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Estimation of REV for fractured rock masses based on Geological Strength Index
International Journal of Rock Mechanics & Mining Sciences 126 (2020) 104179
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International Journal of Rock Mechanics and Mining Sciences journal homepage: http://www.elsevier.com/locate/ijrmms
Estimation of REV for fractured rock masses based on Geological Strength Index Huajie Huang a, Jiayi Shen a, b, c, *, Qiushi Chen c, Murat Karakus d a
Institute of Port, Coastal and Offshore Engineering, Zhejiang University, Hangzhou, 310015, China State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining & Technology, Xuzhou, 221116, China c Glenn Department of Civil Engineering, Clemson University, Clemson, SC, 29634, USA d School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia b
A R T I C L E I N F O
A B S T R A C T
Keywords: GSI REV Synthetic rock mass Hoek-Brown Numerical simulation
The representative elementary volume (REV) of fractured rock masses is a significant index to investigate the rock mass behaviors in the continuum mechanics. In this research, a new indicator to estimate the REV size of fractured rock masses based on the Geological Strength Index (GSI) is proposed. For this purpose, a new method that combines the PFC-based synthetic rock mass (SRM) model with the Hoek-Brown (HB) failure criterion is proposed to investigate the strength and deformation properties of fractured rock masses under biaxial stress conditions. Extensive numerical analyses are carried out to estimate variation of the uniaxial compression strength (UCS), deformation modulus (E) and GSI of the Brunswick mine rock mass with increasing the size of the SRM models up to a REV size. Results show that the GSI-based indicator gives relatively larger REV size compared with the traditional UCS or deformation modulus (E) based indicators. Compared with the traditional indicators, the proposed GSI-based indicator has merits of not only reflecting the geometrical characteristics of rock structures but also containing both geometrical and mechanical properties of discontinuities.
1. Introduction
corresponding results can be applied to represent the overall rock mass properties. The concept of REV was first proposed by Bear6 to study fluid flow in a porous medium. Over the last decades, various indicators have been proposed to estimate the REV size by many researchers. Generally, these indicators can be categorized into three groups, namely, geometrical indicators, mechanical indicators and hydraulic indicators, as shown in Table 1. The geometrical and mechanical indicators can be used for various engineering applications, while the hydraulic indicators are specifically used for projects that consider water flow conditions. In order to obtain realistic mechanical properties of jointed rock masses, large volumes of rock mass configurations should be tested. However, as Bieniawski24 noted, investigating the effect of sample size on the mechanical properties of a large volume jointed rock mass using the in-situ testing approach is not only difficult but also time-consuming and very expensive. Therefore, numerical simulations were widely used by many researchers7,11,15,16 to estimate the large scale rock mass properties using an upscaling procedure which involves estimating the variation of the mechanical properties with increasing the size of the examined rock volumes up to the REV size. After the determination of
The determination of rock mass mechanical properties is critical for the design of many engineering infrastructures, such as slopes, tunnels and foundations.1,2 However, it is not easy to estimate the strength and deformation characteristics of fractured rock masses because of the ex istence of various discontinuities, like joints, fractures, faults and bedding planes in the rock masses. These discontinuities are often different in orientation, trace length, waviness and aperture, which make the rock mass inhomogeneous and anisotropic.3,4 Alejano et al.5 have shown that the strength and deformation prop erties of rock masses are significantly influenced by the rock structure, discontinuity properties and stress states. It is also known that these rock mass properties gradually change with the increase of rock mass sizes. When the rock mass size is relatively small, a slight increase in sample size may result in an obvious change in mechanical characteristics. However, rock mass properties no longer change with an increase of sample size when the sample size is larger than a critical value. Such critical value is named as the representative elementary volume (REV) size. When the sample is greater than or equal to the REV size, the
* Corresponding author. Institute of Port, Coastal and Offshore Engineering, Zhejiang University, Hangzhou, 310015, China. E-mail address: [email protected] (J. Shen). https://doi.org/10.1016/j.ijrmms.2019.104179 Received 15 January 2019; Received in revised form 2 November 2019; Accepted 15 December 2019 Available online 27 December 2019 1365-1609/© 2019 Elsevier Ltd. All rights reserved.
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Table 1 Indicators for estimating the REV of rock masses. Authors
Indicators
Group 1 Geometrical indicators
Esmaieli et al.7 Chae and Seo8 Zhang et al.9 Xia et al.10 Ni et al.11 Li et al.12
Fracture intensity P32 Ratio of the micro-cracks Fracture intensity P32 Blockiness Fracture intensity P32 Geometrical connectivity
Group 2 Mechanical indicators
Min and Jing13
Deformation modulus and Poisson’s ratio Deformation modulus and shear modulus UCS and deformation modulus Deformation modulus and Poisson’s ratio UCS and deformation modulus Damage coefficient UCS and deformation modulus
Pariseau et al.14 Esmaieli et al.7 Khani et al.15 Yang et al.16 Ni et al.11 Farahmand et al.17
Group 3 Hydraulic indicators
Kulatilake and Panda18 Wang et al.19 Min et al.20 Chen et al.21 Rong et al.22 Wang et al.23
�
�a
σ 1 ¼ σ 3 þ σci mb
σ3 þs σci
(1)
where σ1 and σ3 are the major and minor principal stresses respectively. mb, s, and a are HB input parameters, which depend on the fracturing degree of the rock mass and can be estimated from the intact HB con stant mi, GSI and D, given by mb ¼ mi eðGSI s ¼ eðGSI
100Þ=ð28 14DÞ
(3)
100Þ=ð9 3DÞ
1 eð a¼ þ 2
GSI=15Þ
(2)
eð
20=3Þ
6
(4)
The details of calculation and selection of mi and D can be found in the papers by Hoek and Brown,27 Hoek et al.,26 Shen and Karakus,28 Wang and Shen29 and Hoek and Brown.4 The GSI is a rock mass classification system that has been developed in engineering rock mechanics to meet the need for estimating reliable rock mass properties for designing and analyzing tunnels, slopes or foundations. The value of GSI is estimated from visual examination of the rock mass exposed in outcrops, therefore, the heart of the GSI clas sification is a careful engineering geology description of the rock mass which involves two factors, rock structure and discontinuity surface condition. However, due to the lack of measurable parameters and related in terval limits or ratings for describing the rock structure and the surface conditions of the discontinuities, the use of the GSI chart involves some subjectivity. In order to facilitate the use of the classification system especially by inexperienced engineers, many researchers30–32 proposed quantitative GSI charts by introducing measurable parameters. For example, Cai et al.31 made an attempt to provide a more quantitative approach to evaluate GSI. For this purpose, two terms, block volume (Vb) and joint condition factor (Jc) were introduced into the existing GSI classification scheme. In order to assign ratings to these terms, the use of some easily measurable input parameters such as roughness, weath ering, infilling and volumetric joint count were suggested. According to the selected rating intervals, the GSI chart has been modified, as shown in Fig. 1. By adding measurable quantitative inputs for a quantitative output, the system becomes less subjective while maintains its overall simplicity. Once a GSI value has been decided upon, it can be used in conjunction with appropriate empirical equations to calculate the strength and deformation of a rock mass. However, it should be noted that the rock mechanical properties of fractured rock masses are generally anisotropic and stress path depen dent13、33. The HB failure criterion based on the GSI system is used for predicting isotropic rock mass strength which cannot reflect such anisotropic behavior and stress path dependent characteristics. As GSI not only describes the geometrical feature of rock masses but also relates the mechanical properties of the discontinuities, a GSI-based indicator could be ideal to calculate the REV size for a rock mass, overcoming limitations of the traditional indicators based on either geometrical or mechanical characteristics alone.
