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Write the given pressure-dependent ratio of permeability of a porous medium to sponge thickness aswhere $K$ is constant. The fluid pressure drop across the sponge is the hydrostatic pressure of the current,because the air pressures above and below are equal. Therefore Darcy's law gives the downward volume flux per unit horizontal area asThus the sponge drainage obeys a power law in the current thickness, as described by pressure-dependent Darcy drainage.

Suppose a porous layer has thickness $h(p)$ and permeability of a porous medium $k(p)$, and the pressure drop $p$ acts across its thickness. Darcy's law gives the drainage volume fluxIf $k/h=K{p}^{-\beta }$, then $w=(K/\mu )p^{1-\beta }$.