Relationship between Joint Roughness Coefficient and Statistical Roughness Parameters and Its Sensitivity to Sampling Interval
Abstract
:1. Introduction
2. Material and Methods
2.1. Digitization of Barton’s Roughness Profiles
2.2. Joint Sample Preparation
2.3. Joint Sample Digitization
3. Determination of JRC Using Statistical Roughness Parameters
3.1. Study on the Correlation between Statistical Roughness Parameters and JRC
3.2. Determination of JRC Using Statistical Roughness Parameters
4. Correlation between Statistical Roughness Parameters and Sampling Interval
5. Influence of Sampling Interval on Reconstructed Rock Joint Profile
6. Conclusions
- It is observed that there is a good correlation between JRC and statistical roughness parameters Z2, SF, Rp, δ, σi, θ, Ra, Rq, Rz and λ based on the correlation analysis of JRC with statistical roughness parameters with Pearson’s correlation coefficient (γ) method. The coefficient γ values for these roughness parameters exceed 0.7 except for Ra where γ = 0.66. Compared with the amplitude parameters Ra, Rq, Rz and λ (γ ranges from 0.66 to 0.8), a better correlation exists between the textural parameters Z2, SF, Rp, δ, σi and θ and JRC (γ > 0.9).
- Among these parameters, the standard deviation of the roughness angle σi has the strongest correlation with JRC (γ = 0.9923). Further, a linear empirical equation between JRC and the parameter σi is proposed to determine the JRC of the rock joint profile.
- As the sampling interval increases, the Z2, Rp, δ, σi and θ parameter values decrease, and the Ra, Rq, Rz and λ parameter values show slight fluctuations, whereas SF values increase with an increase in the sampling interval. In addition, the evolution in the texture parameters Z2, SF, Rp, δ, σi and θ with the sampling interval can fit a power-law function well.
- Sensitivity analysis has revealed that the texture parameters (Z2, SF, Rp, δ, σi and θ) significantly depend on the sampling interval as a whole. In contrast, the dependence of the amplitude parameters (Ra, Rq, Rz and λ) on the sampling interval is not significant.
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
List of Symbols
JRC | Joint roughness coefficient |
Z2 | Root mean square of the first derivative |
SF | Structure function |
Rp | Roughness profile index |
Rz | Peak asperity height |
Ra | Arithmetic average of the absolute height |
Rq | Root mean square roughness height value |
θ | Average roughness angle |
σi | Standard deviation of the roughness angle i |
λ | The ultimate slope |
δ | Profile elongation index |
θ*max | Angular threshold |
D | Fractal dimension |
γ | Pearson’s correlation coefficient |
SI | Sampling interval |
L | The projected length of fracture profile |
Lt | True length of the profile |
