Groundwater discharge

Groundwater discharge is the volumetric flow rate of groundwater through an aquifer.

Total groundwater discharge, as reported through a specified area, is similarly expressed as:
 * $$Q = \frac{dh}{dl}KA$$

where
 * Q is the total groundwater discharge ([L3·T−1]; m3/s),
 * K is the hydraulic conductivity of the aquifer ([L·T−1]; m/s),
 * dh/dl is the hydraulic gradient ([L·L−1]; unitless), and
 * A is the area which the groundwater is flowing through ([L2]; m2)

For example, this can be used to determine the flow rate of water flowing along a plane with known geometry.

The discharge potential
The discharge potential is a potential in groundwater mechanics which links the physical properties, hydraulic head, with a mathematical formulation for the energy as a function of position. The discharge potential, $\Phi$ [L3·T−1], is defined in such way that its gradient equals the discharge vector.

$$Q_x = -\frac{\partial \Phi}{\partial x}$$

$$Q_y = -\frac{\partial \Phi}{\partial y}$$

Thus the hydraulic head may be calculated in terms of the discharge potential, for confined flow as

$$\Phi = KH\phi$$

and for unconfined shallow flow as

$$\Phi = \frac{1}{2}K\phi^2+C$$

where


 * $H$ is the thickness of the aquifer [L],
 * $\phi$ is the hydraulic head [L], and
 * $C$ is an arbitrary constant [L3·T−1] given by the boundary conditions.

As mentioned the discharge potential may also be written in terms of position. The discharge potential is a function of the Laplace's equation

$$\frac{\partial^2 \Phi}{\partial x^2} + \frac{\partial^2 \Phi}{\partial y^2} = 0$$

which solution is a linear differential equation. Because the solution is a linear differential equation for which superposition principle holds, it may be combined with other solutions for the discharge potential, e.g. uniform flow, multiple wells, analytical elements (analytic element method).