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| LaTeX Math Block |
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| anchor | MatBal |
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| alignment | left |
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\frac{\partial (\rho_m A)}{\partial t} + \frac{\partial}{\partial l} \bigg( A \, \sum_\alpha \rho_\alpha \, u_\alpha \bigg) = 0 |
| LaTeX Math Block |
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| anchor | gradP NS |
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| alignment | left |
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\frac{dp}{dl} = \rho_m \, g \, \sin \theta - \sum_\alpha \rho_\alpha \bigg[ \frac{\partial u_\alpha}{\partial t} + u_\alpha \frac{\partial u_\alpha}{\partial l} \bigg] = - \frac{dp}{dl} + \rho_m \, g \, \sin \theta - \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d} |
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The disambiguation of the properties in the above equation is brought in The list of dynamic flow properties and model parameters.
Equation Equations
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defines the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component
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| body | \{ m_W, \ m_O, \ m_G \} |
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during its transportation along wellbore.
Equation Equations
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define defines the motion dynamics of each phase
(called Navier–Stokes equation), represented as linear correlation between phase flow speed
and
partial pressure
gradient profile of
this phase mutliphase fluid .
The term
| LaTeX Math Inline |
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| body | \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l} |
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represents heat convection defined by the wellbore mass flow.
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