Fractal Characteristics of Micro- and Mesopores in the Longmaxi Shale

: To better understand the variability and heterogeneity of pore size distributions (PSDs) in the Longmaxi Shale, twelve shale samples were collected from the Xiaoxi and Fendong section, Sichuan Province, South China. Multifractal analysis was employed to study PSDs of mesopores (2–50 nm) and micropores ( < 2 nm) based on low-pressure N 2 / CO 2 adsorption (LP-N 2 / CO 2 GA). The results show that the PSDs of mesopores and micropores exhibit a multifractal behavior. The multifractal parameters can be divided into the parameters of heterogeneity (D − 10 –D 10 , D 0 –D 10 and D − 10 –D 0 ) and the parameters of singularity (D 1 and H). For both the mesopores and micropores, decreasing the singularity of the pore size distribution contributes to larger heterogeneous parameters. However, micropores and mesopores also vary widely in terms of the pore heterogeneity and its controlling factors. Shale with a higher total organic carbon (TOC) content may have a larger volume of micropores and more heterogeneous mesopores. Rough surface and less concentrated pore size distribution hinder the transport of adsorbent in mesopores. The transport properties of micropores are not a ﬀ ected by the pore fractal dimension. Investigation, Y.Z. Resources, Y.Z. X.W.; X.W. and Y.Z.; Writing-Original Preparation, X.W.; Writing–Review Y.Z., X.W. and Y.W.; Visualization, Y.Z. and X.W.; Supervision, Y.Z.; Project Administration, Y.Z.; Funding Acquisition, Y.Z. All authors

It is noteworthy that the multifractal theory expands the application of fractal geometry and has been widely used in various fields in recent years [52][53][54][55]. In general, fractals are mainly used to describe irregular geometries or geometric sets, and the spatial distribution of measures can be quantitatively represented by multifractal research [52]. It was found that a multifractal approach has been successfully applied to parameterize the spatial heterogeneity of porous materials [55]. Several multifractal studies were conducted based on two-dimensional image analysis, including X-ray computed tomography (CT) data [54], optical microscopy [56] and environmental scanning electron microscopy (ESEM) images [57]. Recently, multifractal analysis of PSDs determined by Hg injection datasets has been performed [55]. However, they are less frequent with the multifractal analysis of marine shales.
In this study, we mainly carried out fractal analyses on the Lower Silurian Longmaxi shale based on the data of LP-N 2 /CO 2 GA experiments. The relationships between the fractal parameters and adsorbent transport rate are also discussed.

Materials and Methods
For this investigation, twelve shale samples were collected from the Longmaxi Formation in the Xiaoxi and Fendong section, Sichuan Province, South China. The total organic carbon (TOC) content analysis and X-ray diffraction (XRD) were applied to analyze the material composition. In addition, a low-pressure N 2 /CO 2 adsorption experiment (LP-N 2 /CO 2 GA) was employed to analyze the pore structure parameters, including the BET-specific surface area (SSA), micropore volume (PV1), mesopore volume (PV2) and pore size distribution (PSD). Furthermore, multifractal analysis was performed based on the PSD data measured from LP-N 2 /CO 2 GA experiments. Finally, the Frenkel-Halsey-Hill (FHH) and volume-surface area (V-S) fractal models were used to study their relationship with multifractals.

Experiments
The total organic carbon content (TOC) analysis was conducted by using a multi EA 4000 elemental analyzer (Analytik Jena AG, Jena, Germany). The highest temperature of this instrument can reach 1500 • C, and accuracy is ±0.01 ppm. X-ray diffraction (XRD) analysis was performed using X'Pert3 Powder. Both analyses were conducted at the Jiangsu Provincial Mineral Design Institute, China.
The LP-N 2 /CO 2 GA experiment was conducted using an Autosorb-IQ-MP apparatus at the China University of Mining and Technology. Samples were degassed and dried before the experiments. The LP-N 2 GA experiment was performed at temperature of 77.35 K, and the LP-CO 2 GA experiment was conducted at temperature of 273 K. The Brunauer-Emmett-Teller (BET) multilayer adsorption equation was applied to obtain the specific surface area [58]. PSD were calculated using the density functional theory [59].

