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Advances in Water Resources 23 (2000) 825±833
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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, respectively, 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 corresponding to tidal frequencies. As these periodic ¯uctuations 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 thousand meters [7]. In both cases, the tidal ¯uctuations affect largely the groundwater ¯ow and mass transport in
the aquifer [9]. Previous studies on aquifersÕ tidal dynamics have focussed on the inland transmission of tidal
sea level oscillations in the cross-shore direction
*

Corresponding author. Tel.: +44-131-650-5814; fax: +44-131-6505814.
E-mail address: ling.li@ed.ac.uk (L. Li).

[1,8,11,15]. Analytical solutions based on the one-dimensional Boussinesq equation are often used to describe 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 twodimensional problem because the tidal loading varies
along the estuary, i.e., a non-uniform boundary condition
h…x; 0; t† ˆ A exp … ÿ jer x† cos …xt ÿ jei x†;

…1†

where h…x; 0; t† is the ¯uctuation of the water level in the
estuary (the seepage face at the aquifer±estuary interface
has been neglected); A and x are 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

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 1 6 - 6

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L. Li et al. / Advances in Water Resources 23 (2000) 825±833

Notation
A
tidal amplitude [L]
h
head ¯uctuation [L]
L
damping distance [L]
S
aquifer storativity
T

aquifer transmissivity [L2 Tÿ1 ]
tidal period [T]
Tt
t
time [T]
elapsed time for diminished e€ects of initial
tc
conditions [T]
x
cross-shore distance from the shoreline [L]
y
along-shore distance from the estuary [L]
/
phase of tidal wave [rad]

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]

h…x; y; t† ˆ Ref A exp ‰ ÿ …jar ‡ ijai †yŠ

 exp ‰ixt ÿ …jer ‡ ijei †xŠg;

…2†

where h…x; y; t† is the head ¯uctuation; y is the distance
from the aquifer±estuary interface Fig. 1(d);
p`Re' denotes the real part of the function; and i ˆ ÿ1. jar and
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

Fig. 1. Schematic diagram of coastal aquifers, and tidal oscillations in
the ocean, estuary and aquifer.

jar

jai
jaro
jaio
jer
jei
h
x

amplitude damping coecient of
two-dimensional non-interacting tidal wave in
the aquifer [Lÿ1 ]
wave number of two-dimensional
non-interacting tidal wave in the aquifer [Lÿ1 ]
amplitude damping coecient of
one-dimensional tidal wave in the aquifer [Lÿ1 ]
wave number of one-dimensional tidal wave in
the aquifer [Lÿ1 ]
amplitude damping coecient of tidal wave in
the estuary [Lÿ1 ]

wave number of tidal wave in the estuary [Lÿ1 ]
angle between the estuary and coastline [rad]
tidal frequency, 2p/Tt [rad Tÿ1 ]

propagation (in the y direction) of the local tidal oscillations in the estuary (varying in the x direction).
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¯uence 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€erent h. Except for a very small h,
the interaction between the oceanic and estuarine tidal
head ¯uctuations cannot be neglected. SunÕs [14] solution is, therefore, inadequate in describing the tidal dynamics 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 between the estuary and coastline (i.e., h ˆ 90) 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

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L. Li et al. / Advances in Water Resources 23 (2000) 825±833

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 interaction between the inland propagation of oceanic tides
in the cross-shore direction and the transmission of
estuarine tide in the along-shore direction.

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
r
Sx
:
…6b†
jaro ˆ jaio ˆ

2T

3. Analytical solution
2. Problem setup
The two-dimensional groundwater ¯ow equation
averaged over the thickness of the aquifer under the
Dupuit assumption [3] is

Due to the interaction between cross- and alongshore 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

S oh o2 h o2 h
ˆ
‡
;
T ot ox2 oy 2

h…x; y; t† ˆ h0 …x; t† ‡ h1 …x; y; t†;

…3†

where S and T are the storativity and transmissivity of
the aquifer, respectively; and x and y are 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,
S is 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† ˆ A cos…xt†:

…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 h…x; y; t† ˆ 0:

x!1

…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 h…x; y; t† ˆ A exp… ÿ jaro x† cos …xt ÿ jaio x†;

y!1

…6a†

…7a†

with
h0 …x; t† ˆ A exp …ÿjaro x† cos …xt ÿ jaio x†;

…7b†

and
h1 …0; y; t† ˆ 0;

…7c†

h1 …x; 0; t† ˆ A exp … ÿ jer x† cos …xt ÿ jei x† ÿ h0 …x; t†;

…7d†

x!1

lim h1 …x; y; t† ˆ 0;

…7e†

lim h1 …x; y; t† ˆ 0:

…7f†

y!1

Both h0 and h1 satisfy Eq. (3). Using GreenÕs function
method (see Appendix A), we ®nd that h1 is given by the
double integral
h1 …x; y; t†
Z tZ
ˆD
0