zmax | The highest peak |
zmin | The lower valley |
Appendix A
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Profile No. | Rock Type | Typical Roughness Profiles | JRC Back-Calculated |
---|---|---|---|
1 | Slate | 0.4 | |
2 | Aplite | 2.8 | |
3 | Gneiss | 5.8 | |
4 | Granite | 6.7 | |
5 | Granite | 9.5 | |
6 | Hornfels | 10.8 | |
7 | Aplite | 12.8 | |
8 | Aplite | 14.5 | |
9 | Hornfels | 16.7 | |
10 | Soapstone | 18.7 | |
No. | Variable | Equation | R2 | Rang# |
---|---|---|---|---|
E5 | Z2 | 0.9577 | 0.1220–0.4036 | |
E6 | SF | 0.8790 | 0.0037–0.0409 | |
E7 | Rp−1 | 0.8956 | 0.0073–0.0718 | |
E8 | θ | 0.9807 | 3.6151–15.1640 | |
E9 | δ | 0.9006 | 0.0073–0.0718 |
Sample No. | JRC Calculated by σi | |||
---|---|---|---|---|
Equation (1) | Equation (2) | Equation (3) | Equation (4) (This Study) | |
FS1 | 12.9 | 10.6 | 14.4 | 8.1 |
FS2 | 14.9 | 12.4 | 15.7 | 9.9 |
FS3 | 14.0 | 11.6 | 15.1 | 9.0 |
FS4 | 13.2 | 10.9 | 14.6 | 8.4 |
FS5 | 16.3 | 13.7 | 16.5 | 11.1 |
FS6 | 17.0 | 14.3 | 16.9 | 11.6 |
FS7 | 16.9 | 14.3 | 16.9 | 11.6 |
FS8 | 18.1 | 15.4 | 17.5 | 12.7 |
FS9 | 15.8 | 13.3 | 16.3 | 10.6 |
FS10 | 17.5 | 14.8 | 17.2 | 12.1 |
CS1 | 19.0 | 16.2 | 17.9 | 13.4 |
CS2 | 16.6 | 13.9 | 16.7 | 11.3 |
CS3 | 18.1 | 15.4 | 17.5 | 12.7 |
CS4 | 18.0 | 15.3 | 17.4 | 12.6 |
CS5 | 19.1 | 16.3 | 18.0 | 13.5 |
CS6 | 24.9 | 21.6 | 20.3 | 18.6 |
CS7 | 18.3 | 15.5 | 17.6 | 12.8 |
CS8 | 23.9 | 20.6 | 19.9 | 17.7 |
CS9 | 23.4 | 20.2 | 19.7 | 17.3 |
CS10 | 17.7 | 15.0 | 17.3 | 12.3 |
GR1 | 23.0 | 19.8 | 19.6 | 16.9 |
GR2 | 22.0 | 18.9 | 19.2 | 16.0 |
GR3 | 20.5 | 17.5 | 18.6 | 14.7 |
GR4 | 21.9 | 18.9 | 19.2 | 16.0 |
GR5 | 18.9 | 16.1 | 17.8 | 13.3 |
GR6 | 26.3 | 22.9 | 20.7 | 19.8 |
GR7 | 25.0 | 21.7 | 20.3 | 18.7 |
GR8 | 23.2 | 20.0 | 19.6 | 17.1 |
GR9 | 26.3 | 22.9 | 20.7 | 19.8 |
Sample No. | SF | Z2 | ||||
---|---|---|---|---|---|---|
A | B | R2 | A | B | R2 | |
FS1 | 0.0426 | 1.1156 | 0.9851 | 0.1659 | −0.1862 | 0.8517 |
FS2 | 0.0406 | 1.4970 | 0.9975 | 0.1874 | −0.1564 | 0.8935 |
FS3 | 0.0449 | 1.2963 | 0.9970 | 0.1802 | −0.1694 | 0.8769 |
FS4 | 0.0386 | 1.4321 | 0.9996 | 0.1794 | −0.1657 | 0.8893 |
FS5 | 0.0562 | 1.4893 | 0.9997 | 0.2136 | −0.1371 | 0.9029 |
FS6 | 0.0672 | 1.4452 | 0.9967 | 0.2228 | −0.1245 | 0.8693 |
FS7 | 0.0729 | 1.6212 | 0.9998 | 0.2483 | −0.1036 | 0.9013 |
FS8 | 0.0656 | 1.5973 | 0.9977 | 0.2334 | −0.1109 | 0.9114 |
FS9 | 0.0602 | 1.5307 | 0.9976 | 0.2202 | −0.1272 | 0.9034 |
FS10 | 0.0612 | 1.5638 | 0.9998 | 0.2260 | −0.1206 | 0.9055 |
CS1 | 0.0719 | 1.2107 | 0.9943 | 0.2288 | −0.2133 | 0.9367 |
CS2 | 0.0452 | 1.4341 | 0.9983 | 0.2012 | −0.2092 | 0.9645 |
CS3 | 0.0622 | 1.3678 | 0.9963 | 0.2224 | −0.1897 | 0.9464 |
CS4 | 0.0680 | 1.4151 | 0.9944 | 0.2268 | −0.1530 | 0.9313 |
