Experimental and numerical investigation of elastic wave dispersion and attenuation induced by coal particle damping

Abstract

The collision of skeletal matrix particles in dry coal plays a crucial role in the dispersion and attenuation of elastic waves. Using a low-frequency testing system, acoustic velocity dispersion and energy attenuation experiments were performed across a frequency range of 1–250 Hz on two types of dry primary structure coal with varying metamorphic degrees, along with a 3D-printed model. Meanwhile, a discrete element numerical model was developed, incorporating damping particles and different gradations, based on the theory of rock particle damping, to simulate inter-particle collision behavior. The results indicate that rock dispersion and attenuation are affected by tangential damping, normal damping, and particle size distribution. Tangential damping contributes approximately three to four times more to these effects than normal damping. Notably, these effects are minimal when the damping coefficient is zero, and as the damping coefficient increases and particle distribution becomes more heterogeneous, the P-wave velocity decreases while dispersion and attenuation are significantly enhanced. Moreover, both the elastic wave velocity and attenuation coefficient exhibit pronounced frequency dependence in dry coal and 3D-printed models. Low-rank coal, with a broader particle size distribution and a more disordered, looser packing, demonstrates stronger dispersion and attenuation. In contrast, the polylactic acid (PLA) model fabricated by fused deposition modeling shows weaker dispersion and attenuation compared to the photosensitive resin model. This study provides crucial insights into the dispersion and attenuation mechanisms of primary structure coal, optimizing parameters such as particle elastic modulus, damping, and uniformity to improve understanding of the seismic acoustic response of dry coal.

Introduction

Accurate understanding of the physical properties of underground rocks is essential for resource evaluation and the development of strategies in oil and gas exploration. The dispersion and attenuation characteristics of rocks significantly influence seismic wave propagation. They directly impact the accuracy of seismic data interpretation. Moreover, these characteristics are vital tools for in-situ detection. They help detect both solid and fluid properties^1–3. Studying dispersion and attenuation enhances our understanding of the microstructure of underground rocks and the fluid-rock interaction mechanisms⁴, thereby improving the accuracy and reliability of seismic data inversion and boosting the efficiency of oil and gas exploration. Both experimental physical simulation^5,6 and numerical simulation^7–11 are essential tools. They are used for investigating rock mechanical properties. Physical models created using 3D printing technology exhibit high similarity to natural rocks. This is especially true in terms of their mechanical properties¹². The discrete element method (DEM) models the research area as a collection of particles. It focuses on the contact and force transfer between particles while bypassing displacement compatibility between blocks¹³. In this approach, micro-particle mechanics are defined by a set of constitutive equations. The elastic modulus of granular materials is derived from the direct contact behavior of particles. This facilitates extensive study of the elastic characteristics of porous rocks^7,13,14 Compared to continuum models, DEM offers a more accurate representation. It models the contact motion of micro-particles in rock under external forces. By continuously solving particle motion equations, it provides insights into external disturbances. These disturbances influence the internal microstructure and macroscopic physical properties of rock⁸.

Research on rock dispersion and attenuation falls into three main categories. The first category is wave-induced fluid flow. As an elastic wave propagates through rock, it causes periodic deformation of the rock skeleton, leading to compression or expansion of the pore fluid, which generates a pressure gradient. This pressure gradient drives fluid flow within the pores. It results in relative motion between the fluid and the solid skeleton, causing attenuation of wave energy due to internal friction^15–20. The second category is the inherent viscoelasticity of the rock skeleton. The interaction between the particles of the rock skeleton and the fluid leads to energy dissipation and changes in wave velocity^21–24. The third category involves the bound water effect in high-viscosity clay, as proposed by Long et al. This theory suggests that the fracture and formation of hydrogen bonds in bound water within clay, along with complex layer interactions, friction, and sliding of clay particles, contribute to the dispersion and attenuation of waves in shale²⁵. These studies have provided a theoretical foundation for understanding rock dispersion and attenuation. They have also made significant contributions to the evaluation of in-situ formation fluid properties. The in-depth study of attenuation and dispersion in fluid-filled porous rocks has led most scholars to identify wave-induced fluid flow as the primary cause of these phenomena^26,27. However, fully describing the attenuation of broadband elastic waves remains challenging. This is due to the complexity, diversity, and varying properties of real rocks, even when fluid effects are fully accounted for. As a result, it is necessary to separate the pore fluid from the rock skeleton²⁸. Based on an analysis of four rock types’ microstructures, Liu et al. found that the attenuation degree of elastic waves correlates with how tightly mineral particles are bound²⁹. Valdiviezo et al. examined how shale mineral composition and rock maturity affect dispersion and attenuation. Their results showed that both factors exert a significant influence³⁰. Zhao and colleagues developed a viscoelastic model of the coal skeleton based on fractal theory, revealing that fractional order and relaxation time have substantial effects on P-wave velocity dispersion and attenuation²⁴. She et al. investigated the elastic wave dispersion characteristics of different coal structures under dry conditions, finding that all four coal structures exhibited significant dispersion, which became more pronounced as the coal structure fragmentation increased³¹. Additionally, numerous studies have shown that in dry rocks, due to the incomplete elastic properties of the skeleton matrix, the relative movement of particle boundaries and the viscoelastic behavior of clay minerals lead to energy dissipation^32–34. Rock can be viewed as a cemented material. It is composed of complex-shaped particles, and damping effects occur between them. Damping is classified into normal damping and tangential damping. Normal damping primarily affects the compressive and rebound forces between rock particles, while tangential damping influences the relative sliding between the particles³⁵. The organic matter content of coal typically accounts for about 90% of the total mass, with clay minerals constituting a smaller proportion, approximately 10%³⁶. This indicates that coal’s skeletal structure is predominantly organic. This composition forms a unique soft-wrapped-hard structure. Research has shown significant differences in the molecular structure of organic matter particles in coal with varying degrees of metamorphism³⁷. However, while existing studies have revealed energy dissipation caused by non-fluid mechanisms in dry rocks, the research on the dispersion and attenuation characteristics of elastic waves in coal remains incomplete. The influence of key damping factors on dispersion and attenuation is poorly understood. These factors include organic matter type, particle size, and coal skeleton uniformity. In particular, two questions remain underexplored: whether these factors are frequency-dependent within the seismic band, and to what extent they impact the overall energy dissipation in dry rock.

