Guo qiaona¹ hailong^1,2  Xia yuqiang ¹  Li guohui¹

1. School of Environmental Studies & (MOE) Biogeology and Environmental Geology Lab,

China University of Geosciences, Wuhan 430074, P. R. China

2. Dept. of Civil and Env. Enggr, Temple University, 1947 N. 12th Street, Philadelphia, PA 19122

Abstract: The hydraulic relation between groundwater and seawater is close in most coastal areas, because tidal fluctuation is one of the most important driving factor on variation of groundwater head. This paper investigates a coastal aquifer system consisting of a confined aquifer from up and down by two semi-permeable layers. The suboutcrop of the aquifer is covered by a silt layer. A mathematical model is used to describe this system and the analytical solution of this model is derived. The analytical solution contains six parameters: the confined aquifer’s tidal propagation parameter, the dimensionless leakance of the silt layer, the dimensionless storativity ratio and the dimensionless leakage of the upper and lower semi-permeable layers. When these parameters assume special values, the solution becomes several special solutions derived in other situation. The analytical solution indicates that both storage and leakage of the semi-permeable layers and the relative leakance coefficient of the silt layer have great influences on the groundwater-head fluctuation in the confined aquifer. The amplitude of the head fluctuation decreases with the storativity and leakage of both semi-permeable layers and increases with the leakance of the silt-layer. However, the positive phase shift (time-lag) of fluctuation increases with the storativity and leakage of both semi-permeable layers and decreases with the leakance of the silt-layer.

Key words: tide, coastal aquifer system, storage, leakage, silt layer boundary , leakance of the silt layer, analytical solution

 

1 Introduction

The hydraulic relation between groundwater and seawater is close in most coastal areas, because tidal fluctuation is one of the most important driving factor on variation of groundwater head. The hydraulic head or water table in a coastal aquifer system changes in response to tidal fluctuation. Analytical researches have been conducted in this field since the middle period of last century by many hydrogeologists. For example, Jacob (1950) provided the analytical solution of tide-induced groundwater level fluctuation in a coastal confined aquifer; van der Kamp (1972) considered a coastal aquifer which extends under the sea infinitely; Li and Chen (1991a) investigated a finite length of confined aquifer roof extending under the sea; Sun (1997) developed an analytical solution in an estuary using a two-dimensional tidal loading boundary condition. The main limitation of these previous studies is that they only considered a single coastal aquifer configuration. As a matter of fact, most of     the coastal aquifer systems are multi-layered systems, there is usually an unconfined aquifer above one or more confined aquifers and they are separated by semi-permeable layer. Jiao and Tang (1999) discussed a coastal multi-layered aquifer system consisting of a confined aquifer, an unconfined aquifer, and a leaky layer between them. They ignored the leaky layer’s elastic storage. Li and Jiao (2001b) improved the result of Jiao and Tang (1999) by taking the leaky layer’s elastic storage into account. Jeng et al. (2002) presented a complete analytical solution to describe the tidal wave propagation and interference via leakage in the unconfined and confined aquifers separated by a thin, no-storativity leaky layer. Li and Jiao (2002) considered a coastal aquifer system including two aquifers and a semi-permeable layer between them, they differed from previous work in that both the effects of the leaky layer’s elastic storage and tidal wave interference between the two aquifers were considered.

