Achievements
TIDAL GROUNDWATER HEAD FLUCTUATION IN AN AQUIFER OVERLAIN BY A SEMI-PERMEABLE SEABED
Hailong Li，Yakun Ren
(1) School of
Environmental Studies & (MOE) Biogeology and Environmental Geology Lab, China
University of Geosciences, Wuhan 430074, P. R. China
(2) Dept. of
Mathematics, Anshan Normal University, Anshan 114005,
P.R. China
Abstract: Research on the
dynamics interaction between groundwater and seawater has attached much
attention from hydrogeologists. The tide-induced groundwater head fluctuation
in a leaky confined aquifer under the sea is considered in this paper. The
aquifer is assumed to be overlain by a semi-permeable seabed. Both the seabed
and the aquifer are assumed to extend horizontally and infinitely. A
one-dimensional mathematical model is used to describe the problem. An exact
analytical solution is derived and the analytical solution explains that
the hydraulic head fluctuation hardly changes with the variable and contains six
dimensionless parameters: the tidal efficiency of both the confined aquifer and
the semi-permeable seabed and , the two
dimensionless parameters and which are
determined by the hydraulic conductivity and the storativity of the confined
aquifer and the two dimensionless parameters and which are
determined by the hydraulic conductivity and the storativity of the
semi-permeable seabed. We discussed how the hydraulic head fluctuation in the
confined aquifer influenced by the above six dimensionless parameters. The
amplitude of the hydraulic head fluctuation in the confined aquifer
increases strictly with the tidal efficiency of both the confined aquifer and
the semi-permeable seabed, while the phase shift of hydraulic head fluctuation
decreases strictly with the tidal efficiency of the confined aquifer and
increases strictly with the tidal efficiency of the semi-permeable seabed. When ==（is a number
between 0 and 1）, the hydraulic
head relative amplitude equals to approximately
and the phase shift equals to 0 approximately. When the tidal efficiency of the
confined aquifer larger than the tidal efficiency of the semi-permeable seabed,
that is >, the phase
shift of hydraulic head fluctuation is smaller than 0 and exceeds the tide
fluctuation. When the tidal efficiency of the confined aquifer is larger
than the tidal efficiency of the semi-permeable seabed, that is <, the phase
shift of hydraulic head fluctuation is larger than 0 and lags behind the tide
fluctuation. The change of the amplitude of hydraulic head fluctuation in the
confined aquifer relative to the tide fluctuation with the different parameters
is described as follows: Given a fixed , when is given
different values, if is larger than , the relative
amplitude is smaller than ; if is smaller than , the relative
amplitude is larger than . Given a fixed, when is given different
values, if is larger than, the relative
amplitude is larger than ; if is smaller than, the relative
amplitude is smaller than . The change
of the relative amplitude of the hydraulic head fluctuation in the confined
aquifer with the dimensionless parameter is described as
follows: when is larger than, the relative
amplitude is increased strictly; when is smaller than , the relative
amplitude decreased strictly. The change of the phase shift with the
dimensionless parameter is described as
follows: when is larger than , the phase shift
is decreased strictly first and then increased strictly; when is smaller than , the phase shift
is increased strictly first and then decreased strictly. While the impact
of the dimensionless parameterson the hydraulic head
fluctuation is nearly the same as the impact of the dimensionless parameter. The impact
of the dimensionless parameterson the amplitude
of the hydraulic head fluctuation is contrary to the impact of the
dimensionless parameter, while the impact
of the dimensionless parameterson the phase shift
of the hydraulic head fluctuation is the same as the impact of the
dimensionless parameter. The impact
of the parameter on the hydraulic head
fluctuation is very faint relatively and works between 1 and 3 obviously.
Key words: tide, hydraulic
head, analytical solution, semi-permeable seabed, tidal efficiency, leakage,
storativity.
1 Introduction
Ground water is a
most important component of the water resources for human’s lives. With the
quickly development of the economy and city, water resources especially
freshwater resources are needed intensively and aquifers under the ground are
exploited intensively correspondingly. Therefore, people have to be confronted
with various hydrography problems, such as ground water resources dry up,
fountain die away, sea water intrusion, ground sedimentation, karst
sedimentation, soil desert, water quality deterioration and ecosystem
environment ecdysis and so on (Farrell, 1994; Svitil, 1996; Delwyn et al.,
1998; Akpofure et al., 1984; Carr and Van der Kamp, 1969). In order to solve
all of the above problems, people have to evaluate the water resources rationally,
ascertain hydrography condition and make use of water resources under the
ground rationally.
Mineral resources and energy sources in the sea are explored constantly.
