Introduction

Coalbed methane (CBM) serves as a clean and unconventional energy source. The exploitation of CBM plays a crucial role not only in ensuring mining safety and enhancing the economic benefits of coal mines, but also in advancing the coal industry towards carbon peak and ultimately carbon neutrality. CBM is primarily stored in the matrix pores of coal in an adsorbed state. The storage and transport process involves adsorption, desorption, diffusion, and seepage, spanning pores of varying scales. Clarifying the developmental characteristics of full-scale pores is a crucial prerequisite for understanding the mechanisms governing the storage and transport of CBM1,2.

Currently, pore measurement techniques can be classified into two main categories: photoelectric radiation method (PRM) and fluid invasion techniques (FIT). The PRM primarily utilizes the propagation and reflection of physical signals within pores to measure pore size and morphological characteristics. The PRM includes optical microscopy (OM), scanning electron microscopy (SEM), and field emission transmission electron microscopy (FETEM), which offer advantages like high resolution, intuitive visualization, and direct observation3. During the process of sample preparation, the PRM has a tendency to disrupt the pore structure and cause fractures. Additionally, its observation field is limited, which makes it challenging to measure nanometer-scale pores and obtain quantitative parameters such as pore size distribution (PSD)4,5,6. The FIT relies on fluids’ permeation and flow characteristics within pores to measure pore’s structure. The FIT includes mercury intrusion porosimetry (MIP) and gas adsorption method (GAM). The MIP evaluates the PSD by measuring the volume of mercury intruded into the pores under varying pressures. The MIP offers a wide range of PSD and excellent repeatability. However, pore fragmentation can readily occur during the high-pressure testing stage, leading to alterations in the original pore structure. Furthermore, in the mercury withdrawal stage, hysteresis arises due to the interaction between mercury and the pore surfaces, leading to inaccurate PSD measurements7,8. The GAM is employed to analyze the pore structure types and specific surface area by exploiting the adsorption properties of gases on the surface of pores. Primarily utilizing nitrogen and carbon dioxide as adsorbates, the method is known for its high precision and non-destructiveness. However, it is only capable of characterizing pore structure features below 200 nm9,10.

Furthermore, the pore structure of coal exhibits significant heterogeneity, which complicates the storage and transportation of gas, affecting the extraction of CBM11,12,13,14. Mandelbrot15,16 established the theory of fractal geometry, which enables the quantitative characterization of the complexity of pores in coal and sheds light on their impact on the storage and transport processes of CBM. Through LTNA experiments, scholars investigated the fractal characteristics of micropores and performed calculations to determine the fractal dimension of their surface and structure. The findings suggest that a rough pore surface and uniform pore structure facilitate gas adsorption17,18. Zhang19 explained that the heterogeneity of micropores is primarily influenced by metamorphism, microscopic composition, material composition, and specific pore structure characteristics. Some scholars20 have examined the fractal properties of mesopores and macropores through MIP experiments, uncovering a distinct negative correlation between the fractal dimension of these pores and their permeability. Yao et al.21,22,23 found that the heterogeneity of mesopores and macropores was influenced by metamorphic, composition of maceral, and PSD.

Due to the limitations of traditional measurement methods, such as restricted PSD and destruction of the original coal structure, there is an urgent need to find a new method for studying the development characteristics and heterogeneity of full-scale pores. Low-field nuclear magnetic resonance (NMR) reveals the development characteristics of pores by tracking the relaxation behavior of hydrogen-containing fluids within them, offering the advantages of being non-destructive and precise24,25,26. Zhao27 et al. utilized NMR to investigate fractal characteristics of pore and fracture development in loaded coal, introducing a novel method for predicting microcrack propagation based on fractal dimension. Zhou28 established a mathematical relationship between the fractal dimension of pores and permeability by NMR. Ren29,30,31 et al. evaluated the disparity in pore fractal characteristics between NMR and MIP, concluding that the pore heterogeneity obtained through NMR is more in line with reality.

