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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:

${\displaystyle Q={\frac {dh}{dl}}KA}$

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.

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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, ${\textstyle \Phi }$ [L3·T−1], is defined in such way that its gradient equals the discharge vector.[1]

${\displaystyle Q_{x}=-{\frac {\partial \Phi }{\partial x}}}$

${\displaystyle Q_{y}=-{\frac {\partial \Phi }{\partial y}}}$

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

${\displaystyle \Phi =KH\phi }$

and for unconfined shallow flow as

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

where

${\textstyle H}$ is the thickness of the aquifer [L],
${\textstyle \phi }$ is the hydraulic head [L], and
${\textstyle C}$ 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

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

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).