Coalbed methane (CBM) has become an important part of natural gas resources in the world^[1]. The total recoverable coalbed methane reserves in the world are estimated at (1.4-8.5)×10¹⁴m^3[2]. Coal is a kind of porous medium with complex pore structure^[3]. At present, the main methods quantitatively characterizing coal pore structure include mercury intrusion method^[4,5] and nitrogen/carbon dioxide low-pressure adsorption method (LPA)^[6]. Mercury intrusion method may cause damage or deformation to pore structure due to high pressure injection^[4]. The experimental results of LPA need to be interpreted by selecting appropriate analysis model (such as Barrett-Joyner-Halenda (BJH), Dubinin-Astakhov (D-A) and Horvath-Kawazoe (H-K) et al.)^{[7,8,9,10,11]}. But each model is applicable to a specific pore shape and size scope, and coal pore structure has complex geometrical morphology and a wide range of pore diameter, so it is difficult to find a single model to accurately analyze the complete pore size distribution (PSD) in coal. Hence, it is necessary to find new methods to characterize pore structure quantitatively.

The characterization of coal pore structure by small angle X-ray scattering (SAXS) and nano-CT has attracted more and more attention since 2010. The new generation of laboratory nano-CT can reach nanometer resolution^[12]. However, laboratory nano-CT neither has monochromatic beam nor can realize high X-ray coherence, which may affect imaging quality, such as beam hardening^[13]. As conventional SAXS has weak signals, its measurement time must be longer, thus bringing about great difficulties to the experiment. It often takes several hours or tens of hours to measure a sample, and it is difficult to meet the needs of many experiments even using a high-power X-ray source. In comparison, synchrotron radiation source not only has high intensity and good spot size, but also can use a long collimation system (up to 10 m), which greatly improves the resolution and sensitivity of the experiment, saves experimental time (the exposure time of general samples only takes tens of seconds), and simplifies the tedious data correction work. Since the wavelength of synchrotron radiation light is continuously adjustable, the appropriate wavelength can be selected in the experiment to eliminate the problem of multiple scattering in SAXS^[14]. Therefore, compared with conventional SAXS, synchrotron radiation SAXS has some prominent advantages.

Some studies on the pore structure of coal by synchrotron radiation SAXS were carried out abroad. Okolo G N et al.^[15] studied four kinds of coal samples with low-temperature nitrogen adsorption, mercury intrusion and synchrotron radiation SAXS technologies and pointed out that SAXS could detect a wider range of pore sizes, including closed and open pores, so the specific surface area and porosity measured by synchrotron radiation SAXS are the largest. Radlinski A P et al.^[16] measured seven groups of raw coal and corresponding powder samples with SAXS and SANS (Small Angle Neutron Scattering) and found that pellet model fully represented the microscopic pore structure of coal sample. Zhao Yixin et al.^[17] measured six coal samples of different ranks by TEM (Transmission Electron Microscope) and synchrotron radiation SAXS and found that pore size distribution was not related to coal rank, but varied with vitrinite content. Luo et al.^[18] studied the ultra-fine pulverized coal samples of different ranks by synchrotron radiation SAXS and concluded that the fractal dimension of the pore surface decreased with the increase of coal rank. Wang B et al.^[19] conducted researches by synchrotron radiation SAXS and proved that after deashing, low volatile bituminous coal increased in the total number and volume of pores, the proportion of micropores, the fractal dimension and specific surface area of pores, but decreased in average and most probable pore size. Since 2018, the authors' team has studied the experimental method and image processing method characterizing the pore structure of coal by synchrotron radiation nano-CT, and worked out the CFD (Computational Fluid Dynamics) numerical simulation method based on connected pores. The permeability obtained by CFD numerical simulation is in the same order of magnitude as that obtained by other researchers^[20].

