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A perturbation solution for head fluctuations in acoastal leaky aquifer system considering water tableover-heightMo-Hsiung Chuang a , Asadi-Aghbolaghi Mahdi b & Hund-Der Yeh ca Department of Urban Planning and Disaster Management , Ming-Chuan University ,Gweishan District , Taoyuan , 333 , Taiwanb Civil Engineering Department , School of Engineering, Shiraz University , Shiraz , Iranc Institute of Environmental Engineering, National Chiao Tung University , Hsinchu , 300 ,TaiwanPublished online: 20 Jan 2012.

To cite this article: Mo-Hsiung Chuang , Asadi-Aghbolaghi Mahdi & Hund-Der Yeh (2012) A perturbation solution for headfluctuations in a coastal leaky aquifer system considering water table over-height, Hydrological Sciences Journal, 57:1,162-172, DOI: 10.1080/02626667.2011.637496

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162 Hydrological Sciences Journal – Journal des Sciences Hydrologiques, 57(1) 2012

A perturbation solution for head fluctuations in a coastal leaky aquifersystem considering water table over-height

Mo-Hsiung Chuang1, Asadi-Aghbolaghi Mahdi2 and Hund-Der Yeh3

1Department of Urban Planning and Disaster Management, Ming-Chuan University, Gweishan District, Taoyuan 333, Taiwan2Civil Engineering Department, School of Engineering, Shiraz University, Shiraz, Iran3Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, 300, Taiwanhdyeh@mail.nctu.edu.tw

Received 3 June 2010; accepted 13 May 2011; open for discussion until 1 July 2012

Editor D. Koutsoyiannis; Associate editor J. Simunek

Citation Chuang, M.-H., Mahdi, A.-A. and Yeh, H.-D., 2012. A perturbation solution for head fluctuations in a coastal leaky aquifersystem considering water table over-height. Hydrological Sciences Journal, 57 (1), 162–172.

Abstract This paper develops a new analytical solution for the aquifer system, which comprises an unconfinedaquifer on the top, a semi-confined aquifer at the bottom and an aquitard between them. This new solution isderived from the Boussinesq equation for the unconfined aquifer and one-dimensional leaky confined flow equationfor the lower aquifer using the perturbation method, considering the water table over-height at the remote boundary.The head fluctuation predicted from this solution is generally greater than the one solved from the linearizedBoussinesq equation when the ratio of the tidal amplitude to the thickness of unconfined aquifer is large. It is foundthat both submarine groundwater discharges from upper and lower aquifers increase with tidal amplitude–aquiferthickness ratio and may be underestimated if the discharge is calculated based on the average head fluctuation. Theeffects of the aquifer parameters and linearization of the Boussinesq equation on the normalized head fluctuationare also investigated.

Key words perturbation analysis; Boussinesq equation; tidal fluctuation; submarine groundwater discharge

Une solution par la méthode des perturbations pour les fluctuations de charge dans un systèmeaquifère côtier semi-captif tenant compte de la surélévation de la surface libreRésumé Cet article développe une nouvelle solution analytique pour un système aquifère comprenant un aquifèrelibre supérieur, un aquifère semi-captif inférieur, séparés par un aquitard. Cette nouvelle solution est dérivée del’équation de Boussinesq pour l’aquifère libre et de l’équation unidimensionnelle d’écoulement pour l’aquifèresemi-captif, en utilisant la méthode des perturbations et en tenant compte de la surélévation de la nappe phréatiqueà la frontière la plus éloignée. La fluctuation de la charge obtenue à partir de cette solution est généralement plusimportante que celle obtenue par l’équation de Boussinesq linéarisée lorsque le rapport entre l’amplitude de lamarée et l’épaisseur de l’aquifère libre est grande. On constate que les débits des eaux souterraines sous-marinespour les deux aquifères supérieur et inférieur augmentent avec le rapport de l’amplitude des marées à l’épaisseurde l’aquifère et peuvent être sous-estimés si le débit est calculé sur la base des fluctuations de charge moyennes.Les effets des paramètres de l’aquifère et de la linéarisation de l’équation de Boussinesq sur la fluctuation de lacharge normalisée sont également étudiés.