Average block permeability Hydraulic conductivity Permeability tensor Permeability tensor Permeability tensor Permeability coefficient
the REV size, the overall strength and deformation characteristics of rock masses for a specific engineering project could be estimated and the rock mass with the size of REV could be regarded as isotropic materials. Although there are various indicators available as listed in Table 1, one still must take into account the following limitations when using them for the analysis of REV size for different rock projects: (1) Geometrical indicators can only be used for initial estimation for the REV size in a time-efficient manner at an early stage of a project. As geometrical indicators can only reflect the rock mass structures, it cannot provide information of mechanical parameters of discontinuities, such as the mechanical effects of infilling of discontinuities, which reduce its reliability. (2) The existing mechanical indicators for esti mating REV size are based on the uniaxial compression tests, which can only represent the rock mass behavior under the unconfined stress state. However, rock masses are generally in states of triaxial stresses and the properties will be significantly influenced by the triaxial stress states. With the aim of achieving a better understanding of the fundamental rock mass mechanisms and improving the accuracy of the determination of REV size, in this research we propose a new indicator to estimate the REV size of fractured rock masses based on the Geological Strength Index (GSI) classification which has merits of not only reflecting the geometrical characteristic of rock structures but also containing both geometrical and mechanical properties of discontinuities. In order to establish such GSI-based REV indicator, we also propose a new method that combines the PFC-based synthetic rock mass (SRM) model with the Hoek-Brown (HB) failure criterion to investigate the strength and deformation of fractured rock masses under biaxial stress conditions. 2. Hoek-Brown criterion based on Geological Strength Index
3. Synthetic rock mass for numerical analyses
In rock engineering, the Hoek-Brown (HB) criterion based on the Geological Strength Index (GSI) has been widely used for predicting isotropic rock mass strength for many decades. The original HB crite rion, which was proposed by Hoek and Brown25 for estimating the intact rock strength, requires two intact rock properties, namely, the uniaxial compressive strength (UCS or σci) and a constant mi of the intact rock. It was then extended to estimate the rock mass strength using the GSI and disturbance factor D.26 The latest version of the HB criterion presented by Hoek and Brown4 is expressed in Eq. (1).
In this research, the synthetic rock mass (SRM) model of the Bruns wick mine rocks was employed as an example to illustrate the method of estimating REV based on the GSI indicator using an upscaling process. The upscaling procedure involves estimating the variation of the me chanical properties with increasing the size of the rock mass volumes up to a REV in which stationary limits are reached for the properties under study.
2
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104179
Fig. 1. Quantification of GSI chart (Cai et al.31).
3
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3.1. PFC2D-based synthetic rock mass model
Table 2 Fracture set field data for fracture system generation.
The SRM model, which was constructed in PFC2D software in this research, is based on two well-developed numerical techniques, the discrete fracture network (DFN) and the bonded particle model (BPM). A DFN is generally generated using the Monte-Carlo simulation. Several statistical parameters, including the mean and standard devia tion of joint set dip angle and trace length, are required to control the orientation, size and intensity of each joint set in the DFN. In BPM, rigid particles are bonded together to form the basic ele ments of a hard rock. The bond represents the rock cohesive component and breaks when the external stress exceeds the prescribed bond strength. The particle movement is influenced by frictional resistance. As the BPM explicitly considers the particle-scale mechanical behavior, it requires no continuum-scale constitutive model (such as the MohrCoulomb and Hoek-Brown criteria) to control the macroscopic me chanical rock behaviors. The micro-parameters in BPM can be identified following a calibration process to assure the similarity of the macroproperties can be obtained. The SRM is then constructed by inserting a DFN into a BPM assembly and a “smooth joint” logic is developed to assert smooth particle sliding along joints. The smooth joint model (SJM), which resolves the local force-displacement law onto the plane of the joint, could be used to simulate the behavior of such “smooth joint”. By this approach, bonded behavior, Coulomb sliding and tensile opening can be realistically rep resented.34 Fig. 2 shows the main components of a synthetic rock mass.
Fracture characteristic
Set 1
Set 2
Set 3
2D-dip (deg) Trace length (m) P21 (m 1)
90 1.40 0.63
89 1.44 1.80
22 1.16 1.95
scale effect of intact rocks in PFC2D, initial micro defects need to be added into intact rocks,35 which is not considered in this research. Based on the geometrical parameters of fractures listed in Table 2, a big DFN model with the size of 100 � 200 m was generated using the Monte Carlo method. 12 small DFNs with the same sizes corresponding to the 12 intact rock samples (see Fig. 3) were extracted at the center of the big DFN, as shown in Fig. 4. Moreover, in order to avoid the random effect in the DFN model, we generated 10 big DFN models and repeated the procedure of extracting small samples from the big DFN models, resulting in a total of 120 DFN models for subsequent SRM model gen erations and numerical simulations. Finally, the intact rock samples (Fig. 3) were combined with DFN models (Fig. 4) to generate the SRM models. Fig. 5 shows a typical set of 12 SRM models with various sizes. In total, 120 SRM samples were generated in PFC2D. 4. Calibration of synthetic rock mass models The micro-parameters of intact rocks and fractures in the SRM samples need to be identified following a calibration process to assure the rock mass strength and deformation properties estimated from SRM models are the same as that from laboratory and field data.7
3.2. SRM model of the brunswick mine rock mass Rock mass data was collected from Brunswick Mine,7 which is located in northeastern New Brunswick, Canada. The deposit is hosted mainly by metamorphosed volcaniclastic sediments and tuffs that overlie felsic volcanic rocks and have a high specific gravity (4.3) and stiffness (UCS up to 205MPa). The elastic modulus of the intact massive sulphide is 104GPa and the Poisson’s ratio is 0.29. In this case, the mean values of joints dip angles in 2-D were calcu lated from the mean direction of fractures in 3-D onto the East-West plane. The trace length was assumed to follow a normal distribution with a specific fracture intensity P21 (total length of fracture traces per unit area). Table 2 shows the basic information of the identified three sets of fractures used for building the synthetic rock mass models. The shear strength parameters of all fractures are assumed to be cohesionless with cj ¼ 0 and angle of friction ϕj ¼ 30� .7 In addition, the infill of joints was not considered in this case. In order to build the SRM models, firstly, 12 intact rock samples with different sizes were generated based on the BPM, as shown in Fig. 3. The height-width ratio of samples was set to be 2. It should be noted that the scale effect of intact rocks was not considered in this research, which means the UCS is assumed to be the same for all intact rock samples. It is known that the strength of intact rocks is scale-dependent, which is caused by the existence of microcracks and flaws in the rocks. Therefore, in order to investigate the
4.1. Micro-parameters of the bonded particle model for intact rocks Uniaxial compression tests were carried out to calibrate the microparameters of the bonded particle model (BPM) for intact rocks to make sure the BPM model can provide the same mechanical properties as from laboratory tests. In uniaxial compression tests, a symmetric vertical constant loading velocity of 0.02 m/s was applied on top and bottom boundaries of rock models, as shown in Fig. 6. The details of calibration procedure can be found in the paper by Zhou et al.36 Firstly, the Poisson’s ratio v was obtained by calibrating the normal-to-shear stiffness ratio kn/ks; then the effective modulus Ec was modified to acquire the target elastic modulus; the last stage was the modification of the bond tensile strength t and bond cohesion cb to obtain the desired value of UCS. Fig. 6 shows a typical failure pattern and the associated stress and strain responses of the 10 � 20 m intact rock specimen modeled by BPM under the uniaxial compression test. The values of micro-parameters of this intact rock model with ID ¼ k are listed in Table 3. The values of mechanical properties (UCS, elastic modulus and Poisson’s ratio) generated by the BPM using these micro-parameters are shown in Table 4, which are close to laboratory test results in Section 3.2. It should be noted that the value of UCS was obtained directly from the
Fig. 2. Synthetic rock mass components. 4
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104179
Fig. 3. Intact rock samples with different sizes.
