A structural domain represents a volume of a rock mass with similar mechanical and hydrological properties. To demarcate structural domains (or statistically homogeneous regions) in fractured rock masses, this study proposes a threeparameter simultaneous analysis method (3PSAM) that simultaneously considers rock fracture orientation, trace length, and aperture to evaluate statistical homogeneity between two regions. First, a 102patch threedimensional Schmidt net, which represents a new comprehensive classification system, is established to characterize rock fractures based on their orientation and aperture. Two populations of rock fractures can then be projected to the corresponding patches. Second, the Wald–Wolfowitz runs test is used to measure the similarity between the two populations by considering the fracture trace lengths. The results obtained by applying the 3PSAM to seven simulated fracture populations show that the homogeneity is influenced by both the distributions of the fracture parameters and the sequences of the fracture parameters. The influence of a specific combination sequence makes it impractical to analyze the rock fracture parameters individually. Combined with previous methods, the 3PSAM provides reasonable and accurate results when it is applied to a fractured rock slope engineering case study in Dalian, China. The results show that each fracture population should be identified as an independent structural domain when using the 3PSAM. Only the 3PSAM identifies the west exploratory trench 2 and the east exploratory trench as being nonhomogeneous because the difference in the aperture of the two fracture populations is considered. The benefit of the 3PSAM is that it simultaneously considers three parameters in the demarcation of structural domains.
A structural domain represents a volume of a rock mass with similar mechanical and hydrological properties. The demarcation of structural domains in fractured rock masses is a basic step in fractured rock modeling and rock engineering design. Fractures cause rock masses to be discontinuous, nonhomogeneous, and anisotropic, which make the properties of rock masses have randomness and uncertainty. Fortunately, because of their formation process, rock fractures always have some regularities. These regularities are important to rock engineering because they have a significant influence on the mechanical and hydraulic behaviors of rock masses. Therefore, the demarcation of structural domains usually includes two steps:
The first step is to select rock fracture parameters to build a comprehensive classification system. An adequate description of rock fractures is critical to structural domain demarcation. According to ISRM (1978) [
The second step is to demarcate structural domains based on the selected fracture parameters. The demarcation process can be accomplished using discrepancy and similarity measures, which are widely used in applied statistics. For example, Guo et al. [
Structural domains are often demarcated based on fracture orientation. Several classical methods are widely used. Miller [
In some cases, other fracture parameters should be simultaneously considered because fracture orientation is not the only influencing factor. Several studies have used multiple fracture parameters to demarcate structural domains. Kulatilake et al. [
Therefore, this paper presents a threeparameter simultaneous analysis method (3PSAM) to evaluate the homogeneity of fractures from simulated data and field data. The method divides fractures into patches based on the fracture orientation (dip angle and dip direction), trace length, and aperture. Using the Wald–Wolfowitz runs test, the dip angle, dip direction, trace length, and aperture of fractures can be treated simultaneously, which allows the interactions between fracture parameters to be taken into account. The simultaneous analysis of the three parameters is the new consideration in the 3PSAM. The procedures of this approach are as follows:
A 102patch threedimensional (3D) Schmidt net (Section
The fractures projected in each patch are ranked by their trace length, and the structural domains can then be demarcated according to the Wald–Wolfowitz runs test (Section
Fracture orientation is a parameter that includes two variables: dip and dip direction. The orientation can be plotted on a Schmidt net as a pole. The pole coordinate system of the Schmidt net is a 2D plane that can only represent two variables. Therefore, a 3D Schmidt net is proposed to represent three variables in a coordinate system. The importance of using the 3D Schmidt net is that it can simultaneously consider three variables. This feature helps to build a new comprehensive classification system based on fracture orientation and aperture. As shown in Figure
The radial distance
The azimuth
The height
Example of a 3D Schmidt net with cylindrical polar coordinates.
The horizontal section of cylindrical polar coordinates is divided based on Miller’s 34patch network (Figure
From bottom to top
Clockwise, starting from north (0°)
From inside to outside
From 1 to 102
Thirtyfour equalarea patches.
Aperture dimensions [
Aperture  Description  Feature 

