@wikipedia
The momentum balance equation relating a pressure gradient in subsurface reservoir with the induced fluid flow
in porous medium with induced fluid flow (percolation) with velocity :
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{\bf u} = - M \cdot ( \nabla p - \rho \, {\bf g} ) |
where
In single-phase flow the Darcy flow equation takes a following form:
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alignment | left |
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{\bf u} = - \frac{k}{\mu} \cdot ( \nabla p - \rho \, {\bf g} ) |
where
{\bf u} | flow velocity vector | |
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 LaTeX Math Inline |
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body | --uriencoded--%7B\bf u%7D_\alpha |
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and Darcy flow equation is applicable for each phase independently:
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{\bf u}_{\alpha} = - \frac{k_{\alpha}}{\mu_{\alpha}} \cdot ( \nabla p_{\alpha} - \rho_{\alpha} \, {\bf g} ) |
where
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 LaTeX Math Block Reference |
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using the multi-phase mobilityOnly valid for the slow flow: (see also Linear Perrine multi-phase diffusion @model).
See also
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Physics / Fluid Dynamics / Percolation
[ Forchheimer Equation ][ Linear Perrine multi-phase diffusion @model ]
References
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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.{ \rm Re} < 1