## 1 Introduction  
In recent years, many researchers have conducted extensive work on using blasting to address and improve the free coal pillar [1]. In engineering projects, although the blasting method can solve the problem of mine pressure in most cases [2, 3, 4], it still has many limitations [5]: (1) blasting can damage the shaft and drift support, and the surrounding rock mass may suffer secondary impacts if the design is not appropriate for the project; (2) the impact of blasting may damage precision instruments, such as mine pressure detection equipment in intelligent working faces; (3) the toxic gases produced by blasting are harmful to employees and pose a threat to their safety; and (4) the preparation and application methods of CO\({}_{2}\) fracturing blasting are complex, resulting in low construction efficiency. Various methods, such as pre-splitting blasting [6, 7], CO\({}_{2}\) fracturing blasting [8, 9, 10], and static expansive agent fracturing [11, 12, 13], have been used to address and improve the free coal pillar and achieve significant results through on-site industrial tests. However, the research related to hydraulic fracturing in multi-layered regions is limited. The drilling observation results of the roof at the I030902 working face of Limin Coal Mine show that the roof is highly stratified, making it unsuitable for the aforementioned methods. Therefore, the following issues exist in the related domestic and international research:There is a lack of theoretical and experimental basis for addressing and improving free coal pillar roadways. The experimental materials used in the related research are non-transparent materials [14], making it difficult to observe the propagation of the hydraulic fracturing cracks directly. Additionally, the double-seal packer is used to fracture the hard roof of the free coal pillar roadway, while both domestic and international laboratories cannot accurately replicate the actual on-site conditions [15]. Research on hydraulic fracturing in multi-layered regions is limited. Previous methods, such as pre-splitting blasting, CO\({}_{2}\) fracturing blasting, static expansive agent fracturing, and directional long-hole hydraulic fracturing technology, are not suitable for application in Limin Coal Mine, where the roof is severely stratified, especially directional long-hole fracturing. Due to the single fracturing strata and excessive fracturing steps, directional long-hole fracturing cannot fully fracture the roof. As a green, safe, and efficient technological method, hydraulic fracturing has already been successfully applied in the petroleum engineering field [16; 17; 18]. In recent years, the application of hydraulic fracturing technology in the coal mining industry has primarily focused on enhancing the permeability of coal seams by improving the caving ability of roof strata [19]. Hydraulic fracturing is a method that injects high-flow fluids into coal and rock strata to increase artificial fractures within the coal rock [20; 21; 22]. In the coal mining field, hydraulic fracturing technology has become an important approach to enhance the permeability of high-gas coal seams and increase coalbed methane production [23]. Specifically, in response to issues such as the difficulty of natural caving in thick and hard coal seam roofs in some mining areas and the frequent occurrence of rock burst disasters [24], hydraulic fracturing technology provides strong support for achieving green, safe, and efficient coal resource extraction [25]. Based on the micro-seismic monitoring results, two key induced fracture strata were identified in the overlying strata of the free narrow coal pillar: the siltstone layer located 24.65-33.65 m above the coal seam and the coarse-grained sandstone layer located 38.44-48.63 m above the 9\({}^{-1}\) coal seam. There are few research studies on the theory and numerical simulation of irregular hydraulic crack propagation [26].