Re-Calibrating the Mercury-Intrusion-Porosimetry-Measured Pore Size Distribution of Coals: A Novel Method for Calculating the Matrix Compression Coefficient

Abstract

Accurate measurement of the pore size distribution (PSD) in coals is crucial for guiding subsequent coalbed methane (CBM) engineering practice. Currently, mercury intrusion porosimetry (MIP) measurement has been widely used as a PSD testing method due to its effectiveness and convenience. Nevertheless, it is worth noting that the elevated pressure during the MIP experiments can lead to matrix compressibility, potentially causing inaccurate estimations of PSD in coals. Therefore, correction methods are used to modify the PSD in the high-pressure segment to improve the accuracy of MIP data. This study proposed a novel method with higher accuracy and convenience for calculating the matrix compressibility coefficient compared to the traditional calculation methods. Firstly, the matrix compressibility coefficients of six coal samples were calculated by using low-temperature nitrogen adsorption (LTNA) data. Subsequently, by utilizing the mathematical correlation between K_c (the compressibility coefficient of the coal matrix) and R_o,max (the maximum vitrinite reflectance) from prior research, a novel statistical method was designed to determine the matrix compressibility coefficient of the samples. Finally, the statistical matrix compressibility coefficient determination method was used to examine the fractal characteristics of the actual PSD. The results indicate that when the pressure exceeds 24 MPa, the volume obtained from mercury intrusion exceeds the pore volume measurement. The K_c calculated using the traditional correction method is in the range of 0.876–1.184 × 10⁻¹⁰ m²/N, while the K_c values of our proposed statistical correction method range from 0.898 × 10⁻¹⁰ to 1.233 × 10⁻¹⁰ m²/N, with a comparison error rate of ~0.11–5.25%. The MIP data greater than 24 MPa were effectively corrected using the statistical correction method, thus reducing the mercury intrusion volume error by 91.75–96.40%. Additionally, the corrected pore fractal dimension (D₂) values fall within the range of 2.792 to 2.975, which are closer to the actual values than the pore fractal dimension range of 3.186 to 3.339.

1. Introduction

The commercial production of coalbed methane (CBM) truly began after the 1970s. The United States was one of the first countries to commercialize CBM production, experiencing a progression from experimental stages to large-scale commercialization, with its CBM production rapidly increasing, making it a global leader in bed methane development. Other countries, such as Australia, China, and Canada, have also started large-scale development of CBM resources. Currently, the global CBM industry has reached a particular scale, becoming an essential supplement to the natural gas supply [1]. During the extraction of CBM, the pore size distribution (PSD) of coal reservoirs and their variations are vital for the permeation and retention of CBM [2,3,4,5]. Therefore, accurately characterizing the PSD is crucial for improving energy development efficiency. Understanding the PSD helps optimize extraction methods, improve gas recovery rates, and ensure the efficient utilization of coalbed methane resources.

Existing methods for characterizing coal PSD include both direct and indirect approaches [6,7,8,9,10,11]. Low-temperature gas adsorption measurements are only suitable for testing the PSD characteristics of smaller apertures [12,13,14]. Some image-testing techniques (e.g., scanning electron microscope and transmission electron microscope) can visually observe the exterior of pores through image processing, but the PSD data obtained are not representative [15,16]. MIP is a physical, analytical technique employed for the assessment of the pore structure within various materials. This technique measures parameters, such as pore size and distribution, by applying high pressure to force mercury into the material’s pores, and it is considered an important method for analyzing PSD [17,18,19,20,21].

Because of the coal matrix’s high compressibility [22,23,24,25], excessive pressure within the pores can cause the coal matrix to deform or even fracture. This deformation may lead to the mercury intrusion volume surpassing the actual pore volume. This inevitably affects the accuracy of the fractal dimension calculation for the coal PSD. Therefore, when using MIP and fractal dimensions to characterize PSD, it is imperative to comprehensively account for the compressive effects of the coal matrix and implement suitable correction measures to guarantee the precision and validity of the measurement results.

