Binary-Classification Physical Fractal Models in Different Coal Structures

Abstract

Existing theoretical models of wave-induced flow face challenges in coal applications due to the scarcity of experimental data in the seismic-frequency band. Additionally, traditional viscoelastic combination models exhibit inherent limitations in accurately capturing the attenuation characteristics of rocks. To overcome these constraints, we propose a novel binary classification physical fractal model, which provides a more robust framework for analyzing wave dispersion and attenuation in complex coal. The fractal cell was regarded as an element to re-establish the viscoelastic constitutive equation. In the new constitutive equation, three key fractional orders, $\alpha$, $\beta$, and $\gamma$, emerged. Among them, $\alpha$ mainly affects the attenuation at low frequencies; $\beta$ controls the attenuation in the middle-frequency band; and $\gamma$ dominates the attenuation in the tail-frequency band. After fitting with the measured attenuation data of partially saturated coal samples under variable confining pressures and variable temperature conditions, the results show that this model can effectively represent the attenuation characteristics of elastic wave propagation in coals with different coal structures. It provides a new theoretical model and analysis ideas for the study of elastic wave attenuation in tectonic coals and is of great significance for an in-depth understanding of the physical properties of coals and related geophysical prospecting.

1. Introduction

The propagation characteristics of elastic waves in rocks have always been a hot topic in the field of geophysical exploration. Current research focuses mainly on exploring the mechanisms of dispersion and attenuation that occur when elastic waves propagate in rocks [1]. Attenuation refers to the phenomenon where the amplitude of a wave decays exponentially with the propagation distance; dispersion means that the propagation velocity of a wave changes with the frequency. Attenuation and dispersion can be caused by various physical phenomena, which can be mainly divided into elastic processes (scattering attenuation and geometric dispersion, where the total energy of the wave field remains constant) and inelastic processes (inelastic dissipation, where wave energy is converted into heat energy). To cause significant attenuation and dispersion, the degree of heterogeneity between the elastic properties of different regions in the rock must be significant. In recent years, two of the most obvious cases of strong attenuation that have received extensive attention are patch-saturated rocks and fractured reservoirs [2,3,4,5].

If there are two immiscible pore fluids at the mesoscopic scale, with significantly different fluid bulk moduli (such as water and gas) forming a patchy distribution, then significant wave attenuation and dispersion phenomena will occur. The dual-porosity model refers to a porous medium with two types of pore structures: stiff pores and soft pores [6,7]. One of the most well-known is a dual-pore model proposed by Chapman in 2002. This model contains randomly oriented fractures and spherical pores, simulates the squirt-flow effect at the grain scale, and presents a frequency-dependent model from the micro-to meso-scale [8]. However, the attenuation characteristics described by the Chapman model show a strict normal distribution. Judging from the low-frequency experimental results of sandstone and shale in recent years [9,10,11,12], the Chapman model cannot explain their attenuation characteristics well. Pride et al. designed two mesoscopic flow models. One model consisted of a mixture of clay and sand with only one type of fluid filling the pores; the other model had a single lithology but was filled with two immiscible mixed fluids. Both models generated sufficient attenuation, thus proving the existence of mesoscopic flow [13]. Based on the Biot–Rayleigh model, Ba et al. extended the dual-porosity medium to a complex situation considering both the dual-pore structure and fluid patch saturation simultaneously; they proposed a double-dual-porosity medium model to describe the dispersion and attenuation of longitudinal waves [14].

Apart from the wave-induced flow theory, there is another approach that starts with the viscoelasticity of the medium. Without considering the complex pore structure of the medium, multiple mechanical elements are connected in series or in parallel. The constitutive equations of the model under different series-parallel combinations are derived; the complex modulus of the model is obtained through Fourier transform, from which the quality factor $Q$ is calculated [15]. This phenomenological method can accurately simulate the $Q$-value within the effective frequency band and effectively represent the attenuation parameters in rocks, thus reducing the number of parameters. With good applicability, it has broad application prospects.

Table 1. Expressions of complex modulus and quality factor of several viscoelastic models.

