Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue8.2000:

A two-dimensional analytical solution of groundwater responses to

tidal loading in an estuary and ocean

L. Li

a,*

, D.A. Barry

a

, C. Cunningham

a

, F. Stagnitti

b

, J.-Y. Parlange

c a

School of Civil and Environmental Engineering and Contaminated Land Assessment and Remediation Research Centre, The University of Edinburgh, Edinburgh EH9 3JN, UK

b

School of Ecology and Environment, Deakin University, Warrnambool, Victoria 3280, Australia

c

Department of Agricultural and Biological Engineering, Cornell University, Ithaca, NY 14853-5701, USA Received 24 February 2000; accepted 8 March 2000

Abstract

Previous studies on tidal dynamics of coastal aquifers have focussed on the inland propagation of oceanic tides in the cross-shore direction, a con®guration that is essentially one-dimensional. Aquifers at natural coasts can also be in¯uenced by tidal waves in nearby estuaries, resulting in a more complex behaviour of head ¯uctuations in the aquifers. We present an analytical solution to the two-dimensional depth-averaged groundwater ¯ow equation for a semi-in®nite aquifer subject to oscillating head conditions at the boundaries. The solution describes the tidal dynamics of a coastal aquifer that is adjacent to a cross-shore estuary. Both the e€ects of oceanic and estuarine tides on the aquifer are included in the solution. The analytical prediction of the head ¯uctuations is veri®ed by comparison with numerical solutions computed using a standard ®nite-di€erence method. An essential feature of the present analytical solution is the interaction between the cross- and along-shore tidal waves in the aquifer area near the estuaryÕs entry. As the distance from the estuary or coastline increases, the wave interaction is weakened and the aquifer response is reduced, re-spectively, to the one-dimensional solution for oceanic tides or the solution of Sun (Sun H. A two-dimensional analytical solution of groundwater response to tidal loading in an estuary, Water Resour Res 1997;33:1429±35) for two-dimensional non-interacting tidal waves. Ó _{2000 Elsevier Science Ltd. All rights reserved.}

Keywords:Tidal groundwater ¯uctuation; Estuarine tides; Oceanic tides; Two-dimensional solution

1. Introduction

Tide-induced head ¯uctuations are a natural phenomenon in a coastal aquifer. For an uncon®ned aquifer, this leads to water table ¯uctuations corre-sponding to tidal frequencies. As these periodic ¯uctu-ations propagate inland, their amplitude is attenuated and phase-shifts occur. A typical damping distance for tidal water table ¯uctuations in an uncon®ned aquifer is several hundred meters whereas the tidal in¯uence on a con®ned aquifer can extend landward by several thou-sand meters [7]. In both cases, the tidal ¯uctuations af-fect largely the groundwater ¯ow and mass transport in the aquifer [9]. Previous studies on aquifersÕ tidal dy-namics have focussed on the inland transmission of tidal sea level oscillations in the cross-shore direction

[1,8,11,15]. Analytical solutions based on the one-di-mensional Boussinesq equation are often used to de-scribe the tidal head or water table ¯uctuations (also referred to as tidal waves in the aquifer) [2,12,13].

Recently, Sun [14] considered the aquiferÕs responses to tidal oscillations in an estuary, in which case the propagation of the tide in the aquifer becomes a two-dimensional problem because the tidal loading varies along the estuary, i.e., a non-uniform boundary con-dition

h…x;0;t† ˆAexp_ÿjerx†cos…xtÿjeix†; …1_†

whereh…x;0;t†is the ¯uctuation of the water level in the estuary (the seepage face at the aquifer±estuary interface has been neglected);Aandxare the tidal amplitude and frequency, respectively; and x is the distance along the estuary from the entry (Fig. 1(d)). jer and jei are the amplitude damping coecient and wave number of the tidal wave in the estuary, respectively. The former is related to the resistance coecient due to bed friction www.elsevier.com/locate/advwatres

*

Corresponding author. Tel.: 5814; fax: +44-131-650-5814.

