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

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

Before the river disturbance reaches the divide, use the similarity solutionSubstitution gives the nonlinear boundary-value problemIt can be solved numerically. If $C={\lim }_{\eta \downarrow 0}H{H}^{\prime }>0$, then the riverward flux and near-river profile areThe negative sign means flow toward decreasing $x$; the discharge into the river is $-q(0,t)$.

At steady state, $q_x=R$ and $q(L)=0$, soUsing $h(0)=0$ givesNear the river,

After rainfall stops, the late finite-domain solution is the separable aquifer drawdownThe dimensionless profile is determined byIf $C_d={\lim }_{X\downarrow 0}F{F}^{\prime }>0$, thenandThus aquifer thickness decays as $t^{-1}$ and river discharge as $t^{-2}$.