Square-root drainage boundary ($h\sim \sqrt{2Jx},h{h}_x\to J$)

A zero-height outlet can carry finite nonzero volume flux per unit width in a nonlinear Darcy flow model with $q=-Kh{h}_x$. If $h{h}_x\to J>0$ as $x\downarrow 0$, then $(h^2{)}^{\prime }\to 2J$, and integration gives $h\sim \sqrt{2Jx}$ and $h_x\sim \sqrt{J/(2x)}$. Thus the height vanishes while its slope diverges; the flux must be evaluated as a limit of the product. The divergent slope also identifies a region where the outer Dupuit approximation eventually needs a local correction.