A mathematical model for simulation of a water table profile between two parallel subsurface drains using fractional derivatives

By considering the initial and boundary conditions corresponding to parallel subsurface drains, the linear form of a one-dimensional fractional Boussinesq equation was solved and an analytical mathematical model was developed to predict the water table profile between two parallel subsurface drains. The developed model is a generalization of the Glover-Dumm's mathematical model. As a result, the new model is applicable for both homogeneous and heterogeneous soils. It considers the degree of heterogeneity of soil as a determinable parameter. This parameter was called the heterogeneity index. The laboratory and field tests were conducted to study the performance of the proposed mathematical model in a homogenous soil and in an agricultural soil. The optimal values of parameters of the fractional model developed in this study and Glover-Dumm's model were estimated using the inverse problem method. In the proposed inverse model, a bees algorithm (BA) was used. The results showed that in the homogenous soil, the heterogeneity index was nearly equal to 2 and therefore, the developed mathematical model reduced to the Glover-Dumm's mathematical model. The heterogeneity index of the experimental field soil considered was equal to 1.04; hence, this soil was classified as a very heterogeneous soil. In the experimental field soil, the proposed mathematical model better represented the water table profile between two parallel subsurface drains than the Glover-Dumm's mathematical model. Therefore, it appears that the proposed fractional model presented is a highly general and effective method to estimate the water table profile between two parallel subsurface drains, and the scale effects are robustly reflected by the introduced heterogeneity index.

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