Previous studies have demonstrated that various factors influence the measured pore volumes in coals during MIP experiments. At medium- to high-pressure stages, the pore-filling mechanism and matrix-shrinkage effect directly controlled the determined PSD and other pore parameters [26]. Some researchers have determined the relationship between mercury intrusion pressure and coal deformation in a specific pressure range, using relevant methods to calculate the matrix compression coefficient in coals [27,28]. It has been recognized that the coal matrix compression effect becomes significant when the mercury intrusion pressure exceeds 10 MPa [29]. Jin et al. [30] analyzed the changing pattern of coal matrix pore characteristics with pressure based on coal matrix compressibility. After that, the complexity of pore structures was concisely described by using fractal theory. Liu et al. [31] considered the matrix-compression effect and analyzed its impact on pores of different size ranges. Han et al. [32] found that coal samples exhibit significant compressibility during MIP tests and that the pore volume decreases with increasing pressure. Zheng et al. [33] calculated the coal matrix compression coefficient and characterized its PSD by correcting it with fractal theory.

The degree of coal metamorphism is a significant factor affecting the compressibility of the coal matrix. The study by Guo et al. [34] indicates that when R_o,max (maximum vitrinite reflectance) is between 0.65 and 0.99%, the compressibility of the coal matrix shows a rapid decline. In contrast, when R_o,max is between 1.25 and 1.77%, the compressibility shows an increasing trend. Similarly, Ettinger and Zhupakhina’s [35] research also observed this trend and described it as a U-shaped relationship curve. Zheng et al. [33] found that as the degree of metamorphism evolves from medium to high, the compressibility of the coal matrix increases, and the vitrinite and inertinite groups have opposite effects on the compressibility of the coal matrix. Lu’s [36] research indicates that the coal matrix compressibility coefficient exhibits a three-stage change trend with the increase of R_o,max.

In summary, in previous studies on MIP data correction, traditional correction methods require extensive preparation, involve substantial calculations, and rely heavily on test data obtained through LTNA experiments. Without conducting LTNA experiments, MIP data correction cannot be performed. When assessing the impact of coal matrix compressibility on the permeability and porosity of coalbed methane reservoirs, such traditional correction methods result in low MIP data correction efficiency, higher experimental costs, and increased complexity. On the other hand, more studies have shown the relationship between the coal matrix compressibility coefficient (Kc) and the increase in R_o,max, but they have yet to consider utilizing this trend to calculate the coal matrix compressibility coefficient (Kc).

In response to the issues in previous MIP data correction studies, this research proposes a new MIP data correction method (the statistical correction method). The method is based on the coal matrix compressibility coefficient (Kc) and R_o,max values calculated in previous studies [9,17,33,37,38,39,40,41,42,43], fitting their mathematical relationship to calculate the matrix compressibility coefficient of coal samples and correcting the MIP data exceeding 24 MPa. This study further investigates the changes in pore fractal dimensions before and after correction, indirectly validating the accuracy of the proposed method. This method successfully depicts the three-stage variation trend of the matrix compressibility coefficient of coal samples with different degrees of metal as the maximum vitrinite reflectance increases, making it possible to determine the matrix compressibility coefficient based solely on the maximum vitrinite reflectance of coal samples. Therefore, this method simplifies the correction process during MIP data correction, eliminating the need for specialized low-temperature nitrogen adsorption (LTNA) experiments. Compared with traditional correction methods (based on LTNA data), this method significantly reduces experimental preparation and the computational workload, facilitating a deeper understanding and revealing the characteristics of coal pore size distribution, thus making it more practical. Introducing this method helps optimize the complex process of CBM reservoir evaluation, thereby enhancing its applicability.

2. Samples and Experiments

The Qinshui Basin, located in northern China, is rich in coal resources and contains abundant natural gas and coalbed methane resources [44,45,46,47,48,49,50]. Significant progress has been made in developing CBM in this region in recent years. This experiment collected coal samples from six distinct mining areas (DD, QX, TL-2, TL-8, WK, and YY) within the southern Qinshui Basin. Among them, the DD and QX mines are located in the Zhengzhuang block, the TL-2 and TL-8 mines are in the Mabi block, and the WK and YY mines are in the Fanzhuang block (Figure 1). Samples were extracted from the working faces of underground mines to ensure freshness and integrity. The coal samples were wrapped in absorbent paper, sealed with adhesive tape, and placed in sealed bags before being promptly transported to the laboratory for MIP, LTNA, and maximum vitrinite reflectance testing.