Models	Component Combination Mode	Complex Modulus	Quality Factor $\mathit{Q}$
Kelvin-Voigt model [23]		$M\left(\omega \right)=M_R+i\omega \eta$	$Q\left(\omega \right)={\displaystyle \frac{M_R}{\omega \eta }}$
Maxwell model [24]		$M\left(\omega \right)={\displaystyle \frac{\omega \eta }{\omega \tau -i}}$	$Q\left(\omega \right)=\omega {\displaystyle \frac{\eta }{M_U}}$
Standard Linear Solid model [25]		$M\left(\omega \right)=M_R{\displaystyle \frac{1+i\omega {\tau }_{\epsilon }}{1+i\omega {\tau }_{\sigma }}}$	$Q\left(\omega \right)={\displaystyle \frac{1+{\omega }^2{\tau }_{\epsilon }{\tau }_{\sigma }}{\omega \left({\tau }_{\epsilon }-{\tau }_{\sigma }\right)}}$
Burgers model [26]		$M\left(\omega \right)={\displaystyle \frac{1}{\frac{1}{k_1}+\frac{1}{i\omega {\eta }_1}+\frac{1}{k_2+i\omega {\eta }_2}}}$	$Q\left(\omega \right)={\displaystyle \frac{\mathrm{Re}\left(M\left(\omega \right)\right)}{\mathrm{Im}\left(M\left(\omega \right)\right)}}$

presents the expressions for the complex modulus and quality factor of several common viscoelastic models. Among them, the standard linear solid model is the most widely used. The standard linear solid is also known as the Zener model, which is often used to describe the dispersion and attenuation characteristics of elastic waves. McDonal et al. calculated the attenuation of seismic waves in Pierre shale and found that the attenuation factor $Q^{-1}$ has an approximately linear relationship with the frequency within the seismic-frequency band [16]. Since the combination model of viscoelastic elements has a natural advantage in describing this kind of approximately constant-$Q$ relationship, it has been widely applied. Emmerich and Korn established a generalized Maxwell model to describe the dispersion and attenuation characteristics of seismic waves. The research results show that when choosing the two- to three-order approximation, both accuracy and computational efficiency can be taken into account [17]. Blanch conducted in-depth research on the different relaxation times in multiple standard linear solid models and proposed the least-squares $\tau$-value method. The research results showed that an approximately constant-$Q$ relationship can be obtained by simply combining several standard linear solids in parallel, which greatly reduces the computational effort [18]. Moczo and Kristek investigated the generalized Zener model and the generalized Maxwell model. They pointed out that increasing the number of models can improve the approximation accuracy. Therefore, more mechanical elements can be connected in series or in parallel; however, this will reduce the computational efficiency [19]. Du et al. achieved the wave-field simulation of elastic waves propagating in viscoelastic anisotropic media through the Zener model [20]. He et al. proposed an independent characterization $\tau$-value method for the calculation method of relaxation time in the generalized rheological body model. This method removed the unreasonable assumption that all standard linear solids in the generalized rheological body model use the same relaxation time and achieved high-precision inversion of the constant-$Q$ model with a small number of standard linear solids [21]. Based on the high-order generalized standard linear solid model, while considering both computational accuracy and computational cost, Chang et al. proposed that the fifth-order generalized standard linear solid model is the most appropriate model, and established the corresponding velocity-stress equations for three-dimensional viscoelastic waves [22].

Although the combination models of viscoelastic elements have been widely used to describe the attenuation laws of media, these models can only describe the attenuation characteristics within a relatively narrow frequency range. A single model cannot adequately approximate the attenuation characteristics of actual rocks within the seismic-frequency band. Combining multiple models will increase the computational load and may not achieve the desired results. This may be because simulating attenuation laws based on component combination models is usually a process of “guessing”. Most research attempts to fit actual data by continuously increasing the number of components or changing the component combinations; sometimes, this may be less efficient despite greater effort. At the same time, the component combination model does not take into account the medium structure itself. If the internal relationship between the internal structure of the medium and a component combination can be found, it may promote the further development of the viscoelastic medium model.

In recent years, petrophysicists have observed, in experiments, that the attenuation of actual rocks may exhibit complex phenomena such as multiple peaks or obvious asymmetry [27,28,29,30]. Regarding this issue, the above two theories are not very applicable. Therefore, scholars have shifted their attention from integer-order wave equations or viscoelastic constitutive equations to fractional-order ones [31].

