Li Guohui¹ , Li Hailon^{1, 2} , Xia Yuqiang¹ , Guo Qiaona¹ , Ren Yakun^2,3

1. School of Environmental Studies & (MOE) Biogeology and Environmental Geology Lab, China University of Geosciences, Wuhan 430074, P R China

2. Department of mathematics in Anshan Normal College, Anshan 114005 P R China

3. Department of mathematics in Liaoning Normal University, Dalian 116000 P R China

Corresponding author. Corresponding address: Dept. of Civil and Env. Enggr, Temple University, 1947 N. 12th Street, Philadelphia, PA 19122, USA. Tel: +01 215 204 8482; Fax: +01 215 204 6936; e-mail: hailong@temple.edu

Abstract: Analytical studies are carried out to investigate groundwater head changes in a coastal aquifer system in response to tidal fluctuations. The system consists of an unconfined aquifer, a leaky confined aquifer and a semi-permeable layer between them. The leaky confined aquifer and its semi-permeable roof extend under the sea infinitely. An exact analytical solution is derived to investigate the influences of both leakage and elastic storage of the semi-permeable layer on the tide-induced groundwater head fluctuation in the semi-confined aquifer. Given the head fluctuation amplitude and phase shift at the coastline, the head fluctuation in inland aquifer behaves exactly the same as that by Li and Jiao [2001b, Advances in Water Resources, p565-573]. Hence the solution analysis focuses on the impacts of the loading effect, the leakage and elastic storage of the roof on the groundwater head fluctuation at the coastline. The head at the coastline fluctuates with an amplitude approximately half of that, and with a phase shift approximately the same as that in the sea sufficiently far from the coastline. The tidal loading on the roof tends to increase the groundwater head fluctuation at the coastline. Leakage from the roof's offshore part enhances the head fluctuation at the coastline. The roof 's elastic storage tends to reinforce the tidal loading effect within the roof and damp the vertical, downward tidal wave propagation due to the direct hydraulic connection of the pore water in the roof with the tidal water at the seabed boundary. The final impact of the roof 's elastic storage is determined by the superposition of these two factors of opposite effects, which leads to a complicated, non-monotone dependence of the head fluctuation at the coastline on the roof 's elastic storage.

Key words: tide, coastal aquifer system, elastic storage, leakance, analytical solution, tidal loading effect

With social and economic development in coastal areas, various coastal hydrogeological, engineering, and environmental problems arise. Solving these problems very often requires studies on the coastal hydrogeological conditions, among which the study on the dynamic relation between seawater and coastal groundwater is significantly important. The water head in coastal aquifer fluctuates in response to the sea tide wave. Since the 1950s, many researchers have studied this phenomenon by analytical or numerical methods. Jacob [1950] derived an exact analytical solution to describe groundwater head in response to sea tide wave in a single confined aquifer extending infinitely landward. Carr and Van der Kamp [1969] studied estimation of aquifer parameters in coastal areas based on equation to describe the tide-induced groundwater fluctuation in a confined aquifer. Van der Kamp [1972] considered groundwater fluctuation in single aquifer with impermeable roof which extends infinitely both seaward and landward. Li and Chen [1991] considered a confined aquifer with an impermeable roof extending a certain distance under the sea. Sun [1997] derived an analytical solution to understand the groundwater response to tidal fluctuation in a single confined aquifer with two-dimensional tidal loading boundary condition of an estuary. Jiao and Tang [1999] discussed the groundwater head fluctuations in the confined aquifer of a coastal multi-layered groundwater system consisting of a confined aquifer, an unconfined aquifer and a semi-permeable layer between them. All layers end at coastline. They ignored the elastic storage of the semi-permeable layer. Based on the work of Li and Chen [1991] and Jiao and Tang [1999], Li and Jiao [2001a] considered a coastal confined aquifer with an semi-permeable roof extending under the sea. They ignored the roof 's elastic storage. Li and Jiao [2001b, 2002] considered both the elastic storage and the leakage of the semi-permeable layer, but all layers of the aquifer system terminate at the coastline.

