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\sum_\alpha \rho_\alpha \bigg[ \frac{\partial u_\alpha}{\partial t} + u_\alpha \frac{\partial u_\alpha}{\partial l} - \nu_\alpha \Delta u_\alpha\bigg] = - \frac{dp}{dl} + \rho_m \, g \, \sin \theta - \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d} |
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| indicates a mixture of fluid phases |
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| water, oil, gas phase indicator |
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| measure length along wellbore trajectory |  Image Modified
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| cross-sectional average fluid density |
| wellbore trajectory inclination to horizon |
| cross-sectional average pipe flow diameter |
| in-situ cross-sectional area | LaTeX Math Inline |
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| body | A(l) = 0.25 \, \pi \, d^2(l) |
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| Darci flow friction coefficient |
| kinematic viscosity of -phase |
Equations
| LaTeX Math Block Reference |
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| LaTeX Math Block Reference |
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define a closed set of 3 scalar equations on 3 unknowns: pressure
, temperature
and fluid velocity
.
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