Fractal Characteristics of Micro- and Mesopores in the Longmaxi Shale

Abstract

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 N₂/CO₂ adsorption (LP-N₂/CO₂GA). 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–D₁₀, D₀–D₁₀ and D₋₁₀–D₀) and the parameters of singularity (D₁ 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 affected by the pore fractal dimension.

1. Introduction

Black shale, as a nonconventional reservoir, has typical low porosity and low permeability characteristics [1,2,3,4,5]. The nanoscale pores in the shale constitute a complex pore network [6,7,8,9]. In particular, the micropores (pore diameter <2 nm) and mesopores (2–50 nm) have the characteristics of a large specific surface area and high degree of heterogeneity, thereby increasing the complexity of the accumulation and migration mechanism of shale gas [10,11,12,13,14]. The difficulty in the characterization of shale reservoirs is the quantitative description of micropore and mesopore structures. A variety of experiments have been used to study the pore structure, including scanning electron microscopy (SEM) [15,16,17], atomic force microscopy (AFM) [18,19], small-angle X-ray (SAXS) [20,21], small-angle/ultra-small-angle neutron scattering (SANS/USANS) [22,23,24,25,26], nuclear magnetic resonance (NMR) [27,28,29,30], high-pressure mercury intrusion (HMIP) [31,32,33,34] and low-temperature liquid nitrogen/carbon dioxide adsorption (LP-N₂/CO₂GA) [23,24,25,31,35]. Among these, LP-N₂/CO₂GA analysis has been proven to be an effective method in characterizing the pore size distribution of micropores and mesopores [23,24,25,31]. Furthermore, nanoscale pores in shale can be regarded as having a complex fractal geometry. Fractal theory provides a new quantitative method for quantitatively characterizing the heterogeneity of nanoscale pores in shale [36,37,38,39,40]. Several methods have been used to measure fractal dimensions, such as image analysis [41,42], the fractal Frenkel–Halsey–Hill (FHH) model [43,44,45], the fractal BET model [46,47,48], small angle X-ray scattering [48] and HMIP [49]. Among these, the fractal FHH model has been widely used to study pore heterogeneity [50,51].

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-N₂/CO₂GA experiments. The relationships between the fractal parameters and adsorbent transport rate are also discussed.

2. 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 N₂/CO₂ adsorption experiment (LP-N₂/CO₂GA) 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-N₂/CO₂GA experiments. Finally, the Frenkel–Halsey–Hill (FHH) and volume-surface area (V-S) fractal models were used to study their relationship with multifractals.

2.1. 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-N₂/CO₂GA 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-N₂GA experiment was performed at temperature of 77.35 K, and the LP-CO₂GA 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].

2.2. 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 N₂ and CO₂ 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 subinterval contains the measured value. The probability distribution ρ_i(r) of the pore volume in each box is defined as:

$${\rho }_i(r)=M_i(r)/M$$

(1)

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:

$${\chi }_q(r)={\displaystyle \sum _{i=1}^{N(r)}{\rho }_i^q}(r)$$

(2)

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:

$${\chi }_q(r)={\displaystyle \sum _{i=1}^{N(r)}{\rho }_i^q}(r)$$

(3)

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:

$$D_q=\frac{\tau (q)}{1-q}$$

(4)

When q = 1, D₁ can be expressed as:

$$D_1=\underset{r\to 0}{\lim }\frac{{\displaystyle \sum _{i=1}^{N(r)}{\rho }_i(r)\log {\rho }_i(r)}}{\log (r)}$$

(5)

The Dq values correspond to the information dimension D₁ and the associated dimension D₂ 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 (D₁, H, D₋₁₀–D₀, D₀–D₁₀ and D₋₁₀–D₁₀), which can quantitatively characterize the heterogeneity of the pore size distribution from different angles. The information dimension D₁ is a measure of the uniformity of the pore size distribution. The larger the D₁, the greater the uniformity of the pore size distribution. The D₁ 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:

$$H=\frac{D_2+1}{2}$$

(6)

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–D₀, D₀–D₁₀ and D₋₁₀–D₁₀ 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 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].

