Seepage Characteristics and Critical Scale in Gas-Bearing Coal Pores Under Water Injection: A Multifractal Approach

Abstract

To investigate the flow characteristics of movable water in coal under the influence of micro-nano pore fractures with multiple fractal structures, this study employed nuclear magnetic resonance (NMR) and multifractal theory to analyze gas–water seepage under different injection pressures. Then, the scale threshold for mobile water entering coal pores and fractures was determined by clarifying the relationship among “injection pressure-T₂ dynamic multiple fractal parameter seepage resistance-critical pore scale”. The results indicate that coal samples from Yiwu (YW) and Wuxiang (WX) enter the nanoscale pore size range at an injection pressure of 8 MPa, while the coal sample from Malan (ML) enters the nanoscale pore size range at an injection pressure of 9 MPa. During the water injection process, there is a significant linear relationship between the multiple fractal parameters log X(q, ε) and log(ε) of the sample. The generalized fractal dimension D(q) decreases monotonically with increasing q in an inverse S-shape. This decrease occurs in two distinct stages: D(q) decreases rapidly in the low probability interval q < 0; D(q) decreases slowly in the high probability interval q > 0. The multiple fractal singularity spectrum function f(α) has an asymmetric upward parabolic convex function relationship with α, which is divided into a rapidly increasing left branch curve and a slowly decreasing right branch curve with α₀ as the boundary. Supporting evidence indicates the feasibility of a methodology for identifying the variation in multiple fractal parameters of gas–water NMR seepage and the critical scale transition conditions. This investigation establishes a methodological foundation for analyzing gas–water transport pathways within porous media materials.

1. Introduction

The rational use of water injection technology can not only reduce gas emissions in coal mining to effectively prevent gas outbursts, but also decrease reservoir pressure to promote gas desorption through drainage and gas extraction in the process of coalbed methane (CBM) exploitation and utilization [1,2]. Water molecules in coal seams can affect methane migration through adsorption, desorption, displacement, diffusion, and seepage [3,4,5,6]. From a macro perspective, this manifests as changes in gas and water production, with micro-scale changes appearing as alterations in the flow state of gas and water within pore fractures [7,8]. Analyzing changes in the distribution and flow paths of gas and water in the pores and fractures of coal at the micron-scale can support the selection of appropriate mining techniques and facilitate the efficient utilization and development of CBM [9,10].

Due to the presence of both cleats and pores in the coal matrix, water first infiltrates into the cleats and fractures during the coalbed water injection process. Then, the water enters the macropores, micropores, and nanopores under the influence of diffusion [11,12,13]. Water molecules adhere to the surface of the pores via adsorption, thereby displacing some of the gas from the pore surface [14,15,16]. Typically, scholars study the gas migration mechanism in water-bearing coal seams in terms of pore characteristics such as pore size, shape, connectivity, and fractal dimension [17,18,19]. Due to the complex pore structure and good self-similarity of coal rock, it is difficult to accurately characterize coal rock using traditional experimental methods and Euclidean geometry [20,21]. However, fractal theory can be used to quantitatively evaluate the complexity of coal rock [22,23]. In recent years, various methods have been employed to study the fractal characteristics of rock pore distribution [24,25,26], including scanning electron microscopy, small-angle neutron scattering, box counting, mercury intrusion porosimetry, industrial CT scanning combined with the Menger fractal model, and the FHH fractal model [27,28,29,30]. Building upon single-scale fractal theory, researchers have developed fractal Langmuir adsorption models, fractal diffusion models, and fractal seepage models to elucidate how pore structure governs the adsorption and transport behavior of coalbed methane (CBM). Moreover, multifractal theory has been widely applied to characterize the complexity and heterogeneity of nanoporous materials such as coal [22,25,27,31], revealing multifractal features in CBM adsorption and transport processes [32].

