We describe the gravity-driven flow of a viscous fluid in a semi-infinite porous layer, $x>0$, from which fluid can drain freely at $x=0$. New experiments using a Hele-Shaw cell confirm that when the base of the layer is impermeable the motion of the current is self-similar and the dipole moment of the flow is conserved, as proposed theoretically by Barenblatt & Zel'dovich (1957). We extend the model to allow fluid to drain through the base of the porous layer into a thin horizontal layer of lower permeability. In this case we predict that the dipole moment of the current decays exponentially with time. At early times we find that the loss of fluid from the gravity current in the high-permeability layer is dominated by the draining at $x=0$, whereas at long times, the gravity-driven leakage into the underlying low-permeability layer is dominant. We successfully compare these analytic solutions for such draining currents with further laboratory experiments in which fluid drains from the end and through the base of a Hele-Shaw cell. We discuss the implications of these results for the dispersal of chemicals or pollutants injected into a layered porous rock.