Calibrated Multi-Method Fractal Characterization of Full-Scale Pore Structure and Geological Controls in Deep Anthracite: Case Study from Daning–Jixian Block, Ordos Basin

Abstract

Deep coal reservoirs commonly exhibit strong multiscale heterogeneity, which directly affects coalbed methane (CBM) storage, diffusion, and flow. In this study, deep No. 8 coal samples from the Daning–Jixian block, Ordos Basin, were comprehensively and quantitatively characterized using low-pressure CO₂ adsorption, low-temperature N₂ adsorption, mercury intrusion porosimetry (MIP), and nuclear magnetic resonance (NMR). A method-constrained calibration framework was developed to assign reliable fractal dimensions to different pore-size intervals and to calculate a volume-weighted comprehensive fractal index. The scale-dependent pore structure was evaluated, and its relationships with coal quality, maceral composition, and maximum vitrinite reflectance (R_o,max) were analyzed. The results show that deep anthracite has a trimodal pore-size distribution, with micropores dominating both specific surface area and pore volume. Fractal behavior is strongly scale-dependent, and calibrated full-pore-size fractal dimensions provide a more reliable measure of reservoir heterogeneity than single-method interpretations. Pore development and heterogeneity are closely associated with coalification degree, coal quality, and maceral composition. Ash tends to inhibit pore development, whereas fixed carbon and vitrinite promote micropore development; inertinite mainly contributes to macropores and fractures. These findings provide a quantitative basis for evaluating pore-structure heterogeneity and optimizing deep CBM reservoir development.

1. Introduction

Against the global transition toward clean and low-carbon energy, unconventional natural gas resources have attracted increasing attention [1,2]. In this context, CBM stands out as a key unconventional resource, contributing to greenhouse gas mitigation and playing an important role in energy security [3,4]. While China is endowed with abundant CBM resources, the efficient development of deep CBM reservoirs (depths > 1500 m) remains challenging because of their complex structures and low-porosity, low-permeability properties [5]. The adsorption, diffusion, and flow behaviors of CBM are governed by the complexity of the coal reservoir’s pore and fracture system, making pore structure a key geological factor controlling both resource occurrence and reservoir productivity [6,7].

In recent years, advanced characterization techniques such as scanning electron microscopy (SEM), atomic force microscopy (AFM), gas adsorption, MIP, NMR, small-angle neutron scattering (SANS), and micro/nano CT have been widely applied in reservoir pore-structure research [8,9,10,11,12]. Integrated multi-method characterization has become the mainstream approach for describing the multiscale features of pore systems [13,14,15]. According to the International Union of Pure and Applied Chemistry (IUPAC) standards, the pore structure of coal can be classified into micropores (<2 nm), mesopores (2–50 nm), and macropores (>50 nm) [16,17,18]. Previous studies have shown that, with increasing coal maturity, mesopore development shows no significant pattern. In contrast, micropore-specific surface area and pore volume first decrease and then increase with coal maturity [19,20,21], whereas pore-structure evolution is also influenced by reservoir properties, coal macrolithotype, and coal composition [22,23]. Previous studies have provided important insights into pore structures of coal reservoirs, but most full-scale fractal interpretations remain method-dependent because each experimental technique is governed by a different physical principle and applicable pore-size range.

The combined effects of deep high-temperature and high-pressure conditions and continued coalification often lead to a coal reservoir pore structure dominated by micropores and macropores, with poorly developed mesopores [24,25]. Such pore structures may form barriers to gas migration, thereby constraining CBM desorption efficiency and production capacity [26]. Traditional Euclidean geometry cannot adequately describe or quantify such complex pore systems exhibiting self-similarity, nonlinearity, and scale invariance [27,28]. Fractal theory provides an effective approach for quantifying pore surface roughness, structural complexity, and spatial filling heterogeneity [29]. Currently, fractal theory has been extensively applied to coal pore-structure research, yielding various characterization models and methods. These include the Frenkel–Halsey–Hill (FHH) model based on gas adsorption isotherms, the Menger sponge model based on volume filling assumptions, and the box counting method based on digital image processing [30,31,32]. However, most studies have focused on fractal dimensions derived from individual methods, making it difficult to characterize the complexity of full-scale pore structure distribution in coal. Research indicates that fractal dimension is closely related to key physical properties of coal, such as adsorption performance and permeability. These findings provide new insights and technical support for revealing the multiscale pore-structure characteristics of coal from a nonlinear perspective and achieving precise reservoir evaluation [33,34].

Previous multifractal studies have substantially advanced the quantitative characterization of coal pore heterogeneity. Quan et al. established a full-scale multifractal framework for middle- to high-rank coal reservoirs and linked pore heterogeneity to permeability modeling [35]. Li et al. applied multiple fluid-intrusion methods and multifractal theory to evaluate multiscale pore-structure heterogeneity in high-rank coal reservoirs [36]. Zhang et al. compared multifractal parameters derived from MIP and low-pressure N₂ adsorption and found that the parameters obtained from different methods were not directly correlated [37]. However, these studies generally focused on multifractal parameters derived from individual or directly combined datasets, whereas the differences in measurement principles, pore-size accessibility, fitting stability, and physical applicability among different methods were not explicitly constrained before cross-scale integration. In addition, the geological controls of coal quality, maceral composition, mineral content, and thermal maturity on calibrated full-pore-size heterogeneity remain insufficiently clarified.

Therefore, this study integrates low-pressure CO₂ adsorption, low-temperature N₂ adsorption, MIP, and NMR to establish a method-constrained calibration framework for full-pore-size fractal characterization of deep anthracite. Unlike previous multifractal studies that mainly focused on multifractal parameters derived from individual or directly combined datasets, this study first constrains the applicable pore-size interval of each method using fitting quality, physical plausibility, and pore-size applicability, and then calculates a volume-weighted comprehensive fractal index, D_M. This study aims to establish a calibrated full-pore-size fractal framework for deep anthracite. Specifically, we characterize pore-size distribution, calibrate interval-specific fractal dimensions, compare the index with single-method metrics, evaluate the sensitivity of D_M to fractal dimension parameters and pore-volume fractions, and clarify the geological controls of coal quality, maceral composition, porosity, and thermal maturity on full-scale heterogeneity. The results are expected to improve the comparability and physical interpretability of fractal characterization in deep coal reservoirs and provide a reference for deep anthracite reservoir assessment.