Nuclear magnetic resonance (NMR) has emerged as a rapid, non-destructive technique for characterizing the physical properties of coal rocks and fluid flow behaviors in unconventional reservoirs [33]. Through gas–water two-phase NMR experiments, researchers have analyzed the state and distribution of water within rock pores [22,25,33], demonstrating that the drainage efficiency during two-phase CBM extraction significantly affects recovery rates. Other studies have utilized relaxation times to interpret fluid distribution in pores [10,20,34], enabling fluid identification and pore structure characterization [35]. Additional approaches combine NMR with capillary pressure tests to determine relative permeability [34,35,36], while NMR imaging and T₂ spectrum analysis have been applied to classify coal pores and assess pore size uniformity during displacement [37,38].

In summary, although substantial research has been conducted on two-phase micro-seepage, few studies have examined the real-time flow state and fluid distribution under varying water injection conditions in coal rocks. To address this gap, this study investigates the seepage characteristics of movable water in micro-nano fractures of coal under different injection pressures by integrating multifractal theory with NMR experiments. This approach allows us to establish the relationship among injection pressure, T₂ dynamic multi-fractal parameter seepage resistance, and critical pore scale, and identify the corresponding threshold scale for water entry. This investigation can provide new insights into clarifying gas–water transport processes in porous media materials and improving CBM recovery rates.

2. Research Methodology

2.1. NMR Experiment

The coal samples used in this study were obtained from the Yuwu (YW), Wuxiang (WX), and Malan (ML) mines in Shanxi Province. Raw coal blocks were drilled, ground, and processed into cylindrical specimens measuring φ25 mm × 50 mm for subsequent analysis.

The test was conducted using a Meso MR23-060H-I low-field nuclear magnetic resonance instrument (From Nuomei Analytical Instrument Testing Co., Ltd., Suzhou, China), with a resonance frequency of 21.67568 MHz, a magnet temperature controlled at 32 ± 0.1 °C, and a magnetic field strength of 0.5 T (Figure 1). The NMR experimental procedure was to first put the samples into the sample compartment for 5 MPa nitrogen adsorption for 12 h, and then to perform the pressure preservation water injection under 5–9 MPa pressure, respectively, followed by the T₂ spectrum test under nitrogen adsorption and different water injection pressures.

2.2. Calculation of Fractal Parameters

Surface relaxation rates were calculated for the pore and fracture distribution in coal based on the T₂ spectrum data from the NMR experiments [39]. The formula was as follows:

$${\displaystyle \frac{1}{{\mathrm{T}}_2}}=\rho {\displaystyle \frac{S}{V}}$$

(1)

where ρ is the surface relaxation rate, S is the pore-specific surface area, and V is the pore volume.

The cutoff value T_2cutoff was calculated by the geometric mean method, and the movable and bound water in the coal pore space was divided by T_2cutoff, where T_2cutoff was calculated as follows:

$${\mathrm{T}}_{2\mathrm{cutoff}}=\exp \left[{\displaystyle \sum _{T_{2\mathit{S}}}^{T_{2\mathit{max}}}\frac{A_i}{A_T}\ln \left(T_{2i}\right)}\right]$$

(2)

where T_2max is 10,000 ms; A_i is the signal amplitude at T_2i transverse relaxation time; A_T is the maximum amplitude of the NMR spectrum; T_2s is the initial transverse relaxation time of the spectrum of 0.01 ms. The range of movable water aperture: T₂ > T_2cutoff.

Multifractal theory can be used to analyze the fractal characteristics of objects at different scales, with further accurate characterization of the complexity and non-uniformity of pore distribution. Typically, the box counting method is used to characterize multiple fractals in porous media, where the NMR signal is divided into N(ε) boxes. The mass probability of the ith box is defined as [40]:

$$P_{\mathrm{i}}\left(\epsilon \right)={\displaystyle \frac{N_i\left(\epsilon \right)}{{\displaystyle {\sum }_{i=1}^{N\left(\epsilon \right)}{N}_i\left(\epsilon \right)}}}$$

(3)

where N_i(ε) is the NMR signal amplitude of the ith cassette; the relationship between P_i(ε) and cassette size ε follows a power exponential relationship.