Therefore, this paper is based on low-frequency tests of 3D-printed physical models of two types of primary coal with different metamorphic degrees, as well as two distinct materials. A viscoelastic model was developed based on the linear parallel bond theory to characterize the damping effect of particles. Subsequently, the Particle Flow Code in 3 Dimensions (PFC3D) numerical simulation technology was employed to conduct experimental simulations at the same frequency. The study systematically examines the effects of contact damping parameters of rock particles and the gradation of skeleton particles on the dispersion and attenuation characteristics. The impact of these parameters on dispersion and attenuation was further analyzed. Through comparing the results with Zener’s classical viscoelastic model, the applicability and accuracy of the numerical simulation method based on particle damping effects are verified. Compared with previous studies on dry coal dispersion, this study introduces DEM based damping particle model that explicitly incorporates tangential and normal damping coefficients and particle gradation parameters to simulate inter-particle collision and friction. The damping analysis based on DEM provides a micro-mechanical perspective, extending previous phenomenological and statistical approaches.

Theory and method

Sample preparation

The high-grade coal and low-grade coal with primary structure were collected from the Qinshui Basin and the Junggar Basin in China. The selected coal sample is fabricated into a cylinder with a length of 60 mm and a diameter of 38 mm using wire cutting (Fig. 1a), and the low-frequency and Porosity tests are carried out. The remaining broken sample is used for physical property testing and microstructure characterization. The left side of Figure 1a shows the high-order coal sample, while the right side displays the low-order coal sample. The average vitrinite random reflectance of the sample is determined following the national standard of the People’s Republic of China, DB/T 6948–2008. With helium as the reference fluid, the bulk density and porosity of the sample were determined according to DB/T 29172–2012 standard. Industrial composition analysis is carried out in accordance with DB/T 30732–2014 standard. M_ad and V_ad represent moisture and ash, respectively, while the organic matter content OM is the sum of volatile V_ad and fixed carbon FC_ad (Table 1). The core X-ray diffraction experiment shows that the mineral composition of the coal sample is mainly composed of kaolinite, illite, and quartz. Although there are some differences in mineral composition among samples from different study areas, the content of clay minerals in each sample is more than 50% (Table 2). The sample’s bulk modulus K_gr and shear modulus G_gr are determined through the Voigt–Reuss–Hill (VRH) averaging scheme, taking into account the volume fraction and elastic modulus of each constituent material.

Fig. 1.

Plunger test sample: (a) high- and low-rank coal sample, (b) 3D printed physical model.

Table 1.

Physical property parameters of high- and low-rank coal samples.

Coal sample	Ro, ran (%)	(g/cm3)	(%)	Mad (%)	Aad (%)	Vad (%)	FCad (%)	OM (%)	Kgr (GPa)	Ggr (GPa)
H_rank coal	2.82	1.395	5.86	0.74	8.46	10.58	80.22	90.80	6.47	3.68
L_rank coal	0.46	1.256	6.38	1.28	12.69	26.18	59.85	86.03	5.62	2.73

Table 2.

Mineral composition parameters of high- and low-rank coal samples.

Coal sample	Kaolinite (%)	Illite (%)	Quartz (%)	Calcite (%)	Ankerite (%)	Ferricopiapite (%)	Anatase (%)	Anhydrite (%)	Marcasite (%)
H_rank coal	34.6	18.2	7.2	10.8	12.1	6.2	10.9	0	0
L_rank coal	38.4	20.2	6.5	18.2	6.3	0	0	6.8	3.6

In this study, the specifications of the coal samples were compared, and two types of 3D printing equipment were used to fabricate standard cylindrical models with a height of 50 mm and a diameter of 38 mm (Fig. 1b). The first printer, the Objet260 from STRATASYS (USA), utilizes RGD525 photosensitive resin as the printing material (left side of Fig. 1b). The second, the FR650 fused deposition printer from Shandong Fangzun Intelligent Company (China), uses PLA as the printing material (right side of Fig. 1b). The mechanical parameters of the physical models are presented in Table 3.

Table 3.

Mechanical parameters of the 3D printing physical model.

