Simulation of multiphase porous media flows with minimizing movement and finite volume schemes

The Wasserstein gradient flow structure of the partial differential equation system governing multi-phase flows in porous media was recently highlighted in Cances et al. [Anal. PDE 10(8), 1845-1876]. The model can thus be approximated by means of the minimising movement (or JKO after Jordan, Kinderlehrer and Otto [SIAM J. Math. Anal. 29(1), 1-17]) scheme that we solve thanks to the ALG2JKO scheme proposed in Benamou et al. [ESAIM Proc. Surv. 57, 1-17]. The numerical results are compared to a classical upstream mobility finite volume scheme, for which strong stability properties can be established.