INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
Published online 26 September 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2144
Groundwater response to dual tidal fluctuations in a peninsula or an
elongated island
Quanrong Wang1, Hongbin Zhan2,3,*,† and Zhonghua Tang1
1
School of Environmental Studies, China University of Geosciences, Wuhan, Hubei 430074, China
Department of Geology and Geophysics, Texas A&M University, College Station, TX 77843-3115, U.S.A.
3
Faculty of Engineering and School of Environmental Studies, China University of Geosciences, Wuhan, Hubei 430074,
China
2
SUMMARY
Groundwater flow in a peninsula or an elongated island is influenced by tidal fluctuations on both sides of
the peninsula or island, which is named as dual tidal fluctuations. In this study, semianalytical solutions of
transient groundwater flow in response to dual tidal fluctuations in an aquifer–aquitard system were
presented for cases with and without the aquitard storage. These solutions were first derived using the
Laplace transform and subsequently computed by the Fourier series numerical inverse Laplace transform.
The derived solutions were found to agree very well with the results of numerical simulations by
MODFLOW. The solution ignoring the aquitard storage approached the quasi-steady state solution quickly
when the mean sea level initial condition was used. The solutions with and without the aquitard storage
were nearly the same at the early time and were separated from each other during the intermediate time,
and the difference of solutions became constant at late time for small aquifer/aquitard storativity ratio
and large tidal frequency. The propagation bias, which is the departure from the theoretical ratio of
tidal attenuation to tidal lag, was enhanced not only with increase of the dimensionless specific leakage
(aquitard/aquifer hydraulic conductance ratio) but also with decrease of the aquifer/aquitard storativity
ratio and with the increase of the dimensionless tidal frequency. The solution with the aquitard storage
was more sensitive to these three parameters. The newly developed solutions were capable of handling
realistic initial conditions that might be approximated by piecewise linear functions. Copyright © 2012
John Wiley & Sons, Ltd.
Received 29 August 2011; Revised 11 May 2012; Accepted 31 July 2012
KEY WORDS:
coastal aquifer; transient groundwater flow; Laplace transform; modeling
1. INTRODUCTION
Groundwater flow in coastal aquifers has been an interest of research for many decades [1]. It
has become even more important recently because of increasing migration trend of world population
to coastal areas [2, 3]. Groundwater flow in coastal aquifers could be influenced by many factors,
such as lateral recharge from inland; different physical, chemical, and biological characteristics
between fresh water and seawater; and tidal fluctuations [4], to name a few. Among these factors,
the tidal fluctuations are the subject of our interest here. Periodic ocean tides bounding a coastal
aquifer will result in periodically fluctuated hydraulic heads in the aquifer. After a certain lapse of
time for the initial condition effect to die out, frequencies of the hydraulic head fluctuations in the
aquifer will be the same as those of the tidal fluctuations. This is called quasi-steady state condition
*Correspondence to: Hongbin Zhan, Department of Geology and Geophysics, Texas A&M University, College Station,
TX 77843-3115, U.S.A.
†
E-mail: [email protected]
Copyright © 2012 John Wiley & Sons, Ltd.
GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
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(hydraulic head still changes with time and space). Jiao and Tang [5, 6] derived an analytical
solution of groundwater response to tidal fluctuations in a leaky confined aquifer under the quasisteady state condition. Li et al. [7–11] applied similar approaches as Jiao and Tang [5, 6] but to
somewhat different setup of coastal aquifers and derived several analytical solutions for hydraulic
head distributions. Guo et al. [12] also derived an analytical solution of groundwater flow
responding to tidal fluctuation in a two-zone aquifer. Recently, Li et al. [13] and Liu et al. [14]
presented quasi-steady state solutions of groundwater flow in coastal aquifers considering irregular
coastal boundaries.
The studies mentioned earlier only dealt with a coastal aquifer bounded by ocean on one side, which
is referred to as the single tidal fluctuation case. The results of those studies cannot be applied
to groundwater flow in a peninsula or an elongated island, which are bounded by ocean on two
sides, and this is called the dual tidal fluctuations in this study. Rotzoll et al. [15] provided an
analytical solution for quasi-steady state one-dimensional groundwater flow in a single
homogeneous island aquifer subjected to dual tidal fluctuations with application to the island of
Maui, Hawaii. Trefry and Bekele [16] investigated groundwater head fluctuations subject to
asynchronous dual tidal propagation in Garden Island, Australia. They found a phenomenon named
“propagation bias”, which is briefly illustrated as follows. According to the analysis of Trefry and
Bekele [16], for a given harmonic oscillation tide described by A0 cos(ot) with an amplitude of A0
and a frequency of o, the hydraulic head in the inland aquifer can be described as (zA0)cos(ot f),
where z and f are called attenuation and phase difference, respectively. Furthermore, z and f are
found to decline exponentially and increase linearly with distance from the ocean, respectively, that
qffiffiffiffiffiffiffi
pffiffiffiffi
ffi
o
1
is, z ¼ exp x 2D
and f ¼ x 2oD
, where x is the distance from the ocean (positive) and D = T/S
is the aquifer diffusivity, and T and S are aquifer transmissivity and storativity, respectively. One
qffiffiffiffiffiffiffi
1
minor point is that f is always kept in the range of [0, 2p]; thus if 2np≤f ¼ x 2oD
≤2ðn þ 1Þp,
qffiffiffiffiffiffiffi
1
2np. The phase lag,
where n is a positive integer, then the updated f value should be f ¼ x 2oD
f
, and Φ is always in the range of [0, 1].
denoted as Φ, is related to phase difference f as Φ ¼ 2p
According to aforementioned expression of z and f, one can easily see that Φ/ln(z) = 1/(2po) for
0 ≤ f ≤ 2p. It implies that there is a linear relationship between Φ and ln(z). Trefry and Bekele
[16, see Figure 6c and f] found that when plotting the Φ–ln(z) relationship by using the best-fitted
theoretical results and observed data on the same diagram, significant discrepancy always existed.
