Enhanced dispersion in groundwater caused by temporal changes in
recharge rate and lake levels
Kangjoo Kim
*, Mary P. Anderson, Carl J. Bowser
Department of Geology and Geophysics, University of Wisconsin-Madison, Madison, WI 53706, USA
Received 17 October 1999; accepted 4 November 1999
Abstract
Dispersion of solutes in groundwater is caused mainly by spatial variation in aquifer properties (i.e., heterogeneity) but addi-tional dispersion can be induced by temporal ¯uctuations in the ¯ow ®eld. We studied dispersion of an oxygen isotope plume in an aquifer in northern Wisconsin, where signi®cant ¯uctuations in the velocity ®eld are caused by temporal changes in recharge rate and lake levels. The enhanced vertical spreading caused by these transient eects was quanti®ed by tracking pathlines for ap-proximately 32 years of simulated time in a transient cross-sectional model of the groundwater ¯ow system. In this system heter-ogeneity, ¯uctuations in recharge rate, and distance from the transient boundary stresses have a signi®cant in¯uence on the vertical transverse dispersion of the plume, while dispersion caused by ¯uctuations in lake levels alone have a relatively small eect. Ó 2000
Elsevier Science Ltd. All rights reserved.
Keywords:Dispersion; Groundwater±lake interaction; Recharge rate; Transient eects
1. Introduction
Dispersion of solute plumes is generally attributed to velocity variations caused by aquifer heterogeneity. Several researchers, however, have demonstrated that temporal variations in hydraulic gradients can enhance transverse dispersion [1±4]. This eect was noted in an early paper by Sykes et al. [5, p. 1699]: ``From a dis-persion point of view, both the changing velocity di-rections in the horizontal plane and the eect of intergranular ¯ow can be the cause of transverse dis-persion''. Later, Goode and Konikow [3, p. 2339], pointed out that: ``Temporal ¯uctuations in recharge, discharge or boundary conditions will also increase ve-locity variance and thus might also be expected to contribute to plume spreading''. Similarly, Rehfeldt and Gelhar [4] concluded: ``Dispersive mixing due to tran-sient ¯ow is a mechanism that should not be over-looked''.
Kinzelbach and Ackerer [1] calibrated two-dimen-sional, transient and steady-state models to an observed plume and found that a higher value of horizontal
transverse dispersivity was required to simulate the observed plume when using a steady-state model. Na et al. [2] attributed the discrepancy between results from their theoretical three-dimensional steady-state model and an observed plume at the Borden site to small scale transients. Subsequently, Farrell et al. [6] found that almost all of the horizontal transverse spreading at the Borden site was caused by temporal variations in velocity (e.g., see Fig. 3 in [7]). Similarly, Hess et al. [8] found that transverse dispersivities computed from theory were too low to account for transverse spreading at the Cape Cod site and attributed the discrepancy to transient eects. Rehfeldt and Gelhar [4] showed that transient eects in¯uenced transverse dispersion at the Borden, Cape Cod, and MADE sites.
Goode and Konikow [3] studied the eects of sym-metric cyclic boundary stresses on dispersion of an in-stantaneous source using numerical and analytical models of a hypothetical system. They refer to the additional spreading caused by transient eects as
enhanced dispersion and de®neapparent dispersivitiesas ``those values that yield the best match or calibration of the solute transport model under steady-state ¯ow conditions to a plume that developed under transient-¯ow conditions''. They found that the characteristic hydraulic response time for their system was a function
*
Corresponding author. Present address: Department of Environ-mental Engineering, Kunsan National University, Korea.
E-mail address:kangjoo@knusunl.kunsan.ac.kr (K. Kim).
of hydraulic diusivity (the ratio of transmissivity to storativity) and distance to the boundary and that this parameter in¯uenced the magnitude of the additional dispersion caused by transient eects. They also found that enhanced transverse dispersion is a function of the magnitude of the ¯uctuation in ¯ow direction and the ratio of longitudinal to transverse dispersivity. They note that failure to recognize the eects of transience on solute spreading may result in the use of in¯ated values of dispersivity during calibration of a model based on classical solute transport analysis using a steady-state ¯ow ®eld.
