**1. Introduction**

As an extremely important unconventional natural gas energy, coal bed methane (CBM) resources have attracted great attention from the energy community around the world [1–6]. Unlike conventional natural gas reservoirs, the nano-scale adsorption pore structure of coal is the important storage space for gas. And its structural characteristics directly determine the gas adsorption and desorption process, thereby affecting the gas content of coal seams [7,8]. Therefore, studying the characteristics of the coal adsorption pore structure is the basis for revealing the law of the coalbed methane adsorption–desorption–diffusion migration [9,10]. At present, scholars have mainly researched the coal structure based on image analyses, fluid injection, and gas adsorption methods [11–13]. For example, Feng et al. [14] used scanning electron microscopy (SEM) and computer tomography (CT) to analyze the microstructure and deformation of coal samples after the methane adsorption process; Meng and Qiu [15] studied the mechanism of the spatial diffusion of coal samples after supercritical CO2 treatment through SEM, mercury intrusion porosimetry (MIP), and nuclear magnetic resonance (NMR). Simultaneously, due to the wide distribution of coalbed methane reservoirs in China

**Citation:** Wang, W.; Liu, Z.; Zhang, M.; Yang, H. Experimental Study on Fractal Characteristics of Adsorption Pore Structure of Coal. *Processes* **2023**, *11*, 78. https://doi.org/10.3390/ pr11010078

Academic Editors: Feng Du, Aitao Zhou and Bo Li

Received: 2 December 2022 Revised: 24 December 2022 Accepted: 25 December 2022 Published: 28 December 2022

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

and the variety of coal, the degree metamorphism of coal can affect its internal reservoir space. Scholars have also carried out extensive research on this [16]. There are several examples. Chen et al. [17] used the Fourier transform infrared (FTIR) spectrometer and CO2 adsorption methods to study the micropore characteristics of five coal samples with different metamorphism degrees; Niu et al. [18] used the liquid nitrogen method and the methane isotherm adsorption experiment to study the pore structure and adsorption properties of 12 different rank coals. Therefore, based on previous studies, this paper uses the classic experimental methods to carry out multi-scale structural analyses of coal with different degrees of metamorphism to accurately and quantitatively characterize the structural characteristics of coal at the adsorption pore scale.

Coal is a porous medium with a complex structure and the experimental study of its heterogeneity is a key to the microstructure of coal [19,20]. At present, the fractal theory represents the effectiveness of the space occupied by complex shapes. It is a measure of the irregularity of complex shapes that can be introduced instead of characterizing the "special nature" of the coal pore structure [21–23]. Through the research of domestic and foreign scholars, the models for calculating the fractal size of pore structure include FHH model, Menger sponge model, and other applicability models [22,24,25]. In several proposed fractal models, the FHH fractal model is used to discuss the extent to which the internal surface roughness of coal determines the amount of natural gas, especially methane. They can readily adsorb/store, and the gas can easily flow and be produced from coal formations. Cai et al. [26] studied the physical composition and particle size distribution of coal by integrating classical geometric fractal model and thermodynamic fractal model. Lu et al. [27] explored the fitting of Sierpinski, Menger, and other models to coal pore structure at different scales, and summarized the fractal characteristics of coal adsorption pore structure. Hu et al. [28] combined fractal model and image analysis to explore the fractal dimension of coke particles and describe the surface characteristics of coal and coke particles. Liu et al. [29] calculated the adsorption pore volume based on the capillary bubble fractal model and NMR *T*<sup>2</sup> spectrum distribution, thus quantitatively characterizing the physical structure of coal. Scholars have quantitatively characterized the fractal characteristics of coal structures in detail, but have not made a detailed mechanistic explanation for the type of coal structures complexity represented by the fractal model. This paper has carried out certain research based on this.

According to previous studies, scholars have carried out detailed quantitative characterization of the fractal characteristics of coal structure, but there is little research on the type of coal structure complexity represented by the fractal model. On this basis, this paper has carried out some research. The selection of appropriate experimental methods and fractal models, accurate and quantitative analysis of the change law of coal structural parameters with different degrees of metamorphism to clarify the influence mechanism of coal microstructure parameters on fractal characteristics, as well as the detailed physical significance and important influencing factors of fractal models, are the key research contents of this paper. In this paper, NAM and NMR are selected to analyze the multi-scale structural characteristics of adsorption pores (*r* < 50 nm) [30] of coal with different metamorphism degrees according to the metamorphic characteristics of four coal samples. At the same time, the difference and applicability of the fractal dimension of adsorption pore structure combined with FHH model and capillary bundle model are comprehensively analyzed. Finally, the relationships between structural parameters and fractal dimensions are discussed, and based on this, the important factors affecting the pore structure complexity are clarified. The above research is the key to reveal the influence mechanism between the geological evolution process of the coalbed methane reservoir and the distribution law of gas content. It is of great significance for mine gas disaster prevention and coalbed methane research.

