Unconfined aquifer

An unconfined aquifer has a free groundwater table. In a shallow aquifer of thickness $h(x,t)$, hydrostatic pressure and Darcy's law give horizontal flux proportional to $-h{h}_x$.

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.

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 $t^{-2}$.

The Dupuit approximation treats flow in a shallow unconfined aquifer as predominantly horizontal, with hydrostatic pressure $p=\rho g(h-z)$ and horizontal pressure gradient independent of depth. Integrating Darcy's law over the saturated depth then gives a volume flux per unit width depending only on the groundwater height and its slope.

For saturated Darcy flow through a long dam occupying a channel of width $\alpha z^2$, let hydraulic conductivity be $K=k\rho g/\mu$ and free-surface height $h(x)$. The Dupuit approximation gives $Q=-K\alpha h^3{h}_x/3$. Fixed heads $h(0)=H$ and $h(L)=\eta$ give $h^4=H^4-(H^4-{\eta }^4)x/L$ and the displayed steady discharge. Porosity relates pore velocity to superficial Darcy velocity but does not independently multiply this discharge when $k$ is intrinsic permeability. For depth-dependent permeability, integrate the horizontal conductivity over the saturated cross-section first.

With permeability of a porous medium $k(z)=k_0(1+\beta z)$, the Dupuit approximation givesHere $\varphi$ is porosity and $R$ is recharge. The low-height limit has mobility proportional to $h$, whereas large heights have mobility proportional to $h^2$. This changes both the forced filling similarity for power-law diffusion and the separable draining profile for power-law diffusion.

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.

A poroelastic aquifer couples pore-fluid storage and flow to elastic compaction of its solid skeleton. Changes in gravity, surface loading, or pore pressure can therefore change groundwater discharge even without changing rainfall.

For a skeleton with Young modulus $E$, undeformed solid fraction ${\varphi }_0$, grain-fluid density difference ${\rho }_s-\rho$, and reference gravity $g_0$, the hydrostatic poroelastic compaction length isIt is the depth over which self-weight changes the solid fraction appreciably.