1 Introduction

The Hoek–Brown failure criterion is widely used in rock mechanics and rock engineering practice for determining the strength of brittle intact rock and rock masses. This nonlinear semiempirical failure criterion was introduced by Hoek and Brown [33], and the following form was suggested for intact rock (see also [17]) (Eq. 1):

$$\sigma_{1} = \sigma_{3} + \sigma_{c} \left( {m_{i} \frac{{\sigma_{3} }}{{\sigma_{c} }} + 1} \right)^{0.5}$$
(1)

where \(\sigma_{1}\) and \(\sigma_{3}\) are major and minor principal stress at failure, respectively, mi is Hoek–Brown material constant, and \(\sigma_{c}\) is the uniaxial compressive strength of intact rock. According to Eq. (1), two independent parameters are necessary, namely the:

  • Uniaxial compressive strength (\(\sigma_{c}\))

  • Hoek–Brown material constant of the intact rock (mi)

Although several practical, empirical, and probabilistic approaches have been presented in the literature to address uniaxial compressive strength (UCS) [12, 34, 36, 73, 80], the accurate determination of mi is still challenging task and is influenced by many factors such as mineral composition, grain size, and cementation of rock. Generally, mi presents curve-fitting parameter for Hoek–Brown failure envelope [37]. However, researches by Zuo et al. [84] showed that mi is not a curve-fitting parameter but has physical meanings and can be derived from micro-mechanical principles (Hoek and Martin [38]).

Hoek and Brown [33,34,35] suggested that these values should be determined by numerous triaxial tests, applying different confining pressures (\({\sigma }_{3}\)) between zero and 0.5 \({\sigma }_{c}\). These laboratory tests are time-consuming, expensive, and in many cases, there are not enough (or suitable) samples. Singh et al. [70], Peng et al. [55], and Shen and Karakus [65] demonstrated that the reliability of mi values measured from triaxial test analysis depends on the quality and quantity of test data used in the analysis. They concluded that the range of \({\sigma }_{3}\) could have a significant influence on the calculation of mi. It is the reason why several methods were developed for determining the Hoek–Brown constant (mi) [72].

There are several techniques available to calculate mivalues in the absence of triaxial test. These approaches are entitled, Guidelines [32, 33], R-index [10, 33, 52, 59, 60, 62], UCS-based model [65, 74], tensile strength (TS)-based approach [77], and crack initiation stress-based model [10]. These methods are based on the rock lithological classifications and rock properties that can be easily obtained at an early stage of a project, which can be used in preliminary designs of engineering projects when triaxial test data are not available.

Aladejare and Wang [1] published a paper and introduced a new method for the determination of mi value. They developed a Bayesian approach for probabilistic characterization of Hoek–Brown constant mi through Bayesian integration of information from Hoek’s guideline chart, regression model, and site-specific UCS data. The UCS data used in this paper were obtained from testing granite samples collected from the Forsmark site, Sweden. In addition, several sets of simulated data are used to explore the evolution of mi as the number of site-specific data increases.

Wei et al. [78] experimentally investigated the effect of confining pressure \({\sigma }_{3}\) and critical crack parameter on determination of mi value. They applied ultrasonic test and load test results of limestone and verified the exponential impact of \({\sigma }_{3}\) on limestone, while the negative correlation was observed between critical crack parameter and mi.

Recently, Wen et al. [79] developed an empirical relation for estimation mi from \({\sigma }_{3}\) and \(\beta\), where \({\sigma }_{3}\) is minor principal stress and \(\beta\) is an angle between the major principal stress and weak plane based on substantial uniaxial and triaxial compression test data. They considered the effect of anisotropy on the value of Hoek constant (mi).

In the present study, however, we focused on the quasi-isotropic intact rocks and utilized the existing methods in the literature for predicting mi values. Furthermore, the detailed comparison is drawn between the obtained values of each approach to calculate which method provides the best estimation for the investigated rocks based on the calculated error function. Accordingly, new material constants are proposed for different igneous, sedimentary, and metamorphic rocks.

