A Novel Permeability Prediction Model for Deep Coal via NMR and Fractal Theory

Abstract

A quantitative description of the permeability of deep coal is of great significance for improving coalbed methane recovery and preventing gas disasters. The Schlumberger–Doll research (SDR) model is often used to predict rock permeability, but it has inherent defects in characterizing the pore structure of deep coal. In this study, a permeability model with fractal characteristics (FCP model) is established for deep coal based on nuclear magnetic resonance (NMR). The constants in the SDR model are theoretically explained by the relevant parameters in the FCP model. Centrifugation and NMR experiments were performed to determine the optimal centrifugal force and dual T₂ cutoff values. The results show that the coal samples are mainly composed of micrometer and nanometer pores. The adsorption pores account for the largest proportion, followed by the percolation pores and migration pores. In addition, the prediction accuracy of the FCP model is significantly higher than that of the other three models, which provides a fast and effective method for the evaluation of deep coal permeability.

1. Introduction

Due to the decrease in shallow coal resources in China, deep mining is gradually becoming normal. However, deep coal is in a high-stress and high geothermal environment, and the permeability of the coal seam decreases sharply [1,2]. This situation will lead to frequent accidents and difficulty in achieving safe and efficient mining. The magnitude and spatial distribution of permeability represent the ability of the coal seam to allow fluid flow and output [3]. In the mining of deep coal resources, accurate acquisition of coal seam permeability plays an important role in preventing water inrush and preventing gas outbursts.

NMR technology can be effectively applied to the rock strata with complex lithology to accurately measure the permeability of rock mass [4,5]. At present, a variety of NMR permeability prediction models have been established and applied. The SDR model is a classical NMR permeability prediction model. The SDR model mainly estimates permeability based on the T_2g [6,7]. However, as an empirical formula, the SDR model still has many shortcomings. To improve the SDR model, Zhou found that the geometric mean value of T₂ could better characterize the permeability of shale and obtained a single-parameter NMR permeability model [8]. To further reflect the influence of saturated and bound fluids on permeability, a permeability model is proposed based on the dual T₂ cutoff values to better predict shale permeability [9].

The Timur–Coates (Coates) model uses the T₂ cutoff value to determine the distribution of fluid in the pores and porosity to predict permeability [10]. Many scholars have improved the Coates model because the T₂ cutoff values of different rocks are complex and difficult to accurately determine. A permeability model based on dual T₂ cutoff values is established. The model divides the fluid in tight sandstone into three parts to further describe the seepage characteristics of the fluid [11]. The pore connectivity factor was obtained by displacing the fully saturated carbonate core with heavy water. The Coates model was modified according to pore connectivity [12].

In addition to improvements or innovations based on the SDR model or Coates model, some scholars have proposed composite methods to predict permeability. The permeability is predicted by transverse surface relaxivity, logarithmic mean of transverse relaxation time, and formation factor. The obtained results are more accurate than the classical permeability model [13]. The partial least square method is used to determine the relationship between transverse surface relaxivity segments and permeability, and permeability can be predicted [14]. Smith proposed the use of NMR responses to provide equations without any external input, with good results in shale and carbonate reservoirs [15]. The pore network geometry and permeability of the fractured rock are analyzed using OpenFOAM as the main flow solver [16,17,18].

At present, the SDR model is widely used in the prediction of rock permeability, but it still faces the following key problems.

(1) SDR model lacks theoretical support. Whether the constants in the SDR model have practical significance needs to be studied. (2) The SDR model is fitted by measuring the permeability of a large number of rocks, which is uneconomical and time-consuming. (3) The pore structure is assumed to be homogeneous in the SDR model, but it is complex and heterogeneous in deep coal. The application of the SDR model is limited. The equation of the SDR model is as follows [6]:

$$K_{\mathrm{SDR}}=m{\phi }^n{T}_{2\mathrm{g}}^2$$

(1)

where m and n are constants in the model.

