by

Stratamagnetic Software, LLC

Houston, Texas and Beijing, China

Examples of incorrect formulations, 3

Velocity singularities, 3

Fracture flows, 3

Uniform flux fractures, 3

Mudcake buildup, 4

Geometric gridding, 4

Averaging methods, 4

Upscaling techniques, 4

Wells in layered media, 5

Wellbore models, 6

Formation tester multiphase flow, 6

Formation tester pressure transient interpretation, 7

Sweep efficiency and streamline tracing, 7

Book objectives recapitulated, 7

Darcy’s equations for flow in porous media, 8

Differential equations and boundary conditions, 8

Darcy's laws, 10

Logarithmic solutions and beyond, 12

Fundamental aerodynamic analogies, 13

Navier Stokes equations, 13

The Darcy flow limit, 14

The aerodynamic limit, 14

Validity of Laplace’s equation, 15

Different physical interpretations, 17

Meaning of multivalued solutions, 17

Analogies from inverse formulations, 18

Problems and exercises, 19

Example 2-1. Single straight-line fracture in an isotropic circular reservoir containing incompressible fluid, 21

Formulation, 21

Singular integral equation analysis, 23

Specializing Carleman’s results to fracture flow, 24

Physical meaning of f(x), 26

Remark on Muskat’s solution, 27

Velocity singularities at fracture tips, 28

Streamline orientation, 28

Example 2-2. Line fracture in an anisotropic reservoir with incompressible liquids and compressible gases, 29

General formulation, 29

Singular integral equation analysis, 30

The physical meaning of f(x), 32

Velocity singularities at fracture tips, 33

Example 2-3. Effect of nonzero fracture thickness, 34

Practical algebraic issues, 36

Example 2-4. Flow rate boundary conditions, 36

Example 2-5. Uniform vertical velocity along the fracture, 37

Evaluation of singular integrals, 37

Example 2-6. Uniform pressure along the fracture, 39

Example 2-7. More general fracture pressure distributions, 40

Example 2-8. Velocity conditions for gas flows, 41

Example 2-9. Determining velocity fields, 42

Problems and exercises, 43

Example 3-1. Straight-line shale segment in uniform flow, 45

Qualitative problem formulation, 46

The arc tan solution, 46

The elementary vortex solution, 47

Mathematical formulation, 47

Singular integral equation solution, 48

Integral equation solution, 49

Applying the results, 49

Physical significance of vortex strength, 50

Example 3-2. Curved shale segment in uniform flow, 51

Role of circulation in other problems, 51

Example 3-3. Mineralized faults, anisotropy, and gas flow, 51

Problems and exercises, 52

Discussion 4-1. The classical streamfunction, 54

Properties of the "simple" streamfunction, 55

Discussion 4-2. Streamfunction for general fluids in heterogeneous and anisotropic formations, 57

Discussion 4-3. Subtle differences between pressure and streamfunction formulations, 59

More streamfunction properties, 59

The classic streamline tracing problem, 60

The vortex solution, 61

Discussion 4-4. Streamline tracing in the presence of multiple wells, 62

A simple scheme, 63

The final streamline tracing recipe, 64

Discussion 4-5. Streamfunction expressions for distributed line sources and vortexes, 65

Source-like flows, 65

Vortex distributions, 66

Discussion 4-6. Streamfunction solution using complex variables techniques, 67

Obtaining velocity components, 68

Flows of gases in isotropic media, 68

Discussion 4-7. Circle Theorem: Exact solutions to Laplace’s equation, 68

Pressure corollary, 69

Discussion 4-8. Generalized streamline tracing and volume flow rate computations, 70

Boundary value problem types, 71

Streamline tracing in the presence of wells, 71

Discussion 4-9. Streamline tracing in 3D flows, 72

Relationship to streamfunction, 74

Discussion 4-10. Tracer movement in 3D reservoirs, 75

Fluid flow instabilities, 78

Problems and exercises, 80

What is conformal mapping?, 82

A general transformation, 82

Conformal mapping, 83

Using "simple" complex variables, 84

Example 5-1. The classic radial flow solution, 86

Example 5-2. Circular borehole with two symmetric radial fractures, 88

Example 5-3. Circular borehole with two uneven, opposite, radial

fractures; or, a single radial fracture, 90

Example 5-4. Circular borehole with multiple radial fractures, 91

Example 5-5. Straight shale segment at arbitrary angle, 93

Method 1, 94

Method 2, 94

Flow past straight shale segment, 95

More general shale geometries, 95

Example 5-6. Infinite array of straight-line shales, 96

Using the Circle Theorem, 97

More general shale arrays and oncoming fluids, 97

Example 5-7. Pattern wells under aquifer drive, 97

Three-dimensional flows, 98

Example 5-8. Point spherical flow, 99

Example 5-9. Finite line source with prescribed pressure, 99

Integral equation solution, 100

Example 5-10. Finite line source with prescribed flow rate, 101

Example 5-11. Finite conductivity producing fracture having

limited areal extent, 102

Example 5-12. Finite conductivity non-producing fracture having limited areal extent, 103

