  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.
