Water Resources Research

Mixed convection and density-dependent seawater circulation in coastal aquifers

Authors


Abstract

[1] Density-dependent circulation of seawater in coastal aquifers results in submarine groundwater discharge (SGD) across the seabed that is a mixture of terrestrial groundwater and former marine water. In this study, the controls of the relative amount of seawater to freshwater in SGD were investigated numerically using the FEFLOW and SUTRA codes. It was found that the key controls could be expressed in the form of a single nondimensional recirculation number that incorporates the combined effects of free convection, forced convection, and hydrodynamic dispersion on convective overturn within the coastal salt wedge. Anisotropy effects were incorporated into the recirculation number with limited success based on the principle of equivalent isotropic hydraulic conductivity. The type of boundary condition employed along the seabed was shown to be important. Convective overturn was substantially increased if backward dispersion of salt into the aquifer from along the outflowing portion of the seabed boundary was prevented. Overall, the results demonstrated a strong dependence of convective overturn on the aquifer dispersivities, suggesting that results from numerical simulations are problematic to apply to real aquifer systems that typically exhibit uncertain, scale-dependent dispersion properties.

1. Introduction

[2] As water flows from land to sea it transports large amounts of dissolved and suspended materials. River and stream discharges at the coast are conspicuous but groundwater in coastal aquifers also drains into the marine environment across the seabed, mostly unnoticed. This contribution of submarine groundwater discharge (SGD) to nearshore marine processes is thought to be significant but SGD must often be neglected in marine studies because it is poorly described and quantified in many coastal systems. Material fluxes across the sediment-water interface play important roles in biological productivity and geological mineralization processes [Simmons, 1992; Uchiyama et al., 2000; Burnett et al., 2001] and there is a clear need to improve the current state of knowledge by reexamining the topics of saltwater intrusion and SGD from the perspective of nearshore coastal water management.

[3] Defined broadly, SGD is any upward flux of water from seabed sediment into the overlying marine water column. This includes discharges of terrestrial groundwater derived from rainfall as well as discharge of former marine water that previously flowed into sediments. The component of SGD that is induced by hydraulic head differences between sediment pore water and the marine water column is called forced convection since the fluid flow is said to be forced by an external agent that is unrelated to the fluid properties [Nield and Bejan, 1992]. Sea level is dynamic and the hydraulic gradient between the sea and pore waters in coastal sediments fluctuates continuously in response to waves, tides, variation in barometric pressure and longer-term sea level change. Simultaneous variation of inshore groundwater levels results in exchanges of seawater back and forth between marine and submarine environments [Li et al., 1999] over timescales ranging from seconds to many years.

[4] Mixed convection, caused by natural intrusion of seawater along coastal and estuarine margins of groundwater systems, also gives rise to cycling of marine water between the sea and coastal sediments. The term mixed convection is used because the fluid motion involves a combination of forced convection and free or natural convection caused by fluid density variation. A coastal salt wedge is in fact a density-driven convection cell in which saltwater circulates from the sea into the aquifer and back out. The distance inland that seawater intrudes depends on several key factors including the aquifer hydraulic properties, gradient of groundwater pressure and the density difference between groundwater and seawater [Glover, 1959]. Aspects of aquifer geometry, such as shoreline shape, saturated thickness and bottom topography also affect the shape and position of the convection cell and rate that seawater circulates within it.

[5] Estimating relative contributions of terrestrial groundwater and former marine water to measurements of SGD is difficult. Nevertheless, knowing this information is important for developing material budgets, particularly if terrestrial groundwater and marine water have distinct chemistries [Simmons, 1992]. Benthic flux or seepage meters are commonly placed onto the seabed to make direct point measurements of SGD. These devices integrate positive and negative fluxes occurring over dissimilar timescales (e.g., tide and wave periods) and provide values of SGD that represent the net average flux due to forced and free convection. The water quality of the discharge into a seepage meter also must be monitored to distinguish the relative contributions of terrestrial groundwater and marine water. Environmental tracer methods for estimating SGD [e.g., Moore and Moore, 1996; Cable et al., 1996; Hussain et al., 1999] detect the discharge of groundwater that has been in contact with the sediment geological matrix for a sufficiently long period of time to become enriched with naturally occurring geochemical tracers; e.g., radon and radium. Short timescale exchanges of marine water with the aquifer might not be completely detectable by these methods if the residence times for marine water in the aquifer are less that the time required to reach geochemical equilibrium. Model-based estimates of SGD also vary dependent on the physical processes that are incorporated into calculations. An estimation that considers only the terrestrial groundwater balance, for example, will yield different results to one that incorporates seawater circulation, or tidal exchanges, or both. Thus it is important to recognize that care must be exercised when comparing estimates and measurements of SGD obtained by dissimilar methods, since these methods may target different SGD processes in both time and space.

2. This Study

[6] This paper presents the results of numerical experiments to quantify the relative amount of seawater to freshwater in SGD due to mixed convection in coastal aquifers. The numerical code FEFLOW [Diersch, 2002] was used to conduct the simulations and selected results were cross-checked for consistency against results from SUTRA [Voss, 1984]. Key controls of mixed convection were firstly identified and then combined into a single nondimensional recirculation number that characterizes the relative rate of convective overturn. Effects of free convection, forced convection, hydrodynamic dispersion and anisotropy are investigated for two alternate choices of the seabed boundary condition. Forced convection of seawater due to tidal fluctuations, waves and wind setup are beyond the scope of this investigation.

3. Previous Studies

[7] Many coastal aquifers around the world are utilized to supply coastal communities with potable water. Overpumping and exacerbated seawater intrusion has motivated numerous groundwater investigations. A significant number of journal papers on seawater intrusion have been published, including the historical introduction by Reilly and Goodman [1985]. Most groundwater hydrology textbooks also treat this topic [e.g., Bear, 1972; Bouwer, 1978; Freeze and Cherry, 1979; Bear and Verruijt, 1987]. Mathematical models for saltwater intrusion in coastal aquifers were reviewed by Bobba [1993].

[8] It is a relevant observation that previous work on seawater intrusion has focused almost exclusively on describing and predicting saltwater distribution in aquifers, such as the position, shape and thickness of the saltwater-freshwater transition zone. Rates of convective seawater circulation have rarely been reported and it is unknown whether they have been routinely calculated. Convective circulation rates have not been used to compare and benchmark the results from different numerical calculations. Results of density-coupled groundwater flow simulations are typically presented as isochlors [e.g., Lee and Cheng, 1974; Volker and Rushton, 1982; Sherif et al., 1990a; Croucher and O'Sullivan, 1995; Kolditz et al., 1998; Carabin and Dassargues, 1999; Simpson and Clement, 2004] or vector plots of the groundwater velocity field [e.g., Mehnert and Jennings, 1985; Huyakorn et al., 1987; Diersch, 1988; Duffy and Al-Hassan, 1988; Simmons and Narayan, 1997; Sherif et al., 1990b], or both. However, seawater circulation rates cannot be determined from these types of information. Neither do abrupt-interface solutions [e.g., Bobba, 1993] yield estimates of density-driven seawater circulation.

