# Fractal characteristics of the anisotropic microstructure and pore distribution of low-rank coal

## ABSTRACT

Low-rank coal reserves account for nearly 55% of the national coal reserves in China, and it is extremely important to understand their anisotropic microstructure and coalbed methane storage and sorption characteristics. In this study, microstructures that develop parallel to and perpendicular to bedding of low-rank coals are studied through scanning electron microscopy. The micromorphology and microfracture and mineral distributions are examined. Based on fractal theory and digital image processing technology, the fractal box-counting dimensions of microfracture distributions that form parallel as well as perpendicular to the coal bedding planes are calculated, and the anisotropic fractal characteristics of these microfracture distributions are characterized. The pore-size distribution and the fractal characteristics of the microscopic pore structure are systematically examined by means of nitrogen adsorption and mercury intrusion tests. The results of the fractal box-counting dimension calculations indicate that microfractures distributed perpendicular to coal bedding are rougher and more irregular than those distributed along a bedding plane, with the former exhibiting higher space filling and methane adsorption capacity. The fractal dimension of the microscopic pore structure reveals two inflection points measuring approximately 5 and 80 nm, thereby indicating that the low-rank coal exhibits a microscopic pore structure with a three-level fractal characteristic.

## INTRODUCTION

Coal is typically a double-porosity medium that is full of cleats or fractures and pores in the matrix. The macrostructure of coal is extremely complex. Studies on coal microstructure forms the foundation of studies on coalbed methane (CBM) accumulation and migration and gas adsorption/desorption (Tomita et al., 1981; Karacan and Okandan, 2001; Mosher et al., 2013; Mostaghimi et al., 2015; Liu and Nie, 2016; Fu et al., 2017; Liu and He, 2017). The pore structure of coal is a significant factor in CBM storage and diffusion or permeation (Smith and Williams, 1984, 1987; Cai et al., 2013; Nie et al., 2016; Zhao et al., 2017).

In 1974, scanning electron microscopy (SEM) was introduced by Messier (1974), and it has become widely used in the study of rock microscopic morphology. Micro- and meso-imaging technologies significantly contributed to the observation of coal micromorphology characteristics (Neupane et al., 2017). Giffin et al. (2013) examined the morphology and distribution of macro- and mesopores by using broad ion beam milling to prepare low-relief polished surfaces of coal samples for high-resolution SEM imaging. Karacan and Okandan (2001) combined x-ray computed tomography (CT) imaging technology and SEM to observe and evaluate differences in gas migration velocity and adsorption velocity among coals with different types of microstructures. Two different rank coals (the maximum vitrinite reflectance (*R*_{o,max}) of 0.4% and 0.85%, respectively) from the southern Junggar basin in northwestern China were systematically investigated by Zhou et al. (2017) to characterize pore-fracture spaces and their nanoscale connectivity in three dimensions by using focused ion beam-SEM tomography. Liu and He (2017) investigated the influence of pore structure characteristics of different rank coals on CBM adsorption by using Fourier transform infrared spectroscopy, low-pressure nitrogen gas adsorption, and SEM. Zhou et al. (2018) investigated the evolution of fracture networks in coal during loading using x-ray microcomputed tomography with a uniaxial compression facility. In recent years, because of the rapid development of computer technology and digital image processing technology (DIPT), the microstructure characteristics of rock can be quantitatively studied (Xu et al., 2008; Chen et al., 2009). However, the anisotropic characteristics of the micromorphology and microfracture distribution of coals have not been well studied and especially by means of DIPT.

The microscopic pore structure of coal is highly complicated, and pore types and size distribution directly impact the methane adsorption/desorption capability (Faiz et al., 1992; Laxminarayana and Crosdale, 1999; Liu and He, 2017). The microscopic pore structure changes with coal rank, with low-rank coal typically exhibiting higher porosity, more meso- and macropores, and good pore connectivity, thereby leading to better permeability (Thomas and Damberger, 1976; Faiz et al., 1992; Laxminarayana and Crosdale, 1999; Cai et al., 2013; Nie et al., 2015; Rahmat et al., 2015; Liu and He, 2017). Coal generally tends to be nonuniform in terms of both surface and internal structure. Internal pore structures often widely vary in size and shape. This nonuniformity is a decisive factor in gas sorption (Jaroniec, 1983) and, therefore, should be considered apart from factors including the specific surface area and pore-size distribution while interpreting the microstructure of coals.

