Advertisement

Water Resources Management

, Volume 32, Issue 15, pp 5027–5039 | Cite as

An Analytical Solution of Groundwater Flow in Wedge-shaped Aquifers with Estuarine Boundary Conditions

  • Mo-Hsiung Chuang
  • Hund-Der YehEmail author
Article
  • 121 Downloads

Abstract

The issue of the groundwater fluctuation due to tidal effect in a two-dimensional coastal leaky aquifer system has attracted much attention in recent years. The predictions of head fluctuation play an important role in dealing with groundwater managements and contaminant remediation problems in costal aquifers. This article presents a two-dimensional analytical model describing the groundwater flow in a coastal leaky aquifer of wedge shape affected by the tides and bounded by two estuarine rivers with an arbitrary included angle. The solution of the model is derived in the Polar coordinates by the Hankel transform and finite sine transform. The head fluctuation predicted by this new solution is compared with that by an existing solution for groundwater flow in a non-L shaped tidal aquifer. The groundwater fluctuation due to the joint effect of estuarine tides is explored based on the present solution. Moreover, the influences of the parameters such as diffusion (Di), included angle (Ф), and tidal river coefficients (K1, K2) on the head fluctuation in the aquifer are also assessed and discussed. The results demonstrate that those parameters have significant effects on the head fluctuation in the leaky confined aquifer system. Moreover, the effect of Di increases with Ф, and the effects of K1 and K2 on the normalized amplitude and phase lag of the groundwater fluctuation are significant when both parameter values are larger than 10−5.

Keywords

Analytical model Coastal leaky aquifer Wedge-shaped aquifers Tidal river 

Notations

A, A1, A2

Amplitudes of the tidal change [L]

c, c1, c2

Phase shifts of the tidal change [rad]

Di

Diffusion [L2 T−1]

f*

Hankel transform

h(r, θ, t)

Hydraulic head [L]

\( \overline{h}\left(r,\theta \right) \)

Hydraulic head [L]

hA

Normalized amplitude of head fluctuation

hm

Hydraulic head from the datum [L]

i

\( \sqrt{-1} \)

TID

Tidal intrusion distance

\( {J}_{\mu_n}\left(\cdot \right) \)

Bessel function of the first kind of order μn

K1, K2

Tidal river coefficients

K1r, K2r

Damping coefficients of tidal amplitude

K1i, K2i

Separation coefficients

L

Leakage [T−1]

n

Positive integer series

ph

Phase lag

r

Distance for Polar coordinate [L]

S

Storage coefficient

Si

Coefficient given by Eq. 4

T

Transmissivity [L2 T−1]

t

Time [T]

t0

Tidal period [T]

u

Dumb variable for Hankel transform

uc, uk1, uk2

Coefficients defined by Eq. 8

ω,ω1,ω2

Tidal speeds [rad/T]

θ

Angle for Polar coordinate [rad]

Ф

Angle for two boundaries [rad]

μn

Order of Bessel function of the first kind

1 Introduction

Existing studies indicated that the dynamic behavior of groundwater flow induced by tidal fluctuation is an important issue for the management or investigation of coastal aquifer systems. For example, Levanon et al. (2016) adopted numerical models to study the influence of tides on the groundwater fluctuations and seawater interface. There are some articles focusing on the studies of coastal ecosystems. Xiao et al. (2017) conducted field and simulation studies on the potential link between the groundwater flow and the ecological division of the saltwater marshes in the Yangtze River estuary. Wang et al. (2018) studied the changes in the chemical composition of coral reef systems affected by the groundwater discharge and tides. Those studies illustrate the importance of understanding the effects of ocean tides and groundwater interactions on the ecosystems. The groundwater flow problem may be treated as one, two, or quasi-two dimensional model dependents on the reality of the physical situations and the desired accuracy of the problem solutions. The aquifer system may be of a single layer or comprised of multiple layers. In addition, the aquifer may directly contact with the ocean or river and therefore its flow may be affected by oceanic tides and/or estuarine tides. A large number of articles dealing with analytical modeling for groundwater flow in tidal aquifers has been published in recent decades. Table 1 lists some of the articles classified according to the flow dimension, layered system, and type of tidal boundary. Among them, Nielsen (1990) and Jiao and Tang (1999) dealt with 1-D flow, Sun (1997) and Tang and Jiao (2001) studied the quasi 2-D flow, and the articles by Li et al. (2000; Li et al. 2002) and Li and Jiao (2002) presented the solution for flow in L-shaped aquifers with a right angle. Other studies in the field with various types of aquifer configuration may be categorized as: aquifers extending under the sea (e.g., Chuang and Yeh 2007; Fleury et al. 2007; Li et al. 2008; Geng et al. 2009; Guarracino et al. 2012; Wang et al. 2014), aquifers with sloping beaches (Teo et al. 2003) or with sloping bottom (Asadi-Aghbolaghi et al. 2012), inhomogeneous tidal aquifers (Chuang et al. 2010; Monachesi and Guarracino 2011), and island aquifers (e.g., Li et al. 2006; Sun et al. 2008; Chang et al. 2010; Huang et al. 2012).
Table 1

