The motion of instantaneous and maintained releases of buoyant fluid through shallow permeable layers of large horizontal extent is described by a nonlinear advection–diffusion equation. This equation admits similarity solutions which describe the release of one fluid into a horizontal porous layer initially saturated with a second immiscible fluid of different density. Asymptotically, a finite volume of fluid spreads as t1/3. On an inclined surface, in a layer of uniform permeability, a finite volume of fluid propagates steadily alongslope under gravity, and spreads diffusively owing to the gravitational acceleration normal to the boundary, as on a horizontal boundary. However, if the permeability varies in this cross-slope direction, then, in the moving frame, the spreading of the current eventually becomes dominated by the variation in speed with depth, and the current length increases as t1/2. Shocks develop either at the leading or trailing edge of the flows depending upon whether the permeability increases or decreases away from the sloping boundary. Finally we consider the transient and steady exchange of fluids of different densities between reservoirs connected by a shallow long porous channel. Similarity solutions in a steadily migrating frame describe the initial stages of the exchange process. In the final steady state, there is a continuum of possible solutions, which may include flow in either one or both layers of fluid. The maximal exchange flow between the reservoirs involves motion in one layer only. We confirm some of our analysis with analogue laboratory experiments using a Hele-Shaw cell.
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