INCOMPRESSIBLE IMMISCIBLE MULTIPHASE FLOWS IN POROUS MEDIA A VARIATIONAL APPROACH

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 a 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.