= Solution
After recharge stops, the finite interval sets the horizontal scale. Apply the <separable draining profile for power-law diffusion> to each limiting equation
$$
\phi h_t=D_m(h^m h_x)_x,\qquad (m,D_m)=(2,D)\ \hbox{or}\ (1,u_b),\qquad D=u_b\beta/2.
$$
With $\xi=x/L$ and virtual age $\tau=t+t_0$, the <similarity solution> is
$$
h=\left(\frac{\phi L^2}{mD_m\tau}\right)^{1/m}F_m(\xi),\qquad (F_m^mF_m')'+F_m=0,\qquad F_m(0)=0,\qquad F_m'(1)=0.
$$
The positive profile fixes $c_m=\lim_{\xi\downarrow0}F_m^mF_m'=\int_0^1F_m\,d\xi$. The discharge follows either from the boundary <Darcy flux> or integrated <mass conservation>:
$$
Q_m=\frac{D_m}{L}\left(\frac{\phi L^2}{mD_m\tau}\right)^{(m+1)/m}c_m.
$$
In the deep regime this gives
$$
\boxed{h\propto\tau^{-1/2},\qquad Q_2=\frac{c_2\phi^{3/2}L^2}{2\sqrt{u_b\beta}}\tau^{-3/2}.}
$$
In the final shallow regime,
$$
\boxed{h\propto\tau^{-1},\qquad Q_1=\frac{c_1\phi^2L^3}{u_b}\tau^{-2}.}
$$
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 $2/\beta$ gives
$$
\boxed{\tau_{\rm cross}\sim\frac{\phi\beta L^2}{u_b}.}
$$
Using the deep profile's height at the divide makes this $\tau_{\rm cross}\simeq\phi\beta L^2F_2(1)^2/(4u_b)$. 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.
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