1 Introduction

The safety and stability of projects such as mining engineering, underground tunnel construction, and water conservancy engineering are closely related to rock mass seepage (Zeigler et al. 1976; Esaki et al. 1999; Li et al. 2008b; Chen et al. 2017; Xu et al. 2022). A large number of joints, fissures and faults in rock masses are the main channels for fluid flow and play an important role in the permeability characteristics of fractured rock masses (Noorishad et al. 1982; Li et al. 2008a; Zhang et al. 2014). The surface morphology of rock joint is a key factor affecting seepage (Nordstrom et al. 1989; Zimmerman and Bodvarsson 1996; Javadi et al. 2010; Luo et al. 2022), and the flow of water along the surface morphology is very complex. Currently, the separation of two-order morphology is used to simplify the hydraulic calculation problem (Zou et al. 2015, 2020). Therefore, the rough features of the two-order morphology are necessary to understand and quantitatively represent the surface geometric complexity of rock joint, which are important for investigating the energy and mass transport processes in fractured rock masses.

Numerous studies have been carried out on the anisotropy of surface morphology based on roughness parameters in two-dimensional (2D) and three-dimensional (3D) perspectives (Kulatilake et al. 1995; Grasselli and Egger 2003; Tatone and Grasselli 2010, 2012; Kumar et al. 2016). The anisotropy of roughness parameters can increase the uncertainty of seepage in rock joint (Luo et al. 2016; Yin et al. 2019). The anisotropic distribution of joint surfaces by roughness parameters shows an approximate circle (Tatone and Grasselli 2010; Kumar et al. 2016) and irregular arc (Tatone and Grasselli 2012; Grasselli and Egger 2003). Therefore, all these anisotropic distributions present a unified feature in which the undulation directions corresponding to the smallest and largest roughness parameters are basically orthogonal to each other. However, these studies rarely consider the influence mechanism of the two-order roughness (roughness reflected by the waviness and unevenness (ISRM 1978; Belem et al. 2009)) on the anisotropy of the total roughness. Hence, analysis of the anisotropic characteristics of the two-order roughness can establish a theoretical basis for exploring the anisotropic mechanism of shear strength.

The roughness is significantly influenced by the sample sizes (Barton et al. 1985). Many scholars have studied the size effect of the anisotropy of the total roughness. The combination parameter \(D_{{{\text{r1d}}}} \times K_{{\text{v}}}\) was applied to evaluate the anisotropy of the joint surface by Kulatilake et al. (2006) and study the effect of size on the anisotropy in 2D roughness. Tatone and Grasselli (2013) selected \(\theta_{\max }^{ * } /(C + 1)\) to estimate the roughness of natural rock joint in various directions with seven sampling sizes and found that the roughness parameters increased with increasing size. Yong et al. (2020) used \(Z_{2}\), \(R_{P}\), and \(\theta_{\max }^{ * } /(C + 1)\) to calculate the joint roughness coefficients (JRC), exploring the anisotropic distribution of natural rock joint of different sizes. The size effect of the anisotropic variation coefficient (AVC) and its fractal behavior were analyzed in depth by investigating four natural rock joints (Hong et al. 2020). These studies are only based on the total morphology, do not consider waviness and unevenness. Waviness initially refers to large-scale undulations in the field, and unevenness represents small-scale asperities that are observed in the laboratory (Patton 1966). Two-order morphology are themselves affected by the size. Further study on the variation law of the two-order roughness with size is of great significance for understanding the size effect mechanism of roughness anisotropy.

In view of the discussions above, this work uses a wavelet analysis method to separate the waviness and unevenness of three lithological natural joints, and the 3D parameter \(\theta_{\max }^{ * } /(C + 1)\) is adopted to quantify the roughness of the two-order morphology in various directions. The anisotropic distribution of the two-order roughness in large-scale joints is evaluated based on the AVC values, while the anisotropic distribution of the two-order roughness in differently sized joints is further discussed based on the AVC values. Furthermore, the contribution rate of the two-order roughness to the total roughness is estimated in all directions on the basis of the functional relationship between the total roughness and the two-order roughness, and the size effect of the contribution rate of the two-order roughness is explored.

