A generalization of Darcy equation for the flow with account for inertial (kinetic) effects at high velocity reservoir flow:

-  \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 to 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