Classification of analytical solutions involving 1-D, quasi 2-D, and 2-D groundwater flow in tidal aquifers with various types of boundary conditions

References

Aquifer layer

Type of tidal boundary

1-D flow

Nielsen (1990)

single

Oceanic tide

Jiao and Tang (1999)

multiple

Oceanic tide

Quasi 2-D flow

Sun (1997)

single

Estuarine tide

Tang and Jiao (2001)

multiple

Estuarine tide

2-D flow in L-shaped aquifers with a right angle

Li et al. (2000)

single

Oceanic tide in coastline and estuarine tide

Li et al. (2002)

single

Oceanic tide in coastline and estuarine tide

Li and Jiao (2002)

multiple

Oceanic tide in coastline and estuarine tide

For leaky tidal aquifer systems, Jiao and Tang (1999) presented an analytical solution derived from one-dimensional (1-D) groundwater flow equation describing the head fluctuation in the aquifer system. They investigated the effect of leakage on the groundwater behavior in the confined aquifer. Later, Li and Jiao (2001) extended the work of Jiao and Tang (1999) with considering two-dimensional (2-D) flow in confined aquifer and 1-D vertical flow by accounting for the effect of aquitard’s storage. The solution of the model is then employed to investigate the effects of both leakage and aquitard’s storage on the head fluctuation in a semi-confined aquifer.

For tidal aquifers subject to the effect of estuarine tide, Sun (1997) developed a 2-D analytical solution for flow in the aquifer system with a tidal river boundary. The amplitude and phase of the tide vary with position and time and their variations, referred as the damping coefficient for amplitude and separation coefficient for phase, along a shoreline are considered in the analysis. Tang and Jiao (2001) developed a 2-D analytical solution for groundwater fluctuation due to tidal effect from sea or estuarine in a coastal leaky aquifer with an overlying unconfined aquifer. The solutions of Sun (1997) and Jiao and Tang (1999) can be considered as special cases of their solution. Li et al. (2002) proposed an analytical model to simulate groundwater head fluctuations in an L-shaped tidal aquifer bounded by estuarine and oceanic boundaries intersected with a right angle. The solution of the model is derived based on the complex transform and Green’s function method. Li and Jiao (2002) also developed an analytical solution for describing groundwater fluctuation in a coastal leaky aquifer system bounded by an L-shaped coastline with an aquitard between the upper unconfined aquifer and lower confined one. This solution can be simplified to the solutions of Li et al. (2000) and Li et al. (2002). In addition, the authors also presented a simple approximate solution without integral for ease of computing. Recently, Dong et al. (2016) developed an analytical solution for the head fluctuation in 2-D coastal L-shaped and non-L-shaped aquifers. They decomposed the 2D groundwater flow model into two 1D flow problems and then applied the superposition principle to the 1-D solution of Dong et al. (2012) to obtain the 2-D solution. More recently, Chuang and Yeh (2017) developed a 2-D analytical model for describing groundwater flow induced by tidal fluctuations in leaky aquifers bounded by the ocean and an estuarine river with an included angle Ф. They investigated the impacts of included angle, aquitard leakage, and damping and separation coefficients on the head fluctuations in the L-shaped aquifer system.

This study aims at developing an analytical model to describe groundwater flow in a coastal leaky aquifer of wedge shape bounded by two tidal rivers with an arbitrary included angle. The solution of the model expressed in the polar coordinates is developed by the methods of finite sine transform and Hankel transform. Then, this solution is employed to investigate the groundwater head fluctuation due to the effects of the oceanic and estuarine tides. This new solution is also used to predict the head fluctuations in a non-L shaped tidal aquifer and compare with that of Dong et al. (2016). Moreover, this solution is also adopted to assess the effects of diffusion, included angle, and tidal river coefficient on the head fluctuation in a leaky aquifer system. The present solution has been demonstrated that it can be a useful tool to investigate the dynamic groundwater behavior in various scenarios of coastal loading in tidal leaky aquifers with a bending shoreline.

