U.S. Department of the Interior

U.S. Geological Survey

Simulation of the Shallow Hydrologic System in

the Vicinity of Middle Genesee Lake,

Wisconsin, Using Analytic Elements and

Parameter Estimation

Water-Resources Investigations Report 00–4136

Prepared in cooperation with the

Middle Genesee Lake Management District

Wisconsin Department of Natural Resources
Simulation of the Shallow Hydrologic 

System in the Vicinity of Middle Genesee 

Lake, Wisconsin, Using Analytic Elements 

and Parameter Estimation

By R.J. Hunt, Y. Lin, J.T. Krohelski, and P.F. Juckem

U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 00–4136

Prepared in cooperation with the

Middle Genesee Lake Management District

Wisconsin Department of Natural Resources

Middleton, Wisconsin

2000
U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

For additional information write to: Copies of this report can be purchased from:

District Chief U.S. Geological Survey 

U.S. Geological Survey Branch of Information Services

8505 Research Way Box 25286

Middleton, WI 53562-3586 Denver, CO 80225-0286

Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement

by the U.S. Government.
CONTENTS III

CONTENTS

Abstract ................................................................................................................................................................................. 1

Introduction........................................................................................................................................................................... 1

Purpose, scope, and data sources ................................................................................................................................ 3

Methods....................................................................................................................................................................... 3

Physical setting ........................................................................................................................................................... 4

Conceptual model ................................................................................................................................................................. 4

Development of the GFLOW model..................................................................................................................................... 5

Model calibration .................................................................................................................................................................. 6

Hydrologic budget................................................................................................................................................................. 9

Application of model to simulate changes in lake stage due to hydrologic stress................................................................ 11

Scenario number 1—Addition of nine pumping wells ............................................................................................... 11

Scenario number 2—Pumping from Lower Genesee Lake ........................................................................................ 11

Scenario number 3—Reduction in recharge rate ........................................................................................................ 13

Effect on Middle Genesee Lake.................................................................................................................................. 13

Summary and conclusions .................................................................................................................................................... 13

References............................................................................................................................................................................. 16

FIGURES

1. Map showing location of Middle Genesee Lake and study area, Waukesha County, Wisconsin ............................ 2

2. Section showing conceptual model of shallow hydrologic system in the vicinity of Middle Genesee Lake,

Waukesha County, Wisconsin .............................................................................................................................. 4

3. Graph showing lake stage and expected range of fluctuation for Middle Genesee Lake, Wisconsin,

June 1995—June 2000 ........................................................................................................................................ 5

4–8. Maps showing:

4. Simulated hydrologic features with analytic elements, initial water-table elevation and calibration targets ..... 7

5. The ground-water capture zone of Middle Genesee Lake, Wisconsin simulated using the calibrated model

region without development ........................................................................................................................... 10

6. The ground-water capture zone of Middle Genesee Lake region for scenario number 1................................... 12

7. The ground-water capture zone of Middle Genesee Lake region for scenario number 2................................... 14

8. The ground-water capture zone of Middle Genesee Lake region for scenario number 3................................... 15

TABLE

1. Measured and simulated values and weight of calibration targets ........................................................................... 8
IV CONVERSION FACTORS, VERTICAL DATUM, AND ABBREVIATED WATER-QUALITY UNITS

CONVERSION FACTORS, VERTICAL DATUM, AND ABBREVIATED WATER-QUALITY UNITS

Multiply By To Obtain

Length

inch (in.) 25.4 millimeter

foot (ft) 0.3048 meter

mile (mi) 1.609 kilometer

Area

acre (A) 0.4047 hectare

square foot (ft2) 0.09290 square meter

square mile (mi2

) 2.590 square kilometer

Volume

cubic foot (ft3) 7.4805 gallon

Hydraulic conductivity*

foot per day (ft/d) 0.3048 meter per day

Sea level: In this report, “sea level” refers to the National Geodetic Vertical Datum of 1929 (NGVD of 1929)—a geodetic datum derived 

from a general adjustment of the first-order level nets of both the United States and Canada, formerly called Sea Level Datum of 1929.

*Hydraulic conductivity: The standard unit for hydraulic conductivity is cubic foot per day per square foot of aquifer cross-sectional area 

(ft3/d)/ft3. In this report, the mathematically reduced form, feet per day (ft/d), is used for convenience.

Other abbreviations:

gal/min gallons per minute

ft3/s cubic feet per second

in/yr inches per year

ft/d feet per day

The stratigraphic nomenclature used in this report is that of the Wisconsin Geological and Natural History Survey and does not necessarily 

follow usage of the U.S. Geological Survey.
Abstract 1

Abstract

Middle Genesee Lake is a ground-water 

flow-through lake located in a developing area in 

southeastern Wisconsin. Because the lake is in 

good connection with the shallow ground-water 

system, hydrologic stresses to the shallow groundwater system could adversely affect the lake system. In order to assess the effects of potential 

stresses on the lake, a study was completed by the 

U.S. Geological Survey, in cooperation with the 

Middle Genesee Lake Management District. The 

objective of the study was to identify areas that 

contribute ground water to the lake and estimate 

the hydrologic budget of the lake and hydraulic 

parameters affecting ground-water flow. A twodimensional, steady-state analytic element model 

of the lake and surrounding area was developed 

using the computer code GFLOW. A parameter 

estimation model, UCODE, was used to optimize 

the calibration to measured water levels and 

streamflow. 

