# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 3 Formulation of Directional Drilling Trajectory Problem
### Definition of Objective Function for Optimization
#### 3.1.4 Objective Function Definition  
For the objective function, it is necessary to minimize the length of the drilling trajectory and the load on the drilling pipe. For this purpose, the minimum length model of the drilling trajectory can be a model that minimizes Equation (6), which can be employed to calculate the total length of the drilling trajectory as given in Equation (26).  
\[\text{Minimize }f_{1}(x)=TMD. \tag{26}\]  
The load induced on the drilling pipe can be divided into drag and torque. This load can be defined as below. First, the drag of each section can be calculated by dividing the trajectory into a straight section and a curved section based on Equations (21) and (23), and these are expressed in Equations (27)-(33).  
\[F_{6}=F_{7}+\beta wHD(\cos\varphi_{3}+\mu\sin\varphi_{3}) \tag{27}\]  
\[F_{5}=F_{6}e^{\mu[DL_{A5}-DL_{A5}]}+\beta wD_{5}\bigg{(}\frac{\sin\varphi_{3}- \sin\varphi_{2}}{\varphi_{3}-\varphi_{2}}\bigg{)} \tag{28}\]  
\[F_{4}=F_{5}+\beta wD_{4}(\cos\varphi_{2}+\mu\sin\varphi_{2}) \tag{29}\]  
\[F_{3}=F_{4}e^{\mu[DL_{A3}-DL_{A1}]}+\beta wD_{3}\bigg{(}\frac{\sin\varphi_{2} -\sin\varphi_{1}}{\varphi_{2}-\varphi_{1}}\bigg{)} \tag{30}\]  
\[F_{2}=F_{3}+\beta wD_{2}(\cos\varphi_{1}+\mu\sin\varphi_{1}) \tag{31}\]  
\[F_{1}=F_{2}e^{\mu[DL_{A3}-DL_{A1}]}+\beta wD_{1}\bigg{(}\frac{\sin\varphi_{1} -\sin\varphi_{KOP}}{\varphi_{1}-\varphi_{KOP}}\bigg{)} \tag{32}\]  
\[F_{0}=F_{1}+\beta wD_{KOP}(\cos\varphi_{KOP}+\mu\sin\varphi_{KOP}). \tag{33}\]  
The torque in each section can be calculated by dividing it into a straight section and a curved section based on Equations (22) and (24); these are given in Equations (34)-(40).  
\[\tau_{6}=\mu R\beta wHD\sin\varphi_{3} \tag{34}\]  
\[\tau_{5}=\mu RF_{6}|DL_{A5}-DL_{A3}| \tag{35}\]  
\[\tau_{4}=\mu R\beta wD_{4}\sin\varphi_{2} \tag{36}\]  
\[\tau_{3}=\mu RF_{4}|DL_{A3}-DL_{A1}| \tag{37}\]  
\[\tau_{2}=\mu R\beta wD_{2}\sin\varphi_{1} \tag{38}\]  
\[\tau_{1}=\mu RF_{2}|DL_{A1}| \tag{39}\]  
\[\tau_{0}=\mu R\beta wD_{KOP}\sin\varphi_{KOP}. \tag{40}\]  
In each section, the occurrence of sinusoidal buckling or helical buckling was confirmed using Tables 3-5. If the total sum of the final loads in all sections (\(P\)) is minimized, then the load on the entire drilling trajectory can be minimized. Therefore, the objective function can be defined as given in Equation (41).  
\[\text{Minimize }f_{2}(x)=\sum P_{l}. \tag{41}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Load Condition** & **Drill Pipe Behaviors** \\ \hline \(F<F_{cr}\) & No buckling \\ \(F_{cr}\leq F<F_{hd}\) & Sinusoidal buckling \\ \(F_{hd}\leq F\) & Helical buckling \\ \hline \hline \end{tabular}
\end{table}
Table 5: Drill pipe behaviors according to load condition.  
The objective function can be defined by multiplying the length minimization model and the load minimization model by the weight of the drilling trajectory, as given in Equation (42).  
\[\text{Minimize }f(x)=W_{1}\times\left(\frac{f_{1}(x)}{f_{1i}(x)}\right)+W_{2} \times\left(\frac{f_{2}(x)}{f_{2i}(x)}\right) \tag{42}\]  
where \(f_{1i}(x)\) and \(f_{2i}(x)\) are the initial values of each term and are used for the normalization of each term. \(W_{1}\) and \(W_{2}\) are the weights of each term, and the sum of the weights should be one. By adjusting the weight in the objective function, the designer can design the drilling track trajectory by giving more weight to the length of the drilling trajectory or the load on the drilling pipe (see Section 4.1).  
In addition to the design variables, the values required to evaluate the objective function are listed in Table 6, and these values are equally applied to wells A and B.