Multifractal Characterization of Pore Heterogeneity and Water Distribution in Medium- and High-Rank Coals via Nuclear Magnetic Resonance

Abstract

Comprehensive assessment of pore structure and multiphase water distribution is critical to the flow and transport process in coalbed methane (CBM) reservoirs. In this study, nuclear magnetic resonance (NMR) and multifractal analysis were integrated to quantify the multiscale heterogeneity of nine medium- and high-rank coals under water-saturated and dry conditions. By applying the box-counting method to transverse relaxation time (T₂) spectra, multifractal parameters were derived to characterize pore heterogeneity and residual water distribution. The influencing factors of pore heterogeneity were also discussed. The results show that pore structures in high-rank coals (HCs) exhibit a broader multifractal spectrum and stronger rightward spectrum than those of medium-rank coals, reflecting micropore-dominated heterogeneity and the complexity induced by aromatization in HCs. The vitrinite content enhances micropore development, increasing the heterogeneity and complexity of pore structure and residual water distribution. Inertinite content shows opposite trends compared to vitrinite content for the effect on pore structure and water distribution. Volatile yield reflects coal metamorphism and thermal maturity, which inversely correlates with pore heterogeneity and complexity. Residual water mainly distributes to adsorption pores and pore throats, shortening T₂ relaxation (bound water effect) and reducing spectral asymmetry. The equivalence of the multifractal dimension and singularity spectrum validates their joint utility in characterizing pore structure. Minerals enhance pore connectivity but suppress complexity, while moisture and ash contents show negligible impacts. These findings provide a theoretical reference for CBM exploration, especially in optimizing fluid transportation and CBM production strategies and identifying CBM sweet spots.

1. Introduction

Coalbed methane (CBM), a critical component of global energy transitions, offers a cleaner alternative to conventional fossil fuels and contributes to greenhouse gas reductions through methane recovery and CO₂ sequestration [1,2,3]. As a dual-porosity medium, coal reservoirs comprise two distinct yet interconnected pore–fracture networks and are characterized by complex geological conditions, with methane mainly stored in adsorbed and free states. Hodot [4] categorized the pore system as micropores (<10 nm), transitional pores (10–100 nm), mesopores (100–1000 nm), and macropores (>1000 nm) by size. The inherent heterogeneity in the pore system suggests that pores exhibit spatial non-uniformity in terms of random pore size distribution, complex pore structure, pore space connectivity, and pore shape due to coalification degree, mineral components, maceral composition, reservoir characteristics, and brittleness, which strongly control the fluid flow and gas production [5,6,7,8]. The water distribution in pores can affect the mechanical properties, gas evaluation, and CBM production [9]. Therefore, the pore heterogeneity and water distribution in coal reservoirs are pivotal to understanding gas storage, transport mechanisms, and recovery efficiency, particularly for middle- and high-rank coals that dominate many energy-rich basins [10,11,12,13,14].

Traditional characterization techniques include qualitative and quantitative testing methods. Qualitative methods typically employ 2D imaging or 3D reconstruction to visualize pore characteristics, but they have certain limitations, such as a restricted field of view and insufficient resolution for nanopores. Quantitative methods often use fluid injection, gas adsorption, and radiation to analyze pore structure [15]. Mercury intrusion porosimetry (MIP) and low-pressure gas adsorption (LPGA) have been widely employed for characterizing pore size distributions [16,17,18,19,20]. However, these techniques often fail to capture dynamic water distributions or quantify multiscale pore heterogeneity under varying hydration states. Nuclear magnetic resonance (NMR), with its unique capability to noninvasively quantify both pore geometry and water distributions, has emerged as a powerful tool for providing valuable insights into CBM exploration [11,15,21,22,23]. Residual water retention in pores may reduce effective permeability and affect gas migration, while previous studies largely focused on pore structures and water migration through the T₂ spectral area under water-saturated and centrifuged conditions [24,25]. In contrast to the improvements in coal seam reformability methods, the centrifugal method overestimated the amount of bound water due to the poor pore connectivity and the presence of ink bottle pores in middle- and high-rank coals, misunderstanding CBM’s recovery ability. Thus, the residual water distribution and pore structure variations in complex coal matrices need to be further studied.

Monofractal models characterize simple fractal behavior with fractal dimension (D), which simplifies complex pore systems into single scaling exponents, failing to explain types of erratic variation [15,26]. Multifractal analysis has been applied in pore structure characterization due to its capacity to quantitatively characterize the heterogeneity of samples across multiple scales, overcoming the limitations of monofractal methods that oversimplify complex pore systems into single scaling exponents [16,26,27,28]. Unlike monofractal models, multifractal parameters (e.g., Hurst exponent H, dimension span D₋₁₀ − D₁₀, singularity spectrum width α₁₀₋ − α₁₀₊) precisely quantify the coexistence of diverse pore structures: adsorption-dominated micropores, capillary-controlled mesopores, and fracture-dominated macropores. Recent applications in shale and tight sandstones reveal that multifractal parameters effectively correlate with pore heterogeneity, fluid mobility, and storage capacity [5,29,30,31]. Most studies on coal pore systems mostly combine data from MIP, LPGA, and scanning electron microscopy, while informative, pore structure distribution data may be distored due to methodological artifacts (mercury-induced pore deformation, vacuum-drying effects), particularly the distributions of different fluid phases is ignored. Thus, it is necessary to integrate NMR and multifractal theory to explore the pore heterogeneity and multiphase water distribution of coal at different scales.

