Title Details

S. Hossein Mousavizadegan, PhD, is currently on the faculty of marine technology at the Amirkabir University of Technology in Tehran as an assistant professor, specializing in mathematical and numerical modeling of fluid dynamics.

Shabbir Mustafiz, PhD, is a research engineer with the Alberta Research Council in Edmonton, Canada. Shabbir has published over 25 journal articles and has a Ph.D. in Civil Engineering, on the topic of petroleum reservoir simulation, from Dalhousie University and he is the current SPE Scholarship Chair for the Edmonton Section.

Jamal H. Abou-Kassem, PhD, is Professor of Petroleum Engineering at the United Arab Emirates University, where he has taught since 1993. Abou-Kassem is a coauthor of two textbooks on reservoir simulation and an author or coauthor of numerous technical articles in the areas of reservoir simulation and other petroleum and natural gas-related topics.

Foreword xiii

Introduction xv

1. Reservoir Simulation Background 1

1.1 Essence of Reservoir Simulation 1

1.2 Assumptions Behind Various Modeling Approaches 5

1.3 Material Balance Equation 5

1.3.1 Decline Curve 6

1.3.2 Statistical Method 6

1.3.3 Analytical Methods 7

1.3.4 Finite Difference Methods 8

1.3.5 Darcy's Law 11

1.4 Recent Advances in Reservoir Simulation 12

1.4.1 Speed and Accuracy 12

1.4.2 New Fluid Flow Equations 13

1.4.3 Coupled Fluid Flow and Geo-mechanical Stress Model 16

1.4.4 Fluid Flow Modeling Under Thermal Stress 17

1.5 Future Challenges in Reservoir Simulation 18

1.5.1 Experimental Challenges 18

1.5.2 Numerical Challenges 20

1.5.2.1 Theory of Onset and Propagation of Fractures Due to Thermal Stress 20

1.5.2.2 2-D and 3-D Solutions of the Governing Equations 20

1.5.2.3 Viscous Fingering During Miscible Displacement 20

1.5.2.4 Improvement in Remote Sensing and Monitoring Ability 21

1.5.2.5 Improvement in Data Processing Techniques 21

1.5.3 Remote Sensing and Real-time Monitoring 22

1.5.3.1 Monitoring Offshore Structures 23

1.5.3.2 Development of a Dynamic Characterization Tool (Based on Seismic-while-drilling Data) 24

1.5.3.3 Use of 3-D Sonogram 24

1.5.3.4 Virtual Reality (VR) Applications 25

1.5.3.5 Intelligent Reservoir Management 26

1.6 Economic Models Based on Futuristic Energy Pricing Policies 27

1.7 Integrated System of Monitoring, Environmental Impact and Economics 29

2. Reservoir Simulator-input/output 31

2.1 Input and Output Data 32

2.2 Geological and Geophysical Modeling 34

2.3 Reservoir Characterization 37

2.3.1 Representative Elementary Volume, REV 38

2.3.2 Fluid and Rock Properties 41

2.3.2.1 Fluid Properties 42

2.3.2.1.1 Crude Oil Properties 43

2.3.2.1.2 Natural Gas Properties 45

2.3.2.1.3 Water Content Properties 46

2.3.3 Rock Properties 47

2.4 Upscaling 52

2.4.1 Power Law Averaging Method 53

2.4.2 Pressure-solver Method 54

2.4.3 Renormalization Technique 56

2.4.4 Multiphase Flow Upscaling 57

2.5 Pressure/Production data 60

2.5.1 Phase Saturations Distribution 61

2.6 Reservoir Simulator Output 62

2.7 History-matching 65

2.7.1 History-matching Formulation 68

2.7.2 Uncertainty Analysis 71

2.7.2.1 Measurement Uncertainty 71

2.7.2.2 Upscaling Uncertainty 74

2.7.2.3 Model Error 75

2.7.2.4 The Prediction Uncertainty 76

2.8 Real-time Monitoring 77

3. Reservoir Simulators: Problems, Shortcomings, and Some Solution Techniques 83

