Formation Testing

Pressure Transient and Contamination Analysis

Wilson C. Chin, Ph.D., MIT

Stratamagnetic Software, LLC

Yanmin Zhou, Yongren Feng, Qiang Yu and Lixin Zhao

China Oilfield Services Ltd (COSL)

China National Offshore Oil Corporation (CNOOC)

Table of Contents

Opening Message, xiii

Preface, xiv

Acknowledgements, xviii

Part I. Modern Ideas in Job Planning and Execution

- Basic Ideas, Challenges and Developments, 1
- Forward Pressure and Contamination Analysis in Single and Multiphase Compressible Flow, 34
- Inverse Methods for Permeability, Anisotropy and Formation Boundary Effects Assuming Liquids, 56
- Multiphase Flow and Contamination – Transient Immiscible and Miscible Modeling with Fluid Compressibility, 78
- Exact Pressure Transient Analysis for Liquids in Anisotropic Homogeneous Media, Including Flowline Storage Effects, With and Without Skin at Arbitrary Dip Angles, 121
- Permeability Interpretation for Liquids in Anisotropic Media, Including Flowline Storage Effects, With and Without Skin at Arbitrary Dip Angles, 196
- Three-Dimensional Pads and Dual Packers on Real Tools with Flowline Storage in Layered Anisotropic Media for Horizontal Well Single-Phase Liquid and Gas Flows, 274
- Gas Pumping: Forward and Inverse Methods in Anisotropic Media at Arbitrary Dip Angles for Point Source, Straddle Packer and Real Nozzles, 337
^{*} - Three-Dimensional Phase Delay Response in Layered Anisotropic Media with Dip, 385