Physical sample	(g/cm3)	(%)	Tensile strength/MPa	Elastic modulus/GPa	Bend strength /MPa	Flexural modulus/GPa	Water absorption/%
RGD print	1.168	4.21	70	3.2	120	3.2	1.2
PLA print	1.231	3.52	60	2.8	100	3.0	3.5

To ensure the dryness of samples before testing, coal samples and 3D printed samples need to be dried at 70 °C and 50 °C for 48 hours, respectively. After drying, place all samples in indoor air for 48 hours to restore them to room temperature and reach equilibrium state³¹. The two end faces of the sample are bonded to the aluminum probe using epoxy resin, with a small axial force maintained throughout the curing process to ensure no gap between the sample and the probe. To achieve complete sealing, the surface of the dry sample is first evenly coated with epoxy resin, followed by the application of polyimide film, which is then carefully pressed to ensure a tight fit without bubbles. After the epoxy resin had completely set, a strain gauge was mounted onto the surface of the polyimide film. The exterior of the sample was then sealed using potting adhesive, as shown in Figure 1a. Experiments were performed under ambient conditions of 25 °C and 0.1 MPa. The elastic parameters of coal samples and 3D printed samples were measured in the frequency range of 1–250 Hz using a low-frequency stress-strain testing system. To ensure the reliability and repeatability of the measurements, the results for the same sample at each frequency were obtained by averaging five measurements.

The microstructure of the coal sample shows (Fig. 2) that there are significant differences in particle and pore characteristics between high-grade coal and low-grade coal. The particle size distribution of the high-grade coal sample is highly uniform, characterized by a single peak, narrow distribution, complete particle morphology, and close and orderly arrangement, resulting in poor development of its pore system, mainly small and isolated micropores (Fig. 2a). This indicates that high-grade coal experienced strong compaction and coalification during diagenesis. The low-rank coal sample has a wide range of particle size distribution, mixed sizes, a more loose and disorderly arrangement, and the particles are not closely combined, thus forming a pore system with significantly larger quantities, larger sizes, and better connectivity (Fig. 2b).

Fig. 2.

Microstructure of optical thin section of coal sample: (a) high-rank coal, (b) low-rank coal.

Low-frequency test system

This study employs the low-frequency test system based on the axial forced stress-strain method developed by China University of Mining and Technology (Beijing) (Fig. 3). The device is designed based on Batzle. Through structural optimization and signal processing module upgrading, the stability and signal-to-noise ratio are significantly improved to support high-precision measurement of rock dispersion and attenuation^31–38.

Fig. 3.

Forced stress and strain low-frequency test system: ① waveform generator,② Signal acquisition module, ③ power amplifier, ④ computer display screen, ⑤ confining pressure chamber, ⑥ sample table, ⑦ vibrating table.

In the measurement process of the stress-strain method, strain gauges are pasted on the surface of the standard aluminum block and the sample to be measured. The sample and the aluminum block are fixed by glue and aligned along the axial direction to ensure the continuity of the whole stress system in the transverse and axial directions. When sinusoidal stress with a specific frequency is applied to the test sample³⁸, the low-frequency Young’s modulus E and Poisson’s ratio of the tested sample can be obtained based on the strain amplitude recorded on the sample and the standard aluminum block in real time, as shown in Equation (1). Assuming isotropy, the bulk modulus K and shear modulus G of the sample can be calculated from Young’s modulus E and Poisson’s ratio based on the basic theory of elasticity, as shown in Equation (2). Furthermore, with the bulk density known, the P- and S-wave velocities of the measured samples are calculated using Equation (3).

1

2

3

Where is the Young’s modulus of standard aluminum, is the axial strain of standard aluminum, and are the radial and axial strains of the tested sample, respectively.

In the low-frequency seismic testing of viscoelastic media, the complex elastic modulus connects the stress and the strain . When the frequency is , the complex stress can be expressed as , and the complex strain is . In the equation, and are the phase angles of the complex stress and the complex strain, respectively. In a vibration period, the energy dissipation of viscoelastic media is usually characterized by the reciprocal of the quality factor³⁹. Its attenuation is defined as:

4

Here, represents the phase lag between stress and strain resulting from the hysteresis effect in a viscoelastic medium.

LPBM theory

The DEM models the material as a collection of discrete particles, simulating transient behavior during loading by tracking the motion of each particle⁴⁰. As independent rigid bodies, the motion of particles follows Newton’s second law, with interactions occurring only at the contact points. The contact force is calculated by the stress displacement relationship, including normal stress and tangential stress. The indirect contact force and relative displacement of particles usually have a nonlinear relationship, which can simulate the complex mechanical behavior of particles such as crushing, sliding, and rolling.

Coal is a viscoelastic medium, exhibiting both viscous and elastic behaviors. Under dynamic load disturbances, the complex mechanical behaviors between coal skeleton particles can be described using special contact bonds⁴¹. Recent years have seen significant progress in the study of rock mechanical properties using the Linear Parallel Bond Model (LPBM), which accounts for the interaction between rock particles^42–44. Given the effectiveness of the LPBM in studying the mechanical properties of rocks^42,43, we have selected this model to describe the contact mechanical properties between coal particles. In PFC3D, the rock sample model is composed of randomly arranged and combined rigid spherical particles that are tightly packed, allowing for the transfer of force and torque between particles. Assuming that particles are rigid, they interact only at contact points. Collisions or sliding can cause displacement, and there are breakable finite stiffness bonds at the contact points⁴⁰. Each pair of adjacent particles forms contacts, which include both real contacts and virtual contacts. The LPBM consists of a linear part, a parallel bonding part, and a dynamic part that transfers bending moments and bonding forces, as shown in Figure 4.⁴². The linear parallel bonded contact can be modeled as a set of elastic springs with constant normal and shear stiffnesses, uniformly distributed around the contact point on the three-dimensional cross-sectional circle of the contact plane. During compression, the bonded contact will fracture only when the maximum compressive or shear stress exceeds the corresponding bond strength threshold. At this point, the parallel bonding interface and its associated forces, moments, and stiffnesses will be removed from the model, transforming the contact into a non-bonded linear elastic contact.

