Introduction

In recent years, China’s coal industry has undergone a strategic transformation in gas management paradigms. Guided by evolving safety regulations and environmental protection mandates, the nation has introduced a dual-phase gas control protocol: "pre-mining gas extraction" for surface development projects and "pre-drainage gas ventilation" for underground operations. This policy framework prioritizes gas risk mitigation through proactive engineering controls, mandating comprehensive gas drainage systems implementation before mine construction and during coal extraction phases. The systematic approach effectively decouples gas hazards from production cycles while enabling resource utilization of coalbed methane (CBM) as clean energy feedstock1. Due to the depletion of shallow coal resources, deep coal mining has become an unavoidable trend as a result of China’s coal mining continually expanding to the depth at a pace of 10–25 m per year due to the progressive depletion of shallow coal resources2. CBM, sometimes referred to as “gas”, it is linked to the production and transformation of coal and a disaster-causing factor3. It can be effectively reduced by developing a discharge system that is both safe and economically advantageous. China’s deep coal seams have very complicated formation and storage circumstances, and as coal mining depth has risen, so too have the initial ground stress and temperature state of the coal reservoirs. The drainage gas extraction method is currently the main operation method for CBM surface extraction in China. One of the crucial stages of this process is the gas/water two-phase flow stage, and the primary factor influencing this stage is the coal reservoir’s absolute permeability. The size of this stage is influenced by the coal reservoir’s microscopic pore structure, effective porosity, pore connectivity, and other physical parameters4. Therefore, it is crucial to predict this parameter quickly and accurately in order to rationally formulate the surface CBM drainage and mining system and increase the drainage and mining efficiency. On the one hand, Coal is a dual pore medium made up of coal matrix, numerous micropores, and fissures; therefore, the distribution of pore and fissure structures of coal reservoirs under the coupling effect of geothermal heat and stress at the deeper level will also change accordingly. On the other hand, the formation conditions of coal reservoirs in deep coal mines in China are generally characterized by low pressure, low permeability, low saturation, and non-homogeneity5,6. Gas protrusion disaster management is made more difficult by the complex geological structure, which also has a greater impact on the transportation and storage of coal-associated gas products. This further restricts the use of conventional absolute permeability prediction models7. The aforementioned elements have contributed to the intricacy of gas and liquid storage and transportation, making it more challenging to create strategies for gas drainage and extraction and further decreasing the effectiveness of gas recovery.

Predicting the permeability of coal reservoirs requires an effective assessment of their pore structure and pore size distribution. The nuclear magnetic resonance (NMR) technique, which can quantitatively assess both free and bound fluids, offers a distinct advantage in characterizing fluid seepage. NMR techniques have gradually gained popularity since Russell’s 1949 discovery of the phenomenon of free progression of nuclei in the geomagnetic field8; the Inside-out technique, developed by Jaekso et al9, is a significant milestone in the development of this field and lays the theoretical foundation for the study of the pore and fracture structure of porous media by modern NMR techniques; NMR technology was creatively exploited by Battocletti et al.10 to create a blood flow measuring system. They then expanded and used this NMR measurement technique to the detection of multiphase flow in the petroleum sector; Timur11 proposed an empirical formula for permeability based on rock porosity; Coates et al12. discovered that the ratio of movable to bound fluid volume (FFI/BVI) controls the permeability of sandstone. They then modified Timur’s permeability prediction model and proposed the Timur-Coates model, which implies that both movable and bound fluids have an impact on absolute permeability; The SDR model was proposed by Kenyon et al.13. It estimates permeability by merely averaging the pore size distribution and using the geometric mean of T2gm. When combined with the Timur-Coates model, it is regarded as one of the most popular NMR permeability models available today. These NMR permeability models, which employ optimal T2 spectral parameters like T2cut-off and T2gm, which only reflect the seepage information of the connected pores that are similar to these T2 spectral parameters, are generally less helpful in coal reservoirs than in conventional sandstones14. The range of pore size distribution is significantly greater in coal reservoirs due to the complexity and irregularity of pore structure and pore size distribution5,6.