As the world’s largest coal producer, China currently focuses its CBM development efforts primarily on the Qinshui Basin and Ordos Basin. In the future, the anticipated expansion of CBM development is expected to encompass regions such as Zhunnan in Xinjiang, southeastern Henan, and the Huainan-Huaibei in Anhui. In this study, coal samples with varying degrees of metamorphism were collected. Initially, the full-scale pores development characteristics were investigated through LTNA and NMR experiments. Subsequently, fractal theory was employed to demonstrate the heterogeneity of pore structure and elucidate the primary controlling factors that influence this heterogeneity. The research provides scientific evidence to assess CBM reserves and development potential in different regions, optimizing CBM extraction plans, and enhancing CBM production efficiency.

Samples and methods

Sampling and coal analyses

Cubic coal samples with a volume of 30 × 30 × 30 cm3 were collected from the main coal reservoirs of active underground mines in the Liuhuanggou and the No.1890 coal mine in Aiweiergou of Xinjiang province, the Wangzhuang coal mine in the Qinshui basin of Shanxi province, the No. 8 coal mine in Pingdingshan, the No.6 coal mine in Hebi, and the Zhongmacun coal mine in Jiaozuo of Henan province, respectively. Following China National Standards GB/T8899-2013, the maximum reflectance of vitrinite in oil immersion (RO, max) was performed on the polished coal slabs by Precise MY6000A photometer system. According to China National Standards GB/T212-2008, the moisture, volatile matter, and ash yield of coals were obtained by Automatic Proximate Analyzer 5E-6600. The results of RO, max and proximate analysis are shown in Table 1.

Table 1 The results of RO, max and proximate analysis test.
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According to Table 1, the RO, max of coal samples ranges from 0.58 to 3.44%. The relationship between RO, max and the type of coal was studied by Cao32. The coal sample LHG is long flame coal, AWE is gas coal, PDS is 1/3 coking coal, WZ is coking coal, HB is lean coal, and ZM is anthracite, respectively. According to Tao’s research on the relationship between coal type and metamorphism33, long flame and gas coal belong to low-rank coal (LRC), 1/3 coking and coking coal are middle-rank coal (MRC), and lean and anthracite coal are high-rank coal. In summary, coal samples LHG and AWE are LRC, PDS and WZ are MRC, HB and ZM are HRC.

Experimental procedure

LTNA experiments

LTNA experiments are performed by Ultrametrics V-Sorb 2800TP automatic specific surface area and aperture analyzer at the State Key Laboratory cultivation base for gas geology and gas control. The test utilizes 99.999% purity nitrogen, with the temperature set to 77 K, and the measurement aperture range set at 2–200 nm. First, the collected coal samples are prepared as 1–3 mm granular samples by grinding or sieving. Then, the samples are dried at 105 °C for 2 h and vacuumed for 6 h to ensure complete removal of residual gas and moisture. Finally, the experiments are conducted with the samples cooled to approximately 25 °C (room temperature).

NMR experiments

NMR experiments are conducted using Niumag Meso MR23-060H-I analytical instrument at the State Key Laboratory cultivation base for gas geology and gas control. The constant magnetic field is set to 0.5 T, the magnetic resonance frequency is set to 21.67 MHz, and the magnet temperature is controlled at 32 °C. The coal samples are cut into core plugs with a diameter of 25 mm and a height of 50 mm, and then these samples are subjected to NMR measurements.

The specific steps are as follows (Fig. 1):

  1. 1)

    Dry treatment of coal samples. Measure the quality of coal samples in their natural state, and then put the samples into the dryer for at least 12 h at 105 °C. During the drying period, monitor the sample’s quality changes every hour until the samples are completely dried.

  2. 2)

    Saturation treatment of coal samples. Put the dried samples into the vacuum negative pressure saturation device, then saturate the samples with 100% distilled water for 24 h under a pressure of − 0.1 MPa. During saturation, monitor the sample's quality changes every hour until the samples are totally saturated.

