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. 2024 Feb 15;9(8):9686–9701. doi: 10.1021/acsomega.3c09733

Study on the Complex Electrical Response Characteristics of Loaded Coal and the Control Mechanism of the Main Fracture System Structure

Jian Li †, Xiaokai Xu ‡,*, Jie Wu §, Yugui Zhang †, Zhiqi Guo ∥, Zehua Zhang ‡, Zhengzheng Xue ‡
PMCID: PMC10905723  PMID: 38434871

Abstract

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The structure of coal seam fractures is the main physical property of coalbed methane reservoir evaluation, and the complex resistivity method is a potential geophysical evaluation method for coal seam fractures. In this study, cylindrical coal samples with axial directions perpendicular to the bedding, face cleat, and butt cleat were prepared. The complex electrical parameters of the loaded specimens were tested with test frequencies ranging from 1 Hz to 10 kHz. The complex electrical response characteristics of the loaded coal are summarized, and the control mechanism of the main fracture system structure is analyzed. The results indicated that (1) as the loading pressure increased, the resistance R and the absolute values of reactance X(|X|) gradually decreased, especially in the frequency band where R slowly decreased and the characteristic frequency of X, the decreased amplitude was more significant, and the cutoff frequency of R and the characteristic frequency of X all gradually increased. (2) The complex electrical properties of coal show obvious anisotropic characteristics. Both R and |X| decreased sequentially according to the direction perpendicular to the bedding, face cleat, and butt cleat; the cutoff frequency of R and the characteristic frequency of X all increased sequentially. (3) The dispersion phenomenon of the complex electrical properties of coal is attributed to the induced polarization; the elevated loading stress enhances the polarization effects of the molecular-induced moments of the coal skeleton, and the anisotropic difference of the complex electrical properties is due to the difficulty in the degree of transport of charged particles induced by structural differences of the main fracture system. (4) The resistance R3 and capacitance Xc were selected as the complex electrically sensitive parameters of the loaded coal orthogonal fracture structures. A logarithmic inversion model reflecting the main fracture system structure of coal was constructed. This provides a certain theoretical basis for efficient electrical exploration of coal reservoir fracture structures.

1. Introduction

Fractures in coalbed methane reservoirs serve as the places and channels for the occurrence and migration of coalbed methane,1 and are the main factors controlling the permeability of coal seams.2,3 Moreover, heterogeneity greatly constrains the flow and diffusion of gas within the reservoir. Therefore, the fracture structure is the main object for evaluating the physical properties of coal reservoirs.3 Its detection and characterization are the primary prerequisites and key research content for coalbed methane exploration and development.4−7 One of the main effective geophysical methods for detecting and describing the physical properties of coal reservoirs is the application of relevant electrical exploration methods in surface,8 well logging,9 well seismic joint,10 and coal mine underground.11 Many scholars have conducted extensive research on the basic theories and engineering practices related to electrical exploration of coal seams, but most of them have focused on the evaluation of coal seam water content/permeability under coal mining engineering conditions.12,13 However, there is relatively little research on the electrical exploration of coalbed methane reservoir properties, and the basic theories are relatively weak.14 Therefore, the applicability of electrical exploration technology for the physical properties of coal reservoirs urgently needs to be explored, especially for the strengthening of basic theoretical research of experiments. As a fundamental theoretical research method for electrical exploration in the laboratory, the complex resistivity method has been extensively studied in the field of oil and gas geology, focusing on the dispersion characteristics and mechanism of rock resistivity15 and permeability evaluation.16 This indicates that the complex resistivity method has good application prospects for evaluating the physical characteristics of coal reservoirs.

The complex resistivity method is an emerging noncontact and nondestructive detection technology that can achieve high-density measurements in both the frequency and spatial domains over a wide frequency range, considering the influence of frequency variation on the polarization characteristics of coal, and further study the characteristic frequency points of the complex electrical dispersion characteristic curve of coal under different conditions; meanwhile, this method is low-cost and convenient. It has been extensively studied in many fields such as metal deposit exploration,17 distinguishment of the oil layer and water layer,18 pollutant management,19,20 and medical diagnosis.21 At present, there are two theories for the mechanism of rock electrical dispersion: induced polarization caused by electrochemical effects22 and dielectric polarization resulting from displacement currents.23 Many scholars have used conductive models to explain the dispersion mechanism of rock complex resistivity, among which the most common ones are the CPA model,24 Cole–Cole model,25 and double Cole–Cole model.26 There are differences in the extent to which pressure affects different fracture structure media,27,28 and researchers have conducted a large number of experimental studies on the sensitivity response of rock electrical characteristics to pressure. Guo et al.29 simulated the variation of the resistivity of limestone with temperature and pressure to establish the relationship between the formation resistivity and depth variation. Sun et al.30 demonstrated that the resistivity of muddy sandstone decreased with increasing depth. Tong et al.31 proposed a conductivity model based on the variation characteristics of shale complex conductivity under temperature and pressure, which could effectively improve the applicability of experimental data to actual formations. Dou et al.32 studied the effects of temperature and pressure on the complex resistivity of shale and found that, on the basis of ionic conductivity, the complex resistivity of shale decreased with increasing temperature and increased with increasing confining pressure. Tian et al.33 measured the complex resistivity of cylindrical standard samples with different mineral compositions under high pressure, where the graphite composition of the standard sample decreased with increasing pressure difference. However, the complex resistivity method has been extensively studied in rock experimental testing methods and fracture prediction, and further research is required to characterize the complex resistivity of coal reservoirs.

