Boussinesq equation for an unconfined aquifer

For pore fraction $n$, permeability $k$, liquid density $\rho$, viscosity $\mu$, rainfall recharge $R$, and a shallow groundwater thickness $h$,This is a nonlinear diffusion equation with diffusivity proportional to the local thickness.

After recharge stops on a finite aquifer connected to a river, the late solution of the Boussinesq equation for an unconfined aquifer has the form $h\sim t^{-1}F(x/L)$. The profile solves a nonlinear eigenvalue problem, and river discharge decays as $t^{-2}$.