More precisely, the best-fitted curves were always above the observed data. Trefry and Bekele [16]
could not explain such a phenomenon successfully and called it the propagation bias. With
numerical simulations, Trefry and Bekele [16] suggested that the most possible cause of the
propagation bias was horizontal layering in aquifer properties. Sun et al. [17] presented an analytical
solution of groundwater head response to dual tidal fluctuations in a leaky confined aquifer under
the quasi-steady state condition, and their conclusions supported that horizontal layering in aquifer
properties might be responsible for the propagation bias.
The aforementioned studies on dual tidal fluctuations were quasi-steady state solutions. This study,
however, will consider transient groundwater flow in a peninsula or an elongated island considering
dual tidal fluctuations and an aquifer–aquitard system. The rationale of considering the layering
aquifer–aquitard system agrees with the real case application of Garden Island, Australia as
considered by Trefry and Bekele [16]. This study considers both cases with or without aquitard
storage. The case without the aquitard storage is an extension of the quasi-steady state solution of
Sun et al. [17]. The primary focus of this study is to provide a few new transient semianalytical
solutions on groundwater response to dual tidal fluctuations in a peninsula or an elongated island
considering arbitrary initial conditions. These new solutions may serve multiple purposes. For
instance, they provide quick calculation on hydraulic head in a peninsula or an elongated island
without involving complicated and sometimes time-consuming numerical calculation. They provide
a convenient way to investigate the dynamic groundwater response to various dual tidal fluctuation
scenarios. They can be used to inversely calculate the aquifer parameters on the basis of the
observed data with or without the influence of the initial condition.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
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Q. WANG, H. ZHAN AND Z. TANG
2. PROBLEM STATEMENT AND SEMIANALYTICAL SOLUTION
2.1. Problem statement and conceptual model
The physical model comprises a confined aquifer bounded above by a semipermeable aquitard and
below by impermeable bedrock (see Figure 1). All the layers are bounded laterally (along the x-axis)
by ocean, meaning that the situation is most likely occurred in a peninsula or an elongated island,
which is assumed to extend to a very long distance in the along-shore direction. The z-axis is
vertical and positive upward. The origin of the coordinate system is on the left boundary between
the ocean and the peninsula/island and is located at the upper boundary of the aquitard.
Several assumptions are inevitable to make the analytical approach amendable. First, Figure 1
mimics a rectangular shape of peninsula/island whose dimension is much longer in one direction
than the other. Groundwater flow in the aquifer is along the x-axis. Second, boundaries between the
ocean and the aquifer–aquitard are vertical and sharp. Third, both the aquifer and aquitard are
homogenous and horizontally isotropic and each with uniform thickness. Fourth, hydraulic
conductivity of the aquitard is at least two orders of magnitude smaller than that of the aquifer, so
only vertical flow in the aquitard and horizontal flow in the aquifer are considered [18, 19]. An
unconfined aquifer exists above the aquitard whose hydraulic head is constant. These assumptions,
although quite idealized, are rather standard in dealing with flow in an aquifer–aquitard system by
analytical approaches [19–22].
2.2. Mathematical model
The governing equation generally treats the leaky boundary condition as a volumetric source/sink term
by using the Hantush approximation [21], such as Jiao and Tang [5, 6] and Sun et al. [17]. Hantush [23]
recognized the potential problem of such an approximation and pointed out that this assumption might
result in error. The purpose of this section is to study groundwater flow in an aquitard–aquifer system on
the basis of the mass conservation law without using the Hantush approximation. Flows in the aquitard
and aquifer are treated as two systems that are linked through the continuity of flux and head at the
aquifer–aquitard boundary.
Groundwater flow in the aquitard can be expressed as
0
0
Kz
0
@2h
0 @h
0
; 0 < t < 1; B < z < 0;
¼Ss
@z2
@t
0
(1)
h ðx; 0; t Þ ¼ 0;
(2)
0
0
h x; B ; t ¼ hðx; t Þ;
(3)
Figure 1. The schematic diagram of the aquifer–aquitard system with dual tidal fluctuation in a peninsula or
an elongated island.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
0
h ðx; z; 0Þ ¼ 0;
2459
(4)
where h0 (x, z, t) denotes the hydraulic head (L) in the aquitard at the location (x, z) (L) and time t (T); x
and z are horizontal and vertical coordinates, respectively (L); B0 is the thickness of the aquitard (L);
0
0
and S s and K z are the specific storativity (L1) and vertical hydraulic conductivity (LT1) of the
aquitard, respectively.