3.1 Multiple Solutions in Natural Phenomena 85

3.1.1 Knowledge Dimension 88

3.2 Adomian Decomposition 103

3.2.1 Governing Equations 105

3.2.2 Adomian Decomposition of Buckley-Leverett Equation 108

3.2.3 Results and Discussions 110

3.3 Some Remarks on Multiple Solutions 113

4. Mathematical Formulation of Reservoir Simulation Problems 115

4.1 Black Oil Model and Compositional Model 116

4.2 General Purpose Compositional Model 118

4.2.1 Basic Definitions 118

4.2.2 Primary and Secondary Parameters and Model Variables 120

4.2.3 Mass Conservation Equation 123

4.2.4 Energy Balance Equation 126

4.2.5 Volume Balance Equation 132

4.2.6 The Motion Equation in Porous Medium 133

4.2.7 The Compositional System of Equations and Model Variables 138

4.3 Simplification of the General Compositional Model 141

4.3.1 The Black Oil Model 141

4.3.2 The Water Oil Model 143

4.4 Some Examples in Application of the General Compositional Model 146

4.4.1 Isothermal Volatile Oil Reservoir 146

4.4.2 Steam Injection Inside a Dead Oil Reservoir 149

4.4.3 Steam Injection in Presence of Distillation and Solution Gas 150

5 The Compositional Simulator Using the Engineering Approach 155

5.1 Finite Control Volume Method 156

5.1.1 Reservoir Discretization in Rectangular Coordinates 157

5.1.2 Discretization of Governing Equations 158

5.1.2.1 Components Mass Conservation Equation 159

5.1.2.2 Energy Balance Equation 167

5.1.3 Discretization of Motion Equation 170

5.2 Uniform Temperature Reservoir Compositional Flow Equations in a 1-D Domain 172

5.3 Compositional Mass Balance Equation in a Multidimensional Domain 178

5.3.1 Implicit Formulation of Compositional Model in Multi-Dimensional Domain 180

5.3.2 Reduced Equations of Implicit Compositional Model in Multidimensional Domain 183

5.3.3 Well Production and Injection Rate Terms 186

5.3.3.1 Production Wells 186

5.3.3.2 Injection Wells 188

5.3.4 Fictitious Well Rate Terms (Treatment of Boundary Conditions) 189

5.4 Variable Temperature Reservoir Compositional Flow Equations 193

5.4.1 Energy Balance Equation 193

5.4.2 Implicit Formulation of Variable Temperature Reservoir Compositional Flow Equations 197

5.5 Solution Method 201

5.5.1 Solution of Model Equations Using Newton's Iteration 202

5.6 The Effects of Linearization 207

5.6.1 Case I: Single Phase Flow of a Natural Gas 208

5.6.2 Effect of Interpolation Functions and Formulation 214

5.6.3 Effect of Time Interval 215

5.6.4 Effect of Permeability 217

5.6.5 Effect of Number of Gridblocks 217

5.6.6 Spatial and Transient Pressure Distribution Using Different Interpolation Functions 219

5.6.7 CPU Time 222

5.6.8 Case II: An Oil/Water Reservoir 224

6 A Comprehensive Material Balance Equation for Oil Recovery 245

6.1 Background 245

6.2 Permeability Alteration 248

6.3 Porosity Alteration 249

6.4 Pore Volume Change 251

6.5 A Comprehensive MBE with Memory for Cumulative Oil Recovery 252

6.6 Numerical Simulation 255

6.6.1 Effects of Compressibilities on Dimensionless Parameters 257

6.6.2 Comparison of Dimensionless Parameters Based on Compressibility Factor 258

6.6.3 Effects of M on Dimensionless Parameter 259

6.6.4 Effects of Compressibility Factor with M Values 259

6.6.5 Comparison of Models Based on RF 260

6.6.6 Effects of M on MBE 262

6.7 Appendix 6A: Development of an MBE for a Compressible Undersaturated Oil Reservoir 264

