Journal of Hydrology 274 (2003) 211–224 www.elsevier.com/locate/jhydrol

Tide-induced seawater–groundwater circulation in a multi-layered coastal leaky aquifer system

Hailong Lia,b,*, Jiu Jimmy Jiaob

aDepartment of Mathematics, Anshan Normal College, Anshan 114005, Liaoning, People’s Republic of China bDepartment of Earth Sciences, The University of Hong Kong, Pokfulam Road, Hong Kong, People’s Republic of China

Received 23 April 2002; revised 5 December 2002; accepted 6 December 2002

Abstract Mean groundwater levels of a multi-layered coastal leaky aquifer system are considered. The system consists of an unconﬁned aquifer, a conﬁned aquifer and a semi-permeable layer between them. Both exact asymptotic solutions and approximate perturbation solutions are derived for multi-sinusoidal-component sea tide. At inland places far from the coastline, the perturbation solutions show a good agreement with the exact asymptotic solutions. Due to the watertable-dependent transmissivity of the unconﬁned aquifer, the mean groundwater levels of the aquifer system stand considerably above the mean sea level even in the absence of net inland recharge of groundwater and rainfall. These lead to landward positive gradients of both the mean watertable and mean head in the region near the coastline, which consequently results in a seawater–groundwater cycle. Seawater is pumped into the unconﬁned aquifer by the sea tide and divided into two parts. One part returns to the sea driven by the mean watertable gradient. The rest part leaks into the conﬁned aquifer through the semipermeable layer, and returns to the sea through the conﬁned aquifer driven by the mean head gradient. The total discharge through the conﬁned aquifer is signiﬁcant for coastal leaky aquifer system with typical parameter values. This seawater–groundwater cycle has impacts on better understanding of submarine groundwater discharge and exchange of various chemicals such as nutrients and contaminants in coastal areas. If the observed mean water levels in coastal areas are used for estimating the net inland recharge, the enhancing processes of sea tide on the mean groundwater levels should be taken into account. Otherwise, the net inland recharge will be overestimated. q 2003 Elsevier Science B.V. All rights reserved.

Keywords: Coastal leaky aquifer system; Sea tide; Analytical solution; Seawater–groundwater circulation; Submarine groundwater discharge

1. Introduction mation of the submarine groundwater discharge (SGWD) and net inland recharge in coastal areas are Interaction between the seawater and ground- of great importance for the correct assessment of the water in coastal areas is one of the most important role of groundwater in the global water cycle study topics of hydrologists. For example, esti- (Moore, 1996; Li et al., 1999; Church, 1996). The inﬂuences of sea tide on the mean groundwater levels are one of the aspects immediately related to * Corresponding author. Address: Dept. of Mathematics, Anshan Normal University, Anshan 114005, Liaoning. People’s Republic of the SGWD estimation. China. Tel.: þ86-412-5841417; fax: þ86-412-2960111. For a single coastal unconﬁned aquifer, the E-mail address: [email protected] (H. Li). inﬂuences of the sea tide on the mean watertable

0022-1694/03/$ - see front matter q 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S002-1694(02)00413-4 212 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 have been studied by many researchers (e.g. Philip, above the mean sea level, can the unconﬁned aquifer 1973; Smiles and Stokes, 1976; Knight, 1981; Parlange maintain correct water balance (Fig. 1b). et al., 1984; Nielsen, 1990). Philip (1973) derived an Philip’s theoretical prediction was examined and exact asymptotic constant solution to the steady conﬁrmed by a Hele–Shaw experiment conducted by periodic nonlinear diffusion problem. As an appli- Smiles and Stokes (1976). Their experiment curves cation of his analytical solution, he considered the were consistent with Philip’s contention that the root mean watertable in a coastal unconﬁned aquifer mean square should be effectively constant and about bounded by an impermeable bottom and a straight 23% greater than the mean of the reservoir oscillation coastline with vertical beach connected to a sinusoidal for the experiments described there. Parlange et al. tidal water body. Assume that the datum is the mean (1984) used second-order theory to describe the sea level, then, in the absence of net inland recharge of propagation of steady periodic motion of liquid in groundwater and rainfall, the mean water table WðxÞ in porous medium. Their two laboratory experiments, the unconﬁned coastal aquifer satisﬁes (Philip, 1973) together with analytical and numerical analysis also 0sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 support Philip’s (1973) prediction. Nielsen (1990) 1 A2 developed an approximate analytical solution based lim WðxÞ¼D@ 1 þ 2 1A; ð1Þ x!1 2 D2 on a perturbation method to investigate the mean watertable in the inland region near the coastline. where A is the tidal amplitude [L], D is the unconﬁned Under the same conditions used by Philip (1973), the aquifer’s depth [L] below the mean sea level, and x mean watertable in the unconﬁned coastal aquifer can is the landward distance from the coastline. Based be approximated by (see Eq. (25) of Nielsen (1990)) on Eq. (1), Philip (1973) concluded that when 2 2 A ð = Þ ¼ 22aUx ; A D 1 the inland groundwater level lies above WNielsenðxÞ¼ ð1 2 e Þ ð2aÞ the mean sea level by about 23% of the tidal amplitude. 4D

