Yuqiang Xia¹, Hailong Li^{1, 2}, Qiaona Guo¹, Guohui Li¹

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 structure of a coastal aquifer system is generally multi-layered, the confined aquifer usually extends under the sea for a certain distance, and its subcrop is usually covered by a silt layer. Based on these scenarios, this paper considers a leaky aquifer system consisting of an unconfined aquifer, a confined aquifer, and a semi-permeable layer between them. The unconfined aquifer terminates at the coastline, and the aquifer and the semi-permeable layer (aquifer’s roof) extend under the sea for a certain distance. The confined aquifer’s subcrop is covered by a silt layer. The mathematical model for the system is given. The analytical solution to the model is derived. Our solution extends the existing solutions obtained by previous researchers. Detailed analyses are conducted to investigate the influences of aquifer’s extending length, the semipermeable layer’s leakage, the silt-layer’s dimensionless leakance and loading efficiency on tidal propagation in the confined aquifer, with focus on the impacts of leakage through the semipermeable layer, and that of the silt-layer-subcrop on tidal propagation. The existence of the silt layer subcrop may significantly decrease the amplitude and increase the phase shift of the head fluctuation in inland aquifer if the roof length is small. On the other hand, if the roof length is long enough, the impacts of the silt layer on the head fluctuation in inland aquifer becomes negligible. The leakage through the semipermeable layer plays the same role as found by Li and Jiao [2001, Water Resources Research, 27(5), p1165-1171].

Keywords: Coastal multi-layered aquifer system, Aquifer extending under the sea, Silt layer, Analytical solution   Tidal wave propagation, Leaky boundary condition, Leakage

1 Introduction

The coastal area is where the global economy most developed, and also the most complicated zone for the earth environmental dynamical system. The groundwater problems, which have a close connection with the coastal area, for example, seawater intrusion and land subsidence, become the focal point, which people pay close attention to gradually. Thereupon, land-ocean interaction in the coastal zone (LOICZ) has become an active research area in the whole world nowadays [Yang, 2004]. A great deal of previous studies show that the research of the tidal propagation in the coastal aquifer system plays an important and active role in solving various coastal hydrogeological, engineering geology, ecological and environmental problems related to tidal dynamics [e.g., Ferris, 1951; Akpofure et al., 1984; Farrell, 1994; Svitil, 1996].

The existing researches show that the structure of a coastal aquifer system is generally multi-layered [Van der Kamp, 1973; Jiao and Tang, 1999; Li and Jiao, 2001a; b; Li and Jiao, 2002; Jeng et al., 2002]. Groundwater head in inland aquifer will fluctuate with the sea tide, that is, groundwater is provided with tidal effect. Tidal effect of groundwater in different distance from the coast is diversified with various aquiferconstructions [e.g., Jacob, 1950; Van der Kamp, 1972; Li and Jiao, 2001a]. Confined aquifers may exhibit strong tidal effect with fluctuations potentially extending to several kilometers from coast. Significant fluctuations are unlikely to extend to more than several decameters from coast in unconfined aquifers [e.g., White Roberts, 1994; Chen and Jiao, 1999].

The treatment of the aquifer’s boundary condition directly affects the emulation degree of the aquifer system model in the research of the coastal aquifer system [Cheng et al., 2003; 2004]. Most existing studies regard coastline as the boundary of the whole aquifer system [e.g., Jacob, 1950; Ferris, 1951; Jiao and Tang, 1999]. The depth of these models built by previous researchers is not less than several decameters or a hectometer, even much deeper. In reality, however, submarine relief is a flat gradient sloping down to sea instead of an upheaval in coastal area [Cheng et al., 2004; Yang, 2004]. The confined aquifer may extend under the sea with a considerable distance from the coastline [e.g., Li and Chen, 1991a; b; Li and Jiao, 2001a], even can be deemed them to be an infinite extendibility in some sea area [Van der Kamp, 1972].

Furthermore, submarine bank slope lies below the intertidal zone suffered from the effects of waves and ocean currents. With increase of the depth of seawater, owing to the weakness of hydrodynamic force and the diminution of sedimentary particles, silt, clay, and organic oddments may gradually deposit on the seabed, and form a sedimentary band with interbedded silt, clay, and slit deposit, as well as a sandwave zone, that is, exogenous sedimentary band [e.g., Yang, 1986; Zeng et al., 1992]. There is always coastal silt with fuscous and abundant organic material in an anoxia of the environment. Therefore, the subcrop of confined aquifer extending under the sea may be indirectly connected to the seawater, but covered by a thin silt layer; this phenomenon is quite common in muddy coast [e.g., Li and Chen, 1991b; Yang, 2004]. Li et al. [2006](in press) presented the analytical solution for a single aquifer in response to the tidal fluctuation with a thin silt layer boundary, and discussed the influences of the roof length and hydraulic permeability of the silt layer on the groundwater head fluctuation.

