Uniqueness of the Riemann Solution for Three-Phase Flow in a Porous Medium

We solve injection problems for immiscible three-phase flow described by a system of two conservation laws with fluxes originating from Corey's model with quadratic permeabilities. A mixture of water, gas, and oil is injected into a porous medium containing oil, which is partially displaced. We prove that the resulting Riemann problems have solutions, which are unique under technical hypotheses that can be verified numerically. The wave curve method constructs the solutions straightforwardly, despite loss of strict hyperbolicity at an isolated point in state space. This umbilic point induces the separation of the solutions into two types, according to the water/gas proportion in the injection mixture.