The calibrated model was used to evaluate 

the effect of three hypothetical stress scenarios on 

the stage of Middle Genesee Lake; the simulations 

were linked to UCODE, which formally incorporated parameter uncertainty into 95-percent confidence intervals around the simulated value. The 

scenarios included: (1) pumping from upgradient 

irrigation wells, (2) pumping from Lower Genesee 

Lake to lower lake levels, and (3) reduction in 

recharge resulting from development. The results 

of the simulations demonstrated that lake levels 

could be affected by hydrologic stresses in the shallow hydrologic system, with effects ranging from a 

2.7 feet decline in lake stage resulting from pumping in Lower Genesee Lake to a 0.1 feet decline in 

lake stage from development in part of the upgradient recharge area. The range of lake stage decline 

increased when parameter uncertainty was 

included, from a decline of 3.1 feet for pumping 

from Lower Genesee Lake to no reduction in lake 

stage for the development in the recharge area. 

Whereas these simulated effects are within the natural variation in lake stage, they represent a systematic reduction of ground-water flow to the lake. 

Therefore, these hypothetical stresses are expected 

to establish a new, lower, baseline lake stage over 

which the natural variation due to climatic effects 

are added and subtracted.

INTRODUCTION

Middle Genesee Lake is a ground-water flowthrough lake located in a developing area in southeastern Wisconsin (fig. 1). Because the lake is in good connection with the ground-water system, hydrologic 

stresses in the shallow ground-water system could 

adversely affect the lake. In order to assess the effects of 

stresses on the lake, a study was completed by the U.S. 

Geological Survey, in cooperation with the Middle Genesee Lake Management District. The study was initiated 

in 1999 with the following goals: (1) identification of 

the ground-water recharge area that contributes water to 

the lake, (2) estimation of the hydrologic budget of the 

lake and, (3) estimation of hydraulic parameters affecting ground-water flow. In addition to these goals, the 

study also included evaluating effects of potential 

hydrologic stresses to the lake, and formally addressed 

the issue of uncertainty in the hypothetical simulations. 

The hypothetical stresses assessed included: (1) nearby 

pumping of irrigation wells, (2) the artificial lowering 

of lake levels of Lower Genesee Lake and (3) reduction 

in recharge resulting from potential development in the 

basin.

Simulation of the Shallow Hydrologic System in the 

Vicinity of Middle Genesee Lake, Wisconsin, Using 

Analytic Elements and Parameter Estimation

By R.J. Hunt, Y. Lin, J.T. Krohelski, and P.F. Juckem
2 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

88°35'30" 88°30' 88°22'30"

43°07'30"

43°

Middle

Genesee

Lake

Duck

Lake Lower

Genesee

Lake

Silver

Lake

Oconomowoc

Lake

Base from U.S. Geological Survey

1:100,000 Oconomowoc West

Bark River

0

0 1

1

2 3

2 3

KILOMETERS

MILES Pretty

Lake

94

94

Upper

Genesee

Lake

Delafield

Dam

Battle Creek

Oconomowoc River

Location of study area

in Waukesha County.

67

18

Figure 1. Location of Middle Genesee Lake and study area, Waukesha County, Wisconsin.
INTRODUCTION 3

Purpose, Scope, and Data Sources

The purpose of this report is to present the results 

of flow modeling of the hydrologic system in the vicinity of Middle Genesee Lake. Because the ground-water 

and surface-water systems are in good hydrologic connection in this area, the hydrologic system simulated in 

this study encompasses the lakes, streams and shallow 

ground-water system. Only existing geologic and 

hydrologic data were used during this study. The 

ground-water-flow model complexity was commensurate with the extent of the existing data set. That is, the 

data set did not include vertical gradients or groundwater elevation (head) data over time; therefore, the 

system was simulated using an areal, two-dimensional, 

steady-state model. Two-dimensional assumptions are 

appropriate because the shallow ground-water-flow 

system is thin and areally extensive. Steady-state 

assumptions are appropriate because the system has 

high hydraulic conductivity and relatively small distances between surface-water features helping to ensure 

that periodic transient stresses are mitigated quickly 

within the system. In order to describe the largest effect 

on the shallow hydrologic system, hypothetical stress 

simulations also were simulated using steady-state conditions.

Geologic data consisted of interpretive maps presented by Gonthier (1975) and well-construction 

reports (Wisconsin Department of Natural ResourcesBureau of Drinking Water and Groundwater, 1998). 

These data were used to estimate saturated thickness of 

the shallow aquifer and provided estimates of watertable elevation. Streamflow measurements from the 

USGS gaging station on the Bark River at Rome, Wisconsin, were used to estimate flow duration. USGS 

7.5 minute topographic maps were used to locate and 

estimate elevations of surface-water features for model 

development.

Methods

An analytic element ground-water-flow model, 

using the computer program GFLOW (Haitjema, 1995), 

was developed to simulate the shallow ground-water 

system and its interaction with surface-water features. 

A complete description of analytic elements is beyond 

the scope of this report; a brief description is given 

below.

An infinite aquifer is assumed in analytic element 

modeling. The problem domain does not require a grid 

or involve interpolation between cells. To construct an 

analytic element model, features important to groundwater flow (for example, wells) and surface-water features are entered as mathematical elements or strings of 

elements. The amount of detail specified for the features 

depends on distance from the area of interest. Each element is represented by an analytic solution. The effects 

of these individual solutions are superposed or added 

together to arrive at a solution for the ground-waterflow system. Because the solution is not confined to a 

grid, heads and flows can be computed anywhere in the 

model domain without nodal averaging. In the GFLOW 

model used here, the analytic elements are two-dimensional and are used to only simulate steady-state conditions (that is, heads do not vary with time). The analytic 

element method, and the comparison of analytic element to finite-difference numerical model techniques, 

have been discussed by others (Strack, 1989; Haitjema, 

1995; and Hunt and Krohelski, 1996; Hunt and others 

1998, respectively).