Mots clefs analyse de perturbations; équation de Boussinesq; fluctuation de la marée; débit des eaux souterraines sous-marines

1 INTRODUCTION

The study of groundwater in coastal areas is ofpractical interest for many researchers for economicand environmental reasons. Related investigationsare concentrated on several different topics such as

parameter estimation (e.g. Pandit et al. 1991, Li et al.2006) and groundwater head fluctuations (e.g. Jiaoand Tang 1999, Jeng et al. 2002, Li et al. 2007).Among them, tide-induced groundwater head fluc-tuation is an important issue, which is useful instudying other topics. The approaches commonly

ISSN 0262-6667 print/ISSN 2150-3435 online© 2011 IAHS Presshttp://dx.doi.org/10.1080/02626667.2011.637496http://www.tandfonline.com

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A perturbation solution for head fluctuations in a coastal leaky aquifer system 163

used to investigate the groundwater fluctuation in atidal aquifer system include analytical methods (e.g.Jiao and Tang 1999, Chuang and Yeh 2008), numeri-cal methods (e.g. Ataie-Ashtiani et al. 1999, Li andJiao 2002, Chen and Hsu 2004, Liu et al. 2008),and experimental studies (e.g. Nielsen 1990, Turner1993, Ataie-Ashtiani et al. 1999, Uchiyama et al.2000, Cartwright et al. 2004, Robinson et al. 2006).In these studies, two types of problem are consideredfor the coastal aquifer system; one involves singlelayer aquifers (e.g. Parlange et al. 1984, Song et al.2007), while the other is concerned with multiple lay-ered aquifer systems (e.g. Li and Jiao 2001a, 2001b,Jeng et al. 2002).

Many coastal aquifer systems comprise an upperunconfined aquifer, a bottom semi-confined aquiferand an aquitard in between (e.g. White and Roberts1994, Chen and Jiao 1999). The analytical solu-tions developed for the problems commonly rely onsome simplifications and/or assumptions. For exam-ple, the water table fluctuation in the unconfinedaquifer is assumed constant or the Boussinesq equa-tion is linearized. Jiao and Tang (1999) investigatedthe influence of leakage on tidal response in a leakyaquifer system analytically without considering thewater table fluctuation in the unconfined aquifer. Liet al. (2001) used the linearized Boussinesq equationfor the unconfined aquifer and presented an approx-imate analytical solution to investigate the dynamiceffect of unconfined aquifer on groundwater fluc-tuation in confined aquifers. Li and Jiao (2001a)derived an analytical solution to investigate the influ-ence of leakage and storativity of the aquitard ontidal response in a coastal leaky aquifer system. Theyassumed that the water table of the upper unconfinedaquifer remains constant. Also assuming a constanthead for the unconfined aquifer, Li and Jiao (2001b)developed an analytical solution for a coastal leakyaquifer system in which the aquitard extends a finitedistance under the sea. Jeng et al. (2002) developedan analytical solution for a coastal aquifer system,which allows water table fluctuations in responseto the tide. However, their solution was based onthe linearized Boussinesq equation for the uncon-fined aquifer. Chuang and Yeh (2007) developed amathematical model, also based on the linearizedBoussinesq equation, to investigate the effects of tidalfluctuations and aquitard leakage on the groundwaterhead of the confined aquifer extending an infinitedistance under the sea.

The problem of water table over-height, arisenfrom the nonlinear effect, is an interesting issue in the

coastal aquifers. Based on the Boussinesq equation,many researchers (e.g. Knight 1981, Parlange et al.1984, Li and Jiao 2003) studied the over-height prob-lem. Parlange et al. (1984) derived an approximateanalytical solution for groundwater level fluctuationsubject to a tidal boundary condition in an uncon-fined aquifer. They developed a second-order solutionfor the Boussinesq equation using the perturbationapproach. Li and Jiao (2003) developed an exactasymptotic solution and approximate perturbationsolution for multi-sinusoidal components of the seatide in a coastal aquifer system. Their results showedthat the mean groundwater level of the aquifer systemstands considerably above the mean sea level, evenin the absence of net inland recharge of groundwaterand rainfall. Song et al. (2007) presented a perturba-tion solution derived from the Boussinesq equationfor one-dimensional tidal groundwater flow in anunconfined aquifer. Their solution adopted a pertur-bation parameter that was by definition less than one,and thus was applicable to a wide range of physi-cal conditions within the constraint of the Boussinesqapproximation. Their approach avoids a secular termin the third-order perturbation equation of Parlangeet al. (1984).

