4.1. Pumping Wells Layout Scheme Design of the Freshwater Lens
“Sustainable exploitation” is an extension of “allowable exploitation”, based on the concept of sustainable development (UNCED, 1992) [
32]. In 1999, Alley et al. [
33] defined sustainable groundwater development as “long-term and permanent development and utilization of groundwater without serious social, economic and environmental consequences”. The concept of sustainable exploitation highlights environmental factors and emphasizes the renewable and sustainable utilization of groundwater resources. This study proposes that the essence of sustainable exploitation for the groundwater system of coral islands is that after a change in source and sink terms due to human intervention, the freshwater lens system changes from the original dynamic equilibrium to a new dynamic equilibrium, and the latter is acceptable. Because the variation in the groundwater table is at centimeter-level and the variation in the concentration isoline is at meter level in many scenarios of this simulation, this study regards the saltwater up-coning height as the main limiting factor of sustainable exploitation (
Figure 3).
A pumping well extracting from the freshwater lens can cause the saltwater to move upwards towards the well, a phenomenon known as saltwater up-coning [
6,
34]. A schematic diagram of the saltwater up-coning process is shown in
Figure 4. In this study, when the groundwater numerical model reaches the steady-state, the corresponding pumping rate when the rising height just reaches the bottom of the well screen is defined as the critical pumping rate (
QC). The sum of
QC of all pumping wells can then be expressed by the total critical pumping rate (
QT), and the sustainable yield cannot exceed
QT.
Tm is the maximum thickness of the freshwater lens under pumping conditions, and
TC is the central thickness (
Figure 4). The coordinate corresponding to the
Tm changes with pumping conditions, but the position corresponding to the
TC will not change. Due to pumping, the
TC doesn’t coincide with the
Tm. In this study, the length of the well screens, the number of wells, and the distance between wells are selected as the factors of the design scheme of pumping wells, and each factor has three levels (
Table 5). One of the typical well layout schemes is shown in
Figure 4.
In this study, we use a chloride concentration of 250 mg/L as the upper threshold for fresh groundwater, which is the World Health Organization (WHO) [
35] recommendation for drinking water in 2011. However, other researchers have also used 500 mg/L and 600 mg/L as the upper threshold of chloride concentration [
14,
36,
37]. When the model runs stably without pumping, the freshwater flow is stressed by saltwater when the boundary is moved on both sides, whereas saltwater moves upward driven by buoyant effects in the middle of the model [
38]. Three variables are selected as evaluating indicators in the numerical test: total critical pumping rate (
QT), the central thickness of the freshwater lens (
TC), and maximum thickness of the freshwater lens (
Tm) (
Figure 4). The orthogonal experimental design scheme and its corresponding numerical test results are shown in
Figure 5 and
Table 6.
The numerical experiment results show that different pumping well layout schemes have a great impact on
QT. The
QT of test 3 is 4.2 times that of test 8 (
Table 6). Of course,
QT is not the only factor to judge whether the scheme is the best or not. It is also necessary to comprehensively analyze the thickness of the freshwater lens under the pumping scheme. If the thickness of the freshwater lens becomes too shallow, the current pumping scheme is not reasonable, although a large pumping rate may be obtained. In principle, the sustainable yield shall not exceed
QT. The determination of
QT can provide a reference for the decision-making of sustainable yield.
According to the calculation results and difference analysis of orthogonal tests (
Table 7),
RB >
RA >
RC for
QT and
Tm,
RA >
RB >
RC for
TC. The results indicate that the number of wells, and the length of well screens, are the most critical factors in the design of the well layout scheme, followed by the distance between wells. Although the more wells, the greater the
QT, if the number of wells is increased without considering the distance between wells, the
TC will be greatly reduced and the risk of damage to the freshwater lens will increase. Therefore, the number of wells needs to be limited. The longer the well screen and the deeper the buried depth at the bottom of the well screen, the smaller the
QT, so the design of the length of the well screen needs to be cautious. If long-term pumping is implemented under the conditions of short well screens with
QT, it is likely to reduce the
TC to less than 1/3 of the original thickness, which increases the risk of damage to the freshwater lens. Therefore, it is necessary to control the pumping intensity of shallow wells.
In order to reduce
TC and
Tm as little as possible and obtain a larger
QT, the comprehensive balance analysis method is used to select the optimization scheme. Compared with
QT and
Tm, factor A has a more significant influence on
TC. It can be seen from
Table 4. that A2 is the best choice. The influence of A on
QT and
Tm occupies second place among the three factors, with A1 and A3 being the best choices. However, when A1 is selected,
TC and
Tm are very thin, so A1 is not considered; when A2 is selected,
Tm is 2.6% less than that of A3; when A3 is selected, not only is
TC reduced by 4%, but
QT is also reduced by 6% compared with A2, so A2 is considered comprehensively. The influence of factor B on
QT ranks first among A, B, and C. However, when B3 is taken,
TC is less than 1/3 (2.5 m) of the original thickness. When B2 is taken in turn, there is the same problem, so B1 is taken finally, which is also the best level for
TC. According to
Table 4, the influence of factor C on
QT,
TC, and
Tm takes third place, and C1 and C3 can be taken into consideration. When C1 is selected,
TC and
Tm increase by 25.5% and 9.4%, respectively; When C3 is selected,
QT will increase by 39.5% compared with C1, but
TC and
Tm are only reduced by 20.3% and 8.6%, respectively, so C3 is a better choice. As a result, the best scheme is A2B1C3, that is, the linear well layout scheme with a 3 m well screen length, two wells, and 200 m of the distance between wells (Test 9). This distance accounts for about 10% of the island width. Under this scheme,
QT is 0.33 m
3/d.
