**Keywords:**Conservation laws | Riemann problem | umbilic point | WAG injection | petroleum engineering | flow in porous media | permeability Corey model

We focus on a system of two conservation laws representing a large class of models of immiscible flow in porous media relevant for petroleum engineering. The Riemann solutions are found for a range of initial and injection conditions important in applications, including the injection of two fluids (water, gas) into a horizontal reservoir containing a third fluid (oil) to be displaced.

Despite loss of hyperbolicity, there is evidence that the solution for each such data exists and is unique. Also, the solution depends L1 continuously on the Riemann data. Such solutions always present a lead shock involving the fluid already present and one of the other injected fluids. There is a special solution separating solutions according to which of the injected fluids is present in the lead shock, the *separatrix solution*.

This class of solutions was discovered recently for a particular model with quadratic permeabilities, see Azevedo *et al*. (2010). We reveal the nature of the separatrix solution: in general it consists of a 1-rarefaction starting at the left Riemann datum and finishing at the right Riemann datum with a left 1-characteristic generalized 2-Lax shock. The particular case presented by Azevedo *et al*. (2010), where this separatrix lies naturally on a straight line, is very different from the general solution presented here. We give an elegant construction of the effective scalar flow function for this separatrix solution.