In this paper, based on Li and Jiao [2001a] and Li et al. [2006], the authors consider a more common coastal aquifer system, which has three layers: an unconfined aquifer, a confined aquifer, and a semi-permeable layer between them. The confined aquifer extends for a certain distance under the sea, and its subcrop is covered by a thin silt layer with permeability different from that of the main aquifer. Considering leaky boundary condition for the silt layer, the leakage of the semi-permeable layer, and the extending distance of the aquifer under the sea, a new analytical solution will be derived. Compared with the existing solutions [e.g., Li and Jiao, 2001a; Li et al., 2006], an attempt is made to discuss quantitatively the impacts of the roof length, the semipermeable layer’s leakage, the silt-layer’s dimensionless leakance and loading efficiency on tidal propagation in the coastal confined aquifer.

2Model setup and Analytical Solution

2.1 Conceptual mModel and mathematical model

A coastal aquifer system consisting of an unconfined aquifer, a leaky confined aquifer, and a semipermeable layer between them is considered in this paper. Assume that the unconfined aquifer vertically terminates at the coastal line, while the confined aquifer and the semipermeable layer (roof) extend under the sea for a certain distance , the bottom of the leaky confined aquifer is impermeable, all the layers extend landward infinitely. Let the  axis be perpendicular to the coastline, the intersection of the mean sea surface and the beach face be the origin of the axis, and be positive landward (Figure 1). Based on the assumption introduced byJiao and Tang [1999], we assume that the aquifer is horizontal, homogeneous and of constant thickness, and the flow in the confined aquifer is essentially horizontal and there is vertical leakage through the semipermeable layer (roof of the confined aquifer), and the elastic storage of the semipermeable layer is negligible. Further assume that the density difference between the groundwater and the seawater can be neglected, owing to density’s slight impact on groundwater level fluctuation [Hantush and Jacob, 1955; Jiao and Tang, 1999; Li and Chen, 1991a].

Under above assumptions and the theories of leaky and elastic aquifers proposed by Hantush and Jacob [1955] and Jacob [1950], consider the silt-layer-subcrop, the following mathematical model is derived on the basis of Li and Jiao [2001a].

The tidal propagation in the offshore confined aquifer:

     (1)

The tidal propagation in inland confined aquifer:

    (2)

                       (3)

Where ,  and  denote hydraulic head [L], storativity (dimensionless), and transmissivity [L²T^-1] of the aquifer, respectively;  is leakance or specific leakage [T^-1] of the semipermeable layer, which is defined as ,  and  are the thickness [L] and vertical hydraulic conductivity [LT^-1] of the semipermeable layer, respectively;  is loading efficiency (dimensionless), defined as

                         (4)

where  is the compressibility [M^-1LT²] of the confined aquifer skeleton,  is the compressibility [M^-1LT²] of porosity water in the confined aquifer, and  is the porosity (dimensionless) of the aquifer;  are hydraulic head [L] of the sea tide,  is the amplitude [L] of the sea tidal,  ω is angular velocity [T^-1] of tide and equals ,  is the tidal period [T] [Fetter, 1994; Todd, 1980].

The pure hydraulic replenishment is zero at far inland as  approaches infinity,

                       (5)

and at the coastal line, i.e., at , by the continuity of hydraulic head and the continuity of flux, respectively, one has

                 (6)

               (7)

Li et al. [2006] presented leaky boundary condition of silt layer (see equation (8b) of Li et al.), it can be rewritten as

              (8)

                             (9)

The parameter  involved in equation (9) is defined as the relative permeability [L^-1] of the silt layer which describes the leakage of the silt layer relative to the permeability of the confined aquifer.