Based on Li and Jiao [2001a; b], this paper considers a coastal aquifer system consisting of an unconfined aquifer, a leaky confined aquifer, and a semi-permeable layer between them. The leaky confined aquifer and its semi-permeable roof extend infinitely under the sea. Both effects of the elastic storage and the loading efficiency of the semi-permeable layer are taken into account. The mathematic model is given, and the exact analytical solution to the model is derived. The influences of the elastic storage, loading efficiency and leakage of the roof 's offshore portion on the groundwater-head fluctuation in the confined aquifer are discussed.

2 Conceptual Model, Mathematical Model and Analytical Solution

Consider a coastal multi-layer aquifer system consisting of a leaky confined aquifer, an unconfined aquifer and a semi-permeable layer between them (Fig. 1). All the layers extend landward infinitely. The bottom of the leaky confined aquifer is impermeable. The leaky confined aquifer and its semi-permeable roof extend infinitely seaward. The unconfined aquifer has a clear-cut vertical boundary with seawater. The coastline is straight. Both the aquifer and the semi-permeable layer are homogeneous, horizontal, and with constant thickness. Following Hantush [1960] and Neuman and Whiterspoon [1969], the flows in the confined aquifer and in the semi-permeable layer are assumed to be horizontal and vertical, respectively. Let the x-axis be positive landward and coincide with the middle line of the semi-permeable layer with its origin at the coastline. Let the z-axis be vertical, positive upward with its origin coincide with the x-axis (see Fig. 1). Following Li and Jiao [1999] and White and Roberts [1994], assume that the shallow unconfined aquifer had a large specific yield which can damp effectively the loading efficiency so that the tidal fluctuation in the unconfined aquifer is negligible compared to that in the confined aquifer. That is to say, the water table of the unconfined aquifer is a constant. According to these assumptions and the theory of Hantush [1960], the mathematical model for this coastal multi-layer aquifer system can be written as the following boundary-value problem. It includes four unknown functions of groundwater-heads in the semi-permeable layer and the confined aquifer including inland and offshore portions.

1)  Groundwater flow in the semi-permeable layer:

(1.1)

(1.2)

(2.1)

(2.2)

(3)where denotes the hydraulic head [L] of the groundwater in the semi-permeable layer at the location and time ; and are the specific storativity [L^-1] and vertical hydraulic conductivity [LT²] of the semi-permeable layer, respectively; is the loading efficiency of the semi-permeable layer; is the thickness of the semi-permeable layer [L], is the constant water table of the unconfined aquifer overlying the semi-permeable layer [L]. The water table surface is chosen to be the datum of hydraulic-head so that , and is the hydraulic head [L] of the main aquifer at the instant and location .

2) Groundwater flow in confined aquifer control layer:

(4.1)

(4.2)

(5.1)

(5.2)

(6)

       (7)

Here and are the storativity (dimensionless) and the transmissivity [L²T^-1] of the main aquifer, respectively; is the phase shift (dimensionless); and is the angular velocity [T^-1] of tide and equals , where is the tidal period [T].

The derivation of the solutions and to the boundary value system (1)-(7) is presented in detail in the appendix A. The analysis will focus on the groundwater-head in the main aquifer due to space limitation. Hence, only the expression of will be given here. Details about are presented in the Appendix A. For convenience of discussion, three parameters are introduced as in Li and Jiao [2002]. They are the main aquifer's tidal propagation parameter [L^-1], the buffer capacity (dimensionless) and the dimensionless leakage (dimensionless):

(8)

(9)

(10)

where is the specific leakage of the semi-permeable layer. Then, the solution can be written as

(11.1)

(11.2)

where and are dimensionless constants defined as

(12.1)

(12.2)

(12.3)

(12.4)

(12.5)

(12.6)

in which the function and are given by

(13.1)

(13.2)

(13.3)

(13.4)

(13.5)

(13.6)

3 Discussion

Equations (11.1) and (11.2) are the solutions for the groundwater heads in inland and offshore parts of the main aquifer, respectively.

