# Multi-Objective Optimization of Drilling Trajectory Considering Buckling Risk
## 3 Formulation of Directional Drilling Trajectory Problem  
### Definition of Objective Function for Optimization  
#### 3.1.1 Length of Oil Well  
Many methods have been used to calculate the length of the drilling trajectory. In the design of the horizontal well, the length of a three-dimensional drilling trajectory with an azimuth angle (\(\theta\)), inclination angle (\(\varphi\)), and radius of curvature (\(r\)) is the length between two points in space, as shown in Figure 4, and can be calculated using Equation (1) [16]. \[\Delta MD=r\sqrt{\left(\theta_{2}-\theta_{1}\right)^{2}\sin^{4}\left(\frac{ \varphi_{1}-\varphi_{2}}{2}\right)+\left(\varphi_{2}-\varphi_{1}\right)^{2}}. \tag{1}\]  
In Equation (1), the radius of curvature (\(r\)) can be calculated as given in Equation (2), where \(T\) is the dogleg severity (Table 2); this is the angle rate at which the drilling trajectory changes per 100 ft.  
\[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]  
The length of the drilling trajectory can be calculated from Equations (1) and (2) by changing the azimuth angle and the radius of curvature of the drilling trajectory between two points. In addition, the north-south distance between two points can be calculated using Equation (3), the east-west distance using Equation (4), and the vertical distance using Equation (5) [17]. \[\Delta NS=\frac{\Delta MD(\cos(\varphi_{1})-\cos(\varphi_{2})\times(\sin( \theta_{2})-\sin(\theta_{1}))}{(\varphi_{2}-\varphi_{1})\times(\theta_{2}- \theta_{1})} \tag{3}\]  
\[\Delta EW=\frac{\Delta MD(\cos(\varphi_{1})-\cos(\varphi_{2})\times(\cos( \theta_{1})-\cos(\theta_{2}))}{(\varphi_{2}-\varphi_{1})\times(\theta_{2}- \theta_{1})} \tag{4}\]  
\[\Delta Vertical=\frac{\Delta MD(\sin(\varphi_{2})-\sin(\varphi_{1}))}{(\varphi_{ 2}-\varphi_{1})}. \tag{5}\]  
The directional drilling trajectory shown in Figure 3 can be divided into sections: the kickoff section (\(D_{KOP}\)), the build-up and drop off sections (\(D_{1},\ D_{3},\ D_{5}\)), the hold sections (\(D_{2},\ D_{4}\)), and the horizontal section (\(HD\)). If all the sections are summed up, the total  
Figure 4: Surveys’ calculation variables. measured depth (\(TMD\)) can be calculated, where \(\varphi_{KOP}\) is the inclination angle at the kickoff section and is assumed to be \(0^{\circ}\). The relevant equations are Equations (6)-(11). \[TMD=D_{KOP}+D_{1}+D_{2}+D_{3}+D_{4}+D_{5}+HD \tag{6}\]  
\[D_{1}=r_{1}\sqrt{(\theta_{2}-\theta_{1})^{2}\sin^{4}\biggl{(}\frac{\varphi_{1} +\varphi_{KOP}}{2}\biggr{)}+(\varphi_{1}-\varphi_{KOP})^{2}} \tag{7}\]  
\[D_{2}=\frac{(D_{v2}-D_{KOP}-D_{1})\times(\sin(\varphi_{1})-\sin(\varphi_{KOP}) )}{\varphi_{1}-\varphi_{KOP}} \tag{8}\]  
\[D_{3}=r_{3}\sqrt{(\theta_{4}-\theta_{3})^{2}\sin^{4}\biggl{(}\frac{\varphi_{2} +\varphi_{1}}{2}\biggr{)}+(\varphi_{2}-\varphi_{1})^{2}} \tag{9}\]  
\[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]0  
\[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]1  
The radius of curvature in each section is calculated as given in Equation (12). \[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]2  
The casing depth in each section is calculated using Equations (13)-(15). \[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]3  
\[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]4  
\[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]5  
\[r=\frac{180}{\pi}\times\frac{100}{T}. \tag{2}\]6  
where \(F_{2}\) is the force at the top of the section, \(F_{1}\) is the force at the bottom of the section, \(\beta\) is the buoyancy constant, \(w\) is the weight per unit length of the drilling pipe, \(\Delta L\) is the length of the section, \(\varphi\) is the angle of inclination of the section, \(\mu\) is the friction coefficient, and the \(\pm\) in the second term indicates upward and downward of the drilling pipe, respectively. The torque (\(\tau\)) in the hold section is calculated by multiplying the friction coefficient by the radius of the pipe joint tool as given in Equation (22), where \(R\) is the radius of the pipe joint tool.