2.2 Analytical solution

For convenience of discussion, two new parameters, the confined aquifer’s tidal propagation parameter a [L^-1] and the dimensionless leakage , are introduced:

                 (10)

                (11)

The solutions (see Appendix for the derivation) for the boundary value problem (equations (1)-(8)) are derived as follows

(13)

where

     (14)

          (15)

 is presented by equations (A14) and (A16) in the Appendix,  and  are determined by the dimensionless leakage  and defined as

    (16)

and are controlled by the loading efficiency  and the dimensionless leakage , as follows:

       (17)

 and  are two dimensionless constant, defined as

(18)

   (19)

where  is the dimensionless leakance of the silt layer, it can be written as

                       (20)

Equations (12) and (13) are the analytical solution for the tidal propagation in offshore and inland portion of the confined aquifer, respectively,  is comprehensive tidal efficiency of the leaky confined aquifer system,  is the fixed phase shift (radian) and unattached variables x and t. In addition, Equation (13) shows that the groundwater head fluctuation of an observation bore at a fixed inland location  is also sinusoidal. It can be determined by the groundwater head fluctuation amplitude [L] and the time lag [T] of groundwater response to tidal fluctuation, i.e.,

            (21)

Compared with equation (13), yields

              (22)

                      (23)

The detailed discussions for  and  are presented in section 4.

3 Compared with Relative Analytical Solution

Just as above-mentioned, and   depend upon loading efficiency  and dimensionless leakance ,  marked with  and , respectively.  and  defined by equations (18) and (19) are determined by four independent parameters, that is, , labeled with and , respectively. And moreover, the tidal propagations indicated in equations (12) and (13) are controlled by these parameters, a simple functional relation can be used to denote the tidal propagations in inland and offshore parts of the aquifer, viz.

           (24)

The comparisons between with existing analytical solutions are given infra, because people focus all their attention on the tidal propagation in the inland confined aquifer rather than that of in the offshore confined aquifer, so the authors only show the expression of the tidal propagation in the offshore aquifer but no discussion for it because of space limitation.

3.1 Confined Aquifer Extending under the Sea Infinitely

If the extending length is magnified infinitely (), i.e., , in view of (18) and (19), one obtains , then (12) and (13) can be written as

    (25a)

(25b)

where

(25c)

      (25d)

This solution given here is the same as the solution obtained by Li and Jiao [2001a] (equations (21a)-(21c) in Li and Jiao), where the silt-interface is negligible, which implies that the silt-interface of the confined aquifer extending under the sea is inexistent when . Here the tidal propagation is only decided by three parameters, i.e., , the functional relation defined as equation (24) can be written as   (25e)

In this case, if there is no leakage through the semipermeable layer, viz.,, or  , and from (17), (16) and (14) one has , , , , then

(26a)

 (26b)

which is the same as the result derived by Van der Kamp [1972]. This suggests the tidal propagation is determined by the loading efficiency  and  tidal propagation parameter , that is

                     (26c)

3.2 Confined aquifer without extending length under the sea

If the offshore confined aquifer extending length is zero, i.e., , (18) and (19) follows that

        (27a)

            (27b)

in view of equations (14) and (15), yields

  (27c)

then the tidal propagation in confined aquifer can be written as

      (27d)

which is the same as the solution given by equations (7), (8e), and (8f) in Ren et al. [2006](in press). In this case, the tidal propagation rest with three parameters: the confined aquifer’s tidal propagation parameter , semipermeable layer’s leakance  and the silt layer’s dimensionless leakance , that is

                   (27e)

In this case, if there is no leakage (), viz., , one has , , , then yields

    (28a)

  (28b)

      (28c)

which accord with the equations (11) and (12) of Li et al. [2006]. This solution reflects predominantly that the tidal propagation of confined aquifer  is affected by tidal propagation parameter  and the silt layer’s dimensionless leakance , viz.,

                       (28d)

3.3 Offshore confined aquifer without silt-interface layer

Assume that there is no silt layer, the confined aquifer exposes directly itself to the sea, i.e., the thickness of silt layer , which is equivalent to  in view of equation (20); or presume that silt layer’s permeability and other hydrogeologic parameters are not different from that of the confined aquifer, one can regard silt layer as a part of the confined aquifer, that is,  is used in equation (20). Generally speaking, it is reasonable to assume that the thickness of the silt layer approximately has an order of magnitude of O(1) m. On the basis of the nine sets of aquifer parameters obtained by Jiao and Tang [2001] from various pumping tests in real leaky confined aquifer systems, the typical ranges of  is estimated to be  m^-1, in view of (20), the range of  is  , it is not unreasonable to regard   infinity for thisrange, because its magnitude is much greater than that of other parameters () in view of (18) and (19), and follows that

 (29a)

 (29b)

The solution obtained under this term is essentially the same as equations (12)-(15), and equations (29a) and (29b) derived here is uniform with equations (7c) and (7h) presented in Li and Jiao [2001a].

