Un modelo fractal para estimar la conductividad hídráulica no saturada de rocas fracturadas

Abstract

Modeling groundwater flow in unsaturated fractured rocks has received considerable attention in the last two decades. One of the main reasons for focusing on the study of water flow in this type of media is the search for potential safe permanent storage facilities for geological disposal of high-level nuclear wastes. Knowledge of constitutive relations is indispensable for the numerical solution of the equations describing water flow in unsaturated porous media. These relations are saturation (S) and hydraulic conductivity (K) curves, both expressed as functions of the pressure head (h). The experimental determination of K(h) is tedious and time-consuming and the measurements are variables, error-prone, and applicable to only a narrow range of pressure head h. An alternative to direct measurement is to use theoretical models which predict K(h) from the saturation curve S(h) that can be easily measured in laboratory. The majority of the models that predicts K(h) from S(h) have been developed for describing unsaturated flow in sedimentary formations (granular porous media). Predictive models specifically designed for fractured hard rocks are virtually nonexistent. In this study, a hydraulic conductivity model for fractured rocks is derived. The proposed K(h) model is based on the assumption that the fracture pattern is self-similar. The fracture pattern is described using the Sierpinski carpet, a classical fractal object that contains a self-similar geometric pattern of pores. The proposed model has a closed form analytical expression with three independent geometric parameters: the fractal dimension of the Sierpinski carpet and the maximum and minimum fracture apertures. One of the main features of the proposed model that allows its validation is that the expression of K(h) is identical to the one obtained by using the Burdine model with a tortuosity factor equal to one. The proposed model can represent the constitutive relations for fractured rocks obtained by Liu and Bodvarsson (2001) using numerical simulation techniques. The proposed K(h) model is the only existing model which has been completely derived from physical concepts and geometric properties of the fracture pattern.