0

1

h1 …x0 ; 0; t0 †

oG
… x; y; t; x0 ; 0; t0 † dx0 dt0 ;
oy0
…8a†

and



oG 
y
ÿy 2
ˆ
exp
4D…t ÿ t0 †
oy0 y0 ˆ0 4pD2 …t ÿ t0 †2
"
!
!#
2
2
ÿ…x ÿ x0 †
ÿ…x ‡ x0 †
ÿ exp
;
 exp
4D…t ÿ t0 †
4D…t ÿ t0 †
…8b†
where D ˆ T =S and G is the appropriate GreenÕs function (see Appendix A). The initial head ¯uctuation has
been assumed to h0 …x; 0†, i.e., h1 …x; y; 0† ˆ 0. Expressing
as
RefA exp …ixt0 ‡ k1 x0 † ÿ A exp …ixt0
h1 …x0 ; 0; t0 †
‡k2 x0 †g with k1 ˆ ÿ…jer ‡ jei i†, and k2 ˆ ÿ…jaro ‡ jaio i†;
we can reduce the above solution to a single integral, i.e.,

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L. Li et al. / Advances in Water Resources 23 (2000) 825±833

h1 …x; y; t† ˆ ARe

Z

0

t

‰ f …k1 ; x† ÿ f …k1 ; ÿ x†


ÿ f …k2 ; x† ‡ f …k2 ; ÿ x†Š dt0 ;

…9a†

where
y
f …n; f† ˆ p
4 p‰ D…t ÿ t0 †Š3=2

 exp n2 D…t ÿ t0 † ‡ ixt0 ‡ nf ÿ
"

 1 ‡ erf
As
lim
t0 !t

1
…t ÿ t0 †

3=2

2nD…t ÿ t0 † ‡ f
p
2 D…t ÿ t0 †

!#

:

y2
4D…t ÿ t0 †



…9b†

eÿ1=…tÿt0 † ˆ 0;

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.

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 superimposing the solution for each constituent, which has essentially 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.

Table 1
Parameter values used in the numerical simulation and analytical
prediction

3.1. Periodicity of the solution

For large x, Eqs. (9a) and (9b) can be simpli®ed to
Z t
y
p
h1 …x; y; t† ˆ A exp …ixt ‡ k1 x†
2
pD
0
ÿ



2
2
exp ÿ ix ‡ k1 D …t ÿ t0 † ÿ y =4D…t ÿ t0 †
dt0

…t ÿ t0 †3=2
Z t
y
p
ÿ A exp …ixt ‡ k2 x†
0 2 pD
ÿ



2
exp ÿ ix ‡ k2 D …t ÿ t0 † ÿ y 2 =4D…t ÿ t0 †
dt0 ;

3=2
…t ÿ t0 †

Physically, the solution is expected to become periodic 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 solutionÕs
periodicity if t was three times larger than the tidal
period (Tt ).
In Fig. 2(a), we show the head ¯uctuations at a location of the aquifer (x ˆ 600 m and y ˆ 600 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 determined to be three tidal periods (3Tt ).

Parameter

Value

S
T
x
jer
jei
A

0.002
36 m2 hÿ1
0.2618 Rad hÿ1 (i.e., a diurnal tide)
8  10ÿ5 mÿ1
6:67  10ÿ7 mÿ1
1m

3.2. Solutions for large x and y

…10†

where
"

2nD…t ÿ t0 †  x
p
lim erf
x!1
2 D…t ÿ t0 †

#

ˆ 1

has been applied. Furthermore, both integrals in Eq. (10)
can be evaluated explicitly to be
r !
r !
ix ÿ k12 D
ix ÿ k22 D
y
and exp ÿ
y ;
exp ÿ
D
D

L. Li et al. / Advances in Water Resources 23 (2000) 825±833

829

respectively, given that t is large. Therefore, the periodic
solution for large x is
r !
ix ÿ k12 D
y :
…11†
h…x; y; t† ˆ A exp ixt ‡ k1 x ÿ
D
Note that
r
…ix ÿ k22 D†
ˆ 0:
D
Also one can show that
r
…ix ÿ k12 D†
ˆ jar ‡ jai i:
D
Therefore, Eq. (11) is the same solution as that of Sun
[14], i.e., Eq. (2).
For large y; h1 …x; y; t† approaches zero and the solution becomes h…x; y; t† ˆ h0 …x; t†. In summary, the
present analytical solution is reduced to the onedimensional solution for oceanic tides (i.e., Eqs. (6a) and
(6b)) for large y and the solution of Sun [14] for large x.
A comparison of these solutions is given in Section 4.

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

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 locations: 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

Fig. 5. Comparison of the tidal head ¯uctuations in areas far from the
estuary the predicted by the analytical solution and the numerical
simulation. Results from four locations are examined.

