Vertex centred Discretization of Two-Phase Darcy flows on General Meshes

Robert Eymard (1) , Cindy Guichard (2, 3) , Raphaele Herbin (4) , Roland Masson (5, 2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 COFFEE - COmplex Flows For Energy and Environment
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 LATP - Laboratoire d'Analyse, Topologie, Probabilités
5 JAD - Laboratoire Jean Alexandre Dieudonné

This paper concerns the discretization of multiphase Darcy flows, in the case of heterogeneous anisotropic porous media and general 3D meshes used in practice to represent reservoir and basin geometries. An unconditionally coercive and symmetric vertex centred approach is introduced in this paper. This scheme extends the Vertex Approximate Gradient scheme (VAG), already introduced for single phase diffusive problems in \cite{Eymard.Herbin.ea:2010}, to multiphase Darcy flows. The convergence of the VAG scheme is proved for a simplified two-phase Darcy flow model, coupling an elliptic equation for the pressure and a linear hyperbolic equation for the saturation. The ability for the VAG scheme to efficiently deal with highly heterogeneous media and complex meshes is exhibited on immiscible and miscible two phase Darcy flow models.

Domaines: Analyse numérique [math.NA]

Robert Eymard, Cindy Guichard, Raphaele Herbin, Roland Masson. Vertex centred Discretization of Two-Phase Darcy flows on General Meshes. ESAIM: Proceedings, 2012, 35, pp.59-78. ⟨10.1051/proc/201235004⟩. ⟨hal-00706246⟩