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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Fluid Mechanics Based on Continuum Assumption

Abstract
The study of flow problem is very important for many applications and the approach varies across disciplines. The widely used theory is based on the continuum assumption by using macroscopic quantities (e.g., flow velocity, density, pressure and temperature) that can be conveniently measured and are close to our ordinary concepts about the flow problem. Its derivation could be classified into the Eulerian and Lagrangian descriptions, and the later can be presented in a neat mathematical form with clear implementations of the basic physical laws (e.g., mass, momentum and energy conservations) to a closed fluid parcel composed of fixed material particles, as introduced in this chapter. The obtained system of governing equations will be closed by introducing the constitutive equations, which are valid for ordinary flow problems but become inaccurate as the characteristic scale of flow problem or the density of gas media decreases, where the kinetic theory based on a statistical description at the molecular scale can be used as discussed at the end of this chapter. Additionally, the governing equations will be changed in non-inertial reference frame and the component form in orthogonal curvilinear coordinate will be different from that in the Cartesian coordinate, which are also discussed.
Jun Li

Chapter 2. Boltzmann Equation

Abstract
When the mean free path of gas molecules becomes comparable to the characteristic flow length (i.e., Knudsen number is not negligible), the traditional Navier–Stokes–Fourier (N–S–F) equations fail and the kinetic theory could be used to study the gas flows. This will occur due to either large mean free path at low pressure or small characteristic length at micro/nano-scales. The distribution function and Boltzmann equation are introduced in this chapter, as the foundation of the kinetic theory. Different intermolecular collision models are discussed to show the selection of kinetic molecular models according to the macroscopic transport coefficients, which is based on the result of Chapman–Enskog expansion. The derivation of N–S–F-like equation by computing moments of the Boltzmann equation is given to show their correlation at the continuum regime. The entropy and second law of thermodynamics can be better understood by using the H-theorem of the Boltzmann equation. The definitions of mean free path and intermolecular collision frequency can be obtained as properties of the Maxwell equilibrium distribution. Model equation is also introduced as a good approximation to the Boltzmann equation at low speed. At the end, different boundary conditions are discussed together with their implementation algorithms in Monte Carlo molecular simulations.
Jun Li

Chapter 3. Simulation Methods for Rarefied Gas Flows

Abstract
Low-speed gas flows of large Knudsen (Kn) number are characteristic of micro/nano-electro-mechanical systems (MEMS/NEMS), vacuum system and tight/shale gas reservoirs, and can be described by the kinetic theory. The traditional direct simulation Monte Carlo (DSMC) method is the standard technique for modeling gas flows at high Kn but computationally expensive and practically unfordable at low speed due to stochastic noise. The discrete velocity method (DVM) is another traditional method and deterministically solves the Boltzmann kinetic equation or its simplified model equations. It can produce noise-free results that are accurate when fine grid is used in the discretization of high-dimensional phase space, which generally requires high computational cost in terms of memory usage and CPU time. The direct simulation BGK (DSBGK) method was proposed recently to solve the BGK model equation and has been comprehensively validated against the DSMC method and experimental data in several benchmark problems over a wide range of Kn. Although it is also a particle-based approach like the DSMC method, its stochastic noise is very low and independent of the flow speed (or Mach number in general), which is in sharp contrast to the DSMC method, where the stochastic noise is inversely proportional to the square of Mach number. The algorithms of this three simulation methods are introduced in this chapter and several benchmark problems are studied to show their numerical performances in terms of accuracy, efficiency, memory usage, as well as robustness associated with algorithm simplicity.
Jun Li

Chapter 4. Multiscale LBM Simulations

Abstract
For flow problems of the continuum regime, the lattice Boltzmann method (LBM) is a good alternative to the traditional CFD solvers based on the N–S-like equations. It is efficient in modeling dynamic problems and very powerful for pore-scale applications, where the simulation of interface dynamics on the real irregular pore surface is challenging, if not impossible, to most of the traditional CFD solvers. We start in this chapter with the basic LBM algorithm to show its correlation with the N–S equation through the Chapman–Enskog expansion. Then, the widely used Shan–Chen model will be introduced to simulate multiphase multicomponent flow systems, having its applications detailed in the subsequent sections. We also present the extension of LBM to the Darcy-scale simulations, where the LBM works as a unified framework for simulations at different scales, i.e., both pore and Darcy scales, and the detailed results are given at the end of this chapter. In the ordinary application of LBM for computing the absolute permeability, we clarify the prevailed confusion interpreted as viscosity-dependent permeability and reveal the underlying rarefaction mechanism that has been commonly oversighted. Additionally, we also discuss the application of large eddy simulation of turbulence in the LBM framework and the same idea can be extended to model non-Newtonian fluids.
Jun Li

Chapter 5. Summary and Outlook

Abstract
It is our hope that this book has provided a concise but self-contained introduction to the theories of describing various flow problems at different scales, and the difference as well as correlation between different theories. We focused on the kinetic theory of using the Boltzmann equation, which is valid for modeling flows of different regimes from continuum to free molecular flow. Then, various fluid behaviors can be modeled by different simulation methods developed based on the same framework of Boltzmann equation. As summarized in Fig. 5.1, four simulation methods are introduced and they are the DSMC method, DSBGK method, DVM and LBM. The algorithm differences among those simulation methods are noticeable and their performance comparisons are detailed in several benchmark problems of the preceding chapters, which provides some guidelines for those interested in using appropriate methods for their particular applications.
Jun Li
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