Brahim Amaziane

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

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 expansions 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.

Convergence of finite volume schemes for a degenerate convection–diffusion equation arising in flow in porous media

Abstract This paper develops discretizations using the finite volume method for a nonlinear, degenerate, convection–diffusion equation in multiple dimensions on unstructured grids. We will derive three families of numerical schemes. They are classified as explicit, implicit, and semi-implicit. A Godunov scheme is used for the convection term. It is shown that these finite volume schemes (FVS) satisfy a discrete maximum principle. We prove the convergence of these FVS. This is done by means of a priori estimates in L ∞ and weak BV estimates under appropriate CFL conditions. Numerical results for oil recovery simulation are presented.

Homogenization of Immiscible Compressible Two-Phase Flow in Porous Media: Application to Gas Migration in a Nuclear Waste Repository

This paper is devoted to the homogenization of a coupled system of diffusion-convection 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.

Homogenization of a degenerate triple porosity model with thin fissures

We consider the problem of modelling the flow of a slightly compressible fluid in a periodic fractured medium assuming that the fissures are thin with respect to the block size. As a starting point we used a formulation applied to a system comprising a fractured porous medium made of blocks and fractures separated by a thin layer which is considered as an interface. The inter-relationship between these three characteristics comprise the triple porosity model. The microscopic model consists of the usual equation describing Darcy flow with the permeability being highly discontinuous. Over the matrix domain, the permeability is scaled by

Equivalent Permeability and Simulation of Two-Phase Flow in Heterogeneous Porous Media

(\varepsilon \delta)^2

On convergence of finite volume schemes for one-dimensional two-phase flow in porous media

, where

HOMOGENIZATION OF A SINGLE PHASE FLOW THROUGH A POROUS MEDIUM IN A THIN LAYER

\varepsilon

Effective Macrodiffusion in Solute Transport through Heterogeneous Porous Media

is the size of a typical porous block, with

Global behavior of compressible three-phase flow in heterogeneous porous media

\delta

On the homogenization of some double-porosity models with periodic thin structures

representing the relative size of the fracture. We then consider a model with Robin type transmission conditions: a jump of the density across the interface block-fracture is taken into account and proportional to the flux by the mean of a function