At the same time benthal fresh water resources are concerned earnestly. Hence
many researchers pay lots of attention to the relations between the sea water
and ground water. Since the middle period of the last century, they have
presented analytical and numerical research on the hydraulic head fluctuation
in the coastal areas. For example, Jacob [1950] and Ferries [1951] described
the only one coastal aquifer by an equation and derived its analytical
solution; Carr[1969] investigated the tide-induced salt water intrusion;
Carr[1972] estimated the aquifer parameters such as diffusivity, hydraulic
conductivity and specific storage of coastal confined aquifers by using a
harmonic analysis method; Pandit [1991] presented a method to compute the ratio
of the vertical to the horizontal component of the hydraulic conductivity of a
homogeneous aquifer using a finite-element model. Li and Chen [1991a]
considered the confined aquifer with the roof length is finite. Li and Chen
[1991b] further took into account the leakage from the seawater to the offshore
part of the confined aquifer. Sun [1997] used a two-dimensional tidal loading
boundary condition to derive an exact analytical solution of the tide-induced
groundwater fluctuation in an estuary. Jiao and Tang [1999] considered a
multilayer aquifer system consisting of an upper unconfined aquifer, a lower confined
aquifer and a semi-permeable layer between them. Li and Jiao [2001]
further considered the storativity of the semi-permeable layer. All of the
above the researchers considered the coastal aquifer hydraulic head fluctuation
induced by the sea tide. While in this paper on the basis of previous theory on
the dynamics of the groundwater and seawater we considered the tide-induced
hydraulic head fluctuation in a leaky confined aquifer under the sea, and the
aquifer is assumed to be overlain by a semi-permeable seabed. Both the seabed
and the aquifer are assumed to extend horizontally and infinitely. It can
provide reliable theory foundation for understanding the hydraulic head
fluctuation in the confined aquifer under the sea and parameters evaluation. We
established a mathematical model about this system and an exact analytical
solution is derived. Based on this analytical solution we discussed the impact
of some dimensionless parameters on the hydraulic head fluctuation in the
confined aquifer.
2 Conceptual Model
Let the axis be
positive upwards and perpendicular to the seabed (Fig. 1). All the layers
are homogeneous
and with constant thickness. It is further assumed that the density difference
between the groundwater and the seawater can be neglected due to its slight
impact on groundwater level fluctuation (Li and Chen [1991a]). Based on these
assumptions and the theories of leaky, elastic aquifers proposed by Hantush and
Jacob [1955] and Jacob [1950], the mathematical model to describe the
groundwater fluctuations in Fig. 1 is as follows.
(1)
(2)
(3)
(4)
(5)
where , and are
hydraulic head [L], specific storage [L-1] and vertical
hydraulic conductivity [LT-1] of the semi-permeable seabed or the
aquifer, respectively; is the tidal
efficiency (dimensionless) (Jacob, [1950]); and are the
thicknesses [L] of the aquifer and the semi-permeable seabed, respectively; takes the
value of in the
semi-permeable seabed and in the
confined aquifer; equals in the
semi-permeable seabed and in the
confined aquifer; HS(t) is hydraulic head of the
sea tide [L]; is the
amplitude [L] of the tidal change; is the
angular velocity (or frequency) [T-1] of tide and equals with being the
tidal period [T], the time between high and low tides (Todd, [1980]); is the phase
shift (in radian). The mean sea level is set to be the datum of the system.
Eq. (1) indicates that the groundwater level fluctuation in the leaky
confined aquifer under the sea is caused by its elastic compression and
expansion due to the tidal loading rate of the sea water above seabed (), in addition to
the sea tidal fluctuation at the seabed boundary as expressed by
eq. (2). Eqs. (3) and (4) are the continuity conditions of hydraulic head and
groundwater flux, respectively. Eq. (5) states the no-flow boundary condition
on the upper surface of the impermeable bottom.
3 Derivation of
the Analytical Solution
Let be a complex
function of the real variables z and t that
satisfies the equations (1)-(5) after in equations
(1) and (2) is replaced by , where .
