Real-time porosity inversion of rock based on the ultrasonic velocity and its compression-damage coupled model under triaxial compression

Abstract

It is of great significance to evaluate the stress state and mechanical properties of deep rock engineering, by inverting the real-time porosity of rocks. In this study, the triaxial compression tests and real-time ultrasonic velocity tests were conducted on siltstone under different confining pressures. Firstly, a nonlinear model for the evolution of rock ultrasonic velocity with axial strain was proposed, based on the characteristic that the ultrasonic velocity rapidly increases and tends to stabilize. Secondly, a correction coefficient for the ultrasonic velocity of the rock matrix was proposed and the Raymer model was modified. The real-time porosity during triaxial compression of rocks was inverted according to the modified Raymer model and the ultrasonic velocity model. Subsequently, the effective compression coefficient of rocks was calculated based on real-time porosity, and a compaction coefficient was established to describe the compaction effect of cracks and pores. Finally, a new compressive-damage coupled constitutive model for rocks was established, by combining the statistical damage theory and compaction effect.

Introduction

Due to the impenetrability of the Earth’s medium, natural or artificial seismic waves have become a reliable means to study the internal structure, composition, and stress state of the Earth^1,2. Studying the ultrasonic velocity variations of rocks under different pressure conditions is of great significance in determining the composition, structure of deep Earth and evaluating in-situ stress³. In recent years, the exploitation of mineral resources has continuously progressed to deep strata, and the dynamic disaster caused by high stress have become increasingly common⁴. Studying the ultrasonic velocity variations of rocks under different stress levels is an important means to prevent dynamic disasters in rock masses⁵.

The ultrasonic velocity testing is widely used in geotechnical engineering exploration, as a non-destructive monitoring method⁶. On the one hand, the distribution of crack, porosity, and density of rock have a significant impact on the variation of ultrasonic velocity^7,8. On the other hand, the ultrasonic velocity of rocks can characterize many physical parameters of rocks⁹. Makerrji and Dvorkin¹⁰ studied the nonlinear relationship between elastic modulus, density, and ultrasonic velocity of rock. Osman et al.¹¹ calculated the fracture indices of rock mass using ultrasonic velocity, and determined the rock quality specification (RQD) values. The reasons for the differences in ultrasonic velocity among various types of rocks are complex¹². The difference in ultrasonic velocity between the compaction of porous rocks and the local shear fracture of dense rocks is very significant^13,14. Pan et al.¹⁵ found that the formation of corrosive pores inside the rock after acid treatment was the main reason for the significant decrease in ultrasonic velocity. Previous studies have also shown that the porosity of rocks (Simmons)¹⁶ and the irregularity of mineral particles (Senetakis et al.)¹⁷ have the most significant impact on the ultrasonic velocity.

Scholars found an excellent correlation between the ultrasonic velocity and porosity of rocks, and established various models for the relationship between porosity and ultrasonic velocity^18,19. Wyllie²⁰ proposed a classic time averaged model to describe the correlation between ultrasonic velocity and porosity of rock. Raymer²¹ improved the time averaged model and proposed the Raymer-Hunt-Gardner equation. Hamada and Joseph²², and Rahmouni et al.²³ established different methods for estimating porosity. David and Zimmerman^24–26 quantitatively studied the relationship between pore quantity, morphology, and ultrasonic velocity using the equivalent medium theory.

The stress state of rocks in the crust is mainly composed of horizontal structural stress and vertical geostatic stress²⁷. In the present geophysical exploration areas, the ultrasonic velocity variation of rocks under different hydrostatic pressures has been studied²⁸. Birch^29,30 found that the ultrasonic velocity of rocks increases with the increase of hydrostatic confining pressure and then tends to stabilize. Eberhart et al.³¹, Greenfield et al.³², and Wepfer et al.³³ established different functional relationships for hydrostatic pressure and ultrasonic velocity of rock, respectively. However, in the underground rock engineering areas, the stress state of rocks continues to change, such as during mineral resource extraction³⁴. Therefore, a series of ultrasonic velocity tests under uniaxial and multiaxial compression of rocks have been carried out. Zhang et al.³⁵ found that the ultrasonic velocity of rocks shows a three-stage characteristic of nonlinear increase, stabilization, and rapid decrease under triaxial compression. Zhang et al.³⁶ found that the variation of longitudinal ultrasonic velocity is more sensitive than that of transverse ultrasonic velocity under triaxial compression. Nur and Simmons³⁷ found that different biaxial stress combinations can produce the same ultrasonic velocity value. Fortin³⁸ analyzed the reduction of sandstone porosity under triaxial compression through ultrasonic velocity testing. Schulze et al.³⁹ found that the ultrasonic velocity of rocks decreases at the starting point of dilation. Holcomb and Olsson⁴⁰, and Vajdova et al.⁴¹ found that the pore compaction and crack propagation of rocks under stress are the main factors affecting the ultrasonic velocity. Schubnel⁴² revealed the interaction mechanism of the pore compaction and crack propagation by ultrasonic velocity testing. Knippel et al.⁴³ described the nonlinear mechanical behavior of rocks during the yield stage, using ultrasonic velocity measurement results.

