Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation

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

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 JAD - Laboratoire Jean Alexandre Dieudonné
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités
4 IFPEN - IFP Energies nouvelles

The family of the gradient schemes, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and gradient of pressures to the continuous ones, thanks to monotony and compactness arguments. Strong convergence results are issued from the convergence of the norms of the gradients of pressures, which demands to handle the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

Domaines: Analyse numérique [math.NA]

Robert Eymard, Cindy Guichard, Raphaèle Herbin, Roland Masson. Gradient schemes for two-phase flow in heterogeneous porous media and Richards equation. 2012. ⟨hal-00740367v1⟩