The objective of this paper is to develop ananalytical solution using perturbation analysis for acoastal leaky aquifer system, comprising an upperunconfined aquifer, a lower semi-confined aquifer,and an aquitard between them. The one-dimensionalleaky confined flow equation for the lower aquifercoupled with the Boussinesq equation for the upperaquifer is used to describe the groundwater fluc-tuations in the coastal leaky aquifer system. Thesecond-order perturbation method is then applied todevelop the solution for the coupled equations. Themean water table is assumed above the mean sealevel based on the work of Li and Jiao (2003). Thesolution from the Boussinesq equation for the uncon-fined aquifer accounts for the water table over-height,which was generally considered for cases of thinunconfined aquifer and/or large tidal amplitude. Thesolution of Jeng et al. (2002) obtained from the lin-earized Boussinesq equation may therefore be consid-ered as a special case of this newly derived solution.The effects of aquifer parameters and linearizationof the Boussinesq equation on the behaviour of thegroundwater level fluctuations in the coastal leakyaquifer system are examined. Finally, the issues oftide-induced submarine groundwater discharges inthe upper and lower aquifers and leakage flux throughthe aquitard are addressed.

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164 Mo-Hsiung Chuang et al.

2 PROBLEM SET-UP AND BOUNDARYCONDITIONS

Figure 1 shows a coastal leaky aquifer system with anupper unconfined aquifer, a semi-confined aquifer atthe bottom with an impermeable base, and an aquitardbetween them. The formations of both upper andlower aquifers are isotropic and homogenous. Theeffects of tidal fluctuation on the groundwater flowin both aquifers are considered. These two aquifersinteract with leakage through aquitard, which is lin-early proportional to the head difference between twoaquifers. Consider that the storage of the aquitard isvery small and negligible. The origin of the x-axis islocated at the intersection of the mean sea surface andbeach face. The x-axis is horizontal, positive land-ward and perpendicular to the coastal line and bothaquifers are assumed semi-infinite. The flow velocityin the aquifer system is essentially horizontal and thethickness of the unconfined aquifer below the meansea level is considered equal to or larger than the tidalamplitude.

Under those assumptions and using theBoussinesq equation for the upper aquifer andleaky confined flow equation for the lower aquifer,the governing equations of the head fluctuations inboth aquifers are, respectively (Bear 1972, Jiao andTang 1999), expressed as:

Sy∂W

∂t= K1

∂

∂x

(W

∂W

∂x

)+ L(H − W ) (1a)

S2∂H

∂t= T2

∂2H

∂x2+ L(W − H) (1b)

Water table fluctuations

Observation well

Tid

e A

, ω

Unconfined aquifer: Sy, K1

A

W

H

x

h SM

L =

D

Aquitard, Leakage: L

Confined aquifer: S2,T2

Impermeable Layer

Fig. 1 Schematic diagram of a coastal leaky aquifer systemcontaining an upper unconfined aquifer, a lower semi-confined aquifer, and an aquitard in between.

where W and H are the hydraulic heads in the upperand lower aquifers respectively, Sy and K1 are the spe-cific yield and hydraulic conductivity of the upperaquifer, respectively, S2 and T2 are the storativity andtransmissivity of the lower aquifer, respectively, andL is the specific leakage of the aquitard. The tidalboundary conditions for both aquifers are written as:

W (0, t) = H(0, t) = D + Aeiωt (2a)

where D is the thickness of the unconfined aquiferbelow mean sea level and A and ω are the tidalamplitude and frequency, respectively. The boundaryconditions for equations (1) and (2) at the remoteinland side may be written as:

∂W

∂x

∣∣∣∣x→∞

= ∂H

∂x

∣∣∣∣x→∞

= 0 (2b)

Li and Jiao (2003) showed that the mean groundwaterlevel of the aquifer system stands considerably abovethe mean sea level due to the water table over-height. They proved that in both unconfined andsemi-confined aquifers, when the specific leakage ofthe aquitard is larger than zero and x → ∞, the watertable over-height is equal to:

W (∞, t) − D = H(∞, t) − D

=(

D + T2

K1

)(√1 + αL

2− 1

) (2c)

where αL = A2K21

/(DK1 + T2)

2.

3 PERTURBATION SOLUTION

For the convenience of expression, dimensionlessparameters W = W

/D and H = H

/D are adopted

herein. The tidal amplitude-aquifer thickness ratioδ, denoted as the ratio of the tidal amplitude A tothe mean thickness of the unconfined aquifer D, ischosen as the perturbation parameter. Detailed deriva-tion of the perturbation solutions for the governingequations, equations (1a) and (1b), with appropriateboundary conditions, equations (2a)–(2c), is given inAppendix A and the solutions of W (x, t) and H(x, t)for the upper and lower aquifers are, respectively:

W (x, t) = DW

= D[1 + δW 1 + δ2W 2 + O(δ3)](3a)

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A perturbation solution for head fluctuations in a coastal leaky aquifer system 165

and

H(x, t) = DH

= D[1 + δH1 + δ2H2 + O(δ3)](3b)

where the solutions of W 1 and H1 are similar to thesolutions of Jeng et al. (2002) and can be expressed,respectively, as:

W 1 = Re[(a1e−λ1x + a2e−λ2x)eiωt] (4a)

and

H1 = Re[(b1e−λ1x + b2e−λ2x)eiωt] (4b)

and the solutions of W 2 and H2 are, respectively,written as:

W 2 = Re[(c1e−2λ1x + c2e−2λ2x + c3e−λ3x

+ c4eτ1x + c5eτ2x)e2iωt + c6e−αx

+ c7e−βx + B]

(4c)

and

H2 =Re[(d1e−2λ1x + d2e−2λ2x + d3e−λ3x

+ d4eτ1x + d5eτ2x)e2iωt + d6e−αx

+ d7e−βx + B]

(4d)

where Re denotes the real part of the complex expres-sion, i = √−1, and the coefficients a1, a2, b1, b2, c1

to c7, d1 to d7, λ1, λ2, λ3, τ 1, τ 2, α, β and B are,respectively, defined as:

a1 = −−Syiω + K1Dλ22

K1D(λ21 − λ2

2)(5a)

a2 = −Syiω + K1Dλ21

K1D(λ21 − λ2

2)(5b)

b1 = −T2λ22 − S2iω

T2(λ21 − λ2

2)(5c)

b2 = T2λ21 − S2iω

T2(λ21 − λ2

2)(5d)

c1 = −2a21λ

21

(2iS2ω − 4T2λ

21 + L

)E1

(5e)

c2 = −2a22λ

22

(2iS2ω − 4T2λ

22 + L

)E2

(5f)

c3 = −a1a2λ23

(2iS2ω − T2λ

23 + L

)E3

(5g)

c4 = −

{4c1λ21 + 4c2λ

22 + c3λ

23 − τ 2

2

(c1 + c2 + c3) + 2a21λ

21 + 2a2

2λ22

+a1a2λ23}

τ 21 − τ 2

2

(5h)

c5 =

{4c1λ21 + 4c2λ

22 + c3λ

23

−τ 21 (c1 + c2 + c3) + 2a2

1λ21

+2a22λ

22 + a1a2λ

23}

τ 21 − τ 2

2

(5i)

c6 = E4

α2 − β2

(T2 − L

α2

)(5j)

c7 =(

1 − T2β2

L

)[

E4 − B + E4L

α2(α2 − β2

)] (5k)

d1 = 2a21λ

21L

E5(5l)

d2 = 2a22λ

22L

E6(5m)

d3 = a1a2λ23L

E7(5n)

d4 =τ 2

2 (d1 + d2 + d3) − 4d1λ21

−4d2λ22 − d3λ

23

τ 21 − τ 2

2

(5o)

d5 = −τ 2

1 (d1 + d2 + d3) − 4d1λ21

−4d2λ22 − d3λ

23

τ 21 − τ 2

2

(5p)

d6 = E4 − B + E4L

α2(α2 − β2

) (5q)

d7 = −E4 − E4L

α2(α2 − β2

) (5r)

λ1 =√

1

2

(−b +

√b2 − 4c

)(5s)

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166 Mo-Hsiung Chuang et al.

λ2 =√

1

2

(−b −

√b2 − 4c

)(5t)

λ3 = λ1 + λ2 (5u)

τ1 = −√

bb +√

bb2 − 4cc (5v)

τ2 = −√

bb −√

bb2 − 4cc (5w)

α =√

L

T2(5x)

β =√

K1D + T2

K1DT2L(5y)

B = D2

A2

(K1D + T2

K1D

)(√1 + αL

2− 1

)(5z)

Note that the coefficients E1, E2 and E3 in equations(5e)–(5g) are, respectively, defined as:

E1 = 4T2K1Dλ21

(2iωSy + L

K1D+ 2iωS2 + L

T2−4λ2

1

)

+ 2SyS2ω

(2ω − iL

S2− iL

Sy

) (6a)

E2 = 4T2K1Dλ22

(2iωSy + L

K1D+ 2iωS2 + L

T2−4λ2

2

)

+ 2SyS2ω

(2ω − iL

S2− iL

Sy

) (6b)

E3 = T2K1Dλ23

(2iωS2 + L

T2+ 2iωSy + L

K1D−λ2

3

)

+ 2SyS2ω

(2ω − iL

S2− iL

Sy

) (6c)

The coefficient E4 in equations (5j), (5k), (5q) and (5r)is defined as:

E4 = BLβ2

Lβ2 − T2L − L2

T2

(6d)

The coefficients E5, E6, and E7 in equations (5l)–(5n)are, respectively, defined as:

E5 = −4T2λ21

(2iωSy + L

K1D+ 2iωS2 + L

T2−4λ2

1

)

+ 2SyS2ω

K1D

(2ω + iL

Sy+ iL

S2

) (6e)

E6 = −4T2λ22

(2iωSy + L

K1D+ 2iωS2 + L

T2−4λ2

2

)

+ 2SyS2ω

K1D

(2ω + iL

Sy+ iL

S2

) (6f)

E7 = −T2λ23

(2iωSy + L

K1D+ 2iωS2 + L

T2−λ2

3

)

+ 2SyS2ω

K1D

(2ω + iL

Sy+ iL

S2

) (6g)

The coefficients b and c in both equations (5s) and (5t)are, respectively, defined as:

b = −iω

(Sy

T1+ S2

T2

)− L

(1

T1+ 1

T2

)(6h)

c = −SyS2ω2

T1T2+ Liω

(Sy + S2

)T1T2

(6i)

Finally, the coefficients bb and cc in both equations(5v) and (5w) are, respectively, defined as:

bb = 2iω

(Sy

T1+ S2

T2

)+ T2 + K1D

L(6j)

cc = −4SyS2ω2

T1T2(6k)

3.1 Leakage flux and submarine groundwaterdischarges

The mean water levels in the aquifer system aredefined as (Li and Jiao 2003):

W = 1

P

t+P�t

W(x, τ)dτ (7a)

H = 1

P

t+P�t

H(x, τ)dτ (7b)

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A perturbation solution for head fluctuations in a coastal leaky aquifer system 167

where W and H are mean water levels in the upperand lower aquifers and P is the tidal period definedas 2π/ω. Substituting equations (3a) and (3b) intoequations (7a) and (7b), respectively, results in:

W = Aδ(c6e−αx + c7e−βx + B

)(8a)

H = Aδ(d6e−αx + d7e−βx + B

)(8b)

Based on Darcy’s law, the time interval leakageflux, FL(x), through the aquitard is defined as (Li andJiao 2003):

FL(x) = L(

W − H)

= LAδ[(c6 − d7)e

−αx

+ (c7 − d7)e−βx] (9)

The submarine groundwater discharges from theupper and lower aquifers, QU and QL, may, respec-tively, be calculated using following equations:

QU = K1

t+P�t

W (0, t) max

[0,

∂W

∂x

∣∣∣∣x=0

]dt (10a)

QL = T2

t+P�t

max

[0,

∂H

∂x

∣∣∣∣x=0

]dt (10b)

Note that these definitions are different from thosedefined by Li and Jiao (2003).

4 RESULTS AND DISCUSSION

In this section, some examples are used to illus-trate the use of the present solution and investigatethe effects of aquifer and perturbation parameterson the head fluctuation, leakage flux and submarinegroundwater discharges in a coastal leaky aquifersystem. To describe the problem clearly, two newvariables are defined, i.e. w = (W – D)/A and h = (H –D)/A in which w and h represent normalized headfluctuations in the upper and lower aquifers, respec-tively. Jeng et al.’s (2002) solution obtained fromthe linearized Boussinesq equation can be consideredas a special case of the present solution by neglect-ing the second-order perturbation, δ2. The results bythe present solution are compared with those of Jenget al.’s solution to explore the second-order effect.The difference between these two solutions is defined

as Dw = (wp – wJ) for the upper aquifer and Dh =(hp – hJ) for the lower aquifer where the subscriptsp and J denote the present solution and Jeng et al.’ssolution, respectively. For comparison, the values ofaquifer parameters are chosen the same as thosein Jeng et al.’s study, i.e. Sy = 0.3, S2 = 0.001,D = 10 m, K1 = 200 m/d, T2 = 2000 m2/d, L = 1/d,A = 2.5 m and ω = 2π rad/d.

4.1 The effect of tidal amplitude-aquiferthickness ratio on head fluctuation

Figure 2 shows the spatial distributions of w and h att = 24 h for L = 0.2 d-1 and A = 0.1, 1.5 and 2.5 m(i.e. δ = 0.01, 0.15 and 0.25). The temporal fluctua-tions of w and h at x = 100 m are presented in Fig. 3for L = 0.2 d-1 and δ = 0.01, 0.15 and 0.25. Figures 2and 3 indicate that the normalized groundwater headspredicted by Jeng et al.’s (2002) solution are close tothose of the present solution when δ is equal to 0.01.As δ increases, the difference between the presentsolution and Jeng et al.’s (2002) solution increase.The difference between the present solution and Jenget al.’s (2002) solution comes from the term δ2W 2

