*3.1. Experimental Results of NAM*

Through the NAM, the adsorption and desorption curves of coal samples are obtained, as shown in Figure 2. From Figure 2, the adsorption and desorption curves of four coal samples all present the inverted "S" type shape, which is the type II adsorption isotherm. When the adsorption capacity of the desorption branch is greater than that of the adsorption branch at the same relative pressure, the adsorption branch and the desorption branch do not coincide in a certain range of the relative pressure, and the separated loop is called the "hysteresis loop" [33]. Hysteresis loop can be divided into two types according to its shape, the first type of curve is shown in Figure 2a, and the second type of curve is shown in Figure 2b–d. Classification by BDDT and IUPAC [34], the shape of the first kind of hysteresis loop is basically consistent with the H2 (b) curve type, which indicates that it is characterized by the large adsorption loop, obvious hysteresis loop, and inflection point. It is further indicated that the adsorption pore system of low-rank coal is more complex, and there are "ink bottle" pores with small mouths and large bellies. The shape of the second kind of the hysteresis loop is basically consistent with that of H3 type, which indicates that the adsorption loop is small, the hysteresis loop is not obvious, and there is a slight inflection point. Then it shows that the adsorption pore system of medium and high rank coal is relatively simple, and flat slit capillary pores are present. Among them, the size of the second type of hysteresis loop shows the following trend: #2−XLZ < #3−QD < #4−QC, and it can be concluded that it increases with the increase in coal metamorphism degree [35].

**Figure 2.** Adsorption and desorption curves of the four coal samples.

The data analysis module of the adsorption instrument used in the experiment includes many classical model algorithms (such as BET, BJH, H-K, t-plot and NLDFT models).

On this basis, the nonlocal density function theory (NLDFT), which is an effective method to describe the fluid behavior in pores at the molecular level, is selected. It is assumed that the adsorption isotherms are obtained by multiplying numerous individual "single pore" adsorption isotherms by the relative distribution f(W) of their coverage pore size range. As long as the adsorbent and absorbent are given in the system, a set of influence functions (kernels) can be obtained by DFT simulation, and then the pore size distribution curve is obtained by solving the equation by fast non-negative least squares method. The pore size distribution characterized by it can be applied to the entire range of adsorption pores in coal samples [36]. Four pore size distributions (DV/DW, cumulative pore volume, DA/DW, and cumulative specific surface area) are plotted, as shown in Figure 3.

**Figure 3.** The pore size distribution measured with the NAM.

The IUPAC pore classification standard [37] is adopted in this test. The adsorption pore range of coal samples tested includes micropore (<2 nm), mesopore (2 nm ≤ *D* ≤ 50 nm), and macropore (50 nm < *D* < 90 nm). According to Figure 3, the pore size distribution curves of four coal samples are within the microporous scale range, showing the single peak characteristic, with the highest peak point, but the peak values are different, that is, a#1 > a#4 > a#2 ≈ a#3 and b#1 > b#4 > b#2 ≈ b#3. That is to say, within this range, the pore volume and specific surface area decrease first and then increase with the increase of the metamorphic degree, and the micropore volume and surface area of #1−DLT coal sample with the lowest metamorphic degree among four coal samples are the largest. In the range of mesoporous scale, only the #1−DLT coal sample shows the single peak characteristic, and the highest peak point is close to the pore scale boundary of 2 nm. The other three coal samples show complex multi peak characteristics, which are mostly concentrated at 10 nm–50 nm. However, the average peak sizes of four coal samples are different, #1−DLT > #4−QC > #2−XLZ > #3−QD, which indicates that the #1−DLT coal sample has the largest mesoporous pore volume and specific surface area. In the whole range of mesoporous scale, the distribution of mesoporous concentration of the #1−DLT coal sample is close to the boundary of the micro-meso porous scale range, while the distribution trend of the other three coal samples is close to the boundary of the macropore scale range. In the macropore pore scale, due to the limitation of the gas adsorption method, the variation characteristics are not obvious, and the volume and specific surface area change little with the pore width.

#### *3.2. Experimental Results of NMR*

According to the experimental principle of the low field NMR in Section 2.2, there is a quantitative relationship between the *T*<sup>2</sup> relaxation time of the saturated water coal sample and core pore radius *r* (Equation (3)). According to the experimental content, the coal structure is simplified as the columnar pipe [38], so for the parameter in Equation (3), we take *Fs* = 2. And then, the pore size distribution of four kinds of coal samples can be drawn, as shown in Figure 4.

