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 some practical cases the phases are moving in reservoir with similar velocities and have similar phase pressure which allows study of multiphase flow by aggregating them into a single-phase equivalent (1) using the multi-phase mobility M (see also Linear Perrine multi-phase diffusion @model).
See also
Physics / Fluid Dynamics / Percolation
[ Forchheimer Equation ][ Linear Perrine multi-phase diffusion @model ]
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.