Reilly and Pollock [9] also studied the eects of cyclic stresses on a hypothetical groundwater system but they used a particle tracking model rather than a full trans-port model to assess the magnitude of transient ¯uctu-ations in the velocity ®eld on the size of contributing areas to pumping wells. They found that for the system they studied, the magnitude of the mean travel time from the source area to the pumping well relative to the length of the cyclic stress determined whether transient eects in¯uenced the size of the contributing area. When the ratio of travel time to cyclic stress period was greater than 1, transience had little eect but when the ratio was less than 1, transient eects did in¯uence the size of the contributing area.
Transient eects are also important in creating wide mixing zones in coastal aquifers in¯uenced by tidal eects, seasonal recharge ¯uctuations and/or inland pumping [10±13]. In these situations, the zone of diu-sion between fresh water and saltwater is widened by advection of the saltwater front in response to the stress. In this study, we observed a wide mixing zone between two isotopically distinct waters in groundwater ¯owing beneath an isthmus between two lakes in northern Wisconsin (Figs. 1 and 2). The mixing zone (Fig. 2(b)) is created as lake water with an isotopic sig-nature of d18O equal to ÿ3:72& discharges out of Crystal Lake into the groundwater system and mixes with recharge water that has entered the system through the soil zone from precipitation and has an isotopic content of d18O equal to ÿ11:2
&. We speculated that
seasonal ¯uctuations in recharge rate and lake levels cause ¯uctuations in ¯ow direction in the upper part of the aquifer and thereby enhance vertical mixing. Several other investigators have addressed the importance of transient eects on groundwater±lake systems [14±20], suggesting that enhanced transverse dispersion could be a common phenomenon in these systems.
In this paper, we used a numerical groundwater ¯ow model with particle tracking to examine the amount of dispersive mixing caused by transience in the ¯ow ®eld. Calibration of the ¯ow model used here was discussed by Kim et al. [21]. We simulated vertical transverse dispersion while previous investigators focussed on horizontal transverse dispersion. Furthermore, previous
workers [4,9] considered cyclic stresses while our results provide a ®eld example and numerical analysis in sup-port of the theoretical ®ndings of Goode and Konikow [3], who demonstrated the signi®cance of transient ef-fects on horizontal transverse dispersion, and we extend their conclusions to dispersion in the vertical dimension for non-cyclic assymmetric stresses.
2. Study area
2.1. Hydrogeology
The study area lies in a narrow isthmus between Crystal Lake and Big Muskellunge Lake in Northern Wisconsin (Fig. 1). These lakes are located in the Northern Temperate Lakes site in the Long Term Eco-logical Research (LTER) network sponsored by the National Science Foundation (NSF). This area is sparsely populated and heavily forested. The surround-ing region contains more than 3000 kettle lakes, some of which are interconnected, but most of which occur in topographically closed basins.
Crystal Lake is a small seepage lake having no surface inlets or outlets. Approximately 90% of the in¯ow to the lake is from direct precipitation on the lake surface [22,23]. Groundwater ¯ows out of Crystal Lake toward Big Muskellunge Lake (Fig. 2(a)), 130 m to the northwest. The water levels of the two lakes ¯uctuate synchronously, maintaining a dierence in level of about 1.2 m [19].
15 m of aquifer is dominated by sand-sized material and two silt layers, each of which is approximately 1 m thick [25]. The upper silt layer and the upper sandy sediments dip about 7° toward Crystal Lake. Steep hydraulic gradients across the upper silt layer (Fig. 2(a)) indicate that it is continuous and acts as a con®ning unit. The lower silt layer does not seem to be as continuous since only the upper silt layer was encountered during con-struction of a deep multilevel well in 1993. Furthermore, there are no notable head changes across the lower silt layer. Slug tests show that the hydraulic conductivity (K) of the sandy sediment ranges from 0.17 to 17.3 m/day and from 8:610ÿ5 to 3:510ÿ3 m=day for the silt layers [26]. TheKvalues for sandy sediments above the upper silt layer (4.3±17.3 m/day) are generally 4±6 times higher than for the sandy sediments below it (0.17±3.5 m/day). The ratio of horizontal to vertical conductivity KX=KZ, estimated from tracer tests that covered about
one meter of porous material [27], is 3.5±7.8.
Precipitation in the area averages 80 cm/yr and evaporation o the lakes is estimated to be 54 cm/yr [28]. Average groundwater recharge was estimated to be 26.2 cm/yr [21]. Most groundwater recharge takes place during spring snow melt with very little recharge during summer.
