Introduction. The roughness of rock material fractures (called also discontinuities) plays an important role in the mechanical properties and behavior of the rock media [1,2,3]. Therefore, accurately measuring the fracture roughness becomes a fundamental problem in rock mechanics [4,5,6,7,8]. Roughness measurement techniques fall into two primary categories: (1) the contact techniques [9, 10] and (2) non-contact techniques [11, 12].

As an advanced non-contact technique, the three-dimensional laser scanning (3D-LS) is increasingly used to measure the rock material fracture roughness. Previous studies have shown that the sampling interval is important to the characterization of rock material fracture roughness in both 2D and 3D formulations [13,14,15]. Currently, in two-dimensional evaluation methods, most of the research on sampling interval effect has been focused on Barton’s 10 typical profiles (a result of using a profile comb with a 1-mm distance between teeth) [15, 16], while there is very limited attention drawn to the sampling interval effect on 3D geometrical data of rock material fracture surface for the three-dimensional evaluation methods.

Therefore, the sampling interval effect of rock material fracture surface was analyzed in detail in this study, based on the 3D-point-cloud data acquired by the 3D-LS technique. Various concentric circular sampling windows with different diameters were acquired from the 3D surface roughness model, and a series of 2D profiles with different directions at angular increments of 15° were then extracted from each circular sampling window. Each profile was evaluated by three parameters: (1) the mean square first derivate Z2 , (2) the structure function SF, and (3) the roughness profile index Rp at different sampling intervals. These three parameters are widely used to evaluate the rock material fracture roughness and are respectively related to the roughness slope, the degree of change in roughness height, and the actual length of the profile.

1. Methodology of Investigation.

1.1. Specimen and Instrument. A granitic rock material specimen with an irregular size of approximately 4 m3, was considered for the investigation, and a sampling window circle with a diameter of 1.2 m on the specimen was extracted (Fig. 1). The specimen was collected from the Geological Science and Technology Park, which belongs to the National and Provincial Joint Engineering Laboratory for the Hydraulic Engineering Safety and Efficient Utilization of Water Resources of Poyang Lake Basin, China. The sample can be described as a massive blocky, slightly weathered, light red medium-grained granite, with striations on the surface.

Fig. 1.
figure 1

Photo of a scanned sample (a) and the complete 3D surface roughness model acquired by 3D-LS (b). The research area is comprised in the circle with a diameter of 1.2 m.

Full size image

A 3D terrestrial scanner Riegl VZ-1000 from the National and Provincial Joint Engineering Laboratory for the Hydraulic Engineering Safety and Efficient Utilization of Water Resources of Poyang Lake Basin, was mounted on a tripod 1.2 m above the ground and approximately 5.0 m directly in front of the rock face to digitize a rock specimen with the resolution of 0.04°.

1.2. Investigation Procedure. The purpose of this investigation is to (1) clarify the possible sampling interval effect on the measurement accuracy of roughness, and (2) test this effect against the anisotropy effect. The investigation follows a four-step process: Step I – scanning and processing of the 3D point clouds; Step II – extraction of 2D-linear profiles; Step III – extraction of discrete points; and Step IV – evaluation of the rock material fracture roughness.

Step 1. Scanning and Processing of 3D Point Clouds Data. After recording the rock material specimen in the form of point clouds data by 3D-LS technique, the 3D Point clouds data process consists of 4 steps: (1) removing useless point cloud data to focus on the region of interest, as for a 360° panorama was acquired by 3D-LS; (2) reducing the noise in the raw point clouds of the region of interest (ROI); (3) reconstructing the rock surface from the de-noised point clouds data by the Geomagic software; (4) coordinate transformation, including plane fitting and rotation.

Step 2. Extraction of 2D-Linear Profiles. Various concentric circular sampling windows were extracted with a diameter of 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 cm, respectively, and a series of 2D profiles with the directions of 0, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, and 165°, respectively, were then extracted from each circular sampling window. Figure 2 presents the extraction of the sampling windows and 2D profiles. The 2D profiles were finally converted into lines in two-dimensional coordinate system after their alignment.

Fig. 2.
figure 2

(a) A series of concentric circular sampling windows were extracted with a diameter of 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, and 120 cm, respectively; (b) example of twelve2D profiles extracted from a square sampling window (D = 1.2 m).

Full size image

Step 3. Extraction of Discrete Points. For the evaluation of rock fracture roughness, constructed curves should be converted into discrete points. The detailed steps are listed as follows: (1) profiles were extracted through the Geomagic software, and changed into dxf format files; (2) profiles in dxf format were changed into discrete points in txt format files by auto-lisp function. Thus, Z2 , SF, and Rp values can be calculated via the MATLAB software.

