## 2 Establishment and Calculation Process of a Numerical Model
### Establishment of Numerical Model and Parameter Settings  
(1) Establishment of numerical model  
This study focuses on a shaft with an outer diameter of 9.1 m and an auxiliary horsehead roadway inclined at 45 degrees, both reinforced with 0.5 m thick C50 concrete linings. The shaft's underground roadway, located within the shaft bottom parking lot, resides at a depth of 650 m below ground level, subject to an original geo-stress of 17.2 MPa. A computational mesh, accounting for the range of influence and boundary effects, was generated employing the Rhino & Griddle interface. Subsequently, this mesh was imported into FLAC\({}^{\rm 3D}\) (vision 5.00) for the development of a three-dimensional finite difference model, as depicted in Figure 1. The model dimensions are 132 m in height, 70 m in length, and 50 m in width, with horizontally fixed displacement boundaries on all four sides, a vertically fixed displacement boundary at the base, and a boundary condition of vertical stress applied at the top surface. To ensure that the results from the numerical model accurately reflect the actual conditions, the following steps were taken: (1) The orientation and thickness of the constructed rock layers align with the observed exposure of the bottom layer. (2) The lateral pressure coefficient and support form of the model are consistent with real-world conditions, specifically employing a lateral pressure coefficient of 0.75. The stratigraphy and thickness of each geological layer are detailed in Table 1.  
To mirror the complexities of real-world engineering conditions, a large deformation strain approach was adopted for the simulations. Additionally, two types of measurement points were established in the model. The first type was positioned at the center of the horsehead roadway's roof to monitor settlement displacement (as detailed in Section 3). The second type was placed on the inner side of the shaft lining to measure the Mises stress of the shaft unit and assess the extent of shaft damage (as detailed in Section 3.5). (2) Setting of model parameters  
The geological stratification within the study area, as delineated in Table 1, chiefly comprises siltstone, fine sandstone, coal seams, mudstone, and medium sandstone. The mechanical parameters for the numerical model were derived from controlled laboratory tests and modified to reflect in situ conditions of the shaft and horsehead roadway using \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rock Layer** & **Number** & **Strata _l/m_** & **Thickness _t/m_** \\ \hline Weathering zone & 1\# & 407 & 43 \\ Sandy mudstone & 2\# & 450 & 9 \\ Mudstone & 3\# & 459 & 18 \\ Silstone & 4\# & 477 & 30 \\ Mudstone & 5\# & 507 & 7 \\ Silstone & 6\# & 514 & 18 \\ Sandy mudstone & 7\# & 532 & 11 \\ Mudstone & 8\# & 543 & 9 \\ Silstone & 9\# & 552 & 9 \\ Mudstone & 10\# & 561 & 4 \\ Sandy mudstone & 11\# & 565 & 12 \\ Mudstone & 12\# & 577 & 7 \\ Sandy mudstone & 13\# & 584 & 18 \\ Mudstone & 14\# & 602 & 5 \\ Silstone & 15\# & 607 & 7 \\ Medium-grained & 16\# & 614 & 13 \\ Mudstone & 17\# & 627 & 10 \\ Coal & 18\# & 637 & 6 \\ Sandy mudstone & 19\# & 643 & 5 \\ Mudstone & 20\# & 648 & 3 \\ Coal & 21\# & 651 & 2 \\ Sandy mudstone & 22\# & 659 & 8 \\ Mudstone & 23\# & 669 & 10 \\ Silstone & 24\# & 679 & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Strata and thickness of each rock layer in the numerical model. Figure 1: Numerical calculation model. the Hoek-Brown strength criterion and the Geological Strength Index (GSI). These adjusted parameters for select strata and the borehole are detailed in Table 2. The density data of the rock layer are provided directly by the mining party, whereas other parameters are derived from uniaxial and triaxial compression tests on core rock samples. For the remaining layers, the parameters were established using engineering analogy and interpolation methods [8]. Engineering analogy involves referencing geological parameters previously studied and determined within the same mining area.