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 = d h d l K A {\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.
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, Φ {\textstyle \Phi } [L3·T−1], is defined in such way that its gradient equals the discharge vector.
Q x = − ∂ Φ ∂ x {\displaystyle Q_{x}=-{\frac {\partial \Phi }{\partial x}}}
Q y = − ∂ Φ ∂ y {\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
Φ = K H ϕ {\displaystyle \Phi =KH\phi }
and for unconfined shallow flow as
Φ = 1 2 K ϕ 2 + C {\displaystyle \Phi ={\frac {1}{2}}K\phi ^{2}+C}
where
H {\textstyle H} is the thickness of the aquifer [L],
ϕ {\textstyle \phi } is the hydraulic head [L], and
C {\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
∂ 2 Φ ∂ x 2 + ∂ 2 Φ ∂ y 2 = 0 {\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, such as uniform flow, multiple wells, analytical elements (analytic element method).
See also Groundwater flow equation Groundwater energy balance Submarine groundwater discharge Discharge (hydrology) Flux (transport definition) Darcy's law
References
Freeze, R.A. & Cherry, J.A., 1979. Groundwater, Prentice-Hall. ISBN 0-13-365312-9
