Based on the domain theory of hysteresis, the present study rigorously derives a unified scaling transformation for predicting the wetting scanning retention function, y w (c), 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 c 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.
The soil water hysteresis model proposed by Poulovassilis and Kargas (Soil Sci. Soc. Am. J. 64:1947-1950, 2000 is considered in the present study. According to this model, the bivariate domain density distribution function f can be derived by partitioning the slopes of either of two main curves proportionally to the slopes of another. Accordingly, there are two possible ways of deriving function f . The basic claim of Poulovassilis and Kargas is that both possibilities lead to the same resultant function f , which can be evaluated using integral equation presented by them. The present study shows that the above two ways of determining function f actually lead to two incompatible partitioning models yielding different domain density distribution functions. Moreover, none of these two partitioning models can reproduce the measured hysteresis loop used for calibration. Whether the partitioning of the main wetting curve slopes proportionally to the main drying curve slopes or vice versa is applied, most of the predicted primary scanning curves deviate considerably from the measured ones, cross out the measured boundary loop and do not converge at an appropriate edge of the loop. The present study reveals that the above-mentioned integral equation, presented by Poulovassilis and Kargas, appears to be at variance with both partitioning models. It is shown herein that this integral equation unambiguously follows from Mualem (Water Resour. Res. 9:1324-1331, 1973) similarity hypothesis and, accordingly, the correspondent domain density distribution function derived as the unique analytical solution of this equation is evidently identical to that obtained by Mualem (1973). The predicted curves presented by Poulovassilis and Kargas are not obtained when any of the two partitioning models is applied, but when using the integral equation of Mualem's (1973) model.
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