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The term
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| body | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \ mathbf{ u}_\alpha \bigg) \ \bar \nabla T frac{\partial T}{\partial l} |
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represents heat convection defined by the wellbore mass flow.
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The term
| LaTeX Math Inline |
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| body | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_ \alpha \ \mathbf{u} _ \ alpha \bigg) \bar \nabla P |
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represents the heating/cooling effect of the multiphase flow through the porous media. This effect is the most significant with light oils and gases.The term | LaTeX Math Inline |
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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.
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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_\alpha}{\partial t} = 0 |
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This happens during the long-term (usually hours & days & weeks) production/injection or long-term (usually hours & days & weeks) shut-in.
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