## 1 Introduction  
As a trenchless construction method, pipe-jacking technology offers advantages, including precise guidance, remote operation, minimal traffic disruption, and effective control of working face stability [1]. These attributes have rendered it widely applicable in the development and utilization of urban underground spaces. However, considering the surrounding heavy traffic and sensitive facilities, achieving stringent control over soil disturbance and surface settlement poses arduous challenges to the construction of pipe-jacking projects, particularly in cases involving shallow burial and soft soil foundations [2]. Extensive engineering practice has demonstrated that strategically integrating pipe-jacking methods with the pipe-roof pre-construction method (PPM) constitutes a viable countermeasure [3, 4, 5, 6, 7]. Since its initial application in the construction of the Antwerp Metro in Belgium, the PPM has emerged as a critical auxiliary technique for trenchless construction in urban underground spaces, due to its superior control over surface settlement [8; 9; 10; 11]. At present, pipe-roof construction primarily employs the directional follow-pipe drilling technology [9], the auger boring method [1], and the slurry pipe-jacking method [12]. Compared to other pipe-roof construction methods, the slurry pipe-jacking method evinces the following characteristics: (1) considerable pipes are successively jacked at small intervals; and (2) tail voids form around the jacked pipes, necessitating mud injection. Inevitably, repeated soil disturbance and over-excavation precipitate surface settlement. In some cases, settlement occurring during the phase of pipe-roof construction using the slurry pipe-jacking method (PPJM) accounts for up to 75% of the total settlement in underground construction [13; 14]. Therefore, as an auxiliary process intended to alleviate the disturbances to the surrounding stratum induced by trenchless construction, it is crucial to ensure that the impacts associated with PPJM are effectively controlled. Existing studies indicate that surface displacement during the jacking phase of PPJM results from the combined effects of settlement due to soil loss and uplift induced by lubricant injection; in contrast, surface settlement during the pause jacking phase is predominantly attributable to consolidation [14; 15]. Throughout the construction process, lubricants are employed to mitigate soil loss and support tail voids. Consequently, they play a critical role in controlling surface settlement caused by PPJM [16]. In view of this, [PERSON] et al. [17], [PERSON] et al. [18], [PERSON] et al. [19], and [PERSON] et al. [20] focused on developing new lubricants with more prominent friction reduction and better support capabilities. They experimentally evaluated the friction-reduction efficiency and surface settlement inhibition effects of various lubricants, elucidating the influence of condensation force, composition, filter loss, and yield point on surface settlement during pipe-roof construction. In addition, the injection pressure, gel time, and post-gel strength of the lubricant--as well as its permeability characteristics in different soils and distribution patterns within the tail voids--also influence its effectiveness in controlling surface settlement [21; 22; 23]. However, to simplify analysis, lubricants are often modeled as elastic bodies in numerical simulations, where the lubricant Young's modulus characterizes its stiffness and resistance to deformation [24; 25; 26]. A higher Young's modulus indicates that the lubricant deforms less under load, thereby more effectively supporting the tail void and reducing the surface settlement caused by PPJM. Additionally, [PERSON] et al. [12] examined the protection of underground adjacent structures using the PPM during trenchless construction, concentrating on the performance of new lubricants with time-enhanced mechanical properties.