attracted
to the petroleum industry because of its excitement and the opportunities it
offered. That was just five years into my career, as I joined a new industry
undergoing rapid change – a transition requiring me to learn anew the fluid
dynamics of flows as far underground as my prior learning was above ground.
Since then, two decades have elapsed, in which I actively engaged in oil field
research and development. In that time, for example, with leading operating and
service companies like British Petroleum and Halliburton, I was fortunate to
have been continuously challenged by new problems both mathematical and
operational.
xi
This reservoir flow analysis and simulation book is unique because it
brings two decades of perspectives and experience on the fluid mechanics of
Darcy flows. Many commonly accepted “recipes” for flow evaluation are
critiqued, and incorrect underlying assumptions are noted. This volume aims at
a rigorous and scientifically correct approach to reservoir simulation. In each of
dozens of difficult problems surveyed, the state of the art is examined, and
analytical or numerical solutions are offered, with the exact physical
assumptions always stated precisely. Industry “common sense” approaches are
avoided: once the correct model is formulated, the entire arsenal of analysis
tools is brought to bear – we then focus on ways to extract formation
information using the new solution or clever means to exploit the physics
uncovered.
Fortunately, this book does not require advanced mathematics or numerical
analysis to understand. Great care was undertaken to explain and develop very
advanced methods in simple terms that undergraduates can comprehend. For
example, “conforming mapping” usually requires a background course in
complex variables, and complementary subjects like streamfunctions and
streamline tracing in homogeneous media are typically taught in this framework.
Quite to the contrary, our special derivations require just simple calculus but
apply to anisotropic, heterogeneous media. This book addresses “difficult”
flows, such as liquid and gas flows from fractures, general flows past shales,
production from multilateral horizontal wells, multiple well interactions,
rigorous approaches to effective properties, and so on, problems not often
treated in the literature but relevant to modern petroleum engineering. In doing
so, we strive to avoid the simplistic “recipe” approaches our industry often
encourages.
Every effort is made to define and formulate the mathematical problem
precisely and then to solve it as exactly as modern analysis methods will allow.
These include classic differential equation models as well as modern singular
integral equation approaches, all of which were unavailable to Morris Muskat
when he wrote his lasting monographs on Darcy flow analysis. Our techniques
go beyond purely analytical ones. For example, the problem of accurately
modeling flow from interacting multilateral drainholes in anisotropic
heterogeneous media – despite the inefficiencies imposed by nonneighboring
grid point connections – is solved in Chapter 15 (the groundwork for this
research won a Chairman’s Innovation Award at BP Exploration in 1991).
Or consider boundary-conforming, curvilinear grids in Chapter 8. Fast and
accurate mesh generation algorithms are developed in this book, which are
cleverly applied to the solution of complicated reservoir flows. Suppose a
“Houston well” produces from a “Texas-shaped” reservoir. This geometry is
associated with an elementary function as unique as the logarithm is to radial
coordinates. Its “extended log” permits us to instantly write the solution to all
liquid and gas flows for any set of pressure-pressure and pressure-rate boundary
conditions. This work won a prestigious Small Business Innovation Research
Award from the United States Department of Energy in 2000.
xii
Other areas addressed include opened fractures, curved shales, fractured
holes, general heterogeneities, formation invasion, and time-lapse well logging
using drilling data. In terms of techniques, we introduce modern ideas in
singular integral equations, improper integrals, advanced conformal mapping,
perturbation methods, numerical grid generation, artificial viscosity, moving
boundary value problems, ADI and relaxation methods, and so on, developing
these in context with the physics of the problem at hand. These methods, used
by aerodynamicists and theoretical elasticians, can be intimidating. However,
the presentation style adopted is far from difficult: while not exactly easy
reading, there is nothing in this book that could not be grasped by a student who
has taken basic freshman calculus. Whenever possible, Fortran source code is
presented, so that students can test and evaluate ideas old and new without the
trials and tribulations of debugging.
New approaches to old problems are emphasized. For example, how do
mathematical aerodynamicists turned petroleum engineers view the physical
world? Stare up the back end of a rocket lifting off: Is that a fuselage with
stabilizer fins, or is it a circular wellbore with radial fractures? Pry open the
maintenance box of your typical jet engine: Are those cascades of airfoil blades,
or are they distributions of stochastic shales? Can the solutions that describe
brittle failure be repackaged to model arrays of fractures, say, the natural
fracture systems that spur horizontal drilling? Very often, the problems
(inaccurately) crunched by our fastest computers can be solved (accurately)
using closed-form analytical solutions found in other scientific disciplines.
I am indebted to my advisor, Professor Marten T. Landahl at MIT, for
teaching me the subtleties and nuances of aerodynamics and fluid mechanics. I
also thank the faculty at Caltech, where I had learned hands-on applied math
from its most prolific creators, and to the aerodynamics group at Boeing, where
I participated in state-of-the-art research in numerical flow simulation. Much of
this effort would not be possible without the support of my colleagues and
friends at Halliburton Energy Services, who have enabled me to work freely and
productively in areas of personal interest over the past decade. And last but not
least, I wish to acknowledge Phil Carmical, acquisitions editor, for his
continuing support and constructive comments and for his willingness to
introduce new and innovative methodologies into the commercial mainstream –
at Phil’s advice, “Problems and Exercises” are now included in each chapter,
unique challenges that further develop the new ideas introduced, and ideally,
develop the interests and curiosities of satisfied readers.
Wilson C. Chin, Ph.D., MIT
Houston, Texas
E-mail: [email protected]
xiii

1
Motivating Ideas
and Governing Equations
It is no accident that the industry’s first math models for fluid flow in
petroleum reservoirs were developed by analogy to problems in electrostatics
and heat transfer (Muskat, 1937; van Everdingen and Hurst, 1949; Carslaw and
Jaeger, 1959). These solutions reflect well on the investigators; they did not fall
prey to the maxim that “those who refuse to learn from history are doomed to
repeat it.” That the equations for single phase flow are identical to the classical
equations of elliptic and parabolic type facilitated the initial progress; these
similarities also assisted with the design and scaling of experiments, particularly,
those based on electrical and temperature analogies.
To practitioners in reservoir engineering and well test analysis, the state-of-
the-art has bifurcated into two divergent paths. The first searches for simple
closed-form solutions. These are naturally restricted to simplified geometries
and boundary conditions, but analytical solutions, many employing “method of
images” techniques, nonetheless involve cumbersome infinite series. More
recent solutions for transient pressure analysis, given in terms of Laplace and
Fourier transforms, tend to be more computational than analytical: they require
complicated numerical inversion, and hence, shed little insight on the physics.
It seems, very often, that all of the analytical solutions that can be derived,
have been derived. Thus, the second path described above falls largely in the
realm of supercomputers, high-powered workstations, and brute force numerical
analysis: it is the science, or more appropriately the art, that we call reservoir
simulation, requiring industrywide “comparison projects” for validation. There
has been no middle ground for smart solutions that solve difficult problems, that
is, for solutions that provide physical insight and are in themselves useful,
models that can be used for calibration purposes to keep numerical solutions
“honest.” This dearth of truly useful real world examples lends credence to the
1