@article{spayd_shearer_hu_2012, title={Stability of plane waves in two-phase porous media flow}, volume={91}, ISSN={["0003-6811"]}, DOI={10.1080/00036811.2011.618128}, abstractNote={We examine the Saffman–Taylor instability for oil displaced by water in a porous medium. The model equations are based on Darcy's law for two-phase flow, with dependent variables pressure and saturation. Stability of plane wave solutions is governed by the hyperbolic/elliptic system obtained by ignoring capillary pressure, which adds diffusion to the hyperbolic equation. Interestingly, the growth rate of perturbations of unstable waves is linear in the wave number to leading order, whereas a naive analysis would indicate quadratic dependence. 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.}, number={2}, journal={APPLICABLE ANALYSIS}, author={Spayd, Kim and Shearer, Michael and Hu, Zhengzheng}, year={2012}, pages={295–308} }
@article{spayd_shearer_2011, title={THE BUCKLEY-LEVERETT EQUATION WITH DYNAMIC CAPILLARY PRESSURE}, volume={71}, ISSN={["1095-712X"]}, DOI={10.1137/100807016}, abstractNote={The Buckley-Leverett equation for two phase flow in a porous medium is modified by including a dependence of capillary pressure on the rate of change of saturation. This model, due to Gray and Hassanizadeh, results in a nonlinear pseudo-parabolic partial differential equation. 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 and the structures of the various solutions are confirmed with numerical simulations of the partial differential equation.}, number={4}, journal={SIAM JOURNAL ON APPLIED MATHEMATICS}, author={Spayd, K. and Shearer, M.}, year={2011}, pages={1088–1108} }