for unconfined aquifers shown in equation (3a) andδ2H2 for semi-confined aquifers in equation (3b). Forlarge values of δ, δ2 becomes very large and the differ-ence between these two solutions is very significant.Therefore, the error from the solution by linearized

lower aquifer

δ = 0.25δ = 0.15δ = 0.01

Nor

mal

ized

gro

undw

ater

hea

d

upper aquifer

0

-0.2

0

0.2

0.4

0.6

0.8

1

200 400 600Inland distance x (m) from the coastline

Jeng et al.'s (2002) solution

Present solution

Fig. 2 Spatial distributions of normalized groundwaterheads, w = (W – D)/A and h = (H – D)/A and att = 24 h with L = 0.2 d-1 and different values of δ.

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168 Mo-Hsiung Chuang et al.

0

-0.4

-0.2

0

0.2

0.4

0.6

4 8 12 16 20 24

Time t (hr)

Nor

mal

ized

gro

undw

ater

hea

d

lower aquifer

upper aquifer

present solution

δ = 0.25δ = 0.15δ = 0.01

Jeng et al.'s (2002) solution

Fig. 3 Temporal fluctuations of normalized groundwaterheads, w = (W – D)/A and h = (H – D)/A at x = 100 mwith L = 0.2 d-1 and different values of δ.

Boussinesq equation increases with δ. Figure 2 alsodemonstrates that the difference increases with thelandward distance from coastline and approaches aconstant when the landward distance is beyond thethreshold value. Moreover, Fig. 3 shows that the timelag is independent of δ in this case.

4.2 The effects of leakage and transmissivity onhead fluctuation

Figure 4 shows the lines of dimensionless head differ-ence (Dw and Dh) versus δ at x = 100 m and t = 24 hwhen L = 1, 0.5 and 0.05 d-1. This figure indicatesthat both Dw and Dh increase linearly with δ at differ-ent leakages: Dw increases with decreasing leakage inthe upper aquifer, while Dh increases with leakage inthe lower aquifer. In addition, the influence of leakageon Dh is larger than that on Dw. Figure 4 also dis-plays the slopes of the lines in the lower aquifer aresmaller than those in the upper aquifer when the leak-age ranges from 0.05 to 1.0 /day. Based on equation(2c), the value of water table over-height is indepen-dent of the aquifer storativities Sy and S2. Therefore,only the effect of transmissivity on the head fluctu-ations in both upper and lower aquifers is discussedherein. The curves of Dw and Dh versus x for differ-ent transmissivity ratios (Tr = K1D/T2) are shown inFig. 5 for L = 0.5 d-1, t = 24 h and δ = 0.25. Figure 5indicates that Tr has a significant effect on Dw and

0

0

0.01

0.02

Dim

ensi

onle

ss h

ead

diff

eren

ce

0.03

lower aquifer

upper aquifer

L = 1 / dayL = 0.5 / dayL = 0.05 / day

L = 1 / dayL = 0.5 / dayL = 0.05 / day

0.04

0.1Perturbation parameter δ

0.2 0.3

Fig. 4 Dimensionless head differences in the upper andlower aquifers versus perturbation parameter at x = 100 mand t = 24 h for different values of leakage.

0

0

0.01

0.02

Dim

ensi

onle

ss h

ead

diff

eren

ce

0.03

0.04

0.05

0.06

200

lower aquifer

Tr = 5

Tr = 1Tr = 0.5

Tr = 5

Tr = 1Tr = 0.5

upper aquifer

Inland distance x (m) from the coastline

400 600

Fig. 5 Dimensionless head differences in the upper andlower aquifers versus inland distance at t = 24 h withL = 0.2 d-1, δ = 0.25, and different transmissivity ratios(Tr).

Dh. This figure demonstrates that Dh becomes con-stant when x > 500 m for Tr ranging from 0.5 to 5;however, Dw becomes constant when x > 100 m forTr = 0.5, x > 150 m for Tr = 1 and x > 200 m forTr = 5. Figure 5 also shows that both Dw andDh increase with Tr, but the threshold distance is

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A perturbation solution for head fluctuations in a coastal leaky aquifer system 169

0

0

0.01

0.02

0.03

present solution

Li and Jiao's (2003) solution

L = 0.2L = 0.5L = 1

L = 0.2L = 0.5L = 1

100 200

Landward distance x (m) from the coastline

Lea

kage

flu

x F

L (

m/d

ay)

300 400

Fig. 6 Leakage flux FL versus landward distance from thecoastline.

independent of Tr for the lower aquifer; however, itincreases with Tr for the upper aquifer.