between the fracture and the minor principal stress direction. Four rock mass samples with θ ¼ 40� , 45� , 50� and 55� were used for biaxial tests. In these samples, the intact rock was modeled with the BPM using the micro-parameters listed in Table 3, and the strength and deformation behaviors of fractures were modeled with the SJM. During the biaxial test, the lateral stress was loaded to the samples with a confining stress σ3 ¼ 1MPa in horizontal directions. A symmetric vertical constant loading velocity of 0.02 m/s was applied on top and bottom boundaries of rock mass models and the peak strength σ1 was recorded when the sample fails, as shown in Fig. 7. Then, the normal stress σn and shear stress τ acting on the fracture plane can be calculated using the critical stress state (σ1, σ 3). Parametric studies were carried out to get the optimal micro-parameters of the fracture to ensure that the rock sample can generate the same response as that from field data. Fig. 7 presents the final stress-strain curves of four rock samples after the final calibration. These optimal values of input micro-parameters (knj ¼ 2500GPa/m, ksj ¼ 500GPa/m, μj ¼ 0.2) could give proper macro-properties of the fracture with cj ¼ 0 and ϕj ¼ 29.78� , as shown in Fig. 8, which is identical to field data with cj ¼ 0 and ϕj ¼ 30� .
Fig. 4. The big DFN model (right) and the corresponding 12 small DFN models (left) extracted and used for generating SRM samples.
maximum strength of the axial stress-strain curve of the uniaxial compression tests. The elastic modulus was estimated using the tangent modulus at the 50% UCS. The Poisson’s ratio was estimated by taking the slope of the radial strain-axial train curve between 20% and 30% of maximum strength.
5. The proposed method for estimating REV based on GSI 5.1. Quantify fracturing for SRM using equivalent GSI
4.2. Micro-parameters of the smooth joint model for fractures
Properties of the intact rock and discontinuities are necessary and identical input parameters for both the HB criterion and the SRM models. Therefore, if the isotropic condition for the rock mass properties is satisfied, then the jointed SRM model could be treated as a rock mass satisfying the HB criterion. In general, the GSI is used for quantifying the fracturing of large rock masses. In order to quantify the degree of fracturing of these synthetic rock mass samples with the size ranging from 0.05 m to 12 m (Fig. 5), a concept of the equivalent GSI proposed by Alejano et al.5 was used in this research.
In this research, the smooth joint model (SJM) was used to simulate the behavior of a planar interface. The model has three input microparameters: normal stiffness knj, shear stiffness ksj and friction coeffi cient μj. The selection of knj, ksj and μj was based on the inverse-modeling calibration approach based on biaxial compression tests to ensure that the rock mass model can generate the same response as that from site data with cj ¼ 0 and ϕj ¼ 30� . Synthetic rock mass samples used for numerical simulation have a width of 50mm and a height of 100mm. θ is the inclination angle 5
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104179
Fig. 5. Synthetic rock mass samples with different sizes.
The strategy to calculate the equivalent GSI for SRM models is to use the HB failure criterion to fit biaxial data using a regression method. The procedure to calculate the equivalent GSI is introduced as follows: Table 4 Mechanical properties calculated from BPM models.
Fig. 6. The mechanical response of the intact rock under the uniaxial compression test.
Sample ID
Sample length (m)
UCS (MPa)
Ei (GPa)
v
ρ (g/cm3)
a b c d e f g h i j k l
0.05 0.1 0.2 0.5 1.5 3.5 5 7 8 9 10 12
205 205 204 205 205 205 205 205 205 205 205 205
105 106 104 105 105 104 106 106 105 105 105 105
0.28 0.31 0.30 0.31 0.29 0.30 0.29 0.30 0.29 0.30 0.31 0.31
4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3
Table 3 Micro-parameters of the BPM models. Sample ID
Sample length (m)
Rmin (m)
Rmax/Rmin
kn/ks
Ec (GPa)
t (MPa)
ϕb (� )
cb (MPa)
μ
e
a b c d e f g h i j k l
0.05 0.1 0.2 0.5 1.5 3.5 5 7 8 9 10 12
0.0005 0.0010 0.0015 0.003 0.008 0.020 0.026 0.035 0.040 0.045 0.050 0.055
1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66
2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3
73.0 73.2 73.0 72.8 72.9 72.6 73.1 73.0 73.2 73.0 73.0 73.1
116.4 110.5 123.2 126.2 118.2 122.8 118.9 114.8 126.2 114.0 120.2 123.6
20 20 20 20 20 20 20 20 20 20 20 20
116.4 110.5 123.2 126.2 118.2 122.8 118.9 114.8 126.2 114.0 120.2 123.6
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16
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model. 5.2. Comprehensive results of equivalent GSI for SRM models We carried out extensive numerical analyses to identify the equiva lent GSI values of 120 SRM samples and presented results in Fig. 10. The black dots in Fig. 10 represent the mean value of GSI for a given size of SRM models. It is shown that the values of GSI scatter at small sizes while tend to converge with the increase of the sample size. This is because when the sample size is smaller, there might be no fractures or few fractures in the sample, resulting in the obvious differences of GSI values. When the sample size is relatively large, the fracture patterns in the samples tend to be similar. Thus, the value of GSI tends to converge to a stable level. In this research, the Coefficient of Variance (CV) was used to calculate the REV size of rock masses. CV could quantify the random influence of a group of data by calculating the value of Standard Devi ation/Mean7、10、13、33. According to Esmaieli et al.7 and Mahnaz et al.,33 when the value of CV was less than 20%, the sample size could be regarded as REV. The results of REV in Table 6 show that the sample (j) with the size of 7 � 14 m gives the REV size of the Brunswick mine rock masses.
Fig. 7. The stress-strain curves of rock mass samples under biaxial tests.
5.3. Comparison with the mechanical REV size In order to test the reliability of the proposed GSI-based REV size, we also calculated the REV size of this Brunswick mine rock mass using traditional mechanical indicators, the UCS and deformation modulus E. The calculation results are shown in Figs. 11 and 12. The results of the CV are listed in Table 7. It is found that the UCS and E-based indicators give relatively small REV sizes (5 � 10 m) compared with the GSI-based indicator (7 � 14 m). This could be explained by the fact that the UCS and E-based indicators consider only one type of mechanical informa tion, compared with the GSI-based indicator, which contains two types of information: geometrical information of rock structure and mechan ical information of the discontinuity. As stated by Zhang et al.,9 if the REV calculation accounts for more information of fractures, such as density, length, dip direction and dip angle, the REV can have more obvious practical applications. That means, in general, more informa tion included in the indicator for estimating REV values, the better rock mass properties will be achieved. In this regard, the proposed GSI-based indicator would be useful for practical engineering application.
