## 2 HDD Trajectory Designs  
### Geometric Designs  
The catenary trajectory is a chain curve with natural deflection, such as seen with ropes, cables or chains with uniform weight suspended between two points (Figure 1a). In petroleum engineering, the well path in the catenary profile has been proved to reduce the wellbore friction and torque compared to conventional trajectories [22; 23]. The method for planning the catenary well path was proposed in later studies, which avoided a trial-and-error procedure and provided excellent maneuverability of planning requirements [24]. In 1985, [PERSON] [25] proposed an HDD trajectory design method based on a catenary trajectory. However, its gradient curvature made it difficult to be implemented in practice. Therefore, [PERSON] et al. [20] proposed a new method that combined the catenary trajectory with a five-segment trajectory (Figure 1b) to design a drill path that was easier to be implemented using the available technology. The catenary trajectory is actually a hyperbolic cosine, whose standard mathematical formula is written as the following:  
\[z=a\ \times\cosh(x/a) \tag{1}\]  
\[a=Np/q \tag{2}\]  
where, \(a\) is a parameter determined by the unit weight of pipeline \(q\) (\(N/m\)) and pullback force \(Np\) (kgf) [20]. So, the shape of the catenary trajectory is mainly associated with the crossing pipelines and the capacity of the pulling back equipment. By Equation (3), the catenary trajectory can be described in a specific coordinate, where the left point of the catenary trajectory is set as the origin point. Therefore, the depth of each point on the catenary trajectory (\(zc_{i}\)) can be fully calculated with increments. As shown in Figure 2, A and H are the horizontal and vertical distances of the exit and the entry points, respectively. (\(x_{0}\), \(z_{0}\)) is the catenary vertex coordinate. \[\left\{zc_{i}=\ i\ \times\ dx\right. \tag{3}\]  
Another concept for HDD trajectory design is the segment trajectory method, combining the most conventional methods of straight and curvilinear segments. Mostly, the geometric profile of segment trajectory (shown in Figure 1b) commonly consists of an entry straight segment (\(L_{1}\)), an entry curved segment (\(L_{2}\)) with bending radius (\(R_{2}\)), a central horizontal straight segment (\(L_{3}\)), an exit curved segment (\(L_{4}\)) with bending radius (\(R_{4}\)), and an exit straight segment (\(L_{5}\)) respectively. Based on this combination form, the point  
Figure 1: Catenary trajectory (**a**) and five-segment trajectory (**b**). coordinates of each segment can be calculated clearly by geometric constraints. As shown in Figure 3, once the angle \(\gamma\) and radius \(R\) of the curved segments are set, the increments of the trajectory depth \(dz\) and horizontal distance \(dx\) can be determined according to Equation (4). Furthermore, combining the inclination angle \(\theta\) and length \(L\) of the adjacent straight segments, the depth \(z_{i}\) of another adjacent segment can be fully determined with increasing horizontal distance \(dx\) (Equations (5) and (6)). \[ \begin{cases}dx=R\;\times\;(\sin\gamma\;-\;\sin(\gamma\;-\;d\gamma))\\ dz=R\;\times\;(\cos\gamma\;-\;\cos(\gamma\;-\;d\gamma))\end{cases}  \tag{4}\]  
\[\alpha=\theta+\gamma \tag{5}\]  
\[ \begin{cases}x_{i}{=}i\;\times\;dx\\ z_{i}{=}L_{1}\;\times\;\sin\theta+R\;\times\;(\cos\theta\;-\;1)+\frac{x_{i}-L _{1}\;\times\;\sin\theta\;-\;R\;\times\;(\cos\theta+\cos\alpha)}{\tan\alpha} \end{cases}  \tag{6}\]  
According to Equations (4)-(6) above, the designed alternating straight and curvilinear segments can be detailed with tiny segments with coordinates. The whole geometry can be constrained only by four kinds of geometric parameters within the assigned domain, including the inclined angle of the straight segment \(\theta\) (clockwise is positive and counterclockwise is negative), the length of segment \(L_{i}\), the angle of curved segment \(\gamma\), and the curved segment radius \(R\).