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- tubing head pressure which is set by gathering system or injection pump
- wellbore design
- pump characterisits
- fluid friction with tubing /casing walls
- interfacial phase slippage
- heat exchange between wellbore fluid and surrounding rocks
Mathematical Model
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The multiphase wellbore flow is modeled with 1D-model with
axis aligned with well trajectory.defined by the following set of 1D equations: | LaTeX Math Block |
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| anchor | divTMatBal |
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| alignment | left |
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\frac{\partial (\rho_m \,c_{pt})A)}{\partial t} + \frac{\partial T}{\partial tl}
- \sum_\alpha\big( A \, \rho_\alpham \, cu_{pm \alpha}big) \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}
\ = \ \frac{1}{ \pi r_f^2} \ \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial q_\alpha}{\partial l} |
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| LaTeX Math Block |
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\frac{dp}{dl} = \rho_m \, g \, \sin \theta - \rho \, u_m \, \frac{du_m}{dl} - \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d} |
| LaTeX Math Block |
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(\rho \,c_p)_m \frac{\partial T}{\partial t}
- \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}
\ = \ \frac{1}{ \pi r_f^2} \ \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial q_\alpha}{\partial l} |
where
| indicates a mixture of fluid phases |
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| water, oil, gas phase indicator |
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| длина ствола скважины, отсчитываемая вниз от поверхности |  Image Added
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| профиль плотности воды |
| профиль угла наклона скважины к горизонту |
| профиль диаметра скважины, вдоль которого идет поток |
| профиль поперечного сечения ствола скважины | LaTeX Math Inline |
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| body | A(l) = 0.25 \, \pi \, d^2(l) |
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| профиль коэффициента трения Дарси |
| ускорение свободного падения ( = 9.87 м2/сек ) |
Equations
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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 .
The model is set in 1D-model with
axis aligned with well trajectory :
The disambiguation of the properties in the above equation is brought in The list of dynamic flow properties and model parameters.
Equations ??? define
| LaTeX Math Block Reference |
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defines the continuity of the fluid components flow or equivalently represent the mass conservation of each mass component
| LaTeX Math Inline |
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| body | \{ m_W, \ m_O, \ m_G \} |
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during its transportation
in spacealong wellbore.
Equations ???
| LaTeX Math Block Reference |
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define the motion dynamics of each phase, represented as linear correlation between phase flow speed
and partial pressure gradient of this phase
.
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