# Response of Groundwater Levels in a Coastal Aquifer to Tidal Waves and Rainfall Recharge

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

_{z}is the average water depth [L], h is the water depth induced by waves and rainfall [L], x

_{r}is the cover range of rainfall recharge [L], and r is the recharge rate [L/T].

#### 2.1. Governing Equation and Boundary Conditions

_{0}is the horizontal range induced by tides with the following expression:

#### 2.2. Non-Dimensionalization

_{r}might be equal to, larger than, or less than the decay length $\ell $, i.e. x

_{r}= $\ell $, x

_{r}> $\ell $, or x

_{r}< $\ell $. These three scenarios will be discussed in the Results and Discussion section.

_{1}= X − X

_{0}(T) for the moving boundary, the governing equation, Equation (4), subsequently becomes

_{W}and Hr, i.e., one solution for tidal waves effect and the other for rainfall recharge effect, and then both solutions are incorporated by the linear superposition principle.

#### 2.3. Solution for Tidal Waves Effect

_{W}can be easily obtained by separating the solutions into a transient solution and a steady state solution (see Hsieh et al., [11]), and finally we can obtain

#### 2.4. Solution for Rainfall Recharge Effect

## 3. Results and Discussion

_{z}= 200 cm, a = 65 cm, ω = 4π were first employed, and the effects of tidal waves and uniform rainfall recharge are illustrated in Figure 2. From the figure, the present solutions agree very well with the ones of previous research, and thus, the present results are satisfactory, which validates the present analytical solutions.

_{d}= 10 h. It should be noted that the total accumulative rainfall recharge is different, and the former is larger than the latter. The peak value of groundwater levels in Figure 3a occurs at the location (X,T) = (0.7, 3.15), whereas the first peak in Figure 3b occurs at (X,T) = (0.7, 2.1) and the second one at (X,T) = (0.75, 4.7). Although the two peak recharge rates of the double peak case are the same, the maximum water level occurs at the second peak rather than the first one because of the piling-up effect for the double peak case. Moreover, Figure 3 successfully demonstrates that the present solution can simulate the spatiotemporal variation of groundwater levels in a sloping unconfined aquifer under any randomly distributed rainfall recharge, and this was not done in the work of Hsieh et al. [11].

_{r}) and the decay length ($\ell $) on groundwater levels, and considered the representative scenario, x

_{r}= $\ell $, of groundwater level fluctuations under the common effect of tidal waves and rainfall recharge, this study investigated the effect again for x

_{r}> $\ell $, x

_{r}= $\ell $ and x

_{r}< $\ell $, as shown in Figure 5; Figure 6 for different materials (fine sand and sandy clay) of the coastal aquifer, in which θ = 0° and β = 90°, r = 25 mm/h. From Figure 5, the simulated peaks of the cases of x

_{r}> $\ell $ and x

_{r}= $\ell $ are very close to each other, which complies with the discussion in Hsieh et al. [11]. After some more distinct cases of different materials for the aquifer were tested, the same findings were obtained for most cases as were obtained in Hsieh et al. [11]. However, Figure 6 shows the exception that larger fluctuations of groundwater levels might occur for the case of x

_{r}> $\ell $, which indicates the dependence of the alluvial deposits on the texture of the stratum.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Verification of the present solutions with previous results for groundwater levels under (

**a**) rainfall recharge effect; t

_{d}= 5 h, r = 15 mm/h. and (

**b**) tidal waves effect; X = 1.

**Figure 3.**Variation of groundwater levels under different rainfall recharge effects (T > T

_{d}); (

**a**) mid-peak rainfall recharge; (

**b**) double peak rainfall recharge rate.

**Figure 5.**Variation of groundwater levels for different ranges of rainfall recharge (fine sand, S = 0.21, K = 0.0023 cm/s); (

**a**) x

_{r}> $\ell $; (

**b**) x

_{r}= $\ell $; (

**c**) x

_{r}< $\ell $.

**Figure 6.**Variation of groundwater levels for different ranges of rainfall recharge (sandy clay, S = 0.07, K = 0.000217 cm/s); (

**a**) x

_{r}> $\ell $; (

**b**) x

_{r}= $\ell $; (

**c**) x

_{r}< $\ell $.

**Figure 7.**Application of the present solution under randomly temporally distributed rainfall recharge and comparison of groundwater levels between the present solution and field data.

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**MDPI and ACS Style**

Hsieh, P.-C.; Huang, J.-L.; Wu, M.-C. Response of Groundwater Levels in a Coastal Aquifer to Tidal Waves and Rainfall Recharge. *Water* **2020**, *12*, 625.
https://doi.org/10.3390/w12030625

**AMA Style**

Hsieh P-C, Huang J-L, Wu M-C. Response of Groundwater Levels in a Coastal Aquifer to Tidal Waves and Rainfall Recharge. *Water*. 2020; 12(3):625.
https://doi.org/10.3390/w12030625

**Chicago/Turabian Style**

Hsieh, Ping-Cheng, Jing-Lun Huang, and Ming-Chang Wu. 2020. "Response of Groundwater Levels in a Coastal Aquifer to Tidal Waves and Rainfall Recharge" *Water* 12, no. 3: 625.
https://doi.org/10.3390/w12030625