  3. 3)

    Measuring NMR amplitude and Calculating porosity. First, measure the NMR amplitude of porosity calibration samples, then establish the correlation equation between porosity and NMR amplitude. Second, determine the NMR amplitude of 100% water-saturated coal samples and convert the amplitude into porosity using the correlation equation. The mathematical relationship between porosity and NMR amplitude is shown in Fig. 2, and the correlation equation is shown in Formula 1.

    $$\varphi = 90.25 + 720.37A$$
    (1)

    where φ is the porosity, A is the NMR amplitude.

  4. 4)

    Centrifugate treatment of coal samples. Transfer the 100%-saturated samples into the Bioridge TG-21 M high-speed centrifuge at 1.38 MPa and 298 K for 2 h to obtain the irreducibly water-saturated samples. According to Yao et al.34 the centrifugal pressure of 1.38 MPa corresponds to a centrifugal capillary pore radius of 100 nm.

  5. 5)

    The measurement of permeability. Measure the parameters of the transverse relaxation time (T2) spectrum, free fluid volume (FFV), and bound fluid volume (BFV) of irreducibly water-saturated samples. The permeability of coal samples can be calculated according to Coates-Timur35:

    $$\kappa_{c} = \left( {\frac{{\varphi_{T} }}{C}} \right)^{4} \times \left( {\frac{{{\text{FFV}}}}{{{\mathrm{BFV}}}}} \right)^{2}$$
    (2)

    where κc is permeability, φT is total porosity, C is a constant, FFV is free fluid volume, BFV is bound fluid volume.

Figure 1
figure 1

Schematic of NMR experimental.

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Figure 2
figure 2

The relationship between porosity and amplitude.

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Fractal theory

Fractal theory of LTNA

At present, the BET model, the Frenkel-Halsey-Hill (FHH) model, and the thermodynamic model are the primary methods used to calculate pore fractal dimension via LTNA. Among these, the FHH model is widely used due to its convenient and broad application range36:

$$\ln V_{p} = T + K{\text{ln}}\left[ {\ln \left( {\frac{{p_{0} }}{p}} \right)} \right]$$
(3)

where Vp is the gas adsorption volume at equilibrium pressure p, ml; p0 is the gas saturated vapor pressure, MPa; K is a fractal parameter; T is a constant.

Yang et al. postulated that during the nitrogen adsorption process, the van der Waals force acting between the nitrogen molecules and the pore interface can be disregarded10. The behavior is attributed to the capillary condensation effect and is governed by the gas–liquid interfacial tension Therefore, the fractal dimension (DL) can be expressed as:

$$D_{L} = K + 3$$
(4)

Li revealed that coal’s adsorption/desorption curves display hysteresis loops at a relative pressure of 0.5, suggesting disparities in pore size and morphology, as well as variations in nitrogen adsorption behavior, preceding and succeeding this critical pressure threshold37.

Utilizing a relative pressure of 0.5 as the threshold, Eq. 3 was applied to model the data of relative pressure and adsorption capacity in two distinct ranges: p/p0 ≤ 0.5 and p/p0 > 0.5. While both ranges showed satisfactory fits, the varying slopes of the fitted lines suggest the presence of distinct fractal dimensions within these two relative pressure domains. At the relative pressure p/p0 ≤ 0.5, the fractal dimension DL1 characterizes the roughness of the pore surface, whereas for p/p0 > 0.5, the fractal dimension DL2 encapsulates the complexity of the pore structure38.

Fractal theory of NMR

There is a corresponding relationship between capillary pressure and pore diameter, as shown in Eq. (5):

$$P_{c} = \frac{2\sigma \cos \theta }{r}$$
(5)

where r is the pore radius, nm. Pc is the capillary pressure, MPa. σ is the surface tension of liquid, N/m. θ is the wetting contact angle.

The fractal expression of capillary pressure curve is:

$$S_{v} = \left( {\frac{{P_{c} }}{{P_{c\min } }}} \right)^{D - 3}$$
(6)

where Sv is the pore volume fraction of wetting phase when the capillary pressure is Pc, %. Pcmin is the inlet capillary pressure corresponding to the maximum pore diameter, MPa.