The applicability of permeability prediction models based on optimal T2 spectral parameters in coal reservoirs remains constrained due to two inherent limitations. First, these models oversimplify the complex pore structure heterogeneity by relying on selected characteristic pore-throat radii and porosity parameters for permeability estimation, thereby neglecting the comprehensive pore size distribution characteristics15. Second, the current methodology employs static pore-scale parameters that fail to account for dynamic variations under real-time thermal–mechanical coupling conditions, particularly when maintaining prediction accuracy during evolving stress–strain states and thermal-induced matrix alterations becomes a critical requirement for reliable reservoir performance evaluation. Consequently, there exists an urgent requirement to develop a comprehensive petrophysical indicator that simultaneously satisfies three critical criteria: (1) capability to characterize multi-scale pore architecture spanning from nano-scale matrix pores to fracture networks, (2) real-time monitoring feasibility under dynamic thermal–mechanical coupling conditions involving evolving stress fields and temperature gradients, and (3) maintenance of prediction accuracy for coal reservoir permeability through physically-meaningful parameterization of fluid transport mechanisms.

Since Mandelbrot first proposed fractal theory16, it has developed into a potent tool for explaining the physical characteristics of porous media, including their intricate geometrical morphology, fine-grained pore structure, and the irregularity and uniqueness of their surface characteristics17. In addition to expanding our knowledge of the microstructure of porous materials, it encourages the development of new techniques for designing and predicting material properties in related domains17. Low-field nuclear magnetic resonance (LF-NMR) methods in conjunction with fractal theory have been the subject of much research in recent years. Pape18 was the first to forecast permeability using porosity and fractal dimension; In comparison to the traditional NMR permeability prediction model, Lifei et al.19 showed that fractal dimension is more accurate and superior for predicting the absolute permeability of porous media; Jianping et al.20 proposed a method for evaluating the pore structure of sandstones based on NMR pore fractal dimension, which links the microscopic pore structure and macroscopic physical property parameters; Li et al.21 combined fractal and gas microscopic seepage theories to establish a model for the apparent permeability of microfractures under the coupling of medium deformation and slip effects; Chun et al.5 built a correlation model between the relative permeability of the rock cores and the NMR T2 spectra based on fractal theories, which gave a new perspective and a thorough understanding of the two. Haojiang et al.22 employed machine learning techniques to identify and predict critical factors influencing CBM well productivity, subsequently establishing a production prediction model. This model provides a scientific foundation for analyzing dominant controlling factors in typical well blocks, thereby offering methodological support for optimizing CBM extraction efficiency. However, current experimental protocols predominantly investigate isolated stress or thermal field effects, fundamentally neglecting the synergistic thermo-mechanical restructuring of pore networks through coupled field interactions. Furthermore, existing machine learning approaches suffer from two critical constraints: (1) over-reliance on historical datasets that inadequately represent transient thermo-poromechanical responses, and (2) absence of embedded physics-based modules for real-time coupled parameter inputs. This dual limitation significantly impedes the operational effectiveness of permeability prediction systems under actual reservoir conditions, particularly in addressing time-sensitive challenges such as thermal fracturing evolution and stress-dependent permeability hysteresis.

While significant advances in coal reservoir characterization have been achieved through prior investigations, with selected models successfully implemented in engineering practice, the fractal-based thermo-mechanical coupling effects on absolute permeability remain underexplored. Current permeability prediction models demonstrate limited applicability under coupled thermal–mechanical constraints, posing operational challenges in field-scale reservoir management.To determine the mechanism of the seepage performance of the coal reservoir under the action of thermal–mechanical coupling from the perspective of the fractal theory, as well as to establish a permeability prediction model based on the fractal dimensionality, we conducted gas/water two-phase flow displacement and LF-NMR tests on coal samples from the Huanglong Jurassic No. 4 coal seam of the Dafosi Coal Mine in the Binchang Mining Area under the condition of thermal–mechanical coupling. For coal reservoir permeability prediction, the suitability of the fractal dimension-based permeability prediction model and the classical permeability prediction model is contrasted.

The findings of this study theoretically support the development of CBM drainage and extraction regimes on the surface as well as the enhancement of drainage and extraction efficiency by enabling the rapid and precise determination of the absolute permeability of coal reservoirs under thermal–mechanical coupling conditions.