Compared with rock, coal is a material with strong heterogeneity and anisotropy, and the complex resistivity method is of great significance for studying its pore and fracture structure characteristics.34 Wang et al.35 and Meng et al.36 believed that with an increase of stress, the resistivity of coal first decreased significantly and then increased slightly. During the uniaxial compression process of anthracite, its resistivity decreased slowly. Lu and Jia37 pointed out that the significant degree of change in the resistivity of loaded coal increased with the increase of their hardness. Li38 found that the surface potential changes of the coal carrier were influenced by factors such as moisture conditions and loading rate. Wang et al.39 pointed out that before the expansion of the coal, the variation pattern of resistivity was related to the stress and the gradual closure of pores. Chen et al.40 found that the apparent resistivity of primary structural coal was greater than that of structural coal at three frequency points. However, research on the electrical properties of loaded coal was mostly based on the effect of discontinuous frequency electric fields.41

The frequency dispersion characteristics of the complex resistivity of coal are mostly obtained under normal temperature and pressure conditions. The degree of metamorphism of coal,42 structural types,43 measured area,44 anisotropy,45 and water content46,47 all have a certain degree of influence on the electrical characteristics. However, limited research has been conducted on monitoring the complex resistivity of coal reservoirs under continuous loading frequency conditions.

Therefore, on the basis of an independently designed experimental platform, we simulate the in situ stress field of coal reservoirs, test the complex electrical parameters (resistance R and reactance X) of coal with typical main fracture system structures, and calculate the porosity based on the strain method. Meanwhile, we use the conductivity and dielectric theory and conductivity model to invert and optimize the complex electrical-sensitive parameters of loaded orthogonal fracture structure coal and construct a model to determine the relationship between the porosity of main fractures of coal and the sensitive parameters of complex electrical properties. This study provides a new method for predicting the development of coal reservoir fractures in electrical exploration and the effective development of coalbed methane.

2. Experimental Study

2.1. Sample Information

In this study, the II-1 coal seam of the Permian Shanxi Formation in the North Henan coal mine area (Zhaogu No. 2 mine of the Jiaozuo Coal Industry Group) was investigated. The Jiaozuo coalfield is located in the southeastern margin of the Taihang mountain uplift belt and the eastern wing of the front arc of the mountain-shaped structural system. The study area was less affected by geological disturbances, with underdeveloped fold structures and only small folds appearing locally. The coalfield structure in the study area is shown in Figure 1. The main coal seams in the coalfield are mainly block coal, with a small amount of mixed-granular coal. The coal is gray-black to black, and the stripes inside the coal seams are gray-black, resembling a metallic luster. The main fractures are shell-like and some parts are uneven. The thickness of the coal seam changes very little (about 6.16 m), as shown in Figure 2. The minability index Km is 1, and the mean square deviation value S is 0.40, which belongs to the stable thick coal seam that could be mined in the entire region.48

Figure 1.

Figure 1

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Geological overview of the study area.

Figure 2.

Figure 2

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Histogram of the Permian strata in the study area.

The selected samples have a typical main fracture system structure, as shown in Figure 3. The fresh block coal collected from the underground is drilled in the direction perpendicular to the butt cleat plane (x direction), face cleat plane (y direction), and bedding plane (z direction) in the axial direction. The specifications are as follows: Φ50 mm × L100 mm coal pillar, and the two ends of the coal pillar are ground flat using a double-end grinding machine. The unevenness of the coal sample end face is required to be less than 0.02%.49 According to the relevant standards,50,51 the industrial analysis and physical properties of the coal samples are shown in Table 1. The analysis of the mineral composition of the coal samples is shown in Figure 4. Among them, kaolinite, calcite, quartz, and dolomite contents were 74.9, 11.8, 11.3, and 2.1%, respectively.

Figure 3.

Figure 3

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Structural characteristics of the typical main fracture systems in coal.