The research on the theoretical improvement of the SDR model is not enough [19,20], and these problems need further study. In this study, a permeability prediction model for deep coal was established to solve the above problems. The FCP model provides theoretical support for the SDR model. The physical meaning of the constants in the SDR model is explained by the relevant parameters in the FCP model. Meanwhile, the FCP model focuses on predicting the permeability of tight rocks with low porosity. According to the experimental results, the pore size distribution of the samples was determined, and the evolution law of pore water content was analyzed. The predicted permeability value of the FCP model is closer to the measured value. Compared with other permeability prediction models, the accuracy of the FCP model is verified.

2. Permeability Model Derivation

2.1. Principles of NMR

The basic principle of NMR technology is to measure the relaxation properties of hydrogen by arranging hydrogen nuclei under an applied magnetic field. Relaxation responses of various hydrogen-containing components (water, methane, etc.) can be obtained by specific RF scans of hydrogen nuclei. Equation (2) is the expression for the transverse relaxation time T₂ [21]:

$$\frac{1}{T_2}\approx {\rho }_2\left(\frac{S}{V}\right)=F_{\mathrm{s}}\frac{{\rho }_2}{r}$$

(2)

where T₂ is the transverse relaxation time (ms) and ρ₂ denotes transverse surface relaxivity (μm/s); S and V refer to the surface area (μm²) and volume (μm³), respectively. F_s refers to a constant value dependent on geometric pore shape. F_s is 2 for columnar and 3 for spherical pores, respectively.

From Equation (2), there is a proportional relationship between the T₂ value and porosity, as shown in Figure 1. The larger the value of T₂, the larger the corresponding pore radius. The pore structure of porous media can be quantitatively characterized by the T₂ spectrum [22,23,24].

2.2. Permeability Expression

Permeability reflects the capacity of a rock to allow fluid to pass; the connected pores are the prerequisite for seepage. q_i is the flow rate of the capillary with radius r_i by the Hagen–Poiseulle equation [25]:

$$q_i=\frac{v_i{r}_i^2\Delta p}{8\mu L_0^2}$$

(3)

where v_i is the volume of capillary i, v_i = πr_i²L₀, μ is the viscosity of the fluid, Δp is the pressure difference, and L₀ is the actual length of the capillary.

The capillary with different pore sizes in Figure 2 reflects the pore structure of the rock. According to the fractal geometry theory, the cumulative size of pores in porous media is expressed as follows [26]:

$$N\left(\epsilon \ge r\right)={\left(\frac{r_{\max }}{r}\right)}^{D_{\mathrm{f}}}$$

(4)

where N is the cumulative number of pores, ε is the length scale, D_f is the fractal dimension, and r_max is the maximum pore radius.

From Equation (4), the number of pores distributed between r and r + dr is [26]:

$$-\mathrm{d}N=D_{\mathrm{f}}{r}_{\max }{r}^{-(D_{\mathrm{f}}+1)}\mathrm{d}r=f\left(r\right)\mathrm{d}r$$

(5)

The total flow rate Q can be expressed as:

$$Q={\displaystyle \sum _i{q}_i}=\frac{\Delta p}{8\mu L_0^2}{\displaystyle \sum _i{v}_i}{r}_i^2$$

(6)

The expression for v_i is:

$$v_i=\phi AL{\eta }_i$$

(7)

where η_i is the fraction of capillary i in the pore volume, A is the cross-sectional area, and L is the representative length.

The total flow Q is expressed as:

$$Q=\frac{\Delta p\phi AL}{8\mu L_0^2}{\displaystyle \int r_i^2}\mathrm{d}{\eta }_i$$

(8)

The following equation is the expression of dη_i.

$$\mathrm{d}{\eta }_i=\frac{\frac{\pi }{4}{r}^2\left(-\mathrm{d}N\right)}{{\displaystyle \underset{r_{\min }}{\overset{r_{\max }}{\int }}\frac{\pi }{4}{r}^2\left(-\mathrm{d}N\right)}}=\frac{\left(2-D_{\mathrm{f}}\right)r^{1-D_{\mathrm{f}}}}{r_{\max }^{2-D_{\mathrm{f}}}\left[1-{\left(\frac{r_{\min }}{r_{\max }}\right)}^{2-D_{\mathrm{f}}}\right]}\mathrm{d}r$$

(9)

The relation between porosity and D_f is as follows [26]:

$$\phi ={\left(\frac{r_{\min }}{r_{\max }}\right)}^{D-D_{\mathrm{f}}}$$

(10)

where φ is porosity, D is the Euclidean dimension of the space, and D = 2 in two dimensions.

