The momentum balance equation relating a pressure gradient $\begin{array}{l}\nabla p\end{array}$ in porous medium with induced fluid flow (percolation) with velocity $\begin{array}{l}\bf u\end{array}$

(1)	$\begin{array}{l}\displaystyle - \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, \| {\bf u} \| \, {\bf u}\end{array}$

where

(2)	$\begin{array}{l}\displaystyle \beta = \frac{C_E}{\sqrt{k}}\end{array}$

where $\begin{array}{l}C_E\end{array}$ is dimensionless quantity called Ergun constant accounting for inertial (kinetic) effects and depending on flow regime only.

$\begin{array}{l}C_E\end{array}$ is small for slow percolation (thus reducing Forchheimer equation to Darcy equation) and grows quickly with high flow velocities.

Forchheimer equation can be approximated by non-linear permeability model as:

(3)	$\begin{array}{l}\displaystyle {\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p\end{array}$

(4)	$\begin{array}{l}\displaystyle k_f(\|\nabla p\|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big]\end{array}$

(5)	$\begin{array}{l}\displaystyle w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, \|\nabla p\| \, < \, 1\end{array}$

Reference

Philipp Forchheimer (1886). "Über die Ergiebigkeit von Brunnen-Anlagen und Sickerschlitzen". Z. Architekt. Ing.-Ver. Hannover. 32: 539–563.

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