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

4. Discussion of the analytical solution (t > tc )
4.1. Simulation of single tidal constituent

Fig. 3. Comparison of the tidal head ¯uctuations in areas near the
estuary predicted by the analytical solution and the numerical simulation. Results from four locations are examined.

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 crossand along-shore tidal head ¯uctuations in the aquifer,
which are induced by oceanic and estuarine tides,

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L. Li et al. / Advances in Water Resources 23 (2000) 825±833

Fig. 6. Comparison of the tidal head ¯uctuations in areas far from the
coastline predicted by the analytical solution and the numerical simulation. 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-interacting 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 ¯uctuations 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 become dominated by the oceanic tide while the estuarine
tide controls the aquiferÕs responses in areas far from the
shoreline (large x). It has been demonstrated in Section
3 that, for large y, 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, SunÕs [14] solution provides the
prediction of the head ¯uctuation for large x (Fig. 9). As
discussed in Section 1, the weakening of the wave interaction is due to the damping of either cross- or alongshore 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 becomes insigni®cant, is related inversely to the rate of
tidal wave damping in the aquifer, i.e.,
p
4:5 4:5 T
ˆ p ;
…12†
Lˆ
jaro
Sx

where 0.01A has been used as the amplitude of damped
tidal waves at the threshold. For the present simulation
(Table 1), L is 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) x ˆ 300 m and y ˆ
600 m; (b) x ˆ 600 m and y ˆ 600 m and (c) x ˆ 600 m and y ˆ 300 m.

Fig. 8. The behaviour of the solution for a large y ( ˆ 3000 m, i.e., far
from the estuary): (a) x ˆ 300 m; (b) x ˆ 600 m and (c) x ˆ 900 m.

L. Li et al. / Advances in Water Resources 23 (2000) 825±833

Fig. 9. The behaviour of the solution for a large x ( ˆ 3000 m, i.e., far
from the coastline): (a) y ˆ 300 m; (b) y ˆ 600 m and (c) y ˆ 900 m.

831

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 between 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 towards 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†], where A…x; y† is the amplitude of local
tidal head ¯uctuations and (b) three-dimensional plot of the phase of the tidal head ¯uctuations.

832

L. Li et al. / Advances in Water Resources 23 (2000) 825±833

Fig. 11. Simulation results for diurnal and semi-diurnal tides: (a)
x ˆ 600 m and y ˆ 2000 m and (b) x ˆ 8000 m and y ˆ 2000 m. Circles
are from Sim's solution and solid lines from the present solution.

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 SunÕs [14] two-dimensional analytical solutions, respectively. Such simpli®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-diurnal tide. The same parameter values as listed in Table 1
of Sun [14] were used in the calculation. The results are
compared with SunÕs analytical solution at two locations: x ˆ 600 m and y ˆ 2000 m (Fig. 11(a)), and
x ˆ 8000 m and y ˆ 2000 m (Fig. 11(b)). Again, it is
found that SunÕs 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 solution for head ¯uctuations in a coastal aquifer in¯uenced
by both oceanic and estuarine tides. The solution provides ``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 considered in previous research [14].
An important feature of the present solution is its
inclusion of the interaction between the cross- and alongshore tidal head ¯uctuations in the aquifer area near the
estuaryÕs 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 Boussinesq equation or the solution of Sun [14] for twodimensional 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-dimensional depth-averaged groundwater ¯ow equation
Eq. (3) is rewritten in terms of x0 …> 0†; y0 …> 0† and
t0 …> 0†, i.e.,
S oh o2 h o2 h
ˆ
‡
:
T ot0 ox20 oy02

…A:1†

The adjoint equation corresponding to Eq. (A.1) for
calculation of GreenÕs function is
 2

o G o2 G
oG
ˆ ÿd… x ÿ x0 †d… y ÿ y0 †d…t ÿ t0 †;
‡ 2 ÿ
D
ox20
oy0
ot0
…A:2†

where d is the Dirac delta function.
Integrating, by parts, […A1†  G ÿ …A2†  h] over
0 < x0 < 1; 0 < y0 < 1 and 0 < t0 < 1 gives
Z t Z 1 
oG
… x; y; t; x0 ; 0; t0 †
h…x0 ; 0; t0 †
h…x; y; t† ˆ D
oy0
0
0


oh
ÿ G… x; y; t; x0 ; 0; t0 †
…x0 ; 0; t0 † dx0 dt0
oy0
Z t Z 1 
oG
… x; y; t; 0; y0 ; t0 †
‡D
h…0; y0 ; t0 †
ox0
0
0


oh
…0; y0 ; t0 † dy0 dt0
ÿ G… x; y; t; 0; y0 ; t0 †
ox0
Z 1Z 1
‡
G… x; y; t; x0 ; y0 ; 0†h…x0 ; y0 ; 0† dx0 dy0 ;
0