The semi-permeable layers’ elastic storage can’t be neglected generally when they are thick or the elastic yield of these layers is large, which connect with confined aquifer from up and down. The semi-permeable layer’s elastic storage is even more important than that of the confined aquifer sometimes. According to various pumping tests data available (Li and Jiao, 2001b), if the semi-permeable layer is composed of thick and soft sedimentary materials, its storage is much greater than that of the main aquifer. The leakage through semi-permeable layer affects the groundwater head fluctuation in confined aquifer directly. At the same time, it is of great importance to deal with rightly the boundary condition at the submarine outcrop of the aquifer in the model of coastal-seabed confined aquifer. In the previous studies most researchers have assumed that the submarine outcrop of the aquifer connects with the seawater directly, or the coastline is treated as the boundary of the whole aquifer system. However, at its suboutcrop, the aquifer is usually covered by a thin boundary-layer with permeability different from that of the main aquifer. For example, in reality, in the deep seawater below the intertidal zone, owing to the weakness of water waves and to anoxia of the environment, fine sand, clay and organic oddments may gradually aggrade on the seabed. Therefore, the outcrop of an aquifer extending under the sea may not be directly connected to the seawater, but covered by a thin silt-layer (Li and Chen 1991a; b; Li et al., 2006). The boundary condition at the submarine outcrop of the aquifer plays a key role in the hydraulic head fluctuation in response to tide and it has an important effect on groundwater dynamics. Based on this motivation, this paper investigates a coastal three-layered aquifer system with a confined aquifer bounded by two semi-permeable layers from up and down, and the submarine outcrop of the aquifer is covered by a thin silt-layer. A mathematical model has been established to describe this aquifer system, and an attempt is made to derive an exact analytical solution to depict groundwater propagation in response to tidal fluctuation. The analytical solution is compared with several common solutions derived in other specific situations. The influences of both storage and leakage of the semi-permeable layer and the silt layer boundary on the groundwater-head fluctuation behavior in the confined aquifer are discussed and analyzed.

2 Conceptual Model and Analytical Solution

Consider a coastal three-layered aquifer system consisting of an upper and a lower semi-permeable layer and a confined aquifer between them. The suboutcrop of the aquifer is covered by a thin boundary-layer of different permeability, for example, the silt-layer (Fig.1). Assume that 1. the coastline is straight, all the layers are horizontal and extend landward infinitely; 2. both of the two semi-permeable layers and the confined aquifer are homogeneous, isotropic and with constant thickness, they terminate at the coastline; 3. the flows in the confined aquifer and in the two semi-permeable layers are assumed to be horizontal and vertical, respectively (Hantush (1960), Neuman and Whiterspoon (1969)); 4. the submarine outcrop of the aquifer is covered by a thin-silt and the change of water storage in the boundary-layer can be neglected, the silt layer may cover all the outcrops of the three layers as shown in Figure.1, or only cover outcrop of the middle confined aquifer. The different choices will not influence the mathematical model. The leakage of the two semi-permeable layers will not be affected by the existing of the silt layer; 5. the density difference between the groundwater and the seawater can be neglected owing to its slight impact on groundwater level fluctuation (Li and Chen, 1991a). Let the-axis be horizontal, positive landward and perpendicular to the coastline with its origin coinciding at the coastline. Let the -axis be vertical, positive upward with its origin coinciding with the-axis.

According to the assumptions above, the flow governing equations of groundwater in all the layers can be written as:

1. Groundwater flow in the upper semi-permeable layer:

,　　　  (1) (2)　　　                                                            　　       (3)

 

 

 

 

 

 

 

 

 

 

 

 

 

where  denotes the hydraulic head [L] of the groundwater in the upper semi-permeable layer at the location  and time ;  and  represent the specific storativity [L^-1] and vertical hydraulic conductivity [LT^-1] of the semi-permeable layer, respectively;  and  are the thickness of the upper and lower semi-permeable layer, and  is the thickness of confined aquifer.  is the hydraulic head of the confined aquifer at the instant  and location .

2. Groundwater flow in the lower semi-permeable layer:

，,    (4)

　                            (5)

   　　　　　　　 (6)

where  denotes the hydraulic head [L] of the groundwater in the lower semi-permeable layer at the location  and time ;  and  are the specific storativity [L^-1] and vertical hydraulic conductivity [LT^-1] of the lower semi-permeable layer, respectively.

3. Groundwater flow in the middle confined layer:

 ，　　                    (7) 　　   (8) 　　　　　　　　　　　　　　  (9)      where  and  are the storativity (dimensionless) and the transmissivity [L²T^-1] of the confined aquifer.

