Fractal Scaling of Soil Particle-Size Distributions: Analysis and Limitations

Fractal scaling has recently been proposed as a model for soil particle-size distribution (PSD). In this work, the cumulative number of soil grains greater than a characteristic size, N(R > r), and the cumulative mass distribution, M(r < R), are developed and shown to be proportional to R³⁻ᴰ and R³⁻ᴰ, respectively, where r is the grain size, R is a specific measuring scale, and D is the fractal dimension. The cumulative-number approach to estimate D is shown to be sensitive to the assumed grain density and characteristic size, while the mass distribution is less sensitive to the assumed grain density and characteristic size, and therefore more appropriate for the analysis of field soils. These two models of fractal PSD behavior also constrain the fractal dimension to lie between 0 and 3 for field soils. With constraints on the fractal dimension, soils displaying strict fractal scaling in grain-size distribution are shown to be a rather small subset of those soils commonly encountered in the field. Earlier work has shown fractal scaling in many soil PSDs with fractal dimensions exceeding 3.0 using the number-based analysis. The fractal scaling and magnitude of the fractal dimensions found in previous work are shown to be an artifact of the plotting algorithms and assumptions on grain density and size. Although fractal scaling plays an important role in soil water retention and porosity, PSD data alone are not sufficient to fully characterize this scaling.