# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 3 Formulation of Directional Drilling Trajectory Problem
### Definition of Objective Function for Optimization
#### 3.1.3 Buckling of Drilling Pipe  
Buckling refers to a phenomenon in which a drilling pipe is bent or twisted owing to an increase in the load applied to it. Buckling load represents the smallest amount of load in which buckling can occur. It is determined according to the weight and property of the drilling pipe and the shape of the drilling trajectory section. The buckling load can be divided into the critical buckling load (\(F_{cr}\)) and helical buckling load (\(F_{htd}\)). It can be compared with loads such as drag and torque distributed at the point of the drilling trajectory section to determine whether sinusoidal buckling and helical buckling occur.  
Sinusoidal buckling and helical buckling are defined according to the degree of bending of the drilling pipe. Sinusoidal buckling is the bending of the initial borehole, as shown in Figure 7. The borehole has the shape of a sinusoidal wave. If the load continues to increase, the pipe becomes severely bent or twisted in a spiral form and comes into contact with the borehole wall, resulting in helical buckling, as shown in Figure 8. Sinusoidal buckling does not have a significant effect on drilling operations, but helical buckling makes it difficult to control the direction of the drilling pipe and impedes the effective transmission of the load to the drilling pipe. If the degree of deflection is more severe, the drilling pipe can no longer be moved regardless of the amount of load applied, or a lock-up will occur. Directional drilling, such as horizontal drilling, is a risk factor that must be avoided owing to the high level of damage, including delays in drilling work and increased costs from helical buckling.  
Figure 6: Load of the curve section.  
The buckling load model applied in this study can be divided into critical buckling load and helical buckling load based on the shape of the drilling trajectory section, as presented in Tables 3 and 4 [5].  
In Tables 3 and 4, \(E\) is the Young's modulus, \(I\) is the moment of inertia, \(r\) is the radius of curvature of the drilling trajectory, \(d\) is the spatial distance between the bore and the wall of the borehole, \(\varphi\) is the inclination angle, and \(W_{e}\) is the effective weight of the borehole within the drilling trajectory [6].  
\begin{table}
\begin{tabular}{c c} \hline \hline  & **Helical Buckling Load (lbf)** \\ \hline Horizontal wellbore & \(F_{hel}=2\Big{(}2\sqrt{2}-1\Big{)}\sqrt{\frac{EIW_{e}}{d}}\) \\ Inclined wellbore & \(F_{hel}=2\Big{(}2\sqrt{2}-1\Big{)}\sqrt{\frac{EIW_{e}\sin\varphi}{d}}\) \\ Vertical wellbore & \(F_{hel}=5.55\sqrt{EIW_{e}^{2}}\) \\ Build wellbore & \(F_{hel}=\frac{12 EI}{rd}\Big{(}1+\sqrt{1+\frac{R^{2}W_{e}\sin\varphi}{8 EI}}\Big{)}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Helical buckling load for various sections.  
Figure 8: Helical buckling.  
\begin{table}
\begin{tabular}{c c} \hline \hline  & **Critical Buckling Load (lbf)** \\ \hline Horizontal wellbore & \(F_{cr}=2\sqrt{\frac{EIW_{e}}{d}}\) \\ Inclined wellbore & \(F_{cr}=2\sqrt{\frac{EIW_{e}}{d}\sin\varphi}\) \\ Vertical wellbore & \(F_{cr}=2.55\sqrt{\frac{EIW_{e}^{2}}{d}}\) \\ Build wellbore & \(F_{cr}=\frac{4 EI}{rd}\Big{(}1+\sqrt{1+\frac{d^{2}W_{e}\sin\varphi}{4 EI}}\Big{)}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Critical buckling load for various sections.  
Figure 7: Sinusoidal buckling.