The dynamics of two-layer gravity-driven flows in permeable rock

We examine the motion of a two-layer gravity current, composed of two fluids of different viscosity and density, as it propagates through a model porous layer. We focus on two specific situations: first, the case in which each layer of fluid has finite volume, and secondly, the case in which each layer is supplied by a steady maintained flux. In both cases, we find similarity solutions which describe the evolution of the flow. These solutions illustrate how the morphology of the interface between the two layers of fluid depends on the viscosity, density and volume ratios of the two layers. We show that in the special case that the viscosity ratio of the upper to lower layers, V, satisfies V = (1 + F)/(1 + RF) where F and R are respectively the ratios of the volume and buoyancy of the lower layer to those of the upper layer, then the ratio of layer depths is the same at all points. Furthermore, we show that for V > (<)(1 + F)/(1 + RF), the lower (upper) layer advances ahead of the upper (lower) layer. We also present some new laboratory experiments on two-layer gravity currents, using a Hele-Shaw cell, and show that these are in accord with the model predictions. One interesting prediction of the model, which is confirmed by the experiments, is that for a finite volume release, if the viscosity ratio is sufficiently large, then the less-viscous layer separates from the source. We extend the model to describe the propagation of a layer of fluid which is continuously stratified in either density or viscosity, and we briefly discuss application of the results for modelling various two-layer gravity-driven flows in permeable rock.