By combining Equations (9) and (10), the total flow Q is obtained.

$$Q=\frac{\Delta pAL}{8\mu L_0^2}\frac{\left(2-D_{\mathrm{f}}\right)}{\left(4-D_{\mathrm{f}}\right)}\frac{\phi }{1-\phi }{r}_{\max }^2$$

(11)

The average pore radius r_ave can be obtained with the aid of Equation (4) [27,28].

$$r_{\mathrm{ave}}={\displaystyle \underset{r_{\min }}{\overset{r_{\max }}{\int }}r}f\left(r\right)\mathrm{d}r=\frac{D_{\mathrm{f}}{r}_{\max }}{D_{\mathrm{f}}-1}\left[\frac{r_{\min }}{r_{\max }}-{\left(\frac{r_{\min }}{r_{\max }}\right)}^{D_{\mathrm{f}}}\right]$$

(12)

Combined with Equation (10), the maximum pore radius r_max can be expressed as:

$$r_{\max }=\frac{D_{\mathrm{f}}-1}{D_{\mathrm{f}}}\frac{{\phi }^{\frac{1}{D_{\mathrm{f}}-2}}}{1-{\phi }^{D_{\mathrm{f}}}}{r}_{\mathrm{ave}}$$

(13)

Substituting Equation (13) into Equation (11), the expression for Q is obtained.

$$Q=\frac{\Delta pAL}{8\mu L_0^2}\frac{\left(2-D_{\mathrm{f}}\right){\left(D_{\mathrm{f}}-1\right)}^2}{\left(4-D_{\mathrm{f}}\right)D_{\mathrm{f}}^2}\frac{{\phi }^{\frac{D_{\mathrm{f}}}{D_{\mathrm{f}}-2}}}{\left(1-\phi \right){\left(1-{\phi }^{D_{\mathrm{f}}}\right)}^2}{r}_{\mathrm{ave}}^2$$

(14)

The permeability equation is obtained:

$$K=\frac{\mu LQ}{\Delta pA}=\frac{\left(2-D_{\mathrm{f}}\right){\left(D_{\mathrm{f}}-1\right)}^2}{8{\tau }^2{D}_{\mathrm{f}}^2\left(4-D_{\mathrm{f}}\right)\left(1-\phi \right){\left(1-{\phi }^{D_{\mathrm{f}}}\right)}^2}{\phi }^{\frac{D_{\mathrm{f}}}{D_{\mathrm{f}}-2}}{r}_{\mathrm{ave}}^2$$

(15)

where τ is tortuosity.

Τ is given by the following expression [29,30,31]:

$$\tau =\frac{1}{2}\left[1+\frac{1}{2}\sqrt{1-\phi }+\sqrt{1-\phi }\sqrt{{\left(\frac{1}{\sqrt{1-\phi }}-1\right)}^2+\frac{1}{4}}/\left(1-\sqrt{1-\phi }\right)\right]$$

(16)

The relaxation time of rock is in good agreement with its internal aperture distribution [32,33]. Based on Equation (2), r_ave is represented by the geometric mean of T₂.

$$r_{\mathrm{ave}}=F_{\mathrm{s}}{\rho }_2{T}_{2\mathrm{g}}=2{\rho }_2{T}_{2\mathrm{g}}$$

(17)

where T_2g is the geometric mean value of T₂, and F_s is 2 for columnar.