2. Experimental Samples and Methods

2.1. Experimental Samples

The samples used in this study were collected from the Daning–Jixian block on the southeastern margin of the Ordos Basin, all of which are anthracite. The region is characterized by a large monocline dipping from east to west, with complex surface outcrops and prominent fold structures. The study area possesses well developed Paleozoic, Mesozoic, and Cenozoic stratigraphic sequences. The deep No. 8 coal seam in the study area exhibits a gently dipping monocline structure trending northwest with poorly developed faults. It is the main target interval for current CBM exploration and development in the block. Eight representative samples were selected, and their basic properties are listed in

Table 1. Sample physical properties.

Number	Depth/m	Ro,max/%	Mad/%	Aad/%	Vad/%	FCad/%	Vitrinite/%	Inertinite/%	Liptinite/%	Porosity/%
DJ1-6	1880.83	2.86	1.01	30.77	8.43	59.79	67.69	32.21	0.10	6.09
DJ1-8	1881.40	2.84	1.10	19.07	7.55	72.28	72.12	26.58	1.30	6.20
DJ1-11	1882.36	2.88	1.10	12.63	6.39	79.88	74.02	24.34	1.64	7.31
DJ4-4	1961.98	2.83	1.25	5.43	5.87	87.45	85.12	14.26	0.62	8.12
DJ4-7	1962.99	2.92	0.96	13.62	6.38	79.04	85.62	13.27	1.11	7.79
DJ4-9	1963.57	2.90	1.01	9.62	6.03	83.34	78.40	19.44	2.16	7.99
DJ4-13	1964.84	2.91	0.75	33.61	8.16	57.48	77.13	22.35	0.52	5.59
DJ4-17	1965.79	2.95	1.08	19.61	6.99	72.32	73.95	25.39	0.66	8.04

Note: M_ad, moisture; A_ad, ash yield; V_ad, volatile matter; FC_ad, fixed carbon.

.

2.2. Experimental Methods

2.2.1. Pore-Structure Characterization Methods

Adsorption experiments: For low-pressure CO₂ and low-temperature N₂ adsorption experiments, coal samples underwent vacuum degassing at 105 °C for 12 h to remove moisture. CO₂ adsorption measurements were conducted at 273 K in accordance with GB/T 21650.3–2011 [38]. The N₂ adsorption–desorption measurements were conducted at 77 K in accordance with GB/T 21650.2–2008 [39].

MIP experiment: MIP measurements were performed using an AUTOPORE 9505 mercury porosimeter (Quantachrome Instruments, Boynton Beach, FL, USA) in accordance with GB/T 21650.1–2008 [40], after drying at 105 °C for 8 h. The instrument chamber was maintained under vacuum throughout the test. In this study, all samples were inspected after drying, and damaged samples were excluded or retested to minimize potential drying-induced errors.

NMR experiment: Low-field NMR measurements were performed on saturated, piston-shaped coal samples using a Niumag NMR scanning system (Suzhou Niumag Analytical Instrument Co., Ltd., Suzhou, China). By processing and analyzing NMR relaxation signals, the T₂ distribution and signal intensity were obtained [41]. Subsequently, the pore-size distribution curves were derived.

2.2.2. Fractal Models and Calculation Methods

(1)

MIP

In this study, pore radius was related to mercury intrusion pressure using the Washburn equation, whereas the fractal relationship between mercury saturation and intrusion pressure was interpreted using a Menger sponge fractal model [42]. Accordingly, the MIP-derived fractal dimension can be expressed as:

$$\lg \left(1-S\right)=\left(D_{\mathrm{Z}}-3\right)\lg P-\left(D_{\mathrm{Z}}-3\right)\lg P_{\mathrm{m}}$$

(1)

where S is mercury saturation expressed as a fraction. P is mercury intrusion pressure, MPa. P_m is pressure corresponding to the maximum pore size, MPa. D_Z is the MIP-derived pore fractal dimension.

(2)

N₂ adsorption

Methods for calculating fractal dimensions based on low-temperature N₂ adsorption data include thermodynamic methods, fractal Langmuir models, fractal BET models, and fractal FHH models. The fractal dimension of N₂ adsorption calculated using the FHH model satisfies the following equation:

$$\ln \left(V/V_0\right)=\left(D_{\mathrm{Y}}-3\right)\left[\ln \left(\ln \left(P_0/P\right)\right)\right]+C$$

(2)

where V/V₀ is the relative adsorption amount at equilibrium pressure P. C is a constant. P₀ is the saturated vapor pressure of gas adsorption, MPa. D_Y is the fractal dimension of the pore structure for N₂ adsorption data.

(3)

CO₂ adsorption

Previous studies indicate that adsorbate molecules in micropores typically fill the pores. The Sierpinski fractal model was employed in this study to calculate the fractal dimension of CO₂ adsorption, which satisfies the following equation:

$$\ln V=(3-D_{\mathrm{X}})\ln \left(P-P_t\right)+C$$

(3)

where P is the experimental pressure, MPa. P_t is the threshold pressure, MPa. V is the adsorption volume, cm³/g. C is a constant. D_X is the fractal dimension derived from the CO₂ adsorption data.

(4)

NMR

Within the NMR theoretical framework, the T₂ spectrum data of coal samples can be used to calculate the fractal characteristics of pore distribution. The relationship between the cumulative pore-volume fraction and relaxation time can be expressed as:

$$\log V(c)=(3-D_{\mathrm{N}})\log T_2+(D_{\mathrm{N}}-3)\log T_{2,\max }$$

(4)

where V(c) represents the percentage of cumulative pore volume relative to total pore volume when the cumulative transverse relaxation time is less than T₂. D_N represents the pore fractal dimension. T_2,max represents the maximum transverse relaxation time, ms.

