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
6.7.4 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 В ............................. 427
A.3.3 Format Procedure С ............................. 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
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