E-mail address:ling.li@ed.ac.uk (L. Li).

and the celerity of the estuarine tidal wave, whereas the latter can be calculated for given tidal frequency and wave celerity [4]. A two-dimensional analytical solution was obtained for the resulting tidal head ¯uctuations in the aquifer [14]

hx;y;t_{† ˆ}Re_fAexp_‰ÿ…jar‡ijai†yŠ

exp i‰xtÿ…jer‡ijei†xŠg; …2†

whereh…x;y;t†is the head ¯uctuation; yis the distance from the aquifer±estuary interface Fig. 1(d); `Re' de-notes the real part of the function; and i_ˆ 

ÿ1

p

.jarand

jai are the amplitude damping coecient and wave number of the tidal head ¯uctuations in the aquifer, respectively; both are constants (see Eqs. (14) and (15) of Sun [14] for the formulae). The solution, which is clearly in a variables-separable form, indicates that the head ¯uctuations are simply due to the one-dimensional

propagation (in the y direction) of the local tidal oscil-lations in the estuary (varying in the xdirection).

The analytical solution of Sun [14] describes only the aquiferÕs response to the tidal loading in the estuary. In reality, the oceanic tides along the coastline also in¯u-ence the aquifer and thus interfere with the transmission of the estuarine tides in the aquifer (Fig. 1). In other words, the propagation of oceanic and estuarine tidal signals in the coastal aquifer interacts with each other. Li et al. [10] have demonstrated that such interactions lead to more complex patterns of tidal head ¯uctuations in the aquifer than predicted by Eq. (2). Since the damping of both oceanic and estuarine tidal oscillations increases with the distance inland, their interactions are weakened as either the distance from the shore or that from the estuary increases. However, the interaction zone (the area where the tidal wave interaction remains signi®cant) can be very large at a natural coast. For a con®ned aquifer, this area may be of several square kilometers. Obviously, the interaction zone is a€ected by the angle at which the estuary intersects with the coastline (h). In Fig. 1(a±c), sketches of the interaction zone are shown for di€erenth. Except for a very smallh, the interaction between the oceanic and estuarine tidal head ¯uctuations cannot be neglected. SunÕs [14] solu-tion is, therefore, inadequate in describing the tidal dy-namics of coastal aquifers except for large distances from the coastline.

The purpose of this paper is to derive an analytical solution for tidal head ¯uctuations in an aquifer a€ected by both oceanic and estuarine tides. A right angle be-tween the estuary and coastline (i.e., hˆ90) will be assumed for the purpose of simplicity. The present solution will be more general than that of Sun [14]. In particular, the latter can be seen as a solution deduced from the present one for areas far from the coastline.

The paper is organised as follows: ®rst, the analytical solution is derived. We then demonstrate that this solution reduces to the one-dimensional oceanic

Notation

A tidal amplitude [L]

h head ¯uctuation [L]

L damping distance [L]

S aquifer storativity

T aquifer transmissivity [L2_Tÿ1_]

Tt tidal period [T]

t time [T]

tc elapsed time for diminished e€ects of initial conditions [T]

x cross-shore distance from the shoreline [L]

y along-shore distance from the estuary [L]

/ phase of tidal wave [rad]

jar amplitude damping coecient of

two-dimensional non-interacting tidal wave in the aquifer [Lÿ1_]

jai wave number of two-dimensional

non-interacting tidal wave in the aquifer [Lÿ1_]

jaro amplitude damping coecient of

one-dimensional tidal wave in the aquifer [Lÿ1_]

jaio wave number of one-dimensional tidal wave in the aquifer [Lÿ1_]

jer amplitude damping coecient of tidal wave in the estuary [Lÿ1_]

jei wave number of tidal wave in the estuary [Lÿ1]

h angle between the estuary and coastline [rad]

x tidal frequency, 2p/T_t [rad Tÿ1]

tide-driven solution for large distances from the estuary and to the solution of Sun [14] for large distances from the coastline. The solution is veri®ed against numerical simulations using the Crank±Nicolson method. Based on the solution, the characteristics of the tidal head ¯uctuations are examined also, focussing on the inter-action between the inland propagation of oceanic tides in the cross-shore direction and the transmission of estuarine tide in the along-shore direction.

