The momentum balance equation relating a pressure gradient in subsurface reservoir with the induced fluid flow
:
- \nabla p = \frac{\mu}{k} \, {\bf u} + \beta \, \rho \, | {\bf u} | \, {\bf u} |
where
| flow velocity vector | |
| pressure gradient | |
| formation permeability | |
| fluid viscosity | |
| Forchheimer coefficient |
Forchheimer coefficient depends on flow regime and formation permeability as:
\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:
{\bf u} = - \frac{k}{\mu} \, k_f \, \nabla p |
where
k_f(|\nabla p|) = \frac{2}{w} \big[ 1- \sqrt{1-w} \big] |
and
w = 4 \, \left(\frac{k}{\mu} \right)^2 \, \beta \, \rho \, |\nabla p| \, < \, 1 |
Physics / Fluid Dynamics / Percolation