Compared with integer order, the most significant characteristic of fractional order is that it can describe curves with changing convexities or concavities as well as asymmetric curves. The Italian scholars Professor Caputo and Professor Mainiardi proposed a fractional-order Zener model based on fractional-order derivatives. They fitted the attenuation data of elastic waves in aluminum, brass, cupronickel alloy, glass, silver, steel, and soil. The research findings indicated that the fractional-order Zener model can indeed fit asymmetric curves to a certain extent and cover a wider frequency range [32]. On the other hand, Kjartanssoon established a fully constant-$Q$ model based on mathematical pattern assumptions. This fully constant-$Q$ model contains fractional-order time derivatives and can quantitatively describe the dispersion–attenuation relationship of viscoelastic media [33]. Carcione simulated the propagation of acoustic and elastic waves in viscoelastic media based on the fractional-order fully constant-$Q$ model. The Grünwal–-Letnikov definition of the fractional order and central difference were used to calculate the fractional-order time derivatives. The simulation results were compared with the two-dimensional analytical solution of wave propagation in homogeneous Pierre shale, which verified the reliability of this method [34]. However, due to the memory property of the fractional order, the fractional-order derivative, at a certain moment, depends on all the previous values of the variable, which leads to a huge amount of storage.

To solve this problem, one approach is to convert the fractional-order time derivative in the wave equation into a fractional-order Laplace operator [35,36,37,38] and then use Fourier transform to obtain the fractional-order spatial derivative in the wavenumber domain [39,40,41,42]. Another approach is to attempt to truncate the fractional-order differential operator and select appropriate truncation parameters by balancing accuracy, computational efficiency, and memory.

However, most of the applications of fractional order in the viscoelasticity of media by petrophysicists are focused on numerical simulations, which lack the support of attenuation data from actual rock samples. Second, in most studies, the integer order is directly replaced by the fractional order without exploring the reasons for the emergence of the fractional order. The description of the fractional order lacks practical physical meaning, and the fractional order theory is only applied to the calculation of the constant-$Q$ model in the elastic-wave attenuation theory [43]. As more and more low-frequency attenuation data for rocks exhibit complex phenomena, such as multiple peaks and asymmetry, the existing petrophysical models cannot fully describe such phenomena. However, the fractional order theory is naturally capable of depicting these singularities. Therefore, developing new fractional-order models can be regarded as a correct approach. Zhang et al., for the first time, jointly applied the fractal dimension and fractional order to describe the dispersion–attenuation characteristics of tight sandstones and carbonates, successfully explaining the asymmetric attenuation phenomenon [44]. The fractal dimension is a key parameter in fractal theory. This has provided us a new idea with which to explore the internal relationship between fractals and fractional order, and, based on this, to establish a model capable of describing the dispersion–attenuation characteristics of coal.

The physical fractal is a new fractal concept proposed by Professor Yin from Tsinghua University during his research on biological fibers [45,46]. More than a decade ago, Professor Yin’s research group achieved certain progress in the study of the kinematics of fractal growth. They once believed that in a fractal space, the order of the fractional-order derivative depends on the fractal dimension. However, repeated attempts ended in failure. In the past two years, they have abstracted the concept of the fractal unit from the fractal structure of myofibrils [47]. The fractal unit is an infinite-level binary classification structure. Research shows that the time-fractional-order derivative of the viscoelastic motion of myofibrils originates from the motion on its physical fractal space structure. That is to say, it is precisely the fractal structure of myofibrils in physical space that leads to the time–fractional-order derivative of their viscoelastic constitutive equation.

Based on this, Zhao et al. discovered the fractal units existing in primary coal [48]. They constructed an infinite-level, self-similar viscoelastic model based on the macroscopic bedding and cleat system of primary coal that can describe the velocity dispersion and attenuation of elastic waves in coal under dry conditions. However, the model cannot explain dispersion and attenuation when the coal contains fluids. Moreover, it is difficult to observe a clear bedding and cleat system in tectonic coal. Therefore, we need to construct a model with stronger applicability that can explain the singularity of the attenuation peak. In this study, by abstracting the compositional structure of coal, a binary-classification physical fractal model was established and its constitutive equation was derived. Then, through numerical simulation, the velocity dispersion and attenuation characteristics of the binary-classification physical fractal model were studied. The results of the study showed that the model fitted the actual data well.