2. Problem setup

The two-dimensional groundwater ¯ow equation averaged over the thickness of the aquifer under the Dupuit assumption [3] is

S

whereSand Tare the storativity and transmissivity of the aquifer, respectively; andxandyare the cross- and along-shore coordinates Fig. 1(d). Eq. (3) governs the head ¯uctuations and is applicable to both con®ned and uncon®ned aquifers. In the latter application, the equation has been linearised and hence is applicable when the tidal amplitude is relatively small with respect to the mean aquifer thickness [3]. For a con®ned aquifer,

Sis much smaller than that for an uncon®ned aquifer. This leads to a much-enhanced inland propagation of the tidal waves in the con®ned aquifer compared with that in the uncon®ned aquifer.

The boundary conditions along the coastline vary with the ocean tide, i.e.,

h…0;y;t† ˆAcosxt†: …4_†

Only one tidal constituent is considered for the purpose of simplicity; however, others can easily be included subsequently using the principle of superposition [14]. The beach slope and seepage face dynamics are also ignored. Along the coastline, the tidal amplitude and phase vary much less than those of tidal waves in the estuary and thus have been assumed to be spatially constant in Eq. (4) [6]. The boundary conditions along the estuary are speci®ed by Eq. (1). Far inland, the tidal e€ects are diminished and so

lim

x!1h…x;y;t† ˆ0: …5†

Physically, this condition also means that we consider only tidal e€ects and assume that no net ¯ow exists. Away from the estuary, the e€ects of the estuarine tide become negligible and thus the boundary condition there is determined by the cross-shore propagation of the ocean tide alone, i.e., the one-dimensional solution to the Boussinesq equation [10]

lim

y!1h…x;y;t† ˆAexp… ÿjarox†cos…xtÿjaiox†; …6a†

where jaro and jaio are the amplitude damping coef-®cient and wave number of the one-dimensional tidal head ¯uctuations. According to Ferris [5], these parameters are given by

jaroˆjaioˆ

Due to the interaction between cross- and along-shore head ¯uctuations, the separation-of-variables method used by Sun [14] is not applicable here. Instead, we shall employ GreenÕs function method to solve Eqs. (3)±(6b). First, the following decomposition is taken

h…x;y;t† ˆh0…x;t† ‡h1…x;y;t†; …7a† method (see Appendix A), we ®nd thath1is given by the double integral tion (see Appendix A). The initial head ¯uctuation has been assumed toh0…x;0†, i.e.,h1…x;y;0† ˆ0. Expressing h1…x0;0;t0† as Re_fAexpixt0‡k1x0† ÿAexpixt0

h1x;y;t_{† ˆ}ARe

the integrand is non-singular and the integral can be easily evaluated via any standard quadrature scheme.

As mentioned already, the analytical solution for multiple tidal constituents can be obtained by superim-posing the solution for each constituent, which has es-sentially the same form as the general solution, i.e., Eqs. (7a), (9a) and (9b). The phase di€erence between the tidal components can be incorporated directly into the general solution. Alternatively, one can just modify the initial time for the solution of each tidal constituent according to the phase di€erence.

3.1. Periodicity of the solution

Physically, the solution is expected to become per-iodic as t increases. In other words, the e€ects of an initial condition on the solution are diminished after a certain time (memory time, tc) elapses. A dimensional analysis shows that tc/1=x. It is interesting to note that the transmissivity and storativity seem to have no e€ect on the memory time, tc. In our calculation, we found that it was sucient to ensure the solution_Õs periodicity if t was three times larger than the tidal period (Tt).

