# Study on the Vertical Stability of Drilling Wellbore under Optimized Constraints
## 3 Analysis and Discussion  
### Comparison Analyses  
In this part, several representative drilling wellbore projects were selected, and the critical depth values before and after using optimization measures were calculated. The critical depth before adopting optimization measures was calculated based on reference [23]. The critical depth after adopting optimization measures was calculated by the Equation (36) in this article. The selected representative projects included the Panji west wind wellbore, the Keke gai coal mine wind wellbore, the Banji coal mine auxiliary wellbore, and the Banji coal mine main wellbore. The comparison calculation basis and process are shown in Tables 2 and 3. The calculation results are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Traditional Methods** & **Optimized Methods** \\ \hline Restrictive conditions & With both ends hinged & Upper end hinged \\  & Literature [23] & Equation (36) \\ Critical depth formulas & \(H_{cr1}=\sqrt[3]{\frac{2 EI\eta\pi^{2}}{q_{cr2}+4q_{in}\sin^{2}\frac{2\eta\pi }{H}}}\) & \(H_{cr2}=\sqrt[3]{\frac{41 EI\eta^{4}}{(10\pi^{2}-32)q_{cr2}+64q_{in}\sin^{2} \frac{2\eta\pi}{2H}}\sin^{2}\frac{2\eta\pi}{H}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Critical depth equations for different construction methods.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Shaft Walls** & \begin{tabular}{c} **Panji West Wind** \\ **Wellbore** \\ \end{tabular} & \begin{tabular}{c} **Kekegai Coal Mine** \\ **Wind Wellbore** \\ \end{tabular} & \
\begin{tabular}{c} **Banji Coal Mine** \\ **Auxiliary Wellbore** \\ \end{tabular} &
\begin{tabular}{c} **Banji Coal Mine** \\ **Main Wellbore** \\ \end{tabular} \\ \hline Sources & Literature [30] & Literature [31] & Literature [32] & Literature [23] \\ \(E\) (GPa) & 35.5 & 38.89 & 38.15 & 38.85 \\ \(I\) (m\({}^{4}\)) & 126.5 & 68.26 & 203.33 & 139.66 \\ \(D\) (m) & 9 & 7.2 & 9.3 & 8.3 \\ \(d\) (m) & 7 & 6 & 7.6 & 6.5 \\ \(q_{c}\) (N/m) & 457.008 & 328,159 & 660,000 & 469,000 \\ \(q_{w}\) (N/m) & 376,957 & 276,948 & 444,348 & 351,000 \\ \(H_{w}\) (m) & 222.2 & 215.2 & 333.3 & 291.703 \\ Depths of wellbore \(H\) (m) & 508 & 542.5 & 638.084 & 659.675 \\ Critical depths under the & 527.28 & 495.98 & 566.95 & 563.58 \\ traditional method \(H_{cr1}\) (m) & Stability under the & Stable & Unstable & Unstable \\ Traditional method & & & & \\ Critical depths under & 766.11 & 729.06 & 805.52 & 817.55 \\ the optimized method \(H_{cr2}\) (m) & Stability under the & Stable & Stable & Stable \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison calculation process of critical depth before and after optimization measures for representative engineering use.  
By comparing the calculation results of the critical depth of instability before and after the optimized construction process in Table 3, it can be seen that the wellbore stability changed significantly due to the different constraints under different construction processes. The critical depth of the wellbore instability calculated under the optimized construction technology was significantly higher than that under traditional construction technology.  
In the theoretical analysis, the critical depths of instability of the Panji west wind wellbore, Keke gai coal mine wind wellbore, the Banji coal mine auxiliary wellbore, and the Banji coal mine main wellbore were previously increased by 45.29%, 46.99%, 42.08%, and 45.06%, respectively, after adopting the optimized measures. The differences are 3.9%, 5.6%, 0.69%, and 3.67%, respectively, compared with the results of the literature [33], which shows that the optimized construction measures are reasonable and reliable to improve the stability of wellbores by around 41.39 \(\pm\) 5%.