# Analytical Solution of Tidal Loading Effect in a Submarine Leaky Confined Aquifer System

Authors: Zongzhong Song; Hailong Li; Qian Ma; Chunmiao Zheng; Jiu Jimmy Jiao; Shaohong Li
Journal: Geofluids
 (2019)

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## Abstract

Although there are many existing analytical studies of tidal groundwater level fluctuations in coastal aquifer systems, few of them focus on an offshore submarine aquifer. Here, we consider tidal groundwater head fluctuations in a submarine leaky confined aquifer overlain by a semipermeable seabed. Both the seabed and the confined aquifer are assumed to extend horizontally infinitely. A one-dimensional mathematical model is established to describe the problem, and the analytical solution is derived. The impacts of the tidal loading efficiency, hydraulic conductivity and elastic storage of the semipermeable layer and aquifer on the groundwater head fluctuations in the aquifer system are analyzed and discussed. Solution analyses indicated that tidal loading effects tend to enhance the amplitude of the tidal groundwater fluctuation in the confined aquifer system and to reduce the phase shift between the groundwater head and the sea tide fluctuations.


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## Body

## 1. Introduction

The study of tidal groundwater level fluctuations in coastal aquifer systems is beneficial to solving various coast environmental problems. Many analytical solutions have been derived in this field since the 1950s. Jacob [1] and Ferris [2] firstly derived analytical solutions to describe the tide-induced groundwater fluctuation in a confined aquifer which ends at the coastline. The equations have been widely used in field survey and aquifer parameter estimation in coastal areas (e.g., [3–9]). Since then, various analytical solutions related to tidal head fluctuations have been derived for coastal confined aquifers. Van der Kamp [10] investigated the tidal groundwater fluctuation in an aquifer with an impermeable roof extending seaward and landward infinitely. Li and Chen [11] considered a coastal confined aquifer with an impermeable roof extending under the sea for a certain length, neglecting the vertical leakage. Jiao and Tang [12] discussed the tide-induced groundwater head fluctuation in an onshore multilayered coastal aquifer system neglecting the elastic storage of the semipermeable layer. Based on Jiao and Tang’s study [12], Li and Jiao [13] took into account the elastic storage of the semipermeable layer, and they [14] considered a leaky aquifer with the roof extending under the sea. Jeng et al. [15] considered the interference of the tidal groundwater level fluctuations in a multilayered aquifer system with the leakage of a semipermeable layer considered. Li and Jiao [16] improved Jeng et al.’s work [15] in the sense that both the leakage and elastic storage of the leaky layer are considered. Li et al. [17] considered a coastal multilayered aquifer system with the confined aquifer and its semipermeable roof extending infinitely under the sea. All the effects related to the semipermeable layer such as the elastic storage, the leakage, and the tidal loading efficiency were taken into account. Guarracino et al. [18] derived an analytical solution to describe the tide-induced head fluctuation in an aquifer system that extends a finite distance under the sea, with which the extending length may be estimated from groundwater head measurement. Wang et al. [19] presented an analytical study on the loading effect of the water table variations in a coastal aquifer system consisting of an unconfined aquifer, a confined aquifer, and an impermeable layer between them.Due to the tidal fluctuations, the total weight of the pore water (the onshore part) or sea water (the offshore part) in/overlying the aquifers will vary accordingly, which in turn can cause a considerable tidal loading effect on the groundwater flow. Many researchers [11, 14, 17–20] considered the tidal loading effect in their analytical studies, whereas almost all of the existing studies of tide-induced groundwater fluctuations focus on the tide-induced groundwater level fluctuation in onshore aquifer. Although there are some analytical solutions of tide-induced groundwater fluctuations in the submarine confined aquifer system in the offshore part (e.g., [14, 17, 18]), the dependency of the tide-induced groundwater head fluctuation in the offshore submarine aquifer on the model parameter was not analyzed or discussed. The offshore submarine aquifers are important both in water resources and in marine environments (e.g., [21–26]). Therefore, the study of the groundwater flow in offshore submarine aquifer systems is necessary and important.This paper considered a submarine leaky confined aquifer system with the confined aquifer and its semipermeable roof extending infinitely under the sea. All effects related to the submarine leaky confined aquifer system such as the elastic storage, the hydraulic conductivity, and the tidal loading efficiency were taken into account. The mathematic model was given, and the analytical solution was derived. The influences of the tidal loading efficiency, the elastic storage, and the hydraulic conductivity on the tidal groundwater head fluctuations in the confined aquifer and in the seabed were analyzed and discussed.


## 2. Mathematical Model and Analytical Solution

### 2.1. Mathematical Model

The configuration of a submarine leaky confined aquifer system is shown in Figure1. The aquifer is overlain by a semipermeable seabed aquifer/layer. Both the semipermeable seabed and the confined aquifer are assumed to extend horizontally infinitely. This assumption is reasonable when the considered offshore submarine aquifer is far enough from the coastline. There are even many offshore freshwater submarine aquifers extending in all horizontal directions extensively and very far from the coastline [22–26]. In addition, several existing analytical studies assume that the coastal aquifer systems extend seaward infinitely (e.g., [10, 14, 17, 27]). A vertically one-dimensional mathematical model is used to describe the problem.Figure 1
Schematic of a submarine leaky confined aquifer system.Let thez axis be positive upwards (Figure 1). All the layers are homogeneous and with constant thickness. Based on these assumptions and the theories of leaky, elastic aquifers proposed by Hantush and Jacob [28] and Jacob [1], the mathematical model to describe the groundwater fluctuations in Figure 1 is as follows:
(1)SS∂H∂t=K∂2H∂z2+SSLedHSdt,−∞<t<+∞,0<z<b+b′,z≠b,(2)Hb+b′,t=HSt=rρAcosωt+c,(3)limz↓bHz,t=limz↑bHz,t,(4)limz↓bK′∂H∂Z=limz↑bK1∂H∂z,(5)∂H∂z∣z=0=0,where Hz,t, SS, and K are the hydraulic head [L], specific storage [L-1], and vertical hydraulic conductivity [LT-1] of the semipermeable layer or the aquifer, respectively; Le is the tidal loading efficiency (dimensionless) [‒‒]; rρ is the density ratio (dimensionless) of the seawater to the freshwater with the range from 1.02 to 1.03; b and b′ are the thicknesses [L] of the aquifer and the semipermeable layer, respectively; SS takes the value of SS′ in the semipermeable layer and SS1 in the confined aquifer; K equals K′ in the semipermeable layer and K1 in the confined aquifer; HSt is the hydraulic head of the sea tide [L]; A is the amplitude [L] of the tidal fluctuation; ω is the angular velocity (or frequency) [T-1] of tide and equals 2π/t0 with t0 being the tidal period [T]; and c is the phase shift (in radian). The bottom of the confined aquifer is set to be the datum of the system.Equation (1) indicates that the groundwater level fluctuation in the submarine leaky confined aquifer is caused by its elastic compression and expansion due to the tidal loading rate of the sea water above seabed (SSLedHs/dt), in addition to the tidal fluctuation at the seabed boundary z=b+b′ as expressed by equation (2). Equations (3) and (4) are the continuity conditions of the hydraulic head and groundwater flux, respectively. Equation (5) states the no-flow boundary condition on the upper surface of the impermeable bottom.


