Modification of the rock mass rating system (RMR_mbi) considering three-dimensional rock block size

Abstract

Traditional measurements of block size (or degree of jointing), such as the rock quality designation (RQD), are questionable due to certain theoretical limitations, including orientation bias and their weakness in considering the joint persistence and three-dimensional shapes of block sizes. This may lead to inaccurate characterizations of rock mass structures and unreliable classification of rock mass qualities. The modified blockiness index (MB_i) is a three-dimensional measurement of block size which was developed to overcome these problems. In this study, correlations between MB_i and several traditional block size measurements were assessed; based on the MB_i, the rock mass rating (RMR) system was modified, and this version was termed RMR_mbi. In the first part of this work, multiple simulated experiments were conducted using the GeneralBlock software program and 3DEC (three-dimensional distinct element code), and a large volume of MB_i, RQD, joint frequency (JF) and volumetric joint count (J_v) values (artificial data sets) were obtained; subsequently, the correlations between MB_i and RQD, JF and J_v were assessed. In the second part, the combined use of RQD and JF in the RMR system was replaced with MB_i, and hence, the RMR_mbi system was developed; based on the artificial data sets, the viability of RMR_mbi was preliminarily supported. At the end of this study, the correlations between MB_i and RQD, JF and J_v were verified based on actual data; the RMR_mbi was applied to real cases, and a comparison between RMR_mbi and RMR was conducted. The results showed that (i) the MB_i can capture the influence of joint persistence; (ii) the RMR_mbi system can overcome the theoretical limitations caused by RQD and JF; and (iii) simulated experiments showed that the RMR_mbi values are more reliable, thus validating the accuracy of RMR_mbi and revealing its great potential for future application.

Introduction

Rock mass is a type of discontinuous material that often contains a variety of joints, which discretize the rock mass into blocks of various sizes and shapes (Palmstrom 2005; Xia et al. 2016; Yarahmadi et al. 2018). The quantity and sizes of blocks within rock mass (or degree of blockiness) have a major controlling influence on the quality and stability of rock masses. Accurate measurement of block size is a basic task in rock mass characterization and classification, and can also provide robust data for designing rock engineering structures (Celada et al. 2014).

Over the past several decades, a large number of block size measurements have been proposed. Some existing measurements are listed in Table 1. In this table, all the three-dimensional measurements are indirect, because rock mass structure cannot be directly examined in three dimensions (Shang et al. 2018), and these three-dimensional measurements are basically performed from the joint data acquired from rock exposure. In addition, joint frequency (JF) is essentially identical to joint spacing (JS), and the same is true for volumetric joint count (J_v) and P30.

Table 1 An overview of existing measurements of block size (or degree of blockiness)

Actually, only a few of these measurements have been widely accepted, including the RQD, JF, and J_v. The RQD has gained wide acceptance and application compared to the other two indices. In civil and mining engineering worldwide, borehole penetrations occur every day, and professionals always record the degree of rock mass jointing in the form of RQD (Zhang 2016). Also, many rock mass quality classification systems include RQD as a basic input parameter, for example, the geomechanics classification system/rock mass rating system (RMR) (Bieniawski 1989), tunneling quality index (Q-system) (Barton et al. 1974) and the quantified version of the Geological Strength Index chart (GSI) (Hoek and Diederichs 2013). The use of JF and J_v is also popular. In the RMR system, JF has a weighting “score” of 20 points, which is used together with RQD to quantify the degree of rock mass jointing; notably, in the updated version of the RMR system (Celada et al. 2014), the combined use of RQD and JF was replaced by the sole use of JF, but this version is controversial (Koutsoftas 2017). The J_v is often used in conjunction with the Rock Mass index (RMi) (Palmstrom 1996), and in some Chinese geotechnical codes (PRC Ministry of Construction 1995), the intactness index of rock mass (K_v) (Liu et al. 2017) can be estimated using J_v. It should be noted that the previous findings are reviewed only from the perspective of the essential concepts of these indices; some works in which certain techniques such as three-dimensional point cloud (Riquelme et al. 2016) and discrete fracture network (Zhang et al. 2013) were employed to determine the RQD, JF or J_v value are not discussed in this study.

However, various researchers (Palmstrom 2005; Zhang et al. 2012; Chen et al. 2019) have noted that the three measurements (i.e., RQD, JF and J_v) are limited. For example, two major criticisms of RQD are that RQD values are very sensitive to the scanline or borehole direction, and that RQD counts only the core pieces longer than 100 mm and therefore cannot effectively differentiate rock masses with various structures. These two shortfalls have become the basis for arguments against the use of the RQD in rock mass classification systems (Pells et al. 2017). The other two indices also suffer from limitations: JF is also orientation-dependent, and joint persistence is ignored in the J_v method (Lin 2008). Actually, many block size measurements and rock mass quality classification systems fail to consider joint persistence (Kim et al. 2007). In addition, Hoek and Diederichs (2013) reported that some traditional measurements of the degree of rock mass jointing, including RQD, JF and J_v, cannot capture the effect of block scale.