2.2. Mixing zone delineated using oxygen isotopes
dates and the isotopic variation of these replicate sam-ples falls within the level of uncertainty of the analyses (0:1&oxygen).
Isotopic values from the well network shown in Fig. 2 as reported by Kim [29] were converted into percent lake water assuming linear mixing of the end-member waters. The mixing percentages were contoured using the kri-ging option of a commercial software package, äTransform (Fortner Research). Based on the mea-sured isotope values, water along the southern border of the transect shown in Fig. 2 is 100% lake water and water at the water table and near the center of the transect has 0% lake water. The mixing zone (Fig. 2(b)) is the dispersed upper edge of the plume of lake water emanating from Crystal Lake. The plume exhibits a signi®cant amount of vertical transverse dispersion as it moves through the isthmus, an horizontal distance of 130 m. The thickness of the dispersed zone is 10±13 m, where the lower edge of the mixing zone is 100% lake water (shown by the 100% isopleth in Fig. 2(b)) and the top of the mixing zone is near the water table. Samples were also collected in September 1992 and June 1993 and analyzed for oxygen isotopes. A mixing plot using these data was presented by Bullen et al. [30]. The extent of their mixing zone is essentially the same as shown in our Fig. 2(b). We postulated that mixing is enhanced by transient ¯uctuations in recharge rate and/or lake level. Krabbenhoft et al. [31], Bullen et al. [30] and Kim [29] attempted to approximate an average ¯ow path in this
system by assuming that on average a particle of water follows the 100% isopleth along the bottom of the mixing zone. The upper portion of KimÕs ¯ow path, which nearly coincides with the path identi®ed by Krabbenhoft et al. [31] and Bullen et al. [30], is shown in Fig. 2(a). The travel time from Crystal Lake to well K70 along this ¯ow path was estimated to be around 10 years [21]. A precise ¯owpath at the distal end of the plume is more dicult to determine. Isotope samples from deeper wells together with a ¯ow model of a larger domain may resolve the uncertainties; these issues are being pursued in ongoing work.
3. Methods
3.1. Groundwater ¯ow model
The groundwater ¯ow model used in this study is described in detail by Kim et al. [21] and summarized brie¯y below. The model is 450 m long and 45 m deep with 40 layers and 58 columns and covers an area from near the center of Crystal Lake to about 100 m oshore of Big Muskellunge Lake (Figs. 1 and 3). Layer thick-nesses vary from 0.2 to 3.0 m and column widths vary from 5 to 15 m (Fig. 3). The upper 10 layers near the water table are 0.2 to 0.5 m thick, in order to represent the gently sloping bed of Crystal Lake. The computer code MODFLOW [32] was used with the BCF2 package
[33] in order to allow the upper layers to resaturate. The bottom boundary is the crystalline bedrock, which is assumed to be impermeable; the upper boundary is formed by the lakes and the water table, which was represented by a speci®ed ¯ux boundary condition. The two lakes and the side boundaries were represented as speci®ed head boundaries using the general head boundary (GHB) package in MODFLOW in order to allow temporal changes in lake levels. Conductances for the GHB cells under the lakes were calculated assuming that the lake sediments in Crystal Lake are 0±0.3 m thick and those for Big Muskellunge Lake are uniformly 0.2 m thick. Hydraulic conductivity of the lake bed sediments for both lakes was assumed to be 0.04 m/day, which is the value used for silt in zone 4 (Fig. 3, Table 1) of the calibrated ¯ow model [21].
Six conductivity zones were used to delineate the heterogeneity of the aquifer (Table 1, Fig. 3). Zone 6 represents bedding planes within the upper sand layer. These layers were introduced during calibration in order to simulate additional anisotropy within the upper sand aquifer that occurs at an angle to the horizontal coor-dinate axis of the grid [21]. Hence, the upper sand unit is represented in the model by a layered system of aniso-tropic conductivity zones, wherein conductivity zone 1 alternates with conductivity zone 6. Kim et al. [21] showed that the heterogeneity represented in the model by the dipping bedding planes of zone 6 is needed to produce a good calibration to ¯ow. The particle track-ing results discussed in Section 4.1 below show that the bedding plane layers are also important in simulating the vertical transverse spreading of the plume. Based on the work of Kenoyer [27] the ratio of horizontal to vertical hydraulic conductivity (anisotropy) in each
conductivity zone was assigned a value of ®ve. The pa-rameters used in the base model are listed in Table 1. The model was calibrated under both steady-state and transient conditions of ¯uctuating recharge and lake levels [21].
