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 |
|---|---|
\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 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:
| (3) | {\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p |
where
| (4) | k_f(|\nabla p|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big] |
and
| (5) | w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, |\nabla p| \, < \, 1 |
See also
Physics / Fluid Dynamics / Percolation