Integration of homogeneous structural region identification and rock mass quality classification

In rock engineering projects, professionals assess the overall rock mass qualities using a sole value. However, the true qualities of partial rock masses are incompatible with such a value. To address this problem, the idea of regionally classifying rock mass qualities is proposed and the associated procedure presented. To achieve this goal, the probabilistic and deterministic joints within the study area were determined, and a three-dimensional joint network model was created. Then, the three-dimensional joint network model was discretized into interlocking subdomains, and the modified blockiness index (MBi) was used to finely identify the homogeneous structural regions, together with the k-means algorithm and the sum of squared errors (SSE). A synthetic model comprising homogeneous structural regions was developed and validated with respect to the extracted cross-sections. Next, an improved rock mass rating system (RMRmbi) was introduced, and the viability of RMRmbi was supported through a significant amount of theoretical cases and several real cases. Finally, visualization of regional RMRmbi classification results was performed. Results show that: (i) the homogeneous structural regions are finely demarcated in three dimensions, and (ii) the proposed idea can overcome the problem of rock mass quality classification using the conventional approach often leading to ‘overgeneralization’.

QC, 0000-0003-4695-948X; TY, 0000-0002-3079-7976 In rock engineering projects, professionals assess the overall rock mass qualities using a sole value. However, the true qualities of partial rock masses are incompatible with such a value. To address this problem, the idea of regionally classifying rock mass qualities is proposed and the associated procedure presented. To achieve this goal, the probabilistic and deterministic joints within the study area were determined, and a three-dimensional joint network model was created. Then, the three-dimensional joint network model was discretized into interlocking subdomains, and the modified blockiness index (MB i ) was used to finely identify the homogeneous structural regions, together with the k-means algorithm and the sum of squared errors (SSE). A synthetic model comprising homogeneous structural regions was developed and validated with respect to the extracted cross-sections. Next, an improved rock mass rating system (RMR mbi ) was introduced, and the viability of RMR mbi was supported through a significant amount of theoretical cases and several real cases. Finally, visualization of regional RMR mbi classification results was performed. Results show that: (i) the homogeneous structural regions are finely demarcated in three dimensions, and (ii) the proposed idea can overcome the problem of rock mass quality classification using the conventional approach often leading to 'overgeneralization'.
Faults are well developed in the Zinc-polymetallic Mine, and the ore bodies mainly occur in marlstone, calcareous marl, shale and so on. An underground mining stope (with a size of approximately 60 Â 100 Â 50 m) was selected as the testing area (figure 2a), and the field observation was carried out (figure 2b).

Determination of the distribution parameters of probabilistic joints
The existing identification method of homogeneous structural regions mostly focuses on examining the homogeneous degree of joints between two sites, and the tested joints are always small-scale (i.e. mechanical joints with a trace length of approximately 4 m or shorter) and intersect inside the rock mass in a significant amount. The geometrical parameters of small-scale joints can always be said to have a probabilistic distribution, and hence, when a joint network model is created, a large number of small-scale joints are randomly generated via a distribution function, which corresponds to the in situ conditions examined, and are termed 'probabilistic joints' [18]. Regarding that, the small-scale joints in the four cross-cut galleries were measured, and the orientation data are shown in figure 3.
Sufficient samples were obtained in the criss-crossed galleries, which enabled a test to determine the homogeneous degree of all the observed small-scale joints. The method proposed by Li et al. [7] was employed to test such a degree. It can be concluded that all the statistical p-values calculated are greater than 0.05, and hence no significant difference exists between these samples, which indicates that the small-scale joints in the test area are statistically homogeneous.
Overall small-scale joint data are presented in figure 4, and the probabilistic distribution parameters of these joints were determined as shown in table 1, which can be used to generate probabilistic joints.