The GFLOW model was calibrated using parameter estimation techniques. The use of parameter estimation for calibration is a relatively new advancement for 

the science. There are numerous publications detailing 

the advantages of parameter estimation models (for 

example, Hill 1992; Poeter and Hill, 1997; Hill 1998). 

Briefly, the primary benefit of a properly prepared 

parameter estimation model over typical “trial-anderror” calibration is the ability to automatically calculate parameter values (for example, hydraulic conductivity and recharge) that are a quantified best fit between 

simulated model output and observed data (for example, ground-water levels and streamflows). Other benefits also result, such as the quantification of the quality 

of the calibration and a statistically rigorous measure of 

the uncertainty (or confidence interval) of hypothetical 

simulations made using the optimized model. In addition, parameter correlation (for example, hydraulic conductivity and recharge) and parameter sensitivity can be 

quantified and assessed. In this study, the GFLOW 

model was coupled with the parameter estimation code 

UCODE (Poeter and Hill, 1998). This report is among 

the first published application linking an analytic element model to a multi-objective function parameter 

estimation code.
4 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

Physical Setting

Middle Genesee Lake, located in west-central 

Waukesha County, Wisconsin (fig. 1), is a 109-acre 

seepage lake (a lake without stream inlets or outlets). 

Four other lakes are located in the vicinity. Lower Genesee Lake, which is similar in size and shape to Middle 

Genesee Lake, is closest and is located to the south. The 

two lakes are separated by a thin strip of land only 

slightly wider than the width of a road constructed 

between the two lakes. The shorelines of both lakes are 

developed with year-round homes. Rapid development 

—both commercial and residential—is occurring and is 

expected to continue throughout the region. Ground 

water is the source of all domestic and municipal water 

supplies in the area of the lakes. An area of irrigation, 

known locally as Pabst Farms, is located approximately 

1 mi to the north of the lakes. Local streams in the area 

include the Oconomowoc River, Battle Creek, and the 

Bark River (fig. 1).

CONCEPTUAL MODEL

Prior to simulating the ground-water system using 

a flow-modeling code, a conceptualization of the hydrologic system is essential because it forms the framework 

for model development. The conceptualization reduces 

the ground-water system into important component 

parts. This reduction is a necessary simplification of the 

hydrologic system because inclusion of all of the complexities into a model is not feasible. Steps in the development of the conceptual model include: (1) definition 

of the aquifer(s), (2) identification of sources and sinks 

of water, and (3) identification and delineation of hydrologic boundaries present in the area of interest. The conceptualization of the shallow hydrologic system in the 

vicinity of Middle Genesee Lake used in this study is 

shown in figure 2.

The shallow ground-water system consists of a laterally extensive unconsolidated deposit of varying 

thickness. Within these sediments, saturated outwash 

(sand and gravel) about 100 ft thick forms a shallow 

aquifer (Gonthier, 1975). Although the underlying bedrock unit could be considered an aquifer, the ability of 

this unit to transmit water is much less than the overlying unconsolidated sediments. Therefore, the model 

included only the most transmissive upper sediments. 

The depth to ground water (water table) in the vicinity 

of the Genesee Lakes is generally less than 10 ft. The 

depth to water increases as land-surface elevation 

increases. 

Ground water moves from higher to lower potentials (areas of higher ground-water levels to areas of 

lower ground-water levels). As a result, ground water 

generally discharges to surface-water features and 

recharges in areas away from these features. Due to the 

position of Middle Genesee Lake in the ground-water 

basin, it is likely that the lake receives ground-water 

flow on the upgradient side (the side with higher 

ground-water levels) and the lake contributes flow to 

the ground-water system on the downgradient side (the 

side with lower ground-water levels). Therefore, 

ground water flows through the lake. This conceptualization of the interaction of ground water and surface 

water also was observed at nearby Pretty Lake, which is 

located about 6 mi to the south of the Genesee Lakes 

(fig. 1), by Hunt and Krohelski (1996). In the twoLower

Genesee

Lake

Middle

Genesee

Lake

Crooked

Lake

Bark

River Shallow aquifer

Shallow aquifer

Unsaturated

zone

SW

NE

 Recharge

 General direction of ground-water flow

 Pumping well

EXPLANATION

Figure 2. Conceptual model of shallow hydrologic system in the vicinity of Middle Genesee Lake, Waukesha County, 

Wisconsin.
DEVELOPMENT OF THE GFLOW MODEL 5

dimensional model used in this study, it was assumed 

that all upgradient ground water discharges into the 

lake. That is, no ground water flows beneath Middle 

Genesee Lake; thus, the lake acts as a fully penetrating 

boundary.

Middle Genesee Lake is a seepage lake as there are 

no surface-water outlets out of the lake. Lake levels 

have been measured intermittently during the period 

1995–99; the mean lake level was 866.1 ft above sea 

level and fluctuated about 0.9 ft above and below the 

mean (fig. 3). From these data collected to date, it 

appears that Middle Genesee Lake levels fluctuate in 

the range expected for a ground-water flow-through, 

seepage lake. The water level of a typical ground water 

flow-through lake in Wisconsin can be expected to be 

1.4 ft above or below the long-term mean level of the 

lake about 10 percent of the time (House, 1985).

