On the slow draining of a gravity current moving through a layered permeable medium

Abstract

We examine the gravitational dispersal of dense fluid through a horizontal permeable layer, which is separated from a second underlying layer by a narrow band of much lower permeability. We derive a series of analytical solutions which describe the propagation of the fluid through the upper layer and the draining of the fluid into the underlying region. The model predicts that the current initially spreads according to a self-similar solution. However, as the drainage becomes established, the spreading slows, and in fact the fluid only spreads a finite distance before it has fully drained into the underlying layer. We examine the sensitivity of the results to the initial conditions through numerical solution of the governing equations. We find that for sources of sufficiently large initial aspect ratio (defined as the ratio of height to length), the solution converges rapidly to the initially self-similar regime. For longer and shallower initial source conditions, this convergence does not occur, but we derive estimates for the run-out length of the current, which compare favourably with our numerical data. We also present some preliminary laboratory experiments, which support the model.