Studies indicate that microscopic pore structure and surface morphology exhibit a statistical fractal feature and are thereby better depicted by fractal geometry as opposed to Euclidean geometry (Mandelbrot, 1982; Friesen and Mikule, 1987). Fractal geometry, created by Mandelbrot in 1982, investigates irregular objects and phenomena that demonstrate self-similarity. Poulton et al. (1990) suggested that the fractal dimension is an important rock mass mechanics indicator, from which rock mass characteristics can be accurately described. Pfeifer and Avnir (1983) held that the fractal dimension is related to the smoothness of pore surface, and the closer the fractal dimension is to 2.0, the smoother the pore surface is; conversely, the closer the fractal dimension is to 3.0, the rougher the pore surface is. In recent years, the fractal theory has developed rapidly (Wang et al., 2017).

Nitrogen adsorption and mercury intrusion tests are important and widely used techniques to characterize the pore structure and its fractal characteristics in a variety of porous materials (Gan et al., 1972; Clarkson and Bustin, 1999; Mahamud et al., 2003; Yao and Liu, 2006; Mastalerz et al., 2008; Cai et al., 2011; Cardott and Curtis, 2018; Mahamud and García, 2018). Based on the results of mercury intrusion, nitrogen adsorption at 77 K (∼ −196°C or −321°F) and small-angle x-ray scattering, Cai et al. (2013) concluded that an increase in the irregularity of the surface makes the pore structure more inhomogeneous, and this results in the availability of a higher surface area for adsorption.

Different testing methods exhibit their own scope of application based on their test principles. The mercury intrusion test is based on the gradual injection of liquid mercury into an evacuated pore system using external forces. Increases in the pressure make smaller pores accessible to the mercury (Ritter and Drake, 1945). Measurements typically run up to a maximum pressure of 35 MPa (∼5076 psi) at which pore throats as low as 0.018 μm can be penetrated. Li et al. (1999) indicated that coal compressibility significantly affects mercury porosimetry results, especially when the pressure exceeds 20 MPa (∼2901 psi) and the corresponding pore diameter is approximately 60 nm. However, different studies obtained different threshold pressures or pore diameters. Toda and Toyoda (1972) concluded that the evident increase in mercury uptake by coal is entirely due to the compression and crushing of the coal structure when the pressure exceeds 10 MPa (∼1450 psi). Similar conclusions were obtained by Suuberg et al. (1995). Friesen and Ogunsola (1995) and Mahamud et al. (2003) indicated that coal-matrix compression began to occur at a higher pressure (>13 MPa [∼1885 psi]) and that the fractal dimension was greater than 3, and this did not satisfy the fractal characteristics of coal reservoirs under natural conditions. The finding is observed potentially because the coal internal structure is compressed or crushed under high intrusion pressures applied during the mercury intrusion tests, which results in the inaccuracy of pore-size data (Li et al., 1999; Comisky et al., 2011). Specifically, low-rank coal generally exhibits larger porosity when large injection pressures can significantly damage the microscopic pore structure. However, nitrogen adsorption is controlled by capillary condensation at high relative pressures (Brunauer et al., 1940) and thus does not cause significant damage to the microscopic pore structure (Mahnke and Mögel, 2003; Yao et al., 2008; Cai et al., 2013). Therefore, the nitrogen adsorption test is more accurate than mercury intrusion for smaller pore-size distributions (Comisky et al., 2011; Cai et al., 2013). Using both of these complementary techniques allows for accurate measurement of pore-size distributions over a wide range of sizes (Clarkson and Bustin, 1999; Qi et al., 2002; Radlinski et al., 2004; Yao et al., 2008).

Figure 1. The index map (A) and stratigraphic column (B) from 470 to 490 m (1541 to 1607 ft) in depth.

In this study, microstructures that develop parallel to and perpendicular to bedding of low-rank coal obtained from the Hunchun coalfield in the Jilin province of China are examined through SEM. The index map is shown in Figure 1. Specifically, the micromorphology, microfracture, and mineral distributions are investigated. Based on fractal theory and DIPT, the fractal box-counting dimensions of the microfracture distribution are calculated for both the bed parallel and perpendicular directions. The anisotropic fractal characteristics of the microfracture distribution are subsequently analyzed based on the aforementioned calculations. Finally, the pore-size distribution of coal and the fractal characteristics of its microscopic pore structure are studied by means of nitrogen adsorption and mercury intrusion tests. Research methods in this study provide a reference for the microstructure study on other ranks of coal.

