= Solution
Before drainage reaches the divide, most of the aquifer fills locally to height $H=Rt/\phi$. Only a growing <boundary layer> beside the river departs substantially from this height. Write $D=u_b\beta/2$.
At early times $H\ll2/\beta$, the mobility is $u_bh$. The balance between storage and <nonlinear diffusion> gives
$$
\ell_e(t)=\frac{\sqrt{u_bR}}{\phi}t,\qquad h=\frac{Rt}{\phi}F_e(\eta),\qquad \eta=x/\ell_e.
$$
Substitution into the governing equation gives the <forced filling similarity for power-law diffusion> with exponent $m=1$:
$$
(F_eF_e')'+1-F_e+\eta F_e'=0,\qquad F_e(0)=0,\qquad F_e(\infty)=1.
$$
The positive outward <volume flux per unit width> is
$$
\boxed{Q_e=c_e\frac{\sqrt{u_b}\,R^{3/2}}{\phi}t,\qquad c_e=\lim_{\eta\downarrow0}F_eF_e'=2\int_0^\infty(1-F_e)\,d\eta.}
$$
The boundary slope is singular, but $F_e\sim(2c_e\eta)^{1/2}$ gives a finite flux.
At intermediate times $H\gg2/\beta$, the large-depth mobility is $Dh^2$. The corresponding <similarity solution> is
$$
\ell_i(t)=\left(\frac{DR^2t^3}{\phi^3}\right)^{1/2},\qquad h=\frac{Rt}{\phi}F_i(\eta),\qquad \eta=x/\ell_i,
$$
with
$$
(F_i^2F_i')'+1-F_i+\frac32\eta F_i'=0,\qquad F_i(0)=0,\qquad F_i(\infty)=1.
$$
Consequently,
$$
\boxed{Q_i=c_i\frac{\sqrt D\,R^2}{\phi^{3/2}}t^{3/2},\qquad c_i=\lim_{\eta\downarrow0}F_i^2F_i'=\frac52\int_0^\infty(1-F_i)\,d\eta.}
$$
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 $u_bh$ mobility extremely close to the boundary while transmitting this same leading discharge.
The mobility changes when $H\sim2/\beta$, whereas the divide first affects the deep filling solution when $\ell_i\sim L$. Thus
$$
\boxed{t_\beta=\frac{2\phi}{R\beta},\qquad t_{\rm fill}=\phi\left(\frac LR\right)^{2/3}D^{-1/3}.}
$$
A distinct intermediate regime requires $t_\beta\ll t_{\rm fill}$. 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 $\phi L/\sqrt{u_bR}$. 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 $RL$.
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