Homogenization of two-phase immiscible flows in a one-dimensional porous medium

Abstract

We consider the hyperbolic-elliptic coupled nonlinear system describing two-phase immiscible flows, with neglected capillary pressure, through a one-dimensional porous medium. We prove the existence and uniqueness of an entropy solution for data corresponding to the solution with discontinuities in saturations. Then we study the homogenization of this system when the permeability and porosity are rapidly oscillating. We get the stability of our coupled system, due to the very special underlying structure of this system in one dimension. Finally we discuss the behavior of the system with oscillatory initial saturation. This result also contains the homogenization of two-component miscible flow in a one-dimensional porous medium, without any dispersion term, as a special case.