Following Li et al. (2006), the boundary condition of the silt can be seen:

             (8.a)                                       (8.b)  where 、and  are hydraulic conductivity [LT^-1] and thickness of the silt layer [L], and the hydraulic conductivity [LT^-1] of the confined aquifer;  is the relative leakage coefficient of the silt layer [L^-1], it describes the leakage of the silt layer relative to the confined aquifer. In addition, at the coastline ,  is the hydraulic head of the sea tide, where  is the amplitude of the sea tide [L],  is the angular velocity [T^-1] and equals to , where  is the tidal period [T], so the boundary condition of the silt can be written as Eq. (8). Equation    (9) gives the boundary condition of  in the inland side, which states that the flux in the confined aquifer is zero as  approaches infinity.

The analysis will focus on the groundwater-head  in the confined aquifer owing to the limited length of this paper. The derivation of the solution  to the boundary value system (1)-(9) is presented in detail in the Appendix A. For convenience of discussion, six middle parameters are introduced. The are the confined aquifer’s tidal propagation parameter [], the dimensionless leakance of the silt layer , the dimensionless storativity ratio  and  and the dimensionless leakage  and  of the upper and lower semi-permeable layer. They are written as followed:

                       (10.a)                                                                                     (10.b)                                                                      ,               (10.c)                                                                            ,               (10.d)  

where [] and [] are the specific leakage of the upper and lower semi-permeable layer. Using these parameters above, the solution  can be written as

         (11) where  and  are dimensionless constants defined as

(12.a) (12.b) in which the function  and  are given by

   (13.a)         (13.b)            where the function of  and  are defined as

              (14.a)               (14.b) where  and  are dimensionless parameters defined as：

，                       (15)                                   

In Eq.(11), and  are called the amplitude damping coefficient (dimensionless) and constant phase shift caused by the silt layer, they are two constants determined by the parameters of  and defined as

      (16)        (17)

3 Several Common Solutions in Special Situation

3.1 Aquifer with silt layer and without leakage

When there is no leakage from the upper and lower semi-permeable layer flowing into confined aquifer, that is, the vertical hydraulic conductivity of the two semi-permeable layers , in view of (10.d), one obtains . Let  in (13.a) and (13.b), one obtains

                   (18.a)                    (18.b) Substituting (18.a) and (18.b) back into (12.a) and (12.b), yields

，                   (19)                                              

Substituting (19) into (16) and (17), one finds that

，        (20) In this case, the analytical solution (11) can be simplified into

  (21) This is the same as the solution of Eq.(11) expressed by Li et al. (2006, in press). In fact, in the model, the head fluctuation in the aquifer is completely decided by the aquifer’s tidal propagation parameter and the dimensionless leakance of the silt layer .

3.2 Aquifer without leakage and silt

When there is no silt layer, and the leakage through the middle confined aquifer becomes negligible, then the thickness of the silt layer approaches zero, i.e., . In view of (8.b) and (10.b), one has . Let in Eq. (20), one obtains

，                 (22) Substituting (22) into equation (21), the analytical can be simplified into

                 (23) which is essentially the same as the solution introduced by Jacob(1950) with  defined as (10.a), considering that in the original equation of Jacob (1950), only the cosine functions both in solution (23) and boundary condition (8) are replaced by sine functions. In this case, the groundwater fluctuation is determined by the single parameter  related to the confined aquifer. This is why  is called the aquifer’s tidal propagation parameter.

3.3 Aquifer without silt and lower layer

When there is no silt and the lower semi-permeable layer, the aquifer configuration consists of the upper semi-permeable layer and confined aquifer, and they connect with the coastline at the terminal of the aquifer system. Owing to the absence of lower layer, the vertical hydraulic conductivity and the storativity of the lower semi-permeable layer are zero. In view of (10.d), one can obtain that , let  in Eqs.(13.a) and (13.b), one finds that

       (24.a)          (24.b) Substituting (24.a) and (24.b) into equation of (12.a) and (12.b), then

 (25.a)  (25.b)  The amplitude damping coefficient (dimensionless) and constant phase shift caused by the silt layer are followed that

(26.a)    (26.b) Combination of (25.a), (25.b), (26.a) and (26.b), yields

      (27)

In this situation, the number of independent parameters of the aquifer system is reduced to three, i.e., ,  and . This is the special case of the solution (11) if only considering the upper semi-permeable layer and confined aquifer.

