Dendric Fractal Characteristics of Pores in Anthracite and Their Influences on Gas Adsorption Characteristics

The characteristics of pores in anthracite from Guhanshan Coal Mine in the Jiaozuo mining area were tested by the mercury injection method. According to the pore size, pores in the coal sample were classi ﬁ ed into gas seepage pores and gas adsorption pores which were in line with the stochastic fractal model and the dendric fractal model, respectively. Besides, the fractal characteristics of adsorption pores were analyzed by using a dendric fractal model, and the in ﬂ uence law of the tortuosity fractal dimension D t on the gas adsorption capacity was obtained. The research results show that the amount of mercury injected into each experimental coal sample is directly proportional to its porosity, but has no obvious correlation with speci ﬁ c surface area. Seepage pores with sizes over 65nm ﬁ t the stochastic fractal model, and their fractal dimension D increases with the increase of speci ﬁ c surface area. Adsorption pores with sizes of below 65nm ﬁ t the dendric fractal model, and their tortuosity fractal dimension D t is positively correlated with the speci ﬁ c surface area. Through the gas adsorption experiment on coal samples, it is found that a larger D t corresponds to a stronger gas adsorption capacity. The research results can provide a theoretical basis for elucidating the gas adsorption characteristics of anthracite from a microscopic point of view.


Introduction
China is a country that is rich in coal resources and lacks oil and gas resources. Such resource characteristics guarantee coal's strategic position and its role as a basic energy will maintain for a long time [1,2]. However, as coal mining develops into the deep, recurrent gas outburst accidents make gas prevention and control work top priorities [3,4]. Coal, as a typical heterogeneous porous medium, has a pore structure whose characteristics determine the occurrence and flow law of internal gas, and its adsorption capacity for gas is closely related to the pore structure [5][6][7][8]. Therefore, research on the pore structure of coal is of great significance for clarifying the regularity of gas occurrence and migration in coal.
The fractal geometry theory founded by French mathematician Mandelbrot [9] provides a new idea for investigat-ing the complex microscopic pore structure of porous medium. Chen et al., Chen et al., Wei et al., and He et al. [10][11][12][13] have carried out relevant researches on the fractal characteristics of coal pore structure and concluded that coal surface has significant fractal characteristics. Liu et al. [14] discussed the pore structure characteristics and fractal characteristics of different rank coals by low temperature liquid nitrogen adsorption method and believed that the fractal dimension D could quantitatively characterize the roughness of coal surface, and its value increased with the increase of coal rank. In a study of Li et al. [15], the pore structure and coal composition during different coalification stages have specific effects on the fractal characteristics of pore shape, surface, and volume. In a study of Si et al. [16], the variation rule of the coal pore structure before and after water leaching and its influences on gas adsorption characteristics were tested through low-temperature N 2 and CO 2 adsorption experiments. Wang et al. [17] examine the changes in pore structures of different coal samples using liquid nitrogen adsorption, nuclear magnetic resonance, and X-ray photoelectron spectroscopy. Wang et al. [18,19] scanned coal samples based on the industrial CT scanning experimental system, and analyzed the relation between the fractal dimension and the development degree of the fracture structure inside coal samples. Ma et al. and Cai et al. [20,21] discussed the fractal characteristics of the coal pore structure and its influences on permeability. In the studies of Nie et al. [22][23][24], the diffusion coefficient and the permeability were calculated by using fractal characteristics of the pore structure, and a relation between the adsorption characteristics of coal samples and the fractal dimension on the pore surface was established through a cryogenic nitrogen adsorption experiment. Adopting a stochastic fractal model to describe the fractal of pores of the coal body, Li et al. [25] stated that only when P is below 10 MPa, the pores measured by the mercury injection method have the characteristics of the stochastic fractal model, and the corresponding pore radius is about 60 nm. Jia et al. [26] analyzed Yangquan anthracite samples by liquid nitrogen adsorption and found that the specific surface area of anthracite micropores and mesopores increased with decreasing particle size. Lu et al. [27] found that the overall pore volume and specific surface of anthracite coal were positively correlated with the fractal size of the pore structure by fractal calculations of the nitrogen adsorption capacity of the coal.
Gas occurrence and migration are related not only to pore volume but also to pore length and number. On this basis, Wheatcraft and Tyler [28] established the Lagrangian model to depict the relation between pore length and pore size. In the studies of Yu and Cheng and Xu et al. [29,30], a dendric fractal model of the pore structure of a porous medium was established by combining the Lagrangian model and the law of power that describes the number of pores. This model considers the fractal characteristics of the number, length, and pore size of the porous medium and is applied to research the relation between the fractal dimension and the permeability and diffusion coefficient of the porous medium.
At present, the fractal characteristics of coal rock adsorption pores are rarely studied by using the dendric fractal model to analyze mercury injection experimental results. In this paper, relevant experiments were performed on the pore distribution characteristics of anthracite from Guhanshan Coal Mine in the Jiaozuo mining area, China. The pores of coal samples were classified into two categories, i.e., gas seepage pores and gas adsorption pores, according to pore size. The fractal characteristics of seepage pores were analyzed by using the stochastic fractal model, while those of adsorption pores by using the dendric fractal model. Besides, the relation between the tortuosity fractal dimension of the dendric fractal model and the gas adsorption capacity of the coal body was studied through a gas adsorption experiment. The results provide a theoretical basis for elucidating the gas adsorption characteristics of anthracite from a microscopic point of view.