Fig. 4.

Schematic diagram of the structure of the LPBM.

In the linear LPBM⁴⁰, the linear force , damping force , parallel bond force , and friction force are represented in matrix form as follows:

5

Therefore, the contact force is expressed as:

6

On this basis, the parallel bonding moment generated by the parallel bonding force _:

7

Among them, represents normal contact stiffness, represents normal stiffness of parallel bonding; and represent normal displacement and tangential displacement respectively; represents tangential contact stiffness, represents tangential stiffness of parallel bonding; and represent normal and tangential numerical damping coefficients respectively; and represent normal and tangential velocities of particles respectively; represents friction coefficient; represents torsional stiffness, represents bending stiffness; and represent normal and tangential rotation angles, respectively.

Numerical simulation applies an axial dynamic load in the form of a sine wave along the z-axis to the sample through a servo control system (Fig. 5), generating a periodic compression-rebound cycle at a specific effective loading frequency. During this process, the sample undergoes periodic compressive deformation, with dynamic changes in strain and stress monitored and recorded in real-time using the built-in measuring circle. Simultaneously, the damping dissipation within the rock particle system is also recorded. The average stress is calculated as the ratio of the resultant force on the boundary constraint to the acting area. The formulas for axial and radial strain are given as follows:

8

Fig. 5.

Numerical model diagram.

Among them, represents axial strain, represents radial strain; represents the current axial length of the sample, and represents the original axial length of the sample; represents the current radius of the sample, and represents the original radius.

After obtaining the stress-strain relationship, the elastic parameters of the rock sample can be calculated using Equations (2) and (3). Additionally, in granular systems, energy dissipation primarily occurs through the work done by damping forces. Since the damping force is proportional to the rate of change of displacement, the system’s energy dissipation is determined by summing the work done by the damping force at each contact point. The calculation method is as follows:

9

Among them is the time step.

In the context of particle-scale interactions in dry coal, normal damping primarily reflects the energy dissipation associated with the compression and rebound of particles under dynamic loading, corresponding to the viscoelastic behavior in the direction perpendicular to the contact plane. In contrast, tangential damping characterizes the energy loss due to frictional sliding and shear resistance between adjacent particles, capturing the in-plane dissipative mechanisms during relative tangential motion. These two damping components collectively represent the key non-fluid dissipation pathways in a granular coal skeleton under wave propagation.

Numerical model design

To investigate the effects of sine wave loading at different frequencies on the elastic parameters and attenuation characteristics of rock samples, a plunger-like model with a height of 60 mm and a diameter of 38 mm was constructed. The model is composed of randomly shaped spherical particles with varying sizes. To balance computational efficiency and model feasibility, the radius range of the spherical particles was set between 0.5 and 1 mm, resulting in a total of 24,408 spherical particles, as shown in Figure 5.

Based on the aforementioned model, the objective is to investigate the impact of normal and tangential damping on dispersion and attenuation within the model. Three sets of particle samples with different damping coefficients⁴¹ were established. For the first set, the normal damping coefficient was fixed at zero, while the tangential damping coefficient was varied; for the second set, the tangential damping coefficient was fixed at zero, while the normal damping coefficient was varied; for the third set, a particle model was constructed where the normal and tangential damping coefficients were equal, based on the combination of the first two sets and assuming the low-frequency test rock is homogeneous. The damping parameters for each model set are presented in Table 4. It is important to note that, with the particle damping coefficient held constant, the damping effect between particles is further influenced by factors such as particle size, size uniformity, and elastic modulus. These factors are closely related to the grain size distribution of clay minerals, organic matter, and other components in natural rocks. To examine the impact of grain size uniformity on the dispersion and attenuation of elastic waves, three models with varying grain size distributions were constructed (Fig. 6). To facilitate the description of particle uniformity, the uniformity coefficient and curvature coefficient ⁴⁵ are introduced. These two parameters are crucial for characterizing particle gradation. A larger and a smaller indicate a more uneven particle size distribution. The characterization method is as follows:

10

Table 4.

Table of numerical parameters for different model damping values.

Group model	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6
The first set: tangential damping varies	0	0	0	0	0	–
	0.04	0.08	0.2	0.4	0.6	–
The Second set: normal damping varies	0.04	0.08	0.2	0.4	0.6	–
	0	0	0	0	0	–
The third set: Normal & tangential damping co-vary	0	0.04	0.08	0.2	0.4	0.6
	0	0.04	0.08	0.2	0.4	0.6

Fig. 6.

Three numerical models of different grading curves.

Among them, represents the 10% particle size that is smaller than on the gradation distribution curve; similarly for and .

