## Analysis & PDE

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

### Incompressible immiscible multiphase flows in porous media: a variational approach

#### Abstract

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

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

Dates
Revised: 23 May 2017
Accepted: 29 June 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843584

Digital Object Identifier
doi:10.2140/apde.2017.10.1845

Mathematical Reviews number (MathSciNet)
MR3694008

Zentralblatt MATH identifier
1370.35230

#### Citation

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. https://projecteuclid.org/euclid.apde/1510843584

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