# 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
The results are shown in Figure 5.  
As illustrated in Figure 5, a second-order polynomial effectively describes the relationship between the maximum contact pressure increment \(\Delta p\) and each of the related parameters (\(P\), \(E^{*}\), \(R\), \(\sigma_{0}\)). Additionally, compared to linear polynomials, the second-order polynomial fitting results exhibit smaller residuals and a higher R-squared value. Although higher-order polynomials may provide a better fit, they often lead to overfitting. Therefore, the second-order polynomial offers an optimal balance between fitting accuracy and model simplicity. In this study, the second-order polynomial was used to describe the relationship between the maximum contact pressure increment \(\Delta p\) and each of the related parameters. The equation is as follows:  
\[\Delta p=A_{0}+A_{1}E^{*}+A_{2}E^{*2}+A_{3}R+A_{4}R^{2}+A_{5}\sigma_{0}+A_{6} \sigma_{0}^{2}+A_{7}P+A_{8}P^{2}, \tag{16}\]  
where \(A_{i}\) (\(i\) = 1, 2... 8) is the coefficient required to be determined/fitted, \(A_{0}\) represents the theoretical baseline of the system, which is used to adjust and calibrate the overall level of the model; \(A_{1}\), \(A_{3}\), \(A_{5}\), and \(A_{7}\) indicate the linear influence of \(E\), \(R\), \(\sigma_{0}\), and \(P\) on the  
Figure 5: Relationship between contact pressure increment and related parameters: (**a**) external load; (**b**) equivalent elastic modulus; (**c**) relative curvature of the contact; (**d**) drilling fluid pressure. maximum contact pressure increment, respectively; \(A_{2}\), \(A_{4}\), \(A_{6}\), and \(A_{8}\) reflect the nonlinear quadratic effect of \(E\), \(R\), \(\sigma_{0}\), and \(P\) on the maximum contact pressure increment, respectively. Using data shown in Figure 5 and applying the nonlinear regression analysis method, it was found that, for most materials, geometric dimensions, and loads in HDD projects, the values of A\({}_{i}\) (\(i\) = 1, 2,... 8) listed in Table 2 fit the results well. The mean squared error of this fitting is \(2\times 10^{-4}\). The confidence interval is also shown in Table 2.  
The maximum contact pressure \(p_{0}\) can be obtained by substituting Equations (15) and (16) into Equation (14):  
\[p_{0}=\left(\frac{PE^{*}}{\pi R}\right)^{1/2}+A_{0}+A_{1}E^{*}+A_{2}E^{*2}+A_{3 }R+A_{4}R^{2}+A_{5}\sigma_{0}+A_{6}\sigma_{0}^{2}+A_{7}P+A_{8}P^{2}, \tag{17}\]  
where \(P\) can be calculated by subtracting the gravity from the buoyancy acting on the pipe per unit volume, \(E^{*}\) can be obtained by Equation (3), \(R\) can be calculated by Equation (6), and the values of A\({}_{i}\) (\(i\) = 1, 2... 8) are listed in Table 2.  
Figure 6 shows a comparison of the maximum contact pressure \(p_{0}\) between the theoretical result calculated by Equation (17) and the FEM results. It can be seen that the theoretical results are very close to the FEM results when considering the effect of drilling fluid pressure. The average absolute and relative errors for the 51 tests were 0.001 MPa and 0.6%, respectively.