Basic nuclear magnetic proper-ties of coal rock

Principle of NMR testing

The relaxation time (T2) is a key parameter in the nuclear magnetic resonance (NMR) technique, which reflects the amount of water in the sample. By analyzing the distribution of T2 spectra of H atoms present in water molecules in a material’s pores, the fundamental idea of low-field NMR is accomplished. The water content of a rock’s pores and its T2 spectra can be correlated in two different ways. Firstly, the T2 value is positively correlated with pore size; secondly, the signal intensity in the T2 spectrum is positively correlated with the number of pores, i.e., the higher the signal amplitude, the higher the number of pores with the corresponding pore size23,24. The pore size distribution, pore size, and other qualitative and quantitative features of coal that can represent the evolution of the pore structure of coal reservoirs can be obtained from the LF-NMR T2 spectra25,26. The test data can then be processed further to determine the pore fractal dimensions of the test samples. Fluorine oil and nitrogen, which do not interfere with the NMR, were used as thermally coupled environments for the experimental coal rock samples in this study. The LF-NMR T2 spectra were obtained by testing the coal rock samples multiple times in different experimental phases in order to indirectly derive the changes in the distribution of the pore structure and the paths of the fluid transport. The NMR instrument was combined with a peripheral constant temperature and pressure displacement unit. The NMR-derived permeability (kNMR) was calculated as the arithmetic mean of values obtained from the SDR and Timur-Coates models (Eqs. 1 and 2)

$$K_{Timur - Coates} = \left( {\frac{{\varphi_{NMR} }}{{C_{1} }}} \right)^{4} \left( {\frac{FFI}{{BVI}}} \right)^{2}$$
(1)
$$K_{SDR} = C_{2} \varphi_{NMR}^{4} T_{2gm}^{2}$$
(2)

Within the constitutive formulations, the parameters are defined as follows:

ϕNMR (%) represents the NMR-derived porosity calibrated against water-saturated specimens; C1 and C2 denote model constants requiring lithology-specific calibration against coal-bearing formation characteristics; KTimur-Coates (mD) and KSDR (mD) correspond to permeability estimates derived from the Timur-Coates and Schlumberger-Doll Research (SDR) models, respectively; FFI (%) quantifies free fluid index as mobile water saturation; BVI (%) indicates bulk volume irreducible through bound water saturation; T2gm (ms) constitutes the geometric mean of T2 relaxation distributions (Eq. 3).

$$T_{2gm} = {}^{\varphi }\sqrt {T_{21}^{{\varphi_{1} }} + T_{22}^{{\varphi_{2} }} + T_{23}^{{\varphi_{3} }} + \cdots T_{2i}^{{\varphi_{i} }} }$$
(3)

Within the constitutive formulations, the parameters are rigorously defined as: T2gm represents the geometric mean; ϕ denotes the total porosity of the sample; T2i corresponds to individual T2 distribution data points; ϕi indicates the porosity component associated with each T2i distribution.

Test equipment

The perimeter pressure displacement power system, the NMR signaling instrument, and the signal acquisition system were the three main pieces of equipment used in this test, which was conducted at the Key Laboratory of Concrete Structural Safety and Durability in Shaanxi Province (Xijing University).

  1. (1)

    High-temperature high-pressure replacement device

The MR-HTHP high temperature and high-pressure replacement device, constant flow and pressure replacement pump, and other auxiliary equipment make up the majority of the system. The test coal samples are supplied with surrounding pressure and gas/water replacement differential pressure using fluorine oil and compressed nitrogen gas.

  1. (2)

    Nuclear magnetic resonance (NMR) signaling instrument

The MacroMR12-150H-1 NMR instrument, a specialized high-temperature and high-pressure NMR probe, and an NMR test coil make up the majority of the components. The holder’s body, which is composed of non-hydrogen and non-magnetic material, can retain core specimens that are 25 mm in diameter and about 50 mm long.

  1. (3)

    Signal acquisition and analysis system

The system’s central component is a PC mainframe with control and acquisition software installed, which is used to enter the test’s necessary settings, automatically gather test results, and arrange and analyze test data.

A schematic diagram of the test setup connections is shown in Fig. 1.

Fig. 1
figure 1

Schematic diagram of test setup connection.

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The main equipment of the test and the peripheral equipment are physically shown in Fig. 2.

Fig. 2
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Test equipment.

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Test programme

Coal cores were prepared by field sampling for this test, and in compliance with Chinese industry-standard SY/T5336-2006 Core Analysis Methods27, the specimens were placed in the ZYB-II vacuum pressurized saturator, pumped for 3 h at a negative pressure of 0.1 MPa, and then submerged for 24 h in a pressurized environment of 8 MPa to fully saturation. T2 spectrum determination followed by centrifugation, nitrogen displacement test, and variable confining pressure test28,29.