CS5 | 0.0737 | 1.4222 | 0.9989 | 0.2400 | −0.1547 | 0.9252 |
CS6 | 0.1662 | 1.4621 | 0.9949 | 0.3479 | −0.1251 | 0.9143 |
CS7 | 0.0718 | 1.5797 | 0.9998 | 0.2537 | −0.1480 | 0.9543 |
CS8 | 0.1105 | 1.6560 | 0.9999 | 0.3098 | −0.1067 | 0.9385 |
CS9 | 0.1329 | 1.5127 | 0.9959 | 0.3178 | −0.1285 | 0.9096 |
CS10 | 0.0610 | 1.4574 | 0.9999 | 0.2322 | −0.1958 | 0.9695 |
GR1 | 0.0799 | 1.3361 | 0.9994 | 0.2659 | −0.2434 | 0.9472 |
GR2 | 0.0721 | 1.2827 | 0.9991 | 0.2529 | −0.2682 | 0.9682 |
GR3 | 0.0619 | 1.2564 | 0.9988 | 0.2298 | −0.2545 | 0.9539 |
GR4 | 0.0653 | 1.3703 | 0.9997 | 0.2457 | −0.2466 | 0.9501 |
GR5 | 0.0582 | 1.2891 | 0.9996 | 0.2235 | −0.2439 | 0.9518 |
GR6 | 0.1335 | 1.5721 | 0.9996 | 0.3460 | −0.1540 | 0.9675 |
GR7 | 0.1147 | 1.5780 | 0.9996 | 0.3160 | −0.1470 | 0.9681 |
GR8 | 0.0913 | 1.2675 | 0.9950 | 0.2720 | −0.2322 | 0.9490 |
GR9 | 0.1276 | 1.4815 | 0.9999 | 0.3286 | −0.1832 | 0.9585 |
Sample No. | Rp | δ | ||||
A | B | R2 | A | B | R2 | |
FS1 | 1.0154 | −0.0058 | 0.9716 | 0.0140 | −0.0460 | 0.8681 |
FS2 | 1.0187 | −0.0059 | 0.9749 | 0.0173 | −0.0341 | 0.8975 |
FS3 | 1.0176 | −0.0060 | 0.9748 | 0.0166 | −0.0377 | 0.8879 |
FS4 | 1.0174 | −0.0058 | 0.9751 | 0.0161 | −0.0366 | 0.8931 |
FS5 | 1.0236 | −0.0064 | 0.9742 | 0.0224 | −0.0286 | 0.9040 |
FS6 | 1.0254 | −0.0062 | 0.9524 | 0.0247 | −0.0295 | 0.8755 |
FS7 | 1.0307 | −0.0061 | 0.9602 | 0.0301 | −0.0219 | 0.9011 |
FS8 | 1.0276 | −0.0059 | 0.9661 | 0.0266 | −0.0224 | 0.9091 |
FS9 | 1.0251 | −0.0062 | 0.9684 | 0.0239 | −0.0265 | 0.9027 |
FS10 | 1.0261 | −0.0061 | 0.9673 | 0.0252 | −0.0251 | 0.9027 |
CS1 | 1.0293 | −0.0122 | 0.9937 | 0.0260 | −0.0354 | 0.9375 |
CS2 | 1.0231 | −0.0097 | 0.9880 | 0.0204 | −0.0266 | 0.9652 |
CS3 | 1.0269 | −0.0100 | 0.9927 | 0.0242 | −0.0295 | 0.9450 |
CS4 | 1.0269 | −0.0081 | 0.9888 | 0.0254 | −0.0261 | 0.9325 |
CS5 | 1.0300 | −0.0090 | 0.9866 | 0.0275 | −0.0286 | 0.9226 |
CS6 | 1.0537 | −0.0116 | 0.9745 | 0.0522 | −0.0212 | 0.9219 |
CS7 | 1.0330 | −0.0096 | 0.9915 | 0.0313 | −0.0204 | 0.9554 |
CS8 | 1.0471 | −0.0092 | 0.9797 | 0.0453 | −0.0174 | 0.9360 |
CS9 | 1.0468 | −0.0102 | 0.9684 | 0.0451 | −0.0229 | 0.9114 |
CS10 | 1.0291 | −0.0110 | 0.9962 | 0.0262 | −0.0219 | 0.9705 |
GR1 | 1.0396 | −0.0181 | 0.9801 | 0.0366 | −0.0375 | 0.9376 |
GR2 | 1.0374 | −0.0187 | 0.9807 | 0.0314 | −0.0366 | 0.9574 |
GR3 | 1.0308 | −0.0150 | 0.9837 | 0.0262 | −0.0404 | 0.9435 |
GR4 | 1.0344 | −0.0158 | 0.9767 | 0.0298 | −0.0416 | 0.9322 |
GR5 | 1.0287 | −0.0136 | 0.9841 | 0.0247 | −0.0399 | 0.9402 |
GR6 | 1.0585 | −0.0166 | 0.9892 | 0.0560 | −0.0196 | 0.9588 |
GR7 | 1.0494 | −0.0130 | 0.9911 | 0.0467 | −0.0191 | 0.9558 |
GR8 | 1.0408 | −0.0180 | 0.9838 | 0.0358 | −0.0386 | 0.9356 |
GR9 | 1.0557 | −0.0177 | 0.9910 | 0.0518 | −0.0265 | 0.9411 |