To make the calibration procedure reproducible, an explicit acceptance screening criterion was applied prior to the selection of the fractal dimension for each pore-size interval. Fitted results were considered reliable when the coefficient of determination was not lower than 0.90 and the derived fractal dimension fell within the physically meaningful range of 2–3 for a three-dimensional pore system. When several methods covered the same pore-size interval, the method with the highest fitting quality and a physically plausible fractal dimension was selected for the final calibrated framework. Results with a coefficient of determination lower than 0.90 or with fractal dimensions outside the physically meaningful range were retained only for methodological comparison and were not used in the final calculation of D_M.

3. Results and Discussion

3.1. Multiscale Pore Network Characteristics and Heterogeneity in Coal

3.1.1. Pore Development and Fractal Characteristics Based on CO₂ Adsorption Experiments

Figure 1 shows the low-pressure CO₂ adsorption curves of the samples. CO₂ adsorption capacity gradually increases with rising relative pressure. The low-pressure region (P/P₀ < 0.01) exhibits an adsorption-dominant zone with relatively high adsorption rates. At higher relative pressure (P/P₀ > 0.01), the upward trend in the adsorption rate decelerates, whereas the total adsorption capacity exhibits a slight increase. This behavior is consistent with micropore filling by CO₂ molecules. The adsorption curves of all eight samples show a logarithmic trend. Maximum CO₂ adsorption capacities ranged from 14.906 to 26.821 cm³/g. Pore-size analysis revealed that when the tested pore-size intervals were roughly equivalent, the stage adsorption volume curves and differential curves of the coal samples largely overlapped. All samples exhibited similar pore-size distribution patterns, with pore-volume contribution generally decreasing as pore size increased. Pore volume in this range was predominantly concentrated between 0.32 and 0.86 nm. This range accounted for 59.48% to 74.65% of the total pore volume, averaging 68.50%.

The micropore fractal fitting curves are shown in Figure 2. During CO₂ adsorption at 273 K, CO₂ molecules are mainly accommodated in micropores by a pore-filling mechanism. To ensure reliable fractal analysis, subsequent analysis was conducted using CO₂ adsorption data with a relative pressure P/P₀ < 0.02. The pore fractal dimensions calculated based on adsorption experiments are shown in

Table 2. Calculated pore fractal dimensions based on CO₂ and N₂ adsorption experiments.

Number	Low-Pressure CO2 Adsorption		Low-Temperature N2 Adsorption
	DX1	R2	DY1	R2	DY2	R2
DJ1-6	2.408	0.9935	2.763	0.9464	2.397	0.9451
DJ1-8	2.408	0.9932	2.804	0.9156	2.464	0.9183
DJ1-11	2.410	0.9938	2.534	0.9291	2.480	0.9137
DJ4-4	2.399	0.9928	2.664	0.9327	2.608	0.9173
DJ4-7	2.393	0.9930	2.651	0.9791	2.445	0.9764
DJ4-9	2.402	0.9923	2.506	0.9483	2.612	0.9090
DJ4-13	2.411	0.9930	2.572	0.9883	2.515	0.9835
DJ4-17	2.408	0.9933	2.624	0.9900	2.484	0.9455

. The fractal dimension ranged from 2.393 to 2.411, with an average value of 2.400. The coefficient of determination R² ranged from 0.9923 to 0.9938. Among the samples, DJ4-13 exhibited the highest fractal dimension, while DJ4-7 showed the lowest. However, the overall variation in fractal dimension was minimal. This indicates that the micropore systems of different samples have comparable heterogeneity and broadly similar fractal characteristics.

3.1.2. Pore Development and Fractal Characteristics Based on N₂ Adsorption Experiments

Analysis of the low-temperature N₂ adsorption/desorption isotherms (Figure 3) reveals that the N₂ adsorption capacity of coal samples ranges from 1.07 to 2.82 cm³/g. The adsorption curves generally exhibit an inverse S-shaped trend. According to IUPAC classification, these coal samples exhibit Type II isotherms [43]. Due to strong interactions between the adsorbate and surface, the N₂ adsorption capacity increases rapidly with relative pressure when P/P₀ < 0.1, resulting in a convex isotherm. This stage mainly reflects monolayer adsorption and partial micropore filling. As relative pressure increased from 0.1 to 0.9, N₂ adsorption increased more slowly, indicating a transition from monolayer to multilayer adsorption. When relative pressure approached 1.0, adsorption increased rapidly again without reaching saturation. As the relative pressure decreased from 1.0 to 0.45, all coal samples exhibited distinct adsorption and desorption branches. This indicates the presence of nanoscale slit-like pores. Below a relative pressure of 0.45, the desorption branch nearly overlapped with the adsorption branch. Based on low-temperature N₂ adsorption experiments, the pore-volume range of coal from the study area was determined to be 0.006–0.029 cm³/g, with an average of 0.013 cm³/g.

The fractal fitting curves derived from the N₂ adsorption data are shown in Figure 4. At low relative pressures, nitrogen molecules predominantly adsorb as monolayers on pore surfaces, primarily governed by van der Waals forces [44]. As relative pressure increases, adsorption transitions to multilayer and pore-filling adsorption. With increasing relative pressure, multilayer adsorption and capillary condensation become increasingly important. Accordingly, the fractal dimensions D_Y1 and D_Y2 of the pore structure in the coal reservoir were calculated at P/P₀ < 0.5 and P/P₀ > 0.5, respectively. D_Y1 characterizes the fractal dimension of the coal surface, reflecting the roughness of the pore surface. D_Y2 characterizes the fractal dimension of the pore structure, reflecting the irregularity of the pore structure [29]. The fitted curves show good fitting performance under both high and low relative pressure conditions, indicating clear fractal behavior over the selected pressure intervals. The fractal dimension D_Y1 ranged from 2.506 to 2.804, wherein sample DJ1-8 exhibited the highest surface roughness of pores. In contrast, the fractal dimension D_Y2 ranged from 2.397 to 2.612, and sample DJ4-9 showed the strongest heterogeneity of pore structures.