Philip’s (1973) result was derived from Boussinesq’s where aU is the wave number, and x is the landward equation, which is based on the Dupuit–Forchheimer distance from the coastline. The wave number is given (D–F) assumptions that assume groundwater ﬂow to by be essentially horizontal. Knight (1981) considered a qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ = free-surface problem that takes the vertical ﬂow into aU ¼ vSy ð2KUDÞ ð2bÞ account strictly. He proved theoretically that Philip’s 21 result exactly holds independent of the validity of the in which v is the tidal angular velocity [T ], Sy and D–F assumptions. KU are the unconﬁned aquifer’s speciﬁc yield 21 For steady periodic state when there is neither [dimensionless] and hydraulic conductivity [LT ], seawater intrusion nor net inland recharge, to maintain respectively. The sinusoidal sea tide is speciﬁed as ; water balance, water entering a coastal unconﬁned HseaðtÞ¼A cosðvt þ cÞ where t is the time [T] and c aquifer at high tide should be exactly equal to that is the tidal phase shift [Radian]. Eq. (2a) shows that at leaving the aquifer at low tide. The transmissivity inland places where the dimensionless landward q ; of the unconﬁned aquifer is watertable-dependent. If distance aUx 1 the mean watertable will be higher 2= : the mean watertable equalled the mean sea level, the than the mean sea level by A ð4DÞ Using Taylor’s transmissivity of the unconﬁned aquifer at high tide expansion would be greater than that at lower tide (Fig. 1a). pﬃﬃﬃﬃﬃﬃﬃﬃﬃ a a2 Hence, seawater entering the aquifer at high tide 1 þ a=2 ¼ 1 þ þ ; 0 # j # a; ð3Þ 4 32ð1 þ j=2Þ3=2 would be more than that leaving the aquifer at low tide, which would result in water imbalance. If the it follows that Eq. (2a) is an approximation to the mean watertable is higher than the mean sea level, a exact asymptotic constant (1) up to the ﬁrst order greater hydraulic gradient will be generated at low of a ¼ðA=DÞ2: When a # 1; the truncation error is less tide, which will compensate for the low transmissivity than 3.125%. and increase the amount of water recharging to the sea. Nielsen (1990) and Li et al. (2000) also showed Consequently, only when the mean watertable stays that a slope water–land boundary will lead to a much H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 213

Fig. 1. Explanation to the sea tide-induced mean watertable higher than the mean sea level: (a) Hypothetical situation (mean watertable ¼ mean sea level) and (b) Real situation (mean watertable . mean sea level). higher mean water table than a vertical one does. for the whole sea tide. Therefore, generalization of Therefore, the inﬂuence of tide on the mean water their work is necessary by taking into account more table should be considered when SGWD from a than one components of the sea tide in the tidal coastal unconﬁned aquifer is estimated by the mean boundary condition. Although Li et al. (2000) hydraulic gradient determined by comparing the mean considered two tidal constituents, their model seawater and groundwater levels. focused on the effects of the beach slope on tidal The main limitation of the previous work of ﬂow in the unconﬁned aquifer modelled by a Phillip (1973), Smiles and Stokes (1976), Knight linearized Boussinesq equation. Moreover, in many (1981), and Parlange et al. (1984) is that they assume coastal areas, what abuts the sea is usually a multi- the sea tide has only one sinusoidal component. In layered system (e.g. Serfes, 1991; Sheahan, 1977; reality, the sea tide consists of tens of sinusoidal Chen and Jiao, 1999; Jiao and Tang, 1999). It is components that include the effects of the sun, moon interesting to approach the inﬂuences of the sea tide and earth, etc. (e.g. Melchior, 1978; Pugh, 1987). on the mean water levels in such a case. Based Due to nonlinearities of the model equations on such motivations, this paper investigates the describing the unconﬁned aquifer, the solution to a tide-induced mean water levels of a coastal multi- single sinusoidal component cannot be used, in layered groundwater system consisting of a conﬁned general, to superimpose to ﬁnd the complete solution aquifer, an unconﬁned aquifer, and a semi-permeable 214 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 layer between them. A nonlinear mathematical and the boundary condition model is built to describe the relationship between XN the sea tides, water table in the unconﬁned aquifer l ; W x¼0 ¼ WTideðtÞ¼ Aj cosðvjt þ cjÞ ð4bÞ and water head in the conﬁned aquifer. Different j¼1 sinusoidal components of the sea tide are included in ›W the tidal boundary condition. Exact asymptotic lim ¼ 0; ð4cÞ !1 solutions are derived to obtain the information on x ›x 0 ; ; the mean groundwater levels of the multi-layered where K and HSðx z tÞ are the vertical hydraulic aquifer system at inland places far from the coastline. conductivity [LT21] and water head [L] of the semi- Approximate perturbation solutions are also derived permeable layer, respectively; WTideðtÞ is the water to investigate the behaviour of the mean water levels level [L] of the sea tide; N is the number of the of the system in the vicinity of the coastline. The sinusoidal components of the sea tide; Aj; vj and cj solutions are analyzed and discussed. ðj ¼ 1;…;NÞ are the amplitude [L], angular velocity [T21] and phase shift of the jth sinusoidal component, ; ; 2. Mathematical model and deﬁnitions of mean respectively. The angular velocities v1 … vN are not water levels equal to each other. The datum of the system is set to be the mean sea level. Eq. (4b) describes the tidal ð ; Þ 2.1. Mathematical model boundary condition of W x t at the water–land interface along the coastline, while Eq. (4c) gives a Consider a subsurface system consisting of an no-ﬂow boundary condition as x approaches inﬁnity, unconﬁned aquifer, a conﬁned aquifer and a semi- which means that there is no inland recharge far from permeable layer between them (Fig. 1). The system the coastline. satisﬁes the following assumptions: (a) Each layer is Based on the assumption (b), the groundwater head ð ; ; Þ horizontal and homogeneous, and has a vertical HS x z t in the semi-permeable layer satisﬁes boundary with the seawater. The conﬁned aquifer has ›H ›2H b0 b0 S0 S ¼K0 S ; 21,t,1; 2 ,z, ; ð5aÞ an impermeable bottom. (b) The horizontal ﬂow in the S ›t ›z2 2 2 semi-permeable layer and the vertical ﬂow in ; ; l ; ; the conﬁned aquifer are negligible. (c) The ﬂow in the HSðx z tÞ z¼b0=2 ¼Wðx tÞ ð5bÞ ; ; l ; unconﬁned aquifer is horizontal, i.e. the Dupuit– HSðx z tÞ z¼2b0=2 ¼HCðx tÞ; ð5cÞ Forchheimer assumptions (Bear, 1972) apply. (d) The 0 21 unconﬁned aquifer’s depth D below the mean sea level where SS is the speciﬁc storage [L ] of the semi- ; is great enough so that the seawater remains connected permeable layer, and HCðx tÞ is the groundwater head with the unconﬁned aquifer at low tide. of the conﬁned aquifer. Eqs. (5b) and (5c) guarantee Cartesian coordinates x, z will be used, with z taken the water head continuity at the upper and lower to be positive vertically upwards. The two interfaces boundaries of the semi-permeable layer. between the semi-permeable layer and the two Based on the assumptions (a) and (b), the ; ; aquifers are located in the planes z ¼ ^b0=2; where groundwater head HCðx z tÞ in the conﬁned aquifer b0 is the thickness [L] of the semi-permeable layer (see satisﬁes Fig. 2). The water–land vertical boundary is located 2 ›HC › HC 0 ›HS l ; at x ¼ 0. Based on assumption (c) and (d), the water S ¼ T þ K z¼2b0=2 ›t ›x2 ›z ð Þ table Wðx; tÞ in the unconﬁned aquifer satisﬁes the 6a following one-dimensional Boussinesq equation 21 , t , 1; x . 0; (Bear, 1972) XN H l ¼ W ðtÞ¼ A cosðv t þ c Þ; ð6bÞ ›W › ›W 0 ›HS ; C x¼0 Tide j j j Sy ¼ KU ðW þ DÞ 2 K 0 j¼1 ›t ›x ›x ›z ¼ b z 2 ð4aÞ ›H lim C ¼ 0 ð6cÞ 21 , t , 1; x . 0; x!1 ›x H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 215

