-  \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \,  | {\bf u} | \, {\bf u}

 is called Forchheimer coefficient and depends on flow regime and permeability as:

\beta = \frac{C_E}{\sqrt(k)}

where   is called Ergun constant and accounts for inertial (kinetic) effects and depends on flow regime only.

  is small for the small flow velocities (reducing Forchheimer equation t Darcy equation) and grows quickly for high flow velocities.

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

{\bf u} =  - \frac{k}{\mu} \, k_f \, \nabla p

where

k_f(|\nabla p|) =  \frac{2}{w} \big[ 1- \sqrt{1-w}   \big]

and

w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \,  |\nabla p| \, < \, 1