## INTRODUCTION OF THE STUDIED COAL

The low-rank coal (long flame coal) was obtained from the Hunchun coalfield in Jilin province of China. With developed multiple minable coal seams, the regional structural characteristics of the Hunchun coalfield are relatively simple and clear. The stratigraphic column is shown in Figure 1B. The largest proportion of coal in the 26th coal seam is semibright coal. The average bulk density is 1340 kg/m^{3} (∼84 lb/ft^{3}). Different coal quality indicators were obtained by specialized testing methods. The reflected polarizing microscope test is a commonly used method to determine the contents of various macerals (the vitrinite, semivitrinite, inertinite, exinite) by count maceral points under the condition of incomplete orthogonal polarization or single polarization. The moisture (air dry basis) and volatile matter (dry ash-free basis) contents of coal were obtained by the automatic industrial testing instrument. The ash yield (air dry basis) and the gross calorific value (air dry basis) were tested by the ash content testing instrument and coal calorific value measuring instrument, respectively.

The majority of macerals are vitrinite, at 78%, whereas the proportions of inertinite and exinite are 4% and 1%, respectively; the *R*_{o,max} is 0.6%. The *R*_{o,max} of coal increases with buried depth, rising 0.05% every 100 m (∼328 ft). The basic characteristics obtained during the coal quality tests are summarized in Table 1. Figure 2 shows a photograph of the coal in which a variety of cleats or fractures are visible.

Figure 2. Low-rank coal sample photo.

## MICROMORPHOLOGY INVESTIGATION BASED ON SCANNING ELECTRON MICROSCOPY

#### Scanning Electron Microscopy Method

In SEM, focused electron beams are used to scan an image point-by-point on the surface of the sample, and two different imaging modes are possible, namely secondary electron mode (ETD) and backscattered electron diffraction mode (BSED). The contrast of an ETD image is mainly due to surface topography, and thus it exhibits a strong three-dimensional effect. However, ETD images do not provide information on sample composition. It should be noted that the contrast of a BSED image is mainly due to local differences in average atomic weight. Different minerals exhibit different levels of brightness or color in a BSED image from which qualitative compositional information can be extracted. In this study, ETD and BSED are combined to investigate the properties of coal microstructure.

Figure 3. Scanning electron microscopy images observed parallel to coal beddings in (A) secondary electron mode and (B) backscattered electron diffraction mode. σ = speculated in situ principle stress.

The Quanta 250 scanning electron microscope produced by American FEI was used. The resolution is 3.0 nm when the voltage is 30 kV for the ETD mode and 4.0 nm for the BSED mode.

#### Micromorphology Observation Results

Figures 3 and 4 show the SEM images taken parallel to and perpendicular to coal beddings, respectively. The additional delineations shown in Figures 3A and 4A and Figures 3B and 4B denote images observed at the same location but with different SEM electron modes (ETD and BSED, respectively). As shown in Figures 3A and 4A, differences in brightness imply regions of irregularity and unevenness on the coal surface. Conversely, in Figures 3B and 4B, differences in brightness mainly imply regions with different compositions. The white spots in Figures 3B and 4B are mostly minerals that normally fill fractures or bedding planes. As shown in Figure 3, a variety of microfractures are distributed along a bedding plane, with most displaying an X-shaped pattern. The microfractures in direction 1 are denser than those in direction 2, which suggests a connection with the in situ stress state. Specifically, “σ” in Figure 3B is the speculated direction of principle in situ stress. However, the relation between microfracture morphological characteristics and the stress that the coal previously suffered by the coal should be further examined and verified.

Figure 4. Scanning electron microscopy images observed perpendicular to coal beddings in (A) secondary electron mode and (B) backscattered electron diffraction mode.

Figure 5 shows the minerals that fill in the fractures or bedding planes, with Figure 5A being an image along bedding and Figure 5B showing an image that is perpendicular to the bedding. The image is observed with the BSED mode. Most minerals are characterized as having a floccular structure, as shown in the rectangles in Figure 5. It is speculated that the floccular minerals in coal mainly correspond to clay minerals (Gamson et al., 1993 and Giffin et al., 2013), calcite (CaCO_{3}), kaolinite (Al_{2}Si_{2}O_{5}(OH)_{4}), and illite (KAl_{2}(Si_{3}Al)O_{11}(OH)_{2}).

