TIDAL EFFECTS ON GROUNDWATER DYNAMICS IN COASTAL AQUIFER UNDER DIFFERENT BEACH SLOPES*

LIU Yi

College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China, E-mail: yiliu19871017@126.com

SHANG Song-hao

State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

MAO Xiao-min

College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China

TIDAL EFFECTS ON GROUNDWATER DYNAMICS IN COASTAL AQUIFER UNDER DIFFERENT BEACH SLOPES*

LIU Yi

College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China, E-mail: yiliu19871017@126.com

SHANG Song-hao

State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

MAO Xiao-min

College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing 100083, China

(Received August 30, 2011, Revised October 31, 2011)

The tide induced groundwater fluctuation and the seawater intrusion have important effects on hydrogeology and ecology of coastal aquifers. Among previous studies, there were few quantitative evaluations of the joint effects of the beach slope and the tide fluctuation on the groundwater dynamics. In this article, a numerical model is built by using the software FEFLOW with consideration of seawater intrusion, tide effects, density dependent flow and beach sloping effects. The simulation results are validated by laboratory experimental data in literature. More numerical scenarios are designed in a practical scale with different beach slopes. Results show that the groundwater fluctuation decays exponentially with the distance to the beach, i.e.,= A , and our simulation further shows that the beach slope influencecan be expressed in the form of a logarithm function. While for the same location, the amplitude increases logarithmically with the beach angle in the formwhereand γ′ are related with the horizontal distance (x) in the form of a logarithm function. The beach slope has no influence on the phase lag, although the latter increases regularly with the distance from the sea. The beach slope effect on the seawater intrusion is investigated through the quantitative relationship among the relative intrusion length (λ), the relative enhancement of the tide induced seawater intrusion (κ) and the beach angle (α). It is shown that the tide effects on a milder beach is much greater than on a vertical one, and both λ and κ can be expressed in logarithm functions of α. The tidal effect on the flow field in the transition zone for a particular mild beach is also studied, with results showing that the tide induced fluctuation ofis similar to the groundwater table fluctuation whilezV shows a distinct variation along both directions.

tidal effects, salt wedges, saline intrusion, beach slope, finite element method

Introduction

The coastal area is usually the most developed area in the world, where the groundwater would be over-exploited, with serious environmental problems, e.g., the ground surface subsidence and the seawater intrusion into the groundwater. Furthermore, the contaminants released from industrial production, municipal wastes and agricultural activities can be infiltrated into the groundwater, migrate towards the sea and threat the coastal ecology. The groundwater discharge into the sea and the contaminant migration have long been the investigation topics[1]. In this respect, it is important to understand the coastal groundwater dynamics under the tidal fluctuation and the seawater intrusion conditions.

The groundwater dynamics and the seawater intrusion in coastal aquifers started to attract attentionsin the late 19th century. Early researches were mostly based on the Ghyben-Herzberg approach, a sharp interface model assuming that there is no mixing between seawater and freshwater. The seawater intrusion was further investigated under this assumption by extending to two and three dimensions, adding source and sink items[2], or including the wave action[3]. Although the sharp interface approach is convenient, the fact remains that the freshwater and the seawater are mixed. Henry (1964) established the first transition zone model for the seawater intrusion, and deduced an analytical solution at the steady state. In recent years, numerical models are widely used because it can handle complicated and real conditions[4], including the transition theory and various boundary conditions, e.g., the aquifer recharge, tidal effects and sloping beaches.