$$P_i\left(\epsilon \right)\propto {\epsilon }^{{\alpha }_i}$$

(4)

where α_i is the Lipschitz-Holder index indicating the singularity strength of the ith box. Boxes at different locations may have similar singularity indices α_i. Assuming that the number of boxes with singular intensities ranging from α to α + dα is N_α(ε), N_α(ε) increases as the partition scale ε decreases [39]:

$$N_{\alpha }\left(\epsilon \right)\propto {\epsilon }^{-f\left(\alpha \right)}$$

(5)

where f(α) is the fractal dimension of subsets with the same singularity index α. The distribution function is defined as [41]:

$$\chi \left(q,\epsilon \right)={\displaystyle \sum _{\mathrm{i}=1}^{N\left(\epsilon \right)}{P}_i^q\left(\epsilon \right)}\propto {\epsilon }^{-\tau \left(q\right)}$$

(6)

where q is the statistical moment order (−10 ≤ q ≤ 10); τ(q) is the mass index, and τ(q) is monotonically increasing with q.

$$\tau \left(q\right)=-\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\log \chi \left(q,\epsilon \right)}{\log \epsilon }}$$

(7)

α and f(α) can be obtained by the Legendre transformation [42], which is expressed as:

$$\alpha ={\displaystyle \frac{d\tau \left(q\right)}{dq}}$$

(8)

$$f\left(\alpha \right)=q\alpha -\tau \left(q\right)$$

(9)

The multifractal set can also be computed by generalizing the fractal dimension D_q [40]:

$$D_q={\displaystyle \frac{1}{q-1}}\underset{\epsilon \to 0}{\mathit{lim}}{\displaystyle \frac{\mathit{log}{\displaystyle {\sum }_{\mathrm{i}=1}^{N\left(\epsilon \right)}{P}_i^q\left(\epsilon \right)}}{\mathit{log}\epsilon }}$$

(10)

The asymmetry of the multifractal singularity spectrum can be described by the parameter A, which represents the fluctuations of the pore size distribution. The larger the value of A, the stronger the volatility of the pore size distribution. When A > 1, the high probability density region has a significant effect on the pore size distribution, i.e., the pore size fluctuates more. When A < 1, the low probability density region has a significant effect on the pore size distribution, i.e., the pore size fluctuates less. The value A is calculated as follows:

$$A={\displaystyle \frac{{\alpha }_0-{\alpha }_{\mathit{min}}}{{\alpha }_{\mathit{max}}-{\alpha }_0}}$$

(11)

3. Results and Discussion

3.1. Threshold for Water Entering Coal Borehole at Different Injection Pressures

The results of T₂ spectra for 5 MPa nitrogen adsorption and 5~9 MPa holding pressure water injection were obtained by NMR seepage experiments, followed by the surface relaxation rate of T₂ spectra as well as the cutoff point calculated by combining Equations (5) and (6). As shown in

Table 1. T₂ time conversion to equivalent pore size distribution parameters.

Samples	Surface Relaxation Rate (μm/s)	Injection Pressure (MPa)	T2 Cutoff (ms)	Cutoff Pore Size (nm)
YW	18.7	5	0.85	31.79
	18.7	6	0.81	30.29
	18.7	7	0.77	28.8
	18.7	8	0.69	25.81
	18.7	9	0.65	24.31
WX	16.5	5	0.91	30.03
	16.5	6	0.9	29.7
	16.5	7	0.9	29.7
	16.5	8	0.85	28.05
	16.5	9	0.74	24.42
ML	27.5	5	1.14	62.7
	27.5	6	1.03	56.65
	27.5	7	1.28	70.4
	27.5	8	1.28	70.4
	27.5	9	1.26	69.3

, the T₂ cutoffs range from 0.65 to 1.28 ms, indicating that there are large differences in the lithology of the coal samples. The cutoff pore sizes ranged from 24.31 to 70.4 nm, suggesting that there were differences in the adsorption of gases in the coal samples. The T₂ time distribution was converted to an equivalent aperture distribution using the surface relaxation rate in Table 1, with the aperture-converted NMR curves shown in Figure 2.

