Pore structure and its impact on susceptibility to coal spontaneous combustion based on multiscale and multifractal analysis

The relationship between the properties of coal and its tendency to spontaneous combustion is critical for the environment, safety concerns, and economy. In this study, to eliminate the complex influence of moisture; the samples having similar moisture content were selected from Shanxi and Henan provinces. The chemical properties, physical properties, and tendency of coal samples to spontaneous combustion were characterized based on the conventional analysis, mercury intrusion porosimetry, fractal dimensions, and crossing point temperature (CPT). The results confirmed that the coal rank, volatile matter, oxygen contents, and fixed carbon content had a good linear relationship with the CPT. The relationship between the ash content and CPT presented a “U-shaped” non-linear correlation. For the pore size distribution, the total pore volume also possessed a linear positive correlation with the CPT. The fractal curves could be distinctly divided into two stages: low-pressure (<20 MPa) and high-pressure (>20 MPa), from which the fractal dimensions were obtained using the Sponge and Sierpinski models. The relationship between the fractal dimensions (Ds1, Ds2, and Dg1) and CPT could be divided into two distinct stages: a decrease in the CPT with increasing fractal dimensions (2.6–2.85), and then an in increase in the CPT. CPT decreased with increasing parameters of D1, D2, H, and D10, and it gradually increased with increasing D-10-D10, D-10-D0, and D0-D10. The above characteristics are important to comprehensively and systematically reveal the mechanism of spontaneous combustion.

Samples and experiments. Seven coals were collected from Ningwu, Dongqu, Guandi, Longle, Fenghuangshan, and Pingdingshan collieries in Shanxi and Henan provinces. The coal samples were crushed to −200 meshes (~0.074 mm) and vacuum-dried. The ultimate analysis and proximate analysis were conducted in accordance with the Chinese national testing standards (GB/T 212-2008 and GB/T 31391-2015). Vitrinite random reflectance (%Rr) were measured for all the coal samples on the same polished sections using a Leitz MPV-3 photometer microscope, following conventional methods in accordance with the Chinese national standard (GB/T 6948-1998).
The CPT measurements were conducted at the China University of Mining and Technology. The details of the experiment are reported in our previous work 26 . The samples (mass 50 g with a particle size of ~0.18-0.43 mm) were exposed to a dry air flow of 50 mL/min within the reactor with a temperature ramp rate of 1 °C/min with coal and oven temperatures being recorded. When the coal temperature is equal to the oven temperature, the sample is at the CPT 21,52 .
The coal samples were prepared by a vacuum drying for 12 h at 70-80 °C. High pressure mercury injection (HPMI) experiment was performed for the samples using an Auto Pore IV 9510 HPMI instrument at the China University of Mining and Technology. The mercury injection pressure ranged from 0.90 to 4.0 × 10 4 PSIA. The pore diameters obtained were from 5.35 to 2.28 × 10 5 nm.
Fractal dimensions. Sierpinski model. Based on the Sierpinski model 53 , the fractal dimensions (D s ) can be calculated by the following equation: where V is the mercury injection amount at P in mL, D is the volume fractal dimension, P t is the mercury inlet pressure in MPa, and α is a constant. If the slope of the ln(P − P t ) vs. ln (V) curve is K, then D s is 3 − K.