Quantitative evaluation of heterogeneity and anisotropy of coal pore structure is of great significance for revealing the occurrence and migration of gas in coal seam^{[21,22,23,24,25,26]}. According to the literature retrieval, a lot of researches on the quantitative evaluation methods of pore structure heterogeneity and anisotropy have been carried out^{[27,28,29,30,31,32,33,34,35,36]}, but there are few researches on the quantitative characterization of coal pore structure heterogeneity and anisotropy based on synchrotron radiation facility. Both synchrotron radiation SAXS and synchrotron radiation nano-CT have nanometer level resolution, synchrotron radiation SAXS can reach the pore size resolution of 1 nm, and synchrotron radiation nano-CT can reach the spatial resolution of 30 nm, and they have no requirement on pore geometrical morphology, so they can be effective means for quantitative research on coal pore structure. Therefore, we quantitatively characterized the heterogeneity and anisotropy of coal pore structure by synchrotron radiation SAXS and synchrotron radiation nano-CT in this work, in the hope to find a new method to quantitatively evaluate the heterogeneity and anisotropy of coal pore structure.

In the experiment, two coal samples of different ranks were collected from #11 coal seam of Xinzhouyao coal mine in Shanxi Province and #9 coal seam of Tangshan coal mine in Hebei Province. Refer to Table 1 for the proximate analysis results of the coal samples^[37]. The true density of the two coal samples were measured with MDMDY-300 automatic densitometer by the helium replacement method, the true density of Xinzhouyao coal is 1.350 2 g/cm³, and the true density of Tangshan coal is 1.535 5 g/cm³. According to the Chinese Petroleum and Natural Gas Industry Standard SY/T 5124- 2012^[38], the Leitz MPV-3 spectrophotometer was used to measure the vitrinite reflectance and analyze the maceral composition of the polished samples. From the average maximum vitrinite reflectance, Xinzhouyao coal is gas coal and Tangshan coal is fat coal, Xinzhouyao coal has higher vitrinite content and the two coal samples are very close in inertinite content. According to the Chinese Petroleum and Natural Gas Industry Standard SY/T 5163-2010^[39], D/MAX 2500 X-ray diffractometer was used to do X-ray diffraction (XRD) analysis of the two coal samples. The XRD analysis results show that the Tangshan coal contains more clay mineral (Table 1).

Since the conventional pore structure characterization methods can hardly detect closed pores in coal, in order to fully characterize the heterogeneity of pore structure in coal (including closed pores and open pores), a new method to quantitatively characterize the heterogeneity of pore structure in coal by synchrotron radiation SAXS has been explored in this study. The synchrotron radiation SAXS experiment was carried out by 1W2A-small angle X-ray scattering station in Beijing synchrotron radiation facility.

The schematic diagram of synchrotron radiation SAXS optical path is shown in Fig. 1. The synchrotron radiation SAXS has an angle resolution of 0.5×10^-3 rad, energy resolution of about 10^-3, light intensity at the sample of 1×10¹¹ cps,

1×10³-1×10⁴ times higher than that of conventional X-ray machine, wavelength of incident X-ray of 0.154 nm, and spot size of 1.4 mm×0.2 mm. In the synchrotron radiation SAXS experiment, samples are coal particles with a diameter of 0.18-0.25 mm (60-80 mesh), and samples are installed on sample cavity by 3M tape and the sample cavity is installed on the integrated platform. After starting the synchrotron radiation SAXS devices, the exposure time and exposure times are set, and then the readings of photodiode is recorded, and the scattering image data is saved.

When X-ray irradiates the sample, if there is a density inhomogeneous area in nanometer size (2-100 nm) in the sample, scattered X-rays will appear in a small angle range of 2°-5° around the incident X-ray. The scattering intensity of an electron in different directions can be determined by Thomson formula^[40]:

For multi-particle system, if the distance between particles is much larger than the size of themselves, the scattering intensity can be approximately taken as the simple sum of the scattering intensity of each particle. Therefore, for the scattering intensity of M non-interference particles, it is deemed that the scattering intensity approximately conforms to the Guinier's law^[41]:

According to formulas (3)-(5), the scattering intensity decreases with the increase of scattering angle. The relationship between q and I(q) can be obtained from SAXS experimental observation, and structure information of the scattering object, such as the radius of gyration, particle size, shape, fractal dimension and interface layer thickness, can be obtained through data processing.

