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##
Fluid flow in porous media

H. Darcy dicovered in his contribution *Les fontaines publiques de la
ville de Dijon* Dalmont: Paris 1863 the basic equations of porous media:
where
is the velocity,
is the pressure and
the body force. The coeficients
describe the viscosity and permeability. Darcy's equations are only true
in regions where the velocity is not too large. Note that the measure of
porosity is the fraction of fluid per volume devided by the total volume
of the porous body. In the case that this fraction is near to 1 H.C. Brinkman
added in 1957 a further term. The arising system is now called *Brinkman
equations*
Here
is called *effective viscosity*. For situations where the fluid velocity
is large enough the Darcy equations are usually modified by the so-called
*Forchheimer equations*. These equations read as follows:
Note that usually this system with
bears the name Forchheimer equation. The analysis of convective fluid flow
in an porous medium where the viscosity is considerable varying ( with
temperatur or with salt concentration ) it is necessary to use a combination
of both Brinkman and Forchheimer model. We obtain
This last model characterizes a non-slow flow in a saturated porous medium.
The solution depends continuously on the *Forchheimer coefficient *
and also from the *Brinkman coefficient*.
If the effective viscosity tends to zero then the limit model is called
*Darcy-Forchheimer equations*. It is known that the energy decays
exponentially.

**A late reference:** L. E. Payne, J.F. Rodrigues and Straugham B.
*Effect of anisotropic permeability on Darcy's law.*. MMAS 24 ???
- ???.

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collected and worked out by Wolfgang Sprößig, TU-Bergakademie
Freiberg