Figure 5. Minerals that filled in the fractures or bedding planes (backscattered electron diffraction mode): (A) scanning electron microscopy image observed parallel to coal beddings; (B) scanning electron microscopy image observed perpendicular to coal beddings.

#### Digital Image Processing Technology Investigation Results

A digital image is composed of a variety of points with different pixels, and each pixel point exhibits its own color. The pixels in a binary digital image are only black and white and are represented with “1” and “0” in the matrix, respectively, and the corresponding gray values are 0 and 255.

Figure 6. The percentages of the pixel points with different gray values in digital images observed parallel to coal beddings.

The color of minerals significantly differs from that of the coal matrix in BSED mode images. Minerals are white with higher brightness, whereas the coal matrix is dark. Thus, the gray values of the minerals’ pixel points are higher, and those of the matrix’s pixel points are significantly low. The percentages of the pixel points with different gray values in the digital images that are observed parallel to the coal bedding are shown in Figure 6. Figure 7 shows the percentages of the pixel points with different gray values in digital images that are observed perpendicular to the coal bedding.

Figure 7. The percentages of the pixel points with different gray values in digital images observed perpendicular to coal beddings.

The highest percentage of pixel points in a surface that parallels bedding appears near the gray value of 80, and the highest percentage of pixel points in a surface that is perpendicular to bedding appears near the gray value of 155. The mineral content (the gray value is ∼245–255) on a surface that is perpendicular to the bedding is higher than that parallel to bedding. To extract the mineral content, the images observed by SEM are processed into binary digital images using the appropriate gray threshold values. During the actual calculation process, the computer automatically provides a reference gray value based on the maximum entropy thresholding segmentation algorithm to ensure the clarity of target features. The maximum entropy thresholding segmentation algorithm was first proposed by Kapur et al. (1980) and applied to two-dimensional digital images by Abutaleb (1989) and Brink (1992). When the total entropy of the background pixels and the target pixels reaches the maximum, it is considered that the image segmentation reaches the best effect, and the threshold at this time is considered optimal. The mathematical calculation process of the maximum entropy can be found in numerous works (Brink, 1992; Zhang et al., 2012; Bhandari et al., 2015). This paper does not discuss this calculation in detail. Zhou et al. (2018) obtained the grayscale histogram of a CT image of the coal specimen, and the threshold gray values of the pore fracture and coal matrix were obtained using the maximum entropy method. As shown in Figure 8, the white denotes the minerals, whereas the black denotes the coal matrix. The gray threshold values for the binary digital image observed parallel to the coal bedding are 0.5774 and 0.5823, whereas those perpendicular to the coal bedding are 0.7912 and 0.7741. The digital images shown in Figure 8 indicate that the percentages of minerals on the surface that are parallel to coal bedding are approximately 6.14% and 5.76% with an average value of 5.95%; those in surfaces perpendicular to the coal bedding are approximately 8.87% and 8.49%, with an average value of 8.68%.

Figure 8. The binary digital images of minerals and coal matrix. (A, B) Results of images observed parallel to coal beddings. (C, D) Results of images observed perpendicular to coal beddings.

The color of the microfractures also significantly differs from that of the other components (the matrix and minerals) in the BSED mode images. The microfractures are extremely dark because of depressions, and their gray value is extremely low such that it is close to 0. To distinguish the fractures from the matrix and minerals, the binary digital images that are shown in Figure 9 are obtained by selecting appropriate gray threshold values. The gray threshold values of digital images observed parallel to the coal bedding are 0.1497 and 0.1473, whereas the gray threshold values of digital images observed perpendicular to the coal bedding are 0.2134 and 0.2047.

Figure 9. The binary digital images of microfractures. (A, B) Results of images observed parallel to coal beddings. (C, D) Results of images observed perpendicular to coal beddings.

## FRACTAL CHARACTERISTICS OF MICROFRACTURES

The box-counting method is often used for image fractal dimension estimation (Ai et al., 2014; Liu et al., 2015; Nayak et al., 2018). It includes several advantages such as the convenience of implementation, high computing efficiency, reasonable calculation results, and the applicability for the images with or without self-similarity (Di et al., 2007; Tang et al., 2012; Azemin et al., 2015; Liu and Nie, 2016; Nabelek et al., 2018; Nayak et al., 2018).

