Homogenization of compressible two-phase two-component flow in porous media

Abstract

The paper is devoted to the homogenization of immiscible compressible two-phase two-component flow in heterogeneous porous media. We consider liquid and gas phases, two-component (water and hydrogen) flow in a porous reservoir with periodic microstructure, modeling the hydrogen migration through engineered and geological barriers for a deep repository for radioactive waste. Phase exchange, capillary effects included by the Darcy–Muskat law and Fickian diffusion are taken into account. The hydrogen in the gas phase is supposed compressible and could be dissolved into the water obeying the Henry law. The flow is then described by the conservation of the mass for each component. The microscopic model is written in terms of the phase formulation, i.e. the liquid saturation phase and the gas pressure phase are primary unknowns. This formulation leads to a coupled system consisting of a nonlinear parabolic equation for the gas pressure and a nonlinear degenerate parabolic diffusion–convection equation for the liquid saturation, subject to appropriate boundary and initial conditions. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. Under some realistic assumptions on the data, 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 the two-scale convergence.