Fig. 2. Schematic representation of a leaky conﬁned aquifer system near open tidal water. where S and T are the storativity [dimensionless] and each j ¼ 1; …; N; it follows that the transmissivity [L2T21] of the conﬁned aquifer, XN respectively. Eq. (6b) describes the tidal boundary W ðt þ PÞ¼ A cosðv t þ v P þ c Þ condition along the coastline, and Eq. (6c) gives the Tide j j j j j¼1 ! no-ﬂow boundary condition as x approaches inﬁnity, XN nj which means that there is no inland recharge far from ¼ Aj cos vjt þ 2p P þ cj the coastline. j¼1 mj Let Pj ¼ mj=nj ðj ¼ 1; …; NÞ be the period [T] of XN : the jth sinusoidal component of the sea tide, where mj ¼ Aj cosðvjt þ cjÞ¼WTideðtÞ ð7dÞ j¼1 and nj are two positive integers prime to each other 21 and Pj is measured in hours. Then, with unit of h , the angular velocity of the jth sinusoidal component is

2p 2pn 2.2. Deﬁnition of the mean water levels v ¼ ¼ j ; ðj ¼ 1; …; NÞ: ð7aÞ j P m j j Based on Eq. (7d), one can assume that the Therefore, the sea tide water level solutions of the steady-periodic nonlinear system ; ; ; ; ; ; Eqs. (4a)–(6c), Wðx tÞ HSðx z tÞ and HCðx tÞ will XN also be periodic functions of time t with a period of P: WTideðtÞ¼ Aj cosðvjt þ cjÞð7bÞ This assumption is physically reasonable because the j¼1 periodic sea tide (7b) is the only driving force of the system. Therefore, it is reasonable to deﬁne the mean is periodic with respect to the time t with a period of P water levels of the leaky aquifer system by (in hours) given by ðtþP 1 ; ; min ðm ; …; m Þ WðxÞ¼ Wðx tÞdt ð8aÞ ¼ CM 1 N ; ð Þ P t P ; ; 7c maxCDðn1 … nN Þ ðtþP ; 1 ; ; ; ; ; HSðx zÞ¼ HSðx z tÞdt ð8bÞ where minCMðm1 … mN Þ denotes the minimum P t ; ; ; common multiple of m1 … mN and ð ; ; 1 tþP maxCDðn1 … nN Þ the maximum common divisor H ðxÞ¼ H ðx; tÞdt; ð8cÞ ; ; : = C C of n1 … nN In fact, because njP mj is an integer for P t 216 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224