The above investigations greatly enriched the theory of rock mechanics and geophysics. However, the research on real-time inversion of porosity during rock compression is not specific enough, and there are few reports on further applying the evolution characteristics of porosity to rock constitutive models. Therefore, in this study, the triaxial compression tests and real-time ultrasonic velocity tests were conducted on siltstone under different confining pressure. Firstly, a nonlinear theoretical model of real-time ultrasonic velocity with axial strain was been proposed. Secondly, the real-time porosity of rocks under different confining pressures was inverted by improving the Raymer formula. Then, the relative effective compressibility coefficient of the rock during the loading process was derived based on porosity. Finally, a new compressive-damage coupled constitutive model for rocks was established by combining statistical damage theory and compaction effect.

Materials and test methodology

Siltstone materials description

The siltstone used in this study obtained from Gansqing Tunnel of Xining-Chengdu Railway, Gansu Province, China. All rock specimens were processed into Φ 50 mm × 100 mm cylinders, as the International Society for Rock Mechanics, and the non-parallelism of the end faces is below ± 0.02 mm, as shown in the Fig. 1a. The selected specimens for testing were dried for 24 h at a temperature 60°. Under the dry conditions, the average density of the siltstone specimen is 2.27 g/cm³, and the P-ultrasonic velocity is in the range of 2250 m/s to 2350 m/s. Figure 1b shows the micrographs of thin sections and microstructure of siltstone, the main components of siltstone are quartz, plagioclase, mica, and clay, the content is 78.6%, 6.7%, 5.4%, and 2.1%, respectively. The porosity of siltstone measured by mercury intrusion method is 19.89%. Figure 1c shows the micro structure of siltstone by SEM technology, the cement content in siltstone is very low, and the particle skeleton is loose. The mineral sizes of siltstone ranges from 1 μm to 104 μm, and the particles with sizes less than 38 μm accounted for 71.67%⁴⁴.

Fig. 1.

The micrographs of thin sections and microstructure of siltstone. (a) The siltstone specimens. (b) The micrographs of thin section. (c) The microstructure of siltstone.

Testing system and process

The uniaxial and triaxial compression tests on siltstone were performed using a servo-controlled rock mechanics testing system (MTS 815), as shown in Fig. 2a. The axial deformation of specimen was measured by variable differential transformer, and the circumferential deformation was measured using a chain extensometer attached to the middle of the specimen, which an accuracy of 0.001 mm. In this study, the P-ultrasonic velocity was monitored using a PCI-II AE system (Physical Acoustics Corporation, Mistras Group Inc., USA). The P-wave signal transmitter and receiver were placed at the lower and upper ends of the specimen, respectively. The frequency of the P-wave signal was 300 kHz, the breakdown voltage was 150 V, and the interval time was 20s. The preamplifier gain of ultrasonic wave signal receiver was set to 40 dB⁴⁵. Siltstone specimen after the sensor was installed is shown in Fig. 2b.

Fig. 2.

Testing systems. (a) Overview of the stress control system and the P-ultrasonic velocity monitoring system; (b) Siltstone specimen after the sensor was installed.

According to a previous study, the uniaxial compressive strength (UCS) of the siltstone is 35.13MPa⁴⁵. Therefore, the preset value of confining pressure is selected as 2, 4, 6, 8, 10, 12, 16, and 20 MPa, about 7–58% of UCS. Table 1 shows the test scheme and parameters of siltstone specimens. The test process is as follows:

Table 1.

The detailed geometry and test program of the specimens.

