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Effects of aquifer geometry on seawater intrusion in annulus segment island aquifers

Seawater intrusion in island aquifers was considered analytically, specifically for annulus segment aquifers (ASAs), i.e., aquifers that (in plan) have the shape of an annulus segment. Based on the Ghijben–Herzberg and hillslope-storage Boussinesq equations, analytical solutions were derived for steady-state seawater intrusion in ASAs, with a focus on the freshwater–seawater interface and its corresponding watertable elevation. Predictions of the analytical solutions compared well with experimental data, and so they were employed to investigate the effects of aquifer geometry on seawater intrusion in island aquifers. Three different ASA geometries were compared: convergent (smaller side is facing the lagoon, larger side is the internal no-flow boundary and flow converges towards the lagoon), rectangular and divergent (smaller side is the internal no-flow boundary, larger side is facing the sea and flow diverges towards the sea). Depending on the aquifer geometry, seawater intrusion was found to vary greatly, such that the assumption of a rectangular aquifer to model an ASA can lead to poor estimates of seawater intrusion. Other factors being equal, compared with rectangular aquifers, seawater intrusion is more extensive, and watertable elevation is lower in divergent aquifers, with the opposite tendency in convergent aquifers. Sensitivity analysis further indicated that the effects of aquifer geometry on seawater intrusion and watertable elevation vary with aquifer width and distance from the circle center to the inner arc (the lagoon boundary for convergent aquifers or the internal no-flow boundary for divergent aquifers). A larger aquifer width and distance from the circle center to the inner arc weaken the effects of aquifer geometry, and hence differences in predictions for the three geometries become less pronounced.

(2) 164 According to Darcy's law and the Dupuit-Forchheimer approximation, the freshwater flux 165 in the aquifer segment between the seaward boundary and interface tip can be calculated as, 166 where s K [LT -1 ] is the saturated hydraulic conductivity. 168

Interface Tip above the Aquifer Bed 169
We first consider the situation where the interface tip is above the aquifer bed ( Figure 2b).
According to the Ghijben-Herzberg equation, the vertical thickness of the freshwater zone 174 (h) in the interface zone is given by, 175 Rearranging equation (6) produces, 181 Integrating equation (7) where 1 C is the integration constant and can be determined by the sea boundary condition (i.e., 185 Similarly, equation (14) can be adopted to calculate the freshwater-seawater interface Equation (24) can be easily solved by a root-finding method. 237 The freshwater discharge for the aquifer segment between the interface tip and the no-flow 238 boundary can be calculated as, 239 Repeating the steps from equations (4) to (8) gives, 241 where 3 C can be determined by substituting equation (21)    This is because, as L0 increases, the island shape approaches to be rectangular and hence leading 286 to the flow parallel with EH and FG. By comparison, at a given L0 smaller than 10 5 m, the no-287 flow boundary location deviates more from the middle of the ASA with increasing L. 288 Since the no-flow boundary location between the sea and lagoon deviates from the middle 289 of the ASA, we expect aquifer geometry to play a significant role in controlling seawater 290 intrusion. As mentioned previously, ASAs can be convergent (Unit 1) or divergent aquifers 291 (Unit 2) where the extent of seawater intrusion may be different. However, for strip aquifers, 292 both Units 1 and 2 are rectangular with the same extent of seawater intrusion. Therefore, three 293 geometries were compared in this study: convergent, rectangular and divergent ( Figure 5).  Figure 6 shows the freshwater-seawater interface calculated for Cases 1 and 2. As can be 310 seen, the extent of seawater intrusion is greatly different for the three different aquifers. For the 311 high recharge, the interface tip is located at around 500 m for the divergent aquifer, about twice 312 the value of the rectangular aquifer and six times the value for the convergent aquifer ( Figure  313 6a). When the recharge decreases to 3 × 10 -7 m/s, the interface tip moves more landward for 314 three aquifers as expected, but the difference between results of three aquifers is still great 315 ( Figure 6b). The interface tip rises above the aquifer bed for both rectangular and divergent 316 aquifers, while it remains on the aquifer bed for the convergent aquifer. Regardless of the 317 recharge rate, the most landward freshwater-seawater interface occurs in divergent aquifers and 318 vice versa for convergent aquifers. This underlines that aquifer geometry plays a significant 319 role in controlling seawater intrusion and hence it is necessary to account for the effects of 320 aquifer geometry in analytical solutions of seawater intrusion. 321