Other measurements of block size were not mentioned, such as the weighted joint density (WJD) (Terzaghi 1965), RQD/JN (Barton et al. 1974) (RQD/JN denotes the RQD value divided by the joint set number) and block volume (V_b) (Palmstrom 2005). This is due to the following: (i) the WJD is, in fact, a correct approach for joint data, which is mainly used to remedy the one-dimensional joint density for borehole orientation bias (Terzaghi 1965); (ii) the RQD/JN is slightly unreliable, because the determination of the JN is often subjective in nature (Grenon and Hadjigeorgiou 2003); and (iii) the V_b is an interpretation of J_v (Palmstrom 2005).

Inaccurate measurement of block size (or degree of rock mass jointing) may result in unreliable classification of rock mass quality (Palmstrom 2005; Aydan et al. 2014). Therefore, in this study, the modified blockiness index (MB_i) is introduced (for details, see Section 2), and an attempt has been made to incorporate the MB_i into the RMR system. The study consists of two main parts: (i) correlations of MB_i with RQD, JF and J_v are evaluated; and (ii) a modified RMR system (RMR_mbi) with a great application potential is developed and its viability analyzed. To consider all possible cases, the study was carried out based on multiple simulated experiments and some real engineering projects. In addition, multiple simulated experiments were conducted using the GeneralBlock program (a 3DJN model generator) (Xia et al. 2016) and 3DEC (three-dimensional distinct element code) (Itasca 2013) together.

Outline of the modified blockiness index (MB_i)

Joints intersect rock mass in complicated three-dimensional patterns; however, traditional measurements of block size are often performed in one or two dimensions, and they are also unable to consider the influence of joint persistence and block scale. The modified blockiness index (MB_i), proposed by Xia et al. (2015) and revised by Chen et al. (2018), is capable of overcoming these difficulties. The MB_i is a three-dimensional measurement that enables the quantification of the block size distribution of a given rock mass, and its calculation is based on the three-dimensional join network model (3DJN model) generated by stochastic and/or deterministic joints. MB_i value can be calculated by

$$ {MB}_i={B}_1+\frac{1}{2}{B}_2+\frac{1}{3}{B}_3+\frac{1}{4}{B}_4+\frac{1}{5}{B}_5 $$

(1)

where B₁, B₂, B₃, B₄ and B₅ are the ratios of blocks in the ranges of 0–0.008 m³, 0.008–0.03 m³, 0.03–0.2 m³, 0.2–1.0 m³, and > 1.0 m³, respectively, to the total rock mass volume, and the fractions (i.e., 1, 1/2, 1/3, 1/4 and 1/5) are the empirical scale effect coefficients, which are determined based on engineering practice (Wang et al. 2010; Chen et al. 2018). When the MB_i value is close to 1, it suggests that the rock mass is very fractured; when the MB_i value is close to 0, it indicates a very low degree of rock mass jointing. The suggested jointing classes based on MB_i are presented in Table 2.

Table 2 Jointing classes based on MB_i (Chen et al. 2018)

In fieldwork, the MB_i associated with a given rock mass can be determined based on the following steps: (i) obtaining joint measurement data, including joint orientation, trace length and joint density; (ii) demarcating statistically homogeneous structural domains: generally, the domains can be identified using joint orientation data, together with various statistical methods, e.g., Chi-square test (Miller 1983; Martin and Tannant 2004; Li et al. 2014); (iii) establishing a three-dimensional joint network model; (iv) identifying the sizes of all blocks inside the rock mass model; and (v) calculating the MB_i value using Eq. (1). Steps (iii) and (iv) can be carried out using GeneralBlock.

In essence, the MB_i concept can be treated approximately as an extension of RQD and V_b, namely, all the MB_i, RQD and V_b are direct representations of the blockiness of rock mass, which are different from JF and J_v. In other words, MB_i, RQD and V_b all consider block size: MB_i and V_b count blocks of all sizes in three dimensions, and RQD counts rock pieces greater than 100 mm in one dimension; however, JF and J_v quantify rock mass blockiness from the perspective of one- and three-dimensional joint density. In addition, although both MB_i and V_b are indirect measurements, their determination methods differ: V_b is estimated mainly by J_v, and MB_i is determined by certain techniques including three-dimensional joint network representation and rock block identification.

Compared to the RQD and JF, the MB_i index can take into account the joint persistence, because the three-dimensional block size corresponds well with various joint geometries (Kim et al. 2007). However, the RQD is controlled primarily by the one-dimensional joint density (their theoretical correlation is presented in Eq. (7), Section 3), which cannot capture the effect of joint persistence. Therefore, in the following sections, we attempt to replace the RQD and JF with MB_i in the RMR system in order to fully consider the rock mass jointing degree in three dimensions.

Correlations of MB_i with RQD, JF and J_v

To consider all possible cases, multiple simulated experiments were conducted to obtain a large volume of artificial data sets. Under such ideal conditions, the correlations of MB_i with RQD, JF and J_v were evaluated. Validations are presented in Section 5, using a number of real data.

Design of simulated experiment

When the JN value is fixed, the joint density (or frequency, or spacing) has the greatest effect on the degree of rock mass jointing, followed by joint persistence (Bieniawski 1989; Shang et al. 2018), whereas the other joint geometrical parameters, such as joint orientation (JO) and distribution type, have negligible effects (Zhang et al. 2009). If all joint geometrical parameters are considered, the workload is too heavy. Therefore, the joint data in Xia et al. (2015) are employed (Table 3), including the average joint orientation, joint trace standard deviation and distribution type of joint traces. In this section, the joint parameters in Table 3 are fixed, and the JS and JP values are changed. In addition, 3DJN models are constructed using the GeneralBlock program, and the block sizes are identified and recorded; these models are passed on to 3DEC, and a specific code is developed and executed in 3DEC, where RQD, JF and J_v values are determined.

