...
LaTeX Math Block |
---|
|
\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 it's discrete computational scheme will be:
LaTeX Math Block |
---|
|
\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_{\bf r}p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta P ) |
Expand |
---|
|
The wellbore fluid velocity can be expressed thorugh the volumetric flow profile and tubing/casing cross-section area as: LaTeX Math Block |
---|
| u_\alpha = \frac{q_\alpha}{\pi r_f^2} |
so that LaTeX Math Block |
---|
| \bigg( \sum_{a = \{w,o,g \}} \rho_\alpha \ c_{p \alpha} | q\ \mathbf{u}_\alpha \bigg) \ \nabla T
= \frac{\delta E_H}{ \delta V \delta t} |
|
References
...
Beggs, H. D. and Brill, J. P.: "A Study of Two-Phase Flow in Inclined Pipes," J. Pet. Tech., May (1973), 607-617
...