Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities

École Normale Supérieure de Cachan, Antenne de Bretagne, Campus de Ker Lann, Avenue Robert Schuman, 35170 Bruz, France. clement.cances@bretagne.ens-cachan.fr

Received: 9 April 2008
Revised: 1 December 2008

Abstract

We study a one-dimensional model for two-phase flows in heterogeneous media, in which the capillary pressure functions can be discontinuous with respect to space. We first give a model, leading to a system of degenerated nonlinear parabolic equations spatially coupled by nonlinear transmission conditions. We approximate the solution of our problem thanks to a monotonous finite volume scheme. The convergence of the underlying discrete solution to a weak solution when the discretization step tends to 0 is then proven. We also show, under assumptions on the initial data, a uniform estimate on the flux, which is then used during the uniqueness proof. A density argument allows us to relax the assumptions on the initial data and to extend the existence-uniqueness frame to a family of solution obtained as limit of approximations. A numerical example is then given to illustrate the behavior of the model.