Advanced Petroleum Reservoir Simulation

M. R. Islam is Professor of Petroleum Engineering at the Civil and Resource Engineering Department of Dalhousie University, Canada. He has over 700 publications to his credit, including 6 books. He is on the editorial boards of several scholarly journals, and, in addition to his teaching duties, he is also director of Emertec Research and Development Ltd. and has been on the boards of a number of companies in North America and overseas./p>Dr. S. Hossein Mousavizadegan 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.Dr. Shabbir Mustafiz is a research engineer with the Alberta Research Council in Edmonton, Canada. Shabbir has published over 25 journal papers 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 is Professor of Petroleum Engineering at the UAE U. in the United Arab Emirates, 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.

• Preface xv1 Introduction 11.1 Summary 11.2 Opening Remarks 21.3 The Need for a Knowledge-Based Approach 21.4 Summary of Chapters 52 Reservoir Simulation Background 72.1 Essence of Reservoir Simulation 82.2 Assumptions Behind Various Modeling Approaches 102.2.1 Material Balance Equation 112.2.2 Decline Curve 122.2.3 Statistical Method 132.2.4 Analytical Methods 152.2.5 Finite-Difference Methods 162.2.6 Darcy’s Law 192.3 Recent Advances in Reservoir Simulation 192.3.1 Speed and Accuracy 192.3.2 New Fluid-Flow Equations 212.3.3 Coupled Fluid Flow and Geo-Mechanical Stress Model 262.3.4 Fluid-Flow Modeling Under Thermal Stress 292.4 Memory Models 312.4.1 Thermal Hysteresis 322.4.2 Mathematical and Numerical Models 322.5 Future Challenges in Reservoir Simulation 332.5.1 Experimental Challenges 332.5.2 Numerical Challenges 352.5.2.1 Theory of Onset and Propagationof Fractures due to Thermal Stress 352.5.2.2 Viscous Fingering during Miscible Displacement 363 Reservoir Simulator-Input/Output 393.1 Input and Output Data 403.2 Geological and Geophysical Modeling 423.3 Reservoir Characterization 453.3.1 Representative Elementary Volume, REV 463.3.2 Fluid and Rock Properties 493.3.2.1 Fluid Properties 493.3.3 Rock Properties 543.4 Upscaling 583.4.1 Power Law Averaging Method 593.4.2 Pressure-Solver Method 603.4.3 Renormalization Technique 623.4.4 Multiphase Flow Upscaling 633.5 Pressure/Production Data 653.6 Phase Saturations Distribution 663.7 Reservoir Simulator Output 683.8 History Matching 703.8.1 History-Matching Formulation 723.8.2 Uncertainty Analysis 753.8.2.1 Measurement Uncertainty 763.8.2.2 Upscaling Uncertainty 783.8.2.3 Model Error 793.8.2.4 The Prediction Uncertainty 803.9 Real-Time Monitoring 814 Reservoir Simulators: Problems, Shortcomings, and Some Solution Techniques 854.1 Multiple Solutions in Natural Phenomena 874.1.1 Knowledge Dimension 904.2 Adomian Decomposition 1044.2.1 Governing Equations 1064.2.2 Adomian Decomposition of Buckley-Leverett Equation 1084.2.3 Results and Discussions 1114.3 Some Remarks on Multiple Solutions 1145 Mathematical Formulation of Reservoir Simulation Problems 1175.1 Black Oil Model and Compositional Model 1195.2 General Purpose Compositional Model 1205.2.1 Basic Definitions 1205.2.2 Primary and Secondary Parameters and Model Variables 1225.2.3 Mass Conservation Equation 1255.2.4 Energy Balance Equation 1285.2.5 Volume Balance Equation 1335.2.6 The Motion Equation in Porous Medium 1345.2.7 The Compositional System of Equations and Model Variables 1395.3 Simplification of the General Compositional Model 1415.3.1 The Black Oil Model 1415.3.2 The Water Oil Model 1435.4 Some Examples in Application of the General Compositional Model 1465.4.1 Isothermal Volatile Oil Reservoir 1465.4.2 Steam Injection Inside a Dead Oil Reservoir 1485.4.3 Steam Injection in Presence of Distillation and Solution Gas 1506 The Compositional Simulator Using Engineering Approach 1556.1 Finite Control Volume Method 1566.1.1 Reservoir Discretization in Rectangular Coordinates 1576.1.2 Discretization of Governing Equations 1586.1.2.1 Components Mass Conservation Equation 1586.1.2.2 Energy Balance Equation 1666.1.3 Discretization of Motion Equation 1686.2 Uniform Temperature Reservoir Compositional Flow Equations in a 1-D Domain 1706.3 Compositional Mass Balance Equation in a Multidimensional Domain 1756.3.1 Implicit Formulation of Compositional Model in Multidimensional Domain 1786.3.2 Reduced Equations of Implicit Compositional Model in Multidimensional Domain 1806.3.3 Well Production and Injection Rate Terms 1836.3.3.1 Production Wells 1836.3.3.2 Injection Wells 1856.3.4 Fictitious Well Rate Terms (Treatment of Boundary Conditions) 1866.4 Variable Temperature Reservoir Compositional Flow Equations 1906.4.1 Energy Balance Equation 1906.4.2 Implicit Formulation of Variable Temperature Reservoir Compositional Flow Equations 1946.5 Solution Method 1976.5.1 Solution of Model Equations Using Newton’s Iteration 1986.6 The Effects of Linearization 2036.6.1 Case 1: Single Phase Flow of a Natural Gas 2036.6.2 Effect