The Young’s Modulus and Poisson’s Ratio of Hard Coals in Laboratory Tests

Abstract

The Young’s modulus and Poisson’s ratio, parameters reflecting the elastic response of a rock to stress, are the key parameters used in many engineering activities, such as hard coal mining and natural gas extraction. The objective of this paper was to present the results of complex laboratory measurements of the static and dynamic Young’s modulus and Poisson’s ratio for Upper Carboniferous hard coals from the Upper Silesian Coal Basin. The coals differed in geologic age (Mudstone Series—younger coals; Upper Silesian Sandstone Series—older coals) and petrographic structure (vitrain, clarain, and durain lithotype). Elastic parameters of the coals were determined following compression tests under a complex state of stress, as well as using the ultrasonic method in reservoir conditions. On this basis, linear functional dependences between parameters such as UCS, differential stress, confining pressure, strain rate, P- and S-wave velocities, and the static and dynamic Young’s modulus and Poisson’s ratio were determined. These dependences turned out to be linear, with strong and very strong correlation, as indicated by the high coefficients of determination, R². These new results significantly broaden the knowledge of elastic properties of Carboniferous hard coals, especially in the field of geoengineering, underground coal gasification, and reservoir stimulation for coal bed methane extraction.

1. Introduction

Mechanical-related issues of various rock types have been studied for many years. These studies considered sedimentary, igneous, and metamorphic rocks, whereas the best-recognized study objects in these aspects are oil and gas sedimentary reservoir rocks (sandstones, mudstones, shales, and carbonates). Ranges of static and dynamic elastic parameters depending on various pressure, temperature, and saturation conditions, as well as relations of dynamic to static parameters, in these kinds of rocks have been widely presented [1,2,3,4,5,6,7,8].

The elastic parameters of a rock, especially the Young’s modulus (E) and Poisson’s ratio (ν), are used in many engineering activities, such as in construction, tunneling, and mining. In hard coal mining, they are used in designing and maintaining safe mining operations, numerically modeling stress–strain relations, and assessing the possibility of hazards occurring in mining excavations, including pillar instability, rock bursts, as well as outbursts of caves and gasses [3,9,10,11,12,13,14]. They are also useful in assessing the rock mass stability during underground coal gasification.

E and ν are also the some of the fundamental parameters used in designing hydraulic fracturing operations. Elastic parameters, as well as magnitudes and directions of principal stresses, determine the dimensions of the fracture [15]. Young’s modulus defines how much energy is required to accomplish the displacement, which is a classic linear elastic fracture mechanics concept. When increasing E, the height and length of the fracture increases, while the width decreases. Rocks with a large Young’s modulus require more energy to displace. In these formations, fractures tend to be relatively narrow, and the rock is referred to as “hard” [16]. Poisson’s ratio, reflecting rock deformation, occurs perpendicular to the deformation induced by the stress. In the reservoir, ν determines a rock’s ability to fail under stress [17]. Both E and ν parameters allow determining susceptibility for fracking, defined by the brittleness index (BI) [17,18,19,20].

Although dynamic and static E and ν parameters relations for most types of sedimentary rocks are well recognized and described, there are few studies on such functional relationships calculated for organic rocks, especially coal. Hard coal exhibits unique characteristics, distinguishing it from other rock types, including matrix- and cleat-type porosity. Cleat system closure makes the bulk permeability highly stress-dependent. The presence of cleat pressure significantly affects the response of coal to mechanical loading [21]. Due to the extremely friable nature of hard coal, sample preparation is significantly more difficult than for other types of rock. Even if the specimen appears to be intact after preparation, the degree of disturbance and its effect on material properties are unknown [21].

The properties of hard coal from the Upper Silesian Coal Basin (USCB), including elastic parameters, have been studied extensively, but almost all of these studies were conducted for the needs of local coal mines, and the results were not published. Some of the few exceptions are the studies performed and published by researchers from the Central Mining Institute (Poland) [3,22]. Attempts to determine dynamic elastic parameters and brittleness index were also performed. However, these studies included a small number of low-quality samples; thus, the results were not sufficient [20].

Due to the fact that the universal equations used to calculate static moduli based on dynamic moduli presented in the literature are not appropriate for hard coal, the authors of this paper sought to investigate this subject. The aim of this paper was to determine the influence of the stress conditions on the elastic parameters of hard coals and to develop functional relationships of this variability, as well as relations between static and dynamic elastic parameters.