Multifractal Analysis
A number of scholars have carried out multifractal analysis based on pore size distribution data of mercury intrusion experiments, and the generalized dimension spectrum is used to characterize the heterogeneity of pore structures [60]. In this study, the generalized dimension spectrum was calculated based on pore size distribution data of N 2 and CO 2 adsorption experiments. The calculation steps are summarized as follows [55]: The key to the multifractal analysis of the pore size distribution is to define a pore volume probability on multiple sizes scales. First, the aperture range is taken as interval J. According to the dichotomy principle, the interval J of length r is divided into N(r) boxes with scale r, so that the smallest Energies 2020, 13, 1349 3 of 21 subinterval contains the measured value. The probability distribution ρ i (r) of the pore volume in each box is defined as: where M i (r) is the total amount of the study volume in lattice i, and M is the total amount of the study volume in the entire study space. Then, the assignment function χ q (r) can be defined as: where N(r) is the total number of lattice units with side length r, and q is the order of the distribution function (−∞ ≤ q ≤ ∞).
If q > 0, the larger ρ i (r) interval will have a greater contribution to χ q (r), which can reflect the fractal characteristics of the hole-dense region. When q < 0, the region with a small ρ i (r) value will have a large contribution to χ q (r), which can reflect the fractal characteristics of the sparse region. Thus, a single fractal can be extended to a variety of singular degrees of fractal so that the internal structure of the fractal can be fully presented. If ρ i (r) obeys the multifractal pattern, the distribution function has a simple power-law relationship to the grid cell size r: where τ(q) is the power exponent of the q-order moment. If the studied measure satisfies the multifractal pattern, when a q value is given, a line between χ q (r) and r will be formed on a double logarithmic plot. The slope of each line gives a τ(q) value. When q is not equal to 1, D q can be obtained from the power exponent τ(q) of the q-order moment: When q = 1, D 1 can be expressed as: The Dq values correspond to the information dimension D 1 and the associated dimension D 2 respectively, when the q value is equal to 1 or 2. The q-D(q) curve constitutes the generalized dimension spectrum of the pore size distribution. The generalized dimension spectrum has five characteristic parameters (D 1 , H, D −10 -D 0 , D 0 -D 10 and D −10 -D 10 ), which can quantitatively characterize the heterogeneity of the pore size distribution from different angles. The information dimension D 1 is a measure of the uniformity of the pore size distribution. The larger the D 1 , the greater the uniformity of the pore size distribution. The D 1 values of mesopores and micropores are abbreviated as D 1N and D 1C . Most of the pores are distributed in a number of pore ranges [60]. H is the Hurst index, which is calculated as: H can be used as the characterization parameter of porosity autocorrelation, and its value range is within the interval (0.5,1). When H is closer to 1, it indicates that the autocorrelation of the pore size distribution is stronger [52,53]. The H values of mesopores and micropores are abbreviated as H N and H C . D −10 -D 0 , D 0 -D 10 and D −10 -D 10 represent the left branch, the right branch and the full spectrum width of the q-D(q) curve, respectively. For the convenience of expression, the above three Energies 2020, 13, 1349 4 of 21 spectral width parameters were expressed as D NN , D PN and D TN for mesopores and D NC , D PC and D TC for micropores, respectively. In general, pore heterogeneity increases with wider multifractal spectrum [52,53,55,60].

FHH and V-S Fractal Model
The FHH fractal model can be expressed by [61][62][63]: where V is the nitrogen adsorption volume, V 0 represent the monolayer adsorption volume, p 0 is the saturated vapor pressure and A and C are fitting coefficients. The FHH fractal dimension is A + 3.
The V-S model can be expressed by [64,65]: where V is adsorption volume, cm 3 /g, S is the cumulative specific surface area, m 2 /g, and D C is the V-S fractal dimension of the micropores.