2 Model Setup and Boundary Conditions

Consider a conceptual model for a coastal leaky aquifer of wedge shape, which is overlain by an unconfined aquifer, underlain by a confined aquifer, and an aquitard in between. Figure 1 shows the cross-section view of the model for the aquifer system with an arbitrary included angle Φ expressed in the polar coordinates (r, θ). The aquifer can be in different shape with a combination of different tidal boundaries as demonstrated in cases 1–3. In case 1, the aquifer is bounded by two estuarine rivers converging with an acute angle. In cases 2 and 3, they both have an estuarine river in one side and the ocean on the other side, but the estuarine river and the ocean intersect with an acute angle in case 2 and obtuse angle in case 3. The dynamic behavior of groundwater fluctuation in the leaky confined aquifer due to tides is analyzed under the assumption of the water level in the unconfined aquifer maintained constant (Jiao and Tang, 2001). The lower confined aquifer interacts with the upper unconfined one through the middle aquitard. The aquifer is assumed to be homogeneous and isotropic. The thickness of the unconfined aquifer is much larger than the amplitude of the tidal fluctuation, thereby allowing the linearization of the unconfined flow equation. The initial groundwater head in the entire aquifer system is uniform and equal to hm, which represents the distance from a convenient reference datum to the groundwater head. Under those assumptions, the governing equation describing the head fluctuations in the leaky confined aquifer in the polar coordinates can be written as
$$ T\left(\frac{\partial^2h}{\partial \kern0.1em {r}^2}+\frac{1}{r}\frac{\partial h}{\partial r}+\frac{1}{r^2}\frac{\partial^2h}{\partial {\theta}^2}\right)=S\frac{\partial h}{\partial t}+L\left(h-{h}_m\right) $$
(1)
where h(r, θ, t) [L] is the groundwater head; T [L2 T−1] and S [−] are respectively the transmissivity and storage coefficient. The leakage L [T−1] is the ratio of the aquitard hydraulic conductivity to its thickness.
Fig. 1

A cross-section view of 2-D conceptual model in a wedge-shaped aquifer system with an arbitrary included angle (Φ) in the polar coordinates (r, θ). The aquifer can be in different shapes with a combination of different tidal boundaries as demonstrated in cases 1–3

The boundary condition for estuarine river 1 is expressed in terms of a harmonic cosine function to reflect its water level fluctuations induced by the oceanic tides as
$$ h\left(r,0,t\right)={h}_m+A{e}^{-{K}_{1r}r}\cos \left(\omega t+{K}_{1i}r+c\right)={h}_m+A\operatorname{Re}\left({e}^{-i\left(\omega t+{K}_1r+c\right)}\right) $$
(2a)
where h(r, 0, t) is the hydraulic head at θ = 0, A is the amplitude of the tidal change, c is the phase shift and assumed zero for a single tidal constituent, ω is the tidal speed, hm is the hydraulic head and assumed zero for convenience in deriving the analytical solution. Re denotes the real part of the complex expression, \( i=\sqrt{-1} \). Also, ω = 2π/t0 where t0 is the tidal period. Similarly, the boundary condition for estuarine river 2 is denoted as
$$ h\left(r,\phi, t\right)={h}_m+A{e}^{-{K}_{2r}r}\cos \left(\omega t+{K}_{2i}r+c\right)={h}_m+A\operatorname{Re}\left({e}^{-i\left(\omega t+{K}_2r+c\right)}\right) $$
(2b)
where h(r, ϕ, t) is the hydraulic head at θ = ϕ. The K1r and K2r are damping coefficients of tidal amplitude, K1i and K2i are separation coefficients representing the change of phase with distance (Sun 1997). The tidal river coefficients (K1, K2) are defined as the damping coefficient plus the separation coefficient times i, K1 = K1r + iK1i and K2 = K2r + iK2i.
The condition for the remote boundary at the inland side in the r-direction is expressed as
$$ h\left(r\to \infty, \theta, t\right)=0 $$
(2c)