Table 3 Joint geometrical parameters used to construct 3DJN models (Xia et al. 2015)

Pairs of “JS-JP”

Since the degree of jointing is mainly affected by joint spacing (JS) and joint persistence (JP), the schemes of simulated experiments were developed by means of the following steps: (i) select a range of JS and JP values, and two groups of samples (i.e., JS and JP) can be obtained; and (ii) develop pairs of “JS-JP” by the cross joins of JS and JP values, respectively, from the two groups of samples, while other joint geometrical parameters remain fixed.

The joint spacing and persistence can be classified into seven and five intervals, respectively (ISRM 1978), as shown in Table 4. A total of 18 joint spacing values and 10 joint persistence values were selected, and two groups of samples (i.e., JS and JP) were obtained; therefore, 180 pairs of “JS-JP” were developed by the cross join method. It can be observed from Table 4 that most of the selected joint spacing values are in the range of 0.02 to 2 m, which reflects those commonly found in engineering practice (ISRM 1978; Bejari and Khademi Hamidi 2013; Riquelme et al. 2015; Buyer and Schubert 2017; Wong et al. 2018). The other joint geometrical parameters are fixed, as shown in Table 3. Based on these 180 pairs, 180 schemes are established, and 180 3DJN models with different structures are created. Some 3DJN models (established using GeneralBlock) are presented in Fig. 1. It is noted that the distribution types of joint persistence values are in agreement with Table 3, and the joint spacing values are derived for three-dimensional joint densities (Xia et al. 2016):

$$ {d}_3=\frac{4}{\pi \times E\left({JP}^2\right)\times JS} $$

(2)

where d₃ is three-dimensional joint density. This is because in many 3DJN model generators, including GeneralBlock, the input parameters with regard to joint density are of three-dimensions.

Table 4 Classification of joint spacing and persistence (ISRM 1978) and chosen values

Fig. 1

Some 3DJN models (note that JP denotes the joint persistence, JS denotes the joint spacing, and L denotes the side length of the model)

The geometrical and mechanical properties are controlled predominantly by the size of the “sample” or “specimen” under investigation, i.e., the effect of the representative elementary volume (REV). Also, the geometrical REV is not the same as the mechanical REV, and geometrical and mechanical REV represent the geometrical and mechanical properties that have reached a constant behavior, respectively. In this study, all the parameters (e.g., RQD) are related to the structural and geometrical features of models. Thus, all of the 3DJN models have met the size requirement for geometrical REV and are also greater than the joint persistence value. An example of REV determination is presented in Fig. 2. In this figure, the REV of a 3DJN model (JP = 0.5 m and JS = 0.13 m), which is also shown in Fig. 1, was determined: when the model size is greater than 4 m, the coefficients of variation (COV) for RQD, JF, J_v and MB_i (the procedures for determining them are presented in Section 3.1.2) are all within the range of 0 to 0.5, and therefore the REV of this 3DJN model is determined as 4 m. Additionally, in order to reasonably measure RQD values, the sizes of the 3DJN models are all larger than 1.5 m.

Fig. 2

REV determination for the 3DJN model (JP = 0.5 m and JS = 0.13 m)

Data acquisition

Data acquisition was carried out using 3DEC, with exception of the MB_i values. RQD and JF values were determined by setting scanlines in the 3DJN models. To minimize the directional bias, the scanline and model boundary were fixed, and the joint network was rotated. In this way, the RQD and JF values in different directions could be measured, and the scanline length remained unaltered, as shown in Fig. 3. The scanline is perpendicular to XY plane (in a Cartesian coordinate system), and the intersection point is the model’s center (0, 0, 0). Therefore, the scanline can be described as

$$ \left\{\begin{array}{l}{x}_L=0\\ {}{y}_L=0\end{array}\right. $$

(3)

Fig. 3

A two-dimensional example to roughly illustrate the scanline setting in a model

The joint in a model (Fig. 4) can be represented as

$$ \left\{\begin{array}{l}A\left(x-{x}_0\right)+B\left(y-{y}_0\right)+C\left(z-{z}_0\right)=0\\ {}{\left(x-{x}_0\right)}^2+{\left(y-{y}_0\right)}^2+{\left(z-{z}_0\right)}^2={r}^2\end{array}\right. $$

(4)

where A, B and C are the components of a normal vector of a joint, respectively, x₀, y₀ and z₀ are the central coordinate values of a joint, and r is the joint radius. From Eqs. (3) and (4), an equation of the intersection of the joint and scanline can be established; if the solution of this equation is a real number, an intersection point exists between the joint and scanline, and vice versa. After the coordinates of the intersection points are obtained, the RQD and JF values can be easily calculated.