TOC and XRD Analysis
The TOC content and mineral composition are presented in Table 1. The TOC content of the two sections ranges from 1.2% to 9.9% (average = 4.4%), and the dominant mineralogical composition are quartz, clay and calcite. Among these, quartz, ranging from 32% to 83% (average = 53%), displays the highest concentration. Clay ranges from 10% to 42% (average = 21%) and calcite ranges from 0% to 26% (average = 15%).

N 2 and CO 2 Adsorption
Following the Brunauer, Deming, Deming and Teller classification [27], the adsorption isotherms of N 2 and CO 2 are type II and type I, respectively (Figures 1 and 2). Samples with a high TOC content tended to have higher N 2 and CO 2 adsorption capacities. The pore volume of mesopores (PV1) ranged from 0.006 to 0.022 mL/g (average = 0.015 mL/g). Meanwhile, the micropores had a smaller pore volume (PV2) relative to the mesopores, with a range from 0.0016 to 0.0085 mL/g (average = 0.0057 mL/g). Compared to the existing data from the same region [66], the relative error of these two parameters was found to be within 25% ( Figure 3). PV1 had no correlation with the shale composition. However, composition. However, there was a weak positive correlation between PV2 and the content of TOC (R 2 = 0.3) and a certain negative correlation with the clay content (R 2 = 0.4).
SSA ranged from 15.9 to 26.9 m 2 /g (average = 20.1 m 2 /g), and it was substantially consistent with the results of Yang et al. [66]. Meanwhile, the SSA had a positive correlation with the TOC content (R 2 = 0.5) and quartz content (R 2 = 0.3). The pore volume distributions of the mesopores and macropores are illustrated in Figure 4, suggesting a unimodal PSD in the mesopore range and a multimodal PSD in the micropore range. In general, most samples had a broad peak between 10 and 100 nm and a sharp peak between 0.4 and 0.7 nm.      (Yang et.al, 2015) PV1 (Yang et.al, 2015) Average value of PV2 Pore volume (ml/g)

PV1 PV2
Average value of PV1 PV2 (Yang et.al, 2015) a X-4 X-5 X-7 X-9 X-10 X-14 X-15 F-4 F-6 F-8 F-14 F-15 0 SSA ranged from 15.9 to 26.9 m 2 /g (average = 20.1 m 2 /g), and it was substantially consistent with the results of Yang et al. [66]. Meanwhile, the SSA had a positive correlation with the TOC content (R 2 = 0.5) and quartz content (R 2 = 0.3). The pore volume distributions of the mesopores and macropores are illustrated in Figure 4, suggesting a unimodal PSD in the mesopore range and a multimodal PSD in the micropore range. In general, most samples had a broad peak between 10 and 100 nm and a sharp peak between 0.4 and 0.7 nm.

Multifractal Parameters
The plot of log[χ(q, ε)] versus log(ε) is often used to determine whether the observed data have multifractal features [67]. Linear relationships could be observed between log[χ(q, ε)] and log(ε) in all the samples ( Figures 5 and 6), suggesting that the PSDs of micropores and mesopores in shale had multifractal characteristics. In general, the Dq spectra displayed a monotonically decreasing function of q with an anti-S curve (Figures 7 and 8). Although the shapes of the Dq spectra were roughly