Integrating Eqs. (5a)–(5c) in the interval ðt; t þ PÞ aquifer system when the landward distance from with respect to time t, dividing the resulting equations the coastline is so far that the tide-induced by P; and using the periodicity assumption of Wðx; tÞ; oscillations have died out. Exact, asymptotic ; ; ; ; HSðx z tÞ and HCðx tÞ yield solutions when x !1 will be derived and discussed. ›2H ðx; zÞ b0 b0 S ¼ ; ; ð Þ 2 0 2 , z , 9a ›z 2 2 3.1. Exact asymptotic solutions ; l ; ; l : HSðx zÞ z¼b0=2 ¼WðxÞ HSðx zÞ z¼2b0=2 ¼HCðxÞ ð9bÞ as x !1when L . 0 The solution of Eqs. (9a) and (9b) is Adding Eqs. (11) and (12), yields zðWðxÞ2H ðxÞÞ WðxÞþH ðxÞ f 00ðxÞ¼0; x . 0 ð14aÞ H ¼ C þ C : ð10aÞ S b0 2 where K f ðxÞ¼ U W ðxÞþDK WðxÞþTH ðxÞ: ð14bÞ From Eq. (10a), one obtains 2 ms U C : 0ð ð Þ ð ÞÞ The general solution of Eq. (14a) is f ðxÞ¼c1 þ c2x 0 ›HS K W x 2HC x ; K ¼ 0 ¼LðW 2HCÞ ð10bÞ From the no-ﬂow boundary condition Eqs. (4c) and ›z b 0 (6c), it can be easily deduced that limx!1 f ðxÞ¼0; ¼ 0= 0 21 : where L K b is the leakance [T ] of the semi- which implies c2 ¼ 0 Therefore, one has permeable layer (Hantush, 1960). = : Integrating Eqs. (4a) and (6a) in the interval f ðxÞ¼c1 ¼ f ð0Þ¼KU½Wmsð0Þ 2 þ DWð0Þ þ THCð0Þ ðt; t þ PÞ with respect to time t, dividing the resulting ð14cÞ equations by P, and using the equation (10b), the Using the tidal boundary conditions (4b) and (6b), one identical equation obtains ›W 1 ›W2 ›W ðW þ DÞ ¼ þ D ; Wð0Þ¼H ð0Þ¼0: ð14dÞ ›x 2 ›x ›x C and the periodicity assumption of Wðx; tÞ; H ðx; z; tÞ Using Eqs. (4b) and (13), one ﬁnds S and H ðx; tÞ; yield ðtþP C 1 2 ; Wmsð0Þ¼ W ð0 tÞdt 2 P t d 1 ; ; 0 1 KU WmsðxÞþDWðxÞ 2LðW 2HCÞ¼0 x . 0 2 2 ð þ XN dx 2 1 t P @ A ð11Þ ¼ Aj cosðvjtþcjÞ dt P t j¼1 TH00 þLðW 2H Þ¼0; x . 0; ð12Þ ð C C 1 tþP XN ¼ AiAj cosðvitþciÞcosðvjtþcjÞdt where WmsðxÞ is the mean square of the watertable P t 2i;j¼1 deﬁned as ð 1 tþP XN ðtþP ¼ 4 A2 cos2ðv tþc Þ 1 2 ; : j j j WmsðxÞ¼ W ðx tÞdt ð13Þ P t j¼1 P t 3 X In Sections 3 and 4, the behaviours of the mean water 5 þ2 AiAj cosðvitþciÞcosðvjtþcjÞ dt: levels deﬁned as Eqs. (8a)–(8c) will be investigated 1#i,j#N analytically based on the derived Eqs. (10a), (11) ð14eÞ and (12). Substituting the trigonometric identities 2cosðv tþc Þcosðv tþc Þ 3. Asymptotic mean water levels of the system i i j j ¼ cos½ðvitþciÞ2ðvjtþcjÞ This section will focus on the asymptotic behaviours of the mean water levels of the leaky þcos½ðvitþciÞþðvjtþcjÞ H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 217 into Eq. (14e), because vkP ¼ 2pnkP=mk; and nkP=mk it follows that is an integer for each k ¼ 1;…;N according to Eqs. KU 2 1 2 ; : (7a) and (7c), one can easily ﬁnd that lim W þDKUW þTW ¼ KUAS ifL.0 8 x!1 2 4 < ðtþP P=2; ifi¼j; ð16fÞ cosðvitþciÞcosðvjtþcjÞdt¼: ð14fÞ t 0; ifi–j: Eq. (16f) implies that as x!1; Wðx;tÞ tends to a constant c that satisﬁes Substituting Eq. (14f) into Eq. (14e), one obtains 2= 2 = ; : XN KUc 2þðDKU þTÞc2KUAS 4¼0 ifL.0 ð16gÞ 2 def: 2 : 2Wmsð0Þ¼ Aj ¼ AS ð14gÞ j¼1 Neglecting the negative root of Eq. (16g) which is physically unrealistic, one eventually obtains Eventually, from Eqs. (14b), (14d) and (14g) one ﬁnds lim Wðx;tÞ¼lim WðxÞ x!1 x!1 1 2 ; : ð17aÞ f ðxÞ¼ KUAS x$0 ð14hÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 = = ; ¼ðDþT KUÞð 1þaL 221Þ ifL.0 Integrating Eq. (12) in the interval ðx;xþXÞ; let X!1 where a is a dimensionless parameter given by and then use the no-ﬂow boundary condition (6c), one L obtains K2 A2 2 XN U S KU 2: ð aL ¼ ¼ Ai ð17bÞ 1 ðDK þTÞ2 ðDK þTÞ2 0 : U U i¼1 THCðxÞ¼L ½WðjÞ2HCðjÞdj ð15Þ x By means of Eqs. (16a), (16d) and (17a), one has Using Eq. (15) and the no-ﬂow boundary conditions ; lim HCðx tÞ¼lim HCðxÞ (4c) and (6c), one can show that (see Appendix A for x!1 x!1 the proof) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ðDþT=K Þð 1þa =221Þ; ifL.0: ð17cÞ : U L lim ½WðxÞ2HCðxÞ¼0ifL.0 ð16aÞ x!1 If the landward distance x is great enough, all the tide-induced oscillations will die out. This 3.2. Exact asymptotic solutions as x !1when L ¼ 0 implies 2 ; ; Philip’s (1973) exact asymptotic solution (1) lim ½WmsðxÞ2W ðx tÞ¼0 ð16bÞ x!1 considers only one single unconﬁned aquifer with impermeable bottom, which is equivalent to lim ½WðxÞ2Wðx;tÞ¼0; ð16cÞ x!1 the situation in this paper when the semi-permeable layer is replaced by a completely impermeable layer, ; : lim ½HCðxÞ2HCðx tÞ¼0 ð16dÞ : x!1 i.e. L ¼ 0 Substituting L ¼ 0 into Eq. (11), integrat- ing the resulting equation and using the boundary From Eqs. (14b) and (14h), one has conditions (4b) and (4c), yield

1 2 KU ¼ ð Þþ ð Þþ ð Þ 1 1 1 2 KUAS Wms x DKUW x THC x W ðxÞþDWðxÞ¼ W ð0ÞþDWð0Þ¼ A : 4 2 2 ms 2 ms 4 S K ¼ U ðW 2W2ÞþDK ðW 2WÞ ð18aÞ 2 ms U KU 2 By means of Eqs. (18a), (16b) and (16c), one obtains þTðHC 2WÞþTðW 2WÞþ W pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 ; = ; lim Wðx tÞ¼lim WðxÞ¼Dð 1 þ a0 2 2 1Þ : x!1 x!1 þDKUW þTW ð16eÞ Let x!1 in Eq. (16e), and use Eqs. (16a)–(16d), if L ¼ 0 ð18bÞ 218 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 where a0 is a dimensionless parameter given by approximated by

