Requirements for accurate quantification of self affine roughness using the roughness-length method

Abstract Self-affine fractals have the potential to represent rock joint roughness profiles. Fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D , input standard deviation, σ , and data density, d , were generated. For different values of the input parameter of the roughness–length method (window length, w ), D and another associated fractal parameter A were calculated for the aforementioned profiles. The calculated D was compared with the D used for the generation to determine the accuracy of calculated D . Suitable ranges for w were estimated to produce accurate D (within ±10% error) for the generated profiles. The results showed that to obtain reliable estimates for fractal parameters of a natural rock joint profile, it is necessary to choose a unit for the profile length to satisfy a data density ( d ) greater than or equal to 5.1. For roughness profiles having 5.1≤ d ≤51.23 and 1.2≤ D ≤1.7, w values between 2.5% and 10% of the profile length were found to be highly suitable to produce accurate fractal parameter estimates. It is recommended to use at least seven w values between the estimated minimum and maximum suitable w values in estimating fractal parameters of a natural rock joint profile. It was found that σ and a global trend of a roughness profile have no effect on calculated D . The estimated A was found to increase with both D and σ . The parameter D captures the auto-correlation and A captures the amplitude of a roughness profile at different scales. Therefore, the parameters D and A are recommended to use with the roughness–length method in quantifying rock joint roughness. In addition, at least one more parameter is required to quantify the global trend of a roughness profile, if it exist; in many cases just the inclination or declination angle of the roughness profile in the direction considered would be sufficient to estimate the global trend. Calculated cross-over lengths (segment length of a profile at which a self-affine profile becomes self-similar) for the profiles investigated were found to be extremely small (less than 0.6% of the profile length) indicating that laser profilometers are required to make roughness measurements at interval lengths comparable to the cross-over lengths of the natural rock joint profiles. To calculate rock joint roughness parameters accurately using the self-similar techniques, it is necessary to have roughness measurements made at interval lengths comparable to the cross-over length of the profile. This indicate clearly the difficulty of using self-similar techniques such as the divider method in estimating rock joint roughness accurately.

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