U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2012–5108 

Prepared in collaboration with the University of Wisconsin–Milwaukee 

Development and Application of a Groundwater/

Surface-Water Flow Model using MODFLOW-NWT for 

the Upper Fox River Basin, Southeastern Wisconsin 

0

10

50

100

150

200

250

300

350

400

450

88°20' 88°10'

43°10'

43°

Thickness, in feet

Fox

River

Figure 4a. Thickness of shallow aquifer system in model domain -- unconsolidated deposits.

Quarry

Nearfield 
Cover. Map showing thickness of shallow aquifer system in model domain—unconsolidated deposits (see figure 4a in report for details). 
Development and Application of a 

Groundwater/Surface-Water Flow Model 

using MODFLOW-NWT for the Upper Fox 

River Basin, Southeastern Wisconsin 

By D.T. Feinstein, M.N. Fienen, J.L. Kennedy, C.A. Buchwald, and M.M. Greenwood 

Prepared in collaboration with the University of Wisconsin–Milwaukee 

Scientific Investigations Report 2012–5108 

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

KEN SALAZAR, Secretary

U.S. Geological Survey

Marcia K. McNutt, Director

U.S. Geological Survey, Reston, Virginia: 2012 

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Suggested citation:

Feinstein, D.T., Fienen, M.N., Kennedy, J.L., Buchwald, C.A., and Greenwood, M.M., 2012, Development and 

application of a groundwater/surface-water flow model using MODFLOW-NWT for the Upper Fox River Basin, 

southeastern Wisconsin: U.S. Geological Survey Scientific Investigations Report 2012–5108, 124 p. 
iii

Acknowledgments

The development and application of the Upper Fox model depended on contributions from several groups of scientists.

At the U.S. Geological Survey, Richard Niswonger provided indispensable assistance in facilitating the use of the recently released MODFLOW-NWT code. John Walker furnished base-flow 

estimates and statistical support. David M. Ely and David Pollock reviewed the report and it 

benefited greatly from their remarks. Bobbie Louthian provided a thorough editorial review 

supplemented by insightful comments from John Eggleston. 

The funding for the project originated in a grant from the Brico Fund (Milwaukee, Wisconsin). 

Peter McAvoy, now adjunct at the University of Wisconsin–Milwaukee (UWM) School of 

Freshwater Science, helped coordinate the research and shape the research strategy. At UWM, 

Professors Douglas Cherkauer (emeritus) and Timothy Grundl were closely involved with every 

step of model development and application – their participation was essential. As part of her 

graduate work for Prof. Cherkauer, Bonnie Bills tirelessly compiled and interpreted well logs and 

surface-water data. The project also benefited from geophysical investigations conducted by 

Prof. William Kean (emeritus) and his graduate student, Michael Baierlipp.

Several members of the Wisconsin groundwater community provided vital information for the 

project: 

•	John Jansen (Cardno ENTRIX) and Ted Powell (Ruekert and Mielke, Inc.) located and forwarded pumping test and other records; John also offered good suggestions on modeling approaches; 

•	Michael Hahn (Southeastern Wisconsin Regional Planning Commission) and representatives of the Waukesha Water Utility provided important hydrogeologic and water-use 

data; 

•	Researchers at the Wisconsin Geological and Natural History Survey (Kenneth Bradbury, 

Madeline Gotkowitz, and David Hart) were, as always, helpful in talking through ideas. 
iv

Contents

Acknowledgments........................................................................................................................................iii

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

1. Introduction.................................................................................................................................................2

1.1 Purpose and Scope ........................................................................................................................2

1.2 Description of Upper Fox River Basin Study Area ....................................................................2

1.3 Previous Hydrogeologic Investigations and Modeling Studies..............................................6

2. Conceptual Model......................................................................................................................................6

2.1 Hydrogeologic Framework............................................................................................................6

2.2 Hydrologic Framework.................................................................................................................11

2.3 Sources and Sinks of Water .......................................................................................................15

2.4 Code Selection and Model Resolution......................................................................................15

3. Model Construction.................................................................................................................................17

3.1 Model Grid......................................................................................................................................17

3.1.1 Lateral Discretization .......................................................................................................17

3.1.2 Vertical Layering ...............................................................................................................17

3.1.3 Flux Boundary Conditions................................................................................................21

3.2 Hydraulic Conductivity Zones.....................................................................................................22

3.2.1 Unconsolidated Deposits ................................................................................................22

3.2.1.1 Fine-Favored and Coarse-Favored Zonation....................................................26

3.2.2 Silurian Bedrock................................................................................................................40

3.2.3 Initial Hydraulic Conductivity Values.............................................................................42

3.3 Recharge........................................................................................................................................42

3.4 Surface-Water Network ..............................................................................................................43

3.4.1 Outside Upper Fox River Basin.......................................................................................43

3.4.2 Inside Upper Fox River Basin..........................................................................................43

3.4.3 Input to MODFLOW-2005 Surface-Water Packages...................................................46

3.5 Water Withdrawals ......................................................................................................................52

3.5.1 Pumping..............................................................................................................................52

3.5.2 Quarries..............................................................................................................................52

3.6 Time Discretization .......................................................................................................................52

4. Model Solver.............................................................................................................................................54

4.1 MODFLOW-NWT Version of MODFLOW-2005 .........................................................................54

4.2 Application of NWT Formulation ...............................................................................................54

5. Model Calibration.....................................................................................................................................61

5.1 Estimated Parameters..................................................................................................................61
v

Figures

1A. Map showing study area—regional setting ............................................................................3

1B. Map showing study area—Upper Fox River Basin, model domain, and model 

nearfield .........................................................................................................................................4

1C. Schematic diagram showing study area—shallow and deep aquifer systems in 

southeastern Wisconsin..............................................................................................................5

2. Map showing distribution of surficial Pleistocene lithostratigrahic units in 

Wisconsin ......................................................................................................................................7

3. Schematic hydrogeology showing glacial heterogeneity .....................................................8

5.2 Calibration Targets........................................................................................................................61

5.2.1 Water Levels......................................................................................................................61

5.2.2 Vertical Head Differences ...............................................................................................64

5.2.3 Base Flow...........................................................................................................................64

5.2.4 Land-Surface Constraints ...............................................................................................67

5.2.5 Aquifer Test Drawdown...................................................................................................67

5.3 Target Weighting...........................................................................................................................70

5.4 Parameter Estimation Technique...............................................................................................71

5.5 Calibration Results for Fine-Favored and Coarse-Favored Models .....................................72

5.6 Sensitivity Analyses......................................................................................................................85

6. Model Results...........................................................................................................................................88

6.1 Water Levels for Fine-Favored and Coarse-Favored Models ...............................................88

6.2 Water Budgets for Fine-Favored and Coarse-Favored Models............................................95

6.3 Contributing Areas for Groundwater Sinks ..............................................................................97

7. Model Limitations...................................................................................................................................102

7.1 Conceptual Model.......................................................................................................................102

7.2 Model Construction....................................................................................................................103

7.3 Model Calibration........................................................................................................................104

8. Model Application to Hypothetical Well Field...................................................................................104

8.1 Riverbank Inducement...............................................................................................................104

8.2 Hypothetical Pumping Scenario Along 

Fox River................................................................................................................................105

8.3 Sensitivity of Results to Streambed Hydraulic Conductivity Values ..................................114

8.4 Effect of Riverbank Inducement on Drawdown ....................................................................114

9. Summary and Conclusions...................................................................................................................118

9.1 Model Construction....................................................................................................................118

9.2 Model Calibration........................................................................................................................119

9.3 Model Application.......................................................................................................................119

9.4 Concluding Discussion...............................................................................................................120

10. References Cited..................................................................................................................................120 
vi

4A. Map showing thickness (feet) of shallow aquifer system in model domain—

unconsolidated deposits .............................................................................................................9

4B. Map showing thickness (feet) of shallow aquifer system in model domain—

Silurian dolomite .........................................................................................................................10

5A. Map showing land-surface elevation—topography in model domain..............................11

5B. Map showing land-surface elevation—Fox River valley and tributary lowlands ...........12

6A. Map showing surface-water network in model domain—streams, lakes, and 

wetlands.......................................................................................................................................13

6B. Map showing surface-water network in model domain—streamgages, 

wastewater-treatment plants, and nested drainage basins ...............................................14

7. Map showing model grid with nearfield zoom.......................................................................18

8. Model layering: example vertical section ..............................................................................20

9. Map showing well-log database..............................................................................................23

10. Diagram showing logic for assigning facies to layer intervals...........................................25

11A. Map showing classification of log descriptions by dominant facies—layer 1 

(0–20 foot depth) .........................................................................................................................27

11B. Map showing classification of log descriptions by dominant facies—layer 2 

(20–50 foot depth) .......................................................................................................................28

11C. Map showing classification of log descriptions by dominant facies—layer 3 

(50–100 foot depth) .....................................................................................................................29

11D. Map showing classification of log descriptions by dominant facies—layer 4 

(100–150 foot depth) ...................................................................................................................30

11E. Map showing classification of log descriptions by dominant facies—layer 5 

(more than 150 foot depth) ........................................................................................................31

12A. Map showing distribution of coarse deposits based on well logs for example 

layer intervals—layer 1 (0–20 foot depth) ..............................................................................32

12B. Map showing distribution of coarse deposits based on well logs for example 

layer intervals—layer 3 (50–100 foot depth) ..........................................................................33

13. Diagram showing interpolation method for fine- and coarse-favored models................34

14A. Fine- and coarse-favored hydraulic conductivity zones in example vertical 

section—fine-favored zones ....................................................................................................35

14B. Fine- and coarse-favored hydraulic conductivity zones in example vertical 

section—coarse-favored zones ..............................................................................................35

15A. Maps showing fine- and coarse-favored hydraulic conductivity zones by 

model layer for unconsolidated deposits—layer 1 (0–20 foot depth)................................36

15B. Maps showing fine- and coarse-favored hydraulic conductivity zones by 

model layer for unconsolidated deposits—layer 2 (20–50 foot depth)..............................37

15C. Maps showing fine- and coarse-favored hydraulic conductivity zones by 

model layer for unconsolidated deposits—layer 3 (50–100 foot depth)............................38

15D. Maps showing fine- and coarse-favored hydraulic conductivity zones by 

model layer for unconsolidated deposits—layer 4 (100–150 foot depth)..........................39

15E. Map showing fine- and coarse-favored hydraulic conductivity zones by 

model layer for unconsolidated deposits—layer 5 (more than 150 foot depth)...............40
vii

16A. Map showing areas where vertical sequence consists entirely of dominantly 

coarse facies—fine-favored model ........................................................................................41

16B. Map showing areas where vertical sequence consists entirely of dominantly 

coarse facies—coarse-favored model...................................................................................41

17. Map showing model recharge distribution............................................................................44

18. Map showing surface-water cells assigned to MODFLOW-2005 packages....................45

19. Map showing comparison of topographic contours to digital elevation model 

(LIDAR) elevations at stream locations...................................................................................47

20. Map showing streambed hydraulic conductivity zones for routed SFR cells..................49

21. Graph showing Pewaukee Lake outflow as a function of lake level .................................50

22. Map showing high-capacity wells and quarries in model domain ....................................53

23A. Graph showing smoothing curves used in MODFLOW-NWT formulation—

inter-cell conductance and storage........................................................................................55

23B. Graph showing smoothing curves used in MODFLOW-NWT formulation—

pumping........................................................................................................................................55

24A. Simulated water-table configuration using MODFLOW-2005 and 

MODFLOW-NWT—MODFLOW-2005 with PCG2 solver in cross section..........................57

24B. Simulated water-table configuration using MODFLOW-2005 and 

MODFLOW-NWT—MODFLOW-NWT with XMD solver in cross section .........................57

24C. Map showing simulated water-table configuration using MODFLOW-2005 and 

MODFLOW-NWT—MODFLOW-2005 with PCG2 solver in plan view.................................58

25. Map showing calibration targets: water levels.....................................................................63

26. Map showing calibration targets: vertical head difference................................................65

27. Screenshot image showing comparison of base-flow estimate to streamflow 

record at Waukesha gage.........................................................................................................66

28A. Map showing calibration targets: aquifer test location—plan view.................................68

28B. Diagram showing calibration targets: aquifer test location—vertical section ...............69

29A. Calibration scatter plots for water levels—fine-favored model.........................................78

29B. Calibration scatter plots for water levels—coarse-favored model...................................78

30A. Map showing spatial distribution of water-level residuals (feet)—

fine-favored model .....................................................................................................................79

30B. Map showing spatial distribution of water-level residuals (feet)—

coarse-favored model................................................................................................................80

31A. Graphs showing observed and simulated drawdown at pumping-test 

observation wells—fine-favored model .................................................................................82

31B. Graphs showing observed and simulated drawdown at pumping-test 

observation wells—coarse-favored model ...........................................................................83

32A. Graph showing composite sensitivities to calibration targets for fine-favored 

model generated by PEST—initial parameter values ..........................................................86

32B. Graph showing composite sensitivities to calibration targets for fine-favored 

model generated by PEST—calibrated parameter values..................................................86
viii

33A. Graph showing composite sensitivities to calibration targets for 

coarse-favored model generated by PEST—initial parameter values .............................87

33B. Graph showing composite sensitivities to calibration targets for 

coarse-favored model generated by PEST—calibrated parameter values .....................87

34A. Map showing simulated water table in plan view and vertical section—

fine-favored model .....................................................................................................................89

34B. Map showing simulated water table in plan view and vertical section—

coarse-favored model................................................................................................................90

35A. Map showing areas of simulated upward head gradients in unconsolidated 

deposits for model nearfield—fine-favored model...............................................................91

35B. Map showing areas of simulated upward head gradients in unconsolidated 

deposits for model nearfield—coarse-favored model.........................................................91

36A. Map showing areas of upward flow to water table for model nearfield—

fine-favored model .....................................................................................................................92

36B. Map showing areas of upward flow to water table for model nearfield—

coarse-favored model................................................................................................................92

37A. Map showing simulated flooded water-table cells and dry headwater reaches 

for model domain—fine-favored model..................................................................................93

37B. Map showing simulated flooded water-table cells and dry headwater reaches 

for model domain—coarse-favored model............................................................................94

38A. Map showing simulated groundwater basins associated with surface-water 

features, quarries, and high-capacity wells—fine-favored model....................................99

38B. Map showing simulated groundwater basins associated with surface-water 

features, quarries, and high-capacity wells—coarse-favored model ............................100

39. Map showing hypothetical riparian well scenario .............................................................106

40A. Map showing simulated drawdown from riparian pumping in model layer 1 

(water table)—fine-favored model........................................................................................108

40B. Map showing simulated drawdown from riparian pumping in model layer 1 

(water table)—coarse-favored model ..................................................................................109

41A. Map showing simulated drawdown from riparian pumping in model layer 3 

(50 to 100 foot depth)—fine-favored model..........................................................................110

41B. Map showing simulated drawdown from riparian pumping in model layer 3 

(50 to 100 foot depth)—coarse-favored model....................................................................111

42A. Map showing effect of degree of riverbank inducement on drawdown in 

layer 1 (water table) for fine-favored model—riparian simulation allowing 

riverbank inducement..............................................................................................................116

42B. Map showing effect of degree of riverbank inducement on drawdown in 

layer 1 (water table) for fine-favored model—sensitivity simulation limiting 

riverbank inducement..............................................................................................................117 
ix

Tables

1. Layering logic at cell locations for groundwater-flow model of Upper Fox 

River Basin, southeastern Wisconsin .....................................................................................19

2. Fixed fluxes assigned to model at grid boundaries for groundwater-flow model 

of Upper Fox River Basin, southeastern Wisconsin .............................................................22

3. Logic for assignment of unconsolidated facies in model layers 1–5.................................24

4. Initial hydraulic conductivity values for both fine-favored and coarse-favored 

models ..........................................................................................................................................42

5. Correlation of channel width with upstream length by means of fitting curve 

for streamflow-routing (SFR) cells in Upper Fox River Basin..............................................48

6. Input for MODFLOW surface-water packages......................................................................51

7. Suggested and selected input values for the NWT input file .............................................59

8. Parameters estimated in calibration process........................................................................62

9. Calibration target types and assignment of calibration weights........................................70

10. Calibration fit for fine-favored model ......................................................................................73

11. Calibration fit for coarse-favored model.................................................................................75

12. Objective function contributions for initial and final calibrated models ...........................77

13. Calibrated parameter values for fine- and coarse-favored models ..................................84

14. Simulated water budgets for model domain..........................................................................95

15. Simulated water budgets for nested basins in the fine-favored model, 

Upper Fox River Basin................................................................................................................96

16. Simulated water budgets for nested basins in the coarse-favored model, 

Upper Fox River Basin................................................................................................................97

17. Simulated base flow within the Upper Fox River Basin .......................................................98

18. Simulated stages and water budgets for Pewaukee Lake................................................101

19. Sensitivity of simulated stage and water budget for Pewaukee Lake to 

lakebed hydraulic conductivity: fine-favored model ..........................................................101

20. Sources of water to hypothetical riparian wells along the Fox River for 

fine-favored model ...................................................................................................................112

21. Sources of water to hypothetical riparian wells along the Fox River for 

coarse-favored model..............................................................................................................113

22. Sensitivity of scenario results to streambed hydraulic conductivity...............................115 
x

Inch/Pound to SI

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

square foot (ft2

) 0.09290 square meter (m2)

square mile (mi2

) 259.0 hectare (ha)

square mile (mi2

) 2.590 square kilometer (km2)

Volume

gallon (gal) 3.785 liter (L)

gallon (gal) 0.003785 cubic meter (m3)

million gallons (Mgal) 3,785 cubic meter (m3

)

cubic foot (ft3

) 0.02832 cubic meter (m3)

Flow rate

cubic foot per second (ft3

/s) 0.02832 cubic meter per second (m3/s)

cubic foot per day (ft3

/d) 0.02832 cubic meter per day (m3/d)

gallon per minute (gal/min) 0.06309 liter per second (L/s)

million gallons per day (Mgal/d) 0.04381 cubic meter per second (m3

/s)

inch per year (in/yr) 25.4 millimeter per year (mm/yr)

Hydraulic conductivity

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

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

Conversion Factors 

Temperature in degrees Celsius (°C) may be converted to degrees Fahrenheit (°F) as follows:

°F=(1.8×°C)+32

Temperature in degrees Fahrenheit (°F) may be converted to degrees Celsius (°C) as follows:

°C=(°F-32)/1.8

Vertical coordinate information is referenced to the North American Vertical Datum of 1929 

(NAVD 29). 

Elevation, as used in this report, refers to distance above the respective vertical datum.

Horizontal spatial reference for the model grid is in Wisconsin Transverse Mercator projection, 

North American Datum of 1983 (NAD 83).The grid coordinates are in units of feet.

*Transmissivity: The standard unit for transmissivity is cubic foot per day per square foot times foot 

of aquifer thickness [(ft3

/d)/ft2]ft. In this report, the mathematically reduced form, foot squared per 

day (ft2/d), is used for convenience.
Abstract 

The Fox River is a 199-mile-long tributary to the Illinois 

River within the Mississippi River Basin in the states of 

Wisconsin and Illinois. For the purposes of this study the 

Upper Fox River Basin is defined as the topographic basin that 

extends from the upstream boundary of the Fox River Basin 

to a large wetland complex in south-central Waukesha County 

called the Vernon Marsh. The objectives for the study are to 

(1) develop a baseline study of groundwater conditions and 

groundwater/surface-water interactions in the shallow aquifer system of the Upper Fox River Basin, (2) develop a tool 

for evaluating possible alternative water-supply options for 

communities in Waukesha County, and (3) contribute to the 

methodology of groundwater-flow modeling by applying the 

recently published U.S. Geological Survey MODFLOW-NWT 

computer code, (a Newton formulation of MODFLOW-2005 

intended for solving difficulties involving drying and rewetting nonlinearities of the unconfined groundwater-flow equation) to overcome computational problems connected with 

fine-scaled simulation of shallow aquifer systems by means of 

thin model layers. 

To simulate groundwater conditions, a MODFLOW 

grid is constructed with thin layers and small cell dimensions (125 feet per side). This nonlinear unconfined problem 

incorporates the streamflow/lake (SFR/LAK) packages to 

represent groundwater/surface-water interactions, which yields 

an unstable solution sensitive to initial conditions when solved 

using the Picard-based preconditioned-gradient (PCG2) solver. 

A particular problem is the presence of many isolated wet 

water-table cells over dry cells, causing the simulated water 

table to assume unrealistically high values. Attempts to work 

around the problem by converting to confined conditions or 

converting active to inactive cells introduce unacceptable 

bias. Application of MODFLOW-NWT overcomes numerical 

problem by smoothing the transition from wet to dry cells and 

keeps all cells active. The simulation is insensitive to initial 

conditions and the water-table trend is smooth across layers. 

The MODFLOW-NWT code permits rigorous calibration and 

also robust application of the model to transient scenarios. 

Runtimes on a 64-bit computer are kept reasonably short by 

use of updated initial conditions and informed choices of 

solver parameters.

The shallow aquifer system consists of unconsolidated 

material of varying thickness over Silurian dolomite. The 

unconsolidated material, largely of glacial origin, contains 

fine-textured and coarse-textured deposits that vary in 

permeability over short distances. This study at least partly 

encompasses the inevitable uncertainty in the hydraulic 

conductivity zones by developing two models—one favors 

the continuity of fine-grained deposits and a second favors 

the continuity of coarse-grained deposits. The separate 

calibration processes for the fine-favored and coarse-favored 

models using MODFLOW-NWT and the nonlinear regression 

algorithms in the parameter estimation (PEST) code produce 

distinct parameter values for hydraulic conductivity zones, 

storage parameters, and streambed conductance zones. 

Both models are applied to a hypothetical scenario 

involving 27 “riparian” wells completed adjacent to the 

river channel and open to the shallow aquifer systems along 

a 10-mile stretch of the Fox River. The results suggest that 

a riparian well system withdrawing about 9 million gallons 

per day would induce about one-third to one-half its total 

discharge from the river, and that this riverbank inducement 

would appreciably limit drawdown around the hypothetical 

wells.

Development and Application of a Groundwater/SurfaceWater Flow Model using MODFLOW-NWT for the 

Upper Fox River Basin, Southeastern Wisconsin 

By D.T. Feinstein, M.N. Fienen, J.L. Kennedy, C.A. Buchwald, and M.M. Greenwood
2 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

1. Introduction

The Fox River Basin (fig. 1A) spreads over multiple 

counties. Several of these counties, such as McHenry and 

Kane Counties in Illinois and Waukesha County in Wisconsin, 

are undergoing rapid urbanization and consequent stresses on 

their water-supply systems (CH2MHill, 2002; Groschen and 

others, 2004; Kay and others, 2006; Meyer and others, 2009). 

Water-quality considerations that affect groundwater supplies 

also in some instances complicate the outlook for a sustainable 

water supply. The city of Waukesha, for example, in the face 

of radium exceedances in wells tapping the deep CambrianOrdovician aquifer system, is evaluating alternative sources of 

water such as increased withdrawals from shallow wells and 

a possible diversions from Lake Michigan (Southeastern Wisconsin Regional Planning Commission, 2010). Other communities in the Fox River Basin are likely to face similar choices 

in efforts to augment their water-supplies in coming years. The 

possibility that shallow pumping will be increased throughout 

the basin could imply decreased base flow to the Fox River 

and its tributaries and increased stresses on lakes and wetlands, especially in the context of uncertain climate trends. 

Resource-management issues involving water supply 

and environmental impact can be evaluated with quantitative 

tools such as groundwater-flow models. This report describes 

the development and example application of a groundwaterflow model for the uppermost part of the Fox River Basin 

in Waukesha County. The model focuses on groundwater/

surface-water interactions in the shallow part of the groundwater system, including the effect of shallow wells on water-table 

elevations and flows through the surface-water network.

The objectives for the study are: to develop a baseline 

study of groundwater conditions and groundwater/surfacewater interactions in the shallow aquifer system of the Upper 

Fox River Basin as first step in a possible regional study of 

the Fox River Basin in the States of Wisconsin and Illinois; to 

develop a tool for evaluating possible alternative water-supply 

options for communities in Waukesha County in the context 

of the need to substitute or augment deep aquifer water tainted 

by radium exceedances; and to contribute to the methodology of groundwater-flow modeling by applying the recently 

published MODFLOW-NWT code (Niswonger and others, 

2011) in an effort to overcome computational problems connected with fine-scaled simulation of shallow aquifer systems 

by means of thin model layers. Groundwater-flow models that 

focus on shallow conditions commonly are compromised by 

numerical instabilities prompted by dewatering of model cells. 

This study implements a new methodology aimed at overcoming these limitations.

1.1 Purpose and Scope

This report documents the construction and calibration of 

groundwater-flow models used to evaluate groundwater-flow 

patterns in the shallow aquifer system within the Upper Fox 

River Basin in southeastern Wisconsin. The report 

• contributes to the understanding of the hydrogeology 

of the study area by developing two flow models that 

offer distinct interpretations of the unconsolidated 

(mostly glacial) material—one favors the continuity 

of fine-grained sediments and the second favors the 

continuity of coarse-grained deposits;

• integrates hydrogeologic and hydrologic data with the 

hydrogeology to define sources and sinks of groundwater, including recharge, boundary fluxes, interactions 

with surface water, and discharge to wells and quarries; 

• quantifies groundwater/surface-water interactions at 

a scale fine enough to study the effect of pumping on 

flow to and from individual surface-water features such 

as the Fox River and its tributaries; and 

• demonstrates the use of the MODFLOW-NWT code 

(Niswonger and others, 2011) to overcome solver 

limitations and, thereby, improve the capacity of flow 

models to simulate shallow groundwater conditions 

and groundwater/surface-water interactions. 

The Upper Fox River Basin models are designed to provide a platform for studying how development pressures could 

be managed to minimize effects on local recharge and discharge conditions in the Upper Fox River Basin. The models 

could anticipate the effects of climate change on base flows; 

water levels in the basin could be mitigated. The model could 

aid in understanding how groundwater-supply systems in the 

basin take advantage of riverbank inducement to minimize not 

only drawdown but also base-flow reduction. 

1.2 Description of Upper Fox River Basin Study 

Area

The Fox River is a 199-mile (mi)-long tributary to 

the Illinois River within the Mississippi River Basin in the 

states of Wisconsin and Illinois (fig. 1A). The headwaters 

of the Fox River rise at the northern boundary of Waukesha 

County in southeastern Wisconsin. The river flows generally 

southward defining a topographic basin that includes parts 

of Waukesha, Racine, Kenosha, and Walworth Counties in 

Wisconsin and continues southward into northeastern Illinois. 

For the purposes of this study the Upper Fox River Basin

is defined as the topographic basin that extends from the 

upstream boundary of the Upper Fox River Basin to a large 

wetland complex in south-central Waukesha County called the 

Vernon Marsh (fig. 1B). The north/south extent of the Upper 

Fox River Basin is approximately 20 mi and the basin area is 

approximately 207 square miles (mi2

). It is bordered by the 

Lower Rock River Basin to the northwest and west, by the 

Milwaukee River Basin to the east, by the Pike-Root Rivers 

Basin to the southeast, and by the continuation of the Fox 

River Basin to the south. 
1. Introduction 3

Figure 1A. Study area—regional setting. 

Chicago

Milwaukee

89° 88°

43°

42°

DuPAGE

KENDALL

GRUNDY

McHENRY

DeKALB

WILL LA SALLE

LAKE

LEE

COOK

KANE

WASHINGTON

WALWORTH

JEFFERSON

WAUKESHA

KENOSHA

MILWAUKEE

RACINE

OZAUKEE

0 10 20 MILES

0 10 20 KILOMETERS

Fox River

Illinois River

Fox River

L

a

k

e

M i c

h i g

a

n

WISCONSIN

ILLINOIS

ILLINOIS

WISCONSIN

Fox River Basin

EXPLANATION

Figure 1a. Study area -- regional setting.

Base from U.S. Geological Survey 1:000,000 

and 1:250,000 digital data
4 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 1B. Study area—Upper Fox River Basin, model domain, and model nearfield.

The climatic characteristics of the Upper Fox River Basin 

can be estimated by reference to data for Waukesha County 

as a whole. The county receives on average 33.08 inches of 

precipitation annually, with the highest and lowest monthly 

rates in June and February, respectively (Southeastern Wisconsin Regional Planning Commission, 2010). Surface-water 

flow records are reported to indicate that about 7.5 inches of 

precipitation drain through the surface-water network. The 

balance infiltrates the soil or is lost by evapotranspiration 

(Poff and Threinen, 1963). Mean daily temperatures range 

from a low of 19.1 degrees Fahrenheit in January to a high of 

71.9 degrees Fahrenheit in July, averaging 46.9 degrees annually (Southeastern Wisconsin Regional Planning Commission, 

2010). Average snowfall is 40.4 inches and soils are commonly frozen during winter months. The 2000 United States 

census identified 360,767 residents in the county (Southeastern 

Wisconsin Regional Planning Commission, 2006), which is an 

increase of 56,052 residents during the 1990s, exhibiting one 

of the fastest rates of county growth in Wisconsin. Land-use 

inventories for 2000 showed the land cover to be 36.8 percent urban, 4.5 percent surface water, 14.2 percent wetlands, 

7.8 percent woodlands, and 36.7 percent agricultural or open 

land (Southeastern Wisconsin Regional Planning Commission, 

2006).

For convenience, the study area was divided into a 

nearfield (fig. 1B), where model discretization and properties 

are relatively precisely defined, and a surrounding farfield, 

where model inputs are relatively coarsely defined (fig. 1B). 

The model nearfield is centered on the city of Waukesha and 

encompasses much of the Upper Fox River Basin. The model 

is designed to simulate groundwater/surface-water interactions 

at a relatively fine scale in the nearfield.

WISCONSIN

Subbasin (8-digit 

hydrologic unit)

Upper Fox River Basin 

above Vernon Marsh

Model nearfield

Model domain

EXPLANATION

(Farfield area of the model domain is the 

area outside the nearfield)

UPPER FOX RIVER

MILWAUKEE RIVER

FOX

UPPER ROCK RIVER

LOWER ROCK RIVER

PIKE-ROOT RIVERS

88°30' 88°20' 88°10' 88° 87°50'

43°10'

43°

42°50'

DODGE

WASHINGTON OZAUKEE

MILWAUKEE

RACINE

WAUKESHA

WALWORTH

JEFFERSON

Waukesha Milwaukee

0 2.5 5 MILES

0 2.5 5 KILOMETERS

Marsh

Vernon

L

a

k

e

M i c

h i g

a

n

Fox

River

Figure 1b. Study area -- Upper Fox River Basin, model domain and model nearfield.

Base from Natural Resources Conservation Service 

(2007) and modified from Wisconsin Department of 

Natural Resources (2010) digital data, 1:24,000, 

Wisconsin Transverse Mercator, North American 

Datum of 1983
1. Introduction 5

The vertical extent of the model study area corresponds 

to the shallow aquifer system in southeastern Wisconsin (Feinstein and others, 2005a). This system is composed of unconsolidated material (glacial and alluvial deposits) overlying 

Silurian dolomite bedrock (fig. 1C). The bottom of the shallow 

aquifer system corresponds to the contact between the Silurian 

dolomite and the underlying Maquoketa shale confining unit, 

or, if the shale is absent, the uppermost Cambrian-Ordovician 

unit that is present.

Figure 1C. Schematic diagram showing study area—shallow and deep aquifer systems in southeastern Wisconsin. 

Feet below 

land surface

Shallow 

aquifer

system

Maquoketa

Shale

Deep

aquifer

system

0

500

1,000

1,500

Unconsolidated deposits

Dolomite aquifer

Major aquitard

Sandstone aquifer

Leaky aquitard

Crystalline basement

25 miles

Modified figure from Southeastern Wisconsin Regional Planning Commission, 2002

N

EXPLANATION

SHEBOYGAN

JEFFERSON

WASHINGTON

WAUKESHA MILWAUKEE

DODGE

RACINE

KENOSHA

FOND DU LAC

OZAUKEE

Figure 1c. Study area -- Schematic diagram showing shallow and deep aquifer systems in southeastern Wisconsin.
6 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

1.3 Previous Hydrogeologic Investigations and 

Modeling Studies

Southeastern Wisconsin (Kenosha, Milwaukee, Ozaukee, 

Racine, Walworth, Washington and Waukesha Counties) has 

been the object of a number of studies that focus on groundwater flow. Regional investigations, which incorporated southeastern Wisconsin in a larger multistate framework, include 

works of the Illinois State Water Survey (Burch, 1991; Meyer 

and others 2009) and the U.S. Geological Survey (Young 

and others, 1989; Mandle and Kontis, 1992; Young, 1992; 

Feinstein and others, 2010). Studies limited to southeastern 

Wisconsin include efforts restricted to simulating flow in the 

Cambrian-Ordovician units constituting the region’s deep 

sandstone aquifer (Young, 1976; Jansen and Rao, 1998). 

The shallow and deep groundwater resources and geology of the region were investigated in a cooperative effort 

between the Southeastern Wisconsin Regional Planning 

Commission (SEWRPC) and the Wisconsin Geological and 

Natural History Survey (Eaton and others, 1999). This work 

contributed to the development of a groundwater-flow model 

for southeastern Wisconsin (the “SEWRPC model”), which 

integrated the unconsolidated and bedrock aquifers (Feinstein 

and others, 2005a, 2005b). The SEWRPC model represented 

all rock units, including confining beds, from land surface to 

the top of the Precambrian sequence, by means of 18 layers. 

Minimum grid resolution was 2,500 feet (ft) in the model 

nearfield in southeastern Wisconsin. The model was calibrated for predevelopment and pumping conditions by using 

heads and stream base-flow observations for the period 1864 

to 2000. The SEWRPC model was used as a tool to evaluate 

future water supply in southeastern Wisconsin (Southeastern 

Wisconsin Regional Planning Commission, 2010). In that 

effort, pumping rates in the model were updated through 2005. 

Several hydrogeologic investigations have focused on 

areas that overlap the Upper Fox River Basin. Stratigraphic 

and hydraulic data were collected over a 40-mi2

 area along 

the Fox River in Waukesha County as an aid in evaluating 

the potential for expanding water supply (Batten and Conlon, 

1993). Two recent reports describe groundwater-flow models 

that focus on the shallow system in parts of Waukesha County 

(Dunkle, 2008; Jansen and Loughry, 2009). The Dunkle 

model adopts a geostatistical approach to characterize the 

unconsolidated sediments based on records from water-well 

driller logs, whereas, the Jansen and Loughry model interprets 

the system in terms of stratified units consisting of alternating 

outwash/ice contact deposits and till/lacustrine deposits. Also 

available are studies initiated by Waukesha Water Utility along 

the Upper Fox River that contain withdrawal information, 

test boring and geophysical data, as well as records of aquifer 

tests involving pumping of test borings (Aquifer Science and 

Technology, 2004, 2008, 2010).

2. Conceptual Model

The quantitative analysis of the groundwater-flow system 

in the Upper Fox River Basin depends largely on the treatment of three elements: the heterogeneity of the subsurface, 

the exchange between groundwater and surface water, and 

the stresses on the groundwater system expressed as sources 

and sinks of water. The objectives particular to this study also 

require that the model code and model resolution allow precise 

simulation of the interaction between groundwater and surface 

water, including the effect of pumping near streams (riparian 

wells).

2.1 Hydrogeologic Framework

The shallow aquifer system in southeastern Wisconsin consists of unconsolidated sediments overlying Silurian 

dolomite. The unconsolidated sediments are mostly glacial 

in origin, although alluvial sediments also are present along 

stream channels. A key control on local patterns of shallow 

groundwater flow is the heterogeneity of the glacial sediments. 

In some parts of the model study area, the sediment texture is 

predominantly coarse grained or fine grained over the entire 

unconsolidated thickness, whereas in other areas, repeated 

episodes of glacial advance and retreat through the Pleistocene 

Epoch have produced deposits that vary in texture and, therefore, in permeability over relatively short distances. 

It is convenient to roughly correlate the degree of heterogeneity in the unconsolidated deposits with the glacial units 

present in the model study area. The most recent mapping of 

Pleistocene deposits in Wisconsin (Syverson and others, 2011) 

defines three glacial units over the area of interest (fig. 2):

• To the west, the Horicon member of the Green Bay 

Lobe was deposited during the last part of the Wisconsin Glaciation. It includes till, associated sand and 

gravel, and other stratified deposits. Sand content generally is between 60 to 80 percent in this unit, which 

indicates that the deposits are typically coarse grained. 

• To the east, the dominant and most recent unit is the 

Oak Creek Formation of the Lake Michigan Lobe, 

approximately equivalent in age to the Horicon member (15,000 to 17,000 years old). It consists of finegrained till with some lacustrine clay, silt, and sand and 

some glaciofluvial sand and gravel. Generally, the till 

texture is silty clay or silty clay loam, composed of 80 

to 90 percent silt. This unit is largely fine grained.
2. Conceptual Model 7

Figure 2. Distribution of surficial Pleistocene lithostratigrahic units in Wisconsin. 