[9] Circulation rates in steady state mixed convection problems can be estimated from a streamline plot of the results but this information is uncommonly presented. The percent of total SGD that is former marine water can be roughly determined by comparing the number of streamlines that circulate from the sea into the aquifer and back out to the number of other streamlines that discharge but do not originate at the sea. Croucher and O'Sullivan [1995, Figure 7] presented one figure that depicts the steady state stream function distribution for the well-known Henry's problem. Using the above method, seawater contributes approximately 17% of total SGD at the coastal boundary but it is not straightforward to compare this value to results in the present study. First, the saltwater boundary in Henry's Problem is specified differently and is more prescriptive. Second, the Croucher and O'Sullivan result was obtained using a constant scalar dispersion coefficient, whereas results in this study use a simplified velocity-dependent dispersion tensor as implemented in FEFLOW [Diersch, 2002] and SUTRA [Voss, 1984].

[10] Benson et al. [1998, Figure 4] also presented streamline plots for Henry's problem, with constant scalar dispersion coefficient, and for a recast version of Henry's problem incorporating a velocity-dependent dispersion coefficient. The plotted results were obtained using MOCDENSE [Sanford and Konikow, 1985] and a constant dispersion coefficient of 1.86 × 10−5 m2 s−1. The recast problem employed a longitudinal dispersivity of 0.05 m and a molecular diffusion coefficient of 1.6 × 10−9 m2 s−1. All other model parameters were unchanged. In both examples, seawater contributed around 21% of total SGD.

[11] Kohout's [1960] “rough calculation of the seaward movement of salt” in the Biscayne Aquifer in Florida indicated that approximately 10% or more of total SGD was former seawater. Saltwater discharge was estimated as the outflow at the shoreline through a vertical cross section of aquifer and based on field measurements of groundwater salinity. Lee and Cheng [1974] used Kohout's field measurements to calibrate a density-coupled flow and mass transport model of the Biscayne aquifer. While saltwater flows were not reported directly, the streamline pattern in Lee and Cheng's Figure 4 shows that simulated seawater circulation contributed approximately 43% of total simulated SGD for the particular choice of model parameters.

[12] More recently, Smith and Turner [2001] used FEFLOW [Diersch, 2002] to study density-dependent saltwater circulation rates in shallow groundwater adjacent to an estuarine river. Dependent on the hydraulic conductivity and rate of fresh groundwater drainage toward the river from either side, between 0% and 35% of total groundwater discharge to the river was made up of former estuary water. In some circumstances, groundwater discharge to the river from opposing directions along the adjacent riverbanks was sufficiently large to counter buoyancy forces and prevent all saltwater from entering the aquifer. Thus, for these examples, the interaction between groundwater and the river was convectively stable with zero saltwater circulation. The same situation is not possible at a coastal boundary because groundwater discharges from the inshore direction only and cannot everywhere displace seawater beneath the seabed; there will always be a submarine mixing zone at the coast, be it inshore, offshore or stretching across both zones. Other factors that might influence the circulation rate of saltwater in the aquifer, e.g., different choices of the dispersion parameters, were not considered in this study.

[13] Prieto [2001] recently presented results of saltwater intrusion models developed using SUTRA. In the Israel Case Study, the percent of SGD that was former marine water varied between 51 and 62% for the five “temporally constant natural recharge and management scenarios” considered. These scenarios examined the effects on seawater intrusion of natural recharge, artificial recharge and pumping. A notable difference to the present study was the inclusion of groundwater pumping and freshwater injection within the inshore aquifer.

4. Conceptual Model of Density-Dependent Seawater Circulation

[14] Two simultaneous processes give rise to density-dependent seawater circulation in coastal aquifers [Bear, 1972; Reilly and Goodman, 1985; Bobba, 1993]: (1) displacement of relatively fresher groundwater by denser marine water and (2) hydrodynamic dispersion of salt within the saltwater-freshwater mixing zone. The process of seawater intrusion by itself does not cause seawater to circulate in the aquifer. Abrupt interface approximations of seawater intrusion are based on this principle; saltwater within the intruding salt wedge is stagnant and terrestrial groundwater discharges to the sea over the top of the salt wedge. The position and shape of the salt wedge is maintained by the density difference but there is no circulation of seawater.

[15] In reality, the saltwater-freshwater interface is a mixing zone in which salt disperses generally in the direction opposed to the salinity gradient. Former seawater that has entered the aquifer and contacted fresher groundwater becomes diluted and less dense relative to seawater at the coastal boundary. The resulting fluid-density contrast within the aquifer gives rise to buoyancy forces that drive convective overturn. It follows that the rate of convective overturn will depend fundamentally on the dispersive properties of the groundwater flow system and it is expected that saltwater circulation rates calculated using numerical models will be sensitive to the values of the dispersion parameters used, as well as the upward fresh water pressure gradient.

[16] Numerical simulation results in this study are based on the advection-dispersion paradigm and were all obtained using velocity-dependent dispersion coefficients. This required the input of longitudinal and transverse dispersivity values that were used to dynamically calculate the dispersion coefficients during model simulations. While this approach is common in groundwater modeling it is also awkward because the dispersivity parameters are only partial intrinsic properties of the aquifer [Gelhar et al., 1992]. Instead, they are known to depend on a number of factors that include the space and timescales for flow and transport.

5. Numerical Model

[17] Figure 1 depicts the geometry and boundary conditions for the steady flow and density-coupled mass transport problem described in this paper. It represents a vertical section through a coastal aquifer with total length L = LS + LL and saturated thickness B, with the slope of the seabed neglected. The dimension LS is the horizontal length of aquifer beneath the seabed and LL is the inshore length of aquifer. The x-z coordinate origin is located on the top boundary at the position of the seashore.

Figure 1.

Model geometry and boundary conditions.

5.1. Inshore Boundary Condition

[18] Uniform, constant flux was specified along the right vertical model boundary to represent terrestrial groundwater drainage toward the sea

equation image
equation image

where Cf is the salt concentration of groundwater, xR is the x coordinate of the right vertical model boundary, Q is volumetric Darcy flux [L3 T−1 L−1] and qf is uniform, specific discharge per unit length of boundary [L T−1]. The value of xR varied dependent on the inshore extent of seawater intrusion, which was problem dependent. The number of model cells was minimized by keeping LL as small as possible but not so small that the salt wedge geometry was significantly influenced by the boundary.

5.2. Upper and Lower Aquifer Boundaries

[19] The top and bottom of the flow domain were both no-flow boundaries, implying that the aquifer basement was impermeable and that the influence of diffuse groundwater recharge across the upper aquifer boundary on seawater intrusion was negligible. In many practical situations, diffuse groundwater recharge will be small compared to total inflow, Qf = qfB, across the right vertical boundary. Irrespectively, groundwater flow toward the sea in a section of aquifer that is many times longer than it is deep is predominantly horizontal, regardless of whether groundwater enters the model domain across the top boundary or side boundary. Ignoring local recharge in the modeling had the advantage of simplifying the water balance so that the single variable Qf described terrestrial groundwater drainage. The validity of this assumption is examined in more detail in the results.

5.3. Seabed Boundary Conditions

[20] Constant hydraulic head was specified along the seabed portion of the top boundary and along the left vertical boundary

equation image
equation image

where ρS is the density of seawater [M L−3], ρf is the density of terrestrial groundwater and hf is hydraulic head [L] expressed at reference fluid density ρf. Sea level was assumed static and the seabed horizontal. Two alternate forms of mass boundary condition were used in order to compare their effects on the simulated results. Both types of boundary condition are in common use. For convenience, they are referred to hereafter as type A and type B boundaries. Similarly, equivalent simulations with type A and type B boundaries are referred to, respectively, as type A and type B simulations.

