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 = \frac{\delta E_H}{ \delta V \delta t} |
and its discrete computational scheme will be:
(2) | \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) T^{k-1} - \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k = \sum_{a = \{w,o,g \}} \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k ) |
where \delta p^k = p_e^k - p_{wf}^k is drawdown, p_e^k – formation pressure in k-th grid layer, p_{wf}^k – bottom-hole pressure across k-th grid layer, T_r^k – reservoir temperature of k-th grid layer.
The l-axis is pointing downward along hole with (k-1)-th grid layer sitting above the k-th grid layer.
If the flowrate is not vanishing during the stationary lift then T^{k-1} can be calculated iteratively from previous values of the wellbore temperature T^k.
References
Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617