The following mathematic model can govern groundwater flow in the confined aquifer:
0
0
@h
@2h
0 @h x; B ; t
¼ Kx B 2 þ K z
;
S
@t
@x
@z
0≤x≤L0 ;
(5)
hðx; t Þjx¼0 ¼ A1 cosðot Þ;
(6)
hðx; t Þjx¼L0 ¼ A2 cosðot þ θÞ;
(7)
8
a1 x þ b1
>
>
>
>
<
hðx; 0Þ ¼ ai x þ bi
>
>
>
>
:
an x þ bn
0≤x≤L1
⋮
Li1 ≤x≤Li ;
⋮
Li ≤x≤L0
(8)
where h(x, t) is the hydraulic head of the confined aquifer (L); S and Kx are the storativity
(dimensionless) and horizontal hydraulic conductivity (LT1) of the aquifer, respectively; B is the
thickness (L) of the aquifer; A1 and A2 are the amplitudes of the tidal fluctuations (L) of the left and
right oceans, respectively; o is the common tidal frequency for both left and right oceans (T1); θ is
the phase (dimensionless) difference between the left and right ocean tides; L0 is the length of the
peninsula or island along the x-axis (L); ai (dimensionless) and bi (L) are two series of constants used
to describe the initial condition in piecewise linear functions with n subsections in space; and Li (L)
is the right end of the subsection i.
The choice of the initial condition is based on a few considerations. Interaction between ocean and
groundwater in a coastal aquifer generally comprises of two complementary processes: seawater
intrusion and submarine groundwater discharge [24, 25]. As the hydraulic gradient changes slowly [24],
it can be well approximated as a piecewise linear function, which is used in the following discussion.
To make the equations of the initial condition more compact, one can write
a1 x þ b1 ;
if 0≤x≤L1
⋮
ai x þ bi ;
if Li1 ≤x≤Li
⋮
if Li ≤x≤L0
an x þ bn ;
g
¼ ax þ b;
if 0≤x≤L0 ;
(9)
where a (dimensionless) and b (L) are constants in each section but may change with space.
With the dimensionless variables defined in Table I, the mathematic model governing groundwater
flow in the aquitard can be transformed into the following dimensionless forms.
Table I. List of dimensionless variables.
hD ðxD ; tD Þ ¼ hðBx;tÞ, tD ¼ KSBx t, xD ¼ Bx , zD ¼ Bz ,
AD1 ¼ AB1 , AD2 ¼ AB2 , bD ¼ Bb , oD ¼ SBo
Kx ,
0
Þ
LD ¼ LB0 , TD ¼ o2p D , h D ðxD ; zD ; tD Þ ¼ h ðx;z;t
B ,
0
0
D
K z
B
S
aD ¼ Ba
Kx , b ¼ Kx , w ¼ B0 , ¼ S0 B0
s
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
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Q. WANG, H. ZHAN AND Z. TANG
0
0
@2h D
w @h D
1
¼
; 0 < tD < 1; < zD < 0;
2
b @tD
w
@zD
0
h D ðxD ; 0; tD Þ ¼ 0;
0
hD
1
xD ; ; tD
w
(10)
(11)
¼ hD ðxD ; tD Þ;
0
h D ðxD ; zD ; 0Þ ¼ 0;
(12)
(13)
where b is the aquitard/aquifer hydraulic conductivity ratio and w is the aquifer/aquitard thickness ratio
(see Table I).
Similarly, the mathematic model governing groundwater flow in the confined aquifer becomes
0
@hD @ 2 hD
@h D ðxD ; zD ; tD Þ ¼
þ
b
z ¼1w ;
D
@tD
@zD
@x2D
(14)
hD ðxD ; tD ÞjxD ¼0 ¼ AD1 cosðoD tD Þ;
(15)
hD ðxD ; tD ÞxD ¼L D ¼ AD2 cosðoD tD þ θÞ;
(16)
hD ðxD ; 0Þ ¼ axD þ bD :
(17)
2.3. Semianalytical solution
With the Laplace transform, the analytical solution of the mathematical model Equations (10)–(17) in
the Laplace domain is
qffiffiffiffi
sw
sinh zD b
0
qffiffiffiffi HD ðxD ; sÞ;
H D ðzD ; sÞ ¼ sw
sinh 1w b
pffiffiffiffi
W xD
HD ðxD ; sÞ ¼ C1 e
pffiffiffiffi
W xD
þ C2 e þ
a
b
xD þ ;
W
W
(18)
(19)
0
where H D ðzD ; sÞ and HD(zD, s) represent the hydraulic heads in the aquitard and confined aquifer in the
Laplace domain, respectively, and
ffi
s
oD
a
b
s
b LD pffiffiffi
W
L
A
e
AD2 cosðθÞ 2
A
sin
ð
θ
Þ
D2
D
D1
2
2
(20)
W
o D þ s2 W
o D þ s2
o D þ s2 W
pffiffiffiffi
pffiffiffiffi
;
C1 ¼
L
W
L
W
eD
e D
C2 ¼ AD1
Copyright © 2012 John Wiley & Sons, Ltd.
s
b
C1 ;
o2D þ s2 W
(21)
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
sffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffi
sbw
s
;
W ¼sþ
coth
bw
2461
(22)
where s is the Laplace parameter in respect to the dimensionless time tD, sinh() and coth() are
the hyperbolic sine and cotangent functions, respectively, and all other dimensionless parameters
are defined in Table I. According to the definition of the dimensionless parameters w and b in
0
0
Table I and the definition of a ¼ K z =B , one has bw ¼ aD . Substituting bw ¼ aD in Equation (22)
leads to
rffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffi
saD
s
:
W ¼sþ
coth
aD (23)
It seems that it is difficult, if not impossible, to invert Equations (18) and (19) analytically. There are
many developed methods to convert the solution in the Laplace domain to that in the real-time domain.
