There are two goals of this study. The first is to provide an introduction to the wave curve method for finding the analytic solution of a porous medium injection problem. Similar to fractional and chromatographic flow theory, the wave curve method is based on the method of characteristics, but it is applicable to an expanded range of physical processes in porous medium flow. The second goal is to solve injection problems for immiscible three-phase flow, as described by Corey's model, in which a mixture of gas and water is injected into a porous medium containing oil and irreducible water. In particular we determine, for any choice of the phase viscosities, the proportion of the injected fluids that maximizes recovery around breakthrough time. Numerical simulations are performed to compare our solutions for Corey's model with those of other models. For the injection problems we consider, solutions for Corey's model are very similar to those for Stone's model, despite the presence of an elliptic region in the latter; and they are very different from those for the Juanes-Patzek model, which preserves strict hyperbolicity. A nice feature of our analytical method is that it facilitates explaining both differences and similarities among the solutions for the three models considered.