A finite volume scheme for two-phase immiscible flow in porous media

In this paper, we prove the convergence of a numerical method for solving two-phase immiscible, incompressible flow in porous media. The method combines an upwind time implicit finite volume scheme for the saturation equation (hyperbolic-parabolic type) and a centered finite volume scheme for the Chavent global pressure equation (elliptic type). The capillary pressure is not neglected, and we study the case when the diffusion term in the saturation equation is weakly degenerated. Estimates on the approximate solution are proven; then by using compactness theorems we obtain a limit when the size of the discretization goes to zero, and we prove that this limit is the unique weak solution of the problem that we study.