Sample No. | θ | σi | ||||
A | B | R2 | A | B | R2 | |
FS1 | 7.4943 | −0.1746 | 0.8385 | 11.8211 | −0.1782 | 0.8436 |
FS2 | 8.4746 | −0.1492 | 0.8738 | 13.8057 | −0.1456 | 0.8815 |
FS3 | 8.1793 | −0.1591 | 0.8619 | 12.7733 | −0.1624 | 0.8669 |
FS4 | 8.0267 | −0.1574 | 0.8738 | 12.3054 | −0.1634 | 0.8791 |
FS5 | 9.4741 | −0.1234 | 0.9018 | 15.2581 | −0.1258 | 0.8989 |
FS6 | 9.9773 | −0.1142 | 0.8497 | 15.9228 | −0.1146 | 0.8578 |
FS7 | 11.2573 | −0.0901 | 0.8868 | 16.2786 | −0.0995 | 0.8968 |
FS8 | 10.2509 | −0.1062 | 0.8858 | 17.1376 | −0.1004 | 0.8987 |
FS9 | 9.9180 | −0.1113 | 0.9205 | 14.9390 | −0.1211 | 0.9066 |
FS10 | 10.0920 | −0.1060 | 0.8752 | 16.5286 | −0.1073 | 0.8897 |
CS1 | 9.8305 | −0.1873 | 0.9003 | 15.8384 | −0.1941 | 0.9173 |
CS2 | 8.9809 | −0.1792 | 0.9472 | 14.5415 | −0.1850 | 0.9551 |
CS3 | 9.9470 | −0.1560 | 0.9291 | 15.8793 | −0.1672 | 0.9363 |
CS4 | 10.1814 | −0.1333 | 0.9081 | 16.3880 | −0.1368 | 0.9174 |
CS5 | 10.5524 | −0.1309 | 0.8883 | 17.1755 | −0.1363 | 0.9048 |
CS6 | 12.3624 | −0.1168 | 0.9445 | 22.2406 | −0.1103 | 0.9200 |
CS7 | 11.3631 | −0.1314 | 0.9366 | 16.4990 | −0.1423 | 0.9471 |
CS8 | 13.0844 | −0.0851 | 0.9243 | 22.1257 | −0.0868 | 0.9273 |
CS9 | 11.8134 | −0.1101 | 0.9129 | 21.0558 | −0.1120 | 0.9049 |
CS10 | 9.9365 | −0.1708 | 0.9566 | 15.7409 | −0.1808 | 0.9602 |
GR1 | 11.2024 | −0.1981 | 0.9184 | 17.9891 | −0.2136 | 0.9321 |
GR2 | 10.7387 | −0.2154 | 0.9233 | 17.3103 | −0.2332 | 0.9492 |
GR3 | 9.9227 | −0.2086 | 0.9103 | 16.3119 | −0.2209 | 0.9354 |
GR4 | 10.4506 | −0.1922 | 0.9181 | 17.4927 | −0.2068 | 0.9368 |
GR5 | 9.5644 | −0.2057 | 0.9099 | 15.1834 | −0.2223 | 0.9349 |
GR6 | 14.5471 | −0.1268 | 0.9331 | 22.7248 | −0.1322 | 0.9526 |
GR7 | 13.1223 | −0.1144 | 0.9240 | 22.1869 | −0.1190 | 0.9491 |
GR8 | 11.6967 | −0.1853 | 0.9127 | 18.3945 | −0.2034 | 0.9304 |
GR9 | 13.6499 | −0.1380 | 0.9143 | 22.2623 | −0.1506 | 0.9398 |
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Luo, Y.; Wang, Y.; Guo, H.; Liu, X.; Luo, Y.; Liu, Y. Relationship between Joint Roughness Coefficient and Statistical Roughness Parameters and Its Sensitivity to Sampling Interval. Sustainability 2022, 14, 13597. https://doi.org/10.3390/su142013597
Luo Y, Wang Y, Guo H, Liu X, Luo Y, Liu Y. Relationship between Joint Roughness Coefficient and Statistical Roughness Parameters and Its Sensitivity to Sampling Interval. Sustainability. 2022; 14(20):13597. https://doi.org/10.3390/su142013597
Chicago/Turabian StyleLuo, Yong, Yakun Wang, Heng Guo, Xiaobo Liu, Yihui Luo, and Yanan Liu. 2022. "Relationship between Joint Roughness Coefficient and Statistical Roughness Parameters and Its Sensitivity to Sampling Interval" Sustainability 14, no. 20: 13597. https://doi.org/10.3390/su142013597