<0.1 mm  Very tight  “Closed” 
0.1–0.25 mm  Tight  
0.25–0.5 mm  Partly open  
0.5–2.5 mm  Open  “Gapped” 
2.5–10 mm  Moderately wide  
1–10 cm  Very wide  “Open” 
10–100 cm  Extremely wide  
>1 m  Cavernous 
Using cylindrical polar coordinates, any fracture with a dip, dip direction, and aperture can be uniquely plotted as a point in one of the 102 patches of the 3D Schmidt net. It should be noted that the aperture can be described by any classification method for different projects. For other classification methods of the aperture, the key is to determine the cutoff values between the intervals in the
The Wald–Wolfowitz runs test is a method that utilizes a runs approach to examine the similarity between two populations [
The row vectors (fractures) from populations
Use
The number of runs (
Calculate the
Example showing how two populations are ranked.
Patch number  Population 
Population 
Population 
Ranking row vectors (fractures) in ascending order of trace length  Tracking sources  Number of runs 

1 






2 





3 











102 




Therefore, using the usual numerator continuity correction, the cutoff point (critical value)
The onetail
To reduce the subjectivity of the patch orientations, each pair of 3D Schmidt nets is analyzed 18 times as the patch network is rotated from 0° to 180° in 10° increments [
Rotation diagram for the 3D Schmidt net.
Flowchart of the procedures for the program “DOMAINS”.
In this study, seven simulated populations are used to verify the rationality and validity of the 3PSAM. To examine the influence of specific combination sequences of the fracture parameters on the homogeneity, four populations (data 1, 2, 2R4, and 2R5) are assigned to “group one.” To examine the influence of different distributions of the fracture parameters on the homogeneity, five populations (data 1, 2, 3, 4, and 5) are assigned to “group two.” Each population is a 200 × 4 matrix with 200 fractures (row vectors) and 4 variables (column vectors, i.e., dip direction, dip angle, trace length, and aperture). A bivariate normal distribution (BVND), normal distribution (NORMD), and lognormal distribution (LOGD) are used to simulate the fracture orientations, trace lengths, and apertures, respectively [
Parameters of the distribution functions used for the simulated fracture populations.
Population  BVND (orientation)  NORMD (trace length)  LOGD (aperture)  Number of fractures  

Mean (°)  Variance (dip dir., dip)  Mean (m)  Variance  Mean (mm)  Variance  
Data 1  180∠55^{1}  4000, 350  3.5^{1}  1.5  3^{1}  400  200 
Data 2  180∠55^{2}  4000, 350  3.5^{2}  1.5  3^{2}  400  200 
Data 2R4  180∠55^{2}  4000, 350  3.5^{2R4}  1.5  3^{2}  400  200 
Data 2R5  180∠55^{2}  4000, 350  3.5^{2}  1.5  3^{2R5}  400  200 
Data 3  240∠45^{1}  4000, 350  3.5^{2}  1.5  3^{2}  400  200 
Data 4  180∠55^{2}  4000, 350  2.0^{1}  1.5  3^{2}  400  200 
Data 5  180∠55^{2}  4000, 350  3.5^{2}  1.5  10^{1}  400  200 
180∠55^{1 (or2)}: the mean orientation of a population generated by BVND in the
Schmidt plots of the simulated data: (a) data 1; (b) data 2, 2R4, 2R5, 4, and 5; (c) data 3.
As shown in Table
Sequences of the trace length column vector.
Sequences of the aperture column vector.
The main purpose of group two (data 1, 2, 3, 4, and 5) is to examine if the distribution of a column vector (fracture parameter) will affect the identification of structural domains. In group two, the frequency histograms (distributions) of data 3, 4, and 5 are different from those of data 2 in fracture orientation, trace length, and aperture, respectively. The frequency histograms of the trace lengths and apertures of the seven populations are shown in Figures
Frequency histograms of the trace lengths of the simulated populations.
Frequency histograms of the apertures of the simulated populations.
In this study, Miller’s method [
Similarity results of the simulated fracture populations.
Group  Population (data)  Miller’s method  Song’s method  3PSAM  


