# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 2 Definition of Target System  
A typical directional drilling trajectory, which is the main topic of this study, is shown in Figure 3, and Table 1 presents the nomenclature for the calculation of the length of the drilling trajectory.  
Figure 3: Vertical plan of a general 3D horizontal well trajectory. Adapted with permission from Ref. [13]. 2004 [PERSON], [PERSON] et al.  
In this study, we used information on an actual oil well A, which is located in the Gulf of Suez, Egypt, and information on a theoretical oil well B for the analysis.  
Table 2 presents data on the drilling trajectories for wells A and B designed using the conventional method. The definitions of the symbols in the figure are given in Table 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & \multicolumn{2}{c}{**Definition**} \\ \hline \(\varphi_{1}\) & First hold angles (\({}^{\circ}\)) & \\ \(\varphi_{2}\) & Second hold angles (\({}^{\circ}\)) & \\ \(\varphi_{3}\) & Third hold angles (\({}^{\circ}\)) & \\ \(\theta_{1}\) & Azimuth angle at the kickoff point (\({}^{\circ}\)) & \\ \(\theta_{2}\) & Azimuth angle at the end of the first build section (\({}^{\circ}\)) & \\ \(\theta_{3}\) & Azimuth angle at the end of the first hold section (\({}^{\circ}\)) & \\ \(\theta_{4}\) & Azimuth angle at the end of the drop section (\({}^{\circ}\)) & \\ \(\theta_{5}\) & Azimuth angle at the end of the second hold section (\({}^{\circ}\)) & \\ \(\theta_{6}\) & Azimuth angle at the end of the second build section (\({}^{\circ}\)) & \\ \(T_{1}\) & Doglege severity of the first build section (\({}^{\circ}\)/100 ft) & \\ \(T_{2}\) & Doglege severity of the first hold section (\({}^{\circ}\)/100 ft) & \\ \(T_{3}\) & Doglege severity of the drop section (\({}^{\circ}\)/100 ft) & \\ \(T_{4}\) & Doglege severity of the second hold section (\({}^{\circ}\)/100 ft) & \\ \(T_{5}\) & Doglege severity of the second build section (\({}^{\circ}\)/100 ft) & \\ \(TMD\) & True measured depth of the well (ft) & \\ \(TVD\) & True vertical depth of the well (ft) & \\ \(D_{KOP}\) & True vertical depth of the kickoff point (ft) & \\ \(D_{v2}\) & True vertical depth at the end of the first hold section (ft) & \\ \(D_{v3}\) & True vertical depth at the end of the drop section (ft) & \\ \(r\) & Radius of curvature (ft) & \\ \(D\) & Length of the segment (ft) & \\ \(HD\) & Horizontal section (ft) & \\ \(a\) & Curvature of the curve (rad/ft) & \\ C & Casing setting depth (ft) & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nomenclature of variables used to calculate the length of the drilling trajectory.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Application A** & **Application B** \\  & **(Actual Well)** & **(Theoretical Well)** \\ \hline True measured depth (ft) & 15,565 & 13,000 \\ Casing setting depth after 1 st build (ft) & Neglect & Neglect \\ Casing setting depth after 2 nd build (ft) & Neglect & Neglect \\ Casing setting depth after 3 rd build (ft) & Neglect & Neglect \\ \(TVD\) (ft) & 10,848 & 9410.7 \\ Target North–South (ft) & 6284 & 1984.6 \\ Target East–West (ft) & \(-\)3.252 & 3975.4 \\ \(D_{KOP}\) (ft) & 625 & 1000 \\ Depth of 2 nd build point (ft) & 6600 & 5000 \\ Depth of 3 rd build point (ft) & 11,260 & 10,000 \\ \(\varphi_{1}\) (\({}^{\circ}\)) & 15 & 20 \\ \(\varphi_{2}\) (\({}^{\circ}\)) & 45 & 40 \\ \(\varphi_{3}\) (\({}^{\circ}\)) & 90 & 90 \\ \(\theta_{1}\),\(\theta_{2}\), \(\theta_{3}\), \(\theta_{4}\), \(\theta_{5}\), \(\theta_{6}\) (\({}^{