In addition to ground-water flow, other hydrologic 

budget components for Middle Genesee Lake include 

precipitation falling on the lake and water evaporating 

from the lake surface. In southeastern Wisconsin, 

annual precipitation exceeds evaporation by about 

2 in/yr (Novitzki, 1982). Overland flow is assumed to 

be insignificant because infiltration rates of the sandy 

surface deposits are rarely exceeded by precipitation 

rates. Thus, overland flow is not included as a budget 

component in the hydrologic conceptual model.

DEVELOPMENT OF THE GFLOW MODEL

Initial model development included estimating the 

elevation of the base of the shallow aquifer system, a 

global recharge rate, and a horizontal hydraulic conductivity. The base of the model approximates the bottom 

of the high conductivity unconsolidated sediments 

(about 800 ft above sea level). The recharge rate and 

horizontal hydraulic conductivity were considered calibration parameters; thus, these parameters were varied 

during model calibration. Initially, recharge was set to 

864.6

865.1

865.6

866.1

866.6

867.1

867.6

June 1995 June 1996 June 1997 June 1998 June 1999 June 2000

LAKE STAGE, IN FEET ABOVE SEA LEVEL

Mean lake level

10 percent of the time ground-water flow 

through lakes will be 1.4 feet above or 

below mean lake level (House, 1985)

Expected range of natural fluctuation

Figure 3. Lake stage and expected range of fluctuation for Middle Genesee Lake, Wisconsin, June 1995–June 2000.
6 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

5.9 in/yr and horizontal hydraulic conductivity set to 

250 ft/d based on previous modeling results in the area 

by Hunt and Krohelski (1996). 

The ground-water flow model consists of the “farfield” and “near-field” elements. Based on the conceptual model, the location and elevation of far-field surface-water features were added to the model (fig. 4a). 

These are rivers and lakes distant from Middle Genesee 

Lake that are simulated with coarse linesinks and little 

or no resistance between the surface-water features and 

the ground-water system (that is, simulated as having 

good hydraulic connection). The purpose of simulating 

the far-field features is to have the model explicitly 

define the regional ground-water-flow field around 

Middle Genesee Lake area (called the “near field”). The 

near field is the primary area of interest and in this study 

encompasses the Upper, Middle, and Lower Genesee 

Lakes, as well as other nearby features that affect the 

hydrology of the lakes (fig. 4b). 

Streambed sediment resistance in the near and far 

field was set equal to 0.3 days. Resistance in analytic 

element modeling is calculated by dividing the streambed sediment thickness by the vertical hydraulic conductivity. For this model, the value of 0.3 days 

corresponds to a 1-ft sediment thickness and a vertical 

hydraulic conductivity of 3.3 ft/d. The width of the 

stream was assigned according to stream order, and 

ranged from 10 to 50 ft. Parameter sensitivity assessments within UCODE demonstrated that the model 

results are not sensitive to changes in stream resistance 

when varied over reasonable ranges; therefore, the values for specific streams were fixed in all model runs.

Streams in the far field are not used for flux calibration; thus, streams are simply modeled as individual 

linesinks. In near-field streams a special type of linesink 

was used, called a “stream element” (Mitchell-Bruker 

and Haitjema, 1996). This element consists of linked 

linesinks that route water from high elevation linesinks 

to low elevation linesinks. During the routing through 

the stream network, the amount of water captured from 

and lost to the shallow ground-water system by the 

stream is tabulated. This accounting allows easy determination of fluxes from any linesink in the stream network that includes flows from all the upstream 

linesinks. More importantly, the accounting also 

ensures that the amount of stream water lost to the 

ground-water system is restricted to the amount of 

water available (that is, water delivered from upgradient 

linesinks in the network). For streams where the headwaters are not included in the model domain, a headwater inflow term can be specified. This option was 

utilized for the Bark River – a stream with a large flow 

duration data set and one that includes an appreciable 

headwater reach that is not in the model domain. Based 

on field measurements at the Delafield Dam, the 

amount of headwater inflow was set to 28.3 ft3

/s, and 

was added to the stream element immediately downgradient of the dam (fig. 4b).

Lakes where simulation of lake stage is not desired 

(lakes not adjacent to Middle Genesee Lake) were simulated using linesinks with resistance. Drainage lakes in 

the near field were linked to the stream network by 

stream elements based on the methodology of Hunt and 

others (1998). The value of resistance was assigned 

according to the lake’s geologic setting (for example, in 

sandy sediments, near wetlands). The resulting resistance varied over a small range (from 0.3 to 2.0 days). 

Similar to Hunt and Krohelski (1996), simulation of 

Duck Lake, and Upper, Middle, and Lower Genesee 

Lakes used inhomogeneities (change in aquifer properties) rather than the linesinks used to simulate the 

streams and the far-field lakes. This formulation allows 

the model to account for the average rate of precipitation and evaporation and directly solve for lake stage, 

whereas linesinks require the input of a fixed lake stage. 

Use of inhomogeneities required the specification of a 

hydraulic conductivity in the lake three orders of magnitude larger than the aquifer to represent the equipotential surface (Hunt and Krohelski, 1996; Chung, 1998). 

Lakes simulated with inhomogeneities ensure that the 

water table in the vicinity of Middle Genesee Lake is 

allowed to fluctuate and is not overspecified by headdependent flux boundaries.