Table 1. Basic Parameters of the Coal Sample1.

metamorphic degree Mad (%) Aad (%) Vdaf (%) ρtr (t/m3) ρap (t/m3) Ro,max (%)
Anthracite 2.18 17.97 6.56 1.64 1.49 3.16
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1

Mad (%) is the mass fraction of the coal sample moisture; Aad (%) is the mass fraction of air-dried ash; Vdaf (%) is the mass fraction of the volatile matter on a dry ash-free basis; ρtr (t/m3) is the true density of the coal sample; ρap (t/m3) is the apparent density of the coal sample; and R0, max (%) is the maximum reflectance of vitrinite oil immersion.

Figure 4.

Figure 4

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XRD spectrum of the coal sample.

2.2.1. Coal Sample Processing Method

Due to the significant influence of moisture on the conductivity of underground reservoir rocks52 and in order to explore only the complex electrical dispersion response characteristics and control mechanism of the main fracture structure of the coal reservoir, the measured coal samples were dried in a drying oven at 105 °C for 24 h and cooled to normal temperature (28 °C) in the cooling tower (Figure 5). The weight of the samples remained almost unchanged for the last 8 consecutive hours using the gravimetric method, which basically ensured that the interior of the main fracture structure was in a completely dry state, eliminating the interference of moisture on the complex electrical response of the main fracture structure.

Figure 5.

Figure 5

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Coal sample drying oven.

2.3. Experimental System and Scheme

In this article, the coal complex resistivity and deformation test platform under triaxial loading conditions constructed according to the experimental requirements mainly included a triaxial stress servo part (axial pressure pump and circumferential pressure pump), strain test part (strain gauge), complex resistivity measurement part (LCR), data processing part, test tank and constant temperature chamber, as shown in Figure 6(Figure 6a shows the experimental system, and Figure 6b shows the details of the test tank). Among them, the piston of the coal in the test tank, except that the electrode material in contact with the coal sample is copper, is made of highly insulating PEEK material to prevent measurement errors. The LCR tester adopts a four-terminal pair structure to measure the induced polarization effect parameters of the loaded coal under constant current and variable frequency conditions and can measure frequencies ranging from 1 mHz to 200 kHz; the measured current ranges from 0.01 mA to 50 mA/100 mA; the current output accuracy is ±10 μA; the fastest measurement time is 2 ms; the scanning frequency range is 2–801 points; and the internal DC bias is −5–5 V.

Figure 6.

Figure 6

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Experimental testing platform.

The test method is as follows: (1) the LCR instrument is connected to the power supply, preheated for 60 min, set to the ANALYZE mode, and the output external current is controlled to remain unchanged at 0.01 mA; (2) the test repeatability parameters, resistance R and reactance X, are selected, with a measurement frequency range of 1 Hz–10 kHz, and open-circuit compensation, short-circuit compensation, and line compensation are performed to reduce the interference of the cable residue and parasitic admittance on the test results; (3) a standard resistance element with a known resistance value of 2 kΩ is connected in series in the measuring circuit for measurement correction; (4) the coal sample coated with copper foil conductive paper is fixed in the test tank and connected in series in the system. The system temperature is maintained at 28 °C through a constant temperature chamber, and the axial pressure and confining pressure are stabilized. The copper electrode plate is connected to the LCR tester through the Kelvin clamp of the LCR tester and electrical lead, and the strain patch and strain gauge are connected to the strain conductor. The complex electrical parameters of the loaded coal are measured using the LCR tester, and its deformation is monitored using a strain gauge. (5) According to the relationship between the pressure and burial depth53 and the relevant standard,54 the direction of different coal samples (in the x, y, and z directions) is changed, and different stress sizes (the confining pressure at 5 MPa, and the axial pressure with equal gradient changes at 1, 3, 5, and 7 MPa are maintained), repeat steps (2–4), and their corresponding electrical and strain parameters are monitored.

3. Experimental Results

3.1. Influence of Pressure on the Complex Electrical Response Characteristics of Coal

Under the condition of equal gradient varying axial pressures (1, 3, 5, and 7 MPa), the R and X dispersion curves of three orthogonal main fracture structural directions of the coal samples were measured in the 1 Hz–10 kHz frequency band, as shown in Figure 7, 8, and 9.

Figure 7.

Figure 7

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R/X frequency dispersion curves of coal in the x direction under different pressures.

Figure 8.

Figure 8

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R/X frequency dispersion curves of coal in the y direction under different pressures.

Figure 9.

Figure 9

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R/X frequency dispersion curve of coal in the z direction under different pressures.

Regarding the trend of R and X dispersion curves, the summary is as follows: for R, as the frequency f increases, the dispersion curve exhibits a slow decrease followed by an accelerated decrease. For X, as the frequency f increases, the dispersion curve shows a “slow decrease—accelerated decrease—accelerated increase” trend, with values less than zero, indicating capacitive characteristics. Among them, the X increase stage is affected by electromagnetic interference, and there is no slow increase stage.