Results

Experimental result

To verify the reliability and accuracy of the low-frequency test system, the equipment was calibrated using a standard aluminum block and acrylic Plexiglas sample before the experiment. Figure 7 shows the Young’s modulus and Poisson’s ratio of the standard sample measured over the frequency range of 0–200 Hz. Error bars indicate the standard deviation of five repeated measurements performed on the same sample at each frequency, reflecting the measurement precision. The Young’s modulus and Poisson’s ratio of the standard aluminum block were measured to be 71.3 GPa and 0.33, respectively. The Young’s modulus of acrylic Plexiglas increased with frequency, while the Poisson’s ratio showed no significant change.

Fig. 7.

Frequency tolerance of standard sample: (a) Young’s modulus, (b) Poisson’s ratio.

Figure 8 shows the results of elastic parameters measured by forced vibration test of coal samples and 3D printed samples under dry conditions (25 °C and 0.1 MPa). There are obvious differences in Young’s modulus and Poisson’s ratio between different samples. In the test frequency range of 1–250 Hz (Fig. 8a), the Young’s modulus E of high-rank coal samples increased from 4.57 GPa to 4.72 GPa, an increase of about 3.3%; The Young’s modulus E of low-rank coal samples increased from 3.05 GPa to 3.27 GPa, an increase of about 7.2%; The Young’s modulus E of photosensitive resin 3D printing RGD525 increased from 2.05 GPa to 2.32 GPa, an increase of about 13.2%; The Young’s modulus E of melt deposited 3D printing PLA increased from 2.61 GPa to 2.85 GPa, an increase of about 9.2%. The Young’s modulus of all samples showed an increasing trend with the increase of frequency, showing a certain frequency dependence. Poisson’s ratio has no obvious change in the test frequency range, and the frequency dependence is weak (Fig. 8b).

Fig. 8.

Frequency tolerance of the sample under normal temperature and pressure: (a) Young’s modulus, (b) Poisson’s ratio test value.

Simulation results

Model 1 and Model 6 from the third group in Table 4 were selected to test the stress-strain response of the sample under damping and undamped conditions, respectively. Under the undamped condition, the phases of axial stress-strain and radial strain remain consistent, showing a linear relationship, which indicates no energy loss at this stage. However, when damping is introduced, a significant phase difference emerges between stress and strain, resulting in hysteresis in the stress-strain curve (Fig. 9a). During a cyclic loading and unloading process, the stress-strain curve forms a hysteresis loop, and the area of this loop represents the energy loss in one cycle. As the damping coefficient increases, the area of the hysteresis loop also grows, indicating an increase in energy dissipation (Fig. 9b).

Fig. 9.

Stress-strain response of damping effect: (a) damping coefficient is equal to 0.6 N^.s, (b) hysteresis loop under different damping conditions.

Based on the numerical model established in Table 4, Figure 10 simulates the influence of normal damping c_dn and tangential damping c_ds parameters on the dispersion and attenuation characteristics of rock longitudinal waves. Firstly, when the normal damping coefficient c_dn is fixed at zero, the P-wave velocity of the whole rock decreases with the increase of the tangential damping coefficient c_ds. With the increase of elastic wave frequency, the longitudinal wave velocity has a slight upward trend, showing dispersion (Fig. 10a). Within the study frequency range, with the increase of tangential damping coefficient, the inverse quality factor Q^-1 also increases correspondingly, and shows a significant upward trend with the increase of frequency (Fig. 10b). Secondly, when the tangential damping coefficient c_ds is fixed at zero and the normal damping coefficient c_dn changes, the dispersion and attenuation trends are the same as when the tangential damping is changed alone (Fig. 10c, Fig. 10d). However, comparing Figure 10b and Figure 10d, it can be found that when the normal damping and tangential damping are equal, the attenuation caused by normal damping is less than that caused by tangential damping, and the corresponding longitudinal wave velocity is relatively large as a whole. The analysis shows that the attenuation effect of tangential damping is about 3–4 times that of normal damping. Finally, when the damping coefficients in the normal and tangential directions are equal and vary simultaneously, the dispersion and attenuation responses follow the same trend as when damping in a single direction is adjusted independently. However, the P-wave velocity will be smaller and the inverse quality factor will be larger than that when changing the same damping in one direction alone (Fig. 10e and Fig. 10f). Notably, when both the normal and tangential damping coefficients of the rock are zero, the longitudinal wave velocity reaches its maximum value, with no significant dispersion or attenuation.

Fig. 10.

Influence of normal and tangential damping parameters on rock longitudinal wave dispersion and attenuation characteristics: (a) only tangential damping changes longitudinal wave dispersion, (b) only tangential damping changes longitudinal wave attenuation, (c) only normal damping changes longitudinal wave dispersion, (d) only normal damping changes longitudinal wave attenuation, (e) both normal and tangential damping changes longitudinal wave dispersion, and (f) both normal and tangential damping changes longitudinal wave attenuation.

Figure 11 shows the influence of rock particles with a different uniformity coefficient C_u numerical model (Fig. 6) on longitudinal wave dispersion and attenuation. It can be observed that as the uniformity coefficient increases C_u, the P-wave velocity decreases. In the low-frequency range, the variation in P-wave velocity is minimal, and the dispersion curve remains nearly horizontal. As frequency increases, the P-wave velocity gradually rises, with a greater uniformity coefficient corresponding to a more significant increase in P-wave velocity (Fig. 11a). Similarly, regarding attenuation characteristics, a higher uniformity coefficient C_u corresponds to a higher inverse quality factor Q^-1. As the elastic wave frequency increases, the inverse quality factor Q^-1 also exhibits an upward trend (Fig. 11b).

