Definition
Mathematical model of multiphase wellbore flow predicts the temperature, pressure and flow speed distribution along the hole with account for:
- tubing head pressure which is set by gathering system or injection pump
- wellbore design
- pump characterisits
- fluid friction with tubing /casing walls
- interfacial phase slippage
- heat exchange between wellbore fluid and surrounding rocks
Stationary Temperature Model
In stationary conditions are defined as \frac{\partial T}{\partial t} = 0 and \frac{\partial P}{\partial t} = 0 .
This happens during the long-term (days & weeks) production/injection or long-term (days & weeks) shut-in.
The temperature dynamic equation
is going to be:| (1) | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} \ \mathbf{u}_\alpha \bigg) \ \nabla T = \delta({\bf r}) \, T \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} q_\alpha |
References
Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617