Pore Fractal Model
The coal rock porous medium is composed of particles of different sizes, which is quite similar to the construction process of fractal. Hence, the fractal model can be used for quantitative analysis. In a self-similarity system, after i times of iterations, the relation between the number and the size of units can be illustrated by the following equation [25]: where Nð1/b i Þ is the number of units whose length is 1/b i ; K is the number of initial elements per unit length; i is the number of iterations; b is the size-transformation factor, with its value over 1; D is the fractal dimension. Equation (1) is a general equation of the fractal construction, independent of the dimension and the construction method of initial elements.

Stochastic Fractal Model.
The mercury injection experiment is a commonly used method to test characteristics of pores in coal samples. In order to overcome the inner surface pressure between mercury and pore surface, the pressure P required to inject mercury into pores and the pore radius r satisfy the Washburn equation [28]: where P is the mercury injection pressure, MPa; a is the wetting contact angle of mercury, 140°; σ is the surface tension of mercury, 10 -3 N/m; r is the pore radius, nm. During the experiment, the mercury injection volume V p is equal to the sum of the volume of pores whose sizes are over r, under a given pressure condition. The porous medium dV p /dr constructed by Mengar and the fractal dimension D satisfy the following equation: Equation (4) can be obtained by substituting Eq. (2) into Eq. (3): Equation (5) can be obtained by taking the logarithm of both ends of Eq. (4): where V p is the pore volume, cm 3 /g; D is the fractal dimension of coal rock. According to Eq. (5), if pore distribution conforms to fractal characteristics, a linear relation exists between lgðd V p /dPÞ and lgP, and the coal fractal dimension D can be 2 Geofluids expressed as In Eq. (6), k is the slope of the fitting line between lgðd V p /dPÞ and lgP.