The MIP measurements were performed using a Micromeritics Autopore 9510 instrument of the standard SY/T5346-2005 [51]. The sample dimensions were approximated in ~2 mm length, width, and height (Figure 2).

The LTNA test was performed using an Autosorb iQ (Quantachrome, FL, USA). After grinding, the coal powder was sieved with a 40–60 mesh sieve and dried in a vacuum oven at a constant temperature for 24 h before testing.

3. Results and Discussion

3.1. MIP Experimental Results

Various MIP experimental parameters offer insights into pore connectivity, development, and coal matrix compressibility from multiple perspectives. By analyzing MIP data from samples collected across different mining areas, mercury intrusion and extrusion curves, along with the PSDs, were obtained. The MIP curves of coal samples from six distinct mining areas could be categorized into three types (Figure 3). Type I (Samples DD and QX): the mercury intrusion and extrusion curves exhibit similar slopes with no significant hysteresis loop, indicating predominantly semi-open pores with poor connectivity, which is not conducive to gas flow. Type II (Samples TL-8, WK, and YY): the mercury intrusion and extrusion curves have similar slopes at high pressures but demonstrate hysteresis at low pressures, suggesting poor connectivity in the transitional pores segment and better connectivity in the more significant pore segment. Type III (Sample TL-2): the curve indicates that as the pressure increases, the mercury intrusion volume significantly rises at lower pressures, with a wide curve opening and evident hysteresis, indicating predominantly open pores with good connectivity, which is not favorable for gas storage. Notably, as the mercury intrusion pressure increases, all selected samples in this article show a significant increase trend in the cumulative mercury intrusion volume when the pressure exceeds ~10 MPa because of the coal matrix compressibility. Failure to account for this effect during data analysis may result in overestimating the number of micropores.

3.2. LTNA Experimental Results

LTNA curves are commonly used to characterize pore structures by analyzing the hysteresis loops generated during adsorption and desorption. Examining different types of adsorption–desorption curves allows for identifying various pore characteristics. Through horizontal comparison of different coal experimental data with the IUPAC classification of hysteresis loops, the experimental curves could be categorized into three types (Figure 4).

Type I curves (Samples No. DD, WK, and YY) exhibit significant adsorption–desorption hysteresis loops within the P/P0 range of 0.4–1, characterized by a sharp drop, indicating the existence of bottleneck breakthroughs and ink-bottle-shaped pores. Type II curves (Sample No. QX) display nearly parallel adsorption and desorption curves with indistinct hysteresis loops, signifying half-open, plate-like pores. Type III curves (Samples No. TL-2 and TL-8) resemble Type II curves but exhibit more distinct hysteresis loops in the P/P0 range of 0.7–1, which indicates the presence of open plate-like pores. Ink-well-bottle pores and half-open, plate-shaped pores are conducive to the adsorption of coalbed methane. However, they are unfavorable for its desorption, making it difficult for coalbed methane to migrate to seepage channels and coalbed methane wells quickly. In contrast, due to their simple pore structure, open plate-shaped pores are more favorable for the desorption and migration of coalbed methane.

3.3. Considering MIP Data Correction for Matrix Compression

3.3.1. Traditional Correction Method

An analysis of MIP experimental data from six coal samples from various mining regions reveals that when mercury injection pressure surpasses 24 MPa, a strong linear relationship exists between the cumulative mercury injection volume and the injection pressure, with correlation coefficients exceeding 0.99 (Figure 5). This indicates a remarkable compression effect on the coal matrix at pressures above 24 MPa. Assuming the mercury compressibility and instrument are negligible, the coal matrix compression effect can be described as follows [52]:

$$K_c={\displaystyle \frac{d{V}_c}{V_{c}d{P}_c}}$$

(1)

where K_c represents the matrix compression coefficient, 10⁻¹⁰ m²/N; V_c represents the coal matrix volume, and cm³/g; dV_c represents the coal matrix volume increment at a pressure of dP_c, cm³/g.