Under the action of axial pressure with equal gradient variation, the R and X dispersion curves of coal samples in the three main fracture structural directions show a consistent variation pattern: with an increase of pressure, the R value gradually decreases, especially in the frequency range where R slowly decreases (about 1 Hz–1 kHz), and the dispersion curves tend to be consistent in the frequency range where R shows an accelerated decrease. As the pressure increases, the |X| value gradually decreases, especially in the frequency band near the characteristic frequency point (minimum point) of the X dispersion curve. Comparing the complex electrical dispersion curves of R and X of the coal samples, the cutoff frequency of R slowly decreases, and the characteristic frequency of X gradually increases with increasing pressure.

3.2. Influence of the Main Fracture System Structure of Coal on Its Complex Electrical Response Characteristics

The coal samples with different main fracture structural directions exhibit different regular changes in the numerical values of resistance R and reactance X, as well as the degree of deviation of their respective dispersion curves is different. The R and X anisotropic dispersion patterns of the coal under various pressure conditions are shown in Figure 10, 11, 12 and 13:

Figure 10.

Figure 10

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R/X frequency dispersion curves of coal in different main fissure structure directions at 1 MPa pressure.

Figure 11.

Figure 11

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R/X frequency dispersion curves of coal in different main fissure structure directions at 3 MPa pressure.

Figure 12.

Figure 12

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R/X frequency dispersion curves of coal in different main fissure structure directions at 5 MPa pressure.

Figure 13.

Figure 13

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R/X frequency dispersion curves of coal in different main fissure structure directions at 7 MPa pressure.

By comparing the response characteristics of the complex electrical properties of coal samples with the main fracture structure under different pressure conditions, the following conclusion can be drawn. For R, the dispersion curves of coal samples at each pressure point are as follows: in the low-frequency slowly decreasing stage (when induced polarization is fully completed), the change in the magnitude of the value of R is Rz>Ry> Rx, and the dispersion curves gradually shift to the right in the order of z, y, and x directions; that is, the cutoff frequency of R gradually increases with the same order of direction. For X, the difference in the complex electrical properties of the coal samples in the three fracture structural directions is more significant, and the numerical comparison relationship is |Xz| > |Xy| > |Xx|. The dispersion curve also gradually shifts to the right in the z, y, and x directions, and the characteristic frequency gradually increases, especially in the frequency band near the characteristic frequency point where the variation trend is more obvious.

4. Discussion and Analysis

4.1. Analysis of the Influence of Pressure on the Complex Electrical Response Characteristics of Coal

First, the trends of the R and X dispersion curves are analyzed. For R, as the frequency f increases, the R dispersion curve of coal first shows a slow decrease and then an accelerated decrease. Within the frequency domain, the polarization of coal under external electric field excitation generates a secondary electric field, forming an “overpotential”.55 As the reciprocal of time period T, the polarization effect decreases with an increase of frequency f. When the frequency f approaches zero (close to the direct current), the time period T is the longest, and the degree of polarization completion is the highest. From a microscopic perspective, as shown in Figure 14, the electron displacement polarization, ion displacement polarization, orientation polarization, and interface polarization of coal atoms and molecules could be completed in sufficient time, with the total field voltage and secondary voltage reaching their maximum;56 from a macroscopic perspective, the time required for induced polarization of electronic conductors and ion migration in thin film polarization of ion conductors present in the coal could be met, resulting in a high degree of polarization. However, as the frequency f continues to increase, the supply polarization time period T continues to shorten, making polarization completion increasingly difficult. The polarization gradually becomes single, with only oriented polarization and displacement polarization, showing a phenomenon of continuously decreasing voltage and resistance. In addition, the phenomenon of “low frequency-high resistance, high frequency-low resistance” in the R under polarization could also be explained by the characteristics of the dielectric medium in electrical theory, such as “isolate direct current and pass alternating current” and “resist low frequency and release high frequency”, in a circuit system where the coal is composed of multiple equivalent resistors and capacitors. As the frequency f increases, X shows a slow decrease—accelerated decrease—accelerated increase, with values less than zero, indicating capacitive characteristics. The X-increasing stage is affected by electromagnetic interference, and there is no slow-increasing stage. According to the relationship of the reactance of coal with the capacitance and frequency, X = −(2πfC)−1, it can be qualitatively believed that the X dispersion curve of coal is influenced by its capacitance and frequency and is bounded by the curve extreme point. It first decreases with a decrease of capacitance and then increases with an increase of frequency.57

Figure 14.

Figure 14

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Coal polarization type diagram.

