GEOHYDROLOGY AND SIMULATION 

OF GROUND-WATER FLOW IN THE 

AQUIFER SYSTEM NEAR 

CALVERT CITY, KENTUCKY

By J. Jeffrey Starn, Leslie D. Arihood, and Martin F. Rose

U.S. GEOLOGICAL SURVEY 

Water-Resources Investigations Report 94-4239

Prepared in cooperation with the 

KENTUCKY NATURAL RESOURCES AND 

ENVIRONMENTAL PROTECTION CABINET

Louisville, Kentucky 

1995
U.S. DEPARTMENT OF THE INTERIOR 

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY 

Gordon P. Eaton, Director

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

District Chief U.S. Geological Survey

U.S. Geological Survey Earth Science Information Center

District Office Open-File Reports Section

2301 Bradley Avenue Box 25286, MS 517

Louisville, KY 40217 Denver Federal Center

	Denver, CO 80225
CONTENTS

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

Introduction........................................................................................._^ 1

Purpose and scope................................................................................................................................................2

Area of study........................................................................................................................................................2

Methods of investigation .....................................................................................................................................2

Acknowledgments ...............................................................................................................................................4

Geohydrology..............................................................................................................................................................4

Geology................................................................................................................................................................4

Hydrology............................................................................................................................................................8

Ground water................................................................................................................................................9

Withdrawals and availability.................................................................................................................9

Levels.................................................................................................................................................. 14

Recharge..............................................................................................................................................21

Hydraulic properties of the aquifer system.................................................................................................21

Interaction of ground water and Cypress Creek.........................................................................................23

Simulation of ground-water flow..............................................................................................................................25

Description of the mathematical model.............................................................................................................25

Model grid and layers.................................................................................................................................25

Boundary conditions and source terms.......................................................................................................27

Model calibration...........................................................................................»^

Calibration procedure.................................................................................................................................29

Estimates of model parameters................................................................................................................... 31

Hydraulic-head distribution........................................................................................................................ 32

Rate of flow ................................................................................................................................................32

Alternative models......................................................................................................................................36

Reliability of estimates............................................................................................................................... 37

Simulated ground-water flow ............................................................................................................................39

Contributing areas and times of travel........................................................................................................ 40

Average linear ground-water velocities......................................................................................................42

Limitations of particle tracking ..................................................................................................................47

Summary and conclusions ........................................................................................................................................49

References cited........................................................................................................................................................51

PLATE (In Pocket)

1. Diagram showing geologic sections of the Calverl City area, Kentucky

FIGURES

1. Map showing geographic features of the Calvert City area, Kentucky.................................................... 3

2. Diagram showing stratigraphy of the Calverl City area, Kentucky.......................................................... 5

3-4. Maps showing:

3. Generalized geology of the Calvert City area, Kentucky.................................................................. 6

4. Geologic features of the bedrock surface, Calvert City, Kentucky.................................................... 7

5. Schematic of north-south section through the aquifer near Calvert City, Kentucky .............................. 10

6-7. Maps showing:

6. Ground-water flow divides.............................................................................................................. 11

7. Location of ground-water withdrawals greater than 10,000 gallons per day in 1992, and

selected observation wells ...............................................................................................................12

8. Graph showing average daily ground-water withdrawals in 1992, near Calvert City, Kentucky........... 13

Contents Hi
9-10. Maps showing:

9. Potentiometric surface, March 1992................................................................................................ 15

10. Potentiometric surface, June 1992................................................................................................... 16

11-14. Graphs showing:

11. Depth to water in the McNairy Formation at well UCC4 and rainfall, Calvert City, Kentucky..... 17

12. Depth to water in the alluvium at well UCC3 and rainfall, Calvert City, Kentucky....................... 18

13. Selected water levels affected by ground-water withdrawal, Calvert City, Kentucky .................... 19

14. Selected water levels affected by river stage, Calvert City, Kentucky............................................ 20

15. Generalized geologic section at the Calvert City Municipal Water Supply Company,

Calvert City, Kentucky............................................................................................................................22

16-19. Maps showing:

16. Streamflow measured in Cypress Creek, 1992................................................................................24

17. Extent of model grid and model boundary conditions..................................................................... 26

18. Parameter zones used in model........................................................................................................28

19. Potentiometric surface calculated with the calibrated model.......................................................... 33

20. Histogram of weighted differences between calculated and observed heads, Calvert City,

Kentucky.................................................................................................................................................34

21-25. Maps showing:

21. Spatial distribution of weighted residuals........................................................................................35

22. Contributing areas of discharge boundaries computed with the calibrated model,

1992 conditions................................................................................................................................41

23. Contributing areas of discharge boundaries computed with the calibrated model,

1994 conditions................................................................................................................................43

24. Time of travel computed with the calibrated model, 1994 conditions.............................................44

25. Location of nodes for which average linear velocity is calculated, 1994 conditions ......................45

TABLES

1. Model parameters estimated using nonlinear regression, Calvert City, Kentucky................................. 31

2. Optimal parameter values obtained from nonlinear regression, Calvert City, Kentucky....................... 31

3. Parameter values for alternative models computed by nonlinear regression, Calvert City,

Kentucky................................................................................................................................................. 37

4. Water budget computed by alternative models, Calvert City, Kentucky................................................ 37

5. Pseudo-coefficients of variation for alternative models computed by nonlinear regression,

Calvert City, Kentucky............................................................................................................................39

6. Correlation and covariance matrix for optimum values obtained through nonlinear regression

(CAL1), Calvert City, Kentucky............................................................................................................. 39

7. Magnitude and direction of the average linear velocity at selected model cells under 1994

conditions, Calvert City, Kentucky.........................................................................................................46

iv Geohydrology and Simulation of Ground-Water Row in the Aquifer System near Caivert City, Kentucky
CONVERSION FACTORS AND VERTICAL DATUM

Multiply By To obtain

inch (in.)

foot (ft)

mile (mi)

square foot (fr) 

square mile (mi 2 )

gallon per minute (gal/min)

cubic foot per second (ft^/s)

gallon per day (gal/d)

million gallons per day (Mgal/d)

cubic foot per day (ft3/d)

foot squared per day (ft /d)

foot per day (ft/d)

Length

25.4

0.3048

1.609

Area

0.09290

2.590

Flow

0.06308

28.32

5.450

0.04381

0.02832

Transmissivity

0.09290 

Hydraulic conductivity

0.3048

millimeter

meter

kilometer

square meter 

square kilometer

liter per second 

liter per second 

cubic meter per day 

cubic meter per second 

cubic meter per day

meter squared per day

meter per day

Sea level: In this report, "sea level" refers to the National Geodetic Vertical Datum of 1929 (NGVD of 1929) geodetic datum 

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

Datum of 1929.

Contents v
Geohydrology and Simulation of Ground-Water Flow 

in the Aquifer System near Calvert City, Kentucky

By J. Jeffrey Starn, Leslie D. Arihood, and Martin F. Rose

Abstract

The U.S. Geological Survey, in cooperation with the Kentucky Natural Resources and 

Environmental Protection Cabinet, constructed a two-dimensional, steady-state ground-waterflow model to estimate hydraulic properties, contributing areas to discharge boundaries, and the 

average linear velocity at selected locations in an aquifer system near Calvert City, Ky. Nonlinear 

regression was used to estimate values of model parameters and their reliability. The regression 

minimizes the weighted difference between observed and calculated hydraulic heads and rates of 

flow. The calibrated model generally was better than alternative models considered, and although 

adding transmissive faults in the bedrock produced a slightly better model, fault transmissivity 

was not estimated reliably. The average transmissivity of the aquifer was 20,000 feet squared per 

day. Estimated recharge to two outcrop areas, the Cretaceous McNairy Formation and the 

Quaternary alluvium, was 0.00269 feet per day (11.8 inches per year) and 0.000484 feet per day 

(2.1 inches per year), respectively. Contributing areas to wells at the Calvert City Water Company 

in 1992 did not include the Calvert City Industrial Complex. Since completing the field work for 

this study in 1992, the Calvert City Water Company discontinued use of its wells and began 

withdrawing water from new wells that were located 4.5 miles east-southeast of the previous 

location; the contributing area for these new wells moved farther from the industrial complex. The 

extent of the alluvium contributing water to wells was limited by the overlying lacustrine 

deposits. The average linear ground-water velocity at the industrial complex ranged from 0.90 to 

4.47 feet per day with a mean of 1.98 feet per day.

INTRODUCTION

The Department for Environmental Protection (DEP) of the Kentucky Natural Resources and 

Environmental Protection Cabinet has made a comprehensive study of the Calvert City Industrial Complex 

to characterize the environmental conditions of the air, land, and water resources. The study consists of 

21 major components, including ground-water modeling. The DEP study was designed to (1) incorporate 

area-wide and multi-media (air, soil, and water) information to assess the risk to public health and the 

environment caused by past waste disposal practices, (2) assist in the remediation and (or) abatement of 

risk, (3) plan for future protection of the resources of the area, and (4) facilitate the cooperation of the 

public, industry, and government in the management of risk in the area. The U.S. Geological Survey 

(USGS), in cooperation with the DEP, began an investigation of the ground-water component of the 

overall study.

INTRODUCTION 1
Purpose and Scope

This report describes briefly the geohydrology of the aquifer system at Calvert City, Ky., and presents 

results of a regional-scale simulation of ground-water flow. The hydrologic significance of geologic units 

in the study area are discussed using results of previous reports, measurements of streamflow, and 

observations of water levels. The effect of hydrologic factors on the ground-water-flow system were 

represented in a two-dimensional, steady-state, finite-difference ground-water-flow model. A nonlinear 

regression technique was used to calibrate the model. Estimates of the model parameters and their 

reliability are discussed. Contributing areas to discharge boundaries are displayed on maps. The magnitude 

and direction of the average linear velocity is tabulated for selected nodes in the model. Scenarios were 

evaluated with regard to changes in the ground-water flow system by simulating hypothetical locations and 

rates of withdrawal.

Area of Study

Calvert City, Ky., is in Marshall County in western Kentucky. The topography is characterized by flat 

areas and low, gently rolling hills. The area is bounded by Kentucky Lake on the east and the Tennessee 

River on the north. The main area of interest is centered on Calvert City and the Calvert City Industrial 

Complex; however, the area of study encompasses a larger area in order to incorporate the natural 

hydrologic boundaries of the ground-water flow system (fig. 1). The population of Calvert City is 2,531 

(1990 census); the remainder of the area is rural and more sparsely populated. The industrial complex lies 

on the broad, flat terrace south of the Tennessee River. Several industries within the complex produce 

chemicals, including polyvinyl chloride, metal coatings, and paint additives. Other industries include 

production of graphite and metal alloys. Two companies provide waste-handling services to the chemical 

manufacturers. The methods of waste disposal within the complex include landfarming, incineration, 

treatment lagoons, and recycling. Many former waste-treatment lagoons have been closed.

Methods of Investigation

Methods used to describe the geohydrology and to evaluate the ground-water flow system include 

installation of observation wells, measurement of streamflow and ground-water levels, and numerical 

simulation. Six piezometers were constructed near the Calvert City well field for geologic description and 

water-level observation. Two synoptic water-level surveys were done using domestic wells, monitoring 

wells constructed for regulatory purposes, and the newly constructed observation wells. Geologic sections 

were constructed to complete the conceptual model.

Two-dimensional, steady-state ground-water flow in the aquifer near Calvert City, Ky., was simulated 

by use of the finite-difference computer program called MODFLOW (McDonald and Harbaugh, 1988). 

The parameters of the model were estimated using a nonlinear-regression method developed by Cooley 

and Naff (1990) and modified for use with MODFLOW by Hill (1992). The combined ground-water-flow 

model and nonlinear regression method are the basis for the computer program MODFLOWP (the suffix P 

indicates parameter estimation). Other published case studies using nonlinear regression include, for 

example, evaluation of an alluvial aquifer (Yager, 1993) and a fractured carbonate-rock aquifer (Yager, in 

press). Alternative models were evaluated using statistics computed by use of MODFLOWP. The 

alternative models simulated different but equally plausible sets of hydrologic factors. The particletracking technique called MODPATH (Pollack, 1989) was used to compute the zones of contributing areas 

and the average linear velocities of the ground water.