Substituting Equation (17) into Equation (15), the permeability model is:

$$K=\frac{{\rho }_2^2\left(2-D_{\mathrm{f}}\right){\left(D_{\mathrm{f}}-1\right)}^2}{2{\tau }^2{D}_{\mathrm{f}}^2\left(4-D_{\mathrm{f}}\right)\left(1-\phi \right){\left(1-{\phi }^{D_{\mathrm{f}}}\right)}^2}{\phi }^{\frac{D_{\mathrm{f}}}{D_{\mathrm{f}}-2}}{T}_{2\mathrm{g}}^2$$

(18)

The FCP model in Equation (18) provides a theoretical explanation for the SDR model. We found that the constant m in the SDR model is closely related to the fractal dimension D_f, tortuosity τ, porosity φ, and transverse surface relaxivity ρ, and the value of the constant n is affected by the fractal dimension D_f.

3. Samples and Experiments

3.1. Coal Samples

The raw coal, from the Pingdingshan Coal Mine in Henan Province, China, is between 1006 m and 1137 m deep. By official standards, the raw coal was processed into 8 cylindrical coal samples with a diameter of 25 mm and a length of 50 mm, as shown in Figure 3. According to the composition analysis of the X-ray fluorescence spectrum, the coal samples do not contain substances that affect the NMR results.

3.2. Experimental Procedures and Facilities

(1)

Coal samples were weighed, and their length and diameter were measured. The permeability of the coal samples was measured by the gas injection porosimetry method.

(2)

Coal samples were dried at 80 °C for 24 h until the weight no longer changed. After cooling to room temperature, coal samples were placed into the vacuum saturation device. Dry pumping was carried out for 2 h with negative pressure of 0.1 MPa, then soaking in water with a pressure of 15 MPa for 24 h to make coal samples fully saturated with water.

(3)

An NMR experiment was conducted on the coal sample. As shown in Figure 4, the instrument model used is MacroMR12–150 H–I. The magnets used are permanent magnets with a magnetic field intensity of (0.3 ± 0.05) T, main frequency of 12.42 MHz, and temperature of (32.00 ± 0.02) °C. The echo-spacing TE = 0.1 ms, the 90° pulse width P1 = 10 µs, the 180° pulse width P2 = 20 µs, and the echo number NECH = 15,000.

(4)

To determine the optimal centrifugal force of the coal sample, coal sample F was selected to conduct centrifugal experiments with different centrifugal forces.

(5)

The remaining coal samples were centrifuged by optimal centrifugal force and tested by NMR. The experimental parameters were the same as in Step (3) to obtain the NMR porosity and T₂ cutoff value. The NMR data were processed via Numai NMR data analysis software V1.0, and the T₂ spectrum was inverted by the SIRT algorithm.

4. Results

4.1. Determination of the Optimal Centrifugal Force

The purpose of the centrifugation experiment is to obtain the centrifugal force suitable for coal samples. The optimal centrifugal force can achieve accurate calibration of the T₂ cutoff value of NMR. The accurate selection of centrifugal force is very important. If the centrifugal force is small, the T₂ cutoff value will be larger, the bound water saturation will be larger, and the seepage capacity of coal will also be underestimated. To obtain the best centrifugal force of deep coal, centrifugal tests were carried out on the selected sample F under the centrifugal force of 0.53 MPa, 0.72 MPa, 0.94 MPa, 1.19 MPa, and 1.46 MPa. NMR was used to measure the coal sample after each centrifugation.

Figure 5 shows the NMR T₂ spectra of sample F with the characteristics of three peaks. With the increase in centrifugal forces, the cumulative porosity of sample F decreases gradually. According to the pore classification method [34,35,36,37], the peak S1 (T₂ < 2.5 ms) of the T₂ spectrum was defined as adsorption pores. The peak S2 (2.5~100 ms) was defined as the percolation pores. The peak S3 (T₂ > 100 ms) was defined as migration pores. In Figure 6, with the increase in centrifugal force, the water content of each pore decreases gradually. The water content of adsorption pores and total pores has the same trend and similar value. It is higher than percolation pores and migration pores.