Step 4. Evaluation of the Rock Material Fracture Roughness. Various statistical parameters were used to evaluate the rock material fracture roughness in the previous research. It was found that Z2 , SF, and Rp are related to the roughness slope, degree of change in roughness height, and the profile actual length, respectively. In this study, Z2 , SF, and Rp were used as the evaluation indices of the joint roughness coefficient (JRC) to analyze the sampling interval effect. For every circular sampling window, Z2 , SF, and Rp values in different directions ranging from 0 to 165° at angular increments of 15° were calculated with sampling intervals of 0.2, 0.4, 0.8, 1.6, and 3.2 mm, respectively. Values of Z2 , SF, and Rp were determined using the following equations:

$$ {Z}_2={\left[\frac{1}{L}{\int}_{x=0}^{x=L}{\left(\frac{dy}{dx}\right)}^2\right]}^{1/2}={\left[\frac{1}{L}\sum \limits_{i=1}^{n-1}\frac{y_{i+1}-{y}_i}{x_{i+1}-{x}_i}\right]}^{1/2}, $$
(1)
$$ SF=\frac{1}{L}{\int}_{x=0}^{x=L}{\left[f\left(x+ dx\right)-f(x)\right]}^2 dx=\frac{1}{L}\sum \limits_{i=1}^{n-1}{\left({y}_{i+1}-{y}_i\right)}^2\left({x}_{i+1}-{x}_i\right), $$
(2)
$$ {R}_p=\frac{\sum \limits_{i=1}^{n-1}{\left[{\left({x}_{i+1}-{x}_i\right)}^2+{\left({y}_{i+1}-{y}_i\right)}^2\right]}^{1/2}}{L}, $$
(3)

where yi is the height of a rock material fracture profile at xi , dx is the sampling interval between xi and xi+1, and L is the normalized length of the rock material fracture profile.

2. Results.

2.1. Sampling Interval Effect. For the profile in the direction of 0°, Z2 almost does not vary with sampling intervals as L = [2 cm, 120 cm] (Fig. 3a); SF is sensitive to the sampling interval, and SF increases exponentially as the sampling interval increases (Fig. 4a); Rp varies significantly as L ≤ 20 cm (Fig. 5a), while Rp varies very little with sampling interval as L > 20 cm (Fig. 5b). As for the effect of L on parameters Z2 , SF, and Rp , the overall tendency is that at L ≤ 20 cm, values of Z2 , SF, and Rp increase sharply as L increases; Z2 , SF, and Rp are approximately maximal at L = 20 cm; the values of Z2 , SF, and Rp decrease sharply at 20 cm < L < 80 cm, and remain unchanged at L ≥ 80 cm (Figs. 3b, 4b, and 5c).

Fig. 3.
figure 3

(a) Variation of Z2 with the sampling interval in profile 0°; (b) variation of Z2 with L in profile 0°.

Full size image
Fig. 4.
figure 4

(a) Variation of SF with the sampling interval in profile 0°; (b) variation of SF with L in profile 0°.

Full size image
Fig. 5.
figure 5

(a) Variation of Rp with the sampling interval as L ≤ 20 cm in profile 0°; (b) variation of Rp with the sampling interval as L > 20 cm in profile 0°; (c) variation of Rp with L in profile 0°.

Full size image

For the profile in the direction of 90°, Z2 varies very little with sampling interval at the normalized length 2 cm < L < 120 cm (Fig. 6a); SF is sensitive to the sampling interval, and the overall tendency of SF is that it increases exponentially as the sampling interval increases (Fig. 7a); Rp varies for L ≤ 40 cm (Fig. 8a), in contrast to small variation for L > 40 cm (Fig. 8b). As for the influence of L on parameters Z2 , SF, and Rp , the overall tendency is that at L ≤ 40 cm, Z2 and SF increase sharply with L and is consistent at L > 40 cm (Figs. 6b and 7b). Rp is scattered at L ≤ 40 cm (especially at L ≤ 20 cm) and trends to be consistent at L > 40 cm (Fig. 8c).

Fig. 6.
figure 6

(a) Variation of Z2 with the sampling interval in profile 90°; (b) variation of Z2 with L in profile 90°.

Full size image
Fig. 7.
figure 7

(a) Variation of SF with the sampling interval in profile 90°; (b) variation of SF with L in profile 90°.

Full size image
Fig. 8.
figure 8

(a) Variation of Rp with the sampling interval as L ≤ 40 cm in profile of 90°; (b) variation of Rp with the sampling interval at L > 40 cm in profile of 90°; (c) variation of Rp with L in profile of 90°.

