Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 8 (2017), 1845-1876.

Incompressible immiscible multiphase flows in porous media: a variational approach

Clément Cancès, Thomas O. Gallouët, and Léonard Monsaingeon

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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.

Article information

Anal. PDE, Volume 10, Number 8 (2017), 1845-1876.

Received: 13 July 2016
Revised: 23 May 2017
Accepted: 29 June 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K65: Degenerate parabolic equations 35A15: Variational methods 49K20: Problems involving partial differential equations 76S05: Flows in porous media; filtration; seepage

multiphase porous media flows Wasserstein gradient flows constrained parabolic system minimizing movement scheme


Cancès, Clément; Gallouët, Thomas O.; Monsaingeon, Léonard. Incompressible immiscible multiphase flows in porous media: a variational approach. Anal. PDE 10 (2017), no. 8, 1845--1876. doi:10.2140/apde.2017.10.1845.

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