Multifractal Parameters
The plot of log[χ(q, ε)] versus log(ε) is often used to determine whether the observed data have multifractal features [67]. Linear relationships could be observed between log[χ(q, ε)] and log(ε) in all the samples ( Figures 5 and 6), suggesting that the PSDs of micropores and mesopores in shale had multifractal characteristics. In general, the Dq spectra displayed a monotonically decreasing function of q with an anti-S curve (Figures 7 and 8). Although the shapes of the Dq spectra were roughly similar, there were differences in the parameters of the Dq spectra (Table 2). Thus, it was necessary to analyze the correlation of each characteristic parameter. The correlation coefficients of the shale material composition data, pore parameters and fractal parameters are presented in Figure 9. A better correlation was found between the multifractal parameters of mesopores. Specifically, there was a positive correlation between D 1N and D HN , and these two parameters were negatively correlated with D NN , D PN and D TN . However, the correlation between multifractal parameters of the micropores was more complex compared to the mesopores. Similar to the mesopores, D 1C was positively correlated with H C , and D NC was also positively related to D TC . However, unlike the mesopores, D PC had no correlation with D TC .

FHH and V-S Fractal Dimension
The fairly well-fitting results of formula 7 (with correlation coefficients of R 2 > 0.97, see Tables 3  and 4 and Figures 10 and 11) indicated that the mesopores in shale had FHH fractal characteristics.

FHH and V-S Fractal Dimension
The fairly well-fitting results of formula 7 (with correlation coefficients of R 2 > 0.97, see Tables 3  and 4 and Figures 10 and 11) indicated that the mesopores in shale had FHH fractal characteristics.

FHH and V-S Fractal Dimension
The fairly well-fitting results of formula 7 (with correlation coefficients of R 2 > 0.97, see Tables 3  and 4 and Figures 10 and 11) indicated that the mesopores in shale had FHH fractal characteristics. D N1 ranged from 2.66 to 2.77, with a mean of 2.72. D N2 was lower than that of D N1 and ranged from 2.62 to 2.72, with an average of 2.67. There was a positive correlation between D N1 and D N2 (R 2 = 0.5). These results were consistent with previous studies [68]. It is noteworthy that both D N1 and D N2 were negatively correlated with D 1N and H N but were positively correlated with D TN and D PN (Figure 9).               The V-S fractal dimension of micropores (D C ) varied from 2.23 to 2.62, with a mean of 2.72. D C was negatively correlated with D 1C and H C (Figure 9). In addition, there was no significant correlation between D C and the other multifractal parameters of micropores. It is noteworthy that D C had a weak positive correlation with D N1 (Figure 9).

Pore Structure Parameters and Their Controlling Factors
Previous studies have found that the material composition of shale, including the organic matter and inorganic minerals, is an important factor in controlling the pore structure parameters [69,70]. The difference between the PV1-TOC and PV2-TOC covariance (Figure 12a) was consistent with those described by Wang et al. [66]. Some researchers have found that mesopores consisted of elliptical organic matter pores as well, as there were quite a few inorganic matter pores with various morphologies based on FE-SEM [71][72][73][74]. It is well documented that considerable micropores are formed within the macromolecular structure of organic matter [75]. This may be the reason why the micropore volume had a notable positive correlation with the TOC content.
was negatively correlated with D1C and HC (Figure 9). In addition, there was no significant correlation between DC and the other multifractal parameters of micropores. It is noteworthy that DC had a weak positive correlation with DN1 (Figure 9).

Pore Structure Parameters and Their Controlling Factors
Previous studies have found that the material composition of shale, including the organic matter and inorganic minerals, is an important factor in controlling the pore structure parameters [69,70]. The difference between the PV1-TOC and PV2-TOC covariance (Figure 12a) was consistent with those described by Wang et al. [66]. Some researchers have found that mesopores consisted of elliptical organic matter pores as well, as there were quite a few inorganic matter pores with various morphologies based on FE-SEM [71][72][73][74]. It is well documented that considerable micropores are formed within the macromolecular structure of organic matter [75]. This may be the reason why the micropore volume had a notable positive correlation with the TOC content.
Overall, the results of this study are consistent with previous research results indicating that organic matter pores had the largest contribution to the specific surface area (Figure 12b) [76,77]. The specific surface area of the mineral pores was smaller than that of the organic pores [78], which may have resulted in the absence of a positive correlation between the mineral composition and SSA. In addition, the weak positive correlation between SSA and PV1 or PV2 reflects that the micropores and mesopores contribute most of the specific surface area in shale. Therefore, organic pores with a pore size of smaller than 50 nm may provide the majority of the specific surface area of the shale.