The T2 of NMR has the following corresponding relationship with pore radius:

$$\frac{1}{{T_{2} }} = \rho \left( \frac{S}{V} \right) = \frac{{F_{s} \rho }}{r}$$
(7)

where ρ is transverse surface relaxation strength μm/ms. S is the pore surface area, cm2. V is the pore volume cm3. Fs is the pore shape factor (for spherical pores, columnar pores and fractures, the Fs is 3, 2 and 1, respectively.).

According to Eqs. (5) and (7), it can be obtained:

$$P_{c} = M\frac{1}{{T_{2} }}$$
(8)

where M is the conversion factor, \(M = \left| {\frac{2\sigma \cos \theta }{{F_{s} \rho }}} \right|\).

According to Eq. (8), the T2 is related to the capillary pressure Pc:

$$P_{c\min } = M\frac{1}{{T_{2\max } }}$$
(9)

where T2max is the maximum transverse relaxation time.

Substitute Eqs. (8) and (9) into Eq. (6):

$$S_{v} = \left( {\frac{{T_{2\max } }}{{T_{2} }}} \right)^{{D_{N} - 3}}$$
(10)

Take logarithms at both ends of the Eq. (10), and the expression of pore fractal dimension calculated by T2 can be obtained:

$$\lg S_{v} = \left( {3 - D_{N} } \right)\lg T_{2} + \left( {D_{N} - 3} \right)\lg T_{2\max }$$
(11)

where Sv is the percentage of pore accumulate volume when transverse relaxation time reaches T2.

Results

The results of LTNA

Characteristics of adsorption/desorption curves by LTNA

Hodot39 proposed the coal pore classification based on pore size, which holds immense significance in the study of gas storage and migration within coal. Hodot’s classification was utilized in the paper, as outlined in Table 2.

Table 2 Pore size classification.
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Utilizing the Hodot classification, Fu40 precisely categorized the adsorption pores (AP) and seepage pores (SP). Specifically, AP, characterized by diameters below 100 nm, primarily facilitate gas adsorption, storage, and desorption. Conversely, SP, with diameters exceeding 100 nm, serve as the primary conduit for gas seepage and production, thereby playing a pivotal role in gas flow.

The characteristics of coals adsorption/desorption curves obtained by LTNA are shown in Fig. 3. The AP structure type can be evaluated by the adsorption/desorption curves and the morphology of the hysteresis loop.

Figure 3
figure 3

Characteristics of adsorption/desorption curves by LTNA.

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Based on Fig. 3a and b, the large hysteresis loop observed across the entire pressure range for the LRCs indicates that the relative pressure during condensation of the AP exceeds that during evaporation. During evaporation, as the relative pressure decreases, the gas–liquid interface within the pores transitions from a double-plane to a double-concave configuration, leading to the formation of this significant hysteresis loop10. The characteristic suggests that the AP possess a parallel plate structure with openings at both ends.

As depicted in Fig. 3c and d, under low relative pressure (p/p0 ≤ 0.5), the adsorption/desorption curves of MRCs align closely. Conversely, at higher relative pressures (p/p0 > 0.5), a hysteresis loop emerges, albeit with a narrow opening. The results indicates that during evaporation, as relative pressure decreases, the evaporation process transitions from larger pores to smaller pores. The gas–liquid interface in pores transforms from a planar to a concave surface, resulting in a small hysteresis loop41,42. The pore size can be determined from nitrogen adsorption/desorption isotherms using the Kelvin Eq. 43:

$$\frac{RT}{V}\ln \left( {\frac{{p_{0} }}{p}} \right) = \frac{4\gamma \cos (\omega )}{d}$$
(12)

where R is the gas constant, T is the absolute temperature, V is the molar volume of liquid nitrogen, p is the vapor pressure, p0 is the saturated vapor pressure, ω is the nitrogen-solid matrix contact angle, and d = 2r (in which r is the pore radius). Based on Eq. (12), when the relative pressure is 0.5, the corresponding pore diameter is about 4 nm44. Therefore, micropores with a diameter smaller than 4 nm are typically conical in shape, having an opening at one end, whereas micropores and transitional pores with a diameter greater than 4 nm are mostly cylindrical, featuring openings at both ends.