When , or equivalently, from (13.1) and (13.2), one has

(14)

Substituting (14) back into (12.1), (12.2) yields

(15)

(16)

Using (16) and (A17.1)-(A17.2.2), one obtains

(17.1)

(17.2)

(18.1)

(18.2)

Substituting (17.1)-(18.2) back into (11.1) and (11.2), it is follows that

(19.1)

(19.2)

This solution is used to describe this coastal aquifer system ignoring the roof 's elastic storage.

When,

(20)

Using (20) and (17.1)- (17.2.2), one obtains

(21.1)

(21.2)

(22.1)

(22.2)

It is follows that

(23.1)

(23.2)

This is a new solution. For aquifer systems with semi-permeable layer composed of thick, soft sedimentary materials, the loading efficiency is always great between 0.8 and 1.0. So this solution has the practical significance.

Comparisons of the groundwater head amplitude and phase shift in the confined aquifer between in the sea infinitely seaward () and at coastline () are made. To a fixed location, and are the groundwater head amplitude in confined aquifer of the landward proportion and seaward proportion respectively. and are the groundwater head phase shift of the landward proportion and seaward proportion respectively. Basing on (11.1) and (11.2), yields

(24)

Comparing (24) and (11.1) - (11.2), leads

(25)

(26)

(27)

(28)

(27) and (28) demonstrate that the head fluctuation amplitude at coastline () is half of that in the sea infinitely far from the coastline, but the phase shift at the coastline is the same as that in the sea infinitely far from the coastline.

3.2 The influence of loading efficiency, leakage and buffer capacity of the semi-permeable on groundwater-head fluctuation amplitude at the coastline

From equations (8)-(11.2),(12.1)-(12.6),(13.1)-(13.6), it can be seen that five important parameters: the dimensionless leakances of the semi-permeable layer, the buffer capacity of the semi-permeable layer, the confined aquifer's tidal propagation parameter, and loading efficiencies of the semi-permeable layer and leaky confined layer, respectively are involved in the model. For the need of discussion, the values of , and corresponding to some case studies are calculated by Eqs. (8)-(10) for the semidiurnal sea tide whose angular velocity . Based on the data presented in Li and Jiao [2001b] (see Table 1), the discussion ranges of the parameters will be chosen 0.0-25.0 for , 0.0-20.0 for , 0.0-1.0 for , and 0.8-1.0 for .

Figure 2 describes that the amplitude of groundwater head fluctuation at coast line changes with buffer capacity for different loading efficiency and different leakances of the semi-permeable layer. Let the confined aquifer's loading efficiency be a mean value 0.5. The semi-permeable layer's loading efficiency takes two groups of values 0.8 and 1.0, Each group has four curves of leakance (=0.1,1.0,5.0,20 ).

In Fig 2, when the amplitude coefficient increases with the leakage. For , when the buffer capacity is small, the amplitude reduces quickly for a small distance; subsequently, it increases gradually to a certain constant. The physical mechanism of the buffer capacity results in it according to Eq. (9). Small buffer capacity implies that the elastic storage of the leaky layer is very small, and the semi-permeable layer can not counteract the large leakage. Thus minimum amplitude is caused. Meanwhile, the roof 's ability of storing water enhances with the buffer capacity increases, then, weakens the interference induced by tide wave, and amplitude enhances. When the buffer capacity large enough to counteract the leakage, the amplitude will insensitive to the buffer capacity and intends to a certain constant. When , the amplitude raises quickly first, and subsequently increases gradually to a constant. The reason for this behavior is that the change of groundwater head is mainly dominated by leakage from seaward when the leakage is small. Leakances from seaward enhance the change, while leakances from inland weaken it (Li and Jiao [2001a]). So for large leakage, large buffer capacity enhances the amplitude while small buffer capacity behaves as a damping factor. The behaviors of curves are similar to that of . Fig 2 also implies that for the same leakage, larger semi-permeable layer's loading efficiency induces larger amplitude.