### 2.2. Analytical Solution

In order to facilitate analysis, four parameters are introduced: the main aquifer’s tidal propagation parametera [L-1] defined as
(6)a=ωSS12K1,the dimensionless thickness of the seabed defined as
(7)θ=b′ωSS′2K′=s2u,where s is the dimensionless storativity ratio and defined as
(8)s=SS′b′SS1b,and the dimensionless leakageu defined as
(9)u=K′ωSS1bb′.The solution of boundary value problem (1), (2), (3), (4), and (5) is given by the following equations (see Appendix A for the derivation):
(10a)Hz,t=rρAReexpiωtCcoshl+iaz+Le1,0<z<b,(10b)Hz,t=rρAReexpiωtC1el+iθz/b′+C2e−l+iθz/b′+Le′,b<z<b+b′.The analytical solutions (10a) and (10b) of the groundwater heads Hz,t can be deformed as (see Appendix B for the derivation of the analytical solutions)
(11)Hz,t=rρAArcosωt−φ,where ArLe1,Le′,ab,θ,p,τ,z is the amplitude ratio and φLe1,Le′,ab,θ,p,τ,z is the comprehensive tidal phase shift in the aquifer defined as
(12a)ArLe1,Le′,ab,θ,p,τ,z=Ccoshl+iaz+Le1,0<z<b,(12b)ArLe1,Le′,ab,θ,p,τ,z=C1el+iθz/b′+C2e−l+iθz/b′+Le′,b<z<b+b′,(12c)φLe1,Le′,ab,θ,p,τ,z=−argCcoshl+iaz+Le1,0<z<b,(12d)φLe1,Le′,ab,θ,p,τ,z=−argC1el+iθz/b′+C2e−l+iθz/b′+Le′,b<z<b+b′.The three parametersC, C1, and C2 are given by the following equations (see Appendix C for the derivation):
(13a)C=C1expl+iτθ+C2exp−l+iτθ+Le′−Le1coshl+iab,where
(13b)τ=bb′is the thickness ratio (dimensionless) and C1 and C2 are two complex constants determined by equations (A.4) and (A.5) and given by
(13c)C1=1Δe−1+iτθtanh1+iabLe′−Le1e−1+iθ+θpτab+tanh1+iab1−Le′,(13d)C2=1Δe1+iτθ−tanh1+iabLe′−Le1e1+iθ+θpτab−tanh1+iab1−Le′,with
(13e)Δ=2θpτabcosh1+iθ+2tanh1+iabsinh1+iθ,(13f)p=K′K1.


## 2.1. Mathematical Model

The configuration of a submarine leaky confined aquifer system is shown in Figure1. The aquifer is overlain by a semipermeable seabed aquifer/layer. Both the semipermeable seabed and the confined aquifer are assumed to extend horizontally infinitely. This assumption is reasonable when the considered offshore submarine aquifer is far enough from the coastline. There are even many offshore freshwater submarine aquifers extending in all horizontal directions extensively and very far from the coastline [22–26]. In addition, several existing analytical studies assume that the coastal aquifer systems extend seaward infinitely (e.g., [10, 14, 17, 27]). A vertically one-dimensional mathematical model is used to describe the problem.Figure 1
Schematic of a submarine leaky confined aquifer system.Let thez axis be positive upwards (Figure 1). All the layers are homogeneous and with constant thickness. Based on these assumptions and the theories of leaky, elastic aquifers proposed by Hantush and Jacob [28] and Jacob [1], the mathematical model to describe the groundwater fluctuations in Figure 1 is as follows:
(1)SS∂H∂t=K∂2H∂z2+SSLedHSdt,−∞<t<+∞,0<z<b+b′,z≠b,(2)Hb+b′,t=HSt=rρAcosωt+c,(3)limz↓bHz,t=limz↑bHz,t,(4)limz↓bK′∂H∂Z=limz↑bK1∂H∂z,(5)∂H∂z∣z=0=0,where Hz,t, SS, and K are the hydraulic head [L], specific storage [L-1], and vertical hydraulic conductivity [LT-1] of the semipermeable layer or the aquifer, respectively; Le is the tidal loading efficiency (dimensionless) [‒‒]; rρ is the density ratio (dimensionless) of the seawater to the freshwater with the range from 1.02 to 1.03; b and b′ are the thicknesses [L] of the aquifer and the semipermeable layer, respectively; SS takes the value of SS′ in the semipermeable layer and SS1 in the confined aquifer; K equals K′ in the semipermeable layer and K1 in the confined aquifer; HSt is the hydraulic head of the sea tide [L]; A is the amplitude [L] of the tidal fluctuation; ω is the angular velocity (or frequency) [T-1] of tide and equals 2π/t0 with t0 being the tidal period [T]; and c is the phase shift (in radian). The bottom of the confined aquifer is set to be the datum of the system.Equation (1) indicates that the groundwater level fluctuation in the submarine leaky confined aquifer is caused by its elastic compression and expansion due to the tidal loading rate of the sea water above seabed (SSLedHs/dt), in addition to the tidal fluctuation at the seabed boundary z=b+b′ as expressed by equation (2). Equations (3) and (4) are the continuity conditions of the hydraulic head and groundwater flux, respectively. Equation (5) states the no-flow boundary condition on the upper surface of the impermeable bottom.


## 2.2. Analytical Solution

In order to facilitate analysis, four parameters are introduced: the main aquifer’s tidal propagation parametera [L-1] defined as
(6)a=ωSS12K1,the dimensionless thickness of the seabed defined as
(7)θ=b′ωSS′2K′=s2u,where s is the dimensionless storativity ratio and defined as
(8)s=SS′b′SS1b,and the dimensionless leakageu defined as
(9)u=K′ωSS1bb′.The solution of boundary value problem (1), (2), (3), (4), and (5) is given by the following equations (see Appendix A for the derivation):
(10a)Hz,t=rρAReexpiωtCcoshl+iaz+Le1,0<z<b,(10b)Hz,t=rρAReexpiωtC1el+iθz/b′+C2e−l+iθz/b′+Le′,b<z<b+b′.The analytical solutions (10a) and (10b) of the groundwater heads Hz,t can be deformed as (see Appendix B for the derivation of the analytical solutions)
(11)Hz,t=rρAArcosωt−φ,where ArLe1,Le′,ab,θ,p,τ,z is the amplitude ratio and φLe1,Le′,ab,θ,p,τ,z is the comprehensive tidal phase shift in the aquifer defined as
(12a)ArLe1,Le′,ab,θ,p,τ,z=Ccoshl+iaz+Le1,0<z<b,(12b)ArLe1,Le′,ab,θ,p,τ,z=C1el+iθz/b′+C2e−l+iθz/b′+Le′,b<z<b+b′,(12c)φLe1,Le′,ab,θ,p,τ,z=−argCcoshl+iaz+Le1,0<z<b,(12d)φLe1,Le′,ab,θ,p,τ,z=−argC1el+iθz/b′+C2e−l+iθz/b′+Le′,b<z<b+b′.The three parametersC, C1, and C2 are given by the following equations (see Appendix C for the derivation):
(13a)C=C1expl+iτθ+C2exp−l+iτθ+Le′−Le1coshl+iab,where
(13b)τ=bb′is the thickness ratio (dimensionless) and C1 and C2 are two complex constants determined by equations (A.4) and (A.5) and given by
(13c)C1=1Δe−1+iτθtanh1+iabLe′−Le1e−1+iθ+θpτab+tanh1+iab1−Le′,(13d)C2=1Δe1+iτθ−tanh1+iabLe′−Le1e1+iθ+θpτab−tanh1+iab1−Le′,with
(13e)Δ=2θpτabcosh1+iθ+2tanh1+iabsinh1+iθ,(13f)p=K′K1.