2 Determination of Hoek–Brown constant m i

In the absence of triaxial tests, there are several different methods available to estimate the Hoek–Brown constant mi, which are referred to as the R-index method [33], UCS-based model [65, 74], TS-based model [77], and combination method [3, 66] below. By using these methods, the H-B constant mi is obtained for the collected data, the achieved results are compared, and then, the error function was calculated and accordingly considering the lowest error value, the best data fit was presented and finally new material constants introduced.

2.1 Guideline method

Hoek and Brown [33] and Hoek [32] provided values of mi constants for different types of rocks. The values of mi are between 7 and 35; however, several factors that influence these values, such as mineral composition, foliation, grain size (texture), and cementation are among others [10].

The updated values of constant mi for intact rock were collected by [10], using the published values of Hoek [32] (Table 1). Possible data ranges are shown by a variation range value immediately following the suggested mi value. For example, for sandstones, the mi values can vary between 13 and 21, and for slates, between 3 and 11.

Table 1 The Hoek–Brown constant mi for intact rocks, by rock group according to the collection of Cai [10]
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2.2 Calculation the rigidity of the rock (R-index)

This method was introduced by [33] and also developed by [48, 59,60,61]. Cai [10] published the paper and showed that Hoek–Brown’s strength parameter (mi) could be determined from the ratio of the uniaxial compressive strength (\(\sigma_{c}\)) and the tensile strength (\(\sigma_{t}\)), and the suggested relationship is:

$$m_{i} \approx R = \frac{{\sigma_{c} }}{{\left| {\sigma_{t} } \right|}}$$
(2)

As can be seen from Fig. 1, when R > 8, the error for approximating mi (Eq. 2) by R (Eq. 4) is less than 1.6%.

Fig. 1
figure 1

Relationship between the error in mi estimate (using Eq. 2) and the strength ratio \(R\) [10]

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It has been observed that when the R is higher than 8, the value of R is nearly equal to Hoek–Brown material constant (mi). Note, that according to [38] mi is not a curve fitting parameter, but has a physical meaning and can be derived from micro-mechanics principles. The referred mi model is express in Eq. (3):

$$m_{i} = \frac{{\mu \sigma_{c} }}{{\beta \sigma_{t} }}$$
(3)

where \(\sigma_{c}\) and \(\sigma_{t}\) are the uniaxial compressive strength (UCS) and tensile strength (TS) of intact rock, respectively. While \(\mu\) is the coefficient of friction for pre-existing sliding crack surfaces, \(\beta\) is an intermediate fracture mechanics parameter that can be obtained from experimental data. It means when \(R > 8\):

$$\frac{\mu }{\beta } \cong 1$$
(4)

In recent years, Hoek and Brown [34] analyzed several published data and proposed the following approximate relationship between the compressive to tensile strength ratio, \(\frac{{\sigma_{c} }}{{|\sigma_{t} |}}\), and the Hoek–Brown parameter mi (see Fig. 2):

$$\frac{{\sigma_{c} }}{{|\sigma_{t} |}} = 0.81 m_{i} + 7$$
(5)
$$m_{i} = 1.235\frac{{\sigma_{c} }}{{\left| {\sigma_{t} } \right|}} - 8.642$$
(6)

It means that the Hoek–Brown constant (mi) can be calculated using the following relationship, using the uniaxial compressive strength (\(\sigma_{c}\)) and the tensile strength (\(\sigma_{t}\)) data of the intact rock.

Fig. 2
figure 2

Relationship between \(\frac{{\sigma_{c} }}{{|\sigma_{t} |}}\) and mi according to Hoek and Brown [35]

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2.3 UCS-based model

Firstly, Shen and Karakus [65] and later Vásárhelyi et al. [74] also emphasized the difficulties in determining the mi values of rocks. They suggested normalizing the Hoek–Brown constant \(\left( {m_{{{\text{in}}}} = \frac{{m_{i} }}{{\sigma_{c} }}} \right)\) by using the strength of the rock \(\left( {m_{{{\text{in}}}} = \frac{{m_{i} }}{{\sigma_{c} }}} \right)\), i.e.:

$$m_{i} = a_{c}^{b + 1}$$
(7)

where \(a\) and \(b\) are constants that depend on rock types.