2 XN K A2 AS 1 2 U S a ¼ ¼ A : ð18cÞ lim WðxÞ¼lim HCðxÞ < 0 2 2 i x!1 x!1 4ðDK þ TÞ D D i¼1 U K XN Eq. (18b) is a generalization of Philip’s (1973) ¼ U A2 ð19Þ ð þ Þ i solution (1) in the sense that Eq. (18b) considers all 4 DKU T i¼1 the sinusoidal components of the sea tide. Substituting = L ¼ 0 into Eq. (12) and using the boundary conditions with a truncation error relative to D þ T KU being 2= : 2 (6b) and (6c), yield less than aL 32 From assumption (d), aL is usually less than 1, so Eq. (19) has adequate accuracy. For example, consider a coastal leaky aquifer ; ; HCðxÞ ; 0 if L ¼ 0 ð18dÞ 2 system withP parameters TU ¼ T ¼ 10 m /h, N 2 2; D ¼ 1.25 m, j¼1 Aj ¼ 1m the exact solutions which implies that the aquifer’s mean water level (17a)–(17c) shows that the exact asymptotic equals the mean sea level only when the aquifer is mean water level is 0.098076 m, the approximate governed by linear equation and there is no net solution (19) gives a value of 0.10 m. The error is ﬂux. very small. According to solutions (17a)–(17c) or (19), the 3.3. Discussion of the exact asymptotic solutions asymptotic groundwater level of the leaky aquifer system is independent of the magnitude of the Solutions (17a) and (17c) show that, in inland leakance of the semi-permeable layer. This is places far from the coastline, the mean water levels of because the asymptotic groundwater level higher both the unconﬁned aquifer and the conﬁned aquifer than the mean sea level is due to a positive will be higher than the mean sea level by the same difference between the inﬂow and outﬂow in a tidal constant. This phenomenon is due to the watertable- period at the tidal boundary of the unconﬁned dependent transmissivity of the unconﬁned aquifer aquifer, which is independent of the magnitude of and the leakage of the semi-permeable layer. The the leakance of the semi-permeable layer. watertable-dependent transmissivity of the uncon- It is well known that Boussinesq’s equation (4a) ﬁned aquifer, as mentioned in Section 1, will lead to a can be linearized approximately when the tidal mean watertable higher than the mean sea level. Due amplitudes is much less than the aquifer’s depth D to leakage through the semi-permeable layer, the under the mean sea level (e.g. Bear 1972). mean head of the underlying aquifer will increase Solutions (17a)–(18d) show that the linearization accordingly. These lead to landward positive gradi- is valid in the sense that the increase of mean ents of both the mean watertable and mean head in groundwater levels of the aquifer system, whose the region near the coastline. Therefore, a ground- magnitude has the order of AS=D; is negligible for water–seawater cycle is formed. Seawater is pumped small AS=D: into the unconﬁned aquifer by the sea tide. Then, part of it returns to the sea driven by the mean watertable gradient. The rest leaks into the conﬁned aquifer 4. Perturbation solutions for thin semi-permeable through the semi-permeable layer, and returns to the layer and small leakage sea through the conﬁned aquifer driven by the mean head gradient. The leakage from the semi-permeable 4.1. Perturbation solutions layer is the only source of the discharge through the conﬁned aquifer because there is no inland recharge in In order to investigate the mean water levels of the aquifer system. the leaky aquifer system in the vicinity of the Using Taylor’s expansion (3), the exact coastline, approximate perturbation solutions will asymptotic water levels (17a) and (17c) can be be derived in this section. The system (4a)–(6c) is H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 219

2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 3 L L 6 exp 2x þ 7 XN 2 6 = 7 Aj 6 TU T T TU ð2 2 1jTU TÞexpð22ajxÞ 7 WðxÞ < 6 þ 2 7; ð20Þ 4 ð þ Þ ð þ Þ ð ð þ = ÞÞ ð ð þ = ÞÞ 5 j¼1 2D 2 TU T TU T 2 2 1j 1 TU T 2 2 2 1j 1 TU T

2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 3 L L 6 exp 2x þ 7 XN 2 6 7 Aj 6 TU TU T TU 1jTU expð22ajxÞ 7 H ðxÞ < 6 2 þ 7; ð21Þ C 4 ð þ Þ ð þ Þ ð ð þ = ÞÞ ð ð þ ÞÞ 5 j¼1 2D 2 TU T TU T 2 2 1j 1 TU T 2 2T 2 1j T TU

very complex because it includes many aquifer the semi-permeable layer is deﬁned as parameters. For the sake of succinctness, assume 0 ðtþP K ›HS 0 ›HS that the semi-permeable layer’s storage is negli- FLðxÞ¼ dt¼K ¼LðW2HCÞ P ›z ¼ 0= ›z gible (e.g. a thinP semi-permeable layer) and the t z 2b 2 N = p : leakage is small ð j¼1 L ðvjSyÞ 1Þ Under such "# sﬃﬃﬃﬃﬃﬃﬃﬃﬃ! assumptions the following perturbation solutions L L A2 exp 2x þ 2expð22a xÞ XN j j can be obtained (see Appendix B for the LT T TU ¼ : derivation) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2D 2T21 ðTþT Þ ; = j¼1 j U where TU ¼ KUD aj ¼ vjSy ð2KUDÞ is the wave number corresponding to the jth sinusoidal com- ð23Þ ponent, 1j is a small dimensionless constant deﬁned as Because there is no inland recharge, the discharge QC from the conﬁned aquifer equals L u the total leakage through the semi-permeable layer, ¼ ¼ j ð Þ 1j = 22 i.e. vjSy 2ðSy SÞ ð pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 T XN A2ð LTT =ðT þTÞ21 a T Þ Q ¼ F ðxÞdx¼ j U U j j U : C L ð þ Þ 0 2Dj¼1 2T21j T TU with uj ¼ L=ðvjSÞ being the dimensionless leakage (Li and Jiao, 2001). The mean head of the semi- ð24Þ permeable layer HSðxÞ is given by Eq. (10a). 21; One can also calculate For real aquifer systems, one has Sy , 10 S , 25 23 ð10 –10 Þ (Todd, 1980). The major sinusoidal ðtþP T ›HC 0 components of the sea tide are usually semidiurnal QC¼ dt¼THCð0Þ P t ›x x¼0 and diurnal, so the range of uj is from 0 to 20 for real leaky aquifer systems (Li and Jiao, 2001). Therefore, by directly using Eq. (21). The result is the same 23 21 according to Eq. (22), the range of 1j is 10 –10 : as Eq. (24). This validatesP to a certain extent the small leakage To see how the leakage ﬂux FLðxÞ changes with N p : assumption ð j¼1 1j 1Þ landward distance x and how large the discharge QC will be, consider a coastal leaky aquifer system 2= ; ; 4.2. Discussion with typical parameters TU ¼ T ¼ 50 m h D ¼ 2m ; ; : ; : 21 N ¼ 1 A1 ¼ 1m Sy ¼ 0 3 v1 ¼ 0 5h (semidiurnal 21 It may be interesting to have some idea about how sea tide),pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃL ¼ 0:00075–0:0075 h : These data lead to = : 21; = the leakage from the unconﬁned aquifer to a1 ¼ v1Sy ð2TUÞ ¼ 0 03873 m u ¼ 11 ¼ L ðv1SyÞ the conﬁned aquifer changes with the inland distance. ¼ 0:005–0:05: Fig. 3 shows how the leakage ﬂux Based on Darcy’s law and Eqs. (10b), (20) and (21), FLðxÞ changes with the landward distance x for the time-averaged leakage ﬂux FLðxÞ through different values of u. 220 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224

Fig. 3. Change of the leakage ﬂux FLðxÞ in the conﬁned aquifer with the landward distance x for different values of the dimensionless leakage u.