Full size image

2.2. Comparison with the Anisotropy Effect. Furthermore, to examine the sampling interval effect of the 2D roughness parameters Z2 , SF, and Rp in comparison to the anisotropy effect, a series of twelve 2D profiles at L = 120 cm (for 0, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, and 165°, respectively) were extracted at angular increments of 15° and analyzed. However, the profile at the direction of 45° was almost vertical at L = [153.5 cm, 153.9 cm] (Fig. 9), and results in the Z2 value be infinite, and SF and Rp values were much larger than those in other directions. The reason is that the 3D point clouds collected by 3D-LS are much more precise, as compared to the traditional methods such as the Barton comb [17], contour gauge [18], and profile gauge [19]. With the expansion of digital technologies, this profile extraction method will be widely used, and the extreme phenomenon will still be hard to avoid in the future. The suggested solution is to use microtranslation when extracting profiles or local adjustment of the extracted profiles.

Fig. 9.
figure 9

The profile in 45° direction.

Full size image

In this study, Z2 varies very slightly with sampling intervals in any direction, except for that of 45°, which trends to be infinite (Fig. 10a), while Z2 varies between different directions (Fig. 10b). SF increases with sampling intervals in any direction, and SF is sensitive to the sampling interval (Fig. 11a), while SF differs for various directions (Fig. 11b). At the direction of 45°, SF is different from the value in other directions, and it tends to be consistent with them if the sampling interval exceeds 1.6 mm (Fig. 11b). Rp almost does not vary with sampling intervals in any direction, except for of 45° (Fig. 12a), and Rp is almost the same for any direction (Fig. 12b). At 45°, Rp is different from its values in other directions and tends to be consistent when the sampling interval is no less than 1.6 mm (Fig. 12a).

Fig. 10.
figure 10

(a) Variation of Z2 with the sampling interval between different directions; (b) variation of Z2 with the directions between different sampling intervals.

Full size image
Fig. 11.
figure 11

(a) Variation of SF with the sampling interval between different directions; (b) variation of SF with the directions between different sampling intervals.

Full size image
Fig. 12.
figure 12

(a) Variation of Rp with the sampling interval between different directions; (b) variation of Rp with the directions between different sampling intervals.

Full size image

In order to quantify the sampling interval and anisotropy effects, the relative deviation factor (F) was introduced in the following form:

$$ F=\frac{\max (X)-\min (X)}{\mathrm{average}(X)}, $$
(4)

where X refers to Z2 , SF, and Rp .

The relative deviation factor of sampling interval effect means that F is calculated if the direction is certain (Table 1), while the relative deviation factor of anisotropy effect indicates that F was calculated for a certain sampling interval (Table 2). The relative deviation factor of anisotropy effect is much larger than the sampling interval effect.

Table 1. Results on the Relative Deviation Factor for the Sampling Interval Effect
Full size table
Table 2. Results of the Relative Deviation Factor for the Anisotropy Effect
Full size table

By comparing the relative deviation factors of sampling interval and anisotropy, it is seen that the sampling interval has a much weaker influence on the rock material fracture roughness than anisotropy.

3. Discussion and Implications. Most of the traditional methods to measure the surface roughness of rock material fractures are two-dimensional contact methods, and L is often limited within 10–20 cm. However, as indicated in this study, L should be long enough (the effective value of the case in this study is 80 cm). Thus the values of Z2 , SF, and Rp can accurately reflect the overall roughness of the rock surface. The 3D-LS technique is a non-contact, three-dimensional method, which overcomes the size limitation of measurement during field investigation and provides a higher efficiency and accuracy than traditional methods in characterizing the structure surface.

The main finding of this study is that the difference in Z2 and Rp values obtained for different sampling intervals is very small, while SF is very sensitive to the sampling interval. The JRC values are found to be more accurate when calculated via Z2 and Rp , instead of SF, especially when the 3D point cloud data are not dense enough. This complies with results of other researchers. Thus, Ge et al. [15] reported that the amplitude of the variation in the range intervals of 10~110 cm was less than that in the range of 110~1000 cm, and 110 cm could be considered an effective sampling interval. However, sampling intervals in some cases range from 10 to 1000 cm. Therefore, this conclusion is applicable to characterizing large-scale fracture in situ but not under laboratory test conditions. Based on Barton’s 10 typical profiles, with a JRC value ranging from 0.4 to 18.7, Jang et al. [14] implied that Z2 and Rp values dropped with the sampling interval. However, for specific conditions or rock material fracture surfaces, the variation of JRC value will be much smaller; as a result, Z2 and Rp values for different sampling intervals will be consistent. The variation of SF revealed in the present study complies with findings in [14].

Conclusions. Based on the observations and analysis presented in this paper, the major conclusions regarding the sampling interval effect of rock material fracture are as follows:

1. The measurement of rock material fracture surface roughness by 3D-LS can provide a higher efficiency and accuracy in characterizing the rock material fracture surface.

2. SF is very sensitive to the sampling interval, and Z2 and Rp are consistent between different sampling intervals for a specific project or rock material fracture surfaces.

3. It is noted that the sampling interval has a much weaker influence on the rock material fracture roughness than the anisotropy. It is vital to have a good observation for the shear direction of the rock material fracture in engineering practice.