Measurement of basic sample parameters

Three representative coal specimens (DFS-2, DFS-3, DFS-4) were subjected to NMR T2 spectral measurements under saturated and post-centrifugation conditions to characterize critical petrophysical properties: T2cut-off thresholds, bound/free water saturation ratios, and initial permeability correlations.

Experimental workflow:

Saturated-state NMR acquisition

Specimens were positioned in the NMR probe for T2 relaxation profiling using Carr-Purcell-Meiboom-Gill (CPMG) pulse sequences.Sequence parameters were calibrated against reference standards:

Offset frequency (O1) = 242,790.3 Hz; Pulse durations: P1 = 17 μs, P2 = 34 μs; Spectral width (SW) = 250 kHz; Receiver gain (PRG) = 2; Repetition time (TW) = 3000 ms; Scan number (NS) = 16; Echo time (TE) = 0.25 ms (≥ 6P2 constraint); Acquired echoes (NECH) = 12,000.

Post-dehydration NMR validation

Following bench-top centrifugation of saturated specimens (4000 rpm × 15 min), identical CPMG parameterization was systematically reapplied to characterize residual fluid saturation states through NMR T2 spectral.

Test data processing

NMR T2 spectral datasets acquired from saturated and centrifuged specimens were computationally processed through spectral deconvolution and multi-exponential fitting analyses to derive critical petrophysical parameters governing fluid transport behavior.

Determination of sample parameters at different confining pressures

The confining pressure configurations were maintained using an MR-HTHP (Magnetic Resonance-coupled High-Temperature High-Pressure) displacement system, which simultaneously regulated confining pressure (σ3), thermal field (25 °C), and fluid pressure differentials (ΔP). To replicate in situ geostress conditions of the target coal seam, specimen DFS-1 underwent a stepwise confining pressure loading protocol (5, 7.5, 10, 12.5, 15 MPa) under isothermal control. At each pressure tier, pore architecture evolution and absolute permeability responses were systematically quantified through replicating the methodology outlined in "Characterisation of the fractal dimension response to confining pressure" section, enabling preliminary derivation of fractal dimension-permeability constitutive relationships.

Fractal dimension and absolute permeability model validation

To validate the fractal dimension-permeability correlations established in "Pore fractal permeability prediction modelling" section, a factorial experimental matrix comprising nine triaxial configurations was systematically conducted, combining confining pressure (5, 7.5, 10, 12.5, 15 MPa) with thermal gradients (25 °C, 50 °C, 75 °C). Under each thermo-mechanical regime, the experimental protocol as delineated in "Characterisation of the fractal dimension response to confining pressure" section was rigorously replicated. Acquired datasets were subsequently benchmarked against the established fractal-permeability constitutive model through multivariate regression analysis.

Real-time permeability measurement of gas- displacement water

According to the procedure outlined in GBT 28,912–2012 Method for Determination of Relative Permeability of Two-Phase Fluids in Rocks30, the pertinent coal samples and parameter settings for the gas displacement water test are displayed in Table 1. Upon reaching displacement pressure thresholds where nitrogen injection fails to mobilize connate water, the inlet/outlet valves are hermetically sealed to preserve hydraulic integrity. Quiescent-phase T2 relaxation profiling and fractal dimension quantification are subsequently conducted under stabilized pressure conditions. A stepwise displacement pressure escalation protocol is then implemented, with iterative repetition of this hydrodynamic monitoring sequence until experimental termination, thereby enabling dynamic permeability tracking throughout gas displacement water displacement processes. All experimental procedures are maintained under constant confining pressure (σ3 = 5–15 MPa) and isothermal control (25 °C ± 0.5 °C).

Table 1 Replacement pressure and confining pressure setting.
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Test sample

The coal samples originated from the No. 4 Jurassic Huanglong coal seam in the Dafosi Mine, Binchang mining area, characterized as structural-dominant bituminous coal with low-rank metamorphic grade. After field collection, intact samples were machined into cylindrical geometries (nominal dimensions ϕ25 × 50 mm) at the Shaanxi Provincial Key Laboratory (Xijing University) to ensure compatibility with the testing apparatus (Fig. 3). Precise dimensional parameters and mass characteristics are presented in Table 2.

Fig. 3
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Test sample.

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Table 2 Size information of LF-NMR test coal samples.
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The four coal specimens selected for the experimental suite exhibited homogeneous matrix fabric characteristics, with macroscopic inspection confirming structural integrity through the absence of visible fractures or discontinuities.