Multifractal Characteristics of Micropores and Mesopores
The Dq spectrum in the anti-S shaped curve represents a heterogeneous distribution of pore sizes [60]. A larger Dq spectrum width suggests more heterogeneity in the PSDs [52]. The large difference of the multifractal parameters indicates a significant disparity in heterogeneity among the samples. DTN increased with increasing TOC content (Figure 13a). However, there was no correlation between DTC and TOC content (Figure 13b). The thermal evolution degree of the Longmaxi Shale had entered the over-maturation phase, causing the formation of a complex organic pore network [79][80][81][82]. Therefore, shale with a higher TOC content may have more heterogeneous mesopores. In the organic matter with a higher degree of thermal evolution, the molecular structure becomes ordered Overall, the results of this study are consistent with previous research results indicating that organic matter pores had the largest contribution to the specific surface area (Figure 12b) [76,77]. The specific surface area of the mineral pores was smaller than that of the organic pores [78], which may have resulted in the absence of a positive correlation between the mineral composition and SSA. In addition, the weak positive correlation between SSA and PV1 or PV2 reflects that the micropores and mesopores contribute most of the specific surface area in shale. Therefore, organic pores with a pore size of smaller than 50 nm may provide the majority of the specific surface area of the shale.

Multifractal Characteristics of Micropores and Mesopores
The Dq spectrum in the anti-S shaped curve represents a heterogeneous distribution of pore sizes [60]. A larger Dq spectrum width suggests more heterogeneity in the PSDs [52]. The large difference of the multifractal parameters indicates a significant disparity in heterogeneity among the samples. D TN increased with increasing TOC content (Figure 13a). However, there was no correlation between D TC and TOC content (Figure 13b). The thermal evolution degree of the Longmaxi Shale had entered the over-maturation phase, causing the formation of a complex organic pore network [79][80][81][82]. Therefore, shale with a higher TOC content may have more heterogeneous mesopores. In the organic matter with a higher degree of thermal evolution, the molecular structure becomes ordered [83], so the Energies 2020, 13, 1349 13 of 21 microporous structure, related to the macromolecular structure of organic matter, may not become complicated as the organic matter content increases.
Energies 2020, 13, x FOR PEER REVIEW 14 of 22 [83], so the microporous structure, related to the macromolecular structure of organic matter, may not become complicated as the organic matter content increases. The multifractal parameters for q > 0 corresponded to the dominance of a large concentration of the pore volume, and the parameters for q < 0 could be mainly affected by a small concentration of the pore volume [53,60,74]. As mentioned above, the PSDs of mesopores were unimodal. Consequently, the change in DPN may have been due to various distributions of pore sizes larger than 10 nm, and the value of DNN could be assigned to a pore size smaller than 10 nm. However, the PSDs of micropores have a multimodal distribution. Therefore, the change in both DPC and DNC may have been due to a pore volume distribution over multiple ranges of the pore size. Ultimately, the complexity of the multifractal features of micropores increased. It is noteworthy that DNN was larger than DPN, resulting in a notable positive correlation between DNN and DTN. This result is likely because mesopores with pore sizes smaller than 10 nm have a higher degree of heterogeneity. In addition, the TOC content was positively correlated with DNN and had no correlation with DPN. Accordingly, mesopores in the shale sample with a high TOC content showed a heterogeneous structure in the inner distribution of pores smaller than 10 nm. It is noteworthy that there was no correlation between DPC, DNC and DTC of the micropores. Additionally, only the clay mineral content had a weak positive correlation with DNC of the micropores. Considering that the micropores are mostly distributed in the molecular structure, it is presumed that the pores in the molecular structure of the clay had a greater heterogeneity compared to the pores in the molecules of the organic matter.