Dendritic Fractal
Model. The process of gas adsorption and diffusion in a porous medium is rather complex. In this paper, pores of coal particles are regarded as capillaries with certain tortuosity, complying with fractal characteristics [31]. The length of a capillary is much longer than the radius, and the flow of gas in the capillary is onedimensional. The capillary spherical model [32] is shown in Figure 1.
Yu and Cheng [29] argued that the size distribution and structure of pores in a porous medium satisfy the fractal characteristics, and the cumulative pore number distribution function meets: where d (nm) is the pore diameter; N is the number of pores with a diameter of over or equal to d; d max (μm) is the maximum pore diameter; D f is the fractal dimension of pores calculated by the dendritic model, and the values in two and three dimensions are 0 < D f < 2 and 0 < D f < 3, respectively. Equation (7) is regarded as a continuous differentiable function, and the number of pores whose diameters range from d to d + dd could be obtained by taking the derivative of the pore size d through Eq. (7): In Eq. (8), −dN > 0 indicates that the number of pores is negatively correlated with the pore radius.
Meanwhile, according to the dendric fractal model and law, the pore length L satisfies where L (nm) is the pore length; L 0 (nm) is the straight-line distance of gas flow direction; D t is the fractal dimension of pore tortuosity.
In the mercury injection experiment, the pore diameter corresponding to the ith mercury injection pressure point is d i ; that corresponding to the d i+1 mercury injection pressure point is d i+1 = d i + dd; the mercury injection amount from the ith to the i + 1th mercury injection pressure point is dV p . Then, the pore volume V p of pores with a diameter of d i to d i + dd satisfies Equation (11) can be obtained by substituting Eqs. (8) and (9) into Eq. (10): Taking the logarithm of both ends of Eq. (11) leads to where L and d max are known quantities, and the fractal dimension of coal rock pores can be determined through the straight line between lgðdV P /ddÞ and lgd. According to Eq. (12), if pore distribution conforms to the fractal characteristics, lgðdV p /ddÞ and lgd have a linear relation, and the slope k and the intercept b satisfy

Experiments and Results
3.1. Mercury Injection Experiment. The anthracite in the experiment was taken from Guhanshan Coal Mine in the Jiaozuo mining area. The screened coal samples with particulate sizes of 5-6 mm were divided into 4 groups and numbered from GHS-W1 to GHS-W4, respectively. The pore distribution test on coal samples was completed in the Capillary Figure 1: A capillary spherical model of coal particle. Mercury was injected into pores of coal samples by applying external force. The applied external force and the pore radius meet Eq. (2). The pore specific surface area of samples can be further calculated according to the mercury injection pressure. It is obtained from the Young-Duper equation: where S (m 2 /g) is the specific surface area of pores.

Experimental
Results. The mercury intake-withdrawal curves of GHS-W1, GHS-W2, GHS-W3, and GHS-W4 samples were obtained in the experiment (Figure 3).   It can be seen from Figure 3 that with the increase of pressure, the mercury intake-withdrawal curves rise first and then gradually coincide, indicating that the methane adsorption mechanisms of coal differ in different pressure stages. Such a difference is mainly affected by the microscopic pore structure of coal particles. Therefore, the analysis of the microscopic pore structure of coal using the fractal theory conduces to further exploring the methane adsorption characteristics of coal. The mercury injection amount and the pore parameters of each coal sample obtained are listed in Table 1.
As can be observed from Table 1, in descending order, the mercury injection amounts follow GHS-W1, GHS-W3, GHS-W4, and GHS-W2; the porosities follow GHS-W1, GHS-W3, GHS-W4, and GHS-W2 as well; the specific surface areas follow GHS-W2, GHS-W1, GHS-W4, and GHS-W3. The above data suggest that the total mercury injection amount in the experiment is positively correlated with the porosity of coal, but has no clear correlation with the specific surface area. This is mainly because the mercury is injected into pores of coal samples by applying external force in the mercury injection experiment. A larger pore volume corresponds to more injected mercury. The pore volume is directly proportional to porosity, but is not correlated with specific surface area.

Fractal Dimensions of Gas Seepage
Pores. According to the method described in Section 2.1, lgðdV P /dPÞ and lgP were calculated by using the stochastic fractal model. In the rectangular coordinate system, with lgP as the abscissa and lgðdV P /dPÞ as the ordinate, a scatter diagram was plotted and a linear fitting was carried out. The results are displayed in Figure 4.
As can be seen from Figure 4, when P is below 10 MPa (pore radius > 65 nm), lgðdV p /dPÞ is linearly correlated with lgP obviously, and the curve fitting degree is over 0.93. However, when P is above 10 MPa (pore radius < 65 nm), the linear fitting degree is below 0.6. It is a proof that the stochastic fractal model is only applicable to seepage pores whose pore radius is over 65 nm, but cannot depict adsorption pores with a pore radius of below 65 nm. The fractal dimensions of coal samples are calculated according to the slope of the fitting line (Table 2).