4 Influence of various parameters on groundwater head fluctuation

From Eqs.(10)-(17), it can be seen that the model involve six independent parameters, the confined aquifer’s tidal propagation parameter , the dimensionless storativity ratio  and  and the dimensionless leakage  and  of both the two semi-permeable layers, and the dimensionless leakance of the silt layer . Solution (11) shows that groundwater head fluctuation with time at a fixed inland location  is also sinusoidal if the sea tide is a sinusoidal wave. Assume that at a fixed inland location , the ratio of the groundwater-head fluctuation amplitude to the sea tide amplitude is , and the time lag of groundwater response to sea tidal fluctuation is , then in view of (11), one obtains

       (28)          Comparison of (28) with (11), yields

     (29)                               (30)                          One can see that both  and  have correlation with the six parameters . The observed data of  and  at different observation wells can be obtained. Theoretically, if one knows four of the six parameters, then the other two unknowns of the six parameters can be calculated by solving Eqs. (29) and (30).

It is of great importance to know the rough ranges of these six parameters above in real aquifer system. Combining with the testing observation data available (Li and Jiao, 2001b), the six parameters can be obtained under the assumption of semidiurnal sea tide whose angular velocity 0.506h^-1. One can see that  ranges from 9´10^-4 to 9´10^-3 m^-1,  from 5.22 to 117 and  from 0.147 to 8.921. Based on these data, the discussion ranges of the parameters will be chosen 0-100 for  and 0-10 for . For convenience of discussion, a value range from -4 to 4 is chosen for  to make  enough large.

Eqs. (29) and (30) suggest that the groundwater-head fluctuation amplitude decreases with the landward distance from the coastline exponentially, while the time lag increases with it linearly. But each parameter of the aquifer has different influence on the fluctuation attenuation factor and time lag. Owing to the corresponding value range of the dimensionless leakage, the dimensionless storativity ratio of both the upper and lower semi-permeable layer are same, one can finds that the corresponding parameters have the same influences on the groundwater amplitude fluctuation according to symmetry principle. So it is only necessary to discuss the influences of the parameters of one semi-permeable layer on the confined aquifer. 

Fig.2 demonstrates that the different influences of both dimensionless leakage  and storativity ratio  of the upper semi-permeable layer and the dimensionless leakance of the silt layer  on the amplitude of groundwater head fluctuation in confined aquifer. Fig. 2(a) shows the variation of the dimensionless groundwater head amplitude defined by Eq.(29) with dimensionless distance  for different dimensionless leakage  when ， and . Fig. 2(b) shows the variation of the dimensionless groundwater head amplitude  with dimensionless distance  for different dimensionless storativity ratio  when ， and . Fig. 2(c) depicts the variation of the dimensionless groundwater head amplitude  with dimensionless distance  for different dimensionless leakance of the silt layer when  and . Comparison of Figs. 2(a), 2(b) and 2(c) demonstrates that the influence of dimensionless leakage  is similar to the storativity ratio on groundwater amplitude attenuation with dimensionless distance . But the attenuation velocity of the dimensionless leakance of the silt layer  on dimensionless groundwater head amplitude  is more rapid. When  increases from 0 to 10, or  increases from 0 to 100, the dimensionless landward distance from the coastline all disturbed by the sea tide decrease from 1.0 to 0.8, but when  increases from -1 to 4, the dimensionless landward distance from the coastline disturbed by the sea tide increases from 0.4 to 0.7. When the value of  is less than 0, the amplitude attenuation of fluctuation will be near to zero, it demonstrates that the dimensionless leakance of the silt layer have no impacts on the groundwater head amplitude.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.3 shows how the dimensionless groundwater-head  defined by (28) changes with the dimensionless time  at the fixed inland location  from the coastline. Fig. 3(a) is for different values of the dimensionless leakage  when ， and . Fig. 3(b) is for different values of the storativity ratio  when ， and . Fig. 3(c) is for different values of the dimensionless leakance of the silt layer  when  and . Comparing Figs. 3(a), 3(b) and 3(c), it demonstrates that the fluctuation amplitude  decreases as both  and  increase, but increases as  increases. The positive phase shift (time-lag) increases with both  and , but decreases with .