TC is 4.6 m, which accounts for 60% of the original thickness, and
Tm is 9.8 m, which accounts for 86% of the original thickness.
4.2. Uncertainty Analysis of QT and Main Influence Factors
Due to the uncertainty of hydrogeological parameters,
QT in the above-mentioned pumping wells layout scheme is also uncertain. Hydraulic conductivity (
K), vertical dispersion coefficient (
DI), precipitation infiltration coefficient (
α), effective porosity (
ne), and specific yield (
Sy) are the influencing factors:
Sensitivity analysis is used to analyze the uncertainty of
QT and identify the main risk factors. Sensitivity can measure the impact of the change in influencing factors on the indicators. The global sensitivity analysis can fully account for the effect of parameter interaction on the simulation results. In this study, the method proposed by Morris [
39] in 1991 was adopted. By adjusting only one parameter at a time, the influence of each parameter on the overall results is calculated in turn, and the importance of each parameter is effectively prioritized. The influence of parameter changes on the overall results is measured by sensitivity (
Si), expressed as:
where
Si is the sensitivity of function y(x) to variable
Xi;
y0 is the value of
QT before the parameter changes;
X0 is the initial parameter. Among them, the larger the value of
|Si| is, the more sensitive the variable
Xi is, and the greater the influence on the function is.
When K, DI, α, ne, and Sy change from 0 to 30% respectively, the |Si| of α is the largest, ranging from 0 to 2.020, therefore, changes in α will have a greater impact on QT compared with other four parameters with the same amplitude. |Si| of ne is in the second place, ranging from 0 to 0.566, and the influence of ne is much smaller than that of α. The other three parameters have less influence, so the focus should be on the changes in α and ne. When K, DI, α, ne, and Sy change from –30% to 0, the |Si| of α and K are significantly greater than the other three parameters, ranging from 0 to 1.202 and 0 to 0.949, respectively. Therefore, the focus should be on the changes of α and K on the determination of QT. In conclusion, from −30% to 30%, α, K, and ne are the main influencing factors in the determination of QT.
The results of the global sensitivity analysis of
α–
K,
α–
ne,
K–
ne and
α–
K–
ne show that the comprehensive influence of
α–
ne is the largest, and the corresponding
|Si| ranges from 0 to 2.182. When the parameter amplitude is −30% to 0, the comprehensive influence of
α and
ne on
QT is less than the sum of their respective influence, but when the parameter amplitude is 0 to 30%, the comprehensive influence of
α–
ne on
QT is greater than the sum of their respective influence. The analysis results of local sensitivities of all parameters and joint sensitivities of main influencing parameters are shown in
Table 8 and
Figure 6.
The increase in
α and
n has positive impacts on
QT, but the increase in
K has negative impacts (
Figure 7). The reason for this is that
α affects the amount of the net recharge. A larger
α means that the freshwater lens can obtain a larger hydraulic gradient to maintain or even increase the thickness of freshwater, and has a larger maximum permissible rising height of saltwater, resulting in a larger
QT. With the increase in
K, the thickness of the freshwater lens will become thinner, and the maximum allowable rising height of saltwater will decrease, which requires pumping rates to be relatively reduced to protect the wells from salinization.
Under the most unfavorable parameter combination (α and n decrease by 30%, K increases by 30%), QT is 0.174 m3/d, which is only 52.7% of the original (0.33 m3/d), so the original calculated QT is not absolutelyreliable; under the most favorable parameter combination (α and n increases by 30%, K decreases by 30%), QT is 0.573 m3/d, which is 73.63% higher than the original QT. Therefore, the range of QT is from 0.174 to 0.573 m3/d. The identification of the main influencing factors can help identify the key risk sources that affect the determination of QT, from the standpoint of hydrogeological parameters. The results show that the key risk sources of QT are the uncertainties of α, n, and K. The influence of parameters on the determination of QT is related to the variation range of the parameters, within the range of a 30% reduction in parameters, α is the most important risk factor of pumping, and K is the second; within the range of a 30% increase in parameters, α is also the most important risk factor, followed by n and K. The uncertainty of the hydrogeological parameters has a great impact on QT, but their spatial variabilities haven’t been taken into consideration, which may also influence the results, and is meant for relevant analysis.
The 2D model assumes that recharge is stored in the profile and there is no flow movement perpendicular to the profile. This generalization may lead to a slight overestimation of reserves. However, the overestimation is generally within a reasonable range. Numerical models can be used as a powerful tool to predict and evaluate the water resources of coral islands, but it is difficult to avoid errors in comparison to reality, especially in the conceptualization of hydrogeological conditions of coastal aquifers. The determination of sustainable yield should be combined with the specific well layout schemes, and the uncertainty of hydrogeological parameters should also be taken into account, because of the complex hydrogeological conditions of coral islands.