The information dimension (D1) and the Hurst exponent (H) are also commonly used multifractal parameters [52,60]. D1 is considered a measure of the concentration degree of PSD [52]. The smaller the values of D1, the more highly concentrated in the distribution of the pore volume [60]. H indicates the autocorrelation of the distribution of pore volume [54]. The nearer H approximates one, the stronger the existing autocorrelation PSD [54]. Whether considering micropores or mesopores, D1 had a notable positive correlation with H, suggesting that the more concentrated the pore size distribution, the greater the autocorrelation of the micropores and mesopores (Figure 9c). It is noteworthy that both D1N and HN were negatively associated with DPN, DNN and DTN for mesopores (Figure 14a), suggesting that as the singularity of the pore size distribution increased, the pore volume was more concentrated in a certain pore size range, and the heterogeneity of the pores decreased. However, only DPC had a negative correlation with D1C and HC for the micropores (Figure 9). It is hypothesized that due to the multimodal distribution of the PSD in the micropores, the correlation between the singularity and the heterogeneity was weakened compared with the mesopores. Therefore, we can divide the multifractal parameters into the parameters of heterogeneity (D−10-D10, D0-D10 and D−10-D0) and the parameters of singularity (D1 and The multifractal parameters for q > 0 corresponded to the dominance of a large concentration of the pore volume, and the parameters for q < 0 could be mainly affected by a small concentration of the pore volume [53,60,74]. As mentioned above, the PSDs of mesopores were unimodal. Consequently, the change in D PN may have been due to various distributions of pore sizes larger than 10 nm, and the value of D NN could be assigned to a pore size smaller than 10 nm. However, the PSDs of micropores have a multimodal distribution. Therefore, the change in both D PC and D NC may have been due to a pore volume distribution over multiple ranges of the pore size. Ultimately, the complexity of the multifractal features of micropores increased. It is noteworthy that D NN was larger than D PN , resulting in a notable positive correlation between D NN and D TN . This result is likely because mesopores with pore sizes smaller than 10 nm have a higher degree of heterogeneity. In addition, the TOC content was positively correlated with D NN and had no correlation with D PN . Accordingly, mesopores in the shale sample with a high TOC content showed a heterogeneous structure in the inner distribution of pores smaller than 10 nm. It is noteworthy that there was no correlation between D PC , D NC and D TC of the micropores. Additionally, only the clay mineral content had a weak positive correlation with D NC of the micropores. Considering that the micropores are mostly distributed in the molecular structure, it is presumed that the pores in the molecular structure of the clay had a greater heterogeneity compared to the pores in the molecules of the organic matter. The information dimension (D 1 ) and the Hurst exponent (H) are also commonly used multifractal parameters [52,60]. D 1 is considered a measure of the concentration degree of PSD [52]. The smaller the values of D 1 , the more highly concentrated in the distribution of the pore volume [60]. H indicates the autocorrelation of the distribution of pore volume [54]. The nearer H approximates one, the stronger the existing autocorrelation PSD [54]. Whether considering micropores or mesopores, D 1 had a notable positive correlation with H, suggesting that the more concentrated the pore size distribution, the greater the autocorrelation of the micropores and mesopores (Figure 9c). It is noteworthy that both D 1N and H N were negatively associated with D PN , D NN and D TN for mesopores (Figure 14a), suggesting that as the singularity of the pore size distribution increased, the pore volume was more concentrated in a certain pore size range, and the heterogeneity of the pores decreased. However, only D PC had a negative correlation with D 1C and H C for the micropores (Figure 9). It is hypothesized that due to the multimodal distribution of the PSD in the micropores, the correlation between the singularity and the heterogeneity was weakened compared with the mesopores. Therefore, we can divide the multifractal parameters into the parameters of heterogeneity (D −10 -D 10 , D 0 -D 10 and D −10 -D 0 ) and the parameters of singularity (D 1 and H). The parameters are interrelated and have their own independence, which can be used to characterize the fractal regularity of the pore system [55]. Just as there is no correlation between the mesopore volume and micropore volume, the correlation between the multifractal parameters of the mesopores and micropores is not clear, indicating that there may be a large difference in pore types and heterogeneity of pore structures between the mesopores and micropores.
Energies 2020, 13, x FOR PEER REVIEW 15 of 22 H). The parameters are interrelated and have their own independence, which can be used to characterize the fractal regularity of the pore system [55]. Just as there is no correlation between the mesopore volume and micropore volume, the correlation between the multifractal parameters of the mesopores and micropores is not clear, indicating that there may be a large difference in pore types and heterogeneity of pore structures between the mesopores and micropores.

