...
If the flowrate is not vanishing during the stationary lift (
| LaTeX Math Inline |
|---|
| body | \sum_{a = \{w,o,g \}} |q_\alpha^{k-1}| > 0 |
|---|
|
) then
can be calculated iteratively from previous values of the wellbore temperature
as:
| LaTeX Math Block |
|---|
|
T^{k-1} = \frac{\bigg( \sum_\alpha \rho_\alpha^k \ c_{p \alpha}^k \ q_\alpha^k \bigg) T^k + \sum_\alpha \rho_\alpha^k \ c_{p \alpha}^k \ (q_\alpha^{k-1} - q_\alpha^k) \, (T_r^k + \epsilon_\alpha^k \delta p^k )}{\bigg( \sum_\alpha \rho_\alpha^{k-1} \ c_{p \alpha}^{k-1} \ q_\alpha^{k-1} \bigg) }
|
| Expand |
|---|
|
| LaTeX Math Block |
|---|
| (\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 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} \ \mathbf{u}_\alpha \bigg) \ \nabla T
= \frac{\delta E_H}{ \delta V \delta t} |
|
...
