Rain-fed aquifer similarity solution ($h=tf(x/t)$)

For the dimensionless rainfall-driven Boussinesq equation for an unconfined aquifer, $h_t=(h{h}_x{)}_x+1$, balancing every time power in $h=t^{a}f(x/t^b)$ gives $a=b=1$. The similarity profile obeys $(f{f}^{\prime }{)}^{\prime }+\eta f^{\prime }-f+1=0$. A river drained to base level gives $f(0)=0$, while undrained storage far away gives $f(\infty )=1$. The solution approximates a finite aquifer before its drainage-affected length reaches the far boundary. If $J={\lim }_{\eta \downarrow 0}f{f}^{\prime }>0$, the dimensional drainage flux is $\sqrt{K}{R}^{3/2}Jt/\varphi$.