It is generally recognized that the tidal fluctuation can accelerate the seawater intrusion into aquifers. Moreover, the tidal fluctuation would also influence the groundwater dynamics, such as the groundwater table fluctuation and the groundwater discharge towards sea. Based on experimental and numerical simulations, Parlange et al. (1984) deduced a steady periodic solution for the shallow-flow problem under the Boussinesq approximation, with an exponential reduction of the groundwater fluctuation. However, Parlange’s approximation was obtained by using a perturbation technique with the ratio of the tidal amplitude to the mean aquifer thickness as the perturbation parameter. In this way, the third or higher order solutions could not be obtained. Moreover, in that solution, only vertical beach is considered. Nielsen (1990) proposed a new solution for one-dimensional Boussinesq equation for inclined beaches by neglecting variations in the coastline and the beach slope. Parlange and Nielsen’s methods were improved by many subsequent studies. Song et al.[5]derived a new perturbation solution of the non-linear Boussinesq equation for one-dimensional tidal groundwater flow in a coastal unconfined aquifer. Xia et al.[6]analyzed the case where the beach was covered by a layer of less permeable sediments, and obtained a new solution for the tide induced groundwater table fluctuation based on a new perturbation solution. In 2002, Li and Jiao extended the analytical method to the L-shaped leaky coastal aquifer[7], and the coastal two-aquifer system[8]to study the groundwater fluctuation response to the tide. Li et al.[9]investigated the groundwater table response to both ocean and estuary fluctuations and derived a two-dimensional analytical solution with considerations of the interaction between the tidal waves cross and along the shore, instead of only focusing on the inland propagation of oceanic tides in the cross-shore direction as in the previous studies. But the beach slope effect was not considered in their work. This beach slope effect was later considered by Jeng et al.[10], who derived the analytical solution for two-dimensional problems with sloping beaches of different beach angles and multitidal signals.

However, Boussinesq equations oversimplified the real circumstances. Therefore, numerical methods were applied to find relatively accurate solutions. Earlier simulations studied the seawater intrusion using various numerical methods, e.g., method of characteristics, finite element method, boundary integral method, finite difference method, boundary element method or lattice Boltzmann method. Later on, other application fields were considered, such as, the tide/wave induced water table fluctuation[4,5], the interaction among surface water, groundwater and seawater[11], and the tide induced density-dependent contaminant transport and groundwater dynamics in coastal areas[12].

Besides analytical and numerical methods, many laboratory or field experiments were designed and carried out to investigate the coastal groundwater dynamics, with numerical methods being used for further analysis and forecast. Nielsen (1990) conducted field observations of the tide induced water table fluctuation. Li et al. (1997) used a numerical method to explain the field observation results, showing that the simulation can successfully reveal the three features of tide induced water table fluctuations. Mao et al.[13]combined the field monitoring and the numerical method to investigate the effect of the beach slope for a coastal aquifer adjacent to a low-relief estuary. Balugani and Antonellini[14]studied the tide and barometric effects on the groundwater table through field monitoring. Although field experiments and observations are more reliable, the field condition is difficult to control and the monitoring cost is high. Moreover, the field data are relatively difficult to interpret due to the field heterogeneity conditions. On the other hand, the laboratory scaled experiments can be conducted under controlled conditions and the heterogeneity conditions can be avoided. By laboratory experiments, Goswami and Clement[15]investigated the salt-wedge under both steady and transient conditions. Boufadel et al.[16]conducted an experiment to investigate the tidal effects on the solute transport. Wu and Zhuang[17]investigated the tide induced water table elevation. However, the laboratory scaled experiments often simplifies the real situation by neglecting some of influencing factors, such as the density difference in freshwater and seawater[17], or the beach slope[15].

Although the tidal effect on the seawater intrusion and the groundwater dynamics is a much studied issue, the beach slope effects are mostly neglected despite the fact that beach slope vary greatly in nature. In this article, after the calibration with previous laboratory experiments, we use FEFLOW to investigatethe influences of different beach slopes on the tide induced water table fluctuation and the groundwater dynamics in the transition zone based on designed numerical scenarios.