3.1.3. Pore Development and Fractal Characteristics Based on MIP Experiments

Mercury intrusion–extrusion volumes and pore-size distribution curves of the samples are shown in Figure 5. The mercury intrusion–extrusion data indicate that the samples generally exhibit a characteristic pattern of significant decline followed by gradual stabilization during mercury extrusion. However, the mercury intrusion curves exhibited three pressure-dependent stages: a rapid mercury volume increase at low pressures (<0.05 MPa), a deceleration in mercury volume growth within the medium pressure range (0.05–38 MPa), and a renewed increase in mercury volume beyond 38 MPa. The pore volume of samples ranged from 0.008 to 0.059 cm³/g, with a mean value of 0.031 cm³/g. The variation trends of coal pore volume across different pore-size ranges showed significant differences. Within the pore-size range exceeding 30 nm, the increments in coal pore volume and specific surface area exhibited gradual changes. These patterns indicate that macropores are relatively less developed, whereas nanopores constitute the dominant pore type in the samples.

The fractal results and fitting curves of coal samples based on MIP experiments are shown in

Table 3. Calculated pore fractal dimensions based on MIP experiments.

Number	MIP Method
	DZ1	R2	DZ2	R2	DZ3	R2
DJ1-6	0.461	0.7128	2.843	0.8019	2.088	0.9891
DJ1-8	0.225	0.6663	2.987	0.9125	2.006	0.9982
DJ1-11	0.088	0.8914	2.846	0.8760	2.307	0.9649
DJ4-4	0.337	0.6679	2.858	0.8925	2.463	0.9920
DJ4-7	0.093	0.8434	2.980	0.9285	2.895	0.9951
DJ4-9	0.404	0.7185	2.988	0.9828	2.448	0.9881
DJ4-13	0.146	0.7772	2.906	0.8814	2.117	0.9362
DJ4-17	0.491	0.7626	2.995	0.9725	2.599	0.9101

and Figure 6, respectively. The MIP fractal results suggest that the fractal dimensions of pore structures across the eight coal samples were relatively similar. Using critical values at different pore-size scales and inflection points of fractal trends as fractal boundaries, together with established pore-size classification criteria, pores characterized by MIP were categorized into micropores (r < 10 nm), mesopores (10 < r < 100 nm), and macropores (r > 100 nm). Thus, based on MIP experiments, the fractal dimension D_Z1 for micropores varied from 0.088 to 0.491, D_Z2 for mesopores varied from 2.843 to 2.995, and D_Z3 for macropores varied from 2.006 to 2.895. The D_Z1 values obtained for r < 10 nm were far below the conventional physically meaningful range of 2–3 for pore systems. These abnormal values should not be interpreted as true fractal characteristics of micropores. Instead, they indicate that the Menger sponge model combined with MIP data is not applicable to this pore-size interval, probably because mercury cannot effectively access nanoscale micropores and because high intrusion pressure may cause pore compression, ink-bottle effects, or structural disturbance. Therefore, D_Z1 was excluded from the final calibrated fractal framework. The mesopore fractal dimension obtained from the MIP experiment is the highest. This indicates that mesopores exhibit both greater structural complexity and more pronounced heterogeneity.

3.1.4. Pore Development and Fractal Characteristics Based on NMR Experiments

Based on NMR experimental data, the pore spaces were classified into corresponding size intervals: the <10 nm interval represents micropores, the 10 nm–0.1 μm interval corresponds to small pores, the 0.1–1 μm interval to mesopores, and the >1 μm interval to macropores. Additionally, pores smaller than 100 nm are defined as adsorption pores with a critical pore-size of 100 nm [41]. Their development level positively correlates with CBM adsorption/desorption and diffusion capacity. Pores larger than 100 nm are defined as flow pores, whose development directly influences the flow efficiency of CBM. NMR scanning results of study area samples are shown in Figure 7. The T₂ spectra reveal trimodal relaxation patterns in the coal samples, indicating a broad pore-size distribution. The peak area for micropores (adsorption pores) is significantly larger than those corresponding to flow pores and fractures. This suggests that adsorption pores dominate the pore structure of the study area samples.

Because a single pore-size parameter cannot fully characterize the pore and fracture system, piecewise fractal analysis based on NMR data was employed in this study to reveal the heterogeneity of pores at different scales in coal samples. Figure 8 shows that the fractal curve exhibits non-linear behavior with distinct inflection points rather than a linear form. All samples exhibited high fitting accuracy for the fractal dimensions of small pores, mesopores, and macropores, demonstrating typical fractal characteristics. The fractal dimension values for micropores deviated from the theoretical range and showed poor fitting, lacking clear fractal features. This is mainly because adsorption forces dominate within micropores, which constrain free fluid flow and molecular diffusion. Consequently, the influence of diffusion relaxation on NMR relaxation times becomes non-negligible, disrupting the sole control of pore geometry over relaxation signals [45]. Therefore, these values should not be interpreted as true fractal dimensions of micropores. Instead, they indicate that the NMR fractal model is not suitable for reliably characterizing micropore-scale heterogeneity in this interval. Based on NMR experiments, the average fractal dimensions of micropores, small pores, mesopores, and macropores were 1.671, 2.945, 2.869, and 2.867, respectively. Within the same sample, the fractal dimension of small pores was the highest, followed by mesopores. This indicates that small pores have the greatest distributional dispersion, rougher pore-throat surfaces, and the strongest heterogeneity. Mesopores show the next level of heterogeneity. Macropores fall between micropores and mesopores in terms of heterogeneity.

3.2. Full-Pore-Size Fractal Dimension Calibration and Evaluation Methodology

3.2.1. Calibration of Full-Pore-Size Fractal Dimensions

Deep coal reservoirs exhibit pronounced multiscale development of pore and fracture systems. Based on the pore-size distribution characteristics of the study area, the following pore-size classifications were defined: <2 nm as micropores, 2–100 nm as mesopores, and >100 nm as macropores. By converting the T₂ distribution of coal samples into equivalent pore-size distribution curves, the integrated full-pore-size distribution curve was compared with the NMR-derived pore-size distribution curve (Figure 9). The two curves showed highly consistent morphology, both exhibiting typical triple-peak distribution characteristics. The characteristic peaks from left to right were labeled as P₁, P₂, and P₃, respectively, with significant differences in peak intensity. Taking sample DJ1-6 as an example: The pore-size distribution range for peak P₁ is 0.01–3.53 nm, with a pore-size corresponding to the peak of 0.59 nm. The pore-size distribution range for peak P₂ is 17.94–124 nm, with a pore diameter corresponding to the peak of 77.32 nm. The pore-size distribution range for peak P₃ is 2156.5–9033.7 nm, with a pore diameter corresponding to the peak of 7690.52 nm. Overall, micropores dominate in all coal samples from the study area. However, differences in pore-structure characteristics exist among different coal samples. These differences may reflect variations in tectonic modification and in situ stress conditions experienced by coal bodies at different stratigraphic levels [46].