Let is the
solution to (1)-(5), it follows that
(A)
where Re denotes
the real part of the followed complex expression. Now suppose
(A2)
where Z(z)
is an unknown function of . Substituting
(A2) into the five equations which satisfies,
and extending the five resultant real equations into complex ones with respect
to the unknown complex function , then dividing
the results by Aexp(iwt), yield
, (A3)
(A4) (A5) , (A6) , (A7)
(A8)
The solution to
(A3)-(A8) is
(A9)
where
(A10)
(A11)
(A12)
where , C1 and C2 are
two complex constants determined by equations (A4)-(A5) and given by
(A13)
(A14)
where
, (A15)
(A16)
(A17)
(A18)
where is the
specific leakage of the semi-permeable seabed, andare the tidal
efficiencies of both the confined aquifer and the semi-permeable seabed. While , , and are the
dimensionless parameters correlative with the property of both the confined
aquifer and the semi-permeable seabed. Now substituting (A9) back into (A2) to
determine, and finally
calculating the real part of in view of
equation (A1), the solution to (1)-(5) is obtained
(A19)
4 Discussion
Based on the
analytical solution (A19) of the model (1)-(5), we intend to discuss how the
amplitude and the phase shift of the hydraulic head fluctuation in the confined
aquifer change with different dimensionless parameters. Assume that at a fixed
location , the ratio of the
groundwater-head fluctuation amplitude to the sea tide amplitude is (relative
amplitude), and the time lag of the
groundwater response to sea tidal fluctuation is ,then
(A20) In
view of the analytical solution (A19), we can obtain
that is
(A21)
Comparing (A20)
and (A21), we can obtain
(A22)
(A23)
We can see that
there are six dimensionless parameters in the analytical solution by observing
(A19)、(A10)、(A11)、(A16) and (A17): the tidal efficiencies of both the
confined aquifer and the semi-permeable seabedand, the two
dimensionless parameters and which are
determined by the hydraulic conductivity and the storativity of the confined
aquifer and the two dimensionless parameters and which are
determined by the hydraulic conductivity and the storativity of the
semi-permeable seabed. It is important to know the rough ranges of the above
six dimensionless parameters in real aquifer systems. In view of the table1 in Li
and Jiao [2001], one can see that the thickness of the confined aquifer ranges from
6.1~28m, the thickness of the semi-permeable seabed ranges from
3.0~15.2m. Based on these data, we can obtain ranges from
1.22~3.50, in this paper we chose , m, m. Taking
semidiurnal sea tide as an example, we can see that its angular velocity . The hydraulic
head fluctuation in the confined aquifer is more significant than it in the
semi-permeable seabed for us, so in this paper we will mainly discuss how the
six dimensionless parameters influence the hydraulic head fluctuation in the
confined aquifer. Therefore we can see that , that is , we assumed that in this
paper. Here in view of the table 1 in Li and Jiao [2001],
through some simple calculation, we can obtain the vertical hydraulic
conductivity of the confined aquifer ranges from
1.1984~58, while the storativity ranges from 0.00000214~0.000016. In view of
(A10)、(A11)、(A16) and (A17), also through some simple
calculation, we can obtain the rough ranges of the dimensionless parameters ，， and in this paper: ranges from
0.01294~0.1801, ranges from
0.00085~0.00326, ranges from
3.3147~17.3149, ranges from
0.000057~0.00068. Based on the above data, the discussion ranges of the
parameters will be chosen 0.01~1.0 for , 0.0001~0.01
for , 1.0~100.0 for and 0.00001~0.001
for . In this paper we
plot figures which describe how the relative amplitude and the
phase shift of the
hydraulic head fluctuation change with different dimensionless parameters. In
the figures we assume that some fixed parameters are given the middle value of
their ranges, for example =0.1, =50.0, =0.001, =0.0001, =0.5, =0.5.
4.1 Influence of
the variable on the hydraulic
head fluctuation
Fig. 2 shows how
the relative amplitude and the
phase shift of the
hydraulic head fluctuation change with the variable for
different tidal efficiency of the
semi-permeable seabed. One can see that the relative amplitude and the
phase shift is nearly
invariable with the variable.
4.2 Influence of the tidal efficiency on the hydraulic head fluctuation
Fig. 3(a) and (b) shows how the relative
amplitude and the
phase shift of the
hydraulic head fluctuation change with the tidal efficiency of the confined
aquifer for different parameters. Fig. 4(a)
and (b) shows how the relative amplitude and the
phase shift of the
hydraulic head fluctuation change with the tidal efficiency of the
semi-permeable seabed for different parameters.
From the fig. 3 and fig. 4, one can see that the
relative amplitude of the hydraulic head
fluctuation in the confined aquifer increases strictly with the tidal
efficiencies andof both the
confined aquifer and the semi-permeable seabed, while the phase shift of hydraulic head
fluctuation decreases strictly with the tidal efficiency of the confined
aquifer and from lag becomes to exceeding. It increases strictly with the tidal
efficiency of the
semi-permeable seabed and from exceeding becomes to lag. And this critical
point between lag and exceeding is =. When==0.5, one can see
that the relative amplitude is approximately equal to 1/2, while the phase shift is
approximately equal to 0.