5.3.1. Type A Boundary

[21] Constant mass concentration was assigned along the entire saltwater portion of the top boundary and left vertical boundary

equation image
equation image

where C is the aquifer salt concentration [M L−3] and CS is the salt concentration of seawater. Thus equations (3), (4), (5), and (6) specify a hydrostatic seawater boundary in terms of equivalent hydraulic head. The original Henry's Problem was defined with type A boundary [e.g., Croucher and O'Sullivan, 1995] and other numerical studies of saltwater intrusion have also used type A boundaries [e.g., Duffy and Al-Hassan, 1988; Diersch, 1988; Fan et al., 1997; Smith and Turner, 2001; Prieto, 2001]. Backward dispersion of salt along the outflowing (discharging) portion of the boundary is a perceived deficiency and occurs because the dispersive mass flux is always in the direction opposed to the concentration gradient, independent of the flow direction.

[22] The issue of how realistically a type A boundary represents particular field conditions is unstudied, though it is clearly an inappropriate choice of boundary condition if it results in unrealistic accumulation of salt in the aquifer beneath the seabed discharge zone. Although subseabed mixing of groundwater and marine water is known to occur, for example, due to tidal fluctuations, waves, wind setup and convective fingering, it is improbable that accumulation of salt in discharge zone due to these processes would be well mimicked by backward dispersion in numerical models, except possibly in a few rare cases.

5.3.2. Type B Boundary

[23] The alternative approach was to set the dispersive mass flux to zero along the outflowing portion of the boundary, such thatInflowing

equation image

Outflowing

equation image
equation image

where z is defined positive upward. Backward dispersion of salt from the boundary is minimized but there is complication in applying these conditions because the inflowing and outflowing portions on the seabed are a priori unknown. Indeed, this is part of the solution being sought. One approach used to overcome this nonlinearity has been to assume an arbitrary discharge width or ‘window’ on the saltwater boundary where equation (6) applies [e.g., Mehnert and Jennings, 1985; Sherif et al., 1990a, 1990b]. Nevertheless, by fixing this dimension the solution is constrained by not allowing the salt wedge to develop ‘natural’ shape and position. The alternative is to employ a nonlinear coupling that provides dynamic assignment of the seabed boundary condition during a simulation based on the flow direction across the seabed. This approach appears to have been used by several authors [e.g., Volker and Rushton, 1982; Ghassemi et al., 1990; Carabin and Dassargues, 1999] but the numerical implementation for coupling the mass boundary condition with the flow solution was not reported.

[24] In this study, a boundary constraint condition available in the FEFLOW simulator was used to achieve equations (5) and (6). A solution with type B boundary was obtained by advancing a transient simulation to a quasi-steady state using initial conditions from the solution of an identical problem with type A boundary. At each time step, boundary nodes on the seabed were switched dynamically from condition (5) to condition (6) if the hydraulic gradient on the boundary indicated groundwater outflow.

5.4. Approximation of Phreatic Surface

[25] It is worthwhile to note that the coastal boundary of the model represents the contact between a shallow unconfined aquifer and the sea. A horizontal line (z = 0, 0 ≤ xxR) was used to approximate the true geometry of the phreatic surface landward of the coastline. Strictly speaking, this approximation of steady unconfined flow is only reasonable if, within the region of the salt wedge, the maximum elevation of the true water table above z = 0 is relatively small compared to B. Further inshore, the slope and elevation along the free surface are not important because they cannot affect the salt wedge geometry. Because qf was always positive in this study, the true water table elevation was always slightly above the level of the upper aquifer boundary and this meant that the true saturated thickness was slightly underestimated by the model. On the basis of the Gyben-Herzberg principle, the minimum elevation of the water table above sea level required to hold back saltwater intrusion is approximately B/40 (i.e., 0.025B). Thus the true saturated thickness of aquifer within the region of the salt wedge cannot deviate from the assumed constant thickness imposed by the horizontal top boundary by more than approximately 2.5%. In fact, this difference was smaller in this study because the inshore extent of saltwater intrusion in miscible fluid flow models is significantly less than that predicted by abrupt interface approximations. On this principle, it was concluded that approximating the true water table geometry by a horizontal line was a valid approach and did not significantly affect the simulation results.

5.5. Density Difference

[26] The density difference ratio equation image = (ρS − ρf)/ρf relates fluid salt concentration, C, to fluid density, ρ, by the expression ρ = ρ(Cf)[1 + equation image(CCf)/(CSCf)]. It was assigned constant value in this study because groundwater and seawater were assumed to have fixed salt concentrations and densities. For convenience, equation image was grouped with the two concentration terms to form a single nondimensional ratio β = equation image(CSCf)/CS that described the characteristic density contrast between groundwater and seawater. This ratio affects the buoyancy forces that drive free convection within the salt wedge and influences the slope and length of the saltwater-freshwater transition zone.

6. Dimensional Analysis

[27] The mixed convection problem described above involves 11 independent variables {LL, LS, B, Kx, Kz, qf, ne, αL, αT, Dm, β}, which are expressed in three unit dimensions: length [L], time [T], and mass [M]. Variables not defined earlier are the vertical and horizontal components of hydraulic conductivity Kz and Kx [L T−1], aquifer effective porosity ne (nondimensional), longitudinal and transverse components of dispersivity αL and αT [L] and the coefficient of molecular diffusion Dm [L2 T−1]. From Buckingham's Pi Theorem [Buckingham, 1914] the system behavior is characterized by eight (11 minus 3) nondimensional groups. Choices for these in this study are listed and described in Table 1.

Table 1. Nondimensional Groups
RatioPhysical Meaning10-m Aquifer20-m Aquifer50-m Aquifer100-m Aquifer
LL/B, LS/Baspect ratiosvariablevariablevariablevariable
neporosity0.250.250.250.25
βfluid density ratio0.02250.02250.02250.0225
αLTdispersivity anisotropy10101010
V*ratio of free convection velocity to forced convection velocity0.23–1122.25–22.52.25–84.42.25–112
D*dispersion ratio2–404–8010–10020–200
Kx/Kzconductivity anisotropy1–1001–101–101–10

6.1. Geometric Ratios

[28] The two aspect ratios LL/B and LS/B uniquely define the model geometry. Their values were chosen so that the positions of the left vertical and right vertical model boundaries did not significantly influence the shape and behavior of the salt wedge convection cell. When this requirement was satisfied, the specific values of these ratios were redundant and the main objective was to set values that minimized the size of the model domain to reduce the number of elements and computational effort required to solve the problem.

[29] The left vertical model boundary was positioned to avoid convection across the boundary. Density-driven convection was induced if the boundary was positioned too close to the salt wedge convection cell where the vertical gradients of concentration were significant. The relative position of the left vertical boundary was varied in the range 4 ≤ LS/B ≤ 40. A value of 4 was adequate for isotropic aquifers but for anisotropic aquifers LS was increased according to LS = 4B(Kx/Kz)1/2 such that, if Kx/Kz > 1, the model domain was extended further offshore to ensure there was sufficient length of seabed to allow predominantly vertical intrusion of saltwater across the seabed boundary. Conceptually, this provided the correct mechanism for seawater intrusion. The solution streamlines were examined to confirm that saltwater inflow across the left vertical boundary was negligible.

[30] Placement of the right vertical model boundary was determined experimentally and varied in the range 5 ≤ LL/B ≤ 24 dependent on the inshore extent of seawater intrusion. The boundary was placed far enough away from the toe of the salt wedge so that the condition of forced horizontal flow across the boundary did not limit the extent of saltwater intrusion.

