HydroMechanical Behaviors of Rock Fractures and Fracture Networks
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Yingchun Li, Shengyue Sun, Hongwei Yang, "Scale Dependence of Waviness and Unevenness of Natural Rock Joints through Fractal Analysis", Geofluids, vol. 2020, Article ID 8818815, 18 pages, 2020. https://doi.org/10.1155/2020/8818815
Scale Dependence of Waviness and Unevenness of Natural Rock Joints through Fractal Analysis
Abstract
The scale dependence of surface roughness is critical in characterising the hydromechanical properties of fieldscale rock joints but is still not well understood, particularly when different orders of roughness are considered. We experimentally reveal the scale dependence of twoorder roughness, i.e., waviness and unevenness through fractal parameters using the triangular prism surface area method (TPM). The surfaces of three natural joints of granite with the same dimension of are digitised using a 3D laser scanner at three different measurement resolutions. Waviness and unevenness are quantitatively separated by considering the area variation of joint surface as grid size changes. The corresponding fractal dimensions of waviness and unevenness in sampling window sizes ranging from to 1 at an interval of are determined. We find that both the fractal dimensions of waviness and unevenness vary as the window size increases. No obvious stationarity threshold has been found for the three rock joint samples, indicating the surface roughness of natural rock joints should be quantified at the scale of the rock mass in the field.
1. Introduction
Rock masses contain a large body of joints. The mechanical and hydraulic behaviours of rock joints highly affect the hydromechanical properties of rock masses. Due to geological processes, rock joints with natural surfaces occur over a broad scale from millimeters to kilometers. Accurate description of joint roughness at the relevant scale is crucial for predicting the hydromechanical coupling of the rock mass.
The surfaces of natural rock joints exhibit varying degrees of roughness. Roughness refers to the inherent unevenness and waviness of a rock joint surface relative to its mean plane [1]. Initially, waviness and unevenness represent largescale undulations observed in the field and smallscale roughness sampled in the laboratory, respectively [2, 3]. Largescale undulations dominate joint dilatancy since they are too large to be sheared off, and smallscale roughness affects joint shear strength as it is usually damaged under shear. The surface of a laboratorysized rock joint also exhibits twoorder asperities, i.e., firstorder waviness and secondorder unevenness [4–8]. Waviness with comparatively larger wavelength and amplitude primarily contributes to dilation, whereas unevenness of a smaller asperity size is sheared and damaged, providing shear resistance to the shear movement. That is to say, a rock joint surface is characterised by twoorder roughness at various scales [9]. Although numerous empirical and statistical approaches have been proposed to quantify the roughness of rock joints [10–15], they have rarely taken into account the twoorder roughness of a joint surface that plays distinct roles in the mechanical and hydraulic behaviours of rock joints [7, 16, 17].
The roughness of a natural rock joint surface depends on the scale of examination, which is referred to as scale effect. Bandis et al. [18] reported that the value of (joint roughness coefficient) decreased as the rock joint size increased, i.e., negative scale effect. On the other hand, conflicting results including positive and no scale effects have been observed [19–22]. By examining the morphological characteristics of the surface roughness of a largescale rock joint replica (), Fardin et al. [23] also stated that there was a stationarity threshold beyond which the scale dependency of surface roughness vanished, i.e., the roughness remained unvaried once the scale exceeded the size of the stationarity threshold. Due to these controversial findings, the nature of how scale affects the surface roughness remains enigmatic.
Fractal theory [24, 25] has been successfully applied to characterise the roughness of rock joints at varying scales. Many approaches to estimate the fractal dimension of a rock joint profile have been proposed, including ruler length [26], box counting [27], variogram [28, 29], spectral [30, 31], roughness length [9, 23, 32, 33], and line scaling [34]. The triangular prism surface area method (TPM) [35–37], project covering method (PCM) [38–40], and cubic covering method (CCM) [41] are shown to be applicable for determining the fractal geometry of a threedimensional joint surface. However, few of them have considered the individual fractal dimensions of waviness and unevenness since a universal single value was commonly assumed.