These include the Stehfest method [26–28], the Zakian method [29–31], the Fourier series method [32],
and the Schapery method [33]. The Stehfest method works well to solve the numerical inverse Laplace
transform for some groundwater flow problems [34, 35]. Hassanzadeh and Pooladi-Darvish [36]
pointed out that the Stehfest method failed to predict e t type of functions or those with an
oscillatory response, such as sine and wave functions, whereas the Fourier transform technique
may be a powerful tool to handle those cases. For the problem investigated here, we employ
the Fourier series method for the numerical inversion because of the oscillatory nature of the
boundary conditions. The technique is based on choosing the contour of integration in the inversion
integral, converting the inversion integral into the Fourier transform, and then approximating the
transform by a Fourier series. This method approximates the inversion integral by using the following
equation [32]:
es1 tD
hD ðxD ; tD Þ ¼
tD
(
)
n
X
1
kp
k
HD ðxD ; s1 Þ þ Re
HD xD ; s1 þ i
ð1Þ ;
2
tD
k¼1
(24)
pffiffiffiffiffiffiffi
where i ¼ 1, Re means the real part of the function, and n is the number of summation.
The parameters s1 and n must be optimized for increasing accuracy. Lee et al. [37] suggested values
of s1tD between 4 and 5.
2.4. Approximation of the semianalytical solution without aquitard storage
When the aquitard storage is very small, water released from the aquitard into the aquifer can be
neglected. The leakage is now primarily from the upper unconfined aquifer [18, 19], and governing
equation of Equations (1) and (5) can be transformed as
S
0
@h
@2h
¼ Kx B 2 þ aðh0 hÞ; 0≤x≤L0 ;
@t
@x
(25)
0
where a ¼ K z =B is the specific leakage (or aquitard hydraulic conductance) (T1) [38]. The
dimensionless form can be written as
@hD @ 2 hD
¼
aD hD :
@tD
@xD 2
(26)
It is easy to obtain the analytical solution of Equation (26), which is without the aquitard storage, from
analytical solution of Equations (18) and (19). With the dimensionless variables defined in Table I,
0
! 1 when S s ! 0. The solution in Laplace domain is
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
2462
Q. WANG, H. ZHAN AND Z. TANG
pffiffiffiffi
W xD
HD ðxD ; sÞ ¼ C1 e
þ C2 e
pffiffiffiffi
W xD
þ
a
bD
xD þ ;
W
W
(27)
where
AD2 cosðθÞ
C1 ¼
ffi
s
oD
a
bD
s
bD LD pffiffiffi
W
A
L
A
sin
ð
θ
Þ
e
D2
D
D1 2
2
2
2
2
2
W
W
W
oD þ s
oD þ s
oD þ s
pffiffiffiffi
pffiffiffiffi
;
L
W
L
W
D
D
e
e
(28)
C2 ¼ AD1
s
bD
C1 :
o2D þ s2 W
(29)
W ¼ s þ aD ;
(30)
Equation (27) is the analytical solution of the hydraulic head in the confined aquifer in the Laplace
domain without aquitard storage. The mathematic model that treats the leaky boundary condition as
a volumetric source/sink term by using the Hantush approximation [21] is a special case of the
model in this study.
3. MODIFICATION OF SUN ET AL. [2008] SOLUTION
We like to compare the new solution of this study with that of Sun et al. [17]. For the purpose of
comparison, it is necessary to modify the solution of Sun et al. [17] in the same dimensionless
forms defined in Table I. By using the notation of this study, the original solution of Sun et al. [17]
becomes
hðx; t Þ ¼
A1 z cosðot fÞ;
1 A1 sinhðL0 l lxÞ þ A2 eiθ sinhðlxÞ
z¼ ;
A1
sinhðL0 lÞ
f ¼ arg
A1 sinhðL0 l lxÞ þ A2 e sinhðlxÞ
;
sinhðL0 lÞ
iθ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
Kz
S
l¼
;
þ io
Kx BB0
Kx B
(31)
Where z and f are the amplitude attenuation and phase difference of the tidal head fluctuation,
f
. Transforming the original
respectively. The phase lag Φ is related to phase difference f as Φ ¼ 2p
solution of Sun et al. [17] into dimensionless forms by using variables defined in Table I, one has
hD ðx; t Þ ¼ AD1 z cosðoD tD fÞ;
1 AD1 sinhðLD lD lD xD Þ þ AD2 eiθ sinhðlD xD Þ
z¼
;
AD1 sinhðLD lD Þ
f ¼ arg
(32)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
AD1 sinhðLD lD lD xD Þ þ AD2 eiθ sinhðlD xD Þ
; lD ¼ aD þ ioD :
sinhðLD lD Þ
This modified solution of Sun et al. [17] will be compared with our new solutions in the following
discussion.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
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4. RESULTS AND DISCUSSIONS
4.1. Comparison of the semianalytical solutions with numerical solutions of MODFLOW
To test the accuracy and robustness of the semianalytical solutions, numerical simulations using
MODFLOW [39] have been conducted, and the results were compared with the semianalytical
solutions developed earlier. In the numerical simulation, the confined aquifer is discretized into one
row (horizontal) and many columns (vertical), and the aquitard is discretized into many rows and
columns, as shown in Figure 2. The grid spaces used are sufficiently small to make the numerical
cut-off error negligible. The hydraulic conductivities in the aquitard are sufficiently small to reflect
the nearly vertical flow in the aquitard.
