Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue6.2000:

Advances in Water Resources 23 (2000) 625±635

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 additional 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 e€ects was quanti®ed by tracking pathlines for approximately 32 years of simulated time in a transient cross-sectional model of the groundwater ¯ow system. In this system heterogeneity, ¯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 e€ect. Ó 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Dispersion; Groundwater±lake interaction; Recharge rate; Transient e€ects

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 e€ect was noted in an
early paper by Sykes et al. [5, p. 1699]: ``From a dispersion point of view, both the changing velocity directions in the horizontal plane and the e€ect of
intergranular ¯ow can be the cause of transverse dispersion''. Later, Goode and Konikow [3, p. 2339],
pointed out that: ``Temporal ¯uctuations in recharge,
discharge or boundary conditions will also increase velocity variance and thus might also be expected to
contribute to plume spreading''. Similarly, Rehfeldt and
Gelhar [4] concluded: ``Dispersive mixing due to transient ¯ow is a mechanism that should not be overlooked''.
Kinzelbach and Ackerer [1] calibrated two-dimensional, transient and steady-state models to an observed
plume and found that a higher value of horizontal
*

Corresponding author. Present address: Department of Environmental Engineering, Kunsan National University, Korea.
E-mail address: kangjoo@knusunl.kunsan.ac.kr (K. Kim).

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 e€ects. Rehfeldt and Gelhar [4] showed that
transient e€ects in¯uenced transverse dispersion at the
Borden, Cape Cod, and MADE sites.
Goode and Konikow [3] studied the e€ects of symmetric cyclic boundary stresses on dispersion of an instantaneous source using numerical and analytical
models of a hypothetical system. They refer to the
additional spreading caused by transient e€ects as
enhanced dispersion and de®ne apparent dispersivities as
``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

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K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

of hydraulic di€usivity (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 e€ects. 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 e€ects 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 e€ects of cyclic
stresses on a hypothetical groundwater system but they
used a particle tracking model rather than a full transport model to assess the magnitude of transient ¯uctuations 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
e€ects 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 e€ect but when the ratio was
less than 1, transient e€ects did in¯uence the size of the
contributing area.
Transient e€ects are also important in creating wide
mixing zones in coastal aquifers in¯uenced by tidal
e€ects, seasonal recharge ¯uctuations and/or inland
pumping [10±13]. In these situations, the zone of di€usion 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 signature of d18 O 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 d18 O 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 e€ects 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

Fig. 1. The study area is the isthmus between Crystal Lake and Big
Muskellunge Lake shown by the line of section. The inset shows the

location of the study area in northern Wisconsin.

workers [4,9] considered cyclic stresses while our results
provide a ®eld example and numerical analysis in support of the theoretical ®ndings of Goode and Konikow
[3], who demonstrated the signi®cance of transient effects 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 Ecological Research (LTER) network sponsored by the
National Science Foundation (NSF). This area is
sparsely populated and heavily forested. The surrounding 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 di€erence in level of about 1.2 m [19].
Seismic surveys and drilling show that the aquifer is
composed of about 50 m of sandy glacial drift overlying
the crystalline Precambrian bedrock [24]. The upper

K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

627

Fig. 2. (a) Cross-section showing silt layers, well positions (dots), and equipotential lines. Equipotential lines are based on measurements taken on 25
August 1993 by Schindler [48]. The ¯ow path determined by Kim [29] is also shown. (b) Contours in the mixing zone, giving the percent of water
derived from Crystal Lake. Mixing percents were calculated using oxygen isotope measurements reported by Kim [29].

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 construction 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 8:6  10ÿ5 to 3:5  10ÿ3 m=day for the silt
layers [26]. The K values 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
Groundwater in the isthmus is recharged in part by
seepage from Crystal Lake and in part by precipitation
on the upland between the two lakes (Fig. 2). The isotopic contrast between those two sources of water as
determined by Kim [29] (d16 O ˆ ÿ3:72& for Crystal
Lake; d18 O ˆ ÿ11:2& for recharge water) makes it
possible to determine the mixing ratio of water exiting
Crystal Lake. Water samples for isotope analyses were
collected on 29 June 1994 and 13 July 1994 and results
are reported in [29]. Several wells were sampled on both

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K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

dates and the isotopic variation of these replicate samples 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 kriging option of a commercial software package,
äTransform (Fortner Research). Based on the measured 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 dicult 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 o€shore
of Big Muskellunge Lake (Figs. 1 and 3). Layer thicknesses 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

Fig. 3. Hydraulic conductivity zones and boundary conditions for the base model. Numbers in brackets ‰ Š identify each zone. Layers that constitute
zone 6 were inserted into the upper sand unit during model calibration to represent bedding planes dipping at an angle of 7° to the horizontal axis.
The three particle release points used in the particle tracking simulations are designated by
. Hydraulic conductivity values for each zone are given
in Table 1.

