## 2 Theories and Methods
### Fundamentals of the HWS Model  
A novel HWS model (HHU Well-Skin model, abbreviated as the HWS model) is proposed by us for the first time. Meanwhile, a parameter calculation method of the slug test considering the influence of the finite thickness well-skin layer based on the HWS model is proposed. The groundwater continuity equation is established between the confined aquifer and the finite thickness well-skin layer. The water balance equation is established in the contact surface between the confined aquifer and the finite thickness well-skin layer and between the finite thickness well-skin layer and the well. The theoretical model of the slug test under the influence of the well-skin is shown in Figure 3.  
1. Establishment of the theoretical model The governing equation of the groundwater radial flow in a well-skin layer and a confined aquifer is as follows: \[\frac{\partial^{2}h_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{1}}{ \partial r}=\frac{S_{1}}{T_{1}}\frac{\partial h_{1}}{\partial t}r_{w}\leq r \leq r_{s}\] (7) \[\frac{\partial^{2}h_{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{2}}{ \partial r}=\frac{S_{2}}{T_{2}}\frac{\partial h_{2}}{\partial t},r_{s}\leq r\leq\infty\] (8)  
A water balance equation is established in the contact surface between the confined aquifer and the well-skin layer of finite thickness as follows:  
\[K_{1}\frac{\partial h_{1}(r_{s},t)}{\partial r}=K_{2}\frac{\partial h_{2}(r_{ s},t)}{\partial r},t>0 \tag{9}\]  
A water balance equation is established in the contact surface between the well-skin layer of finite thickness and the well as follows:  
\[\pi r_{c}^{2}\frac{dw(t)}{dt}=2\pi r_{w}K_{1}B\frac{\partial h_{1}(r_{w},t)}{ \partial r},t>0 \tag{10}\]  
Among them, subscripts 1 and 2 represent the well-skin layer and the confined aquifer parameters, respectively, \(r_{s}\) is the distance from the center of the well to the outer boundary of the well-skin layer (m) and the rest of the symbols have the same meaning as they do above. Figure 3: Schematic diagram of the slug test model considering the well-skin effect. (2) Solution of the theoretical model  
First, these equations can be converted to a dimensionless form by the following dimensionless variables and parameters:  
\[h_{1D}=\frac{h_{1}}{w_{0}},h_{2D}=\frac{h_{2}}{w_{0}},\rho=\frac{r}{r_{w}},\rho_{ s}=\frac{r_{s}}{r_{w}},\rho_{w}=1,\tau=\frac{K_{2}t}{S_{2}r_{w}^{2}},\alpha=\frac{K_{2 }}{K_{1}},\beta=\sqrt{\frac{S_{1}K_{2}}{S_{2}K_{1}}},\gamma=\frac{r_{c}^{2}}{2S _{2}Br_{w}^{2}} \tag{11}\]  
Then, Laplace transform is used to solve the set of Equations (7)-(10) in turn, as follows:  
\[\frac{\partial^{2}\overline{h_{1D}}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{ \partial\overline{h_{1D}}}{\partial\rho}-\beta^{2}p\overline{h_{1D}}=0,\;1 \leq\rho\leq\rho_{s} \tag{12}\]  
\[\frac{\partial^{2}\overline{h_{2D}}}{\partial\rho^{2}}+\frac{1}{\rho}\frac{ \partial\overline{h_{2D}}}{\partial\rho}-p\overline{h_{2D}}=0,\;\rho_{s}\leq \rho\leq\infty \tag{13}\]  
\[\frac{\partial\overline{h_{1D}}}{\partial\rho}(\rho_{s},p)=\alpha\frac{ \partial\overline{h_{2D}}}{\partial\rho}(\rho_{s},p),\;p>0 \tag{14}\]  
\[\frac{\partial^{2}h_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{1}}{ \partial r}=\frac{S_{1}}{T_{1}}\frac{\partial h_{1}}{\partial t}r_{w}\leq r \leq r_{s}\]0  
According to the general solution of the modified Bessel equation of the order 0, the solutions of Equations (12) and (13) are obtained as follows:  
\[\frac{\partial^{2}h_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{1}}{ \partial r}=\frac{S_{1}}{T_{1}}\frac{\partial h_{1}}{\partial t}r_{w}\leq r \leq r_{s}\]1  
\[\frac{\partial^{2}h_{1}}{\partial r^{2}}+\frac{1}{r}\frac{\partial h_{1}}{ \partial r}=\frac{S_{1}}{T_{1}}\frac{\partial h_{1}}{\partial t}r_{w}\leq r \leq r_{s}\]2  
Among them, \(C_{1}\) and \(C_{2}\), \(C_{3}\) and \(C_{4}\) are constants determined by boundary conditions in the well-skin layer and the aquifer, respectively.