@wikipedia
The momentum balance equation relating a pressure gradient gradient
in
subsurface reservoir with the induced porous medium with induced fluid flow (percolation) with velocity :| LaTeX Math Block |
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- \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
where
| {\bf u} | flow velocity vector | pressure gradient |
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| LaTeX Math Inline |
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body |
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 |
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| {\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p |
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where
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| k_f(|\nabla p|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big] |
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and
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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.
