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Before drainage reaches the divide, most of the aquifer fills locally to height $H=Rt/\varphi$. Only a growing boundary layer beside the river departs substantially from this height. Write $D=u_b\beta /2$.

At early times $H\ll 2/\beta$, the mobility is $u_{b}h$. The balance between storage and nonlinear diffusion givesSubstitution into the governing equation gives the forced filling similarity for power-law diffusion with exponent $m=1$:The positive outward volume flux per unit width isThe boundary slope is singular, but $F_e\sim (2c_e\eta {)}^{1/2}$ gives a finite flux.

At intermediate times $H\gg 2/\beta$, the large-depth mobility is $D{h}^2$. The corresponding similarity solution iswithConsequently,Here $F_i\sim (3c_i\eta {)}^{1/3}$ is the outer outlet profile. The height actually vanishes at the river, so an outlet layer in a deep unconfined aquifer restores the $u_{b}h$ mobility extremely close to the boundary while transmitting this same leading discharge.

The mobility changes when $H\sim 2/\beta$, whereas the divide first affects the deep filling solution when ${\ell }_i\sim L$. ThusA distinct intermediate regime requires $t_{\beta }\ll t_{\mathrm{fill}}$. If the divide is reached while the aquifer is still shallow, the early filling law instead crosses directly toward the steady state, at a time of order $\varphi L/\sqrt{u_{b}R}$. The faster intermediate discharge growth comes from increasing permeability of a porous medium as deeper parts of the aquifer become saturated; once the finite domain is felt, the discharge approaches $RL$.

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.