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On Intrinsic Formulation and Well-posedness of a Singular Limit of Two-phase Flow Equations in Porous Media
Boris Andreianov1, Robert Eymard2, Mustapha Ghilani3, Nouzha Marhraoui3

Starting from a two-phase flow model in porous media with the viscosity of the ''mobile'' phase going to infinity, the Generalized Richards Equation for the ''viscous'' phase: \begin{equation*} \left\{ \begin{array}{l} u_t - \div(k_w(u) \nabla p)&=& \splus - \theta \smoins \char_{[u=1]}, \\ u=1 &\hbox{or}& \grad(p + \Pc(u)) = 0 \hbox{ a.e. in } \O\times(0,T) \end{array} \right. \end{equation*} was derived in the works \cite{MHenry-et-al} and \cite{AndrEymardGhilaniMarhraoui} (see also \cite{Eymard-Ghilani-Marhraoui}). We discuss intrinsic formulations (weak solutions, renormalized solutions) of this singular limit problem, using in particular the techniques developed by Plouvier-Debaigt, Gagneux et al. \cite{PlouvierGagneux,Plouvier,ProuvierEtAl-Cras}. For the no-source case, we justify the equivalence of the Generalized Richards Equation and the classical Richards model.
1 :  LM-Besançon - Laboratoire de Mathématiques
2 :  LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 :  EMMACS
Flow in porous medium – two-phase flow model – Richards model – renormalized solutions