An Improved Fractal Equation for The Soil Water Retention Curve

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Abstract
The water retention curve, θ(ψ), is important for predicting soil physical properties and processes. Until recently, equations for the θ(ψ) were empirical. Advances in fractal geometry have led to the derivation of physical models for the θ(ψ). However, both existing fractal equations have only two parameters and thus are relatively inflexible. We derived a new three‐parameter fractal model for the θ(ψ). This equation was fitted to 36 θ(ψ)'s for a silt loam soil with a wide range of structural conditions. The new equation fitted these data much better than the existing equations. The parameters of the new equation, ψa, ψd and D are physical entities, corresponding to the air‐entry value, tension draining the smallest pores, and fractal dimension, respectively. Estimates of the ψa, ψd and D were physically reasonable, with median values of 2.9 × 10−1 kPa, 1.6 × 104 kPa, and 2.87, respectively. In contrast, the existing equations yielded anomalous estimates of either ψa or D. The new equation was able to fit θ(ψ) for a variety of porous media, including sandstone, glass beads, sands, sieved soil, and undisturbed soils ranging from very fine sandy loam to heavy clay. The ψa and ψd were more sensitive to structural and textural variation than D. The new equation represents an improvement over existing models in terms of both goodness of fit and the physical significance of its parameters.

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