3.2. Recharge and lake level estimation
Recharge rates for 40 years of record (1954±1994) were calculated using Thornthwaite's method [34,35] including snow melt. We assumed that 1 mm of water-equivalent snow melts per day per degree Celsius above the melting point [36;37, p. 7.25]. Temperature and precipitation data were taken from a weather station at Minocqua Dam, which is located 15 km to the south of the study area. The calculation assumed that runo is negligible, following Krabbenhoft et al. [28]. Precipita-tion was classi®ed as rain or snow by the rain-freeze threshold temperature [36]. The monthly recharge rate was then calculated by subtracting the positive value of monthly evapotranspiration from the total monthly percolation.
The annual average recharge for the 40 years of record is 26.2 cm, which is about one-third of annual average precipitation (80.1 cm). Calculated annual re-charge varies from 7.8 to 41.7 cm during this same time period (Fig. 4(a)). The distribution of annual recharge is approximately normal with a standard deviation of 8.77 cm. The average monthly recharge is bimodal with peaks in early spring and late fall (Fig. 4(b)). More than 50% of the annual average recharge occurs during the spring (March and April) and about 30% occurs during fall and early winter (September through November).
Monitoring of Crystal Lake and Big Muskellunge Lake levels began in 1981 under the Long Term Eco-logical Research (LTER) program. Prior to 1981, lake levels were estimated using water-level data for Bualo Lake, a seepage lake (i.e., no surface inlets or outlets) that is located 14 km south of the study area and ¯uc-tuates in a manner similar to Crystal Lake and Big Muskellunge Lake, at least after 1981 (Fig. 5(a)). Water levels in Bualo Lake have been monitored since 1940 by the Wisconsin Valley Improvement (1994, unpub-lished data). Fitting equations, obtained by regression analysis (Fig. 5(b)), were used to calculate lake levels in Crystal and Big Muskellunge Lakes prior to 1981 (Fig. 5(a)).
3.3. Particle tracking
We used a particle tracking code to simulate the spreading, or dispersion, of particles from a point source by advection. Below we review some of the advantages and limitations of using a particle tracking analysis as opposed to a full solute transport model.
Table 1
Parameter values used in the base modela
Kxof sand (m/day) 8.0
Kxof bedding plane layers in the upper sand (m/day) 0.3
Zone [6]
Porosity 0.35
Sand 0.30
Silt
Anisotropy ratiob(K
x/Kz) 5
Storage coecient 0.0001
Speci®c yield 0.27
a
Numbers in brackets correspond to the zone numbers shown in Fig. 3.Kxis horizontal hydraulic conductivity;Kzis vertical hydraulic
conductivity.
3.3.1. Background
According to Bear [38, p. 579], dispersion is the process by which a tracer ``gradually spreads and oc-cupies an ever-increasing portion of the ¯ow domain, beyond the region it is expected to occupy according to the average ¯ow alone''. In classical dispersion theory, the dispersion coecient in the advection±dispersion equation represents spreading of the tracer bydiusion
and bylocal dispersive mixingthat occurs as a result of velocity variations caused by ``smaller-scale erratic mo-tions relative to the bulk movement'' [39, p. 200]. Early workers, however, found that application of classical dispersion theory to ®eld problems required the use of dispersivity values in the dispersion coecient that were much larger than expected based on local dispersive mixing [40]. The consensus emerged that relatively
large-scale geological heterogeneities cause what is generally calledmacroscopic dispersion.Furthermore, as noted in section 1.0 above, it is now widely recognized that transient eects also can enhance transverse dispersion. In current usage, the concept of dispersion, therefore, is broader than originally conceived. For example, Frey-berg [41, p. 2036] observed that ``. . .the termÔdispersionÕ appears to be most commonly used in the physical sense to refer to all deviations in observed or predicted con-centration from that which would be predicted assuming only advection by an average pore water velocity Fig. 4. (a) Monthly recharge rates calculated using the Thornthwaite
method; measured precipitation values are also shown. (b) Monthly average recharge and precipitation based on 40 years of record.
