## 2 Numerical Model for Grout Permeability in Backfilling Layers
### Construction of the Numerical Model  
The segments are assembled inside the shield machine and then pushed out through the shield tail. An annular gap is formed between the surrounding rock and the segmental lining, which is typically filled with gravel first, followed by grout injection. The grout bonds the gravel, surrounding rock, and segmental lining into an integrated structure, forming a complete backfilling layer (Figure 2a) that serves to transmit the surrounding rock pressure and provide impermeability. The pea gravel used in the project typically consists of two types--natural pebbles and artificial crushed stones (Figure 2b)--with particle sizes generally ranging from \(5\times 10^{-3}\) m to \(1\times 10^{-2}\) m. Natural pebbles are sourced from riverbeds, and their surfaces are smooth and rounded; artificial crushed stones are produced by breaking rocks, resulting in rougher surfaces with sharp edges. After being blown into the gap, the gravel accumulates and fills the void, forming the backfilling layer. The gravel backfilling layer is a typical porous medium. Compared to other porous media, the gravel backfilling layer has larger pore sizes and better connectivity between the pores. To estimate the porosity of the pea gravel layer, the filling process of gravel particles was simulated using PFC. First, randomly distributed particles are generated within a cubic region, with particle sizes uniformly distributed between \(5\times 10^{-3}\) m and \(1\times 10^{-2}\) m. Then, a pressure of 1 MPa is applied in all directions to compact the particle packing, transforming it from a loose to a dense configuration and thereby eliminating excess voids. The number of particles generated in this simulation is 827. The particle  
Figure 2: Backfilling layer and pea gravel: (**a**) backfilling layer; (**b**) pea gravel. density is 2500 kg/m\({}^{3}\), with the contact stiffness of particles set to 4 \(\times\) 10\({}^{7}\) N/m and contact stiffness of the walls set to 2 \(\times\) 10\({}^{8}\) N/m. The friction coefficient is 0.6, and the damping ratio is 0.7. The particle generation and consolidation results are shown in Figure 3, with the overall porosity being approximately 0.4. From the grouting port, the grout can flow in three directions: circumferential, axial, and radial. Therefore, the diffusion of grout within the backfilling layer is a complex three-dimensional problem. However, in this study, the grout flow is only considered in the circumferential direction for the following three reasons:  
1. Both segmental lining and surrounding rock can be treated as impermeable layers. 2. The circumferential diffusion speed of the grout is generally much greater than the axial diffusion speed in engineering applications [24]. 3. The circumferential perimeter of the backfilling layer is significantly larger than its thickness and axial length, making circumferential diffusion the dominant mode of grout flow. The simplified backfilling layer is an annular region with an inner diameter of 2 m and an outer diameter of 2.4 m. Since the grouting positions in tunnel construction are symmetrically distributed on the left and right sides, this study analyzes only the left half of the annular region. The process of constructing the backfilling layer computational model is shown in Figure 4. The first step is to generate a discrete set of particles using discrete element software. Due to computational resource limitations, this study scaled up the size of the gravel particles when constructing the porous medium model. Based on the results of the particle packing simulation mentioned earlier, the porosity at the time of particle generation is set to 0.4, with particle diameters ranging from 0.02 to 0.04 m.  
The second step involves importing the particle set data into COMSOL. The built-in functions in PFC (Particle Flow Code) allow the convenient retrieval of data such as the radius and coordinates of each particle. MATLAB R2024b is used to read the exported particle data, and the particles are re-plotted in COMSOL.