In order to compare and verify the quantitative evaluation method of pore structure heterogeneity in coal by synchrotron radiation SAXS, considering the advantages of synchrotron radiation nano-CT in detecting closed pores and obtaining three-dimension geometrical morphology of pore structure, this study explored the quantitative evaluation method of pore structure heterogeneity in coal by synchrotron radiation nano-CT. The synchrotron radiation nano-CT experiment was done with the 4W1A-X-ray imaging experimental station in Beijing synchrotron radiation facility.

The synchrotron radiation nano-CT has an energy range of 5-12 keV, spot size of 10 µm×10 µm, and spatial resolution of better than 30 nm. For more details about the synchrotron radiation nano-CT, readers can refer to the relevant literature [42]. For synchrotron radiation nano-CT imaging, the size of sample particle is required to be smaller than 10 μm. Therefore, it is necessary to firstly crush the coal sample, and then the coal particle close to 10 μm is chosen and fixed on the pin tip with the instrument shown in Fig. 2c. Since reference points are needed for subsequent images alignment, a spherical gold particle with a diameter of 0.5 µm is installed on the sample by the instrument shown in Fig. 2d as a reference point for later images alignment. Finally, the pin with the coal sample is clamped on the sample turntable shown in Fig. 2b. During the imaging, the sample table rotates discontinuously at a preset angle step (rotation speed) to obtain the projection images at different angles.

2.1.1. Quantitative characterization of coal pore structure heterogeneity by synchrotron radiation SAXS

The SAXS scattering images of the two coal samples of different ranks obtained by synchrotron radiation SAXS experiments are shown in Fig. 3.

To facilitate data analysis, FIT2D software was used to convert SAXS scattering images shown in Fig. 3 into the corresponding scattering curves shown in Fig. 4. The abscissa |q| is the scattering vector in the inverse space and the ordinate is the scattering intensity.

The small angle X-ray scattering intensity of fractal object can be expressed by the power law in fractal region:

where α is [0, 4], the slope of ln|I(q)|-ln|q| curve in the linear region is d, and α=-d. When 0<α<3, the scattering object is mass fractal or pore fractal, and its fractal dimension${{D}_{2}}=\alpha $; When 3<α<4, the scattering object is surface fractal, and its fractal dimension${{D}_{1}}=6-\alpha $. Therefore, the occurrence of a linear region in the ln|I(q)|-ln|q| curve indicates that there is fractal phenomenon, and whether the fractal is surface fractal or pore fractal can be judged according to the value of α^[43].

Fig. 5 shows the double logarithm curves of |q| and scattering intensity I of the two coal samples. The pore fractal dimension in the corresponding |q| value range can be obtained by the tangent of the double logarithm curve of the coal sample.

In the region of low |q| value (-3--1 nm^-1), the fractal feature of the curve is surface fractal (D₁), and the smoothness and planeness of the pore surface can be analyzed. The fractal dimension describes the smoothness and planeness of the pore surface, and the larger the value, the more irregular the pore surface is. The fractal feature of the curve in the high |q| value region (-1-1 nm^-1) is pore fractal (D₂) and its fractal dimension can reflect the heterogeneity of pores spatial distribution. The larger the value, the stronger the heterogeneity of pore spatial distribution is. According to the surface fractal (D₁) in Fig. 5, the pore surface of Xinzhouyao coal is more irregular. It can be seen from the pore fractal (D₂) in Fig. 5, the pore spatial distribution heterogeneity of Xinzhouyao coal is stronger.

2.1.2. Quantitative characterization of coal pore structure heterogeneity by synchrotron radiation nano-CT

After the images of coal samples were obtained by synchrotron radiation nano-CT, the image processing procedure mainly included images alignment, 3D reconstruction, selection of region of interest (ROI), noise filtering and image segmentation, as shown in Fig. 6. Details about image processing method has been introduced in literature [20].