2 Methodology

2.1 3D morphology collection

To effectively conduct quantitative research on the two-order roughness of rock joint, three major lithological joints comprising metamorphic rock, magmatic rock and sedimentary rock are selected for analysis. They are slate, tuff and limestone, respectively. They are taken from three typical open-pit mines in Shaoxing, Zhejiang Province, and the joint of the exposed rock mass is well preserved.

A portable laser scanner (MetraSCAN 3D, Creaform, Canada) is used to obtain two groups of digital joints with a size of 1000 mm × 1000 mm for every lithilogical sample. The scanning system includes a dual-camera optical tracker (C-Track), a handheld 3D optical CMM device, and a high-performance PC used for the guidance system. It determines the scanning area by fixing the C-Track and then completes the surface morphology collection by projecting the laser line onto the sample surface, using MetraSCAN 3D to collect the data (see Fig. 1). There is no need to fix the setting during the measurement process, the parts can be moved freely at any time, and the scanning accuracy can be set according to the measurement requirements. To obtain the 3D morphology of the joint with a high degree of reduction, the scanning accuracy is set at 0.5 mm, wherein the six sets of 3D point cloud data of joints are numbered as S1, S2, T1, T2, L1, and L2 (see Fig. 2) for the separation of the two-order morphology.

Fig. 1
figure 1

Scanning of the 3D morphology

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Fig. 2
figure 2

3D morphology and field maps of the six groups of joint surfaces

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2.2 Two-order morphology separation and measurement

Zou et al. (2020) proposed that wavelet analysis stemming from signal processing can achieve better decomposition of the original surfaces into waviness and unevenness. The 3D data of the joint are imported into the workspace, and its original asperity is extracted. The Daubechies' wavelet “db8” producing the best match of the joint surface is chosen as the mother wavelet. The original joint is split into a low-frequency component and a high-frequency component at the decomposition level. When the variance value between the primary joint and the original joint begins to significantly change, compared with the previous levels, the decomposition level is determined as the cut-off level (mc). The approximation joint surface at level mc is determined as the waviness, while the detailed joint surfaces from levels 1 to mc are used to form the unevenness. The decomposition results of the six groups of joint surfaces are presented in Table 1.

Table 1 Surface morphology separation
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Because the morphology of the joint surface exhibits anisotropic characteristics, the separated two-order morphology also exhibits anisotropic characteristics. To further explore the anisotropic characteristics of the two-order morphology, the 3D parameter \(\theta_{\max }^{ * } /(C + 1)\) is selected to quantitatively describe the roughness of the joint surface (Tatone and Grasselli 2009). This parameter has been widely used for characterizing joint roughness in different directions by many scholars in view of the directionality of the geometric morphology. The calculation formula is as follows

$$A_{{\theta^{*} }} = A_{0} \left( {\frac{{\theta_{\max }^{ * } - \theta^{*} }}{{\theta_{\max }^{ * } }}} \right)^{C}$$
(1)

where \(\theta_{\max }^{ * }\) is the maximum apparent dip angle in the shear direction, \(\theta_{{}}^{ * }\) is the apparent dip angle, C is the roughness fitting coefficient, \(A_{{\theta^{ * } }}\) is the potential contact area ratio greater than \(\theta_{{}}^{ * }\), and \(A_{0}\) is the maximum potential contact area ratio. The total roughness \(\theta_{\max }^{ * } /(C + 1)_{T}\), first-order roughness \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I}\), and second-order roughness \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi }\) are defined as the roughness of the total surface, waviness, and unevenness, respectively.

Regarding the anisotropy analysis, 36 directions are selected at angle increment of 10°, wherein \(\theta_{\max }^{ * } /(C + 1)\) is applied to represent the roughness of the joint surface in different directions. The roughness of the two-order morphology for six joints is measured. Table 2 displays the two-order roughness for the S1 joint. The results show that the first-order roughness is larger than the second-order roughness, while the first-order roughness is closer to the total roughness.