The fractal box-counting dimension can be obtained by equation 1 (Fu et al., 2006; Liu et al., 2015)

where *A* is an arbitrary bounded set in space. With respect to an arbitrary *r* greater than 0, the minimum required number of *N*-dimensional boxes with the square of *r* that is used to cover *A* is . Plotting of [,] in the logarithmic coordinate system with as the abscissa and as the ordinate permits the use of the absolute value of the slope of the fitting line of these points to define the fractal box-counting dimension of set *A.*

The fractal box-counting dimensions of microfractures are calculated based on fractal theory (Fu et al., 2006; Liu et al., 2015; Nayak et al., 2018). Microfractures are covered with square boxes with sizes ranging from small to large, and the minimum required number is counted. The covering process of the microfractures distributed on a surface that is perpendicular to coal beddings is shown in Figure 10. Figure 10A–D, respectively, shows different results when the microfractures are covered with boxes with sizes ranging from small to large and when the number of required boxes gradually decreases.

Figure 10. Microfractures are covered with square boxes in varied sizes from small to large (the surface that is perpendicular to coal beddings): (A) box number is 2300; (B) box number is 1500; (C) box number is 800; (D) box number is 300.

The microfractures distributed in a surface that parallels coal bedding are processed in the same manner. Figures 11 and 12 show the points of [,] and fitting lines in the logarithmic coordinate system with as the abscissa and as the ordinate.

Figure 11. >Fractal box-counting dimension fitting process of surfaces that are parallel to coal beddings: (A) result of Figure 9A; (B) result of Figure 9B. *r* = side length of the square box; *N _{r}*(

*A*) = minimum required number of boxes used to cover

*A*(arbitrary bounded set in

*R*space);

^{n}*R*

^{2}= coefficient of determination.

Fractal box-counting dimensions are calculated with equation 1. The results indicate that the fractal box-counting dimensions of microfractures distributed in surfaces that are parallel to coal bedding are 1.22 and 1.10, with an average value of 1.16. Conversely, those of microfractures distributed on surfaces that are perpendicular to coal bedding are 1.33 and 1.31, with an average value of 1.32.

Figure 12. Fractal box-counting dimension fitting process of surfaces that are perpendicular to coal beddings: (A) result of Figure 9C; (B) result of Figure 9D. *N _{r}*(

*A*) = minimum required number of boxes that are used to cover

*A*(arbitrary bounded set in

*R*space);

^{n}*R*

^{2}= coefficient of determination.

Fractal box-counting dimension characterizes the nonuniformity, roughness, and irregularity of microfractures in a surface. The space filling capacity and methane adsorption capacity of coal are also reflected (Liu and Nie, 2016). The larger the fractal box-counting dimension, the more inhomogeneous and irregular the structure is (Pfeifer and Avnir, 1983; Krohn and Thompson, 1986; Xie, 1996). Our results from the fractal box-counting dimension indicate that the microfractures distributed in a surface that is perpendicular to coal bedding are rougher and more irregular when compared to those distributed in a bedding plane, and the former exhibit higher space filling and methane adsorption capacity.

## FRACTAL CHARACTERISTICS OF MICROSCOPIC PORE STRUCTURE

#### Test Methods

**Nitrogen Adsorption Test**

The nitrogen adsorption test is based on the gas adsorption law on the surface of solid pore structure to determine the pore specific surface area and pore-size distribution (Schmitt et al., 2013; Qi et al., 2017).

Pore-size distribution characteristics of less than 89.19 nm were examined through the nitrogen adsorption test. The NOVA1000e-pore diameter distribution testing instrument produced by Quantachrome was used. Based on the classical gas adsorption method and various structural analysis models, the gas is injected into a vacuum and confined space in which the sample was placed by an automatic gas feeding method. The gas isothermal adsorption curve, specific surface area of the sample, porosity, and true density are obtained. The pressure sensor accuracy is less than ±0.1%, and the ultimate vacuum is 1 × 10^{−3} *P*/*P*_{0} (*P* is the adsorption pressure, and *P*_{0} denotes saturated vapor pressure of nitrogen). The samples required for the nitrogen adsorption test are rectangular blocks with a side length of approximately 3.0 mm (∼0.1 in.).

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