## 1 Introduction  
With global urbanization, the adoption of pipe jacking technology has become widespread in the creation of diverse urban infrastructures [1; 2; 3; 4; 5]. Compared with the conventional open-cut method, pipe jacking, which requires a much smaller job site area, could avoid utility diversions, mitigate the risks of damaging existing infrastructure, and reduce traffic disruptions [2; 3; 4]. Yet, the jacking distance, to a great extent, restricts the application of pipe jacking because of friction resistance [1]. When the jacking distance is increased, the jacking force keeps rising, and when the friction resistance increases, more jacking force is also required. As a result, excessive jacking force might damage the pipe segments, reaction walls, and adjoining buildings [1; 2; 3; 4]. In the field of engineering, friction resistance is commonly mitigated using two prevalent strategies: the strategic placement of intermediate jacking stations within pipe segments at regular intervals, and the injection of lubricants into the annular gap that exists between the pipe strings and the rock-soil mass [5; 6]. Of these two methods, lubrication is preferred, since intermediate jacking stations are expensive, environmentally unfriendly, and cumbersome [5; 6; 7; 8]. Thixotropic slurry, a lubricant traditionally based on bentonite, has earned extensive utilization owing to its outstanding efficacy, broad-spectrum suitability, and negligible ecological impact [9]. Scholars and engineers have conducted extensive research on rheological mud [10]. In Ref. [11], in the field of civil engineering, various methodologies have been implemented for the measurement and modeling of thixotropy in fresh concrete. In Ref. [1], a laboratory test was set up to simulate pipe jacking to identify the effects of bentonite pressure on the lubrication, finding that the continuous injection of bentonite slurry can reduce the friction coefficient significantly [1]. In Ref. [12], the dynamics of coarse particle suspensions within a thixotropic cement paste were examined. In Ref. [13], the lubrication performance was evaluated via the reduction in the friction coefficient, and the impacts of the injection type, soil, and lubrication type were investigated on the lubrication performance in a pipe jacking project. However, a change in mud performance alone cannot achieve effective resistance reduction. The contact between the pipe and soil and between the pipe and mud also yields large amounts of friction [14]. Scholars have conducted relevant research on the frictional force generated by contact pressure. In Ref. [15], [PERSON]'s arch model, in its original form, was adapted to facilitate the computation of the vertical pressure exerted on deeply embedded pipes. In Ref. [16], the average friction coefficient (AFC) was obtained in the residual phase for the contact surface under seven contact conditions. In Ref. [4], the frictional mechanism of pipe-soil-slurry and the jacking force prediction of rectangular pipe jacking were studied. Moreover, the frictional properties of the outer pipe interface under various contact conditions were studied along with a direct shear test [8]. In Ref. [7], a novel approach was suggested for estimating friction resistance in the context of rectangular pipe jacking. An experiment on how the slurry concentration and standing time influence the friction characteristics was conducted [6], and it was found that both the contact pressure and pipe stress exhibit significant asymmetry during the jacking process; the magnitudes of contact pressure at the crown and the bottom exhibit relative proximity [17]. Comparisons of the calculation models have been made in terms of the pipe friction resistance and application conditions, resulting in a simplified calculation model to estimate the pipe friction resistance in the rock stratum [18]. In Ref. [19], the grouting pressure was analyzed in slurry pipe jacking in terms of the mechanical characteristics of the surrounding soil stability.