0

…A:3†

where we have taken G…x; y; t; x0 ; y0 ; t0 † ˆ 0 for x0 ˆ 1
or y0 ˆ 1 by de®nition. Similarly, oG=ox0 ; oG=oy0 ;
oh=ox0 and oh=oy0 vanish in the same spatial limits.
Under the following boundary conditions,
h…0; y; t† ˆ W…y; t†;

…A:4a†

L. Li et al. / Advances in Water Resources 23 (2000) 825±833

833

h…x; 0; t† ˆ U…x; t†;

…A:4b†

References

G… x; y; t; 0; y0 ; t0 † ˆ 0;

…A:4c†

G… x; y; t; x0 ; 0; t0 † ˆ 0;

…A:4d†

[1] Baird AJ, Horn DP. Monitoring and modelling groundwater
behaviour in sandy beaches. J Coastal Res 1996;12:630±40.
[2] Barry DA, Barry SJ, Parlange JY. Capillarity correction to
periodic solutions of the shallow ¯ow approximation. In: Pattiaratchi CB, editor. Mixing processes in estuaries and coastal seas,
coastal and estuarine studies 50, Washington, DC: AGU, 1996.
p. 496±510.
[3] Bear J. Dynamics of ¯uids in porous media. New York: Elsevier,
1972.
[4] Dean RG, Dalrymple RA. Water wave mechanics for engineers
and scientists. Singapore: World Scienti®c, 1991;353.
[5] Ferris JG. Cyclic ¯uctuations of water level as a basis for
determining aquifer transmissibility. IAHS Publication
1951;33:148±55.
[6] Ippen AT, Harleman DRF. Tidal dynamics in estuaries. In: Ippen
AT, editor. Estuary and coastline hydrodynamics. New York:
McGraw-Hill, 1966.
[7] Lanyon JA, Eliot IG, Clarke DJ. Groundwater level variation
during semi-diurnal spring tidal cycles on a sandy beaches. Aust
J Mar Freshwater Res 1982;33:377±400.
[8] Li L, Barry DA, Pattiarachi CB. Numerical modelling of tideinduced beach water table ¯uctuations. Coastal Eng 1997;30
(1±2):105±23.
[9] Li L, Barry DA, Stagnitti F, Parlange JY. Submarine groundwater discharge and associated chemical input to a coastal sea.
Water Resour Res 1999;35:3253±9.
[10] Li L, Barry DA, Stagnitti F, Parlange JY. Tidal along-shore
groundwater ¯ow in a coastal aquifer. Env Modelling Assessment
1999;4:179±88.
[11] Li L, Barry DA, Stagnitti F, Parlange JY. Groundwater waves in
a coastal aquifer: a new governing equation including vertical
e€ects and capillarity. Water Resour Res 2000;36:411±20.
[12] Nielsen P. Tidal dynamics of the water table in beaches. Water
Resour Res 1990;26:2127±34.
[13] Parlange JY, Stagnitti F, Starr JL, Braddock RD. Free-surface
¯ow in porous media and periodic solution of the shallow-¯ow
approximation. J Hydrol 1984;70:251±63.
[14] Sun H. A two-dimensional analytical solution of groundwater
response to tidal loading in an estuary. Water Resour Res
1997;33:1429±35.
[15] Turner I. Water table outcropping on macro-tidal beaches: a
simulation model. Mar Geol 1993;115:227±38.

Eq. (A.3) becomes
Z 1Z 1
G… x; y; t; x0 ; y0 ; 0†h…x0 ; y0 ; 0† dx0 dy0
h…x; y; t† ˆ
0
0
Z tZ 1
oG
‡D
… x; y; t; 0; y0 ; t0 † dy0 dt0
W…y0 ; t0 †
ox0
0
0
Z tZ 1
oG
… x; y; t; x0 ; 0; t0 † dx0 dt0 :
‡D
U…x0 ; t0 †
oy
0
0
0
…A:5†
The appropriate GreenÕs Function is
1
G… x; y; t; x0 ; y0 ; t0 † ˆ
4pD…t ÿ t0 †
(
!
2
2
ÿ‰…x ÿ x0 † ‡ …y ÿ y0 † Š
 exp
4D…t ÿ t0 †
2

2

!

2

2

!

2

2

!)

‡ exp

ÿ‰…x ‡ x0 † ‡ …y ‡ y0 † Š
4D…t ÿ t0 †

ÿ exp

ÿ‰…x ÿ x0 † ‡ …y ‡ y0 † Š
4D…t ÿ t0 †

ÿ exp

ÿ‰…x ‡ x0 † ‡ …y ÿ y0 † Š
4D…t ÿ t0 †

:

…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† ˆ 0.