Specimen	σ3/MPa	H/mm	D/mm	m/g	ρ/(kg/m3)	V0/(m/s)
U0	0	99.99	49.98	418.17	2131.64	2245
T2	2	99.98	49.92	417.60	2134.06	2305
T4	4	99.96	49.98	416.16	2122.03	2228
T6	6	99.96	50.02	414.25	2108.91	2325
T8	8	99.98	49.92	417.68	2134.47	2316
T10	10	100.00	49.92	416.23	2126.64	2275
T12	12	99.92	49.90	418.51	2141.71	2296
T16	16	100.00	49.98	415.01	2113.62	2250
T20	20	99.92	49.96	415.24	2119.88	2275

1. The coupling gel was applied between the specimen, signal transmitter, and receiver to eliminate the friction. Then, wrap the specimen tightly with heat-shrinkable sleeves to ensure that the sample is isolated from the hydraulic oil.

2. The circumferential and axial LVDT were installed in sequence according to the design.

3. Firstly, the confining pressure was applied to the specimen at the rate of 0.1 MPa/s. Then, the axial load was applied to 30, 30, 40, 50, 60, 70, 80, and 100 kN at a constant rate of 0.5 kN/s, under different confining pressure. Finally, the axial load was ccontinued applied at a controlled circumferential displacement rate of 0.03 mm/min until the specimen failed.

Test results

Compaction and damage mechanics properties of siltstone

Figure 3a, b, c show the relationship curves between axial strain, circumferential strain, volumetric strain, and axial stress of siltstone under different confining pressures, respectively. As shown in Fig. 3d, , , and are the peak strength, residual strength, and damage stress of siltstone, respectively. E is the elastic modulus of siltstone, obtained by calculating the slope of the axial stress-strain curve at . Table 2 presents the main mechanical parameters of siltstone under different confining pressures.

Fig. 3.

Complete stress-strain curves of siltstone under various confining pressures. (a) Axial strain; (b) Circumferential strain; (c) Volumetric strain; (d) Characteristic parameters of the stress-strain curve.

Table 2.

Strength and strain parameters of argillaceous quartz siltstone under various confining pressures.

Specimen	σ3/MPa	/MPa	/%	E/GPa		/MPa	/	/MPa
U0	0	36.29	0.814	6.04	0.397	15.79	0.449
T2	2	53.21	0.786	7.78	0.454	30.55	0.574	22.70
T4	4	76.30	0.913	10.25	0.367	53.04	0.695	36.97
T6	6	84.24	1.207	10.07	0.360	56.39	0.669	47.30
T8	8	93.93	1.064	11.72	0.315	66.39	0.707	44.04
T10	10	100.97	1.312	10.28	0.274	76.04	0.753	59.24
T12	12	112.15	1.232	11.85	0.228	85.33	0.761	56.01
T16	16	138.27	1.610	10.96	0.177	106.59	0.771	82.76
T20	20	151.02	1.691	11.37	0.151	121.46	0.804	85.93

Figure 4 shows the variation patterns of peak strength, residual strength, damage stress, and elastic modulus of siltstone under different confining pressures. As shown in Fig. 4a, the Hoek Brown (H-B) criterion can more accurately describe the variation of peak strength of siltstone with confining pressure. However, the residual strength of siltstone shows a clear linear relationship with confining pressure. The volume dilatation strength of rocks can represent the beginning of damage. As shown in Fig. 4b, the volume dilatation strength of siltstone increases linearly with the increase of confining pressure. It is indicating that the development of damage of siltstone can be suppressed by the constraint effect of confining pressure. However, the ratio of damage stress to peak strength increases nonlinearly with confining pressure from 0.45 to 0.8 and tends to stabilize. As shown in the Fig. 4c, when the confining pressure is lower than 10 MPa, the elastic modulus of siltstone increases nonlinearly from 6 GPa to 11 GPa with the increase of confining pressure. When the confining pressure is higher than 10 MPa, the elastic modulus tends to stabilize, approximately twice that of uniaxial compression. The results of damage stress and elastic modulus indicate that siltstone shows a significant compaction effect under high confining pressure.

Fig. 4.

Mechanical parameters of siltstone under various confining pressures. (a) Relationships between peak strength, residual strength, and confining pressure; (b) The relationship between the damage stress, normalized damage stress value, and confining pressure; (c) The relationship between the elastic modulus and confining pressure.

