A New Method for Determination of Joint Roughness Coefficient of Rock Joints

The joint roughness coefficient (JRC) of rock joints has the characteristic of scale effect. JRC measured on small-size exposed rock joints should be evaluated by JRC scale effect in order to obtain the JRC of actual-scale rock joints, since field rock joints are hardly fully exposed or well saved. Based on the validity analysis of JRC scale effect, concepts of rate of JRC scale effect and effective length of JRC scale effect were proposed. Then, a graphic method for determination of the effective length of JRC scale effect was established. Study results show that the JRC of actual-scale rock joints can be obtained through a fractal model of JRC scale effect according to the statistically measured results of the JRC of small-size partial exposed rock joints and by the selection of fractal dimension of JRC scale effect and the determination of effective length of JRC scale effect.

Field investigations show that it is usually hard to measure the JRC of actual-scale rock joints directly because the field rock joints are hardly fully exposed or well saved.The estimation of the JRC of actual-scale rock joints by statistically measured results of the small-scale JRC of partial exposed rock joints is critical to the reliability of the shear strength of rock joints empirically estimated by the JRC-JCS model.
Based on the analysis on quantities of rock joints through joint model test, Barton and Bandis [2] proposed a modified curve of JRC scale effect (see Figure 1) and a modified formula defined as where JRC  is the JRC value of rock joints with sampling length   and JRC 0 is the JRC value of standard-size joints (the sampling length is  0 , i.e., 10 cm).Du et al. [9] have statistically measured the JRC values of 1023 profile curves on SSE group joints from Xiaolangdi by the use of the profilograph and roughness ruler and got the regularity of the decrease characteristic of JRC with an increase in sampling length (see Figure 2).
In this paper, a new method is introduced to study the estimation of the JRC of actual-scale rock joints by statistically measured results of the small-scale JRC of partial exposed rock joints through the analysis of JRC scale effect of rock joints.A fractal model of JRC scale effect was established.The physical meaning of the fractal dimension of JRC scale effect was defined.A method to determine the effective length of JRC scale effect was proposed based on the study of the effectiveness of JRC scale effect.

Fractal Dimension of JRC Scale Effect
According to (1) and the regularity of JRC scale effect in Figures 1 and 2, the fractal expression of JRC scale effect can be obtained: where  is the fractal dimension of JRC scale effect, which defines the velocity rate of JRC  decreases with an increase in sampling length   .Statistically measured results show that the fractal dimension of JRC scale effect is comparatively stable.  , calculated from the fractal model of JRC scale effect according to the statistically measured results of JRC  of random sampling length along the same direction of rock joints in the same wall rock, distributes over a stable interval (see Table 1).
From Table 1, it can be seen that the fractal dimension along the tonalite joint ( 1−1 ) trend direction ranges from 0.3307 to 0.4966, with mean value of 0.4404; the range of fractal dimension along 45 ∘ direction is 0.6957-0.8070,and mean value is 0.7466; the range of fractal dimension along the strike direction is 0.6273-0.7431,and mean value is 0.6656; the range of fractal dimension along 135 ∘ direction is 0.5465-0.6341,and mean value is 0.6016.This means that the fractal dimension of JRC scale effect ( 30 ) converted by the JRC measurement results of small-size joint is the same as the fractal dimension of JRC scale effect ( 100 ) converted by the JRC measurement results of large-size joint. 100 and

Effective Length of JRC Scale Effects
In order to discuss the effectiveness of JRC scale effect of rock joints, the sensitivity of JRC  on the sampling length   was analyzed with constant JRC 0 = 12.4 and  = 0.7009.From Figure 3, it can be seen that when sampling length   is smaller than a certain value, JRC  decreases dramatically with an increase in sampling length   , indicating significant JRC scale effect.While the sampling length   is larger than the certain value, JRC  decreases very slowly with the increase of sampling length   ; that is, the JRC scale effect is insignificant.The sampling length corresponding to shift from significant to insignificant JRC scale effect is named  effective length of JRC scale effect, defined as  *  .In other words, only within the length of  *  will the influence of JRC scale effect of rock joints be significant.
The sensitivity analysis between JRC  and fractal dimension of JRC scale effect  is shown in Figure 4 when JRC 0 = 12.40 and   = 80cm.It can be seen that JRC  is very sensitive to the value of fractal dimension of JRC scale effect .
According to the fractal model of JRC scale effect (see (2)), the fractal dimension  can be defined as The relationship curve between  and lg(  ) calculated by (3) is shown in Figure 5 when JRC 0 = 12.40 and JRC 300 = 1.14.From Figure 5, it can be seen that the change of fractal dimension of JRC scale effect is obviously affected by the value of sampling length   of rock joints.
From the aforesaid analyses, it is clear that the change of fractal dimension of JRC scale effect is very sensitive to JRC and effective length of JRC scale effect, which can be used to determine the effective length of JRC scale effect  *  .

