The pore–solid fractal model of soil density scaling

Abstract

SummaryWe have developed the fractal approach to modelling variations in soil bulk density and porosity with scale of measurement or sample size. A new expression is derived for each quantity based on the poresolid fractal (PSF) model of soil structure. This new general expression covers a range of fractal media and accommodates existing fractal models as special cases. Model outputs cover a range of scaling behaviour expressed in terms of monotonic functions, from increasing density and decreasing porosity, through constant porosity and density to decreasing density and increasing porosity with increasing scale of measurement. We demonstrate the link between this new model for the scaling of porosity and bulk density and the water retention model for the PSF. The model for scaling bulk density is fitted to data on aggregate bulk density and shown to yield good fits describing bulk density decreasing with increasing aggregate size. Porosity scaling is also inferred from the fitting of water retention data. Inferred porosities from different fittings are shown to follow decreasing, scale-invariant and increasing values with decreasing size of structural unit, and these theoretical results emphasize the need for further experimental investigation on the basic issue of density scaling in soil science.

Analyzing the Relationships between Processing Parameters and Fractal Dimension of Void Size on Cross-Sections of Oriented Strandboards

“…The "Sierpinski carpet model" (Winslow 1985;Barnsley 1988) was applied to simulate the fractal void size on cross-sections of OSB. Figure 2 shows the Sierpinski carpet approach.…”

“…For materials with complex and irregular porous structure, the fractal theory provides an appropriate mathematical tool to study the relationship between void and density, on the one hand, and material performance (e.g., modulus and strength), on the other. Therefore, fractal geometry has been widely applied to study the internal structure morphology of a great variety of porous materials, including soil (Bird and Perrier 2003), concrete (Diamond 1999), and porous silica (Yamaguchi et al 2008). In fractal theory, the Sierpinski carpet (Winslow 1985;Barnsley 1988;Li et al 2013) can be applied to simulate the void structure.…”

Comportement des sols fins compactés à l’humidification. Apport d’un modèle de microstructure

“…Depuis les travaux précurseurs de Lambe (1958), les connaissances sur l'influence des propriétés de nature et d'état des sols sur la micros tructure ont progressé de manière remarquable (Dia mond, 1969 ;Tessier, 1984 ;Benett et Hulbert, 1986 ;Bruand et Prost, 1997 ;Delage et al, 1996), jusqu'à sa modélisation par l'outil m athém atique des fractales (Bird et Perrier, 2003). D'une manière simplifiée, ces différentes observations sur la m icrostructure et les propriétés physico-chimiques des sols fins compactés permettent de faire ressortir trois points essentiels : -le premier point est la différence fondamentale entre les particules argileu ses et les particules non argileuses.…”

A comparison of fractal analytical methods on 2- and 3-dimensional computed tomographic scans of soil aggregates

“…Many variations on these power laws have been developed, including the conversion of the mass power law to a density power law (Rieu and Sposito, 1991a). Let m r be the mass of a fractal object enclosed in a space of unit length r and m L be the mass of the same fractal object enclosed in a space of reference unit length L. From Bird and Perrier (2003), the mass power law follows as…”

“…Alternatively, Zeng et al (1996) describe how to estimate the lacunarity and surface fractal dimension of CT images without imposing distinct phases. In this study, we adopt another approach in order to preserve structural information within soil images: since density can be inferred from CT image pixels, we utilized the concept of density scaling to describe the soil as a potential fractal (Perfect et al, 2002;Bird and Perrier, 2003). Density scaling refers to the density measured on an object over a range of spatial observation scales.…”