Identification of the deterministic joints and their geometrical properties
Some joints have been observed that are greater than the gallery scale. Investigation shows that these joints mostly belong to incipient discontinuities, minor faults, bedding planes, etc. Because the quantity is relatively small, these large-scale joints can be defined as 'deterministic joints' [19]. It is possible to define the diameter of a large-scale joint as infinite (note that the disc-joint model is used for creating a joint network). However, if all the observed joints that are greater than the gallery scale are generated as an infinite plane, the computer calculation is very time-consuming, and the intensity of jointing will be overestimated because the majority of large-scale joints may be connected. Therefore, a determination method for connected joints was employed, which enables the identification of the connected large-scale joints in an underground mining stope. In this method, the connectivity degree of large-scale joints can be examined with respect to joint categorization, mechanical property, coplanar condition, etc. (more details are provided in the electronic supplementary material).
Following the determination method of connected joints, the connectivity degrees of large-scale joints were investigated, and a total of 42 deterministic joints were identified. Most of them are incipient discontinuities, minor faults and bedding planes, with lithologies of limestone and silicite. The fillings are mainly calcite, and the joint wall is smooth or rough. The groundwater conditions are completely dry. The deterministic joints can be generated using five input parameters: x, y, z, a, and b. (x, y, z)

Three-dimensional joint network model coupling probabilistic and deterministic joints
Typically, when the small-scale joints show a statistically homogeneous feature, the tested samples (galleries) can be treated as identical homogeneous structural regions. However, the large-scale joints should not be ignored, and thus, a further identification of homogeneous structural regions should be conducted. Before proceeding with this further identification, a three-dimensional joint network model coupling probabilistic and deterministic joints (with a size of 60 Â 100 Â 50 m) was developed ( figure 6). Note that the generated range of probabilistic joints was 80 Â 120 Â 70 m, because there are some joints whose disc-model centres are outside the model but intersect the model, i.e. boundary effects were eliminated.
4. Three-dimensionally fine identification of homogeneous structural regions 4.1. Selection of the identification index and rationality Generally, various geometrical parameters, including joint orientation, trace length and joint density, can be used to identify structural region boundaries. In this study, the given rock masses were finely regionalized  in three dimensions using the MB i index. The MB i index is defined as the ratio of the volume of rock blocks, which are fully enclosed by joints, to the total rock mass volume, and it can be calculated as follows: Over the years, joint orientation has typically been used to evaluate homogeneous structural regions, though it is not a prerequisite [20]. For example, Kulatilake et al. [21] used the box fractal dimension as a measure of a statistically homogeneous rock mass; they also reported that block sizes are primarily controlled by joint persistence and density.
The MB i index is a three-dimensional measurement of block size and degree of jointing that can capture the effects of joint persistence and density [22,23]. Meanwhile, the MB i value is produced by   the joint network, which can be exactly adopted for structural region identification based on the three-dimensional joint network model coupling the probabilistic and deterministic joints.

Discretizing the three-dimensional joint network model into subdomains
Before discretizing the three-dimensional joint network model ( figure 6) into subdomains, the size of the subdomain needs to be determined. Referring to the concept of representative elementary volume (REV) [23], the optimal size of the subdomain was derived via the following steps: (i) the children models of the gradually increasing sizes were generated inside the parent model, as shown in figure 7, (ii) the MB i values of all the children models were calculated using equation (4.1), and (iii) the optimal size of 20 m was determined using the coefficient of variation (C v ), as shown in figure 8. It is noted that the deterministic joints are temporarily excluded in the parent model, because if the deterministic joints are considered, the whole joint network model is no longer statistically homogeneous and the determined REV size is meaningless [18].
x y z Figure 6. Three-dimensional joint network generated by the coupling of probabilistic and deterministic joints. The interlocking subdomains that are partially overlapping each other were generated within the three-dimensional joint network model, and the distance between the two adjacent subdomains is 1 m