1. Simulation model for groundwater dynamics in coastal aquifer with sloping beach

1.1 Mathematical formulations

In variably saturated porous media, the governing equation for the variable density groundwater flow can be written as

where θ is the water content, ρ is the fluid density, inρ is the density of the water entering from a source or leaving through a sink,inq is the volumetric flow rate per unit volume of the aquifer representing sources and sinks, t is the time and q is the specific discharge vector (Darcy flux) calculated as

where K is the hydraulic conductivity related to the freshwater, which is a function of θ in the unsaturated zone and can be described by existing models (e.g., the Mualem model (1976)),uf is the ratio of the dynamic viscosity of the fluid to that of the freshwater, h is the hydraulic pressure head with respect to a freshwater of 103kg/m3in density in the saturated zone and the matrix potential of the soil water in the unsaturated zone, z is the vertical axis, positive upward.

In saturated and unsaturated soil water conditions, the left term of Eq.(1) can be further expressed as

where /hθ?? indicates the variation of the water content due to unit change of the soil water matrix potential, which can be calculated from the soil-water retention model (e.g., Van Genuchten (1980)), Ssis the specific storage, indicating the water released from the elastic change of the soil skeleton and the water,concentration.

The commonly used convection and dispersion equation is applied to describe the solute transport. For the two-dimensional vertical model, it is usually expressed as

where D is the hydrodynamic dispersion coefficient, in c is the total aqueous component concentration in the water coming from sources or sinks. Equat

ions (1) and (4) are coupled because the water flow causes the solute transport, especially through advection, and the solute concentration influences the fluid density and thus causes the density dependent flow. For a salt water, the relationship between the concentration and the density can be simply described by a linear empirical relationship[18],

where0ρ is the density of the freshwater, ε is the dimensionless difference between the reference saltwater and the freshwater,sc andsρ are the concentration and the density of the reference saltwater, respectively. In this study, the seawater is taken as the reference saltwater.

1.2 Initial and boundary conditions

The coastal aquifer with a sloping beach is modeled by the polygon ABCDEFGH in Fig.1. The groundwater is assumed to be in a stable state initially. Thus the initial water head and the initial fluid concentration of the fluid are described as

where Ω is the simulation domain (ABCDEFGH in Fig.1),0H is the initial water head,0C is the initial fluid concentration.

Fig.1 Sketch of coastal groundwater flow with sloping beach

For the flow boundary, we have tidal flutuations along the beach, seepages along the upper part of the landward bounday, and no flow through other boundaries. Thus the boundary conditions for the groundwater flow can be written as

wheretH is the tidal signal.

For the mass transport, along the above no-flow boundaries, the concentration gradient is taken to be zero according to the third type of boundary conditions. While for the other types of flow boundaries, either the concentrations (i.e., Dirichlet boundary) or the advective flux (simplified from the third type of boundary) are specified.

where g is the total mass flux (where the advective flux dominates) normal to the boundary,nq is the Darcy flux normal to the boundary.

2. Model validation

Wu et al.[19]carried out a laboratory experiment to investigate the tide-induced water table fluctuation and the over height of the groundwater table. The experiment was conducted by an automatic simulation system for tides, which contains a sand flume, a water storage tank and a control system.

The sand flume was 30 m long, 1.2 m wide and 1.5 m high, with the beach angle =α7o. Sixteen pressure sensors were used to measure the groundwater table change, at the bottom of the sand flume with the distance to the left end of the sand flume (beach) being 2.8 m, 3.4 m, 3.8 m, 4.2 m, 4.6 m, 5.1 m, 5.7 m, 6.3 m, 6.9 m, 7.4 m, 8.1 m, 8.6 m, 9.1 m, 10.3 m, 15 m and 25 m, respectively. The tide signal was the composition of two sine signals

where1A and2A are the amplitudes of the two constituents, with values 0.09 m and 0.045 m, respectively,1ω and2ω are the angular frequencies of 64π d–1and 48π d–1, respectively, δ is the phase lag, andsH is the mean water table of 0.765 m.