Test results analysis

Analysis of basic nuclear magnetic parameters of coal rock samples

By conducting saturation and post-centrifugation T2 spectrometry tests on raw coal samples and by accumulating the amplitude of the T2 spectral signals, it is possible to calculate porosity, movable and bound fluid porosity, movable and bound fluid saturation, T2cut-off, and geometric mean of T2gm31.

The T2cut-off parameter serves as a critical threshold for discriminating pore network architectures, operationally defined as:

Connected pore network (T2 > T2cut-off): These hydraulically continuous pathways dominantly govern fluid transport properties through percolation-controlled mechanisms.

Immobile/bound poro (T2 ≤ T2cut-off): Characterizing dead-end pore structures and surface-adsorbed fluid phases that primarily contribute to matrix storage capacity via surface interaction-dominated retention.

The computed porosity was first calibrated against a standard specimen with respect to the integrated area of the T2 spectrum before the pertinent tests were conducted. Equation (4) displays the derived equation following calibration.

$$\phi_{NMR} = \left( {\frac{s}{v} - 3.65136} \right)/19.44203$$
(4)

where S is the T2 spectral integration area (dimensionless), v is the sample volume (cm3), and ϕNMR is the calculated porosity (%) derived by the LF-NMR technique.

In accordance with the Hodot pore classification framework, the NMR T2 spectra of specimens DFS-2, DFS-3, and DFS-4 exhibit trimodal distributions corresponding to distinct pore regimes (Fig. 4):

Fig. 4
figure 4

Cumulative porosity of saturated and centrifuged rock samples.

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Micropores (T2 < 0.1 ms; pore diameter < 0.01 μm): Representing surface-dominated adsorption sites through capillary condensation and physisorption mechanisms.

Transitional pores (0.1 < T2 < 1 ms; 0.01 < pore diameter < 0.1 μm): Exhibits dual adsorption-permeation behavior and is dominated by Knudsen diffusion at low pressure.

Mesopores (1 < T2 < 10 ms; 0.1 < pore diameter < 1 μm) and Macropores/Fractures (T2 ≥ 10 ms; pore diameter > 1 μm): Governing Darcian flow regimes through laminar-turbulent hydrodynamic transitions32.

Seepage-dominant pores (T2 > 1 ms): CBM desorption kinetics and pressure-driven advection (mobile fluid saturation: 12.14% ± 1.2%); Adsorption-constrained pores (T2 < 1 ms): Regulating Langmuir-type sorption equilibria and surface diffusion processes.

This coal sample’s average T2cut-off value was 6.99 ms (pore diameter ≈ 0.699 μm), and its average moveable fluid saturation was 12.14%.

Figure 5 shows the proportion of each pore section. The coal sample is primarily composed of Micropores and Transitional pores, the T2cut-off value indicates that Mesopores, Macropores, and Fractures are the main effective pores contributing to the seepage performance. Additionally, there are more connected pore and a more developed pore and fissure structure.

Fig. 5
figure 5

Porosity distribution of coal samples in the natural state.

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The detailed basic physical parameters measured by LF-NMR are shown in Table 3, The average value of permeability K0 of coal samples at normal temperature and pressure is 3.2 mD.

Table 3 L4 3 Basic nuclear magnetic parameters of rock samples.
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Fractal dimension of coal rock pores at ambient temperature and pressure

The fractal dimension can quantitatively characterize the complexity of the pore and fracture structures of coal reservoirs33,34, and fractal theory can be introduced for further research and analysis because coal rock bodies have many complex pore and fracture structures with self-similar structures at the same time. The microscopic pore and fracture structures have strong fractal characteristics. The more complicated the pore structure of the pore section and the stronger the pore inhomogeneity, the higher the fractal dimension of the coal reservoir pore structure. This is often thought to decrease the seepage capacity of porous materials35.

The NMR transverse relaxation time T2 can be approximated under a uniform magnetic field as Eq. (5):

$$\frac{1}{{T_{2} }} \approx \frac{1}{{T_{2s} }} = \rho_{2} \left( \frac{S}{V} \right)$$
(5)

where ρ2 is the surface relaxation rate; T2s is the surface relaxation time, m/ms; S is the specific surface area, m2; V is the volume, m3; S/V = 3/r when the pore model is an ideal sphere, S/V = 2/r when the model becomes tubular.