MODEL CALIBRATION

The model solution using the initial parameter values resulted in a water-table configuration (fig. 4b) similar to the water table presented by Gonthier (1975) with 

water flowing from the northeast to the southwest in the 

Middle Genesee Lake area. In order to determine the 

“best fit” of model results to measured values of watertable elevation and streamflows, a more formal calibration process was used (described below).

Calibration targets are measured field data, which 

are used to evaluate how well the model represents the 

hydrologic system. The targets used here include both 

ground-water levels and streamflows (fig. 4b). Only 

existing field data were used in this study; no additional 

data were collected. Ground-water levels for 17 existing 
MODEL CALIBRATION 7

Figure 4. Simulated hydrologic features with analytic elements, initial water-table elevation and calibration targets, 

(a) far field elements (Global recharge is applied for the entire far field); (b) near field elements.

42°52'30" 

88o22’30”

43°05'

88o37’30”

88°

37'30" 88°

22'30"

43°

05'

Base from U.S. Geological Survey

1:100,000 Oconomowoc W 1:100,000 Oconomowoc West

0 1 2 3 KILOMETERS

0 1 2 3 MILES

0 1 2 3 KILOMETERS

Base from U.S. Geological Survey 0 1 2 3 MILES

1:100,000 Oconomowoc West

Middle

Genesee

Lake

Middle

Genesee

Lake

Bark River

Bark River

Pretty

Lake

840

850

840

840

890

JE-3

JE-3 JE-4 JE-5

JE-6

JE-7

WK-9

JE-1

WK-12

WK-7 WK-8 WK-11

WK-5

MIDGEN

LOWGEN

UPGEN

DUCKL

WK-3

WK-4

WK-2

WK-1

MIDGEN 01WK

LOWGEN

UPGEN

DUCKL

Bark River

Near Rome

Bark River

Near Rome

EXPLANATION

Well location and 

identification number

Stream site and

identification 

Analytic element for GFLOW

Simulated water-table contour

at 10 foot intervals. Datum is 

sea level

(a) Far field

(b) Near field

Delafield

Dam
8 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

wells were obtained from well-construction reports. In 

addition to the ground-water levels, calibration targets 

representing the lake levels for Duck Lake as well as 

Upper, Middle, and Lower Genesee Lakes also were 

used. Data from a historical streamflow gaging station 

on the Bark River gaging station at Rome, Wisconsin, 

also were used as a calibration target. This target was 

used to constrain the simulated fluxes and associated 

regional recharge. Average conditions (here defined as 

the “Q50” or flow that occurs 50 percent of the time) 

were used for calibration; the Q50 flow duration was 

estimated to be 77 ft3/s for the Bark River gaging station. The relation of the target to the model calibration 

is such that lower values of the streamflow target result 

in lower rates of recharge and lower corresponding horizontal hydraulic conductivity.

The GFLOW model was coupled to UCODE (Hill 

and Poeter, 1998) to formulize the selection of an optimal set of parameters (a set that best matches observed 

ground-water levels and flows). One of the most important operations in using a parameter estimation technique such as that used in UCODE is assigning weight 

to the observations. The weights assigned to each calibration target and results of the optimized model solution are shown in table 1. Distant calibration targets and 

those with higher uncertainty were assigned lower 

weight for the calibration process. Lake stage and 

ground-water levels are in feet above sea level and 

Table 1. Measured and simulated values and weight of calibration targets

[Lake stage and ground-water levels are in feet above sea level and streamflow is the 50-percent flow

duration in cubic feet per second. Weights for targets are reported as standard deviations. Head targets

are divided into near-field targets (standard deviation equal or less than five feet) and far-field targets 

(standard deviation equal 10 feet). Lake locations are shown in figure 1; ground-water level and flux 

target locations are shown in figure 4b.]

Calibration target Measured – Simulated = Residual Weight

Lake stage (feet above)

Middle Genesee 863 862.62 0.38 1

Lower Genesee 862 861.85 .15 4

Upper Genesee 864 865.07 -1.07 4

Duck Lake 864 864.47 -.47 4

Ground-water level (feet above)

JE_1 845 859.85 -14.85 10

JE_3 839 844.89 -5.89 10

JE_4 830 849.52 -19.52 10

JE_5 836 849.55 -13.55 10

01WK 889 884.29 4.71 5

WK_1 865 863.86 1.15 5

WK_2 869 864.89 4.12 5

WK_3 864 868.71 -4.71 5

WK_4 870 867.17 2.83 5

WK_5 868 868.99 -.99 5

JE_6 834 852.02 -18.02 10

WK_7 868 865.33 2.67 5

WK_8 867 868.95 -1.95 5

JE_7 841 845.95 -4.95 10

WK_9 857 858.30 -1.30 5

WK_11 869 874.51 -5.51 5

WK_12 869 863.23 5.77 5

Streamflow (cubic feet per second)

Bark River near 

Rome, Wisconsin

77 77.3 -0.3 2.3
HYDROLOGIC BUDGET 9

streamflows are in cubic feet per second (ft3/s). Weights 

for ground-water level, lake stage, and streamflow targets are expressed as a standard deviation. Hill (1998) 

gives a detailed explanation of the use of these statistics 

to represent uncertainty. Ground-water levels are 

divided into far-field targets (standard deviation equal 

to 10 ft) and near-field targets (standard deviation equal 

to 5 ft). The assigned standard deviation reflects uncertainty associated with well locations, concerns with 

how well the measurements reflect average conditions, 

and the qualitative desire for a better goodness of fit in 

the near-field targets than far-field targets during 

parameter estimation. Streamflow for the gaging station 

at Rome, Wisconsin was assigned a standard deviation 

of 2.3 ft3

/s (table 1). This value corresponds to a high 

weight for this problem and reflects (1) the long-term 

flow record at the gaging station, and (2) that additional 

weight is needed to offset the higher number of watertable elevation targets. 

