Vol. 10, No. 8, 2017

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Incompressible immiscible multiphase flows in porous media: a variational approach

Clément Cancès, Thomas O. Gallouët and Léonard Monsaingeon

Vol. 10 (2017), No. 8, 1845–1876

We describe the competitive motion of N+1 incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of nonnegative measures with prescribed masses, endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization scheme á la R. Jordan, D. Kinderlehrer and F. Otto (SIAM J. Math. Anal. 29:1 (1998) 1–17). This allows us to obtain a new existence result for a physically well-established system of PDEs consisting of the Darcy–Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure.

multiphase porous media flows, Wasserstein gradient flows, constrained parabolic system, minimizing movement scheme
Mathematical Subject Classification 2010
Primary: 35K65, 35A15, 49K20, 76S05
Received: 13 July 2016
Revised: 23 May 2017
Accepted: 29 June 2017
Published: 18 August 2017
Clément Cancès
Inria Lille – Nord Europe
Thomas O. Gallouët
Département de Mathématiques
Université de Liège
Léonard Monsaingeon
Institut Élie Cartan de Lorraine
Université de Lorraine