No abstract

This paper is devoted to the homogenization of a coupled system of diffusionconvection equations in a domain with periodic microstructure, modeling the flow and transport of immiscible compressible, such as water-gas, fluids through porous media. The problem is formulated in terms of a nonlinear parabolic equation for the nonwetting phase pressure and a nonlinear degenerate parabolic diffusion-convection equation for the wetting saturation phase with rapidly oscillating porosity function and absolute permeability tensor. We obtain a nonlinear homogenized problem with effective coefficients which are computed via a cell problem. We rigorously justify this homogenization process for the problem by using two-scale convergence. In order to pass to the limit in nonlinear terms, we also obtain compactness results which are nontrivial due to the degeneracy of the system.

A mathematically rigorous method of homogenization is presented and used to analyze the equivalent behavior of transient flow of two incompressible fluids through heterogeneous media. Asymptotic e~pansions and H-convergence lead to the definition of a global or effective model of an equivalent homogeneous reservoir. Numerical computations to obtain the homogenized coefficients of the entire reservoir have been carried out via a finite element method. Numerical experiments involving the simulation of incompressible two-phase flow have been performed for each heterogeneous medium and for the homogenized medium as well as for other averaging methods. The results of the simulations are compared in terms of the transient saturation contours, production curves, and pressure distributions. Results obtained from the simulations with the homogenization method presented show good agreement with the heterogeneous simulations.

This article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L ∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions.