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LaTeX Math Block
anchorDarcy_single
alignmentleft
{\bf u}  = -  \frac{k}{\mu}  \cdot ( \nabla p - \rho \, {\bf g} )

where

LaTeX Math Inline
bodyk

formation permeability

LaTeX Math Inline
body\mu

fluid viscosity


Darcy flow only happens for relatively slow percolation

LaTeX Math Inline
body{ \rm Re} < 2,000
.

For a wider range of flow regimes see Forchheimer Equation.


In multiphase flow the different phases 

LaTeX Math Inline
body\alpha
move with different velocities 
LaTeX Math Inline
body--uriencoded--%7B\bf u%7D_\alpha
 and  Darcy flow equation is applicable for each phase independently:

LaTeX Math Block
anchorZLYNK
alignmentleft
{\bf u}_{\alpha}  = -  \frac{k_{\alpha}}{\mu_{\alpha}}  \cdot ( \nabla p_{\alpha} - \rho_{\alpha} \, {\bf g} )

where

LaTeX Math Inline
bodyp_\alpha

phase pressure of the 

LaTeX Math Inline
body\alpha
-phase

LaTeX Math Inline
bodyk_\alpha

phase permeability of the 

LaTeX Math Inline
body\alpha
-phase

LaTeX Math Inline
body\rho_\alpha

fluid density of the 

LaTeX Math Inline
body\alpha
-phase

LaTeX Math Inline
body\mu_\alpha

fluid viscosity of the 

LaTeX Math Inline
body\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:

...

LaTeX Math Inline
body\displaystyle M = \left< \frac{k}{\mu} \right>


multi-phase mobility

Darcy flow only happens for relatively slow percolation

LaTeX Math Inline
body{ \rm Re} < 2,000

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See also

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Physics /  Fluid Dynamics / Percolation

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