• In the middle of the study area and extending over 

most of the Upper Fox River Basin, the unconsolidated 

deposits correspond to the New Berlin Member of the 

Lake Michigan Lobe, equivalent in age to the Oak 

Creek Formation. The New Berlin Member is reported 

to consist of two principal facies, a sand and gravel 

unit typically overlying a till unit (Syverson and others, 

2011). However, comparison of available water-well 

driller logs and geologic logs suggests that unlike the 

area to the west where deposits are predominantly 

coarse grained and the area to the east where deposits 

are predominantly fine grained, the area associated 

with the New Berlin Member shows little continuity 

between sand/gravel and silt/clay deposits. A typical 

stratigraphic section through the unconsolidated material based on the evidence of subsurface logs shows 

a high degree of heterogeneity. Figure 3 contains a 

schematic section, which represents the short distances 

over which the textures of deposits vary as well as 

the overall lack of continuous layering. This scale of 

variation can be attributed to a complicated history of 

glacial advances, which deposited mostly clayey till, 

interrupted by melting, which produced meltwater 

channels that initially eroded the fine-grained material 

and later deposited sandier material. Stacking of erosional and depositional phases would tend to interrupt 

the continuity of clayey till sheets and sandy channel 

deposits (Professor Douglas Cherkauer, University of 

Wisconsin-Milwaukee, oral commun., April 2, 2011). 

Douglas Member

Crab Lak r e Mem be

Florence Member

?

Silver Cliff Mem b er

Nashville Member

Wildcat

Lake

Member

Pokegama

Creek Mbr

Middle

Inlet

Member

Liberty Grove

Member

K E W A U N E E F M

Glenm ore Mem b er

Two Rivers Member

HOL Y HILL

Kirby Lake FM

Member

Chilton

Mbr

Valders Mem ber

Tiskilwa

Mbr

Capron

Mbr

Clinton

Mbr

Ozaukee

Member

New Berlin Mem ber

C O P P E R

F A L L S F O R M A T I O N

Kee ne

Member

TRADE RIVER

FM

S

b

ylvan Lake Mem er ?

Pos m kin e M ber

Mikana

Mem ber

WALWORTH

FM

ZENDA FM

Merrill Me m ber

MARATHON

FM

Edgar

Mbr

Bakerville

Member

Prairie

Farm

Member

R I V E R F AL L S F M

Hersey

Member

Allens Grove

Mbr

M I L L E R C

N

R E I EK FO R M AT O

H o rico n Mem b er

H O L Y H I L L F O R M

A T I O N

OA K C R K E E

HOLY HILL

FM

B I G F L AT S F M

Kinnickinnic

Member

PIERCE

FM PIERCE

FM Branch River Member

Waukesha

County

Modified from Syverson and others, 2011

0 20 40 MILES

0 20 40 KILOMETERS

EXPLANATION

Area of sediment deposited

during last part of Wisconsin

Glaciation, between about 

35,000 and 11,000 years

before present

Boundaries

Area with little or no glacial

sediment

Formation (FM)

Member (Mbr)

Area of sediment deposited

prior to 35,000 years before

present

Figure 2. Distribution of surficial Pleistocene lithostratigrahic units in Wisconsin.
8 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

The “Model Construction” section of this report presents a stratigraphic database for the model domain based on 

information from available logs. The irreducible uncertainty 

in mapping the texture of the unconsolidated deposits because 

of the spacing and vertical penetration of logs, as well as the 

difference in quality and reliability among logs, makes it difficult to definitively assess the degree of continuity of coarsegrained and fine-grained deposits, especially in the parts of the 

Upper Fox River Basin dominated by the New Berlin Member. 

For this reason, the database is used to generate two interpretations of the subsurface—one favors the connection among 

fine-grained deposits and one favors the connection among 

coarse-grained deposits. Of course, neither interpretation is 

a “true” representation of the unconsolidated heterogeneity. 

The intent is to construct and calibrate two Upper Fox River 

Basin groundwater-flow models from the subsurface data in 

an effort to at least partly encompass the range of uncertainty 

in the characterization of the glacial sediments. In this way, 

different subsurface conditions, such as preferential flow paths 

associated with relatively continuous channel deposits and 

confined conditions associated with layering of till sheets, can 

be represented and their effect on groundwater-flow patterns 

(for example, on the sources of water to wells) can be compared. The model that preferentially connects fine-favored 

deposits is called the fine-favored model; the model that 

preferentially connects coarse-favored deposits is called the 

coarse-favored model.

Figure 3. Schematic hydrogeology showing glacial heterogeneity. 

Bottom of model domain

Driller well log

EXPLANATION

Sand; sand and gravel; gravel

Gravel and silt; sand and silt

Silt; clay; silt and clay

Dolomite

Maquoketa shale

Dolomite and sandstone

Figure 3. Schematic hydrogeology showing glacial heterogeneity. 100 feet

1 mile

SCALE

Schematic modified from D.S. Cherkauer, 

Geosciences Department, University of 

Wisconsin–Milwaukee
2. Conceptual Model 9

In addition to marked heterogeneity, the unconsolidated 

deposits display variable thickness across the study area 

(fig. 4A). The deposits are present over more than 99 percent 

of the model domain (in a few places the underlying dolomite 

is at the surface) and is at least 100 ft thick in 50 percent of the 

model domain but only in 34 percent of the model nearfield. 

The thickness is an important control on the transmissivity 

of the glacial deposits and their ability to support pumping. 

However, some zones of greater than average thickness (for 

example, the bedrock valley in the southern part of the study 

area, located in the farfield of the model domain [fig. 4A]) do 

not necessarily correlate with the zones of greatest well yields, 

which also depend on the texture of the deposit and the potential to induce water from streams (Batten and Conlon, 1993).

Figure 4A. Thickness (feet) of shallow aquifer system in model domain—unconsolidated deposits. 

0

10

50

100

150

200

250

300

350

400

450

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

EXPLANATION

Quarry

Thickness, in feet

Nearfield 

Surface-water feature

Fox River

Fox

River

Figure 4a. Thickness of shallow aquifer system in model domain -- unconsolidated deposits.

Aquifer thickness data derived from well logs (Wisconsin Department of Natural Resources, 

2010, and Wisconsin Geological and Natural History Survey, 2009); base modified from 

Wisconsin Department of Natural Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American Datum of 1983
10 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

The Silurian dolomite is the top of bedrock over most 

of the study area, thickening to the east (fig 4B). It is an 

important aquifer in southeastern Wisconsin, supplying many 

domestic and some public and industrial wells (Southeastern 

Wisconsin Regional Planning Commission, 2010). As is typical for carbonate rock in the Upper Midwest, the ability of the 

Silurian dolomite to transmit water is related to the properties of the matrix and to the weathering and development of 

fractures. For instance, numerous investigators have reported 

higher than background matrix hydraulic conductivity values 

in a weathered zone near the carbonate bedrock surface (Carlson, 2001; Stocks, 1998) or in zones of bedding-plane (Eaton, 

2002; Muldoon and others, 2001) and vertical fractures (Jansen, 1995). Following the approach adopted in the SEWRPC 

regional model (Feinstein and others, 2005a) based on field 

studies in southeastern Wisconsin (Rovey, 1990), this study 

assumes that fractures and dissolution because of weathering 

enhance the hydraulic conductivity in the upper 20 ft of the 

Silurian dolomite thickness.

The bottom boundary of the shallow aquifer system is the 

top of the Maquoketa shale (fig. 3), a confining unit that inhibits vertical exchange with the deep aquifer system consisting of Cambrian-Ordovician rocks (Eaton, 2002). However, 

the large rate of withdrawals from the deep aquifer system 

(Feinstein and others, 2005a) have created a regional cone of 

depression at depth that induces some water to leak from the 

shallow to the deep parts of the flow system (Feinstein and 

others, 2005b). Account must be taken of this vertical flux in 

evaluating groundwater conditions for the Upper Fox River 

Basin.

Figure 4B. Thickness (feet) of shallow aquifer system in model domain—Silurian dolomite. 

0

10

50

100

150

200

250

300

350

400

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°

EXPLANATION

Nearfield 

Surface-water feature

Fox River

Thickness, in feet

Fox

River

Aquifer thickness data derived from well logs (Wisconsin Department of Natural Resources, 

2010, and Wisconsin Geological and Natural History Survey, 2009); base modified from 

Wisconsin Department of Natural Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American Datum of 1983
2. Conceptual Model 11

2.2 Hydrologic Framework

The water-table elevation and water levels at depth in the 

shallow aquifer system in southeastern Wisconsin are strongly 

influenced by the undulations of the land-surface topography. 

The most direct control on water levels is the surface-water 

network, which tends to fix the water-table elevation along 

valleys and lowlands (figs. 5A and 5B). The surface-water 

network in the model domain consists of the Fox River, tributary streams, and 22 water bodies (lakes, ponds and wetlands) 

typically connected to streams (fig. 6). The area also contains 

riparian wetlands which drain to streams and lakes as well as 

upland wetlands which are less likely to be integrated in the 

surface-water network. The Vernon Marsh at the south end of 

the Upper Fox River Basin is a complex of wetlands whose 

water level is partly controlled by a system of weirs (operated by the Wisconsin Department of Natural Resources) on 

tributaries to the Marsh such as Pebble Creek and Mill Brook 

(Professor Douglas Cherkauer, University of Wisconsin-Milwaukee, oral commun., September 2009).

Figure 5A. Land-surface elevation—topography in model domain. 

Sussex

Quarries

Waukesha 

Quarries

600

700

800

900

1,000

1,100

1,200

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°

EXPLANATION

Altitude, in feet

Fox

River

Fox

River

Nearfield 

Surface-water 

feature

Fox River

Base from U.S. Geological Survey National Elevation Dataset 30-meter 

digital elevation model showing land-surface elevations shaded at 

20-foot intervals and modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American Datum of 1983
12 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

The main tributaries to the Upper Fox River are Lannon Creek and Sussex Creek in northern Waukesha County; 

Pewaukee River and Poplar Creek immediately north of the 

city of Waukesha; and Pebble Creek, Genesee Creek and 

Pebble Brook between the city of Waukesha and the Vernon 

Marsh (fig. 6A). Pewaukee Lake is a large feature that drains 

to the Pewaukee River and then on to the Fox River. A chain 

of lakes is present in the model farfield in the Bark River 

and Oconomowoc River Basins located northwest of the Fox 

River Basin. Other farfield water bodies include Tamarack 

Swamp near the headwater of the Fox River, and Spring Lake 

and Little Muskego Lake south of the Upper Fox River Basin 

(fig. 6A). 

A primary objective of the Upper Fox River Basin models is to simulate base flow to the surface-water network. For 

this study, base flow is defined not only to include the groundwater contributions to streams and lakes, but also the effluent 

discharge from wastewater-treatment plants (WWTPs). The 

plants in the model domain recirculate water withdrawn from 

shallow and deep wells (chiefly public supply wells for the 

main population centers) to the Fox River. There are three 

WWTPs that discharge to the Fox River, which serve the Cities of Sussex, Brookfield, and Waukesha (fig. 6B). Records 

indicate that the effluent flow from these plants has increased 

over the last 50 years but that in any given year the effluent 

flow is relatively steady. Records also indicate that the combined discharge is not an inconsequential fraction of the Fox 

River streamflow during low-flow periods. The WWTP flux is 

discussed further in the “Model Construction” section.

Figure 5B. Land-surface elevation—Fox River valley and tributary lowlands. 

740

760

780

800

820

840

860

Fox River

Valley

0 2 4 MILES

0 2 4 KILOMETERS

Subcontinental divide 

between Lake Michigan 

Basin (to east) and 

Mississippi River Basin 

(to west)

Fox River Basin upstream 

of Vernon Marsh

Nearfield

Surface-water feature

EXPLANATION

88°20' 88°10'

43°10'

43°

Land-surface elevation in feet 

(not shown above 860 feet)

Pewaukee

Lake

Fox

River

Figure 5b. Land surface elevation -- Fox River valley and tributary lowlands. 

Base modified from U.S. Geological Survey National Elevation Dataset 

30-meter digital elevation model showing land-surface elevations shaded 

at 20-foot intervals and from Wisconsin Department of Natural 

Resources (2010) and Natural Resources Conservation Service (2007), 

1:24,000 digital data, Wisconsin Transverse Mercator, North 

American Datum of 1983 
2. Conceptual Model 13

Figure 6A. Surface-water network in model domain—streams, lakes, and wetlands. 

Vernon Marsh

Pine 

Lake

Beaver 

Lake

North 

Lake

Lake Keesus

Nagawicka 

Lake

Fox

 River

Fox 

River

Tamarack 

Swamp

Little 

Muskego 

Lake

 

Zion

Creek

Saylesville 

Millpond

Tributaries to

Willow Creek 

White

Creek

Spring 

Lake

Tributary to 

Root River

Waukesha

88°20' 88°10'

43°10'

43°

Tributaries to 

Menomonee 

River

Bark River

Woodland Creek

Spring Cr

Pewaukee River

Sussex Creek

Pewaukee

Lake

Lannon Cr

Poplar Creek

Underwood Cr

Oconomowoc R

Mason Cr Brandy Brook

Pebble Creek

Pebble Cr

Scuppernong

Creek

Genesee Cr

Pebble Brook

Spring Brook

Mill Creek

Deer Creek

Muskego

Creek

Mill Brook

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

0 2 4 MILES

0 2 4 KILOMETERS

EXPLANATION

Fox River 

Head- waters

Fox River 

Headwaters

Surface-water feature

Lake

Nearfield

Fox River

Upper Fox River Basin above Vernon Marsh

Figure 6a. Surface water network in model domain -- streams, lakes, and wetlands.
14 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 6B. Surface-water network in model domain—streamgages, wastewater-treatment 

plants, and nested drainage basins. 

Sussex 

WWTP

Waukesha 

WWTP

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Fox

Fox

 River

 River

EXPLANATION

Fox River Basin above Watertown gage

Fox River Basin above Waukesha gage

Fox River Basin above Vernon Marsh (ungaged)

Nearfield

Fox River

U.S. Geological Survey streamflow-gaging station

Wastewater-treatment plant (WWTP)

Brookfield 

WWTP

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American Datum 

of 1983

Figure 6b. Surface water network in model domain -- stream gages, wastewater-treatment plants and 

nested drainage basins. (WWTP, wastewater-treatment plant)

05543800

FOX RIVER AT WATERTOWN ROAD

NEAR WAUKESHA, WI

05543830

FOX RIVER AT

WAUKESHA, WI

USGS STREAMGAGE

USGS STREAMGAGE
2. Conceptual Model 15

There are two U.S. Geological Survey (USGS) gages 

on the Upper Fox River (fig. 6B), one located at Watertown 

(gage 05543800, active from 1992 to 2000) and one located 

at Waukesha (gage 05543830, active from 1963 to present). 

Streamflow records from these gages have been used to estimate long-term average rates of base flow (groundwater contribution plus effluent discharge) for their respective subbasins 

within the Upper Fox River Basin. The estimation and use of 

the base-flow values are described in the “Model Calibration” 

section. 

2.3 Sources and Sinks of Water

The stresses on the Upper Fox River Basin groundwaterflow system consist of fluxes into the model domain (sources) 

and out of the model domain (sinks). The sources of water are:

• the fraction of precipitation that percolates as recharge 

to the water table, 

• losses from surface-water bodies to the subsurface, and

• inflows across the lateral and vertical domain boundaries.

The sinks for water are:

• discharge to surface-water bodies,

• outflows across the lateral and vertical domain boundaries,

• withdrawal from pumping wells, and

• discharge to quarries excavated at the top of the Silurian dolomite.

The evaluation of these flux terms is based on several 

sources: recorded data, estimation techniques, previous modeling at the regional scale, and outputs of the model itself. The 

evaluation, location, and zonation of the fluxes input to the 

model are discussed in the “Model Construction” section.

An additional potential source/sink for the groundwaterflow system is the release of water from or addition of water 

to aquifer storage under transient (time-varying) conditions. In 

this application, the storage flux is only considered as part of 

the calibration process (see “Model Calibration” section).

2.4 Code Selection and Model Resolution

The Upper Fox River Basin models are designed to simulate interactions between the shallow groundwater-flow system 

and the surface-water system at a scale sufficiently refined to 

(1) include and integrate virtually all individual elements of 

the surface-water network and (2) simulate local exchanges 

between the subsurface and surface systems, including flow 

induced from streams to riparian wells. These objectives 

require the use of advanced modeling packages and special 

attention to the discretization of the model grid.

In this study, the groundwater-flow system is simulated 

using packages that are part of the USGS MODFLOW-2005 

numerical code (Harbaugh, 2005), a finite-difference program 

supporting three-dimensional and transient solutions. The 

MODFLOW-2005 code features several advanced packages applied in this study for linking groundwater to surface 

water and for routing water through the surface-water system. 

The streams, lakes, and other water bodies in the Upper Fox 

River surface-water network are simulated by means of the 

Streamflow-Routing (SFR) and Lake (LAK) packages as a 

single integrated system, which routes water from headwater 

features through higher-order features and to the outlet of 

the Upper Fox River Basin at Vernon Marsh. Surface-water 

features outside the Upper Fox River Basin (all located in the 

model farfield) are represented using simpler MODFLOW 

algorithms—the River (RIV) and Drain (DRN) packages—

which simulate groundwater/surface-water interactions but do 

not route water through streams and water bodies or calculate surface-water stages. Other MODFLOW-2005 packages 

employed to represent fluxes include RCH for recharge and 

WEL for pumping wells and boundary inflows and outflows.

The degree of accuracy attained in numerical models 

that simulate groundwater/surface-water interactions is highly 

dependent on the horizontal and vertical grid resolution 

(Haitjema and others, 2001; Hunt and others, 2003; Haitjema 

and others, 2010). A discretization that is too coarse tends 

to group multiple surface-water features in a single model 

cell, which obscures the role of individual streams and lakes. 

Moreover, the numerical code assigns the properties of the 

surface-water features (the stage and bed conductance) to the 

spatial center of the model cell, thereby possibly mislocating 

the origin of the stress. When wells are located near surfacewater bodies, the spatial relation between the features can 

be distorted unless the grid is fine enough both laterally and 

vertically so that the cell centers (where the well withdrawal is 

simulated) are at approximately the correct spatial distances. 

A related set of problems involves errors in modeled 

groundwater/surface-water exchanges that may result from an 

inaccurate representation of the leakage between model layers

in the presence of surface-water features inserted in coarse 

grid cells. One effect of a relatively coarse grid is to smear the 

pattern of upward groundwater discharge to streams over too 

large an area; another is to miscalculate the magnitude of the 

discharge (Haitjema and others, 2010; Feinstein and others, 

2010, appendix 2). To understand these issues it is helpful to 

consider the “leakage factor” λ (feet) (Hantush and Jacob, 

1954; Verruijt, 1970), also referred to as the “characteristic 
16 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

leakage length" (Haitjema and others, 2001; Bakker and 

Strack, 2003). For a two-layer aquifer system, the characteristic leakage length is defined as:

λ = +

T T c

T T

u l

u l

(1)

where 

Tu and Tl are the transmissivities of the upper and lower 

layer, respectively, in feet squared per day, 

and

c is the vertical resistance (thickness divided 

by vertical hydraulic conductivity) of a 

separating confining unit or the aquifer 

material, in days.

The characteristic leakage length determines whether the 

vertical leakage to or from streams or wells is concentrated 

locally or is distributed (“smeared out”) over a larger area. For 

example, it can be shown that the leakage induced by a stream 

nearly vanishes beyond a distance of 3l from the stream 

boundary (Hunt and others, 2003). Thus, for small λ-values, 

the leakage is concentrated near the stream, whereas for large 

λ-values, it occurs over a much larger area. Haitjema and others (2001) found that in order to obtain an acceptable representation of the leakage distribution in finite-difference models, 

such as MODFLOW-2005, the cell size should be less than λ, 

preferably as low as 0.1λ. When the model cell size is larger 

than this threshold value, inaccurate leakage distributions may 

result in inaccurate simulation of heads and streamflows. 

Application of the leakage factor to transmissivity and 

vertical resistance values typical for the shallow system in 

southeastern Wisconsin serves as a guideline for the proper 

grid resolution. Assuming that horizontal hydraulic conductivity is 5 feet per day (ft/d) and vertical hydraulic conductivity is 

0.01 ft/d for unconsolidated and bedrock units (reasonable values for sandy till and dolomite), and the average thicknesses 

are 115 ft and 50 ft for the unconsolidated and bedrock units 

(typical values in Waukesha County) transmissivity values 

are 575 and 250 ft2/d, and the resistance value for the vertical 

thickness between cell centers is 8,250 days. The resulting 

leakage length, l, is about 1,200 ft (Feinstein and others 2010, 

appendix 2). This result implies that a lateral grid spacing 

on the order of 0.1l or about 120 ft will ensure that the grid 

resolution does not introduce inaccuracies into the solution. 

The nearfield grid resolution for the Upper Fox River Basin 

models is 125 ft per side, which should be small enough to 

virtually exclude numerical distortions. The fine grid spacing 

has the added advantage of allowing relatively precise location 

of features such as streams and wells, including riparian wells.

The vertical layer spacing is also made relatively fine in 

the Upper Fox River Basin models in order to allow accurate 

input of partially penetrating wells and also to allow accurate 

simulation of vertical gradients especially in the vicinity of 

surface water. The top layer of the model—designed to host 

the boundary conditions corresponding to surface-water features—extends from the land surface to a depth of only 20 ft. 

(See “Model Construction” section for full explanation of layering scheme.) Static water levels from the database of waterwell driller logs for shallow wells in the study area suggest 

that the median water-table elevation is almost 40 ft below 

the land surface, with depths approaching zero in the valleys 

and depths typically greater than 40 ft under the uplands. (See 

“Model Calibration” section for an account of available waterlevel information.) Accordingly, the models are expected to 

simulate dewatered conditions for the top layer over a large 

fraction of the model domain area. Model layers representing deeper parts of the groundwater system are expected to 

be dewatered under upland areas. MODFLOW-2005 allows 

dewatering of model cells, but the simulation of water-table 

conditions below the top model layer can introduce instabilities to the model solution as well as unwanted sensitivity 

to initial conditions. Use of a recently developed version of 

MODFLOW-2005, called MODFLOW-NWT (Niswonger and 

others, 2011), offers a way to overcome the difficulties associated with thin model layers under unconfined conditions. The 

“Model Solver” section of this report describes in more detail 

the motivation for and the implementation of the NWT formulation to this project.

The final element of the conceptual model involves the 

temporal resolution of the model solutions. The base solutions 

are steady-state simulations and reflect groundwater conditions in equilibrium with the source inflows and sink outflows 

described above. Consideration of transient effects in the 

development of the fine-favored and coarse-favored models is 

limited to the calibration phase (the “Model Calibration” section) involving simulation of an aquifer test.

As discussed in the “Model Construction” section, the 

well withdrawals correspond to 2005 conditions. However, 

other fluxes associated with the model, such as edge boundary fluxes, recharge to the basin, Pewaukee Lake outflow, and 

wastewater-treatment effluent to the Fox River, are referenced 

to separate time frames—that is, long-term average conditions 

in the case of recharge or post 2005 conditions in the case of 

lake outflow and effluent. In this sense, the output of the models is best characterized as reflecting conditions in equilibrium 

with recent flow rates from sources and sinks. Possible errors 

deriving from the approximate nature of the flux estimates and 

the neglect of transient effects corresponding to adjustment of 

the flow system to these time-dependent boundary conditions 

are discussed in the “Model Limitations” section of the report. 
3. Model Construction 17

3. Model Construction

The model construction involves the horizontal and vertical discretization of the model domain into finite-difference 

cells along with the definition of boundary conditions at the 

edge of the grid, the translation of the unconsolidated and bedrock hydrogeology into hydraulic conductivity zones, the distribution of water-table recharge acting as the principal source 

of water to the system, the representation of the principal sinks 

for groundwater corresponding to various surface-water features, and the distribution of local groundwater sinks associated with pumping wells and with flow to dolomite quarries. 

Most elements of the model construction are common to the 

fine-favored and coarse-favored model—the only element that 

differs between the two is the representation of the hydraulic 

conductivity zones.

The platform used to help construct the flow models (as 

well as to visualize input and output) was Groundwater Vistas, 

Version 5 (Rumbaugh and Rumbaugh, 2007).

3.1 Model Grid

The model grid consists of the lateral discretization of the 

model domain into rows and columns and the vertical discretization of the unconsolidated deposits and the Silurian dolomite into layers. At the lateral and bottom edges of the grid, 

groundwater enters and exits the model domain as a function 

of prescribed fluxes.

3.1.1 Lateral Discretization

The MODFLOW finite-difference grid consists of 508 

rows from north to south extending 21.3 mi and 508 columns 

from west to east extending 16.1 mi (fig. 7), yielding a total 

area of 343 mi2. The spacing of the rows and columns is 

nonuniform. The rows and columns at the edge of the domain 

are 2,500 ft in width. The grid spacing is reduced away from 

the edges of the model domain at a ratio of approximately 1.4 

until the nearfield is reached, which roughly coincides with the 

north/south extent between the confluences of Sussex Creek 

and Pebble Brook with the Fox River (about 11.3 mi) and the 

west/east extent between the west side of Pewaukee Lake and 

the east edge of the Upper Fox River Basin (about 11.6 mi). 

Inside the nearfield the row and column widths are uniformly 

set to 125 ft per side. The nearfield grid corresponds to model 

rows 22 to 498 and to model columns 11 to 498 and covers an 

area of 131 mi2

.

3.1.2 Vertical Layering

The model is divided into seven layers. Layers 1 through 

5 represent unconsolidated material and layers 6 through 7 

represent the Silurian dolomite. However, not all unconsolidated layers necessarily participate from a flow standpoint at a 

given row/column location. The number of participating layers 

depends on the total unconsolidated thickness. For example, 

if the unconsolidated material is 10 ft thick at a location, 

then only layer 1 transmits flow whereas layers 2 through 5 

are considered “pinched”; each is assigned a minimal thickness equal to 0.5 ft. In this case, the properties of the pinched 

layers 2 through 5 correspond to the properties assigned the 

active cell in layer 1. As a second example, if the unconsolidated material is 30 ft thick, then layers 1 and 2 transmit 

flow whereas layers 3 through 5 are “pinched”, and each are 

assigned a thickness of 0.5 ft and the properties of the overlying layer 2. All five layers are active only if the unconsolidated material is more than 150 ft thick. The explicit logic for 

determining the number of active unconsolidated layers at a 

location is presented in table 1. The logic for assigning layers 

to the Silurian dolomite is simpler—the upper 20 ft, associated with a weathered interval, correspond to layer 6 and any 

additional thickness is assigned to layer 7.

An important element of the Upper Fox River Basin 

model is the use of the land-surface elevation as a datum for 

the delineation of model layer bottom elevations as a function 

of specific depths. That is, the bottom of layer 1 corresponds 

to a depth of 20 ft below the land surface or, if the unconsolidated material is less than 20 ft, to the bottom of the unconsolidated material. Analogously, the bottoms of layers 2, 3, 

and 4 are associated with depths of 50 ft, 100 ft, and 150 ft, 

respectively, or to the bottom of the unconsolidated material. 

The bottom of layer 5 corresponds to the bottom of the unconsolidated material for locations where the unconsolidated 

material is more than 150 ft thick—otherwise the layer bottom 

corresponds to the bottom of unconsolidated deposits adjusted 

slightly to take account of the nominal thickness of pinched 

cells. This logic implies that the maximum thickness of the 

layers is 20 ft for layer 1, 30 ft for layer 2, 50 ft for layer 3, 

and 50 ft for layer 4. The layering, which results from the 

use of the land surface as the datum, is shown for an example 

model row in figure 8.

The land-surface elevation for the model (fig. 5A) was 

derived from digital terrain models (DTM) of the area. Elevation data for the part of the model domain inside Waukesha 

County (95 percent of the grid area and 100 percent of the 

nearfield) were extracted from Waukesha County 2005 DTM 

files published by Aero-Metric, Inc., Sheboygan, Wisconsin, 

on behalf of Waukesha County Department of Parks and Land 

Use (Waukesha County, 2005). These files are based on a 

Light Detection and Ranging (LIDAR) coverage of the county 

at an extremely fine (feet scale) resolution. The DTM files 

smooth the LIDAR data at a 1-inch-equals-100-ft scale to support land-surface mapping at a 2-ft contour interval. Elevation 

data for Washington County at the north edge of the grid were 

based on 30 m (approximately 100 ft) digital elevation model 

data obtained from the U.S. Geological Survey (2010a). 
18 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 7. Model grid with nearfield zoom. 

Row 298

Maquoketa Shale

absent

Model cell

Lake

Surface-water feature

Nearfield

Pumping well

EXPLANATION

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Figure 7. Model grid with nearfield zoom.

Pumping wells

F o x

R i v e r

Grid cells are 125 feet x 125 feet

1,000 feet

Base from Wisconsin Department of 

Natural Resources digital data, 

2010, 1:24,000, Wisconsin 

Transverse Mercator, North 

American Datum of 1983
3. Model Construction 19

Table 1. Layering logic at cell locations for groundwater-flow model of Upper Fox River Basin, southeastern Wisconsin.

Layer 1 (unconsolidated)

• If unconsolidated deposits are greater than 20 feet (ft) thick, bottom elevation of layer 1 is set 20 ft below land-surface elevation; 

• If unconsolidated deposits are less than 20 ft thick but greater than 0 ft thick, bottom elevation of layer 1 is set to land surface less 

unconsolidated deposit thickness; 

• If unconsolidated deposits are absent, the layer is “pinched” at this location and the bottom elevation of layer 1 is set to 0.5 ft below 

land-surface elevation.

Layer 2 (unconsolidated)

• If unconsolidated deposits are greater than 50 ft thick, bottom of layer 2 is set 50 ft below land-surface elevation; 

• If unconsolidated deposits are less than 50 ft but greater than 20 ft thick, bottom of layer 2 is set to land-surface elevation less 

unconsolidated deposit thickness; 

• If unconsolidated deposits are less than or equal to 20 ft thick, the layer is “pinched” at this location and the bottom of layer 2 is set to 

0.5 ft below bottom elevation of layer 1.

Layer 3 (unconsolidated)

• If unconsolidated deposits are greater than 100 ft thick, bottom elevation of layer 3 is set 100 ft below land-surface elevation; 

• If unconsolidated deposits are less than 100 ft but greater than 50 ft thick, bottom elevation of layer 3 is set to land-surface elevation 

less unconsolidated deposit thickness; 

• If unconsolidated deposits are less than or equal to 50 ft thick, the layer is “pinched” at this location and the bottom elevation of layer 

3 is set to 0.5 ft below bottom elevation of layer 2.

Layer 4 (unconsolidated)

• If unconsolidated deposits are greater than 150 ft thick, bottom elevation of layer 4 is set 150 ft below land-surface elevation; 

• If unconsolidated deposits are less than 150 ft but greater than 100 ft thick, bottom elevation of layer 4 is set to land-surface elevation 

less unconsolidated deposit thickness; 

• If unconsolidated deposits are less than or equal to 100 ft thick, the layer is “pinched” at this locations and the bottom elevation of 

layer 4 is set to 0.5 ft below bottom elevation of layer 3.

Layer 5 (unconsolidated)

• If unconsolidated deposits are greater than 150 ft thick, bottom elevation of layer 5 is set to land-surface elevation less unconsolidated 

deposit thickness; 

• If unconsolidated deposits are less than 150 ft thick, the layer is “pinched” at this location and the bottom elevation of layer 5 is set to 

0.5 ft below bottom elevation of layer 4.

Layer 6 (Silurian dolomite, assumed weathered zone)

• If dolomite thickness is greater than 20 ft, bottom elevation of layer 6 is set to 20 ft below bottom elevation of unconsolidated deposits;

• If dolomite is less than 20 ft but greater than 0 ft thick, bottom elevation of layer 6 is set to bottom elevation of dolomite;

• If dolomite is absent, the layer is “pinched” at this location and the bottom elevation of layer 6 is set to 0.5 ft below bottom elevation 

of layer 5.

Layer 7 (Silurian dolomite)

• If dolomite thickness is greater than 20 ft, bottom elevation of layer 7 is set to bottom elevation of layer 5 (i.e., of unconsolidated 

deposits) less dolomite thickness;

• If dolomite thickness is less than 20 ft, the layer is “pinched” at this location and the bottom elevation of layer 7 is set to 0.5 ft below 

bottom elevation of layer 6.
Figure 8. Model layering: example vertical section. Unconsolidated deposits more than 150 feet thick WEST 

Top of Maquoketa S

Unconsolidated deposits less than Bedrock outcrop 20 feet thick CROSS SECTION, ROW 298 FROM FIGURE 7 Model layer, type, and depth 1234675 EXPLANATION 0–20 feet from land surface 20–50 feet from land surface 50–100 feet from land surface 100–150 feet from land surface More than 150 feet from land surface 0–20 feet from bedrock surface More than 20 feet from bedrock surface Unconsolidated Silurian bedrock

Figure 8. Model layering: example vertical section. 

50 feet

5,000 feet

SCALE
3. Model Construction 21

The elevation data from the DTM files were interpolated 

to 500-ft centers over the model domain. The 500-ft spacing 

is appropriate to the accuracy of the smoothed LIDAR data 

supporting the estimation of the land surface. The interpolation was performed using the “natural neighbor” algorithm 

(Sibson, 1981), which finds the closest subset of data to 

a query point and applies weights based on proportionate 

areas in order to interpolate a value. The scheme guarantees 

interpolated elevations are within the range of the data used; it 

does not infer trends and will not produce peaks, pits, ridges, 

or valleys not already represented by the elevation data. The 

node center locations for a single model cell were paired with 

the closest location of the 500-ft centers and the corresponding 

interpolated land- surface elevation was assigned to the cell. 

Note that clusters of 16 cells on 125-ft centers in the model 

nearfield share a single land-surface elevation from the background interpolation at 500-ft centers.

The top of layer 1 corresponds to the land surface 

assigned to the model cells. Almost the entire surface-water 

network is represented by model boundary conditions associated with layer 1 cells. The few exceptions are limited to 

large-area cells in the model farfield where downcutting of a 

stream produces a streambed that is more than 20 ft below the 

land-surface elevation assigned the cell. It is worth emphasizing that the use of thin layers (that is, 20-ft maximum thickness for layer 1 and 30 ft maximum thickness for layer 2) 

permits a relatively precise calculation of vertical gradients in 

the vicinity of surface-water features.

The bottom of layer 7 corresponds to the bottom of 

the Silurian dolomite (shown schematically in fig. 3). Over 

most of the model, the bottom of the Silurian dolomite is 

equivalent to the top of the Maquoketa shale, the regional 

confining unit separating the shallow and deep parts of the 

flow system. However, over two small areas of the model 

farfield (at the northwestern and southwestern corners of the 

grid), the Maquoketa shale subcrop boundary implies that a 

deeper Paleozoic unit (the Sinnipee dolomite or the St. Peter 

sandstone) lies at the bottom of the Upper Fox River Basin 

models. The bottom of the model at any row/column location 

corresponds not only to a stratigraphic boundary but also to 

an upward or downward flux boundary condition, which is 

described in the next subsection.

3.1.3 Flux Boundary Conditions

From the standpoint of edge boundary conditions, the 

Upper Fox River Basin model can be considered an inset 

model extracted from the SEWRPC regional model for 

southeastern Wisconsin. (Note that properties and boundary 

conditions internal to the model grid have a less direct relation 

to the parent model, as is explained in subsequent subsections). The Upper Fox River Basin model is entirely contained 

within the SEWRPC regional model, both in terms of lateral 

extent (the regional model extends over seven full counties in 

southeastern Wisconsin, including Waukesha and Washington 

Counties) and in terms of vertical extent (the SEWRPC model 

extends vertically (fig. 1C) from land surface to the Mount 

Simon sandstone at the bottom of the Cambrian-Ordovician 

deposits). The regional model contains 18 layers, including 

2 layers representing the unconsolidated material and 3 layers representing the Silurian dolomite. The top layer of the 

regional model is up to 100 ft thick (Feinstein and others, 

2005a).

As part of a recently-completed water-supply study for 

southeastern Wisconsin (Southeastern Wisconsin Regional 

Planning Commission, 2010), the original SEWRPC model 

was updated with 2005 pumping estimates, and the transient 

simulation extended beyond the original endpoint of the 

model in 2000. The output from the regional model corresponding to 2005 pumping rates is extracted as input to the 

Upper Fox River Basin model in the form of edge boundary 

fluxes. To facilitate the transfer of the lateral fluxes, the edge 

rows and columns of the inset Upper Fox models have been 

assigned the same widths as the cells in the parent regional 

model—2,500 ft per side. Layers in the regional model are 

associated with layers in the inset model on the basis of thickness and stratigraphy. Node centers in the regional model also 

are paired with boundary nodes of the inset model—typically 

multiple nodes in the Upper Fox models are associated with a 

single regional node from the SEWRPC model. The fluxes are 

inserted by means of the WEL package in MODFLOW at the 

boundary cells of the Upper Fox models.

The regional model contributes inflows and outflows at 

different layer depth intervals with respect to the inset area. 

When flows are tabulated, unconsolidated and dolomite layers 

along each side of the inset model both lose and gain groundwater from the surrounding region (table 2). However, on balance the net flux derived from the regional SEWRPC model 

is outward across all four sides of the Upper Fox River Basin 

model domain, especially across the east edge of the model 

where both the topography and the dip of the bedrock tends to 

slope eastward.