## 3. Discussion of the Solution

### 3.1. Value Ranges of Model Parameters

The value ranges of dimensionless model parametersab, θ, p, and τ for the leaky confined aquifer system reported in the literature are shown in Table 1, according to the summarization of Li and Jiao ([13], Table 1). One can see that ab ranges from 0.014 to 0.055, θ from 1.37 to 25.87, p from 2.76×10−6 to 6.83×10−3, and τ from 1.22 to 3.50. Based on these data and other situations (for example, the thickness of the submarine confined aquifer could be up to several hundred meters), the ranges of the parameters in the following discussion will be chosen: 0.001~10 for ab, 0~100 for θ, 1×10−5~1×10−1 for p, and 1~10 for τ.Table 1
Ranges of parametersab, θ, p, and τ in the leaky confined aquifer systems reported in the literature.
Leaky aquifer
Semipermeable layer
Model parameters
References
K1 (m·d−1)
SS1 (×10−6 m−1)
b (m)
K′ (m·d−1)
SS′ (×10−4 m−1)
b′ (m)
Ab
θ
p
τ
7.71
2.57
14
0.013
2.5
4
0.020
1.37
1.69E-3
3.50
[31]
11.29
3.21
14
0.013
NA
4
0.018
NA
1.15E-3
3.50
[32]
8.14
0.21
14
0.013
8.75
4
0.018
2.56
1.60E-3
3.50
[32]
579.43
4.00
28
0.0016
3.30
9.1
0.017
10.15
2.76E-6
3.08
[33]
579.43
4.00
28
0.0016
7.89
15.2
0.017
25.87
2.76E-6
1.84
[33]
1.20
16.07
6.1
0.0082
13.33
3
0.055
2.98
6.83E-3
2.03
[34]
45.81
5.63
16
0.0021
0.38
13.1
0.014
4.16
4.58E-5
1.22
[35]
NA means that data are not available.The tidal loading efficiency (Le′) in the semipermeable roof of the coastal confined aquifer system is usually very close to 1 [17]. So the value of Le′ will be fixed at 0.9 in the following discussion. Previous analytical studies and field surveys [7, 29, 30] show that the range of the tidal loading efficiency (Le1) in the confined aquifer changes from 0.05 to 1.


### 3.2. Comparison with Existing Solutions

#### 3.2.1. Case 1: Seabed without Elastic Storage (SS′→0)

When the elastic storage and loading efficiency of the seabed are close to zero (SS′→0; Le′→0), a simpler analytical solution of the confined aquifer system can be obtained. In view of equation (7), one has θ→0. For the confined aquifer, letting Le′→0 and θ→0 in equation (13a) yields
(14a)limθ→0C=1cosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1.Lettingθ→0 in equations (12a) and (12c) yields
(14b)limθ→0Ar=cosh1+iazcosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1+Le1,(14c)limθ→0φ=−argcosh1+iazcosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1+Le1.Assumingab→0az→0 in equations (14b) and (14c) yields
(14d)limab→0Ar=u+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab=u+1+iLe11+i/2u+1+i1+i/2,(14e)limθ→0ϕ=−argu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab=−argu+1+iLe11+i/2u+1+i1+i/2and then yields(14f)Ar=u+iLe1u+i,(14g)φ=−argu+iLe1u+i.Substituting equations (14f) and (14g) back into equation (11), the groundwater head in the confined aquifer can be written as
(15)Hz,t=rρAu+iLe1u+icosωt+argu+iLe1u+i,which is essentially the same as the equation given by Li and Jiao [14] when the roof length L→∞ and x→−∞. Therefore, our analytical solution is a generalization of Li and Jiao’s solution [14] when the roof length L→∞ and x→−∞.For the semipermeable seabed, lettingLe′→0 and θ→0 in equations (12b) and (12d) yields
(16a)limθ→0Ar=pτ+1+iabLe1−1z/b+τ−Le11+itanh1+iabpτ+1+iabtanh1+iab,(16b)limθ→0φ=−argpτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab.Substituting equations (16a) and (16b) back into equation (11), the groundwater head in the semipermeable seabed can be obtained as follows:
(17)Hz,t=rρApτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab×cosωt+argpτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab.


#### 3.2.2. Case 2: Impermeable Seabed (K′→0)

When the semipermeable seabed becomes impermeable (i.e.,K′→0), in view of equation (9), one has p→0. Letting p→0 in equation (13a) yields
(18a)limp→0C=Le−Le1+Le1−Lecosh1+iab=0.Lettingp→0 in equations (12a) and (12c) yields
(18b)limp→0Ar=Le1,(18c)limp→0φ=0.Substituting equations (18b) and (18c) back into equation (11), the groundwater head in the confined aquifer can be written as
(19)Hz,t=rρALe1cosωt.The solution given by equation (19) is essentially the same as the solution of Van der Kamp [10] and Li et al. [17] when K′→0 and x→−∞ except some notation differences.


### 3.3. The Amplitude RatioAr and Phase Shift φ

The amplitude ratioAr and phase shift φ of the groundwater head fluctuation in the confined aquifer and seabed are given by equations (12a), (12b), (12c), and (12d). Here, we will concentrate on the dependency of Ar and φ on the model parameters such as the tidal loading effect of the confined aquifer (Le1) and of the seabed (Le′) and the dimensionless thickness θ of the semipermeable seabed and ab of the confined aquifer. Since the amplitude ratio Ar and the phase shift φ depend on z, we discuss the changes of Ar and φ when z is fixed at the middle of the confined aquifer (z=0.5b) and at the middle of the semipermeable seabed (z=b+0.5b′). The representative values of p and τ in the literature are chosen for the following discussion: i.e., 1×10−3 for p and 3 for τ (see Table 1).


#### 3.3.1. Changes ofAr and φ in the Confined Aquifer

Figure2 shows how the amplitude ratio Ar at the middle of the confined aquifer z=0.5b changes with lgab for different values of θ and Le1 when Le′≡0.9. From Figure 2, one can see that the amplitude ratio Ar decreases from a constant close to 1 to Le1 (which equals 0.8, 0.5, and 0.1, respectively) as lgab increases from −3 to 1. When lgab is small (lgab≤−3), with the θ increasing, Ar decreases from 1 firstly and then increases to 0.9, which is the value of the tidal loading efficiency (Le′) of the semipermeable seabed. When lgab→1, Ar approaches the fixed values of Le1 and becomes independent of θ. For intermediate values of ab (approximately in the range −2.5<lgab<1), Ar decreases significantly with ab. The curves for different θ values may cross each other near lgab=−1.4, indicating that Ar is not always monotonic with respect to θ. In addition, Ar also increases with Le1 for fixed values of ab and θ. This can be observed when one compares the curves for the same values of ab and θ.Figure 2
Changes of the amplitude ratioAr at the middle of the confined aquifer z=0.5b with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)Figure3 shows how the phase shift φ at the middle of the confined aquifer z=0.5b changes with lgab for different values of θ and Le1 when Le′≡0.9. From Figure 3, one can see that the value range of phase shift φ decreases significantly with the tidal efficiency Le1. The value range of φ is about (0, 0.96) for Le1=0.1, (0, 0.34) for Le1=0.5, and (0, 0.11) for Le1=0.8. When ab is very small (lgab→−2.5), with the increasing of θ, φ increases from 0 firstly and then decreases to 0. When ab is very large (lgab→1), the phase shift φ tends to be zero for any values of θ and Le1. From Figure 3, one can also see that φ is nonmonotonic with respect to θ and ab, which is due to the superposition of the multieffects such as the tidal loading, the downward propagation of the tidal signal, and the elastic storages of the semipermeable seabed and the aquifer. According to the curve when θ→0 and Le1=0.1, the maximum phase shift is close to 1.0 (rad) which can lead to a time lag about 2 h for a semidiurnal tide and 4 h for a diurnal tide.Figure 3
Changes of phase shiftφ at the middle of the confined aquifer z=0.5b with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)According to the discussions about the amplitude ratioAr and the phase shift φ in the confined aquifer, one can conclude that tidal loading effects tend to enhance the amplitude and reduce the phase shift of the tidal groundwater head fluctuations in the confined aquifer. Figures 2 and 3 show that the increase of the dimensionless thickness of the upper semipermeable seabed may significantly enhance the tidal head fluctuation in the confined aquifer. Furthermore, Figures 2 and 3 demonstrate that an active range of lgab (−2.5.0<lgab<1.0) exists in which both the amplitude and phase shift change with ab significantly and both become insensitive to lgab outside the range (-2.5, 1).