Table 1 Physical properties of the sand used in experiment of Wu et al.[19]and typical empirical parameters used in the numerical simulations

Fig.2 Comparison of the numerical simulation and experimental measurement of Wu et al.[19]. Points No. 9 to 11 are 7.4 m, 8.1 m and 8.6 m to the left side of the sand flume, respectively

As the freshwater was used for tiding in the experiment, the density difference is not considered in the simulation. However, to investigate the effect of the salt water density on the groundwater fluctuation, we consider two scenarios in the numerical simulation, scenario A with the freshwater as the tide and scenario B with the saltwater of 35 g/L NaCl as the tide.

For simulating the experiment of Wu et al.[19], the hydraulic conductivity K and the effective porosityen are determined according to the experiment measurement, the parameters of the soil-water retention model are based on the empirical estimation for typical sand (Simunek et al. 1999), the saltwater densitysρ, the freshwater density0ρ and the parameters for mass transport are based on values in literature (Ataie-Ashtiani et al. 1999). Values of these parameters used in this model are listed in Table 1.

The simulation results are compared with the experimental data, as shown in Fig.2. It is shown that the simulation results of scenario A (with freshwater as tide) compare reasonably well with the experiment ones. There are some differences for the peaks and troughs, especially at the points of the far inland and in the initial stage. It may be due to the fact that the initial condition and the hydraulic parameters used in the simulation cannot fully represent the real experiment condition. For scenario B with the salt water as the tide, the simulated water table fluctuation is similar with those of the experiment and scenario A, although with a little higher water level systematically.It is mainly because of the density effect, which induces the seawater intrusion and elevates the groundwater table. The comparison indicates that the numerical model designed for the software FEFLOW can properly simulate the tide induced water table fluctuations under inclined beach slopes.

Fig.3 Points in the simulation domain used to analyze the tidal effect on groundwater fluctuation

Fig.4 The simulation results of the tide induced groundwater table fluctuation and the variations of the fluctuation amplitude and phase

3. Investigation of beach slope effect on groundwater table fluctuation and seawater intrusion

To investigate the effects of the beach angle and the tidal fluctuation on the groundwater table and the seawater intrusion, further scenarios are designed in practical scales. The distance from Point C (see Fig.1) to the inner land is assumed to be 500 m, and the aquifer thickness is 45 m.

The types of initial and boundary conditions are similar to those of the previous model, with the main difference in their specified values. The tide is assumed to be the seawater of 35 g/L NaCl. The tidal table is

wheresH is the mean water table, with the value of 37 m, A is the amplitude, with the value of 1 m, ω is the angular frequency with the value of 4π d-1. The corresponding tidal period is 0.5 d or 12 h, which is close to the period of semidiurnal signals M2and S2. The initial groundwater head is 37 m, and the initial concentration of the groundwater is 0 g/l, the same as the landward freshwater.

To investigate the effects of the beach slope, six typical scenarios are designed, i.e., scenarios 1 to 6, with the beach angle (α) taking values of 15o, 30o, 45o, 60o, 75oand 90o, respectively (Fig.1).

The simulation is carried out until a periodically steady state is reached (the simulation time is over 365 d), i.e., the simulation results (e.g., the groundwater dynamics, the solute concentration in the aquifer) will not change for the same tide stage. Note that it is used as the initial point (0 d) for further simulations and analysis.

Simulation results at 12 points are used to analyze the influence of the tide and the beach slope on the groundwater fluctuations (Points 1 to 12 in Fig.3). These points are classified into two groups, group 1 (Points 1 to 6) for all 6 scenarios and group 2 (Points1 and 7 to 12) only for scenario 1. Points in group 1 locate outside the transition zone and are close to the groundwater table, The distances of Points 1-6 in group 1 to Point C are 40 m, 80 m, 120 m, 160 m, 200 m and 240 m, respectively (Fig.3). Points of group 2 locate in the transition zone (Fig.3). In this group, Points 10, 11, 7 and 12 are in the same horizontal level with a spacing of 20 m between them, while Points 1, 7, 8 and 9 in the same vertical line with a spacing of 10 m between them.