Fractal dimension analysis of pores via MIP revealed high fitting accuracy for pore sizes r > 100 nm, but significantly decreased fitting accuracy for r < 100 nm. Furthermore, mercury penetration pressure exceeding 30 MPa (corresponding to approximately 100 nm pore-size) may cause irreversible damage to coal rock pore structures, leading to distorted characterization results. Therefore, the MIP fractal dimension was effectively limited to the r > 100 nm pore-size range. Simultaneously, low-temperature N₂ adsorption and low-pressure CO₂ adsorption methods were employed to characterize fractal dimensions in the 2 < r < 100 nm and r < 2 nm ranges, respectively. A preliminary multi-method fractal dimension characterization system covering the entire pore-size spectrum was thus established. To reveal the intrinsic relationship between the fractal dimension of different pore types and coalification, a quantitative relationship between fractal dimension and coalification degree was further established in this study (Figure 10). The results indicate that the micropore fractal dimension exhibits the highest goodness of fit with coalification degree. This suggests that as thermal evolution progresses, the number of micropores in coal samples continuously increases, accompanied by simultaneous enhancements in pore surface complexity and roughness.

To achieve reliable characterization of the full-scale pore-structure fractal dimension of coal samples, a combined calibration method was proposed in this study, centered on defining dominant pore-size intervals and quantifying fitting accuracy. A schematic of the multi-method calibrated full-pore-size fractal dimension model is shown in Figure 11. This method selects and integrates fractal dimensions across pore-size intervals based on fitting quality and numerical plausibility. By comparing results from NMR with those from adsorption and mercury porosimetry, it prioritizes fractal dimensions with superior fitting accuracy and more reasonable values within the same pore-size range. The results (

Table 4. Fractal dimensions from different methods evaluated by pore-size interval, fitting quality, and physical plausibility.

Number	Characterization of Fractal Dimension D Using Combined CO2, N2, and MIP Methods
	r < 2 nm	R2	2 nm < r < 0.1 μm	R2	0.1 < r < 1 μm	R2	r > 1 μm	R2
DJ1-6	2.408	0.9935	2.397	0.9451	2.933	0.8019	2.088	0.9891
DJ1-8	2.408	0.9932	2.464	0.9183	2.987	0.9125	2.006	0.9982
DJ1-11	2.410	0.9938	2.480	0.9137	2.846	0.8760	2.307	0.9649
DJ4-4	2.399	0.9928	2.608	0.9173	2.958	0.8925	2.463	0.9920
DJ4-7	2.393	0.9930	2.445	0.9764	2.980	0.9285	2.895	0.9951
DJ4-9	2.402	0.9923	2.612	0.9090	2.988	0.9828	2.448	0.9881
DJ4-13	2.411	0.9930	2.515	0.9835	2.960	0.8814	2.117	0.9362
DJ4-17	2.408	0.9933	2.484	0.9455	2.995	0.9725	2.599	0.9101
Number	Characterization of Fractal Dimension D by NMR Method
	r < 10 nm	R2	10 nm < r < 0.1 μm	R2	0.1 < r < 1 μm	R2	r > 1 μm	R2
DJ1-6	1.684	0.7214	2.870	0.8706	2.794	0.9985	2.948	0.9356
DJ1-8	1.653	0.7331	2.969	0.9071	2.863	0.9992	2.973	0.8557
DJ1-11	1.664	0.7341	2.792	0.6703	2.854	0.9978	2.672	0.8809
DJ4-4	1.679	0.7175	2.992	0.8448	2.982	0.9946	2.733	0.8824
DJ4-7	1.637	0.7353	2.984	0.7271	2.783	0.9895	2.896	0.9019
DJ4-9	1.653	0.7235	2.987	0.8033	2.955	0.9816	2.851	0.8912
DJ4-13	1.703	0.7035	2.997	0.4535	2.768	0.9895	2.945	0.8744
DJ4-17	1.699	0.7074	2.974	0.8541	2.958	0.9982	2.919	0.8103

) reveal that NMR exhibits a higher goodness of fit within the 0.1–1 μm pore-size range. Its fractal dimension values are consistently stable within the reasonable range of 2–3. Therefore, NMR is adopted to characterize the fractal dimension for this pore-size range. Finally, the multi-method calibrated full-scale pore-structure fractal dimension results are presented in

Table 5. Final calibrated fractal dimensions for different pore-size intervals.

Number	Calibrated Full-Pore-Size Pore-Structure Fractal Dimensions D
	r < 2 nm	R2	2 nm < r < 0.1 μm	R2	0.1 < r < 1 μm	R2	r > 1 μm	R2
DJ1-6	2.408	0.9935	2.397	0.9451	2.794	0.9985	2.088	0.9891
DJ1-8	2.408	0.9932	2.464	0.9183	2.863	0.9992	2.006	0.9982
DJ1-11	2.410	0.9938	2.480	0.9137	2.854	0.9978	2.307	0.9649
DJ4-4	2.399	0.9928	2.608	0.9173	2.982	0.9946	2.463	0.9920
DJ4-7	2.393	0.9930	2.445	0.9764	2.783	0.9895	2.895	0.9951
DJ4-9	2.402	0.9923	2.612	0.9090	2.955	0.9816	2.448	0.9881
DJ4-13	2.411	0.9930	2.515	0.9835	2.768	0.9895	2.117	0.9362
DJ4-17	2.408	0.9933	2.484	0.9455	2.958	0.9982	2.599	0.9101
Average	2.404	0.9931	2.500	0.9386	2.869	0.9936	2.365	0.9717

