Section: New Results

Modeling and numerics for porous media flows

In [16], C. Cancès and C. Guichard propose a nonlinear Control Volume Finite Elements method with upwinding in order to solve possibly nonlinear and degenerate parabolic equations. This method was designed in order to preserve at the discrete level the positivity and the nonlinear stability of the solutions. In [25], A. Ait Hammou Oulhaj, C. Cancès, and C. Chainais-Hillairet extend the approach of [16] to the more complex case of Richards equation modeling saturated/unsaturated flows in anisotropic porous media. The additional complexity comes from the fact that convective terms and elliptic degeneracy are considered in [25]. The scheme preserves at the discrete level the nonnegativity and the nonlinear stability of the solutions. Its convergence is rigorously proved, and numerical results are provided in order to illustrate the behavior of the scheme.

In [49], C. Cancès, T. O. Gallouët, and L. Monsaingeon show that the equations governing two-phase flows in porous media have a formal gradient flow structure. The goal of the longer contribution [29] is then twofold. First, it extends the variational interpretation of [49] to the case where an arbitrary number of phases are in competition to flow within a porous medium. Second, we provide rigorous foundations to our claim. More precisely, the convergence of a minimizing movement scheme à la Jordan, Kinderlehrer, and Otto  [66] is shown in [29], providing by the way a new existence result for multiphase flows in porous media. The result relies on advances tools related to optimal transportation  [75], [74].