3 Analytical Solution

The solution of the model composed of Eqs. (1) and (2) is developed first by the method of separation of variables. Assume that \( \overline{h}\left(r,\theta \right) \) is a complex function of the real variables r and θ. It is denoted as
$$ h\left(r,\theta, t\right)=\overline{h}\left(r,\theta \right)A \operatorname {Re}\left({e}^{-i\left(\omega t+c\right)}\right) $$
(3)
Substituting Eq. (3) into Eq. (1) and dividing the results byAei(ωt + c) yields
$$ \frac{\partial^2\overline{h}}{\partial \kern0.1em {r}^2}+\frac{1}{r}\frac{\partial \overline{h}}{\partial r}+\frac{1}{r^2}\frac{\partial^2\overline{h}}{\partial {\theta}^2}={S}_i\overline{h} $$
(4)
where Si = (L + iωS)/T. The boundary conditions of Eqs. 2a and 2b can therefore be written as
$$ \overline{h}\left(r,0\right)=\operatorname{Re}\left({e}^{-{K}_1r}\right) $$
(5a)
$$ \overline{h}\left(r,\phi \right)=\operatorname{Re}\left({e}^{-{K}_2r}\right) $$
(5b)
and the remote boundary condition of Eq. (2) becomes
$$ \overline{h}\left(\infty, \theta \right)=0 $$
(5c)
Applying the finite sine transform (\( \overline{H} \)) to Eqs. (4) and (5) yields (Yeh and Chang 2006)
$$ \frac{\partial^2\overline{H}}{\partial \kern0.1em {r}^2}+\frac{1}{r}\frac{\partial \overline{H}}{\partial r}-\frac{\mu_n^2}{r^2}\overline{H}+\left[{e}^{-{K}_1\cdot r}-{\left(-1\right)}^n{e}^{-{K}_2\cdot r}\right]\frac{\mu_n}{r^2}={S}_i\overline{H} $$
(6)
Then the Hankel transform f, defined as \( {f}^{\ast }(u)={\int}_0^{\infty } rf(r){J}_{\mu_n}(ur) dr \) is adopted to eliminate the r coordinate in Eq. (6). The result is (Sneddon 1972)
$$ {\displaystyle \begin{array}{l}{\overline{H}}^{\ast}\left(u,n\right)=\frac{\mu_n}{u^2+{S}_i}{\int}_0^{\infty}\left\{\frac{e^{-{K}_1\cdot r}-{\left(-1\right)}^n{e}^{-{K}_2\cdot r}}{r}\right\}{J}_{\mu_n}(ur) dr={u}_c\left[{u}_{K1}-{\left(-1\right)}^n{u}_{K2}\right]\\ {}\end{array}} $$
(7)
where \( {J}_{\mu_n}\left(\cdot \right) \) is the Bessel function of the first kind of order μn and μn = /ϕ with n = 1, 2, 3, … with
$$ {u}_c=\frac{1}{u^2+{S}_i} $$
(8a)
$$ {u}_{k1}=\frac{{\left(\sqrt{u^2-{K}_1^2}-{K}_1\right)}^{\mu_n}}{u^{\mu_n}} $$
(8b)
$$ {u}_{k2}=\frac{{\left(\sqrt{u^2-{K}_2^2}-{K}_2\right)}^{\mu_n}}{u^{\mu_n}} $$
(8c)
The solution for hydraulic head expressed in the polar coordinates obtained by taking the inverse Hankel transform \( f(r)={\int}_0^{\infty }u{f}^{\ast }(u){J}_{\mu_n}(ur) du \) and the inverse finite sine transform to Eq. (7) is written as
$$ h\left(r,\theta, t\right)=A\operatorname{Re}\left\{\frac{2{e}^{-i\left(\omega t+c\right)}}{\phi}\sum \limits_{n=1}^{\infty}\sin \left[{\mu}_n\theta \right]{\int}_0^{\infty }{u}_c\left[{u}_{K1}-{\left(-1\right)}^n{u}_{K2}\right]u{J}_{\mu_n}(ur) du\right\} $$
(9)

The value of h in Eq. 9 is evaluated using first the Mathematica function NIntegrate to perform the numerical integration for the integral and then the Shanks transformation (Shanks 1955) to accelerate the convergence when evaluating the infinite series. Note Eq. 9 can be considered as a pseudo steady-state solution, or steady-state solution, describing groundwater flow in case 1 for aquifers affected by tidal loadings from two estuaries. On the other hand, Eq. 9 with K1 = 0 or K2 = 0 is the solution for cases 2 and 3. Moreover, the present solution with K1 = K2 = 0 for groundwater fluctuations can simulate the case that the aquifer system has oceanic boundaries in both sides.

