A universal multifractal-based method to model pore size distribution, water retention and hydraulic conductivity of granular green roof substrates

Abstract

Hydrological behaviour of granular substrates is of critical interest in Nature-based solutions (NBS) like green roofs. To simulate this behaviour in a physically realistic manner it is indispensable to model the substrate’s hydraulic conductivity (HC) as it determines infiltration rate at various degrees of saturation. Since HC is directly dependent on water content retained by the substrate, it is necessary to physically model this water retention (WR) behaviour too. Capillary water is stored or retained in pore spaces and this water content that can be retained by a substrate under different suction pressures is therefore dependent upon its pore size distribution (PSD). Since pores in any granular media are spaces where grains are absent, their size distribution too is intrinsically related to the substrate’s grain size distribution (GSD) which provides the probability of finding grains smaller than some threshold diameter dg,t. Although earlier studies have attempted to model PSD, WR and HC, they frequently use simplifying mono-fractal approximations, whereas this study proposes a more generalized multifractal-based approach. Furthermore, while it is quite usual to incorporate pore tortuosity through some indirect parameter l in the HC model, a related ink-bottle effect which even though capable of affecting WR behaviour is commonly ignored. Therefore, this paper attempts to address the aforementioned research gaps in modelling GSD, PSD, WR and HC by i) investigating the somewhat overlooked question of similarity in multifractal behaviour between grain size fields and substrate density fields, and consequently suggesting an improved method for estimating universal multifractal (UM) parameters of grain size fields in a more reliable manner from just conventional GSD measurements in order to be directly used in the multifractal GSD model, ii) proposing a new UM-based PSD model, and subsequently using it to obtain a new UM-based WR model with a parameter to directly represent ink-bottle effect - a consequence of the substrate’s pore configuration or arrangement, iii) using this UM-based WR model to suggest a new UM-based HC model without the necessity for a separate pore tortuosity parameter. Finally, the proposed models have been validated by using experimental measurements from 4 different commercially used green roof substrates.