Fig. 4

Disc-shaped joint

The rotation of a joint network can be achieved by the rotation matrix A (Zheng et al. 2014):

$$ \boldsymbol{A}=\left[\begin{array}{c}\mathit{\cos}{\beta}_1\bullet \mathit{\cos}{\alpha}_1\\ {}\mathit{\cos}{\beta}_1\bullet \sin {\alpha}_1\\ {}-\sin {\beta}_1\end{array}\kern0.5em \begin{array}{c}-\sin {\alpha}_1\\ {}\mathit{\cos}{\alpha}_1\\ {}0\end{array}\kern0.5em \begin{array}{c}\sin {\beta}_1\bullet \mathit{\cos}{\alpha}_1\\ {}\sin {\beta}_1\bullet \sin {\alpha}_1\\ {}\cos {\beta}_1\end{array}\right] $$

(5)

where α₁ is the rotation angle of the joint network in the dip direction, and β₁ is the rotation angle of the joint network in the dip angle. Therefore, the joint central coordinates after rotation (x′, y′, z′) can be obtained by

$$ \left[\begin{array}{l}{x}^{\prime}\\ {}{y}^{\prime}\\ {}{z}^{\prime}\end{array}\right]=\boldsymbol{A}\bullet \left[\begin{array}{l}{x}_0\\ {}{y}_0\\ {}{z}_0\end{array}\right] $$

(6)

Additionally, joint orientation will also change after the rotation of the joint network. Similarly, the joint orientation after rotation can be calculated using A; the detailed processes are not repeated in this section.

Overall, a large number of RQD and JF values can be determined from a 3DJN model (an example is shown in Fig. 5), and their averages are defined as unique representative RQD and JF values. The means and standard deviations of the RQD and JF values for all 3DJN models are shown in Figs. 6 and 7. The two figures indicate that the RQD values fall mainly within the interval of 90–100, and the JF values fall mainly within the interval of 0–20.

Fig. 5

RQD and JF values (in different directions) measured in a model. aRQD values and bJF values. The JS and JP values of the model are 0.17 m and 0.5 m, respectively

Fig. 6

Histogram of RQD (artificial data). N indicates the number of models

Fig. 7

Histogram of JF (artificial data)

Other unsettled questions involve the determination of J_v and MB_i. In fieldwork, the J_v is often measured by the average spacing for well-defined joint sets (Palmstrom 2005), but obviously, this approach is inapplicable in this study, because it leads to identical results produced by various 3DJN models. With regard to the essence of the J_v concept, i.e., the number of joints per unit volume of rock mass, the J_v is found to be equivalent to the P30 (fracture/m³) (Dershowitz et al. 2003). In this work, the P30 values for all 3DJN models were determined using the built-in codes in 3DEC. Additionally, the sizes of blocks inside the 3DJN models were identified using GeneralBlock, as shown in Fig. 8, and the MB_i values were determined according to Eq. (1). The J_v and MB_i values for all 3DJN models are shown in Figs. 9 and 10. It can be seen that the J_v values fall mainly in the interval of 0–200, and the standard deviation is very large (231.37), because several 3DJN models contain very small and dense joints. For example, the J_v of the 3DJN model (JP = 0.5 m and JS = 0.01 m) is 2784.59. The MB_i values fall mainly within the interval of 0–30.

Fig. 8

An example to roughly illustrate the identification of block volume (this model (10 × 10 × 10 m) contains three joint sets which are orthometric)

Fig. 9

Histogram of J_v (artificial data)

Fig. 10

Histogram of MB_i (artificial data)

Validating the effectiveness of artificial data

In Section 3.1, 180 3DJN models were established, and a substantial amount of artificial data was obtained. Before assessing the correlations between MB_i and RQD/JF/J_v, it is necessary to check the validity of the artificial data, i.e., whether these artificial data reflect the objective trend.

The relation between RQD and JF (Priest and Hudson 1976) is

$$ RQD=100{e}^{-0.1 JF}\left(0.1 JF+1\right) $$

(7)

For the JF values ranging from 6 to 16, the linear relation between RQD and JF is

$$ RQD=-3.68 JF+110.4 $$

(8)

Therefore, a scatter plot of artificial RQD and JF values was created, and regression analysis was carried out, which is presented in Fig. 11. The figure shows that the data points are fitted using Eqs. (7) and (8), and the coefficients of determination are 0.99 and 0.93, respectively. Additionally, F tests were carried out, and the F and p values suggested that the null hypotheses were rejected, and that the two regression models, i.e., Eqs. (7) and (8), were significant. This means that the results of the simulation experiment are effective and reliable, and moreover that the artificial data sets produced by the theoretical 3DJN models are in line with the objective trends.

Fig. 11

Correlations between RQD and JF (artificial data)

Correlation between MB_i and RQD

In the field of rock mechanics and rock engineering, RQD has received much attention over the past few decades. It was found that RQD is closely related to JF, J_v, P-wave velocity, deformation modulus and unconfined compressive strength (Feng and Jimenez 2015; Zhang 2016). Since the MB_i is a direct representation of block size, similar to RQD, the MB_i may correlate well with the RQD.

Based on Figs. 6 and 10, a scatter plot of MB_i and RQD was created (note that some data points were deleted because too many RQD values were close to 100%, which can lead to invalid curve-fitting). Linear and nonlinear regression analyses were conducted (Fig. 12), which yielded

$$ {MB}_i=90.14-0.73\times RQD $$

(9)

$$ {MB}_i=80-0.007\times {RQD}^2 $$

(10)

Fig. 12

Correlations between MB_i and RQD (artificial data). Note that the red and blue curves represent Eqs. (9) and (10), respectively

Figure 12 shows that the two coefficients of determination are 0.72 and 0.77, respectively, and the F and p values indicate than the null hypotheses are rejected and the regression models are significant.

