= Vertical imbibition under a draining liquid layer
{title2=$\phi\dot l=(gk/\nu)(1+h/l)$}
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 $\phi\dot l=w$. For a finite column, $h+\phi l=h_0$ and initially
$$
l(t)\sim\left(\frac{2gkh_0t}{\phi\nu}\right)^{1/2}.
$$
The singular initial flux is an idealization; the continuum model requires penetration beyond the pore scale.
Back to article page
