  We consider the release and subsequent gravity-driven spreading of a finite
volume of fluid in an anisotropic porous medium bounded by an impermeable
substrate. When the permeability in the vertical direction is much smaller than
the horizontal direction, as is the case in many real geological reservoirs,
this restricts the spread of the current to a very thin layer near the
impermeable base. Using a combination of asymptotic analysis and finite
difference computations of Darcy flow, we show that there exist two distinct
flow regimes. At early times the bulk of the current descends slowly and
uniformly, injecting fluid into thin finger-like regions near the base. At much
later times the current transitions to the classical gravity-driven solution
and continues to spread with a self-similar shape. One interesting consequence
is that the swept volume of the current grows differently depending on the
anisotropy of the medium. This has important consequences for managing
contaminant spills, where it is important to minimise the contacted volume of
the aquifer, or during geological CO$_2$ sequestration where a larger contacted
volume results in more CO$_2$ being stored.