After image segmentation, the connectivity of pore structure was analyzed. Connected pores refer to the pores continuously connected from one face to the opposite face in ROI cube, and the connected pores in the two coal samples are shown in Fig. 7a and Fig. 7b. Through calculation, the volume of connected pores in the Xinzhouyao coal sample is 1.71 µm³, accounting for 69% of the total pore volume (2.47 µm³), while the volume of connected pores in the Tangshan coal sample is 2.70 µm³, accounting for 83% of the total pore volume (3.24 µm³), which shows that the Tangshan coal sample has more connected pores.

The porosity differences between different sub-blocks in the coal sample can characterize the pore structure heterogeneity. In this study, the size of coal sample ROI is 200 pixel × 200 pixel × 200 pixel, and the pixel size is 0.014 59 μm × 0.014 59 μm × 0.014 59 μm. The porosity difference between different sub-blocks in coal sample ROI can be characterized by calculating the relative standard deviation of each sub- block’s porosity:

As shown in Fig. 8, when the coal sample ROI is divided into s segments in the direction (X, Y, Z), the coal sample ROI is divided into s³ sub-blocks, and the relative standard deviation H of each sub-block’s porosity varies with the number of segments s. Therefore, it is necessary to propose a parameter irrelevant to the number of sub-blocks to characterize the heterogeneity of 3D coal pore structure. Although, the curve rises with the increase of the number of segments s, it becomes more and more gentle. To obtain a constant to characterize the heterogeneity of 3D coal pore structure, the following function was used to fit the data in Fig. 8, and the extremum of the function is a constant.

HV can characterize the heterogeneity of 3D coal pore structure, and the larger the HV, the stronger the heterogeneity of pore structure is. According to Formula (9), the heterogeneity value obtained by this method eliminates the influence of the segmenting number of sub-blocks.

Xinzhouyao Coal: A=0.210 47, B=0.065 62, R²=0.999 41, HV=3.21; Tangshan coal: A=0.175 47, B=0.064 63, R²=0.994 02, HV=2.71.

As shown in Fig. 8, the heterogeneity values of the two coal samples can be well fitted by formula (8). The calculation results show that HV of the Xinzhouyao and Tangshan coal samples are 3.21 and 2.71 respectively. Clearly, Xinzhouyao coal has higher heterogeneity than Tangshan coal in pore structure, and the same conclusion can be drawn from the comparison of the two curves in Fig. 8.

As for the pore structure heterogeneity of the two coal samples, the Xinzhouyao coal has stronger pore structure heterogeneity than the Tangshan coal. The results from synchrotron radiation SAXS and synchrotron radiation nano-CT are consistent, verifying the effectiveness of the two methods in evaluating coal pore structure heterogeneity. Although the establishment of the classification standard of pore structure heterogeneity requires statistics on a large number of coal samples, the pore fractal (D₂) and HV obtained from synchrotron radiation SAXS and synchrotron radiation nano-CT provide new means to describe the pore structure heterogeneity quantitatively. The two indexes can realize the sequencing of pore structure heterogeneity of multiple samples, and also make it possible to establish the classification standard (strong-medium-weak) of pore structure heterogeneity.

According to the literature research, coal pore structure heterogeneity increases with the increase of vitrinite content and decreases with the increase of inertinite content^[44], and presents a linear positive correlation with the ash content^[45]. The vitrinite content of Xinzhouyao coal is higher than that of Tangshan coal, the inertinite and ash content of the two coal samples are similar, and the pore structure heterogeneity of Xinzhouyao coal is stronger than that of Tangshan coal, which is consistent with the previous research results.

