# Examining Pipe-Borehole Wall Contact and Pullback Loads for Horizontal Directional Drilling
## 3 Analytical Model for Calculating Pipe Pullback Loads
### Pipeline Pullback Loads  
During pipeline pullback operation, the pipeline is subjected to the friction force between the pipeline and ground surface or roller \(T_{ig}\), friction force between the pipeline and borehole wall \(T_{is}\), drilling fluid drag force \(T_{if}\), and pipeline's weight \(T_{is}\)[7; 8]. Considering that the drilling fluid pressure at different positions of the borehole may differ, the borehole is divided into \(N\) linear calculation units when calculating \(T_{is}\) (Figure 10). The drilling fluid pressure at the midpoint of the calculation unit was used when calculating the pullback loads. Figure 10: Schematic of borepath in HDD projects. Figure 9: Schematic of the eccentric annulus. When the pipeline is pulled to the end of calculation unit \(i\), the friction force between the pipeline and borehole wall \(T_{is}\) can be obtained by summing the pipe-borehole wall friction force (calculated using Equation (28)) of the previous \(i\) calculation units:  
\[T_{is}=\sum_{k=1}^{i}\Big{[}2 ck\sqrt{R\delta_{k}}S_{k}+P_{k}S_{k}\tan\varphi\ \Big{]}, \tag{34}\]  
where \(\delta_{k}\) represents the compressive displacement of the calculation unit \(k\); \(P_{k}\) denotes the external load per unit length of the calculation unit \(k\); and \(S_{k}\) signifies the length of the calculation unit \(k\). Directional changes in the borehole lead to an increase in the external load, called the capstan effect [44]. Therefore, before calculating the external load, \(P_{k}\), it is necessary to determine whether the direction of the borehole has changed. When the inclination angle of calculation unit \(k\) is the same as that of calculation unit \(k-1\), the external load \(P_{k}\) (per unit length) is calculated as follows:  
\[P_{k}=wg\cos\theta_{k}, \tag{35}\]  
where \(w\) denotes the net weight of the unit length of the pipe; \(\theta_{k}\) represents the inclination angle of the calculation unit \(k\) (the angle between the axis of the calculation unit and the horizontal). When the inclination angle of the calculation unit \(k\) is different from that of the calculation unit \(k-1\), the external load \(P_{k}\) is also related to the pullback force \(T_{k-1}\) and the inclination angle of the calculation unit \(k-1\). The external load increment \(\Delta P_{k}\) caused by the capstan effect can be calculated by the following equation [44]:  
\[\Delta P_{k}=T_{k-1}|\theta_{k}-\theta_{k-1}|, \tag{36}\]  
where \(T_{k-1}\) represents the pullback loads for the pipeline to be pulled back to the calculation unit \(k-1\); \(\theta_{k-1}\) denotes the inclination angle of the calculation unit \(k-1\). It should be noted that if the pipeline has high stiffness and undergoes significant bending when there are directional changes in the borehole, it will also lead to an additional increment of external load, \(\Delta P_{sli}\). In this case, when calculating \(\Delta P_{k}\), this additional increment, \(\Delta P_{sli}\), should be included. The calculation of \(\Delta P_{sli}\) has been addressed by [PERSON] and [PERSON] [8]. Due to the complexity of the calculation method for \(\Delta P_{sli}\), it is not listed here to save space. For details, please refer to reference [8].