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After recharge stops, the finite interval sets the horizontal scale. Apply the separable draining profile for power-law diffusion to each limiting equationWith $\xi =x/L$ and virtual age $\tau =t+t_0$, the similarity solution isThe positive profile fixes $c_m={\lim }_{\xi \downarrow 0}{F}_m^m{F}_m^{\prime }={\int }_0^1{F}_{m}d\xi$. The discharge follows either from the boundary Darcy flux or integrated mass conservation:In the deep regime this givesIn the final shallow regime,The deep profile contains the thin outlet layer in a deep unconfined aquifer described above. The whole aquifer eventually becomes shallow, and its nonzero basal permeability controls the ultimate ${\tau }^{-2}$ discharge.

Estimating the mobility crossover by a typical height $2/\beta$ givesUsing the deep profile's height at the divide makes this ${\tau }_{\mathrm{cross}}\simeq \varphi \beta L^2{F}_2(1{)}^2/(4u_b)$. These are regime-dependent similarity approximations: the virtual origins are set by matching to the initial drainage and crossover, and a deep regime occurs only if the aquifer is initially sufficiently deep.

With permeability of a porous medium $k(z)=k_0(1+\beta z)$, the Dupuit approximation givesHere $\varphi$ is porosity and $R$ is recharge. The low-height limit has mobility proportional to $h$, whereas large heights have mobility proportional to $h^2$. This changes both the forced filling similarity for power-law diffusion and the separable draining profile for power-law diffusion.