Considering the correspondence between pore structure anisotropy and permeability anisotropy, the quantitative evaluation of the pore structure anisotropy can be realized by calculating the permeability tensor with LBM method (Lattice Boltzmann method). Pore structure anisotropy is characterized by the eigenvalue and eigenvector of permeability tensor. The eigenvalue of permeability tensor reflects the permeability in three directions determined by corresponding eigenvectors. The LBM method used in this study adopted a single relaxation time Bhatnagar-Gross-Krook (BGK) model^[46]. This model simulates the variation of the migrating and colliding fluid particles distribution in the lattice nodes by solving a discrete Boltzmann equation, and then the macroscopic fluid flow is approximated by the whole movement of virtual fluid particles. The fluid particle distribution function \({{f}_{i}}\) in the direction\[{{e}_{i}}\] is updated at each time step by the following equation:

where, ${{f}_{\text{eq}}}$is the section equilibrium distribution, and can be defined as,

$\tau $ is a parameter related to dynamic viscosity$\upsilon $, expressed in Chapman-Enskog equation as,

The macro density $\rho $ and velocity \(v\) are determined by the variation of particle distribution:

According to Darcy's law, the permeability tensor can be calculated by the following equation^[47],

Through three independent simulations, three diagonal values of permeability tensor were calculated by applying pressure gradient on two opposite surfaces in three flow directions (X, Y and Z) and no-flow boundary conditions on the remaining four surfaces.

Under the assumption that permeability tensor is symmetric and positive definite, Darcy's law can be extended as:

As shown by Formula (16), if \({{v}_{1}}\), \({{v}_{2}}\) and \({{v}_{3}}\) are obtained, the off-diagonal values of permeability tensor can be calculated by the diagonal values.

To obtain \({{v}_{1}}\), \({{v}_{2}}\) and \({{v}_{3}}\), another set of simulations was carried out. In this set of simulations, the pressure gradient was applied to three orthogonal directions simultaneously. Fig. 9 shows the velocity streamline in four sets of LBM simulations, and it can be seen that the distribution of the streamline is consistent with the direction of the pressure gradient.

In order to analyze the three-dimensional pore structure anisotropy, the eigenvalues and eigenvectors of the two coal samples ROI permeability tensor were calculated.

The calculation result of Xinzhouyao coal permeability tensor is,

$\left[ \begin{matrix} 0.68 & 2.02 & 4.23 \\ {} & 0.87 & 0.40 \\ {} & {} & 0.11 \\\end{matrix} \right]\times {{10}^{-6}}\ \text{ }\!\!\mu\!\!\text{ }{{\text{m}}^{2}}$

The eigenvalues this coal sample are k₁=-4.10×10^-6 μm², k₂=0.42×10^-6 μm², k₃=5.33×10^-6 μm², and the corresponding eigenvectors are,

${{E}_{1}}=\left[ \begin{matrix} -0.70 \\ 0.23 \\ 0.68 \\\end{matrix} \right]\text{ }{{E}_{2}}=\left[ \begin{matrix} 0.12 \\ -0.90 \\ 0.42 \\\end{matrix} \right]\text{ }{{E}_{3}}=\left[ \begin{matrix} 0.71 \\ 0.37 \\ 0.60 \\ \end{matrix} \right]$

The calculation result of Tangshan coal permeability tensor is,

$\left[ \begin{matrix} 0.93 & 6.10 & -0.57 \\ {} & 0.69 & -0.21 \\ {} & {} & 0.57 \\\end{matrix} \right]\times {{10}^{-6}}\ \text{ }\!\!\mu\!\!\text{ }{{\text{m}}^{2}}$

Similarly, the eigenvalues of this coal sample are calculated as k₁=-5.30×10^-6 μm², k₂=0.54×10^-6 μm², k₃=6.69×10^-6 μm², and the corresponding eigenvectors are,

${{E}_{1}}=\left[ \begin{matrix} 0.70 \\ -0.71 \\ 0.04 \\\end{matrix} \right]\text{ }{{E}_{2}}=\left[ \begin{matrix} 0.03 \\ 0.09 \\ 1.00 \\\end{matrix} \right]\text{ }{{E}_{3}}=\left[ \begin{matrix} 0.71 \\ 0.70 \\ -0.09 \\\end{matrix} \right]$

It can be seen that the permeability tensor eigenvalue of Tangshan coal is larger than that of Xinzhouyao coal, which is consistent with the fact that the Tangshan coal sample has more connected pores.

