Conformal fractional derivative-based groundwater flow equation and its exact solution

This study investigates the application of conformable fractional derivatives (CFD) for modeling groundwater flow dynamics. Fractional derivatives offer a powerful alternative to overcome the limitations of traditional models in groundwater management, a critical challenge due to increasing population demands and diminishing water resources. In this context, the conformable fractional time derivative was integrated into the groundwater flow equation to enable more precise and accurate calculations of complex flow dynamics in aquifer systems. The analytical solution of the developed CFD-based groundwater flow equation was established and compared with both the classical Theis solution and experimental data. The proposed solution incorporated the well-known Theis equation, modified to include the fractional-order derivative parameter α , which allowed for greater flexibility in representing the temporal evolution of drawdown. When α = 1, the solution reverted to the classical Theis equation. Especially, the CFD-based solution with α = 0.9 exhibited superior alignment with experimental results, outperforming the classical Theis model. Performance evaluation metrics confirmed the enhanced predictive capabilities of the CFD-based model, with the coefficient of determination value of 0.9806 compared to 0.8781 for the classical model, and reductions in mean absolute error, mean squared error, and root mean squared error by 37%, 45%, and 48%, respectively. These findings highlight the potential of the proposed approach in improving groundwater management and aquifer characterization by accurately modeling storage effects and non-local behaviors in groundwater systems.