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The multiphase wellbore flow in hydrodynamic and thermodynamic equilibrium is defined by the following set of 1D equations:
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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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| in-situ velocity of -phase fluid flow |
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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 |
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| temperature of -phase fluid flowing from reservoir into a wellbore |
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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This usually takes effect in the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation).
The multiphase fluid density
is defined by exact formula:| LaTeX Math Block |
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\rho_m = sum_\alpha \rho_\alpha s_\alpha |
where
– fractional volumes of -phase which are naturally constraint by:| LaTeX Math Inline |
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| body | \sum_\alpha s_\alpha = s_w + s_o + s_g = 1 |
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.The
can be also expressed as fraction of the total flowing cross-sectional area occupied by -phase:| LaTeX Math Block |
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s_\alpha = \frac{A_\alpha}{A} |
The term
defines mass-specific heat capacity of the multiphase mixture and defined by exact formula:| LaTeX Math Block |
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| anchor | rho_cp |
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| alignment | left |
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(\rho \,c_p)_m = \sum_\alpha \rho_\alpha c_\alpha s_\alpha |
The in-situ velocities
are usually expressed via the macroscopic flow velocity using the
| Expand |
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| LaTeX Math Block |
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| (\rho \,c_{pt})_p \frac{\partial T}{\partial t}
- \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ u_\alpha \frac{\partial T}{\partial l}
\ = \ \frac{\delta E_H}{ \delta V \delta t} |
Equation | LaTeX Math Block Reference |
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defines the heat flow continuity or equivalently represents heat conservation due to heat conduction and convection with account for adiabatic and Joule–Thomson throttling effect.The term | LaTeX Math Inline |
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| body | \frac{\delta E_H}{ \delta V \delta t} |
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defines the speed of change of heat energy volumetric density due to the inflow from formation into the wellbore.
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