Table 2 Statistics of the two-order roughness for the S1 joint in 36 directions
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2.3 Anisotropic behavior characterization of two-order morphology

2.3.1 Qualitative analysis

To better embody the anisotropic characteristics of joint roughness, Fig. 3 summarizes the normalized anisotropic distribution map of roughness parameters obtained by different scholars, and the undulation directions of the largest and smallest roughness parameters are approximately consistent and have an obvious orthogonal distribution law. Therefore, the anisotropy variation behavior of the joint can be reflected by the roughness parameters in the orthogonal direction. When the roughness parameters are roughly equal in the orthogonal direction, the shape of the anisotropic distribution is approximately circular; otherwise, it presents an irregular arc. The greater the difference is, the more prominent the irregular variation behavior.

Fig. 3
figure 3

Normalized anisotropic distribution map of the roughness parameters

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2.3.2 Quantitative analysis

Although the anisotropic distribution map can intuitively reflect the changes in the joint roughness in various directions, it cannot provide a quantitative evaluation of anisotropy. According to the results, the variation degree of anisotropy mainly depends on the contribution ratio of the roughness parameters in the orthogonal direction. Huang et al. (2020) proposed the parameter AVC, which reflects the ratio of the roughness parameters in the inferior and superior directions. Combined with the roughness parameter \(\theta_{\max }^{ * } /(C + 1)\), the expression of AVC is proposed based on the joint with orthotropic distribution

$$AVC{ = }\frac{{\left[ {\theta_{\max }^{ * } /(C + 1)} \right]_{Inf}^{F} + \left[ {\theta_{\max }^{ * } /(C + 1)} \right]_{Inf}^{R} }}{{\left[ {\theta_{\max }^{ * } /(C + 1)} \right]_{sup}^{F} + \left[ {\theta_{\max }^{ * } /(C + 1)} \right]_{sup}^{R} }}$$
(2)

To evaluate the anisotropy of the two-order morphology, \(AVC_{T}\), \(AVC_{\rm I}\), and \(AVC_{\Pi }\) are defined as the anisotropy variation coefficients of the total surface, waviness, and unevenness, respectively.

3 Results

3.1 Anisotropy evaluation of two-order roughness parameters

The anisotropic distribution of the two-order morphology for three lithological joints is shown in Fig. 4, and the inferior and superior directions of total roughness are unified as 0°-180° and 90°-270°, respectively. As shown in Fig. 4a, b, the inferior and superior directions of total roughness are the same as the roughness of waviness, but the two directions of the roughness in waviness are offset by 10° and 20° from the total roughness (Fig. 4c), respectively. For the identical lithology, the anisotropic distribution maps of unevenness are similar, which is attributed to the surface morphology of the joint. The unevenness forms angles of 30° and 40° with the total surface for the slate joint, the unevenness forms angles of 90° and 0° with the total surface for the tuff joint, and the unevenness forms angles of 60° and 30° with the total surface for the limestone joint. Because the undulation of the slate joint is in one direction, the anisotropic distribution is more obvious than that of the tuff joint and limestone joint. Compared with the results obtained by other scholars, the curve of the anisotropic distribution maps of the paper is smooth. The reasons can be subdivided into two main components: the parameters selected in the paper are based on the 3D undulation angle, then all angles can be considered, and the smaller the angle increment is, the more accurate and richer the anisotropy map.

Fig. 4
figure 4

Anisotropic distribution of the two-order roughness for three lithological joints: a slate joint, b tuff joint, c limestone joint

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The inferior and superior directions, as well as the AVC values of the two-order morphology with three lithological joints, are listed in Table 3. The results show that the order of AVC values is \(AVC_{\Pi }\) > \(AVC_{T}\) > \(AVC_{\rm I}\). The AVC values of the two-order morphology of the slate joint are smaller than those of the other rock joints. Based on the anisotropic classification proposed by Huang et al. (2020), the total surface of the three lithological joints presents moderately strong, weak and weak anisotropic variations, the waviness presents moderately strong, weak and weak anisotropic variation, and the unevenness presents medium anisotropic, isotropic, and weak anisotropic variation.