Coal’s rank reflects its thermal maturity, determined by geologic conditions (e.g., temperature, pressure, burial history). According to the classification of coal’s metamorphism degree, coal can be divided into low-rank coal (LC: R_o,max < 0.65%), medium-rank coal (MC: 0.65% < R_o,max < 2.00%), and high-rank coal (HC: 2.00% < R_o,max < 4.00%) [32]. Yunnan is one of the regions with the richest CBM resources in China and the coal rank distribution is dominated by medium and high coal ranks [33]. This study systematically investigates pore structure heterogeneity and multiphase water distribution in medium- and high- rank coals under water-saturated and dry conditions through integrated NMR T₂ spectral analysis and multifractal theory. The MIP test was used to verify the reliability of the pore size distribution obtained via NMR. The principal objectives of this study are as follows: (1) to address pore structure heterogeneity and the residual water distribution in pore structure by comparing dual-state (water-saturated vs. dried) NMR characteristics of different rank coal samples; (2) to discuss the relationships between the multifractal parameters and T₂ spectra; and (3) to identify the key factors influencing pore structure and residual water distribution heterogeneity. Our findings contribute to a more comprehensive understanding of the pore structure heterogeneity at different coalification degrees and provide a reference for water migration in the process of CBM extraction.

2. Experimental Methods and Analyses

2.1. Sample Collection and Preparation

The research area is located in the eastern part of Yunnan province, China, with multiple coal seams in the formation. The nine fresh block coal samples (20 × 20 × 20 cm³) were obtained in the Upper Permian Xuanwei and Longtan formation in the research area. Four samples, including coking coal from Qingping mine (M1), Jinxin mine (M2), Xinxin mine (M3), and lean coal from Xiaoheiqing (M4), were taken from Enhong mining area. The other five anthracite samples were from Dage mine (H1), Xiongda mine (H2), Shewu mine (H3), Hongfa mine (H4), and Danshuo mine (H5) in Laochang mining area. The depth of coal seams is from 500 to 800 m, and the tectonic development is relatively stable. The information on samples is shown in

Table 1. Basic physical parameters of coal samples.

Sample ID	Coal Seam	Stratigraphic	Ro,max (%)	Maceral Composition (%)				Proximate Analysis (%)				Porosity (%)
				Vitrinite	Inertinite	Exinite	Minerals	Mad	Ad	Vdaf	FCd
M1	7	P3x	1.05	50.30	37.68	2.90	9.12	0.64	13.77	25.93	63.87	4.19
M2	8	P3x	1.15	66.08	25.67	2.78	5.47	1.12	8.41	24.69	68.98	4.63
M3	8	P3x	1.27	67.40	23.11	/	9.49	1.27	14.55	26.58	62.74	3.87
M4	21	P3x	1.69	65.93	27.03	/	7.04	0.84	10.92	16.47	74.41	2.82
H1	7	P3l	2.30	77.03	19.59	/	3.38	2.61	8.23	7.68	84.73	2.22
H2	8	P3l	2.45	85.17	6.59	/	8.24	1.29	13.17	7.29	80.50	2.75
H3	3	P3l	2.59	87.18	12.18	/	0.64	1.54	9.72	6.88	84.06	3.24
H4	7	P3l	2.65	74.87	18.98	/	6.15	2.67	10.12	7.18	83.43	2.43
H5	3	P3l	2.67	75.81	23.12	/	1.07	1.75	14.94	9.72	76.80	2.56

. The collected samples were then processed into powder (60–80 mesh) in a pulverizer and cylinders (Φ25 mm × 50 mm).

2.2. Experimental Process

The basic properties of the coal samples were determined from proximate and microscopic maceral analysis following the Chinese national standard GB/T 212-2008 [34] and GB/T 8899-2013 [35], respectively. The test results are shown in Table 1.

For the NMR test, the coal samples were firstly dried in a vacuum drying oven at 65 °C until the weight was unchanged. The dried samples were then placed in a core holder cavity of the MesoMR23-60H NMR analyzer (Figure 1). After that, the dry samples were vacuumed in a vacuum device for 12 h and saturated with distilled water under 16 MPa for 48 h. The saturated samples were also placed in the core holder cavity to obtain the transverse relaxation time spectrum. The NMR testing parameters were set as follows: echo time was 0.2 ms, wait time was 3000 ms, number of echoes was 16,000, scanning numbers were 32, and ambient temperature was 22.5 °C. The T₂ spectra of dried and saturated coal samples can be used to analyze the residual water distribution and the variation in the pore–fracture system of different coal ranks. For MIP testing, an Autopore IV 9500 (Micromeritics Instrument, Norcross, GA, USA) mercury intrusion tester was utilized. The samples were crushed into granular grains with dimensions of less than 1 cm³ and were dried at 70 °C for 12 h in a drying oven before testing [12].