The influence of the dimensionless parameter on how the
hydraulic head fluctuation changes with the tidal efficiencies of both two
layers is analyzed in detail through fig. 5 and fig. 6. Fig. 5(a) and (b) show
how the relative amplitude changes with
the dimensionless parameter for
different tidal efficiency and of both the
confined aquifer and the semi-permeable seabed, respectively. Fig. 6(a) and (b)
show how the phase shift changes with
the dimensionless parameter for
different tidal efficiencies and of both the
confined aquifer and the semi-permeable seabed.
From fig. 2, fig.
3, fig. 4, fig. 5 and fig. 6, one can see that the change trend of the relative
amplitude with the different parameters is described as follows: When ==（is a number
between 0 and 1）, the hydraulic
head fluctuation relative amplitude equals toapproximately and
the phase shift equals to 0 approximately. When the tidal efficiency of the
confined aquifer larger than the tidal efficiency of the semi-permeable seabed,
that is >, the phase
shift of hydraulic head fluctuation is smaller than 0 and exceeds the tide
fluctuation. When the tidal efficiency of the confined aquifer larger than
the tidal efficiency of the semi-permeable seabed, that is <,the phase shift
of hydraulic head fluctuation is larger than 0 and lags behind the tide
fluctuation. Given a fixed, when is given
different values, if is larger than, the relative
amplitude is smaller than; if is smaller than , the relative
amplitude is larger than . Given a fixed, when is given different
values, if is larger than, the relative
amplitude is larger than; if is smaller than, the relative
amplitude is smaller than.
Form fig. 5 and
fig. 6, one can see that the change of the relative amplitude of the hydraulic
head fluctuation in the confined aquifer with the dimensionless parameter is described as
follows: when is larger than , the phase shift
is decreased strictly first and then increased strictly; when is smaller than , the phase shift
is increased strictly first and then decreased strictly.
Through the above
analysis, one can see that the influence of the difference of the size
relation between the tidal efficiencies of the two layers on the hydraulic head
fluctuation is intensive enough. Especially when is equal to, the relative
amplitude and phase
shift take special
data. Now in view of the analytical solution (A19), we try to analyze the case
of =.
Substituting ==(is a constant
between 0 and 1) to (A13) and (A14), we can obtain
(B1)
(B2)
now substituting (B1)
and (B2) to (A12), one can see
(B3)
next substituting (B3)
to (A9), then
(B4)
Based on the above
analysis about the rough ranges of the different parameters, we considered when , (B4) can be
predigested as follows
(B5)
From (B5) and the
rough ranges of the parameters, through a series of simple but complex
calculation, we can see that when = =, the relative
amplitude is approximately
equal to, while the phase
shift is
approximately equal to 0.
While the impact of the dimensionless parameterson the hydraulic head
fluctuation is nearly the same as the impact of the dimensionless parameterand the
impact of the dimensionless parameterson thehydraulic head
fluctuation is contrary to the impact of the dimensionless parameter. The impact
of the parameter on the hydraulic head
fluctuation is very faint relatively and the influence ranges of the parameter is only 1 to 3,
when is bigger than 3,
the hydraulic head fluctuation is nearly constant.
5 Summary
In this paper we discussed a leaky confined aquifer system consisting of a
confined aquifer and a semi-permeable seabed overlies it. Both the seabed and
the aquifer are assumed to extend under the sea horizontally and infinitely. We
used a one-dimensional mathematical model to describe this system. An exact
analytical solution is derived to describe how the hydraulic head fluctuates in
the confined aquifer. The analytical solution describes the hydraulic head
fluctuation is nearly invariable in vertical direction. In the analytical
solution we introduced six dimensionless parameters. In view of the
previous researches, we obtained the rough ranges of the above six
dimensionless parameters. Based on the analytical solution we mainly discussed
the influence of the tidal efficiencies of both the confined aquifer and the
semi-permeable seabed on the hydraulic head fluctuation. We also discussed
the impacts of the other parameters on the hydraulic head fluctuation by
taking as an example. The
influences of the on the hydraulic
head fluctuation is nearly the same as the influence of the parameter.While the
influence of is faint and can
be ignored. While the impact of the dimensionless parameterson the hydraulic head
fluctuation is nearly the same as the impact of the dimensionless parameter. The impact
of the dimensionless parameterson the amplitude
of the hydraulic head fluctuation is contrary to the impact of the
dimensionless parameter, while the impact
of the dimensionless parameterson the phase shift
of the hydraulic head fluctuation is the same as the impact of the
dimensionless parameter. The impact
of the parameter on the hydraulic head
fluctuation is very faint relatively.
Acknowledgement
This research is
supported by the National Natural Science Foundation of China (No.40372111).
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