**Figure 4.** The pore size distributions measured by NMR.

The pore size distribution of coal sample measured by nuclear magnetic resonance is shown in Figure 4. The pore size distribution of coal sample adsorption holes in the figure is all single peak, and the maximum peak point is within the range of 2 nm–15 nm. The peak value of gas coal is the largest, followed by anthracite, coking coal, and long flame coal. Peak position from left to right is gas coal, coking coal, anthracite, long flame coal. The largest single peak area is long flame coal, followed by anthracite, gas coal, and coking coal. Due to the change of the test method, the influence mechanism of the metamorphic degree on the whole adsorption pores can be analyzed [39]. That is, for the long flame coal #1−DLT, it is in the low metamorphic stage of coal, it withstands formation pressure less, and the coal body structure is relatively loose.

#### **4. Discussion**

*4.1. The Applicability of Fractal Models of the Coal Adsorption Pore Structure*

Based on the theory proposed by previous scholars [40], the fractal dimension can be calculated by combining the results of the NAM with the FHH model:

$$\ln(V/V\_0) = \mathbb{C} + A \times \ln[\ln(p\_0/p)]\tag{4}$$

In Equation (4), *V* is the volume of adsorbed N2 at equilibrium pressure *p*; *V*<sup>0</sup> is the monolayer adsorption volume. *p*<sup>0</sup> is the saturated vapor pressure of adsorbed N2; *A* is the slope of *ln*(*V*/*V*0) and *ln*[*ln*(*p*0/*p*)] in the double logarithmic coordinates; and *C* is the constant.

When the van der Waals force is the main adsorption force, the relationship between *A* and the fractal dimension *D* is:

$$A = \frac{D - 3}{3} \tag{5}$$

When the capillary condensation occurs in pores, the surface tension is the main force, and the relationship between *A* and the fractal dimension *D* is:

$$A = D - \mathfrak{Z} \tag{6}$$

In Equations (5) and (6), *D* is the fractal dimension calculated by the FHH fractal model; and *A* is the slope of *lnV* and *ln*[*ln*(*p*0/*p*)] in the double logarithmic coordinates.

The fractal dimension of adsorption pore can be calculated by this experimental method. At the same time, when using this model to calculate the fractal dimension, it is usually divided into the low-pressure section (the relative pressure is 0–0.45) and the high-pressure section (the relative pressure is 0.45–1.00) to obtain fractal dimensions of different pressure sections. The fractal dimension of the low-pressure section is set as D1, and the fractal dimension of the high-pressure section is set as D2. Figure 5 shows the fitting diagram of *lnV* and *ln*[*ln*(*p*0/*p*)] in the FHH fractal model.

**Figure 5.** Fitting diagrams of *ln*(*V*/*V*0) and *ln*[*ln*(*p*0/*p*)].

It can be seen from Figure 6 that all fitting coefficients of four coal samples are above 0.89, that is to say, the fitting effects are good. FHH fractal model can be used to calculate the fractal dimension of adsorption pores of four coal samples, which can effectively characterize the complexity of the adsorption pore structure of coal samples. At the same time, because this model can calculate the adsorption capacity of coal, it can be used to characterize the complex pore structure of coal. This complexity represents the degree of irregularity in the coal body surface.

**Figure 6.** Fitting diagrams of *lg*(*T*2) and *lg*(*SV*).

F G

According to an algorithm based on the capillary pressure method to calculate the fractal dimension, Zhang et al. [41] combines the *T*<sup>2</sup> spectrum distribution obtained from the NMR experiment and deduces an algorithm based on the *T*<sup>2</sup> spectrum curve to calculate the fractal dimension:

$$\lg(\mathcal{S}\_{\upsilon}) = (\mathfrak{Z} - D)\lg(T\_2) + (D - \mathfrak{Z})\lg T\_{2\max} \tag{7}$$

In Equation (7), *Sv* is the corresponding cumulative pore volume percentage; *T*<sup>2</sup> is the transverse relaxation time; *T*2*max* is the maximum transverse relaxation time; and *D* is the measured pore volume fractal dimension.

The fractal dimension based on the *T*<sup>2</sup> spectrum can be expressed as follows:

$$D = 3 - k \tag{8}$$

In Equation (8), *k* is the slope of the linear fit of *lg*(*T*2), *lg*(*SV*).

