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

TL;DR

This paper develops and analyzes a finite volume scheme for simulating two-phase flows in heterogeneous porous media with discontinuous capillary pressure functions, proving convergence and illustrating with numerical examples.

Contribution

It introduces a novel finite volume scheme for degenerated non-linear parabolic equations with discontinuous coefficients, and proves its convergence and stability.

Findings

Convergence of the scheme to a weak solution is established.

Uniform flux estimates are obtained under certain initial data conditions.

Numerical examples demonstrate the model's behavior in heterogeneous media.

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 non-linear parabolic equations spatially coupled by non linear 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.