2. Mathematical Model

2.1. The Binary-Classification Physical Fractal Model of Coal

As shown in Figure 1a, hierarchical structures (fractures–cleats–microcleats–pores) can be observed in coal rocks from the macroscopic to nanoscopic scales. The stacked pattern of the fractal coal matrix constitutes the internal structure of coal rocks. At different scales, coal rocks can be regarded as being composed of rough cubic matrixes of different scales stacked together, as shown in Figure 1b (when coal rocks are under stress, the mechanical system can be considered to consist of the elasticity between horizontal matrixes and the viscosity from the friction between vertical matrixes). Based on the geometry and mechanics of coal rock structures, Zhao et al. [48] proposed a self-similar viscoelastic model of dry coal rocks composed of fractal cells (Figure 2b). To deeply investigate the influence of complex pores–fractures and fluids on the dispersion and attenuation of coal rocks, we added a correction unit to the self-similar viscoelastic model of dry coal rocks and proposed a binary physical fractal model (Figure 2c).

2.2. Derivation of the Constitutive Equation for the Binary-Classification Physical Fractal Structure of Coal

Figure 2a shows the simplified model of the fractal cell in the self-similar viscoelastic model of coal. Figure 2b presents the mechanical representation of the infinite-level self-similar viscoelastic model, while Figure 2c gives the simplified model of the binary structural element of coal proposed in this study. The operators of equivalent elements for such binary-classification structures can be expressed as follows [48]:

$$T_3={\displaystyle \frac{T_1+\sqrt{T_1^2+4T_1{T}_2}}{2}}={\displaystyle \frac{T_1}{2}}+\sqrt{T_1{T}_2}{\displaystyle \sum _{k=0}^{\infty }\left(\begin{array}{c}0.5\\ k\end{array}\right){\left(\frac{T_1}{4T_2}\right)}^k}$$

(1)

If $T_1=1$ and $T_2=\tau p$ ($\tau$ represents the relaxation time of the dashpot and $p={\displaystyle \frac{d}{dt}}$ represents the Heaviside operator), then $\sqrt{T_1{T}_2}={\tau }^{\frac{1}{2}}{p}^{\frac{1}{2}}={\tau }^{\frac{1}{2}}{d}^{\frac{1}{2}}/d{t}^{\frac{1}{2}}$. This formula reveals the fractional-order origin of the constitutive equation of the infinite-level self-similar viscoelastic model. By extending this concept, if $T_1=1$ and $T_2={\tau }^{\alpha }{p}^{\alpha }$, at this time $\sqrt{T_1{T}_2}={\tau }^{\frac{\alpha }{2}}{p}^{\frac{\alpha }{2}}={\tau }^{\frac{\alpha }{2}}{d}^{\frac{\alpha }{2}}/d{t}^{\frac{\alpha }{2}}$, where ${\displaystyle \frac{\alpha }{2}}$ is the fractional order of the infinite-level self-similar viscoelastic model.

The constitutive operator $T$ of the equivalent structure in Figure 2c is:

$$T={\displaystyle \frac{\left(\frac{T_1{T}_3}{T_1+T_3}+T_2\right)T_4}{\frac{T_1{T}_3}{T_1+T_3}+T_2+T_4}}$$

(2)

where $T_3={\displaystyle \frac{T_1+\sqrt{T_1^2+4T_1{T}_2}}{2}}$, $T_4={\displaystyle \frac{T_1+\sqrt{T_1^2+4T_1{T}_2}}{2}}$. Although the expressions of $T_3$ and $T_4$ are the same in form, in actual simulations, the value of $\alpha$ in $T_2$ within the expressions is not the same. This is because, compared with $T_4$, $T_3$ still represents a sub-level structure of coal rock. For the sake of easy distinction, let $T_2={\tau }^{\beta }{p}^{\beta }$ in the expression of $T_3$ and $T_2={\tau }^{\gamma }{p}^{\gamma }$ in the expression of $T_4$ and the operator $\sqrt{p+1}f\left(t\right)={}^{C}D^{0.5}f\left(t\right)$ [34], where ${}^{C}D$ represents the fractional-order differential operator under the Caputo definition.