#### 3.3.2. Changes ofAr and φ in the Seabed

Figure4 shows how the amplitude ratio Ar at the middle of the seabed semipermeable layer z=b+0.5b′ changes with lgab for different values of θ and Le1 when Le′≡0.9. It is obviously that no matter how the lgab and Le1 change, the value of Ar becomes constant when θ≥10. However, the changes of Ar become complex when θ<10. When ab is very small (lgab→−3), as θ is increasing from 0 to 10, Ar decreases from 1 firstly and then increases to 0.9, which is the value of the tidal loading efficiency of the semipermeable seabed. As lgab is increasing, the trends of Ar are different with θ values. When θ is small (θ<5), Ar decreases firstly and then becomes a constant independent of ab. However, when θ is in the range 5≤θ<10, Ar increases firstly and then becomes a constant independent of ab. When θ is around 3.0, Ar decreases firstly and then increases with ab and finally becomes a constant. The curves for different θ values cross each other near lgab=−1.4, indicating that the amplitude ratio Ar of the seabed is not always monotonic with respect to θ. Comparing Figures 4(a)–4(c), one can find that when ab is larger, Ar tends to be a constant value of 0.9 in Figure 4(a); however, Ar tends to have different values shown in Figures 4(b) and 4(c). It may be caused due to the tidal loading effect in the confined aquifer.Figure 4
Changes of the amplitude ratioAr at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)Figure5 shows how the phase shift φ at the middle of the seabed semipermeable layer z=b+0.5b′ changes with lgab for different values of θ and Le1 when Le′≡0.9. The value range of φ changes significantly with Le1, which is about (0.00, 0.05) for Le1=0.8, (–0.11, 0.14) for Le1=0.5, and (–0.28, 0.29) for Le1=0.1. Thus, the smaller the tidal efficiency Le1, the wider the value range of the phase shift. In general, the tidal loading effect tends to reduce the phase shift between the groundwater head and the tide fluctuations. No matter how lgab and Le1 change, the value of φ becomes constant when θ≥10. When θ is equal to 0 or 1, the phase shift φ increases firstly and then decreases to a constant with the increasing of lgab. For 2≤θ≤5, the phase shift φ decreases with lgab monotonically to a constant. When θ is equal to or greater than 7, the phase shift φ increases with Le1 monotonically to a constant. Comparing Figures 5(a)–5(c), one can also find that when ab is larger, φ tends to be a constant value of 0.9 in shown Figure 5(a); however, φ tends to have different values shown in Figures 5(b) and 5(c). It may be also caused due to the tidal loading effect in the confined aquifer.Figure 5
Changes of the phase shiftφ at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)


### 3.4. Effect of the Thickness Ratioτ and Hydraulic Conductivity Ratio p

Figure6 shows how the amplitude ratio Ar and phase shift φ at the middle of the confined aquifer z=0.5b change with lgab for different values of the thickness ratio τ. The value range of τ used is from 1 to 10, which is based on Table 2. Figure 6(a) demonstrates that when lgab increases from -3 to 1, the amplitude ratio Ar decreases from Le′ to Le1 for any values of τ. Given any values of lgab in the range of (-2, 0.4), the amplitude ratio Ar increases with τ. Outside the range of (-2, 0.4), the amplitude ratio Ar becomes insensitive to τ. In Figure 6(b), the phase shift is a monopeak function of lgab for any values of τ. The increasing of the thickness ratio τ can decrease the phase shift when −2.4<lgab<−1 and increase the phase shift when −0.4<lgab<0.6. The impacts are complex in other ranges. Overall, the phase shift ranges from 0 to around 0.12, a relatively narrow value range.Figure 6
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the confined aquifer z=0.5b with lgab for different values of τ (τ=b/b′). The loading coefficients Le1=0.5 and Le′≡0.9.
(a)
(b)Table 2
Fixed parameters and their values used in the discussion.
Case no.
z
Le1
θ
Τ
p
1
0.5b
0.5
10
1~10
10-3
b+0.5b′
0.5
2
1~10
10-3
2
0.5b
0.5
10
3
10-6~10-1
b+0.5b′
0.5
2
3
10-6~10-1
The parameters in Case 1 are used to discuss the effect of the thickness ratioτ (τ=b/b′). The parameters in Case 2 are used to discuss the effect of hydraulic conductivity ratio p (p=K′/K1).Figure7 shows how the amplitude ratio Ar and phase shift φ at the middle of the semipermeable seabed layer z=b+0.5b′ change with lgab for different values of τ. Figure 7(a) shows that the amplitude ratio decreases with lgab first and then tends to be a constant with a slight oscillations. The increasing of the thickness ratio τ can lead to the increase of the amplitude for any value of lgab in the range −2.4<lgab<0. Figure 7(b) shows that the phase shift decreases with lgab monotonically from about 0.035 to about -0.11, a very narrow value range. The increasing of the thickness ratio τ can lead to the increase of the phase shift for any value of lgab in the range −2<lgab<0.4.Figure 7
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the semipermeable seabed layer z=b+0.5b′ with lgab for different values of τ (τ=b/b′). The loading coefficients Lei=0.5 and Le′≡0.9.
(a)
(b)Figure8 shows how the changes of the amplitude ratio Ar and phase shift φ at the middle of the confined aquifer z=0.5b with lgab for different values of the hydraulic conductivity ratio p, which ranges from 10-5 to 10-2 according to Table 1. Figure 8(a) shows that the amplitude ratio decreases with lgab monotonically from 0.9 (Le′) to 0.5 (Le1) for any fixed value of p. The increasing of p can lead to the increase of the amplitude ratio for any value of lgab in the range −2.8<lgab<0.4. Figure 8(b) shows that the phase shift is a monopeak function of lgab for any given value of p. The values of lgab where the curve peaks increase with p. The value range of the phase shift is from 0 to less than 0.15.Figure 8
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the confined aquifer z=0.5b with lgab for different values of p (p=K′/K1). The loading coefficients Le1=0.5 and Le′≡0.9.
(a)
(b)Figure9 shows how the amplitude ratio Ar and phase shift φ at the middle of the semipermeable seabed layer z=b+0.5b′ change with lgab for different values of p. Figure 9(a) shows that the amplitude ratio decreases with lgab first and then tends to be a constant with slight oscillations. The increasing of hydraulic conductivity ratio p can lead to the increase of the amplitude for any value of lgab in the range −3<lgab<−0.6. Figure 9(b) shows that the phase shift decreases with lgab monotonically from about 0.035 to about -0.11, a very narrow value range. The increasing of hydraulic conductivity ratio p can lead to the increase of the phase shift for any value of lgab in the range −3<lgab<1.Figure 9
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of p (p=K′/K1). The loading coefficients Le1=0.5 and Le′≡0.9.
(a)
(b)


