## 1 Introduction  
China is one of the most severely threatened nations in terms of the risks of coal and gas outburst in underground coal mines. In recent decades, as coal mining depths have increased, the majority of low-gassy coal mines have become high-gassy and outburst-prone mines that reduce the safety and advantage of mining [1]. Coal mine methane (CMM) extraction is the most fundamental approach to eliminate the risk of coal and gas outburst. It can not only enhance coal output but also capture clean energy and reduce greenhouse gas emission [2; 3; 4]. However, particularly in southwest of China, where the coal seam permeability is lower than 0.001 mD, the flow in single-hole gas drainage from coal seams is quite low and extraction completion takes up to 2-3 years [5; 6]. Hydraulic technology for increasing the permeability of coal seams, including hydraulic slotting, hydraulic fracturing, and hydraulic flushing is an effective way to increase gas drainage efficiency in low permeability coal seams. The hydraulic slotting technology uses a high-pressure waterjet to cut the coal around the borehole and form an elliptic pressure relief zone around the slot. In this area, the crustal stress is relieved and gas can be easily extracted, but the scope of its influence is small. Hydraulic fracturing has become a standard practice applied to low permeability coal seams, and it can transform the structure of coal to enhance gas extraction efficiency. It is generally believed that hydraulic fracture propagation in coal seams is subjected to the horizontal stress difference and coal seam dip angle, elastic modulus, and fracture toughness [7; 8]. These primary factors could lead to disorderly propagation of hydraulic fractures and damage to the roof and floor. As a result, gas drainage from a coal seam stimulated by fracturing can be inefficient. To enlarge the slotting influence scope and resolve the issue of disorderly crack propagation in conventional hydraulic fracturing, a directional hydraulic fracturing method integrated hydraulic slotting and hydraulic fracturing is proposed [9; 10; 11; 12; 13], as shown in Figure 1. This method utilizes a high-pressure water jet to slot in the fracturing borehole and directional boreholes along the coal bed inclination, and then hydraulic fracturing is performed in the fracturing borehole. Through the guidance of directional boreholes, the direction of hydraulic fracture propagation follows the trend of coal seam and therefore averts damage to the roof and floor. In the previous study [11], when hydraulic slotting deviation angle or the horizontal stress difference is small, the hydraulic fracture can be oriented to expand along the direction of the slotting arrangement and connect fracturing borehole to slotting borehole. Coal is typical a dual-porosity reservoirs consisting of matrix and natural fractures, termed cleats [14; 15]. After directional hydraulic fracturing, fractures induced by hydraulic slotting and fracturing can provide highly permeable flow channels for gas transport into boreholes. When CMM extraction process begins, the free methane within the induced fractures will flow out firstly. Then the methane in the coal matrix will desorb into these fractures and the gas pressure will decline rapidly in the coal seam. The whole process contains a series of coupling behaviors, such as coal deformation, gas flow and heat transfer within hydraulic fractures, matrix, and cleat system. Therefore, it is crucial for gas extraction from a coal seam stimulated by directional hydraulic fracturing to consider the different type of gas flow at different scales under thermo-hydro-mechanical (THM) coupling conditions (Figure 2). Figure 1: Directional hydraulic fracturing method. Previously, many models have been derived for CMM extraction or coalbed methane extraction to characterize gas migration in coal seams. [PERSON] (1966) first proposed a coal permeability related to effective stress and coal matrix shrinkage during gas drainage.