4.1.1. Case Designs
For flow and transport simulations, technically improper hydraulic parameter values, such as extremely high or low hydraulic conductivity (K), may cause unreasonable results, especially when considering effective drawdown during pumping and the following recovery process (e.g., no effective drawdown for excessively high K values and unrealistically significant drawdown for excessively low K values). Therefore, a trial-and-error method was used to delineate ‘reasonable values’ for the selected aquifer settings in
Figure 1 and set the values as a base case. The values that were delineated from a coastal aquifer in Gyeongju, South Korea were referenced for the ranges of the actual hydraulic parameters [
10]. Other cases for sensitivity analyses were designed by changing the value of a different target parameter while the other parameters were fixed.
The parameters that were used for the base case are listed in
Table 1. In the base case, the hydraulic conductivity was 5.0 × 10
−7 m/s, the anisotropic ratio was 1, the ratio of the infiltration rate to the precipitation rate was 0.1, the porosity was 0.0035, the longitudinal dispersivity was 100 m, the ratio of the longitudinal to transverse dispersivity was 10, and the specific storage was 1.0 × 10
−4 m
−1.
Other cases were designed based on the base case. For each case, only one parameter was changed; all the other parameters were kept the same as those of the base case. The value of each parameter was 2.0 times larger or smaller than that of the base case. For example, the value of the hydraulic conductivity in the base case was 5.0 × 10
−7 m/s, while this value in case 3 was changed to 5.0 × 10
−7/2 = 2.5 × 10
−7 m/s. However, some exceptions occurred in cases 4 and 9 because of the convergence problem. For these cases, a factor of 1.5 or smaller was applied instead of 2.0 because of numerical instability, and the target minimum head that was caused by pumping at the pumping location, H
min, was changed from −20 to −15 m (
Table 1).
4.1.2. Base Case Study
In the base case study, a steady-state model was first set up to derive the steady distribution of the head and TDS in the model domain. The pumping rate for the hypothetical pumping well was determined inversely by setting the target minimum head as a specified head along the pumping well. With this derived pumping rate, the initial conditions of the head and TDS for the cases were derived and the recovery processes of the head and TDS were simulated by running the transient model. The recovery of head was measured by the time that was required for a certain percentage of head recovery, which is defined as the ratio of the head at the desired time to that in the steady state without pumping at the pumping well.
Figure 2 shows the time that was required for different percentages of head recovery for the base case together with those for the other twelve cases. We found that 80% head recovery was achieved within a relatively short period for all thirteen cases (e.g., 148 day for the base case), and 90% head recovery was also achieved quickly compared to the total simulation period of 100 years. The maximum required time for 90% head recovery was 4078 day in case 6.
Figure 2 shows that less than 10 years was required for 90% head recovery in most cases. The complete recovery of head may take longer, but reached 100% before the end of the simulations.
Figure 3 shows the contour lines of 1000 mg/L TDS as the base case when the percentage of head recovery was 0%, 50%, 60%, 70%, 80%, and 90% alongside the end of the simulation. The contour line for the steady state without pumping is also included. The minimum TDS concentration of the TZ is 1000 mg/L [
19], so these 1000-mg/L TDS contours serve as indicators of the location of the interface between freshwater and saltwater. According to
Figure 3, the TDS recovery was completely different from the head recovery. When the head recovery reached 80%, the interface did not apparently change. Even when the head recovery reached 90%, the upper portion of the interface showed a small recession while the lower portion had no obvious movement. Significant recovery of TDS occurs when the percentage of head recovery is larger than 90%. At the end of the simulation, the 1000-mg/L TDS contour was still detached from the line for the steady state without pumping, suggesting that the recovery of TDS had not finished even after 100 years of simulation time. A much longer time may be required to reach the complete recovery of TDS.