2 Geohydrology and Simulation of Ground-Water Row in the Aquifer System near Calvert City, Kentucky
88°15'

36°57'30

Base from aerial photography by Kentucky Department of Transportation, 1985 

U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

01

5 MILES

\ 

5 KILOMETERS

INDEX MAP

'Study Area jEXPLANATION

Calvert City Industrial 

Complex

Gilbertsville 

Calvert City

Limit of active model 

grid; no-flow boundary

Figure 1. Geographic features of the Calvert City area, Kentucky.

INTRODUCTION 3
Acknowledgments

The authors acknowledge the contribution of industries within the Calvert City area that provided data 

for the project and to Calvert City for allowing the construction of observation wells on city property. Gay 

Brewer (B.F. Goodrich), Jerry Devine (Calvert City Water Company), Richard Nash (Calvert City), and 

Hans Conrad (Calvert City) were especially helpful.

GEOHYDROLOGY

The area lies within the Mississippi Embayment (Fenneman, 1938) physiographic region of western 

Kentucky. The Mississippi Embayment is a concave-upward, trough-shaped geologic structure consisting 

of, in part, Cretaceous and younger sediments. These sediments consist of partially consolidated and 

unconsolidated sand, clay, gravel, and silt. The sediments form a wedge that thickens from the edges of the 

embayment toward the Mississippi River. Ground water in the study area flows from the upland areas 

south of Calvert City and north toward the Tennessee River, whereas flow in other parts of the 

Mississippian Embayment is to the southwest toward the Mississippi River. Ground water in the study area 

discharges to the Tennessee River, to supply wells, and to the upper and lower reaches of Cypress Creek.

Geology

The lithology, structure, and stratigraphy of the study area were described by Olive (1980) and by the 

authors of the 7.5-minute geologic-quadrangle maps (Olive, 1963; Lambert and MacCary, 1964; Amos 

and Wolfe, 1966; Amos and Finch, 1968). The geologic units discussed in this report are Mississippian 

limestones; Cretaceous sand, clay, and gravel deposits; Tertiary gravel deposits; and Quaternary deposits 

of gravel, sand, silt, and clay (fig. 2).

Two bedrock units have hydrologic significance in the study area the Fort Payne Formation and the 

Warsaw Limestone both of which crop out along Kentucky Lake (fig. 3). The Fort Payne Formation 

contains limestone and chert in interbedded layers 0.1 to 2.0 ft thick. It is dense, finely crystalline, and 

locally shaly. The limestone also crops out along the Tennessee River west of the industrial complex and in 

the bed of Cypress Creek 1 mi west of Calvert City (Amos and Finch, 1968). Upon weathering, the Fort 

Payne Formation produces angular fragments of chert in a porous, fine-grained matrix. The Warsaw 

Limestone is a thick-bedded limestone composed of megascopic fossil fragments in a fine-grained 

limestone matrix. Some beds of fine-grained limestone are interspersed throughout the formation. The 

Warsaw Limestone rarely crops out and weathers to a silty clay texture that contains some porous zones of 

chert fragments. Either the Fort Payne Formation or the Warsaw Limestone underlies the unconsolidated 

deposits throughout the study area.

The buried surface of the bedrock (fig. 4) is characterized by the presence of a buried valley and by the 

degree of weathering and fracturing of the rock. The deepest part of the buried valley is located 1.5 mi 

south of the present course of the Tennessee River. This valley is defined approximately by the 220 ft 

topographic contours on figure 4. Although the bedrock units were deposited in a marine environment, 

they were subsequently exposed to weathering for a long period of time perhaps 180 million years (Davis 

and others, 1973). Soil zones formed on the surface of the rocks, and deep leaching of carbonate minerals 

resulted in the formation of karst topography with its characteristic subsurface drainage. Where limestone 

lies beneath unconsolidated surficial sediments, it is fractured and weathered to a depth of 5 to 25 ft. Where 

the Warsaw Limestone overlies the Fort Payne Formation, such as beneath the Calvert City Industrial

4 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
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2.
36°57'30

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

Geology by J.H. Morgan, 1965

I I I I \ 

012345 KILOMETERS

4 5 MILES

_J______I

INDEX MAP

EXPLANATION

DMu--Rocks of Mississippian 

and Devonian age, 

undifferentiated

Qal--Alluvium of Quaternary age 

Km--McNairy formation

Geologic contact between 

Quaternary alluvium and 

McNairy formation

Figure 3. Generalized geology of the Calvert City area, Kentucky.

6 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
88°15'

36°57'30

Base from aerial photography by Kentucky Department of Transportation, 1985 

U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

Geology by Dewey H. Amos 

and Warren I, Finch, 1968

5 MILES

012345 KILOMETERS

INDEX MAP

EXPLANATION

Subsurface contact between 

rocks of late Cretaceous and 

Paleozoic age missing where 

not mapped

Faults

STRUCTURE CONTOUR-Altitude 

of surface of rocks of Late 

Cretaceous, Mississippian, and 

Devonian age. Shown where less 

than 280 feet above sea level. 

Countour interval 20 feet

Figure 4. Geologic features of the bedrock surface, Calvert City, Kentucky.

GEOHYDROLOGY 7
Complex, the Fort Payne Formation is relatively free of fractures (Finch, 1968). The bedrock surface is 

marked by northeast-trend ing near-vertical normal faults that displace Paleozoic and Cretaceous strata less 

than 100 ft vertically (Davis and others, 1973).

Cretaceous sediments overlie the limestone bedrock in the southern part of the study area (fig. 3 and 

pi. 1). The Tuscaloosa Formation is a deeply weathered and broken chert rubble that is present in 

discontinuous patches on the bedrock surface. The McNairy Formation overlies the Tuscaloosa Formation 

or the limestone bedrock and was deposited in a marine to freshwater deltaic environment during the Late 

Cretaceous Period at the northeastern end of the Mississippi Embayment. This formation is composed of 

interbedded sand, clay, and gravel. The sand is generally loose, well-sorted, medium-grained quartz sand, 

although, in places, it can be well-cemented by iron minerals. The clay is present in thin layers and 

irregular masses. The gravel is predominantly chert pebbles in a porous matrix of medium-grained quartz 

sand. The formation thickens from its outcrop near Kentucky Lake to 200 ft in the southwestern part of the 

study area. The base of the McNairy Formation crops out near Kentucky Lake where it is a gravelly sand 

or sandy gravel.

Gravel and sand continental deposits of Pliocene or Pleistocene age overlie the McNairy Formation on 

some hilltops. These sediments were deposited by high-velocity braided streams as channel deposits. The 

gravel generally is composed of fairly well-consolidated chert and sandstone blocks in a matrix of poorly 

sorted coarse sand. A Pleistocene loess deposit overlies the McNairy Formation and the continental 

deposits on hilltops. The loess is a wind-blown deposit composed of silt with variable amounts of clay and 

sand intermixed.

Alluvial deposits are subdivided into upper, middle, and lower units (Finch, 1968) that reflect different 

depositional environments (pi. 1). The lower unit formed during Pliocene or Pleistocene time as 

continental deposits on surrounding hilltops eroded and coarse material was transported into the incised 

bedrock valley. The unit consists of 50 to 70 ft of well-sorted gravelly sand and sandy gravel (Finch, 

1968). In valleys occupied by small rivers and creeks, the lower gravel is thin, absent, or clayey (Davis and 

others, 1973). Although this unit overlies limestone in most places, it overlaps the McNairy Formation in 

the area near their surficial contact. The middle unit was formed as the Ohio River aggraded its valley 

during the Pleistocene Epoch. It is a transition sequence of interbedded sand, silt, and clay that represents 

channel sands, point bar accumulations, and overbank deposits on top of the valley-fill sand and gravel. 

The middle unit consists of 30 to 50 ft of noncohesive silty sand, clayey sand, and clean sand. In a splitspoon sample collected at the Calvert City Water Company, the middle unit consists of alternating layers 

of sand and clay about 2 in. thick. The upper unit is a massive deposit of lake-bed silt and clay that 

accumulated when the river was blocked with sediment during Pleistocene time. These lacustrine deposits 

form the uppermost geologic unit over most of the Tennessee River floodplain and consist of 8 to 10 ft of 

cohesive unconsolidated clayey silt and silty clay.

In the north-central part of the study area adjacent to the Tennessee River, some well logs show a 

distinct, continuous bluish-gray to maroon clay layer from a depth of 40 ft below land surface to the 

bottom of the borehole at a depth of 80 ft. This clay forms a low-permeability zone along the Tennessee 

River. The extent of the clay is unknown.

Hydrology

The dominant hydrologic features of the study area are the Tennessee River and Kentucky Lake. The 

mean annual discharge of the Tennessee River for 1992 was 50,301 ft3/s. In June 1992, when many of the 

data used in this report were collected, the minimum, mean, and maximum discharges were 16,658 ft/s, 

38,282 ft3/s, and 82,664 ft3/s, respectively (Morgan Goranflo, Tennessee Valley Authority, written

8 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
commun., 1994). Normal pool stage in the Tennessee river is 302 ft; flow in the Tennessee River is 

controlled by the Kentucky Dam, which impounds Kentucky Lake. The normal pool stage in Kentucky 

Lake is 359 ft. Major tributaries to the Tennessee River near Calvert City are Cypress Creek and East Fork 

Clarks River. The Tennessee River flows into the Ohio River at Paducah, Ky. about 20 river miles 

downstream from the study area.

The aquifer system underlying the study area comprises two aquifers: one aquifer in the lower unit of 

the alluvium and one aquifer in sands of the McNairy Formation (fig. 5). The upper 5 to 25 ft of the 

limestone bedrock was weathered to a granular texture similar to the overlying sediments and were 

included with the overlying aquifers. The aquifer system is bounded by ground-water flow divides, 

Kentucky Lake, and the Tennessee River (fig. 6) based on the regional potentiometric surface (Davis and 

others, 1973), topography, and geology (MacCary, 1964; Morgan, 1964). The southwestern boundary of 

the aquifer system is a ground-water flow divide that separates discharge to East Fork Clarks River and the 

Tennessee River. The eastern boundary of the aquifer system is the outcrop of consolidated rocks between 

the aquifer and Kentucky Lake. The southern boundary is a topographic high where the aquifer is overlain 

by a confining layer (the zone of "perched water" described by MacCary (1964) and Morgan (1964)). The 

confining layer results in a high plateau on the potentiometric surface from which ground water flows 

southeast to the headwaters of Cypress Creek and north into the aquifer system underlying the study area.

Ground Water

Withdrawals and Availability

Approximately 23 percent of the water used for municipal and industrial water supply in Marshall 

County is ground water (Sholar and Lee, 1988). Permits are required from the Kentucky Division of Water 

for ground-water withdrawals greater than 10,000 gal/d; the following two companies in the area 

(companies A and B in this report), and the Calvert City Water Company, have such permits (figs. 7 and 

8). Since completing the fieldwork for this study in 1992, the Calvert City Water Company discontinued 

use of their wells and began withdrawing water from new wells located 4.5 mi east-southeast of the 

previous location (Jerry Devine, Calvert City Water Company, oral commun., 1994). Also, another 

company (company C) operated 32 wells in June 1992 to remove contaminated ground water from the 

aquifer. The combined rate of pumping from these wells was approximately 650 gal/min during the 

start-up phase of the remediation in June 1992.

Well yields in the area generally are high but depend on local geologic conditions. Yields from the 

lower alluvial unit can be high; one well reportedly yielded a sustained rate of 1,200 gal/min during an 

aquifer test (Davis and others, 1973). Other wells in Ihe lower alluvial unit yield as much as 650 gal/min. 

