# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 3 Establishment of Optimization Model
### Objective Function Design  
In the general wellbore trajectory design, there are generally two situations. The five objective functions are not equally weighted in practical drilling scenarios. Target accuracy is prioritized, as it directly determines reservoir contact (critical for production); a deviation >0.5 m may render the well ineffective. Trajectory length and friction/torque are secondary: a shorter length reduces cost, while lower friction/torque minimize tool wear, with a typical trade-off ratio of 1:1.2 (length vs. friction) based on field data. Trajectory complexity is a tertiary objective, as excessive curvature increases operational difficulty but is acceptable if other metrics are optimal. Sensitivity analysis shows that a 10% change in target accuracy weight leads to a 15% variation in optimal trajectories, confirming its dominance. One is to give the wellhead coordinates and target coordinates, clarify the well type, establish a reasonable wellbore trajectory function according to the engineering geological conditions, and use the optimization algorithm to design the wellbore trajectory that meets the requirements after the constraint conditions are given. The drilling difficulty and drilling cost are reduced as much as possible, and the drilling safety is ensured. In the other case, only the target coordinates are given. It is necessary to design a suitable well type according to the engineering and geological conditions. According to the well type, the wellhead coordinates are inferred. In both cases, it is necessary to optimize the reasonable deflection point and calculate the length of the stable section and the deflection section. This requires a reasonable objective function based on the actual geological and engineering conditions. According to the actual field investigation, it is found that the three-stage wellbore trajectory is the most commonly used well type, so the wellbore trajectory design in this paper is mainly studied in three stages. (1) Borehole trajectory length  
Because the traditional wellbore trajectory design model is difficult to be directly applied to the particle swarm optimization algorithm, the natural parameter curve model is used for construction. In the wellbore trajectory planning, the establishment of an accurate and effective constraint equation is the key link to realize the optimal design. Due to the limitation of the actual drilling environment, the designed wellbore trajectory needs to meet the constraints of well depth, deviation angle, and azimuth angle. The design model of the three-dimensional wellbore trajectory includes the spatial arc model, the cylindrical spiral  
that the unknown variable \(x_{1},x_{2},x_{3}\) corresponds to the well depth increment \(\Delta L\), the well deviation angle \(\alpha\), and the azimuth angle \(\varphi\), then, there is  
\[Y(x)= \begin{cases}f_{1}&=\frac{x_{1}[\sin x_{2}-\sin a_{0}]}{x_{2}-a_{0}}- \Delta Z=0\\ f_{2}&=\frac{x_{1}[\cos(x_{3}-x_{2})-\cos(\varphi_{0}-a_{0})]}{2[(x_{3}-\varphi _{0})-(x_{2}-a_{0})]}-\frac{x_{1}[\cos(x_{3}+x_{2})-\cos(\varphi_{0}+a_{0})]}{2 [(x_{3}-\varphi_{0})+(x_{2}-a_{0})]}-\Delta N=0\\ f_{3}&=\frac{x_{1}[\sin(x_{3}-x_{2})-\sin(\varphi_{0}-a_{0})]}{2[(x_{3}-\varphi _{0})-(x_{2}-a_{0})]}-\frac{x_{1}[\sin(x_{3}+x_{2})-\sin(\varphi_{0}+a_{0})]}{ 2[(x_{3}-\varphi_{0})+(x_{2}-a_{0})]}-\Delta E=0\end{cases} . \tag{19}\]  
It can be seen from the above formula that when \(Y(x)=0\) is equivalent to the minimum sum of square deviations of the increment of the actual coordinate of the wellbore trajectory and the vertical depth coordinate, the north coordinate, and the east coordinate, the established optimal objective function can be written in the form of (20):  
\[ \begin{split}\min& Y(x_{1},x_{2},x_{3})=f_{1}^{2}+f_{ 2}^{2}+f_{3}^{2}\\ s.t.&\left\{\begin{array}{l}\Delta L_{0}\leq x_{1} \leq\Delta L_{2}\\ \alpha_{0}\leq x_{2}\leq\alpha_{1}\\ \varphi_{0}\leq x_{3}\leq\varphi_{1}\end{array}\right.\end{split}