Nonlinear evolution model of ultrasonic velocity

In this study, it is assumed that the pulse signal of ultrasonic wave propagates at a constant speed in the rock specimen. Therefore, the ultrasonic velocity of siltstone can be calculated by the propagation time of the pulse signal and the specimen height, as follows:

1

where, is the propagation time of the pulse signal between the signal exciter and the receiver at time t. L is the height of siltstone specimen. is the real-time ultrasonic velocity of the siltstone at time t.

Figure 5 shows the real-time ultrasonic velocity-strain curve and stress-strain curve of siltstone under uniaxial compression and triaxial compression. Under different confining pressures conditions, the evolution law of ultrasonic velocity of siltstone with axial strain are consistent with that of under uniaxial compression test. When the axial stress is lower than the damage stress , the volume of siltstone is continuously compacted under axial and circumferential pressure, and the ultrasonic velocity increases rapidly. When the axial stress is between the damage stress and peak strength, although the damage occurs in the siltstone, the new micro cracks cannot immediately propagate under high pressure, the ultrasonic velocity tends to stabilize. In the post peak stage, the macro fracture surface is generated by the penetration of micro cracks, the ultrasonic velocity decreases with the drop of axial stress. However, the fracture surface is still in close contact, due to the specimen still being constrained by high pressures. Therefore, the ultrasonic velocity did not suddenly decrease significantly, but stabilized at a constant value during the residual strength stage.

Fig. 5.

Evolution curve of ultrasonic velocity with axial strain of siltstone under different confining pressures. (a) Uniaxial compression; (b) Confining pressure of 6 MPa; (c) Confining pressure of 8 MPa; (d) Confining pressure of 10 MPa; (e) Confining pressure of 16 MPa; (f) Confining pressure of 20 MPa.

The above test results indicate that real-time ultrasonic velocity can accurately reflect the variation of pore under different stress conditions. Therefore, a nonlinear exponential function between ultrasonic velocity and axial strain was proposed, based on the evolution characteristics of the ultrasonic velocity curve. In this study, the theoretical equation between ultrasonic velocity and axial strain in the pre peak stage of rocks is as follows:

2

where, is the initial ultrasonic velocity; is the ultrasonic velocity when the axial strain is , is the maximum increment of ultrasonic velocity during the loading process, is the model index.

The theoretical equations of ultrasonic velocity models under different confining pressures were fitted using the least squares method, and the model’s parameters are shown in Table 3. Figure 6 shows the relationship between the maximum ultrasonic velocity, confining pressure, and elastic modulus during the test process. When the confining pressure is lower than 10 MPa, the maximum ultrasonic velocity of siltstone increases nonlinearly from 3236 m/s to 3688 m/s with the increase of confining pressure. When the confining pressure is higher than 10 MPa, the maximum ultrasonic velocity fluctuates between 3688 m/s and 3768 m/s, which is approximately 1.65 times the natural ultrasonic velocity. As shown in the Fig. 6b, there is a clear linear positive correlation between the maximum ultrasonic velocity and the elastic modulus of siltstone.

Table 3.

Ultrasonic velocity model parameters of siltstone under different confining pressures.

No.	/MPa	/(m/s)	/(m/s)	/%		/(m/s)
U0	0	2245	991	0.814	2.26	3236
T2	2	2305	1031	0.786	2.78	3336
T4	4	2228	1153	0.913	3.11	3381
T6	6	2325	1250	1.207	3.34	3575
T8	8	2316	1355	1.064	4.03	3671
T10	10	2275	1413	1.312	3.50	3688
T12	12	2296	1393	1.232	3.82	3689
T16	16	2250	1397	1.610	4.31	3647
T20	20	2257	1511	1.691	4.77	3768

Fig. 6.

Relationship between maximum ultrasonic velocity, confining pressure, and elastic modulus of siltstone. (a) The nonlinear relationship between maximum ultrasonic velocity and confining pressure of siltstone; (b) Linear relationship between maximum ultrasonic velocity and elastic modulus of siltstone.

Figure 7a and b show the relationship between the ultrasonic velocity model index of siltstone, confining pressure, and damage stress, respectively. As shown in Fig. 7a, the ultrasonic velocity model index increases approximately linearly with the increase of confining pressure. Based on the evolution curve of the ultrasonic velocity in Fig. 5, it can be concluded that the higher the confining pressure, the more significant the pore compaction effect of siltstone. As shown in Fig. 7b, the ultrasonic velocity model index of siltstone shows a significant linear relationship with the increase of damage stress. These results indicate that the ultrasonic velocity model index can represent the extent of porosity closure, and the larger the index, the more complete the pore closure.

