The invasion of one fluid into a fluid occupying the narrow gap between parallel plates is a model for the displacement of a resident fluid in a porous medium by injecting a second fluid. In this paper, I describe equations formulated by Ruben Juanes and Luis Cueto-Felguereso based on a phase-field formulation. Numerical solutions of plane waves exhibit unusual behavior that is explored mathematically. Connections are made to finite speed propagation of a small amplitude wave solution of the porous medium equation, and to traveling waves that in a singular limit would be anomalous shock wave solutions of the equations. This research is documented in the thesis of Melissa Strait.