Sensitivity Analysis 322
A sensitivity analysis was conducted to investigate at which degree of curvature the 323 deviation of seawater intrusion between three different aquifers becomes significant. Since we 324 focus on the effects of aquifer geometry on seawater intrusion, values of L0 and L * were varied, 325 with other parameters kept constant. When conducting the sensitivity analysis of L0, L * was 326 fixed at 1000 m that is a typical value for ASAs (Werner et al., 2017). Figure 7 shows the 327 sensitivity of the freshwater-seawater interface and watertable elevation to changes in L0 (Case 328 3, Table 1). As expected, the freshwater-seawater interface and watertable elevation are 329 independent of L0 for rectangular aquifers. However, the freshwater-seawater interface and 330 watertable elevation differ greatly when varying L0 for both convergent and divergent aquifers, 331 highlighting that L0 plays an important role in affecting seawater intrusion. Specifically, as L0 332 increases, the freshwater-seawater interface moves more landward (Figure 7a and 500 m (191% of that in the rectangular aquifer) for the convergent and divergent aquifers, 345 respectively. As L0 increase, the deviation between three aquifers decreases. When L0 = 2000 346 m, the interface toe location is 262, 209 (80% of that in the rectangular aquifer) and 318 m (121% 347 of that in the rectangular aquifer), respectively. As L0 continues to increase to 6000 m, the 348 freshwater-seawater interface and watertable elevation of both convergent and divergent 349 aquifers tend to those of rectangular aquifers, i.e., geometry effects decrease with increasing L0. 350 These highlight the critical role played by the shape of aquifers. As a result, ignoring geometry 351 effects may lead to an inappropriate management strategy for fresh groundwater resource in 352 atoll islands. 353 The sensitivity of the freshwater-seawater interface and watertable elevation to L * was 354 further conducted with varying L * from 600 to 1600 m while fixing L0 at 200 m (Case 4, Table  355 1). As shown in Figure 8, in contrary to L0, the freshwater-seawater interface and watertable 356 elevation in all three topographies is related to L * . Again, seawater intrusion is greatest in 357 divergent aquifers and least in convergent aquifers for given L * . When L * increases, the 358 freshwater-seawater interface moves more seaward and the watertable elevation increases, 359 regardless of aquifer geometry, i.e., the seawater intrusion decreases (Figure 8a-c). This is 360 because the total freshwater flux increases with increasing L * , leading to a higher hydraulic 361 gradient and hence less seawater intrusion (Figures 8d-f). Moreover, an increase in L * induces 362 a smaller deviation of seawater intrusion between three geometries, i.e., geometry effects on 363 seawater intrusion are more significant at small L * . However, even at the maximum L * given in 364 this study (1600 m), the deviation between three aquifers is significant: the interface toe location 365 is about 148 m for the rectangular aquifer, whereas it is about 32 (22% of that in the rectangular 366 aquifer) and 278 m (188% of that in the rectangular aquifer) for the convergent and divergent 367 aquifers, respectively. Both L0 and L * can greatly impact seawater intrusion for divergent and 368 convergent aquifers, highlighting the necessity to include geometry effects into analytical 369 solutions of seawater intrusion. 370

Limitations of This Study and Future Work 371
The foregoing results show that aquifer geometry affects the freshwater-seawater interface 372 and watertable elevation, and is directly related to the aquifer width, the distance from the circle 373 center to the no-flow boundary, and aquifer shape, i.e., whether the aquifer is convergent or   Note: -means that the parameter is varied. 648