2.4 TS-based model

The tensile strength-based calculation method was suggested by Wang and Shen [77]. In this method, the mi value is determined from the tensile strength (\(\sigma_{t}\)) of the intact rock.

$$m_{i} = A_{t}^{B}$$
(8)

where \(A\) and \(B\) are material constants that depend on rock types. In general, the constants \(A\) and \(B\) are curve-fitting parameters.

However, according to the theory of Zuo et al. [84] the mi value can be estimated from the coefficient of friction for the pre-existing sliding crack surfaces, and fracture mechanics. From this point of view, the constants \(A\) and \(B\) for the TS-based model seem to have its physical meaning, which have possible relations with the micro-mechanics principles. However, investigating the potential relationships between these constants and other rock properties and explaining the possible physical meanings need to carry out further research to get appropriate amount of reliable data.

2.5 Combination method (UCS and TS)

Analyzing the published data of Sheorey [66], Arshadnejad and Nick [3] suggested a new equation to calculate the Hoek–Brown material constant (mi) based on uniaxial compressive strength (\(\sigma_{c}\)) and the tensile strength (\(\sigma_{t}\)) parameters of the intact rock:

$$m_{i} = e^{{\left[ {a\left( {\frac{{\sigma_{c} - b\sigma_{t} }}{{\sigma_{t} }}} \right)^{c} } \right]}}$$
(9)

where \(a\), \(b\), and \(c\) are material constants. The suggested constants are summarized in Table 2, according to Arshadnejad and Nick [3].

Table 2 The suggested constants of different rocks based on Eq. (9) [3]
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2.6 Crack initiation stress-based model

According to [10], the Hoek–Brown constant (mi) depends on the confining pressure. For practical estimate of mi, it was found that for strong, brittle rocks, applicable to high confining zone the following equation could be used:

$$m_{i} = 12\frac{{\sigma_{c} }}{{\sigma_{ci} }}$$
(10)

For low confinement stress to tension zone, especially for the tension zone the suggested equation is:

$$m_{i} = 8\frac{{\sigma_{c} }}{{\sigma_{ci} }}$$
(11)

In these equations, \({\sigma }_{c}\) and \({\sigma }_{ci}\) are the uniaxial compressive strength and the crack initiation stress, respectively.

2.7 Bayesian approach

The proposed approach [1] derives the probability density function (PDF) of mi based on the integration of Hoek’s guideline chart, regression model, and site-specific UCS data, under a Bayesian framework. A large number of equivalent samples of mi are generated from the PDF using Markov Chain Monte Carlo (MCMC) simulation.

3 Experimental data

A wide range of data from literature by [72] were selected and analyzed. The published data of the investigated rocks are illustrated in Appendices 1, 2, and 3 for igneous, sedimentary, and metamorphic rocks, respectively. In these tables, calculated data are also given. Statistical analyses of the measured mi values for different rock types are summarized in Table 3. In this table, parameters such as number of data sets, maximum (Max.), minimum (min.), average (Ave.), and standard deviation (std.) of measured mi for three different groups are presented (see the appendix for more details).

Table 3 Summary of the measured mi for different group of rocks
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4 Analyzing the data

4.1 Comparing the published m i values with the guideline

Comparing the published values of mi by [72] with the guidelines [10, 32, 33] given for igneous rocks, the value of mi for basalt is between 20 and 30, whereas the minimum published value of mi for Basalt is 4.31 (see Appendix 1) which does not fall within the expected range. The minimum published value of mi for basalt is even less than mi value for volcanic tuffs. The latter one is between 8 and 13. The published values of mi for granite are between 11.68 and 34.03, but the expected range based on the guidelines is between 29 and 35. The same discrepancy was observed for gabbro. The expected range based on the guideline values is between 24 and 30, whereas the published results are in between 15.31 and 20.74.