6.2. Free and Forced Convection

[31] The nondimensional group V* = Kzβ/qf is the ratio of the characteristic free convection velocity vβ = Kzβ/ne to the characteristic forced convection velocity vf = qf/ne. It is a logical choice in this study because it provides a measure of the potential for saltwater flow relative to the prescribed freshwater flow. This group is related to the nondimensional discharge parameter “a” used in Henry's problem [e.g., Croucher and O'Sullivan, 1995; Lee and Cheng, 1974]. For isotropic conditions, V* also provides a measure of the slope and length of the saltwater-freshwater transition zone. The ratio qf/K is a characteristic slope along the aquifer free surface and represents the ratio of forced convective flow toward the sea, Qf, to aquifer transmissivity, KB. In turn, the slope along the aquifer free surface and β control the slope and elevation of the saltwater-freshwater transition zone.

6.3. Dispersion and Porosity

[32] The dispersion parameter D* = BT is the ratio of the characteristic aquifer length scale to the characteristic dispersion length scale. It is related to the seepage Peclet number Pe = qfB/D described by Lee and Cheng [1974] for Henry's problem; where D is a constant dispersion coefficient. Small values of D* indicate that the saltwater-freshwater transition zone will be relatively dispersed compared to aquifer thickness, with the expectation that convective overturn will be reduced due to weakening of density gradients. On the other hand, at large values of D* convective overturn also can be limited by reduced salt dispersion across the transition zone. This suggests an intermediate value of D* at which the saltwater circulation rate might reach a maximum value. While solutions for D* < 40 with type A boundary were included in this paper they are considered theoretical and relatively unrealistic. In such cases, αL was greater than 0.25B and there was excessive dispersion of salt from the sea into the aquifer, which was accompanied by excessive freshening of groundwater beneath the seabed.

[33] The value of the dispersivity anisotropy ratio was assumed constant at αLT = 10. In contrast, Gelhar et al. [1992] showed that field-scale estimates of dispersivities were typically in the range 5 ≤ αLV ≤ 500, where αV, the transverse vertical dispersivity, was normally less than 1 meter and sometimes as small as a few millimeters. The value 10 used in this study falls at the lower end of this scale; however, it was problematic to use larger values due to numerical constraints that required small mesh elements when αT was small (see the discussion on grid Peclet numbers below). For very small values of αT, the large number of elements and long simulation times made multiple simulations impracticable to carry out. Furthermore, it was inappropriate to use values of αL that were unrealistically large compared to B or which caused unrealistic dispersion. The effect on convective overturn of varying αLT is examined in the results.

[34] Effective porosity influences the pore water volume and pore water velocity, as well as both the advection and dispersion terms in the solute mass transport equation. The hydrodynamic dispersion tensor employed by FEFLOW [Diersch, 2002, equations (3)–(5)] incorporates molecular diffusion and mechanical dispersion; however, only the molecular diffusion term is dependent on porosity. The mechanical dispersion term of the dispersion tensor is a function of the absolute Darcy flux, rather than the pore water velocity, and therefore accounts for tortuosity effects only. Because mechanical dispersion was assumed large compared to diffusion, the effect of porosity on diffusion and dispersion within the transition zone was neglected in this study.

7. Simulations

[35] Numerical results were generated for the range of nondimensional values listed in Table 1 and using aquifer thicknesses B = 10, 20, 50 and 100 m. Details of the specific parameter values used are provided in the results. Because the values of ne and β were assumed constant (Tables 1 and 2) their effects were also constant and could be neglected. Similarly, the specific values of LL/B and LS/B did not influence the results and could be ignored as long as the requirements discussed in Section 6.1 were observed.

Table 2. Model Parameters With Assumed Constant Values
ParameterValueUnitsPhysical Meaning
Cf0Mg L−1salt concentration of terrestrial groundwater
CS35,000Mg L−1salt concentration of seawater
equation image0.0225 density difference ratio
Dm1.3 × 10−4m2 d−1molecular diffusion coefficient

[36] Most simulations were conducted for 10-m thick aquifers because solutions were quicker to obtain due to the smaller domain sizes and fewer finite elements. Simulations using thicker aquifers were carried out to check whether consistent results were obtained for larger-scale problems and to achieve larger values of D*. Results were generated firstly using type A boundaries and then selected simulations were rerun using type B boundaries. Additional simulations were performed to validate the modeling assumptions and explore the effects of numerical dispersion. All simulations are discussed in the results.

8. Numerical Solution Scheme

[37] Diersch [2002] presents full details of the governing equations and solution methods implemented in the FEFLOW simulator. The accuracy achieved in numerical simulations depends on the specified numerical scheme adopted, including spatial discretization of the flow domain, time stepping procedure and solver options. Guidelines for selecting appropriate numerical schemes for particular classes of problems are considered by Diersch and Kolditz [1996, 1998] and Kolditz et al. [1998]. The solution strategy adopted in this study is listed in Table 3 and discussed below.

Table 3. FEFLOW Solution Strategy
TopicStrategy
Mesh designfour-nodal quadrilateral elements
Finite element schemeGalerkin-FEM, no upwinding
Density couplingBoussinesq approximation
Mass transport equationconvergent form
Time stepping schemeautomatic predictor-corrector method (forward Adams-Bashforth/backward trapezoid rule) to advance simulations to quasi-steady state
Matrix solverproblem dependent

[38] Four-nodal, quadrilateral finite element meshes were constructed with horizontal and vertical grid Peclet numbers in ranges 0.2 ≤ Px ≤ 2 and 0.1 ≤ Pz ≤ 2, respectively. The grid Peclet numbers were evaluated as Px = ΔxT and Pz = ΔzT, where Δx and Δz are the element width and height. The number of elements comprising individual finite element meshes varied between 4000 and 800,000 dependent on the dimensions of the model domain and mesh discretization. Meshes with finer and coarser spatial discretizations were tested to establish whether the above element sizes were adequate to provide numerically converged solutions. Solutions were found to be approximately converged, with no upwinding, if Px ≤ 2 and Pz ≤ 2. The Oberbeck-Boussinesq approximation, which ignores all fluid density dependencies except for the buoyancy term in the momentum (Darcy) equation [Nield and Bejan, 1992; Diersch, 1998; Carabin and Dassargues, 1999] was employed in all FEFLOW simulations.

[39] If an iterative solution for steady flow and mass transport could not be obtained because the solution did not converge, then a steady result was achieved by advancing a transient model run to a quasi-steady state. Most of the results presented in this paper were obtained by this method. Mass concentrations at selected locations in the model domain and the model fluid balance components were monitored during simulations to establish when a quasi-steady state was reached. Run times for transient simulations varied markedly from minutes to many hours dependent upon the initial conditions, number of mesh elements and choice of parameter values. Initial conditions were produced from either a nonconverged steady state solution or a previous simulation obtained for similar choices of the parameter values.

9. Results and Discussion

[40] To provide a simple measure for comparing the results, saltwater circulation rates were calculated as

equation image

where the acronym PSC stands for Percent Saltwater Circulation, Qf is the terrestrial groundwater component of SGD [L2 T−1] and QS is the former seawater component of SGD [L2 T−1]. Simulated PSC was greater than 100% if QS was greater than Qf. Because all results were steady state, QS was obtained from the fluid mass balance as the total inflow across the seabed boundary.