The constitutive equation of the equivalent structure in Figure 2c is as follows:

$$\sigma \left(t\right)=GT\epsilon \left(t\right)$$

(3)

where $G$ represents the elastic modulus, $\sigma$ represents the system stress, $\epsilon$ represents the system strain, and $T$ represents the constitutive operator that describes the stress–strain relationship. When Equation (2) is substituted into Equation (3), it can be rewritten as follows:

$$\sigma \left(t\right)=G{\displaystyle \frac{T_2{T}_3{T}_4+T_3{T}_4+T_2{T}_4}{T_2+T_3+T_4+T_2{T}_3+T_3{T}_4}}\epsilon \left(t\right)$$

(4)

When the expressions of $T_3$ and $T_4$ are substituted into Equation (4), then:

$$\begin{array}{l}\left(T_2^{\beta +\gamma }+{\displaystyle \frac{3}{2}}{T}_2^{\alpha }+{\displaystyle \frac{3}{2}}{T}_2^{{\displaystyle \frac{1}{2}}\beta }+{\displaystyle \frac{3}{2}}{T}_2^{{\displaystyle \frac{1}{2}}\gamma }+T_2^{{\displaystyle \frac{1}{2}}\beta +\alpha }+{\displaystyle \frac{5}{4}}\right)\sigma \left(t\right)=\\ G\left({\displaystyle \frac{3}{4}}{T}_2^{\alpha }+{\displaystyle \frac{1}{2}}{T}_2^{\alpha +{\displaystyle \frac{1}{2}}\beta }+{\displaystyle \frac{3}{2}}{T}_2^{\alpha +{\displaystyle \frac{1}{2}}\gamma }+T_2^{\alpha +{\displaystyle \frac{1}{2}}\beta +{\displaystyle \frac{1}{2}}\gamma }+{\displaystyle \frac{1}{2}}{T}_2^{{\displaystyle \frac{1}{2}}\beta }+{\displaystyle \frac{1}{2}}{T}_2^{{\displaystyle \frac{1}{2}}\gamma }+T_2^{{\displaystyle \frac{1}{2}}\beta +{\displaystyle \frac{1}{2}}\gamma }+{\displaystyle \frac{1}{4}}\right)\epsilon \left(t\right)\end{array}$$

(5)

A Fourier transform is then performed on Equation (5) to obtain the complex modulus $M\left(\omega \right)$:

$$\begin{array}{l}M\left(\omega \right)=\\ G\left({\displaystyle \frac{\frac{3}{4}{(i\omega )}^{\alpha }{\tau }^{\alpha }+\frac{1}{2}{(i\omega )}^{\alpha +\frac{1}{2}\beta }{\tau }^{\alpha +\frac{1}{2}\beta }+\frac{3}{2}{(i\omega )}^{\alpha +\frac{1}{2}\gamma }{\tau }^{\alpha +\frac{1}{2}\gamma }+{(i\omega )}^{\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma }{\tau }^{\alpha +\frac{1}{2}\beta +\frac{1}{2}\gamma }+\frac{1}{2}{(i\omega )}^{\frac{1}{2}\beta }{\tau }^{\frac{1}{2}\beta }+\frac{1}{2}{(i\omega )}^{\frac{1}{2}\gamma }{\tau }^{\frac{1}{2}\gamma }+{(i\omega )}^{\frac{1}{2}\beta +\frac{1}{2}\gamma }{\tau }^{\frac{1}{2}\beta +\frac{1}{2}\gamma }+\frac{1}{4}}{{(i\omega )}^{\beta +\gamma }{\tau }^{\beta +\gamma }+\frac{3}{2}{(i\omega )}^{\alpha }{\tau }^{\alpha }+\frac{3}{2}{(i\omega )}^{\frac{1}{2}\beta }{\tau }^{\frac{1}{2}\beta }+\frac{3}{2}{(i\omega )}^{\frac{1}{2}\gamma }{\tau }^{\frac{1}{2}\gamma }+{(i\omega )}^{\frac{1}{2}\beta +\alpha }{\tau }^{\frac{1}{2}\beta +\alpha }+\frac{5}{4}}}\right)\end{array}$$

(6)

It is important to note that in the construction of the physical fractal model and numerical simulations, all studies are based on the assumption of isotropic media. This physical fractal model attempts to provide a new approach to approximately describe the velocity dispersion and energy attenuation characteristics of coal, thereby building confidence for subsequent research.

3. Significance of Model Parameters

3.1. The Influence of the Value of $\alpha$ on Dispersion and Attenuation

The formulas for calculating the longitudinal wave velocity and attenuation based on the complex modulus are as follows:

$$V_P={\displaystyle \frac{1}{\mathrm{Re}(\frac{1}{\sqrt{M\left(\omega \right)/\rho }})}}$$

(7)

$${\displaystyle \frac{1}{Q}}={\displaystyle \frac{\mathrm{Im}\left(M\left(\omega \right)\right)}{\mathrm{Re}\left(M\left(\omega \right)\right)}}$$

(8)

where $v_c=\sqrt{M\left(\omega \right)/\rho }$ represents the phase velocity, $\rho$ represents the density, and $Q$ represents the quality factor.

