## 1 Introduction  
Coal resource remains one of the primary sources of energy supporting economic development for a long time in the future [1, 2, 3, 4]. Coal gas, as an associated resource of coal, may cause devastating dangers, which become more severe as the mining depth increases [3, 5, 6]. Studying the stress distribution of the coal-rock mass and the associated gas migration evolution is important for proposing efficient gas extraction measures and ensuring the safe production of gas-bearing coal mines [1, 7, 8]. Coal mining unbalances the stress state of the coal-bearing strata, inducing coal-rock mass damage and breakage [9, 10]. During this process, fractures in the coal-rock mass initiate, propagate, and nucleate, forming complex fracture networks. This dramatically increases the permeability of coal-rock mass, providing favorable channels for coal gas migration and accumulation [1]. Numerous researchers have extensively studied the fracture distribution in the overlying rock and induced gas migration through theoretical analysis [11, 12, 13], numerical simulation [9, 14], and field studies [15, 16]. The overlying roof of the coal seam gradually forms three distinctive zones in the vertical direction with the advancement of the working face (Figure 1), namely, the caved zone, the fractured zone, and the continuous zone [10, 14]. The permeability of rock mass in the fractured zone is considered the most appropriate for gas extraction due to numerous horizontal and vertical fractures and poor connectivity with the mining goal [10; 17]. Then, the three-zone model is continuously improved to the four-zone model [12] and the five-zone model [10] based on the deformation and failure of the overlying rock mass [10], aiming to determine the most favorable gas extraction area. [PERSON] et al. [18] investigated coal gas extraction of a working face after pressure relief through protective layer mining, promoting the theory of pressure-relief gas extraction. [PERSON] and [PERSON] [9; 13; 19] revealed the fracture evolution of the overlying rock mass through numerous in situ observations and established the O-shape theory of gas extraction under pressure relief. They proposed that numerous induced fractures in the O-shape area significantly enhance the permeability of the rock mass, which is beneficial for gas extraction. Subsequent studies further promoted these theories by investigating the correlation between gas migration and fracture distribution in the pressure-relief area, optimizing the drilling planning for the pressure-relief gas extraction to ensure acceptable performance. [PERSON] et al. [20] and [PERSON] [21] found that fractures in the overlying rock mass develop continuously, rapidly increasing rock mass permeability as the vertical fractures connect with the separated fractures. [PERSON] et al. [19] investigated gas migration in the overlying rock mass and divided it into hard desorption zone, pressure-relief desorption zone, and gas conduction fractured zone. They suggested that the gas extraction boreholes should be drilled in the pressure-relief desorption zone to achieve a considerable extracting effect. [PERSON] et al. [22] presented that rock mass relieves pressure thoroughly when its expansion rate reaches 3%, which can be used as a critical index for pressure-relief gas extraction. [PERSON] et al. [23; 24; 25] determined the best amount and negative pressure of pressure-relief gas extraction by investigating gas migration and the associated gas concentration evolution in the goal. Previous researchers have made a tremendous progress in investigating the stress and fracture evolution of the overlying rock mass. However, limited efforts have been reported on the mutual effects of fracture distribution, rock permeability, and gas extraction performance. The effect of stress and fracture evolution on the favorable zone for pressure-relief gas extraction has not been thoroughly explored.