Effective permeability of fractal fracture rocks: Significance of turbulent flow and fractal scaling

Abstract

In this study, we develop a new approach to calculate hydraulic gradient dependent effective permeability of a fractal fracture network where both laminar and turbulent flows may occur in individual fractures. The cubic law is used to calculate flow behaviors in fractures where flow is laminar, while the Forchheimer’s law is used to quantify turbulent flow behaviors. A critical fracture length is used to distinguish flow characteristics in individual fractures. While flows in some fractures may be turbulent, we assume that the fractal fracture network can still be treated as a porous medium where Darcy’s law applies and our objective is to determine an effective permeability for the network. The developed new solutions can also be used for the case of a general scaling relationship, an extension to the linear scaling. We examine the impact of fractal fracture network characteristics on the effective permeability of the network. These characteristics include: fractal scaling coefficient and exponent, fractal dimension, ratio of minimum over maximum fracture lengths. The influence of imposed hydraulic gradient and critical length on the effective permeability is also examined and discussed. Results demonstrated that the developed solution can explain more variations of the effective permeability in relation to the fractal dimensions estimated from field observations. At high hydraulic gradient the effective permeability decreases with the fractal scaling exponent, but increases with the fractal scaling exponent at low gradient. The effective permeability increases with the scaling coefficient, fractal dimension, fracture length ratio and maximum fracture length.