4 Results and Discussion

In this section we adopt several hypothetical cases presented in Jiao and Tang (1999) and Li et al. (2002) to investigate the influences of included angle (Ф), diffusivity (Di = T/S), damping coefficient (K1r, K2r), and separation coefficient (K1i, K2i) on the head fluctuation in the leaky confined aquifer system. The parameter values in the case studies are chosen as follows: Di ranging from 104 to 106 m2/h, K1 and K2 ranging from 10−7 to 10−4, ω = 0.2618 rad/h, and hm = 0. The amplitude of head fluctuation is normalized as hA =  ∣ h ∣ /A. The phase lag is defined as ph = tan−1[−1 × Im(h)/ Re(h)] in which Im represents the imaginary part of the complex expression.

4.1 Comparisons with Two Existing Analytical Solutions Using Field Observed Data

Li et al. (2002) reported a case study on the comparison of predicted water table fluctuations from their solution with the water table fluctuation data observed at a pit in the fill material of the Chek Lap Kok (CLK) airport platform, Hong Kong made in 1994. Figure 2a shows the area bounded by the dashed lines reclaimed around CLK island in 1994. The distances from pit 1 to the two sides of the right angle of the sea-land boundary are approximately 80 m and 40 m along the x- and y-directions, respectively. The circles shown in Fig. 2a represent the water table data observed at pit 1 in September 1994 and the hollow square symbol stands for the predicted results from the present solution with considering a L-shaped coastline near the pit. Note that the middle part of the observed data at pit 1 are not available. The parameters used to predict the head fluctuation are determined by least squares fitting to the observed data, which are mainly contributed by semidiurnal and diurnal sinusoidal components. The datum of tidal level they chose is mPD (meter above Principle Datum) which is 1.6 m below the mean sea level. The parameter values they obtained from the least squares fitting were: mean water table hm = 1.77 mPD, tidal amplitudes A1 = 0.36 m and A2 = 0.58 m, tidal frequencies ω1 = 0.507/h and ω2 = 0.237/h, phase shift c1 = 2.138 and c2 = 3.209, and transmissivity T = 854.0 m2/h (Li et al. 2002, Table 1).
Fig. 2

a Locations of piezometer and pit 1 at Chek Lap Kok Airport. (modified from Jiao and Tang (1999, Fig. 5) and Li et al. (2002, Fig. 4)). b Observed head fluctuation in the piezometer and pit 1 and the predicted results by the present solution and the solutions of Jiao and Tang (1999) and Li et al. (2002)

Interestingly, Jiao and Tang (1999) also reported a case study associated with the development of a 1-D analytical solution for the prediction of water table fluctuations in a tidal leaky confined aquifer with a straight coastline. Their solution was mainly developed to investigate the effect of leakage on the head response in a coastal leaky aquifer system. Their solution was also used to predict the head fluctuation at a point, where a piezometer was installed within the CLK airport built in 1996 and indicated in Fig. 2b. The predicted head fluctuations from the solution of Jiao and Tang (1999) represented by triangles and the piezometer data observed in October 1997 denoted by diamonds are also plotted in Fig. 2b. The parameters used for Jiao and Tang (1999) are Di = 4.4 × 103 m2/h, T/L = 9.2 × 105 m2, S/L = 207 h, t0 = 12.2 h, and hm = 0. With Ф = 180o and θ = 90o, the present solution can also predict the head fluctuation in the case study of Jiao and Tang (1999) for a tidal leaky aquifer with a straight coastline. Figure 2b indicates that the present solution and the solutions of Li et al. (2002) and Jiao and Tang (1999) give predictions very close to the head fluctuation observed at pit 1 near a right-angle coastline and at the piezometer near a straight coastline in the CLK airport. Obviously, the solutions of Jiao and Tang (1999) and Li et al. (2002) can be considered as special cases of the present solution for Ф = 180o and θ = 90o, respectively.