The following default parameters were used for the comparison: S = 0.001; KxB = 2000 m2/day;
L0 = 10000 m; θ = 0; A1 = A2 = 0.65 m; o = 2p/24(hr 1), on the basis of the case study of Jiao and
Tang [5]. The initial head is set as the mean sea level. We refer to the parameters of aquitard from
Sheahan [40], where the vertical hydraulic conductivity is 0.013 m/day and the specific storativity is
0.001 m1. To make the aquitard storage effect more obvious, we choose an extremely thick
aquitard with thickness of 40 m. Figure 3 shows the hydraulic head distribution at t = 1 day. The
hydraulic head calculated by MODFLOW is nearly identical to that calculated by the semianalytical
0
0
solution with S s ¼ 103 m1 . If S s is small enough, its effect can be neglected, as expected. For
0
instance, if S s ¼ 109 m1 , the numerical solution from MODFLOW agrees very well with the
semianalytical solution without aquitard storage Equation (27).
4.2. Effects of specific leakage on the semianalytical solutions
For the aquifer–aquitard system investigated here, groundwater flow not only depends on the oceanic
tidal fluctuations but also depends on leakage from the aquitard. From Equations (1)–(8), it can be seen
that the thickness, the vertical hydraulic conductivity, and the specific storativity of the aquitard are
involved in the mathematical model. Because some of these parameters are interdependent, we
prefer to analyze the problem by using dimensionless parameters, which are independent from each
other. From the dimensionless variables in Table I, one can see that the dimensionless specific
leakage aD ðx; t Þ is a primary parameter describing the properties of the aquifer–aquitard system for
both cases with and without the aquitard storage. To eliminate the influence of the initial condition,
we set the mean sea level as the initial condition for this analysis.
With analysis of the observations at various monitoring well locations in Garden Island on the
continental shelf of Western Australia, Trefry and Bekele [16] found there was inconsistency
between the observed strong attenuation and small phase lag and suggested that the analysis of
measured propagation bias had the potential to yield extra information on aquifer properties in the
vertical direction. Sun et al. [17] pointed out that the leakage of the overlying confining layer
enhanced the landward attenuation and shortened the phase lag, on the basis of the quasi-steady
state solution Equation (32), which was derived considering the cyclic tidal boundary condition and
ignoring the initial condition. However, when the effect of the initial condition cannot be ignored,
the hydraulic head will be affected by both the tidal fluctuation and the initial condition at early stage.
Consequently, the solutions will no longer be a single cosine function as shown in Equation (32),
Figure 2. The schematic discretization of the aquifer–aquitard system in MODFLOW.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
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Q. WANG, H. ZHAN AND Z. TANG
Figure 3. Comparison among the numerical solutions from MODFLOW and the new solutions with
and without the aquitard storage effect when the aquitard specific storativity values are 103 (m1) and
109 (m1) at t = 1 day, where S = 0.001, KxB = 2000 m2/d, L0 = 10000 m, θ = 0, A1 = A2 = 0.65 m, and
o = 2p/24(hr 1).
making it difficult to define a clear relationship between the attenuation and phase shift, as can be carried
out by that of Sun et al. [17]. To test the influence of leakage to propagation bias by the Trefry and
Bekele [16] considering the initial condition effect, the alternative approach is to compare the
solutions in this study with the quasi-steady state solution of Sun et al. [17]. For this purpose, the
fitted parameters in the transect A by Sun et al. [17] will be introduced: the average aquifer diffusion
K’
z
¼ 1:082 , o = 0.930 d 1, L = 1420 m, B = 30 m, and
for three transects KSx B ¼ 20149 m2 day1 , BoS
0
B = 10 m. With the definition of the dimensionless variables in Table I, one has aD ¼ 0:135, oD ¼
0:042, and LD ¼ 47:33. Other parameters are given as AD1 = AD2 = 0.01 and θ = 0. To investigate the
influence of different aD values upon the hydraulic head distribution in the confined aquifer, aD of
0.00135 is also used.
Comparison of the new solutions and the quasi-steady state solution of Sun et al. [17] can be seen
easily from the figures of the dimensionless head versus the dimensionless time at a given location,
as shown in Figure 4(a) and (b) when the mean sea level is chosen as the initial condition. The value
of z can be determined by the amplitude, and f can be carried out by the intersection points between
the new solutions and the x-axis. A few interesting observations can be obtained from Figure 4(a) and
(b). First, as expected, the transient solutions are close to the initial head at early time, whereas very
different from the quasi-steady state solution of Sun et al. [17] regardless of the aquitard storage. The
solution ignoring the aquitard storage approaches the quasi-steady state solution soon, whereas the
solution considering the aquitard storage does not for the small aquifer/aquitard storativity ratio .
Second, the f values for solution without the aquitard storage in the quasi-steady state are the same
as with the solution of Sun et al. [17], whereas the f values for solution considering the aquitard
storage are sensitive to the value of aD . In particular, the value of f decreases with the increase of the
aD value. Third, the solutions with and without the aquitard storage are nearly the same at the early
time, they separate from each other during the intermediate time, and eventually the difference
between them becomes constant at late time. Furthermore, the amplitude of the solution considering
the aquitard storage is smaller than the one ignoring the aquitard storage.