Table 3 Anisotropy analysis of two-order roughness for three lithological joints
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3.2 Anisotropic distribution in various sizes

Before obtaining the series sizes of representative samples, choosing an appropriate sampling method that can obtain the representative morphology based on complete coverage of the joint surface is necessary. Hence, we introduce the progressive coverage statistical method and stratified sampling method with a K-medoids clustering algorithm. According to the progressive coverage method, the continuous sampling size \(l_{i}\) is set at 100 mm, 200 mm, 300 mm, and so on, while the corresponding propulsion space \(\Delta d\) is determined as 90 mm, 80 mm, 70 mm, and so on to ensure the same sample capacity \(N\). Finally, the representative sampling with K-medoids clustering sampling is applied to determine the specific location of the sample, while the joint samples corresponding to these center points are representative samples (Table 4).

Table 4 Sample statistics for different sampling sizes
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The 3D directional two-order roughness results for the samples of 100 mm, 400 mm, 700 mm and 1000 mm of the S1, T1 and L1 joints are displayed in Fig. 5. As the size of the joint increases, the shape of the anisotropic distribution becomes more similar. Regarding the S1 joint, the \(\theta_{\max }^{ * } /(C + 1)_{T}\) values increase with increasing sampling size in the direction of 0° to 180°, but the presence of the size effect is less apparent in the direction of 180° to 360°. The relationship between 3D roughness and sampling size in waviness shows an increasing law, but there is a decreasing law in unevenness. As shown in the figures for the T1 joint, the \(\theta_{\max }^{ * } /(C + 1)_{T}\), \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I}\), and \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi }\) values increase with increasing sampling size except for the 100 mm sample, which is mainly attributed to the large dispersion of the roughness parameters in the small sample. There is no obvious orthogonal distribution law for the size of 100 mm with the L1 joint, and the presence of the size effect is variable. Therefore, the first-order roughness parameters mainly reflect the anisotropy of the total roughness.

Fig. 5
figure 5

Anisotropic distribution of two-order roughness for three lithological joints under different sample sizes

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3.3 Size effect on AVC

To better analyze the size effect of the AVC in the joint, the average AVC value of the joint samples under various sampling sizes is counted as the anisotropic variation coefficient representing the joint. Figure 6 displays the variation in AVC values of the two-order roughness under different sample sizes for three lithological joints. As shown in Fig. 6a, the \(AVC_{T}\) and \(AVC_{\rm I}\) values are alike in the change trend, while the AVC values of the two-order roughness remain basically constant other than 100 mm with the S1 joint. The change trends of the total surface, waviness, and unevenness of the S2 joint are similar to those of the S1 joint, but a mutation size of 200 mm in unevenness is observed. When the sample size is larger than 200 mm for the slate joint, the AVC values of the two-order roughness tend to be stable. This is because the structure of the slate joint is dense and the surface undulations are relatively small.

Fig. 6
figure 6

Variation in AVC values of the two-order roughness under different sample sizes: a slate joint, b tuff joint, c limestone joint

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For the tuff joint, the \(AVC_{T}\), \(AVC_{\rm I}\) and \(AVC_{\Pi }\) values all display an increasing trend, while a sudden point is observed in the sample size of 500 mm for the T2 joint, which has a certain correlation with representative sampling results (Fig. 6b). The AVC values of the two-order roughness under varying sample sizes with the limestone joint are presented in Fig. 6c. Although the curve fluctuates greatly, the \(AVC_{\rm I}\) and \(AVC_{T}\) values decrease as the sample size increases, and the \(AVC_{\Pi }\) values increase as a function of the sample size. Thus, when the sample sizes are larger than 700 mm and 500 mm for tuff and limestone, respectively, the AVC values of the two-order roughness tend to be stable.

The results show that the \(AVC_{\rm I}\) and \(AVC_{T}\) values have an analogous change trend, wherein the \(AVC_{\Pi }\) values are larger both. The \(AVC_{\rm I}\) values control the anisotropic characteristics of the total surface, but the \(AVC_{\Pi }\) values reflect the size effect of AVC.

