Earth recharge model with crossover behaviors: application of piecewise differentiation

Abstract The estimation of groundwater recharge is usually done with direct or indirect measurement techniques that are site-specific and derived primarily from flux measurements that due to the flexibility and uncertainty associated with heterogeneous systems has a high margin of error. Fractional derivatives, which are used to reflect the geological heterogeneity and fluid viscoelasticity, may be introduced to solve this challenge. Here, we develop sensitivity analyses to model groundwater recharge in a two-layer (homogeneous upper and highly heterogeneous lower) shallow unconfined aquifer system. The bottom of the model showed a non-Darcian fracture and matrix flow, and associated flow deviation, while the top of the model represented a uniform high-permeability and high-porosity layer, typical for calcite-rich areas. The EARTH model is the governing equation of the system, analytically solved through the use of Laplace and inverse Laplace transforms. A new multi-layer aquifer model for the spatial-parameter, temporal-parameter groundwater flow model established in Earthlab. The model features later flow, gradual layer transitions and appropritate boundary conditions. It also considers evapotranspiration and seasonal recharge, providing a better model for groundwater behavior in heterogeneous aquifer systems. A fractional-order differential equation is proposed to describe recharge dynamics for the homogeneous upper layer and a classical integer-order differential equation for the heterogeneous lower layer with non-Darcy flow. This work applies Caputo and Atangana-Baleanu fractional derivatives to investigate crossover behaviors in the system. The results show unique long-tail exponential growth structures in the homogeneous layer and crossover behaviors at early times that depend on the operator. In contrast, despite the scale invariance property of the Caputo derivative, the Atangana-Baleanu derivative detects alpha-to-alpha crossovers, stressing its ability to reproduce early-time behavior. This study presents a powerful solution for describing groundwater recharge in dual-layered aquifer systems by coupling a fractional differentiation model with the extended Mittag–Leffler law, providing a new insight into understanding heterogeneous and homogeneous formations. This approach may be cross-validated with field data and will provide more evidence for capillary-dominated flow, using fractional derivatives to model subsurface flow.