= Solution
{c}
Set $x=L\tilde x$, $h=H\tilde h$, and $t=T\tilde t$. The <Boussinesq equation for an unconfined aquifer> becomes
$$
\frac{\phi H}{T}\tilde h_{\tilde t}
=\frac{KH^2}{L^2}(\tilde h\tilde h_{\tilde x})_{\tilde x}+R.
$$
Balance rainfall recharge with lateral drainage, and then with storage:
$$
\boxed{H=L\sqrt{\frac RK},\qquad
T=\frac{\phi H}{R}=\frac{\phi L}{\sqrt{KR}}.}
$$
Dropping tildes, the dimensionless equation is
$$
\boxed{h_t=(hh_x)_x+1.}
$$
The horizontal <Darcy velocity>, vertical <Darcy velocity>, and discharge scales are respectively $\sqrt{KR}$, $R$, and $RL$. Thus the vertical thickness scale is set by \b[rainfall balanced against <hydraulic conductivity> over the finite drainage length $L$], rather than by <porosity>. The latter determines the storage time $T$. The <Dupuit approximation> requires $H/L=\sqrt{R/K}\ll1$ for these scales. The initially dry <initial condition> is $h(x,0)=0$.
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