Abstract
In this paper we present a constitutive model to describe unsaturated flow that considers the hysteresis phenomena. This constitutive model provides simple mathematical expressions for both saturation and hydraulic conductivity curves, and a relationship between permeability and porosity. The model is based on the assumption that the porous media can be represented by a bundle of capillary tubes with throats or "ink-bottles" and a fractal pore size distribution. Under these hypotheses, hysteretic curves are obtained for saturation and relative hydraulic conductivity in terms of pressure head. However, a non-hysteretic relationship is obtained when relative hydraulic conductivity is expressed as a function of saturation. The proposed relationship between permeability and porosity is similar to the well-known Kozeny-Carman equation but depends on the fractal dimension. The performance of the constitutive model is tested against different sets of experimental data and previous models. In all of the cases the proposed expressions fit fairly well the experimental data and predicts values of permeability and hydraulic conductivity better than others models.
Highlights
Constitutive models for unsaturated flow provide relationships between saturation, hydraulic conductivity and pressure head
In the particular case of fractured rocks, a physical constitutive model based on fractal geometry has been proposed by Guarracino (2006) and Monachesi and Guarracino (2011)
In the classical models of hysteresis, saturation and relative hydraulic conductivity values are limited by main drying and wetting curves which are obtained for initially fully saturated and dry porous media, respectively
Summary
Constitutive models for unsaturated flow provide relationships between saturation (or water content), hydraulic conductivity and pressure head. These relationships define the hydraulic behavior of soils and are necessary for the numerical resolution of the nonlinear Richards equation (Richards 1931). Van Genuchten proposed an empirical relation for saturation to obtain a closed-form analytical expression for the hydraulic conductivity by using Burdine (1953). The Brooks and Corey model combines a power-law relation for saturation with Burdine model to obtain a simple closed-form analytical expression for the hydraulic conductivity. We derive a constitutive model for unsaturated flow assuming a porous media conceptualized as a bundle of constrictive capillary tubes with a fractal pore size distribution. REV of porous media with a fractal pore size distribution, we obtain expressions for porosity, saturated hydraulic conductivity, and saturation and relative hydraulic conductivity curves
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