The Buckley–Leverett partial differential equation has long been used to model two-phase flow in porous media. In recent years, the PDE has been modified to include a rate-dependent capillary pressure constitutive equation, known as dynamic capillary pressure. Previous traveling wave analysis of the modified Buckley–Leverett equation uncovered non-classical solutions involving undercompressive shocks. More recently, thermodynamically constrained averaging theory (TCAT) has generalized the capillary pressure equation by including additional dependence on fluid properties. In this paper, the model and traveling wave analysis are updated to incorporate TCAT capillary pressure as a generalization of dynamic capillary pressure. Solutions of the corresponding Riemann problem are similar to previous results except in the physically relevant situation in which both phases are pure fluids. The results presented here shed new light on the nature of the interface between one pure fluid displacing another pure fluid, in accordance with TCAT.