6.7.1 Development of a New MBE 265

6.7.2 Conventional MBE 272

6.7.3 Significance of Cepm 274

6.7A Water Drive Mechanism with Water Production 275

6.7.5 Depletion Drive Mechanism with No Water Production 276

7. Modeling Viscous Fingering During Miscible Displacement in a Reservoir 277

7.1 Improvement of the Numerical Scheme 277

7.1.1 The Governing Equation 279

7.1.2 Finite Difference Approximations 281

7.1.2.1 Barakat-Clark FTD Scheme 281

7.1.2.2 DuFort-Frankel Scheme 283

7.1.3 Proposed Barakat-Clark CTD Scheme 284

7.1.3.1 Boundary Conditions 285

7.1.4 Accuracy and Truncation Errors 285

7.1.5 Some Results and Discussion 286

7.1.6 Influence of Boundary Conditions 293

7.2 Application of the New Numerical Scheme to Viscous Fingering 295

7.2.1 Stability Criterion and Onset of Fingering 295

7.2.2 Base Stable Case 296

7.2.3 Base Unstable Case 302

7.2.4 Parametric Study 309

7.2.4.1 Effect of Injection Pressure 309

7.2.4.2 Effect of Overall Porosity 314

7.2.4.3 Effect of Mobility Ratio 317

7.2.4.4 Effect of Longitudinal Dispersion 320

7.2.4.5 Effect of Transverse Dispersion 324

7.2.4.6 Effect of Aspect Ratio 327

7.2.5 Comparison of Numerical Modeling Results with Experimental Results 330

7.2.5.1 Selected Experimental Model 330

7.2.5.2 Physical Model Parameters 331

7.2.5.3 Comparative Study 332

7.2.5.4 Concluding Remarks 336

8. Towards Modeling Knowledge and Sustainable Petroleum Production 339

8.1 Essence of Knowledge, Science, and Emulation 339

8.1.1 Simulation vs. Emulation 340

8.1.2 Importance of the First Premise and Scientific Pathway 342

8.1.3 Mathematical Requirements of Nature Science 344

8.1.4 The Meaningful Addition 348

8.1.5 "Natural" Numbers and the Mathematical Content of Nature 350

8.2 The Knowledge Dimension 354

8.2.1 The Importance of Time as the Fourth Dimension 354

8.2.2 Towards Modeling Truth and Knowledge 362

8.3 Examples of Linearization and Linear Thinking 363

8.4 The Single-Parameter Criterion 365

8.4.1 Science Behind Sustainable Technology 366

8.4.2 A New Computational Method 366

8.4.2.1 The Currently Used Model 366

8.4.2.2 Towards Achieving Multiple Solutions 372

8.5 The Conservation of Mass and Energy 374

8.5.1 The Avalanche Theory 375

8.5.2 Aims of Modeling Natural Phenomena 380

8.5.2 Challenges of Modeling Sustainable Petroleum Operations 382

8.6 The Criterion: The Switch that Determines the Direction at a Bifurcation Point 386

8.6.1 Some Applications of the Criterion 388

8.7 The Need for Multidimensional Study 396

8.8 Assessing the Overall Performance of a Process 399

8.9 Implications of Knowledge-Based Analysis 406

8.9.1 A General Case 407

8.9.2 Impact of Global Warming Analysis 410

8.10 Examples of Knowledge-Based Simulation 413

9. Final Conclusions 421

Appendix A User's Manual for Multi-Purpose Simulator for Field Applications (MPSFFA, Version 1-15) 423

A.l Introduction 423

A.2 The Simulator 423

A.3 Data File Preparation 425

A.3.1 Format Procedure A 426

A.3.2 Format Procedure B 427

A.3.3 Format Procedure C 427

A.3.4 Format Procedure D 427

A.3.5 Format Procedure E 428

A.4 Description of Variables Used in Preparing a Data File 428

A.5 Instructions to Run Simulator and Graphic Post Processor on PC 439

A.6 Limitations Imposed on the Compiled Versions 441

A. 7 Example of a Prepared Data File 442

References 447

Index 463