Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 69 / 2

The required boundary model is a fracture that drains after the storm, with $h(0,t)=0$, and a current of finite extent whose flux vanishes at its advancing nose. Introduce mass density $\rho$, gravity $g$, constant porosity $\varphi$ and dynamic viscosity $\mu$. Hydrostatic Darcy law gives the depth-integrated discharge $q=-k\rho gh{h}_x/\mu$. Fluid-volume conservation then gives the Boussinesq equation for an unconfined aquiferThe fluid volume per fracture length is $V=\varphi {\int }_0^{\infty }hdx$; omitting the constant $\varphi$ does not change its fractional decay rate.

The conserved first moment of a draining porous current follows directly:The finite outlet discharge does not contribute to $[xh{h}_x]$ at $x=0$. This establishes $D=\int xhdx$ constant. An unspecified nonzero fracture head would instead give $\dot{D}=\kappa h(0,t{)}^2/2$; the drained boundary is essential.

For the dipole similarity solution of a draining porous current, set $h=a{t}^{\alpha }f(\eta )$ and $\eta =x/(d{t}^{\beta })$. The conserved first moment requires $\alpha +2\beta =0$, while matching the time powers in the evolution equation gives $\alpha -1=2\alpha -2\beta$. HenceChoose the nose at $\eta =1$ and the normalization $\kappa a/d^2=1$. The profile equation becomesFor $f=b{\eta }^2+c{\eta }^{1/2}$, the two sides have coefficients $-b,-5c/8$ and $6b^2,15bc/4$, respectively. The nonzero positive profile therefore has $b=-1/6$; its zero at $\eta =1$ gives $c=1/6$. Its normalization is determined byThus a complete choice of the constants isSet $h=0$ beyond the nose. The square-root behavior at the outlet allows finite drainage: $f{f}^{\prime }\to 1/72$ as $\eta \downarrow 0$, while $f{f}^{\prime }\to 0$ at the nose.

Since ${\int }_0^{1}fd\eta =1/18$, the volume isThe drainage volume decay in a dipole porous current shows that the printed positive rate cannot describe drainage and its factor is also inconsistent with the conserved-moment scaling. This solution supplies a direct counterexample to that rate and the corrected one. If the initial volume $V_0$ is prescribed at the start of a similarity phase, replace $t$ by $t+t_0$ with $t_0=(\varphi ad/(18V_0){)}^4$. A general post-storm initial shape need not be exactly self-similar; this is the dipole similarity profile and its long-time scaling, not an asserted exact profile for every initial condition. The initial volume alone does not specify $D$.