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

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## 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 (*D*_{i}), included angle (*Ф*), and tidal river coefficients (*K*_{1}*, K*_{2}) 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 *D*_{i} increases with *Ф*, and the effects of *K*_{1} and *K*_{2} 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, A*_{1}*, A*_{2}Amplitudes of the tidal change [L]

*c, c*_{1}*, c*_{2}Phase shifts of the tidal change [rad]

*D*_{i}Diffusion [L

^{2}T^{−1}]*f**Hankel transform

*h*(*r, θ, t*)Hydraulic head [L]

- \( \overline{h}\left(r,\theta \right) \)
Hydraulic head [L]

*h*_{A}Normalized amplitude of head fluctuation

*h*_{m}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

*μ*_{n}*K*_{1},*K*_{2}Tidal river coefficients

*K*_{1r},*K*_{2r}Damping coefficients of tidal amplitude

*K*_{1i},*K*_{2i}Separation coefficients

*L*Leakage [T

^{−1}]*n*Positive integer series

*ph*Phase lag

*r*Distance for Polar coordinate [L]

*S*Storage coefficient

*S*_{i}Coefficient given by Eq. 4

*T*Transmissivity [L

^{2}T^{−1}]*t*Time [T]

*t*_{0}Tidal period [T]

*u*Dumb variable for Hankel transform

*u*_{c}*, u*_{k1}*, u*_{k2}Coefficients defined by Eq. 8

*ω*,*ω*_{1},*ω*_{2}Tidal speeds [rad/T]

*θ*Angle for Polar coordinate [rad]

*Ф*Angle for two boundaries [rad]

*μ*_{n}Order of Bessel function of the first kind

## 1 Introduction

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

*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

*h*

_{m}, 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

*h*(

*r, θ, t*) [L] is the groundwater head;

*T*[L

^{2}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.

*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,

*h*

_{m}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*= 2

*π*/

*t*

_{0}where

*t*

_{0}is the tidal period. Similarly, the boundary condition for estuarine river 2 is denoted as

*h*(

*r*,

*ϕ*,

*t*) is the hydraulic head at

*θ*=

*ϕ*. The

*K*

_{1r}and

*K*

_{2r}are damping coefficients of tidal amplitude

*, K*

_{1i}and

*K*

_{2i}are separation coefficients representing the change of phase with distance (Sun 1997). The tidal river coefficients (

*K*

_{1},

*K*

_{2}) are defined as the damping coefficient plus the separation coefficient times

*i*,

*K*

_{1}

*= K*

_{1r}

*+ iK*

_{1i}and

*K*

_{2}

*= K*

_{2r}

*+ iK*

_{2i}.

*r*-direction is expressed as

## 3 Analytical Solution

*r*and

*θ*. It is denoted as

*Ae*

^{−i(ωt + c)}yields

*S*

_{i}= (

*L*+

*iωS*)/

*T*. The boundary conditions of Eqs. 2a and 2b can therefore be written as

*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)

*μ*

_{n}and

*μ*

_{n}=

*nπ*/

*ϕ*with

*n*= 1, 2, 3, … with

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 *K*_{1} = 0 or *K*_{2} = 0 is the solution for cases 2 and 3. Moreover, the present solution with *K*_{1} = *K*_{2} = 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 (*D*_{i} *= T/S*), damping coefficient (*K*_{1r}, *K*_{2r}), and separation coefficient (*K*_{1i}, *K*_{2i}) on the head fluctuation in the leaky confined aquifer system. The parameter values in the case studies are chosen as follows: *D*_{i} ranging from 10^{4} to 10^{6} m^{2}/h, *K*_{1} and *K*_{2} ranging from 10^{−7} to 10^{−4}*, ω* = 0.2618 rad/h, and *h*_{m} = 0. The amplitude of head fluctuation is normalized as *h*_{A} = ∣ *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

*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

*h*

_{m}= 1.77 mPD, tidal amplitudes

*A*

_{1}= 0.36 m and

*A*

_{2}= 0.58 m, tidal frequencies

*ω*

_{1}= 0.507/

*h*and

*ω*

_{2}= 0.237/

*h*, phase shift

*c*

_{1}= 2.138 and

*c*

_{2}= 3.209, and transmissivity

*T*= 854.0 m

^{2}/h (Li et al. 2002, Table 1).

