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

Authors: Shigui Du; Huicai Gao; Yunjin Hu; Man Huang; Hua Zhao
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

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.


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## Body

## 1. Introduction

Joint roughness coefficient (JRC) is a vital parameter of the JRC-JCS model for estimation of shear strength of rock joints [1–5]. The JRC will decrease with an increase in sampling length due to JRC scale effect [6–8].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(1)
JRC
n
≈
JRC
0
L
n
L
0
-
0.02
JRC
0
,where J
R
C
n is the JRC value of rock joints with sampling length L
n and J
R
C
0 is the JRC value of standard-size joints (the sampling length is L
0, i.e., 10 cm).Figure 1
Scale effect related toJ
R
C
0 [2].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).Figure 2
JRC scale effect of SSE group joint in Xiaolangdi.(
1
) Calcareous packsand along joint strike direction (160°), (
2
) calcareous packsand along joint trend direction (250°), (
3
) silty clay rock along joint strike direction (160°), and (
4
) silty clay rock along joint trend direction (250°).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.


## 2. 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:(2)
JRC
n
=
JRC
0
L
n
L
0
-
D
,where D is the fractal dimension of JRC scale effect, which defines the velocity rate of JRC
n decreases with an increase in sampling length L
n.Statistically measured results show that the fractal dimension of JRC scale effect is comparatively stable.D
n, calculated from the fractal model of JRC scale effect according to the statistically measured results of JRC
n 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 (J
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 (D
30) converted by the JRC measurement results of small-size joint is the same as the fractal dimension of JRC scale effect (D
100) converted by the JRC measurement results of large-size joint. D
100 and D
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 D
30 is usually used to analyze the JRC scale effect in order to facilitate the statistical measurement of the JRC.Table 1
Fractal dimension of JRC scale effect of tonalite jointJ
1
-
1.
Direction/°
L
n/cm
JRC
D
n
Direction/°
L
n/cm
JRC
D
n
Trend (0)
10
10.35
0.0000
Strike (90)
10
11.90
0.0000
20
8.23
0.3307
20
7.11
0.7431
30
6.13
0.4769
30
5.65
0.6781
40
5.20
0.4966
40
4.75
0.6625
50
4.78
0.4800
50
4.00
0.6774
60
4.48
0.4674
60
3.58
0.6705
70
4.58
0.4190
70
3.38
0.6469
80
4.25
0.4281
80
3.05
0.6598
90
4.05
0.4271
90
3.00
0.6273
100
3.78
0.4374
100
2.80
0.6284
Oblique (45)
10
13.83
0.0000
Oblique (135)
10
11.10
0.0000
20
7.93
0.8025
20
7.60
0.5465
30
5.70
0.8070
30
5.80
0.5910
40
5.15
0.7126
40
4.65
0.6277
50
4.23
0.7360
50
4.00
0.6341
60
3.50
0.7670
60
3.58
0.6317
70
3.23
0.7474
70
3.30
0.6234
80
2.88
0.7546
80
3.28
0.5863
90
3.00
0.6957
90
2.98
0.5986
100
2.78
0.6968
100
2.95
0.5755Considering 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 fromJ
R
C
10 and J
R
C
30 of 13529 profile curves of 21 different wall rock joints (see Table 2).Table 2
Fractal dimension of JRC scale effect of typical rock joints.
Joint wall rock
Joint type
D
30
Magmatic rock
Coarse-grained granite
Joint
0.3458
Fine-grained granite
Joint
0.3752
Coarse-grained diorite
Joint
0.3165
Dacite-porphyrite
Joint
0.3877
Fluorite dike rock
Joint
0.3500
Basalt
Joint
0.4883
Sedimentary rock
Rudaceous grit
Joint
0.2578
Arkosic sandstone
Joint
0.1698
Siltstone
Ripple bedding
0.2850
Siltstone
Joint
0.4234
Calcilutite
Joint
0.4758
Calcareous packsand
Joint
0.5093
Nodule clay rock
Joint
0.2662
Calcareous mudstone
Ripple bedding
0.3563
Calcareous mudstone
Joint
0.2578
Volcanic debris rock
Volcanic breccia
Joint
0.2746
Fusion tuff
Joint
0.4234
Crystal tuff
Joint
0.2788
Metamorphic rock
Carbonaceous slate
Foliation
0.2012
Phyllite rock
Phyllite
0.4150
Plagioclase hornblendite
Joint
0.3500


## 3. Effective Length of JRC Scale Effects

In order to discuss the effectiveness of JRC scale effect of rock joints, the sensitivity ofJ
R
C
n on the sampling length L
n was analyzed with constant J
R
C
0
=
12.4 and D
=
0.7009. From Figure 3, it can be seen that when sampling length L
n is smaller than a certain value, J
R
C
n decreases dramatically with an increase in sampling length L
n, indicating significant JRC scale effect. While the sampling length L
n is larger than the certain value, J
R
C
n decreases very slowly with the increase of sampling length L
n; 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 L
n
∗. In other words, only within the length of L
n
∗ will the influence of JRC scale effect of rock joints be significant.Figure 3
Relationship between sampling lengthL
n and J
R
C
0.The sensitivity analysis betweenJ
R
C
n and fractal dimension of JRC scale effect D is shown in Figure 4 when J
R
C
0
=
12.40 and L
n
=
80 cm. It can be seen that J
R
C
n is very sensitive to the value of fractal dimension of JRC scale effect D.Figure 4
Plot ofJ
R
C
n versus D.According to the fractal model of JRC scale effect (see (2)), the fractal dimension D can be defined as(3)
D
=
lg
⁡
JRC
n
/
JRC
0
1
-
lg
⁡
L
n
.The relationship curve betweenD and lg
⁡
(
L
n
) calculated by (3) is shown in Figure 5 when J
R
C
0
=
12.40 and J
R
C
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 L
n of rock joints.Figure 5
Relationship curve ofD
-
lg
⁡
L
n.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 effectL
n
∗.