Graphic Method for Determination of Effective Length of JRC Scale Effect
Define   as the coefficient of JRC scale effect when sampling length is   : where JRC  is the JRC of rock joints when sampling length is   ; JRC −10 is the JRC of rock joints when sampling length is  −10 ,  = 20, 30, 40, 50, . ...Then, the coefficient of JRC scale effect when sampling length is 20 cm can be expressed as Let  =   / 20 as the relative coefficient of JRC scale effect; then Substituting ( 2) into ( 6) enables ( 7) to be determined: According to (7), the relationship curve between   and  can be plotted (see Figure 6).From the figure, it can be found that the initial point whereas the relative coefficient of JRC scale effect  starts to increase with the sampling length   is  = 5%.From error criterion, it can be known that JRC scale effect is insignificant when  ≤ 5%.Then, the value of the effective length of JRC scale effects  *  can be determined.On the surface of the partial exposed rock joints, JRC 10 and JRC 30 of profile curves with sampling length of 10 cm and 30 cm can be statistically measured.Then, the fractal dimension of JRC scale effect can be calculated by the fractal model of JRC scale effect.Substituting the fractal dimension of JRC scale effect into (7), the relationship curve between   and  can be obtained (see Figure 6).The sampling length corresponding to  = 5% on   - relationship curve is the effective length of JRC scale effect  *  .If the value calculated by the graphic method is between   and  +10 , then let  *  =   (e.g.,  *  = 80 cm in Figure 6).Conclusively, the fractal model of JRC scale effect (see (2)) can be used to estimate the JRC of actual-scale rock joints through the statistically measured JRC of small-size rock joints and to determine the fractal dimension of JRC scale effect and the effective length of JRC scale effect.

Case Study
Statistically measured JRC 10 and JRC 30 of 387 profile curves of SSE group joint of the west side slope rock of Xiaolangdi reservoir [10] and the values of  30 converted by the fractal model of JRC scale effect are listed in Table 3.According  to Table 3, relationship curves of   - of calcareous packsand and calcareous nodule clay rock along the joint trend direction and strike direction can be plotted (see Figure 7).From Figure 7, it can be calculated that the effective length of JRC scale effect  *  of calcareous packsand along the joint trend direction and strike direction is 120 cm and 110 cm, respectively; and the effective length of JRC scale effect  *  of calcareous nodule clay rock along the joint trend direction and strike direction is 150 cm and 160 cm, respectively.Figure 7 shows that the effective length of JRC scale effect  *  decreases with an increase in the fractal dimension of JRC scale effect.
According to the effective length of JRC scale effect  *  calculated by the graphic method, the actual-scale joint roughness coefficients JRC  of calcareous packsand and calcareous nodule clay rock along the joint trend direction and strike direction estimated by the fractal model of JRC scale effect are listed in Table 4. JRC  in Table 4 is used in the empirical estimation of shear strength with the JRC-JCS model, which contains the scale effect and has wide application (because the JRC is obtained by directional statistical measurement).Thus, the joint shear strength parameters obtained by empirical estimation can be directly used to evaluate the stability of the rock.

Conclusions
Hence, our conclusions are as follows: (1) Joint roughness coefficient of rock joints has the characteristic of scale effect.The sensitivity analysis of JRC  to the sampling length   shows that JRC  decreases dramatically with an increase in sampling length   when the sampling length   is smaller than a certain value; when the sampling length   is larger than the certain value, JRC  decreases very slowly with an increase in sampling length   .This eigenvalue of JRC scale effect is named the effective length of JRC scale effect.Only within the length of  *  will the influence of JRC scale effect be significant.The sensitivity analysis of the relative coefficient of JRC scale effect  to the sampling length   shows that the sampling length corresponding to  = 5% of   - relationship curve is the effective length of JRC scale effect,  *  .
(2) The sensitivity analysis of the fractal dimension of JRC scale effect  to the sampling length   shows that the change of  strongly affects the value of   .Therefore, the fractal dimension of JRC scale effect can be used to determine the effective length of JRC scale effect,  *  .The case study shows that the effective length of JRC scale effect  *  decreases with an increase in the fractal dimension of JRC scale effect.
(3) The procedures of the graphic method to determine the effective length of JRC scale effect are as follows.
Firstly, JRC 10 and JRC 30 are measured through the statistically measured profile curves with sampling length of 10 cm and 30 cm on the surface of rock joints.Secondly, the fractal dimension of JRC scale effects  30 is determined by the fractal model of JRC scale effect.Thirdly, substituting the fractal dimension into the relative coefficient of JRC scale effect formula (see (7)), the relationship curve of   - can be plotted.Finally, the sampling length corresponding to  = 5% in   - relationship curve is the effective length of JRC scale effect,  *  (if the value calculated by graphic method is between   and  +10 , then let  *  =   ).( 4) Field investigations show that the field rock joints are hardly fully exposed or well saved.The graphic method in some terms solves the problem by promoting the utilization of the fractal model of JRC scale effect.Using this method, the JRC of actual-scale rock joints can be determined by the fractal model of JRC scale effects through the statistically measured JRC of small-size and partial exposed rock joints.

Figure 7 :
Figure 7: Relationship curves of   - of SSE group joint of Xiaolangdi.(a) Calcareous packsand along joint strike direction, (b) calcareous packsand along joint trend direction, (c) calcareous nodule clay rock along joint strike direction, and (d) calcareous nodule clay rock along joint trend direction.

Table 2 :
Fractal dimension of JRC scale effect of typical rock joints.Considering the difference of JRC scale effect of rock joints with different wall rock, fractal dimension of JRC scale effect of typical rock joints was converted by the fractal model of JRC scale effect according to the statistically measured results from JRC 10 and JRC 30 of 13529 profile curves of 21 different wall rock joints (see Table2).
30 signify 100 mm and 30 mm joint sample size.Therefore, the fractal dimension of JRC scale effect can be directly determined according to the JRC statistical measurement results of small-size rock joints.In engineering applications, the value of  30 is usually used to analyze the JRC scale effect in order to facilitate the statistical measurement of the JRC.

Table 3 :
30 of SSE group joint of Xiaolangdi.

Table 4 :
JRC of actual-scale SSE group joint of Xiaolangdi.