9.1. Isotropic Aquifers

[41] Plotted results showing systematic relationships between V*, D* and PSC for isotropic aquifers with type A and type B boundaries are depicted in Figures 2 and 3, respectively. Specific values of K and qf that were used to vary the value of V* are indicated in the matrix in Figures 2 and 3. Hydraulic conductivity was varied by a factor 500 (1 ≤ K ≤ 500 m d−1) and terrestrial groundwater discharge by a factor 50 (0.01 ≤ qf ≤ 0.5 m d−1). Values of transverse dispersivity in the range 0.25 ≤ αT ≤ 5 m were used to achieve values of D* = 2, 5, 10, 20, 40, 50, 100 and 200; noting that αLT = 10 for all examples. The fitted lines in Figures 2 and 3 are simple exponential associations.

Figure 2.

Effects of free convection velocity, force convection velocity, and dispersion on PSC for type A simulations and αLT = 10; specific values of K and qf for all plotted results are indicated in the matrix. In the top plot, D* ≤ 40 and PSC decreases as D* decreases due to weakening of density gradients and break down of convective overturn. In the bottom plot, D* ≥ 40 and PSC decreases as D* increases due to decreased salt dispersion across the saltwater-freshwater transition zone.

Figure 3.

Effects of free convection velocity, force convection velocity, and dispersion on PSC for type B simulations and αLT = 10; specific values of K and qf for all plotted results are indicated in the matrix. PSC decreases as D* increases due to restricted salt dispersion across the saltwater-freshwater transition zone.

9.1.1. Effect of Seabed Boundary Condition

[42] Comparison of Figures 2 and 3 reveals that significantly increased rates of saltwater circulation were obtained for type B simulations compared to equivalent type A simulations. For example, compare the respective curves for D* = 40; simulated PSC using type B boundaries was approximately 1.4–1.6 of PSC using type A boundaries. The explanation for this difference is related to backward dispersion of salt along the outflowing portion of type A boundaries and associated accumulation of salt within the aquifer beneath the seabed. This effect is illustrated in Figure 4, which depicts white streamlines superimposed on shaded concentration fringes. In example 1 with type A boundary, the streamlines exhibit a closed circulation at the center of the convection cell that represents a component of the density-driven saltwater flow that does not contribute to groundwater discharge across the seabed. Thus QS and PSC are both less compared to example 2 with type B boundary. More generally, the accumulation of salt and increased fluid density within the aquifer beneath type A boundaries acts to slow the rate of convective overturn by weakening the density contrast that drives the salt wedge convection cell.

Figure 4.

Example of mixed convection patterns beneath the seabed for equivalent problems with (a) type A boundary and (b) type B boundary; B = 50 m, Kx,z = 200 m d−1, qf = 0.2 m d−1, αL = 10 m, and αT = 1 m. White streamlines (selected values) are superimposed on shaded concentration fringes. For type A boundary the convection cells exhibits a closed circulation at its center that does not contribute to SGD across the seabed.

9.1.2. Effects of Free Convection and Forced Convection

[43] Figures 2 and 3 demonstrate a direct relationship between V* and PSC. This result is sensible and reflects the fact that both ratios are measures of the relative magnitudes of the saltwater (free convection) and freshwater (forced convection) flows. Increasing the free convection velocity (vβ) by increasing the hydraulic conductivity produces an increase in PSC because resistance to convective overturn within the aquifer is reduced. Conversely, reducing the hydraulic conductivity decreases convective overturn and PSC. Variation of the forced convection velocity (vf) has a comparatively smaller effect on convective overturn but directly varies PSC by changing the value of Qf. It follows that the highest values of PSC were obtained for large values of V*, when either vf was small or vβ large, or both. This result confirms logically that coastal aquifers with large hydraulic conductivities and relatively small freshwater discharge rates will have large PSC, while less conductive aquifers that transmit larger volumes of fresh groundwater will have comparably lower PSC.

9.1.3. Effect of Hydrodynamic Dispersion

[44] Two theoretical limits on the rate of convective overturn within coastal aquifers are useful to consider. Firstly, if there is no salt transport by hydrodynamic dispersion then free convection cannot occur and QS must equal zero—as it does in an abrupt interface approximation of seawater intrusion. In a miscible fluid flow model, as used in this study, this limit cannot be reached because there is always some dispersion; hydrodynamic dispersion and, in the modeling exercise, unavoidable numerical dispersion. Secondly, if dispersion is too large, then fluid density gradients and buoyancy forces become too weak to drive convective overturn. In this case, free convection breaks down and the mass transport is achieved entirely by dispersion.

[45] For type A simulations (Figure 2) it was apparent that the maximum rate of PSC was achieved at D* ≈ 40. Below this value the effect of increased dispersion was significant and acted to decrease both QS and PSC. On the other hand, for incremental increases in D* above 40, the value of PSC also decreased, though more gradually, because convective overturn was limited by the rate that salt could disperse across the saltwater-freshwater transition zone. This relationship is depicted schematically in Figure 5.

Figure 5.

Schematic of hydrodynamic dispersion effects on PSC for type A simulations. For D* < 40, PSC decreases as D* decreases due to weakening of density gradients and breakdown of convective overturn; for D* > 40, PSC decreases as D* increases due to decreased salt dispersion across the saltwater-freshwater transition zone.

[46] Solutions for values of D* < 40 with type A boundary were considered potentially unrealistic in this study due to the excessive backward dispersion of salt across the seabed. The issue of how to select appropriate dispersion parameters for saltwater intrusion modeling is difficult because dispersivities are known to be scale-dependent [Gelhar et al., 1992]. Thus, in determining what range of dispersivity values is realistic for particular simulations, consideration must be given to both the scale of the salt wedge and the scale of aquifer heterogeneity that is ‘sampled’ by the saltwater flow within the salt wedge. This leads to nonlinear feedback in numerical simulations because the size and geometry of the simulated salt wedge is strongly dependent of the values of the dispersion parameters used. To further complicate this situation, the range of dispersivity values that are practical to use in modeling studies is often constrained by numerical considerations. The review by Gelhar et al. [1992] indicated that for “scale of test” in the range 10–1000 m, field determinations of longitudinal dispersivity were typically of order 0.1 m to tens of meters. A comparable range, 2.5 ≤ αL ≤ 50 m, was used in this study and is considered reasonable for salt wedges of lengths of tens to hundreds of meters.

[47] Type B simulations (Figure 3) were restricted to examples with D* > 40. Dynamic allocation of the seabed boundary condition often failed for values of D* < 40 due to the difficulty of resolving the direction of flow across the seabed. In such cases, excessive dispersion of salt into the inshore aquifer was accompanied by excessive freshening of groundwater within the offshore aquifer. This smearing of density gradients resulted in weak convective overturn and indistinct flow directions across the seabed. It is possible that extending the flow domain further offshore would overcome this problem; however, because αL > 0.25 B, these examples were considered unrealistic and were not further pursued.

[48] Clearly, the above results demonstrate that dispersion is a key control on PSC for both choices of seabed boundary condition. It is apparent that the values of the dispersivities used in numerical simulations are critical to the result obtained because hydrodynamic dispersion is the fundamental processes that drives convective overturn. This result exposes an inherent problem that comes with trying to apply numerical simulation results to estimate saltwater circulation rates in real groundwater systems. Because estimates of aquifer dispersivities are commonly nonunique, scale-dependent and uncertain [Gelhar et al., 1992], it follows that numerical predictions of PSC based on these estimates will be equally uncertain.