2.3. NMR and Multifractal Theories

The T₂ spectrum is calculated using the SIRT inversion algorithm from the echo attenuation signal. The relaxation time (T₂) consists of three parts: volume relaxation time (T_2B), diffusional relaxation time (T_2D), and surface relaxation time (T_2S). Given that T₂ has a positive correlation with pore size, T_2B and T_2D can be neglected in a single fluid and uniform magnetic field [6,10,36]. T₂ can be expressed as follows:

$${\displaystyle \frac{1}{T_2}}={\displaystyle \frac{1}{T_{2s}}}=\rho {\displaystyle \frac{Fs}{r}},$$

(1)

where ρ is the rapid surface relaxation; F_S is a geometric form factor; and r is the pore radius (μm). Previous studies have classified that pores with a T₂ value of less than 2.5 ms represent adsorption pores (less than 100 nm), those with a T₂ value between 2.5 and 100 ms are seepage pores (100–10,000 nm), and those with a T₂ value greater than 100 ms are fractures (larger than 10,000 nm) [21,37,38,39].

The fractal method is capable of characterizing the morphological features of the pore–fracture system, while the multifractal method offers more pore heterogeneity information than the fractal method [27,40,41]. In this study, the cumulative T₂ spectra curve is divided into N(ɛ) segments, with a scale of ɛ based on the box-counting method, resulting in a mass probability density p_i(ɛ) for the ith section. The details of the multifractal theory can be found in previous studies [26,42]. And the multifractal parameters D_q (multifractal dimension), τ(q) (mass distribution function), α(q) (the singularity strength), and f(α) (the multifractal singular spectrum) can be expressed as follows:

$$\{\begin{array}{cc}{D}_q={\displaystyle \frac{1}{q-1}}\underset{\epsilon \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}{\sum }_{i=1}^{N\left(\epsilon \right)}{p}_i{\left(\epsilon \right)}^q}{\mathrm{l}\mathrm{o}\mathrm{g}\left(\epsilon \right)}}\hfill & q\in [-10,1)\cup (1,10]\\ D_1=\underset{\epsilon \to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{\displaystyle \frac{{\sum }_{i=1}^{N\left(\epsilon \right)}{p}_i(1,\epsilon )\mathrm{l}\mathrm{o}\mathrm{g}\left[p_i\right(1,\epsilon \left)\right]}{\mathrm{l}\mathrm{o}\mathrm{g}\left(\epsilon \right)}}& q=1\end{array},$$

(2)

$$\tau \left(q\right)=(q-1)D_q$$

(3)

$$\{\begin{array}{c}\alpha (q)={\displaystyle \frac{d\tau \left(q\right)}{dq}}\hfill \\ f\left(\alpha \right)=q\alpha \left(q\right)-\tau \left(q\right)\end{array}$$

(4)

$$H={\displaystyle \frac{1}{2}}(D_2+1)$$

(5)

where q represents an order of the box matrix (−∞ < q < +∞). In this study, the value of q ranges from −10 to 10, with an interval of 0.1. N_i(ε) represents the cumulative pore volume of the ith section, and H represents the Hurst exponent, which can be used to specify the pore connectivity.

3. Results

3.1. Coal Characteristics

Four coal samples from Enhong mining area are classified as medium-rank coal, with R_o,max ranging from 1.05 to 1.69%. The other five coal samples from Laochang mining area are classified as high-rank coal, with R_o,max ranging from 2.30 to 2.67% (Table 1). For the maceral composition, vitrinite constitutes the predominant maceral group across all samples, accounting for 50.30–67.40% in MC and 74.87–87.18% in HC. In addition, there is a positive relationship between R_o,max and the vitrinite content (Figure 2a). The content of inertinite varies between 50.30% and 67.40% in MC, whereas in HC, it varies from 74.87 to 87.18%. Minerals range from 0.64% to 9.49%, with an average of 5.62%, predominantly composed of clay minerals, carbonates, and oxides. Notably, some middle-rank coal samples contain exinite, with a minor proportion of approximately 3% (Table 1).

Proximate analyses demonstrate that the moisture content (M_ad), ash yield (A_d), volatile yield (V_daf), and fixed carbon (FC_d) of all samples are in the range of 0.64–2.67% (average 1.53%), 8.23–14.94% (average 11.54%), 6.88–26.58% (average 14.71%), and 62.74–84.06% (average 75.50%), respectively (Table 1). A slight positive relationship (R² = 0.48) exists between the R_o,max and M_ad content (Figure 2b). Volatile matter displays a significant decrease (R² = 0.96) with the increase in the coalification degree (Figure 2c). The porosity content exhibits a bimodal distribution, initially decreasing then slightly increasing at higher maturation stages (Figure 2d), potentially indicating the structural reorganization of micropores during thermal evolution.

3.2. T₂ Spectrum Distribution and Validation

3.2.1. T₂ Characteristics of Saturated and Dried Coal Samples

The T₂ spectrum distributions under water-saturated and dry conditions of different coal samples are shown in Figure 3. In a water-saturated state, T₂ distributions of all samples exhibit three peaks. Among them, peak 1 (P1) from 0.01 to 2.5 ms is dominant and independent of the other two peaks (peak 2 from 2.5 to 100 ms and peak 3 larger than 100 ms), indicating a high content of micropores for gas adsorption and poor connectivity with other pore types. There is no obvious boundary between peak 2 (P2) and peak 3 (P3), demonstrating partial pore connectivity between seepage pores and fractures. In addition, the absolute volume of pore space in high-rank coal samples decreases compared with medium-rank coal samples, especially in seepage space and fractures.