Fig. 7.

The relationship between the index of ultrasonic velocity model and confining pressure and damage stress. (a) The linear relationship between the velocity model index of siltstone and confining pressure; (b) Linear relationship between velocity model index and damage stress of siltstone.

Inversion model of real-time porosity

Previous studies have shown that the ultrasonic velocity in porous media materials can be influenced by various factors⁴⁸. The main component of the siltstone is quartz studied in this paper, and the pore fluid medium is air under the dry state, which has a relatively small impact on the ultrasonic velocity of the siltstone. However, the initial porosity of siltstone is close to 20%, therefore, the main factor affecting the ultrasonic velocity of siltstone can be considered as the change of porosity. Based on the above analysis, a real-time porosity inversion model was established by the real-time ultrasonic velocity model.

In 1958, Wyllie²⁰ equated the rock materials to a solid-liquid two-phase combination model and estimated rock porosity based on ultrasonic velocity. In this model, the rock medium is assumed to be the combination of air and solid skeleton. The total time for elastic longitudinal waves to pass through the rocks is equal to the sum of the time required for elastic longitudinal waves to pass through gas-phase fluids and solid-phase matrices. Therefore, the time averaged model was established to describe the effect of pores and matrix on rock ultrasonic velocity. Assuming the porosity of rock is , the ultrasonic velocity in the rock matrix is , and the ultrasonic velocity in the pore is . The propagation time of waves in rock matrix and pore can be expressed in Eq. (3) and Eq. (4), respectively.

3

4

where, is the equivalent length of the rock matrix when the rock length is L, is the equivalent length of the pore of rock, and represent the time for ultrasonic waves to pass through the rock matrix and pores, respectively.

Therefore, the total ultrasonic velocity of elastic waves in rocks can be calculated using Eq. (3) and Eq. (4):

5

The time averaged model is an experimental formula based on statistical theory, and the accurate inversion results are only available for rocks with relatively uniform pore structure. In 1980, Raymer²¹ established a new ultrasonic velocity model based on the time averaged model, as shown in Eq. (6).

6

The Raymer model has made significant improvements compared to the Wyllie model, and the calculation results are more accurate when the porosity is lower than 30%. Moreover, the relatively accurate results can be obtained for porous media materials without the compaction correction.

According to the range of ultrasonic velocity values of quartz crystals, the ultrasonic velocity of siltstone matrix in this study is selected as 5000 m/s^49,50. In the dry state, the pore medium of siltstone is air, with the ultrasonic velocity of 340 m/s. The initial ultrasonic velocity of siltstone calculated by Eq. (6) is 3276 m/s, which is much higher than the ultrasonic velocity (2250 m/s) of siltstone measured by the test system.

By analyzing the micro structure of siltstone, it is believed that its material composition and pore structure differ significantly from granite, marble, and sandstone. The skeleton composition of siltstone includes not only complete quartz crystal particles, but also several mud cement materials. Therefore, the ultrasonic velocity of rock matrix needs to consider the combined effect of quartz crystals and argillaceous cement, due to the attenuation effect of elastic waves caused by weak cement. For this reason, a rock matrix ultrasonic velocity correction coefficient is introduced based on the Raymer formula, and the improved porosity inversion model is shown in Eq. (7):

7

where, is the correction coefficient for the ultrasonic velocity of the rock matrix. This coefficient is determined by the proportion of quartz crystal in the siltstone skeleton component. According to the analysis results of the siltstone component, is taken as 0.8.

Therefore, the real-time porosity inversion equation of siltstone can be obtained by solving the Eq. (7), as shown in Eq. (8).

8

Fig. 8.

Comparison of real-time porosity of siltstone calculated by Wyllie model, Raymer model, and modified Raymer model in this paper, when the confining pressure is 6 MPa.

Figure 8 shows the comparison of real-time porosity curves calculated by the Wyllie model, Raymer model, and the modified Raymer model, with the confining pressure of 6 MPa. The initial porosity of siltstone calculated by the Wyllie model and Raymer model is 7.9% and 32.5%, respectively, which is significantly different from the mercury intrusion test results. The initial porosity of siltstone calculated by the modified Raymer model is 23.9%, which is more consistent with the measured results. Therefore, the accurately real-time porosity inversion results of rocks can be calculated by the modified Raymer model.

