A Fractal Model for Predicting Water and Air Permeabilities of Unsaturated Fractured Rocks

Abstract

A fractal model to predict water and air permeabilities of unsaturated fractured rocks is presented. The derivation of the model is based on physical and geometric concepts. The pattern of the fracture network is assumed to be fractal and it is described by the Sierpinski carpet. The proposed expressions for the relative water and air permeabilities are closed-form and have five independent parameters: the fractal dimension, the minimum and maximum fracture apertures and the emergence points for water and air flows. The ability of the model to describe experimental data is illustrated by fitting the derived analytical curve to measured data from Grimsel Test Site (Switzerland) and numerical experiments designed by Liu and Bodvarsson (J Hydrol 252:116–125, 2001). In both the cases, the proposed model provides a very good description of water and air permeabilities over several orders of magnitude for the whole range of water saturation.