Let $n=1-\phi$ be the pore fraction and put $K=k\rho g/\mu$. In the <long-wave approximation>, the pore pressure is hydrostatic, $p=\rho g(h-z)$, so <Darcy's law> gives the horizontal <volume flux> per unit width
$$
q=\int_0^h-\frac{k}{\mu}p_x\,dz
=-Khh_x.
$$
Local <mass conservation> $nh_t+q_x=R$ therefore gives the <Boussinesq equation for an unconfined aquifer>
$$
\boxed{(1-\phi)h_t=\frac{k\rho g}{\mu}(hh_x)_x+R.}
$$
The river and drainage-divide conditions are
$$
h(0,t)=0,
\qquad
q(L,t)=0.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 332 / 1 / a / Solution

After recharge stops on a finite aquifer connected to a river, the late solution of the <Boussinesq equation for an unconfined aquifer> has the form $h\sim t^{-1}F(x/L)$. The profile solves a nonlinear eigenvalue problem, and river discharge decays as $t^{-2}$.

Separable aquifer drawdown

For pore fraction $n$, permeability $k$, liquid density $\rho$, viscosity $\mu$, rainfall recharge $R$, and a shallow groundwater thickness $h$,
$$
n h_t=\frac{k\rho g}{\mu}(hh_x)_x+R.
$$
This is a nonlinear diffusion equation with diffusivity proportional to the local thickness.


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