2. Materials and Methods

2.1. Specimens

The coal specimens were collected from currently mined hard coal deposits in the Upper Silesian Coal Basin (USCB) in Poland (Figure 1), from longwalls. However, the studied carboniferous coals were of various geologic ages (within the Pennsylvanian–Upper Carboniferous range) (Table 1). Younger coals belonged to the Mudstone Series (Wesphalian A–B), while the older coals belonged to the Upper Silesian Sandstone Series (Namurian B–C). The coal deposits in these Upper Carboniferous cyclothems in the USCB are surrounded by detrital and argillaceous rock. The contribution of coals to the geological sections of both lithostratigraphic members is presented in Figure 2.

Table 1. Stratigraphic classification of Upper Carboniferous formations of the Polish part of the USCB (simplified version, after [23]).

Chronostratigraphic Division		Lithostratigraphic Units/Beds		No. of Seams
Carboniferous	Westphalian	Krakow Sandstone Series	Libiąż	110–119
			Łaziska	201–215
		Mudstone Series	Orzesze	301–326
			Załęże *	327–406
	Namurian	Upper Silesian Sandstone Series	Ruda	407–419
			Saddle *	501–510
			Jejkowice	-
		Paralic Series	Poruba	601–630
			Jaklovec	701–723
			Hrušov	801–848
			Petřkovice	901–915

* Layers from which samples were taken for laboratory tests.

Table 1. Stratigraphic classification of Upper Carboniferous formations of the Polish part of the USCB (simplified version, after [23]).

Chronostratigraphic Division		Lithostratigraphic Units/Beds		No. of Seams
Carboniferous	Westphalian	Krakow Sandstone Series	Libiąż	110–119
			Łaziska	201–215
		Mudstone Series	Orzesze	301–326
			Załęże *	327–406
	Namurian	Upper Silesian Sandstone Series	Ruda	407–419
			Saddle *	501–510
			Jejkowice	-
		Paralic Series	Poruba	601–630
			Jaklovec	701–723
			Hrušov	801–848
			Petřkovice	901–915

* Layers from which samples were taken for laboratory tests.

The coals of the Mudstone Series were represented by three lithotypes: vitrain, clarain, and durain. The tested coals of the Upper Silesian Sandstone Series were vitrain and clarain.

As a result of the specifics of the geological and environmental conditions for the formation of the sedimentary series of the coal-bearing formation in the USCB, the carboniferous rock, including the coals, is heterogeneous and, therefore, difficult to process using a circular saw and a rock drill. Practical experience has demonstrated that it is often difficult to prepare coal specimens according to international recommendations regarding specimen slenderness and, consequently, to ensure specimen representativeness for mechanical parameter determination. This particularly concerns bright coals, which are characterized by luster and fissures perpendicular to the bedding, known as contraction fissures. These have a significant influence on the mechanical parameter levels and often make it impossible to conduct testing. Furthermore, the low thicknesses of strata in the geological section often make it impossible to prepare representative specimens with a slenderness greater than 1. Such a situation occurred in the case of the studied coals, particularly those forming the deposits belonging to the Mudstone Series in the USCB. The uniaxial compressive strength of coals from this geological formation amounted to a compressive strength of several megapascals. With regard to the above, in order to guarantee result comparability, the experiments were conducted on regular cubic specimens with a base edge of 50 mm or cylindrical specimens with a diameter of 50 mm and slenderness of 1.0 or 2.0.

By applying an empirical factor of 0.89, determined for the carboniferous rock in the USCB, the maximum stress obtained for a slenderness of 1 was referenced to a slenderness of at least 2, as per the recommendations of the ISRM.

The static testing was carried out on 131 regular laboratory specimens in cylindrical or cubic shapes. The dynamic testing was carried out on 45 core specimens, 38.1 mm (1.5 inches) in diameter. The samples were cut out perpendicular to the lamination (Figure 3).

2.2. Experimental Program

The laboratory testing of the Mudstone Series coals and the geologically older Upper Silesian Sandstone Series coals was conducted under the conditions of static loading in a strength testing machine, as well as using the ultrasonic method in reservoir conditions. Static testing of the Young’s modulus (E_st) and Poisson’s ratio (ν_st), as well as of the maximum stress (compressive strength), was conducted in servo-operated testing machines MTS-810 and MTS-815. The coals were tested under the conditions of a complex state of stress, where σ₁ > 0 and σ₂ = σ₃ according to the classic individual test type [24]. The range of values applied in the confining pressure experiments was 0–50 MPa.