Association between fractal dimensions
A strongly positive correlation was found between the fraction dimension of mesopores (DN1 and DN2) and the TOC content (Figures 9a and 15a), suggesting that a higher TOC content in shale may lead to a more complicated pore structure and result in greater FHH fractal dimensions. This conclusion is consistent with the recent studies concerning marine gas shale [39] and lacustrine shale [61]. However, there was no obvious relationship between the value of Dc and the shale composition ( Figure 9b). The controlling factors of the micropore volume fractal dimension require further study. There was a close relationship between the multifractal parameters and FHH fractal dimensions for mesopores (Figure 14b). In particular, the FHH fractal dimension was consistent with the spectral width parameter of the multifractal parameters, so they are all measures of pore heterogeneity. The FHH fractal dimensions were in disagreement with D1N and HN, indicating that as the singularity of the pore size distribution increased, the pore volume was more concentrated in a certain pore size

Association between Fractal Dimensions
A strongly positive correlation was found between the fraction dimension of mesopores (D N1 and D N2 ) and the TOC content (Figures 9a and 15a), suggesting that a higher TOC content in shale may lead to a more complicated pore structure and result in greater FHH fractal dimensions. This conclusion is consistent with the recent studies concerning marine gas shale [39] and lacustrine shale [61]. However, there was no obvious relationship between the value of Dc and the shale composition (Figure 9b). The controlling factors of the micropore volume fractal dimension require further study. H). The parameters are interrelated and have their own independence, which can be used to characterize the fractal regularity of the pore system [55]. Just as there is no correlation between the mesopore volume and micropore volume, the correlation between the multifractal parameters of the mesopores and micropores is not clear, indicating that there may be a large difference in pore types and heterogeneity of pore structures between the mesopores and micropores.

Association between fractal dimensions
A strongly positive correlation was found between the fraction dimension of mesopores (DN1 and DN2) and the TOC content (Figures 9a and 15a), suggesting that a higher TOC content in shale may lead to a more complicated pore structure and result in greater FHH fractal dimensions. This conclusion is consistent with the recent studies concerning marine gas shale [39] and lacustrine shale [61]. However, there was no obvious relationship between the value of Dc and the shale composition ( Figure 9b). The controlling factors of the micropore volume fractal dimension require further study. There was a close relationship between the multifractal parameters and FHH fractal dimensions for mesopores (Figure 14b). In particular, the FHH fractal dimension was consistent with the spectral width parameter of the multifractal parameters, so they are all measures of pore heterogeneity. The FHH fractal dimensions were in disagreement with D1N and HN, indicating that as the singularity of the pore size distribution increased, the pore volume was more concentrated in a certain pore size  There was a close relationship between the multifractal parameters and FHH fractal dimensions for mesopores (Figure 14b). In particular, the FHH fractal dimension was consistent with the spectral width parameter of the multifractal parameters, so they are all measures of pore heterogeneity. The FHH fractal dimensions were in disagreement with D 1N and H N , indicating that as the singularity of the pore size distribution increased, the pore volume was more concentrated in a certain pore size range, and the heterogeneity of the pores decreased. For the micropores, D C was not related to the spectral width parameter of the multifractal parameters but was negatively correlated with D 1C and H C (Figure 15b). This result also suggests the opposite of the singularity and heterogeneity of the pore size distribution. The above studies showed that the fractal features of micropores are more complicated. Thus, more research on this topic needs to be conducted about the association between micropores and shale molecular structures.