9.1.4. Numerical Dispersion

[49] Results in Table 4 illustrate that simulated PSC was significantly increased for values of the grid Peclet numbers greater than two. In these particular examples, PSC increased by a factor 1.3 from 75% to 97% as Px and Pz were increased in value from 2 to 10. Overall, this result is consistent with established guidelines that recommend Px < 2 and Pz < 2 to minimize effects of numerical dispersion [e.g., Ghassemi et al., 1996]. To satisfy these conditions and avoid the truncation errors that cause numerical dispersion [Benson et al., 1998] requires relatively fine spatial discretization if small values of dispersivity are used. This limits the minimum values of the dispersivities that are practical to simulate. In this study, the grid Peclet numbers were kept below the above limits; however, variability in Px and Pz below 2 likely accounts for some of the variability observed in results.

Table 4. Effect of Grid Peclet Numbers on PSC for Type A Simulationsa
Δx, Δz,Px, PzMesh ElementsPSC, %
  • a

    B = 100 m, Kx,z = 50 m d−1, qf = 0.02 m d−1, αL = 5 m, and αT = 0.5 m.

0.51800,00073
1.02200,00075
2.5532,00087
5.0108,00097

9.1.5. Isotropic Recirculation Number

[50] Assuming the relation between D* and PSC to be piecewise monotonic and of the form in Figure 5, the combined effects of free convection, forced convection and hydrodynamic dispersion on PSC can be combined into a single isotropic recirculation number

equation image
equation image

The dispersion function, denoted fd, was derived empirically in two parts to reflect the two theoretical limits on convective overturn described above. Results in Figures 2 and 3 are replotted in Figure 6 as functions of R* with a = 0.11, b = 3.0 and c = 0.3 for type A simulations; and b = 6.3 and c = 0.5 for type B simulations. Both relationships were relatively well fitted by simple exponential associations as shown on Figure 6. Overall, R* provides a relatively consistent measure of saltwater circulation and appears to have useful predictive power within the parameter space considered.

Figure 6.

Recirculation number versus PSC for isotropic aquifers with type A and type B boundary conditions and αLT = 10; a = 0.11, b = 3.0, and c = 0.3 for type A boundary, and b = 6.3 and c = 0.5 for type B boundary.

[51] Importantly, the specific values of a, b, c and d given above are for the particular case αLT = 10. As discussed previously, the values of αLT that were practical to simulate in this study were constrained by numerical considerations; noting, again, that αLT = 10 falls at the lower end of estimates reported from field-scale investigations [Gelhar et al., 1992]. Figure 7 compares results obtained using αLT = 20 for selected type A and type B simulations with D* = 50. While this represents only a modest increase in αLT, simulated PSC was significantly decreased for type A simulations compared to type B simulations. This difference was presumably the result of increased backward dispersion across the seabed boundary in the type A simulations. It is evident that the systematics of the solutions varies as a function of αLT but no attempt was made to quantify this dependence due to the large number of additional simulations that would be required. For example, it is expected that for larger values of D*, increasing αLT would slow the rate of convective overturn by further restricting the rate of dispersive transport across the saltwater-freshwater transition zone. The main purpose of these results is to reinforce the earlier conclusion that simulated PSC is strongly dependent on the presumed values of the dispersivities.

Figure 7.

Recirculation number versus PSC for isotropic aquifers with type A and type B boundary conditions; comparison of selected results for αLT = 10 and 20.

9.2. Anisotropic Aquifers

[52] The effects of anisotropy on PSC were explored by conducting approximately forty additional simulations for various values of both the anisotropy ratio Kx/Kz and D*. Results from these simulations are plotted in Figure 8, with the matrix at the top indicating the values of Kx, Kz and D* used. The upper graph depicts PSC plotted against R*, while the lower graph shows the same data plotted against a modified anisotropic recirculation number, denoted R′*. Results are distinguished by symbols; shaded symbols indicate isotropic aquifers, and solid symbols indicate anisotropic aquifers. The anisotropic recirculation number is considered below but first it is helpful to make a general observation about the usefulness of the conventional anisotropy ratio as a parameter for characterizing convective overturn. Although Figure 8 includes results for 1 ≤ Kx/Kz ≤ 100 (see also Table 1), the value of the anisotropy ratio was found to be unimportant with respect to the influence of Kx and Kz on convective overturn. It is relevant to note that convection within a coastal salt wedge is ‘circular’ and involves both horizontal and vertical flow. The rate that saltwater can circulate within the salt wedge is proportional to both the horizontal and vertical components of hydraulic conductivity; rather than their ratio.

Figure 8.

Effect of anisotropy on PSC for type A simulations and αLT = 10; shaded symbols indicate isotropic simulations, and solid symbols indicate anisotropic simulations. The top plot depicts PSC versus the isotropic recirculation number, and the bottom plot depicts PSC versus a modified anisotropic recirculation number.

[53] To incorporate anisotropy effects into the recirculation number, a modified form of the free convection velocity was sought such that vβ = K′β/ne, V′* = vβ/vf and R′* = V′* × fd, where K′ = f(KxKz) is an equivalent isotropic hydraulic conductivity. Thus R* and R′* differ only in the relative values of K and K′. The other necessary conditions were K′ = K if Kx = Kz and K′ = 0 if either Kx = 0 or Kz = 0. In other words, R′* must equal R* if the aquifer is isotropic and convective overturn must reduce to zero if either the horizontal or vertical component of hydraulic conductivity equal zero. Results depicted in Figure 8 are for type A simulations with the modified free convection velocity calculated as

equation image

From visual inspection of Figure 8 it is clear that significant divergence between isotropic and anisotropic results remains unexplained; however, the approach appears to have some merit in that the major effects of anisotropy on PSC are reasonably well captured. Adjusting the power terms in (13) did not significantly improve the fit. It is perhaps counterintuitive that the form of equation (13) in this study indicates Kx exerts more control on convective overturn than Kz. This result can be understood by realizing that variation in Kx affects both the inshore and offshore geometry of the salt wedge, whereas variation in Kz mostly affects the offshore geometry.

[54] For example, consider the effect on PSC of increasing the value of horizontal conductivity. The offshore extent of the convection cell ‘stretches’ horizontally because horizontal flow is favored over vertical flow, remembering that the length of the seabed in anisotropic simulations was increased to accommodate this effect (see section 6.1). In addition, because the aquifer is more transmissive to horizontal flow, slopes along both the free surface and saltwater-freshwater interface are reduced and the salt wedge convection cell intrudes further inshore within the aquifer. Thus, by increasing Kx the convection cell is made larger in both the inshore and offshore directions, leading to increased saltwater flow. In contrast, increasing Kz does not significantly extend the salt wedge inshore because the position and slope of the saltwater-freshwater interface is controlled independently by qf, Kx and β.

9.3. Diffuse Recharge Assumption

[55] In the earlier description of the mixed convection problem in this study, diffuse groundwater recharge across the top boundary of the model was neglected on the basis that it does not significantly influence convective overturn and PSC. The validity of this assumption was tested by applying uniform groundwater recharge R [L T−1] along the top boundary, landward of the coast and then reducing Qf by an equivalent amount such that the terrestrial component of SGD (QF = Qf + RLL) was kept constant. For the above assumption to hold valid, PSC should be approximately equal in these simulations. Table 5 presents example results for 0 ≤ RLL/Qf ≤ 1, which is equivalent to diffuse recharge contributing between 0% and 50% of terrestrial groundwater discharge at the coast. The other parameter values in this example were B = 10 m, Kx,z = 50 m d−1, qf = 0.1 m d−1, αL = 5 m and αT = 0.5 m.