If there is no leakage through the semipermeable layer, i.e. , then , , , , equations (29a) and (29b) can be written as follows

(30a)

(30b)

According to equations (14) and (15),  one yields

      (30c)

(30d)

the tidal propagation in confined aquifer can be written as

         (30e)

which agrees with equations (16a), (16b) and (16c) derived by Li and Jiao [2001a]. And the groundwater head is impacted by tidal propagation parameter , the dimensionless extending length , loading efficiency   , and the silt layer’s dimensionless leakance , it is follows that

              (30f)

Under this condition, if there is no extending length, that is, , (29a) and (29b) can be simplified as follows:

          (31a)

     (31b)

This solution is exactly the same as the analytical solution introduced by Jacob [1950].

3.4 Offshore confined aquifer with an impermeable silt-interface layer

If the hydraulic permeability of the thin silt layer is quite weak, i.e., the dimensionless leakance =0.0, one obtains

     (32a)

     (32b)

the form of solution obtained under this condition is the same as equations (12)-(15) in this paper.

Further assume that there is no replenishment for confined aquifer, viz., the leakage through the semipermeable layer is zero, that is , or , then , , , one can yield

                 (33a)

       (33b)

         

 (33c)

the tidal propagation in confined aquifer can be presented as

       (33d)

 (33e)

which is exactly the same as equations (13a) and (13b) obtained by Li et al. [2006]. The tidal propagation is enslaved to tidal propagation parameter , the dimensionless extending length , and loading efficiency , viz.,

                   (33f)

If there is no extending length under this condition, that is, , equations (18) and (19) can be simplified to

                     (34)

According to these assumptions, it is obvious to yield , that is, , which indicates that the confined aquifer is enclosed with an impermeable silt layer, so the groundwater head is not tampered withtidal propagation, and is a constant, which is anastomotic with actual fact.

4 Influences of Various Parameters on Tidal Propagation

On the basis of the nine sets of aquifer parameters obtained from various pumping tests in real leaky confined aquifer systems by Jiao and Tang [2001], the typical ranges of   and  are estimated to be  and , respectively [Li and Jiao, 2001a]. For this study, a larger range of  is used, the sea tide is assumed to be diurnal with a period , and the dimensionless distance of the piezometer at the inland area is  from the coastline. An attempt is made to discuss the impacts of dimensionless roof length  , the semipermeable layer’s dimensionless leakage  , the silt-layer’s dimensionless leakance  , and loading efficiency   on tidal propagation in confined aquifer.

4.1 Influences of dimensionless roof length and leakage through the semipermeable layer

As show in Figure 2, for any dimensionless leakage , when the silt-layer’s dimensionless leakance  and , the amplitude is quite small, even equals zero at inland area  when  . This is because the silt-layer can be regarded as impermeable layer, that is, there is no tidal propagation in the coastal confined aquifer under this condition. But the amplitude is raised significantly as the dimensionless roof length  increases when it is small. For the curve of , the increase of the amplitude is induced by the loading efficiency  which acts on the offshore confined aquifer, and the amplitude approaches a constant and is no longer sensitive to the roof length when the dimensionless roof length is greater than a certain value.

Figure 2 also shows that the amplitude increases as  is increased from 0 to 1.5, but it decreases as  is increased further from 1.5 to 15.0. This is because the leakage from the inland portion tends to damp the tidal propagation in the confined aquifer, while the leakage from the offshore part will enhance the tidal propagation. When the dimensionless leakage is small and the roof length is great, the enhancing effect due to the offshore leakage is dominant; Li and Jiao [2001a] have presented the detailed explanations.

Based on above-mentioned assumptions, it is easy to see that the influence of dimensionless leakage  on phase shift is non-monotonic in Figure 3. The phase shift changes from a large positive value to a large negative value when  equals zero and the dimensionless roof length is very small. Then the phase shift increases with , and is no longer sensitive to the roof length when  reaches a certain value.

When  is larger than 0.5, there is only positive phase shift occurring. The change of phase shift is non-monotonic for the curve  of  in Figure 3. When is small, the phase shift reduces significantly begin, and then increases slowly. While there is a minute decrease or no change when  is great, which shown in Figure 3 with two curves and . When the dimensionless roof length is greater than a certain value, these curves tend to different positive values. Figure 3 also shows that the phase shift increases with  when  increases and  is small. On the contrary, the phase shift decreases with  when  increases and   is large.

