On Intrinsic Formulation and Well-posedness of a Singular Limit of Two-phase Flow Equations in Porous Media

Boris Andreianov; Robert Eymard; Mustapha Ghilani; Nouzha Marhraoui

hal-00606948, version 2

On Intrinsic Formulation and Well-posedness of a Singular Limit of Two-phase Flow Equations in Porous Media

Boris Andreianov () ¹, Robert Eymard () ², Mustapha Ghilani () ³, Nouzha Marhraoui () ³

(2011-04-01)

Abstract: 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: Laboratoire de Mathématiques (LM-Besançon)
CNRS : UMR6623 – Université de Franche-Comté
2: Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA)
Université Paris-Est Marne-la-Vallée (UPEMLV) – Université Paris-Est Créteil Val-de-Marne (UPEC) – CNRS : UMR8050 – Fédération de Recherche Bézout
3: EMMACS
Université Moulay Ismail Meknès

hal-00606948, version 2
http://hal.archives-ouvertes.fr/hal-00606948
oai:hal.archives-ouvertes.fr:hal-00606948
From: Boris Andreianov <>
Submitted on: Tuesday, 9 August 2011 13:34:56
Updated on: Tuesday, 9 August 2011 14:35:50

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