This study investigates the surface roughness and fractal properties of failure surfaces at the Mashamba West mine to enhance the understanding of the complex topographical features that influence slope stability in open-pit mining. Traditional methods for quantifying surface roughness, such as the mean absolute deviation (MAD) and standard deviation (SD), often fail to capture the multiscale and correlated behaviors typical of natural surfaces. In this research, advanced techniques, including fractal dimension calculations, semi-variogram and box-counting methods, and power spectral density (PSD) analysis, were employed to provide a more accurate characterization of the failure surface. The results revealed significant micro-roughness, with a mean absolute deviation of 0.83 and a standard deviation of 1.04, indicating a high level of surface irregularity. Geometric irregularities, such as pronounced asperities and depressions, were identified through the analysis of topographical extremes, with a mean peak height of 2.4828 and a mean valley depth of ？4.7747. Fractal analysis, performed using both semi-variogram and box-counting methods, confirmed the self-affine nature of the failure surface with fractal dimensions of 1.5634 and 2.0735, respectively, indicating scale-invariant roughness patterns. The power spectral density analysis revealed a dominant frequency corresponding to periodic geological structures. These findings provide valuable insights into the surface characteristics of failure zones and offer a robust framework for future slope stability analysis. This study contributes to the field by demonstrating the utility of fractal analysis in characterizing complex failure surfaces and lays the groundwork for future research on the relationship between surface roughness and mechanical behavior in open-pit mining contexts.
References
[1]
Boyer, D.D. and Ferguson, K.A. (2000). Important Factors to Consider in Properly Evaluating the Stability of Rock Slopes. In: SlopeStability 2000, American Society of Civil Engineers, 58-71. https://doi.org/10.1061/40512(289)5
[2]
Bye, A.R. and Bell, F.G. (2001) Stability Assessment and Slope Design at Sandsloot Open Pit, South Africa. InternationalJournalofRockMechanicsandMiningSciences, 38, 449-466. https://doi.org/10.1016/s1365-1609(01)00014-4
[3]
Fu, X., Sengupta, M. and Afrouz, A. (1996) Parametric Study of Variable Slope Design for a Multiple Layered Open Pit. InternationalJournalofSurfaceMining, ReclamationandEnvironment, 10, 41-45. https://doi.org/10.1080/09208119608964793
[4]
Xu, N., Zhang, J., Tian, H., Mei, G. and Ge, Q. (2016) Discrete Element Modeling of Strata and Surface Movement Induced by Mining under Open-Pit Final Slope. InternationalJournalofRockMechanicsandMiningSciences, 88, 61-76. https://doi.org/10.1016/j.ijrmms.2016.07.006
[5]
Pan, Q., Jiang, Y. and Dias, D. (2017) Probabilistic Stability Analysis of a Three-Dimensional Rock Slope Characterized by the Hoek-Brown Failure Criterion. JournalofComputinginCivilEngineering, 31, Article ID: 04017046. https://doi.org/10.1061/(asce)cp.1943-5487.0000692
[6]
Yang, J., Tao, Z., Li, B., Gui, Y. and Li, H. (2012) Stability Assessment and Feature Analysis of Slope in Nanfen Open Pit Iron Mine. InternationalJournalofMiningScienceandTechnology, 22, 329-333. https://doi.org/10.1016/j.ijmst.2012.04.008
[7]
Abedini, M.J. and Shaghaghian, M.R. (2009) Exploring Scaling Laws in Surface Topography. Chaos, Solitons&Fractals, 42, 2373-2383. https://doi.org/10.1016/j.chaos.2009.03.121
[8]
Anderson, S.W. (2019) Uncertainty in Quantitative Analyses of Topographic Change: Error Propagation and the Role of Thresholding. EarthSurfaceProcessesandLandforms, 44, 1015-1033. https://doi.org/10.1002/esp.4551
[9]
Lindsay, J.B., Newman, D.R. and Francioni, A. (2019) Scale-Optimized Surface Roughness for Topographic Analysis. Geosciences, 9, Article No. 322. https://doi.org/10.3390/geosciences9070322
[10]
