On the Upstream Mobility Finite Difference Scheme

Incompressible two-phase flow in porous media when capillarity effects are neglected can be modeled as a nonlinear scalar conservation with unknown one of the phase saturations. The standard scheme used in reservoir simulation or in hydrogeology uses the upstream mobility flux. The solution calculated with such a scheme was proven to be converging to the entropy solution. This scheme is used in the homogeneous case as well as in the case of a change in rock types when the mobilities functions are changing from one rock type to the other. However in this latter case, while the scheme is giving the right solution most of the time, we will exhibit some examples of mobility functions for which the scheme does not give the right solution. An alternative scheme is to use an extension of the Godunov flux for which we give a simple formula which can be used when a change of rock types occurs. Thus the solution given by this scheme can be proven to be correct even in cases where the upstream mobility scheme fails.