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A draining gravity current loses fluid through its substrate while spreading horizontally. For a two-dimensional viscous current of thickness $H(x,t)$ that loses fluid according to pressure-dependent Darcy drainage,when the hydrostatic pressure is $p=\rho gH$ and $k/h\propto p^{-\beta }$.

Seek a similarity solution in the dimensionally completed formwhere the fixed scales absorb $A,D$, and $Q_0$. Its inlet volume flux scales asso the prescribed source requiresThe time derivative, horizontal lubrication theory term, and pressure-dependent Darcy drainage term scale respectively asEquating the three exponents givesSolving these linear equations together with the inlet condition yields the similarity exponents of a draining gravity currentEquivalently,and, up to constant dimensional scales,

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.