Introduction of a mathematical model for calculating sub-surface drains spacing using fractional derivatives

One of the limitations of Boussinesq equation is that its parameters (e. g. hydraulic conductivity) are scale-dependent. In this work, a fractional Boussinesq equation was obtained by assuming power-law changes of flux in a control volume and using a fractional Taylor series. Unlike Boussinesq equation, due to the non-locality property of fractional derivatives, the parameters of fractional Boussinesq equation are constant and scale-invariant. The linear form of fractional Boussinesq equation was solved by using spectral representation and an analytical mathematical model was derived to calculate sub-surface drains spacing. The optimal values of parameters of mathematical model developed in this study and Glover-Dumm's model were estimated from inverse modelling. In the inverse methodology, water table data between two sub-surface drains and optimisation method of Bees algorithm were used. The accuracy of proposed model was investigated using water table data between the two sub-surface drains and compared to Glover-Dumm's model. The results indicated that the mathematical model derived in this study predicted the water table profile between two sub-surface drains more exactly than Glover-Dumm's model.