. After calibration, the fractal dimension fitting quality exceeded 0.9 across all pore-size ranges for all samples. This supports the reliability and internal consistency of the calibrated characterization results. The proposed calibration framework is applicable to coal samples for which the pore-size intervals can be independently constrained by complementary methods and for which the fitted fractal dimensions fall within physically meaningful ranges. Its reliability decreases when one method produces poor fitting quality, mechanically induced pore alteration, or fractal dimensions outside the conventional range. The first step of the calibration was to define the dominant pore-size interval of each method according to both theoretical applicability and fitting quality. The second step was to exclude fractal dimensions that were outside the physically meaningful range or were obtained from pore-size intervals vulnerable to experimental artifacts. The third step was to construct a calibrated full-scale fractal profile by selecting CO₂ adsorption for r < 2 nm, N₂ adsorption for 2 nm < r < 0.1 μm, NMR for 0.1 < r < 1 μm, and MIP for r > 1 μm. This selection does not imply that the other methods are invalid over their entire measurement ranges but indicates that they were not used for final fractal calibration where their fitting quality or physical plausibility was weaker.

To compare the overall heterogeneity of the laboratory-measured pore system among samples, the comprehensive fractal dimension D_M was defined in this study as a characterization index. D_M was calculated by weighting the calibrated fractal dimension of each pore-size interval by its corresponding pore-volume fraction, because pore intervals contributing more storage space should exert a larger influence on the bulk heterogeneity index. The weights correspond to the pore-volume fraction of each pore-size interval, calculated as follows:

$$D_M={{\displaystyle \sum D}}_{\mathrm{i}}{T}_{\mathrm{i}}$$

(5)

where D_M represents a volume-weighted comprehensive fractal index for comparing bulk pore heterogeneity among samples; D_i represents the fractal dimension corresponding to the i-th pore-size segment; and T_i represents the pore-volume fraction corresponding to the i-th pore-size segment. Here, T_i was derived from the integrated full-pore-size distribution after cross-calibration of CO₂ adsorption, N₂ adsorption, NMR, and MIP data.

Final calibrated fractal dimensions for different pore-size intervals are shown in Table 5. The average fractal dimensions for the study area samples within the pore-size ranges r < 2 nm, 2 nm < r < 0.1 μm, 0.1 < r < 1 μm, and r > 1 μm were 2.404, 2.500, 2.869, and 2.365, respectively. The fractal dimension was highest in the 0.1 < r < 1 μm pore-size range, indicating the most complex pore structure and strongest heterogeneity. The lowest fractal dimension was observed in the r > 1 μm pore-size range, indicating improved connectivity and simpler pore structure. This scale-dependent pattern may occur because larger pores are more readily interconnected, reducing the apparent complexity of the pore network. Furthermore, reservoir compaction induces deformation of large pores under tectonic squeezing. Numerous irregularly shaped and unevenly distributed mesopores and micropores are generated [47]. The development of such pores significantly increases the complexity of mesopore and micropore structures, leading to an upward trend in micropore fractal dimension.

3.2.2. Evaluation of the Comprehensive Fractal Index and Sensitivity Analysis

A one-at-a-time sensitivity analysis was conducted for the input parameters used in the calculation of D_M. For each sensitivity test, one input parameter X was independently perturbed by ±10%, while the other parameters were kept unchanged:

$$X^{\prime }=X(1\pm 0.1)$$

(6)

When T_i was perturbed, all pore-volume fractions were renormalized to ensure that ∑T_i = 1. The sensitivity coefficient was calculated as:

$$S_{\mathrm{X}}=\left|(D_{\mathrm{M}}^{\prime }-D_{\mathrm{M}})\right|/0.1D_{\mathrm{M}}$$

(7)

where S_X is the sensitivity coefficient of input parameter X; D′_M is the recalculated comprehensive fractal dimension after perturbation; and D_M is the original value.

The sensitivity coefficient of the input parameters to the comprehensive fractal dimension D_M is shown in Figure 12. The sensitivity results indicate that D_M is more sensitive to fractal dimension parameters than to pore-volume fractions. D_X exhibits the highest sensitivity and D_Z shows the second-highest sensitivity, indicating that the macropore/fracture-related interval also contributes substantially to the comprehensive heterogeneity index. In contrast, the sensitivities of T_Y and T_N are relatively low, suggesting that variations in these pore-volume fractions have limited effects on D_M. Overall, the sensitivity analysis confirms that D_M is jointly controlled by scale-dependent fractal complexity and pore-volume allocation.

The calibrated D_M was further compared with conventional single-method fractal dimensions (Figure 13). Qin et al. (2025) [44] reported single-method fractal dimensions based on N₂ adsorption for deep coal samples from the eastern Ordos Basin. For all samples, the average D₁ value was 2.603, and the average D₂ value was 2.125. In comparison, the calibrated mesopore fractal dimension D_Y in this study has an average value of 2.500, which is close to the D₁ value, indicating that the N₂-derived mesopore fractal results in this study are regionally comparable [44]. However, some D₂ values reported by Qin et al. (2025) [44] are lower than 2, suggesting that single-method fractal analysis may have limited physical interpretability and relatively weak fitting stability in certain pore-size intervals. In this study, a calibrated fractal dimension framework was established through three screening criteria: fitting quality, physical plausibility, and pore-size applicability. The comprehensive fractal dimension D_M was then calculated based on this calibrated framework. This approach avoids the simple combination of fractal dimensions derived from different experimental methods and different pore-size intervals. Therefore, D_M is not only comparable with traditional single-method fractal dimensions, but also integrates information from multiple pore-size intervals through pore-volume weighting, thereby characterizing full-pore-size heterogeneity that cannot be fully captured by N₂ adsorption or MIP-derived fractal dimensions alone.