In Fig. 2(a), we show the head ¯uctuations at a lo-cation of the aquifer (x_ˆ600 m andy_ˆ600 m) for the ®rst four tidal periods as predicted by the analytical solution, i.e., Eqs. (7a), (9a) and (9b). The parameter values used in the calculations are listed in Table 1 and a con®ned aquifer was considered. The di€erences of the predicted head ¯uctuations between two subsequent periods are shown in Fig. 2(b). The results indicate that the solution became periodic as t increased and the memory time of the initial conditions may be deter-mined to be three tidal periods (3Tt).

3.2. Solutions for large x and y

For largex, Eqs. (9a) and (9b) can be simpli®ed to

has been applied. Furthermore, both integrals in Eq. (10) can be evaluated explicitly to be

exp _ÿ

Fig. 2. Periodicity of the solution: (a) head ¯uctuation for the ®rst four tidal periods and (b) di€erences of the predicted head ¯uctuations between two subsequent periods.

Table 1

respectively, given thattis large. Therefore, the periodic solution for largexis

h…x;y;t† ˆAexp ixt‡k1xÿ

 ixÿk2

1D

D r

y !

: …11†

Note that



…ixÿk2 2D† D r

ˆ0:

Also one can show that



…ixÿk2 1D† D r

ˆjar‡jaii:

Therefore, Eq. (11) is the same solution as that of Sun [14], i.e., Eq. (2).

For large y; h1x;y;t_† approaches zero and the sol-ution becomes hx;y;t_{† ˆ}h0x;t_†. In summary, the present analytical solution is reduced to the one-dimensional solution for oceanic tides (i.e., Eqs. (6a) and (6b)) for largeyand the solution of Sun [14] for largex. A comparison of these solutions is given in Section 4.

3.3. Comparison between the present analytical solution

and numerical predictions (t>tc)

To validate the present analytical solution, we solved Eq. (3) numerically using the Crank±Nicolson method. The parameter values used in the simulation are as listed in Table 1. Periodic solutions were considered, i.e., t>tc. The results of the water-table ¯uctuations are compared with the analytical solution at various loca-tions: near the estuary (Fig. 3), near the coastline (Fig. 4), far from the estuary (Fig. 5), and far from the coastline (Fig. 6). As discussed in Section 4, the head ¯uctuations behave di€erently in di€erent areas of the

aquifer. In all cases, the analytical and numerical solu-tions agree very well with each other.

4. Discussion of the analytical solution (t_>t_c)

4.1. Simulation of single tidal constituent

Compared with that of Sun [14], the present solution appears to be complicated and, indeed, it exhibits more complex tide-driven dynamics within the aquifer. The complexity re¯ects the interactions between the cross-and along-shore tidal head ¯uctuations in the aquifer, which are induced by oceanic and estuarine tides, Fig. 3. Comparison of the tidal head ¯uctuations in areas near the

estuary predicted by the analytical solution and the numerical simu-lation. Results from four locations are examined.

Fig. 4. Comparison of the tidal head ¯uctuations in areas near the coastline predicted by the analytical solution and the numerical sim-ulation. Results from four locations are examined.

respectively. In Fig. 7, we compare the head ¯uctuations as predicted by Eqs. (7a), (9a) and (9b) with non-inter-acting cross- and along-shore tidal waves, and the sum of both at three locations near the entry of the estuary. Non-interacting cross- and along-shore head ¯uctua-tions were calculated using the one-dimensional solution (i.e., Eqs. (6a) and (6b)) and the solution of Sun [14],

respectively. The same parameter values as listed in Table 1 were used in the calculation. Di€erences exist between the prediction of the present analytical solution and the calculated sum of non-interacting cross- and along-shore tidal waves. This suggests that the tidal wave interactions are non-linear, in the sense that the net e€ect is not simply the sum of the two tide-driven wave signals (due to the boundary conditions at the estuary and coastline).

