Issue ESAIM: M2AN Volume 43, Number 5, September-October 2009 Page(s) 973 - 1001 DOI 10.1051/m2an/2009032 Published online 01 August 2009

ESAIM: M2AN 43 (2009) 973-1001
DOI: 10.1051/m2an/2009032

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

Clément Cancès

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 April 9, 2008. Revised December 1st, 2008. Published online August 1st, 2009.

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.

Mathematics Subject Classification. 35R05, 65M12

Key words: Capillarity discontinuities, degenerate parabolic equation, finite volume scheme


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