## 4 Numerical Examples
### Numerical Example 5  
Previous examples illustrate that our method is appropriately validated using the harmonic boundary conditions. Since the tidal variation is probably the major trigger inducing groundwater level fluctuation in coastal aquifers, the tidal fluctuation data are used to study tide-induced groundwater flow problems. This example is to evaluate the tidal and piezometer data at the Xing-Da port, Kaohsiung, Taiwan. The Xing-Da port is located in the Kaohsiung, southwestern Taiwan, as shown in Figure 15. The tidal data was from Yongan tidal station at \(22^{\circ}49^{\prime}08.00^{\prime\prime}\) N latitude and \(120^{\circ}11^{\prime}51.00^{\prime\prime}\) E longitude. The piezometer data was from Xing-Da groundwater observation well at \(22^{\circ}51^{\prime}30.52^{\prime\prime}\) N latitude and \(120^{\circ}12^{\prime}04.77^{\prime\prime}\) E longitude [38].  
The initial head is described as Equation (24). This study analyzed the tidal fluctuation data recorded at Xing-Da port from 00:00 p.m. on 22 December to 06:00 a.m. on 23 December 2018, by the Ministry of Economic Affairs Water Resources Department, as shown in Figure 16. It is found that the highest hourly tidal fluctuation reached 0.85 m, whereas the lowest hourly tidal fluctuation may reach \(-0.29\) m. Hence, the left tidal boundary condition is described as  
\[h(x=0,t)=T(x), \tag{38}\]  
where \(T(x)\) denotes the measured data by the Water Resources Agency, Ministry of Economic Affairs [38], as shown in Figure 16. The boundary head on the inland is described as  
\[h(x=L_{f},t)=0. \tag{39}\]  
According to the Engineering Geological Investigation Data Bank by the Central Geology Survey of the Ministry of Economic Affairs [38], the property of the confined aquifer and semi-permeable layers are sand and silt, respectively. Figure 17 shows the schematic of the numerical example 5. The thickness of the confined aquifer and semi-permeable layer are 65 m and 15 m, as shown in Figure 17. The parameters used are listed in Table 3. From Table 3, the permeability of the confined aquifer and semi-permeable layer are \(2\times 10^{-3}\) cm/s and \(10^{-5}\) cm/s. The transmissivity is 4.68 m\({}^{2}\)/h. The storage coefficient is \(10^{-4}\). The leakage coefficient is \(6\times 10^{-4}\)/d. The dimension of the length is 3000 m. The final elapsed time is 30 hr. The amplitude of the tidal change is 1 m. The tidal period is 24 h.  
Figure 15: The location of Xing–Da port in southwestern Taiwan.  
We consider 663 boundary points and a source point. The basis function order for the analysis is 15. To obtain the tide-induced groundwater fluctuation, we collocate 18,631 inner points. Figure 18 demonstrates the comparison of measured data with the computed groundwater fluctuation using our method at \(x=0\) m. Red line denotes the computed numerical results and the gray square denotes the measured data. It is found that the computed results of the tide-induced groundwater head using the proposed method may agree with the measured data. Although there may be a shift in time in Figure 18, the overall behavior of the tide-induced groundwater head may be captured using our proposed method.  
Figure 16: The tidal fluctuation and the measured groundwater response.  
Figure 17: Schematic of numerical example 5.  
\begin{table}
\begin{tabular}{l l c c} \hline \hline
**Types of Aquifer** & **Parameter** & \multicolumn{2}{c}{**Value**} \\ \hline \multirow{3}{*}{\
\begin{tabular}{l} confined aquifer \\ (sand) \\ \end{tabular} } & thickness & 65 & m \\  & permeability & \(2\times 10^{-3}\) & cm/s \\  & transmissivity & 4.68 & m\({}^{2}\)/hr \\  & storage coefficient & \(10^{-4}\) & \\ \hline \multirow{3}{*}{
\begin{tabular}{l} semi-permeable layer \\ (silt) \\ \end{tabular} } & thickness & 15 & m \\  & permeability & \(10^{-5}\) & cm/s \\ \cline{1-1}  & leakage coefficient & \(6\times 10^{-4}\) & 1/d \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters used in the numerical example 5.