The Rayleigh-Taylor instability in a porous medium

The classical Rayleigh-Taylor instability occurs when a heavy fuid overlies a lighter one, and the two fuids are separated by a horizontal interface. The confguration is unstable, and a small perturbation to the interface grows with time. Here, we consider such an arrangement for planar fow, but in a porous medium governed by Darcy's law. First, the fully saturated situation is considered, where the two horizontal fuids are separated by a sharp interface. A classical linearized theory is reviewed, and the nonlinear model is solved numerically. It is shown that the solution is ultimately limited in time by the formation of a curvature singularity at the interface. A partially saturated Boussinesq theory is then presented, and its linearized approximation predicts a stable interface that merely difuses. Nonlinear Boussinesq theory, however, allows the growth of drips and bubbles at the interface. These structures develop with no apparent overturning at their heads, unlike the corresponding fow for two free fuids.