In Figure 6, the water content of adsorption pores decreases greatly when the centrifugal force increases from 0.53 MPa to 1.19 MPa. When the centrifugal force increases from 1.19 MPa to 1.46 MPa, the water content of adsorption pores is less than 3%. The water content of percolation pores and migration pores does not change obviously when the centrifugal force is greater than 0.53 MPa. The expected experimental results were achieved at a centrifugal force of 1.46 MPa. In addition, Yang et al. [37] conducted centrifugal experiments on coal samples and selected 1.46 MPa as the best centrifugal force. This result is consistent with the centrifugal force we selected, indicating that the centrifugal experiment is effective. Based on 1.46 MPa centrifugal force, NMR experiments were performed on the remaining seven coal samples before and after centrifugation.

4.2. Calibration of T₂ Cutoff Values

Figure 7 shows the comparison of NMR responses of samples under the saturated state and the bound water state after centrifugation. Coal samples before and after the centrifugal T₂ spectra have the characteristics of the three peaks. In Figure 7, S2 and S3 belong to a whole, and their corresponding pores are well connected. S1 is independent of S2 and S3, and the connectivity between adsorption pores and other pores is poor. After centrifugation, the peak of S1 did not decrease significantly, while the peak of S2 and S3 decreased significantly. The corresponding peak values of S2 and S3 are accompanied by a trend of rightward migration. This is the performance of water flowing from partial percolation pores to migration pores. This indicates that the connectivity between the percolation pores and the transport pores is higher than that of the adsorption pores.

After centrifugation, the water content of the adsorption pores of the samples did not change obviously. Yao [38] gave the T₂ boundary between capillary water and film water, T₂ = 0.25 ms. This indicates that the film water mainly exists in the adsorption pores. The dual T₂ cutoff values method was applied to remove the effect of water film thickness on pore size. In Figure 7, the adsorption pores are divided into two parts by the first cutoff value T_2c1. The water in the adsorption pores between 0 ms and T_2c1 is called immovable film water, and the water between T_2c1 and 0.25 ms is called partially movable film water. Adsorption pores filled with immovable film water are not considered to be involved in water seepage [38,39]. Therefore, we removed this part of adsorption pores to minimize the influence of water film thickness on pore diameter.

Table 1. Petrophysical properties of coal samples in the experiment.

Sample Number	Permeability (10−3 μm2)	Porosity (%)	T2c1 (ms)	T2c2 (ms)	T2g (ms)
A	0.221	3.83	0.130	1084.37	3.22
B	0.021	1.67	0.065	766.34	1.97
C	0.037	1.85	0.080	1762.91	3.00
D	0.025	1.59	0.086	943.79	1.46
E	0.181	2.82	0.122	1162.32	3.37
F	0.027	2.06	0.086	821.43	3.02
G	0.272	3.19	0.150	1011.64	4.41
H	0.123	2.67	0.114	1084.37	3.34

presents the porosity of the eight coal samples ranging from 1.67% to 3.83% with an average of 2.51%, and the permeability ranges from 0.021 × 10⁻³ μm² to 0.272 × 10⁻³ μm² with an average of 0.113 × 10⁻³ μm². The porosity and permeability of the coal samples are complicated and belong to low porosity and low permeability coal.

4.3. Distribution of Coal Pore Size

The pores are assumed to be capillaries in Section 2.2. The coal samples in this paper belong to tight cores, and the empirical value of ρ₂ is 10 µm/s. The T₂ spectra of water-saturated coal samples in Figure 8 correspond to the pore structure of samples. According to Equation (2), the T₂ spectra of samples are converted to pore radius.

In Figure 8a, there are three peaks (S1, S2, S3) in the T₂ spectra of coal samples. The S1 peak distribution of coal samples is similar, and the peak value is around 0.1 ms. However, the peak intensity is slightly different, and the distribution of S2 and S3 peaks is significantly different. In Figure 8b, the S1 peaks of samples B, C, and D almost coincide, indicating that their adsorption pore structures are similar. We find that the porosity, T_2g, and T_2c1 of the samples are indeed similar in Table 1. The difference between the samples is mainly reflected in the S2 peak and S3 peak, which represent the percolation pores and migration pores. The pore structures of samples E and H in Figure 8c also have similar characteristics.

