Triangular Fuzzy Finite Element Solution for Drought Flow of Horizontal Unconfined Aquifers

Abstract

In this paper, a novel approximate triangular fuzzy finite element method (FEM) is proposed to solve the one-dimensional second-order unsteady nonlinear fuzzy partial differential Boussinesq equation. The physical problem concerns the case of the drought flow of a horizontal unconfined aquifer with a limited breath B and special boundary conditions: (a) at x = 0, the water level is equal to zero, and (b) at x = B, the flow rate is equal to zero due to the presence of an impermeable wall. The initial water table is assumed to be curvilinear, following the form of an inverse incomplete beta function. To account for uncertainties in the system, the hydraulic parameters—hydraulic conductivity (K) and porosity (S)—are treated as fuzzy variables, considering sources of imprecision such as measurement errors and human-induced uncertainties. The performance of the proposed fuzzy FEM scheme is compared with the previously developed orthogonal fuzzy FEM solution as well as with an analytical solution. The results are in close agreement with those of the other methods, with the mean error of the analytical solution found to be equal to $1.19·{10}^{-6}$. Furthermore, the possibility theory is applied and fuzzy estimators constructed, leading to strong probabilistic interpretations. These findings provide valuable insights into the hydraulic properties of unconfined aquifers, aiding engineers and water resource managers in making informed and efficient decisions for sustainable hydrological and environmental planning.