These three observations can be explained as follows. The first observation is obvious because the
transient solutions are affected by the initial condition at early stage, whereas the quasi-steady state
solution is not. The quasi-steady state solution of Sun et al. [17] is a special case of the solution
without the aquitard storage after long time until the influence of the initial condition to the head
fluctuation can be ignored, so two curves are the same in the later stage. When the aquitard storage
is considered, the vertical transient flow cannot approach the steady state for small because the
boundary condition at the aquifer–aquitard interface, Equation (3) changes with time periodically.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
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a
b
c
Figure 4. (a) Comparison among the new solutions with and without aquitard storage and the quasi-steady
state solution by Sun et al. [17] when the dimensionless specific leakage aD = 0.135. (b) Comparison among
the new solutions with and without aquitard storage and the quasi-steady state solution by Sun et al. [17]
when dimensionless specific leakage aD = 0.00135. (c) Comparison among the new solutions with and without aquitard storage and the quasi-steady state solution by Sun et al. [17] when dimensionless tidal frequency oD = 0.0042.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
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Q. WANG, H. ZHAN AND Z. TANG
The water released from the aquitard cannot be ignored, so a small difference is still notable at later
time stage.
For the second observation, Sun et al. [17] pointed out that the leakage of the overlying confining
layer could enhance the propagation bias. According to the definition of the specific leakage aD , it
directly represents the aquitard–aquifer water flux for the case without the aquifer storage. As for the
0
case with aquitard storage, a similar parameter a D representing the water flux from the aquitard into
the main aquifer is needed. By substituting Equation (18) into the governing equation in the
confined aquifer in Laplace domain, one has
0
aD
rffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffi
saD
s
:
¼
coth
aD
(33)
0
With the new parameter a D , the governing equation with the aquitard storage in Laplace domain has
0
the similar form as the governing equation without aquitard storage. With Equation (33), a D =aD is
0
larger than 1. A larger aD value results in a larger a D value and stronger leakage effect, thus lead to
stronger propagation bias and a smaller f value.
In respect to the third observation, because it takes time for the leakage water to flow from the
aquitard to the aquifer (or vice versa), the difference between the solutions with and without the
aquitard storage is negligible at early time. Similar observations have been reported in previous
studies of groundwater flow in an aquifer–aquitard system [18, 19]. At late time, water released
from the aquitard storage is mainly influenced by the cyclic tidal boundary condition, which will
cause temporary holding and releasing of water from storage in the aquitard. Such temporary
holding and releasing of water from storage in the aquitard do not exist if ignoring the aquitard
storage. This leads to the eventually small (constant) difference between the solutions of cases with
and without the aquitard storage. As for the case with aquitard storage, the specific leakage is larger
0
than the one without aquitard storage because of a D =aD > 1, so the amplitude of the solution in the
confined aquifer considering the aquitard storage is smaller than the one ignoring the aquitard storage.
Figure 4(a) and (c) compares the new transient solutions and the quasi-steady state solution of Sun
et al. [17] for different oD values of 0.042 and 0.0042 under the strong leakage condition (aD = 0.135).
One can see from Figure 4(c) that the differences are very small for the case with a smaller oD value.
With the definition of oD ¼ SBo=Kx , a small oD value implies a longer period. When the period is
much longer than the time needed for aquitard storage water to dissipate, the solution with the
aquitard storage will approach the one without the aquitard storage. Therefore, the leakage effect to
the propagation bias is more obvious at higher tidal frequencies than lower ones.
4.3. Effects of aquifer/aquitard storativity ratio on the semianalytical solutions
The ratio of the aquifer/aquitard storativity ratio is also a primary parameter for the case with the
aquitard storage in Equation (19). Figure 5(a) and (b) shows the aquifer hydraulic head spatial
distribution for different values for two given dimensionless time tD = 10 and tD = 800. These two
solutions are almost the same when is above two, meaning that the aquitard storage can be
ignored when is greater than two.
In this section, the effect of the aquifer/aquitard storativity ratio to the z and f will be discussed.
Figure 5(c) shows the comparison among the new solutions with and without aquitard storage, and
the quasi-steady state solution by Sun et al. [17] for different at the specific location. It is obvious
that the smaller leads to stronger propagation bias. This observation is easy to explain. A smaller
value implies larger aquitard storativity in respect to the aquifer storativity, so the quantity of the
water released from the aquitard to the aquifer is more remarkable, which results in stronger
propagation bias for the case with aquitard storage.
4.4. Effects of the initial condition on the semianalytical solutions
In aforementioned discussion, the initial conditions are set as the mean sea level for the purpose of
illustration. In fact, the initial condition is often not constant in space. For instance, it may be
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
2467
a
b
c
Figure 5. (a) Comparison among the new solutions with and without aquitard storage and the quasi-steady
state solution by Sun et al. [17] for of 0.01, 0.1, and 2 at tD = 10. (b) Comparison among the new solutions
with and without aquitard storage and the quasi-steady state solution by Sun et al. [17] for of 0.01, 0.1,
and 2 at tD = 800. (c) Comparison among the new solutions with and without aquitard storage and the
quasi-steady state solution by Sun et al. [17] for of 0.01, 0.1, and 2 at xD = 5.
Copyright © 2012 John Wiley & Sons, Ltd.
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DOI: 10.1002/nag
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Q. WANG, H. ZHAN AND Z. TANG
affected by human activity such as groundwater pumping and vertical recharge from precipitation. The
developed solutions are capable of handling realistic initial conditions by using piecewise linear
functions to approximate the initial hydraulic head spatial distributions. The focus of this section is
to test the sensitivity of initial conditions on the derived solutions.
