Plane waves for two phase flow in a porous medium are modeled by the one-dimensional Buckley-
Leverett equation, a scalar conservation law. In the first part of the talk, we study traveling wave solutions of the equation modfied by the Gray-Hassanizadeh model for rate-dependent capillary pressure. The modfication adds a BBM-type dispersion to the classic equation, giving rise to under-compressive waves. In the second part of the talk, we analyze stability of sharp planar interfaces (corresponding to Lax shocks) to two-dimensional perturbations, which involves a system of partial differential equations. The Safman-Taylor analysis predicts instability of planar fronts, but their
calculation lacks the dependence on saturations in the Buckley-Leverett equation. Interestingly, the dispersion relation we derive leads to the conclusion that some interfaces are long-wave stable and some are not. Numerical simulations of the full nonlinear system of equations, including dissipation and dispersion, verify the stability predictions at the hyperbolic level. This is joint work with Kim Spayd and Zhengzheng Hu.