Fig. 11.

Influence of uniformity coefficient on P-wave dispersion and attenuation of rock model: (a) P-wave dispersion, (b) P-wave attenuation.

Discussion

Comparison between simulation results and experimental results

The damping effects are influenced by the density, particle size, and uniformity of the constituent materials in different samples⁴⁶. In this study, the PFC3D numerical simulation selects parameters, such as particle elastic modulus and damping. These are based on a collaborative calibration method that combines low-frequency experimental data with existing literature. First, low-frequency stress-strain tests were conducted on coal samples and 3D-printed models under dry conditions. The tests were performed to obtain frequency-dependent elastic modulus, Poisson’s ratio, P-wave velocity, and attenuation coefficients. A cylindrical particle model, geometrically consistent with the experimental samples, was established in PFC3D. The particle-to-particle contact behavior was described by the LPBM method. The particle elastic modulus was estimated using the Voigt-Reuss-Hill average method. This method takes into account the mineral composition and organic content of the coal samples. The damping coefficient was selected based on Li et al. (2021)⁴¹, who suggested a numerical damping value of 0.07 for coal, with values for c_dn and c_ds chosen based on the simulation results. The value of C_u was qualitatively characterized from microscopic structural images. By comparing the simulated and experimentally measured P-wave velocity and attenuation coefficients (Fig. 13), the damping and uniformity coefficients were iteratively adjusted. The trends matched, ultimately determining the calibrated parameters shown in Table 5.

Fig. 13.

PFC numerical simulation and sample measured longitudinal wave dispersion and attenuation: (a) PFC numerical simulation and measured results longitudinal wave dispersion fitting, (b) PFC numerical simulation and measured results longitudinal wave attenuation fitting.

Table 5.

Petrophysical parameters of the sample particle numerical model.

Smple	E(GPa)	G(GPa)	μ	cdn	cds	Cu
H_rank coal	6.32	2.32	0.13	0.022	0.071	1.28
L_rank coal	5.45	2.04	0.14	0.024	0.091	1.35
RGD print	3.15	1.34	0.18	0.045	0.175	1.00
PLA print	4.64	1.69	0.16	0.030	0.096	1.00

High-rank coal samples were selected to test the response of the damping effect. The focus was on elastic wave velocity. The sinusoidal stress of 20 Hz is taken as the loading frequency, and the sampling frequency is set as 2×10⁷ Hz. The velocity field along the Z axis of the model at different times is obtained (Fig. 12). When the load reaches 1 μs, an obvious velocity concentration band begins to appear in the model. As the loading time increases and reaches 8 μs, the number of velocity concentration bands increases to 4. The vibration gradually forms strong and weak velocity bands in the rock. The stress wave from the lower surface of the sample to the upper surface lasted 26.5 μs. The calculated longitudinal wave velocity of the current numerical rock sample was 2264 m/s, consistent with the experimental results of high-grade coal samples.

Fig. 12.

Internal z-axis velocity field of a numerical rock sample at different times.

Figure 13 presents a comparison of dispersion and attenuation between the measured values and the numerical simulation results for the sample. The numerical simulation results for P-wave velocity in all samples show a slight increase with frequency, and the dispersion characteristics closely align with the measured values (Fig. 13a). Although there is some deviation between the attenuation values from the numerical simulation and the measured results, the simulation still effectively captures the attenuation trend of the dry samples (Fig. 13b).

The discrete element numerical simulation results based on the LPBM exhibit certain discrepancies with the attenuation data obtained from low-frequency experimental measurements (Fig. 13). These discrepancies are not incidental. Instead, they reflect inherent limitations in our current numerical model’s ability to capture the complex dissipative mechanisms of dry coal—limitations with clear physical origins. Firstly, the idealization of particle geometry in the model is a key factor contributing to the potential underestimation of attenuation⁴⁷. In this study, the discrete element model simplifies coal skeleton particles to spherical shapes. This assumption greatly simplifies contact detection and force chain calculations. However, it also disregards the complex, angular morphology of real coal particles. In actual coal, the interlocking effects between irregular particles generate additional contact points and unique force transmission paths, leading to more significant energy dissipation processes such as frictional sliding and particle rotation. The spherical particle model struggles to fully capture these damping mechanisms enhanced by morphological irregularities, and thus, the simulated attenuation, particularly in loosely packed coals where morphological effects are more pronounced, may fall short of the real values. Recent studies also suggest that introducing non-spherical particles into the discrete element framework can more accurately simulate the macro- and micro-mechanical responses of granular media, providing a clear direction for future improvements⁴⁸. Secondly, the simplification of the material’s inherent viscoelasticity in the model may overlook the constitutive dissipation mechanisms⁴⁹. The LPBM used in this study primarily simulates energy dissipation due to relative particle motion by setting damping coefficients at contact points, which essentially captures contact frictional damping. However, coal is a complex, organic-rich medium. Under low-frequency loading, it may exhibit inherent viscoelastic behavior similar to that of polymers. This frequency-dependent constitutive damping mechanism is independent of and superimposed on the particle contact friction. Current particle contact models do not explicitly incorporate such bulk viscoelastic constitutive relationships. Consequently, these models may not fully capture how the inherent properties of the coal matrix itself contribute to attenuation. Integrating macroscopic viscoelastic constitutive models with the discrete element framework is becoming an emerging research topic in rock physics modeling⁵⁰. Finally, the model’s inadequate representation of the coal’s multi-scale heterogeneous structure may overlook the contribution of scattering attenuation⁵¹. The numerical model used in this study approximates the coal’s particle grading by controlling particle size distribution and uniformity coefficients, which effectively reflects the granular nature of the medium. However, natural coal also exhibits more complex multi-scale structural heterogeneity, such as micron-scale cellular pores, mineral inclusion bands, and microcrack networks (Fig. 15). When the wavelength of elastic waves is comparable to the characteristic scales of these heterogeneous structures, significant scattering attenuation occurs. Although the current homogeneous particle packing model incorporates a certain degree of particle size distribution, it does not explicitly construct structures with specific geometries and spatial distributions, thus potentially failing to fully account for the scattering mechanisms’ contribution to overall attenuation.

