## 2 Theories and Methods  
### Fundamentals of the Kipp Model  
[PERSON] [5] derived differential equations characterizing water level variations in the test hole based on the unsteady flow control equation in the aquifer, the mass conservation equation of water between the test hole and aquifer and the momentum balance equation of water motion in the test hole. The solution of the equation was obtained by Laplace transform and its inverse, and then the hydrogeological parameters were determined by fitting the standard curve.  
The groundwater continuity equation in the aquifer is as follows:  
\[\frac{S}{T}\frac{\partial h}{\partial t}=\frac{1}{r}\frac{\partial}{\partial r }\bigg{(}r\frac{\partial h}{\partial r}\bigg{)} \tag{1}\]According to the flow rate balance, the change in the flow in the well caused by oscillation is equal to the radial flow of the aquifer around the well wall:  
\[\pi r_{c}^{2}\frac{dw}{dt}=2\pi r_{w}T\frac{\partial h}{\partial\tau}\Big{|}_{ \tau=\tau_{w}} \tag{2}\]  
The differential equation of the oscillation of water flow in the test hole based on the mass conservation and momentum balance is as follows:  
\[\frac{d^{2}w}{dt^{2}}+\left(\frac{g}{L_{e}}\right)w=\frac{g(h_{w}-h_{0})}{L_{e}} \tag{3}\]  
where \(L_{e}\) is the effective water column length, \(L_{e}=L+\left(\frac{r_{e}^{2}}{r_{e}}\right)\left(\frac{g}{2}\right)\);  
With initial conditions: when \(t=0\), \(h=h_{0}\), \(w=w_{0}\), \(\frac{dw}{dt}=w_{0}^{\prime}\), \(h_{w}=L=h_{0}\);  
With boundary conditions: when \(r\rightarrow\infty\), \(h\to h_{0}\).  
These equations can be converted to a dimensionless form by the dimensionless variables and parameters and solved by the application of Laplace transform and inverse transform. Then, a set of \(w_{D}\sim\tau\) standard curves can be obtained, as shown in Figure 1.  
The aquifer permeability coefficient is calculated as follows:  
\[K=\frac{r_{c}^{2}\tau\ln\beta}{16\zeta tB} \tag{4}\]  
The \(w^{\prime}-t\) measured curve can be plotted according to the slug test data. Keep the vertical coordinate axis of the measured curve and the standard curve aligned at the same horizontal height. Then, the measured curve is shifted along the horizontal axis to make the measured curve fit the standard curve, and the corresponding \(\zeta\) and \(\beta\) values of the fitted standard curve are recorded. At the same time, select any matching point and record the corresponding coordinate values (\([\tau]\) and \([t]\)) of the standard curve and the measured curve. Thus, the aquifer permeability coefficient can be obtained by Equation (4).  
Among them, \(h\) is the aquifer water level variation (m); \(t\) is the time (s); \(\tau\) is the dimensionless time; \(S\) is the storage coefficient; \(T\) is the hydraulic conductivity coefficient (m\({}^{2}\)/s); \(K\) is the permeability coefficient (m/s); \(r_{c}\) is the radius of the well casing (m); \(r_{w}\) is the radius of the well screen (m); \(w\) is the well water level variation (m); \(w_{0}\) is the initial well water level variation (m); \(L\) is the static water column length above the top of the  
Figure 1: Slug test standard curves of the Kipp model.  
aquifer \((\mathrm{m})\); \(B\) is the aquifer thickness \((\mathrm{m})\); \(h_{w}\) is the water head of the screen \((\mathrm{m})\); \(\zeta\) is the dimensionless damping parameter; \(\beta\) is the dimensionless inertial parameter, obtained by iterative calculation; \(w_{D}\) is the dimensionless water level variation of the standard curve; \(w\prime\) is the dimensionless water level variation of the measured curve.