The wave interaction, however, is weakened as the distance from the ocean or estuary increases. Away from the estuary (large y), the aquiferÕs head ¯uctuations be-come dominated by the oceanic tide while the estuarine tide controls the aquiferÕs responses in areas far from the shoreline (largex). It has been demonstrated in Section 3 that, for largey, the head ¯uctuation can be predicted by the one-dimensional analytical solution to the Boussinesq equation subject to oceanic tides, i.e., Eqs. (6a) and (6b). An example of this is shown Fig. 8. On the other hand, Sun_Õs [14] solution provides the prediction of the head ¯uctuation for largex(Fig. 9). As discussed in Section 1, the weakening of the wave in-teraction is due to the damping of either cross- or along-shore tidal head ¯uctuations with the distance from the coastline or the estuary. Thus, the cross- or along-shore distance (L), over which the tidal wave interaction be-comes insigni®cant, is related inversely to the rate of tidal wave damping in the aquifer, i.e.,

Lˆ 4:5 jaroˆ

4:5  T

p

 Sx

p ; …12_†

where 0.01Ahas been used as the amplitude of damped tidal waves at the threshold. For the present simulation (Table 1),Lis calculated to be 1670 m.

Fig. 7. Non-linear interaction between the cross- and along-shore tidal waves as predicted by the analytical solution. Results at three locations near the entry of the estuary are displayed: (a)xˆ300 m andyˆ 600 m; (b)xˆ600 m andyˆ600 m and (c)xˆ600 m andyˆ300 m.

Fig. 8. The behaviour of the solution for a largey(_ˆ 3000 m, i.e., far from the estuary): (a)xˆ300 m; (b)xˆ600 m and (c)xˆ900 m. Fig. 6. Comparison of the tidal head ¯uctuations in areas far from the

From Eqs. (7a), (9a) and (9b), the amplitude and phase of the local tidal waves can be calculated. The contour plot of the amplitude is shown in Fig. 10(a). The results indicate that the aquifer can be divided into four zones of qualitatively di€erent tidal-wave behaviour. The dividing lines are drawn according to the critical distance, L. In zone I, there is a strong interaction be-tween the cross- and along-shore tidal waves as shown by the non-linear contour lines. This is the interaction zone as discussed in Section 1. Cross-shore tidal waves, as induced by oceanic tide, dominate in zone II. Contour lines are straight and parallel to the coastline, indicating a uniform damping of tidal head ¯uctuations una€ected by the along-shore waves. Zone III is controlled by along-shore waves due to the estuarine tide. Since the tidal wave in the estuary is damped with the distance from the entry, the amplitude of the head ¯uctuations decreases with both x and y. The contour lines tilt to-wards the estuary as they extend landward. In zone IV,

Fig. 10. Aquifer zoning based on the characteristics of the head ¯uctuations: (a) contour plot of log [A…x;y†], whereA…x;y†is the amplitude of local tidal head ¯uctuations and (b) three-dimensional plot of the phase of the tidal head ¯uctuations.

tidal waves are small and may be neglected. Note that in zones II and III, the present solution can be reduced to the one-dimensional Bousinessq and Sun_Õs [14] two-di-mensional analytical solutions, respectively. Such sim-pli®cation has been shown analytically and is consistent with the behaviour of the tidal waves in the aquifer.

Similar characteristics of the tidal head ¯uctuations are also shown in the plot of phase variations (Fig. 10(b)). In particular, a strong coupling of cross- and along-shore wave propagation is evident in zone I, the e€ects of which extend to zone IV. In contrast, these two processes become decoupled in zones II and III.