Fig. 5. (a) Observed and calculated lake levels. Calculated lake levels were computed from the relationships in Fig. 5(b). (b) Regression analysis for Crystal (C.) Lake and Big Muskellunge (B.M.) Lake levels against Bualo Lake level, with r20:898 (C Lake) and r20:874
®eld. . .'' Later, Gelhar [39, p. 216] developed a model to
simulate macrodispersion that incorporated the eects of both ``fully three-dimensional heterogeneity'' and local dispersive mixing. Gelhar [39, p. 269] also noted that ``¯uctuations in ¯ow, though quite small, may sig-ni®cantly enhance transverse mixing, which steady-¯ow theory predicts to be very small'' and Rehfeldt and Gelhar [4] developed a model in which the macrodis-persivity tensor is de®ned to be the sum of two com-ponents: one to account for spatial variability caused by geological heterogeneity and another to account for temporal variability caused by transient eects. Hence, as currently understood by most workers, dispersion includes not only the eects of local dispersive mixing and diusion, but also mixing caused by large-scale heterogeneities and transient eects.
We used an advective model to track pathlines and simulate the spreading or dispersion of imaginary par-ticles caused by transient eects. We do not invoke a full solute transport analysis using the advection±dispersion equation because:
1. We do not seek to reproduce the spatial and temporal history of the entire isotope plume. Rather, we con-sider the eects of transience on the vertical trans-verse spreading of a set of particles that originate close to the edge of the dispersed zone. In so doing, we examine mixing within a portion of the zone of dispersion.
2. Both large-scale heterogeneity, which causes macro-dispersion, and transient eects are included directly in our ¯ow model. Therefore, we do not require a full transport analysis with the advection±dispersion equation to represent these eects. Instead, we simu-late the spreading caused by these two eects directly using the particle tracking code PATH3D [42]. The eects of molecular diusion and local dispersive mix-ing (caused by heterogeneities that occur at a smaller scale than those we modeled), which in a full trans-port analysis would be characterized by dispersivity factors, are not considered here. These eects are gen-erally considered to be negligible compared to the spreading caused by macrodispersion. The approach used in our simulations is similar to one used by Reilly and Pollock [9] to generate a set of ¯ow paths to study the eect of cyclic stresses on contributing areas to a pumping well (see Fig. 15).
Use of a particle tracking method, however, does not allow us to calculate isotope concentrations. Thus, we do not compare simulated vs. measured concentrations but instead compare the width of the area covered by the simulated pathlines to the width of the mixing zone in-ferred from ®eld data (Fig. 2(b)). The width of the mixing zone captures the combined eects of mixing induced by large-scale heterogeneities and the cumu-lative eect of transience caused by ¯uctuations in recharge rate and lake levels.
3.3.2. Application
The movement of water within the mixing zone dur-ing a 32 year time period was simulated by trackdur-ing 40 particles as each left the lake at a dierent time starting from a point at the bottom of Crystal Lake (shown by the uppermostin Fig. 3), which is near the edge of the mixing zone. In one set of simulations we also consid-ered spreading of particles that originated at deeper locations within the aquifer (i.e., the intermediate and lower-most Õs in Fig. 3). Each particle started at the same point in space but at a dierent time so that when viewed together the 40 particle paths show the history of water movement from a point source during 32 years of simulated transit time.
The starting times of the particles were staggered by 0.25 year and each particle was tracked for approxi-mately 8000 days (21.9 years). The ®rst particle entered the groundwater system at the beginning of the simu-lation in January 1954 and the last particle entered the system in the last quarter of 1963. Therefore, 32 years 400:2521:9 of transient ¯ow are represented in each simulation set. The changes in ¯ow direction at the particle release point were estimated by recording the movements of each particle at the starting point.
We considered four scenarios. The base case simula-tion used the parameters in Table 1 and included tran-sience caused by ¯uctuations in recharge rate and lake levels. In the second scenario, we removed the large-scale heterogeneities in the form of the bedding plane layers from the upper sand aquifer (zone 6 in Fig. 3). In the ®nal two scenarios we used the base case model with bedding plane layers, but ®rst imposed constant lake levels while including transient recharge and ®nally we simulated ¯uctuating lake levels under constant re-charge. In a ®nal set of simulations, we tested the sen-sitivity of the results to the distance from the boundary stresses.
