## 1 Introduction  
Cool mining constitutes a foundational component of China's energy infrastructure [1]. With progressive depletion of shallow coal reserves, an increasing number of mining operations have advanced into deep geological structures [2]. Deep coal seams are subjected to high geostress and intensive mining induced disturbances, resulting in escalated occurrences of coal rock dynamic disasters such as the compound disaster of the rockburst and outburst, which pose a substantial threat to safe mining [3]. In response to these challenges, coal seam gas drainage has been established as a critical engineering intervention for prevention and control of coal rock dynamic disasters [4]. However, the effectiveness of such operations is frequently constrained by the inherently low permeability of deep coal seams. The permeability of coal seams is regulated by a multifactorial [5], including porosity [6], pore structure [7], coal quality [8], stress state [9], temperature [10], pressure [11], and pore fluid properties [12]. The stress state emerges as the predominant determinant of permeability [13], by inducing pore compression and structure deformation [14]. The permeability around the gas drainage borehole is principally governed by the stress state [15] and the degree broken coal [16]. The stress state plays a key role in controlling the efficiency of coalbed methane drainage. The redistribution of stress arises from spatial variations in stress fields, which are intrinsically linked to the geometric configuration of working face, roadway, and borehole [17]. The stress state within this field impacts the permeability of broken coal mainly by two ways that the stress [13; 18] and the rate of stress loading [19]. As the pressure (including axial pressure and confining pressure) gradually increasing, the coal particles are compacted [20] and the reducing of pore volume inside of the broken coal [21; 22] and the closing of pores [23; 24] lead to narrowing of the seepage channel and an exponential decrease in the permeability [25; 26]. The rapid loading rate and unloading rate result in a sharp reduction in permeability [27] and a relatively low degree of permeability recovery [28], respectively. In addition, the slow loading rate and unloading rate show the opposite effects [29; 30]. In understanding the seepage law of coalbed methane, particle-size distribution plays an important role. The degree of coal broken is generally described using gradation theory [31]. The gradation theory pertains to the distribution law of particles of different sizes within the broken coal body, which is typically quantified by the Talbot power exponent [32]. As the Talbot power exponent increases from 0.1 to 1.0, the proportion of large particles [33] and the number of pores increase [34], and the connectivity of pore are enhanced [35], which promotes the growth of permeability [36; 37]. Collectively, these studies highlight the critical role of particle size, stress state, and gradation in elucidating the complex permeability behaviors of broken coal and gas drainage boreholes. The key parameters of broken coal under multiple permeability conditions were quantified, and the impact of gradation theory on permeability was evaluated. However, intricate mechanisms exist among the stress state, coal seepage, and the degree of coal broken. The spatially heterogeneous stress fields around the boreholes induce progressive broken gradients of coal. Consequently, understanding the permeability law of the continuously broken coal around the borehole remains of paramount importance. As a crucial indicator of the seepage-pressure coupling effect, the influence of effective stress on permeability requires further clarification. In this study, controlled broken coal specimens were made in accordance with gradation theory, followed by a triaxial permeability test under simulated in situ stress conditions to explain the permeability of broken coal around gas drainage boreholes.