As shown in Figure 1, the NMR amplitudes of coal samples YW and BD gradually increased with the increase in water injection pressure during the water injection process (5 to 9 MPa). This indicates that more pores (including small pores) are filled with water as the pressure increases. The T₂ cutoff shifted left during water injection, showing that smaller pores were filled. In the process of water injection in coal samples ML, the NMR amplitude increased significantly in the nanoscale pore size at high pressure of 9 MPa, suggesting that high pressure drove water into the nanopore space [10,21].

It can be seen that the NMR amplitude rises with the increase in water injection pressure, which significantly increases the fluid filling of small pores. High-pressure water injection mainly affects the nanoscale pores, and the difference in amplitude response of YW, WX, and ML is attributed to different pore size distributions.

3.2. Analysis of Multiple Fractal Results by NMR

The generalized fractal parameter spectra and singular fractal spectra derived from multiple fractal calculations based on the multiple fractal theory for the equivalent pore size distribution NMR spectra of movable water from NMR seepage experiments are shown in Figure 3 and Figure 4, respectively. Figure 2 shows the double logarithmic curves of X(q, ε) and ε, i.e., there is a significant linear relationship between log X(q, ε) and log(ε), with a correlation coefficient of R² > 0.979, for coal samples during nitrogen adsorption and pressure-holding water injection from 5 to 9 MPa. This indicates that coal samples show clear multiple fractal features during both nitrogen adsorption at 5 MPa and water injection at holding pressures from 5 to 9 MPa. Log X (q, ε) is negatively correlated with log(ε) in the low-probability interval of q < 0, while it is positively correlated in the high-probability interval of q > 0. The large span of log X (q, ε) values for q < 0 demonstrates that the range corresponds to a wide distribution of apertures, whereas the small span of log X (q, ε) values for q > 0 suggests that q > 0 corresponds to a more concentrated distribution of apertures [11,23,31].

Multiple fractal parameters of movable water nuclear magnetic seepage in coal samples YW, WX, and ML were calculated with the multiple fractal theory (

Table 2. Table of NMR seepage multiple fractal parameters.

Samples	Stage	D−10	D10	ΔD	D0	D1	D2	αmax	α0	αmin	Δα	Δf	A	H
YW	1	1.75	0.58	1.18	1.00	0.81	0.73	1.92	1.27	0.52	1.40	−0.03	1.14	0.87
	2	1.85	0.74	1.11	1.00	0.86	0.81	1.99	1.22	0.71	1.28	0.09	0.65	0.91
	3	1.68	0.75	0.94	1.00	0.87	0.81	1.83	1.18	0.72	1.11	0.30	0.72	0.91
	4	1.89	0.70	1.18	1.00	0.83	0.77	2.07	1.21	0.67	1.40	0.37	0.63	0.89
	5	1.47	0.71	0.75	1.00	0.84	0.78	1.58	1.19	0.69	0.90	0.15	1.27	0.89
	6	1.75	0.77	0.98	1.00	0.87	0.83	1.87	1.25	0.74	1.13	−0.12	0.83	0.92
WX	1	1.58	0.62	0.96	1.00	0.88	0.8	1.73	1.15	0.57	1.16	−0.07	1.01	0.9
	2	1.71	0.73	0.98	1.00	0.86	0.8	1.86	1.18	0.7	1.15	0.5	0.71	0.9
	3	1.89	0.7	1.19	1.00	0.86	0.79	2.09	1.18	0.68	1.41	0.33	0.55	0.9
	4	2.01	0.79	1.22	1.00	0.88	0.85	2.16	1.26	0.76	1.4	0.51	0.55	0.92
	5	1.58	0.78	0.8	1.00	0.9	0.85	1.72	1.14	0.76	0.96	0.08	0.67	0.92
	6	1.51	0.79	0.71	1.00	0.92	0.87	1.65	1.08	0.77	0.88	0.39	0.56	0.94
ML	1	1.35	0.93	0.42	1.00	0.98	0.97	1.47	1.03	0.92	0.56	0.68	0.24	0.98
	2	1.77	0.81	0.96	1.00	0.91	0.88	1.89	1.17	0.78	1.11	0.07	0.53	0.94
	3	1.58	0.78	0.8	1.00	0.89	0.85	1.68	1.15	0.76	0.92	0.01	0.74	0.93
	4	1.64	0.82	0.82	1.00	0.9	0.87	1.74	1.17	0.8	0.95	0	0.64	0.94
	5	1.79	0.83	0.97	1.00	0.91	0.88	1.96	1.15	0.8	1.16	0.51	0.42	0.94
	6	2.08	0.82	1.25	1.00	0.9	0.87	2.25	1.23	0.8	1.44	0.21	0.43	0.94

