Let $\rho_s$ be the grain density and $g'=g/g_0$. Vertical force balance for the <poroelastic aquifer> gives
$$
\frac{d\sigma_e}{dz}=\phi(\rho_s-\rho)g,
$$
with compression taken negative in $\sigma_e=E(\phi_0-\phi)/\phi_0$. Hence
$$
\frac{d\phi}{dz}=-\frac{g'}{\ell_c}\phi,
\qquad
\boxed{\ell_c=\frac{E}{\phi_0(\rho_s-\rho)g_0}.}
$$
The zero-effective-stress condition $\phi(h)=\phi_0$ then gives
$$
\boxed{\phi(z)=\phi_0\exp\left[\frac{g'(h-z)}{\ell_c}\right].}
$$
This identifies the <poroelastic compaction length>.

The water volume per unit horizontal distance is
$$
V=\int_0^h(1-\phi)\,dz.
$$
For $h\ll\ell_c$, a <Taylor expansion> yields
$$
\boxed{V=(1-\phi_0)h-\frac{\phi_0g'h^2}{2\ell_c}+O(h^3/\ell_c^2).}
$$
Storage conservation is $V_t+q_x=R$. Integrating from river to divide and using $q(L,t)=0$ gives the river influx
$$
\boxed{Q_r(t)=-q(0,t)=RL-\frac d{dt}\int_0^L V(x,t)\,dx.}
$$
For $g'=1+\epsilon e^{i\omega t}$ and a groundwater profile that changes little during one tide, the direct storage oscillation has magnitude
$$
|\delta Q_r|\sim\frac{\epsilon\omega\phi_0}{2\ell_c}\int_0^Lh^2\,dx.
$$
With typical thickness $H$ and mean recharge discharge $\overline Q_r=RL$,
$$
\boxed{\frac{|\delta Q_r|}{\overline Q_r}
\sim\frac{\epsilon\omega\phi_0H^2}{2\ell_cR}.}
$$
Using the steady balance $R\sim KH^2/L^2$ gives the equivalent estimate $|\delta Q_r|/\overline Q_r\sim\epsilon\omega\phi_0L^2/(2\ell_cK)$.

Solution

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

{title2=$\ell_c$}

For a skeleton with Young modulus $E$, undeformed solid fraction $\phi_0$, grain-fluid density difference $\rho_s-\rho$, and reference gravity $g_0$, the hydrostatic poroelastic compaction length is
$$
\ell_c=\frac{E}{\phi_0(\rho_s-\rho)g_0}.
$$
It is the depth over which self-weight changes the solid fraction appreciably.


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