The momentum balance equation relating a pressure gradient
in
subsurface reservoir with the induced fluid flow
:
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- \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
where
Forchheimer coefficient depends on flow regime and formation permeability as:
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\beta = \frac{C_E}{\sqrt{k}} |
where
is
dimensionless quantity called
Ergun constant accounting for inertial (kinetic) effects and depending on flow regime only.
is small for the slow flow (thus reducing
Forchheimer equation to
Darcy equation) and grows quickly for 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 |
where
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k_f(|\nabla p|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big] |
and
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w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, |\nabla p| \, < \, 1 |