## 1 Introduction  
With the development of urban underground space, double-shield tunnel boring machines (TBMs) are increasingly applied in various tunnel construction projects due to their advantages, such as minimal impact on the surrounding environment, fast construction progress, and the ability to handle complex geological conditions [1, 2, 3, 4]. Synchronous grouting, as an environmental protection and tunnel stability control technique, plays a crucial role in double-shield TBM construction [5, 6, 7]. However, due to the unpredictable conditions behind the segmental lining, issues such as grout blockage and insufficient grouting are common during the synchronous grouting process. Therefore, exploring the flow mechanisms of grout has become an important research issue that needs to be addressed in double-shield TBM tunnel construction. Currently, the common methods used to study the permeability of grouts include theoretical analytical solutions [8; 9], physical model experiments [10; 11], and numerical research [12; 13]. [PERSON] and [PERSON] [14] conducted laboratory experiments to investigate the permeability of microfine cement grouts in sandy soils and analyzed the impact of particle size on grout permeability. [PERSON] et al. [15] derived a macroscopic model for grouting permeability in porous media based on mass balance equations for different components in the grouting medium and the general phenomenological filtration law. [PERSON] and [PERSON] [16] investigated the effect of the specific surface area of particles on the permeability of grout. The results indicated that finer cement particles have a higher specific surface area and, consequently, a higher surface charge, leading to lower grout permeability. [PERSON] et al. [17] analyzed the stopping mechanism of grout during permeation grouting under different water-to-cement ratios through laboratory sand column experiments combined with field grouting. [PERSON] et al. 's [18] experiments showed that adding oscillatory pressure to the steady-state injection pressure can reduce the adhesion of cement particles, thus improving grouting efficiency. [PERSON] et al. [19] conducted a series of permeability grouting experiments on fractured sandstone samples using low-field nuclear magnetic resonance (LF-NMR) technology, investigating the grout injection volume, effective grouting time, and grout filling velocity under different grouting conditions. [PERSON] et al. [20], based on fractal theory, considered the tortuosity of porous media and the time-dependent viscosity behavior of grout, and derived a two-stage column-hemispherical diffusion model of Newtonian fluid. However, most of these studies employed simplified models of porous media, homogenizing the porous structure and neglecting the influence of the microstructure of the porous medium on grout flow. Many scholars have considered factors such as fluid type and grout viscosity, and developed a variety of analytical models to study grout diffusion patterns under different conditions. [PERSON] et al. [21] treated the shield tail void as a three-dimensional circular thin cake and derived models for the grout filling pressure distribution along the circumferential direction of the shield tail gap under both Newtonian and Bingham fluid conditions. [PERSON] et al. [22] considered the interaction between soil pores and grouting, and developed a hemispherical surface diffusion model to determine the grouting pressure based on soil porosity. [PERSON] et al. [23], based on the characteristics of quasi-rectangular shield tunnels, took into account the viscous resistance during grout filling and diffusion, and derived a three-dimensional spatial distribution model for the diffusion pressure of synchronous grouting in quasi-rectangular shield tunnels. [PERSON] et al. [24] investigated the circumferential, longitudinal, and radial distribution characteristics of grout pressure and developed a pressure distribution and dissipation model that accounts for the time-dependent viscosity of Bingham grout.