For diorite, the value of mi based on the guidelines is between 20 and 30; however, according to published data, the value of mi is between 6.1 and 11. For granodiorite, based on the guideline, the value of mi changes between 26 and 32, while based on published results, it is between 11 and 18.15. For diabase, according to the guidelines, the value of mi is between 10 and 20; nevertheless, based on published data it is between 15.3 and 20.3. For andesite and rhyolite, the value of mi based on the guidelines is between 20 and 30; however, the published values are 6.22 and 5.43 for andesite and rhyolite, respectively. For agglomerate, the value of mi is between 16 and 22, but the published value is 7.92, which does not fit the guideline range.

For sedimentary rocks, the minimum value of mi for shale is 3.76 which is close to the estimated range of mi which is between 4 and 8 (see Appendix 2). Also, the maximum published value of mi for sandstone is 35.11 which is much higher than the estimated range of mi value and based on guideline which varies between 13 and 21. Moreover, for dolomite, the value of mi based on guidelines is between 6 and 12, whereas the value of mi based on published data is between 7.8 and 17.5. For limestone, the value of mi based on guideline is between 7 and 15; however, based on published data, the value of mi is between 5.3 and 14.6.

For metamorphic rocks, it is interesting to know that both maximum and minimum published values are for slate with the amounts of 1.42 and 29.54, respectively (see Appendix 3). However, based on guideline the values of mi for slate changes between 3 and 11. It means that the range of published value for slate does not fit the estimated range given by the guidelines. For gneiss, the value of mi based on guidelines is between 23 and 33; however, based on published data, the value ranges between 5.3 and 27. For schist, the published value of mi is 20.42, but the values based on guideline vary between 9 and 15. For quartzite, the published value of mi is between 7.36 and 22.7; however, based on guideline it is between 17 and 23. Also, the values of mi were calculated according to [10].

The differences between the published values of mi and the guideline values are displayed in Table 4 and Fig. 3. As shown, the published values of mi do not fit well with the guideline. As it is clear from the graph that there is a good consistency between published experimental data for mi and R-index method, whereas other methods have significant errors in estimation of mi for all studied rock samples. The obtained results based on different proposed calculation method are presented in Appendices 4, 5, and 6 for igneous, sedimentary, and metamorphic rocks, respectively. It should be noted that the constant parameters were derived from the existing Eqs. 2, 6, 7, 8, and 9, respectively.

Table 4 Maximum and minimum values of mi based on different methods for different rocks
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Fig. 3
figure 3

Comparison of published values of mi [66] with the guidelines [4, 8, 9], R-index, UCS-based, TS-based, and combination methods: a igneous rocks; b sedimentary rocks; c metamorphic rocks

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4.2 Evaluation of R-index model

According to our analyses for determination of mi in respect to Eq. 2 (R-index method) and Eq. 5, it was observed that mi value is independent of rock type. The correlations between mi and the ratio between uniaxial compressive strength (UCS) and tensile strength (TS), which is called R-index method, for different rocks (sandstone, shale, slate, and gneiss) are presented in Fig. 4. As shown for different rock types, correlation value \(R^{2} = 1\) and it is not influenced by rock type.

Fig. 4
figure 4

Correlation between \(\frac{{\sigma_{c} }}{{\sigma_{t} }}\) and mi for different rock types: a sandstone; b shale; c slate; d gneiss

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4.3 Evaluation of UCS-based model

The relationship between mi and uniaxial compressive strength (UCS) for the investigated rock samples (UCS-based method) shows an opposite result, compared to R-index method. It is evident that based on this approach, no correlation was found for the studied rock samples (Fig. 5).

Fig. 5
figure 5

Correlation between UCS and mi for different rock types: a sandstone; b shale; c slate; d gneiss

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4.4 Evaluation of TS-based model

Based on our analyses for determination of mi value by applying TS-based method, the established correlations for the investigated rock types were weak except for slate where the value of \(R^{2} = 0.66\) (Fig. 6).