Shepard, M.K., Campbell, B.A., Bulmer, M.H., Farr, T.G., Gaddis, L.R. and Plaut, J.J. (2001) The Roughness of Natural Terrain: A Planetary and Remote Sensing Perspective. JournalofGeophysicalResearch: Planets, 106, 32777-32795. https://doi.org/10.1029/2000je001429
[11]
Wang, A., Wang, K. and Tsung, F. (2014) Statistical Surface Monitoring by Spatial-Structure Modeling. JournalofQualityTechnology, 46, 359-376. https://doi.org/10.1080/00224065.2014.11917977
[12]
Alejano, L.R., Gómez-Márquez, I. and Martínez-Alegría, R. (2010) Analysis of a Complex Toppling-Circular Slope Failure. EngineeringGeology, 114, 93-104. https://doi.org/10.1016/j.enggeo.2010.03.005
[13]
Macciotta, R., Creighton, A. and Martin, C.D. (2020) Design Acceptance Criteria for Operating Open-Pit Slopes: An Update. CIMJournal, 11, 248-265. https://doi.org/10.1080/19236026.2020.1826830
[14]
Menezes, D.A.d., Carneiro, S.R.C. and Meireles, B.P. (2019) Map of the Potential Geotechnical Susceptibility for Operational Pit Slopes. REM—InternationalEngineeringJournal, 72, 55-61. https://doi.org/10.1590/0370-44672018720137
[15]
Micklethwaite, S., Sheldon, H.A. and Baker, T. (2010) Active Fault and Shear Processes and Their Implications for Mineral Deposit Formation and Discovery. JournalofStructuralGeology, 32, 151-165. https://doi.org/10.1016/j.jsg.2009.10.009
Khanna, R., Sayeed, I. and Dubey, R.K. (2018) Evaluation of Deformation Modulus for the Jointed Rock Masses Using Equivalent Continuum Approach. AppliedMechanicsandMaterials, 877, 200-213. https://doi.org/10.4028/www.scientific.net/amm.877.200
[18]
Kulatilake, P.H.S.W. (2021) 3-D Rock Mass Strength Criteria—A Review of the Current Status. Geotechnics, 1, 128-146. https://doi.org/10.3390/geotechnics1010007
[19]
Moldabayev, S.K., Sdvyzhkova, O.O., Babets, D.V., Kovrov, O.S. and Adil, T.K. (2021) Numerical Simulation of the Open Pit Stability Based on Probabilistic Approach. NaukovyiVisnykNatsionalnohoHirnychohoUniversytetu, No. 6, 29-34. https://doi.org/10.33271/nvngu/2021-6/029
[20]
Yang, G., Leung, A.K., Xu, N., Zhang, K. and Gao, K. (2019) Three-Dimensional Physical and Numerical Modelling of Fracturing and Deformation Behaviour of Mining-Induced Rock Slopes. AppliedSciences, 9, Article No. 1360. https://doi.org/10.3390/app9071360
[21]
Boyd, D.L., Trainor-Guitton, W. and Walton, G. (2018) Assessment of Rock Unit Variability through Use of Spatial Variograms. EngineeringGeology, 233, 200-212. https://doi.org/10.1016/j.enggeo.2017.12.012
[22]
Fraundorf, P. (2004) Heterogeneity, and the Secret of the Sea.
[23]
Hurich, C.A. and Kocurko, A. (2000) Statistical Approaches to Interpretation of Seismic Reflection Data. Tectonophysics, 329, 251-267. https://doi.org/10.1016/s0040-1951(00)00198-0
[24]
Agbelele, K.J., Houehanou, E.C., Ahlinhan, M.F., Ali, A.W. and Aristide, H.C. (2023) Assessment of Slope Stability by the Fellenius Slice Method: Analytical and Numerical Approach. WorldJournalofAdvancedResearchandReviews, 18, 1205-1214. https://doi.org/10.30574/wjarr.2023.18.2.0874
[25]
Kulatilake, P.H.S.W. and Ankah, M.L.Y. (2023) Rock Joint Roughness Measurement and Quantification—A Review of the Current Status. Geotechnics, 3, 116-141. https://doi.org/10.3390/geotechnics3020008
[26]
Lovejoy, S. and Schertzer, D. (2007) Scaling and Multifractal Fields in the Solid Earth and Topography. NonlinearProcessesinGeophysics, 14, 465-502. https://doi.org/10.5194/npg-14-465-2007
[27]
Andrle, R. and Abrahams, A.D. (1990) Fractal Techniques and the Surface Roughness of Talus Slopes: Reply. EarthSurfaceProcessesandLandforms, 15, 287-290. https://doi.org/10.1002/esp.3290150311
[28]
Burrough, P.A. (1981) Fractal Dimensions of Landscapes and Other Environmental Data. Nature, 294, 240-242. https://doi.org/10.1038/294240a0
Li, Y., Sun, S. and Yang, H. (2020) Scale Dependence of Waviness and Unevenness of Natural Rock Joints through Fractal Analysis. Geofluids, 2020, Article ID: 8818815. https://doi.org/10.1155/2020/8818815
[31]
Ling, F.F. (1989) The Possible Role of Fractal Geometry in Tribology. TribologyTransactions, 32, 497-505. https://doi.org/10.1080/10402008908981918
[32]