The residual water distribution under dry conditions highlights fluid mobility differences. The T₂ spectrum exhibits only two leftmost peaks (P1 and P2), with P1 predominating in the dried samples, indicating the complete evaporation of water from fractures (Figure 3). The persistence of P1 across all samples underscores the dominance of capillary-bound water in adsorption pores, which restricts the fluid flow. Dried T₂ spectra of all samples retain distinct peak boundaries, confirming poor connectivity between adsorption pores and seepage pores.

The proportions of different pore types in all samples in water-saturated and dry states are calculated, as plotted in Figure 4. The adsorption pores, which predominate in all samples in water-saturated conditions, account for 54.27–86.08% (with an average of 72.22%) and 80.86–90.31% (with an average of 88.20%) of the total pore volume in medium- and high-rank coals, respectively. Seepage pores in medium-rank coals occupy 9.22–18.82% (mean of 12.70%) of the total volume, contrasting with a narrower and lower range in high-rank coals (3.77–8.42%, mean of 5.62%). Fractures in medium-rank coals vary widely (4.7–26.91%, mean of 15.08%), whereas high-rank coals show reduced fracture proportions (3.91–11.63%, mean of 6.17%) (Figure 4a). Notably, in dry conditions, fractures and seepage pores retain negligible moisture, with adsorption pores dominating the pore network, particularly in high-rank coals (Figure 4b). The variation in results indicates the heterogeneity and complexity of the coal pore structure and water migration, which are crucial for understanding the occurrence potential and flow characteristics of CBM.

3.2.2. Validation of Pore Size Distribution Between NMR and MIP

Parallel MIP measurements are conducted on four representative samples (M3, M4, H1, and H5) to verify the pore size distribution derived from NMR testing. The hysteresis loops between the injection and ejection curves in samples M3 and M4 are wider than those of H1 and H2 (Figure 5), indicating more open or connected pores in medium-rank coals [12]. The curves of incremental pore volumes are multi-peaked, and the main peak values are located in adsorption pores (Figure 5). In addition, the peaks in the range of seepage pores in M3 and M4 are greater than those of H1 and H2, suggesting that more seepage pores exist in MC.

Figure 6 shows the comparison of the percentages of different pore types obtained via the MIP and NMR methods. Data from MIP testing underestimated the adsorption pore fractions by 3.6–8.9%, while overestimating seepage pores by 1.8–8.8% compared to the NMR results. Fracture proportions showed variable deviations, with MIP underestimating fractures in M3 by 4.8%, but overestimating them in H5 by 4.0%. These variations are mainly caused by two factors: Firstly, MIP testing can only identify pores with a diameter larger than 3.6 nm, leading to an underestimation of adsorption pores. Secondly, granular samples used in MIP testing may disrupt natural fracture networks, while cylindrical samples (2.5 cm × 5 cm) used in NMR preserve in situ pore structures. Overall, the differences between the two methods are less than 9%, which verifies the reliability of the NMR method.

3.3. Multifractal Characteristics

3.3.1. The Generalized Dimensional Spectrum Characteristics

The double logarithm plots between the partition function χ(q,ε) and the box scale ε for samples in different situations are shown in Figure 7. Strong linear relationships (R² > 0.94) between log χ(q,ε) and logε imply that the pore structure distribution and residual water distribution correspond to multifractal features [43]. The characteristics of a negative relationship between log χ(q,ε) and logε when q < 0, and a positive relationship between them when q > 0 under water-saturated conditions indicate that micropores are relatively concentrated in smaller pore spaces (Figure 7a,b). Similar characteristics of log χ(q,ε) and logε in dry conditions suggest that the residual water is more concentrated in smaller pores and struggles to migrate (Figure 7c,d).

Figure 8 illustrates the generalized dimensions D_q of MC and HC under varied conditions, where D_q exhibits a pronounced inverted S-shape, with steep declines in D_q at negative q values (q < 0, low-probability regions corresponding to micropores) and gradual decreases at positive q values (q > 0, high-probability regions linked to macropores) [6,10,40,44]. This asymmetry underscores the enhanced heterogeneity in low-probability regions. In addition, compared with the water-saturated samples, for the dried samples, the value of D_q increases when q < 0 and decreases when q > 0. The increasing trend in D₋₁₀ and the decreasing trend in D₁₀ is due to the increasing proportion of residual water in adsorption pores in dry conditions and the significant difference between residual water in adsorption pores and seepage pores, respectively. Moreover, the trend is more pronounced in HC, indicating that the distribution of residual water in the pores has stronger heterogeneity, especially in HC.

Table 2. Parameters of the generalized fractional dimensional spectrum under different conditions.