1.1 Background and introduction, 1

1.2 Existing models, implicit assumptions and limitations, 6

1.2.1 Exponential tight zone approximation, 7

1.2.2 Permeability and anisotropy from steady-state dual-probe data, 8

1.2.3 Three-probe, vertical well interpretation method, 9

1.2.4 Gas pumping, 10

1.2.5 Material balance method, 10

1.2.6 Conventional three-dimensional numerical models, 12

1.2.7 Uniform flux dual packer models, 13

1.3 Tool development, testing and deployment – role of modeling and "behind the scenes" at CNOOC/COSL, 15

1.3.1 Engineering analysis, design challenges, solutions, 15

1.3.2 From physics to math to engineering – inverse problem formulation, 15

1.3.2.1 Simplified theoretical model, 16

1.3.2.2 More detailed finite element model, 17

1.3.3 Design chronicle – people, places and things, 18

1.3.4 Bohai Bay activities, 25

1.3.5 Middle East operations, 28

1.4 Book objectives and presentation plan, 29

1.5 References, 32

2.1 Single-phase source flow models, 34

2.1.1 Qualitative effects of storage and skin, 37

2.2 Dual packer and dual probe flows, 40

2.2.1 A detailed calculation, 41

2.3 Supercharging, mudcake growth and pressure interpretation, 45

2.3.1 Supercharge numerical simulation, 46

2.3.2 Industry perspectives on "buildup versus drawdown," 46

2.4 Boundary and azimuthal effects in horizontal wells, 48

2.5 Contamination clean-up at the source probe, 49

2.6 Sampling-while-drilling tools and clean-up efficiency, 51

2.6.1 What happens with very short invasion times, 51

2.6.2 What happens with longer invasion times, 52

2.7 References, 55

3.1 New inverse methods summary, 56

3.2 New inverse modeling capabilities, 57

3.2.1 Module FT-00, 58

3.2.2 Module FT-01, 60

3.2.3 Module FT-03, 60

3.2.4 Module FT-PTA-DDBU, 61

3.3 Inverse examples – dip angle, multivalued solutions and skin, 62

3.3.1 Forward model, Module FT-00, 62

3.3.2 Inverse model, Module FT-01 – multivalued solutions, 64

3.3.3 Effects of dip angle – detailed calculations, 65

3.3.4 Inverse "pulse interaction" approach for low permeability zones, 68

3.4 Computational notes on complex complementary error function evaluation, 70

3.5 Source model – analytical and physical limitations, 72

3.6 Full three-dimensional transient Darcy flow model for horizontal wells, 72

3.7 Phase delay inverse method and electromagnetic analogy, 75

3.8 Source model applications to dual packers, 76

3.9 Closing remarks, 76

3.10 References, 77

Part II. Math Models, Results and Detailed Examples

4.1 Invasion, supercharging and multiphase pumping, 79

4.1.1 Invasion and pumping description, 79

4.1.2 Job planning considerations, 82

4.1.3 Mathematical modeling challenges, 83

4.1.4 Simulation objectives, 84

4.1.5 Math modeling overview, 85

4.2 Mathematical formulation and numerical solution, 86

4.2.1 Immiscible flow equations, 86

4.2.1.1 Finite differences, explicit versus implicit, 88

4.2.1.2 Formation tester "ADI" implementation, 89

4.2.1.3 Mudcake growth, formation coupling, supercharging, 90

4.2.1.4 Pumpout model for single-probe pad nozzles, 93

4.2.1.5 Dual-probe and dual packer surface logic, 94

4.3 Miscible flow formulation, 96

4.3.1 Miscible flow numerical solution, 97

4.4 Three-dimensional flow extensions, 97

4.5 Computational implementation for azimuthal effects, 98

4.6 Modeling long-time invasion and mudcake scrape-off, 99

4.7 Software features, 99

4.8 Calculated miscible flow pressures and concentrations, 100

Validation Examples

4.8.1 Example 1. Single probe, infinite anisotropic media, 101

4.8.2 Example 2. Single probe, three layer medium, 107

4.8.3 Example 3. Dual probe pumping, three layer medium, 108

4.8.4 Example 4. Straddle packer pumping, 110

Field Examples

4.8.5 Example 5. Formation fluid viscosity imaging, 112

4.8.6 Example 6. Contamination modeling, 113

4.8.7 Example 7. Multi-rate pumping simulation, 113

4.8.8 Example 8. More detailed clean-up application, 114

4.9 Calculated immiscible flow clean-up examples, 116

4.9.1 Example 9. Higher permeability anisotropic formation, 116

4.9.2 Example 10. Pressure transient modeled, 117

4.10 Closing remarks, 118

4.11 References, 119

5.1 Background and objectives, 122

5.1.1 Detailed literature review and history, 122

5.1.2 Recent 1990s developments, 123

5.1.3 Modeling background and basics, 125

5.1.4 New developments, 127

5.2 Detailed pressure transient examples ( *twenty! *) – competing effects of anisotropy, skin, dip and flowline storage, 130