The tidal loading on the roof tend to increases the groundwater head fluctuation at the coastline. Leakage from the roof 's offshore part enhances the head fluctuation at the coastline. The roof 's elastic storage tends to reinforce the tidal loading effect within the roof and damp the vertical, downward tidal wave propagation due to the direct hydraulic connection of the pore water in the roof with the tidal water at the seabed boundary. The final impact of the roof 's elastic storage is determined by the superposition of these two factors of opposite effects, which leads to a complicated, non-monotone dependence of the head fluctuation at the coastline on the roof 's elastic storage.

4 Conclusion

Analytical studies are carried out to investigate groundwater head changes in a coastal aquifer system in response to tidal fluctuations. The system consists of an unconfined aquifer, a leaky confined aquifer and a semi-permeable layer between them. The leaky confined aquifer and its semi-permeable roof extend under the sea infinitely. An exact analytical solution is derived to investigate the influences of both leakage and elastic storage of the semi-permeable roof on the tide-induced groundwater head fluctuation in the semi-confined aquifer. Given the head fluctuation amplitude and phase shift at the coastline, the head fluctuation in inland aquifer behaves exactly the same as that by Li and Jiao [2001b]. Hence the solution analysis focuses on the impacts of the loading efficiency, the leakage and elastic storage of the roof on the groundwater head fluctuation at the coastline. The head at the coastline fluctuates with an amplitude approximately half of that, and with a phase shift approximately the same as that in the sea sufficiently far from the coastline. The tidal loading on the roof tend to increases the groundwater head fluctuation at the coastline. Leakage from the roof 's offshore part enhances the head fluctuation at the coastline. The roof 's elastic storage tends to reinforce the tidal loading effect within the roof and damp the vertical, downward tidal wave propagation due to the direct hydraulic connection of the pore water in the roof with the tidal water at the seabed boundary. The final impact of the roof 's elastic storage is determined by the superposition of these two factors of opposite effects, which leads to a complicated, non-monotone dependence of the head fluctuation at the coastline on the roof 's elastic storage.

Appendix

To simplify the calculation, let the sea water head noting then

_,  (A1)

Analogously we can write

(A2)

Where and are complex functions, Re denotes the real part of the followed complex expression, and . Substituting (A.1) back into (1.1)-(3), yields

(A3.1)

(A3.2)

(A4.1)

(A4.2)

(A5)

then the solutions to (A3.1)-(A5) are:

(A6.1)

(A6.2)

Where

(A6.1.1)

(A6.1.2)

(A6.2.1)

(A6.2.2)

After substitution (A6.1.1)-(A6.2.2) back to (A6.1) and (A6.2) and simplification, can be written as following

(A6.3) (A6.4) Using (A6.3) and (A6.4), one obtains:

(A7.1)

(A7.2)

Substituting Eqs. (A2), (A7.1) and (A7.2) back into Eqs. (4.1)-(7), yields

(A8.1)

(A8.2)

(A9)

(A10)

(A11)

(A12)

The solution of (A8.1)-(A12) is

(A13.1)

(A13.2)

where

(A13.1.1)

(A13.1.2)

(A13.2.1)

(A13.2.2)

Substitute (A6.3), (A6.4), (A13.1) and (A13.2.2) back into (A1) and (A2) lead to the solution to (1.1)-(3). The expression of the solution is very complex, so it is not given here. Substituting (A13.1) and (A13.2.2) back into (A2) lead to the solution (11.1)-(11.2) to (4.1)-(7).

Reference

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