Borehole interactions, 103

Example 5-13. Producing fracture near multiple wells under aquifer drive, 104

Example 5-14. Producing fractures near multiple wells in solid wall reservoirs, 105

Example 5-15. Straight-line shale segment near multiple wells in uniform flow, 106

Examples 5-16 and 5-17. Non-producing faults in solid wall and aquifer-driven reservoirs, 107

Solid walled aquifer, 107

Aquifer drive, 107

Example 5-18. Highly curved fractures and shales, 108

Problems and exercises, 109

Example 6-1. Steady liquids in homogeneous media, 110

Pressure-pressure formulations, 110

P_W-Q_W formulations, 111

P_R-Q_W formulations, 111

Example 6-2. Simple front tracking for liquids in homogeneous,

isotropic media, 111

Incompressible transient effects, 112

Discontinuous properties, 113

Radial flow streamfunction, 113

Example 6-3. Steady-state gas flows in homogeneous, isotropic media, 113

Pressure-pressure formulations, 113

P_W-Q_W formulations, 114

P_R-Q_W formulations, 114

Transient compressible flows, 115

Example 6-4. Numerical solution for steady flow, 116

Finite difference formulation, 116

Example 6-5. Explicit and implicit schemes for transient compressible liquids, 118

Explicit schemes, 119

Numerical stability, 119

Implicit schemes, 119

Variable grids, 120

Example 6-6. Transient compressible gas flows, 120

Linearity vs nonlinearity, 121

Nonlinear superposition, 121

Choosing variable meshes, 122

Initialization procedures, 122

Flow rate boundary conditions, 122

Problems and exercises, 123

Finite differences: basic concepts, 124

Finite difference approximations, 124

A simple differential equation, 125

Variable coefficients and grids, 128

Formulating steady flow problems, 128

Steady flow problems, 130

Direct versus iterative solutions, 130

Iterative methods, 131

Laplace equation solver, Case 1, 132

Laplace equation solver, Case 2, 134

Convergence acceleration, 137

Wells and internal boundaries, 138

Laplace equation solver, Case 3, 138

Peaceman well corrections, 139

Derivative discontinuities, 141

Point relaxation methods, 142

Laplace equation solver, Case 4, 142

Observations on relaxation methods, 145

Easy to program and maintain, 146

Laplace equation solver, Case 5, 146

Laplace equation solver, Case 6, 148

Laplace equation solver, Case 7, 150

Minimal computing resources, 151

Good numerical stability, 151

Fast convergence, 152

Why relaxation methods converge, 153

Over-relaxation, 154

Line and point relaxation, 154

Isotropy and anisotropy: fluid invasion in cross-bedded sands, 155

Numerical results, 157

Electrical analogy, 159

Problems and exercises, 160

Overview, 162

Problems with idealized grids, 162

Alternative coordinate systems, 163

General coordinate transformations, 164

Thompson’s mapping, 165

Some reciprocity relations, 166

Conformal mapping revisited, 167

Solution of mesh generation equations, 169

Boundary conditions, 170

Fast iterative solutions, 171

Fast solutions for reservoir pressure, 173

Problems and exercises, 174

Overview, 176

Three motivating pressure problems, 176

Reservoir simulation as a topology problem, 177

A practical problem, 177

Governing equations, 178

Steady areal flow: generalized log r solution, 179

Pressure - pressure formulations, 180

Pressure - flow rate formulations, 181

Streamline tracing in curvilinear coordinates, 183

Calculated steady flow examples, 185

Example 9-1. Well in Houston, 186

Example 9-2. Well in Dallas, 191

Example 9-3. Well in center of Texas, 192

Example 9-4. Fracture across Texas, 194

Example 9-5. Isothermal and adiabatic gas flows, 196

Mesh generation: several remarks, 199

Lopsided square grids, 200

Square grid for circles, 200

Grids for odd shapes, 200

Grids for faulted sections, 201

Multiple wells, 202

General stratigraphic grids, internal boundaries, 202

Problems and exercises, 203

Overview, 204

Two-dimensional planar flows, 204

Alternating-direction-implicit (ADI) methods, 204

Solving the mapped equation, 205

Example 10-1. Transient pressure drawdown, 205

Example 10-2. Transient pressure buildup, 209