Sponge model. Based on the Menegr model 53 , the fractal dimensions (Dg) can be calculated by the following equation: where V P(r) is the cumulative injection volume at a given pressure P(r) and α is a constant. Therefore, the pore fractal dimension, D g , can be obtained by: Multifractals. Multifractal analysis is used to measure the statistic Hg pore size distributions. The pore diameter interval (I) ranging from 0.006 nm to initial diameter (responding to the least pressure) was selected to generate a box (N(ε) = 2 k , ε = L × 2 k ) by dyadic partitions in k stages (k = 1, 2, 3, …), where L is the length of the support 45,48,50,51 . The probability, P j (ε), of the pore size can be calculated as 48 , is the volume of a box (j=1, 2, 3…), and N t is the total volume of the system. The probability for each box of size ε unit can be calculated as 48 , www.nature.com/scientificreports www.nature.com/scientificreports/ aj where α j is the coarse Hölder or singularity exponent for the boxes, which theoretically represents how the singularities of a system tend to infinity in the limit ε → 0. The α exponent, N α , was used to evaluate the number of boxes, as follows: where the set of f(α) values represents the spectrum of fractal dimensions that characterize the abundance of the set of points with singularity α. f(α) can be calculated as 48 : where μ j (q, ε) and P j (ε) are the normalized measures, defined as 48 : where χ(q, ε) can be calculated as: where τ(q), qth mass exponent, can be defined as 48 , where D q called the generalized fractal dimensions or Rényi dimensions can be calculated as 48 ,

Results and data analyses
Conventional characteristics. The vitrinite random reflectance (Rr, %) of the coal samples was ranged from 0.58% to 3.43%, corresponding to medium-rank coal (bituminous coals A, B, C, and D) to high-rank coal (anthracite coals B and C) (ISO 11760, 2005) 54 . Proximate analysis showed that the volatility of the coals varied from 5.95% to 43.45%. The moisture content was similar in different coal ranks. The fixed carbon ranged from 27.83% to 80.88%. The total sulfur content changed from 0.42% to 9.36%. The ash yields also had a wide range, varying from 5.47% to 49.09% (Table 1).
Macropores structure from HPMI experiments. Pore structure distribution. Table 2 lists the pore parameters obtained from the HPMI. In this study, the classification standards defined by Yao et al. 28 and the sharpness of the curve were used. The following three ranges are present: V 1 < 100 nm, 100 < V 2 < 1000 nm, and V 3 > 10000 nm. Samples S 2 and S 1 have the largest and smallest cumulative pore volume in V 1 , respectively. Cumulative pore volumes V 2 have a relatively lower distribution than V 1 or V 3 , and among the samples, sample S 3 has the highest distribution. The coal rank ranges from bituminous A to anthracite B when the cumulative pore volume, V 2 , is the same (0.002 cm 3 /g). Samples S 1 and S 3 have the smallest and largest cumulative pore volumes V 3 , respectively, which are in response to the total pore volume. The mercury injection and withdraw curves as well as the pore size distribution are shown in Fig. 1a,b, respectively. For the high-rank coals, the mercury intrusion and extrusion curves (Fig. 2a) display a similar trend (parallel type), indicating the dominance of the parallel plate pores and a good connectivity for gas diffusion. The shape increases in case of mercury intrusion at low pressures reveal a high proportion of V 3 , and the slightly straight lines reveal a poorly developed V 1 , which are in agreement with pore volume distribution. For the medium-rank coals, (2020) 10:7125 | https://doi.org/10.1038/s41598-020-63715-z www.nature.com/scientificreports www.nature.com/scientificreports/ there is larger space between the mercury intrusion and extrusion curves (hysteresis loop) (Fig. 2a) than between those of the high-rank coals (tip-edge type), indicating a larger V 1 , small V 2 and V 3 , and poorer pore connectivity than that in the high-rank coals. The larger hysteresis loop suggests a significant existence of the semi-closed pores. The pore size distribution curves (Fig. 2b) remarkably change with the increase in the coal rank.
Fractal dimensions by Sierpinski model. The fractal curves based on the Sierpinski model are presented in Fig. 3. Fractal dimension D s1 and D s2 are obtained in low-pressure (<20 MPa, responding to the see page pores, >100 nm) and high-pressure (<20 MPa, responding to adsorption pores, <100 nm) stages. The correlation index of D s1 ranges from 0.30 to 0.99. Medium-rank coals S 1 , S 2 and S 3 and high-rank coal S 5 present a better correlation. However, the correlation index of D s2 had a high correlation index (0.80-0.995). The values of D s1 and D s2 ranges from 2 to 3, indicating their power law relationship with the fractal pore surface. The values of D s1 and D s2 are 2.64-2.98 (2.90 in average) and 2.80-2.98 (2.88 in average), respectively.
