| The Buckley-Leverett equation for two phase flow in a porous medium is modified by including dependence of the capillary pressure on the time derivative of saturation. This model, due to Hassanizadeh and Gray, results in a nonlinear pseudoparabolic partial differential equation that includes a mixed third order derivative representing dispersion. Both quadratic and fractional relative permeability functions are considered in the model. Phase plane analysis, including a separation function to measure the distance between invariant manifolds, is used to determine when the equation supports traveling waves corresponding to undercompressive shocks. The Riemann problem for the underlying conservation law is solved in the case of each relative permeability constitutive equation. The structures of the various solutions are confirmed with numerical simulations of the partial differential equation. Specific effects of the mixed third order derivative are investigated in the context of the Benjamin-Bona-Mahony equation modified with a cubic flux function and Burgers term.;Further, the Saffman-Taylor viscous fingering instability is examined and a generalized criterion given for variable saturations. Two dimensional stability of plane wave solutions is governed by the hyperbolic/elliptic system obtained by ignoring capillary pressure, which is diffusive. The growth rate of perturbations of unstable waves is linear in the wave number to leading order. This gives a sharp boundary in the state space of upstream and downstream saturations separating stable from unstable waves. The role of this boundary, derived from the linearized hyperbolic/elliptic system, is verified by numerical simulations of the full nonlinear parabolic/elliptic equations. |