Sample ID	Water-Saturated Conditions						Dry Conditions
	H	D1	D2	D−10 − D10	D−10 − D0	D0 − D10	H	D1	D2	D−10 − D10	D−10 − D0	D0 − D10
M1	0.9703	0.9430	0.9407	2.5849	2.5214	0.0634	0.9286	0.8780	0.8573	2.8602	2.6749	0.1853
M2	0.9701	0.9428	0.9403	2.5956	2.5294	0.0663	0.9283	0.8767	0.8566	2.8632	2.6767	0.1865
M3	0.9700	0.9420	0.9400	2.6055	2.5428	0.0627	0.9263	0.8740	0.8527	2.8842	2.6920	0.1922
M4	0.9703	0.9426	0.9406	2.5977	2.5383	0.0595	0.9276	0.8758	0.8551	2.8719	2.6836	0.1883
H1	0.9643	0.9301	0.9286	2.8695	2.7951	0.0744	0.8970	0.8339	0.7940	3.1232	2.8380	0.2852
H2	0.9644	0.9289	0.9287	2.9091	2.8362	0.0729	0.8672	0.7875	0.7345	3.4536	3.0945	0.3591
H3	0.9607	0.9219	0.9214	3.1475	3.0617	0.0858	0.8686	0.7886	0.7373	3.4632	3.1081	0.3551
H4	0.9639	0.9289	0.9278	2.8986	2.8240	0.0746	0.8859	0.8192	0.7717	3.2151	2.8992	0.3159
H5	0.9636	0.9285	0.9273	2.9170	2.8383	0.0787	0.8930	0.8260	0.7860	3.1621	2.8678	0.2943

lists the multifractal parameters of D_q under varied conditions. When the multifractal matrix q = 0, 1, 2, D_q represents the capacity dimension, information dimension, and the correlation dimension, respectively. Specifically, D₀ offers an overall assessment of the space—filling capacity of the pore system. D₁ is associated with the entropy value of the system and can effectively indicate the heterogeneity of pore distribution. A higher D₁ value suggests a broader range of pore sizes and a more uniform pore distribution [16,45]. Meanwhile, D₂ reflects the inter-relationship between different components of the pore structure. D_min (corresponding to D₋₁₀) and D_max (corresponding to D₁₀) represent the generalized dimensions at the extreme values.

For all samples, D₀ = 1, thus a common relationship D₀ > D₁ > D₂ can be observed, which is a clear indication of the samples’ multifractal features [46]. As illustrated in Table 2, the D₁ values of water-saturated MC range from 0.9420 to 0.9430 (mean of 0.9426), while those of HC range from 0.9219 to 0.9301 (mean of 0.9277), indicating that the pore size distribution of the MC is more uniform than that of HC. The Hurst index (H) calculated from D₂ is mainly used to characterize the degree of pore connectivity [5]. The values of H range from 0.9700 to 0.9703 and 0.9607 to 0.9644 for MC and HC, respectively. The results show a better seepage–fracture connectivity for MC, which is consistent with the T₂ spectrum. The values of spectrum width D₋₁₀ − D₁₀, D₋₁₀ − D₀, and D₀ − D₁₀ provide the heterogeneity of the pore size distribution in the entire range, pore structure within the low-probability measurement area, and pore structure within the high-probability measurement region. The D₋₁₀ − D₁₀ values of the MC range from 2.5849 to 2.6055 (mean of 2.5960) and of the HC range from 2.8695 to 3.1475 (mean of 2.9483) indicate that the heterogeneity of the whole pore distribution in HC is higher than that in MC. In addition, the higher values of D₋₁₀ − D₀ indicate the stronger heterogeneity of seepage pores and fractures compared to adsorption pores in the samples, and this phenomenon is more pronounced in HC.

In the dry state, the values of D₁ for all samples decrease (Table 2), which is primarily attributed to the evaporation of water from seepage pores and fractures. Subsequently, the residual water predominantly accumulates in the smaller pore spaces, leading to worse connectivity. Furthermore, similar to water-saturated samples, the dried HC samples exhibit smaller D₁ values (D₁ = 0.7875–0.8339). The lower D₁ reflects heightened spatial heterogeneity in the residual moisture distribution, which is related to the poor pore connectivity of HC—a consequence of vitrinite-dominated structures and poorly interconnected micropore networks. The decreased value of H (0.9263–0.9286 for MC and 0.8672–0.8970 for HC) provides further evidence of poor connectivity in residual water distribution. The values of D₋₁₀ − D₁₀ for all samples turned to increase in the dry state, with HC demonstrating more pronounced increments (Figure 9), indicative of enhanced residual moisture heterogeneity in HC. For MC, there is little difference between the increased amplitudes of the left branch D₋₁₀ − D₀ and right branch D₀ − D₁₀. In contrast, HC displays significantly larger right branch increments, reflecting preferential micropore heterogeneity amplification (Figure 9, Table 2).

3.3.2. The Multifractal Singular Spectrum Characteristics

The multifractal singularity spectra f(α)-α can also be used to quantify the heterogeneity intensity of the research object, as well as the heterogeneity differences among sparse regions, dense regions, and regions with different densities [47]. The parameter α₀ corresponds to the mean singularity strength of pore size distribution, with larger values reflecting a constricted distribution range and more pronounced heterogeneity.

Table 3. Parameters of multifractal singular spectrum under different conditions.