The McNairy Formation yields sufficient water for domeslic use (Free and Walker, 1952), but the 

maximum potential to yield water has not been explored. Where the top of the Fort Payne Formation is 

overlain by sediments of Cretaceous age and younger, it is weathered and fractured into a chert rubble that 

can yield large amounts of ground water to wells (Davis and others, 1973). In general, yields from the 

weathered limestone can exceed 125 gal/min (Free and Walker, 1952). Localized transmissive faults in the 

limestone also can supply large quantities of water to wells. The spacing of faults that produce large 

quantities of water, however, is not known.

GEOHYDROLOGY 9
South

Recharge

f

McNairy 

Formation

Weathered 

bedrock

Calvert City

Cypress Creek

North

Upper alluvial unit

Middle alluvial unit

Tennessee River

Lower alluvial unit

NOT TO SCALE

EXPLANATION

Direction of ground-water flow

Figure 5. North-south section through the aquifer near Calvert City, Kentucky.

10 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
88°30'

37°02'30

36°57'30" -

36°52'30"

Base from U.S. Geological Survey digital data, 1:100,000, 1983 

Stateplane Projection NAD27

5 MILES

5 KILOMETERS

INDEX MAP

 320-

EXPLANATION

Km-Area of perched aquifer (from 

Morgan, 1964; and MacCary, 1964)

Bedrock outcrop

Ground-water divide

Water-level contour-contour intervals 

10 and 20 feet. Datum is sea level. 

(From Davis and others, 1973; 

plate 10.)

Generalized direction of ground-water 

flow

Figure 6. Ground-water flow divides.

GEOHYDROLOGY 11
88°22'19" 

37°04'02"

88°20'10"

37°01'34" Base from y s Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

0.5 1 1.5

1.5

2 KILOMETERS

2 MILES

EXPLANATION

-f- Wells at company A

D Wells at company B

° Wells at company C

A Calvert City Water Company wells

0ucci uSGS observation wells and 

identifier

J Calvert City

s, j, ,, . . }

1 Calvert City industrial complex

Figure 7. Location of ground-water withdrawals greater than 10,000 gallons per day in 1992 

and selected observation wells.

12 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
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WITHDRAWAL, IN MILLION GALLONS PER DAY

p

CJ1

N>b

p 

bi

N>b

O O

03

O O

omO5m

o OI
The type and location of domestic wells varies with geology, local well-drilling practices, and 

availability of publicly supplied water. Most of the residences over the alluvial deposits west of Calvert 

City depend on ground water. The wells generally are open-ended, 24-in. diameter wells cased with 

concrete in 2-ft sections and are less than 50 ft deep. In contrast, most residences south of Calvert City use 

small-diameter drilled wells ranging from 50 ft deep to more than 100 ft deep. The areas around Calvert 

City, southwest of Calvert City, and Gilbertsville are served by publicly supplied water; however, some 

residences have wells for occasional use. Two golf courses and a State-owned tree farm in the area use 

ground water for irrigation at a rate less than 10,000 gal/d.

Levels

Water levels were measured in monitoring wells at eight industrial plants and in privately owned 

domestic water-supply wells twice in 1992. The first set of 157 measurements was made from March 17 to 

20. A subset of 119 wells was visited again from June 23 to 25. These data were used to produce contour 

maps of the potentiometric surfaces (figs. 9 and 10). The shapes of the 320 ft and 330 ft potentiometric 

contours were similar to the geologic contact between the McNairy Formation and the alluvium (fig. 3), 

indicating that the aquifers on each side of the contact are hydrologically different. The potentiometricsurface maps show that the contours are generally in the same positions as noted by Davis and others 

(1973) (fig. 6); therefore, on the basis of these limited data, there does not seem to be a long-term trend in 

water levels. The long open loop at the western end of the 320-ft contour on figure 5 was drawn by Davis 

and others (1973) on the basis of a single water-level measurement. The 320-ft contours in figures 9 and 10 

differ from figure 5, although not significantly in terms of the general position of the contour.

Water levels in each aquifer in the aquifer system responded slightly different to natural recharge. 

Water levels in the McNairy Formation and the alluvium responded to both daily and seasonal fluctuations 

in recharge (figs. 11 and 12). Water levels in the McNairy Formation fluctuate more to individual rainfall 

events and to seasonal recharge (fig. 11) than water levels in the alluvium. Water levels in bedrock wells 

along the shore of Kentucky Lake coincide with those in the overlying McNairy Formation because the 

two are hydraulically connected in this area (Davis and others, 1973).

Water levels in the aquifer system also may fluctuate because of changes in barometric pressure, 

ground-water withdrawals, and river stage caused by releases of water from Kentucky Dam. The water 

level in wells was reported by Davis and others (1973) to vary by as much as 1 ft over a 2-3 day period in 

response to fluctuations in air pressure; however, water levels in March and June 1992 varied by less than 

0.1 ft in one well that was measured several times during each synoptic measurement period. Water levels 

in the alluvium also respond to ground-water withdrawal at the Calvert City Water Company (fig. 13). 

Water levels in the middle and lower units of the alluvium respond to changes in stage in the Tennessee 

River (Dames and Moore, 1988a) (fig. 14). The duration of the high water levels in wells was 

approximately equal to the duration of the rise in river stage. The change in stage was propagated rapidly 

through the aquifer and was observed at the most distant observation well, 2,000 ft from the river.

14 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
22'30" 88°15'

36°57'30

Base from aerial photography by Kentucky Department of Transportation, 1985 

U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

5 MILES

I I I I I I 

012345 KILOMETERS

EXPLANATION

INDEX MAP

-SOD- POTENTIOMETRIC CONTOUR-- 

Shows altitude at which water 

would have stood in tightly 

cased wells, March 1992. 

Contour interval 10 feet. Datum 

is sea level.

 Ground-water divide

o Data points for potentiometric 

contours

Figure 9. Potentiometric surface, March 1992.

GEOHYDROLOGY 15
22'30"

37°00' 

36°57'30

Base from aerial photography by Kentucky Department of Transportation, 1985 

U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

5 MILES

 I ^ F I 1 

012345 KILOMETERS

INDEX MAP

EXPLANATION

-SOD- POTENTIOMETRIC CONTOUR-- 

Shows altitude at which water 

would have stood in tightly 

cased wells, June 1992. 

Contour interval 10 feet. Datum 

is sea level.

 Ground-water divide

o Data points for potentiometric 

contours

Figure 10. Potentiometric surface, June 1992.

16 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
LU LU

LL

10

cc

LU

fe 12

X 14

Q_ 

LU 

Q

16

18

20

22 L

1 \ \ T I I

EXPLANATION 

Depth to water in well UCC4 

Rainfall

i r

Land-surface datum = 340.4 feet above sea level

,l

MAY JUNE JULY AUG SEP OCT NOV DEC

1992

JAN FEB MAR APR MAY 

1993

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

LU

cc

o I

Figure 11. Depth to water in the McNairy Formation at well UCC4 and rainfall, Calvert City, Kentucky.

GEOHYDROLOGY 17
24

25

26

LLJ

B! 27

CC 

LU

§

o

Q. 

LU 

Q

28

29

30

31

32

I

EXPLANATION 

Depth to water in well UCC3 

Rainfall

Land-surface datum = 342.0 feet above sea level

_L

MAY JUNE JULY AUG SEP OCT NOV DEC

1992

JuijJ Jj

2.0

1.8

1.6

- c/3

UJ I 

o

1.2 ?

1.0

0.8 > 

DC

O 

0.6 X

0.4

0.2

JAN FEB MAR APR MAY 

1993

0

Figure 12. Depth to water in the alluvium at well UCC3 and rainfall, Calvert City, Kentucky.

18 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
LU 

LU

25.0

25.5

26.0

26.5

? 27.0

of 

LU

27.5

28.0

CL 

LU

0 28.5

29.0

29.5

30.0

UCC3

UCC2D

May June

1992

Figure 13. Selected water levels affected by ground-water withdrawal, Calvert City, Kentucky.

GEOHYDROLOGY 19
324

_ _ . well 125 feet from river 

 - well 750 feet from river 

well 2.000 feet from river

300

15 20 25 

TIME, IN DAYS

Figure 14. Selected water levels affected by river stage, Calvert City, Kentucky. (Modified from Dames 

and Moore, 1988b, fig. 8.)

20 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
Recharge

Most recharge occurs south of Calvert City in the area underlain by the McNairy Formation (fig. 5) 

because this area is topographically higher and the surficial deposits are more permeable than in 

surrounding areas. Surficial deposits in this area include loess, continental deposits, or the McNairy 

Formation. Loess covers most hilltops and, based on its geologic description, probably diverts most 

rainfall to surface runoff down the hillsides where it can enter the aquifer through more permeable 

deposits. The extent to which recharge occurs through the loess, possibly through macropores and 

fractures, is unknown. The sand and gravel continental deposits, which underlie the loess and crops out on 

hillsides, allow greater infiltration of rainfall than does the loess. The McNairy Formation crops out on 

hillsides and valley bottoms (pi. 1) and rainfall or surface runoff can enter the aquifer through permeable 

sand layers. Where the McNairy Formation is represented by clay beds, infiltrating water forms local zones 

of perched water. Other than the extensive clay bed and perched water zone (fig. 6) already discussed, 

perched water is not considered further. Based on the geologic description of the upper unit of the 

alluvium, recharge from rainfall directly to the alluvium is less than recharge from rainfall to the McNairy 

Formation.

The timing and the approximate amount of recharge to the aquifer system is indicated on hydrographs. 

Based on hydrographs, recharge to the McNairy Formation occurred from January to April 1993 (fig. 11). 

Davis and others (1973) estimated the amount of recharge to the McNairy Formation to be 7.2 in. per 

year (in./yr). This estimate was made by multiplying the seasonal rise in water level in one well by the 

estimated specific yield of the aquifer. For the following reason, this method of analysis yields an estimate 

of recharge that is too low. By only considering the rise in water level, a constant water level implies no 

recharge; however, for water levels to remain constant, recharge must occur in the same amount as 

discharge from the aquifer system. The hydrograph analysis, applied to ground-water recharge, should 

include an estimate of the rate at which water levels would decline if there were no recharge.

Other sources of recharge include stage changes in the Tennessee River and water from Kentucky 

Lake that enters fractures and open joints in limestone. Recharge from high river stage is a short-lived 

phenomenon that dissipates after the stream stage returns to normal (fig 14); thus, the net recharge to the 

aquifer is zero. This phenomenon, known as bank storage, can be significant in terms of the quality of 

water (Starn, in press; Squillace and others, 1993). The impounding of water in Kentucky Lake caused 

water levels to rise in low-lying areas near the lake (Davis and others, 1973). This indicates that water 

moved from the lake into the aquifer through the fracture system; however, the distance from the lake over 

which the fractures are open is not known.