Figure 9a shows the real-time porosity inversion results of siltstone under different confining pressures. In the initial stage of triaxial compression test, the porosity of siltstone decreases rapidly. When the axial strain accumulates to the value of 0.6%, the porosity of siltstone under different confining pressures gradually stabilizes. Figure 9b shows the relationship between the minimum porosity and confining pressure during the triaxial compression test of siltstone. When the confining pressure is in the range of 0 MPa ~ 8 MPa, the minimum porosity of siltstone decreases from 10.6 to 4.2% with the increase of confining pressure. When the confining pressure is in the range of 10 MPa ~ 20 MPa, the minimum porosity of siltstone gradually tends to stabilize and fluctuating around 4%. It indicates that the closure extent of initial cracks and pores improves with the increase of confining pressure, when the confining pressure is relatively low. Under higher confining pressure levels, the 90% initial cracks and pores are closed. However, the pores between crystals still cannot be completely closed, under the limited axial stress, accounting for approximately 10% of the total porosity.

Fig. 9.

Comparison of real-time porosity inversion results of siltstone under different confining pressures. (a) Comparison of porosity variation curves with axial strain of siltstone under different confining pressures; (b) The variation characteristics of the minimum porosity of siltstone with confining pressure, under different confining pressure conditions.

Compression-damage coupled constitutive model

Compaction coefficient of rock

According to the test result, the rock can be assumed to be a material composed of mineral matrix, cracks, and pores that can be closed. The mineral matrix of rocks is assumed to be an elastic body that undergoes only elastic deformation. The rock matrix is assumed to be the elastic material. The cracks and pores continuously close with the increase of external load. When the axial stress reaches the peak strength, the cracks and pores is completely closed which can be compressed, under the current confining pressure conditions. Therefore, a compaction coefficient is established, in order to evaluate the closure effects of crack and pore on the elastic properties of rocks.

Assuming the total volume of the material is , the total volume of cracks and pores is . The volume increment generated by the matrix material under the condition of strain increment is⁵¹:

9

where, is the compressibility coefficient of the rock matrix.

By integrating both sides of Eq. (9), it can be concluded that:

10

The first term on the left side of Eq. (10) is defined as the compressibility coefficient of the material. For the second term on the right of Eq. (10), when the strain is small enough, can be considered as a constant. Therefore, can be approximately equal to the rate of change in porosity under external loads. According to the convention that porosity increases positively, Eq. (10) can be converted to:

11

According to Eq. (11), it can be concluded that:

12

According to Eq. (12), the relative effective compressibility coefficient of rocks can be obtained by taking the derivative of real-time porosity, which shows in Eq. (8). In this study, the relative effective compressibility coefficient of siltstone is used to describe the extent of porosity closure. When , it indicates that the cracks and pores that can be closed in the siltstone have been completely compacted. Under this condition, the effective compressibility coefficient of the rock is equal to the compressibility coefficient of the matrix material. Figure 10 shows the evolution curve of the relative effective compressibility coefficient of siltstone with axial strain, when the confining pressure is 6 MPa. In the initial stage of triaxial compression, the effective compressibility coefficient rapidly decreases with the axial strain, indicating that cracks and pores are susceptible to compaction. When the axial stress approaches the peak intensity, the rate of decrease in the relative effective compressibility coefficient slows down. When the axial stress reaches the peak strength, the relative effective compressibility coefficient is 0, indicating that the cracks and pores which can be compressed are completely closed.

Fig. 10.

Evolution curve of the relative effective compressibility of rock with axial strain, when the confining pressure is 6 MPa.

Based on the above analysis, we have clarified the mechanical meaning of the relative effective compressibility coefficient of rocks. Therefore, we normalize the relative effective compressibility coefficient and establish a compression coefficient to describe the compression od cracks and pores, as follows:

13

When , , indicating that the cracks and pores of the rock are in an initial state, and the extent of porosity closure is 0. When , , indicating that the extent of porosity closure is 1, and the external load is carried by rock matrix completely.

