Groundwater discharge

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).

• Groundwater flow equation
• Groundwater energy balance
• Submarine groundwater discharge
• Discharge (hydrology) – for rivers
• volumetric flow rate
• flux (transport definition)
• Darcy's Law

• Freeze, R.A. & Cherry, J.A., 1979. Groundwater, Prentice-Hall. ISBN 0-13-365312-9