Considering that the block sizes are controlled mainly by joint spacing and joint persistence (Kim et al. 2007), and that the RQD concept takes into account only the core pieces longer than 100 mm, joint persistence was involved in the assessment of the correlation between MB_i and RQD. Therefore, multiple linear and nonlinear regression analyses were performed (Figs. 13 and 14), which yielded

$$ {MB}_i=82.876-0.769\times RQD+1.370\times JP $$

(11)

$$ {MB}_i=65.28+0.1\times RQD+1.67\times JP-0.008\times {RQD}^2-0.03\times {JP}^2 $$

(12)

Fig. 13

Correlation between MB_i and RQD and JP (artificial data), corresponding to Eq. (11)

Fig. 14

Correlation between MB_i and RQD and JP (artificial data), corresponding to Eq. (12)

Figures 13 and 14 show that the coefficients of determination are 0.85 and 0.92, respectively, and the F and p values indicate that the null hypotheses are rejected and the two regression models are significant. Additionally, Eq. (11) is a multiple linear model, and hence, the collinearity statistics was carried out, which is presented in Table 5. A review of Table 5 indicates that all VIF values for each variable are less than 10, indicating that no strong correlation exists between the two independent parameters.

Table 5 Collinearity statistics of Eq. (11)

From Eqs. (9)–(12), four inverse relations between MB_i and RQD were found, namely, the larger the MB_i, the smaller the RQD. Figure 12 also shows that when the RQD is 0, the MB_i values vary from 60 to 100; when the RQD is 100, the range of MB_i values is approximately 0–30. This may be attributed to the conceptual difference between MB_i and RQD, i.e., (i) the RQD counts only the core pieces longer than 100 mm, while the MB_i consider all blocks; and (ii) joint persistence is neglected with the RQD method.

Correlation between MB_i and JF

Based on Figs. 7 and 10, the MB_i values were plotted with JF, as shown in Fig. 15. The nonlinear regression analysis yields

$$ {MB}_i=-61.77+74.80\times {JF}^{0.15} $$

(13)

Fig. 15

Correlations between MB_i and JF (artificial data), corresponding to Eqs. (13) and (14)

The coefficient of determination is 0.76, and the F and p values are 551.45 and 0.00, respectively, which suggests that the null hypothesis is rejected, and the regression model is significant. For JF values less than 30, a linear relation is found

$$ {MB}_i=9.33+3\times JF $$

(14)

The coefficient of determination is 0.70, and the F and p values are 536.73 and 0.00, respectively. Additionally, when the JF is greater than 30, the MB_i shows a poor relation with JF.

By incorporating the JP into the assessment, multiple linear and nonlinear analyses were conducted (Figs. 16 and 17), which yielded

$$ {MB}_i=0.78+2.94\times JF+0.69\times JP $$

(15)

$$ {MB}_i=-8.09+5.18\times JF+1.69\times JP-0.11\times {JF}^2-0.03\times {JP}^2 $$

(16)

Fig. 16

Correlations between MB_i and JF (artificial data), corresponding to Eq. (15)

Fig. 17

Correlations between MB_i and JF (artificial data), corresponding to Eq. (16)

The corresponding coefficients of determination are 0.84 and 0.91. F tests were carried out, and the F and p values indicate that the null hypotheses were rejected, and the regression models are significant.

Correlation between MB_i and J_v

Based on Figs. 9 and 10, the scatter plot of MB_i vs. J_v is delineated in Fig. 18. Note that the difference in orders of magnitude between various J_v values is too large, so the ln(J_v) is employed, and the data points with different colors denote various joint persistence. It is observed from Fig. 18 that the data points are very discrete, for which regression analysis cannot be properly performed; however, when the color (i.e., the joint persistence) is identical, the MB_i values increase with the increase in J_v. Therefore, it is better to take into consideration the joint persistence.

Fig. 18

Scatter plot of MB_i and J_v (artificial data)

Multiple nonlinear regression analysis (Fig. 19) yields

$$ {MB}_i=\frac{74.383+0.864\times JP}{1+{e}^{-2.489+8.06\times {0.842}^{JP}-0.938\times \ln \left({J}_v\right)}} $$

(17)

Fig. 19

Correlations between MB_i and J_v (artificial data), corresponding to Eq. (17)

Additionally, the coefficient of determination is 0.89, and the F and p values are 603.96 and 0.00, respectively, suggesting that the null hypothesis is rejected, and the regression model is significant.

A comparison between measured MB_i values and estimated values was performed using Eq. (17), which is presented in Fig. 20. Although the coefficient of determination is rather high, the fitted curve is below the 1:1 line, meaning that the measured values are greater than the estimated values.

Fig. 20

Comparison between measured MB_i values (artificial data) and estimated MB_i values using Eq. (17)

RMR_mbi and its preliminary validation

Development of RMR_mbi

Inaccurate measurements of block size may lead to unreliable rock mass classification results (Palmstrom 2005; Aydan et al. 2014). The combined use of RQD and JF in the RMR system is limited, because (i) both RQD and JF values are orientation-dependent; (ii) the RQD method counts only core pieces greater than 100 mm, which is sometimes unreasonable (Harrison 1999); (iii) both the RQD and JF methods neglect joint persistence; and (iv) the ratings of RQD plus JF are a repetition of joint density, resulting in the doubling of the influence of joint density on the final RMR rating. Although some researchers (Koutsoftas 2017) have maintained that these limitations would not reduce the applicability of the RMR system, these theoretical limitations do exist in reality, and thus improvement of the subsystem (i.e., the ratings of RQD plus JF) of the RMR is desired.

