Fractal Fragmentation, Soil Porosity, and Soil Water Properties: I. Theory

TLDR

Recent work on soil water properties has explored fractal representations of soil structure as a promising approach. The paper develops a fully self‑consistent fractal model of aggregate and pore‑space properties for structured soils and calls for experimental validation. The model treats soil as a fragmented fractal porous medium, partitioning volume into self‑similar pore and aggregate classes, assigning uniform fragmentation probabilities, defining fractal dimensions for complete and incomplete fragmentation, and specifying the scale range over which fractal behavior holds. It predicts a fractal dimension ≤3, decreasing aggregate bulk density with larger aggregates, a power‑law aggregate‑size distribution, water potential and retention curves following integer‑power laws, and hydraulic conductivity expressed through single‑size fracture conductivities in a fractal network.

Abstract

Abstract Recent efforts to characterize soil water properties in terms of porosity and particle‐size distribution have turned to the possibility that a fractal representation of soil structure may be especially apt. In this paper, we develop a fully self‐consistent fractal model of aggregate and pore‐space properties for structured soils. The concept underlying the model is the representation of a soil as a fragmented fractal porous medium. This concept involves four essential components: the mathematical partitioning of a bulk soil volume into self‐similar pore‐ and aggregate‐size classes, each of which is identified with a successive fragmentation step; the definition of a uniform probability for incomplete fragmentation in each size class; the definition of fractal dimensions for both completely and incompletely fragmented porous media; and the definition of a domain of length scales across which fractal behavior occurs. Model results include a number of equations that can be tested experimentally: (i) a fractal dimension ≤3; (ii) a decrease in aggregate bulk density (or an increase in porosity) with increasing aggregate size; (iii) a power‐law aggregate‐size‐distribution function; (iv) a water potential that scales as an integer power of a similarity ratio; (v) a power‐law expression for the water‐retention curve; and (vi) an expression for hydraulic conductivity in terms of the conductivities of single‐size arrangements of fractures embedded in a regular fractal network. Future research should provide experimental data with which to evaluate these predictions in detail.