A special case of a complex state of stress was included in experiments conducted under the stress conditions of σ₁ > 0 and σ₂ = σ₃ = 0. This is a case of uniaxial compression, where a graphical visualization of the state of stress is represented by a single Mohr’s circle, tangential to the shear stress axis. The experiments utilized several strain rate values, from low rates characteristic of rock deformation in the working environment to strain rates characteristic of geodynamic phenomena, e.g., rock bursts, which may occur in the mined rock mass [25]. Furthermore, volumetric density values were determined using AccuPyc II 1340 and GeoPyc 1365 analyzers, as per the method described in standard PN-EN 1936:2010. The ultrasonic testing consisted of determining the velocity of compressional (P) and two shear (S) ultrasonic waves (polarized at the angle of 90°) in core specimens, in a complex state of stress, as well as in temperature and pressure conditions similar to those presented in the reservoir at an adopted depth. After stabilizing the temperature and pressure in a specimen placed in the chamber of the AVS 700 apparatus, an ultrasonic signal with a frequency of 500 MHz was generated. The time of flight of the P- and S-waves were used for calculating the velocities of these waves. Then, on the basis of the velocities, the dynamic Young’s modulus (E_dyn) and Poisson’s ratio (ν_dyn) were determined using formulas widely available in the literature (e.g., [26]). The detailed methodology of both conducted rock mechanics laboratory tests are presented in Table 2. This table also provides references to the scientific literature that describes the primary test-related issues, international recommendations, standards, and instructions.

Table 2. Detailed methodology for testing mechanical parameters of coal.

Parameter	Methodology
Applied stress condition: σ1 > 0, σ2 = σ3 Differential stress	Young’s modulus and stress parameters were determined in conventional triaxial compression tests, using a Karman pressure chamber and confining pressures of σ2 = σ3 = 10, 20, 30, and 50 MPa, strain rates of 5 × 10−5 to 10−4 s−1 and 10−1 s−1, and cylindrical specimens with a diameter of 30 mm and a slenderness of 3 [27].
Applied stress condition: σ1 > 0, σ2 = σ3 = 0 Uniaxial compressive strength Static Young’s modulus Static Poisson’s ratio	The specimens were tested in air-dry conditions. Specimen shape: cube with a base edge of 50 mm or cylinder with a diameter of 50 mm. Slenderness of the specimens: 1.0. An empirical factor of 0.89 was used to account for a specimen slenderness of over 2.0, as recommended by the ISRM. Load direction: perpendicular to the lamination. Piston rate: about 0.008 mm/s, i.e., the strain rate of rock in the area of mining excavations. Young’s modulus was determined over the entire height of the compressed rock specimen as a tangent of the inclination angle of the tangent to the x-axis, which is a linear approximation of the stress and strain characteristic. Poisson’s ratio was determined within the longitudinal elastic strains and using a roller chain with a sensor for recording and measuring circumferential strains [3,7,22,25,27].
Dynamic elastic moduli and conditions of testing	Confining pressure, 21.1 MPa; pore pressure, 8.8 MPa; effective pressure, 12.3 MPa; temperature, 37 °C. Core specimens with a diameter of 1.5 inches; a length depending on the condition of individual specimens was cut out perpendicular to the lamination. Before testing, specimens were saturated with a 2% potassium chloride solution in a vacuum chamber, at a pressure of 1 bar, for at least 12 h, before being additionally saturated with the same solution at a backpressure of at least 500 psi, into the core holder of the ultrasonic device [26,28].

Table 2. Detailed methodology for testing mechanical parameters of coal.