It can be seen from Equation (6) that the dispersion and attenuation of the binary-classification physical fractal model are controlled by three fractional orders, $\alpha$, $\beta$, and $\gamma$. Then, according to Formulas (7) and (8), the influence of the fractional orders on the dispersion and attenuation was analyzed by the method of controlling variables.

Table 2. Fitting parameters of Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

Figure Number	$\mathit{\alpha }$ (Dimensionless)	$\mathit{\beta }$ (Dimensionless)	$\mathit{\gamma }$ (Dimensionless)	$\mathit{\tau }$ (ms)
Figure 3 and Figure 4	0.025~0.055	0.2	0.3	1.667
Figure 5 and Figure 6	0.03	0.1~0.2	0.3	1.667
Figure 7 and Figure 8	0.03	0.3	0.1~0.2	1.667

shows the values of each parameter when the variables are controlled.

Dispersion refers to the change in the propagation speed of a wave as the frequency changes. As can be seen from Figure 3, within the frequency range of 1 to 10⁶ Hz, the difference in the variation of the longitudinal wave velocity in the curve increases with the increase in $\alpha$. Therefore, the conclusion can be drawn that dispersion is positively correlated with $\alpha$. By observing Figure 4, it can be seen that when the frequency is fixed, as $\alpha$ increases, the attenuation coefficient curve is almost shifted upward; that is, the attenuation coefficient becomes larger. Therefore, the attenuation coefficient is also positively correlated with $\alpha$. It is worth noting that the attenuation curve under the binary-classification model is not a normal distribution. Before reaching the characteristic frequency, the increase in attenuation is relatively slow; after the characteristic frequency, the attenuation will decrease suddenly. The fractional order $\alpha$ reflects the viscoelastic friction of the macroscopic bedding and cleat system of the coal mass; its magnitude is closely related to the macroscopic fracture density. The attenuation curve shows an overall change with the value of $\alpha$, which verifies its macroscopic-scale property.

3.2. The Influence of the Value of $\beta$ on Dispersion and Attenuation

Figure 5 and Figure 6 show the influence of the fractional order $\beta$ on dispersion and attenuation. $\beta$ originates from the fluid–solid coupling effect of mesoscopic pores. It has little effect on the longitudinal wave velocity in the low-frequency part. The influence of $\beta$ on the longitudinal wave velocity gradually becomes apparent as the frequency increases. The influence of $\beta$ on attenuation is less significant at low frequencies than at high frequencies. Moreover, the larger the value of $\beta$, the faster the decrease in the attenuation rate at high frequencies.

Figure 5. P-wave velocity dispersion curves at different $\beta$ values.

Figure 5. P-wave velocity dispersion curves at different $\beta$ values.

Figure 6. P-wave attenuation curves at different $\beta$ values.

Figure 6. P-wave attenuation curves at different $\beta$ values.

3.3. The Influence of the Value of $\gamma$ on Dispersion and Attenuation

Figure 7 and Figure 8 show the influence of the fractional order $\gamma$ on dispersion and attenuation. $\gamma$ is related to the viscoelastic memory effect of the differential cell. $\gamma$ has an influence on the longitudinal wave dispersion almost only in the ultrasonic-frequency band; it has little impact on the attenuation at low frequencies. The larger the value of $\gamma$ at high frequencies, the faster the rate of decrease in the attenuation. This is consistent with the interactions between the scale of microfissures and high-frequency waves.

Figure 7. P-wave velocity dispersion curves at different $\gamma$ values.

Figure 7. P-wave velocity dispersion curves at different $\gamma$ values.

Figure 8. P-wave attenuation curves at different $\gamma$ values.

Figure 8. P-wave attenuation curves at different $\gamma$ values.

