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 non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schemè a la [R. Jordan, D. Kinder-lehrer & F. Otto, SIAM J. Math. Anal, 29(1):1–17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in 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.
Publié le : 2017-07-04
Classification:
Multiphase porous media flows,
constrained par-abolic system,
minimizing movement scheme,
Wasserstein gradient flows,
35K65, 35A15, 49K20, 76S05,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],
[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC],
[SDU.STU.HY]Sciences of the Universe [physics]/Earth Sciences/Hydrology
@article{hal-01345438,
author = {Canc\`es, Cl\'ement and Gallou\"et, Thomas and Monsaingeon, Leonard},
title = {Incompressible immiscible multiphase flows in porous media: a variational approach},
journal = {HAL},
volume = {2017},
number = {0},
year = {2017},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-01345438}
}
Cancès, Clément; Gallouët, Thomas; Monsaingeon, Leonard. Incompressible immiscible multiphase flows in porous media: a variational approach. HAL, Tome 2017 (2017) no. 0, . http://gdmltest.u-ga.fr/item/hal-01345438/