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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:
\begin{eqnarray*}\frac{\mu}{k}v&=& - {\rm grad} p + f\quad\mbox{(Darcy equations)}\\{\rm div}\;v&=&0\end{eqnarray*}
where $v$ is the velocity, $p$ is the pressure and $f$ the body force. The coeficients $\mu, k$ 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
\begin{eqnarray*}\frac{\mu}{k}v&=& - {\rm grad} p + f + \lambda \Delta v,\\{\rm div}\;v&=&0.\end{eqnarray*}
Here $\lambda $ 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:
\begin{eqnarray*}\frac{\mu}{k}v + a\vert v\vert v + b\vert v\vert^2v&=& - {\rm grad} p + f ,\\{\rm div}\;v&=&0.\end{eqnarray*}
Note that usually this system with $b=0$ 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
\begin{eqnarray*}\frac{\partial v}{\partial t}=\lambda \Delta v - av + b\vert v\vert v&=& - {\rm grad} p + f ,\\{\rm div}\;v&=&0.\end{eqnarray*}
This last model characterizes a non-slow flow in a saturated porous medium. The solution depends continuously on the Forchheimer coefficient $b$ and also from the Brinkman coefficient$\lambda $. 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