Deep-substrate drainage of a gravity current
= Deep-substrate drainage of a gravity current
Combining the <lubrication gravity-current flux> with <vertical imbibition under a draining liquid layer> gives
$$
h_t-\frac{g}{3\nu}(h^3h_x)_x=-\frac{gk}{\nu}(1+h/l)=-\phi l_t.
$$
The injected volume is the integral of $h+\phi l$, not just the surface current thickness. The model neglects horizontal subsurface flow and applies only where the substrate is wetted and the surface current supplies its drainage.