4.2 Joint Effects of Included Angle and Diffusivity

Figure 3 shows the distribution curves of hA and ph of the head fluctuation versus Ф observed at (r, θ) = (300 m, 45o) for L = 0, K1 = K2 = 10−5 /m, and Di ranging from 104 to 106 m2/h. The figure indicates that hA decreases but ph increases significantly when increasing Ф from 50 o to 120 o for this range of Di. The figure, on the other hand, indicates that hA increases but ph decreases significantly when increasing Di from 104 to 106 m2/h for all values of Ф. The figure also displays the effect of Di increasing significantly with Ф varying from 50 o to 120 o. Obviously, both Di and Ф have significant effects on the head fluctuation in the leaky confined aquifer system.
Fig. 3

The curves of hA and ph of the head fluctuation versus Ф happened at ө = 45o for L = 0, K1 = K2 = 10−5 /m with various values of Di at r = 300 m

4.3 Joint Effects of the Tidal River Coefficients of Two Boundaries

Figure 4 demonstrates the contour curves of hA and ph of head fluctuations against negative logarithm values of the tidal river coefficients (K1 and K2) observed at (r, ө) = (3000 m, 22.5o) for Di = 5*105 m2/h, L = 0, Ф = 45o. The dashed line signifies that the values of K1 and K2 are equal along this line. The figure shows that hA decreases markedly but ph increases significantly as K1 and K2 increase from 10−7 to 10−4 for L = 0 /hour. The figure indicates that the joint effects of K1 and K2 on the hA and ph are obvious when the value of K1 or K2 is larger than 10−5. In addition, both K1 and K2 also have significant influences on the head fluctuation in the estuarine leaky aquifer.
Fig. 4

The contour curves of (a) hA and (b) ph of the head fluctuation versus the negative logarithm values of K1 and K2 at Ф =45o, ө = Ф/2 for Di = 5*105 m2/hour, L = 0 at r = 3000 m. The dashed line represents equal values of K1 and K2

4.4 Comparison with Dong et al.’s (2016) Solution for Non-L Shaped Tidal Aquifers

This section discusses and compares the difference in the amplitude of groundwater head fluctuation, hA, predicted by the present solution and Dong et al.’s (2016) solution for non-L shaped tidal aquifers. Note that the interaction zone shown in Fig. 1 represents the region where the groundwater is affected by both the oceanic tides and estuarine tides. The curves of hA versus the included angle Ф observed at ө = Ф/2 and r = 300 m in the interaction zone for L = 0, K1 = K2 = 0 are plotted in Fig. 5(a). The dashed line represents the predicted hA by Dong et al.’s (2016) solution while the solid line denotes that by the present solution. This figure exhibits that the hA decreases significantly as Ф increases from 60o to 180o observed at ө = Ф/2 and r = 300 m. The predicted hA by the present solution with Ф =180o is 0.086 for Di = 2000, 0.177 for 4000, and 0.24 for 8000 m2/h. Figure 5(b) shows the differences between those two solutions versus Ф slightly decrease initially, reaches the lowest value at different Ф for different Di, and then increases markedly with increasing Ф. This figure also indicates that Dong et al.’s (2016) superposition solution largely overestimates the predicted hA in the interaction zone. Figure 5(b) shows that the values of the difference in hA are exactly the same as the predicted hA by the present solution when Ф =180o, indicating that their solution gives twice what the present solution predicts.
Fig. 5

The curves of (a) the normalized amplitude, hA, of the head fluctuation versus Ф predicted by the present solution (solid line) and Dong et al.’s (2015) solution (dashed line) and (b) their differences in hA at ө = Ф/2 and r = 300 m for L = 0, K1 = K2 = 0. (c) the hA along inland distance in the x-direction and at y = 200 m in the L-shaped aquifer for L = 0, K1 = K2 = 0, and different values of Di

For the case that Ф =180o and ө = Ф/2, the 2-D flow in tidal aquifers reduces to 1-D flow. Under this circumstance, the h values predicted by the 2-D flow solution should be equal to those by the 1-D flow solution. However, the h values predicted by Eq. 15 when Ф =180o and ө = Ф/2 equals twice the h values predicted by the 1-D solution of Dong et al. (2015, Eq. 9) (or Dong et al. 2012, Eq. 7). Also, Dong et al.’s solution gives twice the hA predictions by the present solution when Ф =180o as indicated in Fig. 5(a) and 5(b) mentioned above.

Figure 5(c) shows the hA along inland distance from 200 m to 1600 m in the x-direction versus diffusivity for y = 200 m, L = 0, K1 = K2 = 0 in the L-shaped aquifer. The vertical dotted line represents the tidal intrusion distance (TID), defined as the farthest landward distance from the coastline to the location where hA is less than 10−3, in the x-direction. The figure exhibits that the TIDs in the x-direction for Di = 2000, 4000, and 6000 m2/h are about 604, 854, and 1046 m, respectively, revealing a larger Di has a longer TID. The figure also shows that the differences in hA predicted by both solutions are obvious within the interaction zone in which the tidal propagation distance in the x-direction is smaller than the TID. On the other hand, the hA’s predicted by both solutions are the same beyond the interaction zone where the propagation distance in the x-direction is larger than the TID. These results indicate that Dong et al.’s (2015) solution developed based on the 1-D flow solutions and superposition principle is unable to correctly describe the flow in the interaction zone.

