Stationary groundwater flow

The governing equations of stationary groundwater flow are [25]

$$\boldsymbol{ v} = - \boldsymbol{k} \cdot \nabla h$$ (111)

(also called Darcy's law) and

$$\nabla \cdot \boldsymbol{ v} = 0,$$ (112)

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

$$h = \frac{p}{\rho g} + z.$$ (113)

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

$$\nabla \cdot (- \boldsymbol{ k} \cdot \nabla h) = 0.$$ (114)

Accordingly, by comparison with the heat equation, the correspondence in Table  (15) arises. Notice that the groundwater flow equation is a steady state equation, and there is no equivalent to the heat capacity term.

Table 15: 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:

   $$h = \frac{p_0 - \rho g z}{\rho g} + z = \frac{p_0}{\rho g}.$$ (115)

   
   

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

   $$h = \frac{p_0}{\rho g} + z.$$ (116)

   
   

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

   $$h = \frac{p_0}{\rho g} + z.$$ (117)

   
   

   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.