Ouadfeul, S. (2019) Introductory Chapter: Fractal in Sciences. In: Ouadfeul, S.-A., Ed., FractalAnalysis, IntechOpen. https://doi.org/10.5772/intechopen.84368
[33]
Wang, H. and Xu, Z. (2001) A Class of Rough Surfaces and Their Fractal Dimensions. JournalofMathematicalAnalysisandApplications, 259, 537-553. https://doi.org/10.1006/jmaa.2000.7425
[34]
Sanner, A., Nöhring, W.G., Thimons, L.A., Jacobs, T.D.B. and Pastewka, L. (2022) Scale-Dependent Roughness Parameters for Topography Analysis. AppliedSurfaceScienceAdvances, 7, Article ID: 100190. https://doi.org/10.1016/j.apsadv.2021.100190
[35]
Marx, E., Malik, I.J., Strausser, Y.E., Bristow, T., Poduje, N. and Stover, J.C. (2002) Power Spectral Densities: A Multiple Technique Study of Different Si Wafer Surfaces. JournalofVacuumScience&TechnologyB: MicroelectronicsandNanometerStructuresProcessing, Measurement, andPhenomena, 20, 31-41. https://doi.org/10.1116/1.1428267
[36]
Xu, J.D., Chiente, L. and Jacobi, R.D. (2002) Characterizing Fracture Spatial Patterns by Using Semivariograms. ActaGeologicaSinica—EnglishEdition, 76, 89-99. https://doi.org/10.1111/j.1755-6724.2002.tb00074.x
[37]
Maus, S. (1999) Variogram Analysis of Magnetic and Gravity Data. Geophysics, 64, 776-784. https://doi.org/10.1190/1.1444587
[38]
Ai, T., Zhang, R., Zhou, H.W. and Pei, J.L. (2014) Box-Counting Methods to Directly Estimate the Fractal Dimension of a Rock Surface. AppliedSurfaceScience, 314, 610-621. https://doi.org/10.1016/j.apsusc.2014.06.152
[39]
Foroutan-Pour, K., Dutilleul, P. and Smith, D.L. (1999) Advances in the Implementation of the Box-Counting Method of Fractal Dimension Estimation. AppliedMathematicsandComputation, 105, 195-210. https://doi.org/10.1016/s0096-3003(98)10096-6
[40]
Li, J., Du, Q. and Sun, C. (2009) An Improved Box-Counting Method for Image Fractal Dimension Estimation. PatternRecognition, 42, 2460-2469. https://doi.org/10.1016/j.patcog.2009.03.001
[41]
Song, D., Chen, Z., Ke, Y. and Nie, W. (2020) Seismic Response Analysis of a Bedding Rock Slope Based on the Time-Frequency Joint Analysis Method: A Case Study from the Middle Reach of the Jinsha River, China. EngineeringGeology, 274, Article ID: 105731. https://doi.org/10.1016/j.enggeo.2020.105731
[42]
Cox, B.L. and Wang, J.S.Y. (1993) Fractal Surfaces: Measurement and Applications in the Earth Sciences. Fractals, 1, 87-115. https://doi.org/10.1142/s0218348x93000125
[43]
Liu, X., Qu, S., Chen, R. and Chen, S. (2018) Development of a Two-Dimensional Fractal Model for Analyzing the Particle Size Distribution of Geomaterials. JournalofMaterialsinCivilEngineering, 30, Article ID: 04018175. https://doi.org/10.1061/(asce)mt.1943-5533.0002365
[44]
Hai, L., Lv, Y., Tan, S. and Feng, L. (2023) Study on the Influences of the Fractal Dimension of the Root System and Slope Degree on the Slope Stability. ScientificReports, 13, Article No. 10282. https://doi.org/10.1038/s41598-023-37561-8
[45]
Zheng, L., Liu, H., Zuo, Y., Zhang, Q., Lin, W., Qiu, Q., et al. (2022) Fractal Study on the Failure Evolution of Concrete Material with Single Flaw Based on DIP Technique. AdvancesinMaterialsScienceandEngineering, 2022, Article ID: 6077187. https://doi.org/10.1155/2022/6077187
[46]
Bains, R. (1992) Fractal Geometry—Mathematical Foundations and Applications. EngineeringAnalysiswithBoundaryElements, 9, 366-367. https://doi.org/10.1016/0955-7997(92)90028-6
[47]
Oliver, M.A. and Webster, R. (1986) Semi‐Variograms for Modelling the Spatial Pattern of Landform and Soil Properties. EarthSurfaceProcessesandLandforms, 11, 491-504. https://doi.org/10.1002/esp.3290110504
[48]
Mandelbrot, B.B. and Van Ness, J.W. (1968) Fractional Brownian Motions, Fractional Noises and Applications. SIAMReview, 10, 422-437. https://doi.org/10.1137/1010093
[49]
Wu, F.T. (1999) Fractal Geometry in Rock Mechanics: A New Approach to Analyze Rock Mass Properties. EngineeringGeology, 54, 23-38.
[50]
Wu, D.G., Gang, T.Q., Fan, X.G. and Wu, J.L. (2014) Surface Roughness Measuring Based on the Theory of Fractal Geometry. AppliedMechanicsandMaterials, 551, 434-438. https://doi.org/10.4028/www.scientific.net/amm.551.434