To quantitatively evaluate the difference between D_M and single-method fractal dimensions, the correlations between different fractal indices and coal-quality/physical-property parameters were further compared (Figure 14). The coefficients of determination (R²) between D_M and ash yield, volatile matter, fixed carbon content, and porosity are 0.7069, 0.7188, 0.7116, and 0.7867, respectively. These values are generally higher than those of the single-method fractal dimensions D_X, D_Y, D_N, and D_Z. Meanwhile, single-method fractal dimensions still have clear scale-specific significance. For example, the relatively high correlation between D_N and porosity indicates that the pore-size interval characterized by NMR is more sensitive to porosity variation. D_X, D_Y, and D_Z mainly reflect the scale-dependent characteristics of micropores, mesopores, and macropores/microfractures, respectively. These results indicate that single-method fractal dimensions are more suitable for revealing local pore-scale structural responses, whereas D_M has a stronger integrated response to coal-quality evolution and pore-property variation and can more effectively characterize the overall heterogeneity of the full-pore-size system in coal reservoirs.

3.3. Influences on Pore-Structure Evolution in Deep Anthracite

3.3.1. Relationship Between Pore-Structure Parameters and Coal Quality

Coal-quality parameters may influence pore-structure development through physical filling and chemical modification. The relationship between pore structure and coal quality in the study area is shown in Figure 15. Ash yield mainly reflects the abundance of inorganic minerals, such as clay minerals, quartz, and pyrite. In high-ash samples (e.g., DJ1-6, A_ad = 30.77%), mineral particles readily fill coal matrix pores and block fracture pathways, reducing effective pore volume and connectivity [44]. Conversely, low-ash samples (e.g., DJ4-4, A_ad = 5.43%) are dominated by organic components, offering ample space for pore development and higher total porosity. Correlation analysis indicates a negative relationship between ash and total pore volume. Volatile content ranged from 5.87% to 8.43%, decreasing with increasing R_o,max. While volatile release during pyrolysis aids microporous formation, excessive release can trigger pore wall collapse, ultimately reducing total pore volume [48]. In this study, volatile content showed a strong negative correlation with micropore volume (R² = 0.943) and a highly negative correlation with total pore volume. This suggests that lower volatile matter content is associated with more stable micropore development. As the core product of coalification, fixed carbon content correlates positively with microporous development. Higher fixed carbon content promotes more regular aromatic ring arrangements and denser molecular structures within coal, thereby creating favorable conditions for microporous formation [49]. Moisture content in deep coal samples ranged between 0.75% and 1.25%, exhibiting overall low levels with minimal variation. This low moisture content is closely related to the burial depth of the coal seam and the high-temperature, high-pressure reservoir environment. Moisture primarily exists in an adsorbed state on micropore surfaces, thereby suppressing CBM adsorption capacity. Therefore, moisture has a weaker regulatory effect on pore structure than other coal-quality parameters.

The relationship between D_M and proximate-analysis parameters is shown in Figure 16. The ash content of the samples exhibits a negative correlation with the comprehensive fractal dimension D_M, consistent with the correlation characteristics of micropores and total pore volume in pore structure. These results suggest that ash affects the heterogeneity of multiscale pores. In high-ash samples, the hydrophilicity and chemical inertness of mineral surfaces decrease pore surface roughness. This results in a lower macropore fractal dimension (D_Z) compared to low-ash samples. This phenomenon aligns with the findings of Qin et al. (2025) [44], indicating that ash content exerts a more pronounced inhibitory effect on macropore fractal dimension. Volatile content negatively correlates with D_M. At high coalification stages where R_o,max exceeds 2.8%, volatile release enters its terminal phase. The pyrolytic behavior of residual volatiles primarily occurs at the micropore scale. Higher volatile content facilitates the collapse of micropore walls due to small-molecule gas generation during pyrolysis, simplifying the micropore structure and reducing fractal dimension. Conversely, at low volatile matter content, coal molecules undergo complete condensation, resulting in stable nanoscale pores that enhance surface roughness and structural complexity. Fixed carbon, as the core product of coalification, exhibits an apparent positive correlation with D_M. A higher fixed carbon content correlates with more regular arrangements of aromatic condensed rings in coal molecules and greater densification of the organic matrix. This process generates a greater number of micropores with more complex structures during molecular rearrangement. This increases the fractal dimension of micropores and enhances their specific surface area. This pattern corroborates the findings of Sun et al. (2025) [48], who observed that fixed carbon enhances pore fractal dimension by promoting micropore proliferation and structural complexity. The correlation between moisture content and D_M is weakest in the investigated samples. This characteristic contrasts sharply with shallow coal: the presence of free water in shallow coal fills mesopores and macropores, leading to a decrease in fractal dimension. The limited variation in moisture content suggests that its influence on fractal characteristics is weaker than that of ash yield, volatile matter, and fixed carbon.

3.3.2. Relationship Between Pore-Structure Parameters and Macerals

As the major constituents of the coal organic matrix, variations in maceral composition regulate the evolution of coal structure, thereby influencing the development characteristics and spatial distribution of pores [41]. The relationship between pore-structure parameters and macerals in coal is shown in Figure 17. The vitrinite fraction is the dominant maceral component in deep coal reservoirs. During coalification, the loss of oxygen-containing functional groups and molecular condensation reactions can generate numerous nanoscale micropores. Correlation analysis reveals a clear positive correlation between vitrinite content and micropore volume (R² = 0.85). Sample DJ4-4 exhibits the highest vitrinite content, with a higher micropore volume. Inertinite originates from plant remains subjected to fibrous carbonization, retaining relatively intact primary structures and readily developing micrometer-scale macropores and fissures [44]. However, this fraction exhibits dense structures with relatively weak micropore development capacity. Taking sample DJ1-6 as an example, its inertinite content is 32.21%, and its macropore volume is significantly higher than that of the other samples. Correlation analysis indicates that inertinite content positively correlates with macroporous volume and negatively correlates with micropore volume. Liptinite, which is relatively rich in aliphatic structures, can contribute to pore generation during early coalification stages. However, during high-rank coalification, it predominantly transforms into vitrinite or inertinite. Consequently, liptinite content in coal samples from the study area is extremely low, limiting its influence on pore-structure regulation. Nevertheless, the analysis revealed a certain correlation between liptinite and pore structure.

