Page tree


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} )



fluid mobility


gradient operator


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} )



formation permeability


fluid viscosity

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} )



phase pressure of the \alpha-phase


phase permeability of the \alpha-phase


fluid density of the \alpha-phase


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 ]


 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.

  • No labels