Hydraulic Properties of the Aquifer System

The hydraulic character of the lower alluvial unit was inferred from known well yields, lithologic 

description, and aquifer tests. Based on well-yield information, the lower unit of the alluvium is the most 

permeable unit of the aquifer system. The saturated thickness of the alluvium is 80 to 100 ft and includes 

the middle unit of the alluvium. Based on the lithologic description, the horizontal permeability of the 

middle unit is much greater than the vertical permeability, so the middle unit probably forms a confining 

layer over the lower unit; however, some ground water probably leaks through the middle unit into the 

lower unit, particularly when the lower unit is pumped. Water-level data collected during pumping at the 

Calvert City Water Company support this concept. Two pairs of observation wells were installed near 

wells used for water supply at the Calvert City Municipal Water Company (fig. 7). One well of each pair 

was completed in the lower unit of the alluvial aquifer, and one well was completed in the middle unit 

(fig. 15). The pumps were turned off for 12 hours and water levels recovered to equal altitudes in all four 

wells. Pumping began at an unknown and variable rate (approximately 650 gal/min based on the 

requirements of the water treatment plant) and continued for 4 hours. The maximum drawdowns in deep

GEOHYDROLOGY 21
LLJ

LLJ 

_l

IS

CO 

LLJ

§

DO

LLJ 

LLJ

LJJ 

Q

=)

R

340 Pumped well Well UCC1 Well UCC2

_

3?0

300_

-

?80

260_

240_

220_

?00

lt''gfaVet'li'

ill pack |||

:;:;:;:;:7,,.,, ,,,.,..., ,..,w:; w

'^' ^' .'.' ' ' ' "< .. ] J

^ x';:: :; : ;:0 : V:v;- : ;. I /

Upper alluvial unit

llllllllllllllll ' ' ' JMfiddie aif uviai 'unit' ' ' ' 1111111111111111

iiiiiiiiiiiiiiiiiiiii^^^ SiSSMS::::?::-:-:-:^:::::::^^

\ ^CC1 D Lower alluvial unit

'$': i'&'WX&»:m3m$!m

:;UOL»tO iHgJssgg::;:;;;;;

;i|||li

&ji " screened' interval ' /;: ^^^J o^fi ^-Ml^. . .;. '..; I :i : :| ^ - :: ;: -ffS- " f'^^SffKMff^.

I I I I I I I I I

0 10 20 30 40 50 60 70 80 90

DISTANCE FROM PUMPED WELL, IN FEET

Figure 15. Generalized geologic section at the Calvert City Municipal Water Company, Calvert City, 

Kentucky.

22 Geohydrology and Simulation of Ground-Water Flow In the Aquifer System near Calvert City, Kentucky
observation wells UCC1D and UCC2D were 3.17 ft and 0.72 ft, respectively. The corresponding 

drawdowns in the two shallow observation wells were 0.03 ft and 0.02 ft, respectively. In this case, ground 

water flowed more freely to the well through the lower unit, although pumping had some allcct on the 

middle unit.

Several previous investigators have determined the transmissivity of aquifers within the aquifer 

system. Davis and others (1973) reported aquifer tests at companies C and A. The values for transmissivity 

of the lower alluvial unit were 2,940 ft2/d and 174,000 ft2/d, respectively. The first value was affected by 

the lower saturated thickness of the aquifer near the Tennessee River (pi. 1). The second value may reflect 

the influence of localized transmissive faults in the limestone bedrock. A transmissivity of 5,000 ft2/d was 

used in a model of ground-water flow in the industrial complex (Dames and Moore, 1991). A 

transmissivity of 38,200 ft/d was calculated at new wells used for water supply at the Calvert City Water 

Company (W.A. Tanner, Florence and Hutcheson Consulting Engineers, written commun., 1992). 

Transmissivity of the McNairy Formation 5.5 mi south of the study area was calculated as 4,280 ft2/d 

(Davis and others, 1973).

Interaction of Ground Water and Cypress Creek

The relation of Cypress Creek to the aquifer system is best described by dividing the creek into three 

reaches, the boundaries of which are approximated (fig. 16). In the upper reach, Cypress Creek flows 

northward on top of the McNairy Formation. The gradient of the creek is low and the flow is sluggish; the 

upper reach is primarily a swampy lowland. In this reach, water flows from sands in the McNairy 

formation into Cypress Creek. In the middle reach, Cypress Creek flows onto the lacustrine sediments (the 

upper alluvial unit) and is incised into these predominantly clay and silt sediments that confine the creek to 

a narrow channel between steep banks. Also, in the middle reach, Cypress Creek is hydraulically separated 

from the aquifer system by the upper and possibly middle alluvial units. In the lower reach, the creek bed 

decreases in elevation in the downstream direction and is in direct connection with the aquifer in the lower 

alluvial unit.

The potentiometric surfaces indicated different flow patterns in the middle and lower units of the 

alluvium (Ron Yost; Fuller, Mossbarger, Scott, and May Consulting Engineers; written commun., 1989). 

Ground water flowed south toward Cypress Creek in the middle unit and flowed north to the Tennessee 

River in the lower unit. This flow pattern also was inferred from water levels in wells screened in the lower 

unit (wells UCC2 and UCC3) (figs. 12 and 13). The water levels fluctuated on the opposite side of Cypress 

Creek (fig. 13) in response to pumping at the Calvert City Municipal Water Company indicating that 

Cypress Creek is not a hydraulic barrier to flow and that ground water flows beneath Cypress Creek to the 

other side. The middle reach of Cypress Creek receives ground-water discharge from the middle unit of the 

alluvium, particularly during wet weather, but does not receive ground-water discharge from the lower 

alluvial unit.

Measurements of streamflow were made at four sites along Cypress Creek in May-June 1992 (fig. 16). 

The flow velocity in Cypress Creek was too low to measure using spinner-type flow meters except at 

constrictions such as bridge abutments; therefore, a detailed seepage study was not possible. There was 

less rainfall proceeding the streamflow measurement in June and this measurement better represents 

ground-water discharge to the creek at a point in time. The measurement at site CC-1 includes groundwater discharge to Cypress Creek from outside the aquifer system (fig. 6). The large increase in June 

streamflow between sites CC-2 and CC-3 (0.068 ft3/s and 0.236 ft3/s, respectively) was caused by tributary 

flow. No tributaries enter Cypress Creek between sites CC-3 and CC-4 (0.236 fr/s and 0.353 fr/s, 

respectively); the difference between the two (0.117 ft3/s) represents ground-water discharge from the 

middle and lower alluvial units to Cypress Creek.

GEOHYDROLOGY 23
36°57'30

Base from aerial photography by Kentucky Department of Transportation, 

U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

1985

0123

I I ^ I I I T~ 

012345 KILOMETERS

4 5 MILES

J______I

EXPLANATION

INDEX MAP

CC-4.

0.042

Location of streamflow 

measurement and 

identifier

Value above line was 

streamflow on May 21, 

1992. Value below line 

was streamflow on June 

23, 1992. Values in 

cubic feet per second

Direction of flow

Figure 16. Streamflow measured in Cypress Creek, 1992.

24 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
SIMULATION OF GROUND-WATER FLOW

The flow of ground water was simulated to estimate aquifer properties, the rale of recharge to the 

aquifer, the contributing areas to discharge boundaries, and the average linear velocity at selected 

locations. The technique of simulation uses a mathematical model to predict the state of the system 

(hydraulic head and flow rates) from estimates of the hydraulic properties and boundary conditions of the 

aquifer system. The predicted state is then compared to the measured slate of the system to assess the 

accuracy of the estimates. In this simulation, the hydraulic properties and boundary conditions of the 

aquifer were represented by parameters in a nonlinear regression. The regression adjusts the model 

parameters until differences between predicted and measured hydraulic heads and flow rates were 

minimized.

Description of the Mathematical Model

The governing partial-differential equation of two-dimensional, steady-state ground-water flow in a 

heterogeneous, isotropic, confined aquifer is

where

T is the value of transmissivity along the x and y axes that are considered parallel to the

/j _ 1 major axes of hydraulic conductivity (L T );

h is the hydraulic head (L); and

W is a volumetric flux per unit area and represents sources and (or) sinks of water (LT ).

Boundary conditions were selected to represent heads or flow rates at the edges of the model area. The 

governing equation and the boundary conditions were combined in a set of finite-difference equations and 

solved using the computer program MODFLOW (McDonald and Harbaugh, 1988). The parameters of the 

nonlinear regression were estimated using a method developed by Cooley and Naff (1990) and modified 

for use with MODFLOW by Hill (1992 and 1994). The combined ground-water flow model and nonlinear 

regression is called MODFLOWP.

Model Grid and Layers

The model grid (fig. 17) is 55 rows by 80 columns and extends to the natural hydrologic boundaries of 

the area; however, the main area of interest is near Calvert City and the Calvert City Industrial Complex. 

Cells in the area of interest are irregularly spaced with an average size of 300 by 300 ft. Cells outside the 

area of interest have a maximum size of 1,500 by 1,500 ft. Two island-like areas near Kentucky Lake 

(fig. 17) where unsaturated bedrock crops out are represented in the model grid by inactive grid cells. The 

natural hydrologic boundaries of the ground-water basin are the Tennessee River, Kentucky Lake, and 

ground-water divides. Ground-water divides were represented by no-flow boundaries. The no-flow 

boundary on the southern edge of the model grid was defined by the ground-water divide beneath the area 

of perched water (fig. 6). This boundary also coincides with surface drainage boundaries. The no-flow 

boundary on the southwestern edge separates the ground-water basins that discharge into the Tennessee

SIMULATION OF GROUND-WATER FLOW 25
36°57'30

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

5 MILES

012345 KILOMETERS

INDEX MAP

EXPLANATION

River Boundary 

Drain nodes

Well nodes

| I Calvert City Industrial 

u I Complex

! Calvert City

Limit of active model 

grid; no-flow boundary

Figure 17. Extent of model grid and model boundary conditions.

26 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
River and East Fork Clarks River. The affect of Kentucky Lake on the flow system was unknown, so this 

boundary was simulated in alternative models as a constant-head and a no-flow boundary. The base of the 

aquifer system below the fractured and weathered limestone bedrock was simulated as a no-flow boundary.

The aquifer system, although modeled as a single layer, comprises two geologic layers: an upper layer 

of unconsolidated sediments and an underlying layer of weathered and fractured limestone bedrock and 

limestone residuum (fig. 5). In zone 1 (fig. 18), the aquifer in the upper layer is in sands in the McNairy 

Formation. In zone 2, the aquifer in the upper is in the lower alluvial unit. Variations in thickness of the 

aquifer were included in the model by use of a multiplication matrix, the values of which were computed 

from bedrock altitude (Amos and Wolfe, 1966; Amos and Finch, 1968) and the average altitude of the top 

of the aquifer. The hydraulic properties of the upper bedrock were not known, but were considered to be 

similar to the unconsolidated sediments on the basis of water-level data and geologic description.

Boundary Conditions and Source Terms

Boundary conditions and source and (or) sink terms are incorporated in MODFLOW through separate 

modules called packages (McDonald and Harbaugh, 1988). In this simulation the river, drain, generalhead, recharge, and well packages were used. The river, drain, and general-head packages calculate flow 

into the aquifer by multiplying the head difference between an outside source of water (such as a river) and 

the aquifer by the conductance. The conductance, defined below, is calculated by the model depending on 

the type of boundary. Conductance is defined as

C = (KA)/M (2) 

where

C is the conductance of the model cell representing the boundary (LrT );

K is the hydraulic conductivity between the source of water and the model cell (LT' 1 );

A is the area of the boundary in the model cell (L2); and

M is the distance from an outside source of water to the boundary (L).

The recharge and well packages add water to a model cell at a specified rate. Recharge is calculated as 

a linear rate of flow (LT' 1 ) that the model multiplies by the area of the cell. The well package uses the _ *j

volumetric rate of ground-water withdrawal (LT '), which was considered to be known and free of error.

The Tennessee River was simulated using the river package and the leakance (hydraulic conductivity 

divided by thickness) of the riverbed was estimated as a parameter in the regression. The river package 

allows flow to or from the river. The leakance of the riverbed was multiplied by the area of the river in 

each cell to calculate the conductance of the riverbed. The area of the riverbed in each cell was calculated 

by overlaying the grid on a map of the river.

Cypress Creek was simulated using the drain package and the conductance of this boundary was 

estimated as a parameter. The drain package only allows flow into the drain. If the calculated water level 

was below the altitude of the base of the creek, the boundary did not affect the model; thus, the point of 

intersection between the creek bed and the aquifer was determined in the model.

SIMULATION OF GROUND-WATER FLOW 27
88°15'

37°00'

36°57'30"

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

5 MILES

5 KILOMETERS

EXPLANATION

INDEX MAP

Zone 1--Cells used in definition 

of recharge and transmissivity 

that correspond to alluvium

Zone 2--Cells used in definition 

of recharge and transmissivity 

that correspond to the McNairy 

Formation

Limit of active model 

grid; no-flow boundary

Figure 18. Parameter zones used in model.