K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

[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 coordinate axis of the grid [21]. Hence, the upper sand unit is
represented in the model by a layered system of anisotropic 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 tracking 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

Table 1
Parameter values used in the base modela
Kx of sand (m/day)
Zone [1]
Zone [2]
Zone [3]

8.0
2.0
0.7

Kx of silt (m/day)
Zone [4]
Zone [5]

0.04
0.004

Kx of bedding plane layers in the upper sand (m/day)
Zone [6]

0.3

Porosity
Sand
Silt

0.35
0.30

629

conductivity zone was assigned a value of ®ve. The parameters 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 waterequivalent 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]. Precipitation 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 recharge 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 Ecological Research (LTER) program. Prior to 1981, lake
levels were estimated using water-level data for Bu€alo
Lake, a seepage lake (i.e., no surface inlets or outlets)
that is located 14 km south of the study area and ¯uctuates in a manner similar to Crystal Lake and Big
Muskellunge Lake, at least after 1981 (Fig. 5(a)). Water
levels in Bu€alo Lake have been monitored since 1940
by the Wisconsin Valley Improvement (1994, unpublished 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

b

Anisotropy ratio (Kx /Kz )
Storage coecient
Speci®c yield

5
0.0001
0.27

a
Numbers in brackets ‰ Š correspond to the zone numbers shown in
Fig. 3. Kx is horizontal hydraulic conductivity; Kz is vertical hydraulic
conductivity.
b
The anisotropy ratio is the same for each conductivity zone.

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.

630

K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

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.

3.3.1. Background
According to Bear [38, p. 579], dispersion is the
process by which a tracer ``gradually spreads and occupies 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 coecient in the advection±dispersion
equation represents spreading of the tracer by di€usion
and by local dispersive mixing that occurs as a result of
velocity variations caused by ``smaller-scale erratic motions 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 coecient that were
much larger than expected based on local dispersive
mixing [40]. The consensus emerged that relatively large-

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 Bu€alo Lake level, with r2 ˆ 0:898 (C Lake) and r2 ˆ 0:874
(B.M. Lake).

scale geological heterogeneities cause what is generally
called macroscopic dispersion. Furthermore, as noted in
section 1.0 above, it is now widely recognized that
transient e€ects also can enhance transverse dispersion.
In current usage, the concept of dispersion, therefore, is
broader than originally conceived. For example, Freyberg [41, p. 2036] observed that ``. . .the term ÔdispersionÕ
appears to be most commonly used in the physical sense
to refer to all deviations in observed or predicted concentration from that which would be predicted assuming
only advection by an average pore water velocity

K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

®eld. . .'' Later, Gelhar [39, p. 216] developed a model to
simulate macrodispersion that incorporated the e€ects
of both ``fully three-dimensional heterogeneity'' and
local dispersive mixing. Gelhar [39, p. 269] also noted
that ``¯uctuations in ¯ow, though quite small, may signi®cantly enhance transverse mixing, which steady-¯ow
theory predicts to be very small'' and Rehfeldt and
Gelhar [4] developed a model in which the macrodispersivity tensor is de®ned to be the sum of two components: one to account for spatial variability caused by
geological heterogeneity and another to account for
temporal variability caused by transient e€ects. Hence,
as currently understood by most workers, dispersion
includes not only the e€ects of local dispersive mixing
and di€usion, but also mixing caused by large-scale
heterogeneities and transient e€ects.
We used an advective model to track pathlines and
simulate the spreading or dispersion of imaginary particles caused by transient e€ects. 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 consider the e€ects of transience on the vertical transverse 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 macrodispersion, and transient e€ects are included directly
in our ¯ow model. Therefore, we do not require a full
transport analysis with the advection±dispersion
equation to represent these e€ects. Instead, we simulate the spreading caused by these two e€ects directly
using the particle tracking code PATH3D [42]. The
e€ects of molecular di€usion and local dispersive mixing (caused by heterogeneities that occur at a smaller
scale than those we modeled), which in a full transport analysis would be characterized by dispersivity
factors, are not considered here. These e€ects are generally 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 e€ect 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 inferred from ®eld data (Fig. 2(b)). The width of the
mixing zone captures the combined e€ects of mixing
induced by large-scale heterogeneities and the cumulative e€ect of transience caused by ¯uctuations in
recharge rate and lake levels.