The bottom flux boundary condition for the inset model 

also is derived from the SEWRPC regional model. The fluxes 

from the regional model are available at a 2,500-ft spacing 

and, therefore, a single regional flux value is assigned over an 

area represented by multiple bottom Upper Fox model cells 

(equal to 16 cells for the nearfield). Again, the WEL package 

is used to insert the flux and the direction of flow varies by 

location. On balance the vertical flow is outward from the bottom of the Upper Fox models (table 2), which is in line with 

the regional tendency of water to leak downward from the 

shallow to the deep aquifer system as a result of large pumping centers drawing from the Cambrian-Ordovician units in 

Cities such as Waukesha.

The net flux out of the sides and bottom of the shallow 

aquifer system incorporated in the Upper Fox model domain 

is on the order of 7 million gallons per day (Mgal/d). This net 

total is about 10 percent of the total recharge to the top of the 

groundwater system discussed below. 
22 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

3.2 Hydraulic Conductivity Zones

The focus of the Upper Fox River Basin model is the 

most shallow part of the saturated flow system where groundwater/surface-water interactions occur. The water table in 

the model domain is almost always within the heterogeneous 

unconsolidated deposits and, therefore, special attention has 

been given to characterizing their lateral and vertical hydraulic 

conductivity . The assignment of hydraulic conductivity zones 

to the underlying dolomite, by contrast, is relatively simple 

and is inherited from the parent SEWRPC model. 

3.2.1 Unconsolidated Deposits

The assignment of hydraulic conductivity to unconsolidated deposits in this study is based on textural interpretations 

of the subsurface derived from a database consisting mostly of 

water-well driller logs compiled from three sources:

• digital records of well construction logs compiled by 

the Wisconsin Department of Natural Resources (2009) 

for the period from 1988 through December 2008, 

• microfiche records at the University of Wisconsin-Milwaukee library for the period before 1988, and

• records compiled by the USGS as part of the construction of the Lake Michigan Basin model (Arihood, 

2009).

These records are supplemented by logs prepared by 

geologists at the Wisconsin Geological and Natural History 

Survey (2004), and test borings obtained from several consulting reports (GeEx, 1989; Aquifer Science and Technology, 

2004, 2006, and 2010). The total database consists of about 

7,000 logs (fig. 9). In the farfield of the model, the logs are 

located to the nearest quarter-quarter section. In the nearfield 

of the model, street addresses in combination with an online 

geographic database for Waukesha County (Waukesha County 

Internet Mapping site,2009–10) were used to achieve more 

precise locations. The database locations are provided in latitude/longitude and in Wisconsin Transverse Mercator North 

American Datum of 1983 (NAD 83) projection.

Table 2. Fixed fluxes assigned to model at grid boundaries for groundwater-flow model of 

Upper Fox River Basin, southeastern Wisconsin.

[Flux unit is million gallons per day; unconsolidated deposits correspond to model layers 1–5; Silurian dolomite 

corresponds to layers 6 and 7; all layers correspond to layers 1–7]

Lateral model 

boundaries Layers Flux in Flux out Net flux out

North Unconsolidated deposits 0.299 0.313 0.014

Silurian dolomite .153 .189 .036

All layers .452 .502 .050

East Unconsolidated deposits .058 .470 .412

Silurian dolomite .054 1.281 1.227

All layers .113 1.751 1.638

South Unconsolidated deposits .090 1.068 .978

Silurian dolomite .051 .333 .282

All layers .141 1.400 1.259

West Unconsolidated deposits .493 1.719 1.226

Silurian dolomite .108 .113 .005

All layers .601 1.832 1.231

All sides Unconsolidated deposits .940 3.570 2.630

Silurian dolomite .366 1.916 1.550

All layers 1.307 5.485 4.178

Bottom model boundary .31 3.48 3.17

All model boundaries 1.62 8.96 7.35
3. Model Construction 23

Figure 9. Well-log database. (Well-log data interpreted from Wisconsin Department of Natural 

Resources, 2009, and Wisconsin Geological and Natural History Survey, 2004, digital-data sources.) 

(WGNHS, Wisconsin Geological and Natural History Survey.) 

Nearfield 

Lake

Surface-water feature

Water-well driller log not reaching bedrock

WGNHS log not reaching bedrock

Water-well driller log reaching bedrock

WGNHS log reaching bedrock

EXPLANATION

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°

Figure 9. Well log database (WGNHS, Wisconsin Geological and Natural History Survey).

Base from Wisconsin Department of Natural Resources digital data, 

2010, 1:24,000, Wisconsin Transverse Mercator, North American 

Datum of 1983
24 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Most of the logs penetrate the entire unconsolidated 

sequence and extend into the bedrock (fig. 9). All logs provide 

information as a function of depth. For the model database, 

depths were converted to absolute elevations using the log 

location and a 2-ft contour land-surface coverage provided by 

the Waukesha County Internet Mapping site. 

The logs provide descriptions of the texture of the unconsolidated deposits as a function of depth intervals. Water-well 

drillers use many terms to characterize the deposits. The 

initial dataset contained 357 unique identifiers, most applied 

to unconsolidated deposits. These identifiers were grouped in 

three classes:

• descriptions limited to fine-grained textures (clay, silt, 

silty clay, hardpan, and others)

• descriptions combining fine- and coarse-grained 

textures (silty sand, gravelly clay, silt and sand, and 

others)

• descriptions limited to coarse-grained textures (sand, 

gravel, sand and gravel, and others)

Each depth interval in a log corresponding to a single 

description was assigned to one of the three texture classes. 

However, in general, the depth intervals recorded in the logs 

do not match the depth intervals corresponding to the model 

layering scheme. Typically, a layer depth interval contains 

more than one texture class (fig. 10). In order to characterize the layer depth intervals over the vertical extent of each 

log, a set of rules was adopted that depend on the dominant 

texture reported within the model layer depth interval. At 

a log location, the depth interval for a layer is considered 

dominantly fine (facies 1) if 90 percent of the layer thickness contains descriptions reported as fine texture, dominantly coarse (facies 5) if 90 percent of the layer thickness is 

reported as coarse texture, and mixed fine and coarse (facies 3)

if 75 percent of the layer thickness is reported as mixed facies. 

In addition, two transitional facies are applied if none of 

the above criteria is met. If the layer depth interval contains 

more of texture class 1 and 2 than texture class 2 and 3, it is 

assigned the relatively fine (facies 2). Otherwise, it is assigned 

the relatively coarse (facies 4). The logic for the five possible 

facies assignments to model- layer intervals at log locations is 

systematically presented in table 3. An example assignment to 

facies 4 for a depth interval corresponding to model layer 3 is 

illustrated in figure 10.

Table 3. Logic for assignment of unconsolidated facies in model layers 1–5.

[ft, foot; %, percent]

• All driller log intervals are assigned to one of three texture classes:

Class 1 = All fines (clay, silt, silty clay, clayey silt)

Class 2 = Mixed fines and coarse (silt with sand, sandy clay, clayey sand, silt and gravel)

Class 3 = Sand and coarser (sand, gravel, sand and gravel)

• Within each model layer interval (0- to 20-ft depth, 20- to 50-ft depth, 50- to 100-ft depth, 100- to 150-ft depth, more than 

150-ft depth), the three texture classes at driller log locations are converted into five facies according to the following rules:

Step 1:

If at least 90% of the layer interval corresponds to class 1, then the facies is “dominantly fine”= facies 1

If at least 75% of the layer interval corresponds to class 2, then the facies is “dominantly mixed” = facies 3

If at least 90% of the layer interval corresponds to class 3, then the facies is “dominantly coarse” = facies 5

Step 2:

If none of the criteria are meet in Step 1, then

If more than 50% of the layer interval consists of class 1 and class 2, then the facies is “relatively fine” = facies 2

If more than 50% of the layer interval consists of class 2 and class 3, then the facies is “relatively coarse” = facies 4

See figure 10 for visualization.

• The facies assignments at driller log locations are interpolated to node centers on a 500-ft by 500-ft grid for each model layer.

Two methods are used for interpolation—one is fine-favored and one is coarse-favored (fig. 12). The 500-ft by 500-ft grid assignments 

are transferred to the model grid based by associating the model row/column node centers with the nearest node center on the 500 ft by 

500 ft grid. The distribution of facies in the fine-favored and coarse-favored models is shown in figures 14 and 15.
Figure 10. Logic for assigning facies to layer intervals. 

Example facies assignment for model layer 3 (50- to 100-foot dep

Layer 3 top at 50-foot depth

Log interval 1Well log description: clay  texture class 1Thickness: 10 feet (or 20 percent) of total layer thLog interval 2 Well log description: sand  texture class 3Thickness: 20 feet (or 40 percent) of total layer thLog interval 3Well log description: sand and clay  texture claThickness: 20 feet (or 40 percent) of total layer th

Layer 3 bottom at 100-foot depth

Interval assignment

Example layer assignment

Total thickness of layer 3 is 50 feet (100 percent of layer thickness), where tomore than 75 percent mixed fines and coarse-grained materials (excludes facthan 90 percent fines or more than 90 percent coarse-grained materials (excldoes not contain more than 50 percent fines plus mixed fine- and coarse-graibut does contain more than 50 percent coarse- plus mixed fine- and coarse-gassign the hydraulic conductivity for the entire 50−100 foot interval in Note: The well log may extend beyond this layer; therefore, repeat process folog intersects.

Assign as facies 3

YES

NO

Start here to assign hydraulic conductivity facies value to a layer interval based on well log descriptions:

Assign as facies 1

YES

Assign as facies 5

YES

Assign as facies 2

YES

Assign as facies 4

YES

NO

NO

NO

Does the layer interval contain more than 90 percent of texture class 3?Does the layer interval contain more of texture classes 1 and 2 than classes 2 and 3?

Does the layer interval contain more of texture classes 2 and 3 than classes 1and 2?

Does the layer interval contain more than 90 percent of texture class 1?

Value Assignment Lithologic description Examples of corresponding log descriptions

1 dominantly fine more than 90 percent fine clay, silt, clayey till, silty till

2 relatively fine more fine than coarse sandy clay and silt

3 mixed fines and coarse more than 75 percent mixed fines and coarse loamy till, sand and clay

4 relatively coarse more coarse than fine silty sand and gravel

5 dominantly coarse more than 90 percent coarse sand, gravel, sand and gravel

Hydraulic conductivity facies explanation

Figure 10. Logic for assigning facies to layer intervals.

Texture classesClass 1: Log descriptions limited to fine-grained material Class 2: Log descriptions combine fine-grained and coarse-grained materialsClass 3: Log descriptions limited to coarse-grained material

Does the layer interval contain more than 75 percent of texture class 2?
26 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Maps of the distribution of facies at log locations for each 

of the unconsolidated layer depth intervals (figs. 11A–E) show 

fine-grained deposits are more common in the eastern part of 

the model domain (Oak Creek Formation) and coarse-grained 

deposits are more common in the western part (Horicon Member). The plots also show the active and pinched locations 

of the five unconsolidated layers. Note that unconsolidated 

layer 5, corresponding to a depth of more than 150 ft, is almost 

completely pinched inside the model nearfield (fig. 11E).

It is helpful to visualize the textural trends across the 

model domain not only by plotting the distribution of log 

facies but also by interpolating the percent of the dominantly 

coarse facies from the log locations over the domain area. In 

this way, it is evident where the unconsolidated sediments are 

likely to be most permeable and most resistive to flow because 

of high and low sand content, respectively. The interpolation 

results (shown for model layer 1 corresponding to 0–20 ft 

depth in fig. 12A and for model layer 3 corresponding to 

50–100 ft depth in fig. 12B) reveal that in general about onethird of the sediments correspond to sand or gravel texture, 

but that the eastern side of the model domain is poor in coarse 

sediments, the western side is rich in coarse sediments, and the 

central region displays a complicated, interfingered pattern. 

The three texture zones match the distribution of the glacial 

units discussed in the “Conceptual Model” section—the 

clayey Oak Creek Formation to the east, the sandy Horicon 

Member to the west, and mixed New Berlin Member in the 

middle of the study area.

3.2.1.1 Fine-Favored and Coarse-Favored Zonation

The five facies descriptions applied to logs by layer-depth 

interval are the basis for the mapping of five unconsolidated 

hydraulic conductivity (K) zones across all the active cells in 

each layer. However, the necessity to interpolate facies from 

model cells where logs are present to neighboring cells where 

no logs are present adds a crucial element of uncertainty to 

the mapping. In order to explicitly account for the uncertainty, 

an interpolation method is applied which, as explained in the 

“Conceptual Model” section, allows the continuity of finegrained facies (facies 1 and 2) to be favored or the continuity 

of coarse-grained facies (facies 4 and 5) to be favored. The 

method uses a nearest neighbor approach in the following 

way: (a) if a cell in given layer-depth interval contains one 

or more logs within its area, the facies from those logs are 

used to assign a K zone to the cell, such that the finest facies 

encountered in the cell is used for the fine-favored mapping 

and the coarsest facies encountered for the coarse-favored 

mapping; (b) if a cell does not contain a log in a layer-depth 

interval, circles of increasing radii centered at the cell center 

are superimposed on the grid until the circle contains one or 

more logs, such that the finest facies encountered in the circle 

is used for the fine-favored mapping and the coarsest facies 

encountered for the coarse-favored mapping. The first search 

circle is assigned a radius of 708 ft (equal to the diagonal 

length of a cell 500 ft per side); if no log is encountered, the 

search circle is widened by a factor of two to 1,416 ft; if no 

log is encountered, the search circle is widened to 2,124 ft, 

and so on.

The method for interpolating the facies, that is K zones, 

into cells without logs is shown graphically in figure 13. 

The results of the procedure are shown for an example cross 

section in figure 14 and in plain view for each model layer 

in figure 15. In each case, the fine-favored interpretation is 

contrasted with the coarse-favored. 

Note that in model layer 5 (unconsolidated material 

more than 150 ft below land surface, associated with bedrock 

valleys), the configuration of K zones is identical for the finefavored and coarse-favored models (fig. 15E). The zones do 

not correspond to interpolation from logs (of which there are 

very few at this depth), but rather to a broad zonation of coarse 

material in the bedrock valley associated with the Bark River 

in the northwestern part of the model domain and mixed material elsewhere, including the Troy Bedrock Valley, which cuts 

through the southern part of the model domain. The evidence 

for this pattern is derived from previous studies, which treated 

the sediment in the bedrock valleys (Batten and Conlon, 1993; 

Feinstein and others, 2011). The simple zonation of the K 

distribution in layer 5 is almost entirely limited to the farfield 

of the model domain because layer 5 is pinched out in most of 

the nearfield.

Neither the coarse-favored or the fine-favored version is 

correct in the sense of being a true representation of the subsurface at a given depth interval but are useful in the different 

way connectivity is treated between high K zones. For example, the small box near the south end of the model nearfield 

in the plots in figure 15 shows the different proportion of fine 

and coarse sediments along one riparian zone of the Fox River 

generated by the two interpretations. These contrasting patterns have implications for the ability of a well located near 

the river to induce flow from the river. In the example cross 

sections in figure 14, it is evident that vertical flow is facilitated in the coarse-favored case relative to the fine-favored 

case because of the greater lateral and vertical connectivity of 

the coarse-dominated hydraulic conductivity zone (colored red 

in the plots). This outcome is important because the vertical 

hydraulic conductivity (Kv

) of the dominantly coarse K zone 

is expected to be elevated with respect to the other four zones 

given the near or total absence of fine-grained material in 

facies 5 (dominantly coarse). As a result, flow, for example, 

induced from a stream toward a nearby well screened 50 or 

100 ft below land surface would likely be transmitted more 

easily in the coarse-favored case because there are more areas 

where there is vertical continuity of the dominantly coarse 

facies. 
3. Model Construction 27

Figure 11A. Classification of log descriptions by dominant facies—layer 1 (0–20 foot depth). (Well-log data interpreted from 

Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and Natural History Survey, 2004, digital-data 

sources.) 

 Interpolated grid cells are 500 feet on a side 

Number populated grid cells out of 

38,250 = 3,087

 Log facies:

 Percent dominantly fine = 12

 Percent relatively fine = 12

 Percent mixed fine and coarse = 29

 Percent relatively coarse = 32

 Percent dominantly coarse = 15

Number nearfield populated grid cells 

out of 14,518= 1,723

 Log facies:

 Percent dominantly fine = 11

 Percent relatively fine = 12

 Percent mixed fine and coarse = 34

 Percent relatively coarse = 28

 Percent dominantly coarse = 15

Nearfield

Surface-water feature

Area where layer is pinched

EXPLANATION

Summary information

Facies type and percentage of facies type

1 Dominantly fine Greater than 90 percent fine

2 Relatively fine More fine than coarse

3 Mixed Greater than 75 percent mixed fines and coarse

4 Relatively coarse More coarse than fine

5 Dominantly coarse Greater than 90 percent coarse

Symbol Code Assignment Lithologic description

Figure 11a. Classification of log descriptions by dominant facies -- layer 1 (0 to 20 foot depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°
28 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 11B. Classification of log descriptions by dominant facies—layer 2 (20–50 foot depth). (Well-log data interpreted 

from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and Natural History Survey, 2004, digitaldata sources.) 

Figure 11b. Classification of log descriptions by dominant facies -- layer 2 (20 to 50 feet depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

 Interpolated grid cells are 500 feet on a side 

Number populated grid cells out of 

38,250 = 2,761

 Log facies:

 Percent dominantly fine = 11

 Percent relatively fine = 17

 Percent mixed fine and coarse = 22

 Percent relatively coarse = 30

 Percent dominantly coarse = 20

Number nearfield populated grid cells 

out of 14,518= 1,520

 Log facies:

 Percent dominantly fine = 10

 Percent relatively fine = 17

 Percent mixed fine and coarse = 26

 Percent relatively coarse = 27

 Percent dominantly coarse = 20

Nearfield

Surface-water feature

Area where layer is pinched

EXPLANATION

Summary information

Facies type and percentage of facies type

1 Dominantly fine Greater than 90 percent fine

2 Relatively fine More fine than coarse

3 Mixed Greater than 75 percent mixed fines and coarse

4 Relatively coarse More coarse than fine

5 Dominantly coarse Greater than 90 percent coarse

Symbol Code Assignment Lithologic description

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°
3. Model Construction 29

Figure 11C. Classification of log descriptions by dominant facies—layer 3 (50–100 foot depth). (Well-log data interpreted 

from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and Natural History Survey, 2004, digitaldata sources.) 

Figure 11c. Classification of log descriptions by dominant facies -- layer 3 (50 to 100 feet depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

 Interpolated grid cells are 500 feet on a side 

Number populated grid cells out of 

38,250 = 2,012

 Log facies:

 Percent dominantly fine = 12

 Percent relatively fine = 26

 Percent mixed fine and coarse = 21

 Percent relatively coarse = 28

 Percent dominantly coarse = 13

Number nearfield populated grid cells 

out of 14,518= 1,043

 Log facies:

 Percent dominantly fine = 13

 Percent relatively fine = 25

 Percent mixed fine and coarse = 24

 Percent relatively coarse = 25

 Percent dominantly coarse = 13

Nearfield

Surface-water feature

Area where layer is pinched

EXPLANATION

Summary information

Facies type and percentage of facies type

1 Dominantly fine Greater than 90 percent fine

2 Relatively fine More fine than coarse

3 Mixed Greater than 75 percent mixed fines and coarse

4 Relatively coarse More coarse than fine

5 Dominantly coarse Greater than 90 percent coarse

Symbol Code Assignment Lithologic description

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°
30 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 11D. Classification of log descriptions by dominant facies—layer 4 (100–150 foot depth). (Well-log data interpreted 

from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and Natural History Survey, 2004, digitaldata sources.) Figure 11d. Classification of log descriptions by dominant facies -- layer 4 (100 to 150 feet depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

 Interpolated grid cells are 500 feet on a side 

Number populated grid cells out of 

38,250 = 1,087

 Log facies:

 Percent dominantly fine = 19

 Percent relatively fine = 29

 Percent mixed fine and coarse = 25

 Percent relatively coarse = 20

 Percent dominantly coarse = 7

Number nearfield populated grid cells 

out of 14,518 = 514

 Log facies:

 Percent dominantly fine = 20

 Percent relatively fine = 28

 Percent mixed fine and coarse = 27

 Percent relatively coarse = 17

 Percent dominantly coarse = 8

Nearfield

Surface-water feature

Area where layer is pinched

EXPLANATION

Summary information

Facies type and percentage of facies type

1 Dominantly fine Greater than 90 percent fine

2 Relatively fine More fine than coarse

3 Mixed Greater than 75 percent mixed fines and coarse

4 Relatively coarse More coarse than fine

5 Dominantly coarse Greater than 90 percent coarse

Symbol Code Assignment Lithologic description

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°
3. Model Construction 31

Figure 11E. Classification of log descriptions by dominant facies—layer 5 (more than 150 foot depth). (Well-log data 

interpreted from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and Natural History Survey, 

2004, digital-data sources.) 

Figure 11e. Classification of log descriptions by dominant facies -- layer 5 (more than 150 feet depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

 Interpolated grid cells are 500 feet on a side 

Number populated grid cells out of 

38,250 = 75

 Log facies:

 Percent dominantly fine = 13

 Percent relatively fine = 51

 Percent mixed fine and coarse = 15

 Percent relatively coarse = 19

 Percent dominantly coarse = 2 

Number nearfield populated grid cells 

out of 14,518 = 31

 Log facies:

 Percent dominantly fine = 10

 Percent relatively fine = 67

 Percent mixed fine and coarse = 13

 Percent relatively coarse = 10

 Percent dominantly coarse = 0

Nearfield

Surface-water feature

Area where layer is pinched

EXPLANATION

Summary information

Facies type and percentage of facies type

1 Dominantly fine Greater than 90 percent fine

2 Relatively fine More fine than coarse

3 Mixed Greater than 75 percent mixed fines and coarse

4 Relatively coarse More coarse than fine

5 Dominantly coarse Greater than 90 percent coarse

Symbol Code Assignment Lithologic description

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°
32 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 12A. Distribution of coarse deposits based on well logs for example layer intervals—layer 1 (0–20 foot depth). 

(Well-log data interpreted from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and 

Natural History Survey, 2004, digital-data sources.) 

8.1 percent data coverage over cells

Average = 34.8 percent sand 

and coarser

Percent sand and coarser of layer 1 

(0 to 20 foot depth)

0

10

20

30

40

50

60

70

80

90

100

Interpolation:

Percent sand and coarser 

Interpolated grid cells are 500 feet

on a side 

EXPLANATION

Percent sand and coarser

0 – 33

33 – 67

67 – 100

Upper Fox River Basin boundary 

above Vernon Marsh

Nearfield

Area where layer is pinched

Lake

Stream

EXPLANATION

Figure 12a. Distribution of coarse deposits based on well logs for example layer intervals -- layer 1 (0-20 feet depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°
3. Model Construction 33

Figure 12B. Distribution of coarse deposits based on well logs for example layer intervals—layer 3 (50–100 foot depth). 

(Well-log data interpreted from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and Natural 

History Survey, 2004, digital-data sources.) 

5.3 percent data coverage over cells

Average = 32.1 percent sand 

and coarser

Percent sand and coarser of layer 3 

(50 to 100 foot depth)

0

10

20

30

40

50

60

70

80

90

100

Interpolation:

Percent sand and coarser 

Interpolated grid cells are 500 feet

on a side 

EXPLANATION

Percent sand and coarser

0 – 33

33 – 67

67 – 100

Upper Fox River Basin boundary 

above Vernon Marsh

Nearfield

Area where layer is pinched

Lake

Stream

EXPLANATION

Figure 12b. Distribution of coarse deposits based on well logs for example layer intervals -- layer 3 (50-100 feet depth).

Base from Wisconsin Department of Natural Resources 

digital data, 2010, 1:24,000, Wisconsin Transverse 

Mercator, North American Datum of 1983

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS
34 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 13. Interpolation method for fine- and coarse-favored models. 

Figure 13. Interpolation method for fine-favored and coarse-favored models.

Search area (ring)

Well log over layer interval 

Facies 1 (greater than 90 percent fines)

Facies 3 (greater than 75 percent mixed fines 

and coarse)

Facies 5 (greater than 90 percent coarse)

Logic of the interpolation algorithm

Starting at a model cell, if there are:

1. One or more well logs located in the first ring around model cell in question, stop at first ring and assign hydraulic conductivity facies 

based on the log with finest facies for fine-favored model and the log with coarsest facies for coarse-favored model. Otherwise, 

2. Go to second ring. Assign hydraulic conductivity facies based on one or more well logs within the second ring using the following model 

logic: In the fine-favored logic model, assign the hydraulic conductivity of the model cell to the finest facies present within this second ring.

In the coarse-favored logic model, assign the hydraulic conductivity of the model cell to the coarsest facies present within this second ring.

3. If there had been no logs in the first or second ring, the search area would be expanded until an annular ring around the model cell 

contains at least one log.

Example of application

No logs are located in first ring around the cell in question, therefore go to second ring.

Three logs are located in the second ring, therefore stop at second ring.

For the fine-favored model, assign the hydraulic conductivity of the model to value corresponding to the finest facies encountered in the 

second ring: in this example to the value of K1.

For the coarse-favored model, assign the hydraulic conductivity of the model to the value corresponding to the coarsest facies encountered 

in the second ring: in this example to the value of K5.

(Well logs for facies 2 and 4 not present in this example)

EXPLANATION

Model 

cell

Second ring

First ring
3. Model Construction 35

Figure 14A. Fine- and coarse-favored hydraulic conductivity zones in example vertical section—fine-favored zones. 

Figure 14B. Fine- and coarse-favored hydraulic conductivity zones in example vertical section—coarse-favored zones. 

WEST EAST

WEST EAST 50 feet

5,000 feet

SCALE

50 feet

5,000 feet

SCALE

EXPLANATION

Facies 1 (greater than 90 percent fines)

Facies 2 (more fines than coarse)

Facies 3 (greater than 75 percent mixed 

 fines and coarse)

Facies 4 (more coarse than fines)

Facies 5 (greater than 90 percent coarse)

Glacial material absent at this depth

Weathered Silurian Dolomite

Unweathered Silurian Dolomite

Figure 14a. Fine-favored and coarse-favored hydraulic conductivity zones in example vertical section -- 

fine-favored zones.

Figure 14b. Fine-favored and coarse-favored hydraulic conductivity zones in example vertical section -- 

coarse-favored zones

Section trace corresponding to row 298 (see figure 7)

Section trace corresponding to row 298 (see figure 7)
Figure 15A. Fine- and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits—layer 1 (0–20 foot depth). 

1 2 3 4 5

GlaciNearf

Symbol Facies 

Interpolation Number of no(excluded: 3,Median Average Standard devMinimum Maximum 

Figure 15a. Fine-favored and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits — layer 1 (0 to 20 foot 

88°20' 88°10'

43°10'

43°

0 2 0 4 K

2

Fine-favored zonation:Facies Number Percent1 3,019 212 2,378 173 5,149 364 2,639 185 1,222 8

Coarse-favored zonation:Facies Number Percent1 1,142 82 1,139 83 4,460 314 4,351 305 3,315 23

Percent of nearfield that is active (unpinched) = 99.24

LAYER 1

FINE COARSE

88°20' 88°10'

43°10'

43°
Figure 15B. Fine- and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits—layer 2 (20–50 foot depth). 

1 2 3 4 5

GlaciNearf

Symbol Facies 

Interpolation Number of no(excluded: 2,Median Average Standard devMinimum Maximum 

Figure 15b. Fine-favored and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits — layer 2 (x to xx foot 

88°20' 88°10'

43°10'

43°

0 2 0 4 K

2

Fine-favored zonation:Facies Number Percent1 2,502 182 2,938 223 3,987 294 2,453 185 1,828 13

Coarse-favored zonation:Facies Number Percent1 940 72 1,423 103 3,412 254 3,501 265 4,432 32

Percent of nearfield that is active (unpinched) = 94.42

LAYER 2

FINE COARSE

88°20' 88°10'

43°10'

43°
Figure 15C. Fine- and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits—layer 3 (50–100 foot depth). 

1 2 3 4 5

GlaciaNearf

Symbol Facies 

Interpolation Number of no(excluded: 2,Median Average Standard devMinimum Maximum 

Figure 15c. Fine-favored and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits — layer 3 (50 to 100 foo

88°20' 88°10'

43°10'

43°

0 2 0 4 K

2

Fine-favored zonation:Facies Number Percent1 2,449 242 2,538 253 2,389 244 1,649 165 1,092 11

Coarse-favored zonation:Facies Number Percent1 1,009 102 1,775 183 2,393 244 2,663 265 2,277 22

Percent of nearfield that is active (unpinched) = 69.69

LAYER 3

FINE COARSE

88°20' 88°10'

43°10'

43°
Figure 15D. Fine- and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits—layer 4 (100–150 foot depth). 

1 2 3 4 5

GlaciNearf

Symbol Facies 

Interpolation Number of no(excluded: 1,Median Average Standard devMinimum Maximum 

Figure 15d. Fine-favored and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits — layer 4 (100 to 150 fo

88°20' 88°10'

43°10'

43°

0 2 0 4 K

2

Fine-favored zonation:Facies Number Percent1 1,533 312 1,331 273 1,262 254 561 115 276 6

Coarse-favored zonation:Facies Number Percent1 717 142 1,068 223 1,442 294 1,109 225 627 13

Percent of nearfield that is active (unpinched) = 34.19

LAYER 4

FINE COARSE

88°20' 88°10'

43°10'

43°
40 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 15E. Fine- and coarse-favored hydraulic conductivity zones by model layer for unconsolidated 

deposits—layer 5 (more than 150 foot depth). 

A vertical section can be misleading and tends to underestimate the possibility for preferential flow because it represents 

only two spatial dimensions (in the case of figure 14, west to 

east and vertical). Preferential flow along coarse horizons can 

occur along any direction. The presence of locally continuous, 

often nonlinear, high-K features in figure 15 suggests (1) that 

glacial processes of deposition and erosion left some preferential channels intact marked by elevated horizontal hydraulic 

conductivity (Kh ), (2) that these local preferential pathways 

are most common in the nearfield of the model corresponding to the area of New Berlin Member deposition, and (3) that 

the pathways are also represented as more common in the 

coarse-favored than in the fine-favored cases. Pockets of possible preferential vertical flow can be identified by isolating 

areas where the entire unconsolidated thickness is dominantly 

coarse (fig. 16). The resulting maps, for a minimum threshold 

of 60 ft of total unconsolidated thickness, suggest that such 

areas are more extensive in the coarse-favored (4.6 percent of 

the nearfield area) than in the fine-favored cases (1.2 percent 

of the nearfield area) but occupy a small percentage of the 

nearfield area in both cases.

In summary, the fine-favored and coarse-favored interpretations of the unconsolidated hydrogeology are the basis for 

two distinct (although related) models of the Upper Fox River 

Basin—one with hydraulic conductivity zones that favor the 

connectivity of fine-grained deposits and a second with zones 

that favor the connectivity of coarse-grained deposits.

3.2.2 Silurian Bedrock

As noted previously, the Silurian dolomite is assumed 

to be zoned vertically with respect to hydraulic conductivity. 

The upper 20 ft of the dolomite (layer 6) is taken to represent a 

weathered horizon and is assigned a higher hydraulic conductivity than the remaining thickness (layer 7). The dolomite as a 

whole can be divided into several units with differing properties (Rovey, 1990; Carlson 2001). However, for this study, 

which is focused on the overlying unconsolidated material, the 

Silurian units are lumped into a single aquifer whose average 

hydraulic conductivity is assumed to be homogeneous except 

in areas adjacent to bedrock valleys where the glaciers excavated the dolomite and favored the formation of weathered 

zones that worked laterally into the rock (Feinstein and others, 

2005a). The bedrock valleys correspond to areas of thin or 

absent dolomite (see fig. 4B).

1 Greater than 90 percent fine

2 More fine than coarse

3 Greater than 75 percent 

mixed fines and coarse

4 More coarse than fine

5

Glacial material absent at this depth

Nearfield

Greater than 90 percent 

coarse

Symbol Facies Lithologic description

Figure 15e. Fine-favored and coarse-favored hydraulic conductivity zones by model layer for unconsolidated deposits — layer 5 (more th

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 4 KILOMETERS 2

EXPLANATION

LAYER 5

FINE AND COARSE
3. Model Construction 41

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ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ

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ÓÓÓÓÓÓÓÓ

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ÓÓÓÓ ÓÓÓÓ

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ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓ ÓÓÓÓ ÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

EXPLANATION

Nearfield

Fox River

Surface-water feature

Entire vertical sequence 

comprised of Facies 5

Note: The entire vertical sequence consists of Facies 5 for 

1.2 percent of the nearfield where the unconsolidated 

material is more than 60 feet thick.

Unconsolidated material 

at least 60 feet thick

EXPLANATION

Nearfield

Fox River

Surface-water feature

Entire vertical sequence 

comprised of Facies 5

Unconsolidated material 

at least 60 feet thick

Note: The entire vertical sequence consists of Facies 5 for 

4.6 percent of the nearfield where the unconsolidated 

material is more than 60 feet thick.

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ ÓÓÓÓ

ÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓÓ

ÓÓÓÓÓÓÓÓ

ÓÓÓÓ

ÓÓÓÓÓÓÓÓ ÓÓÓÓ ÓÓÓÓ ÓÓÓÓÓÓÓÓÓÓÓÓ

88°20' 88°10'

43°5'

43°

88°20' 88°10'

43°5'

43°

Base modified from Wisconsin Department of Natural Resources 

1:24,000-scale digital data, 2010

Base modified from Wisconsin Department of Natural Resources 

1:24,000-scale digital data, 2010

0 2 4 MILES

0 2 4 KILOMETERS

0 2 4 MILES

0 2 4 KILOMETERS

Figure 16A. Areas where vertical sequence consists entirely of dominantly coarse facies—fine-favored model. 

(Well-log data interpreted from Wisconsin Department of Natural Resources, 2009, and Wisconsin Geological and 

Natural History Survey, 2004, digital-data sources.) 

Figure 16B. Areas where vertical sequence consists entirely of dominantly coarse facies—coarse-favored 

model. (Well-log data interpreted from Wisconsin Department of Natural Resources, 2009, and Wisconsin 

Geological and Natural History Survey, 2004, digital-data sources.)
42 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

3.2.3 Initial Hydraulic Conductivity Values

Hydraulic conductivity values in the SEWRPC model 

were calibrated to conditions across southeastern Wisconsin 

(Feinstein, and others, 2005a). The Kh values for the regional 

model vary between 0.2 and 100 ft/d for the unconsolidated 

material, are set to 4 ft/d in the weathered dolomite, and vary 

between 1 and 4 ft/d for the remaining dolomite thickness 

(with the higher Kh values near the Silurian dolomite subcrops 

adjacent to bedrock valleys). The Kv values for the regional 

model vary between 0.005 ft/d and 1 ft/d for the unconsolidated material, are set to 0.01 ft/d for the weathered dolomite, and vary between 0.001 and 0.01 ft/d for the remaining 

dolomite thickness (with the higher Kv values near the Silurian 

dolomite subcrops adjacent to bedrock valleys). 

The calibrated values from the SEWRPC model are the 

basis for the choice of initial hydraulic conductivity values 

for the Upper Fox River Basin models. In the case of the 

unconsolidated material, five K zones are required for the finefavored and the coarse-favored models. The Kh and Kv

 values 

for the finer facies correspond to the low end of the range 

adopted in the SEWRPC model, and the values for the coarser 

facies correspond to the high end of the SEWRPC range. The 

distribution of initial K values by facies is presented in table 4. 

An important feature of the initial assignments is the implied 

Kh to Kv anisotropy ratio. It is higher for facies 1 to 4 than it 

is for facies 5 because facies 1 to 4 are considered to contain 

at least some fine-grained horizons that inhibit vertical flow, 

whereas facies 5, dominantly coarse, is represented as largely 

free of fine-grained horizons.

It bears emphasizing that the initial K values are identical for the fine-favored and coarse-favored models. Indeed, 

the only difference between the initial versions of these two 

models is the K zonation of the facies across layers, as shown 

by the contrasting plots in figure 15. During the calibration 

process, the K values are allowed to diverge as do several 

other input parameters that yields distinct parameterizations 

for the final versions of the two models. 

The Silurian dolomite K values in the Upper Fox model 

correspond to the zonation in the SEWRPC model. The initial 

Kh

 and Kv values assigned to Silurian dolomite layer 6 correspond to zone 8 in table 4 (4 and 0.01 ft/d, respectively). The 

initial values assigned to most of dolomite layer 7 correspond 

to zone 1 in table 4 (1 and 0.001 ft/d, respectively); however 

values increase gradually to the weathered zone 11 values 

toward the margins of bedrock valleys. 

Table 4. Initial hydraulic conductivity values for both finefavored and coarse-favored models.

[Kh (ft/day), horizontal hydraulic conductivity; Kv (ft/day), vertical hydraulic 

conductivity; Aniso, anisotropic ratio of Kh to Kv]

Unconsolidated facies Initial Kh Initial Kv Aniso

1 = Dominantly fine 0.5 0.005 100:1

2 = Relatively fine 2 .01 200:1

3 = Mixed fine and coarse 10 .02 500:1

4 = Relatively coarse 40 .2 200:1

5 = Dominantly coarse 80 2.0 40:1

Silurian dolomite zones1 Initial Kh Initial Kv

1 1.00 0.001

2 1.25 .002

3 1.75 .003

4 2.25 .004

5 2.75 .005

6 3.25 .006

7 3.75 .007

8 4.00 .01

Streambed2 Initial Kv

Low (characterized as 

silty/muddy)

1.0

Not characterized 5.0

High (characterized as 

sandy/gravelly)

25.0

The mapping of the Silurian dolomite zones in layers 6 and 7 of the 

Upper Fox River Basin model corresponds to the input of the Silurian 

layers in the regional southeastern Wisconsin regional groundwater-flow 

model (Feinstein and others, 2005a).