During the compaction and elastic deformation stages of coal, with a continuous increase of stress, the internal microfractures and pores of the coal matrix gradually close and decrease, and the fluidity of the intermediate medium weakens. As shown in Figure 15, on the one hand, the difficulty of differentiation and migration of cations and anions increases, and the amount and rate of ion migration decrease.33 The solid–liquid two-phase contact area of coal decreases with a decrease of microfractures and pores, resulting in a decrease in the number of ion accumulations at the critical interface. As a result, the conductivity of the coal ions decreases, leading to an increase in coal resistance. On the other hand, coal, as an organic sedimentary rock, not only contains the electrochemical-induced polarization form of ionic conductor conductivity but also includes the form of electronic conductivity. Under the excitation of an external electric field, electrons obtain sufficient energy to overcome the potential barriers between large molecules, forming polarization charges and free charges. Polarization charges and atomic nuclei induced electric moments for directional polarization, and free charges and cations in solution form a double layer of electricity at the solid–liquid interface. The mineral composition of the coal samples was analyzed using an X-ray diffractometer. The XRD spectrum, shown in Figure 4, shows that the mineral composition of anthracite contains as much as 74.9% kaolinite, with a dielectric constant ranging from 28.29 to 49.14,58 which is greater than 10 or 12. Thus, it can regarded as an electronic conductor. With the continuous increase of pressure, the connectivity of the coal skeleton in the direction of loading is enhanced, the contact of charged particles is closer, the electron cloud spacing between molecules within functional groups is smaller, and their ionic activity energy increases.36 Electrons in the captured state are more easily converted into free excited electrons by the electric field.40 The proportion of electrons in the valence band becomes conductive interceptors in the conduction band, and the orientation polarization and displacement polarization of the induced electric moment of the coal skeleton molecules continue to increase. The conductivity of the coal gradually improves and the resistance value greatly decreases. In addition, the gradually shrinking fractures in the coal matrix might allow unconnected water to penetrate, further enhancing the conductivity of coal.59

Figure 15.

Figure 15

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Changes in pore cracks of the loaded coal.

4.2. Analysis of the Influence of the Main Fracture Structure on the Complex Electrical Response Characteristics of Coal

Coal has a layered structure, and its parallel bedding could be considered as multiple resistors in parallel, while the vertical bedding could be considered as multiple resistors in series. According to electrical theory, the resistance value of parallel circuits is smaller than that of series circuits; therefore, the conductivity of parallel bedding is significantly stronger than that of vertical bedding. In addition, the layering phenomenon in the x direction is weaker than that in the y direction, resulting in a weaker conductivity of the latter than the former. Under the action of an external electric field, the migration of charged particles along the bedding direction of coal only needs to overcome the attraction of the macromolecular structure to itself, resulting in the formation of the “fundamental potential barrier”. However, in the vertical bedding direction (z direction), the migration of charge carriers also needs to overcome the “penetrating potential barrier,” which passes through the parallel stacking structure of coal macromolecules. Obviously, the energy consumption of charged particles in the vertical bedding direction is greater than that in the bedding direction. Therefore, the conductivity of coal along the bedding direction is stronger than that along the vertical bedding direction. It can also be seen from Figure 16 that there are fewer fracture structures in the x direction that hinder the migration direction of charged particles than in the y direction, making carrier migration more convenient and smoother. Therefore, the conductivity of coal in the y direction is weaker than that in the x direction.

Figure 16.

Figure 16

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Schematic diagram of coal sample profiles with different fracture structure directions.

Coal with typical orthogonal main fracture system structural characteristics can be simplified, as shown in Figure 17a. By extracting the fracture structural unit from the entire coal, a physical model can be obtained, as shown in Figure 17b, where the interval between adjacent face cleats is Lm, that between adjacent butt cleats is Ld, and that between adjacent beddings is Lc. To characterize the complex electrical dispersion response characteristics of coal reservoir fracture structures, the fracture structure unit is further simplified as an equivalent circuit unit model composed of a series connection of resistance and capacitance, as shown in Figure 17c. Therefore, each fracture structure direction can be regarded as an interval series combination of the equivalent circuit units.

Figure 17.

Figure 17

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Equivalent circuit of the orthogonal main fracture system structure in coal.

According to the basic electrical formula,

4.2. 1

Here, ρ is the coal resistivity, kΩ·m; R is the measured resistance value, kΩ; A is the area of the measuring electrode plate, m2; and L is the distance between the measuring plates, m.

4.2. 2

where C is the capacitance, F; εr is the relative dielectric constant of coal, F/m; A is the area of the measuring electrode plate, m2; k is the constant electrostatic force (=9 × 109 N·m/C); and d is the distance between the measuring plates, m.