The pore distribution frequency of coal samples and the corresponding pore structure diagram are shown in Figure 9. The spherical pores in the pore distribution diagram represent pores with different pore sizes. The pore size distribution and pore connectivity are characterized by the number and arrangement of these spherical pores. Comparing spherical pores between different samples, the complexity of the pore structure can be reflected, and the relationship between pore structure and permeability can be understood. The samples with complex pore structures have lower permeability.

From the pore network composition ratio in Figure 9, adsorption pores account for the largest proportion of total pores in the coal samples. Adsorption pores are specially developed in samples B, C, D, and F. Percolation pores are a small part of the pore system. The development of migration pores in sample G is higher than that in other coal samples. The connectivity between adsorption pores and percolation pores is lower than that between percolation pores and migration pores.

The distribution of adsorption pores in coal samples is similar, and the pore volume proportion is high based on Figure 8 and Figure 9. The coal samples are mainly composed of a relatively uniform distribution of micrometer and nanometer pores. The distribution of percolation pores and migration pores is relatively complex.

5. Discussion

5.1. Water Content of Pores Analysis

Figure 10 shows the pore water content of samples before and after centrifugation at 1.46 MPa. The results in

Table 2. Percentage of water content before and after centrifugation.

Sample Number	Sample Status	The Water Content of Pores
		Adsorption	Percolation	Migration
A	Saturated	67.60%	19.86%	12.54%
	Centrifugal	74.41%	19.62%	5.97%
B	Saturated	85.82%	3.20%	10.98%
	Centrifugal	93.58%	3.38%	3.04%
C	Saturated	78.11%	6.62%	15.27%
	Centrifugal	92.29%	5.49%	2.22%
D	Saturated	75.65%	4.76%	19.59%
	Centrifugal	94.72%	3.09%	2.19%
E	Saturated	69.10%	17.76%	13.14%
	Centrifugal	78.04%	18.97%	2.99%
F	Saturated	77.15%	6.38%	16.47%
	Centrifugal	93.37%	4.82%	1.81%
G	Saturated	68.97%	10.08%	20.95%
	Centrifugal	87.90%	9.90%	2.20%
H	Saturated	71.90%	12.68%	15.42%
	Centrifugal	78.84%	15.13%	6.03%

show that the adsorption pores volume accounts for 67.60~85.82% of the total pore volume, the percolation pores volume accounts for 3.20~19.86%, and the migration pores volume accounts for 10.98~19.59%. The proportion of pore water content shows that the adsorption pores account for most of the total volume of coal samples in Pingdingshan, followed by percolation pores and migration pores. After centrifugation, the water content distribution of the eight samples changed obviously. Residual water mainly exists in the adsorption pores. The water content in the adsorption pores of the samples increased by 6.94% at least and by 19.07% at most. The water content of percolation pores does not change. The decrease in water content in migration pores is most obvious, and the decrease range of water content is between 6.57% and 18.75%. The change in water content in pores is consistent with the conclusion by Shen [40]. The developed migration pores and percolation pores increase the drainage, while the adsorption pores mainly contribute to the residual water content.

5.2. The Permeability Predicted Results

To verify the accuracy of the FCP model for deep coal permeability prediction, it is compared with the predicted values of the other three models based on the measured permeability values of samples.

Different from the FCP model, the Coates model does not consider the effect of pore size on permeability and is an empirical model. The Coates model divides fluid in pore into bound fluid and free fluid by a single T₂ cutoff value. Free fluid, bound fluid, and porosity are considered to be in good agreement with permeability, and the Coates model is established. The extended Coates model uses φ, bound fluid saturation (BVI), and free fluid index (FFI) to predict permeability [10].

$$K_{\mathrm{Coates}}={\left(\frac{FFI}{BVI}\right)}^2{\left(\frac{\phi }{s}\right)}^4$$

(19)

where s is the model constant.

