## 1 Introduction  
During the development and utilization of urban underground space, shield tunneling is widely adopted in tunnel construction owing to its low stratum disturbance and high efficiency [1; 2; 3; 4]. Synchronous grouting is one of the essential processes in shield tunnel construction. The main characteristics of synchronous grouting include preventing large ground settlement, forming uniform force stress on the segment structure, and improving the tunnel impermeability performance [5; 6; 7; 8; 9]. The grouting effect in shield tunnel construction is difficult to monitor owing to the grouting position and lining segment structure. The grouting process is mainly based on engineering experience, which hinders the optimization of the synchronous grouting process. Therefore, it is necessary to study the distribution and dissipation of synchronous grouting pressure, which contributes to optimizing grouting parameters and improving the engineering efficiency of shield tunnels. The main research methods of grout pressure include experiment [10; 11; 12], numerical simulation [13; 14; 15; 16], and theoretical analysis [17; 18; 19]. Grouting materials usually include two types: single-component grout and double-component grout. In the study of single-component grouts, [PERSON] et al. (2017) developed a model to characterize the spatial and temporal distribution of the grout pressure in the shield tail void. The increase in the grout viscosity hindered the grout diffusion. The grout pressure decreased linearly along the longitudinal direction [20]. [PERSON] et al. (2021) established a non-linear spring model to simulate backfill grout consolidation in the shield tail void. The grout diffusion process had a significant effect on the grout pressure dissipation. The distribution of the grout pressure around the tunnel lining was mainly affected by the initial grout injection pressure [21]. The grout transport in the soil around the tunnel tail void was simulated using a coupled hydromechanical model. The results showed that the grout pressure should be increased and the grout time should be reduced in highly permeable soil to limit the ground settlement [16]. Understanding the parameters' influence on the results is important for structure stabilization [22]. [PERSON] et al. (2022) investigated the diffusion mechanism of quick-setting double-component grout, and the results showed that the quick-setting characteristics have an important effect on permeation and diffusion [23]. However, the quick-setting characteristics can be considered negligible when the void space exceeds a certain value. [PERSON] et al. (2017) investigated the influence of mechanical properties of two-component grouting materials on surface settlement and determined the optimal compressive strength range for controlling surface deformation and settlement [24]. The circumferential distribution of grout pressure was widely investigated. Most existing studies on the grout pressure during synchronous grouting in the shield tail void ignore the temporal variation in grout viscosity. In addition, grout penetration into the soil occurs after the grout fills the shield tail void, which has an influence on the grout pressure [25; 26; 27; 28; 29]. The initial pressure of grout penetration into the soil depends on the grout filling pressure in the shield tail void. Most studies simplify the initial pressure of grout penetration to initial injection pressure, which reduces the accuracy of the grout pressure prediction during synchronous grouting in shield tunnels. Few studies on the longitudinal and radial distribution of grout pressure considering the temporal variation in grout viscosity, which need to be studied, exist. The aim of this study was to investigate the circumferential, longitudinal, and radial distribution characteristics of grout pressure considering the temporal variation in grout viscosity.