As shown in Fig. 3e and f, the hysteresis loop is present across the entire pressure range in HRCs. Notably, the desorption curves exhibit a sharp drop at a relative pressure of approximately 0.5, signifying the prevalence of well-developed thin-neck bottle pores. During condensation, as the relative pressure increases, the bottleneck condenses first, followed by the bottle cavity being filled with condensed liquid. Conversely, during evaporation, the liquid in the cavity cannot evaporate due to the blockage at the bottleneck, leading to the formation of a hysteresis loop. As the relative pressure decreases, the liquid at the bottleneck starts to evaporate, and the gas–liquid interface transitions from a planar to a single concave surface10. Once the liquid at the bottleneck evaporates completely, the liquid in the cavity is suddenly released and undergoes rapid evaporation, leading to a sharp descent in the desorption curve.

Characteristics of specific surface area and pore volume of AP

Figure 4 depicts the variation in specific surface area and pore volume with respect to the pore diameter of coals exhibiting varying degrees of metamorphism.

Figure 4
figure 4

The variation of specific surface area and pore volume with pore diameter.

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As shown in Fig. 4a and b, the specific surface area and pore volume of LRCs stand at 0.73 m2/g and 2.22 × 10–3 cm3/g, respectively. The peak value of the specific surface area is concentrated in the range of 2.20–44.91 nm, suggesting that micropores and transitional pores with small diameters contribute significantly to enhancing the pore specific surface area. The pore volume curve exhibits two distinct peaks, averaging out to 25.13 nm and 148.42 nm, indicating that transitional pores and mesopores are the primary contributors to the pore volume. Since micropores possess higher specific surface areas while transitional pores and mesopores offer larger pore volumes, the AP of LRCs exhibit favorable properties for gas adsorption, desorption, and diffusion.

From Fig. 4c and d, the MRCs exhibit a specific surface area of 0.67 m2/g and a pore volume of 2.55 × 10–3 cm3/g. The peak value of the specific surface area is concentrated within the range of 2.20–3.22 nm, indicating a significant contribution from micropores to the specific surface area. The pore volume curve exhibits a single peak, with the peak value occurring at 146.33 nm, suggesting that the pore volume is primarily contributed by transitional pores and mesopores. Due to the higher specific surface area of micropores and the larger pore volume of mesopores, the AP of MRCs exhibit excellent gas adsorption properties but may hinder desorption and diffusion.

As shown in Fig. 4e and f, the specific surface area and pore volume of HRCs are 4.40 m2/g and 8.43 × 10−3cm3/g, respectively. The peak value of the specific surface area is concentrated in the range of 2.20–8.47 nm, indicating that micropores make a significant contribution to specific surface area. The pore volume curve exhibits two peaks, with the highest values occurring at 27.22 nm and 69.65 nm, indicating that the pore volume is primarily contributed by transitional pores. Since HRCs possess numerous thin-neck bottle pores, their AP are favorable for gas adsorption but may pose difficulties in desorption and diffusion.

The results of NMR

T 2 spectrum characteristics by NMR

The T2 spectrum characteristics from NMR experiments are shown in Fig. 5. Based on Hodot’s classification system for coal pores, the T2 spectrum reveals the ranges for various pore types: micropores exhibit a T2 distribution spanning from 0.01 to 25 ms, transitional pores range from 2.5 to 10 ms, mesopores fall within 10–50 ms, and macropores possess a T2 distribution exceeding 50 ms45.

Figure 5
figure 5

T2 spectrum characteristics by NMR.

