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 - \rho \, {\bf g} ) |
where
M | fluid mobility | \nabla | gradient operator |
|---|---|---|---|
\rho | fluid density | { \bf g } | gravity vector pointing along Earth's Gravity Direction |
In single-phase flow the Darcy flow equation takes a following form:
| (2) | {\bf u} = - \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g} ) |
where
Darcy flow only happens for relatively slow percolation: { \rm Re} < 2,000.
For a wider range of flow regimes see Forchheimer Equation.
In multiphase flow the different phases \alpha move with different velocities {\bf u}_\alpha and Darcy flow equation is applicable for each phase independently:
| (3) | {\bf u}_{\alpha} = - \frac{k_{\alpha}}{\mu_{\alpha}} \cdot ( \nabla p_{\alpha} - \rho_{\alpha} \, {\bf g} ) |
where
p_\alpha | phase pressure of the \alpha-phase | k_\alpha | phase permeability of the \alpha-phase |
\rho_\alpha | fluid density of the \alpha-phase | \mu_\alpha | fluid viscosity of the \alpha-phase |
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:
| (4) | {\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 |
|---|
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.