Parameter	Methodology
Applied stress condition: σ1 > 0, σ2 = σ3 Differential stress	Young’s modulus and stress parameters were determined in conventional triaxial compression tests, using a Karman pressure chamber and confining pressures of σ2 = σ3 = 10, 20, 30, and 50 MPa, strain rates of 5 × 10−5 to 10−4 s−1 and 10−1 s−1, and cylindrical specimens with a diameter of 30 mm and a slenderness of 3 [27].
Applied stress condition: σ1 > 0, σ2 = σ3 = 0 Uniaxial compressive strength Static Young’s modulus Static Poisson’s ratio	The specimens were tested in air-dry conditions. Specimen shape: cube with a base edge of 50 mm or cylinder with a diameter of 50 mm. Slenderness of the specimens: 1.0. An empirical factor of 0.89 was used to account for a specimen slenderness of over 2.0, as recommended by the ISRM. Load direction: perpendicular to the lamination. Piston rate: about 0.008 mm/s, i.e., the strain rate of rock in the area of mining excavations. Young’s modulus was determined over the entire height of the compressed rock specimen as a tangent of the inclination angle of the tangent to the x-axis, which is a linear approximation of the stress and strain characteristic. Poisson’s ratio was determined within the longitudinal elastic strains and using a roller chain with a sensor for recording and measuring circumferential strains [3,7,22,25,27].
Dynamic elastic moduli and conditions of testing	Confining pressure, 21.1 MPa; pore pressure, 8.8 MPa; effective pressure, 12.3 MPa; temperature, 37 °C. Core specimens with a diameter of 1.5 inches; a length depending on the condition of individual specimens was cut out perpendicular to the lamination. Before testing, specimens were saturated with a 2% potassium chloride solution in a vacuum chamber, at a pressure of 1 bar, for at least 12 h, before being additionally saturated with the same solution at a backpressure of at least 500 psi, into the core holder of the ultrasonic device [26,28].

3. Results and Discussion

The laboratory tests conducted according to the conventional triaxial and uniaxial compression stress conditions served to determine the following mechanical parameters of hard coals:

• Maximum vertical stress, σ₁ (uniaxial compressive strength, UCS), and differential stress, σ₁–σ₃;
• Young’s modulus: static, E_st, and dynamic, E_dyn;
• Poisson’s ratio: static ν_st and dynamic ν_dyn;
• Bulk density, ρ_o;
• Total porosity, P.

The test results of hard coal mechanical parameters determined on the basis of the mentioned laboratory specimens are presented in

Table 3. Mean physical parameter values of coal specimens: static testing.

Number of Samples	UCS		Est		νst
	Mean Values MPa	SD MPa	Mean Values MPa	SD MPa	Mean Values -	SD -
4	11.2	2.2	1069	327	0.25	0.06
4	5.9	1.3	613	175	0.28	0.04
4	6.3	2.1	638	303	*	-
3	6.6	0.1	836	16	*	-
4	14.2	1.7	1585	151	0.27	0.05
3	11.1	2.9	1120	227	0.24	0.03
1	7.8	-	768	-	*	-
1	5.5	-	536	-	*	-
3	10.5	1.5	1239	246	*	-
4	6.8	2.4	606	331	*	-
3	7.1	5.8	814	766	0.22	0.06
3	8.9	0.7	601	159	0.26	0.06
3	9.0	2.1	942	309	0.32	0.08
2	4.7	1.4	573	276	0.34	0.08
2	4.6	0.6	389	75	0.29	0.09
4	8.0	2.9	945	372	0.22	0.10
4	7.1	2.6	784	421	0.32	0.05
4	8.5	1.2	1031	230	0.29	0.02
2	8.1	2.1	900	324	0.38	0.01
2	5.5	2.8	654	542	0.29	0.04
4	6.3	0.8	619	248	0.28	0.05
4	4.5	2.1	503	127	0.31	0.08
4	4.5	3.0	673	479	0.27	0.10
4	7.4	2.8	948	308	0.25	0.08
3	5.7	0.6	656	68	0.16	0.02
2	4.8	1.4	523	146	0.16	0.06
2	11.0	2.4	1033	394	0.28	0.06
2	9.2	0.2	1066	155	0.39	0.06
4	7.5	2.9	734	470	0.27	0.11
4	26.6	4.4	2399	382	0.30	0.05
3	19.1	1.9	1751	151	0.37	0.02
2	23.2	0.8	1903	96	0.24	0.12
3	15.2	1.1	1233	178	0.23	0.04
3	8.8	2.1	765	251	0.37	0.02
2	10.9	0.4	1061	404	0.24	0.12
3	26.3	5.7	2073	402	0.36	0.05
3	27.8	7.7	2371	340	0.31	0.05
4	28.2	3.0	2312	348	0.30	0.08
3	24.4	2.8	2372	149	0.33	0.02
3	21.1	3.5	2026	367	0.34	0.08
2	23.7	11.1	2234	612	0.30	0.08
4	16.8	2.7	1794	220	0.20	0.08
3	17.6	0.9	1872	125	0.24	0.12

* Weak coal structure, impossible to test.

and

Table 4. Results of ultrasonic tests of the coal samples.

