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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)

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

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): Flow in porous medium – two-phase flow model – Richards model – renormalized solutions

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From: Boris Andreianov <>

Submitted on: Tuesday, 9 August 2011 13:34:56

Updated on: Tuesday, 9 August 2011 14:35:50