Finely identifying homogeneous structural regions using k-means and sum of squared errors
Cluster analysis is a process that partitions objects into a groups, and the yielded groups are generally called clusters. Objects in the same cluster are more similar, in some sense, to each other than to those  in the other clusters. Cluster analysis is essentially similar to homogeneous structural region identification. To measure the similarity degree between different subdomains, a k-means algorithm [24] was used to cluster the MB i values of all the subdomains.
The k-means algorithm is a widely used clustering procedure. Before the k-means algorithm is executed, the cluster number should be determined. A classical k-means algorithm can be performed via the following steps: (i) randomly distribute the sample fx 1 , x 2 , . . . , x n g to k clusters and calculate the initial centre of each cluster fz 1 , z 2 , . . ., z k g using equation (4.2) as follows: where n is the number of objects in cluster i and s i denotes cluster i; (ii) compute the distances between the object x i and Z i (i ¼ 1, 2, . . . , k), and allocate this object to its nearest cluster; (iii) update the centre of the k cluster using equation (4.2); and (iv) calculate the D value using equation (4.3) as follows:   Figure 9. A two-dimensional hypothetical example to illustrate the interlocking ( partially overlapped) subdomains. The three subdomains are highlighted with three colours, the distance between the two subdomains is 1 m, and the side length of a subdomain is 20 m.  In this section, the sum of squared error (SSE) was adopted to determine an optimal cluster number, which can be calculated as follows: When the cluster number increases, the number of objects in a cluster will decrease, and the distances between the objects and the corresponding centres of the clusters will shorten. In this circumstance, SSE values will decrease. However, when the decrease degree of SSE values lessens, i.e. the slopes of the curve (SSE versus cluster number) insignificantly vary, a conclusion can be reached that increasing the cluster number can no longer improve the clustering qualities, and the corresponding data point can be deemed the turning point in the curve. Thus, the associated cluster number can be regarded as optimal. A scatter plot of SSE and cluster number was developed as shown in figure 11, indicating that the optimal cluster number is six. Therefore, a k-means algorithm with a cluster number of six was executed, and six clusters were yielded as follows:

Synthetic model of homogeneous structural regions and its verification
Pre-processing of subdomains was performed: all the subdomains were contracted into elements (or say, sub-subdomains) of 1 Â 1 Â 1 m along the centres of the subdomains, as shown in figure 12. In turn, the model ( figure 6) was discretized again into elements. Subsequently, these elements were stained different colours according to their associated MB i values, i.e. MB i values in the same cluster were highlighted with an identical colour, as shown in figure 13. The histogram of the rock mass volumes in different homogeneous structural regions is shown in figure 14.
As can be seen in figures 13 and 14, the vast majority of rock masses in the study area fall into the intervals [0.002747%, 0.14986%] and [0.014987, 0.28447%]. If the joint network model of the study area is entirely built by probabilistic joints, the MB i values of the children models range from 0.033% to 0.045% (as shown in figure 8), which falls into the interval [0.014987, 0.28447%]. Thus, it can be concluded that the MB i values of the whole study area are majorly affected by probabilistic joints. However, it is improper to evaluate the overall study area as a homogeneous structural region, because the MB i values of the children models in the probabilistic joint network vary within a narrow interval. Figure 14 shows that a substantial proportion of subdomains have higher MB i values than the children models of the REV sizes, and this is a result of a large number of deterministic joints that enhance the degree of jointing. Nevertheless, the existing identification method of structural regions is limited in this respect. Cross-sections were extracted along the three-dimensional joint network model ( figure 6) and synthetic model of the homogeneous structural regions (figure 13) at half length, half width and half height, and comparisons were conducted to validate the accuracy of the fine identifications of homogeneous structural regions ( figure 15). Visual inspections were performed, based on the dense degree of joints; as shown in the closed red (dashed) wireframes in figure 15, where the dense degree of joints is high, the MB i value is high. In short, this synthetic model of homogeneous structural regions can capture the differences in joint densities between subdomains, and its accuracy is supported.