### 3.5. Groundwater Head at Different Depths of the Confined Aquifer

In order to investigate the differences of groundwater head fluctuations at different depths of the confined aquifer, we considered the changes ofAr and φ at the top and the bottom of the confined aquifer. The parameters p and τ are fixed in the following discussion: p=1×10−3 for τ=3. Figure 10 shows how the amplitude ratio Ar at the top and bottom of the confined aquifer changes with lgab for different values of θ and Le1 when Le′≡0.9. The variation patterns of Ar at the top and bottom of the aquifer are similar to that at the middle of the confined aquifer (Figure 2). When θ=1, Figure 10(a) shows that the curves of Ar at the top and bottom of the aquifer coincide completely with each other, and so do the curves for the phase shift φ in Figure 10(b). Figures 10(c) and 10(e) show that when lgab≤−0.4, the curves of Ar at the top and bottom of the aquifer are also entirely coincident; however, Ar at the top of the aquifer becomes larger than that at the bottom of the aquifer with θ when lgab>−0.4. The difference in the value of Ar between the top and bottom of the aquifer decreases gradually as Le1 increases. Figures 10(d) and 10(f) show that the phase shift φ at the top of the aquifer becomes smaller than that at the bottom of the aquifer for large values of θ (10 and 100). The difference in the value of φ between the bottom and top of the aquifer decreases gradually as Le1 increases. It can be concluded that the discrepancy between the tidal groundwater fluctuations at the top and bottom of the confined aquifer becomes larger with the increasing of θ and the decreasing of Le1.Figure 10
Changes of amplitude ratioAr and phase shift φ at the top z=b and bottom z=0 of the confined aquifer with lgab for different values of θ and Le1 when Le′≡0.9. (a, b) Ar and φ for θ=1; (c, d) Ar and φ for θ=10; and (e, f) Ar and φ for θ=100.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.1. Value Ranges of Model Parameters

The value ranges of dimensionless model parametersab, θ, p, and τ for the leaky confined aquifer system reported in the literature are shown in Table 1, according to the summarization of Li and Jiao ([13], Table 1). One can see that ab ranges from 0.014 to 0.055, θ from 1.37 to 25.87, p from 2.76×10−6 to 6.83×10−3, and τ from 1.22 to 3.50. Based on these data and other situations (for example, the thickness of the submarine confined aquifer could be up to several hundred meters), the ranges of the parameters in the following discussion will be chosen: 0.001~10 for ab, 0~100 for θ, 1×10−5~1×10−1 for p, and 1~10 for τ.Table 1
Ranges of parametersab, θ, p, and τ in the leaky confined aquifer systems reported in the literature.
Leaky aquifer
Semipermeable layer
Model parameters
References
K1 (m·d−1)
SS1 (×10−6 m−1)
b (m)
K′ (m·d−1)
SS′ (×10−4 m−1)
b′ (m)
Ab
θ
p
τ
7.71
2.57
14
0.013
2.5
4
0.020
1.37
1.69E-3
3.50
[31]
11.29
3.21
14
0.013
NA
4
0.018
NA
1.15E-3
3.50
[32]
8.14
0.21
14
0.013
8.75
4
0.018
2.56
1.60E-3
3.50
[32]
579.43
4.00
28
0.0016
3.30
9.1
0.017
10.15
2.76E-6
3.08
[33]
579.43
4.00
28
0.0016
7.89
15.2
0.017
25.87
2.76E-6
1.84
[33]
1.20
16.07
6.1
0.0082
13.33
3
0.055
2.98
6.83E-3
2.03
[34]
45.81
5.63
16
0.0021
0.38
13.1
0.014
4.16
4.58E-5
1.22
[35]
NA means that data are not available.The tidal loading efficiency (Le′) in the semipermeable roof of the coastal confined aquifer system is usually very close to 1 [17]. So the value of Le′ will be fixed at 0.9 in the following discussion. Previous analytical studies and field surveys [7, 29, 30] show that the range of the tidal loading efficiency (Le1) in the confined aquifer changes from 0.05 to 1.


## 3.2. Comparison with Existing Solutions

### 3.2.1. Case 1: Seabed without Elastic Storage (SS′→0)

When the elastic storage and loading efficiency of the seabed are close to zero (SS′→0; Le′→0), a simpler analytical solution of the confined aquifer system can be obtained. In view of equation (7), one has θ→0. For the confined aquifer, letting Le′→0 and θ→0 in equation (13a) yields
(14a)limθ→0C=1cosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1.Lettingθ→0 in equations (12a) and (12c) yields
(14b)limθ→0Ar=cosh1+iazcosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1+Le1,(14c)limθ→0φ=−argcosh1+iazcosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1+Le1.Assumingab→0az→0 in equations (14b) and (14c) yields
(14d)limab→0Ar=u+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab=u+1+iLe11+i/2u+1+i1+i/2,(14e)limθ→0ϕ=−argu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab=−argu+1+iLe11+i/2u+1+i1+i/2and then yields(14f)Ar=u+iLe1u+i,(14g)φ=−argu+iLe1u+i.Substituting equations (14f) and (14g) back into equation (11), the groundwater head in the confined aquifer can be written as
(15)Hz,t=rρAu+iLe1u+icosωt+argu+iLe1u+i,which is essentially the same as the equation given by Li and Jiao [14] when the roof length L→∞ and x→−∞. Therefore, our analytical solution is a generalization of Li and Jiao’s solution [14] when the roof length L→∞ and x→−∞.For the semipermeable seabed, lettingLe′→0 and θ→0 in equations (12b) and (12d) yields
(16a)limθ→0Ar=pτ+1+iabLe1−1z/b+τ−Le11+itanh1+iabpτ+1+iabtanh1+iab,(16b)limθ→0φ=−argpτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab.Substituting equations (16a) and (16b) back into equation (11), the groundwater head in the semipermeable seabed can be obtained as follows:
(17)Hz,t=rρApτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab×cosωt+argpτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab.


### 3.2.2. Case 2: Impermeable Seabed (K′→0)

When the semipermeable seabed becomes impermeable (i.e.,K′→0), in view of equation (9), one has p→0. Letting p→0 in equation (13a) yields
(18a)limp→0C=Le−Le1+Le1−Lecosh1+iab=0.Lettingp→0 in equations (12a) and (12c) yields
(18b)limp→0Ar=Le1,(18c)limp→0φ=0.Substituting equations (18b) and (18c) back into equation (11), the groundwater head in the confined aquifer can be written as
(19)Hz,t=rρALe1cosωt.The solution given by equation (19) is essentially the same as the solution of Van der Kamp [10] and Li et al. [17] when K′→0 and x→−∞ except some notation differences.