5.3 Software operational details and user interface, 146

5.4 Closing remarks, 156

5.5 Appendix – Mathematical model and numerical implementation, 159

5.5.1 Isotropic spherical flow with storage and no skin, 159

5.5.1.1 Physical and mathematical formulation, 160

5.5.1.2 General dimensionless representation, 160

5.5.1.3 Exact solution using Laplace transforms, 161

5.5.1.4 Constant rate drawdown and buildup, 162

5.5.1.5 Practical implications, 163

5.5.1.6 Surface plot of exact solution, 164

5.5.1.7 Early time series solution, 165

5.5.1.8 Large time asymptotic solution, 165

5.5.1.9 Arbitrary volume flowrate, 166

5.5.2 Anisotropic ellipsoidal flow with storage and no skin, 168

5.5.2.1 Defining effective permeability, 168

5.5.2.2 Complete physical and mathematical formulation, 168

5.5.2.3 Simplifying the differential equation, 169

5.5.2.4 Total velocity through ellipsoidal surfaces, 170

5.5.2.5 Pressure formulation, 172

5.5.2.6 Volume flowrate formulation, 172

5.5.3 Isotropic spherical flow with storage and skin, 175

5.5.3.1 Mathematical model of skin from first principles, 176

5.5.3.2 Skin extensions to "storage only" pressure model, 176

5.5.3.3 Exact pressure transient solutions via Laplace transforms, 178

5.5.3.4 Explicit and exact time domain solutions, 179

5.5.3.5 More general pressure results away from the source probe, 179

5.5.4 Anisotropic ellipsoidal flow with storage and skin, 180

5.5.4.1 Skin model in multi-dimensional anisotropic flow, 180

5.5.4.2 Implicit assumptions related to formationpermeability, 181

5.5.4.3 General boundary value problem formulation, 183

5.5.5 Numerical issues and algorithm refinements, 184

5.5.5.1 Complex complementary error function, 184

5.5.5.2 Real function methods for FTWD analysis, 187

5.5.5.3 Skin model and mathematical anomalies, 190

5.5.5.4 Multi-rate drawdown schedules, 191

5.5.6 References, 194

6.1 Six new inverse methods summarized, 196

6.2 Existing inverse methods and limitations, 198

6.3 Permeability anisotropy theory without skin (ellipsoidal source), 201

6.3.1 Steady pressure drop formulas at arbitrary dip, 201

6.3.2 Isotropic permeability prediction, 202

6.3.3 Anisotropic media, vertical wells, zero dip angle, 202

6.3.4 Anisotropic media with arbitrary dip angle, 203

6.3.5 Nearly vertical wells, small dip angle approximation, 205

6.3.6 Horizontal wells, large dip angle approximation, 205

6.3.7 General dip angle, k_{h} equation, exact algebraic solution, 205

6.3.8 General dip angles, k_{v}/k_{h} equation, 206

6.3.9 Dip angle and algebraic structure, 207

6.3.10 Azimuthally and generally offset probes, 207

6.3.11 Complementary early time analysis, 208

6.4 Zero skin permeability prediction examples (ellipsoidal source), 209

6.5 Permeability anisotropy with skin effects (ellipsoidal source), 217

6.5.1 Exact steady-state pressure and skin solutions, 217

6.5.2 Exact early time pressure and skin relationship, 218

6.5.3 Numerical algorithm for non-zero skin problems, 219

6.6 Non-zero skin permeability prediction examples (ellipsoidal source), 219

6.7 Low permeability pulse interference testing (ellipsoidal source) – getting results with short test times, 225

6.7.1 Faster pressure testing in the field, 226

6.7.2 Non-zero skin permeability prediction examples, 227

6.7.3 Pulse interaction method for single-probe tools, 232

6.7.4 Dual-probe pulse interaction methods, 232

6.7.5 Zero skin permeability prediction examples, 232

6.8 Fully three-dimensional inverse methods, 238

6.9 Software interface for steady inverse methods (ellipsoidal source), 245

6.9.1 Pumping modes and error checking, 245

6.9.2 Zero-skin and non-zero skin modes, 247

6.9.3 Zero-skin mode, 247

6.9.4 Non-zero skin model, 249

6.10 Formation testing while drilling (FTWD), 251

6.10.1 Pressure transient drawdown-buildup approach, 251

6.10.2 Interpretation in low mobility, high flowline storage environments, 251

6.10.3 Multiple pretests, modeling and interpretation, 253

6.10.4 Reverse flow injection processes, 257

6.10.4.1 Conventional fluid withdrawal, drawdown-then-buildup, 257

6.10.4.2 Reverse flow injection process, buildup-then-drawdown, 261

6.10.5 Best practices – data acquisition and processing, 266

6.11 Closing remarks, 271

6.12 References, 273

7.1 Pad and dual pad models for horizontal well application, 274

7.1.1 Practical modeling applications, 276

7.1.2 Prior pressure transient models, 279

7.1.3 Specific research and software objectives, 279

7.2 Fundamental ideas in finite difference modeling, 280

7.2.1 Finite differencing in space and time, 281

7.2.2 Explicit schemes, 281

7.2.3 Implicit procedures, 282

7.2.4 Tridiagonal matrixes, 283

7.2.5 Grid generation, modern ideas and methods, 283

7.2.6 Detailed math modeling objectives, 285

7.3 Mathematical formulation and geometric transformations, 286

7.3.1 Pressure partial differential equations, 286

7.3.1.1 Geometric domain transformations, 286

7.3.1.2 Alternating-direction-implicit method, 288

7.3.2 Velocity and volume flow rate boundary conditions, 293

7.3.2.1 General velocity transforms, 293

7.3.2.2 Zero flow at solid borehole surfaces, 294

7.3.2.3 Zero flow at horizontal barriers, 294

7.3.2.4 Pad-nozzle boundary conditions, 295

7.3.2.5 Straddle packer or dual packer source

boundary conditions, 296

7.3.2.6 Dual-probe pad boundary conditions, 298

7.3.3 Numerical curvilinear grid generation, 299

7.3.3.1 Fundamental grid generation ideas, 299

7.3.3.2 Fast and stable iterative solutions, 302

7.4 Meshing algorithm construction details, 303

7.5 Three-dimensional calculations and validations, 306

7.5.1 Suite 1. Circular well validations, 306

7.5.2 Suite 2. Modeling zero radial flow at sealed borehole surface, 309

7.5.3 Suite 3. Modeling real pumpouts (high permeability), 311

7.5.4 Suite 4. Modeling real pumpouts (low permeability), 315

7.5.5 Suite 5. Modeling real pumpouts (low permeability and flowline storage), 318

7.5.6 Suite 6. Modeling real pumpouts (variable flow rates), 320

7.5.7 Suite 7. Modeling anisotropy with azimuthally displaced sources, 322

7.5.8 Suite 8. Modeling anisotropy with diametrically opposed probes, 327

7.5.9 Suite 9. Reservoir engineering production forecasting, 329

7.5.10 Suite 10. Straddle packer flow modeling, 329

7.6 User interface and extended capabilities, 330

7.6.1 Extended simulation capabilities, 332

7.7 Closing remarks, 335

7.8 References, 336

*Present chapter and FT-06 software module include time integration algorithm for general variable flowrate pumping for liquids in addition to gases.

