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

Let $n=1-\varphi$ 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 widthLocal mass conservation $n{h}_t+q_x=R$ therefore gives the Boussinesq equation for an unconfined aquiferThe river and drainage-divide conditions are