# Prediction of Delayed Surface Subsidence Based on the Improved Knothe-n Model
## 2 Numerical Simulation Programme
### Numerical Simulation  
The mining area is far from seismic fault zones, and the geological structure is relatively stable, with no history of large scale geological movements. Therefore, the influence of structural stress on numerical simulations is neglected. FLAC3D (Fast Lagrangian Analysis of Continua 3D), chosen for numerical simulation, is a piece of continuum mechanics analysis software. Unlike other finite element software, FLAC3D utilizes dynamic motion equations even for static system simulations, eliminating numerical barriers to simulate physically unstable processes. This enables a better simulation of the entire process from imbalance to equilibrium in the model. Based on relevant survey reports, the modeling process begins by refining the topographic map in CAD, followed by importing it into intermediate software for grid division and preprocessing of the model. This includes establishing the geological strata model within the affected area, dividing rock layers, and partitioning element grids. Finally, the model is imported into FLAC3D for assigning the physical parameters of rock and soil mechanics and conducting model calculations. The model is 4800 m long and 2700 m wide with 5.04 \(\times\) 10\({}^{5}\) elements. The elasto-plastic model serves as the constitutive mechanical model, with the Mohr-Coulomb yield criterion constitutive model chosen for calculation. The stratigraphic lithological parameters are listed in Table 1. The rock strata are regarded as continuous media. The influence of tectonic stress on original rock stress is not considered in the model, and only the self-gravity stress of the rock mass is considered. The model is in a static stress state, with horizontal displacements constrained to the front, back, and sides of the model. The bottom of the model is constrained to horizontal and vertical displacements, and the top of the model is assigned a free boundary. The mining process is divided into three stages: K1, K2, and K3. Profiles I-I', II-II', and III-III' are arranged as shown in Figure 2.  
Figure 1: The geological plan of the Yutianbao mining area. The overall model of the coal seam after mining, as well as the post-mining stress and displacement nephograms of each monitoring profile, are illustrated in Figures 3 and 4, respectively. Figure 3 reveals that the displacement nephogram of the overall model after coal mining can reflect the subsidence condition of the mining area. There are two subsidence basins in the nephogram, in which the subsidence in the basin where monitoring profile II-II' is located is slightly larger than that where monitoring profile III-III' is located, due to the greater subsidence of the mining area. It can be seen from Figure 4 that the displacement and stress fields of the Yutianbao coal mine follow the stress and displacement distribution laws in the roof and floor, as well as the surrounding rocks, after coal mining. The stress field of the mountain has changed greatly after coal seam mining. The boundary stress of the mining area has generally increased, with the monitoring profiles II-II' and III-III' reaching \(-40\) MPa. The stress increase of the monitoring profile I-I' is particularly pronounced in the middle of the concave zig-zag goat, i.e., the two areas of larger subsidence, where it reaches \(-42\) MPa, which exceeds the tensile strength and leads to cracking of the rock. The displacement nephograms confirm that the largest displacements of the I-