## 3 Analysis and Results
### Structural Parameter Optimization Analysis of the Dissolvable Ball Seat  
As previously mentioned, to ensure the safety of the slip structure, the apex angle of the slip teeth was recommended to range between 75\({}^{\circ}\) and 90\({}^{\circ}\), while the slip teeth inclination angle could be set at either 70\({}^{\circ}\) or 75\({}^{\circ}\). When the apex angle of the slip teeth was designed at 75\({}^{\circ}\) or 80\({}^{\circ}\), a higher contact pressure between the slip and the casing was observed, which was found to enhance the solid anchoring of the slip.  
To further study the influence of the structural parameters of the retaining ring on the anchoring, an orthogonal test optimization study on the anchoring performance of the dissolvable ball seats slip was carried out to master the contact pressure distribution between the slip and the casing, the equivalent stress distribution of the slip, and the equivalent stress distribution of the casing during the anchoring process. The parameter combinations are shown in Table 4. The orthogonal experiment results are shown in Figure 13.  
The distribution of contact pressure between the casing and the slip under different slip parameter combinations is illustrated in Figure 13, while the corresponding contact pressure indicators are tabulated in Table 5. It was observed that the contact pressure between the slip and the casing was higher when the slip tooth inclination angle (\(\beta\)) was set at 70\({}^{\circ}\) compared to 75\({}^{\circ}\); the peak contact pressure achieved the maximum when the slip tooth apex angle (\(\alpha\)) was 80\({}^{\circ}\) and the slip tooth inclination angle (\(\beta\)) was 70\({}^{\circ}\)(Figure 14). The average contact pressure achieved the maximum when the slip tooth apex angle (\(\alpha\)  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Scheme** & **Tooth Apex Angle \(\alpha\)** & **Tooth Inclination Angle \(\beta\)** \\ \hline
1 & 75\({}^{\circ}\) & 70\({}^{\circ}\) \\
2 & 80\({}^{\circ}\) & 70\({}^{\circ}\) \\
3 & 85\({}^{\circ}\) & 70\({}^{\circ}\) \\
4 & 75\({}^{\circ}\) & 75\({}^{\circ}\) \\
5 & 80\({}^{\circ}\) & 75\({}^{\circ}\) \\
6 & 85\({}^{\circ}\) & 75\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Orthogonal test scheme table for slip structure parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Slip Tooth Inclination** & **Peak Contact** & **Average Contact** & **Standard Deviation of** \\
**Angle \(\beta\) (\({}^{\circ}\))** & **Pressure (MPa)** & **Pressure (MPa)** & **Contact Pressure** \\ \hline
60 & 1013 & 704.38 & 117.06 \\
65 & 1026 & 717.68 & 105.18 \\
70 & 1068 & 705.11 & 145.23 \\
75 & 1051 & 716.52 & 114.72 \\
80 & 1115 & 735.85 & 119.35 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Contact pressure indicators between casing and slip teeth at varying tooth inclination angles.  
was 85\({}^{\circ}\) and the slip tooth inclination angle (\(\beta\)) was 70\({}^{\circ}\) (Figure 15). The standard deviation of contact pressure was lowest when the slip tooth apex angle (\(\alpha\)) was 80\({}^{\circ}\) and the slip tooth inclination angle (\(\beta\)) was 70\({}^{\circ}\), achieving the most uniform contact pressure between the casing and the slip.  
The stress distribution of the slip under varying parameter combinations is presented in Figure 13. When both the slip tooth apex angle (\(\alpha\)) and the slip tooth  
The equivalent stress distribution of the casing under different slip parameter combinations is displayed in Figure 13. Minimal casing stress was recorded when the slip tooth apex angle (\(\alpha\)) was \(85^{\circ}\) and the slip tooth inclination angle (\(\beta\)) was \(70^{\circ}\), indicating reduced casing damage caused by the slip.