Evaluation of van Genuchten-Mualem model on the relative permeability for unsaturated flow in aperture-based fractures

The relative permeability versus saturation relation is fundamental for modeling unsaturated flow processes in aperture-based fractures. On the basis of the local cubic law and the connectivity probability of aperture distribution within single fractures, an analytical relationship between the relative permeability and saturation is established. When the van Genuchten function on the description of the capillary pressure-saturation relationship is applied, an explicit mathematical expression between the relative permeability and saturation is gained similar to the form of the van Genuchten-Mualem (VG-M) model of porous media. Through the comparisons between the several series of laboratory-measured data available in the literature and theoretical results, good agreements are both presented and they suggest that the proposed VG-M model can be successfully used for aperture-based fractures as well as porous media. To understand aperture variation dependence of the relative permeability properties, a computational method combining fractal theory and the invasion percolation model to determine the empirical parameters associated with the spatial correlation of aperture distribution is developed. Simulated fractures with spatially correlated and uncorrelated aperture distributions corresponding respectively to small and large fractal dimensions are both studied. According to the simulated results, the residual saturation generally increases with fractal dimension increments. However, the value of m exhibits a weak dependence on fractal dimension. The proposed model can be applied to various engineering such as seepage control for rock slopes, petroleum reservoirs, nuclear waste disposal and geologic storage of carbon dioxide.