Thus it can be concluded that (6):

$$T_{2} \propto r$$
(6)

According to the fractal theory, the number of pores with a radius greater than r and the radius of the pore throat is shown in Eq. (7) as follows36:

$${N( > r) = \int_{r}^{{r_{\max } }} S(r){\text{d}}r \propto r^{ - D} }$$
(7)

where N(> r) indicates the number of aperture throats with a radius greater than r, D is the fractal dimension; rmax is the maximum aperture throat radius, μm; S( r) represents the distribution density function of the aperture throat radius.

The cumulative pore volume fraction for pores with radii less than r is shown in Eq. (8)37:

$$S_{v} = \frac{{r^{3 - D} - r_{{{\text{min}}}}^{3 - D} }}{{r_{{{\text{max}}}}^{3 - D} - r_{{{\text{min}}}}^{3 - D} }}$$
(8)

Considering that rmin is much smaller than rmax, Eq. (8) can be simplified to Eq. (9):

$$S_{v} = \left( {\frac{r}{{r_{\max } }}} \right)^{3 - D}$$
(9)

Combining Eq. (6) with Eq. (9) yields Eq. (10):

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

Equation (11) is the commonly used quantitative criteria for the LF-NMR pore-slit fractal dimension. It is derived by taking logarithms on both sides of Eq. (10).

$$Lg(SV) = (3 - D)Lg(T_{2} ) + (D - 3)Lg(T_{2\max } )$$
(11)

where T2 is the transverse relaxation time, T2max is the maximum transverse relaxation time, D is the fractal dimension, and SV is the cumulative pore volume % for which the transverse relaxation time is smaller than T2.

The slope of the fitted straight line was obtained by linear fitting of Lg(SV) to Lg(T2) as (3-D) and then the fractal dimension D was obtained19. The fractal diagram of the normal temperature and normal pressure coal samples is shown in Fig. 6, and the fractal dimension is shown in Table 4.

Fig. 6
figure 6

Initial state fractal of coal samples.

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Table 4 Fractal dimension of coal samples in the initial state.
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The fractal dimension analysis of pristine coal specimens revealed nonlinear scaling behavior, as evidenced by the multi-regime morphological signature depicted in Fig. 6. First-derivative quantification of the fractal curve identified a statistically significant inflection point (critical slope threshold = 0.01), demarcating distinct geometric regimes: a curvilinear segment (slope < 0.01) indicative of surface roughness-dominated scaling, and a quasi-linear ___domain (slope ≥ 0.01) characteristic of bulk matrix-controlled fractal architecture38,39. As a result, the fractal map must be divided into two stages, which correspond to the fractal dimensions of the connected pore segment and the bound pore segment, respectively40 and the formula has been shown to be used for both linear fitting of the curve19. Table 4 displays the two-stage fractal dimensions. The T2cut-off value is the turning point. It can be measured quickly by locating the inflection point in the fractal diagram and is quite similar to the data obtained by centrifugation. This allows one to avoid the time-consuming experimental process of centrifugation.

The coal samples’ pore and fracture structures were comparatively uniform, with the fractal dimensions of the bound pore segments ranging D1 from 1.038296 to 1.038970 and those of the connected pore segments D2 from 2.996173 to 2.996174 at normal temperature and pressure.

Pore fractal dimension permeability modelling

Characterisation of the fractal dimension response to confining pressure

Further analysis of the evolution of the fractal characteristics of the coal samples with the confining pressure, the fractal map of the coal sample DFS-1 under different confining pressures, and the fractal dimension of the coal sample DFS-1 under different confining pressure and the absolute permeability change of the predictive model of NMR are shown in Fig. 7 and Table 5, respectively, and the NMR permeability is selected from the averages of the Timur-Coates and the SDR models, respectively12.

Fig. 7
figure 7

Fractal diagram of DFS-1 coal samples under different confining pressures.

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Table 5 Fractal dimensions of DFS-1 coal samples under different confining pressure states.
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As the external circumferential pressure increases step by step from 5 to 15 MPa under the constant temperature of 25 °C, the fractal dimension of the pore section before the T2cut-off value, D1, rises gradually from 1.042650 to 1.053609. This is evident from Fig. 7 and Table 5, and the fractal dimension of the pore section after the T2cut-off, D2, rises to a lesser extent. During the fixed-temperature variable-pressure tests, the coal samples exhibited an upward trend in the T2cut-off value before the segment fractal dimension, indicating a complex pore composition, decreased pore cleavage permeability, and deteriorated seepage performance20. This phenomenon is typically attributed to the coal samples being subjected to the combined action of the external perimeter pressure and pore water pressure. The pore gaps that were once linked to one another and may have contributed to the seepage performance are thought to have split and closed as a result of the combined action of the external surrounding pressure and the pore water pressure. Additionally, the water in the pore spaces finds it harder to resist the binding force and is unable to be moved outside when the flow pore spaces gradually close into bound pore spaces.

