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The term
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| body | \ \phi \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.
Stationary Flow Model
Stationary wellbore flow is defined as the flow with constant pressure and temperature:
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| body | \frac{\partial T}{\partial t} = 0 |
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and
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| body | \frac{\partial P}{\partial t} = 0 |
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