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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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{\bf u} = - \frac{k}{\mu} \cdot ( \nabla p |
where
where
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| body | \displaystyle M = \frac{k}{\mu} |
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Darcy flow only happens for relatively slow percolation:
.For a wider range of flow regimes see Forchheimer Equation.
In multiphase flow the 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.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:
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| anchor | I7FARZLYNK |
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{\bf u}_{\alpha} = - M\frac{k_{\alpha}}{\mu_{\alpha}} \cdot ( \nabla p =_{\alpha} - \left< \frac{k}rho_{\mualpha} \right, > {\cdotbf \nabla p |
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where
displaystyle M = \left< \frac{k}{\mu} \right>multi-phase mobility
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 mobilityDarcy flow only happens for relatively slow percolation:
{ \rm Re} < 2,000 | (see also Linear Perrine multi-phase diffusion @model). For a wider range of flow regimes see Forchheimer Equation.
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.
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