Fractal dimensions by sponge model. The fractal curves are divided into two stages based on the classic geometry model (sponge model) (Fig. 4), and thus two fractal dimensions (D g1 , low-pressure and D g2 , high-pressure) are obtained using the mercury intrusion data. D g1 exhibits a good linear relationship (correlation index R 2 ,  Table 2. Results of the HPMI and pore distribution for different coal ranks. Note: V 1 , V 2 , and V 3 are the cumulative pore volumes of <100 nm, 100-10000 nm, and >10000 nm respectively, and www.nature.com/scientificreports www.nature.com/scientificreports/ 0.82-0.96), whereas D g2 has a wide range of R 2 (0.02-0.91). The values of D g1 are widely distributed (2.04-3.14, 2.67 in average), indicating significant differences in the discontinuities and roughness of different coal ranks. However, the value of D g2 are 2.77-3.93 (3.56 in average). Most of the values of D g2 are close to 3, indicating that the surface is extremely rough and the pore structure is irregular 55 . The values of D g1 in S 1 and S 3 are >3, and all the values of D g2 are more than 3, except for sample S 1 . For the fractal dimensions of 3, numerous explanations have been provided in previous research 28,44 .
Multifractal analysis. The spectrum curves of logχ(q, ε) versus logε show a linear relationship (correlation index R 2 , 0.78-1), indicating a multifractal distribution of the pore sizes 56 (Fig. 5). The spectrum curves of the generalized dimensions, D(q), versus q present a sigma-shaped curve and follow a monotone decreasing function of q (Fig. 6a). The characteristic parameters of D(q), dimensions D 0 , D 1 , and D 2 , Hurst exponent H (2 H =D 2 +1),

Effect of evolution of coal petrology on spontaneous combustion. Relationship between Rr and
CPT and pore structure. Coal rank has a significant influence on the propensity of coal to spontaneous combustion (Fig. 8a). A good linear relationship (R 2 =0.74) is exhibited between the coal rank and the CPT. The CPT increases with increasing coal rank, which is in agreement with previous studies [58][59][60][61] . However, sample S 6 exhibits    Table 3. Multifractal parameters from the generalized dimension spectrum. D 0 , the capacity dimension; D 1 , the entropy dimension; D 2 , the correlation dimension; H, Hurst exponent; D 10 and D −10 are the generalized dimensions responding to q = 10 and q = −10, respectively.
Scientific RepoRtS | (2020) 10:7125 | https://doi.org/10.1038/s41598-020-63715-z www.nature.com/scientificreports www.nature.com/scientificreports/ a deviation compared to the other samples. If this sample data are removed, a better linear (R 2 = 0.89) is obtained, as shown in Fig. 8a. For the high-rank sample S 6 , the low CPT may be attributed to its different chemical (low fixed carbon) and physical (low total pore volume) structure, which is in agreement with previous studies 11,12 . Coal rank is an important index of coalification, influencing the structures of coal pores and fractures. When the vitrinite random reflectance ranges from 0.58 to 2.13% (Fig. 8b), the mean volume increases with the coal rank, but the mean surface area decreases. When the vitrinite random reflectance ranges from 2.13 to 3.43% (Fig. 8c), the mean volume decreases with coal rank, but the mean surface area increases with coal rank. When the coalification ranges from 0.5 to 2.1%, the aromatic structures including non-protonated aromatic carbons (fa N ), nuclear magneton resonance (NMR) aromaticity (fa'), and aromatic carbon ratio (fa) increase linearly, whereas the aliphatic structures decreases linearly 62 . Because the volume is mainly affected by the aliphatic parts of the chemical structure, it may be the cause of the displayed mean volume and surface area trends 63,64 .

Relationship between coal composition and CPT.