#### **2. Materials and Experiments**

#### *2.1. Samples*

This laboratory experiment selects raw coal from four coal mines in Shaanxi, Shandong, Anhui, and Shanxi provinces of China. Bulk raw coal samples obtained on site are transported to the laboratory for the industrial analyses, Table 1 shows the analysis results. And raw coals are used in both experiments. Four lump coal samples with a mass slightly greater than 0.1 g were used in NAM experiment, with a total area of 2–50 m2/g. In the NMR experiment, four kinds of lump coal samples are made into experimental coal samples with a diameter of 25 mm and a height of 50 mm by using equipment.



Note: No. 1 is long flame coal; No. 2 is gas coal; No. 3 is coking coal; and No. 4 is anthracite. The metamorphic degree ranking order of the four kinds of coal samples is #1−DLT < #2−XLZ < #3−QD < #4−QC. Mad is the moisture; Ad is the content ash; Vda is the volatile matter; FCdaf is the fixed carbon; Romax is the maximum reflectance of vitrinite.

#### *2.2. NAM and NMR*

For the NAM analyses, the four kinds of coal samples are tested with a Micromeritics ASAP2020 BET specific surface area and pore diameter analyzer. The samples are heated, dehydrated and degassed, cooled, and weighed in turn. Then, the samples are put into the instrument for measurement and analyses, and the adsorption and desorption isotherms of coal samples are obtained. Finally, the pore structure parameters of coal samples are obtained through the BET, BJH, and DFT models. For the NMR analyses, four kinds of coal samples are tested with the MesoMR23-060H low-field NMR equipment. The magnet of the instrument is a permanent magnet, the magnetic field strength is 0.5 T, the main frequency is 11 MHz, the diameter of the probe coil is 60 mm, the echo time (TE) is 0.1 ms, the number of echoes is 6000, the number of scanning is 64, and the experimental environment temperature is 25.5 ◦C. First, the coal samples are dried at 60 ◦C to remove the impurities; second, the samples are vacuumized, and the coal samples are placed in water for 48 h for the saturated water treatment. The coal sample before the NMR experiment does not have additional pressure on it during the 48-h saturated water treatment process, and because the water is an incompressible fluid, it will not destroy the pore structure of the original coal sample, so it will not affect the NMR experimental results. Finally, the samples are taken out to test the saturated water samples, and the *T*<sup>2</sup> relaxation distributions are calculated by using the simultaneous iterative reconstruction technique, and the number of iterations is 10,000. Because the NMR is based on the spin motion of the hydrogen nucleus, the amplitude of the *T*<sup>2</sup> spectrum of coal samples measured in this experiment can reflect the total hydrogen content of a certain pore diameter. Three different relaxation mechanisms affecting *T*<sup>2</sup> spectrum are: the free relaxation, the surface relaxation, and the diffusion relaxation. When the three relaxation mechanisms exist simultaneously, the *T*<sup>2</sup> of the fluid in the pore can be expressed as [31]:

$$\frac{1}{T\_2} = \frac{1}{T\_{2B}} + \frac{1}{T\_{2S}} + \frac{1}{T\_{2D}}\tag{1}$$

In Equation (1), *T*2*<sup>B</sup>* is the free relaxation time; *T*2*<sup>S</sup>* is the surface relaxation time; *T*2*<sup>D</sup>* is the diffusion relaxation time.

The experiment is that pure fluid (water) is carried out under the uniform magnetic field, so the effect of diffusion and free relaxation is not considered. Then only the surface relaxation is considered. The surface relaxation is from the relaxation action of coal particle surface to fluid, which is related to the ratio of specific surface area to pore volume in the coal sample. Then Equation (1) can be simplified as follows:

$$\frac{1}{T\_2} = \frac{1}{T\_{2S}} = \rho\_2(\frac{S}{V})\_{\text{porc}}\tag{2}$$

In Equation (2), *T*<sup>2</sup> is the relaxation time; *ρ*<sup>2</sup> is the surface relaxation rate of *T*2; it can be directly related to capillary force and pore diameter, *S* is the pore surface area; *V* is the pore volume; ( *<sup>S</sup> <sup>V</sup>* )porosity is the specific surface area of pores, the relationship between the pore radius and pore size is ( *<sup>S</sup> <sup>V</sup>* )*pore* <sup>=</sup> *Fs <sup>r</sup>* ; *Fs* is the shape factor, which is determined by the pore model. Therefore, Equation (3) can be expressed as [32]:

$$T\_2 = \frac{1}{\rho\_2 F\_s} r \tag{3}$$

Let <sup>1</sup> *<sup>ρ</sup>*2*Fs* = *<sup>C</sup>*, then the Equation (3) can be expressed as *<sup>T</sup>*<sup>2</sup> = *<sup>C</sup>* × *<sup>r</sup>*, and the coefficient *C* is the constant. After the value of *C* is obtained, the NMR *T*<sup>2</sup> spectrum can be converted into the pore radius distribution. Therefore, the pore size distribution of coal samples can be obtained according to the *T*<sup>2</sup> spectrum distribution. Figure 1 is the experimental equipment and experimental schematic diagram.

**Figure 1.** Experimental equipment and principles.