Geofluids
According to Table 2, when the pressure P is below 10 MPa, fractal dimensions of coal samples lie in the range of 2-3; when it is over 10 MPa, those are over 3. From the perspective of classical geometry, 2 < D < 3 in a threedimensional space, that is, D > 3 is meaningless, which is consistent with the research results of Li et al. [25]. This further suggests that the use of the stochastic fractal model to analyze mercury injection experimental results is only applicable to gas seepage pores whose radii are over 65 nm.

Fractal Dimensions of Gas Adsorption
Pores. According to the method described in Section 2.2, lgðdV P /ddÞ and lg d were calculated by using the dendric fractal model. In the rectangular coordinate system, with lgd as the abscissa and lgðdV P /ddÞ as the ordinate, a scatter diagram was plotted, and linear fitting was carried out. The tortuosity fractal dimension was calculated by combining Eqs. (13) and (14) ( Figure 5).
As shown in Figure 5, a good linear relation exists between lgðdV P /ddÞ and lgd, and the fitting degree of the curve is over 0.99, that is, adsorption pores with the sizes of below 65 nm conform to the dendric fractal model.
The slope k and intercept b of the line could be obtained by linearly fitting the relation between lgðdV P /ddÞ and lgd. The pore characteristic length and the maximum pore diameter were taken as L 0 = 3 mm and d max = 10 μm, respectively [33]. The fractal dimensions D t and D f of GHS-W1-GHS-W4 could be obtained from Eqs. (13) and (14), and the results are presented in Table 3.
As can be observed from Table 3, the tortuosity fractal dimension D t increases as the specific surface area increases,  6 Geofluids and they show a positive correlation. However, the porosity does not have direct relations to the fractal dimensions D f and D t . This is because the porosity is mainly determined by mesopores and macropores whose sizes are over 100 nm, while the fractal dimensions D f and D t mainly describe the fractal characteristics of adsorption pores whose sizes are below 65 nm.

4.3.
Relation between Pore Volume dV p and Pore Size d. In order to facilitate the comparison with experimental results, the pore size corresponding to each pressure point in the mercury injection experiment was taken as the pore size d, and the fractal dimension D f of adsorption pores was put into Eq. (7) to obtain the amounts (N i and N i+1 ) of pores whose diameters are over d i and d i+1 . Afterwards, dN was obtained by a subtraction of N i from N i+1 . The fractal dimension D t of adsorption pores was substituted into Eq. (9) to obtain the diameter d and the corresponding pore length L. By substituting dN and L into Eq. (10), the curve of pore volume dV p variation with the diameter d could be obtained. The curves of the pore volume dV p of adsorption and seepage pores of GHS-W1 variation are with the diameter d, and their experimental and theoretical values are compared ( Figure 6). According to the curve of GHS-W1 pore volume dV p variation with the diameter d in Figure 6, the theoretical value of dendric fractal highly matches the experimental results when the pore size is below 1 μm. When the pore size is over 1 μm, the experimental value is greater than the theoretical value, because a certain number of cracks exist in coal rock. These cracks are regarded as pores during the mercury injection experiment. Therefore, as mercury is injected into the cracks rapidly at the beginning of the exper-iment, the amount of mercury during this time should be subtracted to calculate the real porosity of coal rock pores. Furthermore, according to the dV p variation curve of GHS-W1 (pore sizes 5-500 nm) with the diameter d, the dendric theoretical value of adsorption pores is in good agreement with the experimental results, with a fitting degree of 0.991, which is consistent with the conclusion that the pore specific surface area of coal samples is positively correlated with the tortuality fractal dimension D t in Section 3.2. It is a mark that the distribution of adsorption pores of coal rock conforms to the dendric fractal model.

Influences of the Tortuality Fractal Dimension D t on Gas Adsorption Characteristics
The specific surface area of coal determines the gas adsorption capacity of coal. When the specific surface area is larger, more adsorption sites are provided by coal, and more gas can be absorbed by coal. The number of micropores makes a decisive contribution to the specific surface area of pores. Hence, the influences of the tortuality fractal dimension D t on gas adsorption characteristics can be explored through the gas adsorption experiment on coal samples. Since the specific surface area of coal is positively correlated with the tortuosity fractal dimension D t , it can be inferred that V L is also positively correlated with D t . Therefore, the authors verified the above prediction through a gas adsorption experiment on coal samples.