During calibration, horizontal hydraulic conductivity and recharge were adjusted in the model to obtain the 

best fit to the observed heads and streamflows. 

Although initial model runs included combinations of 

hydraulic conductivity zonation and recharge zonation, 

UCODE parameter sensitivity evaluations demonstrated that the field data did not support this level of 

complexity and the model could be adequately calibrated using one global hydraulic conductivity and one 

global recharge rate. This parsimonious design also corresponds to the homogeneous geological structure 

expected in the outwash sediments and uniform precipitation conditions expected in the study area. The optimized model indicated horizontal hydraulic conductivity and recharge rate of 112 ft/d and 6.7 in/yr, 

respectively. Unweighted statistics comparing measured ground-water levels to calibrated modeled values 

include an average difference of –3.2 ft, a mean absolute error of 5.2 ft and the root mean square error of 

7.7 ft. The streamflow in the Bark River also was well 

simulated (77.3 ft3

/s simulated flow versus 77.0 ft3/s 

measured flow). The optimized values for horizontal 

hydraulic conductivity and recharge (112 ft/d and 

6.7 in/yr) are similar to the values used near the southern boundary of the model at Pretty Lake (250 ft/d and 

5.9 in/yr, respectively) by Hunt and Krohelski (1996).

HYDROLOGIC BUDGET

An annual hydrologic budget for Middle Genesee 

Lake can be described by

∆S = P – E + GWin - GWout,

where

∆S is change in lake storage,

 P is volume of precipitation falling directly on

the lake, 

 E is volume of water evaporated from the lake

surface,

GWin is volume of ground-water flow into lake,

and

GWout is volume of flow out of the lake to the

ground-water system.

Various assumptions were made in the application 

of this equation. Overland flow is assumed to be negligible because infiltration rates of the sandy surface 

deposits are rarely exceeded by precipitation rates. 

Annual evaporation is assumed to be about 2 in/yr less 

than precipitation (Novitzki, 1982). Average precipitation is 32 in/yr for the period 1961 to 1990; climatological data are available at 

http://www.crh.noaa.gov/mkx/data/wipcpn.gif. It is 

assumed that lake storage does not change with time; 

that is, the hydrologic system is at a steady state. In 

addition, it is assumed that Middle Genesee Lake acts as 

a fully penetrating boundary; thus, ground water does 

not flow under the lake. 

The calibrated model results (fig. 5) were used to 

estimate GWin; the GWout term is estimated as a residual. The width of the simulated contributing area was 

used, in conjunction with Darcy’s law, to estimate the 

volume of ground-water flow into the lake. Groundwater flow into the lake is equal to the cross-sectional 

area, which is the width of the contributing area multiplied by the saturated aquifer thickness, multiplied by 

the model-simulated gradient and the horizontal 

hydraulic conductivity. The contributing area of the 

lake is the land area with the same horizontal extent as 

that part of the aquifer from which ground-water flow is 

diverted to the lake. The contributing area of the lake 

was delineated by backward particle tracking from the 

lake edge. Mathematical particles of water were placed 

around the edge of the lake at the bottom of the aquifer 

and traced backwards to the water table. Using a porosity value typical of sand (0.25 – Freeze and Cherry, 

1979), estimates of the time required for a particle of 

water located in the contributing area to reach the lake 

also can be calculated (fig. 5). 
10 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

80°30' 80°27'30" 80°25'

43°05'

43°02'30'

Middle

Genesee

Lake

Lower

Genesee

Lake

Oconomowoc

Lake

94

67

Bark River

200

100

50

10

10

Base from U.S. Geological Survey

1:100,000 Oconomowoc West 0

0 1

1

2 3

2 3

KILOMETERS

MILES

Oconomow oc River

EXPLANATION

Undeveloped capture zone

Travel time, in years

MIDDLE GENESEE LAKE LEVEL

Average measured lake level:

Simulated level:

863.00 feet above sea level

862.62 feet above sea level 

Silver Lake

Figure 5. The ground-water capture zone of Middle Genesee Lake, Wisconsin simulated using the calibrated model

region without development.
APPLICATION OF MODEL TO SIMULATE CHANGES IN LAKE STAGE DUE TO HYDROLOGIC STRESS 11

Using this approach, estimates of the hydrologic 

budget components of Middle Genesee Lake are:

P = 32 in/yr, 

E = 30 in/yr,

GWin = 25.2 in/yr, and

GWout = P – E + GWin = 27.2 in/yr.

APPLICATION OF MODEL TO SIMULATE 

CHANGES IN LAKE STAGE DUE TO 

HYDROLOGIC STRESS

Three hypothetical hydrologic stress scenarios 

were assessed with the calibrated model. These include 

the effects on the lake stage of Middle Genesee Lake 

due to: (1) pumping of nearby irrigation wells, (2) the 

artificial lowering of lake levels of Lower Genesee 

Lake, and (3) reduction in recharge to the shallow 

ground-water system resulting from development in the 

basin. These three hypothetical scenarios were simulated using the calibrated ground-water flow model by 

adding nine pumping wells in the vicinity of the contributing area with each well pumping at a steady-state rate 

of 50 gal/min (scenario number 1), pumping from 

Lower Genesee Lake at a steady-state pumping rate of 

500 gal/min (scenario number 2), and reducing recharge 

rate by 25 percent within an area to the northeast of 

Middle Genesee Lake (scenario number 3).

