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

Clément Cancès (1, 2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Mots clés: capillarity discontinuities; degenerate parabolic equation; finite volume scheme

Domaines: Equations aux dérivées partielles [math.AP]

Clément Cancès. Finite volume scheme for two-phase flows in heterogeneous porous media involving capillary pressure discontinuities. ESAIM: Mathematical Modelling and Numerical Analysis, 2009, 43 (5), pp.973 - 1001. ⟨hal-00360292v2⟩