For , at inland area  when the silt-layer’s dimensionless leakance  and , as shown in Figure 4, the amplitude decreases with the dimensionless roof length  when  is small. This is because the tidal propagation in the confined aquifer is mostly damped when the dimensionless roof length  increases. But when  reaches a certain value, the amplitude tends to a constant. This accords with the discussion in section 3.3 of Li and Jiao [2001a]: there is a threshold value for roof length, and when the roof length is greater than threshold value, the water level fluctuation will behave as if the roof length were infinite.

In addition, if the silt-layer’s dimensionless leakance  is very large, it is equivalent to the permeability of silt layer is same as the confined aquifer, one can regard that there is no silt layer as discussed in section 3.3. The effect of the dimensionless leakage  on the amplitude is same as the Figure 2 in Li and Jiao [2001a].

The phase shift is positive value at inland area  when  and , as shown in Figure 5, for the curve , the change of phase shift is complex. To begin with, there is a rapid increase, and then is a slow decrease. Once there is leakage through semipermeable layer, the change of phase shift is monotone increasing with . And there is no variation for phase shift when  is large enough. In addition, the phase shift decreases as  increases.

 

4.2 Influence of silt-layer’s dimensionless leakance

At inland area , forany dimensionless leakance , assume  and , as shown in Figure 6, the amplitude increases significantly with the dimensionless roof length  when  is small and . But the amplitude decreases significantly when . And when the dimensionless roof length is greater than a certain value, the amplitude is a constant. Figure 6 also shows that the amplitude increases as  increases. It is close to the permeability of the confined aquifer when the silt-layer’s dimensionless leakance  is very large, and the damping effect of the silt layer may be small, which is of great benefit to the tidal propagation in confined aquifer.

According to above-mentioned assumptions, Figure 7 shows that the curves of phase shift are non-monotonic functions. When   equals zero and  is terribly small, the phase shift falls rapidly from 3.87 to -1.56, after that, it increases quickly with   and then slowly. When the dimensionless roof length is greater than a certain value, these curves tend to a positive value. Phase shift only appears positive values when  is lager than 0.4. As shown in Figure 7with the curve , phase shift decreases significantly begin and then laggardly when  is small. When  is large, as shown in Figure 7 with the curves  and , the phase shift increases significantly begin and then decreases laggardly. Finally, these curves reach a positive value.

The general trend shown in Figure 7 is: when both dimensionless roof length  and dimensionless leakance  are small , phase shift decreases significantly begin and then rises laggardly to a constant. However, for a large , phase shift rises significantly begin and then decreases laggardly to a constant. Figure 7 also shows that the initial phase shift decreases as  increases; this also suggests that the increase of the silt-layer’s dimensionless leakance is benefit for tidal propagation in confined aquifer.

If there is a large leakage through, assume and , for any dimensionless leakance , at inland area , Figure 8 shows that theamplitude raises rapidly with  when it is small as  changes from 0.1 to 1.0. But the amplitude decreases significantly when  under the same assumptions. As shown in Figure 7, Figure 8 also shows that the initial phase shift decreases as  increases, this suggests that the increase of the silt-layer’s dimensionless leakance is benefit for tidal propagation. Compared Figure 6 with Figure 8, it is obvious that the semipermeable layer’s leakage will damp the enhancing effect of the silt-layer’s dimensionless leakance  on tidal propagation, this result has been pointed out in section 4.1.

The phase shift is positive value as shown in Figure 9 when we use the same conditions considered in Figure 8. When dimensionless roof length  is small and dimensionless leakance is very small , the phase shift decreases significantly begin and then rises laggardly to a constant. But when a large  is used, phase shift rises significantly begin and then rises laggardly to a constant. Figure 9 also shows that the initial phase shift decreases with  ; this suggests that the increase of the silt-layer’s dimensionless leakance is benefit for tidal propagation again.

If Figure 7 was compared with Figure 9, one can see that the phase shift changes with a positive range in Figure 9. The two figures indicate that the initial phase shift and amplitude variation of tide waves are decreased significantly, under the comprehensive action of the semipermeable layer’s leakage andthe silt-layer’s dimensionless leakance.