The scenarios described above were assessed by 

comparing calibrated lake stages and lake contributing 

areas to simulated lake stages and contributing areas 

after the hypothetical stresses were added. A range in 

the simulated lake stage is determined by applying 

parameter-estimation techniques, and is provided to 

describe the uncertainty present in the model due to 

uncertainty in selected underlying parameters (horizontal hydraulic conductivity and recharge). The uncertainty is reported as a 95-percent confidence interval 

around the simulated value; if the underlying assumptions are met, this range can be considered a good representation of the extreme values that are expected 

given the uncertainty in the model parameters. It should 

be noted that the range of uncertainty could be reduced 

by lowering the uncertainty within the model. Lowering 

the uncertainty would require refining existing data 

such as improving well locations, elevations, and average water levels and average flows, or collecting additional long-term data in areas near Middle Genesee 

Lake.

Scenario Number 1—Addition of Nine 

Pumping Wells

The addition of nine pumping wells to the north of 

Middle Genesee Lake with a pumping rate of 

50 gal/min would reduce the lake stage by 0.4 ft with a 

possible range of no effect (no lowering of lake level) to 

a maximum of 2.0 ft. Because the lake and wells draw 

water from the same source, the amount of lake stage 

decrease is sensitive to the pumping rate assigned to the 

pumping wells. The 50 gal/min rate used in this simulation is based on withdrawals reported to the Wisconsin 

Department of Natural Resources for 1988, a drought 

year when irrigation water use was extremely high. The 

annual withdrawal rate for the nine irrigation wells 

when averaged was equal to 70 gal/min. The consumptive use (the amount of water not returned to the groundwater system) was assumed to be 70 percent (Weeks 

and Strangland, 1971). Applying the consumptive use 

rate to the annual irrigation rate for the nine wells in 

1988 gives a withdrawal rate of about 50 gal/min. This 

rate should be considered a high end of expected withdrawals because pumping is not expected to be this high 

in non-drought years. 

The effect of pumping the irrigation wells is to shift 

the lake contributing area to the south (fig. 6). Averaging of higher growing-season pumping over the year to 

obtain an average annual pumping rate for the steadystate model may underestimate short-term effects near 

the pumping wells. That is, the shifting of the contributing area and the amount of lake stage reduction may be 

greater when the wells are pumping during the irrigation season than the effects reported by a model simulating steady-state conditions. However, the hydrologic 

system recovers during the non-growing season when 

the wells are not pumping, and the long-term average 

effect on the hydrologic system is expected to be that 

simulated by the steady-state model.

Scenario Number 2—Pumping from Lower 

Genesee Lake

A proposal to lower the stage of Lower Genesee 

Lake also was evaluated using the model. A pumping 

station on Lower Genesee Lake was simulated using a 

well in the lake pumping at a steady-state rate of 

500 gal/min. In this simulation, Middle Genesee Lake 

stage declined by 2.7 ft, with a range of 2.2 to 3.1 ft at 

the 95-percent confidence interval. An important 
12 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

80°30' 80°27'30" 80°25'

43°05'

43°02'30"

Oconomowoc

Lake

94

67

Bark River

Oconomow oc River

Silver Lake

0

0 1

1

2 3

2 3

KILOMETERS

MILES

EXPLANATION

Undeveloped capture zone

Developed capture zone

Travel time, in years

Pumping well

MIDDLE GENESEE LAKE LEVEL CHANGE

Undeveloped lake level:

Developed lake level:

95% C.I. (+)lake level:

95% C.I. (-) lake level:

862.6 feet

862.2 feet (∆h = 0.4 feet)

863.7 feet (∆h = 0.0 feet)

860.6 feet (∆h = 2.0 feet)

Base from U.S. Geological Survey

1:100,000 Oconomowoc West

10

10

50

100

200

Middle

Genesee

Lake

Lower

Genesee

Lake

∆h is change in lake stage;

95% C.I. is 95-percent confidence interval

Figure 6. The ground-water capture zone of Middle Genesee Lake region for scenario number 1 (9 wells pumping 

at 50 gallons per minute each well).
SUMMARY AND CONCLUSIONS 13

assumption in this scenario is that the water pumped for 

Lower Genesee Lake is conveyed away from the lakes 

so that it cannot re-enter the lake systems (that is, conveyed to a down gradient surface-water body by a nonleaking channel or storm sewer). The contributing area 

is in about the same position with or without pumping 

but is larger during pumping (fig. 7).

Scenario Number 3—Reduction in Recharge 

Rate

A 0.1-ft decline in lake levels was simulated by 

reducing recharge rate by 25 percent within an area of 

potential development to the north of Middle Genesee 

Lake, with a range of 0.0 to 2.0 ft at the 95-percent confidence interval. A reduction in recharge is assumed 

because paved areas due to development will limit 

recharge and increase runoff, which may not be infiltrated or recharged to the shallow ground-water system. 

The contributing areas have a similar shape with or 

without the reduction in recharge (fig. 8). The larger 

capture zone resulting from reduced recharge reflects 

the need to encompass a larger area to offset the lower 

recharge rate. If the reduction in recharge encompasses 

more of the contributing area, a greater reduction in lake 

stageand shifting of the contributing area is expected.