In this paper, we examine the fractal characteristics of waviness and unevenness of three natural granite rock joints dimensioned up to . We find that eachorder roughness possesses individual fractal dimension at varying sizes from to . The waviness and unevenness of a rock joint surface are separated by considering the surface area variation as grid size changes. The fractal dimension of eachorder roughness is calculated using TPM (triangular prism surface area method). Evident scale dependency of fractal dimension of eachorder roughness has been observed. However, the stationarity threshold of the joint surface roughness is possibly absent.
2. Data Acquisition
We scanned and reconstructed the threedimensional surfaces of three natural granite joints 70 (labeled _{1}, _{2}, and _{3}, respectively) sourced from a quarry in Fujian Province, China (Figure 1).
The dimension of each rock joint surface was . The rock joint surfaces were initially covered by a very thin layer of dust. We carefully cleaned the dust with wipes to avoid damaging the surface roughness. Visual observation suggested that the granite joints are light gray and unweathered.
When the rock joint surfaces were naturally dried in the laboratory, a Creaform MetraSCAN 3D 750 system was employed to digitise the joint surface at three measurement resolutions with point spacings being 0.5 mm, 1.0 mm, and 2.0 mm, respectively. The optical scanning system consists of a HandyPROBE for scanning, a CTrack sensor to locate the position of the HandyPROBE, a CTrack controller for data acquisition, and a laptop for image processing and display (Figure 1). It ideally can acquire the topographic information of an object up to several meters at a minimum point spacing of 0.05 mm. During data acquisition, we scanned the rock joint surface region by region very slowly and carefully to fully capture the morphological properties of the rock joint surfaces. The digitised surface of the rock joint was visualised simultaneously over scanning through the laptop monitor, which ensured that each surface was thoroughly reconstructed without small empty areas remained. We attempted to obtain a more detailed joint surface at the point spacing of 0.2 mm but failed due to memory limitation of the laptop. The graphic processing software, Geomagic Studio, was employed to coordinate the data acquired through the scanner. The graphic processing software, PolyWorks, converted the format of the data imported from Geomagic Studio to the format that is readable by the data processing software, MATLAB.
3. Fractal Dimensions of Waviness and Unevenness
We used the wellestablished triangular prism surface area method (TPM) [35, 37] to estimate the fractal dimensions of waviness and unevenness. The principle of TPM is that the true surface area of a joint surface is measurable once the heights of all points on the joint surface above a base reference plane are established. For a square grid with a side length of (Figure 2), the elevation at the center of the grid cell () is determined by the elevations of its four points: where , , , and are the elevations of the four points, respectively (Figure 2).
The area of one of the triangles, _{1}, is as follows: where
Similarly, the areas of the other three triangles, i.e., _{2}, _{3}, and _{4}, are calculated, respectively. The true area of a joint surface in a given grid cell sized of is as follows:
The joint surface area is as follows: where denotes the number of total grid cells. The joint surface area is a function of grid size () by [37]: where is the fractal dimension of a joint surface, and is a coefficient. Note that the original approach of Clarke [35] estimated the fractal dimension () through the relationship between joint surface area and grid size square (), i.e., . However, Equation (6) using grid size is proven to be mathematically correct and experimentally reliable [36, 37, 42] as the use of the grid size square underestimated fractal dimension [43].
To doublelogarithmise Equation (6), we have: where and are estimated from the slope and intercept of the plot, respectively.
