@wikipedia
The momentum balance equation relating a pressure gradient
in porous medium with induced fluid flow (percolation) with velocity A generalization of Darcy equation for the flow with account for inertial (kinetic) effects at high velocity reservoir flow:| LaTeX Math Block |
|---|
|
- \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
where
{\bf u} | pressure gradient |
|---|
| LaTeX Math Inline |
|---|
body | flow velocity vector |
Forchheimer coefficient depends on flow regime and formation permeability as:
...
where
is
Dimensionless dimensionless quantity called
Ergun constant accounting for inertial (kinetic) effects and
depends depending on
flow regime only only.
is small for
the slow flow slow percolation (thus reducing
Forchheimer equation to
Darcy equation) and grows quickly
for with high flow velocities.
Forchheimer equation can be approximated by non-linear permeability model as:
| LaTeX Math Block |
|---|
| {\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p |
|
...
|
| LaTeX Math Block |
|---|
| k_f(|\nabla p|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big] |
|
and
|
| LaTeX Math Block |
|---|
| w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, |\nabla p| \, < \, 1 |
|
See also
...
Physics / Fluid Dynamics / Percolation
[ Darcy Flow Equation ]
Reference
...
Philipp Forchheimer (1886). "Über die Ergiebigkeit von Brunnen-Anlagen und Sickerschlitzen". Z. Architekt. Ing.-Ver. Hannover. 32: 539–563.
