## 1 Introduction  
Nowadays, methane is the main threat to the increase of productivity in the coal industry [1]. The high gas pressure in coal seams may lead to coal and gas outburst accidents [2]. The rational use of methane is as efficient and clean energy-coalbed methane (CBM), which mainly contains methane and is an economical and promising solution for the global energy crisis [3]. Therefore, there is an urgent need for developing CBM resources, which are abundant. China has the most coal seams with low permeability, which make it difficult to achieve rapid extraction compared with other countries (such as Australia and the United States) [4, 5]. Low permeability severely restricts the extraction of methane in coal seams and also restricts the speed of coal mining. Hydraulic fracturing is an efficient stimulation method of pressure relief and permeability enhancement for strengthening the effect of methane drainage. Moreover, hydraulic fracturing has been applied to coal seam pressure relief to prevent coal mine gas outburst accidents [6; 7]. Presently, existing research on hydraulic fracturing in coal mines mainly concerns on-site practical applications. The occurrence of coal seams in different mines is variable, and a single theory cannot explain the complicated and variable coal seam hydraulic fracturing process. Researchers have investigated theoretical and technological approaches toward enhancing the permeability of coal seams. In the last century, researchers proposed the theory of tensile failure induced by hydraulic fracturing caused by stress concentration in the wall. The analysis of the circumferential tensile stress of the borehole wall has indicated that the circumferential tensile stress of the borehole wall increases with the liquid pressure in the wellbore. The cracks initiate when the tensile stress exceeds the tensile strength of the hole wall rock [8; 9; 10]. The fracture condition of the coal around the borehole wall of the horizontal bedding layer is established based on the maximum tensile stress failure criterion, and the formula for the critical value of the coal fracturing crack initiation pressure can be derived [11; 12]. Several theoretical models have been proposed for predicting the fracture initiation pressure for different pore pressure conditions and rock properties [13; 14]. Through hydraulic fracturing experiments in soft rock, it has been found that the confining pressure and fracturing fluid properties determine the effect of hydraulic fracturing [15]. Many factors such as high injection pressure exert a certain influence on the hydraulic fracturing efficiency [16]. The influence of the pre-fracture shear strength on the propagation of hydraulic fracture was investigated [17]. Over the past decades, considerable research effort has been directed toward investigating the mechanics of hydraulic fracturing [18]. Computer simulation technology has been used to simulate hydraulic fracturing in coal seams [19; 20; 21]. For example, the effect of natural fractures on hydraulic fracturing has been investigated by analyzing the characteristics of the crack radius, crack number, and growth rate of cracks versus the injection time using PFC2D [22]. Hydraulic fracturing technology of inclined coal seams has been extensively investigated, and many theories of hydraulic fracturing initiation have guided the safe production of coal mines, which effectively supports economic development. [PERSON] studied the effect of the maximum principal stress or intermediate principal stress on initiation and propagation of fractures [23]. The borehole stress has been analyzed through coordinate transformation, revealing the initiation pressure, initiation location, and initiation direction of inclined well [24; 25]. The crack initiation mechanism of penetrating boreholes in vertical seams was studied by analyzing the horizontal principal ground stresses [26]. However, many mining areas in China, such as Songzao, have multiple coal seam layers and a large burial depth and dip angle range.