5 Concluding Remarks

A novel analytical model has been developed to describe the groundwater fluctuation in a coastal wedge-shaped leaky aquifer system. The aquifer may be bounded by two estuaries or surrounded by the ocean and one estuary at the other side with an arbitrary included angle. The solution of the model expressed in the polar coordinate system is developed based on the finite sine transform and Hankel transform. This solution is adopted to assess the impacts of the hydraulic diffusivity, included angle, damping coefficient, and separation coefficient on the head response in a wedge-shaped leaky confined aquifer. The results of this study demonstrate that the normalized amplitude of the head fluctuation decreases notably but the phase lag of the head fluctuation increases with the increasing included angle. Contrarily, the normalized amplitude increases but the phase lag decreases apparently with the increasing diffusivity. In addition, the normalized amplitude decreases but the phase lag increases with increasing tidal river coefficients. Apparently, the parameters such as the included angle, diffusivity, and tidal river coefficients have significant effects on the groundwater head fluctuations in coastal leaky aquifers. Finally, two case studies indicate that the present solution gives very good predictions as compared with the observed head fluctuation data and the results predicted by the solutions of Jiao and Tang (1999) and Li et al. (2002).

Notes

Acknowledgements

This study was supported by the Taiwan Ministry of Science and Technology under the grant MOST 107-2221-E-009-019-MY3 and 106-2621-M-130-002-.

Compliance with Ethical Standards

Conflict of Interest

None.

