Prepared in cooperation with the

U.S. Army Corps of Engineers, Memphis District and the 

Arkansas Soil and Water Conservation Commission

CONJUNCTIVE-USE OPTIMIZATION MODEL AND 

SUSTAINABLE-YIELD ESTIMATION FOR THE SPARTA 

AQUIFER OF SOUTHEASTERN ARKANSAS AND NORTHCENTRAL LOUISIANA

Water-Resources Investigations Report 03-4231

1990-1997

WITHDRAWAL RATE

OPTIMAL

WITHDRAWAL RATE

U.S. Department of the Interior

U.S. Geological Survey
Cover: Oblique view of the resultant steady-state simulated hydraulic heads using baseline 1990-1997 withdrawal rate and the optimal withdrawal rate from scenario 3A.
CONJUNCTIVE-USE OPTIMIZATION MODEL AND 

SUSTAINABLE-YIELD ESTIMATION FOR THE SPARTA 

AQUIFER OF SOUTHEASTERN ARKANSAS AND NORTHCENTRAL LOUISIANA

By Paul W. McKee, Brian R. Clark, and John B. Czarnecki

U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 03-4231

Prepared in cooperation with the

U.S. Army Corps of Engineers, Memphis District and the 

Arkansas Soil and Water Conservation Commission

Little Rock, Arkansas

2004
U.S. DEPARTMENT OF THE INTERIOR

GALE A. NORTON, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

For additional information Copies of this report can be

write to: purchased from:

District Chief U.S. Geological Survey

U.S. Geological Survey, WRD Branch of Information Services

401 Hardin Road Box 25286

Little Rock, Arkansas 72211 Denver Federal Center

Denver, Colorado 80225

The use of firm, trade, and brand names in this report is for identification purposes only and does not constitute endorsement by the U.S. Geological Survey.
Contents III

CONTENTS

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

Introduction........................................................................................................................................................ 2

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

Previous Studies........................................................................................................................................ 4

Study Area Description............................................................................................................................. 4

Hydrogeology of the Sparta Aquifer......................................................................................................... 4

Sparta Aquifer Development and Associated Effects ............................................................................... 5

Acknowledgments..................................................................................................................................... 5

Conjunctive-Use Optimization Model ............................................................................................................... 5

Ground-Water Flow Model ....................................................................................................................... 5

Optimization Model .................................................................................................................................. 6

Problem Formulation ................................................................................................................................ 8

Objective Function........................................................................................................................... 8

Streamflow Constraints.................................................................................................................... 8

Hydraulic-Head Constraints...........................................................................................................11

Specified Limits on Ground-Water Withdrawals...........................................................................11

Specified Limits on Surface-Water Withdrawals...........................................................................11

Sustainable-Yield Estimation...........................................................................................................................11

Determination of Sustainable Yield and Unmet Demand.......................................................................12

Scenario 1................................................................................................................................................17

Scenario 2................................................................................................................................................17

Scenario 3................................................................................................................................................17

Analysis of Scenario Sustainable Yield Estimation................................................................................19

Infeasible Scenarios ................................................................................................................................23

Verification of Optimized Withdrawals ..................................................................................................24

Model Assumptions and Limitations ......................................................................................................24

Summary ..........................................................................................................................................................28

Selected References .........................................................................................................................................29

ILLUSTRATIONS

Figure 1. Map showing extent of Sparta aquifer and location of model area with Arkansas Soil

and Water Conservation Commission Critical Ground-Water Areas shaded ............................ 3

2. Flow chart of optimization modeling process ................................................................................. 7

3. Map showing control point locations for Sparta aquifer hydraulic-head and streamflow

constraints ................................................................................................................................10

4. Map showing baseline 1990-1997 ground-water withdrawal rates used in conjunctiveuse optimization model............................................................................................................13

5. Map showing difference between Sparta aquifer 1997 simulated heads and top of

the Sparta Sand ........................................................................................................................18

6. Map showing ratio of optimal ground-water withdrawal to baseline 1990-1997

withdrawals (in percent) for (A) scenario 3A, (B) scenario 3B, and (C) scenario 3C including

locations where hydraulic-head constraints are reached..........................................................20
IV Contents

7. Graph showing scenario results of estimated sustainable yield as a function of the

maximum allowable ground-water withdrawal ....................................................................... 23

8. Map showing difference between steady-state simulated hydraulic heads and the top of

the Sparta Sand for (A) scenario 3a, (B) scenario 3b, and (C) scenario 3c using

optimized withdrawals in the Sparta aquifer flow model for verification............................... 25

