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

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 $t^{-2}$.

Let ${\rho }_s$ be the grain density and $g^{\prime }=g/g_0$. Vertical force balance for the poroelastic aquifer giveswith compression taken negative in ${\sigma }_e=E({\varphi }_0-\varphi )/{\varphi }_0$. HenceThe zero-effective-stress condition $\varphi (h)={\varphi }_0$ then givesThis identifies the poroelastic compaction length.

The water volume per unit horizontal distance isFor $h\ll {\ell }_c$, a Taylor expansion yieldsStorage conservation is $V_t+q_x=R$. Integrating from river to divide and using $q(L,t)=0$ gives the river influxFor $g^{\prime }=1+ϵe^{i\omega t}$ and a groundwater profile that changes little during one tide, the direct storage oscillation has magnitudeWith typical thickness $H$ and mean recharge discharge ${\overline{Q}}_r=RL$,Using the steady balance $R\sim K{H}^2/L^2$ gives the equivalent estimate $|\delta Q_r|/{\overline{Q}}_r\sim ϵ\omega {\varphi }_0{L}^2/(2{\ell }_{c}K)$.