= Solution
{c}
The condition $f(0)=0$ means the river maintains the <water table> at the impermeable base level: $h(0,t)=0$. The liquid is exposed to <atmospheric pressure> there. It does not imply zero drainage, because $h_x$ may be singular at the outlet.
Far from the river, lateral drainage has not significantly affected storage. Rainfall simply fills pore volume, so the dimensional height is $h\sim Rt/\phi$. In the dimensionless variables this is $h\sim t$, giving $f(\infty)=1$.
Although the physical <aquifer> is finite, these <boundary conditions> describe its early <similarity solution> before the drainage-affected region reaches $x=L$. In dimensional variables its horizontal extent is of order $\sqrt{KR}\,t/\phi$, so the semi-infinite approximation requires $t\ll T=\phi L/\sqrt{KR}$.
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