The purpose of pore structure anisotropy analysis is to find out the most permeable direction. However, it should be noted that there are negative values in the permeability tensor eigenvalues of both coal samples, which indicates that there are U-shaped connected pores in the direction determined by the corresponding eigenvector. The direction determined by eigenvector corresponding to the largest value of the permeability tensor eigenvalues is the most permeable, and the cross-section area of the connected pores in this direction is also the largest. The direction of eigenvector corresponding to the second largest value of the permeability tensor eigenvalues is the second most permeable, and the cross-section area of the connected pores in this direction is also the second largest. In Fig. 7a, 7b, the directions determined by the corresponding eigenvectors are marked. It can be found that there is an obvious correlation between the permeability anisotropy and the pore structure geometrical morphology anisotropy. In the direction at which eigenvalue is larger, the cross-section area of connected pores is also larger.

In order to reveal the characteristics and advantages of synchrotron radiation SAXS and synchrotron radiation nano- CT, the pore size distributions (PSD) measured by synchrotron radiation SAXS and synchrotron radiation nano-CT were compared with the results measured by low-temperature N₂ adsorption/desorption and NMR cryoporometry (Fig. 10). The low-temperature N₂ adsorption/desorption and NMR cryoporometry experiment and data processing methods were introduced in details in the author’s papers published previously^[48]. It can be seen from Fig. 10a, when the pore size is less than 100 nm, the pore volumes measured by low-temperature N₂ adsorption/desorption and NMR cryoporometry are higher than that measured by synchrotron radiation nano-CT, which can be explained by the limitation of synchrotron radiation nano-CT resolution. Some pores less than 100 nm cannot be detected by synchrotron radiation nano-CT but can be detected by low-temperature N₂ adsorption/desorption and NMR cryoporometry. When the pore size is larger than 100 nm, the pore volume measured by synchrotron radiation nano-CT is larger than that measured by the other two methods, the reason is that the closed pores cannot be detected by low-temperature N₂ adsorption/desorption and NMR cryoporometry, but can be detected by synchrotron radiation nano-CT. Apparently, synchrotron radiation nano-CT has advantages in detecting closed pores, but due to the limitation of resolution, it has deficiency in detecting pores with small diameter.

It can be seen from Fig. 10b, the test results from synchrotron radiation SAXS and low-temperature N₂ adsorption/desorption are very consistent, with two peaks in the pore size ranges of 0-40 nm and 40-80 nm, respectively. In the pore size range of 20-25 nm, the pore volume measured by synchrotron radiation SAXS is larger than that measured by low-temperature N₂ adsorption/desorption, which can be explained by the fact that the closed pores can be measured by synchrotron radiation SAXS, but cannot be measured by low-temperature N₂ adsorption/desorption. In the rest range, the pore volume measured by synchrotron radiation SAXS is smaller than that measured by low-temperature N₂ adsorption/desorption, and the reason for this phenomenon still needs to be further studied. In the author’s measurement experiments of shale pore structure by synchrotron radiation SAXS and NMR cryoporometry, the pore volume measured by synchrotron radiation SAXS is larger than that measured by NMR cryoporometry in some ranges, but smaller than that measured by NMR cryoporometry in other ranges^[41]. Although the pore size range measured by synchrotron radiation SAXS can’t cover the whole pore size range, but according to the test results from NMR cryoporometry shown in Fig. 11, the volume of pores less than 80 nm in size accounts for 74% of the volume of pores less than 500 nm. Therefore, the pore size range measured by synchrotron radiation SAXS is of certain significance for the analysis of coal pore structure.

In conclusion, compared to conventional pore structure characterization methods, although synchrotron radiation SAXS is smaller in the range of pore size detected and synchrotron radiation nano-CT can’t detect pores smaller than its resolution, the advantages of the two methods are that they are able to detect closed pores, and synchrotron radiation nano-CT can obtain 3D geometrical morphology of pore structure. Besides, the combination of the two methods can mutually make up their respective disadvantages in the detecting pores of larger and smaller sizes.