There exists a non-linear relationship between the ash content and the CPT, which shows an inverted 'U-shape' (Fig. 9a). The CPT is a weakly correlated to the total sulfur content (Fig. 9b). The coal components (volatile matter, oxygen contents, and fixed carbon contents) display good relationships with the CPT (Fig. 9c-e), which is in agreement with Zonguldak coals 65 . The CPT decreases with increasing volatile matter, illustrating that low volatile matter is prone to spontaneous combustion. It may be that low volatile matter content can increase the difficulty of ignition and result in an unstable combustion flame 66,67 . The relationship between the oxygen content and the CPT also shows the same trend as the volatile matter, indicating that coals with high oxygen content have a high tendency to chemically bind moisture, thereby rendering the surface, highly susceptible to autogenous heating 68 . The CPT increases linearly with the fixed carbon content, indicating that less fixed carbon content is prone to spontaneous combustion. This is because a small amount of fixed carbon in coal requires a low activation energy to initiate combustion 69 . Relationship between pore structure and CPT. Distribution of pore structure and CPT. It is necessary to discuss the relationship between the pore size and the CPT. Fig. 10a shows a weak negative linear correlation between the cumulative pore volume V 1 and the CPT, which shows an inverted "U-shape". Cumulative pore volume V 2 shows a "U-shape" (Fig. 10b). The relationship between cumulative pore volume V 3 and the CPT exhibit a linear positive correlation (Fig. 10c). Moreover, the total pore volume also has a good linear positive correlation with the CPT (R 2 up to 0.71), indicating that pore sizes of more than 10000 nm (V 3 ) play a main role in coal spontaneous combustion (Fig. 10d). Specifically, the pore structure is a dominant factor causing the coal spontaneous combustion, particularly in pore sizes more than 10000 nm.
Relationship between coal rank, pore volume, and multifractal parameters. The entropy dimension (D 1 ) reveals the concentration degree of the porosity distribution 48 . The values of D 1 are less than or equal to D 0 . When D 1 is close to D 0 , the porosity is evenly distributed. Otherwise, most particles are concentrated in a small area and appear as a high peak on the graph 70 . Among of all sample S 1 has the highest homogeneous pore size distribution. D 1 decreases with increasing coal rank (Fig. 11a), indicating an increase in the heterogeneity. However, the fitting results do not a high correlation. www.nature.com/scientificreports www.nature.com/scientificreports/ This may be owing to joint action factors such as maceral content, ash content, volatile matter, and tectonic deformations. The relationship between cumulative pore volume V 1 and D 1 is not clear, indicating that V 1 (adsorption-pores) may have little influence on the entropy dimension (D 1 ) (Fig. 11b). The D 1 increases with increasing cumulative pore volume V 2 (Fig. 11c) and decreases with increasing cumulative pore volume V 3 (Fig. 11d), indicating that seepage-pores have an important influence on the entropy dimension (D 1 ).
The Hurst exponent (H) is used to quantify the degree of correlation on the logarithmic scale 71 . If H > 0.5, the increments are correlated 71 . The Hurst exponent of all the samples exceeds more than 0.5 (H, 0.57-0.97), indicating the increments are correlated ( Table 3). The Hurst exponents of samples S 1 and S 3 are 0.97 and 0.90, respectively, which two data are close to the 1, reflecting the presence of strong persistence or positive autocorrelations 49,71 . The Hurst exponent has the same characteristics as entropy dimension (Fig. 12).
The width of D(q) reflects the heterogeneity in the porosity distribution 45,51 . Sample S 7 has the highest D −10 − D 10 (the widest spectrum), indicating the highest heterogeneity over the entire pore size range among all the coal samples 45,51 . In contrast, sample S 1 has the lowest D −10 -D 10 (the narrowest spectrum), reflecting the lowest heterogeneity in the porosity distribution over the entire pore size range 45,51 . Samples S 2 and S 6 have wider right side D(q) spectra than the left side D(q) spectra, indicating high dominance of the high porosity concentrations 45 . However, samples S 1 , S 4 , S 5 , and S 7 have wider left side D(q) spectra than right side D(q) spectra, indicating a small porosity concentration 45 . The widths of the D(q) (D −10 -D 10 , D 0 -D 10 , and D −10 -D 0 ) spectra increase with increasing vitrinite random reflectance (Rr, %) (Fig. 13a). The relationship between cumulative pore volume V 1 and width of the D(q) spectra is not clear, indicating that V 1 (adsorption-pores) may have little influence on the width of the D(q) spectra (Fig. 13b). The width of the D (q) spectra logarithmically decreases with increasing cumulative pore volume V 2 (Fig. 13c) and logarithmically increases with increasing cumulative pore volume V 3 (Fig. 13d). The above indicates that seepage-pores have an important influence on the heterogeneity in the porosity distribution.
