# Non-Vertical Well Trajectory Design Based on Multi-Objective Optimization
## 3 Establishment of Optimization Model
### Constraint Conditions  
(1) Separation coefficient  
In the analysis of adjacent well anti-collision, in order to quantify the collision probability between the wellbore axes of two wells, the separation coefficient is usually used to evaluate and judge. The separation coefficient \(S_{f}\) is evaluated by the distance between the ellipsoids, as shown in Figure 6. The separation coefficient of the anti-collision decision between adjacent wells is calculated by distance scanning. Considering the safety distance of each wellbore and the distance between two wellbores, the calculation formula is as follows [34]:  
\[S_{f}=\frac{R_{1,2}}{R_{1}+R_{2}} \tag{31}\]  
(2) Build slope  
The build-up rate is the core parameter of wellbore trajectory designs, which directly affects drilling efficiency, drilling cost, and drilling safety. A high build-up rate can shorten the build-up length, thereby reducing the well depth and drilling cost, but it may increase the loss of drilling tools. On the contrary, it is easier in construction. According to the build-up capacity and geological characteristics of the tool, the build-up rate is generally controlled at 1.8-6\({}^{\circ}\)/30 m.  
\[K_{\text{imin}}\leq K_{i}\leq K_{\text{imax}} \tag{32}\]  
This range is derived from field practice: rates exceeding 6\({}^{\circ}\)/30 m increase directional tool wear by over 30% in on-site operations, while rates below 1.8\({}^{\circ}\)/30 m extend the build-up section unnecessarily, raising drilling costs. (3) Stabilizing angle  
In the design of the wellbore trajectory, the inclination angle refers to the constant degree of the inclination angle in the stable section, which directly affects the stability of the wellbore trajectory, drilling efficiency, and target accuracy. A reasonable inclination angle can make a smooth transition between the build-up section and the stable section and reduce the dogleg degree. Generally, the inclination angle is controlled between 60\({}^{\circ}\) and 90\({}^{\circ}\), preferably 90\({}^{\circ}\), and the constraint range is as follows:  
\[\alpha_{\text{imin}}\leq\alpha_{i}\leq\alpha_{\text{imax}} \tag{33}\]  
Field experience shows angles below 60\({}^{\circ}\) reduce reservoir contact area, lowering productivity, while angles above 90\({}^{\circ}\) increase the risk of wellbore collapse in shale formations. (4) Inclination point  
In the design of the wellbore trajectory, the build-up point is the initial control point of directional drilling, which will affect the feasibility, economy, and safety of drilling. When the build-up point is shallow, the smaller the build-up rate is, the more flexible the drilling is, the stronger the controllability is, and the lower the construction difficulty is. Different geological conditions have different requirements for the build-up point. The  
Figure 6: Separating coefficient. data of adjacent wells should be referred to. In this paper, the build-up point is controlled at 300 m-500 m in numerical simulation. Its constraint range is as follows:  
\[D_{\text{min}}<D_{\text{{k}op}}<D_{\text{max}} \tag{34}\]  
Shallower points (<300 m) are prone to trajectory deviation due to loose surface layers, while deeper points (>500 m) limit the adjustment space for subsequent sections, as verified in multiple drilling projects. (5) Stabilizing slope length  
In the three-stage wellbore trajectory design, the length of the stabilizing section refers to the distance of the well section that maintains the constant extension of the well deviation angle, which directly affects the safety and drilling efficiency of the wellbore trajectory control. In this paper, the numerical simulation is controlled within 2200 m. Longer lengths exceed the friction and torque limits of standard drilling equipment, based on operational data from extended-reach horizontal wells. In addition to the above constraints, according to the engineering and geological conditions, other linear and nonlinear constraints may be designed.