Fig.5 The tidal effects on the salt concentration distribution in the transition zonehH is the horizontal distance to Point 10,vH is the vertical distance to Point 1

3.1 Tide induced groundwater table fluctuation under different beach slopes

The simulation results of the tide induced groundwater table fluctuation and the variations of its amplitude and phase are shown in Fig.4. Obviously, the amplitude and the phase vary with the distance from the beach (see Fig.4(a)). With the increase of the distance from the beach, the amplitude of the groundwater fluctuation decreases (Fig.4(b)), the mean groundwater table increases (Fig.4(c)), with phase lags (Fig.4(d)). The results also demonstrate that the tide induced groundwater fluctuation has a skew shape in the inland area, i.e., the groundwater table rises faster and falls more slowly. This asymmetry is in accordance with the observation results and the numerical solution of Mao et al.[13].

As shown in Fig.4(b), the amplitude of the groundwater fluctuation decreases exponentially with the distance to the beach, which can be expressed as

where x represents the horizontal distance to the beach face, and1β and γ are regression coefficients. It is in accordance with the results obtained by Parlange et al. (1984), who derived the same relationship through the analytical solution for a two-dimensional flow problem without considering the density influence.

Our simulation further demonstrates the influence of the beach slope on the coefficients in Eq.(12). The results show that γ does not change with the beach angle, which can be explained by the fact that γ represents the inverse wave length (Parlange et al. 1984) and is independent of the beach slope. As for 1β, it increases with the beach angle and their rela-0.5751 with the squared correlation coefficient2=0.9955 R. This result indicates that a milder beach tends to have a greater resistance for the growth of the amplitude, i.e., a sloping beach has a more important influence on the amplitude of the water fluctuation than a vertical beach. Jeng et al.[10]also found that water fluctuations vary inversely with the beach slope, although they have not derived a particular form of the relationship between the amplitude and the beach angle.

As shown in Fig.4(c), the amplitude at the same point increases logarithmically with the beach angle, i.e.,

where2β and γ′ are regression coefficients related to the horizontal distance to the beach (x). The regre-ssion results indicate that2β and γ′ are related with the horizontal distance (x) in the form of a loga-ln()cxd+ with both squared correlation coefficients greater than 0.99.

Figure 4(d) shows that the phase lags linearly with the distance for beach angle =α15o. In fact, this linear lag is the case for all beach slopes, with the same slope value of -0.0087 and all with squared correlation coefficients greater than 0.98. This indicates that the phase lags regularly with the increase of the distance from the sea and is independent of the beach slope.

3.2 Tidal effects on groundwater dynamics of transition zone under different beach angles

3.2.1 Tidal effects on salt concentration distribution and intrusion length

From the simulation results, the contour of the salt concentration at 1 d for beach angle of 15ois shown in Fig.5(a). In both the horizontal direction and the vertical direction, the concentrations in the transition zone are distributed linearly (see Fig.5(b)). There is no distinct difference for the concentration contours at different tide stages. However, the salt concentration of a particular point in the transition zone (e.g. Point 7) vibrates with the tidal fluctuation (Fig.5(c)), although the amplitude is small.

From the contour of the salt concentration, the intrusion length with the tidal effect is defined as the length from the beach to the innermost point of the concentration contour of 0.15sc. The relative intrusion length λ is defined as the ratio of the intrusion length to the mean water table, i.e.,

whereIL is the intrusion length under the tidal effect. Moreover, the relative enhancement of the tide induced seawater intrusion (κ) is used to describe the influence of the tide on the seawater intrusion, which is defined as

wheresL is the intrusion length (for 0.15sc) without the tidal effect.