The 1000-mg/L TDS contours in
Figure 3 suggest that the recovery of TDS was fast in the upper aquifer, especially near the shore area, and that the recovery process was quite slow in the deep portion of the aquifer. These results occurred because the shallower aquifer near the shore area is the main submarine groundwater-discharge area, where changes in groundwater flow because of pumping are dominant. In the deep portion of the aquifer, the hydraulic gradient is too low to push the interface back to the original position.
4.1.3. Sensitivity Analysis
For the other twelve cases in
Table 1, a steady-state model with corresponding parameter settings was set up to derive the steady distribution of the head and TDS, which acted as the initial conditions of the transient model for each case. The same simulation procedures as in the base case study were applied to these twelve cases. The TZ is a diffuse zone, so quantitatively expressing differences in the TZ for the above cases is difficult. Centroids of the TZ at different stages of head recovery for different cases were computed, and the coordinates of these centroids were rescaled for further analysis. The distances of the rescaled centroids for the base case and the other twelve cases at different stages of head recovery were computed. If the distance was large, the displacement of the TZ centroids from the base case was large, suggesting that the TZ was sensitive to the parameter that was adjusted in the corresponding case.
Figure 4 shows the distances of the rescaled centroids between the base case and the other twelve cases at different stages of head recovery.
The centroid analyses of the TZ revealed that the TZ was most sensitive to the hydraulic conductivity, corresponding to cases 3 and 4 (
Figure 4b), if long-term pumping was conducted in the coastal aquifer with large drawdown, corresponding to 0% head recovery in
Figure 4. The second-most sensitive parameter was recharge, corresponding to cases 9 and 10 (
Figure 4e). The anisotropy ratio (
Figure 4c) played an important role in the movement of the TZ in cases 5 and 6, especially when the vertical hydraulic conductivity was higher than the horizontal conductivity (case 5). The dispersivity (cases 1 and 2 in
Figure 4a) was a relatively less sensitive parameter for determining the position and movement of the TZ. The flow system had reached a steady state at this stage, so the effect of the specific storage in cases 11 and 12 (
Figure 4f) and the porosity in cases 7 and 8 (
Figure 4d) was almost negligible, which was expressed by zero displacement in
Figure 4d,f.
When the hypothetical well stopped pumping, the recovery process of head and the TDS began. When the head recovery was less than 80%, the displacements did not change. Therefore, the sensitivity of the TZ to different parameters was not obvious. Even when the head recovery was 80%, the displacements of the centroids for all twelve cases were not discernable. At this stage, the displacement of the centroids in cases 7 and 8 (
Figure 4d) became nonzero, which suggests that the porosity started to affect the position of the TZ. When the head recovery was larger than or equal to 90%, the displacements of the centroids became large. The hydraulic conductivity, represented in cases 3 and 4 (
Figure 4b), was still the most important factor that controlled the TZ’s movement. The recharge rate, represented in cases 9 and 10 (
Figure 4e), was the second-most important factor, and the anisotropic ratio, represented in cases 5 and 6 (
Figure 4c), was the third-most important. At this stage, the sensitivity of the TZ to the porosity in cases 7 and 8 (
Figure 4d) and specific storage in cases 11 and 12 (
Figure 4f) increased, even becoming larger than that for the dispersivity in cases 1 and 2 (
Figure 4a). When the simulation was run for 100 years, the results were similar to when the head recovery was 90%. However, the porosity seemed to be more sensitive than the specific storage and dispersivity. Finally, when the flow and TDS fields reached the new steady state, the sensitivity rank of the TZ to the parameters did not significantly change, with the exception of the porosity and specific storage, whose contribution to the TZ became zero again.
Figure 4 also suggests that larger dispersivity values were linked to larger displacement when the head was slightly recovered. However, this trend was reversed when the head recovery was 90% and at the end of simulation. This result occurred the flow field approached the normal gravity-driven flow in the domain, especially in areas that were originally occupied by freshwater, when the percentage of head recovery was larger. Larger dispersivity values are linked to quicker TDS recession. When a new balance of freshwater and saltwater was reached, the displacement of the centroids for cases with high dispersivity values became larger, as shown in
Figure 4.
