# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 3 Establishment of Optimization Model
### Selection of Optimization Index
The force analysis is shown in Figure 4. In Figure 4, M is the bending moment on the cross section of the drill string, N/m; \(n\) is the axial tension of a drill string element segment, N; \(q\) is shear force, N; \(q\) is the weight per unit length of drill string in mud, N/m; \(n\) is the wellbore support force per unit length of drill string, N;\(f\) is the friction on the drill string per unit length; \(\rho\) is the radius of borehole curvature, m; \(D_{w}\) is the deviation angle, \({}^{\circ}\); \(D_{c}\) is the borehole diameter, m; \(D_{c}\) is the outer diameter of the drill string, m; and \(d\) is the inner diameter of the drill string, m. The friction coefficient between the drill string and the borehole wall is set as \(\mu\). According to the equilibrium condition and the friction law of the micro-element, differential equations can be obtained:  
\[ \begin{array}{c}-\frac{\partial Q}{\partial\alpha}+n(\rho+\frac{D_{w}}{2})= (\rho+\frac{D_{w}}{2}-\frac{D_{c}}{2})q\sin\alpha\\ \frac{\partial P}{\partial\alpha}\pm n\mu(\rho+\frac{D_{w}}{2})=-(\rho+\frac {\bar{D}_{w}}{2}-\frac{\bar{D}_{c}}{2})q\cos\alpha\\ \frac{\partial M}{\partial\alpha}+(\rho+\frac{D_{w}}{2}-\frac{D_{c}}{2}) \frac{\partial P}{\partial\alpha}\pm n\mu(\rho+\frac{D_{w}}{2})^{2}=-(\rho+ \frac{D_{w}}{2}-\frac{D_{c}}{2})^{2}q\cos\alpha\end{array} . \tag{5}\]  
In the formula, '\(\pm\)', the upper drilling tool is taken as '\(+\)', and the lower drilling tool is taken as '\(-\)', which is similar to the following. Figure 4: Friction force analysis of drill string unit. Figure 3: Schematic diagram of segmental calculation of drill string friction. It is assumed that the cross section of the drill string is only subjected to tensile and compressive loads, that is, the drill string is simplified as a soft column. At this time, it is obtained from Equation (5). \[ \begin{array}{c}T+nR_{1}=R_{2}q\sin\alpha\\ \frac{\mathrm{d}T}{\mathrm{d}\alpha}\pm n\mu R_{1}=-R_{2}q\cos\alpha^{\prime} \end{array}  \tag{6}\]  
where \(R_{1}=\rho+D_{w}/2;R_{2}=\rho+D_{w}/2-D_{c}/2\). The pulling force of drill string can be solved by the formula  
\[T=Ae^{\pm\mu\alpha}+\frac{qR_{2}}{1+\mu^{2}}[(\mu^{2}-1)\sin\alpha\pm 2\mu\cos \alpha]. \tag{7}\]  
Drill string line friction  
\[f=-\frac{2q\mu R_{2}}{(1+\mu^{2})R_{1}}(-\sin\alpha\pm\mu\cos\alpha)-\frac{\mu A }{R_{1}}e^{\pm\mu\alpha}. \tag{8}\]  
In the formula, A is an undetermined coefficient, which is determined by the boundary conditions. (3) Torque  
Torque has many constraints in well trajectory optimization design. In terms of drilling efficiency, excessive torque will make the contact pressure between the bit and the bottom hole rock uneven, resulting in accelerated bit wear, a reduced penetration rate, and may also increase the non-productive time and extend the drilling cycle. From the perspective of drilling cost, too much torque can easily damage the drill bit, and the frequent replacement of the drill bit will increase the cost. At the same time, it may also cause fatigue damage to drill pipes, drill collars, and other drilling tools, shorten their service life, and increase the cost of replacement and maintenance. In terms of downhole safety, excessive torque will increase the friction between the drill string and the wellbore, which easily causes sticking accidents, and may also affect the stress distribution of the formation around the wellbore, resulting in wellbore collapse. From the perspective of well trajectory control, excessive torque will cause a bending deformation or offset of the drilling tool in the well, resulting in the actual well trajectory deviating from the design trajectory, reducing the accuracy of the target, and affecting the stability and control accuracy of the tool face angle. For different well types and complex formations, the torque problem is more prominent, and the torque must be strictly controlled to meet the drilling requirements of special well types and adapt to the characteristics of complex formations.