28 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
Recharge to the lower alluvial unit from the middle alluvial unit was simulated using the general-head 

package and the conductance of this boundary was estimated as a parameter. This package allows flow to 

or from the boundary. The head in the middle alluvial unit was considered to be constant at 325 ft based on 

limited water-level data collected in June 1992. In reality, head in the middle alluvial unit was probably not 

constant, so this assumption is a source of uncertainty in the model. Recharge to the alluvium also was 

modeled as a specified flow in an alternative model discussed later in this report.

Recharge to the McNairy Formation was simulated using the recharge package and the rate of recharge 

was estimated as a parameter. Flow from the boundary in this package is an areally distributed quantity 

that does not depend on head in the aquifer; it depends on factors such as rainfall, soil type, and 

topography. The parameter is an estimate of the average over zone 1 and does not include differences 

between recharge from sand and gravel or from loess. Recharge to the McNairy Formation from Kentucky 

Lake was simulated as a constant-head boundary in an alternative model.

Model Calibration

The model was calibrated to 55 head measurements and 3 streamflow measurements made in 

June 1992 that represent average flow conditions. Many of the 157 original head measurements were 

omitted from the calibration data set because (1) they were made in wells completed in shallow zones 

above the main aquifer system, (2) there was recent pumping in or near the well, or (3) because there was a 

large number of wells in a small area, as was the case with observation wells in the industrial complex. 

Hydrographs (figs. 11 and 12) indicated that June 1992 marked the end of seasonal recharge to the aquifer 

system and the beginning of the gradual decline in water levels as the aquifer system drained; therefore, 

based on this limited evidence, the water levels in June 1992 were assumed to approximate mean annual 

water levels.

Streamflow measurements were used to compute gains in streamflow along Cypress Creek (fig. 16). 

Flow to Cypress Creek at CC-1 (0.042 ft3/s) included streamflow from a drainage area outside the model 

area (fig. 5). The contribution from within the model area was estimated to be one-half the measured flow 

(0.021 ft3/s) based on the approximately equal sizes of the two drainage areas. Between CC-2 and CC-3 

the flow to Cypress Creek was from tributary flow and was not included in the calibration data set. The 

difference between CC-3 and CC-4 (0.117 ft/s) was equal to the ground-water discharge to Cypress Creek 

in this reach. It was assumed that this value mainly represented contributions to discharge from the lower 

alluvial unit and not the middle alluvial unit, which was not simulated in the model.

Calibration Procedure

Ground-water flow models are typically calibrated by the trial-and-error method, in which a series of 

adjustments to the model is made until a reasonable match between calculated and observed heads and 

flows is produced. Nonlinear regression makes calibration more efficient and objective by adjusting 

parameters automatically, using the response of the model to changes in parameter values as a guide, until 

the weighted sum-of-squares error (SSE) is minimized. The statistical framework of this process can be 

used to determine the validity of the regression, the reliability of the parameter estimates, and the 

likelihood that a given model more accurately represents the system than an alternative model. The method 

is described by Cooley and Naff (1990) and Hill (1992 and 1994).

SIMULATION OF GROUND-WATER FLOW 29
The nonlinear regression minimizes the weighted SSE, given by

n ry

ccr V t 1/2 x 2

SSE = > (w. e.) (3)

where

et is the difference between the observed and calculated values of measurement / (L);

1 /2 Wj ' is the square root of the weight assigned to the error in the observed value of 

measurement i (L); and

n is the number of observations.

Weights were defined and calculated according to procedures in Hill (1992) to account for 

measurement error in the observed values and for the difference in units between heads and flows (ft and 

ft3/d, respectively). Weights of the heads and flows were calculated as a2/ a2,, where a2 was the common 

error variance and c2, was the variance of measurement error of the /th head or flow. Measurement errors 

for heads and flows were considered to be normally distributed. The standard deviations (the square root of 

the variance) of measurement errors were considered to be equal to the accuracy of the measurement, 

stated below, divided by the critical value at the 95-percent confidence interval (1.96). The accuracy of 

head measurements was considered to be the accuracy of the altitude of the measuring point, which was 

1 ft for surveyed altitudes and 5 ft for altitudes estimated from a 1:24,000-scale topographic map. The 

accuracies of each streamflow measurement, which were between 5 and 10 percent, were added to obtain 

the accuracies of gains in streamflow. The common error variance must equal the error variance computed 

by the model, so the final step in model calibration was to set the common error variance equal to the 

computed error variance. This procedure did not change weights of the heads, but did change weights of 

the flows. The common error variance used in the final model was 9.27 ft2 .

The abbreviations used for parameters are listed in table 1. Not all parameters were estimated in every 

model; some parameters were estimated in alternative models. All parameters except recharge were 

estimated as their log-transformed values, which ensured that the parameter value was always positive. 

The criteria to determine the final parameter set were (1) parameter estimation converged, (2) the weighted 

SSE was minimized, (3) correlation between parameters was minimized, (4) variance of the estimates was 

minimized, and (5) as many parameters were estimatedt as possible. Satisfying all these criteria 

simultaneously generally was not possible and some judgment was required to determine the set of 

parameters that was finally estimated in the calibrated model. Parameters of the calibrated model were Tl, 

T2, RCH1, KRB, KDR, and GHB1.

30 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
Table 1. Model parameters estimated using nonlinear regression, Calvert City, Kentucky

Parameter Abbreviation

Transmissivity of the McNairy Formation Tl

Transmissivity of the lower alluvial unit T2

Transmissivity of faults in the bedrock T3

Recharge to the McNairy Formation RCH1

Recharge to the lower alluvial unit RCH2

Leakance of the riverbed KRB

Conductance of the bed of Cypress Creek KDR 

Conductance of the general-head recharge boundary over the alluvium GHB1

Estimates of Model Parameters

The optimal parameter values obtained from nonlinear regression (table 2) were close to expected 

values. The estimated transmissivities were within the range of reported values. The transmissivity of the 

McNairy Formation was expected to be lower than the transmissivity of the lower alluvial unit; however, 

they were estimated to be almost equal. Although the hydraulic conductivity of the lower alluvial unit is 

probably higher than of individual sand beds of the McNairy Formation, the McNairy Formation can be 

thicker; therefore, the transmissivity of the two aquifers, which is the product of hydraulic conductivity and 

aquifer thickness, can be similar.

Table 2. Optimal parameter values obtained from 

nonlinear regression, Calvert City, Kentucky

2 [ft /d, square feet per day; ft/d, feet per day]

Parameter

Tl

T2

KDR

KRB

RCH1

GHB

Estimate

19,000

20,000

143

.00592

.00269

.391

Units

ft2/d

ft2/d

ft2/d

d- 1

ft/d

ft2/d

The rate of recharge to the lower alluvial unit was much lower than recharge to the McNairy 

Formation. Although recharge to the lower alluvial unit was simulated as a general-head boundary, the 

quantity of interest was the average rate of recharge. The rate of recharge to the alluvium was 

0.000484 ft/d (2.1 in./yr) under simulated conditions. This rate would change if head in the aquifer

SIMULATION OF GROUND-WATER FLOW 31
changed, as would occur if pumping increased or decreased. The rate of recharge to the McNairy 

Formation was expected to be higher than 7 in./yr (Davis and others, 1973) as discussed in the section on 

recharge. The rate of recharge to the McNairy Formation was 0.00269 ft/d (11.8 in./yr).

Previous estimates of streambed leakance were not available, but geologic descriptions of the texture 

and thickness of the deposits can be used to determine a reasonable value. The riverbed is composed of 

sands with lenses of organic-rich fine material and is an average of 20 ft thick (Dames and Moore, 1991). 

In vertical flow through layers, the hydraulic conductivity of the sequence is dominated by the least 

permeable layer; therefore, the mean vertical hydraulic conductivity of silt from 39 samples reported by 

Morris and Johnson (1967), 0.08 ft/d, was used to compute a reasonable leakance value. Dividing 0.08 ft/d 

by the average thickness yields a reasonable estimate of leakance of 0.004 d' 1 . The range of reasonable 

vertical hydraulic conductivity is large considering the limited data available for the site, but the modelcalculated value of 0.00592 d" 1 seems reasonable.

Hydraulic-Head Distribution

The values of hydraulic head computed by the calibrated model (fig. 19) were similar to those 

observed. The apparent difference with the observed June 1992 potentiometric surface (fig. 10) in the 

upper reach of Cypress Creek was because of the coarseness of the model grid. The hydraulic gradient on 

either side of the upper reach of Cypress Creek was steep and occurred within a model cell; thus, contours 

based only on calculated heads at cell centers did not show the effects of ground-water discharge to 

Cypress Creek.

The model should not be biased; that is, it should not systematically overpredict or underpredict head. 

Based on their distribution, the weighted residuals seem to be unbiased (fig. 20). The weighted residuals 

ranged from -9.92 to 8.4 ft. Only five values were less than -5 ft or greater than 5 ft and 71 percent of all 

weighted residuals were between -3 and 3 ft. The residuals also were plotted (fig. 21) on a map to 

determine if there was a spatial bias in the model. The weighted residuals of the calibrated model seem 

randomly distributed over the modeled area with the exception of the western part of the modeled area 

north of Cypress Creek. In this area, there was a tendency for the model to overpredict head. This could be 

a result of unknown variation of aquifer properties that was not simulated.

For the regression to be valid, the distribution of the weighted residuals should resemble a standard 

normal distribution. The correlation coefficient between the ordered weighted residuals and random 

independent, normal deviates was used to test for independent, normally distributed weighted residuals 

(Hill, 1992). The critical value of the correlation coefficient at the 5-percent significance level is 0.961. 

The correlation coefficient computed from the regression, 0.965, was greater than the critical value and the 

residuals were considered to result from a random normal distribution.

Rate of Flow

Flow to the upper reach of Cypress Creek was calculated to be 0.00251 f^/s compared to an estimated 

flow of 0.00211 fr/s; thus, the calculated flow was 19 percent too high. Flow to the lower reach of Cypress 

Creek between measurement sites CC-3 and CC-4 (fig. 16) was calculated to be 0.116 ft3/s compared to a 

measured flow of 0.117 ft3/s. The model predicted, as expected, that ground water did not discharge from 

the aquifer to the middle reach of the creek. The total rate of flow to Cypress Creek for the calibrated 

model was 0.597 ft3/s, most of which occurred downstream of measurement site CC-4. The total flow of 

ground water to the Tennessee River computed by the calibrated model was 22.3 ftVs.

32 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
37°00' - -

36°57'30"

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

5 MILES

012345 KILOMETERS

EXPLANATION

INDEX MAP

 320 CALCULATED POTENTIOMETRIC 

CONTOUR-Shows altitude at 

which water would have stood 

in tightly cased wells, June 1992. 

Countour interval 10 feet. Datum 

is sea level.

Limit of active model 

grid; no-flow boundary

Figure 19. Potentiometric surface calculated with the calibrated model.

SIMULATION OF GROUND-WATER FLOW 33
WEIGHTED DIFFERENCE BETWEEN PREDICTED AND OBSERVED HEADS, IN FEET

Figure 20. Weighted differences between calculated and observed heads, Calvert City, 

Kentucky.

34 Geohydrology and Simulation of Ground-Water Flow In the Aquifer System near Calvert City, Kentucky
22'30"

36°57'30

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

234 SMILES

 I ^ r I 1 I . I.

012345 KILOMETERS

INDEX MAP

EXPLANATION

Weighted residual greater than 5 feet 

Weighted residual 1 to 5 feet 

Weighted residual 0 to 1 feet 

Weighted residual -1 to 0 feet 

Weighted residual -5 to -1 feet

Weighted residual less than -5 feet

Limit of active model 

grid; no-flow boundary

Figure 21. Spatial distribution of weighted residuals.