4.3 Investigation on the changes of the leakageflux and submarine groundwater discharge

Figure 6 shows the leakage flux, FL versus landwarddistance, x, with different values of the leakage L forthe present solution and Li and Jiao’s (2003) solu-tion. This figure depicts that the leakage flux is zeroat both aquifers and increases gradually with the land-ward distance, reaches a maximum value at somedistance, and then decreases predicted by both solu-tions. The maximum leakage flux occurs almost atthe same landward distance for all curves. In addi-tion, the present solution predicts lower leakage fluxesthan those of Li and Jiao’s (2003) solution whenx < 100 m, implying that the leakage flux may beoverestimated if based on the average head fluctua-tions in the upper and lower aquifers. It is interestingto note that the leakage flux increases with L near thecoastal line and decreases with increasing L as thedistance increases.

Figure 7 shows the submarine groundwater dis-charge from the lower aquifer, i.e. QL versus L withdifferent δ estimated from the present method andLi and Jiao’s (2003) method. Those results shownin the figure indicate that the present method giveshigher estimations than those of Li and Jiao’s (2003)method. This figure also shows that the difference ofQL values predicted by these two solutions increases

0

0

5

10

QL

(m2 /d

ay)

15

20

25present solution

δ = 0.25δ = 0.15δ = 0.1

δ = 0.25δ = 0.15δ = 0.1

Li and Jiao's (2003) solution

0.2 0.4

Leakage L (1/day)0.6 0.8 1

Fig. 7 Submarine groundwater discharge from the loweraquifer, QL versus leakage L with different values of δ forthe present solution and Li and Jiao’s (2003) solution.

with L. In addition, QL estimated from both solutionsincreases with δ and the rate of increase in QL alsoincreases with δ.

Figure 8 shows the discharge from the upperaquifer, i.e. QU versus L with different δ for thepresent method and Li and Jiao’s (2003) method. Thisfigure demonstrates that QU decreases with increasing

80

present solution Li and Jiao's (2003) solutionδ = 0.25

δ = 0.15δ = 0.1

δ = 0.25

δ = 0.15δ = 0.1

60

40

QU

(m2 /d

ay)

20

0

0 0.2 0.4Leakage L (1/day)

0.6 0.8 1

Fig. 8 Submarine groundwater discharge from the upperaquifer, QU versus leakage L with different values of δ.

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170 Mo-Hsiung Chuang et al.

L for all δ and increases with δ for all leakage.However, the rate of decrease in QU is higher for thepresent solution than that for Li and Jiao’s (2003)solution. At all δ and L, the present solution giveshigher predictions for QU than Li and Jiao’s (2003)solution.

5 CONCLUDING REMARKS

This paper presents a nonlinear mathematical model,with the Boussinesq equation for the upper uncon-fined aquifer and leaky confined flow equation forthe lower aquifer, to simulate head fluctuation fora coastal leaky aquifer system, which contains anupper unconfined aquifer, an aquitard, and a lowersemi-confined aquifer. The solution of the modelcan describe the tide-induced head fluctuation in acoastal leaky aquifer system for the case that the tidalamplitude–aquifer thickness ratio is large. This solu-tion can be considered as a generalization of Jenget al.’s (2002) solution, which was derived from thelinearized Boussinesq equation. In both upper andlower aquifers, the head fluctuation predicted by thepresent method is generally greater than that of Jenget al.’s (2002) solution. The difference between thesetwo solutions increases with both the inland dis-tance and the magnitude of tidal amplitude–aquiferthickness ratio when the distance is smaller than athreshold. However, the difference increases as theleakage decreases in the upper aquifer while the dif-ference increases with leakage in the lower aquifer.The effect of the transmissivity ratio (Tr) on the dif-ference is significant and the differences increase withTr in both upper and lower aquifers. Both the max-imum leakage flux and the submarine groundwaterdischarge from the lower aquifer may be overesti-mated if the discharge is calculated based on thedifference of average head fluctuation between theupper and lower aquifers. In addition, both subma-rine groundwater discharges from upper and loweraquifers increase with the tidal amplitude–aquiferthickness ratio. However, the discharge from theupper aquifer decreases with increasing leakage whilethe one from the lower aquifer increases with leakage.

Acknowledgements This study was partly sup-ported by the Taiwan National Science Council underthe grants NSC 98-3114-E-007-015, NSC 99-NU-E-009-001, and NSC 99-2221-E-009-062-MY3 and the“Aim for the Top University Plan” of the NationalChiao Tung University and Ministry of Education,Taiwan. The authors would like to thank the Associate

editor and two reviewers for their valuable and con-structive comments. In addition, we are grateful to thereviewer Dr Hailong Li for proposing equations (10a)and (10b) and suggesting comparisons between ournew results and those of Li and Jiao (2003).

