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