4. Experimental Results and Discussion

4.1. Attenuation Experiments Across Frequency Bands for Different Coal Mass Structures

Primary coal and tectonic coal were collected from the 3# coal seam in the southern part of Qinshui Basin for the attenuation test across frequency bands. The structure of the primary structured coal is quite well preserved, with coherent and natural bedding, and there are no damaged parts as a whole. In contrast, the tectonic coal is mostly developed with fracture and cleat systems (Figure 9c) and its strength is significantly reduced. Dong et al. carried out uniaxial mechanical property test experiments on primary structured coal and tectonic coal at the small particle size scale (0.2–4.0 mm). The results showed that the elastic parameters of the primary coal were 2.72–4.57 times those of the tectonic coal [49]. Peng et al. (2004) [50] analyzed the relationship between density and acoustic characteristics through the logging curves in the Huainan Coalfield. The acoustic velocity of the primary coal in the same layer is about 13% higher than that of the tectonic coal [50]. The basic parameter information of the samples is shown in

Table 3. Basic parameters of coal samples.

Sample	Coal Structure	Length (mm)	Diameter (mm)	Velocity (m/s)	Porosity (%)	Permeability (mD)
DT-1	Primary coal	64.980	37.207	2394.28	7.41	0.36 × 10−3
DT-4	Tectonic coal	65.198	36.932	2108.13	11.23	60.7 × 10−3

. The porosity of the samples was measured using Boyle’s law double-chamber method; the permeability was determined through the pulse-decay technique. The results showed that the ultrasonic velocity of tectonic coal is about 300 m/s lower than that of primary coal, which is in line with the classification of primary coal and tectonic coal.

To achieve cross-band analysis, it is necessary to obtain the low-frequency velocity and ultrasonic velocity at the same time during the test. The ordinary aluminum probe can only obtain the velocity in the low-frequency band; therefore, we added a piezoelectric ceramic sheet to stimulate the ultrasonic wave on the contact surface of the aluminum probe and the sample to obtain the ultrasonic velocity. After the sample was dried, the aluminum probe was attached at both ends; the strain gauge was attached to the coal sample and the aluminum probe, respectively. The strain gauge was drawn out by welding the wire. Finally, the coal sample to be tested was sealed with epoxy resin after the wire was welded and a complete test sample was made (Figure 9b).

In this experiment, the rock low-frequency test system (Figure 9a), introduced and improved by China University of Mining and Technology (Beijing), was used. After the instrument was connected, the low-frequency elastic wave test was carried out under the pressure of the greenhouse. The axial strain and radial strain signals of standard samples and rock samples, tested in the frequency range of 5–500 Hz, were obtained.

4.2. Fitting Results and Analysis of Partially Saturated Coal Samples Under Variable Confining Pressure Conditions

To verify that the two-classification fractal model can characterize the attenuation characteristics of the measured coal samples, MATLAB (2019a) was used to fit the laboratory measured data and the model. The relaxation time was calculated by the formula given by Batzle [51]:

$$\tau =C{\displaystyle \frac{x^2}{K_f}}{\displaystyle \frac{\eta }{k}}$$

(9)

where x is the equilibrium penetration distance of the fluid; K_f represents the bulk modulus of the fluid; η denotes the dynamic viscosity of the fluid; k is the permeability; and C is a dimensionless constant that (a) incorporates unit conversion factors between parameters, and (b) allows for empirical calibration through laboratory measurements.

The research in Section 3 found that the fractional orders $\alpha$, $\beta$, and $\gamma$, respectively, control the dispersion and attenuation in the seismic-frequency band (1~10³ Hz), the logging-frequency band (10³~10⁵ Hz), and the ultrasonic-frequency band (>10⁵ Hz). These three frequency bands correspond to the macroscopic, mesoscopic, and microscopic scales, respectively. The research in Section 3 reveals that the fractional orders α, β, and γ govern the dispersion and attenuation in the seismic-frequency band (1–10³ Hz), logging-frequency band (10³–10⁵ Hz), and ultrasonic-frequency band (>10⁵ Hz), respectively. These three bands correspond to the macroscopic, mesoscopic, and microscopic scales. During low-frequency fitting (Figure 10, Figure 11, Figure 12 and Figure 13), β and γ are fixed because α dominates the viscoelastic response and controls bulk energy dissipation in this band; β and γ primarily influence high-frequency (>10³ Hz) scattering effects. The values of β and γ were determined based on coal structural characteristics: for primary coal (DT-1), dominated by the matrix structure. A high β value (0.6) reflects the dominant role of pore-scale heterogeneity in energy loss; a low γ value (0.1) indicates weak fracture-induced scattering. For tectonically deformed coal (DT-4), characterized by a fracture-dominated network, a low β value (0.2) corresponds to reduced pore-related dissipation, while a high γ value (0.8) captures strong wave scattering from cleats/voids. As intrinsic parameters representing the pore-fracture architecture, β and γ remain relatively stable under the tested conditions (0.1–12 MPa, 25–42 °C): β correlates with pore types (intergranular vs. fracture) and shows no significant variation under moderate pressure–temperature conditions, whereas γ depends on fracture connectivity and is unaffected within the elastic deformation regime. This stability is further validated by the inverse proportionality between porosity and β in Table 3: DT-1 (low porosity, ~4%) exhibits a high β value due to localized energy dissipation in isolated pores, while DT-4 (high porosity, ~12%) shows a low β value because percolative flow through connected fractures reduces viscous losses. Additionally, all samples were tested at 60% water saturation, where preferential water-filling in fractures (cleats) of DT-4 enhances the scattering center effect, consistent with its high γ value.

