A prefractal model for predicting soil fragment mass-size distributions

Models are needed to parameterize and predict soil fragment mass-size distributions created by different tillage operations. Assuming untilled soil can be represented as a random prefractal porous medium, we present a homogeneous fragmentation model for quantifying the cumulative mass-size distribution of fragments produced by a given energy input. The model contains three physically based parameters: the scale-invariant probability of failure (P), the length of the porous medium (x₀) and the scale factor (b). The model was fitted to 144 mass-size distributions obtained by drop shattering and dry sieving samples collected from two soil types (Maury silt loam and McAfee clay loam) under conditions of varying management, initial water content, and core diameter, with x₀ set equal to the depth of sampling (10 cm), and P and b estimated inversely. The relationship between the predicted and observed distributions was indistinguishable from a 1:1 line with an R² = 0.97. The estimates of P and b ranged from 0.29 to 0.67, and from 1.76 to 6.25, respectively. The P parameter controls the slope of the cumulative mass-size distribution, while the b parameter determines the length of the largest fragments. The Maury soil had a higher mean b value and a lower mean P value relative to the McAfee soil, indicating differences in fragmentation due to soil type. The P values were more sensitive to the imposed treatments than the b values. For both soils, the most important sources of variation in P were the water content at sampling, and an interaction between core diameter and management. The new prefractal equation has a distinct advantage over other fragment mass-size distribution models in that its parameters are physically based and can potentially be obtained from independent measurements. This predictive capability warrants further research.