Considering the insufficiency in quantitative evaluation of coal pore structure heterogeneity and anisotropy, this paper explores the quantitative evaluation methods of coal pore structure heterogeneity and anisotropy by synchrotron radiation SAXS and synchrotron radiation nano-CT. In quantitative evaluation of coal pore structure heterogeneity by synchrotron radiation SAXS, the pore structure heterogeneity of two coal samples was quantitatively evaluated by pore fractal D₂, the pore fractal dimension of Xinzhouyao coal is 2.74, and the pore fractal dimension of Tangshan coal is 1.69. In quantitative evaluation of coal pore structure heterogeneity by synchrotron radiation nano-CT, the coal sample ROI was divided into a certain number of sub-blocks, and the pore structure heterogeneity was quantitatively evaluated by calculating the extremum of relative standard deviation of each sub-block’s porosity. The pore structure heterogeneity value of the Xinzhouyao coal is 3.21, and the heterogeneity value of the Tangshan coal is 2.71. The pore structure heterogeneity values obtained by the two methods are consistent, namely, the coal pore structure heterogeneity of Xinzhouyao coal is stronger than that of Tangshan coal. In quantitative evaluation of coal pore structure anisotropy by synchrotron radiation nano-CT, considering the correspondence between pore structure anisotropy and permeability anisotropy, the quantitative evaluation of pore structure anisotropy was realized by calculating the permeability tensor of the pore structure by LBM method, and the pore structure anisotropy was quantitatively evaluated by the eigenvalues and eigenvectors of the permeability tensor. The proposed method for quantitatively evaluating coal pore structure anisotropy was verified by the pore structure geometrical morphology.

Nomenclature

a—distance from sample to receiver, m;

A—fitted parameter, dimensionless;

B—fitted parameter, dimensionless;

c—velocity of light, 299,792,458 m/s;

c_s—velocity of sound, 346 m/s;

d—slope, dimensionless;

D—pore diameter, nm;

D₁—surface fractal dimension, dimensionless;

D₂—mass or pore fractal dimension, dimensionless;

e—electron charge, 1.6×10^-19 C;

e_i—direction vector;

E₁, E₂, E₃—eigenvector, dimensionless;

f_i—fluid particle distribution function;

f_eq—section equilibrium distribution function;

H—relative standard deviation of each sub-block’s porosity, dimensionless;

HV—heterogeneity value of three-dimensional coal pore structure, dimensionless;

i_e—constant, 7.9×10^-20 cm²;

I_e—scattering intensity of a single electron to photon;

I_e(θ)—scattering intensity of a single electron to photon in different directions;

I(q)—scattering intensity;

I₀—incident X-ray flux, phs/(mm²·s);

k₁, k₂, k₃—eigenvalue of permeability tensor, μm²;

k_ij—permeability tensor, μm²;

m—mass of the electron, 9.109 56×10^-31 kg;

M—the number of non-interference particle systems, pcs;

n—the number of electrons in a particle, pcs;

N—the number of sub-blocks, pcs;

p—fluid pressure, Pa;

q—scattering vector, nm^-1;

r—grain radius, nm;

R—particle radius, nm;

R_g—particle radius of gyration, nm;

s—the number of sections that coal sample ROI is divided into in (X, Y, Z) direction, dimensionless;

t—time, s;

Δt—time increment, s;

2θ—scattering angle, rad;

v—macro velocity, m/s;

$v{}_{i}(x)$—velocity distribution function;

V—irradiation volume of SAXS, mm³;

V_p—pore volume, cm³/g;

V_Ω—volume of integral domain, m³;

w_i—weight coefficient, dimensionless;

x—position of fluid granule, m;

α—constant, dimensionless;

γ₀—correlation function;

λ—wavelength, nm;

μ_v—kinematic viscosity of fluid, m²/s;

ρ—macro density, kg³/m³;

ρ_m—electron density of matrix, eA^-3;

ρ_p—electron density of micropore, eA^-3;

τ—relaxation time, s;

υ—dynamic viscosity, Pa·s;

ϕ_s—porosity of sample, %;

ϕ_i—porosity of the i^th sub-block, %;

ϕ—porosity of the whole coal sample, %;

Ω—integral domain;

_x—gradient operator in direction x, Pa/m.