Number of Samples	VP m/s	VS m/s	VP/VS -	Edyn GPa	νdyn -
1	2409	1200	2.01	5.21	0.34
1	2399	1259	1.91	5.63	0.31
1	2435	1250	1.95	5.41	0.32
1	2392	959	2.49	3.37	0.40
1	2488	1299	1.92	6.15	0.31
1	2421	1249	1.94	5.71	0.32
1	2441	1104	2.21	4.36	0.37
1	2441	1271	1.92	5.50	0.31
1	2481	1283	1.93	5.61	0.32
1	2425	1154	2.10	4.75	0.35
1	2098	1034	2.03	3.74	0.34
1	2389	1273	1.88	5.56	0.30
1	2441	1509	1.62	7.09	0.19
1	2504	1377	1.82	6.33	0.28
1	2530	1537	1.65	7.77	0.21
1	2491	1501	1.66	7.24	0.21
1	1982	1150	1.72	4.34	0.25
1	2500	1175	2.09	5.50	0.35
1	2446	1289	1.90	5.89	0.31
1	2457	1416	1.73	6.52	0.25
1	2503	1332	1.88	6.12	0.30
1	2533	1364	1.86	6.41	0.30
1	2472	1130	2.19	4.52	0.37

.

The tested coal specimens from the Mudstone Series were primarily represented by vitrain and clarain, and secondarily represented by durain. The vitrain specimens exhibited clear fissures perpendicular to the lamination, sometimes filled with carbonates, which is characteristic of this lithotype. The tested coals of the Upper Silesian Sandstone Series included vitrain and clarain. Figure 4A,B present example stress–strain charts obtained during the recorded course of static loading of selected regular coal specimens over the full course of deformation (pre-critical loading stage and post-critical deformation stage) until the achievement of a stable (residual) stress as a result of the applied load transferred by the testing machine plate.

The variability thresholds of the (mean) strength, elastic, and deformation parameters of coals sampled from mine workings, determined on the basis of laboratory testing, are presented in

Table 5. Variability thresholds of the (mean) mechanical parameters of coal: static testing.

Lithostratigraphic Series	Coal Lithotype	UCS stress Condition σ1 > 0, σ2 = σ3 = 0 MPa	Static Young’s Modulus MPa	Static Poisson’s Ratio -	Bulk Density kg/m3
Mudstone Series	Vitrain	4.5–11.2	389–1069	0.16–0.34	1112–1376
	Clarain	5.5–11.1	601–1239	0.24–0.39	1220–1302
	Durain/vitrain	8.0–14.2	942–1585	0.22–0.32	1229–1366
Upper Silesian Sandstone Series	Vitrain	8.8–19.1	765–1751	0.23–0.37	1268–1490
	Clarain	21.1–28.2	1903–2399	0.24–0.36	1262–1384

. A strength assessment of the tested coals was conducted (

Table 6. Coal uniaxial compressive strength assessment.

Coals	Mudstone Series	Upper Silesian Sandstone Series
Vitrain	Very low and low strength	Very low and low strength
Clarain		High strength
Durain/vitrain		Not present in the rock mass in the sampling regions

) according to a classification formulated for sedimentary carboniferous rock in the USCB in Poland [3].

For the conditions of the USCB, it is characteristic that coals with very low compressive strength (below 10 MPa) have a contribution of about 13% in the geological section, whereas clarain with a low compressive strength (10–19.9 MPa) has a contribution of about 20%. The clarain from the Upper Silesian Sandstone Series was characterized by high compressive strength. Such coals have a contribution of about 32% in the geological section, compared to the other tested coals of various lithostratigraphic groups in the USCB [3] (Figure 5).

The uniaxial compressive strength (stress condition σ₁ > 0, σ₂ = σ₃ = 0) of coals representing the Mudstone Series, as well as the older strata belonging to the Upper Silesian Sandstone Series, had a correlation with the static Young’s modulus. The coals of the Mudstone Series and the Upper Silesian Sandstone Series analyzed separately exhibited high R² values in the functional dependence E_st = f(UCS) of 0.76 and 0.93, respectively (Figure 6A,

Table 7. Linear regression analysis results of E_st = f(UCS) for coals of various geologic age and petrographic structure.