The equation of the extended SDR model in coal is as follows [41,42]:

$$K_{\mathrm{SDR}}=c{\phi }^4{T}_{2\mathrm{g}}^2$$

(20)

where c is the model constant.

Based on measured permeability, the extended Coates model is fitted, and the equation is obtained:

$$K_{\mathrm{Coates}}={\left(\frac{FFI}{BVI}\right)}^2{\left(\frac{\phi }{2.6}\right)}^4$$

(21)

The extended SDR model is given by the following expression:

$$K_{\mathrm{SDR}}=\mathrm{12,667}\cdot {\phi }^4{T}_{2\mathrm{g}}^2$$

(22)

Liu obtained the permeability equation by the pore size ratio λ [43]:

$$K_{\mathrm{D}}=\frac{\phi }{8{\tau }^2}{r}_{\max }^2\left(\frac{3-D_{\mathrm{f}}}{5-D_{\mathrm{f}}}\right)\left(\frac{1-{\lambda }^{5-D_{\mathrm{f}}}}{1-{\lambda }^{3-D_{\mathrm{f}}}}\right)$$

(23)

Table 3. Parameters of the Coates model and the SDR model.

Sample Number	Porosity (%)	BVI (%)	FFI (%)	T2g (ms)
A	3.83	80.88	19.12	3.22
B	1.67	84.73	15.27	1.97
C	1.85	76.51	23.49	3.00
D	1.59	89.51	10.49	1.46
E	2.82	81.83	18.17	3.37
F	2.06	67.30	32.70	3.02
G	3.19	74.99	25.01	4.41
H	2.67	82.67	17.33	3.34

shows the parameters needed to calculate the SDR model and Coates model. The permeability calculation results of the three models are shown by comparing them with the measured permeability in Figure 11a–c. The results show that the correlation coefficients with measured permeability are 0.61, 0.83, and 0.89, respectively. Liu’s model has the lowest prediction accuracy, its confidence interval is larger, and the number of samples in the confidence interval is the least.

The calculation effect of the FCP model is shown in Figure 11d, and its correlation coefficient reaches 0.98. The correlation coefficient between measured permeability and predicted permeability of the FCP model is the highest, which indicates that the prediction accuracy of the FCP model is better than that of other models. The confidence interval of the FCP model is the narrowest, and the number of samples in the confidence interval is also the largest. The results show that the stability and reliability of the FCP model are higher than those of the three models. The accuracy and validity of the FCP model for deep coal permeability prediction are verified on this basis.

6. Conclusions

In this study, we propose a permeability model with fractal characteristics to improve the SDR model. The parameters in the FCP model were obtained through centrifugation and NMR experiments, and the FCP model was validated through comparison with other models. The following conclusions were obtained:

(1)

A permeability prediction model suitable for deep coal is established in this paper. The FCP model is an improvement of the SDR model. The constants in the SDR model are given theoretical explanations based on the fractal dimension D_f, tortuosity τ, porosity φ, and transverse surface relaxivity ρ in the FCP model.

(2)

The optimum centrifugal force of the samples was determined to be 1.46 MPa by the evolution of pore water content. After centrifugation, the adsorption pores volume accounts for 67.60~85.82% of the total pore volume, the percolation pores volume accounts for 3.20~19.86%, and the migration pores volume accounts for 10.98~19.59%. This indicates that the adsorption pores account for most of the total volume of coal samples, followed by percolation pores and migration pores.

(3)

Based on the gas injection porosimetry method, the permeability ranges from 0.021 × 10⁻³ μm² to 0.272 × 10⁻³ μm² with an average of 0.113 × 10⁻³ μm². The correlation coefficients between the predicted values and the measured values of Liu’s model, Coates model, and SDR model are 0.61, 0.83, and 0.89, respectively. The correlation coefficient between the predicted permeability and the measured permeability of the FCP model is 0.98. It shows that the prediction results of the FCP model are better than those of other models. and its accuracy is verified.