## 3.2.1. Case 1: Seabed without Elastic Storage (SS′→0)

When the elastic storage and loading efficiency of the seabed are close to zero (SS′→0; Le′→0), a simpler analytical solution of the confined aquifer system can be obtained. In view of equation (7), one has θ→0. For the confined aquifer, letting Le′→0 and θ→0 in equation (13a) yields
(14a)limθ→0C=1cosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1.Lettingθ→0 in equations (12a) and (12c) yields
(14b)limθ→0Ar=cosh1+iazcosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1+Le1,(14c)limθ→0φ=−argcosh1+iazcosh1+iabu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab−Le1+Le1.Assumingab→0az→0 in equations (14b) and (14c) yields
(14d)limab→0Ar=u+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab=u+1+iLe11+i/2u+1+i1+i/2,(14e)limθ→0ϕ=−argu+1+iLe1tanh1+iab/2abu+1+itanh1+iab/2ab=−argu+1+iLe11+i/2u+1+i1+i/2and then yields(14f)Ar=u+iLe1u+i,(14g)φ=−argu+iLe1u+i.Substituting equations (14f) and (14g) back into equation (11), the groundwater head in the confined aquifer can be written as
(15)Hz,t=rρAu+iLe1u+icosωt+argu+iLe1u+i,which is essentially the same as the equation given by Li and Jiao [14] when the roof length L→∞ and x→−∞. Therefore, our analytical solution is a generalization of Li and Jiao’s solution [14] when the roof length L→∞ and x→−∞.For the semipermeable seabed, lettingLe′→0 and θ→0 in equations (12b) and (12d) yields
(16a)limθ→0Ar=pτ+1+iabLe1−1z/b+τ−Le11+itanh1+iabpτ+1+iabtanh1+iab,(16b)limθ→0φ=−argpτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab.Substituting equations (16a) and (16b) back into equation (11), the groundwater head in the semipermeable seabed can be obtained as follows:
(17)Hz,t=rρApτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab×cosωt+argpτ+1+iabLe1−1z/b+τ−Le11+τtanh1+iabpτ+1+iabtanh1+iab.


## 3.2.2. Case 2: Impermeable Seabed (K′→0)

When the semipermeable seabed becomes impermeable (i.e.,K′→0), in view of equation (9), one has p→0. Letting p→0 in equation (13a) yields
(18a)limp→0C=Le−Le1+Le1−Lecosh1+iab=0.Lettingp→0 in equations (12a) and (12c) yields
(18b)limp→0Ar=Le1,(18c)limp→0φ=0.Substituting equations (18b) and (18c) back into equation (11), the groundwater head in the confined aquifer can be written as
(19)Hz,t=rρALe1cosωt.The solution given by equation (19) is essentially the same as the solution of Van der Kamp [10] and Li et al. [17] when K′→0 and x→−∞ except some notation differences.


## 3.3. The Amplitude RatioAr and Phase Shift φ

The amplitude ratioAr and phase shift φ of the groundwater head fluctuation in the confined aquifer and seabed are given by equations (12a), (12b), (12c), and (12d). Here, we will concentrate on the dependency of Ar and φ on the model parameters such as the tidal loading effect of the confined aquifer (Le1) and of the seabed (Le′) and the dimensionless thickness θ of the semipermeable seabed and ab of the confined aquifer. Since the amplitude ratio Ar and the phase shift φ depend on z, we discuss the changes of Ar and φ when z is fixed at the middle of the confined aquifer (z=0.5b) and at the middle of the semipermeable seabed (z=b+0.5b′). The representative values of p and τ in the literature are chosen for the following discussion: i.e., 1×10−3 for p and 3 for τ (see Table 1).


### 3.3.1. Changes ofAr and φ in the Confined Aquifer

Figure2 shows how the amplitude ratio Ar at the middle of the confined aquifer z=0.5b changes with lgab for different values of θ and Le1 when Le′≡0.9. From Figure 2, one can see that the amplitude ratio Ar decreases from a constant close to 1 to Le1 (which equals 0.8, 0.5, and 0.1, respectively) as lgab increases from −3 to 1. When lgab is small (lgab≤−3), with the θ increasing, Ar decreases from 1 firstly and then increases to 0.9, which is the value of the tidal loading efficiency (Le′) of the semipermeable seabed. When lgab→1, Ar approaches the fixed values of Le1 and becomes independent of θ. For intermediate values of ab (approximately in the range −2.5<lgab<1), Ar decreases significantly with ab. The curves for different θ values may cross each other near lgab=−1.4, indicating that Ar is not always monotonic with respect to θ. In addition, Ar also increases with Le1 for fixed values of ab and θ. This can be observed when one compares the curves for the same values of ab and θ.Figure 2
Changes of the amplitude ratioAr at the middle of the confined aquifer z=0.5b with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)Figure3 shows how the phase shift φ at the middle of the confined aquifer z=0.5b changes with lgab for different values of θ and Le1 when Le′≡0.9. From Figure 3, one can see that the value range of phase shift φ decreases significantly with the tidal efficiency Le1. The value range of φ is about (0, 0.96) for Le1=0.1, (0, 0.34) for Le1=0.5, and (0, 0.11) for Le1=0.8. When ab is very small (lgab→−2.5), with the increasing of θ, φ increases from 0 firstly and then decreases to 0. When ab is very large (lgab→1), the phase shift φ tends to be zero for any values of θ and Le1. From Figure 3, one can also see that φ is nonmonotonic with respect to θ and ab, which is due to the superposition of the multieffects such as the tidal loading, the downward propagation of the tidal signal, and the elastic storages of the semipermeable seabed and the aquifer. According to the curve when θ→0 and Le1=0.1, the maximum phase shift is close to 1.0 (rad) which can lead to a time lag about 2 h for a semidiurnal tide and 4 h for a diurnal tide.Figure 3
Changes of phase shiftφ at the middle of the confined aquifer z=0.5b with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)According to the discussions about the amplitude ratioAr and the phase shift φ in the confined aquifer, one can conclude that tidal loading effects tend to enhance the amplitude and reduce the phase shift of the tidal groundwater head fluctuations in the confined aquifer. Figures 2 and 3 show that the increase of the dimensionless thickness of the upper semipermeable seabed may significantly enhance the tidal head fluctuation in the confined aquifer. Furthermore, Figures 2 and 3 demonstrate that an active range of lgab (−2.5.0<lgab<1.0) exists in which both the amplitude and phase shift change with ab significantly and both become insensitive to lgab outside the range (-2.5, 1).