The relationship between D_M and maceral composition in the study area coal samples is illustrated in Figure 18. As the dominant component in deep coal reservoirs, vitrinite exhibits an apparent positive correlation with the comprehensive fractal dimension D_M. It serves as the primary carrier of micropore development and an important factor affecting fractal characteristics. The relatively homogeneous organic matrix of vitrinite may help preserve nanoscale pore networks, thereby enhancing the self-similar characteristics of micropores. Inertinite, formed from plant remains via fibrous carbonization, shows a negative correlation with D_M. The dense structural characteristics of inertinite inhibit micropore development. The highly polymerized aromatic condensed rings within its organic matrix reduce micropore formation space, leading to decreased D_M. Although the liptinite content in coal samples from the study area is extremely low, it still shows a weak positive correlation with D_M. Overall, the regulatory effect of liptinite on fractal dimension is limited. Its contribution is significantly lower than that of vitrinite and inertinite.

3.3.3. Relationship Between Pore-Structure Parameters and R_o,max

Previous studies have shown that R_o,max is an important maturity indicator for coal physicochemical evolution [50]. As shown in Figure 19, the micropore, mesopore, and total pore-structure parameters of the samples exhibit U-shaped correlations with R_o,max (Figure 19a–c). In contrast, both the specific surface area and pore volume of macropores exhibit inverted U-shaped relationships with R_o,max (Figure 19d). Furthermore, the correlations between the specific surface areas of micropores, mesopores, and total pores with R_o,max generally align with the evolution patterns of their respective pore volumes as R_o,max increases. Notably, except for one outlier, the specific surface areas and pore volumes of macropores in all coal samples exhibit an inverted U-shaped relationship with R_o,max. This pattern contrasts with the evolution of micropore and total pore-volume parameters. This discrepancy arises because the total pore volume of coal samples is primarily contributed by micropores and mesopores, while the total specific surface area is predominantly controlled by micropores. Consequently, as R_o,max increases, the evolution trend of micropore-specific surface area reflects the variation characteristics of total pore-specific surface area. This phenomenon further highlights the core controlling role of micropores in the evolution process of pore-specific surface area within deep coal reservoirs.

The correlation between the comprehensive fractal dimension of coal samples from the study area and the degree of thermal evolution is shown in Figure 20a. Sample DJ4-13 was identified as an outlier in the D_M and R_o,max relationship. This sample has the highest ash yield, a relatively low fixed carbon content, and the lowest porosity among the studied samples, suggesting stronger mineral filling and greater local petrographic heterogeneity. These features may weaken the maturity-controlled pore-evolution trend and cause DJ4-13 to deviate from the overall relationship. Therefore, this sample was marked as an anomalous point in the trend analysis but was retained in the dataset and in the overall discussion. The comprehensive fractal dimension tends to exhibit a U-shaped relationship with R_o,max. This trend closely aligns with the evolution patterns of micropores and the total pore structure. These results suggest that the comprehensive fractal dimension effectively characterizes the evolution trend of pore structure with increasing thermal evolution, and further indicate that micropores are likely the dominant contributors to pore-structure variations in the investigated samples.

To further examine the contribution of different pore-size intervals to coal reservoir heterogeneity, this study calculated the calibrated fractal dimensions D_X, D_Y, D_N, and D_Z. A calibrated fractal dimension system covering the entire pore-size range was established. Correlation diagrams between the fractal dimensions of different pore-size intervals and R_o,max were plotted (Figure 20). D_X, D_Y, and D_Z all exhibit positive correlations with R_o,max. In contrast, D_N shows a U-shaped relationship with R_o,max (Figure 20d). This trend aligns with the evolution patterns of D_M and is primarily attributed to differences in the formation mechanisms and evolutionary processes of pores across various pore-size ranges. From a molecular structural evolution perspective, coal macromolecules undergo profound reorganization: residual aliphatic side chains continue to cleave, while aromatic rings form highly ordered graphite-like layered structures through polycondensation, driving fundamental restructuring of the pore network system [51]. The formation and evolution of mesopores and macropores are controlled by multiple factors, including tectonic setting, depositional environment, reservoir pressure, and hydrocarbon generation, resulting in more complex formation mechanisms. This leads to relatively weaker correlations between their fractal dimensions and R_o,max. Microporosity formation and evolution are strongly associated with the thermal evolution of coal macromolecular structures, although mineral matter, maceral composition, and local structural heterogeneity may also contribute. Aromatization and polycondensation can induce secondary microporosity formation, thereby enhancing microporosity heterogeneity.

4. Conclusions

(1)

The deep No. 8 anthracite samples from the Daning–Jixian block exhibited a trimodal full-pore-size distribution, with CO₂-derived micropore volume mainly concentrated at 0.32–0.86 nm, accounting for 59.48–74.65% of the measured micropore volume. A comprehensive calibration and evaluation system for fractal dimensions across the full-scale pore-structure range was established, based on dominant pore-size intervals and goodness of fit. After calibration, all samples exhibited a goodness of fit exceeding 0.9 for fractal dimensions across the pore-size ranges. Sensitivity analysis showed that D_M is more sensitive to calibrated fractal dimensions than to pore-volume fractions, with D_X and D_Z exerting the strongest effects.

(2)

Pore-structure parameters and fractal dimensions vary systematically with R_o,max. The specific surface area and pore volume of micropores and mesopores, together with the comprehensive fractal dimension D_M, show a U-shaped trend with increasing R_o,max. By contrast, the specific surface area and pore volume of macropores show an inverted U-shaped trend. This difference indicates that micropores and mesopores are more closely related to maturity-controlled organic-matter evolution, whereas macropores are more strongly affected by mineral filling, compaction, and local structural heterogeneity.

(3)

The potential factors influencing pore development in the investigated deep anthracite samples were discussed, including ash yield, fixed carbon content, vitrinite content, inertinite content, and thermal maturity. Ash content inhibits overall pore development, whereas fixed carbon and vitrinite dominate micropore formation. Inertinite promotes macropore and fracture development. The thermal evolution of coal macromolecular structures is a vital factor controlling pore heterogeneity in deep reservoirs and variations in coal quality. These findings provide a quantitative basis for deep coal reservoir evaluation and CBM development optimization.