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 discontinuities, degenerate parabolic equation, finite volume scheme

@article{M2AN_2009__43_5_973_0, author = {Canc\`es, Cl\'ement}, title = {Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {973--1001}, publisher = {EDP-Sciences}, volume = {43}, number = {5}, year = {2009}, doi = {10.1051/m2an/2009032}, mrnumber = {2559741}, zbl = {1171.76035}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2009032/} }

TY - JOUR AU - Cancès, Clément TI - Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 973 EP - 1001 VL - 43 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2009032/ DO - 10.1051/m2an/2009032 LA - en ID - M2AN_2009__43_5_973_0 ER -

%0 Journal Article %A Cancès, Clément %T Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 973-1001 %V 43 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2009032/ %R 10.1051/m2an/2009032 %G en %F M2AN_2009__43_5_973_0

Cancès, Clément. Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. ESAIM: Modélisation mathématique et analyse numérique, Volume 43 (2009) no. 5, pp. 973-1001. doi : 10.1051/m2an/2009032. http://archive.numdam.org/articles/10.1051/m2an/2009032/

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