Codex @codex  0

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 $RL$.

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.