# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 3 Establishment of Optimization Model  
### Selection of Optimization Index  
The optimization target indexes of the wellbore trajectory used in this paper include five aspects: wellbore trajectory length, friction, torque, wellbore trajectory complexity, and target accuracy (Appendix A). (1) Borehole trajectory length  
The length of wellbore trajectory is an important index to evaluate the drilling cost. The deeper the well is drilled, the more materials are needed, the more time is spent, and the higher the energy is consumed. The wellbore trajectory is the actual path of the drill bit in the three-dimensional geological space. It is a continuous and smooth curve, which is generally divided into multiple straight lines and curves. In this paper, the wellbore trajectory is considered to be composed of straight lines and deflecting sections. The minimum curvature method is used to describe the well section. The calculation of the minimum curvature method is to form a smooth curve connection transition by ensuring that the curvature of each wellbore trajectory changes very little, so that the drilling process is more stable, which can reduce the friction and torque in the drilling process. The calculation model of well depth \(L_{i}\) in stable section is  
\[L_{i}=(D_{B}-D_{A})/\cos\alpha, \tag{2}\]  
where \(D_{A}\) and \(D_{B}\) represent the vertical depth of point \(A\) and \(B\), respectively, and \(\alpha\) is the deviation angle at \(A\) point. The calculation model of well depth \(S_{i}\) in the build-up section is  
\[S_{i}=R\theta,(\theta=\cos^{-1}[\cos\alpha_{1}\cos\alpha_{2}+\sin\alpha_{1} \sin\alpha_{2}\cos(\varphi_{2}-\varphi_{1})]). \tag{3}\]  
Among the variables, \(R\) is the radius of curvature (m), \(\theta\) is the bending angle of the measuring section (dogleg angle), \(\alpha_{1}\) and \(\alpha_{2}\) are the well deviation angle (_rad_) at the beginning A and the end B of the deflecting section, respectively, and \(\varphi_{1}\) and \(\varphi_{2}\) are the azimuth angles (_rad_) at the beginning A and the end B of the deflecting section, respectively. The total length of the wellbore trajectory composed of \(m+n\) well sections is as follows:  
\[L=\sum_{i=1}^{m}L_{i}+\sum_{i=1}^{n}S_{i}. \tag{4}\]  
(2) Friction  
The friction is the friction between the drilling tool and the borehole wall during the movement of the drilling tool in the well. The high friction will not only bring energy consumption to the rotation of the drilling tool, but also easily damage the drilling tool and cause the drilling tool to jam. The calculation model of drill string friction can usually be divided into a hard model, soft model, and finite element model [32; 33]. According to the design of the wellbore trajectory, a soft model with a simple calculation is adopted in this paper. The model assumes that the drill string is a flexible rod that can withstand torque but cannot withstand bending moments. Considering that the radius of curvature and friction coefficient of the horizontal well vary with the depth of the well, and the weight per unit length of the drill string also varies with the depth of the well, the tension and friction of the drill string should be calculated in sections. The segmentation method is shown in Figure 3. Each unit length is called a micro-segment. In each micro-segment, the wellbore curvature is considered to be a constant, and the drill string axis remains the same as the wellbore axis,regardless of the dynamic effect of the drill string in the well. Based on this assumption, the force analysis of an arbitrary drill string unit with a length of \(d_{l}\) is shown in Figure 4.  
Here, the friction in the process of tripping is mainly considered. It is assumed that the drill string is close to the lower wall during the tripping process, and the drill string micro-element \(dl\) is taken as the research object.