4.2. Simulation of multiple tidal constituents

In this section, we consider a diurnal and semi-diur-nal tide. The same parameter values as listed in Table 1 of Sun [14] were used in the calculation. The results are compared with Sun_Õs analytical solution at two loca-tions: xˆ600 m and yˆ2000 m (Fig. 11(a)), and xˆ8000 m and y ˆ2000 m (Fig. 11(b)). Again, it is found that Sun_Õs solution fails to describe the aquifer's tidal responses in areas near the estuaryÕs entry due to the lack of tidal wave interactions. Away from the coastline, such interactions are weakened and both solutions become equivalent to each other.

5. Concluding remarks

We have derived a two-dimensional analytical solu-tion for head ¯uctuasolu-tions in a coastal aquifer in¯uenced by both oceanic and estuarine tides. The solution pro-vides ``exact'' predictions of the tidal dynamics in an aquifer that is adjacent to a cross-shore estuary. It can also be used to describe approximately the tidal

re-sponses of aquifers next to estuaries intersecting the coastline at large angles. These aquifers are common occurrence at natural coasts but have not been consid-ered in previous research [14].

An important feature of the present solution is its inclusion of the interaction between the cross- and along-shore tidal head ¯uctuations in the aquifer area near the estuary_Õs entry. Far from the river or the estuary, the wave interaction becomes weakened and the solution is reduced to the one-dimensional solution to the Bous-sinesq equation or the solution of Sun [14] for two-dimensional non-interacting tidal waves in the aquifer.

Compared with that of Sun [14], the present solution is more general. Apart from the tidal wave interaction, it includes the e€ects of initial conditions. The scenario considered here is more complicated than that in Sun [14]. It can be applied directly, or be used as a test case for testing numerical models.

Appendix A. GreenÕs function solution of the two-dimen-sional depth-averaged groundwater ¯ow equation

Eq. (3) is rewritten in terms of x0…>0_†;y0…>0_† and

The adjoint equation corresponding to Eq. (A.1) for calculation of Green_Õs function is

D o

whered is the Dirac delta function.

Integrating, by parts, [A1_† G_ÿA2_† h] over Under the following boundary conditions,

h0;y;t_{† ˆ}Wy;t_†; …A:4a_† Fig. 11. Simulation results for diurnal and semi-diurnal tides: (a)

hx;0;t_{† ˆ}Ux;t_†; …A:4b_†

G x… ;y;t;0;y0;t0† ˆ0; …A:4c†

G x… ;y;t;x0;0;t0† ˆ0; …A:4d†

Eq. (A.3) becomes

h…x;y;t† ˆ

Z 1

0

Z 1

0

G x… ;y;t;x0;y0;0†h x0… ;y0;0†dx0dy0

‡D Z t

0

Z 1

0

W…y0;t0†_ooG

x0…x;y;t;0;y0;t0†dy0dt0

‡D Z t

0

Z 1

0

U…x0;t0†_ooG

y0…x;y;t;x0;0;t0†dx0dt0:

…A:5†

The appropriate GreenÕs Function is

G x… ;y;t;x0;y0;t0† ˆ 1

4pD t… ÿt0†

exp ÿ‰…xÿx0† 2

‡ …yÿy0†2Š

4Dtÿt0†

! (

‡exp ÿ‰…x‡x0† 2

‡ …y‡y0†2Š

4D…tÿt0†

!

ÿexp ÿ‰…xÿx0† 2

‡ …y‡y0†2Š

4Dtÿt0†

!

ÿexp ÿ‰…x‡x0† 2

‡ …yÿy0†2Š

4Dtÿt0†

!)

: …A:6_†

Eq. (A.5) is a generic solution for arbitrary initial and ®rst-type boundary conditions. For periodic boundary conditions, the solution will become periodic as the elapsed time increases. In other words, the e€ects of the initial condition are diminished after a certain critical time. From Eq. (A.5), the solution for h1 can then be obtained as expressed by Eqs. (8a)±(9b) assuming that h1…x;y;0† ˆ0.

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