). From Figure 3 and Table 2, the mass indices q-τ(q) of coal samples YW, WX, and ML monotonically increase with a nonlinear relationship, indicating that the T₂ spectra of coal samples during both nitrogen adsorption and pressure-retaining water injection from 5 to 9 MPa are characterized by multiple fractals. As q increases, τ(q) tends to be convex upwards. The main parameters of the generalized fractal spectrum (q-D(q)) include D₋₁₀, D₁₀, ΔD, D₀, D₁, D₂, and H, where D₋₁₀ is affected by the low probability measurement region and D₁₀ by the high probability measurement region. ΔD is the difference between D₋₁₀ and D₁₀, with a larger ΔD indicating greater heterogeneity of the pores. The larger the value of D₀, the wider the range of pore size distribution and the more non-uniform the distribution. The smaller the value of D₁, the more concentrated the pore size distribution, and the larger the value of D₂, the more uniform the pore size distribution. The larger the value of H, the stronger the pore connectivity.

The generalized fractal dimension D(q) exhibits a decreasing inverse S-shape with increasing q, i.e., the value of D(q) decreases rapidly at low probability intervals of q < 0, while the value of D(q) decreases slowly at high probability intervals of q > 0. YW and WX in the q > 0 region, the curves keep shifting upward with time, indicating that the pore space changes significantly in the high probability region. The greater the curvature of the multifractal spectrum, the stronger the heterogeneity of the pore distribution [18,20,29].

The curvature of the coal samples increases sequentially with increasing water-driven pressure, suggesting that the increase in localized pore size distribution non-uniformity leads to an increase in pressure [2,17]. ML in the region of q > 0, the curve keeps moving downward with increasing pressure, showing that the heterogeneity of the pore distribution is weaker in the high probability region [10,21]. The curvature of the coal samples decreases sequentially as the water-driven pressure increases, demonstrating that the increase in the non-uniformity of the local pore size distribution leads to a decrease in the pressure [12], reflecting the transport process of water in the coal [30]. In the low probability interval of q < 0, the curve of ML with increasing pressure shifts upward continuously, implying that the heterogeneity of the pore distribution in the low probability region of ML is enhanced with increasing pressure [17,25,30].

The multifractal singularity spectral function f(α) is related to α as a convex function of an asymmetric upward parabolic curve, that is, a curve with a sharply increasing left branch bounded by α₀ and a slowly decreasing right branch. The main parameters of α-f (α) include α_min, α₀, α_max, Δα, Δf, and A. α_min is related to the distribution of small pores, while α_max is related to the distribution of large pores. The larger α₀, the stronger the local fluctuation, and the narrower the distribution interval. Δα is the difference between αmax and α_min, and the larger the value is, the greater the inhomogeneity of pore size distribution. Δf is the difference between f(α_min) and f(α_max), reflecting the relative proportions of small-volume and large-volume pores. Δf > 0 means that the small pores dominate the seepage resistance with a large tortuous path. Conversely, Δf < 0 means that large pores dominate with uniform seepage paths [43,44,45,46,47]. In addition, the asymmetry of the multifractal singularity spectrum can be characterized by the parameter A, that is, the larger the value of A, the stronger the fluctuation of the aperture distribution [16,23]. A > 1 indicates that the high probability density region has a significant influence on the aperture distribution, with larger fluctuations in the aperture. A < 1 implies that the low probability density region has a significant influence on the aperture distribution, resulting in smaller fluctuations in the aperture [9,35].

