## 1 Introduction  
Groundwater, in nearly all geological formations, is visible and in movement. Movement is very sluggish in some materials such as rock or clay, as opposed to the groundwater pouring in sand and gravel patterns. However, highly eroded rock may display discharges of groundwater identical to or in surplus of sand and gravel aquifers [1]. In almost every part of the globe, aquifers are essential mineral resources for irrigated agriculture, commercial and drinkable water supply. The study of aquifers is germane to chemical engineers, agricultural engineers, civil engineers, forestry engineers, ecologists, geologists, geographers, geotechnical engineers, irrigation engineers, mining engineers, hydraulic engineers, hydro-geologists, hydrologists, petroleum engineers, soil scientists, and others [2]. In groundwater research, the central issue faced so far is the actual configuration of the geological structure in which water flows into the aquifer under analysis. However, there are many fractured rock aquifers where groundwater flow does not conform to traditional geometries [3], such as the Karoo aquifers in South Africa, described by the existence of a few parallel bedding fractures that act as the primary water channels in aquifers [4]. For the Karoo aquifer, [PERSON] et al. [4] established a three-dimensional prototype and demonstratedthat the dominant flow field in these aquifers is vertical and linear and not, as generally thought, horizontal and radial. However, subsequent investigations [5] indicate that the flow is often affected by the geometry of bedding parallel fractures. It was concluded that the most appropriate modification is to generalize the flow dimension to non-integer values after considering a number of potential variations on the models, while maintaining the premises of radial flow and homogeneity [6]. Analogously, in order to investigate a radially symmetric model for the groundwater flow, [PERSON] and [PERSON] suggested the idea of a non-integer fractional derivative by changing the conventional first order derivative of a piezometric head with a complementary fractional derivative [7]. To investigate groundwater flow, [PERSON] and [PERSON] [7] have considered the Riemann-Liouville fractional derivatives ([8], chapter 1, Equation (1.25)) to formulate the fractional diffusion-wave equations for the radial solute transport in aquifers, and this formulation is similar to the equation suggested by [PERSON] et al. [9]. During aquifer analyses in fractured networks, [PERSON] et al. [10] analyzed the impact of flow dimensions and irregular diffusion of groundwater. Quite recently, [PERSON] and [PERSON] [11] reused a time-fractional partial differential equation (PDE) proposed by [PERSON] et al. [9] and three variants of fractional partial differential equations (fPDEs) in a radial coordinate that were considered by [PERSON] and [PERSON] [12], [PERSON] [13]. More recently, the study of groundwater flows in aquifers using fPDEs for groundwater transport in unsaturated soils has been addressed [14; 15; 16; 17; 18]. In several scientific fields, such as geoscience, demography, physics, mathematics, biology, and bioengineering, the simulation of complex systems is made with fractional order differential/integral operators [19]. Many fractional-differential operators have been studied by researchers. We recall here some of them: namely, the fractional Caputo-Fabrizio operator [20], the integral/derivative Riemann-Liouville operator [21], fractional Caputo derivative [22], the fractional Yang-Srivastava-Machado derivative [23], the Atangana-Baleanu fractional derivative [24; 25], etc. The space-fractional diffusion with power-law super diffusivity formulated with the Riemann-Liouville fractional derivative was analyzed by [PERSON] [26]. The time-fractional Caputo-Fabrizio derivative was implemented by [PERSON] et al. [27] to examine the advective diffusion process with low-range memory and concentrated source.