Moreover, the inflection point of the curve in the relationship curve between the logarithmic value *lg*(*T*2) of the transverse relaxation time *T*<sup>2</sup> and the logarithmic value *lg*(*SV*) of the corresponding cumulative pore volume percentage *Sv* in the NMR data is the boundary point between the fractal dimension of the adsorption pore volume and the fractal dimension of the seepage pore volume. The fractal dimension of adsorption pores is the volume fractal dimension, which indicates the complexity of the pore size distribution. Generally, the pore size in porous media satisfies the pore diameter *<sup>λ</sup>min λmax D* <sup>≈</sup> 0 [42,43], so for the *<sup>λ</sup>min <sup>λ</sup>max* of four kinds of test samples selected 2.764 nm/95.36 nm, 1.48 nm/95.36 nm, 1.318 nm/96.46 nm, 1.134 nm/96.38 nm as the pore diameter range to calculate the fractal dimension of the overall structure of adsorption pores, and thus obtained the fractal dimension D3. The fitting curves of *lg*(*T*2) and *lg*(*SV*) is shown in Figure 6.

Table 2 shows the calculation results of fractal dimension obtained from the above fitting curve and algorithm.


**Table 2.** The calculation results of fractal dimensions.

From the range distribution of mesoporous fractal dimensions, there are no effective values in D1, which are lower than 2, and all values of D2 are between 2 and 3, indicating that D2 is more effective. This is because in the FHH fractal model, the calculation model of the high-pressure section is based on the capillary condensation process, and the effect of surface tension is considered. For the mesopores, the capillary condensation process is a characteristic process within the pore size range, which does not occur in the micropores. Therefore, the effective fractal dimension can be obtained by using the adsorption data of the high-pressure section. However, the data of low-pressure section contain some microporous adsorption data, which affect the calculation results of fractal dimensions. Among them, D2 mainly ranges from 2.5 to 2.9, and the value of D2 gradually decreases with the increase of the metamorphism degree, which indicates that the complexity of mesopores decreases with the increase of the metamorphism degree [44]. The D3 is between 2.004 and 2.0369 as the #1−DLT long flame coal has a lower D3 fractional dimensional value due to its looser coal structure and more connected macropore network. With the exception of the #1−DLT long flame coal, the D3 values gradually decrease with increasing metamorphism, which could indicate that the complexity of the pores throughout the adsorption pore space gradually decreases with increasing metamorphism.

The average value of D3 is less than that of D2, which is due to the difference between test methods and fractal models used. First, D2 is calculated by using the FHH model and N2 adsorption capacity, while D3 is calculated by using the capillary beam fractal model and *T*<sup>2</sup> spectrum. Therefore, when the pore diameter is in the range of 0–100 nm, gas can enter into smaller pores more than water, which results in more complex pore structure and smaller D3 value. Second, there are some differences in the physical meaning of characterization. The FHH model is based on the adsorption capacity to characterize the surface irregularity of coal body, while the capillary beam fractal model represents the complexity of the pore size distribution. The complexity of pore size distribution in coal is composed of micropores, mesopores, and macropores, so the D3 change of four coal samples is not obvious. Third, except for the #1−DLT coal sample, D2 and D3 of the other three coal samples gradually decrease with the increase of metamorphic degree, and the variation laws are consistent, that is, the complexity of adsorption pore structure generally decreases with the increase of metamorphic degree [45].

#### *4.2. The Relationships between Adsorption Pore Structural Parameters and Fractal Dimensions*

Table 3 shows the experimental results based on the specific surface area and total pore volume of adsorption pores of coal samples obtained with NAM:


**Table 3.** Specific pore surface area and total pore volume of adsorption pores of 4 experimental samples.

From Table 3, with the increase in the metamorphism degree, the specific surface area and total pore volume of adsorption pores show the trend of first decreasing and then increasing, especially for the long flame coal. Compared with the other three coal samples, the specific surface area and total pore volume differ by 100 and 10 orders of magnitude, respectively, with significant differences. In order to further discuss the quantitative relationship between the pores and fractal dimensions of coals with different metamorphic degrees, Figure 7 shows the relationship curves between specific surface area, pore volume, and D2 and D3.

**Figure 7.** Relationships between structural parameters and fractal dimensions.

Figure 7 shows that the fractal dimensions of the two types first increase, then decrease, and finally increase with the increase of total pore volume and specific surface area. From these four groups of curves, it can be concluded that there are no direct quantitative relationships between the two structural parameters and the fractal dimensions D2 and D3, which represent the surface irregularity and the complexity of the pore size distribution, respectively [46]. In other words, the factors that determine the fractal dimension are not only determined by the pore volume or specific surface area. In order to verify this conclusion, the calculation results of the proportion of the pore volume and specific surface area of micropores, mesopores, and macropores to the whole adsorption pores are arranged in Table 4. The curves of D2 and D3 change with the proportion of each scale, as shown in Figure 8.