The Relationship between Fractal Parameters and Shale Transport Properties
A previous study found that the multifractal parameters of tectonic coals are closely related to permeability [55]. A negative correlation between fractal dimension (FHH model) and permeability (measure by pulse-decay) was observed in an earlier study [39]. The relationships between permeability (K) and porosity (ϕ) and fractal dimensions (D) were also acquired based on the Kozeny-Carman equation [84]: where r is the average radius and τ is tortuosity.
Due to the complex mesopore and micropore features of both tectonic coals and shales [23,25,55,70], shale fractal parameters may be related to the shale permeability or the diffusion coefficient. Here, we characterized the transport properties of shale based on the cumulative adsorption and corresponding equilibrium time measured by low-pressure N 2 /CO 2 adsorption. Since the adsorption process is considered transient, the adsorption time is closely related to the transport rate of the adsorbate molecules in the connected pore system. Therefore, we characterized the transport rate of the adsorbent according to the ratio of the adsorption amount to corresponding equilibrium time ( Table 5). The results show that for mesopores, the transport rate has a positive correlation with the information dimension D N1 , and a negative correlation with the FHH fractal dimension (D 1N ) ( Figure 16). Therefore, it is suggested that the adsorbent transport rate is higher with the decrease of the pore surface heterogeneity and the increase in the concentration of the pore size distribution inside the connected mesoporous system. There is no obvious correlation between transport rate and the structure parameters (including fractal dimension) of micropores. Controlling factors for fluid transport inside microporous pores need further study. As connectivity, tortuosity and constrictivity are classical parameters used to characterize the pore morphology, it might be possible to find a causal connection between these parameters to fractal characteristics for deepening the understanding of shale transport properties in future investigations. Table 5. Adsorbent transport rate of Longmaxi shale based on low-pressure N 2 /CO 2 adsorption (LP-N 2 /CO 2 GA).

Sample No.
Transport Rate of N 2 (cm 3 /(g·min)) Transport Rate of CO 2 (cm 3 /(g·min)) distribution inside the connected mesoporous system. There is no obvious correlation between transport rate and the structure parameters (including fractal dimension) of micropores. Controlling factors for fluid transport inside microporous pores need further study. As connectivity, tortuosity and constrictivity are classical parameters used to characterize the pore morphology, it might be possible to find a causal connection between these parameters to fractal characteristics for deepening the understanding of shale transport properties in future investigations. Figure 16. Relationship between the fractal parameter and transport rate of the adsorbate of mesopores.

Conclusions
(1) The PSDs of mesopores are unimodal, whereas micropores have a multimodal distribution. Organic matter pores with a pore size smaller than 50 nm may provide the majority of the specific surface area of a shale and have strong heterogeneity.
(2) The multifractal parameters can be divided into the parameters of heterogeneity (D −10 -D 10 , D 0 -D 10 and D −10 -D 0 ) and the parameters of singularity (D 1 and H). As the singularity of the pore size distribution increases, the pore volume is more concentrated in a certain pore size range, and the heterogeneity of the pores decreases. The roughness of the mesoporous surface has a positive correlation with the heterogeneity of its pore size distribution.
(3) A rough mesoporous surface hinders the transport of adsorbent. Mesopores with more concentrated pore size distribution have a higher adsorbate transport rate. The transport properties of micropores are not affected by the pore fractal dimension. The main controlling factors of the transport properties of micropores need further study.