The impedance expressions of the three orthogonal fracture structural units are as follows:

4.2. 3

Equation 3 indicates that the anisotropy of the impedance real part (resistance) and imaginary part (reactance) between the orthogonal fracture structural units of coal depends mainlyon the spacing Lm, Ld, and Lc values between the fracture structures. Through extensive statistical analysis of the fracture structure of coal in the study area, the statistical average values of Lm, Ld, and Lc of the reservoir coal in the area were obtained and concentrated around 4.76, 3.51, and 8.83 mm, respectively. The specific observations and statistics are shown in Table 2.

Table 2. Statistics of the Interval of Orthogonal Fracture Structures in Coal.

observation structural unit Lm/mm Ld/mm Lc/mm
1 4.51 3.04 9.62
2 4.33 3.35 7.03
3 4.37 3.06 8.81
4 5.21 3.57 9.85
5 3.96 3.52 9.26
6 4.22 2.98 8.84
7 4.56 4.43 9.25
8 4.98 2.96 7.26
9 4.72 3.68 9.28
10 5.62 3.41 10.47
11 5.02 3.71 8.44
12 5.83 4.02 7.94
13 4.77 3.88 8.22
14 4.53 3.49 9.35
mean value 4.76 3.51 8.83
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Therefore, substituting the values of Lm, Ld, and Lc into eq 3, the following equation is obtained after derivation and calculation:

4.2. 4

Equation 4 shows that the proportional coefficients of the coal resistance in the z, y, and x directions are 0.528, 0.154, and 0.083, respectively, showing a decreasing trend in sequence. The proportional coefficients of coal reactance are 2.112, 0.616, and 0.332, which also decrease in the same direction of the fracture structure, further verifying the anisotropic characteristics of the measured resistance and reactance of coal.

4.3. Optimization of Complex Electrical Sensitivity Parameters for Loaded Coal Orthogonal Fracture Structures

When the power supply current density through the rock (ore) is not high, the induced polarization effect of rock is linear and can be described by invariant parameters within a time range, making it a linear time-invariant system.60 A popular method for describing this linear time-invariant system is the circuit model equivalence method, which simplifies the induced polarization effect by replacing it with a circuit constructed in series and parallel with electronic components. An equivalent circuit analysis is performed on it. We use the Cole–Cole conductivity model to invert and calculate the measured complex electrical dispersion data and further explain its electrical response characteristics through the corresponding model parameters. The structure shown in Figure 18 includes fracture channels blocked by metal minerals and unobstructed fracture channels. In this equivalent circuit, the resistance R1 is equivalent to the solution resistance in the unblocked fracture channel; the resistance R2 is equivalent to the sum of the solution resistance in the blocked fracture channel and the metal mineral resistance; and the parallel combination of complex impedance (jωXc)−c and resistor R3 effectively simulates the interface impedance of the metal mineral ion solution.

Figure 18.

Figure 18

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Basic structural units and equivalent circuits of mineralized rocks.

The expression for the complex impedance of the equivalent circuit is

4.3. 5

In q 5, Zs(f) is the interface impedance of the metal ion solution.

4.3. 6

When f →0,

4.3. 7

When f → ∞,

4.3. 8

Through calculation and deduction, and to reflect the electrical characteristics of coal, based on the relationship between the resistivity and size parameters of coal, the model is calculated in the form of resistivity as follows:

4.3. 9

In eq 9, ρ(0) is the overall resistivity of the coal when the frequency is zero, kΩ·m; Xc is the capacitance exhibited by the induced polarization of coal seam (capacitance on fractures and solution contact surfaces), μF; R3 is the resistance exhibited by the induced polarization of coal seam (resistance on fractures and solution contact surfaces), kΩ·m; and c is the frequency correlation coefficient (without units).

Based on the above equivalent circuit model, the measured complex electrical dispersion curve of coal is inverted and processed, as shown in Figure 19

Figure 19.

Figure 19

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Inversion of the measured complex electrical dispersion curves using the Cole–Cole model.

Table 3 shows the inversion parameters of the Cole–Cole model for coal samples in different fracture structure directions under different pressures, where R3 and Xc are the changes in resistance and capacitance caused by the induced polarization effect of coal. Therefore, these two model parameters are associated with the pressure and the main fracture structure, as shown in Figure 20.