Results 

Results 

Results  
Max  Min  Ave  Max  Min  Ave  Max  Min  Ave  
One  1 and 2  0.677  0.067  0.321  Accepted  1.000  0.176  0.641  Accepted  0.341  0.064  0.211  Accepted 
1 and 2R4  0.677  0.067  0.321  Accepted  5.83 
1.39 
6.97 
Rejected  1.06 
1.65 
1.38 
Rejected  
1 and 2R5  0.677  0.067  0.321  Accepted  1.000  0.176  0.641  Accepted  0.051  1.77 
0.007  Rejected  


Two  1 and 3  4.44 
3.50 
7.40 
Rejected  0.031  5.05 
0.006  Rejected  0.024  1.11 
0.005  Rejected 
1 and 4  0.677  0.067  0.321  Accepted  0.014  1.06 
0.002  Rejected  0.146  0.002  0.041  Rejected  
1 and 5  0.677  0.067  0.321  Accepted  1.000  0.176  0.641  Accepted  0.008  1.06 
0.002  Rejected 
In group two, data 1 and 3 are not similar according to the three methods because the distributions of the fracture orientations in data 1 and 2 are different. This change implies that the three methods are capable of distinguishing the change of the orientation distribution. Data 1 and 4 are not similar according to Song’s method and the 3PSAM because the distributions of fracture trace lengths are different in data 1 and 2. This change implies that Song’s method and the 3PSAM are capable of distinguishing the change in the trace length distribution. Data 1 and 5 are not similar according to the 3PSAM because the distributions of the fracture trace apertures are different in data 1 and 2. This change implies that the 3PSAM is capable of distinguishing the change in the aperture distribution.
Therefore, three conclusions are obtained from the comparison of the results shown in Table
The real data are collected from a planned excavation slope located in the Southeast Dalian Port, Dalian City, China (Figure
Location map of the study area.
Sketch map of the outcrops.
To avoid the influence of sampling bias, only the populations that were collected from windows with similar strikes are paired for the similarity test. Therefore, the test is divided into two groups according to the window orientations. Group one includes the south exploratory trench, north exploratory trench, and north outcrops 1 and 2, which strike nearly eastwest. Group two includes the west exploratory trenches 1 and 2 and the east exploratory trench, which strike nearly northsouth.
Because the 3PSAM considers the fracture orientation, trace length, aperture, and their combinations, other fracture parameters (e.g., infilling, roughness, weathering) are not discussed in this study. The fracture orientations in each window are plotted on Schmidt nets, as shown in Figure
Schmidt plots of the fracture populations. Group one: (a) south exploratory trench; (b) north exploratory trench; (c) north outcrop 1; (d) north outcrop 2. Group two: (e) east exploratory trench; (f) west exploratory trench 1; (g) west exploratory trench 2.
Frequency histograms of the trace lengths of the fracture populations: (a) group one: south exploratory trench (ST), north exploratory trench (NT), north outcrop (NO) 1 and 2; (b) group two: east exploratory trench (ET), west exploratory trench (WT) 1 and 2.
Frequency histograms of the apertures of the fracture populations: (a) group one: south exploratory trench (ST), north exploratory trench (NT), north outcrop (NO) 1 and 2; (b) group two: east exploratory trench (ET), west exploratory trench (WT) 1 and 2.
It should be noted that aperture quantification is a very complex topic. In this study, the apertures are represented using three categories. This is because the hydrological properties of the rock masses are neglected, and only the perpendicular distance between the adjacent rock walls of an open fracture is used to represent the aperture. The aperture in Figure
The 3PSAM, Miller’s method [
Similarity results of the real fracture populations.
Group  Regions  Miller’s method  Song’s method  3PSAM  