Fig. 8. The σn-τ curve of the fracture under biaxial tests.
Firstly, substitute Eqs. (2) to (4) into Eq. (1), the HB failure criterion can be expressed as σ1 ¼ f (GSI, mi, σ ci, D and σ3). The value of blasting damage factor D was assumed to be zero in this research. Then, the values of σci and mi were estimated from the numerical biaxial tests of the intact rocks (Fig. 3) over a confining stress σ3 ranging from 0 to 0.5σci by using regression analyses suggested by Hoek and Brown27. The values of mi and σci_fitted of various scales of intact rocks (Fig. 3) are listed in Table 5. Finally, the equivalent GSI for SRM models can be calculated from the HB failure criterion, which can be expressed as σ1 ¼ f (GSI, mi, σ ci_fitted, D and σ3), to fit biaxial data using a regression method. Take the SRM model (ID ¼ k) as an example, the values of mi ¼ 3.28, σci_fitted ¼ 211.56MPa and D ¼ 0. The values of biaxial data (σ3, σ 1) and the final fitting result are shown in Fig. 9, which gives GSI ¼ 50 for this SRM
6. Conclusions In this research, we propose a new indicator to estimate the repre sentative elementary volume (REV) of jointed rock masses based on the
Table 5 mi and σci_fitted of the intact rock models. Sample ID
Sample size (m)
mi
σci_fitted (MPa)
a b c d e f g h i j k l
0.05 0.1 0.2 0.5 1.5 3.5 5 7 8 9 10 12
2.84 2.95 3.06 3.37 2.83 3.22 3.56 2.79 3.36 3.14 3.28 3.03
211.75 212.85 217.51 218.02 214.04 212.30 203.72 214.03 210.44 209.10 211.56 217.32
Fig. 9. GSI-equivalent result for the SRM model (ID ¼ k). 7
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Table 7 CV results for estimating REV size based on UCS and E.
Table 6 CV results for estimating GSI-based REV. Sample size (m)
CV of GSI
a b c d e f g h
0.05 0.1 0.2 0.5 1.5 3.5 5 7
0.20 0.33 0.37 0.57 0.34 0.31 0.22 0.13
Sample size (m)
CV of UCS
CV of E
a b c d e f g h
0.05 0.1 0.2 0.5 1.5 3.5 5 7
0.51 0.73 0.88 1.93 1.11 0.56 0.19 0.18
0.16 0.25 0.30 0.29 0.55 0.62 0.19 0.17
Geological Strength Index (GSI). Compared with the existing two geometrical and mechanical categories, the proposed GSI-based indi cator has merits of reflecting the geometrical characteristics of rock structures, but also containing both geometrical and mechanical prop erties of discontinuities. For this purpose, the synthetic rock mass (SRM) models of the Brunswick mine rocks, based on the discrete fracture network (DFN) and the bonded particle model (BPM) techniques, were constructed in PFC2D to investigate the strength of fractured rock masses with increasing the size of the rock mass models up to a REV size. The microparameters of the SRM model were calibrated using the published experimental intact rock and field joint data to make sure the SRM model is capable of capturing the mechanical behaviors of fractured rock masses. Biaxial tests on the 120 SRM models with different sizes were carried out to get their failure envelopes, and then, the Hoek-Brown (HB) failure criterion was used to fit the biaxial data to get the values of equivalent GSI of the SRM models. Based on the evaluation of CV value of equiv alent GSI values, GSI-based REV size (7 � 14 m) is obtained, which is relatively larger compared with the traditional UCS and E-based REVs (5 � 10 m).
Fig. 10. Relationships between GSI and sample size of SRM models.
Sample ID
Sample ID
Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgment This research has been funded by the National Natural Science Foundation of China (No.51504218) and the State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining and Technology (SKLGDUEK1808) and the Australian Endeavour Research Fellowship.
Fig. 11. Relationships between UCS and sample size of SRM models.
References 1 Shen J, Priest SD, Karakus M. Determination of Mohr-Coulomb shear strength parameters from generalized Hoek-Brown criterion for slope stability analysis. Rock Mech Rock Eng. 2012;45:123–129. 2 Zheng H, Li T, Shen J, Xu C, Sun H, Lü Q. The effects of blast damage zone thickness on rock slope stability. Eng Geol. 2018;246:19–27. 3 Zhang G, Karakus M, Tang H, Ge Y, Zhang L. A new method estimating the 2D joint roughness coefficient for discontinuity surfaces in rock masses. Int J Rock Mech Min Sci. 2014;72:191–198. 4 Hoek E, Brown ET. The Hoek-Brown failure criterion and GSI-2018 edition. J Rock Mech.Geotech Eng. 2019;11(3):445–463. 5 Alejano LR, Arzúa J, Bozorgzadeh N, Harrison JP. Triaxial strength and deformability of intact and increasingly jointed granite samples. Int J Rock Mech Min Sci. 2017;95: 87–103. 6 Bear J. Dynamics of Fluids in Porous Media. Amsterdam: Elsevier; 1972. 7 Esmaieli K, Hadjigeorgiou J, Grenon M. Estimating geometrical and mechanical REV based on synthetic rock mass models at Brunswick Mine. Int J Rock Mech Min Sci. 2010;47(6):915–926. 8 Chae BG, Seo YS. Homogenization analysis for estimating the elastic modulus and representative elementary volume of Inada Granite in Japan. Geosci J. 2011;15(4): 387–394.
Fig. 12. Relationships between deformation modulus and sample size of SRM models.
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104179 23 Wang Z, Li W, Bi L, Qiao L, Liu R, Liu J. Estimation of the REV size and equivalent permeability coefficient of fractured rock masses with an emphasis on comparing the radial and unidirectional flow configurations. Rock Mech Rock Eng. 2018;51(5): 1457–1471. 24 Bieniawski Z. Engineering classification of jointed rock masses. Civ Eng South Afr. 1973;15(12). 25 Hoek E, Brown ET. Empirical strength criterion for rock masses. J Geotech Geoenviron Eng. 1980;106:1013–1035. 26 Hoek E, Carranza-Torres C, Corkum B. Hoek-Brown failure criterion-2002 Edition. In: Proceedings of the Fifth North American Rock Mechanics Symposium. vol. 1. 2002: 18–22. 27 Hoek E, Brown ET. Practical estimates of rock mass strength. Int J Rock Mech Min Sci. 1997;34(8):1165–1186. 28 Shen J, Karakus M. Simplified method for estimating the Hoek-Brown constant for intact rocks. J Geotech Geoenviron Eng. 2014;140(6), 04014025. 29 Wang W, Shen J. Comparison of existing methods and a new tensile strength based model in estimating the Hoek-Brown constant mi for intact rocks. Eng Geol. 2017;224: 87–96. 30 Sonmez H, Ulusay R. Modifications to the geological strength index (GSI) and their applicability to stability of slopes. Int J Rock Mech Min Sci. 1999;36(6):743–760. 31 Cai M, Kaiser PK, Uno H, Tasakab Y, Minami M. Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system. Int J Rock Mech Min Sci. 2004;41(1):3–19. 32 Hoek E, Carter TG, Diederichs MS. In: Quantification of the Geological Strength Index Chart. 47th US Rock Mechanics/geomechanics Symposium. American Rock Mechanics Association; 2013. 33 Laghaei M, Baghbanan A, Hashemolhosseini H, Dehghanipoodeh M. Numerical determination of deformability and strength of 3D fractured rock mass. Int J Rock Mech Min Sci. 2018;110:246–256. 34 Poulsen B, Adhikary D, Elmouttie M, Wilkins A. Convergence of synthetic rock mass modelling and the Hoek-Brown strength criterion. Int J Rock Mech Min Sci. 2015;80: 171–180. 35 Zhang Q, Zhu H, Zhang L, Ding X. Study of scale effect on intact rock strength using particle flow modeling. Int J Rock Mech Min Sci. 2011;48(8):1320–1328. 36 Zhou C, Xu C, Karakus M, Shen J. A systematic approach to the calibration of microparameters for the flat-jointed bonded particle model. GeoMech Eng. 2018;16(5): 471–482.