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As depicted in Fig. 5, the coal sample LHG exhibits two distinct peaks. The first peak spans from 0.01 to 1.72 ms, comprising 34.50% of the total peak area. The second peak ranges from 1.72 to 666.99 ms, accounting for 65.50% of the peak area. Similarly, the coal sample AWE also displays two peaks. The first peak occurs between 0.01 and 5.54 ms, making up 64.95% of the peak area. The second peak extends from 5.54 to 541.59 ms, representing 35.05% of the total peak area. The coal sample PDS exhibits three distinct peaks. The first peak spans from 0.01 to 2.77 ms, comprising 72.96% of the total peak area. The second peak ranges from 2.77 to 135.10 ms, accounting for 26.41% of the peak area. The third peak extends from 135.10 to 541.59 ms, making up only 0.63% of the total. The coal sample WZ also displays three peaks. The first peak occurs between 0.01 and 3.18 ms, representing 81.69% of the peak area. The second peak spans from 5.54 to 38.72 ms, accounting for 8.76% of the total. Finally, the third peak ranges from 38.72 to 766.34 ms, comprising 9.55% of the peak area. The coal sample HB exhibits two peaks. The first peak ranges from 0.01 to 2.59 ms, comprising 91.65% of the total peak area. The second peak spans from 7.32 to 51.11 ms, accounting for 8.35%. And the coal sample ZM also has two peaks. The first peak occurs between 0.01 and 3.65 ms, representing 99.87% of the total peak area. The second peak spanning from 382.75 to 1335.45 ms, only accounts for 0.13% of the total.

The PSD law is illustrated in Fig. 6. Within LRCs, pores of varying diameters are well developed. The transition between the first and second T2 peaks is seamless, suggesting a robust interconnectivity among the diverse pores. In MRCs, micropores account for 73.65%, while transitional pores make up 12.68%. For the coal sample WZ, there is a distinct gap between the first and second peaks, indicating a narrow throat channel connecting micropores and transitional pores. This suggests poor connectivity between these pore types. However, the continuous nature of the second and third peaks signifies good connectivity between transitional pores, mesopores, and macropores. In HRCs, micropores constitute a significant 92.74% of the pore volume, while transitional pores account for a mere 4.66%. There exists a distinct zero-value region between the first and second peaks, pointing to a poor interconnectivity between micropores and transitional pores.

Figure 6
figure 6

PSD law.

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Porosity and permeability

The T2 spectrum and cumulative porosity distribution of coals under 100% water-saturated and irreducible-watered conditions are presented in Fig. 7. As indicated in Table 3, various parameters such as total porosity (φT), effective porosity (φE), T2 cutoff value (T2C), free fluid volume (FFV), bound fluid volume (BFV), and permeability (κc) have been determined using the “saturation-centrifugation” method45.

Figure 7
figure 7

The T2 spectrum and cumulative porosity distribution.

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Table 3 The parameters obtained by NMR.
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The T2C divides the T2 spectrum into two parts: free fluid and bound fluid, in which the free fluid is located in connected and open pores or fractures and can be completely removed by centrifugation. Bound fluid resides in narrow, small, or closed/semi-closed pores. Its confinement is significantly influenced by capillary force and surface adsorption, preventing its escape during centrifugation.

As the level of metamorphism rises, the φT gradually increases, whereas the proportions of φE, FFV, and κc systematically decrease, as evident from Table 3. The average φT of HRCs is 1.95%, the T2C is 0.81 ms, the φE accounts for 62.07%, and the κc is 4.75 × 10–5 μm2. These findings suggest that the interconnected and accessible pores within the HRCs are extensively developed, favoring gas flow yet not favoring gas storage. The average φT of MRCs is 3.72%, the T2C is 2.35 ms, the φE accounts for 20.31%, and the κc is 1.54 × 10–5 μm2. The average φT of HRCs is 5.03%, the T2C is 1.73 ms, the φE accounts for 7.55%, and the κc is 5.67 × 10–6 μm2. That indicates that the closed/semi-closed pores within the HRCs are quite developed, which is advantageous for gas adsorption but disadvantageous for gas flow.