Sample ID	Water-Saturated Conditions						Dry Conditions
	α0	α10− − α0	α0 − α10+	α10− − α10+	Rd	Δf	α0	α10− − α0	α0 − α10+	α10− − α10+	Rd	Δf
M1	1.2591	2.6145	0.3312	2.9457	−2.2833	0.8501	1.4749	2.5674	0.6773	3.2447	−1.8902	0.6445
M2	1.2616	2.6207	0.3384	2.9591	−2.2824	0.8284	1.4759	2.5685	0.6797	3.2482	−1.8887	0.6406
M3	1.2656	2.6314	0.3339	2.9653	−2.2975	0.8816	1.4838	2.5774	0.6938	3.2711	−1.8836	0.6298
M4	1.2638	2.6283	0.3263	2.9547	−2.3020	0.9093	1.4794	2.5726	0.6850	3.2576	−1.8876	0.6393
H1	1.3437	2.8309	0.4219	3.2528	−2.4091	0.8882	1.5715	2.6502	0.8890	3.5392	−1.7613	0.3931
H2	1.3520	2.8678	0.4279	3.2957	−2.4399	0.8970	1.7139	2.7901	1.1089	3.8990	−1.6813	0.2815
H3	1.4225	3.0454	0.5140	3.5593	−2.5314	0.8576	1.7172	2.8017	1.1082	3.9099	−1.6936	0.2868
H4	1.3523	2.8541	0.4300	3.2840	−2.4241	0.8948	1.6089	2.6802	0.9596	3.6397	−1.7206	0.3368
H5	1.3551	2.8670	0.4410	3.3080	−2.4260	0.8500	1.5884	2.6662	0.9148	3.5810	−1.7513	0.3847

shows that under water-saturated conditions, α₀ values for MC range more narrowly (1.2591–1.2656) than that of HC (1.3437–1.4225), reflecting homogeneous pore networks in MC. Drying induces a systematic α₀ increase for all samples (1.4749–1.4838 for MC and 1.5715–1.7172 for HC), which reflects the stronger heterogeneity in residual water distribution. This phenomenon is consistent with the result of D_q observed in Section 3.3.1. Greater values of the f(α) spectra α₁₀₋ − α₁₀₊ can be found in HC (Table 3). The broader widths of the f(α) spectra indicate a complex pore structure, indicating greater variability in pore sizes [18,27]. The spectral width also expands in dried samples, particularly for HC. The lengths of the left branch (α₀ − α₁₀₊) and the right branch (α₁₀₋ − α₀) describe the pore structure in the low-probability and high-probability measurement zone, respectively. Drying amplifies the left branch widths and reduces the right branch widths (Table 3, Figure 10). The asymmetry index R_d = [(α₀ − α₁₀₊) –(α₁₀₋ − α₀)] quantifies the deviation degrees of the spectrum to the center. When R_d equals 0, the singularity spectra exhibit symmetry around α₀. Table 3 and Figure 10 show that all samples exhibit R_d < 0 and that the f(α) spectrum deviates to the right, reflecting preferential control via sparsely distributed regions which corresponded to negative values of q [26,30].

The value of Δf = f(α_min) − f(α_max) was used to represent the vertical difference between the two branches of the spectrum. Δf > 0 indicates that the left branch of the spectrum is shorter than the right branch, suggesting a smaller variation in the geometrical size of points with minimum α exponents [26]. The values of R_d and Δf in water-saturated samples are much larger than those in dry samples (Figure 11), indicating that the rightward deviations of the singular spectrum are stronger, and that the right branches are longer under water-saturated conditions. Overall, the high-rank coal samples are more heterogeneous than the middle-rank coal samples, and the residual water distribution is more complicated and heterogeneous.

4. Discussion

4.1. The Distribution of Multiphase Water Based on NMR

Water in coal can be divided into free water, mainly existing in seepage pores or fractures, and bound water, dominating in adsorption pores [3,48,49]. Figure 12 shows the multiphase water distribution of samples M1 and H1 according to the NMR T₂ spectra distributions in water-saturated and dry conditions. After drying, the distributions of the two rightmost T₂ spectra show a noticeable reduction (P2) or even disappearance (P3) (

Table 4. T₂ spectrum area and variation in M1 and H1.

Sample ID	P1			P2			P3
	Water-Saturated Conditions	Dry Conditions	Variation	Water-Saturated Conditions	Dry Conditions	Variation	Water-Saturated Conditions	Dry Conditions	Variation
M1	12,804.83	7396.46	−42.24%	4294.02	196.35	−95.43%	6502.29	7.50	−99.88%
H1	8391.26	8110.20	−3.35%	384.85	25.20	−93.45%	572.50	0	−100%

), indicating that almost all the water in the seepage pores and fractures has evaporated (the orange district in Figure 12). Compared with the reduction in P2 and P3, the peaks of adsorption pores (P1) in medium-rank coals demonstrate a declining trend, whereas P1 in high-rank coals remains predominantly unchanged (Figure 3, Figure 12). On the one hand, the distributions of residual water are primarily composed of adsorbed water within micropores due to the strong retention of bound water, and the effect of bound water is stronger in high-rank coals. This is mainly due to the advanced thermal maturation in HC, which enhances capillary forces (retaining bound water via strong surface interactions) and increases the pore-specific surface area for water adsorption, while MC shows better pore structure connectivity, promoting the migration of movable water. On the other hand, the evaporated water from seepage pores and fractures increases the fluid volume to fluid surface area ratio in unsaturated samples, leading to the surface relaxation enhancement effect and making the residual water have a shorter relaxation time [9], which is more obvious in the more heterogeneous HC.