Gas Adsorption Experimental
Results. In order to analyze the difference of the gas adsorption capacities of the four groups of coal samples, the selected coal samples were tested  At present, the Langmuir equation V = V L P/ðP + P L Þ is the most commonly used method to describe gas adsorption characteristics of coal. In the equation, V L (ml/g) is the Langmuir volume, representing the maximum adsorption capacity of single molecular layer; P L (MPa) is the Langmuir pressure, representing the pressure when the adsorption amount reaches half of V L . The results of the gas adsorption experiment on each coal sample were fitted and calculated according to the Langmuir equation for the purpose of obtaining relevant adsorption characteristic parameters ( Figure 7 and Table 4).
It can be observed from Figure 7 and Table 4 that the fitting degrees of gas adsorption experiment results of all coal samples are all over 0.99 according to the Langmuir equation, and the gas adsorption characteristic curves of all coal samples conform to the Langmuir equation. The Langmuir pressures P L of the four groups of coal samples follow the order W2 < W1 < W4 < W3; the limit gas adsorption capacities V L follow the order W3 < W4 < W1 < W2. Meanwhile, when the pressure is low (below 2 MPa), the gas adsorption capacities of the four groups of coal samples differ insignificantly. With the further increase of the pressure, a significant difference appears in the gas adsorption capacity, which is mainly caused by the difference in the internal pore microstructure.

Influences of the Tortuosity Fractal Dimension D t on Gas
Adsorption. In order to figure out the relations between the dendric tortuality fractal dimension D t and the Langmuir volume V L and Langmuir pressure P L , the gas adsorption characteristic parameters of the four groups of coal samples were fitted with D t , and the results are exhibited in Figure 8.
It can be found from Figure 8 that the dendric tortuality fractal dimension D t of experimental coal samples is positively correlated with V L and negatively correlated with P L , which is attributed to the fact that the dendric tortuality fractal dimension D t characterizes the complexity of pores of coal samples. To be specific, it depicts the bending effect of pores rather than the size of pores, reflecting the roughness of pore surface and the difficulty of the flow through the solid phase. Generally, a larger D t corresponds to a coarser coal pore surface, a more complex pore structure, and a larger specific surface area of the corresponding coal sample, and thus more points available for gas adsorption, which is manifested by the larger V L of the coal sample.  Meanwhile, the stronger the gas adsorption capacity of the coal sample is, the smaller the energy required to absorb a certain amount of gas is, which is manifested as a larger tortuality fractal dimension D t corresponding to a smaller P L .

Conclusions
Though the mercury injection experiment and the gas adsorption experiment on anthracite from Guhanshan Coal Mine in the Jiaozuo mining area, the pore dendric fractal characteristics of anthracite and their influences on gas adsorption characteristics are discussed in this paper. The following conclusions were drawn through research and analysis: (1) The total mercury injection amount of anthracite is directly proportional to the porosity, but bears no obvious correlation with the specific surface area (2) The mercury injection experiment on anthracite reveals that seepage pores whose sizes are over 65 nm conform to the stochastic fractal model, and the fractal dimension D increases with the increase of specific surface area; adsorption pores whose sizes are below 65 nm conform to the dendric fractal model, and the tortuosity fractal dimension D t is positively correlated with the specific surface area of coal (3) According to the gas adsorption experiment on various coal samples, the gas adsorption characteristic curves of anthracite all agree with the Langmuir equation, and the Langmuir pressures P L of the four groups of coal samples follow the order W2 < W1 < W4 < W3; the limit gas adsorption capacities V L follow the order W3 < W4 < W1 < W2 (4) The dendric tortuosity fractal dimension D t of anthracite is positively correlated with V L and negatively correlated with P L . The reason is that the tor-tuosity fractal dimension D t characterizes the complexity of internal pores. To be specific, it depicts the bending effect of pores rather than the size of pores, reflecting the roughness of the pore surface and the difficulty of the flow through the solid phase

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.   Geofluids