The leakage ﬂux FLðxÞ is zero at the coastline the increase of the leakage ﬂux. On the contrary, x ¼ 0 because the water levels in the unconﬁned inland recharges into the conﬁned aquifer will and conﬁned aquifers equal the sea level. It reaches result in the decrease of leakage ﬂux. Sufﬁciently its maximum at a certain point and then great inland recharge into the conﬁned aquifer will decreases to zero as x !1: The maximum leakage lead to negative leakage ﬂux-leakage ﬂux from the ﬂux decreases with u. When u is smaller, the conﬁned aquifer into the unconﬁned aquifer at leakage will occur over a greater range of inland some inland places. distance. Let the landward distance x approach inﬁnite in As the dimensionless leakage u increases from Eqs. (20), (21) and (10a), one obtains 0.005 to 0.05, the discharge QC from the conﬁned 2 21 XN aquifer increases from 0.38 to 1.06 m d , which TU 2 lim W ¼ lim HS ¼ lim HC ¼ Aj means that the groundwater entering the sea from the x!1 x!1 x!1 ð þ Þ 4D TU T j¼1 conﬁned aquifer per day per meter of the coastline is 3 K XN 0.38–1.06 m . Considering that most coastal aquifers ¼ U A2: ð25Þ ð þ Þ j have long coastlines, this is a considerable amount. 4 DKU T j¼1 One of the assumptions of this study is that there are neither rainfall inﬁltrations into the Comparison of Eqs. (25) and (19) reveals that at unconﬁned aquifer nor net groundwater recharges inland places far from the coastline, the pertur- into the aquifers from inland places far from the bation solutions equal the approximate Taylor coast. In reality, however, there usually exist expansion of the exact asymptotic solutions with 2= the rainfall inﬁltration and other inland recharges a truncation error less than aL 32 relative to D þ = ; to the unconﬁned and conﬁned aquifers, which T KU where aL is deﬁned as Eq. (17b). From may inﬂuence the direction of the leakage ﬂux and assumption (d), aL is usually less than 1. So the the circulation, i.e. the sign of FL.Inthiscase, relative error of Eq. (25) is less than 3.125% and because the local leakage ﬂux direction is can be neglected. For example, for the above determined by the difference of the local water- aquifer system discussed in Fig. 3, the exact table of the unconﬁned aquifer and the local solutions (17a) and (17c) show that the hydraulic head of the conﬁned aquifer, the sign of asymptotic mean water level far inland is the leakage ﬂux may be negative in some inland 0.06202 m, the perturbation solutions (20), (21) places and positive at other places. It is obvious or (25) give a value of 0.0625 m. that the rainfall inﬁltration and other inland Solution (2a) by Nielsen (1990) assumed that the recharges into the unconﬁned aquifer will lead to unconﬁned aquifer has an impermeable bottom, i.e. H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 221 the leakance L ¼ 0: Substituting L ¼ 0 into Eqs. (20) the sea tide and divided into two parts. One part and (21) yields returns to the sea driven by the mean watertable gradient. The rest part leaks into the conﬁned aquifer XN 2 Aj l 22ajx ; l : through the semi-permeable layer, and returns to the WðxÞ L¼0 < ð12e Þ HCðxÞ L¼0 ;0 ð26Þ j¼1 4D sea through the conﬁned aquifer driven by the mean head gradient. The perturbation solutions show that Eq. (26) is a generalization of Nielsen’s the discharge through the conﬁned aquifer is signiﬁ- solution (2a) in the sense that Eq. (26) cant for typical aquifer parameter values. The considers all the sinusoidal components of the sea seawater–groundwater circulation described in this tide. paper has impacts on the exchange and movement of chemicals such as nutrients and contaminants in the coastal areas. This circulation provides also insights 5. Summary into better understanding the mechanism regarding SGWD. If the observed mean water levels in coastal This paper investigates the inﬂuences of the sea areas are used for estimating the net inland recharge, tide on the mean water levels in a multi-layered the enhancing processes of sea tide on the mean coastal aquifer system with a conﬁned aquifer, an groundwater levels should be taken into account. unconﬁned aquifer, and a semi-permeable layer Otherwise, the net inland recharge will be between them. Exact asymptotic solutions and overestimated. approximate perturbation solutions are derived for According to Nielsen (1990) and Li et al. (2000),a multi-sinusoidal-component sea tide. At inland places slope water–land boundary of the coastal aquifer will where the distance from the coastline is so far that all lead to much higher watertable in the unconﬁned the tide-induced oscillations have died out, the mean aquifer than a vertical water–land boundary does. In water levels in both the unconﬁned and conﬁned reality, the water–land boundary of the coastal aquifer aquifers approach the same constant considerably systems is usually a slope. Hence, the phenomenon higher than the mean sea level. Exact asymptotic revealed in this paper may be reinforced in multi-- solutions show that this constant water level depends layered coastal aquifer systems with sloping beach. on the amplitudes of the sinusoidal components of the sea tide, the conﬁned aquifer’s hydraulic transmissiv- ity, the unconﬁned aquifer’s permeability and the Acknowledgements depth below the mean sea level, and that it is independent of the magnitude of the leakance of the This research is supported by Dr Stephen S.F. Hui semi-permeable layer. As the landward Trust Fund at the University of Hong Kong. The distance approaches inﬁnite, the perturbation sol- authors are grateful for the helpful comments of Dr utions in both aquifers tend to the same constant that Ling Li. is found to be a Taylor-expansion approximation to the exact asymptotic constant with a relative trunca- tion error less than 3.125%. Nielsen’s mean water- table solution (2a) is the special case for single- Appendix A. Proof of Eq. (16a) sinusoidal-component sea tide when the middle layer becomes impermeable. Suppose Eq. (16a) does not hold, then there exists a ; ; ; Due to the watertable-dependent transmissivity of ﬁxed number c0 . 0 and a series of xj ðj ¼ 1 2 …Þ the unconﬁned aquifer, the mean water levels of the that tends to inﬁnite, such that unconﬁned and conﬁned aquifer higher than the mean l l ; ; ; ; sea level lead to positive landward gradients of the EðxjÞ . c0 ðj ¼ 1 2 …Þ ðA1Þ mean water levels, which result in a seawater- : groundwater cycle in the region near the coastline. holds, where EðxÞ¼WðxÞ 2 HCðxÞ From the no- Seawater is pumped into the unconﬁned aquifer by ﬂow boundary conditions (4c) and (6c), there exists 222 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 an integer J, such that for each j $ J; xJ is great Appendix B. Derivation of perturbation solutions enough for the following inequalities to hold

