Analytical solutions of space fractional Boussinesq equation to simulate water table profiles between two parallel drainpipes under different initial conditions

Abstract

This work presents the first attempt to derive analytical solutions of a space fractional Boussinesq equation (SFBE) for parabola type 1, parabola type 2, and elliptical initial water table profiles, denoted as SFBE-P1, SFBE-P2, and SFBE-E, respectively. Laboratory and field data published in literature were used to evaluate the performances of the fractional models in homogeneous and heterogeneous soils. Besides, the performances of the proposed fractional models were compared with that of a fractional analytical solution developed in the literature for a flat initial water table condition (SFBE-F). According to the results of the sensitivity analysis, heterogeneity degree of soil (α) had a more effect on outputs of the fractional models. Also, the results indicated that the α parameter in the SFBE-P1, SFBE-E, and SFBE-F was capable of describing well the heterogeneity degree of soil. The α value was estimated almost 2 in relatively homogenous soil, while its values were respectively appraised almost 1.3 and 1.1 in soils with average and relative heterogeneity degrees. The fractional models reduced to their counterpart classical ones (BE-P1, BE-P2, BE-E, and BE-F) in the relatively homogenous soil. The measurement and prediction results demonstrated the best and similar performance for the SFBE-E and SFBE-F models, while the SFBE-P2 models was the weakest. This behavior was also observed by using the classical models. Compared to the BE-E, the SFBE-E provided better prediction results in the heterogeneous soils. Overall, the SFBE-E and SFBE-F can be applied as practical models to simulate water table profile between two parallel drainpipes for both homogeneous and heterogeneous soils.