9 Zhang W, Chen J, Liu C, Huang R, Li M, Zhang Y. Determination of geometrical and structural representative volume elements at the Baihetan dam site. Rock Mech Rock Eng. 2012;45(3):409–419. 10 Xia L, Zheng Y, Yu Q. Estimation of the REV size for blockiness of fractured rock masses. Comput Geotech. 2016;76:83–92. 11 Ni P, Wang S, Wang C, Zhang S. Estimation of REV size for fractured rock mass based on damage coefficient. Rock Mech Rock Eng. 2017;50(3):555–570. 12 Li Y, Chen J, Shang Y. Determination of the geometrical REV based on fracture connectivity: a case study of an underground excavation at the Songta dam site, China. Bull Eng Geol Environ. 2018;77(4):1599–1606. 13 Min KB, Jing L. Numerical determination of the equivalent elastic compliance tensor for fractured rock masses using the distinct element method. Int J Rock Mech Min Sci. 2003;40(6):795–816. 14 Pariseau WG, Puri S, Schmelter SC. A new model for effects of impersistent joint sets on rock slope stability. Int J Rock Mech Min Sci. 2008;45(2):122–131. 15 Khani A, Baghbanan A, Norouzi S, Hashemolhosseini H. Effects of fracture geometry and stress on the strength of a fractured rock mass. Int J Rock Mech Min Sci. 2013;60 (8):345–352. 16 Yang J, Chen W, Yang D, Yuan J. Numerical determination of strength and deformability of fractured rock mass by FEM modeling. Comput Geotech. 2015;64: 20–31. 17 Farahmand K, Vazaios I, Diederichs MS, Vlachopoulos N. Investigating the scaledependency of the geometrical and mechanical properties of a moderately jointed rock using a synthetic rock mass (SRM) approach. Comput Geotech. 2018;95: 162–179. 18 Kulatilake PHSW, Panda BB. Effect of block size and joint geometry on jointed rock hydraulics and REV. J Eng Mech. 2000;126(8):850–858. 19 Wang M, Kulatilake P, Um J, Narvaiz J. Estimation of REV size and threedimensional hydraulic conductivity tensor for a fractured rock mass through a single well packer test and discrete fracture fluid flow modeling. Int J Rock Mech Min Sci. 2002;39(7):887–904. 20 Min KB, Jing L, Stephansson O. Determining the equivalent permeability tensor for fractured rock masses using a stochastic REV approach: method and application to the field data from Sellafield, UK. Hydrogeol J. 2004;12(5):497–510. 21 Chen SH, Feng XM, Isam S. Numerical estimation of REV and permeability tensor for fractured rock masses by composite element method. Int J Numer Anal Methods Geomech. 2008;32:1459–1477. 22 Rong G, Peng J, Wang X, Liu G, Hou D. Permeability tensor and representative elementary volume of fractured rock masses. Hydrogeol J. 2013;21:1655–1671.
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Contents lists available at ScienceDirect
International Journal of Rock Mechanics and Mining Sciences journal homepage: http://www.elsevier.com/locate/ijrmms
Estimation of REV for fractured rock masses based on Geological Strength Index Huajie Huang a, Jiayi Shen a, b, c, *, Qiushi Chen c, Murat Karakus d a
Institute of Port, Coastal and Offshore Engineering, Zhejiang University, Hangzhou, 310015, China State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining & Technology, Xuzhou, 221116, China c Glenn Department of Civil Engineering, Clemson University, Clemson, SC, 29634, USA d School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA, 5005, Australia b
A R T I C L E I N F O
A B S T R A C T
Keywords: GSI REV Synthetic rock mass Hoek-Brown Numerical simulation
The representative elementary volume (REV) of fractured rock masses is a significant index to investigate the rock mass behaviors in the continuum mechanics. In this research, a new indicator to estimate the REV size of fractured rock masses based on the Geological Strength Index (GSI) is proposed. For this purpose, a new method that combines the PFC-based synthetic rock mass (SRM) model with the Hoek-Brown (HB) failure criterion is proposed to investigate the strength and deformation properties of fractured rock masses under biaxial stress conditions. Extensive numerical analyses are carried out to estimate variation of the uniaxial compression strength (UCS), deformation modulus (E) and GSI of the Brunswick mine rock mass with increasing the size of the SRM models up to a REV size. Results show that the GSI-based indicator gives relatively larger REV size compared with the traditional UCS or deformation modulus (E) based indicators. Compared with the traditional indicators, the proposed GSI-based indicator has merits of not only reflecting the geometrical characteristics of rock structures but also containing both geometrical and mechanical properties of discontinuities.
1. Introduction
corresponding results can be applied to represent the overall rock mass properties. The concept of REV was first proposed by Bear6 to study fluid flow in a porous medium. Over the last decades, various indicators have been proposed to estimate the REV size by many researchers. Generally, these indicators can be categorized into three groups, namely, geometrical indicators, mechanical indicators and hydraulic indicators, as shown in Table 1. The geometrical and mechanical indicators can be used for various engineering applications, while the hydraulic indicators are specifically used for projects that consider water flow conditions. In order to obtain realistic mechanical properties of jointed rock masses, large volumes of rock mass configurations should be tested. However, as Bieniawski24 noted, investigating the effect of sample size on the mechanical properties of a large volume jointed rock mass using the in-situ testing approach is not only difficult but also time-consuming and very expensive. Therefore, numerical simulations were widely used by many researchers7,11,15,16 to estimate the large scale rock mass properties using an upscaling procedure which involves estimating the variation of the mechanical properties with increasing the size of the examined rock volumes up to the REV size. After the determination of
The determination of rock mass mechanical properties is critical for the design of many engineering infrastructures, such as slopes, tunnels and foundations.1,2 However, it is not easy to estimate the strength and deformation characteristics of fractured rock masses because of the ex istence of various discontinuities, like joints, fractures, faults and bedding planes in the rock masses. These discontinuities are often different in orientation, trace length, waviness and aperture, which make the rock mass inhomogeneous and anisotropic.3,4 Alejano et al.5 have shown that the strength and deformation prop erties of rock masses are significantly influenced by the rock structure, discontinuity properties and stress states. It is also known that these rock mass properties gradually change with the increase of rock mass sizes. When the rock mass size is relatively small, a slight increase in sample size may result in an obvious change in mechanical characteristics. However, rock mass properties no longer change with an increase of sample size when the sample size is larger than a critical value. Such critical value is named as the representative elementary volume (REV) size. When the sample is greater than or equal to the REV size, the
* Corresponding author. Institute of Port, Coastal and Offshore Engineering, Zhejiang University, Hangzhou, 310015, China. E-mail address: [email protected] (J. Shen). https://doi.org/10.1016/j.ijrmms.2019.104179 Received 15 January 2019; Received in revised form 2 November 2019; Accepted 15 December 2019 Available online 27 December 2019 1365-1609/© 2019 Elsevier Ltd. All rights reserved.
