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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:

- 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) |

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

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

- unpermeable boundary:
**v_n = 0**

- 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.