To investigate the scale dependency of surface roughness, the fractal dimensions of waviness and unevenness of the three rock joint samples at varying window sizes from to are estimated (Figure 3). The square window of different sizes is selected from the central part of a rock joint surface. We first plot the relationship between and based on Equations (1), (2), (4), (5), (6), and (7). The joint surface area is computed at varying grid sizes through Equation (5). Table 1 shows the grid size used in calculating the joint surface area at varying window sizes of different measurement resolutions from to at the interval of . As illustrated in Figure 4, the principle of grid size determination is to ensure that the side length of the sampling window is divisible by the grid size () that is a multiple of the point spacing. Figure 5 demonstrates the doublelogarithmic relationship between surface area and grid size of the three rock joint samples at the dimension of under the resolution of point spacing at 1.0 mm. The surface areas of the three joint samples are calculated through TPM with each grid size ranging from 1 mm to 400 mm, i.e., . Waviness and unevenness are separated by considering the area variation of a joint surface at varying grid sizes. Specifically, as the grid size decreases, the joint surface area increases to approximately the real surface area. When the grid size exceeds 30 mm, the slope of the plot decreases remarkably. Under this circumstance, the joint surface area is primarily contributed by waviness, whereas the surface area of unevenness is excluded. For all the three rock joint samples, the slopes of the curves vary noticeably at the grid size of 30 mm at which waviness and unevenness are separated. Figure 6 illustrates the decomposition of a rock joint surface into profiles of waviness and unevenness.

The unevenness is acquired by subtracting the waviness from the whole joint surface. The fractal dimensions of waviness () and unevenness () of a rock joint surface are determined from the two slopes of each bilinear curve, respectively (Figure 4). Actually, similar bilinearity of the plot of tensioninduced rock joint surfaces has been reported by several researchers [40, 41]. They found that the rock joint surface has nonuniversal fractal dimensions, depending on the measurement scale. However, the nature of the twoorder fractal dimensions that are explained above was not unveiled by the authors.
4. Results
4.1. Scale Effect
Figure 7 shows that the fractal dimensions of twoorder roughness are scaledependent for the three rock joint samples digitised at three measurement resolutions. Tables 2–4 list the fractal dimensions of waviness and unevenness of rock joint samples _{1} to _{3} at varying sizes, respectively. For joint sample _{1} at a fixed point spacing, the fractal dimension of waviness is the highest at the sampling window of , followed by a decrease once the sampling window grows to . As the side length of the sampling window increases to 400 mm, the fractal dimension of waviness peaks with a value smaller than that at the sampling window of . When the side length of the sampling window increases from 400 mm to 1000 mm, the fractal dimension of waviness generally decreases with slight fluctuations at the side lengths of 700 mm and 900 mm. The fractal dimension of unevenness of joint sample _{1} at a certain point spacing, however, is the smallest at the window size of . The fractal dimension of unevenness rises continuously to a peak value as the sampling window size increases to , followed by an overall decrease as the sampling window size is increased to the maximum value of .
(a) Twoorder fractal dimensions of surface roughness of joint sample 1
(b) Twoorder fractal dimensions of surface roughness of joint sample 2
(c) Twoorder fractal dimensions of surface roughness of joint sample 3
 
Note: and are fractal dimensions of waviness and unevenness, respectively. and are the coefficients during linear correlations for estimating and , respectively. and represent the coefficients of determination during linear correlations for estimating and , respectively. 
 
Note: and are fractal dimensions of waviness and unevenness, respectively. and are the coefficients during linear correlation for estimating and , respectively. and represent the coefficients of determination during linear correlations for estimating and , respectively. 
 
Note: and are fractal dimensions of waviness and unevenness, respectively. and are the coefficients during linear correlation for estimating and , respectively. and represent the coefficients of determination during linear correlations for estimating and , respectively. 
For rock joint sample _{2} under a certain measurement resolution, the fractal dimension of waviness fluctuates slightly as the window side length increases from 100 mm to 400 mm, preceding a gradual decrease when the window size grows to . As the sampling window size increases from to , the fractal dimension of waviness increases marginally. The fractal dimension of unevenness seemingly exhibits no general tendency. The magnitude of the fractal dimension of unevenness roughly levels off with several unremarkable fluctuations at different sampling window sizes.
For rock joint sample _{3} at a fixed point spacing, the fractal dimension of waviness is maximum at the window size of and then predominantly declines as the window side length increases to 1000 mm with slight fluctuations. The variation of fractal dimension of unevenness resembles that of rock joint sample _{2} without noticeable trend.