The migration of moisture in coal represents a complex and continuous process, significantly affected by coal pore structures. There are overlapping T₂ distributions of absorbed and free water in coal. When samples are undergoing the desiccation process, the free water within specific pore size ranges initially moves towards the coal surface and then evaporates. Once the majority of the free water has been released, the bound water starts to evaporate in succession [49]. Ma et al. [50] found that the adsorption pores exhibit stronger spontaneous imbibition, and that the capillary force enhances the binding effect of water. HC has a relatively high proportion of adsorption pores and significant size variability (evidenced by higher multifractal parameters like D₋₁₀ − D₁₀) than MC, leading to the preferential migration of water through larger throats, and the retention of water in constricted regions, thus, the reduction in P1 in HC decreases less. In contrast, pores with larger sizes, corresponding to higher T₂ values, possess a larger pore volume and weaker capillary forces. As a result, these larger pores have less impact on the movement of moisture. Consequently, it is relatively easier for moisture to migrate through larger pores compared to smaller ones. According to the research results of this study and previous studies, the distributions of saturated and residual water in coal are shown in Figure 13.

4.2. The Relationship Between T₂ Spectra and Multifractal Parameters

Figure 14 demonstrates the correlations between the T₂ spectrum area of different pore types and multifractal parameters (H, D₋₁₀ − D₁₀, α₁₀₋ − α₁₀₊, R_d). In water-saturated conditions, the proportion of the adsorption pore spectrum area (S_a) shows a weak negative linear correlation with pore connectivity and the heterogeneity index (R_d), while exhibiting positive correlations with D₋₁₀ − D₁₀ and α₁₀₋ − α₁₀₊. These trends suggest that an increase in adsorption pores may weaken the pore connectivity and enhance pore heterogeneity. The positive correlations with D₋₁₀ − D₀, D₀ − D₁₀, α₁₀₋ − α₀, and α₀ − α₁₀ indicate that adsorption pores simultaneously enhance the heterogeneity of low- and high-probability measurement zones (

Table 5. Impact of T₂ spectra area on pore heterogeneity.

Parameter		D−10 − D0	D0 − D10	α10− − α0	α0 − α10+
Water-saturated conditions	Sa	0.34	0.18	0.34	0.29
	SS+f	−0.34	−0.18	−0.34	−0.29
Dry conditions	Sa	0.15	0.24	0.14	0.20
	SS+f	−0.15	−0.24	−0.13	−0.20

). Additionally, the influence of S_a on D₋₁₀ − D₀ and α₁₀₋ − α₀ is slightly stronger than that on D₀ − D₁₀ and α₀ − α₁₀₊, reflecting the dominant role of low-value regions in controlling pore structure heterogeneity. The influencing effects of the sum of the seepage pores and fractures (S_s+f) show an opposite trend compared to the adsorption pores (Figure 14a,c). This indicates that the existence of seepage pores and fractures increases pore connectivity and reduces pore heterogeneity.

In dry conditions, the correlations between the proportions of adsorption pores, the sum of seepage pores and fractures, and multifractal parameters are significantly weaker (Figure 14b,d; Table 5). This can be attributed to inhomogeneous residual water distribution in seepage pores and fractures after evaporation. As discussed in Section 4.1, residual water in coal under dry conditions primarily consists of immobile and bound water stored in unconnected adsorption pores and pore throat corners (Figure 13), resulting in complex and heterogeneous pore structures.

4.3. Influencing Factors of Multifractal Parameters in Coal Samples

4.3.1. Correlation Analysis of Multifractal Parameters

Spearman correlation analysis was employed to investigate the relationships between multifractal parameters under different conditions (Figure 15). Significantly stronger correlations among multifractal parameters were observed under dry conditions, indicating the complexities of residual water distribution. Moreover, the high degree of consistency among these parameters suggests that they share the same physical meaning in characterizing pores’ structural features. In water-saturated conditions, H exhibits a strong positive correlation with R_d (R² = 0.80), indicating that systems with good pore connectivity display homogenizing fluid pathways and reduced heterogeneity. D₋₁₀ − D₁₀ shows a negative correlation with R_d (R² = −0.95), suggesting that broader pore size distributions and irregular spatial clustering remarkably enhance the heterogeneity of pore structures. Notably, D₋₁₀ − D₀ exhibit stronger correlations to R_d than D₀ − D₁₀, implying that the heterogeneity is mainly controlled by the low-probability regions. D₋₁₀ − D₁₀, D₋₁₀ − D₀, and D₀ − D₁₀ correlate strongly with α₁₀₊ − α₁₀₋, α₁₀₋ − α₀, and α₀ − α₁₀₋, respectively, reflecting the equivalence of the multifractal dimension spectrum q-D(q) and multifractal singularity spectrum α-f(α) in describing the multifractal characteristics of coal’s pore structure. These insights advance the mechanistic understanding of fluid trapping and transport limitations in low-permeability reservoirs, providing a quantitative basis for predicting gas storage capacity and water migration in heterogeneous coal seams.

