Two-dimensional viscous gravity currents flowing over a deep porous medium

The spreading of a two-dimensional, viscous gravity current propagating ove r and draining into a deep porous substrate is considered both theoreticall y and experimentally. We first determine analytically the rate of drainage of a one-dimensional layer of fluid into a porous bed and find that the the oretical predictions for the downward rate of migration of the fluid front are in excellent agreement with our laboratory experiments. The experiments suggest a rapid and simple technique for the determination of the permeabi lity of a porous medium. We then combine the relationships for the drainage of liquid from the current through the underlying medium with a formalism for its forward motion driven by the pressure gradient arising from the slo pe of its free surface. For the situation in which the volume of fluid V fe d to the current increases at a rate proportional to t(3), where t is the t ime since its initiation, the shape of the current takes a self-similar for m for all time and its length is proportional to t(2). When the volume incr eases less rapidly, in particular for a constant volume, the front of the g ravity current comes to rest in finite time as the effects of fluid drainag e into the underlying porous medium become dominant. In this case, the runo ut length is independent of the coefficient of viscosity of the current, wh ich sets the time scale of the motion. We present numerical solutions of th e governing partial differential equations for the constant-volume case and find good agreement with our experimental data obtained from the flow of g lycerine over a deep layer of spherical beads in air.