Streambed characterization derived from Poff and Threinen (1963).

3.3 Recharge

The most important source for groundwater in southeastern Wisconsin is natural recharge to the water table. The 

SEWRPC regional model used recharge rates estimated as 

a function of stream base flow for watersheds in southeastern Wisconsin (Cherkauer, 2004; Cherkauer and Ansari, 

2005). The approach correlates base flow estimated for 

gaged basins (assumed to be as a proxy for recharge) with 

variables related to topography, climate, land use, and soil 

type in order to develop a regression equation applicable to 

neighboring ungaged basins. The underlying method, detailed 

in Cherkauer and Ansari, 2005, equates stream base flow to 

recharge (R), which is normalized to observed annual precipitation (P). Regression analysis shows that R/P is controlled by 

three dimensionless ratios: (1) infiltrating to overland water 

flux, (2) vertical to lateral distance water must travel, and 

(3) natural to developed land cover. The individual watershed 
3. Model Construction 43

properties that indicate or comprise these ratios are commonly 

available in Geographic Information System (GIS) databases. 

The empirical relation for predicting R/P was first developed 

for selected watersheds and then tested outside the study area 

against other methods for calculating recharge. The method 

produces values that agree with base-flow separation from 

streamflow hydrographs (to within 15 to 20 percent), groundwater budget analysis (4 percent), well hydrograph analysis 

(12 percent), and the distributed-parameter watershed model, 

PRMS, calibrated to total streamflow (18 percent). 

The recharge rates produced by the method are estimates 

of long-term average rates (rather than rates associated with 

any given year such as 2005). It yields a mean recharge rate 

for the region of southeastern Wisconsin equal to 4.5 inches/

yr (Feinstein and others, 2005a). For this study, the recharge 

regression equation developed for southeastern Wisconsin was 

applied to 27 drainage subbasins within the model domain 

(Professor Douglas Cherkauer, University of Wisconsin-Milwaukee, written commun., December 6, 2009). The resulting 

rates vary between 1.6 and 9.5 inches per year (in/yr) (fig. 17).

Note, however, that zero recharge is assigned to areas covered 

by Pewaukee Lake and Vernon Marsh.

The recharge rates are partly correlated to the type of 

glacial material (mapped in fig. 2). The lowest rates tend to be 

in the eastern part of the model domain (in areas underlain by 

the Oak Creek Formation) and the highest rates in the western 

part (in areas underlain by the Horicon Member). Recharge 

rates are most variable in the central part of the study area 

(mostly underlain by the New Berlin Member). The average 

recharge rates for the model domain and nearfield are 4.43 

and 4.16 in/yr, respectively. It is worth noting that although 

the recharge zonation in the parent SEWRPC model derived 

from the regression equation was based on larger basins than 

those used for mapping recharge in the Upper Fox model, the 

average rate in the part of the SEWRPC model corresponding 

to the Upper Fox model domain was also 4.43 in/yr, implying 

that the boundary fluxes transferred from the SEWRPC model 

to the Upper Fox model are consistent with the recharge rates 

applied inside the Upper Fox model.

3.4 Surface-Water Network

The surface-water network for the model domain (fig. 6A) 

consists of streams, lakes, and wetlands. These features are 

represented by four MODFLOW packages (fig. 18), two of 

which are applied outside the Upper Fox River Basin (RIV 

and DRN) and three inside the Basin (SFR, LAK and DRN). 

3.4.1 Outside Upper Fox River Basin

Streams outside the Upper Fox River Basin are represented by the RIV package and water bodies (lakes and 

wetlands) are represented by the DRN package. From an input 

standpoint, these packages are populated by relatively simple 

inputs—estimates of stage and estimates of streambed conductance—which in turn are a function of the streambed Kv , 

thickness, width, and length inside a model cell. Base flow to a 

stream occurs from the groundwater to the stream in accordance with Darcy’s Law whenever the simulated water-table 

elevation is higher than the prescribed stream stage in a cell. 

The RIV package also allows loss of water from the stream 

to the groundwater when the stream stage is higher than the 

ambient water table. Maximum inflow from the stream to the 

groundwater is reached when the water-table elevation is at or 

below the bottom of the streambed elevation. The DRN package only allows outflow from groundwater to the simulated 

water body. By assigning lake and wetlands to the DRN package, the model does not allow water bodies that are simulated 

as perched above the water table from influencing the model 

solution.

3.4.2 Inside Upper Fox River Basin

Streams inside the Upper Fox River Basin are represented 

by the SFR package and water bodies are represented by the 

LAK or DRN package. The SFR algorithm in MODFLOW 

permits more sophisticated simulation of groundwater/surfacewater interactions than does the RIV package (Niswonger and 

Prudic, 2006). The streamflow in the channel is propagated 

downstream through the network from lower order to higher 

order tributaries. For example, the amount of simulated flow 

is calculated in such a way that if a well is pumping at a 

rate high enough to dewater a part of the channel, the water 

available to the well from the stream will be limited. In the 

Upper Fox River Basin models, the streamflow in the channel is coincident with the contribution from accumulated base 

flow (including inflows from the Sussex, Brookfield, and 

Waukesha WWTPs) , but the overland runoff component is 

not considered. Consequently, the model output for streamflow corresponds conceptually to low flow conditions typical 

of late summer or early fall when runoff is minimal. Because 

the entire Upper Fox River Basin is incorporated in the model 

domain, including the headwaters of all tributaries to the Fox, 

there is no need to add water to channels to account for accumulated base flow originating outside the model. 
44 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 17. Model recharge distribution. (Recharge values based on Cherkauer and Ansari, 2005, p. 102–112.) 

88°20' 88°10'

43°10'

43°

Fox River

Surface-water

feature

Nearfield

0 2 4 MILES

0 2 4 KILOMETERS

0.00

1.58

1.84

2.54

2.59

3.02

3.11

3.20

3.29

3.68

3.73

3.94

4.08

4.12

4.34

4.38

5.17

5.52

5.65

5.70

5.74

6.40

6.44

6.84

6.97

9.42

9.47

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

Zone

Recharge,

in inches

per year

EXPLANATION

Figure 17. Model recharge distribution.

Base modified from Wisconsin Department 

of Natural Resources 1:24,000-scale 

digital data, 2010
3. Model Construction 45

Figure 18. Surface-water cells assigned to MODFLOW-2005 packages. 

EXPLANATION

Figure 18. Surface-water cells assigned to MODFLOW-2005 packages.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Surface-water 

boundary conditions

River

Stream

Drain

Lake

Surface-water feature

Nearfield

F o x

R i v e r

Base modified from Wisconsin Department 

of Natural Resources 1:24,000-scale 

digital data, 2010
46 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Another capability of the SFR package is to calculate 

stream stage (rather than accept it as an input) by virtue of the 

net accumulated upstream base flow and the wetted channel 

geometry, which, through the channel width, also is partly a 

function of the accumulated base flow. In this study, the capability to calculate stage is only used for the headwater reaches 

of the stream network inside the Fox River Basin; that is, 

for the “first order” reaches above the first confluence with a 

second tributary. If the model simulates a water-table elevation 

below the top of the streambed for a headwater reach, then 

the reach is reported dry and is, by implication, ephemeral. If 

the model accumulates base flow in a headwater reach, then 

it is simulated as perennial. About one-half the total length of 

streams in the Upper Fox River Basin models corresponds to 

headwater reaches; the reaches that are second order or above 

(and are unlikely to be ephemeral) are assigned a fixed stage. 

Like the RIV package, the SFR package allows water to flow 

from the stream channel to the groundwater when the simulated water table is below the stage (fixed or calculated).

The Upper Fox River Basin contains one large water 

body—Pewaukee Lake. This feature is simulated with the 

LAK package (Merritt and Konikow, 2000) that simulates 

lake level in terms of the lake geometry and the balance of 

inflow and outflow of water. The water budget for the lake 

takes account of precipitation on the lake surface, any runoff 

from the lake basin area, direct groundwater base flow, and 

tributary streamflow as inflows. The budget also takes account 

of evaporation from the lake surface, losses to groundwater, 

streamflow into an outlet, and any direct pumping of lake 

water as outflows. In the case of Pewaukee Lake, direct runoff 

is neglected (as it is for the stream network) and no pumping 

is reported to occur, but the other budgets terms, including 

contributions from inflowing streams (such as Zion Creek) and 

discharge to an outlet stream (Pewaukee River and then to the 

Fox River) are simulated.

The topography in the Upper Fox River Basin includes 

valleys and lowlands (fig. 5B) where wetlands in connection 

with the groundwater system are present. These features are 

often in close proximity to streams. Several riparian wetlands 

associated with the Fox River are explicitly simulated in the 

models by means of the DRN package with the largest occurring near the confluence of Sussex Creek with the Fox River, 

upstream from the confluence of Pebble Creek with the Fox, 

and just north of Vernon Marsh where Pebble Brook joins 

the Fox. Most of the groundwater discharge to these riparian features is expected to flow a short distance overland into 

the river channel and, therefore, for the purposes of model 

accounting, the DRN flow to these features is added to the 

accumulated base flow in the Fox River. 

3.4.3 Input to MODFLOW-2005 Surface-Water 

Packages

The mapping of the surface-water network was constructed from spatial information available in GIS format in 

the National Hydrography Dataset (U.S. Geological Survey, 

2010b) that was clipped to the model study area and overlain 

on the model grid. Each model cell intersected by a surfacewater feature requires a number of inputs to characterize the 

connection of the feature to the groundwater system. An input 

common to the RIV, DRN, and SFR packages for second 

order and higher streams is the surface-water stage (whereas 

the stage is calculated for headwater SFR streams and for 

the application of the LAK package to Pewaukee Lake). The 

fixed stages are linked to the land-surface elevations derived 

from Waukesha County and USGS DTM files discussed in the 

“Model Construction” section under “Vertical Layering”. To 

interpolate from the equally-spaced estimates of land-surface 

elevation to the midpoints of stream stretches inside model 

cells, a triangular irregular network (TIN) consisting of the 

three land-surface elevations points nearest the stage location was constructed and the stage elevation determined using 

weights calculated from the natural neighbor algorithm referenced in the “Vertical Layering” subsection Each resulting 

stage elevation was inspected to ensure that stream elevations 

decreased in the downstream flow direction and, if not, a linear 

correction was applied across the reach to enforce downstream 

routing. 

A likely bias arising from this procedure is estimated 

stages are too high given that the TIN surface is unlikely to 

reflect channel depressions and is more likely to reflect the 

elevation of the first terrace bordering a stream. To account 

for the bias, a comparison was made between the TIN-derived 

stages and stages recovered from USGS 7.5-minute topographic maps (scale: 1 inch equals 2,000 ft) in the Waukesha 

area. Comparison of elevations were calculated as interpolated 

TIN elevation minus contour elevation at about 350 locations (fig. 19) where the map contours cross streams (varying 

between 717 ft and 1,069 ft elevation). It was found that:

• on average the TIN elevations are higher than the contour elevations; the mean difference is +3.2 ft and the 

median difference is +2.7 ft;

• 49 percent of the differences are between +3 and −3 ft 

and 83 percent are between +6 and −6 ft, but only 

2.8 percent of the comparisons are more negative than 

−3 ft;

• the biggest differences are almost always associated 

with headwater reaches (fig. 19). 
3. Model Construction 47

Figure 19. Comparison of topographic contours to digital elevation model (DEM) (LIDAR) elevations at 

stream locations. Biases calculated by subtracting the elevation where a topographuic contour crosses a 

stream from the elevation obtained from the interpolated DEM at the same location.

Figure 19. Comparison of topographic contours to DEM (LIDAR) elevations at stream locations. Bias is calculated

by subtracting the elevation where a topographic contour crosses a stream from the elevation obtained from the 

interpolated digital elevation model (DEM) at the same location.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

−18 to −6

−6 to −3

−3 to 3

3 to 6

6 to 12

12 to 30

Nearfield Bias (as defined below), in feet

Lake

Surface-water feature

EXPLANATION

DEM elevation point 

Base modified from Wisconsin Department of Natural Resources 

1:24,000-scale digital data, 2010; topographic contour values 

derived from 1:100 digital data, 1993–2000, projected to 

Wisconsin Transverse Mercator, North American 

Datum of 1983 
48 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

To correct the bias, two actions were taken. All TIN-generated stream elevations in the farfield and the nearfield were 

lowered by 2.7 ft (corresponding to the median bias); in areas 

of large difference, the stream elevations were also manually 

corrected to equal contour elevations. After the corrections, 

more than 70 percent of the estimated stages are within 3 ft of 

the stages implied by the contour maps.

The stages of lakes are directly imported from the original land-surface databases. Little fitting was needed to make 

the elevations of lakes integrated in the surface-water network 

consistent with the stage elevations of inlet and outlet streams. 

The stages of riparian wetlands included as groundwater discharge areas were set 1 ft below the land surface assigned to 

the occupied model cells. 

A second input common to surface-water packages is 

the conductance term mediating the flux exchange between 

groundwater and the stream, lake, or wetland. The conductance term is equal to the product of streambed Kv

, width, and 

length inside the model cell divided by assumed streambed 

thickness. For the streams inside the Upper Fox River Basin 

represented by the SFR package, a compilation of the properties of surface-water features for Waukesha County (Poff 

and Threinen, 1963) served to separate the reaches into three 

zones (fig. 20) based on bed descriptions as silty/muddy (low 

Kv zone), sandy/gravelly (high Kv zone), or the absence of a 

description (middle Kv zone). The bed of the Fox River falls 

into the unknown, that is middle, category. To quantify the Kv

values , notice was taken of a study conducted in Dane County 

(located about 50 mi west of the study area), where measured 

riverbed Kv values (assuming bed thickness of 1 ft) ranged 

between 1.6 ft/d and 37 ft/d, averaging 8 ft/d (Krohelski and 

others, 2000). For this study, 1 ft/d was assigned as the initial 

Kv for the low zone, 5 ft/d was assigned for the middle zone, 

and 25 ft/d for the high zone (table 4). 

A simple equation (table 5) was used to correlate the 

width component of the Upper Fox stream conductances with 

the upstream length of the longest trunk of the stream upgradient from the model cell in question. The intent of the equation 

is to assign widths closely corresponding to the trial widths 

listed in table 5 for selected upstream lengths. The application 

of this equation assigns widths less than 8 ft for reaches less 

than 1 mi in length, typically those associated with headwaters. For larger features, the width can exceed 100 ft as is 

observed for the Upper Fox River toward its entry into the 

Vernon Marsh at the downstream end of the model nearfield. 

The stream length corresponds to the superposition of the GIS 

National Hydrography Dataset arc coverages on the model 

grid. Finally, the thickness of the streambed was assumed to be 

equal to 1 ft for all Upper Fox River Basin stream reaches.

Inputs particular to the headwater (first order) reaches 

represented by the SFR package are the slope and Manning 

coefficients of the streambeds that are used in the calculation 

of the stream stage by means of the Manning equation (Niswonger and Prudic, 2006). In the absence of data to quantify 

these inputs for individual reaches in the domain, the models 

apply a uniform slope of 0.002 ft/ft corresponding to a moderate slope, and a uniform dimensionless Manning coefficient 

of 0.037 corresponding to a bed with some gravel and cobbles 

(Barnes, 1967).

For streams outside the Upper Fox River Basin represented by the RIV package, the bed Kv

 is assumed to be equal 

to 1 ft/d for all reaches; the stream width for these features is 

set to 30 ft, the length corresponds to the intersection of GIS 

arcs with the model grid; and the bed thickness is uniformly 

set to 1 ft. For lakes outside the Upper Fox River Basin as well 

as riparian wetlands inside the basin, both represented by the 

DRN package, the bed Kv is assumed to be equal to 0.01 ft/d 

(reflecting the assumption that the lake and wetland beds are 

predominantly silty), the area corresponds to the GIS coverage 

intersection with model cells, and the bed thickness is set to 

2 ft.

Table 5. Correlation of channel width with upstream length by 

means of fitting curve for streamflow-routing (SFR) cells in Upper 

Fox River Basin.

[The main trunk of the Fox River near the southern edge of the nearfield has 

an upstream length of about 24 miles, which implies a fitted width of 99 feet, 

roughly equal to the observed channel width at that location, equal to 89 feet]

Upstream length

(miles)1

Target channel width 

(feet) assumed for 

upstream length

Fitted channel width 

from equation (feet)2

0.1 1 1.3

.3 3 3.1

1 8 7.9

5 30 28.2

10 50 48.7

25 100 100.2

1

Upstream length corresponds to length of main trunk of each stream. 

2

Fitting equation: Width_ft=0.0092*(Upstream_Length_ft)0.7884. 
3. Model Construction 49

Figure 20. Streambed hydraulic conductivity zones for routed SFR cells. Figure 20. Streambed hydraulic conductivity zones for routed SFR cells.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

EXPLANATION

Streambed hydraulic

conductivity zone

Low

Middle

High

Surface-water feature

Nearfield

Base modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, 

Wisconsin Transverse Mercator, North 

American Datum of 1983
50 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

The LAK package representation of Pewaukee Lake 

requires special input. Precipitation to and evaporation from 

the lake surface are assumed to be 32 and 29 in/yr, respectively, on the basis of average long-term observations in 

northwestern Waukesha County (Linsley and others, 1982, 

p. 78, 154). Lakebed conductance and thickness are assumed 

to be equal to 0.01 and 1 ft/d, respectively. Lake stage is a 

function of inflows, outflows, and the volume of the lake, 

which in turn depends on the lake stage, lake area, and lake 

bottom. The geometry and bathymetry of Pewaukee Lake 

have been mapped (Wisconsin Conservation Department, 

1966). The survey indicates that the average area of the lake 

is on the order of 2,425 acres and the volume is on the order 

of 35,507 acre-ft. The 7,059 LAK cells (each 125 ft on a side) 

approximate the area, and a flat bottom assigned to the lake 

cells set at 738 ft elevation approximates the desired volume 

for the expected stage between 852.5 and 853 ft.

A key input for calculating lake stage is the relation 

between lake stage and outflow at the outlet to Pewaukee 

River. Data made available by the engineer who controls the 

weir at the spillway outlet (David White, Director of Public 

Works/Village Engineer, Village of Pewaukee, written commun., September 2009) allows a rating curve to be constructed 

between lake stage and outflow using the relation from 

Streeter (1966):

Q w = 3 09 h1 5 . * * ( ) . (2)

where

Q is approximated outflow, in cubic feet per 

second;

w is width, in feet, of weir at outlet equals 12; 

and

h is calculated lake stage above spillway 

elevation, in feet.

The spillway elevation varies over time with adjustment 

of the weir but averaged about 852.35 ft between January 2007 and August 2009. The median recorded difference 

between the spillway elevation and lake stage (h) was 0.51 ft, 

with the lake level below the spillway about 20 percent of the 

time and as much as 3.61 ft above the spillway elevation at 

maximum lake stage. Applying the weir equation to the period 

of record and adjusting it to the average spillway elevation 

yields the relation shown graphically in figure 21. For example, a lake stage of 852.75 ft (corresponding to h equals 0.4 ft) 

corresponds to an outflow 9.4 cubic feet per second (ft3/s). 

Using this relation, MODFLOW-2005 iterates to a lake stage 

solution reflecting long-term low-flow conditions that balances 

all inflows and outflows including the discharge across the 

weir to the Pewaukee River.

Figure 21. Pewaukee Lake outflow as a function of lake level.

0.0

4.0

8.0

12.0

16.0

20.0

24.0

28.0

32.0

852.15 852.35 852.55 852.75 852.95 853.15 853.35 853.55

Lake outflow , in cubic feet per second

Lake level, in feet

Spillway elevation 

is 852.35 feet
3. Model Construction 51

The inputs to the LAK, SFR, RIV, and DRN packages 

are summarized in table 6. One additional input to the SFR 

package is the effluent added to the Upper Fox River Basin 

surface-water network from the three WWTPs. The effluent originates as water pumped from the shallow and deep 

aquifer systems and is used for public supply and industrial/

commercial purposes before treatment and discharge. In 

low-flow periods, the combined effluent flow represents about 

one-third of the streamflow in the Upper Fox River above the 

Vernon Marsh (see “Model Results” section). Effluent flows 

recorded by the Sussex, Brookfield, and Waukesha water 

utility agencies at the WWTPs (Professor Timothy Grundl, 

University of Wisconsin-Milwaukee, written commun., February 15, 2010) show daily and seasonal rates vary around the 

average as a function of water demand, and there has been a 

long-term upward trend since the 1960s. Model input flows 

between September 2008 and July 2009 are averaged to yield 

a single value at each WWTP—Sussex, 2.44 ft3/s; Brookfield, 

10.85 ft3/s; and Waukesha, 13.83 ft3/s. These average rates are 

added to the model-calculated streamflow at the WWTP locations (fig. 6B).

Table 6. Input for MODFLOW surface-water packages.

[NHD, U.S. Geological Survey National Hydrography Dataset ( 2010a); land-surface dataset from Waukesha County (2005); 

LAK, lake; ft/d, foot per day; ft, foot; in/yr, inch per year; SFR, streamflow routing; ft/ft; foot per foot; RIV, river; 

DRN, drain]

Value or data source

LAK7 package for Pewaukee Lake

Lakebed hydraulic conductivity 0.01 ft/d

Lakebed thickness 1 ft

Precipitation on lake 32 in/yr

Evaporation from lake 29 in/yr 

Runoff to lake 0.0 

Pumping from lake 0.0 

Lake sill elevation corresponding to outlet into Pewaukee River 852.35 ft 

Lake stage/lake discharge relation for outlet see figure 21 

SFR2 package for streams inside Upper Fox River Basin

Streambed hydraulic conductivity Zones: low, middle, high (figure 20)

Streambed thickness 1 ft

Channel width in cell related to upstream length (table 5)

Channel length in cell NHD coverage intersected with grid

Streambed slope for headwater reaches 0.002 ft/ft

Streambed Manning Coefficient for headwater reaches 0.037

Stream stage in cell for reaches downgradient from headwater reaches interpolated from land-surface data

RIV package for streams outside Upper Fox River Basin

Riverbed hydraulic conductivity 1 ft/d

Riverbed thickness 1 ft

Channel width in cell 30 ft

Channel length in cell NHD coverage intersected with grid

River stage in cell interpolated land surface

DRN package for water bodies (wetlands and lakes other than Peawaukee Lake)

Bed hydraulic conductivity 0.01 ft/d

Bed thickness 2 ft

Area in cell NHD coverage intersected with grid

Stage land surface for lakes; land surface 

minus 1 ft for wetlands

DRN package for Sussex and Waukesha Silurian dolomite quarries

Quarrybed hydraulic conductivity 10 ft/d

Quarrybed thickness 1 ft

Quarry stage 10 ft below top of dolomite in layer 6
52 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

3.5 Water Withdrawals

Development of the Upper Fox River Basin has been 

accompanied by stresses on the groundwater-flow system. The 

models consider two activities that withdraw groundwater—

pumping from high-capacity wells and discharge to dolomite 

quarries. The addition of water by artificial recharge is not 

reported for the area and inflow from irrigation, a rare practice 

in Waukesha County, is neglected.

3.5.1 Pumping

The economic growth and suburban expansion in southeastern Wisconsin have been because of, in part, the abundant 

water supplies available for public, domestic, and industrial 

uses. Lake Michigan is the source for about 70 percent of 

all water used in the region, mainly in the lakeshore counties (Ozaukee, Milwaukee, Racine, and Kenosha), which lie 

mostly within the Great Lakes drainage basin. Farther inland, 

Washington, Waukesha, and Walworth Counties are principally in the Mississippi River Basin, and, because of international limitations on diversion of water out of the Great Lakes 

Basin, rely on groundwater for over 99 percent of their needs 

(Southeastern Wisconsin Regional Planning Commission and 

Wisconsin Geological and Natural History Survey, 2002). In 

2005, about 95 Mgal/d of groundwater were withdrawn for 

public, domestic, industrial, commercial, and agricultural uses 

(Southeastern Wisconsin Regional Planning Commission, 

2010). The withdrawal in Waukesha County is estimated to be 

34 Mgal/d. About 25 percent of the total is pumped by private 

domestic wells penetrating shallow aquifers. The remaining 

withdrawal is extracted from high-capacity wells (defined as 

withdrawing on average more than 0.1 Mgal/d) penetrating the 

shallow and deep aquifer systems, chiefly for public supply 

and industrial purposes.

Recent studies of water use (Buchwald and others, 2010; 

Feinstein and others, 2010; Southeastern Wisconsin Regional 

Planning Commission, 2010) indicate that shallow aquifer 

pumping from high-capacity wells in the model domain for 

2005 was on the order of 6.7 Mgal/d. The fine-favored and 

coarse-favored models include wells open to the unconsolidated material, which pump 1.62 Mgal/d, and wells open to 

the Silurian dolomite, which pump 5.07 Mgal/d. The greatest 

concentration of pumping is from dolomite wells in the eastern 

one-third of the study area (fig. 22). The pumped interval of 

most wells penetrate multiple model layers, and withdrawal is 

distributed among the layers according to the relative magnitude of the transmissivity in each layer (equal to the initial 

hydraulic conductivity values multiplied by the total cell 

thickness).

Domestic pumping is not included as a model stress 

(see discussion in “Model Limitations” section). The models 

also exclude pumping wells installed after 2005—notably 

three public supply wells located at the south end of the city 

of Waukesha, which withdraw about 1.6 Mgal/d from the 

shallow aquifer system (Jeff Detro, Waukesha Water Utility, 

written commun., May 2010). However, two of these wells are 

included in the demonstration application of the fine-favored 

and coarse-favored models used to evaluate the possibility of 

augmenting water supply by means of riparian pumping (see 

“Model Application to Hypothetical Well Field” section).

3.5.2 Quarries

The Silurian dolomite is quarried as a construction stone 

at several sites in Waukesha County. Complexes inside the 

Upper Fox River Basin include quarries adjacent to Sussex 

Creek and the Fox River (fig. 5A). The excavations remove 

the unconsolidated overburden and penetrate the top of the 

bedrock. These features are represented in the model by the 

DRN package (table 6). At the quarry locations model layers 1 

through 5 are pinched (owing to the absence of unconsolidated 

material), and the drain cells are inserted in model layer 6 at 

an elevation 10 ft below the top of the bedrock in adjacent 

unexcavated areas. The conductance term corresponds to the 

product of the area of the quarry cells and bed Kv is assumed 

to be equal to 10 ft/d divided by an assumed bed thickness of 

1 ft. The resulting conductance value is set deliberately high 

because little resistance to groundwater discharge is expected 

through the quarry walls. The fine-favored and coarse-favored 

models simulate the rate of discharge as part of model output 

(see “Model Results” section).

3.6 Time Discretization

The base model is a steady-state solution corresponding 

to recent flux conditions and, therefore, model construction 

does not involve changes to input over time. However, as part 

of the calibration process, a pumping test is simulated based 

on 2005 initial conditions. The transient simulation involves a 

2-year runup to the pumping test (during which flow conditions responded to pumping from two wells installed in 2006), 

followed by the 20-hour test period (during which a third new 

well was introduced and simulated drawdown around the new 

well was compared to observed drawdown). The details of the 

transient simulations of the pumping test for the fine-favored 

and coarse-favored models are given in “Model Calibration” 

section under “Aquifer Test Drawdown.”
3. Model Construction 53

Figure 22. High-capacity wells and quarries in model domain.

1 - Unconsolidated deposits

3 - Unconsolidated deposits (installed after 2005)

6 - Silurian Dolomite 

EXPLANATION

Nearfield

Lake

Fox River

Surface-water feature

Quarries in Silurian Dolomite

High-capacity well type

Figure 22. High-capacity wells and quarries in model domain.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American Datum of 

1983; high-capacity well data from Wisconsin 

Department of Natural Resources, 2010
54 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

4. Model Solver

A new version of MODFLOW-2005 incorporating the 

Newton formulation of the groundwater-flow equation was 

recently released by the U.S. Geological Survey (Niswonger 

and others, 2011). This version is designed to provide a solution for difficult unconfined groundwater-flow problems. In 

this section, the key features of the computer code are presented and the motives and manner of applying the code in the 

present study are described.

4.1 MODFLOW-NWT Version of MODFLOW-2005

MODFLOW-NWT is a stand-alone version of MODFLOW-2005 that is intended to overcome instabilities arising 

from drying and rewetting nonlinearities of the unconfined 

groundwater-flow equation. It treats nonlinearities of cell drying and rewetting by use of a continuous function of groundwater head, rather than the discrete approach of drying and 

rewetting used with previous versions of MODFLOW, which 

resulted in inactive cells when heads fell below the bottom 

elevation of unconfined cells during any solver iteration. Conversion of cells between active and inactive tends to create discontinuities in calculated flow and can result in convergence 

failures in many situations. MODFLOW-NWT keeps all active 

cells that are active at the start of the simulation. It assigns a 

head to unconfined cells even when the head falls below the 

cell bottom. Dry cells, although kept active, no longer participate in horizontal aquifer flow. 

The Newton formulation calculates intercell conductances in a different way than past versions of MODFLOW in 

order to maintain continuity in the head field. MODFLOWNWT uses upstream weighting in the conductance calculation, 

which allows the formulation of conductance derivatives to 

remain smooth over the full range of head for a model cell. 

Instead of converting to zero as a break in a linear slope as 

the unconfined cell dries, conductance values approach zero 

along a gradual slope at small saturated thicknesses (fig. 23A). 

A similar smoothing function is enforced for pumping from 

wells—the discharge does not change abruptly to zero when 

the head falls below the unconfined cell bottom. At low saturations, the pumping rate is gradually decreased to zero such that 

the derivatives associated with pumping remain smooth over 

the full range of heads (fig. 23B). The change to the conductance calculation requires use of the Upstream Weighting 

(UPW) package in MODFLOW-NWT instead of the BlockCentered Flow (BCF), Layer-Property Flow (LPF), or Hydrogeologic-Unit Flow (HUF) packages in MODFLOW-2005. It 

is worth noting that the input to the UPW package is virtually 

identical to the LPF package input. One input change of consequence relative to MODFLOW-2005 is an option (in version 

1.0.1 of MODFLOW-NWT at the beginning of the WEL 

package) that allows the user to specify the saturated-thickness 

threshold at which well discharge is gradually decreased.

The NWT linearization approach generates an asymmetric matrix, which is different from the symmetric matrix generated by the standard Picard formulation of the groundwaterflow equation (Niswonger and others, 2011). The asymmetric 

matrix cannot be resolved with the solvers accompanying 

previous versions of MODFLOW. The NWT version contains 

asymmetric-solver options, including the Orthomin/stabilized 

conjugate gradient (χMD) solver used in this application. 

The solution method is made more stable by under-relaxation 

(a method for calculating the head solution for a particular 

nonlinear iteration that weights the solution from previous 

iterations with the present iteration) and, at the discretion of 

the user, by residual control (a backtracking scheme that slows 

the solution if the Newtown method overshoots a solution 

when derivatives change abruptly as a function of head). More 

solver input is required for the χMD solver relative to packages such as PCG2, but the code allows the user to select a 

broad set of options with a single choice by specifying the 

degree of model complexity (simple, moderate, or complex). 

4.2 Application of NWT Formulation 

The layers in the Upper Fox River Basin models are all 

defined as unconfined so that the transmissivity of cells is 

properly adjusted to reflect the degree of saturated thickness. 

The fine-favored and coarse-favored models share a discretization scheme characterized by thin layers “suspended” from 

the land surface. For example, the bottom of layer 1 is 20 ft 

below the land surface; the bottom of layer 2 is 50 ft below 

land surface. Whenever the simulated water table is more 

than 20 ft below the land surface, which is expected to occur 

over much of the model domain, cells in the top layer are dry; 

simulated water tables more than 50 ft deep (which is expected 

to occur under hillsides) yield dry cells in both layers 1 and 2. 
4. Model Solver 55

Fraction of specified pumping rate

0.0 0.5 1.0

0.0

0.5

0.25

1.0

(H-BOT)/(TOP-BOT)

 Curve used to smoothly reduce 

specified pumping to zero when cell dewaters, 

Φ = 0.25

0

1

0 1

Quadratic

Quadratic

Linear

X

Y(X)

 Combined quadratic and linear functions used to smooth conductance and 

storage in MODFLOW-NWT (black), and a linear function that is used by MODFLOW-2005 

(blue). X is the saturated thickness divided by the cell thickness, Y is the value of the 

smoothing function, and Ω is the interval of X where the quadratic equation is applied, 

and is equal to 0.1 in this example (NWT input file variable THICKFACT). 

Ω

Ω

Figure 23a. Smoothing curves used in MF-NWT formulation -- inter-cell conductance and storage

(from Niswonger and others, 2011).

Figure 23b. Smoothing curves used in MF-NWT formulation -- pumping (from Niswonger and others, 2011).

Note: Φ is saturated thickness threshold below 

which pumping is reduced.

Figure 23A. Smoothing curves used in MODFLOW-NWT formulation—inter-cell conductance 

and storage (from Niswonger and others, 2011).

Figure 23B. Smoothing curves used in MODFLOW-NWT formulation—pumping (from Niswonger and 

others, 2011).
56 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

The high frequency of dewatered cells creates discontinuities 

in the head solution and convergence problems with MODFLOW-2005. Initial attempts to resolve the groundwater-flow 

equation for the Upper Fox models using MODFLOW-2005 

and the PCG2 solver resulted not only in many dry cells 

(which is inevitable given the layer structure), but also severe 

numerical problems associated with the presence of active 

cells over inactive cells (fig. 24). For thousands of row/column 

locations during the solver iterations, layer 1 would remain 

saturated but layer 2 would become dry, and, therefore, inactive. The layer 1 cells would, in many cases, be isolated from 

surrounding cells (the intercell conductances falling to zero), 

and recharge applied to these active cells would accumulate, 

yielding perversely high water-table elevations. The unacceptable solution shown in the cross section in figure 24 occurred 

throughout the model domain and was not amenable to stable 

correction by manipulating input to the PCG2 or other Picard 

solvers. A partial remedy could be attained by switching more 

than 5,000 cells from active to inactive status, but the ad 

hoc changes bias the solution and severely restrict the ways 

the model can be applied to test, for example, management 

alternatives involving hypothetical pumping. Attempts to work 

around the problem, by converting to confined conditions, 

introduces unacceptable bias in transmissivity assigned to 

model layers. 

The alternative formulation offered by the MODFLOWNWT program resolves the problem of active over inactive 

cells and produces a reasonable and smooth water-table 

surface over the entire model domain as well as reasonable 

patterns of horizontal and vertical flow “Model Results” section). Virtually all model cells remain active throughout the 

MODFLOW-NWT solution, although the head in many cells, 

especially in layer 1, falls below the cell bottom. The only 

inactive cells in the model are those occupied by Pewaukee 

Lake because they are handled through the LAK package.

The NWT solver options selected for this study (table 7) 

correspond to choices associated with each of the three levels 

of model complexity recognized by the code (for example, see 

solver input, classified as simple, moderate, or complex, under 

“Underrelaxation,” “Residual Control,” and “Linear Solution 

Options for χMD” in table 7). The head tolerance for achieving a model solution is set to 0.05 ft for both models; the flux 

tolerance is set to 5,000 ft3

/day for the fine-favored model and 

to 10,000 ft3/day for the coarse-favored model (the latter turns 

out to be somewhat more difficult to solve). These criteria 

yielded a small overall mass balance error and small local 

mass balance errors (see “Model Results” section).

The ability of MODFLOW-NWT to produce robust solutions under a variety of steady-state and transient stress conditions for a challenging unconfined problem outweighs several 

disadvantages from use of the new program. One disadvantage 

is the number of cells (a total of 1,806,448 active cells in the 

model domain) requires use of a 64-bit instead of a 32-bit 

computer architecture to execute the code. 

A second disadvantage is that runtimes (averaging about 

an hour for a steady-state simulation) are longer than for a 

MODFLOW-2005 run using the PCG2 solver. Part of the 

increased runtime is because flux convergence is calculated 

in terms of the root-mean-squared (RMS) flux difference 

among cells instead of simply the maximum change as it is for 

PCG2. The RMS criteria was necessary in MODFLOW-NWT 

because it is required for residual control (backtracking). RMS 

can be a better criteria for convergence because it is an indicator of the residuals of all cells rather than the error for the cell 

with the largest residual. For example, if all the cells have an 

error slightly less than the largest error, then this solution will 

have a larger RMS than the case where nearly all the cells 

have a low residual and one or a few have larger residuals. 

For the Upper Fox simulations, the RMS criterion was always 

more strict than the maximum flux difference and, therefore, 

required more iterations.