Steady three-dimensional flow, 211

Transient 3D flow, ADI methods, 212

Example 11-1. Constant density liquid in steady linear flow, 214

Effective permeability and harmonic averaging, 215

Cores arranged in parallel, 215

Effective porosity and front tracking, 216

The lessons learned, 216

Example 11-2. Lineal multiphase flow in two serial cores, 217

Darcy's laws, 217

Mass conservation, 217

Fractional flow functions, 217

Saturation equations, 218

Solving the saturation equations, 218

Characteristic speeds in reservoir analysis, 219

The multiphase pressure field, 220

Example 11-3. Effective properties in steady cylindrical flows, 221

Example 11-4. Steady, single-phase, heterogeneous flows, 221

Example 11-5. Time scale for compressible transients, 221

Problems and exercises, 223

Observations on existing models, 224

Dual porosity models, 224

Geostatistical vs direct modeling, 225

Mathematical connections, 225

A mathematical strategy, 226

Permeability modeling, 226

Physical implications, 227

Mathematical approaches, 227

Example 12-1. Contractional fractures, 228

Methods from heat transfer, 228

Pressure solution, 229

Alternative solutions for permeability, 229

Problems and exercises, 230

Real viscosity and shockwaves, 231

Low-order nonlinear wave model, 231

Singularities in the low-order model, 232

Existence of the singularity, 232

Entropy conditions, 233

Artificial viscosity and fictitious jumps, 234

Problems and exercises, 236

Borehole invasion modeling, 237

Example 14-1. Thin lossy muds (that is, water), 238

Pressure-pressure formulation, 238

Example 14-2. Time-dependent pressure differentials, 239

Example 14-3. Invasion with mudcake effects, 239

Time lapse logging, 240

Lost circulation, 245

Problems and exercises, 246

Overview, 247

Formulation errors, 247

I/O problems, 248

Synopsis, 248

Fundamental issues and problems, 249

Numerical stability, 249

Inadequacies of the von Neumann test, 250

Convergence, 251

Physical resolution, 251

Direct solvers, 252

Modern simulation requirements, 252

Pressure constraints, 254

Flow rate constraints, 254

Object oriented geobodies, 255

Plan for remaining sections, 255

Governing equations and numerical formulation, 256

Steady flows of liquids, 256

Difference equation formulation, 256

The iterative scheme, 258

Modeling well constraints for liquids, 258

Steady and unsteady nonlinear gas flows, 260

Steady gas flows, 261

Well constraints for gas flows, 262

Transient, compressible flows, 264

Compaction, consolidation and subsidence, 266

Boundary conforming grids, 267

Stratigraphic meshes for layered media, 268

Modeling wellbore storage, 269

Group 1, basic example calculations, 270

Simulation capabilities, 270

Data structures and programming, 271

Example 15-1. Convergence acceleration, two

deviated horizontal gas wells in a channel sand, 271

Example 15-2. Dual-lateral horizontal completion

in a fractured, dipping, heterogeneous, layered formation, 275

Example 15-3. Stratigraphic grids, drilling dome-shaped structures, 278

Example 15-4. Simulating-while-drilling horizontal gas wells through a dome-shaped reservoir, 280

Example 15-5. Modeling wellbore storage effects and compressible borehole flow transients, 286

Run 1. Production well, no wellbore storage effects, 287

Run 2. Production well, with some wellbore storage effects, 290

Run 3. Production well, with more wellbore storage effects, 290

Run 4. Injector well, without wellbore storage effects, 291

Run 5. Injector well, with wellbore storage effects, 291

Group 2, advanced calculations and user interface, 292

Overview, 292

Multisim^TM software features, 294

Reservoir description, 294

Well system modeling, 294

Additional simulator features, 294

Example 15-6. Multilateral and vertical wells in multilayer media, 295

Example 15-7. Dual lateral with transient operations, 319

Example 15-8. Producer and injector conversions, 330

Example 15-9. Production with top and bottom drives, 347

Example 15-10. Transient gas production from dual horizontal with wellbore storage effects, 355

