(1) | - \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
\beta is called Forchheimer coefficient and depends on flow regime and permeability as:
(2) | \beta = \frac{C_E}{sqrt(k)} |
where C_E is called Ergun constant and accounts for inertial (kinetic) effects and depends on flow regime only.
C_E is small for the small flow velocities (reducing Forchheimer equation t Darcy equation) and grows quickly for high flow velocities.
Forchheimer equation can be expressed in terms of non-linear permeability model as:
(3) | {\bf u} = - k_f \frac{k}{\mu} \nabla p |
where
(4) | k_f(w) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big], |
and
(5) | w = 4 \big(\frac{k}{\mu} \big)^2 \beta \rho |\nabla p| \, < \, 1 |