Results 

Results 

Results  
Max  Min  Ave  Max  Min  Ave  Max  Min  Ave  
One  ST and NT  6.68 
8.20 
2.19 
Rejected  4.29 
4.38 
6.11 
Rejected  1.89 
1.50 
2.64 
Rejected 
ST and NO 1  0.002  7.15 
7.89 
Rejected  0.0015  8.73 
4.00 
Rejected  0.002  2.18 
1.69 
Rejected  
ST and NO 2  9.03 
1.38 
2.80 
Rejected  8.10 
7.85 
1.68 
Rejected  5.82 
3.52 
1.15 
Rejected  
NT and NO 1  0.002  1.80 
2.61 
Rejected  0.031  3.33 
0.005  Rejected  4.90 
4.57 
6.82 
Rejected  
NT and NO 2  0.611  1.88 
0.221  Accepted  0.002  4.57 
2.13 
Rejected  8.39 
1.05 
9.10 
Rejected  
NO 1 and 2  0.007  1.04 
0.002  Rejected  0.040  8.15 
0.005  Rejected  1.19 
1.11 
1.56 
Rejected  


Two  ET and WT 1  0.001  1.61 
7.15 
Rejected  0.013  8.21 
0.002  Rejected  0.006  8.21 
0.001  Rejected 
ET and WT 2  0.146  0.019  0.090  Accepted  0.203  0.020  0.085  Accepted  0.083  0.004  0.030  Rejected  
WT 1 and 2  3.16 
1.73 
4.00 
Rejected  5.90 
1.37 
6.12 
Rejected  4.54 
1.37 
4.83 
Rejected 
ST = south exploratory trench; NT = north exploratory trench; NO = north outcrop; ET = east exploratory trench; WT = west exploratory trench.
In group two, only the fracture populations from the west exploratory trench 2 and the east exploratory trench are identified as statistically homogeneous regions according to Miller’s method and Song’s method. This result is reasonable when only considering the orientation and trace length. However, according to the 3PSAM, the two populations are not homogeneous because of their differences in aperture. The fracture populations from the other populations in group two are not homogeneous according to the three methods. Hence, when simultaneously considering the fracture orientation, trace length, and aperture, all of the fracture populations in group two should not be considered statistically homogeneous regions.
The results obtained by applying the 3PSAM to seven simulated fracture populations (Table
The results obtained by applying the 3PSAM to seven real fracture populations (Table
This study proposes a threeparameter simultaneous analysis method (3PSAM) that can simultaneously consider rock fracture orientation, trace length, and aperture to demarcate structural domains in fractured rock masses. First, a 3D Schmidt net, which represents a new comprehensive classification system, is established to characterize rock fractures based on their orientation and aperture. Two populations of rock fractures can then be projected to the corresponding patches. Second, the Wald–Wolfowitz runs test is used to perform a similarity test between two populations by ranking the row vectors (fractures) according to the trace length. Thus, the interactions between fracture parameters are taken into account. The simultaneous analysis of the three parameters is a new consideration in the 3PSAM.
The analysis showed that Miller’s method considers the fracture orientation, Song’s method considers the fracture orientation and trace length, and the 3PSAM simultaneously considers the fracture orientation, trace length, and aperture. Therefore, the 3PSAM can be seen as an extension of Song’s method and Miller’s method. All three methods exhibit reasonable and accurate performance when different fracture parameters are considered. In particular, when a combination of these methods is used, structural domains can be adequately demarcated.
For the future research of structural domain demarcation in fractured rock masses, we have three suggestions. First, the simultaneous analysis of other important fracture parameters such as spacing should be considered because fracture spacing also affects the mechanical and hydrological properties of rock masses. Second, the quantification of aperture can be more detail except considering only the mean perpendicular distance in the context (e.g., considering roughness, infilling, and water condition). This quantification is a complex topic. Third, when simultaneous analysis of multiple parameters, other similarity measures used in applied statistics (e.g., Euclidean distance, kernel method) may be introduced to demarcate structural domains.
Data supporting this research article are available from the corresponding author on request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (grant nos. U1702441, 41330636, 41402242, and 41702301), the Opening Fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology) (grant no. SKLGP2018K017), and the Graduate Innovation Fund of Jilin University.