### 3.3.2. Changes ofAr and φ in the Seabed

Figure4 shows how the amplitude ratio Ar at the middle of the seabed semipermeable layer z=b+0.5b′ changes with lgab for different values of θ and Le1 when Le′≡0.9. It is obviously that no matter how the lgab and Le1 change, the value of Ar becomes constant when θ≥10. However, the changes of Ar become complex when θ<10. When ab is very small (lgab→−3), as θ is increasing from 0 to 10, Ar decreases from 1 firstly and then increases to 0.9, which is the value of the tidal loading efficiency of the semipermeable seabed. As lgab is increasing, the trends of Ar are different with θ values. When θ is small (θ<5), Ar decreases firstly and then becomes a constant independent of ab. However, when θ is in the range 5≤θ<10, Ar increases firstly and then becomes a constant independent of ab. When θ is around 3.0, Ar decreases firstly and then increases with ab and finally becomes a constant. The curves for different θ values cross each other near lgab=−1.4, indicating that the amplitude ratio Ar of the seabed is not always monotonic with respect to θ. Comparing Figures 4(a)–4(c), one can find that when ab is larger, Ar tends to be a constant value of 0.9 in Figure 4(a); however, Ar tends to have different values shown in Figures 4(b) and 4(c). It may be caused due to the tidal loading effect in the confined aquifer.Figure 4
Changes of the amplitude ratioAr at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)Figure5 shows how the phase shift φ at the middle of the seabed semipermeable layer z=b+0.5b′ changes with lgab for different values of θ and Le1 when Le′≡0.9. The value range of φ changes significantly with Le1, which is about (0.00, 0.05) for Le1=0.8, (–0.11, 0.14) for Le1=0.5, and (–0.28, 0.29) for Le1=0.1. Thus, the smaller the tidal efficiency Le1, the wider the value range of the phase shift. In general, the tidal loading effect tends to reduce the phase shift between the groundwater head and the tide fluctuations. No matter how lgab and Le1 change, the value of φ becomes constant when θ≥10. When θ is equal to 0 or 1, the phase shift φ increases firstly and then decreases to a constant with the increasing of lgab. For 2≤θ≤5, the phase shift φ decreases with lgab monotonically to a constant. When θ is equal to or greater than 7, the phase shift φ increases with Le1 monotonically to a constant. Comparing Figures 5(a)–5(c), one can also find that when ab is larger, φ tends to be a constant value of 0.9 in shown Figure 5(a); however, φ tends to have different values shown in Figures 5(b) and 5(c). It may be also caused due to the tidal loading effect in the confined aquifer.Figure 5
Changes of the phase shiftφ at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)


## 3.3.1. Changes ofAr and φ in the Confined Aquifer

Figure2 shows how the amplitude ratio Ar at the middle of the confined aquifer z=0.5b changes with lgab for different values of θ and Le1 when Le′≡0.9. From Figure 2, one can see that the amplitude ratio Ar decreases from a constant close to 1 to Le1 (which equals 0.8, 0.5, and 0.1, respectively) as lgab increases from −3 to 1. When lgab is small (lgab≤−3), with the θ increasing, Ar decreases from 1 firstly and then increases to 0.9, which is the value of the tidal loading efficiency (Le′) of the semipermeable seabed. When lgab→1, Ar approaches the fixed values of Le1 and becomes independent of θ. For intermediate values of ab (approximately in the range −2.5<lgab<1), Ar decreases significantly with ab. The curves for different θ values may cross each other near lgab=−1.4, indicating that Ar is not always monotonic with respect to θ. In addition, Ar also increases with Le1 for fixed values of ab and θ. This can be observed when one compares the curves for the same values of ab and θ.Figure 2
Changes of the amplitude ratioAr at the middle of the confined aquifer z=0.5b with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)Figure3 shows how the phase shift φ at the middle of the confined aquifer z=0.5b changes with lgab for different values of θ and Le1 when Le′≡0.9. From Figure 3, one can see that the value range of phase shift φ decreases significantly with the tidal efficiency Le1. The value range of φ is about (0, 0.96) for Le1=0.1, (0, 0.34) for Le1=0.5, and (0, 0.11) for Le1=0.8. When ab is very small (lgab→−2.5), with the increasing of θ, φ increases from 0 firstly and then decreases to 0. When ab is very large (lgab→1), the phase shift φ tends to be zero for any values of θ and Le1. From Figure 3, one can also see that φ is nonmonotonic with respect to θ and ab, which is due to the superposition of the multieffects such as the tidal loading, the downward propagation of the tidal signal, and the elastic storages of the semipermeable seabed and the aquifer. According to the curve when θ→0 and Le1=0.1, the maximum phase shift is close to 1.0 (rad) which can lead to a time lag about 2 h for a semidiurnal tide and 4 h for a diurnal tide.Figure 3
Changes of phase shiftφ at the middle of the confined aquifer z=0.5b with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)According to the discussions about the amplitude ratioAr and the phase shift φ in the confined aquifer, one can conclude that tidal loading effects tend to enhance the amplitude and reduce the phase shift of the tidal groundwater head fluctuations in the confined aquifer. Figures 2 and 3 show that the increase of the dimensionless thickness of the upper semipermeable seabed may significantly enhance the tidal head fluctuation in the confined aquifer. Furthermore, Figures 2 and 3 demonstrate that an active range of lgab (−2.5.0<lgab<1.0) exists in which both the amplitude and phase shift change with ab significantly and both become insensitive to lgab outside the range (-2.5, 1).


## 3.3.2. Changes ofAr and φ in the Seabed

Figure4 shows how the amplitude ratio Ar at the middle of the seabed semipermeable layer z=b+0.5b′ changes with lgab for different values of θ and Le1 when Le′≡0.9. It is obviously that no matter how the lgab and Le1 change, the value of Ar becomes constant when θ≥10. However, the changes of Ar become complex when θ<10. When ab is very small (lgab→−3), as θ is increasing from 0 to 10, Ar decreases from 1 firstly and then increases to 0.9, which is the value of the tidal loading efficiency of the semipermeable seabed. As lgab is increasing, the trends of Ar are different with θ values. When θ is small (θ<5), Ar decreases firstly and then becomes a constant independent of ab. However, when θ is in the range 5≤θ<10, Ar increases firstly and then becomes a constant independent of ab. When θ is around 3.0, Ar decreases firstly and then increases with ab and finally becomes a constant. The curves for different θ values cross each other near lgab=−1.4, indicating that the amplitude ratio Ar of the seabed is not always monotonic with respect to θ. Comparing Figures 4(a)–4(c), one can find that when ab is larger, Ar tends to be a constant value of 0.9 in Figure 4(a); however, Ar tends to have different values shown in Figures 4(b) and 4(c). It may be caused due to the tidal loading effect in the confined aquifer.Figure 4
Changes of the amplitude ratioAr at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)Figure5 shows how the phase shift φ at the middle of the seabed semipermeable layer z=b+0.5b′ changes with lgab for different values of θ and Le1 when Le′≡0.9. The value range of φ changes significantly with Le1, which is about (0.00, 0.05) for Le1=0.8, (–0.11, 0.14) for Le1=0.5, and (–0.28, 0.29) for Le1=0.1. Thus, the smaller the tidal efficiency Le1, the wider the value range of the phase shift. In general, the tidal loading effect tends to reduce the phase shift between the groundwater head and the tide fluctuations. No matter how lgab and Le1 change, the value of φ becomes constant when θ≥10. When θ is equal to 0 or 1, the phase shift φ increases firstly and then decreases to a constant with the increasing of lgab. For 2≤θ≤5, the phase shift φ decreases with lgab monotonically to a constant. When θ is equal to or greater than 7, the phase shift φ increases with Le1 monotonically to a constant. Comparing Figures 5(a)–5(c), one can also find that when ab is larger, φ tends to be a constant value of 0.9 in shown Figure 5(a); however, φ tends to have different values shown in Figures 5(b) and 5(c). It may be also caused due to the tidal loading effect in the confined aquifer.Figure 5
Changes of the phase shiftφ at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of θ when Le1 is equal to (a) 0.8, (b) 0.5, and (c) 0.1. The loading coefficient Le′≡0.9.
(a)
(b)
(c)