Well modeling and productivity indexes, 367

Radial vs 3D modeling – loss of wellbore resolution, 367

Analogies in computational aerodynamics, 367

Curvilinear grids in reservoir simulation, 369

Productivity index modeling, 371

Applications to unconventional resources, 372

Problems and exercises, 373

Overview, 374

Qualitative ideas on formation invasion, 376

Background literature, 380

Darcy reservoir flow equations, 383

Single-phase flow pressure equations, 383

Dynamically coupled lineal flow, 384

Problem formulation, 386

Eulerian versus Lagrangian description, 386

Constant density versus compressible flow, 387

Steady versus unsteady flow, 387

Incorrect use of Darcy's law, 388

Moving fronts and interfaces, 389

Use of effective properties, 390

Problems and exercises, 391

Simple flows without mudcake, 392

Homogeneous liquid in a uniform linear core, 393

Homogeneous liquid in a uniform radial flow, 394

Homogeneous liquid in a uniform spherical domain, 395

Gas flow in a uniform linear core, 396

Flow from a plane fracture, 397

Flows with moving boundaries, 398

Lineal mudcake buildup on filter paper, 398

Plug flow of two liquids in linear core without cake, 401

Coupled dynamical problems: mudcake and formation interaction, 402

Simultaneous mudcake buildup and filtrate invasion in a linear core (liquid flows), 402

Simultaneous mudcake buildup and filtrate invasion in a radial geometry (liquid flows), 405

Fluid compressibility, 409

Dynamic filtration and borehole flow rheology, 411

Erosion due to shear stress, 412

Dynamic filtration in Newtonian fluids, 413

Modifications for drillpipe rotation, 418

Effect of solids concentration, 419

Turbulent versus laminar flow, 420

Concentric power law flows without pipe rotation, 420

Concentric power law flows with pipe rotation, 422

Formation invasion at equilibrium mudcake thickness, 423

Dynamic filtration in eccentric boreholes, 424

Problems and exercises, 426

Problems and exercises, 472

Experimental model validation, 473

Static filtration test procedure, 473

Dynamic filtration testing, 474

Measurement of mudcake properties, 474

Formation evaluation from invasion data, 474

Field applications, 475

Characterizing mudcake properties, 477

Simple extrapolation of mudcake properties, 477

Radial mudcake growth on cylindrical filter paper, 478

Porosity, permeability, oil viscosity and pore pressure determination, 481

Simple porosity determination, 481

Radial invasion without mudcake, 481

Time lapse analysis using general muds, 485

Examples of time lapse analysis, 488

Formation permeability and hydrocarbon viscosity, 488

Pore pressure, rock permeability, and fluid viscosity, 491

Problems and exercises, 493

Finite difference modeling, 494

Basic formulas, 495

Model constant density flow analysis, 496

Transient compressible flow modeling, 499

Numerical stability, 500

Convergence, 501

Multiple physical time and space scales, 501

Example 20-1. Lineal liquid displacement without mudcake, 502

Example 20-2. Cylindrical radial liquid displacement without cake, 507

Example 20-3. Spherical radial liquid displacement without cake. 510

Example 20-4. Lineal liquid displacement without mudcake, including

compressible flow transients, 512

Example 20-5. Von Neumann stability of implicit time schemes, 514

Example 20-6. Gas displacement by liquid in lineal core without mudcake, including compressible flow transients, 516

Incompressible problem, 516

Transient, compressible problem, 517

Example 20-7. Simultaneous mudcake buildup and displacement front motion for incompressible liquid flows, 520

Matching conditions at displacement front, 523

Matching conditions at the cake-to-rock interface, 523

Coding modifications, 524

Modeling formation heterogeneities, 526

Mudcake compaction and compressibility, 527

Modeling borehole activity, 527

Problems and exercises, 528

Immiscible Buckley-Leverett lineal flows without capillary pressure, 530

Example boundary value problems, 532

Mudcake-dominated invasion, 534

Shock velocity, 534

Pressure solution, 535

Molecular diffusion in fluid flows, 537

Exact lineal flow solutions, 538

Numerical analysis, 538

Diffusion in cake-dominated flows, 539

Resistivity migration, 540

Immiscible radial flows with capillary pressure and prescribed mudcake growth, 545

Governing saturation equation, 545

Numerical analysis, 547

Fortran implementation, 548

Typical calculations, 548

Mudcake-dominated flows, 553

Unshocking a saturation discontinuity, 556

Immiscible flows with capillary pressure and dynamically coupled mudcake growth, 559

Flows without mudcakes, 559

Modeling mudcake coupling, 566

Unchanging mudcake thickness, 567

Transient mudcake growth, 569

General immiscible flow model, 572

Problems and exercises, 573