Homogenization of two-phase fluid flow in porous media via volume averaging

Abstract

A technique of local volume averaging is employed to obtain general equations which depict mass and momentum transport of incompressible two-phase flow in porous media. Starting from coupled Navier-Stokes-Cahn-Hilliard equations for incompressible two-phase fluid flow, the averaging is performed without oversimplifying either the porous medium or the fluid mechanical relations. The resulting equations are Darcy's law for two-phase flow with medium parameters which could be evaluated by experiment. The Richards' equation of the mixed form can be deduced from the resulting equations. The differences between the resulting equations and the empirical model of two-phase flow adopted in oil industry are discussed by several numerical examples.

Computational multiscale methods for parabolic wave approximations in heterogeneous media

“…In this section, we present the convergence analysis of the proposed POD-based multiscale method for the parabolic wave equation. Let v CEM ∈ V CEM be the semi-discretized solution which solves the problem (10). Throughout this section, we assume that the coercivity and boundedness of this bilinear form hold.…”

“…Proposition 1 in [39]). Let v CEM ∈ V CEM be the semi-discretized solution which solves the problem (10) and K ij := 1 3K+2 (y j , y i ) H 1 be the correlation matrix and λ k be the corresponding eigenvalues sorted ascending. For any integer with 0 < ≤ N , the following error formula holds:…”

“…We are now able to analyze the error. Assume that {v n k } ⊂ V POD solves (11), {U k } K k=0 ⊂ V POD solves (19), and the function v CEM ∈ V CEM solves (10). We decompose the error into three parts:…”

“…Therefore, many methods which solve the multiscale problems on coarser mesh have been proposed. These include homogenization-based approaches [8,9,10,29,40,41,47], multiscale finite element methods [31,35,37,38], generalized multiscale finite element methods (GMsFEM) [14,15,18,19,21,25,30], constraint energy minimizing GMsFEM (CEM-GMsFEM) [16,17], nonlocal multi-continua (NLMC) approaches [20], metric-based upscaling [45], heterogeneous multiscale method [1,24], localized orthogonal decomposition (LOD) [33,42], equation free approaches [46,48,49], computational continua [26,27,28], hierarchical multiscale method [7,34,50], and so on. Some of these approaches, such as homogenization-based approaches, are designed for problems with scale separation.…”

Nonlocal multicontinua with Representative Volume Elements. Bridging separable and non-separable scales

“…These methods typically explore the entire coarse block and some nearby regions to biuld effective properties. Some original methods in this direction include Multiscale Finite Element Method (MsFEM) [25], Generalized Multiscale Finite Element Method (GMsFEM) [16,21,13,8,14,7], Multiscale Finite Volume [22,26,27], Constraint Energy Minimizing GMsFEM (CEM-GMsFEM) [9], Nonlocal Multicontinua Approaches (NLMC) [10], metric-based upscaling [34], Heterogeneous Multiscale Method [15,1], LOD [23], equation free approaches [35,38,37], computational continua [19,18,17], hierarchical multiscale method [24,3,39], homogenization-based approaches [4,30,20,29,6,5,36], and so on. In this paper, we focus on high-contrast problems, where approaches CEM-GMsFEM [12,9] and NLMC [10] are used to achieve an optimal convergence independent of contrast and scales.…”