@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 |
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- \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
where
Forchheimer coefficient depends is called Forchheimer coefficient and depends on flow regime and formation permeability as as:
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\beta = \frac{C_E}{\sqrt{k}} |
where
is
dimensionless number called quantity called Ergun constant accounting for inertial (kinetic) effects and
depends depending on
flow regime only only.
is small for
the small flow velocities (reducing Forchheimer equation to 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:
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| {\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p |
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| k_f(|\nabla p|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big] |
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| w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, |\nabla p| \, < \, 1 |
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See also
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Physics / Fluid Dynamics / Percolation
[ Darcy Flow Equation ]
Reference
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Philipp Forchheimer (1886). "Über die Ergiebigkeit von Brunnen-Anlagen und Sickerschlitzen". Z. Architekt. Ing.-Ver. Hannover. 32: 539–563.