Rock damage variables

When the axial stress exceeds the volumetric dilatation stress, new cracks initiate in the siltstone, causing in the accumulation of damage to the rock matrix. It is generally believed that the damage and fracture of rocks show the characteristic of typical non-uniform. Assuming that the initiation of new cracks in rocks shows the random characteristic, the distribution of micro cracks transitions to ordered with the develop of damage. Due to the Weibull function satisfying the statistical distribution characteristics of micro crack, and has the advantages of easy integration and reasonable range of values⁵². Assuming that the strength of each unit in the rock follows the Weibull distribution, the probability density function is:

14

where, F is the distribution variable of rock unit strength; n and are distribution parameters of the Weibull model, which can reflect the mechanical properties of rock materials.

The damage of rock materials is caused by the continuous destruction of these units. Assuming the number of damaged units in the rock is , when the stress increment is . Then, the damage variable of rocks can be defined as the ratio of the number of damaged units to the total units N.

15

where, D is the damage variable of rock.

Therefore, the total damage variable inside the rock can be obtained by integrating Eq. (14), under the condition of . The expression of the damage variable is shown in the Eq. (16).

16

For the distribution variables of unit strength in Eqs. (14) and (16), we choose the Mohr-Coulomb strength criterion to represent them, as shown in the Eq. (17).

17

where, is the internal friction angle of the rock.

Establishment and validation of models

1. Establishment of constitutive model

The rocks can be divided into two parts: undamaged and damaged, according to the assumption of Lemaitre’s strain equivalence. Therefore, the improved compression-damage coupled constitutive model which can describe the residual strength characteristics of rock was constructed based on these two parts. The schematic diagram of the new established constitutive model is shown in Fig. 11. The mechanical relationship between the damaged, undamaged, and compacted parts in the model is shown in Eq. (18).

Fig. 11.

Schematic diagram of the new established constitutive model.

18

where, A, , and are the cross-sectional area of the rock material, undamaged matrix, and damaged. ,, and are the nominal axial stresses of the rock, the axial stresses borne by the undamaged matrix and the damaged part, respectively. Similarly, , , and are the nominal axial strains of the rock, as well as the axial strains of the undamaged matrix and damaged parts, respectively.

The elastic deformation only generates at the matrix of the undamaged part of the rock. the stress-strain relationship follows to the generalized Hooke criterion. The axial stress borne by this part is:

19

The axial stress borne by the damaged part is equal to the residual strength of the rock:

20

Substituting Eq. (19) and Eq. (20) into Eq. (18), it can be obtained:

21

Based on the stage characteristics of strain-stress curves of siltstone, the constitutive model of the rock can be divided into the following three intervals.

• ①When the axial stress is lower than the damage stress, only cracks and compaction occur inside the rock, and no damage occurs

22

• ②When the axial stress is between the damage stress and the peak strength, damage begins to occur in the rock matrix, and the closure of the cracks and pores continues

23

• ③In the post peak stage, the axial stress decreases with the failure of rock, and it is considered that the closure of the cracks and pores has stopped at this stage, with the compaction coefficient .

24

• (2)Solving of the model parameters

The compaction coefficient in the new constitutive model can be obtained through the theoretical model of rock ultrasonic velocity established in this paper and the modified porosity inversion model. Parameters E, , and can be obtained from the results of triaxial compression tests. The damage parameters and n need to be obtained using the key point method. When the axial stress reaches the peak strength, , and . By substituting Eq. (24), it can be concluded that:

25

If , then Eq. (25) can be transformed into:

26

At the point of the peak strength, the slope of the stress-strain curve is 0. Taking the derivative of Eq. (24), it can be obtained:

27

By substituting Eqs. (17), (19), and (26) into Eq. (27), it can be concluded that:

28

The can be obtained by substituting Eq. (28) into Eq. (25). The parameter n can be obtained by substituting into Eq. (26).

• (3)Validation of model

Table 4 shows the damage parameters of siltstone under different confining pressure, calculating by the Eqs. (25) and (28). Figure 12 shows the comparison of stress-strain theory curve and test results under different confining pressure, calculating by the compaction coefficient and damage variable of siltstone. As shown in Fig. 12, when the axial stress is lower than the damage stress, no damage occurs to the rock, ; The crack compaction coefficient rapidly increases with the axial strain, and the axial stress-strain curve shows significant initial compaction characteristics. When the axial stress reaches the damage stress, the growth rate of crack compaction coefficient significantly decreases; The damage begins to develop and accumulates exponentially. When the axial stress reaches the peak strength, the cracks and pores which can be closed inside the siltstone have been completely compacted, . At the post peak failure stage, the damage variable of the rock accumulates linearly with the decrease of axial stress, and the rock gradually failed. When the damage variable , the macro fracture surface of the rock has formed, and the strain-stress curves entered the residual strength stage.