8.1 Gas reservoir pumping basics and modeling objectives, 338

8.1.1 Single-phase sampling, 338

8.1.2 Pad nozzle versus dual packer usage, 338

8.1.3 General transient flowrate pumping, 339

8.2 Direct and inverse formulations for ellipsoidal source, 340

8.2.1 Governing gas flow equations, 340

8.2.2 Similarity transform, 342

8.3 Ellipsoidal source – exact steady forward and inverse solutions, 343

8.3.1 Exact, steady, forward formulation, 343

8.3.2 Exact, steady, forward solution at source and observation points, 344

8.3.3 Exact, steady, inverse formulation and solutions, 346

8.4 Special analytical results, 347

8.4.1 Liquid flow, check limit, 347

8.4.2 Isothermal gas expansion, all dip angles, 347

8.4.3 Vertical wells, all "m" (thermodynamic) values, 348

8.4.4 Horizontal wells, all "m" (thermodynamic) values, 348

8.5 Direct solver, solution procedure, 349

8.6 Forward model gas calculations, 350

8.7 Second-order schemes, 353

8.8 Inverse solver, solution software, 353

8.9 Inverse gas calculations, 355

8.10 Ellipsoidal source – fully transient numerical solutions for gases and liquids, 358

8.10.1 Transient flow modeling, 359

8.10.2 Finite difference equation, 360

8.10.3 Boundary conditions – modeling flowline storage with and without skin effects, 361

8.10.4 Detailed time integration scheme, 362

8.10.5 Observation probe response, 362

8.10.6 Software interface and example calculations, 363

8.10.7 Source formulation limitations, 368

8.11 Transient source pulse interaction inverse method, 369

8.11.1 Pulse interaction, procedure at nonzero dip angle, 369

8.12 Ring source, layered model for vertical wells, 372

8.12.1 Source model limitations and refinement, 372

8.12.2 Finite difference method, 372

8.12.3 Alternating-direction-implicit integration, 373

8.12.4 Formation tester nozzle as a simple ring source, 375

8.12.5 Pad nozzle pumpout boundary condition, 376

8.12.6 Dual probe and dual packer surface logic, 377

8.12.7 Detailed boundary condition implementation, 377

8.12.8 Example calculations, 378

8.13 Pad nozzle and dual packer sources for horizontal wells, 381

8.14 Application to modern gas reservoir characterization, 383

8.15 References, 383

9.1 Basic phase delay and amplitude attenuation ideas, 385

9.1.1 Isotropic uniform media, 385

9.1.2 Anisotropic homogeneous media, 386

9.2 Layered model formulation, 387

9.2.1 Homogeneous medium, basic mathematical ideas, 387

9.2.2 Boundary value problem for complex pressure, 389

9.2.3 Iterative numerical solution to general formulation, 389

9.2.4 Successive line over relaxation procedure, 390

9.2.5 Advantages of the scheme, 391

9.2.6 Extensions to multiple layers, 391

9.2.7 Extensions to complete formation heterogeneity, 392

9.3 Phase delay software interface, 392

9.3.1 Output file notes, 394

9.3.2 Special user features, 395

9.4 Detailed phase delay results in layered anisotropic media, 396

9.5 Closing remarks – extensions and additional applications, 404

9.5.1 Inverse model in uniform anisotropic media, 404

9.5.2 Inverse model in layered media, 404

9.5.3 Variable gridding, 405

9.5.4 Other physical models, 405

9.6 References, 406

Part III. Consulting Services and Advanced Software

Consulting services and advanced software, 407

Module FT-00, 408

Module FT-01, 410

Module FT-02, 412

Module FT-03, 414

Module FT-04, 418

Module FT-05, 420

Module FT-06, 421

Module FT-07, 423

Module FT-PTA-DDBU, 425

Part IV. Cumulative References, Index and Author Contact

Cumulative References, 426

Index, 431

About the Authors, 439

W.C. Chin, lead author resume, 440

- Honors and awards, 440
- Scientific book publications, 441
- United States patents, 442
- International and domestic patents, 443
- Journal and conference publications, 445