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 *D*_{i} = 4.4 × 10^{3} m^{2}/h, *T*/*L* = 9.2 × 10^{5} m^{2}, *S*/*L* = 207 h, *t*_{0} = 12.2 h, and *h*_{m} = 0. With *Ф* = 180^{o} and *θ* = 90^{o}, 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* = 180^{o} and *θ* = 90^{o}, respectively.

### 4.2 Joint Effects of Included Angle and Diffusivity

*h*

_{A}and

*ph*of the head fluctuation versus

*Ф*observed at (

*r*,

*θ*) = (300 m, 45

^{o}) for

*L*= 0,

*K*

_{1}=

*K*

_{2}= 10

^{−5}/m, and

*D*

_{i}ranging from 10

^{4}to 10

^{6}m

^{2}/h. The figure indicates that

*h*

_{A}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

*h*

_{A}increases but

*ph*decreases significantly when increasing

*D*

_{i}from 10

^{4}to 10

^{6}m

^{2}/h for all values of

*Ф*. The figure also displays the effect of

*D*

_{i}increasing significantly with

*Ф*varying from 50

^{o}to 120

^{o}. Obviously, both

*D*

_{i}and

*Ф*have significant effects on the head fluctuation in the leaky confined aquifer system.

### 4.3 Joint Effects of the Tidal River Coefficients of Two Boundaries

*h*

_{A}and

*ph*of head fluctuations against negative logarithm values of the tidal river coefficients (

*K*

_{1}and

*K*

_{2}) observed at (

*r*,

*ө*) = (3000 m, 22.5

^{o}) for

*D*

_{i}

*=*5*10

^{5}m

^{2}/h,

*L*= 0

*,*

*Ф*

*=*45

^{o}. The dashed line signifies that the values of

*K*

_{1}and

*K*

_{2}are equal along this line. The figure shows that

*h*

_{A}decreases markedly but

*ph*increases significantly as

*K*

_{1}and

*K*

_{2}increase from 10

^{−7}to 10

^{−4}for

*L*= 0 /hour. The figure indicates that the joint effects of

*K*

_{1}and

*K*

_{2}on the

*h*

_{A}and

*ph*are obvious when the value of

*K*

_{1}or

*K*

_{2}is larger than 10

^{−5}. In addition, both

*K*

_{1}and

*K*

_{2}also have significant influences on the head fluctuation in the estuarine leaky aquifer.

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

*h*

_{A}, 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

*h*

_{A}versus the included angle

*Ф*observed at

*ө*=

*Ф*/2 and

*r*= 300 m in the interaction zone for

*L*= 0,

*K*

_{1}=

*K*

_{2}= 0 are plotted in Fig. 5(a). The dashed line represents the predicted

*h*

_{A}by Dong et al.’s (2016) solution while the solid line denotes that by the present solution. This figure exhibits that the

*h*

_{A}decreases significantly as

*Ф*increases from 60

^{o}to 180

^{o}observed at

*ө*=

*Ф*/2 and

*r*= 300 m. The predicted

*h*

_{A}by the present solution with

*Ф*=180

^{o}is 0.086 for

*D*

_{i}= 2000, 0.177 for 4000, and 0.24 for 8000 m

^{2}/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

*D*

_{i}, and then increases markedly with increasing

*Ф*. This figure also indicates that Dong et al.’s (2016) superposition solution largely overestimates the predicted

*h*

_{A}in the interaction zone. Figure 5(b) shows that the values of the difference in

*h*

_{A}are exactly the same as the predicted

*h*

_{A}by the present solution when

*Ф*=180

^{o}, indicating that their solution gives twice what the present solution predicts.

For the case that *Ф* =180^{o} 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* =180^{o} 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 *h*_{A} predictions by the present solution when *Ф* =180^{o} as indicated in Fig. 5(a) and 5(b) mentioned above.

Figure 5(c) shows the *h*_{A} along inland distance from 200 m to 1600 m in the *x*-direction versus diffusivity for *y* = 200 m, *L* = 0, *K*_{1} = *K*_{2} = 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 *h*_{A} is less than 10^{−3}, in the *x*-direction. The figure exhibits that the TIDs in the *x-*direction for *D*_{i} = 2000, 4000, and 6000 m^{2}/h are about 604, 854, and 1046 m, respectively, revealing a larger *D*_{i} has a longer TID. The figure also shows that the differences in *h*_{A} 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 *h*_{A}’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.

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