Fig. 6
figure 6

Correlation between tensile strength (TS) and mi for different rock types: a sandstone; b shale; c slate; d gneiss

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In order to examine the relationship between mi and rigidity of rock (R-index method) for different rock groups, analyses were also carried out based on three different rock types (igneous, sedimentary, and metamorphic) and the results are illustrated in Fig. 7. Additionally, the error function was calculated for each rock type. The calculated amount of error function represents the value of \(\left( {\frac{\mu }{\beta }} \right)\) as mentioned earlier in Eq. (3).

Fig. 7
figure 7

Correlation between the ratio of uniaxial compressive strength (UCS) and tensile strength (TS) and mi for: a igneous; c sedimentary; e metamorphic rocks and calculated error function respected (b, d, f)

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The obtained results of regression analyses based on proposed approaches for determination of mi are summarized in Table 5 (according to rock group) and Table 6 (according to different investigated rock samples), and the statistical results are illustrated in Table 7.

Table 5 Calculated values of mi by different methods (Eqs. 2, 5, 7, 8, 9, 10)
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Table 6 Determination of mi based on existing methods
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Table 7 Statistical results of predicted mi values using different approaches
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In the proposed method by Arshadnejad and Nick [3], (combination method), (i), (j), and (k) are presented as material constants. Specifically explaining, based on their analyses for different rock groups (igneous, sedimentary, and metamorphic), there are different material constants. See Table 2 for their obtained results.

4.5 Evaluation of combination method (UCS and TS methods)

We analyzed the data set based on Eq. 9 as proposed by [3]. The results are depicted in Fig. 8. The graphs clearly illustrate that the Hoek constant (mi) is independent of rock type. Accordingly, it is worth to mention the point that the intact rocks are known to contain a wide-ranging uncertainty owing to inherent heterogeneities. For rock mass classification in respect to uncertainties, GSI (Geological Strength Index) provides good classification of rock mass category as it relies on structure as well as surface conditions of existing discontinuities which are highly associated with high uncertainties in rock mass. Ván and Vásárhelyi [72] investigated the sensitivity of Hoek material constant for rock mass (mb) in relation to GSI. Based on their measurement, they calculated that 10% deviation in the GSI value; the relative sensitivity of mb is at least double the uncertainties of the GSI values and may be 7 times higher in case of large disturbance parameters and low and high GSI values.

Fig. 8
figure 8

Calculated values of mi based on combination method for: a igneous rocks; b sedimentary rocks; c metamorphic rocks; d all rocks

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5 Discussion

When triaxial test data are not available, mi values can be estimated by six existing methods entitled, guidelines [32, 33], R-index [10, 33, 52, 59, 60, 62], UCS-based model [65, 74], TS-based model [77], crack initiation stress-based model [10], combination based model [3, 66], and Bayesian approach [1] to evaluate the prediction performance of existing methods, large database of triaxial tests for eight common rock types (slate, shale, limestone, quartzdiorite, sandstone, quartzite, gneiss, and granite) from literature was selected and analyzed as indicated in Appendices 1, 2, and 3.

In the present research, to ensure the quantity and quality of data, the statistical analysis with the help of SPSS software was performed on quasi-isotropic rocks. The results are presented in Fig. 9. As shown, normal distribution function and standard deviation were calculated for each rock group. According to Fig. 9, for igneous rock, the calculated standard deviation is 7.21 with the mean value of 15.36, for sedimentary rock is 6.32 with the mean value of 12, and for metamorphic rocks is 10.07 with the mean value of 13.37. Interestingly enough, metamorphic rocks exhibit higher standard deviation, which can be can associated with the foliated texture of the investigated metamorphic rocks.