2.3. FHH and V-S Fractal Model

The FHH fractal model can be expressed by [61,62,63]:

$$\ln (V/V_0)=C+A\cdot \ln (\ln (p_0/p))$$

(7)

where V is the nitrogen adsorption volume, V₀ represent the monolayer adsorption volume, p₀ is the saturated vapor pressure and A and C are fitting coefficients. The FHH fractal dimension is A + 3. In this study, the FHH fractal dimension was divided into D_N1 (p₀/p < 0.5) and D_N2 (p₀/p > 0.5).

The V-S model can be expressed by [64,65]:

$$\ln (V)=\frac{3}{D_C}\ln S+k$$

(8)

where V is adsorption volume, cm³/g, S is the cumulative specific surface area, m²/g, and D_C is the V-S fractal dimension of the micropores.

3. Results

3.1. TOC and XRD Analysis

The TOC content and mineral composition are presented in

Table 1. Total organic carbon (TOC) and X-ray diffraction (XRD) analysis results of shale samples.

Sample No.	TOC (%)	Mineralogical Compositions (%)
		Quartz	Orthoclase	Plagioclase	Calcite	Dolomite	Pyrite	Clay
X-4	8.26	37.00	4.00	12.00	22.00	8.00	-	17.00
X-5	9.93	77.00	3.00	4.00	-	-	-	16.00
X-7	5.81	44.00	2.20	1.00	13.40	5.10	2.30	32.00
X-9	1.62	36.80	1.10	6.40	6.70	5.20	1.60	42.20
X-10	1.19	51.00	6.00	17.00	-	-	-	26.00
X-14	1.64	34.00	10.00	10.00	20.00	-	-	26.00
X-15	1.34	32.00	6.00	10.00	26.00	-	-	26.00
F-4	4.95	76.10	0.50	2.70	5.80	3.80	0.90	10.20
F-6	4.60	52.40	0.50	3.10	16.00	7.80	4.30	15.90
F-8	5.26	42.50	0.50	4.10	18.40	14.90	3.60	16.00
F-14	4.20	62.00	-	5.00	6.00	13.00	-	14.00
F-15	3.60	83.00	-	1.00	-	-	-	16.00

. 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%).

3.2. N₂ and CO₂ Adsorption

Following the Brunauer, Deming, Deming and Teller classification [27], the adsorption isotherms of N₂ and CO₂ are type II and type I, respectively (Figure 1 and Figure 2). Samples with a high TOC content tended to have higher N₂ and CO₂ 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, there was a weak positive correlation between PV2 and the content of TOC (R² = 0.3) and a certain negative correlation with the clay content (R² = 0.4).

SSA ranged from 15.9 to 26.9 m²/g (average = 20.1 m²/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 (R² = 0.5) and quartz content (R² = 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.

3.3. 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 (Figure 5 and Figure 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 (Figure 7 and Figure 8). Although the shapes of the Dq spectra were roughly similar, there were differences in the parameters of the Dq spectra (

Table 2. Multi-fractal parameters of Longmaxi Formation shale.

Sample No.	Mesopores							Microporous
	D1N	HN	DTN	DPN	DNN	DN1	DN2	D1C	HC	DTC	DPC	DNC	DC
X-4	1.01	0.99	0.81	0.19	0.63	2.76	2.70	0.97	0.97	0.52	0.22	0.30	2.23
X-5	1.01	0.98	0.97	0.22	0.75	2.77	2.72	0.95	0.96	0.54	0.16	0.38	2.56
X-7	1.01	0.99	0.74	0.23	0.50	2.75	2.65	0.96	0.96	0.53	0.19	0.34	2.62
X-9	1.00	0.98	0.90	0.25	0.65	2.73	2.66	0.97	0.97	0.66	0.18	0.49	2.30
X-10	1.04	1.01	0.45	0.12	0.33	2.71	2.65	0.98	0.97	0.51	0.15	0.36	2.23
X-14	1.02	0.99	0.62	0.21	0.41	2.70	2.62	0.97	0.97	0.58	0.16	0.42	2.29
X-15	1.01	0.98	0.68	0.24	0.44	2.72	2.62	0.95	0.95	0.48	0.23	0.25	2.52
F-4	1.02	0.99	0.74	0.22	0.52	2.71	2.70	0.98	0.97	0.51	0.20	0.32	2.29
F-6	1.02	1.00	0.65	0.15	0.50	2.75	2.70	0.98	0.97	0.57	0.18	0.39	2.31
F-8	1.02	0.99	0.69	0.23	0.47	2.73	2.70	0.98	0.97	0.47	0.18	0.28	2.27
F-14	1.02	0.99	0.63	0.21	0.42	2.72	2.68	0.98	0.97	0.57	0.18	0.39	2.26
F-15	1.03	1.00	0.59	0.18	0.41	2.66	2.63	0.94	0.94	0.62	0.31	0.32	2.32

). 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.