of Interpolation Functions and Formulation 2106.6.3 Effect of Time Interval 2106.6.4 Effect of Permeability 2126.6.5 Effect of Number of Gridblocks 2146.6.6 Spatial and Transient Pressure Distribution Using Different Interpolation Functions 2146.6.7 CPU Time 2186.6.8 Case 2: An Oil/water Reservoir 2207 Development of a New Material Balance Equation for Oil Recovery 2397.1 Summary 2397.2 Introduction 2417.3 Mathematical Model Development 2437.3.1 Permeability Alteration 2437.3 Porosity Alteration 2447.4 Pore Volume Change 2467.4.1 A Comprehensive MBE with Memory for Cumulative Oil Recovery 2477.5 Numerical Simulation 2507.5.1 Effects of Compressibilities on Dimensionless Parameters 2517.4.2 Comparison of Dimensionless Parameters Based on Compressibility Factor 2527.4.3 Effects of M on Dimensionless Parameter 2537.4.4 Effects of Compressibility Factor with M Values 2557.4.5 Comparison of Models Based on RF 2557.4.6 Effects of M on MBE 2577.5 Conclusions 258Appendix Chapter 7: Development of an MBE for a Compressible Undersaturated Oil Reservoir 2598 State-of-the-art on Memory Formalism for Porous Media Applications 2718.1 Summary 2718.2 Introduction 2728.3 Historical Development of Memory Concept 2738.3.1 Constitutive Equations 2748.3.2 Application of Memory in Diffusion in Porous Media 2748.3.3 Definition of Memory 2778.4 State-of-the-art Memory-Based Models 2778.5 Basset Force: A History Term 2848.6 Anomalous Diffusion: A memory Application 2878.6.1 Fractional Order Transport Equations and Numerical Schemes 2888.7 Future Trends 2978.8 Conclusion 2989 Modeling Viscous Fingering During Miscible Displacement in a Reservoir 3019.1 Improvement of the Numerical Scheme 3029.1.1 The Governing Equation 3039.1.2 Finite Difference Approximations 3059.1.2.1 Barakat-Clark FTD Scheme 3059.1.2.2 DuFort-Frankel Scheme 3079.1.3 Proposed Barakat-Clark CTD Scheme 3079.1.4 Accuracy and Truncation Errors 3099.1.5 Some Results and Discussion 3099.1.6 Influence of Boundary Conditions 3169.2 Application of the New Numerical Scheme to Viscous Fingering 3179.2.1 Stability Criterion and Onset of Fingering 3189.2.2 Base Stable Case 3189.2.3 Base Unstable Case 3249.2.4 Parametric Study 3309.2.4.1 Effect of Injection Pressure 3319.2.4.2 Effect of Overall Porosity 3359.2.4.3 Effect of Mobility Ratio 3369.2.4.4 Effect of Longitudinal Dispersion 3419.2.4.5 Effect of Transverse Dispersion 3439.2.4.6 Effect of Aspect Ratio 3479.2.5 Comparison of Numerical Modeling Results with Experimental Results 3509.2.5.1 Selected Experimental Model 3509.2.5.2 Physical Model Parameters 3509.2.5.3 Comparative Study 3519.2.5.4 Concluding Remarks 35510 An Implicit Finite-Difference Approximation of Memory-Based Flow Equation in Porous Media 35910.1 Summary 35910.2 Introduction 36010.3 Background 36110.4 Theoretical Development 36410.4.1 Mass Conservation 36510.4.2 Composite Variable, η 36610.4.3 Implicit Formulation 36710.6 Numerical Simulation 36910.7 Results and Discussion 37010.8 Conclusion 38111 Towards Modeling Knowledge and Sustainable Petroleum Production 38311.1 Essence of Knowledge, Science, and Emulation 38411.1.1 Simulation vs. Emulation 38411.1.2 Importance of the First Premise and Scientific Pathway 38611.1.3 Mathematical Requirements of Nature Science 38811.1.4 The Meaningful Addition 39211.1.5 “Natural” Numbers and the Mathematical Content of Nature 39411.2 The Knowledge Dimension 39711.2.1 The Importance of Time as the Fourth Dimension 39811.3 Aphenomenal Theories of Modern Era 40011.3.1 Examples of Linearization and Linear Thinking 40811.3.2 The Knowledge-Based Cognition Process 40911.4 Towards Modeling Truth and Knowledge 41211.5 The Single-Parameter Criterion 41311.5.1 Science Behind Sustainable Technology 41311.5.2 A New Computational Method 41511.5.3 Towards Achieving Multiple Solutions 42011.6 The Conservation of Mass and Energy 42211.6.1 The Avalanche Theory 42311.6.2 Aims of Modeling Natural Phenomena 42811.6.3 Challenges of Modeling Sustainable Petroleum Operations 43011.6.4 The Criterion: The Switch that Determines the Direction at a Bifurcation Point 43311.6.4.1 Some Applications of the Criterion 43611.7 The Need for Multidimensional Study 44211.8 Assessing the Overall Performance of a Process 44511.9 Implications of Knowledge-Based Analysis 45211.9.1 A General Case 45211.9.2 Impact of Global Warming Analysis 45511.9.3 Examples of Knowledge-based Simulation 45812 Reservoir Simulation of Unconventional Reservoirs 46512.1 Introduction 46512.2 Material Balance Equations 46612.3 New Fluid Flow Equations 47612.4 Coupled Fluid Flow and Geo-mechanical Stress Model 47812.5 Fluid Flow Modeling under Thermal Stress 48012.6 Challenges of Modeling Unconventional Gas Reservoirs 48112.7 Comprehensive Modeling 48912.7.1 Governing Equations 48912.7.2 Darcy’s Model 49012.7.3 Forchheimer’s Model 49112.7.4 Modified Brinkman’s Model 49412.7.5 The Comprehensive Model 49613 Final Conclusions 501References and Bibliography 505Appendix A 545Index 569