Research indicates that when the mercury intrusion pressure exceeds 20 MPa, the compression effect on the coal matrix becomes significant [53,54,55,56]. At this stage, the variation in the mercury injection volume of the coal sample is attributed to the combined effects of pore filling and matrix compression [52].

$$\mathsf{\Delta }{V}_d=\mathsf{\Delta }{V}_p+\mathsf{\Delta }{V}_c$$

(2)

where ΔV_d represents the mercury intrusion volume; ΔV_p denotes the pore-filling volume; and ΔV_c is the change in the volume of matrix compression, all measured in cm³/g.

Combining MIP test data with LTNA test data, the appropriate correction range is 24 MPa–200 MPa (the corresponding pore size range is 3–25 nm). Before the correction, the volume and pressure of mercury intrusion within the correction range of the coal sample exhibit a linear relationship. Therefore, ΔV_d/ΔP is considered a constant C, resulting in ΔV_c/ΔP [52]:

$${\displaystyle \frac{\mathsf{\Delta }{V}_c}{\mathsf{\Delta }P}}\approx C-{\displaystyle \frac{{\sum }_{3\mathrm{nm}}^{25\mathrm{nm}}\mathsf{\Delta }{V}_p}{\mathsf{\Delta }P}}$$

(3)

where ΔP represents the pressure increment in MPa and ΔV_p represents the pore volume within the 3–25 nm range obtained from LTNA test data, in mL/g.

Because ΔV_c/ΔP can be approximated as a constant, using ΔV_c/ΔP instead of dV_c/dP_c, the K_C can be obtained by combining Equations (1) and (3).

3.3.2. Statistical Correction Method

The traditional correction method corrects MIP data through LTNA data, but explicitly testing LTNA data to achieve the purpose of correcting MIP data has the disadvantages of high cost, a long time, and low efficiency, with certain limitations. In response, a statistical correction method for MIP data correction is proposed. The specific method is as follows. Based on the statistical analysis of K_C data from 176 different coal samples, the samples are divided into three groups according to their degrees of metamorphism: low metamorphism (0.35% ≤ R_o,max < 0.65%), medium metamorphism (0.65% ≤ R_o,max ≤ 2.00%), and high metamorphism (R_o,max > 2.00%). For each group, the K_C data are averaged based on R_o,max intervals of 0.05%, and this method is used to fit the relationship between K_C and R_o,max for coal samples of different metamorphic degrees. First, different coal samples’ matrix compressibility coefficients (K_C) are calculated using the fitting Equation (4). Then, based on the obtained K_C values, Equation (5) is applied to calculate the matrix compressibility volume for each sample. Finally, Equation (6) is used to subtract the incremental mercury intrusion volume from the original MIP data to obtain the corrected actual pore volume increment, thus completing the correction of the MIP data. Figure 6 shows the trend of K_C increasing with R_o,max with a low metamorphism degree (0.35% ≤ R_o,max < 0.65%), decreasing with a medium metamorphism degree (0.65% ≤ R_o,max ≤ 2.00%), and increasing with a high metamorphism degree (R_o,max > 2.00%). The K_C calculation formula is obtained through a fitting analysis of the data with different metamorphism degrees:

$$K_c=\left\{\begin{array}{c}91.509R_{o,\mathit{max}}^3-144.07R_{o,\mathit{max}}^2+77.029R_{o,\mathit{max}}-10.791 (0.35\%\le Ro,\mathit{max}<0.65\%)\\ -1.5392R_{o,\mathit{max}}^3+6.8265R_{o,\mathit{max}}^2-11.108R_{o,\mathit{max}}+7.8239 (0.65\%\le Ro,\mathit{max}\le 2.00\%)\\ -2.3236R_{o,\mathit{max}}^3+17.964R_{o,\mathit{max}}^2-45.084R_{o,\mathit{max}}+37.674 (Ro,\mathit{max}>2.00\%)\end{array}\right.$$

(4)

The formula’s fitting degrees for different metamorphic grades are 0.9821, 0.9274, and 0.984, respectively, demonstrating the formula’s strong capability to represent the K_C value accurately.

Using the traditional correction method, the K_C1 value was calculated through MIP test data and LTNA test data in conjunction with Formulas (1) and (3). Simultaneously, using the proposed method, the K_C2 value was calculated by combining the R_o,max values of different collected samples with Formula (4). Figure 7 visually demonstrates the comparison of K_C values calculated using the two methods.

