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The term
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| body | \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t} |
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represents the heating/cooling effect of the fast adiabatic pressure change.
This usually takes effect in and around the wellbore during the first minutes or hours after changing the well flow regime (as a consequence of choke/pump operation). This effect is absent in stationary flow and negligible during the quasi-stationary flow and usually not modeled in conventional monthly-based flow simulations.
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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} |
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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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