A fractal analysis of permeability for fractured rocks

Rocks with shear fractures or faults widely exist in nature such as oil/gas reservoirs, and hot dry rocks, etc. In this work, the fractal scaling law for length distribution of fractures and the relationship among the fractal dimension for fracture length distribution, fracture area porosity and the ratio of the maximum length to the minimum length of fractures are proposed. Then, a fractal model for permeability for fractured rocks is derived based on the fractal geometry theory and the famous cubic law for laminar flow in fractures. It is found that the analytical expression for permeability of fractured rocks is a function of the fractal dimension Df for fracture area, area porosity ϕ, fracture density D, the maximum fracture length lmax, aperture a, the facture azimuth α and facture dip angle θ. Furthermore, a novel analytical expression for the fracture density is also proposed based on the fractal geometry theory for porous media. The validity of the fractal model is verified by comparing the model predictions with the available numerical simulations.