# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 5 Comparison of Optimization Results  
Table 12 presents the values of the optimal design variables when the weights (\(W_{1},~{}W_{2}\)) that are multiplied by each term in the objective function are set to 0.5 during the optimization. The optimization results, including the total length, load, and buckling of the drilling trajectories, were evaluated based on Table 12. The corresponding values are summarized in Tables 13 and 14 by comparing them with the results in existing research (refer to Table 8). In the case of the load, it was divided into drag and torque after examining the occurrence of buckling. The drag that was applied to the entire borehole that constitutes the drilling trajectory is \(F_{0}\), and the torque was calculated using the following equation:  
\[\tau_{total}=\sum_{i=0}^{6}\tau_{i}. \tag{43}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Application A (Actual Well)** & & \\  & **Existing** & **New** & **New/Existing Ratio** \\ \hline Total measured depth (ft) & 15,127 & 14,768 & 0.976 \\ Total drag force (lbf) & 62,567 & 61,116 & 0.976 \\ Total torque (lbf.ft) & 16,950 & 14,413 & 0.850 \\ Horizontal section & No buckling & No buckling & - \\ Second build-up section & No buckling & No buckling & - \\ Second hold section & No buckling & No buckling & - \\ First drop off section & Sinusoidal buckling & No buckling & - \\ First hold up section & Helical buckling & No buckling & - \\ First build section & No buckling & No buckling & - \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Comparison of existing design and new design for application A. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Application A** & **Application B** \\ \hline \(\varphi_{1}\) (\({}^{\circ}\)) & 10 & 10 \\ \(\varphi_{2}\) (\({}^{\circ}\)) & 40 & 50 \\ \(\varphi_{3}\) (\({}^{\circ}\)) & 90 & 95 \\ \(T_{1}\) (\({}^{\circ}\)/100 ft) & 0.1657 & 0.1989 \\ \(T_{2}\) (\({}^{\circ}\)/100 ft) & 2.5443 & 0.3005 \\ \(T_{3}\) (\({}^{\circ}\)/100 ft) & 3.0187 & 3.3597 \\ \(T_{4}\) (\({}^{\circ}\)/100 ft) & 4.0363 & 0.5631 \\ \(T_{5}\) (\({}^{\circ}\)/100 ft) & 5 & 10 \\ \(\theta_{1}\) (\({}^{\circ}\)) & 274.2409 & 318.9605 \\ \(\theta_{2}\) (\({}^{\circ}\)) & 273.1937 & 307.9697 \\ \(\theta_{3}\) (\({}^{\circ}\)) & 280 & 360 \\ \(\theta_{4}\) (\({}^{\circ}\)) & 330 & 360 \\ \(\theta_{5}\) (\({}^{\circ}\)) & 340 & 431.93 \\ \(\theta_{6}\) (\({}^{\circ}\)) & 355 & 431.93 \\ \(D_{KOP}\) (ft) & 1000 & 2000 \\ \(D_{v2}\) (ft) & 7000 & 7000 \\ \(D_{v3}\) (ft) & 10,000 & 8000 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Design variables of the optimization result. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Application A (Actual Well)** & \\  & **Existing** & **New** & **New/Existing Ratio** \\ \hline Total measured depth (ft) & 15,127 & 14,768 & 0.976 \\ Total drag force (lbf) & 62,567 & 61,116 & 0.976 \\ Total torque (lbf.ft) & 16,950 & 14,413 & 0.850 \\ Horizontal section & No buckling & No buckling & - \\ Second build-up section & No buckling & No buckling & - \\ Second hold section & No buckling & No buckling & - \\ First drop off section & Sinusoidal buckling & No buckling & - \\ First hold up section & Helical buckling & No buckling & - \\ First build section & No buckling & No buckling & - \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Comparison of existing design and new design for application A. First, Table 13 presents the results of the optimization of well A. Compared with results from the previous design, the total length of the drilling trajectory was reduced by approximately 2.4% and the drag was reduced by approximately 2.4%. In addition, the torque applied to the drilling pipe was reduced by 15% from 16,950 lbf-ft to 14,413 lbf-ft. There are sections where helical buckling and sinusoidal buckling occurred in the existing design; however, buckling did not occur in all sections owing to reduction in the load on the drilling pipe as a result of the optimization.