# Study on the Vertical Stability of Drilling Wellbore under Optimized Constraints
## 3 Analysis and Discussion
### Discussion on Critical Depth Influence Factors  
From Equation (29), it can be seen that the factors affecting the critical depth of wellbores are elastic modulus \(E\), the self-weight of the wellbore \(q_{c}\), the thickness of the wellbore \(\triangle\), and the height of counterweight water \(H_{w}\). In this section, a two-by-two combination of single-factor analysis of the effects of the elastic modulus, the self-weight of the wellbore, the thickness of wellbore, and the height of counterweight water on the critical depth was performed by using the control variable method. Taking the Banji coal mine auxiliary wellbore as an example, the depth of the wellbore is 638.084 m, the minimum height of the counterweight water is 333.3 m, the external diameter is 9.3 m, the internal diameter is 7.6 m, the elastic modulus is 38.15 Gpa, the moment of inertia of the cross-section is 202.65 m\({}^{4}\), the self-weight per unit length is 6.6 \(\times\) 10\({}^{5}\) N/m, and the weight of counterweight water per unit length is 4.44 \(\times\) 10\({}^{5}\) N/m. We use the above parameters as a baseline to distribute the change in the elastic modulus of the wellbore \(E\) from 33.65 Gpa to 43.15 Gpa at intervals of 0.5 Gpa, the self-weight of the wellbore \(q_{c}\) from 570,000 N/m to 760,000 N/m at intervals of 10,000 Gpa, the thickness of the wellbore \(\triangle\) from 760 mm to 950 mm at intervals of 10 mm, and the height of counterweight water \(H_{w}\) from 333.3 m to 618.3 m at intervals of 15 m to calculate and analyze the change of the critical instability depth of the deep wellbore structure after adopting the optimized construction measure. To briefly analyze the trend of any two influencing factors on the  
Figure 4: Analysis diagram of critical depth growth after optimization of each wellbore. critical depth of wellbore instability \(H_{cr}\), the analytical diagram of influencing factors is arranged and drawn according to Table 4.  
As can be seen in Table 4, the combinations \(E\)-\(E\), \(q_{c}\)-\(q_{c}\), \(H_{w}\)-\(H_{w}\), and \(\triangle\)-\(\triangle\) in the table are considered duplicates because they are the same influencing factors. In the blank section above the duplicates, the combinations of any two influencing factors are duplicated with all those below the duplicates in this section. It is sufficient to consider analyzing only the following six factor combinations: \(E\)-\(q_{c}\), \(E\)-\(H_{w}\), \(E\)-\(H_{w}\), \(q_{c}\)-\(H_{w}\), \(q_{c}\)-\(\triangle\), and \(H_{w}\)-\(\triangle\). The calculation results are shown in Figure 5.  
Figure 5: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline  & \(E\) & \(q_{c}\) & \(H_{w}\) & \(\triangle\) \\ \hline \(E\) & \(E\)-\(E\) & & & \\ \(q_{c}\) & \(E\)-\(q_{c}\) & \(q_{c}\)-\(q_{c}\) & & \\ \(H_{w}\) & \(E\)-\(H_{w}\) & \(q_{c}\)-\(H_{w}\) & \(H_{w^{\prime}}\)-\(H_{w}\) & \\ \(\triangle\) & \(E\)-\(\triangle\) & \(q_{c}\)-\(\triangle\) & \(H_{w^{\prime}}\)-\(\triangle\) & \(\triangle\)-\(\triangle\) \\ \hline \end{tabular}
\end{table} 
Table 4: Influence factor combination analysis matrix. From Figure 5a-c, it can be seen that the values of the critical depths of the wellbore increase with the increase in elastic modulus regardless of any self-weight of the wellbore, the height of counterweight water, and the thickness of the wellbore. Under a different elastic modulus, the critical depths of the wellbore instability decrease with the self-weight of wellbore and increase with the increase in the thickness of the wellbore, and the degree of increase tends to be slow. When the injection heights of counterweight water are 483.3 m, 453.3 m, 468.3 m, 483.3 m, and 498.3 m, the critical depths of the wellbore instability increase with the increase of counterweight water height, and the degree of increase tends to become urgent.