General scaling rules of the hysteretic water retention function based on Mualem's domain theory

Based on the domain theory of hysteresis, the present study rigorously derives a unified scaling transformation for predicting the wetting scanning retention function, θ_w ( ψ ), following any sequence of wetting and drying processes, from the measured main wetting curve. It is proved theoretically that a shape-similarity exists among the different wetting curves of a given soil. Each wetting scanning curve and the main wetting curve are described by the same normalized equation, over the common interval of ψ values. This shape-similarity is a direct result of the Mualem domain theory. On this basis, a general unique equation is formulated for prediction of all wetting scanning curves, of all orders, which is compatible with the Mualem dependent, as well as independent, domain theory. On the same theoretical basis, it is proved that no similarity exists among the drying scanning curves. Thus, the use of the scaling transformation for prediction of the drying scans is subjected to inconsistency with the physical principles underlying the dependent domain theory. As a result, scaling in this case would inevitably lead to inaccuracies in the calculated drying curves. A significant step forward has been made in the present study, from Mualem's dependent domain model (1984) with its implicit predictive formulae of the drying scanning curves. A unique explicit equation was theoretically derived, applicable for describing the drying scanning curves of all orders. This unique equation of the drying scanning curves, together with the general single equation of the wetting scanning curves, provided closure of all theoretical wetting and drying scans within the main hysteresis loop. Consequently, the retention function following any sequence of successive wetting and drying processes can be predicted by these two equations determined by the measured boundary curves.