Vertical imbibition under a draining liquid layer ($\varphi \dot{l}=(gk/\nu )(1+h/l)$)

A liquid layer of depth $h$ above a dry deep porous substrate drives a Darcy flux $w=(gk/\nu )(1+h/l)$ through its saturated depth $l$, assuming negligible capillary pressure and negligible resistance from the displaced phase. The pressure head contributes $h/l$ and gravity contributes one. Volume conservation gives $\varphi \dot{l}=w$. For a finite column, $h+\varphi l=h_0$ and initiallyThe singular initial flux is an idealization; the continuum model requires penetration beyond the pore scale.