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

Published online by Cambridge University Press:  01 August 2009

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

Keywords

Capillarity discontinuitiesdegenerate parabolic equationfinite volume scheme
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 5 , September 2009 , pp. 973 - 1001
Copyright
© EDP Sciences, SMAI, 2009

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