# Numerical Analysis of Friction Reduction of Grouting in Long-Distance Pipe Jacking
## 1 Introduction
The combined structure of the slurry and the permeation layer is referred to as the grout sheath [12], as shown in Figure 1.  
The grout sheath transmits the liquid pressure, which exceeds the underground water pressure to the soil particles, and the effective stress generated by this process compacts the soil. Meanwhile, the hydraulic pressure of the grout also supports the borehole wall, maintaining its stability and preventing collapse. Therefore, the mechanism of grouting mainly involves two aspects: on one hand, the grout provides lubrication between the pipe and the surrounding soil, thereby reducing the frictional resistance during the jacking process; on the other hand, grouting fills the  
Figure 1: Formation of the grout sheath. gaps between the pipe and the soil, and the grouting pressure supports the soil, reducing its deformation, thus enhancing the stability of the soil. In the calculation of jacking force considering grout friction reduction, the most common method is presented in Section 2[13]. This method assumes that the circumferential friction of the pipe is uniformly distributed and uses an imprecise range of average unit frictional resistance between the pipe's outer surface and the soil to account for the friction reduction. This method cannot consider the depth at which the pipe is buried, which is a significant parameter that influences the friction values. To address these limitations, some scholars have proposed new methods to enhance the accuracy. [PERSON] et al. [14] derived a calculation formula for the additional stress in soil caused by grouting pressure under self-weight conditions based on [PERSON]'s theory of earth pressure. [PERSON] et al. [15] proposed a method to determine the three common slurry ring shapes for a circular pipe and introduced a segmental method to calculate the corresponding friction resistance. However, in practice, the distribution of friction resistance around the pipe is continuous and more complex, which can affect the calculation results, especially in the analysis of friction resistance for ultra-long pipe jacking, leading to non-negligible errors. The friction is influenced not only by the grouting, but also by the depth. It is necessary to improve the common method through a numerical analysis, especially by accounting for both the grouting and the depth at which the pipe is buried. This paper first presents the common theoretical calculation method adopted in China. Then, finite element analysis is employed to simulate the jacking force in long-distance pipe jacking, obtaining the jacking force values under different interface friction angles to simulate the effect of the grouting. The common theoretical method about the friction resistance is improved based on the simulation results to take into account both the grouting and the depth. The improved theoretical method evaluates the friction by calculating interface friction angles through the friction-reduction effect of grouting and normal stress distribution. The numerical and improved theoretical results are compared with the common method that assumes an imprecise range of average unit frictional resistance.