3.3. Characterization of Multiple Fractal Features of Seepage

According to the data in Table 2, the relationship between the multiple fractal parameters was obtained as shown in Figure 5. The multiple fractal parameter D₋₁₀ of YW shows a significant positive relation to ΔD, α_max, and Δα, with correlation R² above 0.764, indicating that the heterogeneity of the pore size distribution affected by the low-probability measurement region is positively correlated with the macropores [1,20]. The parameter D₁₀ exhibits a significant positive correlation with D₁, D₂, α_min, and H, showing that the influence of the high probability measurement region is positively correlated with the pore size distribution, the degree of homogeneity, and the pore connectivity [10,25,27].

During the boost injection stage, the Δf of YW gradually increases from 0.09 at 5 MPa to 0.37 at 7 MPa, demonstrating that water is difficult to enter the adsorption pores with high seepage resistance at this stage [7]. Δf = 0.15 at 8 MPa, showing that the water starts to enter the nanoscale adsorption pores at 8 MPa, with significantly lower percolation resistance [33]. Δf = −0.12 at 9 MPa, meaning that the water enters the nanoscale adsorption pores smoothly with smooth channels and uniform percolation paths [28]. The parameter A = 1.27 > 1 at 8 MPa pressure with large pore size fluctuations indicates that the critical pressure for water to enter the nanoscale adsorption pores is 8 MPa.

The fractal parameter D₋₁₀ of WX shows a significant positive correlation with ΔD, α_max, and Δα, demonstrating that the influence of the low-probability measurement region has a significant positive correlation with the distribution of micropores [7,8]. H, D₁, and D₂ showed obvious positive correlations, with correlation R² above 0.824, indicating that there is an obvious positive correlation between pore connectivity and the degree of dispersed uniformity of pore size distribution. During the boosting water injection and gas displacement stage, the Δf of WX is mostly above 0.3, reflecting that the pore and fracture channels of WX are more tortuous and rough. The gradual decrease in Δf from 0.5 at 5 MPa to 0.33 at 6 MPa indicates that the resistance to seepage decreases as the pressure rises. Water seepage from the millimeter-scale fracture into the micron-scale pore fracture at 5 MPa until Δf rises to 0.51 at 7 MPa, suggesting that water in the pore fracture is difficult to enter the nanoscale adsorption pores due to the increase in seepage resistance [5,8,24]. The Δf = 0.08 at 8 MPa indicates that water enters the nanoscale pores with less resistance to filling and larger pore volumes. Subsequently, Δf = 0.39 at 9 MPa shows that water has filled the nanoscale adsorption pores that continue to flow through the micrometer-sized pores as well as the millimeter-sized fractures [19,25,36]. The parameters A at 5 MPa and 8 MPa are 0.71 and 0.67, respectively, much higher than the values of A at other water-driven pressure points, demonstrating that the critical pressure for water to enter the micrometer-sized pores is 5 MPa, whereas the critical pressure for water to enter the nanosized adsorption pores is 8 MPa.

ML’s D₋₁₀ shows a significant positive correlation with ΔD, α_max, and Δα, revealing that the unevenness of the pore size distribution is positively correlated with the distribution of macropores affected by low-probability measurement areas [20,37]. Parameter D₁₀ indicates a significant positive correlation with D₁, D₂, α_min, and H, demonstrating that the influence of high-probability measurement regions is positively correlated with pore size distribution, uniformity, micropore distribution, and pore connectivity [9,16]. During the pressure-boosting water injection gas displacement phase, ML’s Δf gradually decreased from 0.07 at 5 MPa to 0 at 7 MPa, showing that the flow resistance gradually decreased as the pressure increased. The Δf values for 8 MPa and 9 MPa are 0.51 and 0.21, respectively, indicating that the excessive pressure at 8 MPa causes some pores to be compressed, resulting in a more complex flow path and increased flow resistance [17,25].

3.4. Analysis and Verification of Gas–Water Seepage Characteristics

The T₂ spectrum area was calibrated by injecting water of different masses into the nuclear magnetic resonance chamber, as shown in Figure 6. According to the results of the nuclear magnetic resonance calibration group experiment, the relationship between different water masses and T₂ peak areas can be obtained as shown in Figure 6a. The T₂ peak area (y) of coal samples in the study area is linearly related to the water mass (x) as y = 0.0114x − 0.3668 (Figure 6b), with a correlation coefficient R² = 0.981. Based on the fitting curves obtained from the calibration results of different water quality standards, the movable water content in coal pores under different gas–water conditions for YW, WX, and ML can be calculated separately. As shown in

Table 3. Movable water content in coal pores.