4. Simulation results and discussion
4.1. Eects of heterogeneity
In additional simulations that are not shown here, we tested the eect of the angle of the bedding plane layers [29]. These results show that the plume is most dispersed when the bedding planes are oblique to the direction of ¯ow as they are for the simulation shown in Fig. 6(a), where the angle between the bedding plane layers (zone 6) and the ¯ow direction is 13°as shown by Kim [29]. There is very little transverse spreading when the zone 6 layers are oriented parallel to the ¯ow direction. There is also little transverse spreading when the zone 6 layers are aligned perpendicular to the ¯ow direction because in this case the vertical hydraulic gradients (and the vertical velocities) are small.
4.2. Eects of ¯uctuations in recharge rate and lake levels
The next set of simulations was designed to test the signi®cance of transient eects on the pathlines. Fig. 7(a), which shows the same set of pathlines as in Fig. 6(a), was produced in a simulation that included the eects of modeled heterogeneities as well as transience in both recharge rates and lake levels. A simulation that used transient recharge rates but constant average lake levels
produced a set of pathlines (Fig. 7(b)) that was only slightly less dispersed than in the base case simulation (Fig. 7(a)). When the recharge rate was set equal to the average annual value for the period of record (26.2 cm/ yr), while still using transient lake levels, the pathlines were much less dispersed (Fig. 7(c)) than when transient recharge was included in the simulation (Figs 7(a) and (b)). These results suggest that ¯uctuations in recharge rate rather than ¯uctuations in lake levels create tran-sient eects that enhance transverse dispersion in this system.
The key factor in creating enhanced dispersion is ¯uctuation in the direction of velocity; changes in magnitude alone do not cause signi®cant enhanced dis-persion [3]. Seasonal ¯uctuations in recharge cause Fig. 7. Eects of transience on pathlines. Circles represent well points. The 100% mixing isopleth from Fig. 2(b) is shown by the dashed line. (a) Transient lake levels; transient recharge. (b) Average lake levels; transient recharge. (c) Transient lake levels; average recharge. Fig. 6. Eect of heterogeneity in the upper sand aquifer on pathlines.
transience in this system by causing the formation of groundwater mounds between the lakes [19] with ac-companying changes in the direction of groundwater velocity (Table 2). Groundwater mounds rarely formed in response to ¯uctuations in lake levels, i.e., during the simulation represented in Fig. 7(c), and consequently changes in ¯ow direction owing to lake level ¯uctuations alone are relatively small (35°; Table 2). The change in ¯ow direction is as much as 160° (Table 2) when ¯uc-tuations in both lake levels and recharge are considered together. It is also noteworthy that when the bedding plane layers are omitted (Fig. 6(b)), the mixing zone is relatively narrow even though transience is present (Table 2). Hence, in our system transient eects alone did not produce signi®cant dispersion but transience did signi®cantly enhance dispersion that was caused by heterogeneity.
Goode and Konikow [3] found that the length of the cyclic stress period relative to the characteristic response time for their system was important in determining whether transience was signi®cant in their problem. Reilly and Pollock [9] identi®ed the ratio of the mean travel time to the length of the cyclic stress as important in determining whether transient eects would be sig-ni®cant in their system. In both of these studies, the imposed stress was cyclic, whereas the ¯uctuations in recharge rate in our system are not cyclic in that they do not form a ®xed repetitive pattern (Fig. 4(a)). The length of the ``cycle'' in our system is essentially the entire simulation period of 32 years, while each particle travels approximately 22 years. Therefore, the ratio of travel time to stress period for our simulation is 22±32, which is less than 1 and suggests that transient eects will be signi®cant [9].
4.3. Distance from the boundary stresses
In order to examine the eect of near-surface tran-sience on deeper particle paths, we simulated ¯ow paths
issuing from two other points under conditions of transient recharge and transient lake levels. The shallow set of pathlines shown in Fig. 8 is the same as those shown in Figs. 6(a) and 7(a), where the starting position of the particles was at the bottom of Crystal Lake. A set of pathlines was also generated at an intermediate lo-cation with particles starting from a point in the middle of the upper sand layer and a deep set of pathlines was generated by releasing particles from a point located below the lower silt layer (Fig. 3).
