## 2 The Formulation of the Generalized Problem  
The flow to or from a well is described by the following diffusion equation [1; 36]  
\[\frac{\partial H}{\partial t}=\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}Dr \frac{\partial H}{\partial r}\bigg{)}, \tag{1}\]  
which can be written equivalently as  
\[C_{1}\frac{\partial H}{\partial t}=\bigg{(}\frac{\partial^{2}H}{\partial r^{2 }}+\frac{1}{r}\frac{\partial H}{\partial r}\bigg{)}, \tag{2}\]where \(H\) being the normalized groundwater depth and is given by  
\[H=\frac{h-h_{0}}{h_{m}-h_{0}}, \tag{3}\]  
with \(h\) is the groundwater height above the datum, \(h_{0}\) is the original height of groundwater above the datum before the test, \(h_{m}\) is the extreme groundwater height following the injection or pumping, \(D\) is the aquifer diffusivity, and \(r\) is the radial distance from the well center. \(C_{1}=\frac{1}{D}\) or \(D=\frac{1}{C_{1}}\) for both unconfined and restricted aquifers described by [PERSON] [36] as:  
1. for an unconfined aquifer, the diffusivity \(D\) is \[D=\frac{1}{C_{1}}=\frac{T}{n_{e}},\] (4)
2. for a confined aquifer, the diffusivity \(D\) is \[D=\frac{1}{C_{1}}=\frac{T}{S},\] (5) where \(T=Kh_{m}\) is called the transmissivity with \(K\) being the hydraulic conductivity, \(n_{e}\) being the effective porosity in an unconfined aquifer and \(S\) being the specific storage coefficient [37; 38]. Equation (4) can also be written \[D = K_{e}h_{m}\] (6) with \(K_{e}\) being called the effective hydraulic conductivity given as \[K_{e}=\frac{K}{n_{e}}\] (7) where \(T=Kh_{m}\), which occurs in Equation (4) suggests that Equation (1) is a linearization of Boussinesq's radial coordinate equation [39]. Equation (6) results when the height perturbation of injected water or pumped drawdown is small enough to assume that \(H\approx h_{m}\). To take into account the impact of pores of various sizes on saturated flow as a complement of the distributed-order fractional partial differential equation for water flow in an unsaturated medium [18], we incorporate distributed-order in Equation (1) to construct the distributed-order fractional partial differential equation of diffusion. We rewrite Equation (1) with the Atangana-Balean fractional derivative in time with the temporal distributed orders in the following form,  
\[a^{ABC}\mathcal{D}_{t}^{A}H(r,t)+b^{ABC}\mathcal{D}_{t}^{B}H(r,t)=\frac{D_{f}} {r}\frac{\partial}{\partial r}\left(r\frac{\partial H(r,t)}{\partial r}\right) \tag{8}\]  
where: \(a\) and \(b\) are, respectively, the ratio of immobile and mobile section porosities with the total porosity and satisfy the relation \(a+b=1\); \(0<\alpha\leq 1\) and \(0<\beta\leq 1\) are, respectively, the fractional derivative parameters for the mobile and the immobile sections; and \(D_{f}\) is the fractional diffusivity, which also takes two forms for unconfined and confined aquifers as their traditional complements in Equations (4) and (5). As this can be seen in Equations (4) and (5) for two significant groundwater kinds, i.e., unconfined and confined aquifers, Df has specific meaning, but the major component of Equation (8) remains mathematically the same. An alternative approach to describe the differentiated drainage from the two main forms of pores is the use of the distributed orders in Equation (8). Of course, further orders to account for main types of pores can be put in temporal-fractional derivatives, but the determination of their values will become a technical issue. The Atangana-Baleanu fractional derivative in the Caputo sense is defined as [24; 25; 40]  
\[{}^{ABC}{\cal D}_{t}^{\beta}g(r,t)=\left\{ \begin{array}{cc}\frac{N(\beta)}{1- \beta}\int_{0}^{t}\dot{g}(r,\tau)E_{\beta}(\frac{-\beta}{1-\beta}(t-\tau)^{ \beta})d\tau,\ 0<\beta<1,\\ \frac{\partial g(r,t)}{\partial t},\ \beta=1.\end{array} \right. \tag{9}\]  
where \(N(\beta)\) is the normalization function such that \(N(0)=N(1)=1,\dot{g}(r,\tau)=\left.\frac{\partial g(r,t)}{\partial t}\right|_{t =\tau}\) and \(E_{\beta}(\cdot)\) is the Mettag-Leffler function [41].