Stationary groundwater flow

The governing equations of stationary groundwater flow are [28]

$\displaystyle \boldsymbol{ v} = - \boldsymbol{k} \cdot \nabla h$ (378)

(also called Darcy's law) and

$\displaystyle \nabla \cdot \boldsymbol{ v} = 0,$ (379)

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

$\displaystyle h = \frac{p}{\rho g} + z.$ (380)

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

$\displaystyle \nabla \cdot (- \boldsymbol{ k} \cdot \nabla h) = 0.$ (381)

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:

    $\displaystyle h = \frac{p_0 - \rho g z}{\rho g} + z = \frac{p_0}{\rho g}.$ (382)

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

    $\displaystyle h = \frac{p_0}{\rho g} + z.$ (383)

  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

    $\displaystyle h = \frac{p_0}{\rho g} + z.$ (384)

    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.