Motivated by the need to improve the RMR system, an attempt was made to substitute the combined use of RQD and JF with MB_i, while keeping the other input parameters unchanged; this version is referred to as RMR_mbi. As shown in Sections 2 and 3, the main advantages of using MB_i to replace RQD and JF are as follows: (i) the MB_i is a three-dimensional measurement and is not orientation-dependent; (ii) the MB_i counts blocks of all sizes; (iii) joint persistence is considered, as shown in Section 3; and (iv) the MB_i does not repeatedly calculate the joint density. The rating of MB_i (R_M) can be determined using a continuous function, as described by Eq. (18).

$$ {R}_M=40-0.4\times {MB}_i $$

(18)

The greater the value of the MB_i, the smaller the degree of rock mass jointing, and the poorer the rock mass quality, and vice versa. Table 6 can be used to determine the R_M value. Therefore, R_M is inversely proportional to MB_i. The other input parameters, including, intact rock uniaxial compressive strength, are kept unchanged.

Table 6 Determination of R_M

Preliminary validation of the viability of RMR_mbi

Since the RMR_mbi and traditional RMR systems are different from the characterization of rock mass structure, only the R_M and R_RQD + JF were compared. R_RQD + JF (Warren et al. 2016) can be calculated by

$$ {R}_{RQD}=0.2\times \mathrm{RQD} $$

(19)

$$ {R}_{JF}=-3.767\times \ln (JF)+16.482 $$

(20)

Therefore, R_RQD + JF = R_RQD + R_JF.

Actually, the application of RMR_mbi in real cases is inadequate, because it is a new modification with regard to RMR. To confirm its viability, the results of simulated experiments considering many circumstances were used. The R_M and R_RQD + JF were applied to the artificial data sets (Figs. 6, 7 and 10), and the results are presented in Fig. 21, which shows that most of the R_M and R_RQD + JF values fall within the interval 30–40. The cumulative frequency curves are delineated in Fig. 22. The figure shows that for the interval of 8–40, the two curves are very smooth and have extremely similar trends, meaning that the two subsystems (i.e., R_M and R_RQD + JF) share an ability to distinguish the rock masses of various structures.

Fig. 21

Histograms of R_M and R_RQD + JF (artificial data)

Fig. 22

Comparison of R_M and R_RQD + JF based on the cumulative frequency curves (artificial data)

Figure 22 also indicates that for the interval of 0–8, the R_RQD + JF shows poor “performance”, because the cumulative frequency curve of R_RQD + JF does not start from the origin; however, in this interval, the curve of R_M grows from the origin and shows a more natural transition, which means that the R_M is more sensitive to rock masses with a high degree of jointing and can properly distinguish them. If one classification system is able to differentiate objects with a greater degree of refinement than another system, then it can be said that this system is better (Warren et al. 2016). Therefore, the R_M is viable.

The correlation between R_M and R_RQD + JF values was evaluated based on the 1:1 line, which is presented in Fig. 23. It is found that although the MB_i can be treated as an extension of RQD, the R_M and R_RQD + JF yield slightly different results. As shown in Fig. 23, when the R_RQD + JF is close to 0, the R_M values vary from 0 to 15; when the R_RQD + JF is near 40, the R_M values vary in the range of 35–40. However, the best-fitted curve is very close to the 1:1 line, with a rather high coefficient of determination.

Fig. 23

Comparison between R_M and R_RQD + JF based on the 1:1 line (artificial data)

Additionally, for the RQD and JF values determined in the simulated experiments, the directional bias has been the removed (i.e., they properly reflect the degrees of rock mass jointing), but in fieldwork, it is very difficult to guarantee that the RQD and JF values will not be biased by direction. This is why the substitution was performed. This problem is completely overcome in the RMR_mbi system, which is a particular advantage of this RMR_mbi.

Validations based on actual data

Validating the correlations of MB_i with RQD, JF and J_v based on actual data

Validation studies were performed based on real case histories (Guangxi University and Huaxi Group 2013; Central South University et al. 2011; Wang 2011; Pan 2012). The MB_i values were plotted with the RQD, JF and J_v values in Fig. 24. Based on these real data (Fig. 24), the assessed correlations (i.e., Eqs. (9)–(17)) were verified:

• (i) The fitting analyses were applied to the actual MB_i values and estimated MB_i values using Eqs. (9)–(12), and the fitted curves are presented in Fig. 25. The figure shows that all fitted curves are very close to the 1:1 line, indicating that the four correlation equations (i.e., Eqs. (9)–(12)) properly reflect the relationship of MB_i and RQD. Considering the coefficients of determination and the proximity degree of the 1:1 line and fitted curves, it can be concluded that Eq. (9) estimates the MB_i value more accurately.

