Accurate estimation of rock joint roughness using variogram method

Fractional Brownian profiles with known values of fractal dimension, D, input standard deviation, a, and data density, d, were generated. For different values of the input parameter of the variogram method (lag distance, h), D and another associated fractal parameter K_v were calculated for the aforementioned profiles. The estimated K_y was found to relate to D, a and d according to K_v=2.0×10-⁵d^0.35σ^1.95D^14.5. The parameter K_v can be used to capture the scale effect of roughness profiles. Suitable ranges for h were estimated to obtain computed D within ±10% of the D used for the generation and also to satisfy a power functional relation between the variogram and h. Results indicated the importance of removal of global trend for roughness profiles with global trend to obtain accurate estimates for the fractal parameters. The parameters D and K_v are suggested to use with the variogram method to quantify stationary roughness. In addition, at the basic level, the mean inclination/declination angle of the surface in the direction considered can be used to represent the global trend. Procedures are given in the paper to estimate suitable h range for a given natural rock joint profile to use with the variogram method to estimate D and K_v accurately for the profile.