 

 

 

 

Fig.3 also shows that the amplitudeof the groundwater head in a coastal aquifer can be affected largely by the leakage or storativity of the semi-permeable layer when they change. In the traditional explanation, it was believed that the quicker damping of tide-induced groundwater head fluctuation in the confined aquifer was caused by a greater tidal propagation parameter  of the aquifer. The analysis of this paper indicates that the traditional understanding is simple, because the decreasing of the amplitude may have relation with significant leakage and elastic storage. A similar finding was also obtained by Jiao and Tang (1999) when they considered only the leakage from the semi-permeable layer. Especially, when there is a boundary silt layer at the submarine outcrop of the aquifer, the amplitude fluctuation of groundwater head may increase with the relative leakage coefficient of the silt layer at a fixing location from the coastline.

5 Summary

A coastal three-layered aquifer system is considered which consists of an upper semi-permeable layer, a lower semi-permeable layer and a confined aquifer between them. Each layer extends landward infinitely and terminates at the coastline, with the suboutcrop of the aquifer covered by a thin silt layer. A mathematical model is built to describe the tidal wave propagation in the confined aquifer and an analytical solution is derived. Our analytical solution can be simplified into several common special solutions when these parameters assume special values. Especially this solution is generalizations of the solutions obtained by Jacob (1950) and Li et al. (2006). The discussion of these parameters indicates that both storage and leakage of the upper and lower semi-permeable layers and the relative leakance coefficient of the silt layer play an important role in the groundwater-head fluctuation in the confined aquifer. The influence of dimensionless leakage  is similar to the storativity ratio on groundwater amplitude attenuation with dimensionless distance . But the attenuation velocity of the dimensionless leakance of the silt layer  on dimensionless groundwater head amplitude  is more rapid. The fluctuation amplitude decreases with the storage and leakage of both the upper and lower semi-permeable layers, and the phase shift (time-lag) increases with them. If there exists the silt layer, the fluctuation amplitude increases with the relative leakance coefficient of the silt layer and phase shift (time-lag) decreases with it.

Appendix: Derivation of the solution (11)

Now suppose that

                (A.1)            (A.2)               (A.3) where , ,  are complex functions, Re denotes the real part of the followed complex expression, .Substituting (A.2) back into (1)-(3), and then extending the three resultant real equations into complex ones with respect to the unknown function , yields

                 (A.4)                   (A.5)　　                          (A.6) The solution of (A.4)-(A.6) is：                                                       

  (A.7)                                          where                          (A.8) Using (A.7), one can obtain

(A.9) where 

         (A.10)               where the functions of  and  are expressed by Eqs. (14.a) and (14.b).

Substituting (A.3) back into (4)-(6), and then extending the three resultant real equations into complex ones with respect to the unknown function, yields

               (A.11)                           (A.12)                            (A.13) The solution of (A.11)-(A.13) is

(A.14) where                       (A.15)

Using (A.14), one can obtain

(A.16) where

          (A.17) Now substituting (A.1), (A.9) and (A.16) back into Eqs. (7)-(9)，and extending the three resultant real equations into complex ones with respect to the unknown function ,yields

      (A.18)                 (A.19)                               (A.20) The solution of (A.18)-(A.20) is

                       (A.21) where , , are given by Eqs. (10.a), (12.a) and (12.b), respectively. In view of (A.19)one can know that 　　

i.e.                     (A.22) Substituting (A.22) back into (A.21) leads to

　　     (A.23) So the complex head function can be expressed

             (A.24) Then the real head function can be seen

          (A.25) i.e., Eqs. (11).      

Wher

                      (A.26)  (A.27)          

where is the leakance of the silt layer 

So in view of (A.27), it leads to

   (A.28)                     

 (A.29)

 

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