# Research on Intelligent Optimization of Wellbore Trajectory in Complex Formation
## 3 Borehole Trajectory Optimization Model  
Because the traditional wellbore trajectory design model is difficult to directly apply to the particle swarm optimization algorithm, the natural parameter curve model is used to construct it. 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, well deviation angle, and azimuth angle. The design model of the three-dimensional wellbore trajectory includes the spatial arc model, the cylindrical spiral model, and the natural curve model. It is of great significance to select the appropriate model and establish the corresponding constraint equation. The spatial arc model and the cylindrical spiral model are suitable for the drilling trajectory of the general steering tool. The natural curve model is used to establish the constraint equation in this study because it can better adapt to the needs of wellbore trajectory planning with azimuth drift. In the natural parameter curve model, it is assumed that the well deviation change rate \(K_{\alpha}\) and the azimuth change rate \(K_{\varphi}\) of the arc section of the design trajectory are constant values. When \(K_{\alpha}=0\), the wellbore trajectory presents a straight line shape on the vertical profile; conversely, it is an arc. On the horizontal projection, it is a straight line when \(K_{\alpha}=0\) and \(K_{\varphi}=0\), and an arc when \(K_{\alpha}=0\) and \(K_{\varphi}\
eq 0\). This characteristic gives the natural parameter curve model unique advantages in dealing with the wellbore trajectory with azimuth drift, and it can describe the shape and change rule of wellbore trajectory more accurately. According to the theory of wellbore trajectory geometry, under the natural parameter curve model, the well deviation angle change rate \(K_{\alpha}\) and the azimuth angle change rate \(K_{\varphi}\) are defined by the following formulas:  
\[K_{\alpha}=\frac{\alpha-\alpha_{0}}{\Delta L},\;K_{\varphi}=\frac{\varphi- \varphi_{0}}{\Delta L} \tag{3}\]  
Here, \(\alpha_{0}\) and \(\varphi_{0}\) are the well deviation angle and azimuth angle at the initial position of the curve segment, \(\alpha\) and \(\varphi\) are the well deviation angle and azimuth angle at the end position of the curve segment, \(\Delta L\) is the well depth increment, \(K_{\alpha}\) and \(K_{\varphi}\) are the well deviation change rate and azimuth change rate, respectively. These two parameters quantitatively describe the change rate of the deviation angle and azimuth angle of the wellbore trajectory in the well depth direction, which are of great significance in accurately characterizing the bending degree and direction change of the wellbore trajectory. The assumption of constant deviation and azimuth change rates simplifies the mathematical model, making it more computationally tractable. However, in dynamic geological environments with abrupt stress fields or fracture zones, this assumption might not fully capture real-world complexities. Future research could explore variable-rate models to better adapt to these scenarios. Based on the geometric theory of wellbore trajectory, it can be concluded that there is an integral relationship between vertical depth coordinate increment \(\Delta Z\), north coordinate increment \(\Delta N\), east coordinate increment \(\Delta E\) and well depth increment \(\Delta L\), well deviation angle \(\alpha\) and azimuth angle \(\varphi\). The specific expression is as follows:  
\[\Delta Z=\int_{L_{0}}^{L}\cos\alpha dL,\;\Delta N=\int_{L_{0}}^{L}\sin\alpha \cos\varphi dL,\;\Delta E=\int_{L_{0}}^{L}\sin\alpha\sin\varphi dL \tag{4}\]  
The definition formulas of well deviation change rate and azimuth change rate are substituted into the above integral, and the calculation formula of coordinate increment of natural parameter curve is obtained by integral operation.