To quantify the variation of fractal dimensions of waviness and unevenness as the window size is enlarged, the percent error relative to the value of window size of is calculated as follows: where and represent the percent error and fractal dimension of waviness or unevenness at a window size between to , respectively. is the fractal dimension of waviness or unevenness at the window size of .
Figure 8 presents the percent errors of waviness and unevenness of the three rock joint samples at window sizes from to at three measurement resolutions. Generally, the effect of window size on the fractal dimension of waviness is more pronounced than that of unevenness. Particularly for rock joint samples _{2} and _{3}, the percent errors of fractal dimension of unevenness are lower than 0.1%. The variations of fractal dimensions of both waviness and unevenness of rock joint sample _{1} are overall larger than those of rock joint samples _{2} and _{3} at the same window size and resolution. For rock joint sample _{1}, the percent errors of the fractal dimension of waviness at varying window sizes under the resolution of 1.0 mm are unanimously the highest, and 2.0 mm the lowest. The percent error of the fractal dimension of unevenness under the resolution of 0.5 mm is the highest, and 2.0 mm the lowest except at the window size of where the percent error of the fractal dimension of unevenness under the point spacing of 1.0 mm is marginally smaller than that under 2.0 mm. For rock joint sample _{2}, the percent errors of the fractal dimension of waviness at varying window sizes under the resolution of 0.5 mm generally are the highest, and 2.0 mm the lowest except at the window side lengths of 300 mm and 800 mm. The percent errors of the fractal dimension of unevenness of rock joint samples _{2} and _{3} at the resolution of 1.0 mm are the highest.
(a) Percent errors of fractal dimensions of waviness and unevenness of joint sample 1
(b) Percent errors of fractal dimensions of waviness and unevenness of joint sample 2
(c) Percent errors of fractal dimensions of waviness and unevenness of joint sample 3
Figures 7 and 8 show that the fractal dimension of eachorder roughness varies from 2.001 to 2.014, and the percent errors of fractal dimensions of waviness and unevenness of the three rock joint samples are numerically small, which are less than 1%. One may draw the conclusion that the scale effect of the fractal dimensions of waviness and unevenness could be neglected. Actually, the low values of percent errors result from the low values of the fractal dimensions (Figure 7) which are common for the rough surfaces of naturally formed rock joints [39, 41, 44]. Many studies reported that the fractal geometry of the surface roughness of threedimensional rock joints is slightly larger than 2.0 [38–41]. Zhou and Xie [41] showed that the fractal dimensions of the surface roughness of tensioninduced rock joints of varying degrees of roughness are all smaller than 2.07. Similarly, several rock joints collected from in situ also exhibited surface roughness of fractal dimensions around 2.05, whereas the (joint roughness coefficient) values [10] of these joint surfaces reached as high as 14.0 [38]. That is to say, a naturally surfaced rock joint possesses a fractal dimension varying in a very narrow band. A small variation in fractal dimension likely leads to noticeable change of surface roughness [26, 45].
The low values of the percent errors of fractal dimensions do not necessarily mean that the variation of surface roughness is negligible as window size change, because the widely used indicators of surface roughness such as and asperity slope can be mathematically related with fractal dimension through certain relationships [45, 46]. These relationships commonly involve scaling coefficients of quite high values [45].
To illustrate the effect of fractal dimension variation on the joint roughness change, the wellestablished relationship linking fractal dimension () and is employed [26]:
Equation (9) was originally proposed to estimate through the fractal dimension () of a twodimensional joint profile. Since the relationship between and the fractal dimension () of a threedimensional joint surface is unavailable, the above formulation is directly adopted by extending the twodimension to threedimension by replacing () with (). Additionally, our purpose is not to quantify through fractal dimension (), but to demonstrate the significant joint roughness variation due to the small change of fractal dimension. Considering the twoorder roughness separation, values of waviness and unevenness ( and ) are, respectively,
Figure 9 shows the effect of window size on the percent errors of values of waviness and unevenness (