## 3.4. Effect of the Thickness Ratioτ and Hydraulic Conductivity Ratio p

Figure6 shows how the amplitude ratio Ar and phase shift φ at the middle of the confined aquifer z=0.5b change with lgab for different values of the thickness ratio τ. The value range of τ used is from 1 to 10, which is based on Table 2. Figure 6(a) demonstrates that when lgab increases from -3 to 1, the amplitude ratio Ar decreases from Le′ to Le1 for any values of τ. Given any values of lgab in the range of (-2, 0.4), the amplitude ratio Ar increases with τ. Outside the range of (-2, 0.4), the amplitude ratio Ar becomes insensitive to τ. In Figure 6(b), the phase shift is a monopeak function of lgab for any values of τ. The increasing of the thickness ratio τ can decrease the phase shift when −2.4<lgab<−1 and increase the phase shift when −0.4<lgab<0.6. The impacts are complex in other ranges. Overall, the phase shift ranges from 0 to around 0.12, a relatively narrow value range.Figure 6
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the confined aquifer z=0.5b with lgab for different values of τ (τ=b/b′). The loading coefficients Le1=0.5 and Le′≡0.9.
(a)
(b)Table 2
Fixed parameters and their values used in the discussion.
Case no.
z
Le1
θ
Τ
p
1
0.5b
0.5
10
1~10
10-3
b+0.5b′
0.5
2
1~10
10-3
2
0.5b
0.5
10
3
10-6~10-1
b+0.5b′
0.5
2
3
10-6~10-1
The parameters in Case 1 are used to discuss the effect of the thickness ratioτ (τ=b/b′). The parameters in Case 2 are used to discuss the effect of hydraulic conductivity ratio p (p=K′/K1).Figure7 shows how the amplitude ratio Ar and phase shift φ at the middle of the semipermeable seabed layer z=b+0.5b′ change with lgab for different values of τ. Figure 7(a) shows that the amplitude ratio decreases with lgab first and then tends to be a constant with a slight oscillations. The increasing of the thickness ratio τ can lead to the increase of the amplitude for any value of lgab in the range −2.4<lgab<0. Figure 7(b) shows that the phase shift decreases with lgab monotonically from about 0.035 to about -0.11, a very narrow value range. The increasing of the thickness ratio τ can lead to the increase of the phase shift for any value of lgab in the range −2<lgab<0.4.Figure 7
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the semipermeable seabed layer z=b+0.5b′ with lgab for different values of τ (τ=b/b′). The loading coefficients Lei=0.5 and Le′≡0.9.
(a)
(b)Figure8 shows how the changes of the amplitude ratio Ar and phase shift φ at the middle of the confined aquifer z=0.5b with lgab for different values of the hydraulic conductivity ratio p, which ranges from 10-5 to 10-2 according to Table 1. Figure 8(a) shows that the amplitude ratio decreases with lgab monotonically from 0.9 (Le′) to 0.5 (Le1) for any fixed value of p. The increasing of p can lead to the increase of the amplitude ratio for any value of lgab in the range −2.8<lgab<0.4. Figure 8(b) shows that the phase shift is a monopeak function of lgab for any given value of p. The values of lgab where the curve peaks increase with p. The value range of the phase shift is from 0 to less than 0.15.Figure 8
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the confined aquifer z=0.5b with lgab for different values of p (p=K′/K1). The loading coefficients Le1=0.5 and Le′≡0.9.
(a)
(b)Figure9 shows how the amplitude ratio Ar and phase shift φ at the middle of the semipermeable seabed layer z=b+0.5b′ change with lgab for different values of p. Figure 9(a) shows that the amplitude ratio decreases with lgab first and then tends to be a constant with slight oscillations. The increasing of hydraulic conductivity ratio p can lead to the increase of the amplitude for any value of lgab in the range −3<lgab<−0.6. Figure 9(b) shows that the phase shift decreases with lgab monotonically from about 0.035 to about -0.11, a very narrow value range. The increasing of hydraulic conductivity ratio p can lead to the increase of the phase shift for any value of lgab in the range −3<lgab<1.Figure 9
Changes of (a) amplitude ratioAr and (b) phase shift φ at the middle of the seabed semipermeable layer z=b+0.5b′ with lgab for different values of p (p=K′/K1). The loading coefficients Le1=0.5 and Le′≡0.9.
(a)
(b)


## 3.5. Groundwater Head at Different Depths of the Confined Aquifer

In order to investigate the differences of groundwater head fluctuations at different depths of the confined aquifer, we considered the changes ofAr and φ at the top and the bottom of the confined aquifer. The parameters p and τ are fixed in the following discussion: p=1×10−3 for τ=3. Figure 10 shows how the amplitude ratio Ar at the top and bottom of the confined aquifer changes with lgab for different values of θ and Le1 when Le′≡0.9. The variation patterns of Ar at the top and bottom of the aquifer are similar to that at the middle of the confined aquifer (Figure 2). When θ=1, Figure 10(a) shows that the curves of Ar at the top and bottom of the aquifer coincide completely with each other, and so do the curves for the phase shift φ in Figure 10(b). Figures 10(c) and 10(e) show that when lgab≤−0.4, the curves of Ar at the top and bottom of the aquifer are also entirely coincident; however, Ar at the top of the aquifer becomes larger than that at the bottom of the aquifer with θ when lgab>−0.4. The difference in the value of Ar between the top and bottom of the aquifer decreases gradually as Le1 increases. Figures 10(d) and 10(f) show that the phase shift φ at the top of the aquifer becomes smaller than that at the bottom of the aquifer for large values of θ (10 and 100). The difference in the value of φ between the bottom and top of the aquifer decreases gradually as Le1 increases. It can be concluded that the discrepancy between the tidal groundwater fluctuations at the top and bottom of the confined aquifer becomes larger with the increasing of θ and the decreasing of Le1.Figure 10
Changes of amplitude ratioAr and phase shift φ at the top z=b and bottom z=0 of the confined aquifer with lgab for different values of θ and Le1 when Le′≡0.9. (a, b) Ar and φ for θ=1; (c, d) Ar and φ for θ=10; and (e, f) Ar and φ for θ=100.
(a)
(b)
(c)
(d)
(e)
(f)


## 4. Conclusion

This paper investigated the tidal groundwater head fluctuation in a submarine confined leaky aquifer system. A vertical one-dimensional mathematical model is given with the analytical solution derived. Both the leakage and elastic storage of the semipermeable seabed are considered in this model. The derived analytical solution is different from previous analytical solutions by Li and Jiao [14] and Guarracino et al. [18] which focus on the analysis in the inland portion. Although Li et al. [17] give the analytical solutions in the offshore part, they did not do sensitivity analysis of parameters of the aquifers in the offshore part to head fluctuations of the aquifers there. Our solution quantifies the tidal groundwater head fluctuations in an offshore submarine aquifer system, which contains a confined aquifer and a semipermeable seabed.The effects of several dimensionless model parameters, including the tidal loading efficiencies (Le1 and Le′) of the confined aquifer and the semipermeable seabed, the dimensionless thickness ab of the confined aquifer, and the dimensionless thickness θ of the seabed, on the tidal groundwater fluctuations are quantitatively discussed and analyzed. The following conclusions can be drawn based on the solution analyses and discussion: (1) The tidal loading effect tends to enhance the amplitude of the tidal groundwater fluctuation in the confined aquifer system, and it can also tend to reduce the phase difference between the fluctuation of the groundwater head in the aquifer (or an aquitard) and the tidal fluctuations. (2) The increase of the thickness ratio or the hydraulic conductivity ratio can not only result in a larger amplitude in the confined aquifer but also lead to a larger phase shift. (3) The discrepancy between the tidal groundwater fluctuations at the top and bottom of the confined aquifer increases when θ increases and/or Le1 decreases.



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*Source: 8017164-2019-06-20.xml*