Formulation about the potential

In a saturated aquifer with constant density, the differential equation for calculating a steady state flow condition can be calculated by coupling Darcy's law:

with the continuity condition:

Which gives:

With:

h = Potential head [m],

v_f = Darcy velocity [m/s],

K = symmetrical tensor for the K-value [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 quantity. In the case of an anisotropic aquifer, the tensor property of the K value must be taken into account, as the hydraulic conductivity values at one location will vary in different directions.

In addition to the mathematical formulation of the steady state flow equation via the potential head (h), the pressure (p) or the pressure head (ψ) can also be used as primary variables. This is particularly necessary if density-dependent flow conditions are considered or a variably saturated calculation is to be carried out. The variables "saturation" and "relative K-value" for variably saturated conditions are also formulated in the literature as variables dependent on the pressure or pressure head.

Formulation using the pressure equation