MULTISCALE AND MULTIPHYSICS FLOW SIMULATIONS OF USING THE BOLTZMANN EQUATION : applications to ... porous media and mems

This book provides a comprehensive introduction to the kinetic theory for describing flow problems from molecular scale, hydrodynamic scale, to Darcy scale. The author presents various numerical algorithms to solve the same Boltzmann-like equation for different applications of different scales, in which the dominant transport mechanisms may differ. This book presents a concise introduction to the Boltzmann equation of the kinetic theory, based on which different simulation methods that were independently developed for solving problems of different fields can be naturally related to each other. Then, the advantages and disadvantages of different methods will be discussed with reference to each other. It mainly covers four advanced simulation methods based on the Boltzmann equation (i.e., direct simulation Monte Carlo method, direct simulation BGK method, discrete velocity method, and lattice Boltzmann method) and their applications with detailed results. In particular, many simulations are included to demonstrate the applications for both conventional and unconventional reservoirs. With the development of high-resolution CT and high-performance computing facilities, the study of digital rock physics is becoming increasingly important for understanding the mechanisms of enhanced oil and gas recovery. The advanced methods presented here have broad applications in petroleum engineering as well as mechanical engineering , making them of interest to researchers, professionals, and graduate students alike. At the same time, instructors can use the codes at the end of the book to help their students implement the advanced technology in solving real industrial problems. Preface 6 Contents 7 Acronyms 9 1 Fluid Mechanics Based on Continuum Assumption 11 1.1 Continuum Assumption 11 1.2 Macroscopic Descriptions of Flow Phenomenon 13 1.3 Derivations of Macroscopic Governing Equations 15 1.3.1 Force and Heat Interactions Between Fluid Parcels 15 1.3.2 Continuity Equation 19 1.3.3 Momentum Equation 20 1.3.4 Energy Equation 21 1.3.5 Entropy Equation 22 1.3.6 Convection-Diffusion Equation 23 1.4 Transformation to Arbitrary Non-inertial Reference Frame 24 1.5 Component Form in Arbitrary Orthogonal Curvilinear Coordinate 28 1.6 Limitations of Continuum Assumption 31 References 33 2 Boltzmann Equation 34 2.1 Distribution Function in the Phase Space 34 2.2 Intermolecular Collision Dynamics 35 2.3 Molecular Models for Transport Coefficients 42 2.4 Derivation of the Boltzmann Equation 47 2.5 Calculations of Macroscopic Properties 50 2.6 Moment Equation 54 2.7 H-Theorem 56 2.8 Properties at Equilibrium State 59 2.9 Simplification to Model Equation 62 2.10 Boundary Conditions 65 2.10.1 Specular Reflection Model 67 2.10.2 Diffuse Reflection Model 68 2.10.3 CLL Reflection Model 70 References 73 3 Simulation Methods for Rarefied Gas Flows 75 3.1 Direct Simulation Monte Carlo Method 75 3.2 Direct Simulation BGK Method 80 3.2.1 Initialization Process 82 3.2.2 Molecular Motion and Intermolecular Collision 82 3.2.3 External Body Force 88 3.2.4 Boundary Conditions 88 3.2.5 Summary of the DSBGK Algorithm 91 3.2.6 Extension to Gas Mixtures 92 3.3 Discrete Velocity Method 95 3.4 Simulation Results 98 3.4.1 Couette Flows 98 3.4.2 Lid-Driven Cavity Flows 100 3.4.3 Thermal Transpiration Through Micro-channel 108 3.4.4 Apparent Permeability of Shale Gas 112 3.4.5 Scaling Law of Gas Permeability in the Slip Regime 121 References 124 4 Multiscale LBM Simulations 126 4.1 Basic Algorithm 126 4.2 Chapman–Enskog Expansion 129 4.3 Shan–Chen Model for Multiphase Multicomponent Systems 133 4.3.1 Two-Phase Flows of Single Component 135 4.3.2 Flows of Two Immiscible Components 138 4.4 Upscaled Lattice Boltzmann Method 139 4.5 Simulation Results 141 4.5.1 Large Eddy Simulation of Turbulence 141 4.5.2 Intrinsic Permeability 145 4.5.3 Buoyant Flow 149 4.5.4 Two-Phase Flow Inside Real Digital Rock 155 4.5.5 Upscaled Simulations 158 4.6 Introduction to Fortran Code 167 References 168 5 Summary and Outlook 170 Front Matter ....Pages i-xii Fluid Mechanics Based on Continuum Assumption (Jun Li)....Pages 1-23 Boltzmann Equation (Jun Li)....Pages 25-65 Simulation Methods for Rarefied Gas Flows (Jun Li)....Pages 67-117 Multiscale LBM Simulations (Jun Li)....Pages 119-162 Summary and Outlook (Jun Li)....Pages 163-164