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Jiao, J.J. and Tang, Z., 1999. An analytical solution of groundwaterresponse to tidal fluctuation in a leaky confined aquifer. WaterResources Research, 35 (3), 747–751.

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Li, H.L. and Jiao, J.J., 2001a. Analytical studies of groundwater-headfluctuation in a coastal confined aquifer overlain by a leaky layerwith storage. Advances in Water Resources, 24 (5), 565–573.

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Li, H.L. and Jiao, J.J., 2002. Numerical studies of tide-induced groundwater level change in a multi-layered coastalleaky aquifer system. In: XIV International Conference onComputational Methods in Water Resources, Delft, TheNetherlands.

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APPENDIX A

Derivation of the solutions

Define the dimensionless parameters W = W/

D,H = H

/D, and δ = A/D. Substituting these param-

eters into equations (1) and (2), the governing equa-tions and boundary conditions can then be obtainedas:

Sy∂W

∂t= K1D

∂

∂x

(W

∂W

∂x

)+ L

(H − W

)(A1)

S2∂H

∂t= T2

∂2H

∂x2+ L

(W − H

)(A2)

W(0, t) = H (0, t) = 1 + δeiωt (A3)

W(∞, t) = H (∞, t)

= 1 + 1

D

(D + T2

K1

)(√1 + αL

2− 1

) (A4)

The perturbation method is adopted herein to developthe solution of equations (A1) and (A2). The follow-ing two assumptions are used to obtain W and H :

W = W 0 + δW 1 + δ2W 2 + O(δ3) (A5)

H = H0 + δH1 + δ2H2 + O(δ3) (A6)

where δ = A/D is the perturbation parameter.Substituting both equations (A5) and (A6) into equa-tions (A1) and (A2) results in following equations for(δ0

):

Sy∂W 0

∂t= K1DW 0

∂2W 0

∂x2+ K1D

(∂W 0

∂x

)2

+ L(H0 − W 0

) (A7)

S2∂H0

∂t= T2

∂2H0

∂x2+ L

(W 0 − H0

)(A8)

W 0 (0, t) = H0 (0, t) = W 0 (∞, t)

= H0 (∞, t) = 1(A9)

Accordingly, we get the results of W 0 = H0 = 1. For(δ1

)the governing equation and associated boundary

conditions are:

Sy∂W 1

∂t= K1D

∂2W 1

∂x2+ L

(H1 − W 1

)(A10)

S2∂H1

∂t= T2

∂2H1

∂x2+ L

(W 1 − H1

)(A11)

W 1 (0, t) = H1 (0, t) = δ eiωt (A12)

W 1 (∞, t) = H1 (∞, t) = 0 (A13)

The solution for equations (A10)–(A13) is obtainedusing the method of separation of variables (Jenget al. 2002); the results are expressed in equations (4a)and (4b). For

(δ2

)the related governing equation and

its associated boundary conditions are:

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172 Mo-Hsiung Chuang et al.

Sy∂W 2

∂t= K1D

∂2W 2

∂x2

+ K1D

[W 1

∂2W 1

∂x2+

(∂W 1

∂x

)2]

+ L(H2 − W 2

) (A14)

S2∂H2

∂t= T2

∂2H2

∂x2+ L

(W 2 − H2

)(A15)

W 2 (0, t) = H2 (0, t) = 0 (A16)

W 2 (∞, t) = H2 (∞, t) = B (A17)

Based on the method of separation of variables, thevariables W 2 and H2 are assumed as:

W 2 = F1 (x) e2iωt + F2 (x) (A18)

H2 = G1 (x) e2iωt + G2 (x) (A19)

Substituting equations (A18)–(A23) into equations(A14) and (A15) and using boundary condition, equa-tions (A16) and (A17), leads to the following results:

F1 (x) = c1e−2λ1x + c2e−2λ2x + c3e−λ3x

+ c4eτ1x + c5eτ2x (A20)

F2 (x) = c6e−αx + c7e−βx + B (A21)

G1 = d1e−2λ1x + d2e−2λ2x + d3e−λ3x

+ d4eτ1x + d5eτ2x (A22)

G2 (x) = d6e−αx + d7e−βx + B (A23)

where coefficients c1 to c7, d1 to d7, λ1, λ2, λ3, τ 1,τ 2, α, β and B are defined as equations (5e)–(5z),respectively, in the text.

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