A two-fluid hyperbolic model in a porous medium

¹ EDF, R&D, Fluid Dynamics, Power Generation and Environment, 6 quai Watier, 78400 Chatou, France. jean-marc.herard@edf.fr
² Centre de Mathématiques et Informatique, LATP, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France.

Received: 22 July 2008
Revised: 20 July 2009

Abstract

The paper is devoted to the computation of two-phase flows in a porous medium when applying the two-fluid approach. The basic formulation is presented first, together with the main properties of the model. A few basic analytic solutions are then provided, some of them corresponding to solutions of the one-dimensional Riemann problem. Three distinct Finite-Volume schemes are then introduced. The first two schemes, which rely on the Rusanov scheme, are shown to give wrong approximations in some cases involving sharp porous profiles. The third one, which is an extension of a scheme proposed by Kröner and Thanh [SIAM J. Numer. Anal. 43 (2006) 796–824] for the computation of single phase flows in varying cross section ducts, provides fair results in all situations. Properties of schemes and numerical results are presented. Analytic tests enable to compute the L¹ norm of the error.