A third issue is less a disadvantage than a particular 

feature of the solution arising from the Newton linearization 

method. As discussed above, the MODFLOW-NWT formulation reduces the well discharge when the saturated thickness 

in a cell hosting a pumping well falls below a prescribed 

saturation threshold. In the Upper Fox River Basin models, the 

saturation threshold is set to the default value of 20 percent of 

the total layer thickness in MODFLOW-NWT (for example, 

in the case of a well pumping from layer 1, which normally 

has a thickness of 20 ft unless the unconsolidated sequence 

is very thin, discharge begins to be reduced when the water 

table falls more than 16 ft below land surface, the top of 

layer 1). As in the case of the Picard formulation and MODFLOW-2005, the loss of pumping is a function of the head 

solution, which in turn is a function of parameter input. Unlike 

MODFLOW-2005, the well discharge supported by the solution is not at its full rate or at zero but can attain any rate at or 

between the full rate and zero as a function of simulated saturated thickness. The summary MODFLOW-NWT output LST) 

file tabulates the simulated discharge in wells that experience 

reduction from the desired input rate. This new paradigm adds 

a complication to the interpretation of the output, but it has the 

compensating advantage that the simulated withdrawals from 

wells under unconfined condition reflect the capacity of the 

unconfined aquifers to support target or prescribed discharge 

rates on the basis of the values of input parameters. For the 

Upper Fox River Basin models, the NWT code does reduce 

pumping from reported rates for some production wells, but 

the overall reduction is small (see “Model Results” section). 

Reduction in pumping because of cell drying can be an indication of calibration error (neglecting the uncertainty in reported 

rates) because the simulation aquifer is unable to provide the 

amount of water that the real aquifer provides. 
4. Model Solver 57

EXPLANATION

Dry cell

Active cell

Simulated water table

EXPLANATION

Dry cell

Active cell

Simulated water table

Figure 24a. Simulated water table configuration using MODFLOW-2005 and MODFLOW-NWT -- 

MODFLOW-2005 with PCG2 solver. Example results in cross-section are for calibrated coarse-favored inputs along 

Row 298 (see fig. 7 for section trace)

Figure 24b. Simulated water table configuration using MODFLOW-2005 and MODFLOW-NWT -- 

MODFLOW-NWT with XMD solver. Example results in cross-section are for calibrated coarse-favored inputs along 

Row 298 (see fig. 7 for section trace)

100 feet

2,000 feet

SCALE

100 feet

2,000 feet

SCALE

Figure 24B. Simulated water-table configuration using MODFLOW-2005 and MODFLOW-NWT—MODFLOW-NWT with 

XMD solver in cross section. Example results in cross section are for calibrated coarse-favored inputs along Row 298 

(see fig. 7 for section trace).

Figure 24A. Simulated water-table configuration using MODFLOW-2005 and MODFLOW-NWT—MODFLOW-2005 with 

PCG2 solver in cross section. Example results in cross section are for calibrated coarse-favored inputs along Row 298 

(see fig. 7 for section trace).
58 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 24C. Simulated water-table configuration using MODFLOW-2005 and MODFLOW-NWT—MODFLOW-2005 with PCG2 

solver in plan view. Red cells are locations where active cells overlie inactive cells for calibrated coarse-favored model, 

producing excessively high simulated water-table elevations (see fig. 24a). (Total number of cell locations with excessively 

high simulated water-table elevations is 6,578.)

Figure 24c. Simulated water table configuration using MODFLOW-2005 and MODFLOW-NWT -- 

MODFLOW-2005 with PCG2 solver in plan view. Red cells are locations where active cells overlie inactive cells for 

calibrated coarse-favored model, producing excessively high simulated water-table elevations (see figure 24a). 

(Total number of cell locations with excessively high simulated water-table elevations is 6578.)

EXPLANATION

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Area where active cell 

overlies inactive cell

Nearfield

Surface-water feature

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983
4. Model Solver 59

Table 7. Suggested and selected input values for the NWT input file.—Continued

[m, meter; ft, foot; m3/d, cubic meters per day; ft3/d, cubic feet per day; --, indicates that values are not necessarily 

applicable; selected values are in 

section titled “NWT input file”]

bold-face type. For explanation of variables, see Niswonger and others, 2011, 

Input variable 

name

Default values Selected values

Iteration control

HEADTOL *1x10-4 (m) 0.05 ft

FLUXTOL *500 (m3

/d) 5,000 ft3/d for fine-favored; 10,000 ft3/d for 

coarse-favored

MAXITEROUT 100 300 for fine-favored; 1,000 for coarse-favored

Dry-cell tolerance

THICKFACT 0.00001 0.00001

NWT options

LINMETH 1 2 → χMD linear solution control

IPRNWT 0 1 → print max head change

IBOTAV **0 1 → corrections will be made to groundwater head relative 

to the cell-bottom altitude if the cell is surrounded by 

dewatered cells

Underrelaxation input

Model complexity

Simple Moderate Complex

DBDTHETA 0.97 0.7 0.4 0.5

DBDKAPPA .0001 .0001 .00001 .00001

DBDGAMMA .0 .0 .0 .0

MOMFACT .0 .1 .1 .0

Residual control

Model complexity

Simple Moderate Complex

BACKFLAG ***0 0 0 1 → Residual 

control active

MAXBACKITER -- -- -- 20

BACKTOL -- -- -- 1.05

BACKREDUCE -- -- -- .6

Linear solution control and options for GMRES – not used
60 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Table 7. Suggested and selected input values for the NWT input file.—Continued

Input variable 

name

Default values Selected values

Linear solution control and options for χMD – used

Model complexity

Simple Moderate Complex

IACL 2 2 2 2

NORDER 1 1 0 0

LEVEL 0 1 3 8

NORTH 2 2 7 2

IREDSYS 0 0 0 0 

RRCTOLS .0 .0 .0 .0

IDROPTOL 1 1 1 1

EPSRN 1x10-3 1x10-3 1x10-4 1x10-3

HCLOSEχMD 1x10-4 1x10-4 1x10-4 1x10-3 ft

MAXITINNER 50 50 50 100

*

These values are dependent upon the units specified in the MODFLOW-2005 discretization input file. Values 

given are for units of meters and days.

**The optimal value for IBOTAV is problem specific. Values of 0 and 1 should be tested for each problem.

***BACKFLAG should be set to 0 (residual control set to inactive) unless there are convergence problems. 

“Options” must be set to “Specified” if the residual control option is used.

The implementation of the NWT solution method to the 

Upper Fox River Basin models was subjected to three sensitivity tests: 

• An alternative solver algorithm, called GMRES, was 

substituted for χMD; 

• Residual control was deactivated; and

• Starting heads equal to the land-surface elevation at 

each row/column location were substituted for starting 

heads corresponding to the preliminary solution with 

initial parameter values (for which the average watertable depth from land surface is on the order of 35 ft). 

A variety of metrics—including head distribution, average depth to the water table, number of dry cells, and water 

budget terms—for comparing the base results for the finefavored and coarse-favored models to the solutions for the 

three sensitivity simulations show negligible difference in 

output. The major difference among the simulations was runtime—the GMRES solver required more execution time than 

the XMD solver and, not surprisingly, a solution starting from 

heads equivalent to the land surface required more execution 

time than a solution starting from heads corresponding to 

preliminary output. At least for the sensitivity tests conducted, 

the application of the NWT formulation is robust.

The version of the MODFLOW-NWT program used in 

this application is not the final published code (which was 

not released until the modeling was completed), but a development code made available to USGS researchers several 

months before the May 2011 release. The development code 

differs from the final version in two ways:

• Several packages (UPW, SFR, NWT and WEL) have 

slightly different input formats although the options in 

the development and published versions are identical; 

and

• The LAK package algorithm is updated in the final version, partly to reduce simulation time.

As a check on the stability of the code, the fine-favored 

model calibrated with the development version also was run 

with the published version of MODFLOW-NWT. The results 

of the published version are very similar to those of the development version. For example, in comparing the global water 

budgets for the simulations, the biggest difference occurs for 

the net flow to streams represented by the SFR package, and 

the difference is less than 0.04 percent. 

Because the development version of the program was 

used to generate the results discussed in this report, it is 

included in the model archive. The source FORTRAN files 

and compiled executable correspond to the version of the code 

available on January 7, 2011.

3 3 [m, meter; ft, foot; m /d, cubic meters per day; ft /d, cubic feet per day; --, indicates that values are not necessarily 

applicable; selected values are in bold-face type. For explanation of variables, see Niswonger and others, 2011, 

section titled “NWT input file”]
5. Model Calibration 61

5. Model Calibration

The calibration process involves updating the initial estimates for selected input values in order to improve the match 

between simulated groundwater levels and flows and corresponding field observations called targets. Several techniques 

are employed to enhance the calibration process and make it 

tractable for parameter estimation (that is, estimate of independent variables—parameters—through observed values of 

dependent variables—head and flow targets). The techniques 

include selection of parameters subject to estimation; assembly of diverse target types to better explore the parameter 

space; setting of target weights; and, after preliminary adjustment of parameters by manual trial-and-error, application of 

nonlinear regression by means of singular value decomposition with regularization. For this study, the nonlinear regression is implemented by use of the code PEST (Doherty, 2008a, 

2008b). In the end, the calibration process not only improves 

the fit between measured and simulated targets, but also adds 

insight into the major controls on the flow system through 

identification of parameters to which model results are most 

sensitive. A key element of the calibration process is the application of the parameter estimation methods to the fine-favored 

and coarse-favored models.

5.1 Estimated Parameters

Several sets of model input parameters are estimated as 

part of the calibration process. As discussed in the “Model 

Construction” section, the unconsolidated material in model 

layers 1 through 5 is mapped to five facies, with distinct 

facies zones for the fine-favored and coarse-favored models 

(fig. 15). The Kh assigned each facies constitutes one set of 

five estimated parameter values; the Kv constitutes a second 

set of five values. In the calibration process, only the Kv value 

of the zone corresponding to the dominantly fine facies is 

directly estimated. The values for the other zones are estimated as ratios relative to the neighboring finer zone. A third 

set of Kv values is associated with the three streambed zones 

applied in the Upper Fox River Basin (fig. 20). A fourth set of 

inputs corresponds to the Kh and Kv of the Silurian dolomite 

in model layers 6 and 7. A single multiplier is applied to all 

the Kh and to all the Kv zones of the dolomite copied from the 

SEWRPC regional model. In effect, the calibration process 

is used to improve the estimate of the bedrock unit’s overall 

transmissivity and vertical resistance. The last parameter set 

estimated is particular to the use of the Upper Fox River Basin 

models in transient mode—that is, the storage properties of 

the unconsolidated deposits in response to a time-dependent 

stress—expressed as uniform values for specific storage 

and specific yield. One parameter set that is not subject to 

the calibration process is the zonation of recharge (fig. 17). 

The zonal values were estimated by a method dependent on 

matching recharge to base flow in tributary basins (see “Model 

Construction” section under “Water Withdrawals”). As a 

result only limited information is gained by reestimating these 

values on the basis of base-flow targets applied to the model 

calibration process and discussed below. One motive for fixing 

the recharge values in the models is to simplify the calibration process and make it more sensitive to the uncertainty of 

the various hydraulic conductivity values. A second motive 

for fixing the input is to insure that the recharge values are 

consistent with the flux boundary conditions derived from the 

regional SEWRPC model and applied at the lateral edges and 

bottom of the Upper Fox River Basin model.

A listing of estimated parameters along with initial values 

can be found in table 8. It is important to emphasize that the 

parameters estimated and the initial values assigned to them 

are the same for the fine-favored and coarse-favored models. 

However, the calibration process ultimately yields distinct 

parameter values for the two models.

5.2 Calibration Targets

The Upper Fox River Basin models are calibrated to five 

target types. In the estimation process, different parameter sets 

are expected to be sensitive to different target types. The target 

values themselves are all affected by error and uncertainty.

5.2.1 Water Levels

Water levels simulated by the model are compared to a 

map of the generalized water table in southeastern Wisconsin 

prepared by researchers at the Wisconsin Geological and Natural History Survey (Southeastern Wisconsin Regional Planning 

Commission and Wisconsin Geological and Natural History 

Survey, 2002, Map 21). Calibration targets are obtained by 

sampling the published contours at a 2,000-ft interval within 

the model domain. This process yields 3,492 water-level targets: 2,099 in the farfield and 1,393 in the nearfield (fig. 25). 

The median depth of the water levels below land surface at the 

target locations is 36 ft. Taking the targets as approximations 

of the water-table elevation implies that layer 1, bottoming 

20 ft below land surface, will be dewatered over much of the 

model domain.
62 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Table 8. Parameters estimated in calibration process.

[ft/d, foot per day; kx, horizontal hydraulic conductivity; kv, vertical hydraulic conductivity; ksfr, streambed hydraulic 

conductivity]

Parameter Initial value Lower bound Upper bound

Horizontal hydraulic conductivity (ft/d)

kx2 (dominantly fine) 0.50 0.10 1.00

kx4 (relatively fine) 2.00 1.00 4.00

kx12 (mixed fine and coarse) 10.00 4.00 20.00

kx13 (relatively coarse) 40.00 20.00 60.00

kx14 (dominantly coarse) 80.00 60.00 200.00

Vertical hydraulic conductivity (ft/d)

kv2 (dominantly fine) .01 .002 .02

Ratio kv4 to kv2 (for relatively fine) 2.00 1.00 100.00

Ratio kv12 to kv4 (for mixed fine and coarse) 2.00 1.00 100.00

Ratio kv13 to kv12 (for relatively coarse) 10.00 4.00 100.00

Ratio kv14 to kv13 (for dominantly coarse) 10.00 4.00 100.00

Streambed hydraulic conductivity (ft/d)

ksfr_low 1.00 .10 5.00

ksfr_middle 5.00 .50 25.00

ksfr_high 25.00 2.50 100.00

Multiplier on dolomite hydraulic conductivity zones

kx 1.00 .02 4.00

kv 1.00 .02 4.00

Multiplier on storage values of unconsolidated deposits

specific storage (ft-1) 1.00 .10 10.00

specific yield (--) 1.00 .10 10.00

Multiplier on recharge zones (fixed)

recharge 1.00 1.00 1.00
5. Model Calibration 63

Figure 25. Calibration targets: water levels.

700

720

740

760

780

800

820

840

860

880

900

920

940

960

980

1,000

1,020

Water table elevation, 

in feet

Water-level contours derived from:

Southeastern Wisconsin Regional Planning Commission and Wisconsin 

Geological and Natural History Survey, 2002, Groundwater resources 

of southeastern Wisconsin: Southeastern Wisconsin Regional 

Planning Commission Technical Report 37, 203 p., Map 21

Water level and 

calibration targets

Contour interval is 20 feet. 

Water-level calibration 

targets are placed at 

2,000-foot intervals along 

contour lines.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

EXPLANATION

Nearfield 

Comparison of points along contour lines 

to land surface at same location:

Median depth ........................... 36.1 feet

Average depth ........................... 41.6 feet 

Standard deviation depth ...... 30.9 feet

Minimum depth ........................ 0.1 foot

Maximum depth ..................... 213.3 feet 

Figure 25. Calibration targets: Water levels.
64 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

The water-level contours in the map are based chiefly 

on static water levels recorded by drillers and geologists in 

wells open to the shallow aquifer, on elevations corresponding to groundwater outcrops at surface-water bodies, and on 

topography. The 2002 Wisconsin and Natural History Survey report states that “it was beyond the scope of this study 

to field-check the locations and water level measurements 

reported on DNR well constructor’s reports that were used to 

construct the water-table map. The map does not reflect local 

details in the water because the contour lines were generalized and interpolated from the nearest data points, and natural 

fluctuations in the water table can be as much as 15 feet.” 

Another, although smaller, source of uncertainty arises from 

the fact that the static water levels do not correspond exactly 

to the water table itself, but to the elevations below the water 

table where the wells are open. Given the presence of even 

small downward or upward vertical gradients within the shallow aquifer system, the implied water-table surface will fall 

below or above the recorded water levels. (For example, if the 

average vertical gradient is 0.01 ft/ft downward and the well 

is open 100 ft below the water table, then the recorded water 

level is 1 ft below the implied water table). Inspection of the 

well database used to construct the map (Peter Schoephoester, 

Wisconsin Geological and Natural History Survey, written 

commun., January 12, 2010) indicates that the median depth of 

the 1,679 shallow wells listed is on the order of 92 ft and that 

the median depth of the casing interval is on the order of 89 ft 

(water wells typically have short screens), with 80 percent of 

the casing depths between 42 ft and 200 ft. On the basis of this 

information, the calibration targets were assigned to unconsolidated model layer 4, which when active is 100 to 150 ft 

below land surface but when pinched is less than 100 ft below 

land surface. To account for the influence of surface-water 

elevations and topography on mapped contours, especially 

in low-lying areas, the water-level targets were assigned to 

model layer 2 instead of model layer 4 where the land surface 

is below 860 ft, an area corresponding to the Upper Fox River 

Valley and confluent valleys (see fig. 5B). These layer adjustments are made to better align the contour-derived targets with 

the underlying water-level data, but it is recognized there is 

uncertainty around target values on the order of as much as 

15 ft because of natural fluctuations in head.

5.2.2 Vertical Head Differences

The estimation of Kv in the calibration process is greatly 

enhanced by the availability of targets linked to the vertical 

head difference between layers. Public water-supply projects for the city of Waukesha included the installation of test 

borings at depth and the collection of water levels measured 

relative to land surface. In particular, data collected as part 

of studies conducted near a wetland between the Fox River 

and Pebble Brook (Aquifer Science & Technology, 2010) 

and along the Fox River upgradient from the confluence of 

Genesee Creek (John Jansen, Aquifer Science & Technology, 

written commun., December 1, 2010) allow the estimation of 

vertical gradient targets at the two locations (fig. 26). 

The open interval for the test borings was around 100 ft 

depth. Depth to water was measured relative to the top of 

casing, noted to be about 2.5 ft above land surface. Six of the 

eight test borings registered water levels above land surface of 

1 to 3 ft, implying “flowing conditions” and upward gradients 

with respect to the water table located below land surface. 

Only test borings labeled as “lath1” and “lath10” (fig. 26) 

registered water levels below land surface. No direct measurements of the water-table elevation were available at the 

test boring locations because they are typically open 100 ft 

below land surface and tens of feet below the water table. In 

order to construct a vertical head difference target, the depth 

of the water table was assumed to be 1 ft below land surface 

at all the test boring locations except for “lath1,” because it is 

farther from surface water, and the water table was assumed 

to be 3 ft below land surface. The observed target corresponds 

to the difference between the assumed water table and the 

measured water level in the test borings; the simulated target 

corresponds to the difference between the head in layer 1 and 

layer 4. The uncertainty of the targets is high given all the 

assumptions incorporated in estimating the observed values, 

but one qualitative test of the model fit is its ability to reproduce “flowing conditions” at the appropriate locations. The 

simulation of strong upward gradients and “flowing conditions” depends not only on the proximity of the test borings 

to surface-water features but also on the vertical resistance to 

flow imposed by the Kv

 assigned the unconsolidated facies.

5.2.3 Base Flow

Base-flow estimates offer a calibration target linked to 

groundwater flow instead of groundwater levels. Even though 

the main source of water to the groundwater system, the 

recharge input, is fixed for the fine-favored and coarse-favored 

models, base-flow targets still afford a way to check the 

model’s ability at specific locations to properly partition flow 

between circulation to shallow discharge zones represented by 

surface water as opposed to deeper movement toward wells 

and model boundaries.
5. Model Calibration 65

Figure 26. Calibration targets: vertical head difference.

Streamflow records are used to estimate base flow 

U.S. Geological Survey, 2010c). Records from two USGS 

streamgages are available along the Upper Fox River 

(fig. 6B)—the Waukesha gage for the period January 1, 1963, 

to September 30, 2008, and the Watertown gage for the period 

December 1, 1992, to September 30, 2000 (this gage was 

discontinued after 2000). It is useful to relate base flow to the 

flow-duration curve, a plot that shows the percentage of time 

that flow in a stream is likely to equal or exceed some specified value (accordingly, the 50-percent value (Q50 ) defines the 

flow exceeded 50 percent of the time, the 75-percent value 

(Q75 ) defines the flow exceeded 75 percent of the time, and the 

90-percent value (Q90 ) defines the flow exceeded 90 percent of 

the time). A method developed for Wisconsin by Gebert and 

others (2007) estimates annual base flow for the basin upgradient from a gage in terms of basin area and a base-flow factor 

proportional to the 90-percent flow-duration value according 

to the equation:

Q A b f = 0 906 B 1 02 0 52 . . . (3)

where

Qb is average annual base flow, in cubic feet per 

second, 

A is drainage area, in square miles, and 

Bf is the base-flow factor, defined as the 

90-percent flow-duration value (Q90, in 

cubic feet per second) divided by the 

drainage area, in square miles.

Pebble Brook

EXPLANATION

Surface-water boundary conditions

River

Stream

Drain

Surface-water feature

LATH1 LATH9

LATH8

LATH3

WK-12

LATH10

WK-11

LATH7

88°20' 88°15'

42°58'

42°56'

Calibration target and name LATH8

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

Figure 26. Calibration targets: vertical head difference.

0 .5 1 MILE

0 .5 1 KILOMETER

Fox

 River

Fox River

Vernon

Marsh

Genesee Creek
66 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

For this statistically significant regression (p is less than 

0.001), the statewide average standard error of estimate was 

12 percent, and the statewide R2 (a measure of how well a 

regression line approximates the data points, varying between 

0 and 1, with 1 representing a perfect fit) was 0.992. Application of the equation to the Waukesha gage yields a base-flow 

estimate of 49.6 ft3/s. This value is typical of total streamflow 

in the low-flow period between July and November (fig. 27), 

although, as expected, it is only a fraction of the typical 

streamflow in other months when overland flow from precipitation and snowmelt dominates the base-flow component. 

The average base flow calculated from the Gebert formula 

is intended to reflect long-term conditions subject to average 

recharge over the basin rather than the actual 2005 low-flow 

conditions at the Waukesha gage, which averaged 33.7 ft3/s 

between July and November 2005, reflecting precipitation patterns in the spring and summer of 2005.

Application of the equation to the Watertown gage was 

biased by the short period of record. To overcome the bias, the 

Q90 for the Watertown and Waukesha gages were calculated 

for the 1992–2000 period over which the Watertown gage 

was active, and then the ratio of these values was multiplied 

by the Q90 for the full period of the Waukesha gage to generate a corresponding full period Q90 for the Watertown gage. 

This last value then was applied to the base-flow factor in the 

regression equation. The resulting base-flow estimate for the 

Watertown gage is 33.6 ft3/s. The implied base-flow gain from 

the Watertown to the Waukesha location (including base flow 

from Pewaukee Lake and Pewaukee River) is about 16 ft3/s.

Figure 27. Comparison of base-flow estimate to streamflow record at Waukesha gage.

Image downloaded on April 4, 2011, from: 

http://waterwatch.usgs.gov/new/index.php?sno=Waukesha&yr=2011&dt=dv28d&btnGo=GO&m=sitedur&s=&sno2

Plot shows flows at Waukesha gage 05543830 from January 2010 to April 2011 superimposed on long-term

streamflow pattern.

Observed 

streamflow

in 2010–2011
5. Model Calibration 67

The base-flow estimates for the gages are derived from 

records that include not only groundwater discharge to surface 

water but also the addition of effluent at two WWTPs upgradient from the target locations of Sussex and Brookfield 

(fig. 6B). The combined average inflow from these WWTPs, 

corresponding to rates for 2008–9, is 13.29 ft3/s (see “Model 

Construction” section under “Input to MODFLOW-2005 

Surface-Water Packages”), equal to 40 percent of the baseflow estimate at Watertown and 27 percent of the estimate at 

Waukesha. However, the effluent flux was probably smaller 

than the 2008–9 rates for most of the 1963–2008 period of 

streamflow record because the area developed gradually, opening the possibility that the model will tend to oversimulate 

observed base flow for 1963–2008 by adding recent rates of 

effluent discharge to simulated groundwater discharge. Conversely, development over time led to more pumping, implying that the model will tend to undersimulate average base 

flow by applying 2005 rates of well discharge. These biases 

add to the uncertainty inherent in the base-flow estimation by 

regression.

5.2.4 Land-Surface Constraints

The surface-water network input to the model does not 

contain all groundwater-discharge areas active in the domain. 

Most notably, groundwater tends to discharge along waterway 

corridors at various locations that are difficult to map. It is 

convenient to conceptualize this outflow as part of base flow 

because of its tendency to exit the groundwater system and 

immediately discharge as runoff to streams. In the absence 

of explicit discharge zones inserted at many locations in the 

model along surface-water bodies, this discharge will be 

simulated as base flow to the adjacent stream rather than as 

outflow to the adjacent land surface, but often these riparian locations are marked by water-table conditions simulated 

above land surface. In other words, “groundwater flooding” in 

riparian cells is an expected model outcome, which ordinarily 

represents an acceptable simplification of reality. In contrast, 

simulated flooding elsewhere in areas of higher topographic 

elevation is likely to indicate areas of poor model fit. For this 

reason, two land-surface constraints have been added to the 

calibration targets: 

• a target to minimize average flooding in the group of 

model cells where water-table elevations are simulated 

above land surface; and

• a target to minimize the percent of nearfield model 

cells reporting simulated water-table elevations more 

than 3 ft above land surface.

These targets are constructed in a way that tends to allow 

riparian groundwater flooding but discourages it elsewhere in 

the model domain.

5.2.5 Aquifer Test Drawdown

The water-level, vertical-head difference, base flow, and 

land-surface targets are all applied to the steady-state model 

intended to approximately reproduce 2005 base-flow conditions. A last target type refers to the drawdown response to a 

transient stress in the form of an aquifer test conducted on a 

candidate public supply well installed in the unconsolidated 

material about one-half mile west of the Fox River (fig. 28). 

The aquifer test occurred in November 2007 at the site of the 

future well WK-13 (which began operation in 2009), about 

2 years after withdrawals began from nearby consolidated 

wells WK-11 and WK-12 at a combined average rate of 

1,235 Mgal/d over 2006–7 (Jeff Detro, Waukesha Water Utility, written commun., January 18, 2010). 

The withdrawal at WK-13 was superimposed on the 

2006–7 pumping from wells WK-11 and WK-12, which 

in turn was superimposed on 2005 conditions. In order to 

simulate the aquifer-test response to pumping, two transient 

stress periods were added to the base steady-state simulation, 

the first of 2-years duration and the second corresponding to 

the drawdown phase of the aquifer test. Simulated drawdown 

during the second transient stress period is calculated relative 

to the water levels at the end of the preceding 2-year stress 

period. One aquifer test, conducted by Aquifer Science and 

Recovery (2008), involved 20 hours of pumping WK-13 at 

600 gallons per minute (gal/min) and recording of waterlevel changes at several test borings acting as observation 

wells, including ENGL-1 and ENGL-5 located 70 and 280 ft, 

respectively, from WK-13 (fig 28A). Shallow well OW-1, 

located near WK-13 but screened at a depth corresponding 

approximately to model layer 1 (0 to 20 ft below land 

surface), also was monitored during the test but showed no 

detectable response (John Jansen, formerly Aquifer Science 

& Technology, oral commun., February 2010). Stratigraphic 

and well-construction records indicate that WK-13 drew water 

from an open interval corresponding to sandy deposits present 

from about 85 to 105 ft below land surface (fig. 28B). This 

depth intersects the lower part of model layer 3 (extending 50 

to 100 ft below land surface) and the thin vertical extent of 

model layer 4 (extending 100 ft to 105 ft below land surface at 

this location). In the fine-favored and coarse-favored models, 

the layer 3 cell at the WK-13 location is identified with the 

mixed facies and layer 4 with the coarse-dominated facies. 
68 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 28A. Calibration targets: aquifer test location—plan view.

WK-12

WK-13

OW-1

ENGL-1

ENGL-5

WK-11

WK-12

WK-13

OW-1

ENGL-1

ENGL-5

WK-11

Pumping test well

Observation well

Well type of calibration target

Calibration target and name

EXPLANATION

WK-11

0 500 1,000 FEET

PLAN VIEW

Figure 28a. Calibration targets: aquifer test location -- plan view.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

EXPLANATION

Surface-water feature

Nearfield

Lake

F o x

R i v e r

Base modified from Wisconsin Department 

of Natural Resources digital data, 2010, 

1:24,000, Wisconsin Transverse 

Mercator, North American 

Datum of 1983

Fox River
5. Model Calibration 69

Clearly, the model layering is not perfectly aligned with 

the local stratigraphic and well-construction conditions at 

WK-13, which limits the ability of the models to reproduce the 

drawdown trends. Moreover, it is unlikely that the mapping 

of only five facies to represent the flow properties of the 

unconsolidated aquifer over the entire Upper Fox River Basin 

is detailed enough to capture local features around a single 

well. Despite these obstacles, the aquifer test was added to the 

calibration process as a check on model construction in terms 

of the transmissivity implied by the response to the pumping 

at the WK-13 site. The transmissivity, which, along with 

vertical resistance and storage release, controls the drawdown 

response, is a function of the assemblage of facies near the 

pumping well at different depths for the two models. Also 

of interest was the models’ ability to simulate the absence of 

response at OW-1, which serves as a constraint on the vertical 

hydraulic conductivity properties assigned the unconsolidated 

facies. Finally, simulation of the aquifer test in the calibration 

process allows estimation of storage properties of the shallow 

aquifer system at this one location within the basin. The 

simulated response is expected to be somewhat sensitive to 

the elastic aquifer properties represented by specific storage at 

the very beginning of the test and more sensitive to drainage 

properties represented by specific yield later in the test, but 

the ability to estimate specific yield is compromised by the 

delay in vertical leakage at the water table, which occurs 

when pumping is of short duration, and at screened intervals 

some distance below the water table. Starting values for the 

unconsolidated specific storage (1x10-6 1/ft) and specific yield 

(0.05) are based on preliminary analysis of the aquifer test 

data associated with pumping of WK-13 using curve-matching 

methods (Aquifer Science and Technology, 2008).

The model stress period devoted to the WK-13 aquifer 

test, lasting 20 hours, is divided into 12 time steps. The duration of the time steps increases geometrically by a factor of 

1.4, such that the first step simulates the initial 0.1436 hours of 

the test and the last step simulates the final 5.917 hours of the 

test. For calibration, the drawdown at the times corresponding to the end of each time step is compared to the observed 

drawdown interpolated from the drawdown curve recorded 

during the test (based on approximately 250 observations at 

each observation well) The observed drawdown trends for the 

aquifer test at the two observation wells along with the match 

to simulated drawdown are shown in the “Model Calibration” section under “Calibration Results for Fine-Favored and 

Coarse-Favored models.”

Figure 28B. Calibration targets: aquifer test location—vertical section.

Pumping test well

Observation well

Well type of calibration target

Calibration target and name

EXPLANATION

WK-11

50 FEET

150 FEET

SCALE

VERTICAL SECTION

Model layer, type, and depth 

0–20 feet from land surface

20–50 feet from land surface

50–100 feet from land surface

100–150 feet from land surface

More than 100 feet from land surface

0–20 feet from bedrock surface

More than 20 feet from bedrock surface

Unconsolidated

Silurian bedrock

1

2

3

4

6

7

5

Figure 28b. Calibration targets: aquifer test location -- vertical section.

ENGL-1 WK-13 OW-1
70 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

5.3 Target Weighting

Weights are used in the calibration process to assign 

importance to individual targets based on factors such as relative uncertainty or desired influence. Weights also are used to 

reconcile targets with different units (such as water levels and 

fluxes) so mathematical comparisons can be made. The parameter-estimation program PEST uses algorithms to modify 

parameter values to minimize the objective function; that is, 

the sum of squares of the weighted target residuals. The target 

residual is equal to the measured value minus the simulated 

value at the target location for the appropriate model time step. 

The modeler-assigned target weights are a primary mechanism 

for translating the modeler’s relative ranking of target fit used 

to assess calibration quality; consequently, the weights are 

important for guiding the search for optimal parameters. 

For this study, target weights were selected to normalize the influence of the target types by accounting for their 

uncertainty because of measurement uncertainty and structural 

uncertainty that impacts the reasonable expectations of how 

well the model is expected to reproduce them. This approach 

also serves to balance the contribution to the objective function among the target types to inform the estimation of different parameter sets (unconsolidated Kh

 and Kv zones, bedrock 

transmissivity and vertical resistance, streambed K, and storage terms). The method adopted was to

• choose the desired percent contribution of each target 

type to the objective function (sum of squares of 

weighted target residuals); 

• select an expected residual for each target type (based 

on preliminary simulations with the initial parameters); 

• calculate the contribution by multiplying the residual 

guess by the trial weight, squaring the product, and 

multiplying the result by the number in the target type; 

and

• adjust the trial weights so the calculated percent contributions approximately match the desired contributions. 

The weights applied to the regression process (listed in 

table 9) were selected to yield the following ranking of target 

types in terms of their target contribution to the objective 

function:

Approximate contribution

Nearfield water levels 40 percent

Vertical head differences 20 percent

Pumping test drawdown 15 percent

Flooding constraints 15 percent

Farfield water levels 10 percent

Nearfield water-level targets are collectively ranked most high 

in importance because they help estimate all the parameters 

(except storage terms) in the focus area of the study. Vertical head difference targets are ranked second highest because 

they guide the calibration process in optimizing the values of 

Kv parameters, otherwise difficult to estimate. The weights 

assigned the target types (for example, 1.0 in the case of 

nearfield water levels, 110 in the case of vertical head differences) reflect the relative importance and number of targets by 

set.

Table 9. Calibration target types and assignment of calibration weights.

[ft, foot; ft3/s, cubic foot per second; >, greater than; %, percent]

Target type Group Target 

value

Number 

of targets

Trial 

average 

absolute 

residual

Trial 

weight

Trial sum 

of squares 

of weighted 

residuals

Expected 

percent 

contribution 

to objective 

function

Nearfield water levels (ft) 1 variable 1,393 18 1 451,332 40.6

Farfield water levels (ft) 2 variable 2,099 18 .4 108,812 9.8

Base flow at Watertown and Waukesha gages (ft3

/s) 3 33.6 and 

49.6

2 2 0 0 .0

Vertical head difference (ft) 4 variable 8 1.5 110 218,800 19.6

Pumping test drawdown at observation wells ENG-1 

and ENGL-5 (ft)

5 variable 14 4 27 163,296 14.7

Pumping test drawdown at shallow well OW-1 (ft) 6 0.1 1 .5 0 0 .0

Minimize average flooding in flooded water table 

cells (ft)

7 0 1 7 42 86,436 7.8

Minimize percent nearfield cells without flooding 

> 3 ft (%)

8 0 1 9 32 82,944 7.5

Total 1,110,620 100.0
5. Model Calibration 71

Two of the target types are assigned zero weight in the 

calibration process and make no contribution to the objective 

function: the base-flow estimates at the Watertown and Waukesha gages and the expected negligible drawdown at the end 

of the 20-hour aquifer test in shallow observation well OW-1. 

The base-flow targets primarily are affected by variations in 

recharge, and because the recharge zone values are kept fixed 

(see “Model Calibration” section under “Estimated Parameters”), it is convenient to simplify the inversion mathematics by assigning them zero weight. However, it is important 

to compare the simulated base flows to the estimated values 

after the optimization is complete as an independent check on 

the parameter estimation results. Similarly, the drawdown at 

OW-1 is neglected in the inversion to simplify the mathematical burden, but the quality of the calibration is evaluated with 

that target included. There is one other group of zero weights 

corresponding to drawdowns recorded at the two observation 

wells after 3.5 hours of pumping; the drawdown curves flatten 

considerably. This trend is due in large measure to the horizontal and vertical flow within the unconsolidated aquifer, but it 

is possible that it is also due in part to enhanced flow toward 

the well through fractures at the top of the Silurian dolomite. 

The pumping well screen is open to an elevation just above 

the top of the weathered dolomite, so that any conduits in the 

neighborhood could readily influence the test. Because of the 

uncertainty about these local conditions, it was judged more 

efficient to limit the active targets to the first 3.5 hours of the 

test; however, the comparison of simulated to observed drawdown is reported for the entire 20-hour test.

The fine-favored and coarse-favored models are calibrated using the same initial parameter values, the same solver 

input, and the same target types and weights. As discussed 

in previous sections, the calibration process yields distinct 

optimized values for the estimated parameters between the two 

models because they represent the geology differently.

5.4 Parameter Estimation Technique

Calibration of the Upper Fox River Basin models is 

tractable only if the parameter estimation process operates on 

an input structure that simplifies the complexity of the natural 

world. The required simplification operates at several levels. 

The initial model construction already radically simplifies 

real-world complexity by means of zones inside where values 

of hydraulic conductivity, recharge, streambed hydraulic 

conductivity, and storage are uniform. Zonation of inputs is 

a common simple form of “regularization” applied during 

model construction and calibration (Hunt and others, 2007a). 

However, there are other forms of regularization that can 

facilitate the inversion process. These other forms can be particularly important when, as in this study, the many parameters 

(even after zonation) are difficult to estimate. The use of the 

parameter-estimation computer code PEST (Doherty, 2008a; 

Doherty and Hunt and others, 2010) facilitates the application 

of sophisticated regularized inversion tools.

A groundwater-flow model with many parameters is 

commonly affected by parameter insensitivity and correlation, which in turn leads to solution non-uniqueness and an 

ill-posed inverse problem. In this study the “preferred value” 

form of Tikhonov regularization was used to counter these 

problems (Tikhonov 1963a, 1963b). The preferred values 

were set at the initial parameter values. Deviations from these 

values were penalized by increasing the sum of squares of 

weighted residuals, that is, the objective function. Therefore, 

deviations only occur if the regularization penalty is more 

than offset by a better match to targets and an overall smaller 

objective function. The amount of “preferred value” regularization applied to the fine-favored and coarse-favored models 

was relatively modest but served to exclude PEST results that 

minimized the objective function at the cost of unrealistic 

parameter values.

