# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 4 Drilling Trajectory Optimization
### Optimization Process  
Existing research results [13] were used to specify the initial values of the drilling trajectory length term of the objective function and the load term of the drilling pipe. The design variable values derived from the results obtained in existing research are summarized in Table 8 and were used in the objective function to confirm the initial length (\(f_{1i}(x)\)), initial load (\(f_{2i}(x)\)), and the occurrence of buckling for wells A and B.  
Figure 9: Flowchart of the particle swarm optimization algorithm.  
Using the values in Table 8, the occurrence of buckling in wells A and B was verified. The results are summarized in Table 10. The results indicate that helical buckling and sinusoidal buckling occurred.  
A formulation process was performed to define the objective function, design variables, and constraints based on the given drilling trajectory data, the length model, and the load model. Optimization was performed using MATLAB(r) and the particle swarm optimization algorithm. The setting of the particle swarm optimization algorithm is presented in Table 11. The parametric setting values were optimized with various numerical values, and the value with the shortest calculation time and best result among those obtained was selected.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Application A** & **Application B** \\ \hline Horizontal section & No buckling & No buckling \\ Second build section & No buckling & No buckling \\ Second hold section & No buckling & No buckling \\ First drop section & Sinusoidal buckling & No buckling \\ First hold section & Helical buckling & Sinusoidal buckling \\ First build section & No buckling & No buckling \\ \hline \hline \end{tabular}
\end{table}
Table 10: Verification of buckling of various sections.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Application A** & **Application B** \\ \hline \(\phi_{1}\) (\({}^{\circ}\)) & 13.77 & 10.73 \\ \(\phi_{2}\) (\({}^{\circ}\)) & 42.131 & 34.773 \\ \(\phi_{3}\) (\({}^{\circ}\)) & 90 & 90 \\ \(T_{1}\) (\({}^{\circ}\)/100 ft) & 1.675 & 1.5195 \\ \(T_{2}\) (\({}^{\circ}\)/100 ft) & 0.047 & 0.00355 \\ \(T_{3}\) (\({}^{\circ}\)/100 ft) & 1.431 & 1.708 \\ \(T_{4}\) (\({}^{\circ}\)/100 ft) & \(-\)0.0102 & 0.01258 \\ \(T_{5}\) (\({}^{\circ}\)/100 ft) & 2.413 & 1.9978 \\ \(\theta_{1}\) (\({}^{\circ}\)) & 279.976 & 332.156 \\ \(\theta_{2}\) (\({}^{\circ}\)) & 279.72 & 322.22 \\ \(\theta_{3}\) (\({}^{\circ}\)) & 275.02 & 322.15 \\ \(\theta_{4}\) (\({}^{\circ}\)) & 332.3 & 402.355 \\ \(\theta_{5}\) (\({}^{\circ}\)) & 334.32 & 457.04 \\ \(\theta_{6}\) (\({}^{\circ}\)) & 355 & 437.234 \\ \(D_{KOP}\) (ft) & 987.975 & 934.2557 \\ \(D_{v2}\) (ft) & 6801.37 & 6651.251 \\ \(D_{v3}\) (ft) & 10,001.48 & 8090.88 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Design variables of the initial design. Adapted with permission from Ref. [13]. 2004 [PERSON], [PERSON] et al.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Application A (Actual Well)** & **Application B (Theoretical Well)** \\ \hline Total measured depth of the initial design, \(f_{\text{1i}}(x)\) (ft) & 15,127 & 12,829 \\ Total load of the initial design, \(f_{\text{2i}}(x)\) (lbf) & 59,946 & 50,812 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Calculation results of the initial design.