Fig. 9
figure 9

Histograms of the predicted mi values for: a igneous; b sedimentary; c metamorphic rocks considering the new material constants; d equivalent samples for Hoek–Brown constant mi from Bayesian approach

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Similarly, Aladejare and Wang [1] investigated the values of mi for 30,000 equivalent granite samples based on Bayesian approach and realized that the histogram peaks at a mi value of around 30, and roughly about 27,479 out of the total 30,000 samples representing about 91.5% are within mi range of 18 and 48. Therefore, the 90% interperceptual range of mi is roughly between mi values of 18 and 48 (Fig. 9d). In comparison with our results, however, Wen et al. [79] published an interesting paper about anisotropic rocks, which has drawn our attention to their new empirical relation. The large number of the data set (738) of different rock types was analyzed. They observed that the more anisotropic the rock is, the more significant decrease in mi value is evident. Using this relation, the H–B strength criterion is modified, and its performance is evaluated in two examples. The results illustrate the improved ability of the modified criterion to determine the strength of metamorphic and sedimentary rocks.

Based on the empirical model by Wen et al. [79], most of the estimated values of mi are within the upper/lower limit of 90%, which accounts for 96.6% of all the data, and that all the estimated values are within the upper/lower limit of 80%. This result indicates that the developed relation estimates of mi agree well with the experimental observations, demonstrating that the developed relation has an excellent ability to estimate mi accurately. The comparison graph between our results and recently published paper by Wen et al. [79] is presented in Fig. 10.

Fig. 10
figure 10

Comparison between mi values estimated via the proposed relation and measured values for different rock types (red rhombus symbol is related to quasi-isotropic intact igneous rocks; green triangle symbol is related to quasi-isotropic intact sedimentary rocks; blue cross symbol is related to quasi-isotropic intact metamorphic rocks; and small orange square symbol is related to anisotropic intact rocks (Wen et al. [79])) (color figure online)

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Moreover, Wen et al. [80,81,82] thoroughly investigated the impact of minor principal stress (σ3) on rock ductile–brittle behavior. Based on their measurement, under a low minor principal stress, rocks experience dilatancy and brittleness that causes the exciting microcrack inside the rock to coalesce, which result in an increase in volume; In contrast, applying the high minor principal stress, the rock undergoes ductility which leads to microcrack opening.

In the recently published paper by He et al. [27], they suggested that the constant mi of the H–B criterion can be continuously estimated during drilling. Therefore, a method to estimate the constant mi from drilling data was proposed. Based on the proposed method for mi determination, for the granite, slate, the obtained values of mi in their work were lower than the suggested values for granite and sandstone from [33], and they were higher than the suggested value for slate and limestone.

To establish the correlations for specific rock types, Shen and Karakus [65], Wang and Shen [77] considered five of the most common rock types (coal, granite, limestone, marble, and sandstone) from the database in the Rocscience [61], in which there are at least 12 groups of data with 115 triaxial tests available. Results illustrate that four rock types have trends of decreasing mi with increasing \(\sigma_{ci}\); however, such a correlation for limestone was not observed. It is worth noting that this finding is also in agreement with our obtained results for limestone. This may be due to the fact that limestone has a wide array of test data with different compositions and cementations; for example, there are different guidelines-based mi values for three types of limestone, crystalline (mi = 12 ± 3), sparitic (mi = 10 ± 2), and micritic (mi = 9 ± 2), in the Hoek guidelines [32]. Therefore, the data sets for limestone are quite widely scattered compared with those for the other four rock types.

The observed value of mi for limestone is between 5.3 and 14.6; In contrast, according to guideline, the mi value of limestone is between 7 and 15, according to R-index method mi is between 5.32 and 14.67, according to UCS-based method mi is between 19.8 and 22.1, based on TS-based method mi is between 6.18 and 13.05, and based on combination method mi is between 4.9 and 13. Thus, R-index method gives the best estimate of mi value among the others. However, based on [65], UCS-based method provides a better prediction of mi value than the guidelines-based and R-index methods and according to [77], TS-based model exhibits the best performance for granites, limestone, and marble.

Additionally, the observed value of mi for slate is between 1.42 and 30.97; however, based on guideline, the mi value of slate is between 3 and 11, based on R-index method mi is between 1.59 and 31.23, based on UCS-based method mi is between 7.2 and 22.3, based on TS-based method mi is between 3.82 and 48.42, and based on combination method mi is between 1.98 and 32.44. So, R-index method gives the best estimate of mi value among the others. The achieved value of mi is inconsistent with the obtained value of mi by [16], which they used the internal friction angel method for estimating the value of mi for dolomite, slate, dike, and granodiorite. The obtained values of mi were 6.28, 5.1, 12.8, and 13.7, respectively.

