## 1 Introduction  
Coal is highly heterogeneous and porous [1; 2; 3; 4], and gas adsorption and desorption primarily occur on its surface. Studying the pore size, pore structure, and pore fractal of coal is key to predicting, evaluating, and developing coalbed methane, as well as controlling coal mine gas disasters. At present, two main research methods are applied to the pore structure of coal, destructive and nondestructive testing. The destructive testing method involves injecting a fluid medium at high pressure or adsorbing it at a low temperature into the pores of coal and determining the pore volume of coal by calculating the volume of the fluid medium; examples include the widely used mercury intrusion method, liquid nitrogen adsorption method, and low-temperature carbon dioxide adsorption method. However, after the high-pressure or low-temperature fluid medium enters the sample, some of the original pores are destroyed. Nondestructive testing involvestechniques such as microscope observation and X-ray detection to study the pores inside the coal body without intervening, such as optical microscopy, atomic force microscopy, field emission electron microscopy, environmental scanning electron microscopy, X-ray computed tomography, nuclear magnetic resonance, and small-angle X-ray scattering. However, this method suffers from low accuracy, large errors, and poor adaptability and needs to be combined with traditional methods. Traditional Euclidean geometry can describe objects with smooth and homogeneous surfaces, but accurately characterizing the degree of roughness is difficult [5]. It was not until 1975 that [PERSON] established fractal geometry theory [6; 7], which accurately characterized rough-shaped objects and became an important method for describing the complexity of the coal pore structure. [PERSON] et al. [8] used sponge models to analyze the fractal characteristics of the coal pore structure at different degrees of metamorphism. [PERSON] et al. [9] used a thermodynamic fractal model to obtain the fractal dimension of coal seepage pores. [PERSON] et al. [10] quantitatively explored the pore structure characteristics and influencing factors of permeable pores in coal reservoirs using the Scherpinski model. [PERSON] et al. [11] used capillary models to analyze the complexity of pore structures in the 5-2 coal seam of the Haiwan Coal Mine and the 1 central coal seam of the Gubei Coal Mine. [PERSON] et al. [12] used the Frenkel-Halsey-Hill (FHH) model to study the relationship between the micropore fractal characteristics of prominent coal in western Guizhou and its adsorption performance and permeability. [PERSON] et al. [13] analyzed the adsorption pore characteristics of low-rank coal through fractal analysis using the Brunauer-Emmett-Teller (BET) model. [PERSON] et al. [14] compared and analyzed the fractal characteristics of coal and shale microporous structures based on the pore size distribution density function model. Progress has been made in the research on the pore structure of coal, but some shortcomings persist. (1) Existing research results focus on shallow coal seams that are currently being mined, but the pore structure of deep coal bodies, especially those buried at a depth of thousands of meters, has received little attention, and achievements are few. (2) The current pore size structure testing methods have certain limitations and applicability, and the ability of a single pore size characterization method to accurately determine the pore size structure parameters of coal is limited. Meanwhile, the pore system inside the coal body is very complex, with significant differences in pore width, making it easier to characterize macropores and mesopores. However, research on the characterization of the full pore-size structure of coal, including micropores, mesopores, and macropores, remains scarce. (3) The current research on the fractal law of pores often adopts one or two fractal methods for analysis.