Fractal dimensions and CPT. 1) Multiscale fractal dimensions and CPT. The relationships between the fractal dimensions of the Sierpinski model (D s1 and D s2 ) and the CPT are shown in Fig. 11. The relationship between D s1 and the CPT can be divided into two distinct stages (Fig. 14a). In the first stage, the CPT first decreases,  www.nature.com/scientificreports www.nature.com/scientificreports/ and then increases with increasing D s1 (when D s1 > 2.8). With increasing D s2 , the CPT decreases first (from 2.75 to 2.85), and then increases (Fig. 14b). Fractal dimensions D g1 of samples S 1 and S 3 are >3; however, most of the fractal dimensions, D g2 , are >3, except of sample S 1 . Fractal dimensions D g1 and D g2 are >3, which are not suitable to characterize the pore heterogeneity 72,73 . To prevent the interference of abnormal points, the data for fractal dimensions >3 were removed 37 . The relationship between D g1 and the CPT (without samples S 1 and S 2 )  www.nature.com/scientificreports www.nature.com/scientificreports/ displays a similar trend as that between Ds and the CPT obtained from the Sierpinski model (Fig. 14c). When D g1 ranges from 2 to 2.6, the CPT decreased significantly. When D g1 becomes larger than 2.6, the CPT increases with increasing D g1 . The above results demonstrate that the heterogeneities obtained from the Sierpinski and Sponge  www.nature.com/scientificreports www.nature.com/scientificreports/ models do not present linear relationship with the tendency of coal to spontaneous combustion. However, a high heterogeneity (fractal dimensions >2.8) is associated with a low tendency of spontaneous combustion. 2) Multifractal fractal dimensions and CPT. Fig. 15 displays the correlation between the multifractal fractal dimensions (D 1 , D 2 , H, D 10 , D− 10 , D −10 -D 10 , D −10 -D 0 , D 0 -D 10 ) and the CPT. It can be found that the CPT decreases with increasing parameters D 1 , D 2 , H, and D 10 , suggesting that a high degree of the distribution of the porosity quantifies the degree of correlation on the logarithmic scale. Further, the heterogeneity in the porosity distribution decreases the tendency of coal spontaneous combustion. The correlation between D −10 , D −10 -D 10 , D −10 -D 0 , and D 0 -D 10 and the CPT is clear in that the CPT gradually increases with increasing D −10 -D 10 , D −10 -D 0 , and D 0 -D 10 . This indicates that the complexity of the local characterization pore structure decrease the spontaneous combustion propensity.

conclusions
The conventional analysis and CPT measurements were conducted to obtain the properties of coal petrology and spontaneous combustion. Coal rank, volatile matter, oxygen content, and fixed carbon content were found to play important roles in spontaneous combustion.
The pore structure properties obtained from HPMI provided a direct measurement of coal physical properties. The cumulative pore volume of V 3 (>10000 nm) and total pore volume have positive correlation with CPT.  www.nature.com/scientificreports www.nature.com/scientificreports/ Multiscale and multifractal analyses were conducted to evaluate the pore size distribution. From the multiscale analysis, the relationship between the fractal dimensions (D s1 , D s2 , and D g1 ) and the CPT basically displayed a "U-shaped" tendency, with the minimum occurring at 2.6-2.85. Based on the multifractal analysis, a high degree of porosity distribution, quantified the degree of correlation on the logarithmic scale. Furthermore, the heterogeneity in the porosity distribution decreased the tendency of coal spontaneous combustion; therefore, a more complex local characterization pore structure lowered the spontaneous combustion propensity.