Sanford and Pope [41] recently developed a three-dimensional model to simulate seawater intrusion
to a peninsula in Eastern Shore of Virginia, U.S.A. to reproduce historical water levels and to forecast
the potential for seawater intrusion. In their model, groundwater flowed mostly laterally from the
recharge areas in the central upland and discharged to the Atlantic Ocean and Chesapeake Bay [42];
a small percentage will move vertically downward through the upper Yorktown-Eastover confining
unit into the confined aquifer system. With the study of Sanford and Pope [41], we choose the
following initial condition for our study:
hD ðxD ; 0Þ ¼
0:02xD ;
0:02xD þ 1;
if xD ≥ 25
:
if xD > 25
(34)
When considering this initial condition, the spatial distribution of the hydraulic head is shown in
Figure 6 at given times of tD = 1, tD = 1 + TD, and tD = 1 + 20TD, where TD is dimensionless period
of tidal fluctuations defined in Table I. The quasi-steady state solution of Sun et al. [17] is
also included in this figure. As can be seen from this figure, when the realistic initial condition
Equation (34) is considered, the solution considering the aquitard storage approaches the quasisteady state solution after 20TD, whereas the solution ignoring the aquitard storage cannot approach
the quasi-steady state solution because the aquifer/aquitard storativity ratio is very small, and the
reason has been explained in Section 4.3.
5. CONCLUSIONS
In this study, semianalytical solutions of transient groundwater response to dual tidal fluctuations in an
aquifer–aquitard system in a peninsula or an elongated island are derived using Laplace transform, for
both cases with and without aquitard storage effect. The solutions in real-time domain are subsequently
obtained through the Fourier series numerical inverse Laplace transform. The robustness of the
solution is tested by comparison with numerical simulations using MODFLOW for two hypothetical
cases with and without the aquitard storage. The developed solutions are capable of handling the
realistic initial condition using piecewise linear functions.
With this study, the following conclusions can be drawn.
Figure 6. Comparison among the new solutions with and without aquitard storage and the quasi-steady state
solution by Sun et al. [17] for the piecewise linear initial condition at different time.
Copyright © 2012 John Wiley & Sons, Ltd.
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GROUNDWATER RESPONSE TO DUAL TIDAL FLUCTUATIONS IN A PENINSULA
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1. The transient solutions are very different from the quasi-steady state solution of Sun et al. [17] at
early time regardless of the aquitard storage. The solution ignoring the aquitard storage
approaches the quasi-steady state solution soon when the initial condition is as the mean sea
level, whereas the one considering the aquitard storage cannot for the small aquifer/aquitard
storativity ratio and large oD.
2. The solutions with and without the aquitard storage are nearly the same at the early time, they
separate from each other during the intermediate time, and eventually the difference becomes
constant at late time.
3. The solutions are sensitive to the values of aD and , where aD is the aquitard/aquifer hydraulic
conductance ratio and is aquifer/aquitard storativity ratio (see Table I). Our solutions
demonstrate that the large leakage of the overlying confining layer enhances the landward
attenuation of the tidal head fluctuation and shortens the time lag between the head and tide
fluctuations [16, 17].
4. The contribution of leakage phenomena to the propagation bias is more important at higher tidal
frequencies than lower ones. The attenuation of the tidal head in the aquifer increases with the
increase of the tidal frequency, whereas the time lag decreases with the tidal frequency increase.
ACKNOWLEDGEMENTS
This research was partially supported by the program of research of groundwater resources and
corresponded environmental problems in Jianghan-Dongtin plain (Grant Number: 1212011121142) and
the National Basic Research Program of China (973) (2011CB710600). We thank two anonymous reviewers
for their constructive comments, which help improve the quality of the manuscript.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
Jacob CE. Flow of groundwater. In Engineering Hydraulics, Rouse H (ed.). Wiley: New York, USA, 1950.
Deboudt P. Towards coastal risk management in France. Ocean & Coastal Management 2010; 53(7):366–378.
Vasey-Ellis N. Planning for climate change in coastal Victoria. Urban Policy and Research 2009; 27(2):157–169.
Trefry MG, Johnston CD. Pumping test analysis for a tidally forced aquifer. Ground Water 1998; 36:427–433.
Jiao JJ, Tang ZH. An analytical solution for groundwater response to tidal fluctuation in a leaky confined aquifer.
Water Resources Research 1999; 35(3):747–751.
Jiao JJ, Tang ZH. Comment on “An analytical solution of groundwater response to tidal fluctuation in a leaky confined
aquifer” by Jiu Jimmy Jiao and Zhonghua Tang – Reply. Water Resources Research 2001; 37(1):187–188.
Li HL, Jiao JJ. Tide-induced groundwater fluctuation in a coastal leaky confined aquifer system extending under the
sea. Water Resources Research 2001; 37(5):1165–1171.
Li HL, Jiao JJ. Analytical studies of groundwater-head fluctuation in a coastal confined aquifer overlain by a
semi-permeable layer with storage. Advances in Water Resources 2001; 24:565–573.
Li HL, Jiao JJ. Analytical solutions of tidal groundwater flow in coastal two-aquifer system. Advances in Water
Resources 2002; 25(4):417–426.
Li HL, Jiao JJ, Luk M, Cheung KY. Tide-induced groundwater level fluctuation in coastal aquifers bounded by
L-shaped coastlines. Water Resources Research 2002; 38:361–368.
Li HL, Li L, Lockington D, Boufadel MC, Li GY. Modeling tidal signals enhanced by a submarine spring in a coastal
confined aquifer extending under the sea. Advances in Water Resources 2007; 30:1046–1052.