**Table 4.** Calculation results of the proportion of the pore volume and specific surface area of micropores, mesopores, and macropores to the whole adsorption pores.


**Figure 8.** Relationships between proportions of pore parameters and fractal dimensions.

It can be seen from Figure 8 that there are no direct quantitative relationships between the proportions and fractal dimensions, indicating that single structural parameters do not directly determine the values of fractal dimensions [47]. Combined with Figures 3 and 4 obtained from two experiments, it can be found that for D2, in the mesoporous scale range, the peak value of pore diameter distribution is concentrated at 2–10 nm, and the closer the highest peak point is to the range of 2–5 nm, the larger the fractal dimension value is. For D3, in the whole adsorption pores, the single peak value of the pore diameter distribution is mainly in the range of 2–30 nm (except for the #1−DLT coal sample). Similarly, the closer the pore diameter distribution is to the range of 2–5 nm, the larger the fractal dimension value is, the higher the complexity of pore structure is.

At the same time, the influence mechanism of fractal characteristics of coal structure on its adsorption properties is discussed. Therefore, according to the results of the NAM experiment, the relationships between two fractal dimensions and the maximum nitrogen adsorption properties are drawn.

First of all, it can be seen from Figure 9a that there is a good linear positive correlation between the fractal dimension D2 calculated by the FHH fractal model and the maximum N2 adsorption capacity, and the correlation coefficient is 0.87, which indicates that the more complex the inner surface irregularity of coal adsorption pore structure is, the better the adsorption properties of coal is, which also indicates that the main determinant of adsorption properties of coal is the specific surface area [48–50]. Second, it can be seen from Figure 9b that there is no obvious quantitative relationship between the fractal dimension D3 calculated by the capillary beam fractal model and the maximum N2 adsorption capacity, which also confirms that D3, which represents the complexity of the pore size distribution, is not the main factor affecting the adsorption performance of coal. Therefore, according to the relationships between the two fractal dimensions and the maximum adsorption properties, the physical meaning of two fractal dimensions can also be distinguished.

#### **5. Conclusions**

This paper uses the NAM and low-field NMR to conduct the multistage analyses of parameters such as the pore size distribution, pore shape, pore volume, and specific surface area of four coal samples with different metamorphism degrees. Based on the fractal models, multiple fractal dimensions are calculated. The relationships between fractal dimensions and various quantitative parameters of micropores, mesopores, macropores and whole adsorption pore structure are obtained.

(1) Combined with the metamorphism characteristics of coal, the formation pressure and temperature are constantly changing, and the rapid pyrolysis fracture of aliphatic rings in coal macromolecules results in disappearing first and then forming a large number of pores between coal molecules. Therefore, the pore volume and specific surface area of adsorption pores first decrease and then increase with the increase of the metamorphism degree.

(2) Due to the difference in fractal models, among them, the effective fractal dimensions based on the FHH fractal model are concentrated in the range of 2.5–2.9, which are suitable for the pore size range of the capillary condensation process; the effective fractal dimensions based on *T*<sup>2</sup> spectrums and the capillary bundle fractal model are concentrated in the range of 2.004–2.037.

(3) It is found that the single structure parameter of adsorption pores has no direct quantitative relationship with the fractal dimension. However, the fractal dimension has a certain relationship with the range of pore diameter distribution, that is, the more the pore size distribution is concentrated in 2–5 nm, the larger the fractal dimension value, the higher the complexity of pore structure.

(4) There is a good linear positive correlation between the fractal dimension D2 which characterizes the degree of surface irregularity in the coal body and the maximum nitrogen adsorption capacity, with a correlation coefficient of 0.87, indicating that the greater the degree of surface irregularity in the coal body, the better the coal body's adsorption performance.

**Author Contributions:** Writing—original draft preparation, W.W.; methodology, data curation, writing, Z.L.; methodology, data processing, validation, English polish, M.Z.; methodology, data processing, H.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to acknowledge the support of the National Natural Science Foundation of China (52274213, 52074173, 51934004), Natural Science Foundation of Shandong Province (ZR2022YQ52), and Taishan Scholars Project Special Funding (TS20190935).

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to acknowledge the support of the National Natural Science Foundation of China (52074173, 51604168, 51934004), the Key Research and Development Plan of Shandong Province, China (2019GSF111033), Major Program of Shandong Province Natural Science Foundation (ZR2018ZA0602) and Taishan Scholars Project Special Funding (TS20190935).

**Conflicts of Interest:** All authors declare that there is no conflict of interest.

## **References**


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