The momentum balance equation relating a pressure gradient \nabla p in porous medium with induced fluid flow (percolation) with velocity {\bf u}:
| (1) | {\bf u} = - M \cdot \nabla p = - \frac{k}{\mu} \cdot \nabla p |
where
\displaystyle M = \frac{k}{\mu} | fluid mobility |
|---|---|
k | formation permeability |
\mu | fluid viscosity |
In multiphase flow the Darcy flow equation is applicable for each phase independently.
In most popular case of a 3-phase Oil + Gas + Water fluid model with relatively homogeneous flow (phases may move at different velocities but occupy the same reservoir space and have the same phase pressure) the Darcy flow equation can be approximated with Perrine model of Multi-phase Mobility:
| (2) | {\bf u} = - M \cdot \nabla p = - \left< \frac{k}{\mu} \right > \cdot \nabla p |
where
\displaystyle M = \left< \frac{k}{\mu} \right> | multi-phase mobility |
|---|
Darcy flow only happens for relatively slow percolation: { \rm Re} < 2,000
For a wider range of flow regimes see Forchheimer Equation.
See also
Physics / Fluid Dynamics / Percolation
References
Jules Dupuit (1863). Etudes Théoriques et Pratiques sur le mouvement des Eaux dans les canaux découverts et à travers les terrains perméables (Second ed.). Paris: Dunod.