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.