Numerical simulation and homogenization of two-phase flow in heterogeneous porous media

Abstract

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.

JHomogenizer: a computational tool for upscaling permeability for flow in heterogeneous porous media

“…The results for the various homogenization methods for the 10:1 ratio are given in table 2. These results and the results for the other ratios can be found in [2]. It is not practical to include all the results here.…”

“…Each section describes results from a single test problem along with the results from homogenization methods. The first two sections contain standard periodic test problems using a symmetric cell and the inverted-L cell (see [2]) and are included for completeness. The rest of the sections give results for more realistic problems from porous media flow and one example of a heat diffusion problem wherein results related to optimal design for an optimized heating element for a microarray.…”

Taking into account general boundary conditions in upscaling absolute permeability

“…We will look for the tensor H, which minimizes the discrepancy between the energy dissipation in problems (1) and (2). This is consistent with classical methods considered in [6,11,27], where the energy dissipated in the upscaled field is equal to that dissipated in the original one.…”

“…8 1 i m, we assume that f i ¼ 0 in problems (1) and (2). Then the second derivative of I have the following form 8 1 ; ; ; d:…”

“…We give now some details about the functional framework we adopt to solve problems (1) and (2). Very often, the heterogeneous permeability field KðxÞ is given on a fine grid using geostatistical tools.…”