Relationship	R2 Coefficient	Estimated SSE Standard Error
Mudstone Series Est = 94.824UCS + 107.18	0.7623	128.79
Upper Silesian Sandstone Series Est = 79.592UCS + 216.47	0.9278	159.44
Vitrain Est = 80.56UCS + 179.52	0.8842	106.55
Clarain and durain Est = 78.905UCS + 255.48	0.9386	170.55

). The linear dependence between the uniaxial compressive strength and the static Young’s modulus was also clearly visible for individual lithotypes of the studied coal. The R² coefficients for this correlation were 0.88 for vitrain and 0.94 for clarain and durain (Figure 6B, Table 7). The functional dependences determined on the basis of static testing conducted under uniaxial compression (as a special case of a complex state of stress) indicated a better E_st = f(UCS) dependence for geologically older coal and for clarain and durain, compared to vitrain. These dependences can be explained by the significantly lower porosity of geologically older clarain, compared to the porosity of younger vitrain.

According to experiments in the conditions of conventional triaxial compression (stress condition σ₁ > 0, σ₂ = σ₃ ≠ 0), Figure 7 and

Table 8. Linear regression analysis results of E_st = f(σ₁–σ₃).

Relationship	R2 Coefficient	Estimated SSE Standard Error
Est = 11,471(σ1–σ3) + 2417.9 strain rates = 10−5 to 10−4 s−1; confining pressures of 10, 20, 30, and 50 MPa	0.9543	83.78
Est = 5932(σ1–σ3) + 2878.1 strain rates = 10−1 s−1; confining pressures of 10, 20, 30, and 50 MPa	0.8455	106.62

present the static Young’s modulus as a function of differential stress. Laboratory experiments were conducted at confining pressures of 10, 20, 30, and 50 MPa and at various strain rates: 5 × 10⁻⁵ to 10⁻⁴ s⁻¹ (rock strain rates in the working environment), and 10⁻¹ s⁻¹ (rock strain rate representative for characteristic values, e.g., for geodynamic phenomena occurring in the mined rock mass and earthquakes (10⁻² to 10² s⁻¹) [25]).

The linear dependences determined on the basis of tests conducted under the conditions of various confining pressures and strain rates were characterized by a very high correlation between the differential stress and the static Young’s modulus.

The static Poisson’s ratio for the Mudstone Series and the Upper Silesian Sandstone Series coals, as well as the volumetric density and porosity, conformed with the variability thresholds of these parameters determined for USCB coals [3]. However, no dependences were determined between these parameters and Young’s modulus or uniaxial compressive strength. Poisson’s ratio and the volumetric density of hard coal varied within a very narrow threshold (first parameter: up to 0.5; second parameter: fluctuating around 1.25–1.35 g/cm³). Determining functional dependences was impossible due to the low variability. The authors would like to emphasize that scientific publications providing the results of experiments conducted under complex states of stress and at various strain/load rates are rare. The results of such tests are important for explaining the reasons for the occurrence of various geodynamic phenomena in the rock mass, including rock bursts. Therefore, studies such as these may be helpful in the planning of extraction, in terms of the face advance rate selection, in order to limit the rock burst hazards in the workings.

The dynamic Young’s modulus and Poisson’s ratio were determined in laboratory tests under conventional triaxial compression (reservoir conditions) for the coals (

Table 9. Variability thresholds of the mechanical parameters of coal: dynamic testing.

Parameter	SI Unit	Min. Value	Max. Value
Compressional (P) wave velocity	m/s	1982	2533
Shear (S) wave velocity	m/s	959	1537
VP/VS	-	1.62	2.49
Dynamic Young’s modulus	GPa	3.370	7.770
Dynamic Poisson’s ratio	-	0.19	0.40
Porosity	%	1.96	8.72
Bulk density	kg/m3	1290	1390

), similarly to the case of static testing. However, given the very low quality of the majority of coals (numerous fissures, particularly in vitrain) and the harmful influence of post-test transport to successive test stands on the condition of the coal specimens, it was impossible to conduct most static and dynamic tests on the same laboratory specimens. The tests were conducted on successive cylindrical specimens cut out from blocks sampled from the same location in the mine workings.