In contrast to Tikhonov regularization, which adds information to the calibration process in order to achieve numerical stability, subspace Singular Value Decomposition (SVD) 

regularization achieves numerical stability through subtracting 

parameters. This subtraction also is interpreted as the creation 

of a subset of parameters, each made up of a partial linear 

combination of all the parameters. As stated in Doherty and 

Hunt (2010, p. 4):

“As a result of the subtraction, the calibration 

process is no longer required to estimate either 

individual parameters or combination of correlated 

parameters that are inestimable on the basis of the 

calibration dataset. These combinations are automatically determined through singular value decomposition (SVD) of the weighted Jacobian matrix. ...The 

Jacobian matrix consists of the sensitivities of all 

specified model outputs to all adjustable parameters. 

Individual parameters, or combinations of parameters, that are deemed to be estimable on the basis 

of the calibration dataset constitute the “calibration 

solution space.” Those parameter/parameter combinations that are deemed to be inestimable (these 

constitute the “calibration null space”) retain their 

initial values.” 
72 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

In this application, the Jacobian Matrix contains 17 columns 

corresponding to the estimated parameters listed in table 8, 

noting that the 18th entry, recharge, is fixed. Some of the 

parameters correspond not to a single zone but to multipliers on zones (for example, dolomite hydraulic conductivity). 

The rows in the Jacobian Matrix correspond to the individual 

targets summarized as types in table 9. Within PEST, the 17 

estimated parameters (zone values or multipliers on initial values) were grouped into mathematical constructs representing 

combinations of parameters by means of the SVD algorithm. 

The fine-favored and coarse-favored models were each subjected to four PEST parameter-estimation updates after which 

the reduction in the objective functions became negligible or 

reversed. For both models, the number of estimated parameter 

combinations decreased from 17 to 12 in the course of the 

updates because of insensitivity of some combinations.

The fine-favored and coarse-favored updates were influenced not only by Tikhonov regularization and SVD subspace 

regularization, but also by lower and upper bounds placed on 

the values that parameters were allowed to assume (table 8). 

The Kh

 parameters are bound in such a way that during the 

PEST inversions, the fine-dominated facies would be assigned 

the smallest Kh value, the relatively fine facies the next smallest value (or a value equal to the fine-dominated facies), the 

mixed facies the next smallest value (or a value equal to the 

relatively fine facies), and so forth. This ordering ensures 

that the conceptual model underlying the computer model 

is enforced. A similar logic is used for the Kv parameters, 

except that only the fine-dominated facies Kv value is directly 

estimated. Because of the bigger spread in Kv values between 

zones, upper- and lower-bound ratios rather than values are 

used to enforce the ordering from lowest to highest K facies. 

A lower ratio bound equal to one allows the coarser facies 

in the ratio pair to be assigned the same Kv value as the finer 

facies, whereas, the upper ratio bound, always set to 100 for 

this parameter group, controls how much higher the coarse 

Kv can be relative to the finer Kv in the ratio pair. Upper and 

lower bounds determine acceptable values of the streambed 

hydraulic conductivity in such a way that, unlike the Kv ordering, allows overlap between the values of the three streambed 

zones (table 8). The bounds assigned the dolomite K and 

the storage terms are applied to the estimation of multipliers 

on initial values. Note that the exact same lower and upper 

bounds are applied to the inversions of the fine-favored and 

coarse-favored models.

The parameter-estimation problem was run on two 

64-bit computers with multiple processors using the specialized parallel-processing version of PEST called BEOPEST 

(Schreüder, 2009; Hunt and others, 2010). Given the large 

number of times the models must be solved to implement 

the PEST algorithms, care was taken to reduce runtimes by 

(1) conducting preliminary runs with the fine-favored model 

to choose initial parameters that reproduce water-level targets 

reasonably well, (2) relaxing head and flux tolerances to the 

point where global mass balance errors for the runs equals 

about 0.1 percent, and (3) updating starting heads so that they 

correspond to preliminary simulations with initial parameter 

values. The total runtime for the fine-favored and coarsefavored inversions was 2 to 4 days. 

5.5 Calibration Results for Fine-Favored and 

Coarse-Favored Models

The inversion routines in PEST yield improved congruence between observed and simulated values at targets 

by means of optimized parameter values. Comparison of 

observed to simulated targets shows improved fit with updates 

from initial and calibrated parameter values over most targets 

and target types for the fine-favored (table 10) and coarsefavored (table 11) models. The improved fit can be summarized by comparing the objective function contribution of each 

target type before and after optimization (table 12). The least 

target improvement occurs for the nearfield and farfield water 

levels for the fine-favored model—an expected result because 

preliminary manual calibration used to choose the initial 

parameter values was focused on the fine-favored model and 

was guided by the fit to water levels. Good target improvement for both models occurs for the vertical head difference 

and the aquifer test drawdown. The target types related to 

groundwater flooding show improvement for the fine-favored 

model (because of optimized parameter values that tend to 

lower the water table) but deterioration for the coarse-favored 

model (because of optimized parameter values that tend to 

raise the water table). Both models converge on thresholds 

that largely limit the flooding to riparian areas (see “Model 

Results” section).

The quality of fit for water levels is visualized by means 

of scatter plots showing the agreement over the range of head 

values (fig. 29) and by means of residual plots showing the 

spatial distribution of the errors (fig. 30). The vertical grouping of points (fig. 29) for the fine-grained and coarse-grained 

scatter plots is because of the use of water-level contours 

at 20-ft intervals as calibration targets. In figure 30, there is 

some spatial banding in the residuals for the fine-favored and 

coarse-favored models, which is linked to calibration error 

but may also be because of inaccuracies in the mapping of the 

water-level contours underlying the target values. Trial runs 

with the water-level targets moved from layers 2 and 4 to layer 

1 reduced water-level residuals on an average of about 1 ft.
Table 10. Calibration fit for fine-favored model.—Continued[Initial, simulation with initial parameter values; calibrated, results from optimal PEST iteration; residual, observed or estimated value less the simulated value; ft, fft3/s, cubic foot per second; >, greater than; %, percent]

Water levels – statistics Group

Mean error 

(ft)

Mean absolute 

error (ft)

R

squ

Initial Calibrated Initial Calibrated Initia

3,492 active targets All −5.17 −1.66 14.06 14.08 18.79 

2,099 active targets Farfield only −6.77 −3.35 15.31 15.43 20.68 

1,393 active targets Nearfield only −2.75 .89 12.12 12.05 15.52 

Pumping test drawdown (ft) Observed Initial Calibrated Residual

Observation well, ENG-1; Time steps 1–7

active target ENG-1-1 20.701 28.019 26.368 −5.667

active target ENG-1-2 26.795 31.409 31.572 −4.777active target ENG-1-3 29.671 33.192 32.735 −3.064active target ENG-1-4 31.847 34.549 33.528 −1.681active target ENG-1-5 33.021 35.814 34.236 −1.215active target ENG-1-6 34.396 37.14 34.993 −.597active target ENG-1-7 35.096 38.639 35.851 −.755

Observation well, ENG-1; Time steps 8–11

inactive target ENG-1-8 35.625 40.323 36.842 −1.217inactive target ENG-1-9 36.035 42.503 37.923 −1.887inactive target ENG-1-10 36.309 45.108 39.065 −2.757inactive target ENG-1-11 36.39 48.18 40.138 −3.748

Observation well, ENGL-5; Time steps 1–7

active target ENGL-5-1 13.236 7.169 12.529 0.707active target ENGL-5-2 20.014 12.076 17.932 2.082active target ENGL-5-3 23.354 15.192 20.148 3.206

active target ENGL-5-4 25.587 17.405 21.416 4.171

active target ENGL-5-5 26.847 19.149 22.274 4.573

active target ENGL-5-6 28.166 20.627 22.969 5.197

active target ENGL-5-7 28.895 21.968 23.607 5.287
Table 10. Calibration fit for fine-favored model.—Continued [Initial, simulation with initial parameter values; calibrated, results from optimal PEST iteration; residual, observed or estimated value less the simulated value; ft, fft3/s, cubic foot per second; >, greater than; %, percent] Pumping test drawdown (ft) Observed Initial Calibrated Residual Observation well, ENGL-5; Time steps 8–11 inactive target ENGL-5-8 29.485 23.093 24.28 5.205 inactive target ENGL-5-9 29.897 24.511 24.971 4.926 inactive target ENGL-5-10 30.184 26.011 25.722 4.462 inactive target ENGL-5-11 30.384 27.697 26.416 3.968 Shallow well inactive target OW-1 Assumed 0.1 0.309 0.036 0.064 Base flow (ft3/s) Estimated Initial Calibrated Fraction_target inactive target Watertown Gage 33.6 32.1 32.0 0.95 inactive target Waukesha Gage 49.6 45.1 45.4 .91 Vertical head difference (ft) Estimated Initial Calibrated Residual active target lath1 2.6 2.2 0.9 1.7 active target lath3 −2.7 −1.2 −1.3 −1.4 active target lath7 −2.4 −.8 −2.3 −.1 active target lath8 −3.5 −1.1 −2.6 −.9 active target lath9 −2.5 −.9 −2.3 −.2 active target lath10 1 .8 −.4 1.4 active target wk11 −4 −4.7 −4.8 .8 active target wk12 −4 −5.1 −4.6 .6 Flooding Target Initial Calibrated active target Average flooding for flooded cells (ft) minimize 7.7 6.5 active target Percent nearfield cells with flooding > 3 ft (%) minimize 11.8 7.6
Table 11. Calibration fit for coarse-favored model.—Continued[Initial, simulation with initial parameter values; calibrated, results from optimal PEST iteration; residual, observed or estimated value less the simulated value; ft, fft3/d, cubic foot per day; >, greater than; %, percent]

Water levels – statistics Group

Mean error 

(ft)

Mean absolute 

error (ft)

R

squ

Initial Calibrated Initial Calibrated Initia

3,492 active targets All 1.76 −0.60 14.41 14.03 19.72 

2,099 active targets Farfield only −1.40 −3.02 14.96 15.05 20.40 

1,393 active targets Nearfield only 6.52 3.04 13.60 12.48 18.65 

Pumping test drawdown (ft) Observed Initial Calibrated Residual

Observation well, ENG-1; Time steps 1–7

active target ENG-1-1 20.701 27.382 24.382 −3.681

active target ENG-1-2 26.795 30.676 30.424 −3.628active target ENG-1-3 29.671 32.35 33.151 −3.479active target ENG-1-4 31.847 33.537 34.487 −2.64active target ENG-1-5 33.021 34.563 35.122 −2.101active target ENG-1-6 34.396 35.611 35.282 −.885active target ENG-1-7 35.096 36.823 35.314 −.219

Observation well, ENG-1; Time steps 8–11

inactive target ENG-1-8 35.625 38.263 35.332 0.293inactive target ENG-1-9 36.035 40.144 35.35 .686inactive target ENG-1-10 36.309 42.418 35.372 .937inactive target ENG-1-11 36.39 45.096 35.401 .988

Observation well, ENGL-5; Time steps 1–7

active target ENGL-5-1 13.236 7 10.777 2.459active target ENGL-5-2 20.014 11.757 16.212 3.803active target ENGL-5-3 23.354 14.699 19.026 4.328

active target ENGL-5-4 25.587 16.658 20.481 5.106

active target ENGL-5-5 26.847 18.038 21.198 5.649

active target ENGL-5-6 28.166 19.083 21.453 6.713

active target ENGL-5-7 28.895 19.982 21.542 7.353
Table 11. Calibration fit for coarse-favored model.—Continued [Initial, simulation with initial parameter values; calibrated, results from optimal PEST iteration; residual, observed or estimated value less the simulated value; ft, fft3/d, cubic foot per day; >, greater than; %, percent] Pumping test drawdown (ft) Observed Initial Calibrated Residual Observation well, ENGL-5; Time steps 8–11 inactive target ENGL-5-8 29.485 20.771 21.581 7.904 inactive target ENGL-5-9 29.897 21.819 21.61 8.287 inactive target ENGL-5-10 30.184 22.985 21.642 8.542 inactive target ENGL-5-11 30.384 24.328 21.68 8.704 Shallow well inactive target OW-1 Assumed 0.1 0.067 0.105 −0.005 Base flow (ft3/d) Observed Initial Calibrated Fraction_target inactive target Watertown Gage 33.6 31.9 32.5 0.97 inactive target Waukesha Gage 49.6 46.5 46.5 .94 Vertical head difference (ft) Observed Initial Calibrated Residual active target lath1 2.6 1.4 0.9 1.7 active target lath3 −2.7 .3 −1.2 −1.5 active target lath7 −2.4 .1 −2.6 .2 active target lath8 −3.5 −.3 −3.0 −.5 active target lath9 −2.5 .2 −2.6 .1 active target lath10 1 1.3 −.3 1.3 active target wk11 −4 −2.8 −4.0 .0 active target wk12 −4 −3.4 −4.5 .5 Flooding Target Initial Calibrated active target Average flooding for flooded cells (ft) minimize 5.8 6.7 active target Percent nearfield cells with flooding > 3 ft (%) minimize 4.1 6.1
5. Model Calibration 77

Table 12. Objective function contributions for initial and final calibrated models.

[Initial values correspond to contributions before PEST inversion; final values correspond to contributions for calibrated model after successive PEST 

inversions; contributions to objective function are dimensionless; NA, not applicable; >, greater than; ft, foot]

Target type Group

Initial 

contribution 

to weighted 

objective 

function

Final 

contribution 

to weighted 

objective 

function

Percent change in 

objective function 

contribution

Fine-favored model

Nearfield water levels 1 335,431 351,750 5 (increase)

Farfield water levels 2 143,638 145,930 2 increase

Base flow at Watertown and Waukesha gages 3 0 0 NA

Vertical head difference 4 185,387 104,200 −44 (reduction)

Pumping test drawdown at observation wells ENG-1 and ENGL-5 5 378,990 129,710 −66 (reduction)

Pumping test drawdown at shallow well OW-1 6 0 0 NA

Minimize average flooding in flooded water table cells 7 105,093 75,076 −29 (reduction)

Minimize percent nearfield cells with flooding > 3 ft 8 142,119 58,693 −59 (reduction)

TOTAL 1,290,660 865,380 −33 (reduction)

Target type Group

Initial 

contribution 

to weighted 

objective 

function

Final 

contribution 

to weighted 

objective 

function

Percent change in 

objective function 

contribution

Coarse-favored model

Nearfield water levels 1 484,524 394,070 −19 (reduction)

Farfield water levels 2 139,802 136,840 −2 reduction

Base flow at Watertown and Waukesha gages 3 0 0 NA

Vertical head difference 4 430,446 91,321 −79 (reduction)

Pumping test drawdown at observation wells ENG-1 and ENGL-5 5 421,234 180,340 −57 (reduction)

Pumping test drawdown at shallow well OW-1 6 0 0 NA

Minimize average flooding in flooded water table cells 7 59,611 78,248 31 (increase)

Minimize percent nearfield cells with flooding > 3 ft 8 17,052 37,828 122 (increase)

TOTAL 1,552,670 918,650 −41 (reduction)
78 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 29B. Calibration scatter plots for water levels—coarse-favored model.

650

750

850

950

1,050

650 750 850 950 1,050

Line of perfect fit.

Simulated water levels, in feet

Contoured water levels, in feet above mean sea level

Contoured water levels, in feet above mean sea level

650

750

850

950

1,050

650 750 850 950 1,050

Line of perfect fit.

Simulated water levels, in feet

Figure 29a. Calibration scatter plots for water levels -- fine-favored model.

Figure 29b. Calibration scatter plots for water levels -- coarse-favored model.

Figure 29A. Calibration scatter plots for water levels—fine-favored model.
5. Model Calibration 79

Figure 30A. Spatial distribution of water-level residuals (feet)—fine-favored model. (Residual = observed − simulated; 

blue residuals indicate observed greater than simulated; red residuals indicate observed less than simulated.)

EXPLANATION

−67.0 to −50.1

−50.0 to −30.1

−30.0 to −20.1

−20.0 to −10.1

−10.0 to −5.1

-5.0 to 0.0

0.1 to 5.0

5.1 to 10.0

10.1 to 20.0

20.1 to 30.0

30.1 to 50.0

Surface-water feature

Nearfield

Fox River

Residual value

Figure 30a. Spatial distribution of water level residuals (feet) -- fine-favored model. (Residual = observed – simulated; 

blue residuals indicate observed greater than simulated; red residuals indicate observed less than simulated)

Base modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, 

Wisconsin Transverse Mercator, North 

American Datum of 1983

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS
80 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 30B. Spatial distribution of water-level residuals (feet)—coarse-favored model. (Residual = observed − simulated; 

blue residuals indicate observed greater than simulated; red residuals indicate observed less than simulated.)

EXPLANATION

−67.0 to −50.1

−50.0 to −30.1

−30.0 to −20.1

−20.0 to −10.1

−10.0 to −5.1

-5.0 to 0.0

0.1 to 5.0

5.1 to 10.0

10.1 to 20.0

20.1 to 30.0

30.1 to 50.0

Surface-water feature

Nearfield

Fox River

Residual value

Figure 30b. Spatial distribution of water level residuals (feet) -- coarse-favored model. (Residual = observed – simulated; 

blue residuals indicate observed greater than simulated; red residuals indicate observed less than simulated)

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Base modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, 

Wisconsin Transverse Mercator, North 

American Datum of 1983
5. Model Calibration 81

The calibration scatter plots and the residual distribution 

figures both indicate that low water levels associated with valleys tend to be overestimated in the simulations and high water 

levels associated with uplands tend to be underestimated. 

Despite the calibration process this bias indicates lingering 

deficiencies in the model. However, it is also possible that part 

of the bias is due to the target values themselves, which might 

reflect perched conditions under uplands. It is also possible 

that the water-level contour map (from which the targets are 

derived) systematically underestimates the water table elevation in valleys (where data density is low) because assumed 

contours neglect the tendency for groundwater discharge and 

high water-table elevations to occur over riparian areas larger 

than the width of the stream channels.

The congruence between the observed and simulated 

drawdown at aquifer test observation wells (fig. 31) shows 

fairly good agreement for both models in the early part of the 

test (first 3.5 hours). It is speculated that local heterogeneity 

in the dolomite could be affecting the drawdown slope in the 

latter part of the test. In any event, the comparison indicates 

that the overall transmissivity and vertical resistance applied 

to both models in the vicinity of pumping well WK-13 is not 

greatly in error even though the model was not constructed to 

precisely match the hydrogeology in this area. Another check 

on model performance is the simulated drawdown at shallow 

well OW-1, located in layer 1 of the same row and column as 

the well pumping 600 gal/min from layers 3 and 4 . For the 

optimized parameter values, the simulated drawdown at OW-1 

after 20 hrs of pumping is 0.04 and 0.1 ft in the fine-favored 

and coarse-favored models, respectively (tables 10 and 11), 

whereas, the simulated drawdown adjacent to the pumping 

well opening, located less than 100 ft away, is more than 60 ft. 

Spot measurements during the test showed no drawdown at 

OW-1 (as opposed to drawdown on the order of 30 ft at the 

observation wells), which indicates that the stress from the 

well does not propagate in the vertical direction nearly as easily as it does in the lateral direction. The model appears to be 

in agreement with this finding.

The match for the vertical head difference targets 

(tables 10 and 11) can be summarized in terms of how well 

observed “flowing” conditions at six of the eight target test 

wells (open on average 100 ft below land surface but reporting heads above land surface) are reproduced by the models. 

Measured water levels in the six flowing wells were 1 to 3 ft 

above land surface. In the fine-favored and coarse-favored 

model, five of these six wells are simulated as flowing with 

water levels 0.5 to 2.5 ft above land surface. 

The second dimension of the calibration results are 

the optimized parameter values generated by the inversion 

routines (table 13). Partly because of the imposed bounds, the 

results honor the conceptual model in terms of the ordering 

of Kh

 and Kv values by facies zones. The coarse-favored K 

values are generally lower than the fine-favored K values. This 

is reasonable because both models are calibrated to the same 

targets, but the coarse-favored model contains higher K material than the fine-favored model and, in compensation, requires 

lower K values to obtain similar results, whereas, the calibration of the fine-favored model, given its relatively larger volume of fine-grained material, tends to yield relatively higher 

K values to meet target values. For both models,facies 1, 2, 

and 3 have appreciably lower Kv values than facies 4, which, 

in turn, is markedly lower than facies 5. This result is welcome 

because it restricts vertical preferential flow to areas where 

facies 4 and, especially, facies 5 are present. The implied 

vertical anisotropy results are a useful check on the calibration 

process. For the initial values, vertical anisotropy was set lowest for the dominantly fine and dominantly coarse facies and 

set highest for the mixed fine and coarse facies. This ordering 

follows the logic that the contrast between Kh and Kv should be 

weakest for facies dominated by a single texture and strongest 

for the facies with the most varied textures. The calibrated K 

values generally imply the same ordering for the fine-favored 

and coarse-favored models. The most striking departure from 

the initial settings is the large size of the implied vertical 

anisotropy for the mixed facies (table 13).

The optimized results for streambed K are higher in most 

cases than the initial values across the two models but still 

respect the ordering from low to middle to high K zones. Calibration of both models yielded lower transmissivity and higher 

vertical conductivity for the dolomite bedrock by factors 

averaging about 1.5 (table 13). The optimized specific storage results from the WK-13 aquifer test are very close to the 

global specific storage assigned the subsurface in the regional 

SEWRPC model, 2.6e-7 1/ft (Feinstein and others, 2005a). 

The optimized specific yield values varied greatly between 

the fine-favored and coarse-favored models (table 13), which 

suggests that limiting the analysis to the first 3.5 hrs of the 

aquifer test was insufficient time to generate reliable estimates 

of specific yield by means of the optimizations.

One test of the quality of the calibration process is to 

examine the number of parameters where calibrated values 

equal lower or upper bounds. The less constrained the optimization is by bounds, the more it is informed by mathematical 

criteria of best fit, and the less it is informed by prior information (that is judgment) imposed by the user. Among the 17 

parameters estimated, the fine-favored calibration returns three 

upper bounds and two lower bounds, whereas, the coarsefavored calibration returns five lower bounds (all involving 

K) and one upper bound (table 13). These results suggest that 

without the imposition of bounds, the objective function could 

have been lower in both numerical models but at the expense 

of violating some elements of the underlying conceptual 

model.
82 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 31A. Observed and simulated drawdown at pumping-test observation wells—fine-favored model.

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

Observed Simulated

Observed Simulated

Drawdown at Observation Well ENG1

Drawdown at Observation Well ENG5

Test duration, in hours

Test duration, in hours

Drawdown, in feet Drawdown, in feet

Figure 31a. Observed and simulated drawdown at pumping-test observation wells -- fine-favored model.
5. Model Calibration 83

Figure 31B. Observed and simulated drawdown at pumping-test observation wells—coarse-favored model.

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25

Drawdown at Observation Well ENG1

Drawdown at Observation Well ENG5

Drawdown, in feet Drawdown, in feet

Test duration, in hours

Test duration, in hours

Figure 31b. Observed and simulated drawdown at pumping-test observation wells -- coarse-favored model.

Observed Simulated

Observed Simulated
84 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Table 13. Calibrated parameter values for fine- and coarse-favored models.

[ft, foot; ft/d, foot per day; kh, horizontal hydraulic conductivity; kv, vertical hydraulic conductivity; ksfr, streambed hydraulic conductivity]

Parameter set Zone or variable Initial 

value

Calibrated value

Fine-favored 

model

Coarse-favored 

model

Horizontal hydraulic conductivity (ft/d) Facies 1 – dominantly fine 0.5 2

1 0.75

Facies 2 – relatively fine 2 2.26 2

1

Facies 3 – mixed fine and coarse 10 2

20 19.75

Facies 4 – relatively coarse 40 51.31 2

20

Facies 5 – dominantly coarse 80 183.09 2

60

Vertical hydraulic conductivity (ft/d)1 Facies 1 – dominantly fine 0.005 0.00955 2

0.002

Facies 2 – relatively fine .01 2

.00955 .00245

Facies 3 – mixed fine and coarse .02 2

.00955 2.00245

Facies 4 – relatively coarse .2 .24361 .12658

Facies 5 – dominantly coarse 2 1.66107 7.08675

Implied vertical anisotropy (kh

:kv ) Facies 1 – dominantly fine 100 105 376

Facies 2 – relatively fine 200 237 408

Facies 3 – mixed fine and coarse 500 2,095 8,062

Facies 4 – relatively coarse 200 211 158

Facies 5 – dominantly coarse 40 110 8

Streambed hydraulic conductivity (ft/d) ksfr_low 1 0.8 2.0

ksfr_middle 5 9.6 2

25

ksfr_high 25 2

100 63.6

Multiplier on dolomite hydraulic conductivity kh 1 0.77 0.42

kv 1 1.48 1.55

Storage of unconsolidated deposits specific storage (ft-1) 1.00E-06 2.86E-07 3.68E-07

specific yield (--) .05 .028 .306

Multiplier on recharge (fixed) recharge 1 1 1

1

For vertical hydraulic conductivity, facies 1 was estimated directly and the remaining parameters were estimated using sequential multipliers bounded 

between 1 and 100.

2

Calibrated value at bound set in PEST inversion.
5. Model Calibration 85

Another common check on the quality of the calibration 

process involves determining that the amount of correlation 

between parameters is not large enough to subvert the ability 

to independently estimate their values. However, the use of 

SVD subspace methods in the inversion routines invalidates 

calculation of correlation coefficients, so that check cannot be 

made.

Finally, it is noteworthy that the calibrated Kh values are 

systematically higher for the fine-favored model than for the 

coarse-favored model. In fact, if all the nearfield hydraulic 

conductivity values weighted by the saturated thickness in 

glacial cells are summed and then divided by the sum of the 

saturated thicknesses, the resulting global Kh value across the 

five glacial layers for the fine-favored model is about 20‐percent higher than the global value for the coarse-favored model 

(26 and 22 ft/d, respectively). The difference in the average 

values does not mean that groundwater is transmitted more 

easily to surface water and wells in the fine-favored than the 

coarse-favored model given the crucial role that the facies distribution plays in controlling flow. Because the coarse model 

imposes more continuity among the coarse facies, it supports 

more preferential flow in the horizontal and vertical directions 

even though the Kh values assigned to those facies are moderately lower than the values assigned in the fine-favored model. 

Results from simulations of a hypothetical pumping from 

wells along the Fox River are consistent with this conclusion 

(see “Model Application to Hypothetical Well Field” section).

5.6 Sensitivity Analyses

The PEST code generates composite sensitivity values 

between all observations and individual estimated parameters 

for each update of the parameter values. These sensitivity 

values indicate how well parameters can be estimated from the 

available information. Composite sensitivities compiled at the 

beginning of the inversion and for the final update (fig. 32 for 

fine-favored model and fig. 33 for coarse-favored model) indicate that the targets inform Kh

, Kv , dolomite transmissivity and 

vertical resistance, and specific storage better than streambed 

K or specific yield. The parameter most sensitive to observations is the Kh

 of the mixed facies. The relative composite 

sensitivity by parameter set for the final inversion update is as 

follows:

Parameter set

Finefavored 

model 

(percent)

Coarsefavored 

model 

(percent)

Kh 39 51

Kv 46 36

Streambed K 3 2

Storage terms 6 6

Dolomite K 6 5

This tabulation indicates that for the final update Kv was 

the most sensitive target type for the fine-favored calibration, 

but Kh was the most sensitive for the coarse-favored calibration. The analysis also shows that streambed K values are 

insensitive to the assembled targets. It is interesting to note 

that initial PEST runs, which included the base-flow targets along with recharge as a parameter, produced even less 

sensitivity with respect to streambed conductance than did 

the final PEST implementation without any base-flow targets 

or recharge parameters. Given the uncertainty attached to the 

streambed K values, further analysis is needed to explore how 

changing the values can influence model results (see “Model 

Application to Hypothetical Well Field” section). 

PEST output also can be used to look at the sensitivity 

relation to determine the most informative target types by 

parameter set. The rankings are similar for the two models 

and show that all the active target types participated in the 

optimizations:

Parameter set Most informative target types

Kh most sets, especially water levels and 

aquifer-test drawdowns

Kv vertical head differences

Streambed K flooding constraints and nearfield water levels

Storage terms aquifer-test drawdowns

Dolomite K water levels
86 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 32B. Composite sensitivities to calibration targets for fine-favored model generated by 

PEST—calibrated parameter values.

0

0.2

0.4

0.6

0.8

1

kx2kx4

kx12kx13kx14

kz2

rat4_2

rat12_4

rat13_12rat14_13

ksfrl

ksfrm

ksfrh

ssmsym

kxsimul

kzsimul

Estimated parameters

EXPLANATION

Abbreviation Description

kx2 horizontal hydraulic conductivity of unlithified facies 1 (fine dominated) 

kx4 horizontal hydraulic conductivity of unlithified facies 2 (relatively fine)

kx12 horizontal hydraulic conductivity of unlithified facies 3 (mixed)

kx13 horizontal hydraulic conductivity of unlithified facies 4 (relatively coarse)

kx14 horizontal hydraulic conductivity of unlithified facies 5 (coarse dominated)

kz2 vertical hydraulic conductivity of unlithified facies 1 

rat4_2 vertical hydraulic conductivity of unlithified facies 2 expressed as ratio to facies 1

rat12_4 vertical hydraulic conductivity of unlithified facies 3 expressed as ratio to facies 2

kx13 vertical hydraulic conductivity of unlithified facies 4 expressed as ratio to facies 3

kx14 vertical hydraulic conductivity of unlithified facies 5 expressed as ratio to facies 4

ksfrl hydraulic conductivity for streambeds characterized as muddy, silty

ksfrm hydraulic conductivity for uncharacterized streambeds 

ksfrh hydraulic conductivity for streambeds characterized as sandy and gravelly

ssm specific storage of unlithified deposits (model layers 1–5) expressed as single multiplier

sym specific yield of unlithified deposits (model layers 1–5) expressed as single multiplier

kxsimul horizontal hydraulic conductivity of Silurian dolomite expressed as single multiplier on 8 zones

kzsimul vertical hydraulic conductivity of Silurian dolomite expressed as single multiplier on 8 zones

Figure 32a. Composite sensitivities to calibration targets for fine-favored model generated by PEST --

initial parameter values.

Figure 32b. Composite sensitivities to calibration targets for fine-favored model generated by PEST -- 

calibrated parameter values.

0

0.2

0.4

0.6

0.8

1

kx2kx4

kx12kx13kx14

kz2

rat4_2

rat12_4

rat13_12rat14_13

ksfrl

ksfrm

ksfrh

ssm

sym

kxsimul

kzsimul

Estimated Parameters

Figure 32A. Composite sensitivities to calibration targets for fine-favored model generated by 

PEST—initial parameter values.
5. Model Calibration 87

Figure 33B. Composite sensitivities to calibration targets for coarse-favored model generated by 

PEST—calibrated parameter values.

EXPLANATION

Abbreviation Description

kx2 horizontal hydraulic conductivity of unlithified facies 1 (fine dominated) 

kx4 horizontal hydraulic conductivity of unlithified facies 2 (relatively fine)

kx12 horizontal hydraulic conductivity of unlithified facies 3 (mixed)

kx13 horizontal hydraulic conductivity of unlithified facies 4 (relatively coarse)

kx14 horizontal hydraulic conductivity of unlithified facies 5 (coarse dominated)

kz2 vertical hydraulic conductivity of unlithified facies 1 

rat4_2 vertical hydraulic conductivity of unlithified facies 2 expressed as ratio to facies 1

rat12_4 vertical hydraulic conductivity of unlithified facies 3 expressed as ratio to facies 2

kx13 vertical hydraulic conductivity of unlithified facies 4 expressed as ratio to facies 3

kx14 vertical hydraulic conductivity of unlithified facies 5 expressed as ratio to facies 4

ksfrl hydraulic conductivity for streambeds characterized as muddy, silty

ksfrm hydraulic conductivity for uncharacterized streambeds 

ksfrh hydraulic conductivity for streambeds characterized as sandy and gravelly

ssm specific storage of unlithified deposits (model layers 1–5) expressed as single multiplier

sym specific yield of unlithified deposits (model layers 1–5) expressed as single multiplier

kxsimul horizontal hydraulic conductivity of Silurian dolomite expressed as single multiplier on 8 zones

kzsimul vertical hydraulic conductivity of Silurian dolomite expressed as single multiplier on 8 zones

Figure 33a. Composite sensitivities to calibration targets for coarse-favored model generated by PEST --

initial parameter values.

0

0.2

0.4

0.6

0.8

1

kx2kx4

kx12kx13kx14

kz2

rat4_2

rat12_4

rat13_12rat14_13

ksfrl

ksfrm

ksfrh

ssm

sym

kxsimul

kzsimul

Estimated parameters

0

0.2

0.4

0.6

0.8

1

kx2kx4

kx12kx13kx14

kz2

rat4_2

rat12_4

rat13_12rat14_13

ksfrl

ksfrm

ksfrh

ssm

sym

kxsimul

kzsimul

Estimated parameters

Figure 33b. Composite sensitivities to calibration targets for coarse-favored model generated by PEST --

calibrated parameter values.

Figure 33A. Composite sensitivities to calibration targets for coarse-favored model generated by 

PEST—initial parameter values.
88 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

6. Model Results

The calibrated versions of the fine-favored and coarsefavored models simulate water levels including the water-table 

elevation, water budgets, and contributing areas for discharge 

zones such as streams, pumping wells, and quarries. These 

results are based on steady-state simulations intended to 

approximate 2005 conditions. In both models, the numerical solutions achieve favorable balance between sources and 

sinks. The global mass balance error for the steady-state stress 

period is 0.05 percent for the fine-favored model (convergence 

after 173 outer iterations of the solver) and 0.04 percent for the 

coarse-favored model (convergence after 771 outer iterations 

of the solver). Checks of the mass balance errors in subdomains of the models suggest that local errors are of the same 

order as the global error.

6.1 Water Levels for Fine-Favored and CoarseFavored Models

As pointed out in the “Model Solver” section of this 

report, it is the merit of the MODFLOW-NWT formulation 

that dry cells remain active, so that abrupt changes in the vertical flow profile leading to numerically isolated cells and spikes 

in water-table elevation are avoided (fig. 24B). Accordingly, 

the water-table solutions are acceptably smooth (fig. 34). The 

average depth from land surface to the water table across 

the model domain is 39.12 ft for the fine-favored model and 

38.7 ft for the coarse-favored model (within model layer 2). 

The percent distribution of the water table by layer depth interval is similar for the two models:

Layer depth 

interval

Finefavored 

model

(percent)

Coarsefavored 

model

(percent)

1 (0 to 20-ft depth) 50.6 51.5

2 (20.1- to 5-ft depth) 21.7 22.2

3 (50.1- to 100-ft depth) 19.9 20.0

4 (100.1- to 150-ft depth) 6.5 5.6

5 (more than 150.1-ft depth) .7 .4

6 (upper 20 ft of bedrock) .6 .3

The percentage of water-table cells simulated as perched 

(locations where a water-table cell is underlain by a dry cell) 

is very small: 0.04 percent for the fine-favored model and 

0.02 percent for the coarse-favored model. All the water levels 

simulated in perched cells are reasonable elevations (unlike 

the unacceptably high elevations simulated for many more 

perched cells using PCG2).

Areas of vertically upward gradients in the unconsolidated deposits are generally simulated along surface-water 

bodies (fig. 35). The deeper parts of the unconsolidated 

material are pressurized to a fairly high degree in places and 

upward head differences over 10 ft are common. This result 

sorts well with the observation that water levels in test wells 

open to the bottom of the unconsolidated material are often 

flowing (John Jansen, formerly Aquifer & Technology, oral 

commun., February 2010) when it is assumed that the location of flowing wells is typically in low-lying areas and that 

the water table is close to the land surface. Heterogeneity in 

the hydraulic conductivity field combines with the geometry 

of discharge zones and the simulated head distribution to produce a complicated pattern of upward flow to the water table 

(fig. 36). Most areas of upward flow are simulated to transmit 

groundwater at a rate between 0.0001 and 0.001 ft/d, but the 

upward flow rate transmits 0.01 ft/d or more in some riparian 

reaches. These results demonstrate the ability of the models to 

simulate preferential flow associated with natural discharge. 

Note that pumping stresses inserted in the models can also 

induce preferential flow away from surface-water features into 

the groundwater system. 

Groundwater flooding is simulated mostly along waterways (fig. 37). As discussed in the “Model Calibration” section 

of this report under “Land Surface Constraints”, this outcome 

is expected because not all existing riparian discharge zones 

are represented in the models, causing the groundwater in the 

models to discharge directly to SFR and LAK cells rather than 

to adjacent riparian zones. By contrast, in upland areas of relatively low water-table levels, discharge to the stream channel 

is limited and, as a result, base flow does not accumulate along 

the channel. In this case, the stream can be simulated as dry 

for low-flow conditions (ephemeral). For headwater reaches 

this condition commonly occurs when the water-table elevation is below the top of the streambed. Ephemeral streams are 

simulated to occur in about 20 percent of the stream reaches 

inside the Upper Fox River Basin, representing approximately 

40 percent of the headwater reaches (fig. 37). 
6. Model Results 89

Figure 34A. Simulated water table in plan view and vertical section—fine-favored model. Vertical cross section 

along row 298 of the model. Vertical exaggeration is 35 times.

