# Examining Pipe-Borehole Wall Contact and Pullback Loads for Horizontal Directional Drilling
## 2 Analysis of Pipe-Borehole Wall Contact
### HDD Pipe-Borehole Wall Contact Model
#### 2.2.1 Maximum Contact Pressure in HDD  
Compared to analytical methods, FEM is not limited by the assumptions of contact pressure distribution and geometric relationships and is widely used in contact analysis [31; 34; 35; 36; 37; 38; 39]. This study used FEM to investigate the pressure distribution and external load-compressive displacement relationship for the pipe-borehole wall contact in HDD projects. The details of the FEM model are provided in Appendix A. The calculation parameters are listed in Table 1, and the range of parameters is based on the HDD operating conditions. [PERSON] et al. [34] showed no significant difference between the elastic and elastic-plastic analysis results when the external load \(P\) was small. Taking the 900 mm diameter HDPE pipe as an example, the order of magnitude of the external load, \(P\), was approximately 1 N/mm. In addition, we conducted an elastic-plastic analysis for No. 1-3 test conditions (Table 1). These results were consistent with those of the elastic analysis, and no plastic deformation occurred. Therefore, this contact analysis assumes that the pipe and borehole walls are isotropic elastic bodies. Figure 3 shows the change in the maximum contact pressure when only the drilling fluid pressure \(\sigma_{0}\) is changed (No. 40-52 tests in Table 1). It can be observed that the drilling fluid pressure affects the maximum contact pressure; the greater the drilling fluid pressure, the more significant the impact. Therefore, it is necessary to consider the influence of the drilling fluid pressure in HDD pipe-borehole wall contact analysis. Figure 3 also shows  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Test** & \(\mathbf{E_{1}}\) & \(\mathbf{E_{2}}\) & \(\mathbf{R_{1}}\) & \(\mathbf{R_{2}}\) & **Overcut Ratio** & **1/2\(P\)** & **Drilling Fluid Pressure** \\
**No. ** & **/MPa** & **/MPa** & **v\({}_{1}\)** & **v\({}_{2}\)** & **/mm** & **/mm** & **Overcut Ratio** & **1/N-mm\({}^{-1}\)** & **/MPa** \\ \hline
1–7 & 750 & 10–70 (with an interval of 10) & 0.3 & 0.3 & 600 & 900 & 1.5 & 2.11 & 0.1 \\
8–11 & \(2.1\times 10^{5}\) & \(10,30,50,70\) & 0.3 & 0.3 & 600 & 900 & 1.5 & 2.19 & 0.1 \\  & & & & & &  
that the drilling fluid pressure \(\sigma_{0}\) results in a maximum contact pressure increment \(\Delta p\) of \(q_{0}\) (the maximum contact pressure without the drilling fluid pressure). The maximum contact pressure \(p_{0}\) in HDD pipe-borehole wall contact can be represented as follows:  
\[p_{0}=q_{0}+\Delta p, \tag{14}\]  
where \(\Delta p\) stands for the maximum contact pressure increment caused by the effect of drilling fluid pressure. Figure 4 shows the maximum contact pressure without drilling fluid pressure (\(q_{0}\)) calculated by different methods. The equations of the classical contact models have been introduced in the previous section. As depicted in Figure 4, the Hertz model had the smallest error in calculating \(q_{0}\), and the \(q_{0}\) variation law of the Hertz model was also consistent with the FEM results. Therefore, the Hertz model was used to calculate the maximum contact pressure without drilling fluid pressure \(q_{0}\):  
\[q_{0}=\left(\frac{PE^{*}}{\pi R}\right)^{1/2}. \tag{15}\]  
Figure 4: Comparison of maximum contact pressure without drilling fluid pressure calculated by different models. Figure 3: Effect of drilling fluid pressure on maximum contact pressure. According to Equations (14) and (15), the maximum contact pressure \(p_{0}\) is related to the external load \(P\), equivalent elastic modulus \(E^{*}\), equivalent radius \(R\), and drilling fluid pressure \(\sigma_{0}\). To study the relationship between the maximum contact pressure increment \(\Delta p\) and the above parameters, the maximum contact pressure increment \(\Delta p\) under different parameters was calculated using FEM.