Gravity currents in a porous medium at an inclined plane

We consider the release from a point source of relatively heavy fluid into a porous saturated medium above an impermeable slope. We consider the case where the volume of the resulting gravity current increases with time like $t^\alpha$ and show that for $\alpha<3$, at short times the current spreads axisymmetrically, with radius $r\sim t^{(\alpha+1)/4}$, while at long times it spreads predominantly downslope. In particular, for long times the downslope position of the current scales like $t$ while the current extends a distance $t^{\alpha/3}$ across the slope. For $\alpha>3$, this situation is reversed with spreading occurring predominantly downslope for short times. The governing equations admit similarity solutions whose scaling behaviour we determine, with the full similarity form being evaluated by numerical computations of the governing partial differential equation. We find that the results of these analyses are in good quantitative agreement with a series of laboratory experiments. Finally, we briefly discuss the implications of our work for the sequestration of carbon dioxide in aquifers with a sloping, impermeable cap.