Table 3. Inversion Parameters of the Cole–Cole Model for Complex Electrical Dispersion Curves of Coal in Different Directions.

fracture structure axial pressure/MPa confining pressure/MPa ρ(0)/kΩ·m R1 kΩ·m R2/kΩ·m R3/kΩ·m Xc/μF c R2
x 1 5 1.671 0.241 1.56 × 10–12 0.368 1.331 × 10–1 0.584 0.996
3 5 1.555 0.31 6.836 × 10–32 0.312 9.901 × 10–2 0.545 0.996
5 5 1.471 0.356 5.593 × 10–29 0.268 8.588 × 10–2 0.514 0.996
7 5 1.434 0.397 1.588 × 10–132 0.226 8.223 × 10–2 0.423 0.988
y 1 5 1.834 0.282 1.068 × 10–12 0.506 1.776 × 10–1 0.422 0.986
3 5 1.754 0.322 6.515 × 10–33 0.406 1.301 × 10–1 0.411 0.99
5 5 1.707 0.386 2.25 × 10–67 0.354 1.058 × 10–1 0.448 0.993
7 5 1.673 0.433 1.673 × 10–156 0.336 9.955 × 10–2 0.437 0.995
z 1 5 2.256 0.326 6.47 × 10–14 0.842 4.515 × 10–1 0.537 0.989
3 5 2.148 0.468 2.881 × 10–29 0.682 2.736 × 10–1 0.525 0.992
5 5 2.036 0.772 2.253 × 10–56 0.613 1.926 × 10–1 0.507 0.987
7 5 1.976 0.988 1.671 × 10–121 0.589 1.584 × 10–1 0.505 0.992
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Figure 20.

Figure 20

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Relationship between inverted parameters and the pressure of the Cole–Cole model for coal in different directions.

The inversion parameters of the Cole–Cole model show that R3 and Xc of the coal samples in three directions gradually decrease with increasing pressure. In addition, by comparing and analyzing the inversion parameter data of the coal sample models in different directions under the same pressure, the parameters R3 and Xc decrease sequentially in the z, y, and x directions. These two electrical model parameters have patterns consistent with the measured data, which can characterize the development characteristics and directionality of the coal fractures. Therefore, the model inversion parameters R3 and Xc are selected as the sensitive parameters for the complex electrical properties of loaded coal with orthogonal fracture structures.

4.4. Relationship between the Porosity of the Main Fracture of the Loaded Coal and the Complex Electrical Sensitivity Parameters

The change of electrical parameters of the loaded coal is due to the shrinkage and closure of the internal fractures of the coal sample. The degree of fracture change in coal can be calculated by measuring the strain values using a strain gauge. The strain generated by the change of fracture volume is influenced by its size, which in turn leads to changes in the total volume of the coal. The relationship between the porosity and strain of coal is that the volume of fracture before loading minus the overall deformation and temperature expansion of the coal sample equals the volume of fracture after deformation. Therefore, the expression for the relationship between the porosity and strain is shown in eqs 10–1013.61

4.4. 10

In the above formulas, VT is the total volume of the coal sample; VP is the total volume of the coal sample fractures; ΔVS is the volume change of the coal sample skeleton; VS is the volume of the coal skeleton; ΔVP is the change in the fracture volume of the coal sample; VT0 is the initial total volume of the coal sample; ΔVT is the total volume change of the coal sample; VS0 is the initial skeleton volume of the coal sample; VP0 is the original fracture volume of the coal sample; φ is the porosity; and φ0 is the original porosity of the coal sample.

Under the action of stress, ΔVS only considers the change in the temperature effect caused by the temperature changes ΔVST. Therefore, the strain increment can be written as

4.4. 14

The porosity expression can be written as

4.4. 15

The thermal expansion deformation is written as62

4.4. 16

In eq 16, ΔT is the absolute temperature change (T–T0) and β is the volumetric thermal expansion coefficient of the coal. After derivation,

4.4. 17

Electricity and strain experiments were conducted on coal samples under triaxial loading without changing the temperature state. We simplify the porosity model and ignore the influence of temperature on the porosity of the coal.

Based on the above formula, the porosity of coal at different pressures and main fracture structure directions is calculated, as shown in Figure 21:

Figure 21.

Figure 21

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Changes in the porosity of coal under different main fracture structure directions and pressures.

In Figure 21, Inline graphic is the degree of porosity change of coal (φ0 is the porosity of coal under initial conditions, and φj is the porosity of coal under a certain pressure).

The porosity of coal is sensitive to pressure and fracture structure. The porosity of the coal samples in all directions gradually decreases with an increase of pressure, and the degree of porosity change φε continuously increases. As the pressure increases, the decrease in the rate of change between adjacent pressure points shows a decreasing trend. Under the same pressure conditions, the porosity of the coal exhibits anisotropic characteristics, increasing sequentially in the z, y, and x directions, that is, φx > φy > φz. However, the direction of the degree of porosity change is opposite, that is, φεz > φεy > φεx, and the difference in the degree of porosity change between adjacent pressure points also follows this pattern, which is closely related to the anisotropic development characteristics of the fractures, leading to consistent changes in coal permeability. Therefore, there is a strong correlation between the porosity of coal and the changes in R3 and Xc values caused by the induced polarization.

