The multiphase wellbore flow in hydrodynamic and thermodynamic equilibrium is defined by the following set of equations:
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\frac{\partial (\rho_m A)}{\partial t} + {\nabla } \bigg( A \, \sum_\alpha \rho_\alpha \, { \bf u}_\alpha \bigg) = 0 |
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\rho_\alpha \bigg[ \frac{\partial {\bf u}_\alpha}{\partial t} + ({\bf u}_\alpha \cdot \nabla) \ {\bf u}_\alpha - \nu_\alpha \Delta {\bf u}_\alpha\bigg] = - \nabla p + \rho_\alpha \, {\bf g} - \frac{ f_\alpha \, \rho_\alpha \, u_\alpha}{2 d} \ {\bf u}_\alpha |
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(\rho \,c_{pt})_p \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} \ ({\bf u}_\alpha \cdot \nabla) \ T
\ = \ \frac{\delta E_H}{ \delta V \delta t} |
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Projecting the above equations to the well trajectory
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\frac{\partial (\rho_m A)}{\partial t} + \frac{\partial}{\partial l} \bigg( A \, \sum_\alpha \rho_\alpha \, u_\alpha \bigg) = 0 |
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\sum_\alpha \rho_\alpha \bigg[ \frac{\partial u_\alpha}{\partial t} + u_\alpha \frac{\partial u_\alpha}{\partial l} - \nu_\alpha \Delta u_\alpha\bigg] = - \frac{dp}{dl} + \rho_m \, g \, \sin \theta - \frac{ f_m \, \rho_m \, u_m^2 \, }{2 d} |
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(\rho \,c_p)_m \frac{\partial T}{\partial t}
- \bigg( \sum_\alpha \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha}\bigg) \ \frac{\partial p}{\partial t}
+ \bigg( \sum_\alpha \rho_\alpha \ c_{p \alpha} \ u_\alpha \bigg) \frac{\partial T}{\partial l}
\ = \ \frac{1}{A} \ \sum_\alpha \rho_\alpha \ c_{p \alpha} T_\alpha \frac{\partial q_\alpha}{\partial l} |
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The disambiguation of the properties in the above equation is brought in The list of dynamic flow properties and model parameters.
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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).
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(\rho \,c_p)_m = \sum_\alpha \rho_\alpha c_\alpha s_\alpha |
Stationary wellbore flow is defined as the flow with constant pressure and temperature:
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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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(\rho \,c_{pt})_p \frac{\partial T}{\partial t}
- \ \phi \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \eta_{s \alpha} \ \frac{\partial P_\alpha}{\partial t}
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \epsilon_\alpha \ \mathbf{u}_\alpha \bigg) \nabla P
+ \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T
- \nabla (\lambda_t \nabla T) = \frac{\delta E_H}{ \delta V \delta t} |
The wellbore fluid velocity
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u_\alpha = \frac{q_\alpha}{\pi r_f^2} |
so that
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\bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T
= \frac{\delta E_H}{ \delta V \delta t} |