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.

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.

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 $\sqrt{K}{R}^{3/2}Jt/\varphi$.

For $h_t=\kappa (h{h}_x{)}_x$ with a drained boundary $h(0,t)=0$ and a finite current vanishing at its nose, integration by parts gives $\dot{D}=-\kappa [h^2{]}_0^{\infty }/2=0$. The outlet can have finite volume flux because its contribution to the first-moment flux is multiplied by $x=0$. A positive prescribed boundary height instead gives $\dot{D}=\kappa h(0,t{)}^2/2$, so drainage is essential to this invariant.

The conserved first moment of a draining porous current fixes $\alpha +2\beta =0$ in a self-similar solution $h=a{t}^{\alpha }f(x/(d{t}^{\beta }))$. The Boussinesq equation for an unconfined aquifer supplies $\alpha =2\beta -1$, giving $\alpha =-1/2$, $\beta =1/4$. Normalize $\kappa a=d^2$ and the nose at $\eta =1$. Then $-f/2-\eta f^{\prime }/4=(f{f}^{\prime }{)}^{\prime }$ is solved byThe square-root outlet has finite drainage flux and the nose has zero flux. This is a particular similarity solution and its scaling family, not an exact description of arbitrary initial data.

For the dipole similarity solution of a draining porous current, $V=\varphi \int hdx=\varphi ad{t}^{-1/4}/18$. Thus $\dot{V}/V=-1/(4t)$. A shifted similarity origin replaces $t$ by $t+t_0$. Neither a positive rate nor a rate $-1/(2t)$ is compatible with this two-dimensional first-moment-conserving profile.

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 $t^{-2}$.