As shown in Figure 22, the porosity of the coal samples in the three orthogonal main fracture directions shows a good correspondence with R3 and Xc. As R3 increases, the porosity gradually increases, and the relationship between the two follows a logarithmic function. There is a similar correspondence between the capacitance parameter Xc and the porosity of the coal. This is related to the continuous deformation of coal and the shrinkage and closure of fractures under the action of pressure.

Figure 22.

Figure 22

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Porosity of coal in three main fracture structural directions at different pressures and a fitting diagram with R3 and Xc.

4.4. 18
4.4. 19

Here, φ is the porosity; R3 and Xc are the resistance and capacitance parameters caused by the preferred induced polarization effect in the Cole–Cole model; and A, B, γ, and ξ are the model fitting parameters.

As shown in Figure 23, there is also a certain variation pattern between the porosity of coal samples under the same pressure and the R3 and Xc values in the fracture structure; the R3 and Xc values decrease in the order of the z, y, and x directions, while the porosity values gradually increase in the order of the same direction. This is because the fracture in the z direction of the coal is relatively complex. In addition, the fracture structure is perpendicular to the direction of the external electric field, which takes a long time to complete the induced polarization effect and has a large capacitance value. The development characteristics and directionality of fractures in the x direction of coal are more conducive to the completion of the induced polarization effect than in the y direction. The correspondence between the porosity with R3 and Xc also follows a logarithmic function (eqs 18 and 19). The above inversion formula has a certain degree of guiding significance for using the complex resistivity method to evaluate the numerical value of the coal porosity and the directionality of the main fracture structure. This initially formed an electrical method for evaluating porosity. First, the Cole–Cole model is used to invert and analyze the measured complex electrical dispersion curve, and the sensitive characteristic parameters R3 and Xc are selected. Then, based on the derived logarithmic function relationship, the porosity of the coal is further obtained, and the porosity varies in different directions. This has certain significance for the efficient development of coalbed methane and accurate extraction of gas.

Figure 23.

Figure 23

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Fitting diagram of the porosity and R3 and Xc values of coal in different directions under the same pressure.

5. Conclusions

This article summarizes the response characteristics of the complex resistivity of coal and analyzes its mechanism by conducting complex electrical experiments on loaded coal with a main fracture structure. The following conclusions can be drawn:

  • (1)

    The complex resistivity of coal has a good response to the loading pressure. As the pressure increases, R and |X| of the coal gradually decrease, especially in the frequency band, where R slowly decreases near the characteristic frequency of X. The cutoff frequency of R and the characteristic frequency of X show a gradual upward trend. This is because the electron cloud spacing between molecules within the functional groups of coal is smaller, and the orientation polarization and displacement polarization of the induced electric moment of the coal skeleton molecules continue to increase.

  • (2)

    The complex resistivity of coal varies regularly due to differences in the main fracture system structure. R and |X| in the coal decrease sequentially in the z, y, and x directions, and the dispersion curve shifts to the right. The cutoff frequency of R and the characteristic frequency of X increase in the same order. The extracted fracture structural unit is equivalent to the unit circuit model of the resistor and capacitor series connection, further verifying the reliability of the anisotropic characteristics of R and |X|.

  • (3)

    The complex electrical sensitivity parameters of the loaded coal with an orthogonal fracture system structure are optimized. The Cole–Cole model is used to invert the induced polarization dispersion curves of coal under different pressures and fracture structures. R3 and Xc are selected to characterize the induced polarization effect of coal. As the pressure increases, R3 and Xc values continue to decrease, and the two values also decrease sequentially in the order of the z, y, and x directions.

  • (4)

    There is a good logarithmic function relationship between the porosity of the main fracture of the loaded coal and the complex electrical sensitivity parameters R3 and Xc. The porosity gradually decreases with a decrease of R3 and Xc, and the anisotropic characteristic of coal also results in a negative correlation between its porosity and the complex electrical sensitivity parameters.

This article systematically characterized the complex electrical resistivity response characteristics of typical coal within the main experimental testing frequency range and comprehensively revealed the corresponding control mechanism of the main fracture system structure. The application of efficient electrical exploration for the fracture structure of coalbed methane reservoirs has certain fundamental theoretical significance.

Acknowledgments

We are grateful for financial support from the National Natural Science Foundation of China (2016ZX00504007-007), National Natural Science Foundation of China (41904118), and the Open Research Fund of the Key Laboratory of “Structure and Oil and Gas Resources” of the Ministry of Education (TPR-2019-05).

Author Present Address

⊥ Henan Polytechnic University, Century Ave. 2001, Hich-Tech Campus, Jiaozuo City, Henan Province, P.R. China, 454003

The authors declare no competing financial interest.

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