The tested hard coals of the Mudstone Series were characterized by P- and S-wave velocity within thresholds of 2098–2488 m/s and 959–1299 m/s, respectively. The values of Young’s modulus E_dyn ranged within 3.37 to 6.15 GPa, whereas the range of Poisson’s ratio ν_dyn was 0.30–0.40. The V_P/V_S values were within a range of 1.88–2.49. The porosities for coals of this geologic age were at a level of 3.84–8.72%, whereas the volumetric density was 1290–1390 kg/m³.

The hard coals of the Upper Silesian Sandstone Series were characterized by slightly greater ultrasonic velocities, for both P- and S-wave, but the latter was particularly prominent, with ranges of 1982–2533 m/s and 1130–1537 m/s, respectively. The E_dyn moduli for these coals ranged within 4.34 to 7.77 GPa. The specimens exhibited relatively low V_P/V_S coefficients, from 1.62 to 2.19, as reflected in the ν_dyn ratios, within a range of 0.19–0.37. The porosities for coals of this geologic age were at a level of 1.96–6.18%, whereas the volumetric density was 1300–1360 kg/m³.

Data analysis (Figure 8) revealed the difference between coals belonging to the two different cyclothems. The Mudstone Series coals were characterized by greater porosity, compared to the coals of older geologic age (the Upper Silesian Sandstone Series). Porosity, as an original parameter with direct relation to the formation conditions of the coal-bearing series, proved to be significant for the variability of the dynamic Young’s modulus and the compressional wave propagation velocity.

The laboratory test results allowed determination of the linear dependence between the compressional wave propagation and the dynamic Young’s modulus (Figure 9A) and Poisson’s ratio (Figure 9B). The obtained coefficients of determination indicated a very strong correlation in the case of Young’s modulus and a strong correlation for Poisson’s ratio (

Table 10. Linear regression analysis results of E_dyn = f(V_S, V_P/V_S) and ν_dyn = f(V_S, V_P/V_S).

Relationship	R2 Coefficient	Estimated SSE Standard Error
Edyn = 0.0073VS − 3.637	0.9643	0.2100
Edyn = −4.3609(VP/VS) + 14.017	0.6485	0.6590
νdyn = 0.0003VS + 0.7072	0.7579	0.0270
νdyn = 0.2524 (VP/VS) − 0.1825	0.8990	0.0170

). Furthermore, the test results showed no relationship between P-wave velocity and the dynamic moduli of both investigated hard coals. The lack of dependence between the P-wave velocity and the dynamic Young’s modulus is also characteristic of claystone and limestone [29]. The results of tests conducted on limestone, mudstone, and sandstone by Puskarczyk et al. [30] also confirmed a lack of dependence between the compressional wave velocity and the dynamic Young’s modulus and Poisson’s ratio.

The determined V_P/V_S values for carboniferous coals were within the range of 1.62–2.49 and exhibited strong correlation with Young’s modulus and very strong linear correlation with the dynamic Poisson’s ratio (Figure 10). The regularities indicated above can also be demonstrated for other sedimentary rock (limestone, mudstone, and sandstone), according to data presented in [30].

A comparison of static and dynamic Young’s moduli and Poisson’s ratios determined for hard coal on the basis of a number of regular laboratory specimens is shown in Figure 11. Due to technical complications, the Poisson’s ratios of three samples (marked in Table 3 and Figure 11A) could not be designated.

The results presented above confirm the point voiced most often by researchers that the values of the dynamic moduli are greater than the static ones [5]. This also concerns materials other than rock, e.g., polymer matrix composite materials [6]. Studies have demonstrated that this is visible in rock characterized, e.g., by low elasticity during static testing, due to differences in porosity and the spatial orientation of structural defects, as well as other factors [31].

Literature reports have indicated that the differences in values between the static and dynamic moduli of brittle rock may amount to about 10% [32]. However, depending on the experimental conditions, the dynamic and static modulus quotients may differ. For example, at low loads, V_P/V_S may be close to unity [33]. Other researchers have claimed that the dynamic modulus may be 1.5–3.0 times greater than the static one [8]. When interpreting this contradiction, it is important to state that our tests did not concern sedimentary (detrital, argillaceous, and organochemical), igneous, or metamorphic rock, but an organogenic sedimentary rock, i.e., hard coal. However, there was no functional dependence between the static and dynamic Young’s modulus for the studied hard coal specimens. Studies conducted on small statistical samples by other researchers confirmed this dependence to be weak or lacking (e.g., [4,34]). Other researchers provided nonlinear or linear E_st = f(E_dyn) dependences, with different ones for igneous and metamorphic rock, compared to sedimentary rock [35]. By analyzing the numerous E_st = f(E_dyn) functional dependences determined by various researchers, Davarpanah et al. [35] concluded that linear dependences offer the best fitting to the data used in rock engineering.