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Table 1 Indicators for estimating the REV of rock masses. Authors
Indicators
Group 1 Geometrical indicators
Esmaieli et al.7 Chae and Seo8 Zhang et al.9 Xia et al.10 Ni et al.11 Li et al.12
Fracture intensity P32 Ratio of the micro-cracks Fracture intensity P32 Blockiness Fracture intensity P32 Geometrical connectivity
Group 2 Mechanical indicators
Min and Jing13
Deformation modulus and Poisson’s ratio Deformation modulus and shear modulus UCS and deformation modulus Deformation modulus and Poisson’s ratio UCS and deformation modulus Damage coefficient UCS and deformation modulus
Pariseau et al.14 Esmaieli et al.7 Khani et al.15 Yang et al.16 Ni et al.11 Farahmand et al.17
Group 3 Hydraulic indicators
Kulatilake and Panda18 Wang et al.19 Min et al.20 Chen et al.21 Rong et al.22 Wang et al.23
�
�a
σ 1 ¼ σ 3 þ σci mb
σ3 þs σci
(1)
where σ1 and σ3 are the major and minor principal stresses respectively. mb, s, and a are HB input parameters, which depend on the fracturing degree of the rock mass and can be estimated from the intact HB con stant mi, GSI and D, given by mb ¼ mi eðGSI s ¼ eðGSI
100Þ=ð28 14DÞ
(3)
100Þ=ð9 3DÞ
1 eð a¼ þ 2
GSI=15Þ
(2)
eð
20=3Þ
6
(4)
The details of calculation and selection of mi and D can be found in the papers by Hoek and Brown,27 Hoek et al.,26 Shen and Karakus,28 Wang and Shen29 and Hoek and Brown.4 The GSI is a rock mass classification system that has been developed in engineering rock mechanics to meet the need for estimating reliable rock mass properties for designing and analyzing tunnels, slopes or foundations. The value of GSI is estimated from visual examination of the rock mass exposed in outcrops, therefore, the heart of the GSI clas sification is a careful engineering geology description of the rock mass which involves two factors, rock structure and discontinuity surface condition. However, due to the lack of measurable parameters and related in terval limits or ratings for describing the rock structure and the surface conditions of the discontinuities, the use of the GSI chart involves some subjectivity. In order to facilitate the use of the classification system especially by inexperienced engineers, many researchers30–32 proposed quantitative GSI charts by introducing measurable parameters. For example, Cai et al.31 made an attempt to provide a more quantitative approach to evaluate GSI. For this purpose, two terms, block volume (Vb) and joint condition factor (Jc) were introduced into the existing GSI classification scheme. In order to assign ratings to these terms, the use of some easily measurable input parameters such as roughness, weath ering, infilling and volumetric joint count were suggested. According to the selected rating intervals, the GSI chart has been modified, as shown in Fig. 1. By adding measurable quantitative inputs for a quantitative output, the system becomes less subjective while maintains its overall simplicity. Once a GSI value has been decided upon, it can be used in conjunction with appropriate empirical equations to calculate the strength and deformation of a rock mass. However, it should be noted that the rock mechanical properties of fractured rock masses are generally anisotropic and stress path depen dent13、33. The HB failure criterion based on the GSI system is used for predicting isotropic rock mass strength which cannot reflect such anisotropic behavior and stress path dependent characteristics. As GSI not only describes the geometrical feature of rock masses but also relates the mechanical properties of the discontinuities, a GSI-based indicator could be ideal to calculate the REV size for a rock mass, overcoming limitations of the traditional indicators based on either geometrical or mechanical characteristics alone.
Average block permeability Hydraulic conductivity Permeability tensor Permeability tensor Permeability tensor Permeability coefficient
the REV size, the overall strength and deformation characteristics of rock masses for a specific engineering project could be estimated and the rock mass with the size of REV could be regarded as isotropic materials. Although there are various indicators available as listed in Table 1, one still must take into account the following limitations when using them for the analysis of REV size for different rock projects: (1) Geometrical indicators can only be used for initial estimation for the REV size in a time-efficient manner at an early stage of a project. As geometrical indicators can only reflect the rock mass structures, it cannot provide information of mechanical parameters of discontinuities, such as the mechanical effects of infilling of discontinuities, which reduce its reliability. (2) The existing mechanical indicators for esti mating REV size are based on the uniaxial compression tests, which can only represent the rock mass behavior under the unconfined stress state. However, rock masses are generally in states of triaxial stresses and the properties will be significantly influenced by the triaxial stress states. With the aim of achieving a better understanding of the fundamental rock mass mechanisms and improving the accuracy of the determination of REV size, in this research we propose a new indicator to estimate the REV size of fractured rock masses based on the Geological Strength Index (GSI) classification which has merits of not only reflecting the geometrical characteristic of rock structures but also containing both geometrical and mechanical properties of discontinuities. In order to establish such GSI-based REV indicator, we also propose a new method that combines the PFC-based synthetic rock mass (SRM) model with the Hoek-Brown (HB) failure criterion to investigate the strength and deformation of fractured rock masses under biaxial stress conditions. 2. Hoek-Brown criterion based on Geological Strength Index
3. Synthetic rock mass for numerical analyses
In rock engineering, the Hoek-Brown (HB) criterion based on the Geological Strength Index (GSI) has been widely used for predicting isotropic rock mass strength for many decades. The original HB crite rion, which was proposed by Hoek and Brown25 for estimating the intact rock strength, requires two intact rock properties, namely, the uniaxial compressive strength (UCS or σci) and a constant mi of the intact rock. It was then extended to estimate the rock mass strength using the GSI and disturbance factor D.26 The latest version of the HB criterion presented by Hoek and Brown4 is expressed in Eq. (1).
In this research, the synthetic rock mass (SRM) model of the Bruns wick mine rocks was employed as an example to illustrate the method of estimating REV based on the GSI indicator using an upscaling process. The upscaling procedure involves estimating the variation of the me chanical properties with increasing the size of the rock mass volumes up to a REV in which stationary limits are reached for the properties under study.
2
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Fig. 1. Quantification of GSI chart (Cai et al.31).
3
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3.1. PFC2D-based synthetic rock mass model
Table 2 Fracture set field data for fracture system generation.
The SRM model, which was constructed in PFC2D software in this research, is based on two well-developed numerical techniques, the discrete fracture network (DFN) and the bonded particle model (BPM). A DFN is generally generated using the Monte-Carlo simulation. Several statistical parameters, including the mean and standard devia tion of joint set dip angle and trace length, are required to control the orientation, size and intensity of each joint set in the DFN. In BPM, rigid particles are bonded together to form the basic ele ments of a hard rock. The bond represents the rock cohesive component and breaks when the external stress exceeds the prescribed bond strength. The particle movement is influenced by frictional resistance. As the BPM explicitly considers the particle-scale mechanical behavior, it requires no continuum-scale constitutive model (such as the MohrCoulomb and Hoek-Brown criteria) to control the macroscopic me chanical rock behaviors. The micro-parameters in BPM can be identified following a calibration process to assure the similarity of the macroproperties can be obtained. The SRM is then constructed by inserting a DFN into a BPM assembly and a “smooth joint” logic is developed to assert smooth particle sliding along joints. The smooth joint model (SJM), which resolves the local force-displacement law onto the plane of the joint, could be used to simulate the behavior of such “smooth joint”. By this approach, bonded behavior, Coulomb sliding and tensile opening can be realistically rep resented.34 Fig. 2 shows the main components of a synthetic rock mass.