The realtionships between the relative intrusion length (λ), the relative enhancement of the tide induced seawater intrusion (κ) and the beach angle (α) are shown in Fig.6. It is indicated that both λ and κ decrease logarithmically with the beach angle α (Fig.6), and one would have more intrusion range and more enhancement of the tide on the intrusion for a milder beach. Ataie-Ashtiani et al. (1999) also found that the influence of the tide on the intrusion length is much greater at a sloping beach than a vertical one.

Fig.6 Realtionships between relative intrusion length λ, relative enhancement of tide induced seawater intrusion (κ) and beach angle (α)

3.2.2 Tidal effects on the velocity field for a typical sloping beach (=α15o)

Figure 7 shows the simulation results of the tide induced groundwater velocity in x and z directions. It is shown that the velocity fluctuates similar to the tidal signal (sine function), as a response to the tidal fluctuation. The horizontal velocity,xV, does not attenuate in the vertical direction (Fig.7(a)), but it attenuates exponentially with the horizontal distance to the beach (Fig.7(b)), similar to the attenuation of the groundwater table (Eq.(12)). The fitting results show that the attenuation coefficients (γ) for the groundwater table andxV in the horizontal direction are the same, which indicates they are both caused by the tide fluctuation because γ represents the attribute of the tidal fluctuation. Moreover, there is an obvious phase lag forxV in the horizontal direction (Fig.7(b)), because it takes time for the tidal influence to spread inland.

The vertical velocity, Vz, is much smaller than Vx, and there is no distinct phase lag in both horizontal and vertical directions, mainly because the wave spreads along the horizontal direction, perpendicular to the vertical direction. As shown in Fig.7(c), at the higher points in the transition zone one usually hasa larger upward velocity, because the groundwater above the transition zone tends to discharge upward to the sea.zV near the center of the transition zone (e.g., Points 7 and 8) has a larger fluctuation, which indicates the significant influence of the tidal fluctuation on the transition zone. Figure 7(d) shows that the amplitude ofzV decreases quickly in the direction towards the inland area similar toxV, because the tidal influence is weakened towards the inland direction.

Fig.7 Periodic change of velocity at Points 1, 7 to 12

From Fig.7, it is seen that the flow velocity at the top of the aquifer, especially, near the discharge face (e.g., Point 12), is much greater than that at the bottom. Ataie-Ashtiani et al. (1999) and Mao et al.[13]also investigated the velocity of the seawater intrusion through their numerical models, but those studies mainly focused on the overall velocity field at a particular tidal stage. Here by analyzing the simulation data at different locations, the influence of the tide and the beach angle on the groundwater dynamics is revealed quantitatively.

4. Conclusions

To investigate the tide induced groundwater dynamics under different beach slopes, a numerical model is built with the software FEFLOW. The model is firstly examined by comparing with the laboratory experiment of Wu et al.[19], who monitored the water table fluctuation induced by the tide on inclined beaches. It is shown that the numerical model can describe the tide induced water table fluctuations under an inclined beach, and it is also indicated that the water table is elevated when considering the salt water intrusion.

More numerical scenarios are designed with different beach slopes in the practical scale. The results have not only confirmed the previously established relationships between the tide induced groundwater table fluctuation and the distance to the sea, but have also further revealed the influence of the beach slope on the relationship coefficients quatitatively. The tidal effects on the seawater intrusion and the velocity field for cases with sloping beaches are also quantified.

Despite the fact that some quantitative relationships among beach slope, tide signal, seawater intrusion and groundwater dynamics are obtained, the numerical scenarios we have designed are based on a specific case, with simplified tidal signal, constant beach slope and homogenous aquifer. To understand the influence of the beach slope thoroughly, more complex cases should be studied in future researches and the results need to be verified under the real circumstances.

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10.1016/S1001-6058(11)60223-0

* Project supported by the Program for New Century Excellent Talents in Universities (Grant No. 07-0814).

Biography: LIU Yi (1987- ), Male, Master Candidate

MAO Xiao-min,

E-mail: maoxiaomin@cau.edu.cn

2012,24(1):97-106