CROSS SECTION ALONG ROW 298

Vertical exaggeration = 35X

West East Simulated water table

Water-table elevation, in feet

700 – 725

726 – 750

751 – 775

776 – 800

801 – 825

826 – 850

851 – 875

876 – 900

901 – 925

926 – 950

951 – 975

976 – 1000

1001 – 1025

1026 – 1050

EXPLANATION

Nearfield

Fox River

Surface-water feature

Figure 34a. Simulated water table in plan view and vertical section -- fine-favored model. Vertical cross-section along 

row 298 of the model. Vertical exaggeration is 35 times.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

ROW 298

Base modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, 

Wisconsin Transverse Mercator, North 

American Datum of 1983
90 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 34B. Simulated water table in plan view and vertical section—coarse-favored model. Vertical cross section 

along row 298 of the model. Vertical exaggeration is 35 times.

CROSS SECTION ALONG ROW 298

Vertical exaggeration = 35X

West East Simulated water table

Water-table elevation, in feet

700 – 725

726 – 750

751 – 775

776 – 800

801 – 825

826 – 850

851 – 875

876 – 900

901 – 925

926 – 950

951 – 975

976 – 1000

1001 – 1025

1026 – 1050

EXPLANATION

Nearfield

Fox River

Surface-water feature

Figure 34b. Simulated water table in plan view and vertical section -- coarse-favored model. Vertical cross-section along 

row 298 of the model. Vertical exaggeration is 35 times.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

ROW 298

Base modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, 

Wisconsin Transverse Mercator, North 

American Datum of 1983
6. Model Results 91

Figure 35B. Areas of simulated upward head gradients in unconsolidated 

deposits for model nearfield—coarse-favored model. White area is where 

head gradient is downward in unconsolidated deposits.

Upward head difference, in feet,

between layers 4 and 1

0.004 − 0.05

0.051 − 0.5

0.51 − 5

5.1 − 10

10.1 − 15

15.1 − 20

20.1 − 25

25.1 − 30

30.1 − 35

35.1 − 40

88°20' 88°10'

43°5'

43°

88°20' 88°10'

43°5'

43°

Figure 35a. Areas of simulated upward head gradients in unconsolidated 

deposits for model nearfield -- fine-favored model. White area is where 

upland head difference is less than 0.005 feet (see minimum of scale). 

Figure 35b. Areas of simulated upward head gradients in unconsolidated 

deposits for model nearfield -- coarse-favored model. White area is where 

upland head difference is less than 0.005 feet (see minimum of scale). 

EXPLANATION

Nearfield

Surface-water feature

0 2 4 MILES

0 2 4 KILOMETERS

Figure 35A. Areas of simulated upward head gradients in unconsolidated 

deposits for model nearfield—fine-favored model. White area is where 

head gradient is downward in unconsolidated deposits.
92 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 36B. Areas of upward flow to water table for model nearfield—

coarse-favored model. White area is where flow is downward from water 

table.

88°20' 88°10'

43°5'

43°

88°20' 88°10'

43°5'

43°

Upward flow rate, in feet per day,

to unperched water-table layer

0.00001 − 0.0005

0.00051 − 0.001

0.0011 − 0.005

0.0051 − 0.01

0.011 − 0.05

0.051 − 0.1

0.11 − 0.15

Figure 36a. Areas of upward flow to water table for model nearfield --

fine-favored model. White area is where upward flow rate is less than 

0.00001 feet per day (see minimum of scale).

Figure 36b. Areas of upward flow to water table for model nearfield --

fine-favored model. White area is where upward flow rate is less than 

0.00001 feet per day (see minimum of scale). 

EXPLANATION

Nearfield

Surface-water feature

0 2 4 MILES

0 2 4 KILOMETERS

Upward flow to water table corresponds 

to flow to unperched water-table cell 

(whatever layer water table is in) from 

underlying cell

a) Areas of upward flow to water table = 

34.6 percent of nearfield

b) Areas of upward flow to water table = 

33.8 percent of nearfield

Figure 36A. Areas of upward flow to water table for model nearfield—

fine-favored model. White area is where flow is downward from water 

table.
6. Model Results 93

Figure 37A. Simulated flooded water-table cells and dry headwater reaches for model domain—fine-favored model. 

[18.8 percent (1,457 out of 7,731) of stream reaches are dry; all are headwater reaches.]

Dry stream reach

(ephemeral streams)

Flooded model cell

(where simulated water table 

is more than 3 feet above land 

surface)

Figure 37a. Simulated flooded water-table cells and dry headwater reaches for model domain -- fine-favored model.

[18.8 percent (1,457 out of 7,731) stream reaches are dry; all are headwater reaches]

EXPLANATION

Surface-water feature

Nearfield

Fox River

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983
94 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 37B. Simulated flooded water-table cells and dry headwater reaches for model domain—coarse-favored model. 

[21.2 percent (1,638 out of 7,731) of stream reaches are dry; all but 10 are headwater reaches.]

Dry stream reach

(ephemeral streams)

Flooded model cell

(where simulated water table 

is more than 3 feet above land 

surface)

Figure 37b. Simulated flooded water-table cells and dry headwater reaches for model domain -- coarse-favored model.

[21.2 percent (1,638 out of 7,731) of stream reaches are dry; all but 10 are headwater reaches]

EXPLANATION

Surface-water feature

Nearfield

Fox River

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983
6. Model Results 95

6.2 Water Budgets for Fine-Favored and CoarseFavored Models

The water budgets simulated by the calibrated Upper 

Fox models combine prescribed inputs (such as boundary 

fluxes and recharge) with calculated outputs (such as base 

flow and quarry discharge). The steady-state fluxes associated 

with the sources (recharge, inflow across model boundaries, 

and inflow to the groundwater system from surface-water 

features) balance the steady-state fluxes associated with sinks 

(outflow across model boundaries, well discharge, quarry discharge, and base flow to surface-water features). The relative 

magnitudes of the simulated inflows and outflows are very 

similar for the fine-favored and coarse-favored models. For the 

model domain as a whole, recharge is the dominant inflow and 

base flow is the dominant outflow with each term representing over 80 percent of the sources and sinks (table 14). The 

amount of groundwater discharge to streams within the Upper 

Fox River Basin is about nine times greater than the amount 

of water induced from streams to the groundwater system for 

the fine-favored model and about seven times greater for the 

coarse-favored model. Pumping from the high-capacity wells 

included in the models is responsible for most of the induced 

flow and totals about 7 percent of the simulated domain 

outflow.

Table 14. Simulated water budgets for model domain.

[Mgal/d, million gallons per day; for flows related to surface water, MODFLOW package simulating flow is 

indicated in parentheses]

Fine-favored model Coarse-favored model

Flux

(Mgal/d)

Percent 

of total

Flux

(Mgal/d)

Percent 

of total

Inflows = sources

Recharge 72.3 82.9 72.3 82.9

Boundary fluxes 1.6 1.8 1.6 1.9

Lateral (model grid edges) 1.3 1.5 1.3 1.5

Bottom (Silurian/Maquoketa contact) .3 .3 .3 .5

Induced flow from surface water 13.4 15.3 13.3 15.2

Upper Fox River Basin streams (SFR) 3.9 4.5 5.1 5.8

Other streams (RIV) 9.4 10.7 8.1 9.3

Pewaukee Lake (LAK) .1 .1 .1 .1

Other lakes and wetlands (DRN) .0 .0 .0 .0

TOTAL 87.3 100.0 87.3 100.0

Outflows = sinks

Boundary fluxes 8.9 10.2 8.9 10.2

Lateral (model grid edges) 5.4 6.2 5.4 6.2

Bottom (Silurian/Maquoketa contact) 3.5 4.0 3.5 4.0

High Capacity Wells 6.4 7.3 6.1 7.0

Unconsolidated (13 wells) 1.6 1.8 1.6 1.8

Dolomite (68 wells) 4.8 5.5 4.5 5.2

Sussex and Waukesha Quarries 1.3 1.5 1.0 1.2

Base flow to surface water 70.8 81.0 71.2 81.6

Upper Fox River Basin streams (SFR) 34.7 39.8 37.3 42.8

Other streams (RIV) 23.0 26.2 23.3 26.6

Pewaukee Lake (LAK) 1.6 1.9 1.7 1.9

Other lakes and wetlands (DRN) 11.5 13.2 9.0 10.3

TOTAL 87.4 100.0 87.2 100.0
96 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

For this application, the NWT solver decreases pumping when the saturated thickness of a cell hosting a well 

falls below 20 percent of its total thickness. From a total of 

6.7 Mgal/d input to the models, the loss of pumping because 

of this condition is less than 4 percent for the fine-favored 

model and less than 8 percent for the coarse-favored model. 

Almost all the loss is from Silurian dolomite wells. The 

discrepancy between what withdrawals the calibrated parameters can support and the pumping rates the public-supply 

systems have reported might point to calibration errors, but it 

could also be a function of overestimated pumping rates. It is 

interesting to note that preliminary simulations using MODFLOW-2005 with the PCG2 solver yielded losses of pumping 

discharge because of dry (inactive) cells equal to almost twice 

the amount lost in the MODFLOW-NWT simulations as a 

result of reduced saturated thickness.

Water budgets calculated for nested basins show a trend 

in the relative magnitude of outflows to sinks with size of 

basin. The areas corresponding to the basins upgradient from 

the Watertown and the Waukesha streamgages (fig. 6B) define 

nested subbasins within the Upper Fox River Basin. For the 

fine-favored (table 15) and coarse-favored (table 16) models, 

the proportion of outflow that circulates to surface-water features increases with basin size, accompanied by a decrease in 

the proportion of well discharge. This outcome is partly linked 

to the concentration of pumping wells northeast of Waukesha 

inside the Watertown subbasin nested inside the other two 

basins. The simulated budget for the entire Upper Fox River 

Basin for the two models indicates that about 2 percent of the 

groundwater flow leaks downward from the basin into the 

deep part of the flow system, 12 to 15 percent moves laterally across the basin boundaries, 77 to 80 percent discharges 

to Upper Fox surface waters, and about 7 percent to pumping 

wells within the basin.

Table 15. Simulated water budgets for nested basins in the fine-favored model, Upper Fox River Basin.

[Mgal/d, million gallons per day; ΣIN, sum of inflows; mi2, square mile; see figure 7 for outlines of basins]

Inflow

(Mgal/d)

Outflow

(Mgal/d) OUT percent of ΣIN

Upper Fox River Basin upgradient of Watertown gage (area is 86 mi2

)

Recharge 16.63

Vertical flux across Silurian/Maquoketa boundary .00 0.27 1.4

Lateral flux across basin boundaries 1.83 5.12 25.7

Surface-water exchange1 1.43 11.58 58.2

Pumping 2.99 15.0

100.3

[mass balance error = 0.3]

Upper Fox River Basin upgradient of Waukesha gage (area is 135 mi2

)

Recharge 24.34

Vertical flux across Silurian/Maquoketa boundary .10 0.57 1.9

Lateral flux across basin boundaries 4.09 6.28 20.6

Surface-water exchange1 2.00 20.19 66.1

Pumping 3.55 11.6

100.2

[mass balance error = 0.2]

Upper Fox River Basin upgradient of Vernon marsh (area is 207 mi2

)

Recharge 41.89

Vertical flux across Silurian/Maquoketa boundary .23 0.94 1.8

Lateral flux across basin boundaries 6.17 7.67 14.7

Surface-water exchange1 3.99 40.08 76.7

Pumping 3.68 7.0

100.2

[mass balance error = 0.2]

1

Budget includes exchanges with streams, wetlands, lakes, and quarries but excludes wastewater-treatment plant effluent 

added to Fox River. 
6. Model Results 97

6.3 Contributing Areas for Groundwater Sinks

The input and output of the Upper Fox models can be 

used as input for the particle tracking program MODPATH 

(Pollock, 1994) in order to delineate groundwater basins based 

on the simulated flow systems (as opposed to surface-water 

drainage basins based on topography). For this study, 35,719 

particles were released at a 500-ft spacing over the entire 

model domain at a vertical starting position just below the 

simulated water table. The effective porosity, an input that 

affects travel times simulated by MODPATH, was assumed to 

be 0.2 for the unconsolidated layers and 0.02 for the dolomite 

layers. Particles were tracked forward under steady-state 

(2005) conditions until their pathlines terminated at a sink—a 

surface-water feature, quarry, pumping well, or model boundary. The starting locations of particles discharging to each 

tributary system within the Upper Fox River Basin were tabulated, as were particles discharging directly to the main trunk 

of the Fox River. Particles discharging to minor unnamed 

tributaries connected directly to the Fox River (fig. 6A) and to 

riparian wetlands along the Fox River (fig. 18) were grouped 

with particles terminating at the Fox River itself. Finally, the 

starting locations of particles discharging to quarries and the 

high-capacity pumping wells inserted in the models were 

separately tabulated. 

Table 16. Simulated water budgets for nested basins in the coarse-favored model, Upper Fox River Basin.

[Mgal/d, million gallons per day; ΣIN, sum of inflows; mi2, square mile; see figure 7 for outlines of basins]

Inflow

(Mgal/d)

Outflow

(Mgal/d) OUT percent of ΣIN

Upper Fox River Basin upgradient of Watertown gage (area is 86 mi2

)

Recharge 16.63

Vertical flux across Silurian/Maquoketa boundary .00 0.27 1.4

Lateral flux across basin boundaries 1.43 4.48 22.9

Surface-water exchange1 1.51 12.06 61.6

Pumping 2.81 14.4

100.3

[mass balance error = 0.3]

Upper Fox River Basin upgradient of Waukesha gage (area is 135 mi2

)

Recharge 24.34

Vertical flux across Silurian/Maquoketa boundary .10 0.57 1.9

Lateral flux across basin boundaries 3.03 4.80 16.2

Surface-water exchange1 2.07 20.94 70.9

Pumping 3.29 11.1

100.2

[mass balance error = 0.2]

Upper Fox River Basin upgradient of Vernon marsh (area is 207 mi2

)

Recharge 41.89

Vertical flux across Silurian/Maquoketa boundary .23 0.94 1.8

Lateral flux across basin boundaries 5.57 6.33 12.0

Surface-water exchange1 5.17 42.17 79.8

Pumping 3.42 6.6

100.1

[mass balance error = 0.1]

1

Budget includes exchanges with streams, wetlands, lakes, and quarries but excludes wastewater-treatment plant effluent 

added to Fox River.
98 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

By mapping the starting locations of the particles for each 

kind of sink, it is also possible to map areas where groundwater contributes to the sink discharge. The implied groundwater 

contributing basins to surface-water features as well as the 

contributing areas to man-made features (wells and quarries) 

are very similar for fine-favored (fig. 38A) and coarse-favored 

(fig. 38B) models. The groundwater contributing basins tend 

to extend upgradient from the Upper Fox River drainage basin 

boundary to the west and fall short of the drainage boundary 

to the east, probably because of tendency of groundwater to 

flow from west to east, especially in the Silurian. Quarries and 

wells divert groundwater from its natural surface-water sinks 

in selected areas.

The calculated median travel time across the groundwater 

contributing basins associated with surface water is 11.4 years 

for the fine-favored model and 9.9 years for the coarse-favored 

model. Simulated travel times are longer to quarries (median 

time equals 25.4 years for the fine-favored model) and to highcapacity wells (median time equals 63.4 years for the finefavored model). These results are reasonable given that flow 

paths are longer to the man-made features. The quarries are 

Table 17. Simulated base flow within the Upper Fox River Basin.

[ft3/s, cubic foot per second]

Fine-favored model

(ft3

/s)

Coarse-favored model

(ft3/s)

Tributary base flow at confluence with Fox River

Fox Headwaters 2.1 2.1

Lannon Creek 1.2 1.2

Sussex Creek 5.6 5.8

Poplar Creek 2.5 2.9

Pewaukee Lake and Pewaukee River 10.6 11.3

Pebble Creek 5.9 6.0

Genesee Creek 7.1 7.0

Pebble Brook 7.6 8.0

Sum of base flow from major tributaries 42.5 44.1

Sum of base flow from minor tributaries .2 .3

Sum of tributary flow to Fox River 42.7 44.5

Net gain of base flow along main trunk of Fox River 7.6 8.1

Contribution of riparian wetlands 3.7 3.4

Sum of return flow from quarries 2.0 1.6

Added flow from wastewater-treatment plants 31.9 31.9

Fox River base flow above Vernon Marsh 88.0 89.4

excavated at the top of the Silurian dolomite, and most of the 

well withdrawal is from the dolomite, forcing the discharging 

groundwater to travel not only laterally but downward.

In addition to the spatial insight derived from the particle 

tracking results, insight can be gained into the quantity of 

water flowing to the surface-water network by collecting 

water-budget terms (table 17). For the Upper Fox River Basin, 

the total base flow simulated by the fine-favored model is 

88.0 ft3/s and 89.4 ft3/s for the coarse-favored model. The 

source of the base flow is partitioned among eight tributary 

systems (including the headwater portion of the Fox River) 

as well as minor direct tributaries to the Fox, the main trunk 

of the Fox, return flow from riparian wetlands, return flow 

from quarries, and added inflow from WWTPs. The tabulated 

quantities for both models indicate that about 36 percent of the 

total base flow to the Fox River is derived from the WWTPs. 

Keeping in mind that the steady-state simulations correspond 

to low-flow conditions, one can say that in periods where the 

supply of water to the Fox River surface-water network is 

dominated by groundwater flow, about one-third of the flow 

draining toward the Vernon Marsh is treated wastewater.
6. Model Results 99

Figure 38A. Simulated groundwater basins associated with surface-water features, 

quarries, and high-capacity wells—fine-favored model.

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

Figure 38a. Simulated groundwater basins associated with surface water features, quarries, and pumping wells -- 

fine-favored model.

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°

Genesee

Creek

Pebble Creek

Pebble Brook

Pewaukee 

River Pewaukee 

Lake

Poplar 

Creek

Vernon 

Marsh

Fox

River

Fox

River

Lannon 

Creek

Sussex

Creek

Fox River Basin upgradient 

of Vernon Marsh

Fox 

Headwaters

Waukesha

Stream

Lake

Contributing areas to pumping wells

Contributing areas to quarries

EXPLANATION

Fox River
100 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 38B. Simulated groundwater basins associated with surface-water features, 

quarries, and high-capacity wells—coarse-favored model.

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

Figure 38b. Simulated groundwater basins associated with surface water features, quarries, and pumping wells -- 

coarse-favored model.

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°10'

43°

Genesee

Creek

Pebble Creek

Pebble Brook

Pewaukee 

River Pewaukee 

Lake

Poplar 

Creek

Vernon 

Marsh

Fox

River

Fox

River

Lannon 

Creek

Sussex

Creek

Fox 

Headwaters

Waukesha

Fox River Basin upgradient 

of Vernon Marsh

Stream

Lake

Contributing areas to pumping wells

Contributing areas to quarries

EXPLANATION

Fox River
6. Model Results 101

One special component of the integrated surface network 

is Pewaukee Lake. The calculated groundwater contributing 

basin for the lake extends far beyond its boundaries (green 

area in fig. 38A and 38B). The models also simulate the stage 

of the lake, which partly controls the rate of surface-water 

outflow to the Pewaukee River. The surface-water outflow is 

fed by streams flowing into the lake, by groundwater discharge 

to the lake, and by precipitation on the lake. These inflows are 

lost to evaporation as well as to surface-water outflow. The 

tabulated fluxes indicate that all these terms are important in 

the lake budget (table 18). The surface-water outflow is about 

7.52 ft3/s for the fine-favored model (representing 47 percent 

of the simulated flow through the lake) and about 7.9 ft3/s for 

the coarse-favored model (representing about 48 percent of the 

simulated flow through the lake). This outflow component is 

almost one-half of the increase in base flow estimated between 

the Watertown and Waukesha gages (about 16 ft3/s). The 

remainder is derived from base flow to the Pewaukee River 

and directly to the Fox River.

The lakebed hydraulic conductivity was fixed for all 

simulations at a value of 0.01 ft/d. Varying its value has very 

little effect on the simulated lake stage or lake budge. For 

example, in the case of the fine-favored model, varying its 

value between 0.001 ft/day to 1 ft/day caused simulated lake 

stage to change from its calibrated stage of 852.820 ft by no 

more than 0.004 ft and simulated surface-water outflow to 

change from its calibrated rate of 7.52 ft3/s by no more than 

0.13 ft3/s (table 19).

Table 18. Simulated stages and water budgets for Pewaukee 

Lake.

Simulated results Fine-favored 

model

Coarse-favored 

model

Stage (feet) 852.820 852.833

Inflow (cubic feet per second)

Precipitation 9.33 9.33

Groundwater inflow 2.53 2.60

Surface-water inflow 4.28 4.54

TOTAL 16.13 16.47

Outflow (cubic feet per second)

Evaporation 8.45 8.45

Groundwater outflow .16 .08

Surface-water outflow 7.52 7.94

TOTAL 16.13 16.47

Table 19. Sensitivity of simulated stage and water budget for Pewaukee Lake to lakebed hydraulic 

conductivity: fine-favored model.

[Kbed, hydraulic conductivity of lakebed; ft/d, foot per day; ft, foot; ft3/s, cubic foot per second; results for calibrated model 

repeated from table 18 to facilitate comparison of sensitivity results]

Simulated results Calibrated model 

(Kbed = 0.01 ft/d)

Sensitivity model 1 

(Kbed = 0.001 ft/d 

reduced by 0.1x)

Sensitivity model 2 

(Kbed = 0.1 ft/d 

increased by 10x)

Sensitivity model 3 

(Kbed = 1.0 ft/d 

increased by 100x)

Stage (ft) 852.820 852.819 852.818 852.816

Inflow (ft3

/s)

Precipitation 9.33 9.33 9.33 9.33

Groundwater inflow 2.53 2.20 2.69 2.75

Surface-water inflow 4.28 4.45 4.20 4.17

TOTAL 16.13 15.98 16.21 16.25

Outflow (ft3

/s)

Evaporation 8.45 8.45 8.45 8.45

Groundwater outflow .16 .06 .27 .41

Surface-water outflow 7.52 7.47 7.49 7.39

TOTAL 16.13 15.98 16.21 16.25
102 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

7. Model Limitations

Model limitations arise from aspects of the conceptual 

model as well as simplifications inherent in the construction 

and calibration of the numerical model. This section describes 

these limitations and related uncertainties.

7.1 Conceptual Model

A major conceptual issue involves the use of water-well 

driller logs (along with geologic logs) to describe the hydrogeology. Water-well logs are of mixed quality because of 

variability in the quality of the data collected by drillers. Some 

investigators prefer to rely on a small sample of high-quality 

logs only,such as those prepared by geologists (Jansen, 2009), 

at the risk of introducing a high degree of homogeneity into 

the geologic interpretation. The large number of available 

water-well logs partly offsets the mixed quality (Arihood, 

2009), but given their uneven spatial distribution, it is inevitable that in some areas few logs are used to interpolate subsurface properties over relatively large distances. In this application, the distribution of logs is comparatively sparse in some 

focus areas (for example, in the valley of the Fox River where 

the density of housing is less than elsewhere) (fig. 7). 

In order to partly overcome this limitation, two interpolation schemes for mapping hydrogeologic facies in the 

unconsolidated deposits have been employed (1) connectivity among fine-grained facies is favored and (2) connectivity 

among coarse-grained facies is favored (figs. 15 and 16). This 

approach was adopted to allow for a range in the degree to 

which preferential flow is expressed in the models. Preferential flow, too often neglected in modeling studies or treated 

by scaling up K parameters to meet calibration targets, is an 

inevitable feature of highly heterogeneous material such as the 

glacial deposits characteristic of the Upper Fox River Basin. 

The detailed cell-by-cell representation of the unconsolidated 

hydrogeology in this study (along with the spatially refined 

model mesh in the nearfield) allows direct simulation of 

preferential flow paths along laterally and vertically oriented 

horizons to natural discharge zones such as streams as well as 

man-made discharge points such as pumping wells. However, 

it is conceded that the fine-favored nor the coarse-favored 

model incorporates a true representation of the hydrogeology 

and, that at any specific location, especially in areas of sparse 

data, the distribution of preferential flow pathways could be 

exaggerated or suppressed. The use of the Upper Fox models 

for trial simulations to characterize the likely average response 

of the aquifers to hypothetical stresses is probably reliable, but 

without ways to precisely map the subsurface including zones 

of preferential flow—through site-specific geologic studies 

and/or other lines of evidence (for example, a dense array of 

water-level measurements or chemical tracers)—the results of 

any application for site-specific or design purposes should be 

treated with caution.

Several other limitations are associated with the conceptual model:

• The lumping of the log descriptions into three texture 

classes (all fines, mixed, and all coarse) and the lumping of the layer intervals into five facies (dominantly 

fine, relatively fine, mixed fine and coarse, relatively 

coarse, and dominantly coarse) are simplifications that 

might blur differences in hydraulic conductivity properties. For example, assigning gravelly clay and silty 

sand to the same mixed texture class could conflate 

relatively impermeable with permeable sediments;

• Individual fracture zones in the Silurian dolomite 

underlying the unconsolidated material can exert a 

major control on local conditions of shallow groundwater flow in southeastern Wisconsin (Jansen, 1995), 

but there is no attempt to characterize them. The Upper 

Fox models included a weathered horizon at the top 

of the dolomite (layer 6), but discrete preferential flow 

zones associated with fracture traces are absent from 

the models because they are difficult to map without 

site-specific investigations.

• Stresses on groundwater from domestic pumping are 

neglected. Although the number of domestic wells is 

large compared to those in other water-use categories, 

households generally use relatively small amounts 

of groundwater and pump it from shallow aquifers; 

the drawdown cone around each well is commonly 

buffered by nearby surface water unless recharge is 

very small (Bradbury and Rayne, 2009). In addition, 

most of the pumped water is returned to groundwater through onsite septic systems. In Wisconsin, it is 

estimated that domestic wells account for 23 percent 

of total groundwater withdrawals (Lawrence and 

Ellefson, 1982) and about 25 percent of withdrawals in 

Waukesha County (Southeastern Wisconsin Regional 

Planning Commission, 2010), but estimates of return 

rate have been as high as 80 to 90 percent (Topper, 

2007; Cherkauer, 2007). These estimates of withdrawal 

and return suggest omission of domestic wells from 

the database underestimates total withdrawals by about 

5 percent. 

• The base simulations of the Upper Fox River Basin 

models assume steady-state conditions incorporating 

recharge estimates and edge boundary fluxes corresponding to average conditions for the basin over 

the last several decades, pumping rates corresponding 

to 2005, and estimates of WWTP effluent and lake 

discharge based on data collected after 2005. It is difficult and perhaps not feasible to construct the models 

based on fluxes corresponding to a single year. Some 

error arises from the time mismatch in fluxes, although 

the fact that all the fluxes correspond to recent years 

probably means the error is small. A related error is 

introduced by using a steady-state solution to simulate 
7. Model Limitations 103

basin behavior, thereby neglecting the role played by 

changes in storage. In fact, the modeled groundwater 

system is always responding to changes in stresses 

inside and outside the model domain, including not 

only boundary fluxes and pumping, but also variations in recharge. One way to assess the importance 

of transient effects is to examine the relative importance of storage release simulated by regional models, 

which include the Upper Fox River Basin area for the 

timeframe around 2005. The SEWRPC model was 

used to simulate transient conditions involving changes 

in pumping (but not recharge) from 2005 to 2010. 

For the area corresponding to the Upper Fox River 

Basin models, simulated storage release to groundwater represented 1.76 percent of total sources, and 

simulated storage gain from groundwater represented 

0.05 percent of total sinks. The Lake Michigan Basin 

model also was used to simulate transient conditions 

responding to changes in pumping and recharge for 

the Upper Fox River domain between 2000 and 2005, 

the last stress period considered in that regional study 

(Feinstein and others, 2010). For the area of concern, 

storage fluxes represented 1.42 percent of sources and 

0.13 percent of sinks. These results suggest that the 

neglect of transient effects is a minor source of error.

• Application of the Upper Fox River Basin models to 

scenario simulations, which change stresses such as 

pumping or recharge, could invalidate the specified 

flux boundary conditions at the domain edges. These 

conditions correspond to 2005 conditions (as reproduced by the SEWRPC regional model). One check 

on the severity of the stress on the boundary for a 

scenario simulation is to consider the drawdown and 

change in flux at the lateral boundaries of the domain. 

If the simulated response at the boundaries is very 

small, then the use of fixed lateral boundary fluxes 

does not compromise the forecast (for the analysis of 

an example scenario, see the “Model Application to 

Hypothetical Well Field” section). Vertical exchange 

across the bottom of the Upper Fox domain also are 

fixed in the models, partly reflecting the magnitude 

of Cambrian-Ordovician aquifer pumping in 2005. 

Forecasting scenarios that change the balance between 

deep and shallow pumping could imply a change in the 

vertical exchange at the boundary between the shallow and deep parts of the flow system. Care should be 

taken to evaluate the effect of any boundary condition 

change on the integrity of the simulations. Forecasting 

simulations could benefit from updating the boundary 

conditions by adding the new stresses to the regional 

SEWRPC model and re-extracting the implied boundary fluxes.

7.2 Model Construction

Many of the limitations associated with model construction involve the representation of the surface-water network. 

Although the fine model mesh enhances the numerical reliability of the simulation, little information is available regarding 

the streambed properties and how they change from point to 

point. The use of only three zones to characterize the streambed conductance (as a function of three zones of streambed 

hydraulic conductivity) is a major simplification and the 

resulting uncertainty could have implications (for example, 

the reliability of simulating stream losses and gains over local 

stretches). Streambed conductance can even be considered a 

transient property, subject to flooding events, which seasonally 

change the texture and thickness of the bed material in different stretches. However, model results generally are only sensitive to the streambed conductance when it is low, reflecting the 

accumulation of fine-grained material. More attention is given 

to this issue in the “Model Application to Hypothetical Well 

Field” section of the report.

Stream stage has been calculated only through the SFR 

package for the headwater reaches accounting for about onehalf the total length of channel. Fixing the stages in reaches of 

stream order two or higher on the basis of interpolation from 

GIS coverages is not an exact process, but it does have the 

advantage of stabilizing the model solution. Another uncertain input is the outlet elevation of the spillway for Pewaukee 

Lake. The chosen elevation reflects the reported average elevation of the weir at the outlet, but it is recognized that the LAK 

package results are very sensitive to this choice because the 

spillway elevation controls the simulated stage and outflow. 

The simulated outflow from the lake (averaging 7.7 ft3

/s for 

the two models) probably carries an uncertainty, based on the 

constructed rating curve and uncertainty about the spillway 

elevation of plus or minus 2 ft3/s.

The presence of simulated groundwater flooding (fig. 37) 

is partly an artifact of the absence of riparian discharge zones 

in the model. The Upper Fox River Basin models could be 

made more complex by using the MODFLOW-2005 UZF 

package (Niswonger and others, 2005). This package, which 

simulates flow between the land surface (or bottom of the root 

zone) and the water table, can be used to quantify flow conditions near streams and lakes, which results in overland flow 

of groundwater discharge to surface water (Hunt and others 

2007b). It is likely that eventual implementation of the method 

would yield further insight into groundwater/surface-water 

interactions along the Upper Fox River and its tributaries.
104 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

A second set of issues arises from simplifications inherent 

in the discretization of the model. For example, the representation of the land surface, which is used as a datum for defining the layer bottoms, is discretized on 500-ft centers so that 

clusters of 16 cells in the model nearfield, where spacing is 

125 ft on a side, share common top and bottom elevations. A 

related issue stems from the nonhorizontal slope of the layer 

bottoms. The discretization of the unconsolidated material 

into layers on the basis of the undulations of the land-surface 

trend does influence the solution by encouraging the simulated 

flow to follow the land-surface trend along the surface of least 

resistance represented by the horizontal hydraulic conductivity. However, the average change in elevation in the model 

domain is on the order of 200 ft to over 40,000 ft or only about 

0.3 degrees (even though very evident in cross sections with 

vertical exaggeration). Bottom slopes less than 15 degrees are 

not expected to distort the MODFLOW solution (Henry and 

others, 1998; Mary Hill, U.S. Geological Survey, written commun., May 25, 2011).

A third type of limitation arises from the time period 

associated with estimates of model inputs that determine 

sources and sinks. Whereas the inflow from recharge reflects 

long-term average conditions, the model pumping rates correspond to 2005, the model edge boundary conditions reflect 

both long-term recharge and 2005 pumping, the available lake 

spillway data are for the 2007–9 period and the WWTP effluent was estimated using 2008–9 data. It is clear that the fluxes 

associated with the model are not tied tightly to 2005 but are 

better characterized as approximations of recent conditions. 

7.3 Model Calibration

The calibration targets all contain error, which contributes 

to uncertainty in parameter estimation. These sources of error 

are discussed for each target type in the “Model Calibration” 

section of the report under ”Calibration Targets”. It is difficult 

to quantify the relative uncertainty among target types, but 

the availability of multiple sets is important for expanding the 

number of parameters that can be informed by the calibration 

process.

The distribution of pumping from wells penetrating 

multiple layers, assigned as a function of layer transmissivity based on initial Kh

 values, was not updated to reflect the 

calibrated Kh values. However, sensitivity simulations for the 

fine-favored and coarse-favored models part way through 

the calibration process indicate that the models results are 

almost completely insensitive to the update in the distribution of pumpage between layers. Of more serious concern, 

the calibrated parameter values do not support the total target 

withdrawal rates for the high-capacity wells inserted in the 

models. The loss (4 to 8 percent of withdrawals, amounting to 

0.3 to 0.6 percent of simulated total outflows) suggests some 

calibration error, especially because the models do not include 

the additional loss because of drawdown induced by well 

inefficiencies. However, there is no guarantee that the reported 

discharge rates are fully accurate (it is possible pumping 

capacity is reflected rather than actual pumping) and, therefore, no effort has been made to reinterpret calibrated values to 

support the total reported rates.

An important limitation of the calibration process is the 

use of bounds to enforce the ordering of K values from the 

finest to the coarsest unconsolidated facies. This technique 

balances the power of the PEST algorithms to find a best fit 

against the geological judgment embedded in the underlying 

conceptual model. However, the presence of some calibrated 

values at the upper and lower bounds for the fine-favored and 

coarse-favored models does confirm that the mapped facies in 

the Upper Fox River Basin only approximate the subsurface 

reality.

8. Model Application to Hypothetical 

Well Field

This section describes a demonstration application of the 

Upper Fox River Basin models. The models serve as a tool 

for exploring the possibility of augmenting municipal water 

supply by means of riverbank inducement from hypothetical 

shallow wells as a way of minimizing drawdown coupled with 

recirculation of the water to upstream WWTPs as a way of 

enhancing sustainability. 

8.1 Riverbank Inducement

Over two-thirds of the residents of Wisconsin use groundwater as their source of supply; the proportion is 86 percent in 

Waukesha County (Southeastern Wisconsin Regional Planning Commission, 2010, table 29). As populations grow, the 

expanding demand for water will cause increased competition 

among users and may exceed the groundwater system’s supply. Riverbank inducement (RBI) is a process used to augment 

groundwater supplies in many nearby States and in numerous 

midwestern urban areas. 

Wells are commonly installed adjacent to rivers (or 

other surface-water bodies) in order to draw water from two 

sources: (1) intercepting groundwater that previously would 

have flowed into the river, and (2) inducing water to flow from 

the river into the aquifer, which is induced infiltration, and 

RBI is accomplished by reversing the natural hydraulic gradient by pumping groundwater levels below river level. The 

induced water augments the original groundwater supply and 

proportionally reduces the amount of drawdown caused by the 

wells in the pumped aquifer. The reduced drawdown mitigates 

the effect of pumping on (1) reducing base flow to nearby 

wetlands and other local surface-water features, and (2) interference with nearby household and high-capacity wells.
8. Model Application to Hypothetical Well Field 105

Flow in the source river also is reduced by the amount of 

inducement to a well field, and the magnitude of this impact 

will depend upon the river’s total flow and where the treated 

effluent is returned. If the wastewater return is downstream 

from the inducing well field, flows in the river will be reduced 

between the well field and the return point by the sum of the 

induced and intercepted water. Alternatively, if the well field’s 

water is transferred to another watershed, the river from which 

it is induced will have flows reduced below the well field by 

the sum of the induced and intercepted water. On the other 

hand, there will be little impact on low flows and the volume 

of induced water is actually recirculated past the well field 

if the well field discharge is returned to an upstream WWTP. 

When the sources of water to the pumping wells include riverbank inducement, then the recirculation of the pumped water 

from the river to the aquifer to the WWTP and back to the 

river enhances the sustainability of the system.

8.2 Hypothetical Pumping Scenario Along 

Fox River

For the demonstration application, areas north and south 

of the city of Waukesha along the Fox River are selected to 

test how a riverbank-inducement strategy could be implemented. It is important to emphasize that this scenario is not 

supported by site-specific evaluations of well capacity, but 

the hydrogeologic conditions reflect the interpolation methods from compiled logs described in previous sections. For 

this reason, the application only offers a general idea of the 

response of the system to riparian wells and is not the basis for 

an actual well-field design. However, the results of the simulations should be useful for judging the prospects for riparian 

systems generally in glaciated areas in the Upper Midwest 

with subsurface conditions similar to the Upper Fox River 

Basin.