Samples	Injection Pressure (MPa)	Water Volume (g)
YW	5	0.451
	6	0.749
	7	0.787
	8	0.891
	9	0.934
WX	5	3.002
	6	3.417
	7	5.012
	8	4.642
	9	4.556
ML	5	0.005
	6	0.028
	7	0.104
	8	0.104
	9	0.111

and Figure 7, the water intake of coal sample YW at 9 MPa was 0.934 g, while that of WX was 4.556 g, and that of coal sample ML was 0.111 g. This result indicates that the well-developed micron-scale fractures in coal sample WX facilitate water intrusion, leading to the highest water intake [15,26,33]. In contrast, the abundance of nanoscale pores in sample ML hinders water penetration, resulting in the lowest intake. Furthermore, the water content within the pores and fractures of all coal samples increases with injection pressure. However, due to variations in pore-fracture structures, the water injection volumes under the same pressure differ significantly among the samples [17,24].

Combining Figure 3 and Figure 8, water enters the micron-scale drainage network at 5 MPa, with a slow increase in water volume between 6 MPa and 7 MPa, and injection into the nanoscale reservoir units only occurs at 8 MPa. This indicates that the nanoscale reservoir units in YW are relatively complex, with high injection resistance, making injection difficult at lower pressures, consistent with the analysis in the preceding text. The micron-scale flow channels of WX and ML are relatively tortuous, with high seepage resistance, causing the flow to be difficult for water to enter micron-scale pore fractures at low pressures. Both WX and ML exhibit a sharp increase in injection volume once pressure reaches 6 MPa, consistent with the multi-fractal parameter analysis of gas–water seepage characteristics discussed earlier. From the injection volume of ML, water has difficulty entering micron-scale pores and can only occupy a small portion of the fractures.

In summary, the gas–water transport characteristics obtained through variable water flow rates are basically consistent with the gas–water dynamic seepage characteristics analyzed via T₂ dynamic multifractal seepage resistance parameters. This indicates that quantifying the gas–water dynamic seepage characteristics under different pressures through the relationship between “injection pressure-T₂ dynamic multifractal parameter seepage resistance- critical pore scale” is accurate.

4. Conclusions

This paper analyzes the relationship between “micron-scale conductivity-nanoscale adsorption capacity-T₂ dynamic multifractal parameter flow resistance” through nuclear magnetic resonance flow experiments and multifractal principles, followed by quantifying the gas–water dynamic seepage characteristics under different pressures. Subsequently, the scale threshold for movable water to enter coal pore fractures is determined. The main conclusions are as follows:

There is a significant linear relationship between log X(q, ε) and log(ε), with a negative correlation in the low probability interval of q < 0 and a positive correlation in the high probability interval of q > 0. Since the log X(q, ε) values for q < 0 exhibit a large range, this means that the pore size distribution for q < 0 is broad, while the log X(q, ε) values for q > 0 reveal a small range, indicating that the pore size distribution for q > 0 is more concentrated.

The quality index q-τ(q) increases monotonically in a nonlinear relationship, and τ(q) increases strictly, also showing an upward convex trend as q increases. The generalized fractal dimension D(q) decreases monotonically as q increases, forming an inverse S-shape. This can be divided into two stages: D(q) decreases rapidly in the low-probability interval of q < 0; D(q) decreases slowly in the high-probability interval of q > 0.

The multifractal singularity spectrum function f(α) exhibits an asymmetric convex function relationship with α, divided into a rapidly increasing left branch curve and a slowly decreasing right branch curve with α₀ as the boundary. The T₂ dynamic fractal parameters can be obtained by combining low-field NMR seepage experiments with multifractal theory, followed by determining the critical pressure values for water entering micron-scale fractures and nano-scale pores in coal samples from the study area through seepage resistance parameters.