Table 1. R_o,max values and coal matrix compression coefficients for different coal samples.

Sample No.	Ro,max/%	Kc/(10−10 m2/N)		Error Rate
		Kc1	Kc2
QX	1.83	0.876	0.925	5.25%
DD	1.59	1.184	1.233	3.97%
YY	2.47	0.897	0.898	0.11%
WK	2.54	0.984	0.980	0.35%
TL-2	1.82	0.911	0.940	3.12%
TL-8	1.64	1.137	1.178	3.50%

presents the R_o,max for six selected coals from six mining regions alongside the K_C1 values determined through the traditional correction method and the K_C2 values calculated using the statistical correction method, which are consistent with previous research results of 0.67 × 10⁻¹⁰ m²/N–3.06 × 10⁻¹⁰ m²/N [57,58], proving the rationality of Formula (4). It can be seen that the error rate range between K_C1 and K_C2 is 0.11–5.25%, indicating that the K_C values calculated using the statistical correction method have good accuracy and reliability. From this, it can be seen that the statistical correction method proposed in this study has the following advantages over traditional correction methods: when calculating the coal matrix compression coefficient K_C, only the maximum vitrinite reflectance parameter is needed. This significantly reduces experimental preparation and the computational workload, simplifies the MIP data correction process, and avoids the traditional method’s over-reliance on LTNA test data for correcting MIP data. It is particularly noted that within the high metamorphism range (R_o,max > 2.00%), the K_C1 value calculated using this method has minimal error compared to the value calculated using traditional methods, and it is very close to the actual value (K_C2). In the preliminary evaluation stage of the compressibility of coal in coalbed methane reservoirs, using the method proposed in this study dramatically improves the evaluation efficiency (it only requires substituting the maximum vitrinite reflectance into Formula (4) for evaluation), and it can directly serve as a reference.

3.4. Analysis of Calibration Results

Research indicates that during the mercury intrusion process, when the compressibility coefficient of the coal matrix is considered a constant, the volume of the coal sample can be corrected using Formula (5) [52]:

$$V_{c\left(P_i\right)}=V_c-{\displaystyle \frac{\mathrm{d}{V}_c}{\mathrm{d}{P}_c}}\left(P_i-P_0\right)$$

(5)

where V_c(Pi) represents the volume of the coal matrix under pressure P_i, measured in cm³/g; r denotes the serial number of the pressure collection point; and P₀ is set to 24 MPa.

The pore-filling volume of the coal sample under pressure P_i can be calculated using Equation (6) [52]:

$$\mathsf{\Delta }{V}_{\mathit{pi}}=\mathsf{\Delta }{V}_d-K_c{V}_{c\left(P_i\right)}\left(P_i-P_0\right)$$

(6)

where ΔV_pi represents the volume change of the coal matrix at the mercury intrusion pressure P_i, cm³/g; ΔV_d represents the mercury intrusion volumes at the pressures P_i and P₀, respectively, in cm³/g; and K_C is the K_C2 calculated using the statistical correction method (see Table 1), 10⁻¹⁰ m²/N.

By integrating Equations (5) and (6), this study calculated the coal’s cumulative mercury intrusion volumes before and after correction. As shown in Figure 8, the data reveal a significant discrepancy in cumulative mercury intrusion volumes before and after correction. The theoretical analysis results suggested that the corrected MIP data show substantial improvement, thus facilitating a more accurate characterization of the coal samples’ pore structure. The PSD of the coals determines their pore structure characteristics. According to Hodot’s (1966) pore-classification methodology, pores are categorized into micropores (<10 nm), transitional pores (10–100 nm), mesopores (100–1000 nm), and macropores (>1000 nm).

Figure 9 presents the PSD and the volume proportions of various pores in the corrected coal samples. The pore structure of the coal samples predominantly comprises macropores, followed by mesopores and transitional pores, with micropores constituting the most minor proportion. Compared to the mercury intrusion volumes before correction, the 3–25 nm pore radius range decreases the mercury intrusion volume by 91.75–96.40% after correction, indicating a successful correction effect. This implies that the coal matrix’s compressibility significantly impacts the PSD measurement in MIP experiments.