Fracture characteristic
Set 1
Set 2
Set 3
2D-dip (deg) Trace length (m) P21 (m 1)
90 1.40 0.63
89 1.44 1.80
22 1.16 1.95
scale effect of intact rocks in PFC2D, initial micro defects need to be added into intact rocks,35 which is not considered in this research. Based on the geometrical parameters of fractures listed in Table 2, a big DFN model with the size of 100 � 200 m was generated using the Monte Carlo method. 12 small DFNs with the same sizes corresponding to the 12 intact rock samples (see Fig. 3) were extracted at the center of the big DFN, as shown in Fig. 4. Moreover, in order to avoid the random effect in the DFN model, we generated 10 big DFN models and repeated the procedure of extracting small samples from the big DFN models, resulting in a total of 120 DFN models for subsequent SRM model gen erations and numerical simulations. Finally, the intact rock samples (Fig. 3) were combined with DFN models (Fig. 4) to generate the SRM models. Fig. 5 shows a typical set of 12 SRM models with various sizes. In total, 120 SRM samples were generated in PFC2D. 4. Calibration of synthetic rock mass models The micro-parameters of intact rocks and fractures in the SRM samples need to be identified following a calibration process to assure the rock mass strength and deformation properties estimated from SRM models are the same as that from laboratory and field data.7
3.2. SRM model of the brunswick mine rock mass Rock mass data was collected from Brunswick Mine,7 which is located in northeastern New Brunswick, Canada. The deposit is hosted mainly by metamorphosed volcaniclastic sediments and tuffs that overlie felsic volcanic rocks and have a high specific gravity (4.3) and stiffness (UCS up to 205MPa). The elastic modulus of the intact massive sulphide is 104GPa and the Poisson’s ratio is 0.29. In this case, the mean values of joints dip angles in 2-D were calcu lated from the mean direction of fractures in 3-D onto the East-West plane. The trace length was assumed to follow a normal distribution with a specific fracture intensity P21 (total length of fracture traces per unit area). Table 2 shows the basic information of the identified three sets of fractures used for building the synthetic rock mass models. The shear strength parameters of all fractures are assumed to be cohesionless with cj ¼ 0 and angle of friction ϕj ¼ 30� .7 In addition, the infill of joints was not considered in this case. In order to build the SRM models, firstly, 12 intact rock samples with different sizes were generated based on the BPM, as shown in Fig. 3. The height-width ratio of samples was set to be 2. It should be noted that the scale effect of intact rocks was not considered in this research, which means the UCS is assumed to be the same for all intact rock samples. It is known that the strength of intact rocks is scale-dependent, which is caused by the existence of microcracks and flaws in the rocks. Therefore, in order to investigate the
4.1. Micro-parameters of the bonded particle model for intact rocks Uniaxial compression tests were carried out to calibrate the microparameters of the bonded particle model (BPM) for intact rocks to make sure the BPM model can provide the same mechanical properties as from laboratory tests. In uniaxial compression tests, a symmetric vertical constant loading velocity of 0.02 m/s was applied on top and bottom boundaries of rock models, as shown in Fig. 6. The details of calibration procedure can be found in the paper by Zhou et al.36 Firstly, the Poisson’s ratio v was obtained by calibrating the normal-to-shear stiffness ratio kn/ks; then the effective modulus Ec was modified to acquire the target elastic modulus; the last stage was the modification of the bond tensile strength t and bond cohesion cb to obtain the desired value of UCS. Fig. 6 shows a typical failure pattern and the associated stress and strain responses of the 10 � 20 m intact rock specimen modeled by BPM under the uniaxial compression test. The values of micro-parameters of this intact rock model with ID ¼ k are listed in Table 3. The values of mechanical properties (UCS, elastic modulus and Poisson’s ratio) generated by the BPM using these micro-parameters are shown in Table 4, which are close to laboratory test results in Section 3.2. It should be noted that the value of UCS was obtained directly from the
Fig. 2. Synthetic rock mass components. 4
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Fig. 3. Intact rock samples with different sizes.
between the fracture and the minor principal stress direction. Four rock mass samples with θ ¼ 40� , 45� , 50� and 55� were used for biaxial tests. In these samples, the intact rock was modeled with the BPM using the micro-parameters listed in Table 3, and the strength and deformation behaviors of fractures were modeled with the SJM. During the biaxial test, the lateral stress was loaded to the samples with a confining stress σ3 ¼ 1MPa in horizontal directions. A symmetric vertical constant loading velocity of 0.02 m/s was applied on top and bottom boundaries of rock mass models and the peak strength σ1 was recorded when the sample fails, as shown in Fig. 7. Then, the normal stress σn and shear stress τ acting on the fracture plane can be calculated using the critical stress state (σ1, σ 3). Parametric studies were carried out to get the optimal micro-parameters of the fracture to ensure that the rock sample can generate the same response as that from field data. Fig. 7 presents the final stress-strain curves of four rock samples after the final calibration. These optimal values of input micro-parameters (knj ¼ 2500GPa/m, ksj ¼ 500GPa/m, μj ¼ 0.2) could give proper macro-properties of the fracture with cj ¼ 0 and ϕj ¼ 29.78� , as shown in Fig. 8, which is identical to field data with cj ¼ 0 and ϕj ¼ 30� .
Fig. 4. The big DFN model (right) and the corresponding 12 small DFN models (left) extracted and used for generating SRM samples.
maximum strength of the axial stress-strain curve of the uniaxial compression tests. The elastic modulus was estimated using the tangent modulus at the 50% UCS. The Poisson’s ratio was estimated by taking the slope of the radial strain-axial train curve between 20% and 30% of maximum strength.
5. The proposed method for estimating REV based on GSI 5.1. Quantify fracturing for SRM using equivalent GSI
4.2. Micro-parameters of the smooth joint model for fractures
Properties of the intact rock and discontinuities are necessary and identical input parameters for both the HB criterion and the SRM models. Therefore, if the isotropic condition for the rock mass properties is satisfied, then the jointed SRM model could be treated as a rock mass satisfying the HB criterion. In general, the GSI is used for quantifying the fracturing of large rock masses. In order to quantify the degree of fracturing of these synthetic rock mass samples with the size ranging from 0.05 m to 12 m (Fig. 5), a concept of the equivalent GSI proposed by Alejano et al.5 was used in this research.
In this research, the smooth joint model (SJM) was used to simulate the behavior of a planar interface. The model has three input microparameters: normal stiffness knj, shear stiffness ksj and friction coeffi cient μj. The selection of knj, ksj and μj was based on the inverse-modeling calibration approach based on biaxial compression tests to ensure that the rock mass model can generate the same response as that from site data with cj ¼ 0 and ϕj ¼ 30� . Synthetic rock mass samples used for numerical simulation have a width of 50mm and a height of 100mm. θ is the inclination angle 5
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Fig. 5. Synthetic rock mass samples with different sizes.
The strategy to calculate the equivalent GSI for SRM models is to use the HB failure criterion to fit biaxial data using a regression method. The procedure to calculate the equivalent GSI is introduced as follows: Table 4 Mechanical properties calculated from BPM models.
