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Abstract
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.
Acknowledgements
We thank Dr Steven Wise (Department of Mathematics, University of Tennessee, Knoxville) for providing the open source codes that solve the original Cahn–Hillard equations as discussed in Citation16. The first author's research was supported by NSF Grants DMS 0968258, DMS 0636590 RTG, the second author's research by NSF Grant DMS 0968258 and the third author's research by NSF Grant DMS 0636590 RTG.