Pore fractal permeability prediction modelling

Additionally, as seen in Fig. 8a and b, the pore fractal dimensions D1 and D2 can be fitted and displayed against the coal samples’ absolute permeability, respectively.

Fig. 8
figure 8

Relationship between fractal dimension and absolute permeability of coal samples.

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It is evident that absolute permeability changes exponentially with D2 and has a linearly negative correlation with D1. Using the fractal dimension of the pore structure to forecast permeability may be considered a practical and efficient method41. To accurately forecast permeability, a crucial physical feature, this technique relies on the intricate and irregular pore structure, which is quantitatively analyzed using fractal theory. Thus, as demonstrated by Eqs. (12), (13), and (14), a permeability prediction model based on the fractal dimension of coal rock NMR was built for this investigation.

$$K_{{D_{1} }} = K_{0} (a - b \times D_{1} )$$
(12)
$$K_{{D_{2} }} = 0.1K_{0} \left\{ {1 + c \times \exp \left[ {(D20 - D2)/d} \right]} \right\}$$
(13)
$$\overline{K}_{D} = 0.5 \times (K_{{D_{1} }} + K_{{D_{2} }} )$$
(14)

where \({K}_{0}\) is the permeability of the coal rock at ambient temperature and pressure (mD millidarcy); \({D}_{{2}_{0}}\) is the fractal dimension of the pore NMR after T2cut-off at ambient temperature and pressure; \({\overline{K}}_{D}\) is the average predicted permeability (mD millidarcy); a, b, c, and d are constant terms, respectively, determined by fitting; and \({K}_{D1}\) and \({K}_{D2}\) are the predicted permeability (mD millidarcy) of D1 and D2, respectively; D1 and D2 are the fractal dimensions of the pore NMR before and after T2cut-off, respectively. In this investigation, a, b, c, and d were determined to be 203.62469, 192.92641, 6.56653, and 2.05407e-5, respectively.

Validation of pore fractal dimension permeability models

The evolution of the fractal characteristics of the coal samples with temperature was further analyzed to confirm the accuracy of the NMR fractal dimension-based permeability prediction model for coal rock established in the previous paper. The fractal maps of DFS-1 at various temperatures and the change of the fractal dimension of DFS-1 coal samples at various temperature states are displayed in Fig. 9 and Table 6, respectively.

Fig. 9
figure 9

Fractals of DFS-1 coal samples at different temperatures.

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Table 6 Fractal dimension of DFS-1 coal samples at different temperature states.
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Figure 9 and Table 6 show that when the temperature is maintained at a constant confining pressure, the fractal dimensions of the pore section before the T2cut-off value, D1, and after the T2cut-off value, D2, both decrease as the external confining pressure gradually rises from 25 to 75 °C. It is evident that when coal samples undergo fixed- confining pressure-variable temperature testing, their pore composition tends to become simpler, their pore and fracture permeability improves, and their seepage performance improves somewhat as the temperature rises20.

The agreement between datasets \({\overline{K}}_{D}\) and \({\overline{K}}_{NMR}\) was evaluated through descriptive statistics, paired t-tests, correlation analysis, and Bland–Altman methodology. Key findings are summarized below:

Descriptive statistics

Mean: Both groups yielded identical mean values of 1.3893.

Median: The median of \({\overline{K}}_{D}\) was 1.08, while \({\overline{K}}_{NMR}\) showed a median of 1.12, indicating close alignment.

Standard Deviation: Dispersion metrics were highly comparable, with \({\overline{K}}_{D}\)= 1.34 and \({\overline{K}}_{NMR}\)= 1.34, reflecting equivalent variance magnitudes.

Paired t-test

Mean Difference = 0.01 (no systematic bias observed).

Standard Deviation of Differences = 0.02.

Test Results: t-statistic = 0.01, p-value = 0.998; the null hypothesis was retained, confirming no statistically significant difference between means (α = 0.05).

Correlation analysis

Pearson Correlation Coefficient = 0.998, demonstrating an exceptionally strong linear relationship between the two datasets.

