The momentum balance equation relating a pressure gradient in porous medium with induced fluid flow (percolation) with velocity
:
{\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g} ) |
where
| fluid mobility | gradient operator | ||
| fluid density | gravity vector pointing along Earth's Gravity Direction |
In single-phase flow the Darcy flow equation takes a following form:
{\bf u} = - \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g} ) |
where
| formation permeability | |
| fluid viscosity |
Darcy flow only happens for relatively slow percolation: .
For a wider range of flow regimes see Forchheimer Equation.
In multiphase flow the different phases move with different velocities
and Darcy flow equation is applicable for each phase independently:
{\bf u}_{\alpha} = - \frac{k_{\alpha}}{\mu_{\alpha}} \cdot ( \nabla p_{\alpha} - \rho_{\alpha} \, {\bf g} ) |
where
phase pressure of the | phase permeability of the | ||
fluid density of the | fluid viscosity of the |
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 using the multi-phase mobility
(see also Linear Perrine multi-phase diffusion @model).
Physics / Fluid Dynamics / Percolation
[ Forchheimer Equation ][ Linear Perrine multi-phase diffusion @model ]
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.