The E_st = f(E_dyn) dependence provided in the scientific literature cannot be applied to carboniferous hard coal. To substantiate this argument, the authors of this article calculated the static Young’s modulus using selected formulas concerning sedimentary rock, while also using equations factoring in bulk density, porosity, and UCS [31]. It was demonstrated that the static Young’s modulus values, calculated using the selected formulas, differed significantly relative to the values determined under static specimen loading in the testing machine.

Compared to the aforementioned rock groups, hard coal is characterized by very low uniaxial compressive strength (UCS). The E_dyn/E_st values for the hard coals tested as part of this article were 3–9, whereas the E_dyn values were within 3.37–6.15 GPa, similar to the publication by Wu et al. [36], where the authors achieved an E_dyn value of 3.884–5.959 GPa for anthracite by means of conventional triaxial compression. Morcote et al. [37] presented V_P and V_S values of bituminous coal, from which similar values of E_dyn could be calculated (3.36–5.16 GPa). Our tests also revealed no dependences between the velocities of the compressional and shear wave and the UCS of coals, even though linear dependences were provided by researchers in the literature for sedimentary rock other than hard coal (e.g., [38]).

For a small statistical sample, a weak linear correlation was found between UCS and the dynamic Young’s modulus (R² = 0.05, r = 0.22; SSE = 0.89), in contrast to the E_st = f(UCS) dependence, which, despite the size of the set (n = 8), was characterized by a very strong linear correlation (R² = 0.8959; SSE = 0.09) (Figure 12), similar to that presented in Figure 6A,B for a large statistical sample.

Furthermore, it was demonstrated for the studied carboniferous coals that both the small statistical sample (n = 8) and the sample of greater size (n = 23) exhibited weak linear correlations between porosity and the static and dynamic Young’s moduli and Poisson’s ratios. The coefficients of correlation for the E = f(P) dependence were 0.26–0.28 (SSE = 1.04–1.07). The determined coefficients of correlation for the ν = f(P) dependence were 0.21–0.38 (SSE = 0.03–0.05). Despite the weak correlation between the aforementioned parameters, a tendency for individual parameter value variations coinciding with the increase in porosity can be observed, i.e., a decrease in Young’s modulus and an increase in Poisson’s ratio. This concerns testing under both static and dynamic loading.

4. Conclusions

The authors were interested in mechanical properties of the carboniferous hard coals belonging to different cyclothems deposited in Namurian B and C, as well as younger ones belonging to Westfalian A and B. The tests were conducted in varied stress conditions, taking into account temperature and pressure at the target zone, as well as the deformation velocities, which are characteristic for the rocks surrounding underground excavation. Such complex testing conditions reflect the natural conditions of the rock mass and exploited mining area, bringing new light to the elastic properties of hard coals.

According to the static and dynamic laboratory testing of these hard coals, the following conclusions can be formulated:

• The geologic age of coal, as well as its petrographic structure, had a significant influence on the dependence between uniaxial compressive strength and the static Young’s modulus determined on the basis of the stress and strain curve (R² = 0.76–0.96).
• The functional dependences determined on the basis of tests conducted under a complex state of stress and at various confining pressures and high strain rates were characterized by a very high correlation between the differential stress and the static Young’s modulus.
• A strong and very strong correlation between the S-wave velocity and the elastic parameters (R² = 0.76–0.96) were obtained on the basis of the conducted laboratory ultrasonic tests. However, no correlation was observed between the P-wave velocity and the elastic parameters, which also confirms the results of experiments conducted by other researchers. The determined values of V_P/V_S for carboniferous coals revealed a strong correlation with the dynamic Young’s modulus (R² = 0.65) and a very strong linear dependence with the dynamic Poisson’s ratio (R² = 0.89).
• Testing of the studied carboniferous coals demonstrated a weak linear correlation between porosity and the Young’s modulus and Poisson’s ratio.

In the future, the authors will extend this study, especially in the terms of the relationship between the static and dynamic Young’s moduli and Poisson’s ratios, using a larger statistical sample.