A rock mass quality classification system: RMR mbi
The failure to accurately assess rock mass qualities may be a result of an unreliable measurement of block size [13]. Some drawbacks of the simultaneous use of RQD and joint spacing in the RMR system have been widely acknowledged as follows: (i) both RQD and joint spacing are anisotropic, (ii) the RQD concept ignores the block size effect [15], (iii) joint persistence is neglected, and (iv) the simultaneous use repeatedly calculates the joint density [25].
In this instance, the RMR mbi was introduced, in which the simultaneous use of RQD and joint spacing is replaced with MB i and the other input parameters remain unchanged. The RMR mbi has several advantages compared to the RMR [17,23] as follows: (i) MB i is a three-dimensional quantification of jointing degree and is not anisotropic, (ii) MB i counts blocks of all sizes, (iii) joint persistence is considered, and (iv) the RMR mbi does not repeatedly calculate the joint density. The rating of MB i (R M ) can be determined using a continuous function as follows: The higher the MB i value, the higher the degree of jointing, and the poorer the rock mass quality.

Preliminarily analysing the viability of RMR mbi based on theoretical discrete fracture network (DFN) models
As described in this section, a significant amount of theoretical DFN models were created to preliminarily analyse the viability of RMR mbi . Because the degree of jointing (or block size) is primarily influenced by joint spacing and persistence [23,26], 20 joint spacing values and 10 joint persistence values were chosen (table 2)   Because the RMR mbi and RMR only differ in the characterization of the degree of jointing, the RQD values and the joint spacing values of all theoretical DFN models were determined referring to [27], as shown in figures 17 and 18. Additionally, all the measured MB i values are shown in figure 19. Based on figures 17-19, the ratings of RQD plus joint spacing (R RQDþJS ) and MB i (R M ) were calculated, according to [28] and equation (5.1), respectively, as shown in figure 20. This figure shows that for the majority of theoretical DFN models, the R RQDþJS and R M share similar evaluation results, e.g. in the intervals of 10 to 30. However, for a number of theoretical DFN models, the two rating standards yield different results, which may be because of the implication of joint persistence on MB i . The correlation coefficient r [29] between R M and R RQDþJS was determined to be 0.93, which is very close to 1, and this suggests that (i) R M and R RQDþJS share a similar sensibility to distinguish rock mass structures, and (ii) the evaluation results of R M are in close proximity to those of R RQDþJS , which has been substantially applied, and therefore, R M is potentially adoptable.

5.2.
Assessing the viability of RMR mbi based on several real data As described in this section, some real data (including all the input parameters of RMR mbi and RMR) collected from [30 -33] were used to again support the viability of RMR mbi , and the final rating results    are shown in figure 21. It can be seen that the fitting line is near the 1 : 1 line, suggesting that they have similar abilities to evaluate the qualities of real rock masses. However, the fitting line is under the 1 : 1 line, which indicates that most RMR mbi values are greater than those of RMR, and in turn, when appraising the rock mass qualities in the 'Fair' category or higher, the RMR mbi may not be as conservative, similar to RMR.
Overall, according to the results of the simulated experiments and real applications, it is considered that RMR mbi is a workable classification method with great application potential. RMR mbi not only overcomes the theoretical limitations of RQD and joint spacing but also produces practical and reasonable rating results. Figures 20 and 21 imply that RMR mbi is slightly different from RMR, and this circumstance is inevitable because the subsystem of RMR mbi (R M ) can capture the influence of joint persistence on the degree of jointing.
6. Visualization of the regional rock mass quality classification results

15
RMR mbi classification and its visualization can be effectively implemented. During the geological investigation, a large number of stations were sited in the main and cross-cut galleries, and the geological data were measured. The Kriging method [11] was used to estimate the RMR mbi values of the untouchable rock masses, based on geological data measured in galleries. Then, the visual model of RMR mbi classification was constructed, which was stained with different colours according to the RMR mbi values ( figure 22). As shown in figure 22, the rock masses of the study area are in Classes I, II and III, respectively. The volumes of the rock masses of the different classes were counted, as shown in figure 23. The great majority of rock masses are in Class II followed by Classes III and I as follows: 25.9857 Â 10 4 m 3 (Class II), 3.4568 Â 10 4 m 3 (Class III) and 0.5575 Â 10 4 m 3 (Class I). The rock masses in the three classes differ greatly in volume, indicating that Class II is the dominant quality of the study area. Additionally, the rock mass qualities gradually decrease with an increase in depth.