Table 5. Effect of Diffuse Recharge on PSC for Type A and Type B Simulationsa
Qf, m2 d−1RLL, m2 d−1QF, m2 d−1PSC Type A Boundary, %PSC Type B Boundary, %
  • a

    B = 10 m, Kx,z = 50 m d−1, αL = 5 m, and αT = 0.5 m.

  • b

    Zero-concentration condition along landward portion of the top boundary.

1.00.01.027.361.3
1.0b0.0b1.0b30.7b67.8b
0.950.051.029.163.6
0.90.11.029.264.1
0.80.21.029.564.4
0.70.31.029.765.0
0.60.41.030.366.5
0.50.51.030.466.1

[56] Results for both type A and type B simulations indicate that simulated PSC was slightly increased when diffuse recharge was introduced along the top boundary. The magnitude of this increase was not strongly correlated to the magnitude of R, indicating that the effect was related to the change in solute concentration imposed along the top boundary, rather than the recharge rate. Applying a freshwater flux reduces concentrations along the upper boundary and increases the vertical density gradient above the salt wedge, particularly at the shoreline where the top boundary and saltwater-freshwater transition zone converge. In turn, this promotes a slightly increased rate of convective overturn. To illustrate this effect, Table 5 includes an additional simulation for R = 0 but with constant concentration C = 0 enforced along the landward portion of the top boundary. For both type A and type B boundaries, simulated PSC was increased by amounts consistent with results for R ≠ 0, even though no diffuse recharge was applied. The magnitude of this effect is also likely to be a function of D*, which controls the width of the saltwater-freshwater transition zone.

[57] It was concluded that neglecting diffuse recharge across the top boundary of the model had a small effect on simulated PSC but that this effect did not change or invalidate the general results and conclusions in this study.

9.4. Comparison of Results From FEFLOW and SUTRA

[58] A selected number of model runs were undertaken to crosscheck the numerical results from FEFLOW against results from SUTRA [Voss, 1984] and to test whether consistent rates of PSC were computed by both codes. Details of these examples are provided in Tables 6 and 7. All examples were for 10-m thick, isotropic aquifers. Initial conditions for the SUTRA simulations were generated from the FEFLOW solutions, which significantly reduced the simulation times required by SUTRA. No attempt was made to exhaustively reexamine the full set of results that were obtained using FEFLOW. The main objective was to identify any obvious discrepancies between numerical codes. The results are depicted in Figure 9.

Figure 9.

Comparison of simulation results from this study and other studies with type A boundary condition and αLT = 10.

Table 6. Numerical Specifications of FEFLOW and SUTRA Models
Setting/OptionFEFLOWaSUTRAb
  • a

    Version 4.802, March 2000; version 5.01, June 2003.

  • b

    Version V09972D, 19 September 1997.

Problem classsaturated media, flow and mass transport, vertical problem projectionSUTRA solute transport, saturated flow, transient flow, transient transport
Fluid viscosity dependenciesneglected; hydraulic conductivity assumed to be independent of viscosityViscosity is assumed to be constant at μ = 10−3 kg.m−1 s−1 (20°C)
Boussinesq approximationassumed [Diersch, 1998, equation (134)]not assumed [Voss, 1984, equation (2.24)]
Evaluation of element integralGauss quadratureGauss quadrature
Upwindingnone (Galerkin FEM)none (Galerkin FEM)
Method of resolving nonlinearityiterative; convergence based on Euclidian L2 integral (RMS) normiterative; convergence based on maximum pressure and concentration changes from previous iteration
Velocity calculationdiscontinuousconsistent [Voss, 1984]
Matrix solverdirectdirect
Finite element meshuniform, quadrilateral elementsuniform, quadrilateral elements
Time steppingpredictor-corrector AB/TR time integration scheme (explicit predictor, implicit corrector)explicit finite difference; constant time step to match stabilized time step from FEFLOW AB/TR scheme
Density-concentration (ρ-C) relationfirst order (linear): ρ(C) = γC + ρ(Co) [kg/m3], γ = 0.6429 (nondimensional); achieved by setting equation image = 0.0225, Co = 0 kg/m3, Cmax = 35 kg/m3first order (linear), as FEFLOW; achieved by setting the coefficient of density change to DRWDU = 642.9
Initial conditionsFEFLOW nonconverged steady flow and mass transport solutionfinal (quasi-steady) pressures and solute concentrations from FEFLOW transient flow and mass transport solution
Table 7. SUTRA Model Inputs and Results for Selected Type A Simulationsa
RunKx,z, m d−1qf, m d−1αL, mαT, mPSC, %V*D*R*
  • a

    B = 10 m, and ne = 0.25.

R49200.0250.548.222.52020.0
R691000.0510169.445.01030.0
R76500.0250.510256.22050.0
R681000.021011401121075.0
R125000.150.516411220100

[59] Consistent but slightly smaller values of PSC were calculated using SUTRA compared to FEFLOW. In the five examples considered, PSC calculated using SUTRA was 0.93–0.95 of PSC calculated using FEFLOW. Overall, the results indicate a reasonable level of consistency between codes with respect to simulation of saltwater intrusion and calculation of fluid mass balances.

[60] To ensure that an appropriate comparison of results was being made, care was taken to adopt consistent solution strategies for both codes. Details are listed in Table 6. Consistent attributes from both sets of simulations include the specified problem class, neglecting of fluid viscosity dependencies, method of evaluating element integrals, degree of upwinding, design of finite element mesh, density-concentration relation and matrix solver. Differences include the degree of density coupling employed, the error criteria used for resolving nonlinearity, method of calculating velocity, and time steeping procedure.

[61] Because quasi-steady solutions were sought, the error criteria for resolving nonlinearity and the employed time stepping procedures should not have significantly affected the results, provided reasonable choices were made. Constant time steps were used in the SUTRA simulations to approximately match the stabilized time step size selected by FEFLOW's predictor-corrector scheme.

[62] A potentially significant difference is the degree of density coupling employed in the simulations. The Oberbeck-Boussinesq approximation was assumed in the FEFLOW simulations but not in the SUTRA simulations (this option is not available in SUTRA). In saltwater intrusion problems, the Oberbeck-Boussinesq assumption is only reasonable if (1) temporal change in density is very small (i.e., ∂ρ/∂t ≅ 0) and (2) density gradients are effectively orthogonal to flow (i.e., q • ∇ρ ≅ 0, where q is the velocity vector). Diersch [1998] discusses these requirements in more detail (see Diersch's equation 138). The first condition is satisfied in the present study because all solutions are advanced to quasi-steady states. The second condition is approximately satisfied because, in areas where Δρ is significant, flow is mainly parallel to the saltwater-freshwater interface, which is more or less orthogonal to the density gradient. Thus application of the Oberbeck-Boussinesq approximation in this study was reasonable and should not have caused significant divergence in the results from FEFLOW and SUTRA.

[63] Other intrinsic differences between the model codes and their implementations might also be responsible for some of the observed differences in results. These factors are beyond the direct control of users and were not explored in this study.

9.5. Comparison With Numerical Results From Previous Studies

[64] Two investigations of saltwater intrusion that directly report rates of saltwater circulation are know to the author. Table 8 and Figure 9 depict the simulations from these studies that on the basis of model geometry and boundary conditions, provide valid comparisons with the results from the modeling in this study. Both sets of results were obtained using type A boundaries and αLT = 10.

