# Software for Evaluating Pumping Tests on Real Wells
## 1 Introduction
### Well Storage (WBS)  
Well storage significantly impacts the initial phase of the pumping test (see Figure 1) in the order of seconds from commencing the test. The reason is that, at the start of the pumping test, water is pumped from the well itself. As the pumping test proceeds, inflow from the surrounding aquifer increases until it equals the pumped amount. The amount of water pumped from the well volume, on the other hand, decreases with time until it becomes negligible. The degree of this impact depends on the well radius. [PERSON] and [PERSON] [8] addressed this issue and defined a per-unit well storage factor, as presented in Equation (5):  
\[C=Q\frac{t_{j}}{t_{j}}, \tag{5}\]  
where \(C\) is the per-unit well storage factor (m\({}^{2}\)) and \(t_{j}\) and \(s_{j}\) are time and drawdown, respectively, from the unit slope log-log graph, where WBC dominates (Figure 1) [8]. This article uses the following dimensionless parameters (see [16; 42; 43]):  
* Dimensionless time \[t_{D}=\frac{Tt}{r_{w}^{2}S}\] (6)
* Dimensionless radius \[r_{D}=\frac{r}{r_{w}}\] (7)
* Dimensionless drawdown \[s_{D}(r_{D},t_{D})=\frac{2\pi T}{Q}(H-h(r,t))\] (8)
* Dimensionless drawdown at a well \[s_{WD}(r_{D}=1,\ t_{D})=\frac{2\pi T}{Q}(H-h_{W}(r_{W},t))\] (9)
* Dimensionless well storage \[C_{D}=\frac{C}{2\pi Sr_{w}{}^{2}}\] (10)  
where \(r\) is the distance from the pumped well (m), \(H\) is the initial hydraulic head (m), \(h(r,t)\) is the hydraulic head at distance r and time \(t\) (m), and \(h_{w}\) is the hydraulic head in the well (m). The method of determining the size of additional resistances used in this study was published by [PERSON] et al. [16] in relation to the oil sector. This solution is derived from a drop in pressure across the impacted area surrounding the well, given an unsteady flow. Several assumptions have been made for modeling [34; 36]:  
* The aquifer is confined and has a seemingly infinite areal extent. * The aquifer is homogeneous and isotropic and of uniform thickness over the area. * The flow is horizontal. * Prior to pumping, the piezometric surface is horizontal over the area. * The pumping rate is constant throughout the pumping test. * The well penetrates the entire thickness of the aquifer. * The well has a finite volume, and the well storage coefficient is constant throughout the pumping test. * Additional resistances in the well and its surroundings are non-zero. For groundwater flow, in terms of dimensionless parameters, the well-known diffusivity equation in the radial coordinates has the form [16; 34; 42; 44]  
\[\frac{\partial^{2}s_{D}}{\partial r_{D}^{2}}+\frac{1}{r_{D}}\frac{\partial s_ {D}}{\partial r_{D}}=\frac{\partial s_{D}}{\partial t_{D}}. \tag{11}\]  
Initial and boundary conditions:  
At time \(t=0\) and when applying the dimensionless parameters of Equations (6) to (8),  
\[s_{D}(r_{D},t_{D}=0)=0\;. \tag{12}\]  
In the case of dimensionless well drawdown  
\[s_{wD}(r_{D}=1,\;t_{D}=0)=0\;. \tag{13}\]  
For \(r_{D}\to\infty\), dimensionless drawdown is as follows:  
\[s_{D}(r_{D},\;t_{D})=0\;(\ref{eq:2}) \tag{14}\]  
The marginal condition for additional resistances in the well and its surroundings is as follows [14]:  
\[t_{D}=\frac{Tt}{r_{w}^{2}S}\]0  
The outer condition for a pumped well when factoring in the impact of the well volume on the pumping test in relation to Equation (15) is (see [24])  
\[t_{D}=\frac{Tt}{r_{w}^{2}S}\]1  
Equation (11) is solved using a one-dimensional Laplace transform.