; 2 ; 2 Neglecting the storage of semi-permeable layer, dWðx tÞ c0 dHCðx tÞ c0 , ; , ; from Eqs. (5a)–(5c) one obtains dx 2 dx 2 0 0 ›HS K ; ; ; ; : K ¼ 0 ½Wðx tÞ2HCðx tÞ ¼ LðW 2HCÞðA5Þ ;t [ ð21 1Þ x $ xJ ðA2Þ ›z b

Using Eq. (A2) and deﬁnitions of mean water Let d ¼ AS=D be the perturbation parameter, and let levels (8a) and (8c), it follows that ð ; Þ¼ ð ; Þþ ð ; Þþ 2 ð ; Þ ; ð Þ W x t W0 x t dW1 x t d W2 x t … A6

dE dW dHC ; ; ; 2 ; ; # þ HCðx tÞ¼HC0ðx tÞþdHC1ðx tÞþd HC2ðx tÞ… ðA7Þ dx dx dx inserting Eqs. (A6) and (A7) into Eqs. (4a)–(4c) and ðtþP (6a)–(6c), and separating terms of equal order in d; 1 ›W ›HC 2; : # ðj jjþ jÞ t # c0 x $ xJ ðA3Þ one ﬁnds that W0 ¼HC0 ;0 from equations of order P t ›x ›x 0; ; d while Wi and HCi ði¼1 2Þ satisfy the following = ; Using Eqs. (A1) and (A3), when xj $ xJ þ 1 c0 one has system of equations. ð = 1 xjþ1 c0 Order d : ðW 2HCÞdx = 2 xj21 c0 ›W › W 1 ¼ 1 ð Þ; Sy KUD 2 2 L W1 2 HC1 ð = ›t ›x xjþ1 c0 ðA8:1Þ ¼ EðxÞdx xj21=c0 21 , t , 1; x . 0; "# XN ð = ð D xjþ1 c0 x ð Þ l ; : dE j W1 x¼0 ¼ Aj cosðvjt þ cjÞ ðA8 2Þ ¼ djþEðxjÞ dx A = dj S j¼1 xj21 c0 xj

"# ›W1 ðx þ1=c ðx ¼ 0; ðA8:3Þ j 0 dEðjÞ ›x ¼1 . 2þ dj dx x = dj 2 xj21 c0 xj ›H › H C1 ¼ C1 þ ð Þ; S T 2 L W1 2 HC1 ð = ð ›t ›x ðA8:4Þ xjþ1 c0 x ð Þ dE j $ 22 dj dx ; ; xj21=c0 xj dj 21 , t , 1 x . 0

›HC1 ðx ðx ¼ ; ð : Þ j j dEðjÞ lim 0 A8 5 $ 22 dj dx x!1 ›x = xj21 c0 x dj XN l D : " # HC1 x¼0 ¼ Aj cosðvjt þ cjÞ; ðA8 6Þ ð þ = ð xj 1 c0 x dEðjÞ AS j¼1 2 dj dx dj 2 xj xj Order d : ! ð ð xj xj 2 2 2 2 ›W2 › W2 › W1 ›W1 $ 22 c0 dj dx Sy ¼ KUD þ KU W1 þ = 2 2 xj21 c0 x ›t ›x ›x ›x "# ; ; ; : ð ð 2 LðW2 2 HC2Þ 21 , t , 1 x . 0 ðA9 1Þ xjþ1=c0 x 2 : 2 c0 dj dx ¼ 1 ðA4Þ l ; : W2 x¼0 ¼ 0 ðA9 2Þ xj xj l : : HC2 x¼0 ¼ 0 ðA9 3Þ Eq. (A4)Ð is in contradiction with the convergence of the 1 integral x ½WðjÞ2HCðjÞdj in the right-hand side of The other three equations satisﬁed by W2 and HC2 are Eq. (15). not written out here because they can be obtained H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224 223 directly by substituting W1 and HC1 in Eqs. (A8.3)– on Eqs. (A10.2)–(A10.4), the source term of ; (A8.5) with W2 and HC2 respectively. the time-averaged system is ! Although the exact solution W1 to Eqs. (A8.1)– ð 1 tþP ›2W ›W 2 (A8.6) has been found (Li and Jiao, 2002), here 1 þ 1 KU W1 2 dt it will be simpliﬁed for the sake of succinctness. P t ›x ›x = p ; Using the condition 1j ¼ L ðvjSyÞ 1 the exact XN solution W of Eqs. (A8.1)–(A8.6) can be simpli- D 1 ¼ A2v S ð1þOð1 ÞÞexpð22a xÞ: ðA11Þ ﬁed into 2 j j y j j 2AS j¼1 D XN ; 2ajx : ð Þ; W1ðx tÞ¼ Aj e ð1þOð1jÞÞcosðvjt2ajxþcjÞ Neglecting terms of O 1j the time-averaged AS j¼1 system is eventually simpliﬁed into ðA10:1Þ L ; W00 2 ðW 2 H Þ Using Eq. (A8.1) and the periodicity of W1 one 2 T 2 C2 has U XN ðtþP 2 ðtþP 2 D 1 › W Sy ›W 2 2 ; ; : 1 ¼ 1 ¼ 2 Aj aj expð22ajxÞ x . 0 ðA12 1Þ KUW1 2 dt dt 2 P t ›x 2PD t ›t AS j¼1 ð L tþP 00 L þ W1ðW1 2HC1Þdt H þ ðW 2 H Þ¼0; x . 0; ðA12:2Þ PD t C2 T 2 C2 ð L tþP l ; 0 l ; : ; W2 x¼0 ¼ 0 W2 x¼1 ¼ 0 ðA12 3Þ ¼ W1ðW1 2HC1Þdt PD t l ; 0 l ; : ðA10:2Þ HC2 x¼0 ¼ 0 HC2 x¼1 ¼ 0 ðA12 4Þ substituting Eq. (A10.1) into the above equation, : where TU ¼ KUD For the jth single source term using W 2H ¼ A Oð1Þ; yield = 2 2 2 ; 1 C1 S 2ðD ASÞAj aj expð22ajxÞ the solutions W2j and ð H to the system (A12) are 1 tþP ›2W XN 2j K W 1 dt ¼ A v S Oð1 Þe2ajx: U 1 2 j j y j 2 2 P t ›x ¼ DA a p L 2 q L j 1 W ¼ j j j T j U : 2j 2 þ ðA10 3Þ AS LT LU pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Using Eq. (A10.1) and the trigonometric identities L ðp þ q Þexpð2x L þ L Þ þ U j j T U 2sinf cosc ¼ sinðfþcÞ2sinðf2cÞ; LT þ LU 2cos fcos c ¼ cosðfþcÞþcosðf2cÞ; 2 pj expð22ajxÞ ; ðA13:1Þ one obtains ðtþP 2 DA2a2 p L 2 q L 1 ›W1 H ¼ j j j T j U KU dt 2j 2 þ P t ›x AS LT LU pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ XN LTðpj þ qjÞexpð2x LT þ LUÞ D 2 : : 2 ¼ Aj vjSyð1þOð1jÞÞexpð22ajxÞ ðA10 4Þ 2A2 LT þ LU S j¼1