Fig. 15.

SEM micrographs of coal samples: (a) low-rank coal, (b) high-rank coal.

Comparison between the classical phenomenological model and experimental results

The Zener model integrates the Maxwell and Kelvin-Voigt models. This integration addresses a key limitation of the Maxwell model in describing creep, and of the Kelvin-Voigt model in capturing stress relaxation⁵². Exhibiting spring-like elastic behavior under both low-frequency and high-frequency conditions, the Zener model is widely used to phenomenologically describe the dynamic mechanical behavior of viscoelastic media⁵³. The expression for its complex modulus is given by

11

Among them, represents the angular frequency, represents the characteristic frequency, and represents the elastic moduli at the low-frequency and high-frequency limits, respectively.

The characteristic frequency influences the relaxation time of the medium. Using the measured velocity of the sample under in-situ pressure as a benchmark, we compare the model results and determine the characteristic frequency that most closely aligns with the experimental data. Additionally, the modulus under the lowest frequency condition of 4 Hz in this experiment is considered the elastic modulus at the low-frequency limit, while the modulus under ultrasonic conditions at a frequency of 2.5 MHz is regarded as the elastic modulus at the high-frequency limit. The dispersion attenuation curve described by the Zener model was obtained (Fig. 14). Within the 1–250 Hz frequency range, it is evident that the Zener model accurately captures the dispersion of the sample (Fig. 14a), whereas the standard linear bulk model fails to reproduce the measured attenuation data, regardless of how the relaxation time is adjusted (Fig. 14b). In dry coal samples, the complex interplay between material composition and pore structure leads to a nonlinear stress-strain relationship. This complexity makes it challenging for the Zener model to accurately capture the mechanical behavior⁵⁴. DEM characterizes energy dissipation in samples. It does so by representing the nonlinear relationship between discrete inter-particle contact forces and relative displacements, which arise from microscopic frictional sliding and particle dislocation. This approach provides a reliable explanation for the fluctuation response of dry coal samples.

Fig. 14.

Comparison of theoretical values from the Zener model with measured longitudinal wave dispersion and attenuation of the sample: (a) Fitting of theoretical values from the Zener model with measured longitudinal wave dispersion, (b) Fitting of theoretical values from the Zener model with measured longitudinal wave attenuation.

This study assumes that all coal particles are spherical. Although numerical models for different particle size distributions have been developed, the actual shapes of coal particles are diverse. Therefore, constructing a numerical model to characterize non-spherical particles and studying their impact on dispersion and attenuation represents the next direction for future research.

Frequency dependence of dispersion and attenuation of dry coal

Numerical simulations show that three factors collectively determine wave propagation and energy dissipation: the tangential and normal damping of particles, and particle gradation. As the damping coefficient increases, the collisions and friction between medium particles intensify, leading to a decrease in the overall longitudinal wave velocity of the rock and greater attenuation (Fig. 10). More uneven particle size distributions lead to more complex elastic wave propagation paths. Consequently, dispersion and attenuation phenomena become more pronounced. (Fig. 11). This is highly consistent with the research findings of Wu et al⁵⁵.

In the early stages of coalification, coal’s molecular structure is characterized by long alkyl side chains. These chains contain oxygen-containing functional groups. Thin-section microscopy reveals that low-rank coal has a relatively loose particle structure and an uneven particle size distribution. (Fig. 2b). These structures function as dampers⁵⁶. As the frequency of elastic waves increases, the vibration speed of particles within the structure accelerates, leading to intensified friction and collisions between particles. Furthermore, a large number of proto-compliant pores with significant size variations, such as cell cavities and intercellular pores, are present (Fig. 15a). This makes scattering phenomena more likely to occur during the propagation of elastic waves³¹, thereby affecting their frequency-dependent dispersion and attenuation. In contrast, high-metamorphic coal, due to intense coalification and compaction, experiences the closure of primary fractures, leaving only a small number of micropores formed by the contraction of organic matter (Fig. 15b). Its molecular structure is primarily composed of tightly and orderly arranged aromatic clusters, with continued carbonization of organic matter³⁷. The particles are tightly packed and more uniformly distributed (Fig. 2a), resulting in a lower damping effect compared to lower-rank coal. As a result, the scattering effect is greatly suppressed, and the energy loss caused by particle collisions and sliding is relatively low, leading to a relatively low degree of dispersion and attenuation in high-rank coal (Fig. 13), with weaker frequency dependence.