The hypothetical riparian scenario consists of 27 pumping wells distributed along the Fox River over the north/south 

extent of the model nearfield between the confluence of Sussex Creek and Pebble Brook (fig. 39). The design includes two 

public wells already installed by the city of Waukesha: WK-11 

and WK-12. All 27 wells are located in the model at a lateral 

distance 125 ft from the Fox River in the riparian zone. The 

vertical interval for withdrawal is determined by the following 

logic:

• if 60 ft of unconsolidated material is present, pumping 

is assigned to the lowest unconsolidated layer with at 

least 10 ft thickness; and

• if 60 ft of unconsolidated material is not present, pumping is distributed among the combined unconsolidated 

and Silurian dolomite material in layers 2 through 7.

This filter results in 9 wells extracting from layer 3 (50.1- to as 

much as 100-ft depth), 8 wells extracting from layer 4 (100.1- 

to as much as 150-ft depth), and 10 wells extracting from 

layers 2 to 7 (20.1-ft to variable depth). The unconsolidated 

sequence is thinner north of Pebble Creek than to the south; 

consequently most of the 15 wells north of Pebble Creek 

extend to the dolomite in layers 6 and 7, whereas, the 12 wells 

south of the creek are limited to the unconsolidated layers 3 or 

4.

The target rate for each well is 0.667 Mgal/d, amounting 

to a total of 18 Mgal/d for the system. Based on the distribution of calculated transmissivity using the calibrated K 

parameters and layer thickness, about two-thirds of the target 

pumping is from the unconsolidated material and one-third is 

from the dolomite bedrock. However, it is an important feature 

of the MODFLOW-NWT formulation that the simulated

discharge rate from each well is a function of how much the 

aquifers in the vicinity of the well can support. Consequently, 

one of the important outputs of the demonstration exercise is 

the sustainable pumping achieved relative to the 18 Mgal/d 

target. 

The background pumping from high-capacity wells is 

maintained at 6.67 Mgal/d. The flux boundary conditions of 

the model also are not changed in the simulations. If withdrawals from riverbank-inducement wells were intended to 

substitute for pumping from deeper wells, a more rigorous 

approach would be to regenerate flux conditions from the 

regional SEWRPC model with the new pumping distributions. However, simulations conducted as part of the 2010 

Southeastern Wisconsin Regional Planning Commission study 

suggest that the effect of shifting 18 Mgal/d or less of deep 

pumping to shallow pumping would have only a minor effect 

on the magnitude of the vertical leakage between the shallow 

and deep parts of the flow system. 

There are three WWTPs in the area of interest. In 

principle, 5 of the wells could recirculate their water to the 

upstream Sussex plant, 9 to the upstream Brookfield plant, and 

13 to the upstream Waukesha plant. This element of the hypothetical design was not explicitly simulated (by increasing the 

inflow to the SFR nodes adjacent to the WWTPs) because, 

given that the stages of the Fox River are fixed in the model, it 

would have little or no influence on the solution. However, the 

recirculation of water could be an important element of a real 

design of a RBI system that would serve to sustain base flow 

in the river, notably during low-flow periods.

The fine-favored and coarse-favored models were run 

in steady-state mode to determine the long-term response of 

the groundwater to the riparian system in terms of drawdown, 

sustainable pumping, and sources of water to riparian wells, 

including water induced from the Fox River.
106 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 39. Hypothetical riparian well scenario. (There are 15 riparian wells (orange) located in the area upstream of the 

confluence between the Fox River and Pebble Creek. There are 12 riparian wells (green) located in the area downstream 

of the confluence between the Fox River and Pebble Creek. The riparian zone (purple) was used for mass balance 

analysis—see tables 19 and 20.) 

Sussex WWTP

WK-12

WK-11

Waukesha WWTP

Brookfield WWTP

Figure 39. Hypothetical riparian well scenario. (There are 15 riparian wells (orange) located in the area upstream of 

the confluence between the Fox River and Pebble Creek. There are 12 riparian wells (green) located in the area 

downstream of the confluence between the Fox River and Pebble Creek. The riparian zone (purple) was used for 

mass balance analysis—see tables 19 and 20.)

0 2 4 MILES

0 2 4 KILOMETERS

88°20' 88°10'

43°5'

43°

42°55'

EXPLANATION

Downstream riparian well

Nearfield

Surface-water feature

Riparian zone

Surface-water feature

Fox River

Wastewater-treatment plant (WWTP)

Upstream riparian well

Base modified from Wisconsin Department of 

Natural Resources digital data, 2010, 1:24,000, 

Wisconsin Transverse Mercator, North 

American Datum of 1983

Fox

 River

Fox 

River

Pewaukee

Lake

Pebble Creek

Pebble Cr

Pebble Brook

Sussex Creek
8. Model Application to Hypothetical Well Field 107

The simulated drawdown response at the water table in 

layer 1 (fig. 40) and at a depth of 50–100 ft in layer 3 (fig. 41) 

is similar for the two models. Little drawdown occurs outside 

the Upper Fox River Basin and the drawdown at the edges 

of the domain is localized and small. Near the Fox River, the 

maximum drawdown is on the order of 20 ft in layer 1 (fig. 40) 

and 30 ft in layer 3 (fig. 41). At the water table, drawdown 

occurs away from the river (because the water level in the 

stream tends to fix the level of groundwater adjacent to the 

stream) in areas determined by the relative capacity of the 

aquifer to transmit water. Note that the simulated drawdown 

when RBI is ineffective is much larger (see discussion under 

“Effect of Riverbank Inducement on Drawdown”).

The coarse-favored model, as expected, supports more 

pumping from the riparian system than the fine-favored 

model. The total pumping sustained from the 27 wells in the 

fine-favored model is 9.13 Mgal/d (table 20), whereas it is 

9.65 Mgal/d in the coarse-favored model (table 21). It is noteworthy that these values are only slightly above one-half the 

total target rate of 18 Mgal/d. The simulated sources of water 

to the wells also vary somewhat between the two models. In 

the fine-favored model, 30.8 percent of the sustained pumping is induced from the Fox; whereas, 34 percent is derived 

from groundwater that would have discharged as base flow to 

the Fox in the absence of the riparian system. In the coarsefavored simulation, the water induced from the Fox by the 

riparian system jumps to 41.3 percent; whereas, 31.7 percent 

is derived from diverted base flow. This comparatively large 

difference in the induced component results from the presence of more preferential flow pathways in the coarse-favored 

model than the fine-favored model, which allow more water 

in the coarse-favored case to be conveyed from the Fox River 

laterally and vertically down to the open intervals of the hypothetical wells. The two models taken together, by exploring the 

uncertainty in our representation of the subsurface, possibly 

bracket the overall capacity of a real-world riparian system to 

induce river water to the shallow wells.

Other simulated sources of water for the fine-favored 

and coarse-favored models include water induced from other 

streams within the Upper Fox River Basin (5.8 and 2.8 percent 

in the two models, respectively), base flow diverted from 

other Upper Fox River Basin streams (15.5 and 14.2 percent), 

water induced and diverted from wetlands, lakes and quarries 

in the basin (10.7 and 7.2 percent), and lateral inflow across 

the boundaries of the Upper Fox River Basin (3.6 and 2.5 percent). The mass balance error (calculated by comparing the 

sustained pumping to the combined sources of water to the 

wells) is small (0.2 and 0.3 percent, respectively).

A common element of the two simulations is the 15 hypothetical wells in the northern part of the riverbank-inducement 

system are less successful in inducing water from the Fox 

River than the 12 wells in the southern part of the system. For 

the fine-favored simulation, the induced flux represents only 

15 percent of the source water for northern wells, whereas, it 

constitutes 38.6 percent of the source water for southern wells 

(table 20). The corresponding induced fluxes simulated by 

the coarse-favored model are higher (27.7 and 46.5 percent) 

but also indicate relatively more favorable conditions for 

RBI in the southern part of the system (table 21). The most 

important reason for this geographic difference, common to 

both interpretations, is the presence of more sandy facies in 

the riparian zone to the south accompanied by higher overall 

transmissivity.

No effort has been made in the hypothetical exercise 

presented in this report to optimize pumping by extracting 

more than 0.667 Mgal/d from RBI wells than could support 

the increase. Alternative simulations with the two models suggest that some of the wells in the southern area could support 

higher pumping rates, and at those higher rates, on the order of 

50 percent of the source water could be induced from the Fox 

River.
108 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 40A. Simulated drawdown from riparian pumping in model layer 1 (water table)—fine-favored model. 

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

Figure 40a. Simulated drawdown from riparian pumping in model layer 1 (water table) -- fine-favored model.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Fox River

Headwaters 

Nearfield

Fox River

Surface-water

feature

Riparian well

EXPLANATION

Drawdown, in feet

0.05 – 0.50

0.51 – 1.00

1.01 – 2.00

2.01 – 3.00

3.01 – 4.00

4.01 – 5.00

5.01 – 7.00

7.01 – 10.00

10.01 – 15.00

15.01 – 30.00
8. Model Application to Hypothetical Well Field 109

Figure 40B. Simulated drawdown from riparian pumping in model layer 1 (water table)—coarse-favored model. 

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Fox River

Headwaters 

Nearfield

Fox River

Surface-water

feature

Riparian well

EXPLANATION

Drawdown, in feet

Figure 40b. Simulated drawdown from riparian pumping in model layer 1 (water table) -- coarse-favored model.

0.05 – 0.50

0.51 – 1.00

1.01 – 2.00

2.01 – 3.00

3.01 – 4.00

4.01 – 5.00

5.01 – 7.00

7.01 – 10.00

10.01 – 15.00

15.01 – 30.00
110 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 41A. Simulated drawdown from riparian pumping in model layer 3 (50 to100 foot depth)—fine-favored model. 

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

Figure 41a. Simulated drawdown from riparian pumping in model layer 3 (50 to100 feet depth) -- fine-favored model.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Nearfield

Fox River

Surface-water

feature

Riparian well

EXPLANATION

Drawdown, in feet

0.05 – 0.50

0.51 – 1.00

1.01 – 2.00

2.01 – 3.00

3.01 – 4.00

4.01 – 5.00

5.01 – 7.00

7.01 – 10.00

10.01 – 15.00

15.01 – 30.00
8. Model Application to Hypothetical Well Field 111

Figure 41B. Simulated drawdown from riparian pumping in model layer 3 (50 to100 foot depth)—coarse-favored model. 

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

Figure 41b. Simulated drawdown from riparian pumping in model layer 3 (50 to100 feet depth) -- coarse-favored model.

88°20' 88°10'

43°10'

43°

0 2 4 MILES

0 2 4 KILOMETERS

Nearfield

Fox River

Surface-water

feature

Riparian well

EXPLANATION

Drawdown, in feet

0.05 – 0.50

0.51 – 1.00

1.01 – 2.00

2.01 – 3.00

3.01 – 4.00

4.01 – 5.00

5.01 – 7.00

7.01 – 10.00

10.01 – 15.00

15.01 – 30.00

30.01 – 45.00
112 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Table 20. Sources of water to hypothetical riparian wells along the Fox River for fine-favored model.

[Mgal/d, million gallons per day]

Response of all wells

Number riparian wells simulated 27

Riparian pumping input (Mgal/d) 18.00

Riparian pumping sustained (Mgal/d) 9.13

Mgal/d Sustained pumping 

(percent)

Source of water to wells:

Induced flow from Fox River 2.81 30.8

Diverted flow from Fox River 3.10 34.0

Induced flow from other streams .53 5.8

Diverted flow from other streams 1.41 15.5

Net induced and diverted flow from wetlands, lakes, quarries .98 10.7

Lateral flow across Upper Fox River Basin boundary .33 3.6

TOTAL 9.15 100.2

Response of upstream wells

Upstream area is north of 

Pebble Creek confluence 

with Fox River

Number upstream riparian wells 15

Upstream riparian pumping input (Mgal/d) 10.00

Upstream riparian pumping sustained (Mgal/d) 3.04

Induced streamflow to riparian wells (Mgal/d) .46

Induced from Fox River (percent) 15.0

Response of downstream wells

Downstream area is south 

of Pebble Creek confluence 

with Fox River

Number downstream riparian wells 12

Downstream riparian pumping input (Mgal/d) 8.00

Downstream riparian pumping sustained (Mgal/d) 6.09

Induced streamflow to riparian wells (Mgal/d) 2.35

Induced from Fox River (percent) 38.6
8. Model Application to Hypothetical Well Field 113

Table 21. Sources of water to hypothetical riparian wells along the Fox River for coarse-favored model.

[Mgal/d, million gallons per day]

Response of all wells

Number riparian wells simulated 27

Riparian pumping input (Mgal/d) 18.00

Riparian pumping sustained (Mgal/d) 9.65

Mgal/d Sustained pumping 

(percent)

Source of water to wells:

Induced flow from Fox River 3.99 41.3

Diverted flow from Fox River 3.06 31.7

Induced flow from other streams .27 2.8

Diverted flow from other streams 1.37 14.2

Net induced and diverted flow from wetlands, lakes, quarries .70 7.2

Lateral flow across Upper Fox River Basin boundary .24 2.5

TOTAL 9.62 99.7

Response of upstream wells

Upstream area is north of 

Pebble Creek confluence 

with Fox River

Number upstream riparian wells 15

Upstream riparian pumping input (Mgal/d) 10.00

Upstream riparian pumping sustained (Mgal/d) 2.68

Induced streamflow to riparian wells (Mgal/d) .74

Induced from Fox River (percent) 27.7

Response of downstream wells

Downstream area is south 

of Pebble Creek confluence 

with Fox River

Number downstream riparian wells 12

Downstream riparian pumping input (Mgal/d) 8.00

Downstream riparian pumping sustained (Mgal/d) 6.98

Induced streamflow to riparian wells (Mgal/d) 3.24

Induced from Fox River (percent) 46.5
114 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

8.3 Sensitivity of Results to Streambed Hydraulic 

Conductivity Values

In this study, the calibration process did not generate 

reliable estimates for the zonal values of streambed hydraulic 

conductivity because of insensitivity to the available target 

types. In general, it is difficult to estimate this parameter (or 

the related parameter, streambed conductance) from model 

calibration because most of the head loss in the groundwater 

system occurs within the aquifer itself; the streambed typically 

does not offer enough resistance to strongly influence the overall flow system. However, the response to some stresses in the 

vicinity of surface water (for example, the drawdown response 

to riparian pumping) are potentially more likely to be affected 

by the nature of the streambed. In the absence of extensive 

field data, one way to test hypotheses about its influence is 

to conduct a sensitivity analysis by varying its properties and 

recording the effect on drawdown around riparian wells.

The conductance (inverse to the resistance) of the streambed is a function of its area inside the model cell (corresponding in this study to the mapped channel length multiplied 

by the width as a function of upstream distance), thickness 

(assumed to be 1 ft everywhere), and hydraulic conductivity 

(SFR cells grouped into three zones). To simplify the analysis, 

only the hydraulic conductivity zonal values are varied (so 

that streambed conductance is a linear function of hydraulic conductivity values), and only the fine-favored model 

is tested. The base zonal values generated by the calibration process are reduced by 0.001, 0.01, and 0.1 times, and 

increased by 3 times (table 22). For example, the calibrated 

value of 9.56 ft/d (characteristic of a fine to medium sand) for 

the middle streambed Kv zone is changed in the sensitivity 

simulations to 0.0095, 0.0956, 0.956 and 28.68 ft/d. The effect 

of the changes on simulated base flow to the Waukesha gage, 

groundwater flooding, and calibration statistics for nearfield 

water levels are calculated with all other inputs equal to the 

base 2005 model, including the high-capacity pumping rate. 

The effect on the amount of water induced from the Fox River 

is calculated with the added withdrawals from the 27-well 

riparian scenario. The simulations with streambed K set at 3 

times and 0.1 times the base streambed Kv

 values show little 

difference from the results for the base calibrated fine-favored 

model. The simulation with streambed Kv set to 0.01 times 

the base value shows sensitivity in terms of a small decrease 

in base flow, more than doubling of the number of flooded 

cells, and some deterioration of the calibration statistics; for 

the scenario simulation, riverbank inducement as a source 

of water to riparian wells decreases from 31 to 25 percent 

(table 22). The effects are considerably more severe when 

the streambed K is set to 0.001 times the base value. For 

example, rate of riverbank inducement decreases to 6 percent 

of well sources, less than one-fifth its base scenario value. It is 

noteworthy that for the most sensitive simulation, the Kv value 

assigned the middle streambed zone is almost exactly equal to 

the Kv

 assigned the fine-dominated facies for the fine-favored 

model (0.0095 ft/d). The Fox River channel is assigned to the 

middle Kv zone (fig. 20). Therefore, this sensitivity simulation 

effectively treats the Fox River bed as fine grained everywhere 

(perhaps the texture of a clayey silt). If that were the makeup 

of the bed for the entire channel, then the analysis implies that 

RBI would be largely ineffective.

Observation and geophysical evidence indicates that 

at least in the vicinity of water-supply wells WK-11 and 

WK-12 (fig. 28), the streambed is a mixture of fine-grained 

and coarse-grained sediment, with sandy and gravelly material dominant in some stretches (Baierlipp and Kean, 2011). 

This direct evidence suggests it is unrealistic to argue that 

the For River streambed generally poses strong resistance to 

groundwater/surface-water exchange. However, it does open 

the possibility that the exchange is uneven and associated with 

pathways of preferential flow.

8.4 Effect of Riverbank Inducement on 

Drawdown 

A riverbank-inducement system produces less drawdown than an extraction system isolated from surface water 

where riverbank inducement does not occur. To quantify 

the difference, it is convenient to compare the drawdown 

for the hypothetical 27-well riparian scenario with the base 

calibrated parameters to the drawdown for the same scenario 

with the streambed hydraulic conductivity reduced by 0.001 

times. Although the latter simulation is out of calibration and 

although a small amount of water is still induced from the Fox 

River, it serves as an approximation of drawdown conditions 

that would be obtained if the system had no access to streamflow as a source of water to the hypothetical wells. Consider 

the sustained withdrawal and drawdown simulated for the 

downstream part of the RBI system (fig. 42). The withdrawal 

is close to 6 gal/min for the set of 12 wells for the two cases, 

but the drawdown pattern is considerably different between 

the two cases. In the base scenario simulation, the maximum 

drawdown is on the order of 25 ft, and the drawdown cone 

greater than 5 ft is roughly a mile from east to west. In the 

sensitivity scenario simulation, the maximum drawdown is on 

the order of 90 ft, and the drawdown cone expands to a width 

of more than 3 mi. In the base case, riverbank inducement 

contributes nearly 40 percent of the well water, but it is only 

about one-fifth of that for the sensitivity case. In the end, this 

comparison suggests that if site-specific conditions are favorable, riparian pumping is likely to minimize reduction of base 

flow to surface-water features in the valley and limit drawdown interference between public-supply wells and nearby 

household wells.
Table 22. Sensitivity of scenario results to streambed hydraulic conductivity.[Scenario results are compared to calibrated simulation for fine-favored model, shown in boldface; Kv, vertical hydraulic conductivity; ft/d, foot per day; ft3/s, cubic foot per 

million gallons per day; ---, not applicable; x, multiplied by]

Run Fox riverbed K

v (ft/d)

Change to 

riverbed K

v

Simulated 

base flow at 

Waukesha gage 

(target = 49.6 ft3/s)

Percent of 

nearfield 

with more than 

3 ft flooding

Calibration: 

mean error (ft) 

Water levels in 

model nearfield

Calibration: 

mean absolute 

error (ft) 

Water levels in 

model nearfield

Sustaine

riparian pum

input = 18 Mg

Calibrated-coarse 7.75 --- 46.48 6.1 3.04 12.48 9.65 Calibrated-fine 9.56 --- 45.38 7.6 .89 12.05 9.13 SEN00-fine .0095 reduced by 0.001 41.11 35.3 −15.16 21.23 8.73 SEN0-fine .0956 reduced by 0.01x 44.25 17.9 −4.26 13.83 9.04 SEN1-fine .956 reduced by 0.1x 45.17 9.8 −.25 12.31 9.11 SEN2-fine 28.68 increased by 3x 45.41 7.4 1.05 12.03 9.13 
116 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

Figure 42A. Effect of degree of riverbank inducement on drawdown in layer 1 (water table) for fine-favored model—

riparian simulation allowing riverbank inducement. 

88°20' 88°15'

43°

42°55'

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

5.0 – 15.0

15.1 – 25.0

25.1 – 35.0

35.1 – 45.0

45.1 – 55.0

55.1 – 65.0

65.1 – 75.0

75.1 – 85.0

85.1 – 95.0

Figure 42a. Effect of degree of riverbank inducement on drawdown in layer 1 (water table) for fine-favored model --

riparian simulation allowing riverbank inducement.

0 2 4 MILES

0 2 4 KILOMETERS

Fox River

Surface-water

feature

Riparian well

EXPLANATION

Drawdown, in feet
8. Model Application to Hypothetical Well Field 117

Figure 42B. Effect of degree of riverbank inducement on drawdown in layer 1 (water table) for fine-favored model—

sensitivity simulation limiting riverbank inducement. 

88°20' 88°15'

43°

42°55'

Base modified from Wisconsin Department of Natural 

Resources digital data, 2010, 1:24,000, Wisconsin 

Transverse Mercator, North American 

Datum of 1983

5.0 – 15.0

15.1 – 25.0

25.1 – 35.0

35.1 – 45.0

45.1 – 55.0

55.1 – 65.0

65.1 – 75.0

75.1 – 85.0

85.1 – 95.0

0 2 4 MILES

0 2 4 KILOMETERS

Fox River

Surface-water

feature

Riparian well

EXPLANATION

Drawdown, in feet

Figure 42b. Effect of degree of riverbank inducement on drawdown in layer 1 (water table) for fine-favored model --

sensitivity simulation limiting riverbank inducement.
118 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

9. Summary and Conclusions

This report discusses the construction, calibration and 

application of the Upper Fox River Basin groundwater-flow 

model. It also offers a demonstration of some advanced modeling techniques for simulating unconfined aquifer systems.

9.1 Model Construction

The Upper Fox River Basin model simulates groundwater flow in the area of the Upper Fox topographic basin 

north of the Vernon Marsh in Waukesha County with the 

U.S. Geological Survey (USGS) code MODFLOW-NWT. The 

model domain extends from the source of the Fox River in the 

southernmost area of Washington County to where the Fox 

River passes through the northern part of the Vernon Marsh in 

south-central Waukesha County. The nearfield of the model, 

characterized by more detailed input and a refined lateral grid 

spacing 125 feet (ft) on a side, extends north to south between 

the confluences of Sussex Creek and Pebble Brook with the 

Fox River and extends west to east from Pewaukee Lake to the 

tributaries to Poplar Creek near the eastern Waukesha County 

boundary. All the surface-water features in the nearfield of the 

model are within the Upper Fox River Basin, including (in 

addition to the features mentioned above) Pewaukee Creek, 

Pebble Brook, and Genesee Creek and their tributaries. The 

model also includes wetland areas and quarries in the Cities of 

Sussex and Waukesha.

The model, consisting of seven layers, extends vertically 

from land surface to the top of the Maquoketa shale confining 

unit which divides the shallow and deep aquifer systems in 

southeastern Wisconsin. The top five layers represent alluvial 

and glacial deposits. The bottom two layers represent the 

Silurian dolomite. The layering is organized by depth. Layer 1 

incorporates the top 20 ft of unconsolidated material, layer 2 

extends from 20 to 50 ft below land surface, layer 3 from 50 

to 100 ft below land surface, layer 4 from 100 to 150 ft below 

land surface, and layer 5 from 150 ft to the bottom of the 

unconsolidated material. The number of unconsolidated active 

layers depends on the thickness of the glacial sequence, which 

varies from 0 ft in isolated areas where the dolomite is at the 

land surface to more than 400 ft in the Troy Bedrock Valley in 

the southern part of the model domain. The weathered top of 

the dolomite is represented by layer 6 (set to 20 ft thick) and 

the remainder of the dolomite thickness is assigned to layer 7.

The boundary conditions at the edges of the model correspond to groundwater fluxes, which vary by boundary model 

cell and by layer. The fluxes into or out of the model domain 

are derived from the output of the Southeastern Wisconsin 

Regional Planning Commission (SEWRPC) regional flow 

model for southeastern Wisconsin developed by the USGS 

and the Wisconsin Geological and Natural History Survey. 

The model was updated to 2005 pumping conditions as part 

of SEWRPC’s Water Supply Plan; the results of this 2005 

simulation are used to set the flux boundary conditions. In 

particular, the flux condition at the bottom of the Upper Fox 

River Basin model, mostly downward, represents the vertical exchange between the shallow groundwater-flow system 

represented by the sequence of glacial and dolomite sediments 

(simulated by the Upper Fox River Basin models) and the 

deep groundwater-flow system incorporating the CambrianOrdovician aquifer (represented by the lower boundary condition of the Upper Fox River Basin model). 

The Upper Fox River Basin model uses advanced USGS 

modeling techniques to simulate groundwater circulation 

from the water table to streams, lakes, wetlands, and quarries. 

The model also is designed to simulate reduced base flow to 

surface-water features and induced flow from surface-water 

features to the groundwater in the presence of pumping. The 

surface water in the Upper Fox River Basin was routed into 

a single network within the model by means of the specialized LAK (Merritt and Konikow, 2000) and SFR2 packages 

of MODFLOW-2005. These packages allow the lake level 

and the stream stage (particularly headwater stages) to be 

simulated along with groundwater levels. The inputs to these 

packages includes LIDAR land-elevation data , characterization of stream channel and streambed features, average rates of 

effluent inflow reported by the Sussex, Brookfield, Waukesha 

wastewater-treatment plants (WWTPs), and the bathymetry 

of Pewaukee Lake as well as lake outlet data provided by the 

village of Pewaukee. In addition to the advanced surface-water 

packages, this project takes advantage of the recently released 

MODFLOW-NWT code, which provides numerical stability 

in the presence of dry cells and generates a smooth water-table 

solution. The Newton solver has been applied to allow for 

more accurate simulation of groundwater-surface-water interactions and the short- and long-term responses of the system 

to pumping.

The modeling approach focuses not only on incorporating 

the complete surface-water network, but also on reproducing, at the 125-ft by 125-ft scale of the grid, the heterogeneity 

of the unconsolidated deposits. Generally, these deposits fall 

into three bands within the model domain – to the east they 

are dominated by the clay-rich Oak Creek Formation of the 

Lake Michigan Lobe; to the west by the more sandy Horicon 

Member of the Green Bay Lobe; and in the center, over most 

of the model nearfield, by the mixed deposits of the New 

Berlin Member of the Lake Michigan Lobe. It is particularly 

challenging to represent the heterogeneity of the New Berlin 

Member because repeated episodes of deposition and erosion 

associated with multiple glacial advances and retreats, along 

with deposition and downcutting of modern streams, has 

produced a setting in which the sediment texture changes over 

very short lateral and vertical distances. A dataset of about 

7,000 well logs is the basis of the mapping of five unconsolidated zonal textures or “facies” by model layer—(1) dominantly fine, (2) relatively fine, (3) mixed fine and coarse, 

(4) relatively coarse, and (5) dominantly coarse, where fine 

deposits are silts and clays and coarse deposits are sands and 

gravels. The key point is that this modeling approach does not 

adopt a single interpretation of the unconsolidated facies but 
9. Summary and Conclusions 119

attempts to encompass the inevitable uncertainty by developing two facies models—one which favors the continuity of 

fine-grained deposits and a second which favors the continuity 

of coarse-grained deposits. Both the fine-favored and coarsefavored models with their distinct zonation of the unconsolidated deposits are carried forward in the calibration and 

application phases of the model development.

Other important model inputs are recharge and pumping. A single recharge value is associated with each of 27 

subbasins within the Upper Fox River Basin and surrounding watersheds based on studies conducted in southeastern 

Wisconsin. The values range from 1.5 to 9.5 inches/year. The 

total recharge applied to the domain of the Upper Fox River 

Basin model is equal to the total recharge applied to the same 

area in the parent SEWRPC regional model, implying that the 

flux edge boundary conditions translated from the regional 

model to the Upper Fox River Basin model are consistent 

with the applied recharge. The distribution of pumping from 

unconsolidated and dolomite deposits is equivalent to the 

input to the revised 2005 version of the SEWRPC regional 

model; the dataset amounts to 99 wells pumping 6.7 million 

gallons/day (1.6 Mgal/d from the glacial aquifer, 5.1 Mgal/d 

from the dolomite aquifer). Domestic pumping, most of which 

is returned to the shallow aquifer, is neglected in this model. In 

the application phase, additional wells are inserted to represent 

withdrawals added after 2005 or to simulate a hypothetical 

pumping system.

9.2 Model Calibration

Calibration of the fine-favored and coarse-favored versions of the model was performed by means of nonlinear 

regression using the program PEST. A variety of target types 

allowed the calibration process to be sensitive to different 

parameter sets and to enforce controls on vertical as well 

as lateral flow. More than a thousand targets sampled from 

contoured water levels mapped by the Wisconsin Geological 

and Natural History Survey provided information on the horizontal and vertical hydraulic conductivity assigned to the five 

unconsolidated facies as well as on the underlying dolomite; 

estimated base flow derived using a regression method from 

streamflow statistics at the Watertown and Waukesha gages 

on the Fox River acted as a check on recharge; eight vertical head difference targets at two locations south of the city 

of Waukesha derived from site investigations for Waukesha 

wells helped estimate vertical hydraulic conductivity of the 

unconsolidated facies; transient drawdown reported for an 

aquifer test conducted on Waukesha well 13, located about 

one-half mile west of the Fox River between the confluence 

of Pebble Creek and Genesee Creek, yielded information not 

only on hydraulic conductivity but also on storage parameters; and, finally, the land surface was used as a check to 

minimize unrealistic groundwater “flooding”, and thereby, 

as a way to constrain hydraulic conductivities and recharge. 

One parameter set that proved relatively insensitive to the 

calibration targets was streambed conductance, proportional to 

the assumed vertical hydraulic conductivity of the streambed. 

The effect of the uncertainty of the streambed conductance can 

be evaluated; however, in this case it was done by varying the 

streambed hydraulic conductivity in the application phase of 

the modeling.

9.3 Model Application

The separate calibration processes for the fine-favored 

and coarse-favored models produce distinct parameter inputs 

for hydraulic conductivity zones, storage parameters, and 

streambed conductance zones. Given their different structures 

and inputs, the two models also yield somewhat different 

results with respect to the pattern of groundwater flow and 

its connection to surface water in the Upper Fox River Basin 

above the Vernon Marsh. For example, the water budget for 

the fine-favored model indicates that groundwater outflow 

from the Upper Fox River Basin is approximately 77 percent 

to streams, 15 percent to lateral flow across the basin boundaries, 2 percent to downward leakage to the deep part of the flow 

system, and 7 percent to wells, whereas, the water budget for 

the coarse-favored model simulates approximately 80 percent 

to streams, 12 percent to lateral flow, 2 percent to vertical 

leakage, and 7 percent to wells. 

The application is a steady-state long-term simulation 

of a hypothetical scenario involving 27 riparian wells open 

to the shallow aquifer systems, which are located along the 

Fox River between the confluences of Sussex Creek and 

Pebble Brook with the Fox River and situated within the 

model nearfield. The objective of this scenario is to test the 

concept that a system of riparian wells located downgradient 

from WWTPs could induce enough water from the river to 

limit drawdown at the water table away from the river (and 

the effects of drawdown on surface-water bodies and nearby 

wells) and also allow for recycling of the water back to the 

river through the WWTP, thereby supporting a more sustainable water-supply system at least from the standpoint of the 

physical source of water. It must be emphasized that the hypothetical locations of 25 of the riparian wells are not based on 

any site-specific investigations but are spaced roughly equally 

apart and are located 125 ft from the river. 

The target withdrawal rate for each well is 0.667 Mgal/d, 

amounting to a total of 18 Mgal/d for the system. However, an 

important feature of the MODFLOW-NWT solver is that the 

simulated withdrawal rate from each well is a function of how 

much the aquifer in the neighborhood of the well can support. In the event, the models support only a little more than 

one-half the target rate, but the coarse-favored model sustains 

more pumping (9.65 Mgal/d) from the hypothetical system 

than the fine-favored model (9.13 Mgal/d). The presence of 

more connected preferential flow zones in the coarse case 

probably accounts for the difference. The simulated sources of 

water to the riparian wells also vary between the two models. 
120 Development and Application of a GW/SW Flow Model using MODFLOW-NWT, Upper Fox River Basin, Wisconsin

In the fine-favored case, 31 percent of the sustained pumping is induced from the Fox River. For the coarse-favored 

simulation, the water induced from the Fox River increases to 

41 percent. Among other sources of water to wells, the most 

prominent is diverted base flow that in the absence of pumping 

would have discharged to the river.

Sensitivity runs were performed on the fine-favored 

model to test the robustness of the results for the hypothetical 

application as a function of changes to streambed hydraulic 

conductivity (assuming a uniform bed thickness of 1 ft). The 

calibrated (but insensitive) value in the Fox River channel is 

equal to about 10 ft/d and corresponds to the hydraulic conductivity of sand. When the streambed hydraulic conductivity 

values were reduced everywhere by three orders of magnitude 

so that the Fox riverbed hydraulic conductivity corresponds 

to the permeability of the dominantly fine facies, riverbank 

inducement falls from 31 percent to 6 percent of the total 

discharge from the hypothetical riparian well system. In this 

connection, it is useful to refer to geophysical data recently 

collected from the Fox River adjacent to Waukesha wells 11 

and 12 that suggests the streambed in the vicinity of the wells 

is not uniformly fine- or coarse-grained, but rather a heterogeneous mixture of textures, which vary over short distances and 

allow for the presence of preferential pathways favorable to 

riverbank inducement.

The modeling techniques employed in this study of the 

Upper Fox River Basin groundwater-flow system can be 

applied elsewhere in areas where unconfined glacial aquifers 

are in connection with surface water. The use of a refined 

mesh and advanced surface-water MODFLOW packages in 

conjunction with the MODFLOW-NWT formulation offers a 

powerful approach for capturing key quantitative features of 

groundwater/surface-water interactions. 

9.4 Concluding Discussion

A key feature of this study is the construction and calibration of two models for the same domain based on different 

conceptual models of the unconsolidated hydrogeology. The 

intent is to generate a range of outputs that increase understanding of the uncertainty inherent in model results and 

predictions. Several lessons can be drawn from the application 

of this approach to the Upper Fox River Basin. 

First, the application of distinct interpolation schemes—

one favoring the connectivity of fine-grained sediments and 

one favoring the connectivity of coarse-grained sediments—

yields markedly different realizations of the hydraulic conductivity fields in the unconsolidated layers. 

Second, the calibration process tends to blunt the difference between the model results despite the contrasting inputs. 

Because the results from both models are necessarily compared to the same calibration targets, the parameter estimation 

algorithm tends to match the targets by balancing the greater 

volume of coarse material in the coarse-favored model with 

relatively low hydraulic conductivity values, whereas, the 

greater volume of fine material in the fine-favored model is 

offset by relatively high hydraulic conductivity values. As a 

consequence of this “feedback mechanism,” many integrating 

measures of output are similar for the two models, including 

water budgets, rates of groundwater/surface-water interactions, areas of simulated upward gradients and upward flows, 

and simulated areas of groundwater basins associated with 

tributaries to the Fox River. 

Third, probably the greatest difference in the simulated 

output for the two models occurs in analyzing the response to 

local stresses, particularly, in calculating the source of water 

for pumping wells. Directions and rates of flow to wells can be 

especially sensitive to the presence or absence of preferential 

flow paths linking a source area to a point of discharge, and 

probably is the reason the responses of the two models differ 

somewhat when applied to a hypothetical system of riparian 

wells aligned along the Fox River. The fine-favored model 

simulates a more modest connection between the wells and the 

river, inducing 31 percent of the sustained discharge from the 

river, than does the coarse model, which induces 41 percent of 

its sustained discharge from the river. 

It is undeniable that carrying two groundwater models 

forward in this instance to better characterize the range of 

results and their uncertainty constitutes a large burden in terms 

of cost and time. Many of the quantitative simulated results 

are similar for the two models and many aspects of the overall 

basin system could have been achieved with a single model 

realization, possibly a compromise between the fine-favored 

and coarse-favored realizations. However, it is also clear that 

some applications, such as forecasting local possibilities for 

water supply, benefit from the use of more than one scheme 

to eliminate possible bias. Neither model is a true representation of the subsurface, but there is value in taking both sets of 

results into consideration.

10. References Cited

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results and recommendations: Letter to Mr. Jeff Detro, City 

of Waukesha Water Utility, September 10, 2004, 5 p. and 

attachments.

Aquifer Science & Technology, 2006, Geophysical investigation and test boring activities, Lathers site, Waukesha 

County, Wisconsin: Letter to Mr. Jeff Detro, City of Waukesha Water Utility, May 25, 2006, 14 p. and attachments.

Aquifer Science & Technology, 2008, Well 13 constant rate 

pumping test analysis: Letter to Mr. Jeff Detro, City of 

Waukesha Water Utility, August 11, 2008, 6 p. and attachments.
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parameters of the glacial sediments in a ground-water flow 

model of the Lake Michigan Basin: U.S. Geological Survey 

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American Water Resources Association, Wisconsin Section, 

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213 p.

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Carlson, D.A., 2001, Dependence of conductivities and anisotropies on geologic properties within the near-surface aquifer 

in Milwaukee, Wisconsin: Unpublished Ph.D. dissertation 

(Geosciences), University of Wisconsin-Milwaukee, 768 p.

CH2MHill in association with Rukert and Mielke, 2002, 

Report on future water supply: prepared for Waukesha 

Water Utility, Executive Summary and eight sections.

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