3.4. FHH and V-S Fractal Dimension

The fairly well-fitting results of formula 7 (with correlation coefficients of R² > 0.97, see

Table 3. Fractal dimensions derived from the fractal Frenkel–Halsey–Hill (FHH) model.

Sample No.	P/P0 > 0.5			P/P0 < 0.5
	Fitting Equation	R2	D1	Fitting Equation	R2	D2
X-4	y = −0.239x + 2.097	0.999	2.761	y = −0.299x + 2.092	0.996	2.701
X-5	y = −0.233x + 2.049	0.999	2.767	y = −0.282x + 2.042	0.997	2.718
X-7	y = −0.254x + 2.151	0.997	2.746	y = −0.347x + 2.105	0.998	2.653
X-9	y = −0.273x + 1.705	0.998	2.727	y = −0.344x + 1.691	0.998	2.656
X-10	y = −0.295x + 1.734	0.998	2.705	y = −0.355x + 1.725	0.999	2.645
X-14	y = −0.304x + 1.729	0.999	2.696	y = −0.382x + 1.719	0.998	2.618
X-15	y = −0.282x + 1.690	0.999	2.718	y = −0.378x + 1.671	0.996	2.622
F-4	y = −0.289x + 2.177	0.998	2.711	y = −0.305x + 2.185	0.998	2.695
F-6	y = −0.248x + 1.816	0.997	2.752	y = −0.296x + 1.806	0.997	2.704
F-8	y = −0.275x + 1.656	0.999	2.725	y = −0.304x + 1.661	0.998	2.696
F-14	y = −0.285x + 1.889	0.998	2.715	y = −0.319x + 1.889	0.996	2.681
F-15	y = −0.336x + 1.990	0.997	2.664	y = −0.372x + 1.980	0.999	2.628

and

Table 4. Fractal dimensions derived from the volume-surface area (V-S) model.

Sample No.	Fitting Equation	R2	D	Sample No.	Fitting Equation	R2	D
X-4	y = 0.742x + 6.941	0.979	2.229	X-15	y = 0.840x + 7.159	0.999	2.522
X-5	y = 0.852x + 7.487	0.998	2.556	F-4	y = 0.762x + 7.037	0.994	2.288
X-7	y = 0.872x + 7.451	0.999	2.617	F-6	y = 0.770x + 6.978	0.995	2.310
X-9	y = 0.767x + 6.874	0.996	2.304	F-8	y = 0.757x + 6.979	0.994	2.271
X-10	y = 0.764x + 6.916	0.995	2.227	F-14	y = 0.752x + 6.872	0.995	2.257
X-14	y = 0.762x + 6.872	0.995	2.286	F-15	y = 0.772x + 7.013	0.997	2.316

and Figure 10 and Figure 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 (R² = 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).

4. Discussion

4.1. 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.

4.2. 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 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 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 (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 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–D₁₀, D₀–D₁₀ and D₋₁₀–D₀) and the parameters of singularity (D₁ 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.

4.3. 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 (Figure 9a and Figure 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 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.

4.4. 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]:

$$K=\frac{r^2}{8}\frac{\phi }{\tau }{[\frac{2\phi }{3\tau \left(1-\phi \right)}]}^{\frac{2}{D-1}}$$

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 N₂/CO₂ 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. Adsorbent transport rate of Longmaxi shale based on low-pressure N₂/CO₂ adsorption (LP-N₂/CO₂GA).

Sample No.	Transport Rate of N2 (cm³/(g·min))	Transport Rate of CO2 (cm³/(g·min))
X-4	0.037	0.009
X-5	0.034	0.017
X-7	0.029	0.008
X-9	0.037	0.003
X-10	0.059	0.009
X-14	0.049	0.003
X-15	0.047	0.004
F-4	0.052	0.005
F-6	0.033	0.008
F-8	0.031	0.009
F-14	0.042	0.005
F-15	0.071	0.020

). 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.

5. 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–D₁₀, D₀–D₁₀ and D₋₁₀–D₀) and the parameters of singularity (D₁ 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.