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The Dupuit approximation treats flow in a shallow unconfined aquifer as predominantly horizontal, with hydrostatic pressure $p=\rho g(h-z)$ and horizontal pressure gradient independent of depth. Integrating Darcy's law over the saturated depth then gives a volume flux per unit width depending only on the groundwater height and its slope.

Consider a nonlinear diffusion equation with uniform supply, $\varphi h_t=D_m(h^m{h}_x{)}_x+R$, on $x>0$, with an absorbing boundary $h(0,t)=0$ and initially $h=0$. Far from the boundary, $h=Rt/\varphi$. Balancing the time derivative, supply, and diffusion givesThe similarity solution satisfiesThe outward boundary volume flux per unit width is $Q=R\ell c_m$, where $c_m={\lim }_{\eta \downarrow 0}{F}^m{F}^{\prime }$. Integrating the equation givesThus the growing region of depleted storage fixes the discharge prefactor, and $Q\propto t^{(m+1)/2}$.

For a thin Newtonian fluid film on a conical slope at angle $\alpha$, neglecting the thickness-gradient contribution to the driving pressure gives volume flux per unit width $q=D{h}^3$, where $D=g\sin \alpha /(3\nu )$. Radial mass conservation isThe distance $x$ is measured along the slope; the horizontal radius is $x\cos \alpha$, so total volume is $2\pi \cos \alpha \int xhdx$.

Even when most of an unconfined aquifer with depth-dependent permeability is deep, its height vanishes at an absorbing outlet. Near that outlet, finite outward volume flux per unit width $Q$ gives $u_{b}h{h}_x\simeq Q$ and hence $h\simeq (2Qx/u_b{)}^{1/2}$. The high-depth outer approximation instead has $h\simeq [6Qx/(u_b\beta ){]}^{1/3}$. Their crossover lies at height of order $1/\beta$ and distance of order $u_b/({\beta }^{2}Q)$; the thin inner boundary layer transmits the same leading discharge.

The volume flux per unit width isThereforeThe first is the stress-free film; the second has an effectively immobilized free surface.

In the steady state, mass conservation reduces to $q_x=R$. The zero-flux condition at the divide gives $q=R(x-L)$. Substituting the Darcy flux and integrating from the absorbing river boundary yieldsThus the nonnegative groundwater height is the unique nonnegative root ofThe height increases toward the divide, where $h_x=0$. The river receives all the rainfall, so its steady volume flux per unit width is $\boxed{Q=RL}$.

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 $RL$.

Use the Dupuit approximation: the saturated region is shallow enough that hydrostatic pressure is $p=\rho g(h-z)$ and flow is predominantly horizontal. Darcy's law then gives the horizontal Darcy velocity $u_x=-u_b(1+\beta z)h_x$, where $u_b=k_0\rho g/\mu$. Integrating through the saturated depth gives the volume flux per unit widthThe stored water volume per unit area is $\varphi h$, so mass conservation yields the unconfined aquifer with depth-dependent permeability equationDefine the positive river discharge by $Q(t)=-q(0,t)$. A useful check on all subsequent results is the integrated mass conservation law

For $\varphi h_t=D_m(h^m{h}_x{)}_x$ on $0<x<L$, with $h(0,t)=0$ and zero right-hand volume flux, a separation of variables giveswhere the positive profile satisfiesIts boundary volume flux per unit width isThese profiles are exact separated solutions and describe the leading long-time discharge for a broad class of positive initial data. The virtual origin $t_0$ depends on the initial profile; it does not make the separated solution an exact representation of every initial condition.