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

The Wasserstein gradient flow structure of the PDE system governing multiphase flows in porous media was recently highlighted in [C. Canc`es, T. O. Gallou"et, and L. Monsaingeon, {\it Anal. PDE} 10(8):1845–1876, 2017]. The model can thus be approximated by means of the minimizing movement (or JKO) scheme, that we solve thanks to the ALG2-JKO scheme proposed in [J.-D. Benamou, G. Carlier, and M. Laborde, {\it ESAIM Proc. Surveys}, 57:1–17, 2016]. The numerical results are compared to a classical upstream mobility Finite Volume scheme, for which strong stability properties can be established.