Discussion

Fractal characteristics of AP

Based on the LTNA results, at pressures relatively low (p/p0 ≤ 0.5), nitrogen adsorption is primarily governed by van der Waals forces. However, at pressures relatively high (p/p0 > 0.5), nitrogen adsorption is predominantly influenced by capillary condensation, and the adsorption process becomes closely associated with the development of pore structure. Equation (3) models the mathematical relationship between the relative pressure (p/p0) and nitrogen adsorption volume across low and high pressures, as depicted in Fig. 8. The fractal dimension of AP is determined using Eq. (4), which is presented in Table 4.

Figure 8
figure 8

The mathematical relationship between the relative pressure (p/p0) and nitrogen adsorption volume.

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Table 4 The fractal dimension of AP.
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According to Table 4, the fractal dimension DL1 varies between 2.13 and 2.45 at relatively lower pressure (p/p0 ≤ 0.5), with an average R2 value of 0.98. Conversely, at higher pressure levels (p/p0 > 0.5), the fractal dimension DL2 ranges from 2.56 to 2.77, with an average R2 value of 0.99. It is noteworthy that the DL1 values are generally lower than the DL2 values, yet there appears to be no discernible correlation between the two fractal dimensions. The shape of the hysteresis loop is closely linked to the fractal dimension DL2, as evidenced by the analysis of nitrogen adsorption and desorption curves.

The HRCs exhibit a pronounced hysteresis loop opening, characterized by a distinct inflection in the desorption curve and well-formed thin-neck bottle pores. This results in a maximized DL2 value, ranging from 2.72 to 2.77. Similarly, the LRCs display a wide hysteresis loop opening, with a DL2 value between 2.60 and 2.63. In contrast, the MRCs exhibit a narrower hysteresis loop opening, resulting in a minimized DL2 value ranging from 2.53 to 2.56. In conclusion, a larger and more intricate hysteresis loop opening correlates with an increased heterogeneity of the AP structure.

Study on the heterogeneity of AP

By examining the variations in fractal dimension across different metamorphic coals, the heterogeneous evolution traits of AP during coalification can be elucidated. The correlation between fractal dimension and the pore structure of AP is illustrated in Fig. 9.

Figure 9
figure 9

The correlation between the fractal dimension and the pore structure of AP.

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As shown in Fig. 9a, during the long flame coal stage (RO,max < 0.65%), hydroxyl and carboxyl functional groups exhibit robust hydrophilic properties due to their pronounced development. This extensive water filling within pores contributes to a complex pore surface, leading to elevated DL1 and DL2 values. As the coal transitions from long flame to 1/3 coking coal (0.65% ≤ RO,max < 1.20%), following the initial coalification phase, hydrophilic groups decrease, resulting in a reduced water content within pores. Simultaneously, thermally induced pore formation intensifies. Consequently, the complexity of the pore surface and structure diminishes progressively, leading to a gradual reduction in DL1 and DL2. During the progression from 1/3 coking coal to lean coal (1.20% ≤ RO,max < 2.10%), secondary hydrocarbon generation occurs during the second and third leaps of coalification, which prompts the formation of large aromatic clusters and significant metamorphic pore development, leading to a pronounced increase in the quantity and volume of micropores and transitional pores. Consequently, the complexity of the pore surface and structure gradually intensifies, resulting in elevated DL1 and DL2 values. From lean coal to anthracite (2.10% ≤ RO,max), the high metamorphic degree maximizes the amount and volume of micropores and transitional pores, thereby saturating the complexity of the pore surface and structure. Accordingly, DL1 and DL2 values remain consistently high.

As depicted in Fig. 9b, the specific surface area is predominantly determined by micropores and transitional pores. When the specific surface area increases, the proportion of AP and the roughness of the pore surface also gradually augment, and once the specific surface area reaches a critical threshold, the roughness of the AP surface attains its maximum, thus the DL1 exhibits a power-law growth pattern (R2 = 0.88) with increasing pore specific surface area. Furthermore, the pore volume exhibits a positive correlation with specific surface area, and the development of micropores and transitional pores enhances the volume of AP. Consequently, with an increase in specific surface area, DL2 also follows a power-law growth pattern (R2 = 0.75).

