**Keywords:**XV International Conference on Hyperbolic Problems -- Hyp2014 | Conservation laws | Riemann problem | petroleum engineering | flow in porous media | permeability Corey model

We utilize the wave curve method to prove the existence and give a characterization of a certain bifurcation locus in the saturation triangle. Such structure arises, for instance, when water and gas are injected in a mature reservoir either to dislodge oil or to sequestrate CO2. The proof takes advantage of a certain wave curve to ensure that the waves in the flow are a rarefaction preceded by a shock, which is in turn preceded by a constant two-phase state (*i.e.*, it lies at the boundary of the saturation triangle). For convex permeability models of Corey type, the analysis reveals further details, such as the number of possible two-phase states that correspond to the above mentioned shock, whatever the left state of the latter is within the saturation triangle.