Fractal applications to soil hydraulic properties
AuthorTyler, Scott Woodman
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Three manuscripts are presented describing the application of fractal mathematics to soil hydraulic properties. The results indicate that the pore space of many field soils can best be represented by a self-similar or fractal geometry. The solid phase of the soil, i.e. the soil grains, is less amenable to fractal scaling and only a narrow range of typically encountered field soils are likely to be well described by fractal geometry. In the first manuscript, a simple self-similar model, the Sierpinski carpet is used to represent the pore number and pore size distribution of typical field soils. The soil water retention function is theoretically developed and shown to be equivalent to the power law empirical model for retention developed by Brooks and Corey (1964) and Campbell (1974). These results are extended to data reported on field soils and a relation between the fractal dimension and soil texture is presented. The results indicate an increasing fractal dimension with finer soil texture. In the second manuscript, fractal scaling arguments are used to develop a theoretical basis for Arya and Paris’ (1981) curve fitting coefficient, Oi. This term is shown to be equivalent to the fractal dimension of the pore trace and is consistent with the ideas of a scale-dependent tortuosity (Wheatcraft and Tyler, 1988). To estimate the fractal dimension, the fractal scaling often seen in particle size distributions (PSD), is utilized as a surrogate measure of the pore space fractal dimension. Ten soils were analyzed for fractal behavior, five reported by Arya and Paris (1981) and five reported by Mualem (1976). Of these ten soils, nine clearly showed fractal scaling in their particle number verses size. Good agreement between measured and predicted water retention was observed in nine out of the ten soils when the fractal dimension of the pore trace was estimated from the particle size distribution. The third manuscript examines, in detail, the concepts of self-similarity in particle size distributions. It is shown that self-similarity in grain number verses grain size may be limited to a narrow range of typically encountered field soils. Theoretical results show that the fractal dimension for PSDs must range between 0.0 and 3.0, with typical soils ranging from 2 to 3. The analysis suggests that most soils do not behave as fractal porous media, rather only the void space (porosity) of the soil displays fractal behavior. This conclusion suggests that fractal will play an important role in the estimation of hydraulic and transport properties of soils.
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soil hydraulic properties
soil water retention function
power law empirical model for retention
fractal scaling arguments
particle size distributions
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