Modeling the Primary Drainage Curve of Prefractal Porous Media

Fractal models for the soil water retention curve have largely ignored hysteresis. Such models generally require that all pores be connected to the atmosphere via a network of similar‐sized or larger pores. In random mass fractals and natural porous media, however, not all pores of a given size class empty during drainage due to incomplete pore connectivity and/or the presence of surrounding smaller pores. A modification of Rieu and Sposito's (1991a) prefractal water retention equation is proposed to accommodate this hysteresis during monotonic drying. The resulting expression is identical in form to the empirical Brooks and Corey (1964) model and contains three physically‐based parameters: the proportion of nondraining pores ( P d ), the scale factor ( b ), and the apparent fractal dimension of the primary drainage curve ( D d ), which is related to the underlying mass fractal dimension ( D ) of the porous medium by D d = D + [log( P d )]/[log( b )]. Model testing consisted of fitting the modified equation to previously published water retention data for 30 randomized Sierpinski carpets and seven soils. Error sums of squares ranged from 0.001 to 0.093 for the carpets, and from 0.015 to 0.047 for the soils. In contrast, the unmodified Rieu and Sposito (1991a) equation performed very poorly. Best fit estimates of D d ranged from 0.295… to 1.640… for the two‐dimensional carpets, and from 2.467… to 2.902… for the three‐dimensional soils. Prediction of D , based on estimates of D d derived from the primary drainage curve, will require additional research on how to obtain P d and b . Based on the analytical approach outlined here, it should also be possible to model the primary wetting curve, thus providing a more complete fractal description of soil water hysteresis.