The motion of instantaneous and maintained releases of buoyant fluid t
hrough shallow permeable layers of large horizontal extent is describe
d by a nonlinear advection-diffusion equation. This equation admits si
milarity solutions which describe the release of one fluid into a hori
zontal porous layer initially saturated with a second immiscible fluid
of different density. Asymptotically, a finite volume of fluid spread
s as t(1/3). On an inclined surface, in a layer of uniform permeabilit
y, a finite volume of fluid propagates steadily alongslope under gravi
ty, and spreads diffusively owing to the gravitational acceleration no
rmal to the boundary, as on a horizontal boundary. However, if the per
meability varies in this cross-slope direction, then, in the moving fr
ame, the spreading of the current eventually becomes dominated by the
variation in speed with depth, and the current length increases as t(1
/2). Shocks develop either at the leading or trailing edge of the flow
s depending upon whether the permeability increases or decreases away
from the sloping boundary. Finally we consider the transient and stead
y exchange of fluids of different densities between reservoirs connect
ed by a shallow long porous channel. Similarity solutions in a steadil
y 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 ma
ximal exchange flow between the reservoirs involves motion in one laye
r only. We confirm some of our analysis with analogue laboratory exper
iments using a Hele-Shaw cell.