Table 8. Model Inputs and Results From Previous Studies With Type A Boundary Conditions
RunB, mLS, mLL, mne,Kx,z, m d−1qf, m2 d−1αL, mαT, mPSC, %V*D*R*
Swan R1a15502000.25400.0450.554.622.53021.7
Israel 1b12050040000.3615.10.00212.51.2511333.29625.3
Israel 2b12050040000.3615.10.00412.51.2513441.69631.7
Israel 3b12050040000.3615.10.00612.51.2516855.19642.0
Israel 4b12050040000.3615.10.00812.51.2523183.29663.5
Israel 5b12050040000.3615.10.0112.51.2540916396124

[65] The study by Smith and Turner [2001] of density-dependent groundwater circulation adjacent to an estuarine river channel was previously described. In the example denoted “Swan R1” the rates of terrestrial groundwater discharge to both sides of the river (i.e., on opposite banks of the river) were equal and the pair of convection cells beneath the riverbed was symmetric. Despite the obvious difference compared to a single convection cell along the coastal boundary of an aquifer, simulated PSC was relatively consistent with the results in this study. This level of agreement is not surprising, given that both studies used FEFLOW and employed similar mesh designs, density differences and solution strategies. Comparable grid Peclet numbers also were used.

[66] Prieto [2001] used SUTRA to investigate three case studies of seawater intrusion in coastal aquifers in Israel, Rhodes and Cyprus, with the Israel case study having most similarity to the aquifer geometry in this study. Figure 9 reveals that the relation between R* and PSC from the Israel case study is different to the present results. This difference is due in part to the inclusion of inshore groundwater pumping that caused up coning and partial capture of seawater by the pumping bore. In addition, a coarser spatial discretization and larger grid Peclet numbers were used. In the Israel case study, Px = 20 and Pz = 8 compared to the limits Px ≤ 2 and Pz ≤ 2 conformed to in the present study. As discussed above, significant increases in PSC were observed for values of the grid Peclet numbers greater than these limits. This effect likely accounts for a significant part of the discrepancy between results.

10. Summary and Conclusions

[67] Theoretical patterns of mixed convection and seawater circulation rates in coastal aquifers can be investigated numerically using simulation codes for density-coupled flow and mass transport. This study has shown that the relative amount of seawater to freshwater in simulated SGD is predictable and controlled by two key nondimensional quantities: V*, the ratio of the free convection velocity to the forced convection velocity; and D*, the ratio of the aquifer length scale to the dispersion length scale. These controls can be expressed in a single nondimensional recirculation number, R*, which incorporates the combined effects of free convection, forced convection and hydrodynamic dispersion on convective overturn within the coastal salt wedge.

[68] A monotonic relationship was observed between V* and PSC, whereas the relationship between D* and PSC involved nonlinear feedback between flow and mass transport. If D* was too small and dispersion excessive, convective overturn was shown to breakdown due to weakening of density gradients. Conversely, if D* was too large, convective overturn was limited by restricted salt transport across the saltwater-freshwater transition zone. Maximum rates of density-driven seawater circulation were therefore achieved for large values of V* and intermediate values of D*.

[69] For the range of parameter values explored in this study, simulated SGD contained from 0% up to 75% former seawater. While the seawater component of discharge, QS, was sometimes larger than the freshwater component, Qf, the results suggest that QS is unlikely to exceed Qf by more than half an order of magnitude. Therefore it is unexpected that estimates of SGD based only on Qf could be more than an order of magnitude incorrect through neglecting the contribution of density-driven seawater circulation.

[70] Significantly larger values of PSC were obtained using type B boundaries than type A boundaries. Accumulation of salt beneath the outflowing portion of type A boundaries weakened the density gradients within the salt wedge and slowed the rate of convective overturn. Nevertheless, it remains unclear to which range of field conditions each type of boundary condition is best applied, if at all. Site-specific differences in tidal fluctuations, waves, wind setup and associated beach face processes result in diverse and site-specific submarine salinity profiles. Comprehensive field data from a variety of field sites are required to better assess the applicability and usefulness of seabed boundary conditions commonly used in numerical models.

[71] Effects of anisotropy on PSC and the systematics of these effects were quantified with only moderate success based on the concept of equivalent isotropic hydraulic conductivity. The rate of convective overturn was shown to be directly related to the magnitudes of both Kx and Kz, revealing that the conventional anisotropy ratio, Kx/Kz, is not a useful group for characterizing convective overturn.

[72] Applying the results from this study to estimate saltwater circulation rates in real groundwater systems is considered problematical because determinations of aquifer transport parameters are seldom unique or unambiguous. It was found that simulated PSC was strongly dependent on the values of the dispersivities used, such that uncertainty in these parameters would convey equal uncertainty to the model predictions of PSC. Further work is required to establish whether results from numerical simulations can be applied reliably to estimate PSC in real aquifers that exhibit scale-dependent transport properties. This problem in nontrivial and requires improved field methods for estimating the relative amount of seawater to freshwater in SGD to provide data sets that are adequate to compare with numerical results. For example, using resistivity profiles to optimize a method of characteristic (MOC) model of a saltwater inversion beneath a beach at the French-Belgian border, Lebbe [1999] obtained estimates of transverse and longitudinal dispersivities of 1.59 mm and 7.01 mm, respectively (i.e., αLT = 4.4). These values, which are around one order of magnitude less than the smallest value used in the present study, are not practical to simulate using the current method. To achieve Px < 2 and Pz < 2 in a FEFLOW or SUTRA simulation would require a mesh discretization with Δx = Δz = 3.2 mm and D* = 6,250–62,500 for aquifer thicknesses B = 10–100 m. Nevertheless, it is clear that salt dispersion across the saltwater-freshwater transition zone and PSC would be significantly reduced for such large values of D*. Determining what values of D* best represent real aquifers is an important issue that must be addressed before the results from numerical simulations can be applied usefully to the real world.

[73] In conclusion, it is worthwhile to reiterate that this study was concerned only with density-dependent seawater circulation arising from the density contrast between seawater and fresher groundwater in the coastal aquifer. Forced convection of seawater into and out of coastal groundwater systems in response to fluctuation in hydraulic gradients between sediment pore water and the marine water column is also an important process for seawater circulation. In turn, seawater that enters the aquifer by one of these mechanisms is then subject to density-dependent flow. Li et al. [1999] estimated from theoretical considerations that up 96% of SGD could be seawater discharge resulting from wave setup and tide. On the basis of field measurements of geochemical tracers, Garrison et al. [2003] estimated that approximately 20% of average SGD into the inner part of Kahana Bay in Hawaii was terrestrial groundwater (i.e., PSC = 400%), implying 80% of SGD was former marine water. To the author's knowledge, interaction between free and forced components of seawater flow in coastal aquifers has not yet been investigated in detail. This suggests a promising and important topic for future research.

Acknowledgments

[74] The research described in this paper was supported by CSIRO Land and Water (CLW) Australia. Salient advice and constructive comment from Mike Trefry from CLW during the course of conducting this research were invaluable. Kumar Narayan from CLW and Irene Pestov from Bureau of Rural Sciences, Australia, reviewed the original manuscript prior to submission to Water Resources Research (WRR). Their critiques lead to significant improvements in the paper and are gratefully acknowledged. The manuscript was also reviewed on two separate occasions by anonymous reviewers for WRR. The author is indebted to the reviewers for providing detailed and thoughtful comments that led to crucial improvement and expansion of the paper. Their contributions are thankfully acknowledged.

Ancillary