4. Fractal Dimension Characteristics before and after Calibration

The MIP pore structure exhibits fractal characteristics, which can be used to illustrate the influence of pre- and post-correction of MIP data on the fractal properties of the pores using fractal geometry concepts, thereby characterizing the pore morphology of coal. The Menger sponge model, a universal fractal calculation curve with a topological dimension of 1 [59], has been extensively applied to porous media. After repeated verification by scholars, it has ultimately been simplified into a general model:

$$D_2=4+\lg \left({\displaystyle \frac{\mathrm{d}V}{\mathrm{d}P}}\right)/\lg P$$

(7)

where P represents the pressure during the mercury intrusion process, measured in MPa; V represents the incremental volume of mercury intrusion in the interval, measured in mL/g; and D₂ denotes the pore fractal dimension, which is dimensionless.

Figure 10 compares the fractal dimension D₂, calculated from MIP data within the pressure range of 24 MPa to 200 MPa, before and after correction. The corrected fractal dimension D₂ shows a substantial improvement over the uncorrected values. Figure 11 shows that before correction, the D₂ values range from 3.186 to 3.339, indicating fractal dimensions between 3 and 4, which suggests suboptimal fractal characteristics of the overall pore structure. At this stage, the filling effect coexists with the matrix compression effect, with the latter predominating. This predominance results in significant measurement deviations, hindering an accurate reflection of the true pore structure.

Figure 12 illustrates the relationship between pressure and dV/dP for the six groups of corrected coal samples. The figure shows that the adjusted fractal dimension D₂ ranges from 2.792 to 2.975 within the expected range of 2 to 3, demonstrating robust fractal characteristics. This indicates that the correction process is highly effective, providing a more accurate and reliable depiction of the pore structure. The corrected fractal dimension values show that the pore sizes of the six coal samples collected in this study are well-developed at all stages. Compared to mesopores and macropores, the surfaces of micropores and transitional pores are rougher. The pore size distribution is complex, with irregular shapes and firm heterogeneity, making it easier to store free gas.

5. Research and Prospects

This study innovatively proposes a method for calculating the coal matrix compressibility coefficient based on maximum vitrinite reflectance. However, a specific deviation exists between the empirical formula derived from data statistics and the actual calculated values. On the other hand, as the coal samples used in this study do not belong to the low metamorphic degree (0.35% ≤ R_o,max < 0.65%), no error comparison was conducted for K_C value calculation at a low metamorphic degree. Therefore, with the enrichment of research data on K_C and R_o, max in the industry, future studies using this method should incorporate the calculation of the compressibility coefficient for a low-metamorphic-degree coal matrix and further refine the accuracy of the empirical formula to reduce calculation errors. In the practice of coalbed methane work, the compressibility coefficient of the coal matrix in the reservoir is calculated to evaluate certain parameters, such as permeability and porosity, thereby guiding coalbed methane development.

6. Conclusions

(1)

The analysis of MIP data reveals significant differences in the pore structure and the connectivity of coal samples from different mining areas. Type I coal samples predominantly exhibit semi-open pores with poor connectivity. Type II coal samples show poor connectivity in the transitional pores segment but good connectivity in the large pore segment. In contrast, Type III coal samples mainly feature open pores with good connectivity. Furthermore, data from LTNA experiments indicate notable pore morphology variations among coal samples. Ink-bottle, semi-open, and open plate-shaped pores primarily characterize these morphologies.

(2)

Considering the matrix compression effect of coal, a novel statistical correction method was devised based on statistical analysis. Compared to K_C values calculated using the traditional correction method, the statistical correction methods have high practical applicability. This statistical correction method was applied to correct MIP data under pressures ranging from 24 to 200 MPa, resulting in a decrease in the corrected mercury intrusion volume by 91.75% to 96.40%. The correction effect was significant, enabling a more precise characterization of the PSD.

(3)

The D₂ of MIP-measured initial uncorrected data values ranged from 3.186 to 3.339, with fractal dimensions between 3 and 4, indicating the suboptimal fractal characteristics of the overall pore structure. After PSD correction, D₂ values ranged from 2.792 to 2.975 within the 2 to 3 range, reflecting improved fractal characteristics. Correcting the MIP data significantly enhances the fractal representation of the coal’s pore structure, aligning the fractal dimensions more closely with their true values and thus offering a more accurate depiction.