Existing groundwater recharge estimation methods appear to mainly generate site-specific groundwater recharge estimates. In doing so, they fail to give reliable recharge estimates on a regional to global scale. This is due to failure of taking into account the concepts of heterogeneity and viscoelasticity. Accordingly, this study was aimed at developing a new approach to groundwater recharge estimation by means of taking into account these concepts. The methodological approach entailed obtaining an exact solution to a selected groundwater recharge equation by applying the Laplace and inverse Laplace transform. Upon doing an uncertainty analysis and statistical analysis of the parameters within the solution, it was found that storativity and drainage resistance both require accurate estimation when estimating recharge from the selected equation. Following this, the Caputo derivate, Caputo-Fabrizio derivative, and the Atangana-Baleanu derivative were applied and an exact solution was obtained for each derivative. Upon doing a numerical simulation for each of these solutions, the results depict the behaviour of a particular real-world problem. The recharge within a heterogeneous and viscoelastic geological formation is well described with the concept of differentiation with the generalised Mittag-Leffler law or the Atangana-Baleanu fractional derivative. The recharge via elastic geological formation can be modelled via the Caputo and Caputo-Fabrizio.