= Solution
Write the given pressure-dependent ratio of <permeability of a porous medium> to sponge thickness as
$$
\frac{k(p)}{h(p)}=Kp^{-\beta},
$$
where $K$ is constant. The <fluid pressure> drop across the sponge is the <hydrostatic pressure> of the current,
$$
p=\rho gH,
$$
because the air pressures above and below are equal. Therefore <Darcy's law> gives the downward <volume flux> per unit horizontal area as
$$
w=\frac{k(p)}{\mu h(p)}p
=\frac K\mu p^{1-\beta}
=\boxed{D H^{1-\beta}},
\qquad
D=\frac K\mu(\rho g)^{1-\beta}.
$$
Thus the sponge drainage obeys a <power law> in the current thickness, as described by <pressure-dependent Darcy drainage>.
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