Substituting Eq. (A10.1) into Eq. (A9.1), then þ qj expð22ajxÞ ; ðA13:2Þ integrating Eqs. (A9.1)–(A9.3) and other three equations satisﬁed by W2 and HC2 with respect to where t from t to tþP; one obtains the time-averaged : L ¼ L=T; L ¼ L=T ; linear system with respect to W2 and HC2 Based T U U 224 H. Li, J.J. Jiao / Journal of Hydrology 274 (2003) 211–224

2 a semi-permeable layer with storage. Advances in Water 4aj 2 LT ¼ ; Resources 24 (5), 565–573. pj 2 2 4aj ð4aj 2 LU 2 LTÞ Li, H., Jiao, J., 2002. Analytical solutions of tidal groundwater ﬂow in coastal two-aquifer system. Advances in Water Resources 25 L q ¼ T : (4), 417–426. j 2 2 Li, L., Barry, D.A., Stagnitti, F., Parlange, J.-Y., 1999. 4aj ð4aj 2 LU 2 LTÞ Submarine groundwater discharge and associated chemical Using Eqs. (A8.1)–(A8.6), it follows that input to a coastal sea. Water Resources Research 35 (11), 3253–3259. : W1ðxÞ¼HS1ðxÞ¼HC1ðxÞ ; 0 ðA14Þ Li, L., Barry, D.A., Stagnitti, F., Parlange, J.-Y., Jeng, D.-S., 2000. Beach water table ﬂuctuations due to spring-neap tides: Implementing the time-averaged transform to both moving boundary effects. Advances in Water Resources 23, sides of Eqs. (A6) and (A7), then substituting Eqs. 817–824. (A13.1), (A13.2) and (A14) back into the transform Melchior, P., 1978. The Tides of the Planet Earth, Pergamon Press, results, one ﬁnally obtains the time-averaged London. ; Moore, W.S., 1996. Large groundwater inputs to coastal waters solutions W HC and HS given by Eqs. (20), (21) 226 and (10a), respectively. revealed by Ra enrichment. Nature 380 (18), 612–614. Nielsen, P., 1990. Tidal dynamics of the water table in beaches. Water Resources Research 26 (9), 2127–2134. Parlange, J.-Y., Stagnitti, F., Starr, J.L., Braddock, R.D., 1984. Free- References surface ﬂow in porous media and periodic solution of the shallow-ﬂow approximation. Journal of Hydrology 70, 251–263. Bear, J., 1979. Hydraulics of Groundwater, McGraw-Hill, New Philip, J.R., 1973. Periodic nonlinear diffusion: an integral relation York, pp. 74–78, 121,93. and its physical consequences. Australian Journal of Physics 26, Chen, C.X., Jiao, J., 1999. Numerical simulation of pumping tests in 513–519. multilayer wells with non-darcian ﬂow in the wellbore. Ground Pugh, D.T., 1987. Tides, Surges and Mean Sea-level, Wiley, New Water 37 (3), 465–474. York. Church, T.M., 1996. A groundwater route for the water cycle. Serfes, M.E., 1991. Determining the mean hydraulic gradient of Nature 380 (18), 579–580. ground water affected by tidal ﬂuctuations. Ground Water 29 Hantush, M.S., 1960. Modiﬁcation of the theory of leaky aquifers. (4), 549–555. Journal of Geophysical Research 65 (11), 3713–3716. Sheahan, N.T., 1977. Injection/extraction well system—a unique Jiao, J., Tang, Z., 1999. An analytical solution of groundwater seawater intrusion barrier. Ground Water 15 (1), 32–49. response to tidal ﬂuctuation in a leaky conﬁned aquifer. Water Smiles, D.E., Stokes, A.N., 1976. Periodic solutions of a nonlinear Resources Research 35 (3), 747–751. diffusion equation used in groundwater ﬂow theory: examin- Knight, J.H., 1981. Steady period ﬂow through a rectangular dam. ation using a Hele–Shaw model,. Journal of Hydrology 31, Water Resources Research 17 (4), 1222–1224. 27–35. Li, H., Jiao, J., 2001. Analytical studies of groundwater-head Todd, D.K., 1980. Groundwater Hydrology, Wiley, New York, ﬂuctuation in a coastal conﬁned aquifer overlain by pp. 45.