Fig. 6. The mechanical response of the intact rock under the uniaxial compression test.
Sample ID
Sample length (m)
UCS (MPa)
Ei (GPa)
v
ρ (g/cm3)
a b c d e f g h i j k l
0.05 0.1 0.2 0.5 1.5 3.5 5 7 8 9 10 12
205 205 204 205 205 205 205 205 205 205 205 205
105 106 104 105 105 104 106 106 105 105 105 105
0.28 0.31 0.30 0.31 0.29 0.30 0.29 0.30 0.29 0.30 0.31 0.31
4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3 4.3
Table 3 Micro-parameters of the BPM models. Sample ID
Sample length (m)
Rmin (m)
Rmax/Rmin
kn/ks
Ec (GPa)
t (MPa)
ϕb (� )
cb (MPa)
μ
e
a b c d e f g h i j k l
0.05 0.1 0.2 0.5 1.5 3.5 5 7 8 9 10 12
0.0005 0.0010 0.0015 0.003 0.008 0.020 0.026 0.035 0.040 0.045 0.050 0.055
1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66 1.66
2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3 2.3
73.0 73.2 73.0 72.8 72.9 72.6 73.1 73.0 73.2 73.0 73.0 73.1
116.4 110.5 123.2 126.2 118.2 122.8 118.9 114.8 126.2 114.0 120.2 123.6
20 20 20 20 20 20 20 20 20 20 20 20
116.4 110.5 123.2 126.2 118.2 122.8 118.9 114.8 126.2 114.0 120.2 123.6
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4
0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16
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model. 5.2. Comprehensive results of equivalent GSI for SRM models We carried out extensive numerical analyses to identify the equiva lent GSI values of 120 SRM samples and presented results in Fig. 10. The black dots in Fig. 10 represent the mean value of GSI for a given size of SRM models. It is shown that the values of GSI scatter at small sizes while tend to converge with the increase of the sample size. This is because when the sample size is smaller, there might be no fractures or few fractures in the sample, resulting in the obvious differences of GSI values. When the sample size is relatively large, the fracture patterns in the samples tend to be similar. Thus, the value of GSI tends to converge to a stable level. In this research, the Coefficient of Variance (CV) was used to calculate the REV size of rock masses. CV could quantify the random influence of a group of data by calculating the value of Standard Devi ation/Mean7、10、13、33. According to Esmaieli et al.7 and Mahnaz et al.,33 when the value of CV was less than 20%, the sample size could be regarded as REV. The results of REV in Table 6 show that the sample (j) with the size of 7 � 14 m gives the REV size of the Brunswick mine rock masses.
Fig. 7. The stress-strain curves of rock mass samples under biaxial tests.
5.3. Comparison with the mechanical REV size In order to test the reliability of the proposed GSI-based REV size, we also calculated the REV size of this Brunswick mine rock mass using traditional mechanical indicators, the UCS and deformation modulus E. The calculation results are shown in Figs. 11 and 12. The results of the CV are listed in Table 7. It is found that the UCS and E-based indicators give relatively small REV sizes (5 � 10 m) compared with the GSI-based indicator (7 � 14 m). This could be explained by the fact that the UCS and E-based indicators consider only one type of mechanical informa tion, compared with the GSI-based indicator, which contains two types of information: geometrical information of rock structure and mechan ical information of the discontinuity. As stated by Zhang et al.,9 if the REV calculation accounts for more information of fractures, such as density, length, dip direction and dip angle, the REV can have more obvious practical applications. That means, in general, more informa tion included in the indicator for estimating REV values, the better rock mass properties will be achieved. In this regard, the proposed GSI-based indicator would be useful for practical engineering application.
Fig. 8. The σn-τ curve of the fracture under biaxial tests.
Firstly, substitute Eqs. (2) to (4) into Eq. (1), the HB failure criterion can be expressed as σ1 ¼ f (GSI, mi, σ ci, D and σ3). The value of blasting damage factor D was assumed to be zero in this research. Then, the values of σci and mi were estimated from the numerical biaxial tests of the intact rocks (Fig. 3) over a confining stress σ3 ranging from 0 to 0.5σci by using regression analyses suggested by Hoek and Brown27. The values of mi and σci_fitted of various scales of intact rocks (Fig. 3) are listed in Table 5. Finally, the equivalent GSI for SRM models can be calculated from the HB failure criterion, which can be expressed as σ1 ¼ f (GSI, mi, σ ci_fitted, D and σ3), to fit biaxial data using a regression method. Take the SRM model (ID ¼ k) as an example, the values of mi ¼ 3.28, σci_fitted ¼ 211.56MPa and D ¼ 0. The values of biaxial data (σ3, σ 1) and the final fitting result are shown in Fig. 9, which gives GSI ¼ 50 for this SRM
6. Conclusions In this research, we propose a new indicator to estimate the repre sentative elementary volume (REV) of jointed rock masses based on the
Table 5 mi and σci_fitted of the intact rock models. Sample ID
Sample size (m)
mi
σci_fitted (MPa)
a b c d e f g h i j k l
0.05 0.1 0.2 0.5 1.5 3.5 5 7 8 9 10 12
2.84 2.95 3.06 3.37 2.83 3.22 3.56 2.79 3.36 3.14 3.28 3.03
211.75 212.85 217.51 218.02 214.04 212.30 203.72 214.03 210.44 209.10 211.56 217.32
Fig. 9. GSI-equivalent result for the SRM model (ID ¼ k). 7
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104179
Table 7 CV results for estimating REV size based on UCS and E.
Table 6 CV results for estimating GSI-based REV. Sample size (m)
CV of GSI
a b c d e f g h
0.05 0.1 0.2 0.5 1.5 3.5 5 7
0.20 0.33 0.37 0.57 0.34 0.31 0.22 0.13
Sample size (m)
CV of UCS
CV of E
a b c d e f g h
0.05 0.1 0.2 0.5 1.5 3.5 5 7
0.51 0.73 0.88 1.93 1.11 0.56 0.19 0.18
0.16 0.25 0.30 0.29 0.55 0.62 0.19 0.17
Geological Strength Index (GSI). Compared with the existing two geometrical and mechanical categories, the proposed GSI-based indi cator has merits of reflecting the geometrical characteristics of rock structures, but also containing both geometrical and mechanical prop erties of discontinuities. For this purpose, the synthetic rock mass (SRM) models of the Brunswick mine rocks, based on the discrete fracture network (DFN) and the bonded particle model (BPM) techniques, were constructed in PFC2D to investigate the strength of fractured rock masses with increasing the size of the rock mass models up to a REV size. The microparameters of the SRM model were calibrated using the published experimental intact rock and field joint data to make sure the SRM model is capable of capturing the mechanical behaviors of fractured rock masses. Biaxial tests on the 120 SRM models with different sizes were carried out to get their failure envelopes, and then, the Hoek-Brown (HB) failure criterion was used to fit the biaxial data to get the values of equivalent GSI of the SRM models. Based on the evaluation of CV value of equiv alent GSI values, GSI-based REV size (7 � 14 m) is obtained, which is relatively larger compared with the traditional UCS and E-based REVs (5 � 10 m).
Fig. 10. Relationships between GSI and sample size of SRM models.
Sample ID
Sample ID
Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgment This research has been funded by the National Natural Science Foundation of China (No.51504218) and the State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining and Technology (SKLGDUEK1808) and the Australian Endeavour Research Fellowship.
Fig. 11. Relationships between UCS and sample size of SRM models.
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