On the application of methods used to calculate the fractal dimension of fracture surfaces

This paper presents an evaluation of the methods applied to calculate the fractal dimension of fracture surfaces. Variogram (applicable to 1D self-affine sets) and power spectral density analyses (applicable to 2D self-affine sets) are selected to calculate the fractal dimension of synthetic 2D data sets generated using fractional Brownian motion (fBm). Then, the calculated values are compared with the actual fractal dimensions assigned in the generation of the synthetic surfaces. The main factor considered is the size of the 2D data set (number of data points). The critical sample size that yields the best agreement between the calculated and actual values is defined for each method. Limitations and the proper use of each method are clarified after an extensive analysis. The two methods are also applied to synthetically and naturally developed fracture surfaces of different types of rocks. The methods yield inconsistent fractal dimensions for natural fracture surfaces and the reasons of this are discussed. The anisotropic feature of fractal dimension that may lead to the correlation of fracturing mechanism and multifractality of the fracture surfaces is also addressed.