## 1 Introduction  
As shallow coal resources are gradually depleted, mining operations are increasingly extending into deeper strata [1], and the increase in in situ stress in deep strata complicates the management of mining disasters, especially accidents caused by the large-scale collapse of thick-hard roofs in the goaf area, which pose severe challenges to the safe production of coal mines [2; 3]. Hard roofs can lead to large-scale hanging roofs in working faces, posing the risk of rock bursts and seriously threatening the safe production of coal mines [4; 5; 6]. Currently, the management of hard roofs can be divided into passive protection and active pressure relief methods, including conventional techniques such as hydraulic fracturing, deep-hole blasting, and pressure relief drilling [7; 8; 9]. Passive protection methods, such as coal pillar support and backfill mining, are less commonly used in coal mines due to the low mining efficiency and high input costs [10]. The deep-hole blasting method is a traditional approach for controlling hard roof strata. However, it has several drawbacks that make it unsuitable for safe and efficient coal mine production [11]. The method requires large amounts of explosives, which not only pose significant safety risks but also contribute to underground air pollution. Moreover, in high-gas mines or coal seams, blasting increases the risk of gas or coal dust explosions, posing a severe safety hazard. As a technique for enhancing oil and gas permeability, hydraulic fracturing has been increasingly applied in the coal industry, especially in weakening hard roofs and relieving stress in high-stress roadways [12; 13; 14]. This technology can reduce the strength and integrity of hard roofs and promote the layered collapse of hard roofs, thereby reducing the periodic weighting interval and mitigating the negative impacts of large-scale roof collapse on mining operations [15]. Therefore, it is widely used in the management of hard roofs and has achieved good results [16; 17; 18]. However, conventional hydraulic fracturing typically uses short boreholes [19; 20], which have limited effectiveness in managing strong mining pressure induced by high-position hard roofs. It cannot address the problem at the source in high-position strata, and can only indirectly mitigate the negative effects of high-position strata by controlling low-position strata, making it difficult to ensure prevention and control effects. Thus, there is an urgent need to develop a preemptive and regional pressure relief method for high-position thick hard roofs [21; 22]. Determining the influence range of segmented hydraulic fracturing is a critical aspect of evaluating roof weakening effectiveness. Various numerical simulation methods have been used to study the extent of segmented fracturing and its influencing factors, providing guidance for optimizing hydraulic fracturing design and reducing associated risks. The main numerical calculation methods for hydraulic fracturing include the finite element method (FEM), extended finite element method (XFEM), and boundary element method (BEM). [PERSON] and [PERSON] [23] used FEM to simulate the fracture propagation process in porous elastic materials. [PERSON] et al. [24] simulated hydraulic fracture propagation in impermeable elastic media considering fluid lag using traditional FEM. [PERSON] et al. [25; 26] combined stiffness condensation technology with FEM to propose a fully coupled method for simulating hydraulic fracturing. [PERSON] et al. [27] used FEM to investigate the interaction between hydraulic fractures and natural fractures. [PERSON] and [PERSON] [28] first proposed the extended finite element method, solving the problem of remeshing during numerical simulation. [PERSON] and [PERSON] [29] proposed a new coupled XFEM algorithm and a new implicit level set algorithm to handle the singularity at the crack tip of hydraulic fractures. [PERSON] [30], [PERSON] et al. [31], and [PERSON] et al. [32] used XFEM to simulate non-planar hydraulic fracture propagation.