The shallow set of pathlines exhibits the most trans-verse dispersion. The intermediate set of pathlines en-counters almost the same amount of heterogeneity but exhibits much less transverse dispersion because these pathlines are less in¯uenced by near-surface temporal changes in boundary stresses and seasonal formation of groundwater mounds. The deep pathlines show essen-tially no transverse dispersion because they encountered less heterogeneity and are protected from the eects of near-surface transience by the silt layers, which act as semi-con®ning units. The average annual change in ¯ow direction is 160°at the shallow release point, 95°at the intermediate release point and less than 1° at the deep release point (Table 2). These changes in ¯ow direction occur near the surface because of the formation of sea-sonal groundwater mounds that form in response to ¯uctuations in recharge rate. Hence, like other re-searchers including Freeze [43] and Goode and Koni-kow [3], we ®nd that transient eects, as expected, diminish with distance from the boundary stress.
5. Summary and conclusions
A 10±13 m thick mixing zone (Fig. 2(b)) is created as lake water with an isotopic signature of d18O equal to
ÿ3:72& discharges out of Crystal Lake into the
groundwater system and mixes with recharge water that has entered the system through the soil zone from Table 2
Average annual change in ¯ow direction at the particle release point for the pathlines shown in Figs. 6,7 and 8a;b
Simulation Din ¯ow direction (degrees)
Recharge Lake level Particle starting point
Fig. 6(a)c T T S 160
Pathlines in Figs. 6 and 7 and the shallow set of pathlines shown in Fig. 8 originate from the shallow release point at x;z 245;42shown in Fig. 3. The intermediate set of pathlines shown in Fig. 8 had a release point at (245,36) and the deep pathlines had a release point at (245,19). The simulations in Figs 6 and 8 used transient lake levels and transient recharge. The ratio of horizontal hydraulic conductivity in zone 1 to that in zone 6 (K1/K6) was 8.0/0.3 for all simulations except for the results in Fig. 6(b), whereK1/K6was 2.0/2.0;Kis given in m/day.
b
T is transient; A is average. S, I and D stand for shallow, intermediate and deep, respectively.
precipitation and has an isotopic content ofd18O equal
to ÿ11:2&. The vertical transverse dispersion of this
isotope plume is enhanced by ¯uctuations in the direc-tion of velocities in the shallow aquifer caused mainly by seasonal ¯uctuations in recharge rate.
We used a transient groundwater ¯ow model with particle tracking to show that mixing caused by ¯uctu-ations in recharge rate and lake levels causes vertical transverse dispersion. The mixing zone was simulated in cross-section by tracking particles during 32 years of ¯ow. Results (Figs. 6±8) showed that:
· In a transient ¯ow ®eld, heterogeneity in the form of bedding planes present in the upper sand aquifer caused signi®cant vertical transverse dispersion (Fig. 6). Furthermore, vertical transverse dispersion was enhanced by transience in the ¯ow ®eld induced by ¯uctuations in recharge rate and lake levels (Fig. 7). · Fluctuation in recharge rate caused more vertical transverse dispersion than ¯uctuations in lake levels (compare Figs. 7(b) and (c)).
· The eects of transience diminished with depth (Fig. 8) as the distance from the recharge boundary was increased.
· Dispersion was signi®cantly enhanced by transient ef-fects, but transience alone did not create signi®cant dispersion in our system. (Compare Fig. 6(a), which includes the eects of heterogeneity and transience, with Fig. 6(b), which assumes a homogeneous upper sand unit and transience.)
Our results show that vertical transverse dispersion can be signi®cantly enhanced by transience in the ¯ow ®eld caused by ¯uctuations in recharge rates. In the groundwater±lake system we studied, ¯uctuations in recharge rate caused the formation of seasonal groundwater mounds which in turn caused average an-nual changes in ¯ow direction up to 160°. The formation of seasonal groundwater mounds near lakes is a
well-known phenomenon, which was reported at our site by Anderson and Cheng [19] and at other sites by Anderson and Munter [15], Mills and Zwarich [44], Cherkauer and Zager [16], Phillips and Shedlock [45], Shedlock et al. [46], and Lee and Swancar [47]. Hence, conditions that might cause signi®cant enhanced vertical transverse dispersion could be common near lakes.
Acknowledgements
This research was funded by a National Science Foundation (NSF) grant in support of the Long Term Ecological Research program (LTER-DEB #9011660) and by NSF-EAR (#9304811). We thank Daniel Fein-stein, Randy Hunt, David Krabbenhoft, and Kathy Webster for helpful discussions and comments and John Schindler for ®eld assistance. Isotope analyses were done by Susanah Michaels and Mike Spicuzza using the isotope facilities under the direction of Professor John Valley, UW-Madison.
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