# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 3 Establishment of Optimization Model
### Objective Function Design
 \tag{20}\]  
In this optimization problem, the objective function aims to minimize the sum of squares of the coordinate increment difference to achieve the optimal design of the wellbore trajectory. The constraint conditions limit the range of the well depth increment, well inclination angle, and azimuth angle. These ranges can be obtained according to engineering experience. Through this transformation, the wellbore trajectory planning problem is transformed into a standard optimization problem, which is conducive to the application of intelligent algorithms to solve, thus providing theoretical and methodological support for finding the optimal wellbore trajectory. \[Y_{1}=(\Delta N_{ab}+\Delta N_{bc}-N_{c})^{2}+(\Delta E_{ab}+\Delta E_{bc}-E _{c})^{2}+(D_{kap}+\Delta D_{ab}+\Delta D_{bc}-D_{c})^{2} \tag{21}\]  
\[ \begin{split} L_{ab}&=R_{i}*\alpha_{i}\\ L_{bc}&=\frac{D_{c}-D_{kap}-R_{i}*\sin\alpha_{i}}{\cos a_{i}}\\ \Delta E_{ab}&=R_{i}(1-\cos\alpha_{i})\sin\varphi_{i}\\ \Delta N_{ab}&=R_{i}(1-\cos\alpha_{i})\cos\varphi_{i}\\ \Delta D_{ab}&=R_{i}\sin\alpha_{i}\\ \Delta N_{bc}&=L_{bc}\sin\alpha_{i}\cos\varphi_{i}\\ \Delta E_{bc}&=L_{bc}\sin\alpha_{i}\sin\varphi_{i}\\ \Delta D_{bc}&=L_{bc}\cos\alpha_{i}\end{split}  \tag{22}\]  
(2) Friction  
It is assumed that the wellbore trajectory is a smooth curve, which is composed of a vertical section, a build-up section, and a stable section. The parameters of the initial point and the target point are \(N_{0},E_{0},D_{0},\alpha_{0},\varphi_{0}\) and \(N_{e},E_{e},D_{e},\alpha_{T},\varphi_{T}\), respectively. The length of the arc \(i\) section is \(s_{i}\), the radius of the arc element is \(R_{i}\), the length of the straight section \(j\) is \(l_{j}\), and the Pareto optimal solution set of friction resistance has a significant correlation between the lifting process and the lowering process. Therefore, the above lifting process is taken as an example to establish the objective function, which can simplify the whole calculation process. \[F(\alpha,\varphi_{i},R_{i},l_{j})=\sum_{i=1}^{n}\int_{0}^{s_{i}}F_{s}+\sum_{j =1}^{m}\int_{0}^{l_{j}}F_{l} \tag{23}\]\[ \begin{array}{l}F_{s}=T_{i+1}-T_{i}+qd_{l}R\left(\cos\beta_{i+1}-\cos\beta_{i} \right)\quad(\alpha_{i+1}>\alpha_{i})\\ F_{s}=T_{i+1}-T_{i}+qd_{l}R(\sin\alpha_{i+1}-\sin\alpha_{i})\quad(\alpha_{i+1}< \alpha_{i})\\ F_{l}=\mu qd_{l}\cos\alpha\\ \theta_{i}=\arccos(\cos\alpha_{i-1}\cos\alpha_{i}+\sin\alpha_{i-1}\sin\alpha_ {i}\cos(\varphi_{i}-\varphi_{i-1}))\\ T_{i+1}=(T_{i}-A\sin\beta_{i}-B\cos\beta_{i})e^{-\mu(\beta_{i+1}+\beta_{i})}+A \sin\beta_{i+1}+B\cos\beta_{i+1}\;(\alpha_{i+1}>\alpha_{i})\\ T_{i+1}=(T_{i}+A\sin\alpha_{i}+B\cos\alpha_{i})e^{-\mu(\alpha_{i+1}+\alpha_{i} )}-A\sin\alpha_{i+1}-B\cos\alpha_{i+1}\;(\alpha_{i+1}<\alpha_{i})\\ A=\frac{2\mu}{1+\mu^{2}}dq_{l}R\\ B=\frac{p^{2}-1}{1+\mu^{2}}dq_{l}R\\ \end{array}  \tag{24}\]  
During the rotation of the drill string, the friction resistance between the drill string and the borehole wall is generated, and the resulting torque causes the drilling pressure on the well to not be fully transmitted to the bottom of the well, which affects the drilling efficiency and increases the safety risk. Total torque:  
\[M=M_{0}+\frac{\mu qD_{o}}{2\sum\limits_{i=1}^{m}\left(L_{i}\sin\alpha_{i} \right)}+\sum\limits_{j=1}^{n}M_{j} \tag{25}\]  
Among the variables, \(M_{0}\) and \(M_{j}\) are the torque of the drill bit and the micro-segment torque of the build-up section or the drop section, respectively. Stabilized section drill string unit:  
\[M_{i+1}=M_{i}+\frac{\mu q\sin\alpha\Delta LD_{o}}{2} \tag{26}\]  
where \(D_{o}\) is the diameter of the pipe string (m), which is the length of the micro-segment. Axial force and torque of drill string unit in build-up section:  
\[ \begin{array}{l}T_{i+1}=T_{i}+qR(\cos\beta_{i}-\cos\beta_{i+1})\\ M_{i+1}=M_{i}+\frac{\mu[q\cos\beta_{i+1}-\frac{T_{i+1}}{R}]D_{o}}{2}R(\beta_{ i+1}-\beta_{i})\end{array}