Table 4.

The parameters of compaction coefficient and damage variable of the new constitutive model.

Specimen		Compaction coefficient					Damage variable
	/MPa	/(m/s)	/(m/s)	/%				n
T2	2	2305	1031	0.786	2.78	0.8	28.75	10.61
T4	4	2228	1153	0.913	3.11		33.24	9.96
T6	6	2325	1250	1.207	3.34		43.71	9.89
T8	8	2316	1355	1.064	4.03		52.33	6.27
T10	10	2275	1413	1.312	3.50		52.63	5.81
T12	12	2296	1393	1.232	3.82		63.52	4.63
T16	16	2250	1397	1.610	4.31		78.86	3.32
T20	20	2257	1511	1.691	4.77		77.62	3.82

Fig. 12.

Comparison of stress-strain theory curve and test results under different confining pressure. (a) Confining pressure of 6 MPa; (b) Confining pressure of 8 MPa; (c) Confining pressure of 10 MPa; (d) Confining pressure of 20 MPa.

As shown in Fig. 12, at the points of damage stress, peak strength, and residual strength in the strain-stress curve of rock, the key information of compaction and damage on rock can be reflected clearly. The initial compaction stage of rocks can be described by the compaction coefficient, which calculating according to the ultrasonic velocity model and real-time porosity inversion model. The nonlinear deformation characteristic of rock in the yield stage and residual strength stage can be describe accurately by the damage variables established in this paper.

Figure 13 shows the relationship between damages and confining pressure. With the confining pressure increases, the damage parameters and n show clear linear relationships, increasing and decreasing respectively. According to the evolution of the compaction coefficient and damage parameters with confining pressure, the stress-strain behavior of rocks under different stress states can be scientifically predicted. These conclusions indicate that the compression-damage coupled constitutive model established in this study can reflect the complete deformation process of rocks, and has clear mechanical meanings. It has important theoretical guidance significance for evaluating the stability of underground rock engineering.

Fig. 13.

The relationship between damage parameters and confining pressure.

Conclusions

In this study, the nonlinear ultrasonic velocity model, real-time porosity inversion method, and compression damage coupled constitutive relationship on rock were researched, based on the triaxial compression test on siltstone. The main conclusions are as follows:

1. At the initial deformation stage of siltstone under triaxial compression test, the ultrasonic velocity rapidly increases with axial strain. When the axial stress reaches the damage stress, the ultrasonic velocity gradually stabilizes. In the post peak stage, the wave speed decreases slightly. A nonlinear model of ultrasonic velocity with strain was established, and there is a clear correspondence between the parameters and elastic modulus, damage stress, and confining pressure.

2. A correction coefficient for the ultrasonic velocity of the rock matrix was introduced to modify the Raymer formula, a real-time porosity inversion model was established. According to the proportion of quartz in the siltstone, the correction coefficient for matrix ultrasonic velocity is set to 0.8. The initial porosity of siltstone calculated using the modified real-time porosity inversion model is 23.9%, verifying the accuracy of this model. According to the porosity inversion results, the minimum porosity of siltstone is about 4%, under the triaxial compression test.

3. The relative effective compressibility coefficient of rocks was derived, based on the real-time porosity inversion model. A compaction coefficient to describe the closure extent of cracks and pores was established, by normalizing the relatively effective compressibility coefficient. When the compaction coefficient is 0, it indicates that the cracks and pores of the rock have not been compacted. When the compaction coefficient is 1, it indicates that the cracks and pores in the rock that can be compacted have been completely compacted.

4. A damage variable for rocks was established based on statistical damage theory. When the axial stress reaches the volume dilatation strength, the damage begins to develop. At the residual strength stage, the cumulative damage variable reaches 1. The established rock compaction-damage coupled model can accurately describe the nonlinear deformation characteristics of the initial compaction stage and pre-peak yield stage of rocks.

Author contributions

Yang Pengjin: Conceptualization, Methodology, Writing - original draft, Investigation, Data curation. Miao Shengjun: Supervision, Conceptualization, Writing-review & editing. Cai Meifeng: Conceptualization, Methodology. Du Shigui: Conceptualization, Methodology. Hu Yunjin: Methodology.

Data availability

The data are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.