At the beginning of recovery, upward flow near the pumping well was dominant because of the depression cone that was caused by long-term pumping. Because of gravity effects, the TZ had difficulty moving inland and towards the depression cone for case 6, where the vertical hydraulic conductivity was higher than the horizontal conductivity. Therefore, the distance in case 5 was larger than that in case 6 for 0% head recovery. When the recovery process started, the TZ receded in both the downward and seaward directions. However, when the percentage of head recovery was small, the recession of the TDS was not obvious. The distances in case 5 remained larger than those in case 6. As head recovery progressed, the downward movement of the TZ became more important and the displacement increased accordingly, although the absolute value was still smaller than that in case 5. At the end of the simulation, the distance in case 6 was larger than that in case 5. When the percentage of head recovery increased and the process finally completed, the TZ finally moved towards the sea. Therefore, when the flow and TDS fields reached a new balance, the horizontal hydraulic conductivity became more important; the distances of the rescaled centroids for the steady state in case 6 were again smaller than those in case 5.
The porosity and specific storage are two special parameters, because they are not involved in the governing equations for a steady-state model. However,
Figure 4 demonstrates that these two parameters did affect the TZ’s position/movement. When the percentage of head recovery was small, the distances of the rescaled centroids did not change. When the head recovery reached 90%, the distances of the rescaled centroids in cases 7, 8, 11, and 12 (
Figure 4d,f) greatly increased. In cases 7 and 8 (
Figure 4d), smaller porosity (case 7) and higher seepage velocity expedited the recovery process of the TDS, which explains why the distances in case 7 were smaller than those in case 8. The storage in case 11 was much lower than that in case 12 (
Figure 4f), so case 12 took much more time to reach 90% head recovery compared to case 11. Therefore, when the head recovery reached 90% for both cases, the displacement of the TZ in case 12, which had high storage, was reasonably larger than that in case 11. The distances between case 12 and the base case should have been smaller than those between case 11 and the base case. However, when the head recovery was almost completed, the distances between case 12 and the base case at the end of the simulation were larger than those between case 11 and the base case.
The above arguments are based on the displacement of the centroids between the base case and different cases at different stages during head recovery. To understand the recovery of the TDS with time, the distances between different stages of head recovery were computed for the same cases. The accumulated distances from the beginning of recovery to different stages of head recovery are plotted in
Figure 5.
According to
Figure 5, when the head recovery is less than 80%, the TDS recovery was very small compared to the displacement during groundwater extraction. When 80% of the head was recovered, the TDS recovery was still not obvious. The maximum TDS recovery was observed in case 2, where the accumulated distance comprised approximately 10% of the total TDS recovery. Case 2, in which the dispersivity was larger than in the base case, suggests that dispersion was a dominant process in TDS recovery compared to advection when the percentage of head recovery was low. The process of TDS recovery accelerated when the head recovery reached 90%, and the differences in the amount of TDS recovery for different cases were more peculiar. For example, the accumulated distances comprised more than 30% of the total TDS recovery in cases 1, 7, and 12. In cases 4 and 9 and the base case, the accumulated distances were between 20% and 30% of the total TDS recovery. In cases 3, 5, 8, 10, and 11, the accumulated TDS recovery was still less than 10%, which suggests that the TDS recovery was still not obvious. Significant TDS recovery occurred in all cases during the period from 90% head recovery to the end of the simulation. In cases 3, 7, and 10, the TZ mostly recovered to its non-pumping steady-state location. However, in cases 4, 6, 8, and 9, the progress of the TZ’s recovery was still slow and may have taken much more time to reach 100% TDS recovery. According to the above analyses, most TDS recovery is confined to the period when the head recovery is more than 90%, and complete TDS recovery may take much more time after complete head recovery.