= Solution
{c}
Let $J_X=\lim_{x\downarrow0}X(x)X'(x)>0$. The spatial <ordinary differential equation> gives $(XX')'=-\lambda X$, so its leading outlet behavior is
$$
\boxed{X(x)\sim\sqrt{2J_Xx},\qquad
X'(x)\sim\sqrt{\frac{J_X}{2x}}.}
$$
Again this is a <square-root drainage boundary>. For the <separable aquifer drawdown>, the dimensional drainage flux is
$$
\boxed{Q_0(t)=KT(t)^2J_X
=\frac{KJ_X}{[1+(K\lambda/\phi)(t-t^*)]^2}.}
$$
Equivalently, with $\tau_d=\phi/(K\lambda)$, it is $Q_0(t^*)[1+(t-t^*)/\tau_d]^{-2}$ for an exact separated profile. Integrating the spatial <ordinary differential equation> gives $J_X=\lambda\int_0^LX\,dx$. This verifies the global <mass conservation> check
$$
\frac d{dt}\left(\phi T\int_0^LX\,dx\right)=-KJ_XT^2=-Q_0(t).
$$
Thus the late drainage flux follows an inverse-square <power law>, while stored water decays inversely with time.
The <aquifer drainage hydrograph> starts at zero, rises linearly in the early rainfall <similarity solution>, and approaches the total recharge $RL$ if rainfall lasts long enough for a finite-length <steady state>. Indeed, steady <mass conservation> gives $q_x=R$ and $q(L)=0$, hence $Q_0=RL$. At $t^*$, the outlet flux stays continuous for a regular inherited flux profile, then recedes as water drains; the eventual decay is proportional to $(t-t^*+\text{constant})^{-2}$. An arbitrary transition requires the full time-dependent <partial differential equation>, so matching an early rainfall formula directly to a separated drought formula is only a schematic approximation.
\Image[/past-exam-of-the-topics-in-fluid-mechanics-course-of-the-university-of-oxford/2023-aquifer-drainage.png]
{title=Schematic aquifer drainage response during rainfall and drought}
The curve illustrates these regimes and a possible $t^*$. Its numerical shape and the relative duration of the regimes are schematic; the early rainfall and late drought formulas apply in their respective asymptotic ranges.
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