SIMULATION OF GROUND-WATER FLOW 35
Alternative Models

Values of the parameters of the calibrated model cannot be changed without increasing the SSE; 

therefore, if an improvement to the model is desired, the number of parameters, the si/e and shape of 

parameter /ones, or model design must be changed. Although the calibrated model was a reasonable 

representation of flow in the aquifer, four other plausible models were posed to determine if they were 

better than the calibrated model. A particular model can be demonstrated to be better than alternative 

models, but it cannot be proven to be the best model. The si/e and shape of parameter /ones were based on 

geologic maps and were considered to be correct, so alternative models were created by changing how 

boundaries were represented and by adding additional geologic features.

Two of the alternative models were used to test different representations of boundary conditions. In 

alternative model CALO, recharge to the alluvium was simulated as a specified-flow boundary; in other 

words, the amount of recharge did not depend on head in the aquifer. The model CAL1 was the calibrated 

model, with recharge to the alluvium represented as a head-dependent flow. Kentucky Lake supplies water 

to wells near Kentucky Lake through weathered bedrock. To simulate the possible effect of Kentucky Lake 

as a source of water to the aquifer system in areas not near the lake, CAL2 was modified from CAL1 by 

representing the lake as a constant-head boundary.

Two alternative models simulated the effects of change in transmissivity in the aquifer system. Several 

faults that could hypothetically represent zones of high transmissivity relative to the surrounding alluvium 

underlay the area (fig. 4). A well at ISP that was used in an aquifer test to compute an anomalously high 

transmissivity of 174,000 ft2/d, is near a fault. To test the possibility that the fault is transmissive, a third 

transmissivity zone (T3) along the fault was simulated in the alternative models CAL3 and CAL4. In 

CAL3, recharge to the alluvium was a specified flow (RCH2). In CAL4, recharge to the alluvium was a 

head-dependent flow. Neither CAL3 nor CAL4 included a constant-head boundary at Kentucky Lake.

Parameters for the alternative models were estimated through the regression procedure (table 3). The 

alternative models were compared using the SSE that was computed by MODFLOWP. In some 

circumstances, smaller values of the SSE indicate better models (Hill, 1992). On this basis, models that 

simulate head-dependent recharge to the alluvium (CAL1 and CAL4) are better models than the others. 

Residuals from the alternative models are similar, indicating that in general the alternative models have the 

same source of model uncertainty as the calibrated model. Correlation between parameters was similar in 

the alternative models. In all models tested, recharge to the McNairy Formation and the transmissivity of 

the alluvium have a correlation coefficient of greater than 85 percent.

Water budgets computed by the regression procedure for the alternative models are presented (table 4) 

to illustrate their differences. Total flow through the aquifer was 2.31 to 2.78 Mgal/d in all alternative 

models. When transmissive faults were simulated (CAL3 and CAL4), there was less recharge to the lower 

alluvial unit because pumping wells were able to capture water from a larger area. When recharge to the 

lower alluvial unit was treated as a head-dependent boundary (CAL1 and CAL4), the average rate of 

recharge was less than when it was treated as a specified flow (CAL 0 and CAL3). In CAL1 and CAL4, 

more water was supplied near pumping wells the head difference between the aquifer and the external 

source of water was greater and the average rate of recharge was not increased by the regression to 

compensate, as was the case in CALO and CAL3. The regression procedure adjusted parameters of the 

alternative models so that lower recharge correlated with lower flow to the Tennessee River, rather than as 

less flow to Cypress Creek.

36 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
Table 3. Parameter values for alternative models computed by nonlinear regression, Calvert City, Kentucky

[CAL, calibration; ft2/d, square feet per day, NE, not estimated; d-1, inverse days; ft/d, feet per day; CAL1 is calibrated model]

Parameter

Tl,ft2 /d

T2, ft2 /d

T3, ft2/d

GHB, ft2/d

KDR, ft2/d

KRB, d' 1

RCH1, ft/d

RCH2, ft/d

Sum-ofsquares

error

CALO

21,000

26,000

NE

NE

114

.00592

.00274

.000807

513

CAL1

19,000

20,000

NE

.391

143

.00564

.00269

NE

473

CAL2

32,000

17,000

NE

.0874

48

.00258

.00184

NE

762

CAL3

19,000

22,000

270,000

NE

113

.00436

.00243

.000774

509

CAL4

17,000

18,000

160,000

.376

144

.00504

.00250

NE

467

Table 4. Water budget computed by alternative models, Calvert City, Kentucky

[CAL, calibration; NE, not estimated; in millions of cubic feet per day]

Source or sink

Flow from Kentucky Lake

Net recharge

Total in

CALO

NE

2.79

2.79

CAL1

Inflow

NE

2.46

2.46

CAL2

0.41

1.45

1.86

CAL3

NE

2.45

2.45

CAL4

NE

2.31

2.31

Outflow

Wells

Cypress Creek

Tennessee River

Total out

.48

.048

2.26

2.79

.48

.052

1.93

2.46

.48

.027

1.36

1.87

.48

.052

1.93

2.46

.48

.052

1.77

2.30

Reliability of Estimates

Parameter estimates obtained through nonlinear regression are likely to be reliable if the estimates are 

precise and uncorrelated, the residuals are random and normally distributed, and the model is correct. 

Statistical techniques used to analyze linear regression can be used for nonlinear regression if the model is 

linear near the estimated parameter values. In particular, confidence intervals can be computed from the 

standard deviation of the estimates, and are a convenient means of graphically displaying the precision of 

the estimates. The modified Beale's measure presented by Cooley and Naff (1990) and adapted for use

SIMULATION OF GROUND-WATER FLOW 37
with MODFLOWP by Hill (1994) was used to determine whether this criterion was met. The critical value 

of Beale's measure was computed using the F distribution for 57 observations, 6 parameters, and a 

95-percent confidence interval. Beale's measure, 2.71, computed from the calibrated model (CAL1) is 

greater than the critical value of 0.45; therefore, the calibrated model is highly nonlinear near the estimated 

parameter values and confidence intervals were not calculated.Because confidence intervals were not 

calculated, the precision of the estimates was assessed by using pseudo-coefficients of variation 

(M.C. Hill, U.S. Geological Survey, written commun., 1994). Pseudo-coefficients of variation were 

needed because log-transformed values were estimated, and their coefficients of variation were not directly 

comparable. Pseudo-coefficients of variation were computed as follows:

eb -ea

X = - (4)

\f

where

X is the pseudo-coefficient of variation (dimensionless); 

e is the base for natural logarithms (dimensionless);

b is the upper 95-percent linear confidence-interval boundary calculated for the 

log-transformed parameter (ft);

a is the lower 95-percent linear confidence-interval boundary calculated for the 

log-transformed parameter (ft); and

p is the parameter estimate (ft).

In comparing parameter estimates, a smaller pseudo-coefficient of variation indicated a more precisely 

estimated parameter. Estimates of parameters to which the model was insensitive had high pseudocoefficients of variation. The pseudo-coefficients of variation of the calibrated model ranged from 0.17 to 

13.1 (table 5). All parameters were estimated with approximate equal reliability except the transmissivity 

of faults and GHB in CAL2. The precision of the estimates could be reduced by collection of additional 

data from areas in which the model is sensitive to the parameter. For example, model sensitivity to 

hydraulic conductivity in an area where there is a steep hydraulic gradient and few head measurements 

could be improved by additional head measurements.

Recharge to the McNairy Formation and the transmissivity of the alluvium were correlated at r = 0.91 

in the calibrated model (table 6). Correlation between recharge and transmissivity was expected because 

the ground-water-flow equation can be written in terms of their ratio (Hill, 1992). Unless independent 

information on recharge or transmissivity is available, the regression cannot be used to distinguish each 

individual component of the ratio.

The effects of correlated parameters of the calibrated model were investigated by holding the 

transmissivity of the alluvium constant at minimum and maximum reasonable values, 5,000 and 

50,000 ft2/d, respectively, and the other parameters were estimated. Estimates of recharge (RCH1) were 

0.00119 ft/d (5 in./yr) and 0.00559 ft/d (24.5 in./yr) for the two models, respectively, and were consistent 

with the correlation between recharge and transmissivity. All but one of the other estimated parameter 

values differed slightly from the parameter values in table 3 because the correlation was not perfect 

(r = 1.00). One estimated parameter, KRB in the first simulation, differed greatly from the parameter 

values in table 3 (by a factor of 21,000). This increase was possibly caused by nonlinearity in the model.

38 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
Table 5. Pseudo-coefficients of variation for alternative models computed by nonlinear regression, 

Calvert City, Kentucky

[CAL, calibration; NE, not estimated; CAL1 is calibrated model]

Parameter

Tl

T2

T3

GHB

KDR

KRB

RCH1

RCH2

Sum-of-squares 

error

CALO

0.23

.26

NE

NE

.25

.40

.30

.63

513

CAL1 CAL2

0.23 0.39

.21 .23

NE NE

.37 13.1

.31 .24

.39 .48

.17 .23

NE NE

472 763

CAL3

0.24

.33

2.01

NE

.24

.49

.31

.65

509

CAL4

0.24

.28

4.53

.43

.31

.49

.20

NE

467

Table 6. Correlation and covariance matrix for optimum values obtained through nonlinear regression (CAL1), 

Calvert City, Kentucky

[Shaded portion is scaled variance/covariance matrix; unshaded portion is the correlation matrix]

Parameter

KRB

RCH1

Tl

KDR

T2

GHB

KRB

IftMfi

.45

.24

.71

.30

.67

RCH1 T1

7.507 x 1 0'5 1 .886 x Ifr2

'l;i^l^iiJ$^MjjS

.55 '" ' gg^lg|

.16 -.16

.91 .26

-.30 -.33

KDR

: :;:;:;:;, 7 534:'i;:103 ::;.-;

2.I64XM

-1.037x10^

liPII^I

.23

.70

T2 GHB

2,288 xlO4 8.109 x 10"2

9 152 x 10 ^ -4.732 x 10 5

§ii^i^§ji^;j^^

1.477 x 10"2 6.99 x 10^

^^^^jyij^^ff^

-.30 1151

Simulated Ground-Water Flow

A particle-tracking technique was used to illustrate the simulated ground-water flow system. First, the 

volumetric flow rates generated by MODFLOW were used to define a continuous velocity field within 

each model cell by computing the average linear velocity vectors. Particles, which represent a tiny volume 

of ground water, then were selectively placed in model cells and allowed to move within the velocity field. 

The location and cumulative time of travel were recorded for each particle; these characteristics were used 

to depict the contributing areas to discharge boundaries and the time required for a particle to reach a 

discharge boundary. Knowledge of the direction and rate of flow beneath waste-disposal units within the

SIMULATION OF GROUND-WATER FLOW 39
industrial complex could be used in the management and abatement of the ground-water resource; 

therefore, the average linear velocity was computed and tabulated at selected locations in the industrial 

complex. The average linear velocity is defined as

0 = -T- ' (5) 

An

where

v is the average linear velocity (ft/d);

Q is the volumetric flow, (ft3/d);

A is the area of flow (ft2); and

n is the porosity of the aquifer, dimensionless.

The location of particles at the start of the simulation must be specified by the investigator. In this 

study, one particle was placed at the center of the top face of all cells. As each particle moved to a 

discharge boundary, the beginning and ending coordinates were recorded. The beginning locations of 

particles were grouped by discharge boundary to produce maps showing the contributing areas (fig. 22).

In this study, it was specified by the investigator that particles would not discharge to weak sinks. 

Model cells representing the Tennessee River and wells at company A and the Calvert City Water 

Company were strong sinks because all water that flowed into the model cell discharged through the 

boundary. In a model cell that is a weak sink, some water discharges through the boundary, but some water 

flows into an adjacent model cell. For example, Cypress Creek does not extend to the base of the aquifer, 

so water near the top of the aquifer can discharge to the creek and water at the base of the aquifer can flow 

beneath the creek to the other side. The weak-sink effect also was evident at model cells representing wells 

of companies B and C (fig. 22). Whether a well is a weak sink or not is solely a function of the model grid 

design and does not indicate the efficiency of a particular well in regard to capturing ground-water flow.

