## 1 Introduction  
The structure of the 3 rd coal seam in the Shanxi Formation of the Permian System in the western wing of the Dongtan mining area is highly complex, splitting into two distinct layers: the upper 3 rd coal seam (3 upper) and the lower 3 rd coal seam (3 lower). These seams are separated by a 4 m siltstone interlayer, which represents an ultra-close coal seam with significant implications for mining activities. In the global coal mining process, multi-seam mining is extremely common, and the spacing between coal seams and the mining sequence has a significant impact on the stability of the surrounding rock. This can easily lead to issues such as roof subsidence, rib spalling, and stress concentration, which contribute to rock layer damage and pose challenges to the safety of coal mine production. In the later stage of lower seam mining in the Dongtan mining area, roof support can only be limited to a 4 m layer of siltstone. Therefore, it is necessary to study the damaging effects of mining the 3 upper and 3 lower coal seams on the surrounding rock. In particular, the integrity of the thin sandstone layer and its associated stress distribution should be thoroughly examined. Understanding the damage characteristics of these layers provides crucial insights for supporting the surrounding rock during the mining of the lower 3 lower coal seam. The mining of multiple coal seams, particularly in close proximity, is inherently more complex than the extraction of a single seam. Numerous studies, both domestic and international, have explored these challenges through field monitoring, theoretical analysis, physical modeling, and numerical simulation. Some researchers have attempted to establish observation stations on the surface to monitor long-term strata movement, analyzing the influence of mining speed and distance on the displacement of overlying strata at different stages of coal seam extraction [1]. Other studies have utilized boreholes that penetrate multiple coal seams to monitor the induced strata subsidence and interlayer shear during mining activities [2; 3; 4]. However, field monitoring data alone can only provide a superficial understanding of the phenomena. To more accurately characterize the fracturing behavior induced by multi-coal seam mining, theoretical models have been proposed for calculation [5; 6], with analytical solutions or computational criteria derived from mathematical and mechanical theories. In addition, to assess the specific structural evolution, stress redistribution, and strata movement caused by multi-coal seam mining, physical models are employed for similarity simulations [7; 8; 9]. Nevertheless, with the increasing complexity of these issues, both approaches face certain limitations. The complexity of theoretical models often leads to cumbersome analytical processes, while physical model construction can be challenging, as it is difficult to replicate the exact field conditions. Compared to theoretical analysis, numerical simulation offers higher levels of visualization and more intuitive results. Moreover, numerical simulations provide better engineering compatibility and greater ease of variable control and monitoring than physical models. For these types of engineering challenges, finite element method (FEM) and discrete element method (DEM) are commonly used for modeling and analysis. FLAC\({}^{\text{3D}}\), a widely employed numerical simulation software based on the finite difference method, is particularly effective for simulating stress and deformation in the surrounding rock induced by multi-coal seam mining [10; 11; 12; 13; 14; 15]. FLAC\({}^{\text{3D}}\) has been used to analyze the impacts of multi-layer mining on strata stability [17], stress unloading [18], and the nonlinear superposition of stress [19]. While finite element-based simulations focus primarily on stress and deformation [20], discrete element methods more effectively visualize the evolution of rock mass structures during mining [21; 22; 23; 24; 25].