• (ii) According to the actual JF values presented in Fig. 24, the estimated MB_i values were produced using Eqs. (13)–(16). Fitting analyses were carried out based on the actual MB_i values and estimated MB_i values by JF, which are illustrated in Fig. 26. It can be seen from Fig. 26 that almost all the actual MB_i values are less than the estimated values. Additionally, the coefficient of determination of the red curve is larger, but the associated curve is far from the 1:1 line; the blue and brown curves are closer to the 1:1 line, with lower coefficients of determination. Considering the coefficients of determination and the proximity degree of the 1:1 line and fitted curves, it can be concluded that Eq. (15) estimates the MB_i value more accurately.

• (iii) From Eq. (17), the estimated MB_i values were obtained by the J_v values (Fig. 24). Fitting analysis was performed, and the fitted curve is presented in Fig. 27. It is observed that the fitted curve is very close to the 1:1 line, with a coefficient of determination of 0.60, which is not very high. Also, the mean absolute percentage error (MAPE) and root-mean-square error (RMSE) of the measured and estimated values are 39.31% and 15.97, respectively. The lower coefficient of determination and larger errors can be explained by the relatively weak reliability of Eq. (17) or the lack of enough real data.

Fig. 24

MB_i, RQD, JF and J_v values acquired from real cases

Fig. 25

Comparison of the four regression models, i.e., Eqs. (9)–(12) (real cases). Note that in the legend, the red, blue, brown and green straight lines are the fitted curves of the measured MB_i values (real cases) and the estimated MB_i values using Eqs. (9)–(12), respectively

Fig. 26

Comparison of the four regression models, i.e., Eqs. (13)–(16) (real cases). Note that in the legend, the red, blue, brown and green straight lines are the fitted curves of the measured MB_i values (real cases) and estimated MB_i values using Eqs. (13)–(16), respectively

Fig. 27

Comparison of measured MB_i values (real cases) and estimated MB_i values using Eq. (17)

Figures 25, 26 and 27 show that none of the fitted curves are too far away from the 1:1 line, and in fact, some curves are very close to the line. Moreover, the corresponding coefficients of determination are rather high, with the smallest being 0.60. It can be concluded that the correlations of MB_i with RQD, JF and J_v, which were obtained from the artificial data sets, are capable of being applied to the real data and are extremely sound.

Application of RMR_mbi and its comparison with RMR

Comparative analysis of RMR_mbi and RMR

In Section 4, the viability of RMR_mbi was preliminarily verified based on the artificial data sets, which confirm that the ability of RMR_mbi to differentiate rock masses with various structures is better than that of RMR (Fig. 22). In this section, the real data values (presented in Fig. 24) and the other input parameters (i.e., intact rock uniaxial compressive strength, joint condition and groundwater condition) are used to determine the final RMR_mbi and RMR values, and a comparison between RMR_mbi and RMR is subsequently carried out, as shown in Fig. 28. As seen in Fig. 28, the fitted curve is fairly close to the 1:1 line, suggesting that in some cases they have similar a ability to differentiate the rock mass quality.

Fig. 28

Comparison between RMR_mbi and RMR based on real cases

Figure 28 shows few data, which means that at times, RMR_mbi is the same as RMR. Actually, because the real application of RMR_mbi is inadequate, simulated experiments were conducted in this study. Figure 22 indicates that R_M is better from the perspective of assessing rock mass structures with a rating of < 8 (Section 4.2). Therefore, the R_M and RMR_mbi are viable, because RMR_mbi is just different from RMR in evaluating rock mass jointing degree.

Some typical case studies of RMR_mbi

The Tongkeng Mine project was launched by a subsidiary corporation of the Huaxi Group Co., Ltd., Guangxi, China, which is owned by the Chinese government, and has a yearly capacity of 2 million tons or more. Three large ore bodies occur in this mine, i.e., Ximaidai, and no. 91 and 92 ore bodies. The exploitation of ore body no. 91 has been completed, and ore body no. 92 is under production, with probable mineral reserves of 30 million tons.

The no. 92 ore body occurs in silicalite and limestone, and is a large, thick ore body with a gentle dip angle. The minerals vary, and mainly include cassiterite, pyrite, marmatite, pyrrhotite and arsenopyrite, and quartz is the typical gangue mineral. When the caving method is used, instabilities (e.g., ground fall, sidewall spalling and roof collapse) frequently appear due to the incipient joint swarm, high ground stress and blasting disturbance, and some installed supports always fail.

Four study sites were selected in which to perform rock mass quality classification. Study site 1 is in the T214 stope at the 494 level, which is the intersection between the no. 14 production drift and ventilation gallery; study site 2 is in the T201 panel at the 434 level; study site 3 is in the no. 2 panel at the 405 level, whose center is the intersection between the no. 3 drill access and ventilation drift; and study site 4 is in the undercut chamber at the 355 level. The locations and ground conditions of these sites are illustrated in Fig. 29. Additionally, joint mapping and core drilling were carried out, which is shown in Fig. 30. The joint orientation data are presented in Fig. 31.

Fig. 29

Locations and ground conditions of the four study sites. a Study site 1; b study site 2; c study site 3; and d study site 4

Fig. 30

Field investigation. a Joint mapping and b core drilling

Fig. 31

Contour plots of joint poles collected from the four study sites. a Study site 1; b study site 2; c study site 3; and d study site 4

Based on the joint information collected, the joint network models of the four study sites were established, and are shown in Fig. 32. The values of all input parameters of RMR_mbi and RMR were measured, and the projected features were also considered, as shown in Table 7. In addition, the MB_i values were calculated by identifying the blocks inside the cubic sub-joint network models to arrive at the REV sizes (Fig. 33), which are extracted from the models presented in Fig. 32.