Eliminating weak sinks from the simulation was reasonable because the primary concern was the 

contributing area to the public-water supply wells at the Calvert City Water Company. Additional data on 

the vertical hydraulic gradients would have been necessary to determine the nature of particle discharge to 

weak sinks. If these data are incomplete and particles were allowed to discharge to weak sinks, some 

particles that would otherwise have discharged to the water-company wells would have discharged 

incorrectly to the weak sink and the contributing area to the wells would be too small.

Contributing Areas and Times of Travel

Contributing areas for the wells at ISP and the Calvert City Water Company (fig. 22) were 

1.27 x 108 ft2 (4.54 mi2) and 7.14 x 107 ft2 (2.56 mi2), respectively. The two areas are contiguous and 

extend from the wells to the southern boundary of the model. Model results indicate that particles 

discharge to Cypress Creek from a small area along its lower reach but not along its upper reach. Particles 

did not discharge to the upper reach of Cypress Creek or to wells at B.F. Goodrich and Estron because they 

are weak sinks. Particles in most of the area discharge to the Tennessee River.

40 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
22'30"

37°00' - *

36°57'30

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

5 MILES

I 

5 KILOMETERS

INDEX MAP

EXPLANATION

Cells in which a particle at the 

center discharges to Tennessee 

River

Cells in which a particle at the 

center discharges to Cypress 

Creek

Cells in which a particle at the 

center discharges to wells at 

Calvert City Water Company

Cells in which a particle at the 

center discharges to wells at 

company A

Calvert City industrial complex

Limit of active model 

grid; no-flow boundary

Pumped wells

Figure 22. Contributing areas of discharge boundaries computed with the calibrated model, 

1992 conditions.

SIMULATION OF GROUND-WATER FLOW 41
An additional particle-tracking analysis was performed to simulate the effects of pumping from new 

wells drilled by the Calvert City Water Company (fig. 23). For this simulation, which approximated 1994 

conditions (Jerry Devine, Calvert City Water Company, oral commun., 1994), the number of wells and 

rates of pumping were kept the same as for the calibrated model and only the locations were changed. 

Contributing areas to the wells at ISP and the Calvert City Water Company under predicted 1994 

conditions were 1.31 x 108 ft2 (4.71 mi2) and 9.79 x 1C)7 ft2 (3.51 mi2), respectively. The main effect of the 

new wells was to split the contributing areas between wells at the Calvert City Water Company and ISP. 

This separation reduced the hydraulic gradient and thus reduced the rate of flow causing the 

contributing areas to increase relative to the calibrated model. The contributing area to the wells at the 

Calvert City Water Company also increased because the bedrock outcrop south of the new location caused 

particles from a large area to be channeled toward the wells.

The contributing area to wells at the Calvert City Water Company in the calibrated model did not 

include the Calvert City Industrial Complex, under simulated conditions. The contributing area to wells at 

the Calvert City Water Company under predicted 1994 conditions was farther removed from the industrial 

complex than for the calibrated model.

The time required for a particle to move through the aquifer from its point of recharge to its point of 

discharge depends on where the particle enters the aquifer. The cumulative times of travel, computed by 

MODPATH, range from a few days to more than 100 years. Bands of equal time of travel are parallel to 

surface-water discharge boundaries and are approximately concentric about well discharge boundaries 

(fig. 24). The time of travel increases with distance from the discharge boundary. The time required for 

water to move through the unsaturated zone above the aquifer is not included in these estimates.

Average Linear Ground-Water Velocities

The average linear velocity was computed at the center of model cells spaced evenly throughout the 

industrial complex area (fig. 25 and table 7). MODPATH computes the orthogonal components of the 

average linear velocities in the x and y directions. The rate of horizontal flow in the aquifer is

where

1 V is the rate of horizontal flow in the aquifer (LT),

Vx is the average linear velocity in the x direction (LT ),

V is the average linear velocity in the y direction (LT1 ).

42 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
36°57'30

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

SMILES

5 KILOMETERS

INDEX MAP

"Study Area

_r -

EXPLANATION

Cells in which a particle at the 

center discharges to Tennessee 

River

Cells in which a particle at the 

center discharges to Cypress 

Creek

Cells in which a particle at the 

center discharges to wells at 

Calvert City Water Company

Cells in which a particle at the 

center discharges to wells at 

company A

Calvert City

[___j Calvert City industrial complex

 Limit of active model

grid; no-flow boundary

Pumped wells

Figure 23. Contributing areas of discharge boundaries computed with the calibrated model, 

1994 conditions.

SIMULATION OF GROUND-WATER FLOW 43
22'30"

37°00' - -

36°57'30"

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

0 1 2

I______I______I

5 MILES

INDEX MAP

5 KILOMETERS

EXPLANATION

ZONE OF EQUAL TIME OF TRAVEL

Zone with particle time of travel 0-2 

years to discharge area.

Zone with particle time of travel 2-5 

years to discharge area.

Zone with particle time of travel 5-10 

years to discharge area.

Zone with particle time of travel 10-20 

years to discharge area.

Zone with particle time of travel greater 

than 20 years to discharge area. 

Maximum is 123 years.

Limit of active model grid; no-flow 

boundary

LINE OF EQUAL TIME OF TRAVELBased on velocities calculated by 

calibrated model. Contour interval 

variable

Pumped wells

Figure 24. Time of travel computed with the calibrated model, 1994 conditions.

44 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
88°15'

36°57'30

Base from U.S. Geological Survey, digital data, 1:100,000, 1983 

Stateplane projection, NAD27

0 1

 I L

5 MILES

012345 KILOMETERS

INDEX MAP

EXPLANATION

Location of row and column 

(nodes) for which average 

linear velocity is calculated

Calvert City

Calvert City industrial complex

Limit of active model 

grid; no-flow boundary

Figure 25. Location of nodes for which average linear velocity is calculated, 1994 conditions.

SIMULATION OF GROUND-WATER FLOW 45
Table 7. Magnitude and direction of the average linear velocity at selected model cells under 1994 conditions, 

Calvert City, Kentucky

[ALV, average linear velocity; ft/d, feet per day; 0 degrees or 360 degrees refers to north on the State Plane projection, South Zone, North 

American Datum, 1987]

Row

5

10

15

20

25

30

35

5

10

15

20

25

30

35

5

10

15

20

25

30

35

10

15

20

25

30

35

10

Column

30

30

30

30

30

30

30

35

35

35

35

35

35

35

40

40

40

40

40

40

40

45

45

45

45

45

45

50

ALV 

(ft/d)

1.94

1.50

1.53

2.28

2.52

2.60

2.64

2.02

1.21

1.17

4.47

3.24

2.74

2.70

1.95

1.08

.90

3.21

2.91

2.65

2.76

1.44

1.51

2.03

2.10

2.34

2.73

1.94

Azimuth 

(degrees)

177.1

4.2

20.7

23.6

15.3

9.5

7.6

4.9

15.6

62.9

44.0

13.7

9.5

7.9

13.8

11.7

105.0

145.7

176.5

8.0

10.4

15.9

1.4

174.7

2.4

13.0

18.6

31.7

Row

15

20

25

30

35

10

15

20

25

30

35

15

20

25

30

35

15

20

25

30

35

20

25

30

35

30

35

Column

50

50

50

50

50

55

55

55

55

55

55

60

60

60

60

60

65

65

65

65

65

70

70

70

70

75

75

ALV 

(ft/d)

2.02

1.98

1.83

2.06

2.77

2.96

2.62

2.05

1.74

1.79

2.37

2.92

2.15

1.69

1.49

1.43

2.69

1.83

1.54

1.51

1.28

1.65

1.71

1.67

1.42

1.76

1.83

Azimuth 

(degrees)

22.4

13.2

16.4

24.4

31.8

56.5

22.0

20.0

25.6

36.4

56.6

12.2

20.7

30.4

44.8

66.5

6.1

24.9

38.2

49.6

61.0

41.8

50.0

57.3

62.4

62.0

60.4

46 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
The direction of flow with respect to the positive x axis is

V

6 = atan , (7) 

^ x'

where

6 is the direction of flow with respect to the positive x axis, in degrees, and 

atan x is the arctangent of argument x.

The azimuth is obtained by adding 90° to 8. The model grid was oriented parallel to the axes of the State 

Plane Projection, South Zone, North American Datum, 1987. The azimuth is reported with respect to north 

in this projection. The average linear velocities ranged from 0.90 to 4.47 ft/d.

Limitations of Particle Tracking

Particle tracking is subject to uncertainty that stems from variance in the parameters of the model. The 

particle tracking presented here was deterministic; that is, one set of parameters was used resulting in one 

outcome of the particle tracking. Although the optimal parameter values (the ones that minimize the SSE) 

were used in the model, uncertainty in the parameters is translated into uncertainty in the size and shape of 

the contributing area. Some statistical measures can be derived from the regression that indicate the 

uncertainty of the computed parameters and thus of the contributing area. The variance-covariance matrix 

is a "measure of the range over which the respective parameter may be varied to produce a similar solution 

for the dependent variable as that obtained using . . . [the optimal parameter set. The variance-covariance 

matrix also] . . . gives an estimate of the degree of linear dependence of one parameter on another 

throughout the course of repeated experiments if such experiments were to be carried out" (Cooley and 

Naff, 1990). This type of analysis is beyond the scope of this report.

An additional parameter not represented in the variance-covariance matrix, porosity, was required to 

compute the average linear velocity. Porosity was estimated at 20 percent based on geologic description of 

the lower alluvial unit and sand layers in the McNairy Formation. The actual value of porosity is not a 

source of uncertainty, it is simply a constant factor in the calculation of average linear velocity. The 

uncertainty stems from the assumption that one value characterizes the average porosity in both the 

alluvium and the McNairy Formation. Based on the geologic description of each aquifer one is sand and 

gravel and the other is sand the porosities are probably similar. Because of the interbedded character of 

the McNairy Formation, true flow paths would be more complex in the vertical direction than was 

simulated in the calibrated model.

The resolution of contributing areas was limited by the number of particles used. Using more particles 

would more closely approximate the true boundaries that lie between the center of cells on either side of 

the boundary of the contributing areas (figs. 22 and 23). A method that would produce higher resolution of 

the contributing areas would be to place particles in each cell of the model grid that is a discharge point. 

The particles would be assigned to sides of the grid cell in proportion to the amount of flow through that 

side, as determined by MODFLOW. In this method, the path of the particle would be tracked backwards to 

the point of recharge to delineate the contributing area. The resolution of the contributing areas in the 

calibrated model was deemed sufficient for the purposes of this study and represents a compromise 

between higher resolution and increased computer time.

SIMULATION OF GROUND-WATER FLOW 47
The movement of particles differs from the movement of a conservative solute in ground water, such 

as a contaminant. A discrete mass of solute (for example, a contaminant plume) can he represented by a 

group of particles, the distribution of which will be affected by variation in the velocity of ground-water 

flow. This variation causes solute dispersion on a wide range of scales: from within individual pores in the 

aquifer, within geologic units, and between geologic units. The net effect of dispersion is that the solute 

mixes with uncontaminated ground water and becomes diluted. Apparent dispersion of a solute also can 

occur through temporal changes in the velocity of ground-water flow at boundaries, such as changes in 

river stage. Change in river stage causes more spreading and dilution of the solute as it moves closer to the 

river (Starn, in press). Particle tracking by itself does not account for dispersion. The average linear 

velocity is the rate at which the center of mass of the group of particles moves, provided there are no 

biological, physical, or chemical processes that alter the concentrations of the solute.

48 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
SUMMARY AND CONCLUSIONS

The aquifer at Calvert City, Ky., consists of alluvial sediments of Quaternary age, unconsolidated 

sediments of the McNairy Formation of Cretaceous age, and weathered limestone bedrock of 

Mississippian age. The alluvial sediments are composed of three units: the lower gravel and sand unit, the 

middle interbedded sand and clay unit, and the upper clayey lacustrine deposits. The McNairy Formation is 

composed mostly of unconsolidated sand interbedded with silty clay. The alluvium and the McNairy 

Formation are underlain by fractured and weathered bedrock. Several faults displace strata of the McNairy 

Formation and the bedrock.