4 Discussion

4.1 Effect of two-order roughness on anisotropy

Joint anisotropy is mainly reflected by the waviness, while the anisotropic variation degree is less than the total roughness. This indicates that the roughness of waviness in the superior direction is more susceptible to unevenness than that in the inferior direction, which is mainly attributed to the larger wavelength of waviness. In theory, the total surface comprises waviness and unevenness, in which the two-order roughness exhibits anisotropic properties, so there is a certain correlation between the \(\theta_{\max }^{ * } /(C + 1)\) values of the total surface, waviness and unevenness, expressed as

$$\theta_{\max }^{ * } /(C + 1)_{T} = c_{\rm I} \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I} + c_{\Pi } \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi }$$
(3)

where \(c_{\rm I}\) and \(c_{\Pi }\) denote the conversion factor of waviness and unevenness, respectively. In every direction, the contribution rates of the waviness and unevenness to the total surface can be quantified as

$$cr_{\rm I} = \frac{{c_{\rm I} \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I} }}{{c_{\rm I} \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I} + c_{\Pi } \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi } }} \times 100{\text{\% }}$$
(4)
$$cr_{\Pi } = \frac{{c_{\Pi } \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi } }}{{c_{\rm I} \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I} + c_{\Pi } \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi } }} \times 100{\text{\% }}$$
(5)

Taking the S1 joint as an example, Fig. 7 shows the fitting results of \(\theta_{\max }^{ * } /(C + 1)_{T}\), \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I}\), and \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi }\) in 36 directions. The model formula is as follows

$$\theta_{\max }^{ * } /(C + 1)_{T} = 1.054 \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I} + 0.3886 \times \theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi } ,\;{\text{R}}^{{2}} = 0.{999}$$
(6)
Fig. 7
figure 7

Fitting plane chart of \(\theta_{\max }^{ * } /(C + 1)_{T}\), \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\rm I}\), and \(\theta_{\max }^{ * } /\left( {C + 1} \right)_{\Pi }\) in 36 directions for the S1 joint

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The contribution rates of the two-order roughness in every direction are measured and listed in Table 5. The waviness has the highest contribution rate to the total roughness in the 120° and 300° directions, but the unevenness has the highest contribution rate to the total roughness in the 10° and 190° directions. The average contribution rates of waviness and unevenness are 93.17% and 6.83%, respectively. The total roughness is affected by the roughness parameters of waviness, while it is less affected by the roughness parameters of unevenness.

Table 5 Statistics of the contribution rate of two-order roughness in 36 directions
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Figure 8 shows the distribution of the cr values of the two-order roughness in 36 directions for the S1 joint, in which the distribution of \(cr_{\rm I}\) values is isotropic, which is between 80 and 100%. The distribution of \(cr_{\Pi }\) values is anisotropic, and the variance range is 0–20%. Hence, the contribution rates of the waviness to the total surface are similar in 36 directions, but the contribution rates of the waviness to the total surface are different in 36 directions.

Fig. 8
figure 8

Distribution of cr values of two-order roughness in 36 directions for the S1 joint

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4.2 Size effect on the contribution rate

To analyze the size effect of the contribution rate in the two-order roughness, the average cr values in 36 directions are calculated. The variations in the cr values of the two-order roughness under different sample sizes with three lithological joints are shown in Fig. 9. As shown in Fig. 9a, the \(cr_{\rm I}\) and \(cr_{\Pi }\) values of waviness and unevenness remain basically stable with the increase in size. For the T1 joint, the \(cr_{\rm I}\) values continue to fluctuate up and down at 70%, and the \(cr_{\Pi }\) values remain at 30%. The \(cr_{\rm I}\) values for the T2 joint decrease with increasing size, but the \(cr_{\Pi }\) values increase with increasing size (Fig. 9b). The cr values of the two-order roughness under different sample sizes of the limestone joint are presented in Fig. 9c, although the curve fluctuates greatly, and the \(cr_{\rm I}\) values decrease as the sample size increases, while the \(cr_{\Pi }\) values increase as a function of sample size. Therefore, the contribution rate of waviness to the total surface decreases or does not change as the size increases, while the contribution rate of unevenness to the total surface increases as the size increases. The \(cr_{\rm I}\) values are approximately 95%, 70%, and 67% for the three lithologies, and the \(cr_{\Pi }\) values are approximately 5%, 30%, and 23%.