Fig. 32

Joint network models of the four sites. a Study site 1 (with drift length of 20 m and height of 2.8 m); b study site 2 (with drift length of 10 m and height of 3.2 m); c study site 3 (with drift length of 10 m and height of 3.2 m); and d study site 4 (with drift length of 10 m and height of 3.2 m). Note that only the large-scale joints are exhibited, and the small-scale joints are unobservable

Table 7 RMR_mbi and RMR values for the four study sites

Fig. 33

Sub-joint network model of the study site 4 (5 m × 5 m × 5 m)

Table 7 shows that with exception of study site 1, the RMR_mbi and RMR ratings are divergent. To compare the two systems, the field situations are presented in Table 5. The rock mass classes by stand-up span time are shown in Table 9. From Tables 7, 8 and 9, it can be observed that the RMR_mbi is more reliable; for example, for study site 3, the RMR rating is IV, and in Table 9, the rock mass stand-up time is expected to be approximately 10 h for a span of 2.5 m. However, with respect to the field situation, when the gallery is formed, a large-scale collapse occurs immediately; very long and dense rock bolts are systematically installed, with wire mesh and shotcrete of high thickness. Obviously, the RMR_mbi rating (V) is more reasonable. In short, therefore, based on the field situation, the following can be concluded: for study site 1, both the RMR_mbi and RMR ratings are rational; for study sites 2 and 3, the RMR ratings are overestimated, because the field situations are worse, and the expected stand-up time derived by RMR_mbi ratings are in keeping with the actual conditions. For study site 4, the RMR rating is conservative, and the RMR_mbi rating is appropriate.

Table 8 Field situations for the four study sites

Table 9 Rock mass ratings by stand-up time (mines) (Aksoy 2008)

A review of Table 7 also shows that the discrepancy between the R_RQD + JF and R_M values for the study sites (aside from study site 1) are very significant. Palmstrom (2005) found that when one-dimensional measurements of rock mass jointing degree (e.g., RQD) were applied to a blocky rock mass, various results could be experienced owing to different directions of the core drillings and scanlines, as shown in Fig. 34. Zheng et al. (2018) also used an example to demonstrate that for a stratified rock mass, the RQD values rely heavily on borehole direction. Therefore, differences between R_RQD + JF and R_M are inevitable. However, three-dimensional measurements of rock mass jointing degree are not orientation-dependent (Palmstrom 2005). The MB_i is indeed a three-dimensional index, which is also a critical advantage of RMR_mbi compared to RMR.

Fig. 34

Different RQD values with various directions (Palmstrom 2005). Note that S1, S2 and S3 are the average spacing of joint sets 1, 2 and 3, respectively

Conclusions

Traditional measurements of block size or degree of jointing (e.g., RQD, JF and J_v) have been questioned and criticized, due to their inherent limitations, and the same is true for rock mass classification systems. Therefore, in this study, the MB_i was introduced, and correlations of MB_i with RQD, JF and J_v were evaluated; the combined use of RQD and JF was replaced with MB_i in the RMR system, and this version was termed RMR_mbi. From the results, the following conclusions can be drawn:

• (i) Close correlations were found between MB_i and RQD, JF and J_v, and when joint persistence was considered, the correlations were more significant, which shows that the MB_i can fully capture the influence of joint persistence. Based on these correlations, the reasonableness and advantages of MB_i are highlighted.

• (ii) Because of the lack of sufficient real applications of RMR_mbi (which is also a limitation of this study), simulated experiments were conducted to confirm the viability of RMR_mbi, and the experimental results were in line with the objective trends (Section 3.2). Experimental simulation results suggest that the sensibility of R_M and R_RQD + JF is similar, but R_M is better because it can properly distinguish rock masses whose jointing ratings are lower than 8 (Fig. 22). Comparative analysis of RMR_mbi and RMR shows that in some cases, RMR_mbi is the same as RMR. The typical applications of RMR_mbi and RMR show that the RMR_mbi ratings are more in line with common practice, based on observations of field situations. For the relatively fractured rock masses, the RMR_mbi may be more reliable. Undoubtedly, the theoretical limitations of the combined use of RQD and JF are addressed, and the three-dimensional rock block sizes and joint persistence can be considered; therefore, the RMR_mbi system has strong application potential.

• (iii) Compared to the RMR system, the RMR_mbi is simplified, because the RQD and JF are replaced with MB_i. To determine MB_i, the joint data and computer technology (presented in Section 2) are needed. However, in the short term, it is necessary to take time to complete the simulation of the joint system and calculation of block sizes, and the amount of time spent will be entirely dependent on the complexity of the joint system.

Additionally, the RMR_mbi developed in this study is a new offering. In future work, significant effort should be made to further justify it, including (1) relations between RMR_mbi and various rock mass properties (e.g., rock mass strength); (2) stability assessment of different rock engineering properties (e.g., slope and tunnel) by supplementing the RMR_mbi; (3) determining how to select proper supporting schemes based on RMR_mbi; and (4) determining how to rapidly simulate the rock joint system and calculate the rock block sizes for determining an RMR_mbi value within a short period of time.