Bland–Altman analysis

95% Limits of Agreement (-0.039, 0.039). All data points resided within this range, with no systematic deviations detected.

These results robustly validate the interchangeability of the two experimental methods in practical applications, supporting their concurrent use for permeability characterization.

In "Fractal dimension of coal rock pores at ambient temperature and pressure" section, a permeability prediction model was developed by integrating nuclear magnetic resonance (NMR)-derived fractal dimensions of coal pores. Leveraging the scaling principles of magnetic fractal theory, this model establishes a robust correlation between the NMR fractal dimension and pore structure heterogeneity, enabling accurate estimation of absolute permeability in coal rocks. The methodology not only simplifies experimental workflows but also provides a theoretical framework for characterizing permeability evolution under static reservoir conditions.

Pore fractal dimension permeability application example

Under certain unique working conditions, the traditional NMR permeability prediction model is not appropriate for precisely determining the real-time permeability of coal rock samples because it necessitates laborious experimental procedures and mainly depends on the centrifugal method to obtain experimental parameters. This section will analyze the features of the absolute permeability response during the coal rock gas–water two-phase replacement test using the previously developed pore fractal dimensions absolute permeability prediction model.

Figure 10 displays the fractal maps at various displacement differential pressures, while Table 7 displays the fractal dimensions at various displacement differential pressures as well as the absolute permeability that the fractal theory permeability prediction model predicts.

Fig. 10
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DFS-3 fractal diagrams for each displacement differential pressures.

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Table 7 DFS-3 fractal dimension permeability for each displacement pressure difference.
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As the replacement pressure difference was gradually increased from 1 to 11 MPa during the constant-peripheral-pressure replacement process, the fractal dimension D1 of the pore section before the T2cut-off value gradually decreased from 1.01235 to 0.98677, and the fractal dimension D2 of the pore section after the T2cut-off value gradually decreased from 2.99892 to 2.99868. It is evident that during the replacement process, the fractal dimension of the two-pore segments decreases as the replacement pressure difference increases, the coal rock’s absolute permeability increases from 4.29 mD to 6.76 mD, the pore permeability increases, and the seepage performance improves20. Because the pore space was initially disconnected and unable to contribute to the seepage performance due to the combined action of the replacement pressure and the pore water pressure. This is because the initially disconnected pores that are unable to contribute to seepage performance are enlarged and connected by the combined action of the displacement pressure and pore water pressure. At the same time, water molecules in the bound pores gradually overcome the binding force to transport, while water in the flowing pores is continuously transported outward.

Conclusions

The response characteristics of pore and fracture structure under thermal coupling were examined in this study using the LF-NMR technique and fractal theory on coal samples extracted from the No. 4 coal seam of Huanglong Jurassic of Dafosi Coal Mine in Binchang Mining Area. Additionally, the relationship between the fractal dimension of bound and connected pore segments and absolute permeability was examined, which resulted in the following findings:

  1. (1)

    To avoid the drawn-out centrifugal operation process of the centrifugal method, the fractal diagram of coal samples is not a straight line in the initial state. There is a clear inflection point at the relaxation time cut-off value T2cut-off to distinguish the curved section from the nearly straight line section, and T2cut-off can be quickly determined by searching for the inflection point in the fractal diagram.

  2. (2)

    In the initial state, the bound pore section’s fractal dimension is significantly smaller than the connected pore section’s. The bound pore section’s fractal dimension, D1, ranges from 1.040 to 1.063, while the connected pore section’s fractal dimension, D2, ranges from 2.99091 to 2.99308; this indicates that the bound pore section’s structure is relatively simple, while the connected pore section’s pore structure is relatively complex.

  3. (3)

    The bound pore section’s fractal dimension D1 exhibits a linearly negative correlation with the absolute permeability, while the connected pore section’s fractal dimension D2 exhibits an exponentially negative correlation.

  4. (4)

    The fractal dimension-based permeability prediction model utilizes nuclear magnetic resonance NMR T2 spectra of water-saturated coal samples to enable real-time estimation of absolute permeability through simplified experimental procedures. This approach facilitates dynamic permeability monitoring in CBM wells by integrating downhole NMR logging data during field operations, optimizing gas drainage strategies, and enhancing recovery rates. The model’s incorporation of pore structure evolution dynamics and fractal scaling principles establishes theoretical foundations for enhancing CBM extraction systems. Furthermore, its methodological framework demonstrates potential applicability for permeability prediction in other porous media types, extending its utility beyond coal reservoir characterization.