Fig. 9
figure 9

Variation in the cr values of the two-order roughness under different sample sizes: a slate joint, b tuff joint, c limestone joint

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4.3 Size effect mechanism of anisotropy

The results show that the first-order roughness parameters mainly reflect the anisotropy of the total roughness, while the second-order roughness parameters are mainly isotropic. To explore the influence of different sampling sizes in each direction on the two-order roughness, Fig. 10 shows a schematic diagram of the fluctuation amplitude of the waviness and unevenness under sampling sizes. The figure shows that the undulation amplitude of the unevenness is basically identical on the same average surface, the undulation amplitude of the waviness is greatly affected by the sampling size, and the variation in the first-order roughness values is larger than that of the second-order roughness values.

Fig. 10
figure 10

Fluctuations of the two-order morphology under different sampling sizes

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The anisotropic distribution of the second-order roughness is nearly isotropic, but the angle accounts for a relatively large amount, and the sampling size has a great influence on the relative change range of the roughness parameter. Regarding the waviness, the anisotropic distribution is more obvious. When the sampling size is small, the roughness of the joint surface is dominated by the second-order roughness. The proportion of waviness gradually increases as the sampling size increases. When the joint surface exhibits large roughness, the joint surface itself contains more waviness. The waviness contained in the joint surface gradually increases as a function of sampling size, and the undulation height also gradually increases. However, the roughness of the large size gradually becomes less than that of the small size, and it is smoother than the unevenness, in which the first-order roughness parameter gradually decreases. When the undulation degree of the joint surface is small, the waviness contained in the joint surface is much less. As the sampling size increases, the effective contact area ratio of the waviness gradually decreases, and the roughness parameter gradually increases. Thus, when the sampling size is increased to the point where the waviness is taken into account, the roughness parameter is no longer affected by the sample size and reaches a stable value.

5 Conclusion

The wavelet analysis method is applied to separate the waviness and unevenness in three dimensions of different lithological natural joints. The anisotropic distribution of two-order roughness is evaluated based on the AVC values, and their contribution to the total roughness is estimated in all directions. The size effect of anisotropy in the two-order roughness is further explored. The following conclusions can be drawn,

  1. 1.

    The waviness and unevenness of 6 natural joint surfaces are extracted by the wavelet analysis method. The 3D parameter \(\theta_{\max }^{ * } /(C + 1)\) can be used to measure the surface roughness and express the anisotropy of the two-order morphology. The anisotropy variation of the two-order roughness is quantified based on the AVC value.

  2. 2.

    By analyzing the anisotropic distribution of the two-order roughness for a large size, the results show that the first-order roughness mainly reflects the anisotropy of the total roughness, while the second-order roughness is mainly isotropic. The size effect law of AVC values is different in the three lithologies, in which the stable sizes of the slate, tuff, and limestone are 200 mm, 700 mm, and 500 mm, respectively.

  3. 3.

    According to the functional relationship between the total roughness and the two-order roughness, the contribution rate of the two-order roughness is estimated in 36 directions. In addition, we discover that the influence of waviness and unevenness on the total surface decreases or does not change and increases as the size increases. The waviness in different sizes contributes approximately 95%, 70%, and 67% for the three lithologies, in which unevenness contributes approximately 5%, 30%, and 23%.

In summary, this work uses a wavelet analysis method to separate the waviness and unevenness of three natural lithological joints, and the anisotropy of the two-order roughness and its variation with size are studied deeply. Further work should be done as follows: The effect of sampling interval on roughness anisotropy of rock joint; extending the field of study of parameter AVC to more general anisotropic distributions; mechanical test will be conducted to further explore the contribution of two-order roughness in hydraulics and understand the relationship between two-order roughness and hydraulic parameters.