Conclusion
To address the problem that rock mass quality classification is often one of 'overgeneralization' when a traditional evaluation approach is used, the idea of regionally classifying rock mass qualities, i.e. identifying rock mass homogeneous structural regions then classifying rock mass qualities, was proposed and the associated procedure presented. Visualizations of homogeneous structural region  Figure 22. Visualization of the regional rock mass quality classification results of the study area. identification were performed and rock mass classification results obtained. This work allows for several conclusions as follows: (i) An existing evaluation method of structural regions was employed to determine the distribution parameters of probabilistic joints, and the deterministic joints were identified through a determination method of connected joints. Subsequently, a three-dimensional joint network model coupling probabilistic and deterministic joints was established. (ii) The blockiness level of rock masses are primarily controlled by joint spacing and persistence; thus, these two joint properties can be considered in three dimensions if the MB i index is used to finely identify homogeneous structural regions. (iii) The k-means and SSE were used together to cluster the measured MB i values of all subdomains, and optimal cluster numbers and clustering results were obtained. The MB i values in the same cluster can be said to have originated from an identical source (i.e. homogeneous structural region). (iv) The three-dimensionally fine identification of homogeneous structural regions performed in this study can consider the implications of joint persistence and spacing, which have a strong identification ability. The identification results (synthetic model of homogeneous structural regions) can reflect the discontinuous and inhomogeneous features of natural rock masses. (v) The consolidated use of RQD and joint spacing in the conventional RMR system was replaced with the MB i index, and this version was termed RMR mbi . Based on a large volume of theoretical DFN models and several real data, it is proved that R M and R RQDþJF share a similar sensibility to differentiate rock mass structures, and RMR mbi has a similar applicability compared to the RMR system. In other words, for the majority of rock masses, RMR mbi and RMR yield similar results, indicating that RMR mbi is adoptable; but owing to the difference in rock mass structure characterization, it is apparent that a slight discrepancy exists between the two systems. However, undoubtedly, the theoretical limitations caused by the combined use of RQD and joint spacing were tackled in the RMR mbi system that, as a consequence, have a strong application potential. (vi) Based on the identification results of homogeneous structural regions, the RMR mbi system was applied to the study area, and a visual model of classification results of rock mass qualities was established. In this way, the complicated data (i.e. rock mass quality classification results) were converted to a three-dimensional digital model, which are beneficial in representing the spatial distribution of rock mass qualities and determining the engineering support schemes. (vii) Based on the homogeneous structural region identification and rock mass quality classification system, we proposed a new method of regionally classifying rock mass qualities. However, it is still a generalization but with improvements, because it is not a true representation of the rock mass.
Additionally, the development of RMR mbi is a new attempt. Although the conventional RMR system has some limitations, it has been successfully applied for more than 40 years. The experience of its application has been substantially accumulated, which is exactly what the RMR mbi of the current version lacks. Furthermore, completing a conventional RMR task requires a few minutes or an hour in a local area; however, to obtain a final RMR mbi value, professionals may spend more time constructing joint network models and calculating rock block sizes (maybe several hours or days). Therefore, future studies are needed to further verify the applicability of RMR mbi (e.g. applying the RMR mbi to more real cases and developing a method to rapidly determine the MB i value of jointed rock masses).
Ethics. Research ethics: We are not required to complete an ethical assessment prior to conducting our research. Because the research in this paper is a rock engineering problem, no ethical problem is related to this paper. Animal ethics: We are not required to complete an ethical assessment prior to conducting our research. No animal is used or related to this paper.
Authors' contributions. Q.C. conceived of the study, helped draft the manuscript and reviewed the manuscript. T.Y.
collected and analysed the data, constructed the models and wrote the manuscript. All authors gave final approval for publication.