## 4. Graphic Method for Determination of Effective Length of JRC Scale Effect

Definef
n as the coefficient of JRC scale effect when sampling length is L
n:(4)
f
n
=
JRC
n
-
JRC
n
-
10
L
n
-
L
n
-
10
,where J
R
C
n is the JRC of rock joints when sampling length is L
n; J
R
C
n
-
10 is the JRC of rock joints when sampling length is L
n
-
10, n
=
20,30,40,50
,
….Then, the coefficient of JRC scale effect when sampling length is 20 cm can be expressed as(5)
f
20
=
JRC
20
-
JRC
10
L
20
-
L
10
=
JRC
20
-
JRC
0
L
20
-
L
0
.Letf
=
f
n
/
f
20 as the relative coefficient of JRC scale effect; then(6)
f
=
JRC
n
-
JRC
n
-
10
/
L
n
-
L
n
-
10
JRC
20
-
JRC
0
/
L
n
-
L
0
.Substituting (2) into (6) enables (7) to be determined:(7)
f
=
0.1
L
n
-
D
-
0.1
L
n
-
1
-
D
2
-
D
-
1
.According to (7), the relationship curve between L
n and f 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 f starts to increase with the sampling length L
n is f
=
5
%. From error criterion, it can be known that JRC scale effect is insignificant when f
≤
5
%. Then, the value of the effective length of JRC scale effects L
n
∗ can be determined.Figure 6
Relationship curve ofL
n
-
f.On the surface of the partial exposed rock joints,J
R
C
10 and J
R
C
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 L
n and f can be obtained (see Figure 6). The sampling length corresponding to f
=
5
% on L
n
-
f relationship curve is the effective length of JRC scale effect L
n
∗. If the value calculated by the graphic method is between L
n and L
n
+
10, then let L
n
∗
=
L
n (e.g., L
n
∗
=
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.


## 5. Case Study

Statistically measuredJ
R
C
10 and J
R
C
30 of 387 profile curves of SSE group joint of the west side slope rock of Xiaolangdi reservoir [10] and the values of D
30 converted by the fractal model of JRC scale effect are listed in Table 3. According to Table 3, relationship curves of L
n
-
f 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 L
n
∗ 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 L
n
∗ 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 L
n
∗ decreases with an increase in the fractal dimension of JRC scale effect.Table 3
D
30 of SSE group joint of Xiaolangdi.
Joint wall rock
Direction
L
n
=
10 cm
L
n
=
30 cm
D
30
Sample
J
R
C
10
Sample
J
R
C
30
Calcareous packsand
Strike
35
8.61
34
5.50
0.4079
Trend
31
12.12
30
7.43
0.4454
Calcareous nodule clay rock
Strike
88
15.83
84
11.86
0.2628
Trend
82
20.48
77
15.46
0.2560Figure 7
Relationship curves ofL
n
-
f 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.
(a)
(b)
(c)
(d)According to the effective length of JRC scale effectL
n
∗ calculated by the graphic method, the actual-scale joint roughness coefficients J
R
C
n 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. J
R
C
n 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.Table 4
JRC of actual-scale SSE group joint of Xiaolangdi.
Joint wall rock
Direction
J
R
C
0
L
n
∗/cm
D
30
J
R
C
n
Calcareous packsand
Strike
8.61
120
0.4079
3.12
Trend
12.12
110
0.4454
4.17
Calcareous nodule clay rock
Strike
15.83
150
0.2628
7.77
Trend
20.48
160
0.2560
10.07


## 6. Conclusions

Hence, our conclusions are as follows:(1)
Joint roughness coefficient of rock joints has the characteristic of scale effect. The sensitivity analysis ofJ
R
C
n to the sampling length L
n shows that J
R
C
n decreases dramatically with an increase in sampling length L
n when the sampling length L
n is smaller than a certain value; when the sampling length L
n is larger than the certain value, J
R
C
n decreases very slowly with an increase in sampling length L
n. This eigenvalue of JRC scale effect is named the effective length of JRC scale effect. Only within the length of L
n
∗ will the influence of JRC scale effect be significant. The sensitivity analysis of the relative coefficient of JRC scale effect f to the sampling length L
n shows that the sampling length corresponding to f
=
5
% of L
n
-
f relationship curve is the effective length of JRC scale effect, L
n
∗.
(2)
The sensitivity analysis of the fractal dimension of JRC scale effectD to the sampling length L
n shows that the change of D strongly affects the value of L
n. Therefore, the fractal dimension of JRC scale effect can be used to determine the effective length of JRC scale effect, L
n
∗. The case study shows that the effective length of JRC scale effect L
n
∗ 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,J
R
C
10 and J
R
C
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 D
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 L
n
-
f can be plotted. Finally, the sampling length corresponding to f
=
5
% in L
n
-
f relationship curve is the effective length of JRC scale effect, L
n
∗ (if the value calculated by graphic method is between L
n and L
n
+
10, then let L
n
∗
=
L
n).
(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.



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*Source: 634573-2015-11-09.xml*