Frequency dependence of dispersion and attenuation of the 3D printing model

To assess the suitability of different materials and printing methods for studying coal dispersion and attenuation, and to provide a foundation for the physical simulation of coal with varying structural characteristics, this study compares two materials: the photosensitive resin RGD525 and PLA. The samples were fabricated using two 3D printing processes: photopolymerization printing (with a precision of 16 microns) and fused deposition modeling (with a precision of 0.1 millimeters). As an additive manufacturing technology, 3D printing creates objects by depositing materials in successive layers. During the forming process, it typically generates about 5% inherent porosity⁵⁷. These characteristics resemble, to some extent, the diagenetic processes of sedimentary rocks in nature. Therefore, by adjusting the size and distribution of particles in each layer, the characteristics of rock particle size distribution and grain composition can be simulated. Research indicates that the photosensitive resin printing model is a highly cemented polymeric structure, making it a typical viscoelastic medium⁵². The photosensitive resin (RGD525) model, fabricated via photopolymerization printing, exhibits a dense structure with minimal and uniform porosity, demonstrating a high Poisson’s ratio and near-incompressible rubber-like viscoelastic characteristics⁵⁸. It has a high compressional wave velocity, but under dynamic loading, it exhibits volumetric deformation hysteresis, with a low shear wave velocity reflecting strong shear damping characteristics. Intense friction occurs within its microstructure, significantly inhibiting relative sliding between particles or segments, converting strain energy into heat through internal friction. Low-frequency shear deformation induces a pronounced viscoelastic response, resulting in a strong frequency dependence of wave velocity and attenuation, with a notable energy dissipation effect (Fig. 10). In contrast, the PLA model, fabricated via fused deposition modeling, exhibits some heterogeneity, more closely resembling a particle-accumulated structure, with a lower Poisson’s ratio and a more rigid structure⁵⁹. It displays solid-like elastic characteristics with lower internal friction damping, weaker shear damping, higher shear wave velocity, and minimal tangential damping effects. At low frequencies, the PLA model behaves similarly to a linear elastic body, leading to weaker dispersion and attenuation phenomena (Fig. 13). This suggests that PLA, as a printing material using fused deposition modeling, is better suited for simulating the dispersion and attenuation of coal.

The internal structure of natural coal is highly diverse. The aim of this study is not to precisely replicate natural coal but to utilize this controllable ideal model to qualitatively investigate the energy dissipation mechanisms of different materials. This approach provides a physical simulation foundation and mechanistic insights for understanding the complex behavior of natural coal. Future research could further explore the potential of 3D printing technology in rock physics simulations, developing models that more closely replicate the natural structures of coal and rock. This would provide strong support for studying dispersion and attenuation phenomena under the coupling of multiple physical fields, such as fluid, temperature, and pressure.

Conclusion

To reveal the mechanism of relative collisions between dry coal skeleton matrix particles, this paper constructs a discrete element numerical model of rock with damping particles and varying particle gradations, based on the theory of rock particle damping effects. Using the PFC3D numerical simulation software, the relative collisions of granular media were simulated through frictional damping, exploring the energy dissipation and dispersion phenomena associated with granular damping effects in the low-frequency range. Key elastic parameters, such as Young’s modulus and Poisson’s ratio, for coal samples and 3D-printed physical models were accurately measured using a low-frequency measurement system based on the stress-strain method. The micro-mechanical behavior of coal was then simulated using the LPBM, leading to the following conclusions.

(1) The damping effect of skeleton particles is the primary factor responsible for the dispersion and attenuation phenomena in dry rocks. The magnitude of damping and particle gradation in the medium collectively influence the propagation characteristics and energy dissipation mechanisms of elastic waves. The contribution of tangential damping to energy dissipation is approximately 3–4 times greater than that of normal damping, and particle gradation also plays a significant role in shaping the dispersion and attenuation characteristics.

(2) The dispersion and attenuation phenomena of elastic waves in dry coal and 3D-printed physical models show significant frequency dependence. Differences in the microstructure of high-rank and low-rank coal particles lead to stronger dispersion and attenuation characteristics in low-rank coal. The fused deposition 3D-printed PLA model demonstrates weaker dispersion and attenuation compared to the photosensitive resin RGD525 model.

(3) Experimental and simulation results indicate that by selecting appropriate parameters such as elastic modulus, damping, and particle gradation, the linear parallel bonded contact model effectively captures the dispersion and attenuation trends of the samples.

Author contributions

In this study, Hao Chen performed the experiments, analyzed the data, and authored the manuscript. Guangui Zou provided guidance and reviewed the manuscript. Xiaolei Feng funded the experimental project. Suping Peng directed the study. Zhu Gao assisted with the derivation and verification of the formulas. Jianhua Wang contributed to the experimental work.

Funding

This research was partially supported by the National Natural Science Foundation of China (Grant No. 42274165), Guizhou Provincial Science and Technology Program ([2023]131), and Guizhou Provincial Department of Education University Scientific and Technological Innovation Team ([2023]092).

Data availability

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.