Effect on Middle Genesee Lake

The range of the simulated effects shown above is 

similar to natural variations in lake stage (a 2-ft fluctuation or approximately one foot above or below the 

mean lake stage). Whereas it might be concluded that 

the simulated reductions in lake stage are within the natural fluctuation of lake stage, the hydrologic stresses 

simulated here differ from normal climatic variation in 

one important way. Pumping stresses and recharge 

reduction are systematic changes to the hydrologic system. That is, whereas normal changes in climate include 

both wet and dry years, the hydrologic stresses simulated here always reflect a one-directional stress toward 

less water entering the lakes. Therefore, although the 

normal climatic variation would still occur and lake levels would continue to fluctuate, the normal fluctuations 

would be overlain on a new, lower, average lake level. 

Thus, both wetter and drier years would have lower lake 

levels than if the hydrologic stress was not present. 

SUMMARY AND CONCLUSIONS

Middle Genesee Lake is a ground-water flowthrough seepage lake located in a developing area in 

southeastern Wisconsin. The lake is in good connection 

with the ground-water system; thus, hydrologic stresses 

in the shallow ground-water system could adversely 

affect the lake system. A study was completed by the 

U.S. Geological Survey, in cooperation with the Middle 

Genesee Lake Management District, to identify areas 

that contribute ground water to the lake and estimate the 

hydrologic budget of the lake and hydraulic parameters 

affecting ground-water flow.

A two-dimensional, steady-state, analytic element 

ground-water model of the shallow hydrologic system 

was developed and calibrated using the computer code 

GFLOW. The model provides identification of the area 

that contributes water to Middle Genesee Lake and estimates the hydrologic budget of the lake and the hydraulic parameters affecting ground-water flow. The model 

was calibrated to an existing data set that included 

ground-water levels measured in 17 wells, four lakes, 

and the average (Q50) flow duration for a stream gaging 

station on the Bark River located in Rome, Wisconsin. 

The parameter estimation model UCODE was coupled 

to the GFLOW model to automate and optimize the calibration. This report represents one of the first efforts to 

link an analytic element code to a multi-objective function parameter estimation code.

Parameter estimation techniques also allowed estimation of uncertainty in model simulations of hypothetical hydrologic stressors. The stresses simulated 

included withdrawals from nearby pumping of irrigation wells, the artificial lowering of lake levels of Lower 

Genesee Lake, and reduction in recharge in an area of 

potential development. The simulation results indicate 

that Middle Genesee Lake stage would be reduced—

ranging from 0.1 for the reduction in recharge to 2.7 ft 

by pumping from Lower Genesee Lake. Decreases in 

lake stage resulting from these hydrologic stresses 

ranged between 0 and 3.1 ft when parameter uncertainty 

was included. The range in simulated values represents 

the uncertainty in the underlying model construction 

and calibration data, and can only be quantified using 

parameter estimation techniques. Whereas these simulated effects are within the natural variation in lake 

stage, they represent a systematic reduction of groundwater flow to the lake. Therefore, these hypothetical 

stresses are expected to establish a new, lower, baseline 

lake stage over which the natural variation due to climatic effects are added and subtracted. 
14 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

Oconomowoc

Lake

94

67

Bark River

Oconomow oc River

Silver Lake

EXPLANATION

Undeveloped capture zone

Developed capture zone

Travel time, in years

Pumping well

MIDDLE GENESEE LAKE LEVEL CHANGE

Undeveloped lake level:

Developed lake level:

95% C.I. (+)lake level:

95% C.I. (-) lake level:

862.6 feet

859.9 feet (∆h = 2.7 feet)

860.4 feet (∆h = 2.2 feet)

859.5 feet (∆h = 3.1 feet)

Base from U.S. Geological Survey

1:100,000 Oconomowoc West

Middle

Genesee

Lake

Lower

Genesee

Lake

0

0 1

1

2 3

2 3

KILOMETERS

MILES

10

10

50

100

200

∆h is change in lake stage;

95% C.I. is 95-percent confidence interval

80°30' 80°27'30" 80°25'

43°05'

43°02'30"

Figure 7. The ground-water capture zone of Middle Genesee Lake region for scenario number 2 (500 gallons per

minute pumping from Lower Genesee Lake).
SUMMARY AND CONCLUSIONS 15

Middle

Genesee

Lake

Lower

Genesee

Lake

Oconomowoc

Lake

94

67

Bark River

Oconomow oc River

Silver Lake

EXPLANATION

Original capture zone

Developed capture zone

Travel time, in years

Recharge reduced area

MIDDLE GENESEE LAKE LEVEL CHANGE

Original lake level:

Developed lake level:

95% C.I. (+)lake level:

95% C.I. (-) lake level:

862.6 feet

862.5 feet (∆h = 0.1 feet)

864.4 feet (∆h = 0.0 feet)

860.6 feet (∆h = 2.0 feet)

Base from U.S. Geological Survey

1:100,000 Oconomowoc West

80°30' 80°27'30" 80°25'

43°05'

43°02'30"

10

10

50

100

200

∆h is change in lake stage;

95% C.I. is 95-percent confidence interval

0

0 1

1

2 3

2 3

KILOMETERS

MILES

Figure 8. The ground-water capture zone of Middle Genesee Lake region for scenario number 3.
16 Simulation of the Shallow Hydrologic System in the Vicinity of Middle Genesee Lake, Wisconsin, Using Analytic Elements and 

Parameter Estimation

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