## 3 Numerical Analysis
### Generation of the Numerical Analysis Model  
Here, we examine a numerical analysis model that simulates a gravel-mixed ground. To create a model in which the size, arrangement, shape, and content of the gravel and the sandy ground around it can be varied, the rectangular division model and Voronoi division model shown in Figure 3 were examined. The analytical areas of the model here all have a height, width, and depth of 50 m, 50 m, and 0.25 m, respectively, for exhibiting two-dimensional plane strain behavior. To simulate the gravel using the voxel method, the analytical model was divided into rectangle grids of 0.2 m \(\times\) 0.2 m \(\times\) 0.25 m (length \(\times\) width \(\times\) depth). Each model in Figure 3 shows the case where the gravel content is approximately 40% as an example. Here, the gravel content represents the value obtained by dividing the gravel area by the model cross-sectional area; the black-colored area represents gravel, and the brown-colored one represents sandy ground. First, the rectangular division model shown in Figure 3 was set as follows:  
(1) Areas that imitate gravel of 0.2 m \(\times\) 0.2 m (length \(\times\) width), 1.0 m \(\times\) 1.0 m (length \(\times\) width), and 2.0 m \(\times\) 2.0 m (length \(\times\) width) were randomly placed in a rectangular grid of 0.2 m \(\times\) 0.2 m (length \(\times\) width) (Table 1)  
(2) The input for each of the gravel and sandy ground was set as constant as shown in Table 2 (Oshima, [16]). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Square Division Model** & **Voronoi Division Model** \\ \hline \multirow{3}{*}{Gravel dimensions} & 0.2 m \(\times\) 0.2 m & 500 \\ \cline{2-3}  & 1.0 m \(\times\) 1.0 m & Voronoi points & 600 \\ \cline{2-3}  & 2.0 m \(\times\) 2.0 m & 800 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Gravel simulation method and division method. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Unit Weight** & **Elastic Modulus** & **Poisson Ratio** \\  & **(kN/m3)** & **E (MPa)** & **Poisson Ratio** \\ \hline Sandy soil & 16 & 10 & 0.35 \\ \hline Medium hard rock & 23 & 1000 & 0.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input value of the ground. Figure 3: Numerical analysis model simulating ground mixed with gravel. However, most gravel causing problems in tunnel excavation has a huge size with rounded corners of rock, which is difficult to reproduce with a rectangular division model. Figure 4 is a photograph of terrace deposits exposed on the ground surface, and it is considered that the gravel-mixed ground used in this study contains a large amount of gravel of this shape. In particular, when a tunnel is excavated in such ground, gravel is present in the vicinity of the tunnel, and if gravel appears around the face, top end, and side wall due to tunnel excavation, the ground collapses and the risk in terms of occupational safety increases. For example, Figure 5 is an image of the front of the face when gravel appeared on the face due to tunnel excavation. Therefore, it is preferable to elucidate the behavior of these gravels in advance when they appear around the face and the crown/side wall. However, the rectangular division model cannot simulate the model. There are many differences from the gravel shape of the actual site, which makes tunnel construction difficult. Therefore, to reproduce the shape of the gravel seen in the field, we used the Voronoi partition model of the gravel, mixed ground. In a previous study in the rock field using Voronoi division, [PERSON] and [PERSON] [17] used this method to perform simulated calculations of rock tests and analysis of rock behavior considering cracks in the rock and further analyzed the behavior of discontinuous rock. Although it is said that the qualitative tendency can be expressed, no study exists where the strengths of both are significantly different in the model, such as the gravel-mixed ground, which is the target of this study, using Voronoi division.