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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-332/1/a/solution