Multiscale Modeling of Fluid Transport in Heterogeneous Materials Using Discrete Boltzmann Methods

The lattice Boltzmann method is a promising approach for modeling single and multicomponent fluid flow in complex geometries like porous materials. Here, we review some of our previous work and discuss some recent developments concerning fluid flow in multiple pore size materials. After presenting some simple test cases to validate the model, results from large scale simulations of single and multicomponent fluid flow through digitized Fountaine sandstone, generated by X-Ray microtomography, are given. Reasonably good agreement was found when compared to experimentally determined values of permeability for similar rocks. Finally, modification of the lattice Boltzmann equations, to describe flow in microporous materials, is described. The potential for modeling flows in other microstructures of interest to concrete technology will be discussed. INTRODUCTION Diffusive and moisture transport in porous materials like ceramics, concrete, soils, and rocks plays an important role in many environmental and technological processes [1]. For example, the service life and durability of concrete can depend on the rate of ingress of chloride ions while the diffusion of carbon dioxide controls the rate of carbonation of the cementitious matrix. Further, such processes depend on the degree of saturation of the porous medium. The detailed simulation of such transport phenomena, subject to varying environmental conditions or saturation, is a great challenge because of the difficulty of modeling fluid flow in random pore geometries and the proper accounting of the interfacial boundary conditions. In this paper, we will review [2] some recent advances in the modeling of fluid flow in complex geometries using the discrete Boltzmann methods. Discrete or lattice Boltzmann methods (LB) have emerged as a powerful technique for the computational modeling of a wide variety of complex fluid flow problems including single and multiphase flow in complex geometries. These methods naturally accommodate a variety of boundary conditions such as the pressure drop across the interface between two fluids and wetting effects at a fluid-solid interface. Since the LB method can be derived from the Boltzmann equation, its physical underpinnings can be understood from a fundamental point of view. Indeed, discrete Boltzmann methods serve as an ideal mesoscopic approach that bridges microscopic phenomena with the continuum macroscopic equations. Further, it can be directly implemented as a numerical method to model the time evolution of such systems. Finally, the LB method generally needs nearest neighbor information, at most, so is well suited to take advantage of parallel computers. An outline of the paper goes as follows. After a brief review of the theory of the LB method, results are presented to validate predictions of fluid flow through a few simple pore geometries. Large scale simulations of fluid flow through a Fontainebleau sandstone microstructure, generated by X-ray microtomography, will then be presented. Single phase flow calculations were carried out on systems containing 5103 computational elements. We also calculate relative permeability curves as a function of fluid saturation and driving force. The next section describes solution of the Brinkman equation using a lattice Boltzmann based approach. Finally, a comparison of the performance of such codes on different computational platforms is presented. A LATTICE BOLTZMANN MODEL OF MULTICOMPONENT FLUIDS The LB method of modeling fluid dynamics is actually a family [3] of models with varying degrees of faithfulness to the properties of real liquids. These methods are currently in a state of evolution as the models become better understood and are corrected for various deficiencies. The approach of LB is to consider a typical volume element of fluid to be composed of a collection of particles that are represented in terms of a particle velocity distribution function at each point in space. The particle velocity distribution, na(x, t), is the number density of particles at node x, time t, with velocity, ea, where (a = 1, ..., b) indicates the velocity direction and superscript i labels the fluid component. The time is counted in discrete time steps, and the fluid particles can collide with each other as they move under applied forces. For this paper we use the D3Q19 (3 Dimensional lattice with b = 19) lattice[4,5]. The microscopic velocity, ea, equals all permutations of (±1,±1, 0) for 1 ≤ a ≤ 12, (±1, 0, 0) for 13 ≤ a ≤ 18, and (0, 0, 0) for a = 19. The units of ea are the lattice constant divided by the time step. Macroscopic quantities such as the density, n(x, t), and the fluid velocity, u, of each fluid component, i, are obtained by taking suitable moment sums of na(x, t). Note that while the velocity distribution function is defined only over a discrete set of velocities, the actual macroscopic velocity field of the fluid is continuous. The time evolution of the particle velocity distribution function satisfies the following LB equation: na(x + ea, t + 1) − na(x, t) = Ωa(x, t) − g a, (1) where Ωa is the collision operator representing the rate of change of the particle distribution due to collisions and g a is body forcing term. The collision operator is greatly simplified by use of the single time relaxation approximation[6,7] Ωa(x, t) = − 1 τ i [ na(x, t) − n a (x, t) ]