Recharge to ground water is from the uplands south of Calvert City underlain by the McNairy 

Formation. Ground water discharges to the Tennessee River, to the upper and lower reaches of Cypress 

Creek, and to pumped wells. The saturated thickness of the alluvium is 80 to KK) ft, which includes the 

middle unit that forms a leaky confining layer over the aquifer. Transmissivity of the alluvium determined 

from aquifer tests ranges from 2,940 to 174,000 ft2/d (Davis and others, 1973). The maximum value may 

reflect the presence of localized transmissive faults in the bedrock.

Ground-water flow in the aquifer was simulated in a two-dimensional, steady-state model. The 

hydraulic properties and boundary conditions of the aquifer were represented by parameters in a nonlinear 

regression. The parameters of the regression were the transmissivity of the alluvium, the transmissivity of 

the McNairy Formation, conductance of a general-head boundary representing recharge to the alluvium, 

specified flow representing recharge to the McNairy Formation, conductance of the bed of Cypress Creek, 

and leakance of the bed of the Tennessee River. The regression procedure calibrated the model to 

55 measurements of head in the aquifer and 2 measurements of streamflow in Cypress Creek. The optimal 

parameters of the model were computed such that the sum of the weighted difference between observed 

and calculated heads and flows was minimized. Weights used in the regression account for error in the 

measurements of heads and flows and for the difference in units between heads and flows (ft and fr/d).

Transmissivity of the alluvium and the McNairy Formation were estimated to be approximately equal 

(20,000 and 19,000 ft2/d). The rate of recharge to the McNairy Formation was estimated to be 0.00269 ft/d 

(11.8 in./yr). Although the rate of recharge to the alluvium was estimated using a head-dependent flow 

boundary, the average rate of recharge was 0.000484 ft/d (2.1 in./yr). Higher estimated recharge to the 

McNairy Formation than to the alluvium supported the conceptual model that the upper alluvial unit limits 

recharge to the alluvium.

The calculated pattern of hydraulic head conformed to the expected pattern. The weighted residuals 

(difference between observed and predicted values) ranged from -9.92 to 8.4 ft. Only 5 values (out of the 

total of 55) were less than -5 ft or greater than 5 ft and 71 percent of all weighted residuals were between 

-3 and 3 ft. The weighted residuals of the calibrated model were distributed randomly over the modeled 

area except in the northwestern part, where simulated heads were consistently greater than observed heads. 

This may have been the result of unknown variations in aquifer properties.

Alternative models were tested to determine if some model other than the calibrated model provided a 

better fit to the observed data or provided more reliable parameter estimates. In general, the parameter 

estimates were similar for the calibrated model and the alternative models. The weighted residuals also 

were similar for the calibrated and alternative models; therefore, bias in the calibrated model was not 

corrected through different boundary conditions in the alternative models. Models that simulate headdependent recharge to the alluvium were better models than those that simulated specified recharge on the 

basis of lower sum-of-squares error. Models that simulated transmissive faults resulted in slight 

improvements in model fit; however, estimates of fault transmissivity had a large variance and were 

considered unreliable.

SUMMARY AND CONCLUSIONS 49
Within the limits discussed in the report, the parameters of the calibrated model were reliably 

estimated. The regression model was found to be nonlinear through application of Beale's measure; thus, 

no confidence intervals were computed for the model. Recharge to the McNairy Formation and the 

transmissivity of the alluvium were correlated at r = 0.91. This is expected because these two parameters 

are the primary control on inflow to and outflow from the model. To determine the effect of the correlation, 

transmissivity was fixed at 5,000 ft2/d, which produced a recharge estimate of 0.00119 ft/d (5 in./yr) and at 

50,000 ft2/d, which produced a recharge estimate of 0.00559 ft/d (24.5 in./yr).

The contributing areas to discharge boundaries were computed using a particle-tracking technique. 

The contributing area to wells at the Calvert City Water Company did not include the Calvert City 

Industrial Complex, under simulated conditions. The contributing area to the wells at the Calvert City 

Water Company changed when pumping was shifted to a new location. The new contributing area, 

although larger than that simulated with the calibrated model, was farther from the industrial complex. The 

time of travel of particles ranged from several days to 123 years depending on the distance from the 

discharge point. The average linear velocity of ground water ranged from 0.90 to 4.47 ft/d with a median of 

1.98 ft/d. The average linear velocity is the rate at which the center of a mass of contaminated ground water 

will move; thus, some contaminated water will move more slowly and some will move more rapidly. The 

alternative models in which transmissive faults were simulated was not used to compute contributing 

areas; however, new wells at the Calvert City Water Company are near a potentially transmissive fault that 

may affect the contributing area.

Reliability of estimates from the calibrated model could be increased through additional data 

collection. Head measurements in the middle alluvial unit would increase the reliability of recharge 

estimates. Aquifer tests at more locations that include head measurements in the middle and upper alluvial 

units would provide independent information on aquifer properties and more information on possible 

spatial changes in aquifer properties. Aquifer tests and (or) head measurements in or near faults would 

allow more reliable simulation of the effect of transmissive faults on contributing areas. Recharge to the 

McNairy Formation is complex; more detailed mapping of clay beds and investigation of the unsaturated 

flow in loess and continental deposits would allow more accurate estimates of the rate and distribution of 

recharge. Vertical-head data, especially near discharge boundaries and in the McNairy Formation, would 

allow better representation of particle flow paths.

The results of this study could be used in future studies to answer more detailed questions about 

ground-water flow in the area. The cell-by-cell flow rates computed by MODFLOW could be used in a 

model with a smaller grid cell size to eliminate the weak-sink problem and to provide higher resolution 

contributing areas. Models of smaller areas could used in simulations of solute transport to determine the 

effect of dispersion on contaminant concentrations. Also, statistical measures computed by the regression 

could be used to assess the effect of uncertainty in the parameter estimates on contributing areas.

50 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky
REFERENCES CITED

Amos, D.H., and Finch, W.I., 1968, Geology of the Calvert City quadrangle, Livingston and Marshall Counties, 

Kentucky: U.S. Geological Survey Geologic Quadrangle Map GQ-731, scale 1:24,000.

Amos, D.H., and Wolfe, E.W., 1966, Geology of the Little Cypress quadrangle, Kentucky-Illinois: U.S. Geological 

Survey Geologic Quadrangle Map GQ-554, scale 1:24,000.

Cooley, R.L., and Naff, R.L., 1990, Regression modeling of ground-water flow: U.S. Geological Survey Techniques 

of Water-Resources Investigations, book 3, chap. B4,232 p.

Dames and Moore, 1988a, Report: Remedial investigation, B.F. Goodrich/Airco site, Calvert City, Ky., 85 p., 

60 tables, 37 figs., 9 appendices.

____1988b, Report: Evaluation of aquifer response during flooding, B.F. Goodrich/Airco site, Calvert City, Ky.,

16 p., 3 tables, 16 figs.

_1991, Report: Remedial environmental investigation, groundwater assessment, B.F. Goodrich Company,

Calvert City, Ky., 67 p., 15 tables, 34 figs., 3 appendices.

Davis, R.W., Lambert, T.W., and Hanson, A.J., Jr., 1973, Subsurface Geology and Ground-Water Resources of the 

Jackson Purchase Region, Kentucky: U.S. Geological Survey Water-Supply Paper 1987, 66 p., 11 pis., 9 figs.

Fenneman, N.M., 1938, Physiography of the Eastern United States: New York, McGraw-Hill Book Company, 714 p.

Finch, W.I., 1968, Engineering geology of the Calvert City Quadrangle, Livingston and Marshall Counties, Kentucky: 

Kentucky Geological Survey Information Circular 16, Lexington, Ky., 28 p.

Hill, M.C., 1992, MODFLOWP, A computer program estimating parameters of a transient, three-dimensional groundwater flow model using non-linear regression: U.S. Geological Survey Open-File Report 91-484, 358 p.

____1994, Five computer programs for testing weighted residuals and calculating linear confidence intervals on 

results from the ground-water parameter-estimation computer program MODFLOWP: U.S. Geological Survey 

Open-File Report 93-481, 81 p.

Lambert, T.W., and MacCary, L.M., 1964, Geology of the Briensburg quadrangle, Kentucky: U.S. Geological Survey 

Geologic Quadrangle Map GQ-327, scale 1:24,000.

MacCary, L.M., 1964, Availability of ground water in the Briensburg quadrangle, Kentucky: U.S. Geological Survey 

Hydrologic Atlas HA-114, scale 1:24,000.

McDonald, M.G., and Harbaugh, A.W., 1988, A modular three-dimensional finite-difference ground-water flow 

model: U.S. Geological Survey Techniques of Water-Resources Investigations, book 6, chap. Al, 528 p.

Morgan, J.H., 1964, Availability of ground water in the Elva quadrangle, Kentucky: U.S. Geological Survey 

Hydrologic Investigations Atlas HA-117, scale 1:24,000.

____1965, Availability of Ground-Water in the Parts of the Little Cypress and Calvert City Quadrangles in the 

Jackson Purchase Region, Kentucky: U.S. Geological Survey Hydrologic Investigations Atlas HA-155, 1 sheet, 

scale 1:24,000.

Morris, D.A., and Johnson, A.I., 1967, Summary of hydrologic and physical properties of rock and soil materials, as 

analyzed by the Hydrologic Laboratory of the U.S. Geological Survey, 1948-1960: U.S. Geological Survey 

Water-Supply Paper 1839-D, 42 p.

Olive, W.W., 1963, Geology of the Elva quadrangle, Kentucky: U.S. Geological Survey Geologic Quadrangle Map 

GQ-230, scale 1:24,000.

____1980, Geologic maps of the Jackson Purchase Region, Kentucky: U.S. Geological Survey Miscellaneous 

Investigations Map 1-1217, scale 1:250,000.

Pollack, D.W., 1989, Documentation of computer programs to compute and display pathlines using results from the 

U.S. Geological Survey modular three-dimensional finite-difference ground-water flow model: U.S. Geological 

Survey Open-File Report 89-381, 188 p.

REFERENCES CITED 51
Free, H.L., Jr., and Walker, W.H., 1952, Memorandum on the geology and ground-water resources of the Calvert CityGilbertsville area, Marshall County, Kentucky: U.S. Geological Survey Open-File Report, 13 p.

Sholar, C.J., and Lee, V.D., 1988, Water use in Kentucky, 1985: U.S. Geological Survey Water-Resources 

Investigations Report 88-4043, 53 p.

Squillace, P.J., Thurman, E.M., and Furlong, E.T., 1993, Groundwater as a nonpoint source of atrazine and

deethylatrazine in a river during base flow conditions: Water Resources Research, v. 29. no. 6, p. 1719-1729.

Starn, J.J., in press, Apparent solute dispersion in an alluvial-aquifer/stream system, in Morganwalp, D.W.. and 

Aronson, D.A., eds., U.S. Geological Survey Toxic Substances Hydrology Program Proceedings of the 

Technical Meeting, Colorado Springs, Colorado, September 20-24, 1993: U.S. Geological Survey WaterResources Investigations Report 94-4015.

Yager, R.M., 1993, Estimation of hydraulic conductivity of a riverbed and aquifer system on the Susquehanna River 

in Broome County, New York: U.S. Geological Survey Water-Supply Paper 2387,49 p.

____in press, Simulated three-dimensional ground-water flow in a fractured dolomite aquifer of the Lockport Group 

near Niagara Falls, New York: U.S. Geological Survey Open-File Report 94-386,43 p.

. GOVERNMENT PRINTING OFFICE: 1995-0-652-643

52 Geohydrology and Simulation of Ground-Water Flow in the Aquifer System near Calvert City, Kentucky