A fractal constitutive model for unsaturated flow in fractured hard rocks

A physical conceptual model for water retention in self-similar fractured hard rocks is presented. The geometric pattern of the fracture network is described using a classical fractal object known as the Sierpinski carpet. Assuming this type of geometry, a relatively simple closed-form expression for the water content is obtained. All model parameters can be calculated from the density of the main fractures, the maximum and minimum values of the fracture aperture, and the residual water content. The resulting water content expression is then used to estimate the unsaturated hydraulic conductivity of the fractured medium based upon the well-known model of Burdine. It is found that for large enough ranges of fracture apertures the new constitutive model converges to the empirical Brooks–Corey model.