Guo HP, Jiao JJ, Li HL. Groundwater response to tidal fluctuation in a two-zone aquifer. Journal of Hydrology 2010;
381(3–4):364–371.
Li HL, Jiao JJ, Tang ZH. Semi-numerical simulation of groundwater flow induced by periodic forcing with a
case-study at an island aquifer. Journal of Hydrology 2006; 327:438–446.
Liu S, Li HL, Boufadel MC. Numerical simulation of the effect of the sloping submarine outlet-capping on tidal
groundwater head fluctuation in confined coastal aquifers. Journal of Hydrology 2008; 361:339–348.
Rotzoll K, El-Kav AI, Gingerich SB. Analysis of an unconfined aquifer subject to asynchronous dual-tide propagation.
Ground Water 2008; 46:239–250.
Trefry MG, Bekele E. Structural characterization of an island aquifer via tidal methods. Water Resources Research
2004; 40:645–656.
Sun PP, Li HL, Boufedal MC, Geng XL, Chen S. An analytical solution and case study of groundwater head
response to dual tide in an island leaky confined aquifer. Water Resources Research 2008; 44:1–7.
Sun DM, Zhan HB. Pumping induced depletion from two streams. Advances in Water Resources 2007; 30:
1016–1026.
Sun DM, Zhan HB. Flow to a horizontal well in an aquitard–aquifer system. Journal of Hydrology 2006; 321:
364–376.
Bear J. Hydraulics of Ground Water. McGraw-Hill: New York, USA, 1979.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag
2470
Q. WANG, H. ZHAN AND Z. TANG
21. Hantush MS. Hydraulic of wells. In Advances in Hydroscience, Chow VT (ed.). Academic Press: New York, USA,
1964.
22. Volker RE, Zhang Q. Comment on “An analytical solution of groundwater response to tidal fluctuation in a leaky
confined aquifer” by Jiu Jimmy Jiao and Zhonghua Tang. Water Resources Research 2001; 37(1):185–186.
23. Hantush MS. Flow of groundwater in relatively thick leaky aquifers. Water Resources Research 1967; 3(2):583–590.
24. Kaleris V, Lagas G, Marczinek S, Piotrowski JA. Modelling submarine groundwater discharge: an example from the
western Baltic Sea. Journal of Hydrology 2002; 265(1–4):76–99.
25. Kaleris V. Submarine groundwater discharge: effects of hydrogeology and of near shore surface water bodies.
Journal of Hydrology 2006; 325(1–4):96–117.
26. Gaver DP. Observing stochastic processes and approximate transform inversion. Operations Research 1966;
14(3):444–459.
27. Stehfest H. Algorithm 368 numerical inversion of Laplace transforms. Communications of the ACM 1970a;
13(1):47–49.
28. Stehfest H. Remark on algorithm 368 numerical inversion of Laplace transforms. Communications of the ACM
1970b; 13(10):624–625.
29. Halsted DJ, Brown DE. Zakian’s technique for inverting Laplace transform. Chemical Engineering Journal 1972;
3:312–313.
30. Zakian V. Numerical inversion of Laplace transform. Electronics Letters 1969; 5(6):120–121.
31. Zakian V. Properties of IMN approximants. In Pade Approximants and their Applications, Graves-Morris PR (ed.).
Academic Press: London, UK, 1973.
32. Dubner H, Abate J. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform.
Journal of the ACM 1968; 15(1):115–123.
33. Schapery RA. Approximate methods of transform inversion for viscoelastic stress analysis. Proceedings 4th US
National Congress of Applied Mechanics, 1962; 1075–1085.
34. Wen Z, Huang GH, Zhan HB. An analytical solution for non-Darcian flow in a confined aquifer using the power law
function. Advances in Water Resources 2008; 31(1):44–55.
35. Wen Z, Huang GH, Zhan HB, Li J. Two-region non-Darcian flow toward a well in a confined aquifer. Advances in
Water Resources 2008; 31(5):818–827.
36. Hassanzadeh H, Pooladi-Darvish M. Comparison of different numerical Laplace inversion methods for engineering
applications. Applied Mathematics and Computation 2007; 189(2):1966–1981.
37. Lee ST, Chien MCH, Culham WE. Vertical single-well pulse testing of a three-layer stratified reservoir. SPE Annual
Technical Conference and Exhibition, 16–19 September, Houston, Texas, 1984, SPE 13249.
38. Datta B, Vennalakanti H, Dhar A. Modeling and control of saltwater intrusion in a coastal aquifer of Andhra Pradesh,
India. Journal of Hydro-Environment Research 2009; 3(3):148–158.
39. McDonald MG, Harbaugh AW. A modular three-dimensional finite-difference groundwater flow model. US
Geological Survey Techniques of Water Resources Investigations Book 6, Chapter A1. United States Government
Printing Office, Washington, DC, 1988.
40. Sheahan NT. Injection/extraction well system-A unique seawater intrusion barrier. Ground Water 1977; 15(1):32–49.
41. Sanford WE, Pope JP. Current challenges using models to forecast seawater intrusion: lessons from the eastern shore
of Virginia, USA. Hydrogeology Journal 2010; 18(1):73–93.
42. Speiran GK. Geohydrology and geochemistry near coastal ground-water-discharge areas of the Eastern Shore,
Virginia. Reston. US Geological Survey Water-Supply Paper 2479, Department of the Interior, 1996.
Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2013; 37:2456–2470
DOI: 10.1002/nag