Figure 10 and Figure 11 are the attenuation fittings of the primary coal and the tectonic coal under different confining pressure conditions when saturated with ~60% water. The attenuation of both decreases with the increase in confining pressure. It can be seen that the attenuation of tectonic coal is not only numerically larger than that of primary coal, but is also more sensitive to confining pressure.

Table 4. Fitting parameters of attenuation in 60% water-saturation primary and tectonically deformed coal under different confining pressure conditions.

Sample	Confining Pressure (MPa)	$\mathit{\tau }$ (ms)	$\mathit{\alpha }$ (Dimensionless)	$\mathit{\beta }$ (Dimensionless)	$\mathit{\gamma }$ (Dimensionless)
DT-1	0.1	0.33	0.038	0.6	0.1
DT-1	7	0.19	0.036	0.6	0.1
DT-1	12	0.14	0.032	0.6	0.1
DT-4	0.1	0.55	0.065	0.2	0.8
DT-4	7	0.31	0.054	0.2	0.8
DT-4	12	0.23	0.044	0.2	0.8

is the fitting parameter of the model. The fractional order $\alpha$ in the tectonic coal is significantly larger than that in the primary coal; $\alpha$ decreases with the increase in confining pressure. For the broken coal structure (similar to tectonic coal), with the increase in confining pressure, the pore and fracture gradually closed; the coal structure was more compact (similar to primary coal), and the attenuation was gradually weakened; therefore, it can be inferred that $\alpha$ is greatly affected by the coal structure.

4.3. Fitting Effects and Analysis of Partially Saturated Coal Samples Under Variable Temperature Conditions

Figure 12 and Figure 13 are the attenuation fittings of primary coal and tectonic coal under different temperature conditions when saturated ~60% water. In the range of 10–100 Hz, the attenuation of both increases with the increase in temperature. It can be seen that temperature mainly affects the attenuation of the middle-frequency band. Within 10 Hz, the effect of temperature on attenuation is very weak.

Table 5. Fitting parameters of attenuation in 60% water-saturation primary and tectonically deformed coal under different temperature conditions.

Sample	Temperature (°C)	$\mathit{\tau }$ (ms)	$\mathit{\alpha }$ (Dimensionless)	$\mathit{\beta }$ (Dimensionless)	$\mathit{\gamma }$ (Dimensionless)
DT-1	25	0.33	0.038	0.6	0.1
DT-1	30	0.28	0.039	0.6	0.1
DT-1	42	0.22	0.040	0.6	0.1
DT-4	25	0.55	0.065	0.2	0.8
DT-4	30	0.33	0.0652	0.2	0.8
DT-4	42	0.22	0.0654	0.2	0.8

is the fitting parameter of the model. The difference of fractional order $\alpha$ under different temperature conditions is small, indicating that the influence of temperature on attenuation is significantly smaller than that of the confining pressure. The increase in temperature led to the decrease in relaxation time and the shift of the attenuation peak to a high frequency. However, the increase in temperature did not change the selection of fractional order $\beta$ and $\gamma$.

5. Conclusions

This paper used primary coal and tectonically deformed coal as research objects and studied the velocity dispersion and energy attenuation properties of coal. A binary physical fractal model was established and the constitutive equation of the binary physical fractal model was determined. Three key fractional orders, $\alpha$, $\beta$ and $\gamma$, were discussed.

The fractional orders $\alpha$, $\beta$ and $\gamma$ control the velocity dispersion and energy attenuation in the seismic-frequency band, the logging-frequency band and the ultrasonic-frequency band, respectively, corresponding to the attenuation response mechanisms at the macroscopic, mesoscopic, and microscopic scales, respectively.