Formulation of the potential

Assuming a saturated aquifer, constant density and steady state conditions, the flow equation is based on Darcy's Law:

 

and the fluid mass balance

 

that can be combined to

 

With:

 

h = Potential head [m],

v_f = Darcy velocity [m/s],

K = symmetric tensor for the permeability [m/s],

q = source/sink term [1/s].

 

For the flow in a homogeneous, isotropic medium, the K-value can be regarded as a scalar variable. In an anisotropic aquifer the tensor property of the K-value must be considered, since the permeabilities in one point are different in size in different directions.

Besides the mathematical formulation of the steady state flow equation with the potential h, the pressure p or the pressure head ψ can be used as the primary variable. This is particularly necessary when density-dependent flow conditions are considered, or a saturated/unsaturated calculation should be carried out. The sizes of "saturation" and "relative K-value" of saturated/unsaturated conditions are formulated in the literature as well as from the pressure or the pressure head dependent variables.

 

Formulation of the pressure equation