 Outlet layer in a deep unconfined aquifer

Even when most of an unconfined aquifer with depth-dependent permeability is deep, its height vanishes at an absorbing outlet. Near that outlet, finite outward volume flux per unit width $Q$ gives $u_{b}h{h}_x\simeq Q$ and hence $h\simeq (2Qx/u_b{)}^{1/2}$. The high-depth outer approximation instead has $h\simeq [6Qx/(u_b\beta ){]}^{1/3}$. Their crossover lies at height of order $1/\beta$ and distance of order $u_b/({\beta }^{2}Q)$; the thin inner boundary layer transmits the same leading discharge.