4.3 Influence of loading efficiency

The loading efficiency reflects the groundwater level fluctuation caused by compression of both the aquifer skeleton and groundwater, due to the loading rate of seawater above the roof of the confining layer. Solution obtained by previous researchers shows that the impact of the loading efficiency on fluctuation is significant only when the leakage of the semipermeable layer is small and the roof length is great [Li and Jiao, 2001a]. According to section 4.2, one can easily receive the information that the influence of the silt-layer’s dimensionless leakance  on tidal propagation is small when  is very large; on the other hand, when  is enough large, this condition is similar to the case that there is no silt-layer-subcrop considered by predecessor. In view of these results, we discuss the effect of the loading efficiency on fluctuation only when , dimensionless leakage , and  are very small, , ,  are used in lower section.

When , compared three curves , , and  in Figure 10,  the amplitude is largest and the phase shift is smallest when  , and per contra for curves with  and . That is, the amplitude oscillation and the time lag of the groundwater head fluctuation responses to sea tide increases and decreases with loading efficiency, respectively.

Contrasted  with  when we keep the same assumptions used in Figure 10, as shown in Figure 11, three curves, , and are close to each other. Just as above-mentioned, the increase of the silt-layer’s dimensionless leakance is favorable for tidal propagation. It is seen intuitively that the amplitude oscillation for  changes from 0.05 to 0.13 as  is increased from 0.1 to 1.0. Corresponsively, the amplitudes for  and change from 0.18 and 0.30 to 0.22 and 0.32, respectively, which also incarnate the rule that the amplitude oscillation and the time lag of the groundwater levels response to sea tide increases and decreases with loading efficiency, respectively.

5. Conclusions

The structure of a coastal aquifer system is generally multi-layered, and the confined aquifer usually extends for a certain distance under the sea. This has been observed and studied analytically and numerically by many previous researchers. An analytical solution is derived in this paper to investigate the tidal propagation in a coastal leaky confined aquifer with a thin silt-layer-interface. Several solutions obtained by previous researchers are obtained when give some certain values to roof length , the semipermeable layer’s leakage , and the silt-layer’s dimensionless leakance ,this shows that analytical solution presented here is more general than the traditional solution presented by previous researchers. And the detailed discussions for the influences of thedimensionless roof extending length, the semipermeable layer’s leakage, the silt-layer’s dimensionless leakance, and loading efficiency on the amplitude and the phase shift of the groundwater head fluctuation in inland confined aquifer were analyzed. The analyses show that the effects of the leakage from the inland and offshore portions of the semipermeable layer are different. The leakage from the inland portion tends to damp the water level fluctuation in the confined aquifer, while the leakage from the offshore part will enhance the fluctuation. When the dimensionless leakage is small and the roof length is great, the enhancing effect due to the offshore leakage is dominant. The amplitude oscillation increases with loading efficiency, butthe time lag of the groundwater head fluctuations response to sea tide decreases. Owing to the existence of the silt layer subcrop, which is equivalent to a certain extension for the roof length of the confined aquifer, the amplitude oscillation and the phase shift of the tidal waves are held down and increased, respectively. And the increase of the silt-layer’s dimensionless leakance is favorable for tidal propagation, but the leakage of the semipermeable layer will damp this promoter action, it always makes phase shift positive, and no negative phase shift occurs. Compared with the condition that there is no leakage through the semipermeable layer, the initial phase shift and amplitude variation of tide waves is decreased significantly under the comprehensive action of the semipermeable layer’s leakage and the silt-layer’s dimensionless leakance.

 

Appendix: Derivation of the solution to (1)-(8)

Assume is a complex function, let  and are independent variables for convenience using separation of variables to solve the equations, and use  as its real part, that is

                 (A1)

                      (A2)

where  is an unknown function of ,  denotes the real part of the complex expression, .

Based on these assumptions, substituting (A1) into the equations (1)-(8) which  satisfies, yields

  (A3)

 (A4)

                           (A5)

                      (A6)

                     (A7)

                     (A8)

             (A9)

The general solutions of (A3) and (A4) are followed

   (A10)

   (A11)

where , and , and  are defined by (8), (10) and (11), respectively.  and are unknown complex coefficients. Substituting the boundary condition equations (4)-(7) into (A10) and (A11), one yields

(A12)

                     (A13)

 (A14)

                                   (A15)

make some routine calculations for (A12) and (A14), they can be written as follows

  (A16)

 and  are constants defined by (18) and (19), respectively. Substituting (A13), (A15) and (A16) into (A10) and (A11) can obtain the solution of the , then substitute  into, and acquire the real part of, finally leads to the solutions for the boundary value problems (1)-(8) given by (12) and (13).

 

Acknowledge

This research is supported by the National Natural Science Foundation of China (NSFC No. 40372111). The authors are very grateful to Ms. Ying Yang for her much help.

References

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