4.3.2. Influence Factors

The degree of coal metamorphism plays a critical role in determining the pore development characteristics, while the pore structure significantly influences the moisture distribution within coal. Figure 16 shows the impacts of R_o,max, vitrinite content, and inertinite on the multifractal parameters (H and α₁₀₋ − α₁₀₊). A strong negative relationship exists between R_o,max and H (R² = 0.89 under water-saturated conditions; R² = 0.81 under dry conditions), while R_o,max positively correlates with α₁₀₋ − α₁₀₊ (R² = 0.83 under water-saturated conditions; R² = 0.74 under dry conditions)(Figure 16a,b). In medium- and high-rank coals, increasing coal metamorphism enhances aromatization and cycle polycondensation, thereby increasing the proportion of micropores [51,52,53]. As discussed in Section 4.2, the proportion of adsorption pores has a positive impact on the heterogeneity of pore structures. High-rank coals exhibit more complex pore structures and reduced connectivity, trapping gas in isolated pores. During the process of CBM production, enhanced recovery techniques are needed. In contrast, medium-rank coals show more developed seepage pores and fractures (Figure 4), where microfractures will increase pore connectivity and reduce heterogeneity, favoring gas diffusion. Under dry conditions, complex pore structures and strong effects of bound water in HC reduce pore connectivity and increase the heterogeneity of residual water distribution (Figure 16b). The vitrinite contents show negative correlations with H (R² = 0.84 under water-saturated conditions; R² = 0.90 under dry conditions) and positive correlations with α₁₀₋ − α₁₀₊ (R² = 0.78 and 0.94, respectively) (Figure 16c,d). The impact of inertinite content on H and α₁₀₋ − α₁₀₊ is opposite to that of vitrinite (Figure 16e,f). Therefore, as the vitrinite content increases, the connectivity of the pore structure and residual water decreases, and the heterogeneity of pore size distribution increases. A high content of vitrinite in coal promotes the development of micropores, increasing the complexity of the pore structure and reducing pore connectivity, and thus enhancing the heterogeneity [36].

V_daf exhibits relatively strong positive correlations with multifractal parameters (Figure 17a,b), confirming coal metamorphism as the primary control on multifractal behavior. The impacts of minerals on multifractal parameters are consistent with those of V_daf (Figure 17c,d), meaning that the minerals have a positive effect on pore connectivity and a negative effect on pore heterogeneity and complexity. This may be caused by pore agglomeration and the connection of pores and microfractures [12]. There are few or no obvious correlations observed between M_ad, A_d, and multifractal parameters (

Table 6. Relationships between Mad, Ad, pore connectivity, and heterogeneity of coal.

Parameter		H	D−10 − D10	α10− − α10+	Rd
Water-saturated conditions	Mad	−0.4246	0.3224	0.3189	−0.3052
	Ad	0.0303	−0.0295	−0.0295	0.0245
Dry conditions	Mad	−0.2666	0.1892	0.1941	0.3676
	Ad	0.0076	−0.0065	−0.0067	−0.0106

), suggesting the negligible impacts of M_ad and A_d on multifractal parameters in this study.

5. Conclusions

To investigate the multifractal characteristics of pore heterogeneity and multiphase water distribution in medium- and high-rank coals, nuclear magnetic resonance experiments were performed on nine medium- and high-rank coal samples under water-saturated and dry conditions. The relationships between the T₂ spectra, multifractal parameters, and the influencing factors are also discussed. The conclusions are as follows:

(1)

The T₂ spectra of water-saturated coal samples exhibit three peaks: an isolated P1 peak (adsorption pores, 0.01–2.5 ms) dominates all samples, followed by overlapping P2 (seepage pores, 2.5–100 ms), and P3 (fractures, >100 ms). Drying preferentially removes free water in P2 and P3 regions, leading to significant reductions in the T₂ spectrum, especially in medium-rank coals;

(2)

Multifractal characteristics derived from NMR reveal that coal rank dictates pore heterogeneity evolution. High-rank coals exhibit greater pore heterogeneity and complexity and lower pore connectivity due to aromatization-driven micropore dominance. Drying intensifies heterogeneity asymmetrically, with high-rank coals showing pronounced right-branch spectral broadening;

(3)

Free water dehydration in coal follows a size-dependent sequence: fractures/seepage pores → adsorption pores. Under dry conditions, multifractal analysis shows that the distribution of residual water is more complex, primarily occupying the adsorption pores and pore throat corners, which are controlled by the retention of bound water and the surface relaxation enhancement effects;

(4)

Strong correlations between the multifractal dimension spectrum (q-D_q) and singularity spectrum (f(α)-α) validate dual-spectrum consistency in characterizing pore fluid systems. The degree of coalification, vitrinite content, inertinite content, and V_daf are the primary factors affecting pore heterogeneity, connectivity, and water distribution. Additionally, no significant correlations exist between multifractal parameters and M_ad or A_d, excluding these as key heterogeneity drivers.