Moreover, the observed value of mi for shale is between 3.76 and 25.31, but, according to guideline, the mi value of shale is between 4 and 8, according to R-index method mi is between 3.7 and 24.9, according to UCS-based method mi is between 17.82 and 22.1, based on TS-based method mi is between 6.52 and 29.87, and based on combination method mi is between 3.65 and 24.3. Therefore, R-index method gives the best estimate of mi value among the others.

Furthermore, the observed value of mi for sandstone is between 3.97 and 35.1; nevertheless, according to guideline, the mi value of sandstone is between 13 and 21, according to R-index method mi is between 3.9 and 35, according to UCS-based method mi is between 17.82 and 22.23, based on TS-based method mi is between 5.48 and 29.87, and based on combination method mi is between 3.83 and 38.5. As a result, R-index method gives the best estimate of mi value among the others. Nevertheless, based on [77], UCS-based model gives the best prediction for sandstone.

As well as that, the observed value of mi for quartzite is between 7.36 and 30.1, while, according to guideline, the mi value of quartzite is between 17 and 23, according to R-index method mi is between 7.36 and 30.14, according to UCS-based method mi is between 8.45 and 21.1, based on TS-based method mi is between 5.66 and 10.43, and based on combination method mi is between 6.55 and 31.10. Hence, R-index method gives the best estimate of mi value among the others.

In addition, the observed value of mi for gneiss is between 5.34 and 32.33, whereas, according to guideline, the mi value of gneiss is between 23 and 33, according to R-index method mi is between 5.3 and 32.3, according to UCS-based method mi is between 8.07 and 22.46, based on TS-based method mi is between 4.68 and 11.44, and based on combination method mi is between 4.92 and 34.28. Therefore, R-index method gives the best estimate of mi value among the others.

Finally, the observed value of mi for granite is between 11.68 and 34.03; on the other hand, according to guideline, the mi value of granite is between 29 and 35, according to R-index method mi is between 11.7 and 34.1, according to UCS-based method mi is between 8.92 and 22.8, based on TS-based method mi is between 7.42 and 15.42, and based on combination method mi is between 10.22 and 36.95. Thus, R-index method gives the best estimate of mi value among the others.

6 Conclusion

This paper comprehensively reviewed the proposed methods for determination of mi value using the triaxial data set published by [66]. New linear and nonlinear correlations were found. New material constants were suggested for igneous, sedimentary, and metamorphic rocks such as granite, quartzdiorite, sandstone, shale, limestone, gneiss, quartzite, and slate. Furthermore, the error function was calculated for each rock type. According to our analyses, R-index approach with new material constants provides the best fit for all the studied lithotypes including igneous, sedimentary, and metamorphic rocks. Moreover, it is worth mentioning that comparison graph between out achieved results for quasi-isotropic rocks and anisotropic rocks was developed. The data scattering of quasi-isotropic rocks differs from the anisotropic rocks in that quasi-isotropic rocks exhibit normal distribution in narrower range between 1.43 and 35.11, whereas anisotropic rocks display wider range of data distribution between 1 and 80.

The combination methods (UCS) and (TS) provide the best fit for investigated rock data, when the determination coefficient and error function are considered. For various rock types such as igneous, sedimentary, and metamorphic rocks, the different approaches resulted in non-uniform correlations. For instance, TS-based approach works well for the granite and gives estimation fit with \(R^{2} = 0.89\). For quartzites, UCS-based approach displays the best approximation with \(R^{2} = 0.91\), and for slate, TS-based model exhibits the best correlation with \(R^{2} = 0.67\).

R-index method, the value of mi is not influenced by rock type. In other words, this method gives the best estimation for all the investigated rock types and is independent of rock type. However, in all the other approaches, the rock type plays a significant role in correlation values and results.