Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium

Boris Andreianov (1) , Clément Cancès (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LJLL - Laboratoire Jacques-Louis Lions

For the hyperbolic conservation laws with discontinuous flux function there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley-Leverett equations, each notion of solution is uniquely determined by the choice of a ''connection'', which is the unique stationary solution that takes the form of an undercompressive shock at the interface. To select the appropriate connection, one may use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in previous works that the ''optimal'' connection and the ''barrier'' connection may appear at the vanishing capillarity limit, in this paper we show that any connection may appear. We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley-Leverett equation with interface coupling that retains information from the vanishing capillarity model. We illustrate the theoretical result with numerical examples.

Mots clés: scalar conservation laws with discontinuous flux functions; two-phase flows in heterogeneous porous media; finite volume schemes

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Sciences de l'environnement

Boris Andreianov, Clément Cancès. Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium. 2011. ⟨hal-00631584v1⟩