Stationary groundwater flow

The governing equations of stationary groundwater flow are [28]

\boldsymbol{ v} = - \boldsymbol{k} \cdot \nabla h (360)

(also called Darcy's law) and

\nabla \cdot \boldsymbol{ v} = 0, (361)

where \boldsymbol{v} is the discharge velocity, \boldsymbol{k} is the permeability tensor and h is the total head defined by

h = \frac{p}{\rho g} + z. (362)

In the latter equation p is the groundwater pressure, \rho is its density and z is the height with respect to a reference level. The discharge velocity is the quantity of fluid that flows through a unit of total area of the porous medium in a unit of time.

The resulting equation now reads

\nabla \cdot (- \boldsymbol{ k} \cdot \nabla h) = 0. (363)

Accordingly, by comparison with the heat equation, the correspondence in Table  () arises. Notice that the groundwater flow equation is a steady state equation, and there is no equivalent to the heat capacity term.

Table: Correspondence between the heat equation and the equation for groundwater flow.

heat	groundwater flow
T	h
\boldsymbol{q}	\boldsymbol{v}
q_n	v_n
\boldsymbol{\kappa}	\boldsymbol{k}
\rho h	0
\rho c	-

Possible boundary conditions are:

1. unpermeable surface under water. Taking the water surface as reference height and denoting the air pressure by p_0 one obtains for the total head:

   h = \frac{p_0 - \rho g z}{\rho g} + z = \frac{p_0}{\rho g}. (364)

2. surface of seepage, i.e. the interface between ground and air. One obtains:

   h = \frac{p_0}{\rho g} + z. (365)

3. unpermeable boundary: v_n = 0

4. free surface, i.e. the upper boundary of the groundwater flow within the ground. Here, two conditions must be satisfied: along the free surface one has

   h = \frac{p_0}{\rho g} + z. (366)

   In the direction \boldsymbol{n} perpendicular to the free surface v_n = 0 must be satisfied. However, the problem is that the exact location of the free surface is not known. It has to be determined iteratively until both equations are satisfied.