631

3.3.2. Application
The movement of water within the mixing zone during a 32 year time period was simulated by tracking 40
particles as each left the lake at a di€erent time starting
from a point at the bottom of Crystal Lake (shown by
the uppermost
in Fig. 3), which is near the edge of the
mixing zone. In one set of simulations we also considered 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 di€erent 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 approximately 8000 days (21.9 years). The ®rst particle entered
the groundwater system at the beginning of the simulation in January 1954 and the last particle entered the
system in the last quarter of 1963. Therefore, 32 years
…ˆ 40
0:25 ‡ 21:9† 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 simulation used the parameters in Table 1 and included transience caused by ¯uctuations in recharge rate and lake
levels. In the second scenario, we removed the largescale 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 recharge. In a ®nal set of simulations, we tested the sensitivity of the results to the distance from the boundary
stresses.

4. Simulation results and discussion
4.1. E€ects of heterogeneity
In the base case simulation, the pathlines spread out
over a vertical distance of about 10 m (Fig. 6(a)), which
is close to the maximum vertical transverse dispersion
exhibited in the mixing zone inferred from ®eld data
(Fig. 2(b)). De¯ections in the pathlines are caused as
particles encounter the bedding plane layers of zone 6
(Fig. 3). When the bedding plane layers were removed in
the second simulation, there was little transverse
spreading (Fig. 6(b)), even with transience in the ¯ow
®eld. Hence, the heterogeneity of the bedding plane
layers is important in causing vertical transverse dispersion in this system.

632

K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

Fig. 6. E€ect of heterogeneity in the upper sand aquifer on pathlines.
Circles represent well points. The 100% mixing isopleth from Fig. 2(b)
is shown by the dashed line. (a) Base case simulation using zonation
shown in Fig. 3, parameter values in Table 1, and assuming ¯uctuations in both recharge and lake levels. (b) Same as (a) except the
horizontal hydraulic conductivity values for zones 1 and 6 were set to
2 m/day so that the upper sand was represented as a homogeneous unit
without bedding plane layers.

In additional simulations that are not shown here, we
tested the e€ect 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. E€ects of ¯uctuations in recharge rate and lake levels
The next set of simulations was designed to test the
signi®cance of transient e€ects 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 e€ects 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

Fig. 7. E€ects 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.

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 transient e€ects 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 dispersion [3]. Seasonal ¯uctuations in recharge cause

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K. Kim et al. / Advances in Water Resources 23 (2000) 625±635
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
Fig.
Fig.
Fig.
Fig.
Fig.
Fig.

6(a)c
6(b)
7(a)a
7(b)
7(c)
8

D in ¯ow direction (degrees)

Recharge

Lake level

Particle starting point

T
T
T
T
A
T
T
T

T
T
T
A
T
T
T
T

S
S
S
S
S
S*
I
D

160
152
160
134
35
160
95
0.8

a

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† ˆ …245; 42† shown 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), where K1 /K6 was 2.0/2.0; K is given in m/day.
b
T is transient; A is average. S, I and D stand for shallow, intermediate and deep, respectively.
c
Identical set of pathlines.

transience in this system by causing the formation of
groundwater mounds between the lakes [19] with accompanying 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 ¯uctuations 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 e€ects 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 e€ects would be signi®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 e€ects will be
signi®cant [9].
4.3. Distance from the boundary stresses
In order to examine the e€ect of near-surface transience 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 location 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 transverse dispersion. The intermediate set of pathlines encounters 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 essentially no transverse dispersion because they encountered
less heterogeneity and are protected from the e€ects 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 seasonal groundwater mounds that form in response to
¯uctuations in recharge rate. Hence, like other researchers including Freeze [43] and Goode and Konikow [3], we ®nd that transient e€ects, 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 d18 O 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

634

K. Kim et al. / Advances in Water Resources 23 (2000) 625±635

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

Fig. 8. Pathlines generated from the shallow, intermediate and deep
particle release points shown in Fig. 3 under transient recharge and
transient lake levels. Circles represent well points. The 100% mixing
isopleth from Fig. 2(b) is shown by the dashed line.

precipitation and has an isotopic content of d18 O equal
to ÿ11:2&. The vertical transverse dispersion of this
isotope plume is enhanced by ¯uctuations in the direction 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 ¯uctuations 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 e€ects of transience diminished with depth
(Fig. 8) as the distance from the recharge boundary
was increased.
· Dispersion was signi®cantly enhanced by transient effects, but transience alone did not create signi®cant
dispersion in our system. (Compare Fig. 6(a), which
includes the e€ects 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 annual changes in ¯ow direction up to 160°. The formation
of seasonal groundwater mounds near lakes is a well-

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 Feinstein, 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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