As illustrated in Fig. 9c, the gradual increase in AP development with growing pore volume and the progressive enhancement in pore structure complexity result in the emergence of a power-law growth rhythm (0.90 ≤ R2) for both DL1 and DL2.

Fractal characteristics of SP

NMR results indicate that the T2 distribution of SP exceeds 10 ms. The T2 and proportion of cumulative pore volume was fitted by Eq. (11), and the derived results are presented in Fig. 10, while the fractal dimension of SP is documented in Table 5.

Figure 10
figure 10

The mathematical relationship between T2 and proportion of cumulative pore volume.

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Table 5 The fractal dimension of SP.
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As evident from Table 5, the DN of SP gradually increases with increasing metamorphism. In the stage of long flame coal (RO,max < 0.65%), the structure of SP simplifies due to sedimentary compaction and pore hydrophilicity, leading to a relatively low DN. However, as the coal transitions from long flame to 1/3 coking coal (0.65% ≤ RO,max < 1.20%), the first coalification jump attenuates the influence of sedimentary compaction and pore hydrophilicity, and temperature-induced hydrocarbon generation leads to the formation of gas and pores, enhancing pore structure complexity and progressively increasing the DN. During the transition from 1/3 coking coal to lean coal (1.20% ≤ RO,max < 2.10%), the second and third leaps of coalification induce secondary hydrocarbon generation, leading to a significant rise in the complexity of the pore structure and a continuous increase in the DN. From lean coal to anthracite (2.10% ≤ RO,max), the coal matrix undergoes a significant densification following the fourth leap of coalification, the complexity of SP attains its peak, and the DN stabilizes at a high value, approaching 3.

Study on the heterogeneity of SP

Figure 11 illustrates the correlation between the DN and the structure of SP. As the proportion of SP and FFV increase, a linear decline in DN is observed. When the φT is constant, a higher proportion of SP signifies ample gas flow space and a simpler SP structure, ultimately leading to a decrease in DN. Alternatively, when the amount of SP is determined, a higher proportion of FFV signifies a superior development of interconnected and open pores, resulting in exceptional connectivity among pores of varying radius and stable uniformity of the pore structure. Consequently, an increase in the FFV proportion leads to a reduction in DN.

Figure 11
figure 11

The relationship between DN and the structure of SP.

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The mathematical fitting result between the DN and κc is depicted in Fig. 12. The data reveals a robust negative power law relationship between these two parameters as DN increases from 2.92 to 2.99. Based on previous analysis, the heterogeneity of SP is primarily influenced by PSD and pore connectivity. Therefore, the increase of DN indicates a reduction in the gas flow space or an enhancement in gas flow resistance, leading to a corresponding decrease in κc.

Figure 12
figure 12

The mathematical fitting results between DN and κc.

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Conclusion

The pore development characteristics and heterogeneity properties of full-scale pores in various metamorphic coals (0.58% ≤ RO,max ≤ 3.44%) were investigated through LTNA and NMR experiments.

  1. 1)

    The pores with varying diameters in LRCs were well-developed, and the connectivity of pores was favorable. In MRCs, the micropore content was 73.56%, whereas the connectivity among transitional pores, mesopores, and macropores was suboptimal. As for HRCs, the proportion of micropores reached 92.74%, with a significant number of micropores being closed or semi-closed. Additionally, the connectivity between micropores and transitional pores in HRCs was poor.

  2. 2)

    As coal metamorphism progressed, the roughness of the AP surface and the complexity of the AP structure initially declined but later increased due to coalification. The augmentation in pore specific surface area and pore volume contributed to an escalation in both pore surface roughness and the complexity of the pore structure.

  3. 3)

    The heterogeneity of the SP structure increased as coal metamorphism intensified, while the uniformity of pore structure was enhanced with the increase in SP content. When the proportion of SP was determined, the increase in FFV indicated a greater connectivity between pores and a decreased complexity of pore structure.