Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 5, p. 973-1001

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.

DOI : https://doi.org/10.1051/m2an/2009032
Classification:  35R05,  65M12
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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {5},
     year = {2009},
     pages = {973-1001},
     doi = {10.1051/m2an/2009032},
     zbl = {1171.76035},
     mrnumber = {2559741},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_5_973_0}
}
Cancès, Clément. Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 5, pp. 973-1001. doi : 10.1051/m2an/2009032. http://www.numdam.org/item/M2AN_2009__43_5_973_0/

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