The momentum balance equation relating a pressure gradient \nabla p in subsurface reservoir with the induced fluid flow {\bf u}:
(1) | - \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
where
{\bf u} | flow velocity vector |
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\nabla p | pressure gradient |
k | formation permeability |
\mu | fluid viscosity |
\beta | Forchheimer coefficient |
Forchheimer coefficient depends on flow regime and formation permeability as:
(2) | \beta = \frac{C_E}{\sqrt{k}} |
where C_E is dimensionless quantity called Ergun constant accounting for inertial (kinetic) effects and depending on flow regime only.
C_E is small for slow percolation (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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See also
Physics / Fluid Dynamics / Percolation