References

  1. Asadi-Aghbolaghi M, Chuang MH, Yeh HD (2012) Groundwater response to tidal fluctuation in a sloping leaky aquifer system. Appl Math Model 36(10):4750–4759CrossRefGoogle Scholar
  2. Chang YC, Jeng DS, Yeh HD (2010) Tidal propagation in an oceanic island with sloping beaches. Hydrol Earth Syst Sci 14(7):1341–1351CrossRefGoogle Scholar
  3. Chuang MH, Huang CS, Li GH, Yeh HD (2010) Groundwater fluctuations in heterogeneous coastal leaky aquifer systems. Hydrol Earth Syst Sci 14:1819–1826CrossRefGoogle Scholar
  4. Chuang MH, Yeh HD (2007) An analytical solution for the head distribution in a tidal leaky aquifer extending an infinite distance under the sea. Adv Water Resour 30(3):439–445CrossRefGoogle Scholar
  5. Chuang MH, Yeh HD (2017) A two-dimensional analytical model for tide-induced groundwater fluctuation in leaky aquifers. Eur Water J 57:435–441Google Scholar
  6. Dong L, Chen J, Fu C, Jiang H (2012) Analysis of groundwater-level fluctuation in a coastal confined aquifer induced by sea-level variation. Hydrogeol J 20:719–726CrossRefGoogle Scholar
  7. Dong L, Cheng D, Liu J, Zhang P, Ding W (2016) Analytical analysis of groundwater responses to estuarine and oceanic water stage variations using superposition principle. J Hydrol Eng 21(1):04015046CrossRefGoogle Scholar
  8. Fleury P, Bakalowicz M, Marsily GD (2007) Submarine springs and coastal karst aquifers: a review. J Hydrol 339:79–92CrossRefGoogle Scholar
  9. Geng XL, Li H, Boufade MC, Liu S (2009) Tide-induced head fluctuations in a coastal aquifer: effects of the elastic storage and leakage of the submarine outlet-capping. Hydrogeol J 17(5):1289–1296CrossRefGoogle Scholar
  10. Guarracino L, Carrera J, Vazquez-Sune E (2012) Analytical study of hydraulic and mechanical effects on tide-induced head fluctuation in a coastal aquifer system that extends under the sea. J Hydrol 450-451:150–158CrossRefGoogle Scholar
  11. Huang CS, Yeh HD, Chang CH (2012) A general analytical solution for groundwater fluctuations due to dual tide in long but narrow islands. Water Resour Res 48:W05508.  https://doi.org/10.1029/2011wr011211 CrossRefGoogle Scholar
  12. Jiao JJ, Tang ZH (1999) An analytical solution of groundwater response to tidal fluctuation in a leaky confined aquifer. Water Resour Res 35(3):747–751CrossRefGoogle Scholar
  13. Levanon E, Shalev E, Yechieli Y, Gvirtzman H (2016) Fluctuations of fresh-saline water interface and of water table induced by sea tides in unconfined aquifers. Adv Water Resour 96:34–42CrossRefGoogle Scholar
  14. Li GH, Li H, Boufadel MC (2008) The enhancing effect of the elastic storage of the seabed aquitard on the tide-induced groundwater head fluctuation in confined submarine aquifer systems. J Hydrol 350(1–2):83–92CrossRefGoogle Scholar
  15. Li H, Jiao JJ (2001) Analytical studies of groundwater-head fluctuation in a coastal confined aquifer overlain by a semi-permeable layer with storage. Adv Water Resour 24:565–573CrossRefGoogle Scholar
  16. Li H, Jiao JJ (2002) Tidal groundwater level fluctuations in L-shaped leaky coastal aquifer system. J Hydrol 268:234–243CrossRefGoogle Scholar
  17. Li H, Jiao JJ, Luk M, Cheung K (2002) Tide-induced groundwater level fluctuation in coastal aquifers bounded by L-shaped coastlines. Water Resour Res 38(3):1024CrossRefGoogle Scholar
  18. Li H, Jiao JJ, Tang ZH (2006) Semi-numerical simulation of groundwater flow induced by periodic forcing with a case-study at an island aquifer. J Hydrol 327:438–446CrossRefGoogle Scholar
  19. Li L, Barry DA, Cunningham C, Stagnitti F, Parlange JY (2000) A two-dimensional analytical solution of groundwater responses to tidal loading in an estuary and ocean. Adv Water Resour 23(8):825–833CrossRefGoogle Scholar
  20. Monachesi LB, Guarracino L (2011) Exact and approximate analytical solutions of groundwater response to tidal fluctuations in a theoretical inhomogeneous coastal confined aquifer. Hydrogeol J 19(7):1443–1449CrossRefGoogle Scholar
  21. Nielsen P (1990) Tidal dynamics of the water table in beaches. Water Resour Res 26(9):2127–2134Google Scholar
  22. Shanks D (1955) Non-linear transformations of divergent and slowly convergent sequences. J Math Phys 34:1–42CrossRefGoogle Scholar
  23. Sneddon IN (1972) The use of integral transforms. McGraw-Hill, Inc, pp 298–352Google Scholar
  24. Sun H (1997) A two-dimensional analytical solution of groundwater response to tidal loading in an estuary. Water Resour Res 33(6):1429–1435CrossRefGoogle Scholar
  25. Sun PP, Li H, Boufadel MC, Geng XL, Chen S (2008) An analytical solution and case study of groundwater head response to dual tide in an island leaky confined aquifer. Water Resour Res 44:W12501.  https://doi.org/10.1029/2008WR006893 CrossRefGoogle Scholar
  26. Tang Z, Jiao JJ (2001) A two-dimensional analytical solution for groundwater flow in a leaky confined aquifer system near open tidal water. Hydrol Process 15(4):573–585CrossRefGoogle Scholar
  27. Teo HT, Jeng DS, Seymour BR, Barry DA, Li L (2003) A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches. Adv Water Resour 26:1239–1247CrossRefGoogle Scholar
  28. Wang C, Li H, Wan L, Wang X, Jiang X (2014) Closed-form analytical solutions incorporating pumping and tidal effects in various coastal aquifer systems. Adv Water Resour 69:1–12CrossRefGoogle Scholar
  29. Wang G, Wang S, Wang Z, Jing W, Xu Y, Zhang Z, Tan E, Dai M (2018) Tidal variability of nutrients in a coastal coral reef system influenced by groundwater. Biogeosciences 15:997–1009CrossRefGoogle Scholar
  30. Xiao K, Li H, Wilson AM, Xia Y, Wan L, Zheng C, Ma Q, Wang C, Wang X, Jiang X (2017) Tidal groundwater flow and its ecological effects in a brackish marsh at the mouth of a large sub-tropical river. J Hydrol 555:198–212CrossRefGoogle Scholar
  31. Yeh HD, Chang YC (2006) New analytical solutions for groundwater flow in wedge-shaped aquifers with various topographic boundary conditions. Adv Water Resour 29(3):471–480CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Urban Planning and Disaster ManagementMing-Chuan UniversityTaoyuanTaiwan
  2. 2.Institute of Environmental EngineeringNational Chiao Tung UniversityHsinchuTaiwan

Personalised recommendations