TABLES

Table 1. Characteristics of the Sparta aquifer ground-water flow model ....................................................... 6

2. Flow and optimized surface-water withdrawals for major rivers within the Sparta aquifer

outcrop area specified within the conjunctive-use optimization model for all three

scenarios (3A-3C) ...................................................................................................................... 9

3. Summary of optimization scenarios and associated constraints including sustainable 

yield estimates and total 1997 withdrawals for Arkansas and Louisiana ................................ 14

4. Estimated sustainable yield and unmet demand by county and parish for each scenario

in Arkansas and Louisiana ....................................................................................................... 15

In this report, vertical coordinate information is referenced to the National Geodetic Vertical Datum of 1929 

(NGVD of 1929). Horizontal coordinate information is referenced to North American Datum of 1927 (NAD 27).

Altitude, as used in this report, refers to distance above or below NGVD of 1929.

CONVERSION FACTORS AND ABBREVIATIONS

Multiply By To obtain

Length

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

 square foot (ft2) 0.09290 square meter (m2)

square mile (mi2) 2.590 square kilometer (km2)

Flow rate

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

cubic foot per day (ft3/d) 0.02832 cubic meter per day (m3/d)

gallon per minute (gal/min) 0.06308 liter per second (L/s)

gallon per day (gal/d) 0.003785 cubic meter per day (m3/d)

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

billion gallons per day (Bgal/d) 43.81 cubic meter per second (m3/s)

Hydraulic conductivity

foot per day (ft/d) 0.3048 meter per day (m/d)
Abstract 1

CONJUNCTIVE-USE OPTIMIZATION MODEL AND 

SUSTAINABLE-YIELD ESTIMATION FOR THE SPARTA 

AQUIFER OF SOUTHEASTERN ARKANSAS AND NORTHCENTRAL LOUISIANA

By Paul W. McKee, Brian R. Clark, and John B. Czarnecki

ABSTRACT

Conjunctive-use optimization modeling was 

done to assist water managers and planners by estimating the maximum amount of ground water that 

hypothetically could be withdrawn from wells 

within the Sparta aquifer indefinitely without violating hydraulic-head or stream-discharge constraints. The Sparta aquifer is largely a confined 

aquifer of regional importance that comprises a 

sequence of unconsolidated sand units that are 

contained within the Sparta Sand. In 2000, more 

than 35.4 million cubic feet per day (Mft3/d) of 

water were withdrawn from the aquifer by more 

than 900 wells, primarily for industry, municipal 

supply, and crop irrigation in Arkansas. Continued, heavy withdrawals from the aquifer have 

caused several large cones of depression, lowering 

hydraulic heads below the top of the Sparta Sand 

in parts of Union and Columbia Counties and several areas in north-central Louisiana. Problems 

related to overdraft in the Sparta aquifer can result 

in increased drilling and pumping costs, reduced 

well yields, and degraded water quality in areas of 

large drawdown. 

A finite-difference ground-water flow 

model was developed for the Sparta aquifer using 

MODFLOW, primarily in eastern and southeastern 

Arkansas and north-central Louisiana. Observed 

aquifer conditions in 1997 supported by numerical 

simulations of ground-water flow show that continued pumping at withdrawal rates representative 

of 1990 - 1997 rates cannot be sustained indefinitely without causing hydraulic heads to drop 

substantially below the top of the Sparta Sand in 

southern Arkansas and north-central Louisiana. 

Areas of ground-water levels below the top of the 

Sparta Sand have been designated as Critical 

Ground-Water Areas by the State of Arkansas. A 

steady-state conjunctive-use optimization model 

was developed to simulate optimized surfacewater and ground-water withdrawals while maintaining hydraulic-head and streamflow constraints, thus determining the “sustainable yield” 

for the aquifer. 

Initial attempts to estimate sustainable yield 

using simulated 1997 hydraulic heads as initial 

heads in Scenario 1 and 100 percent of the baseline 

1990-1997 withdrawal rate as the lower specified 

limit in Scenario 2 led to infeasible results. Sustainable yield was estimated successfully for scenario 3 with three variations on the upper limit of 

withdrawal rates. Additionally, ground-water 

withdrawals in Union County were fixed at 35.6 

percent of the baseline 1990-1997 withdrawal rate 

in Scenario 3. These fixed withdrawals are recognized by the Arkansas Soil and Water Conservation Commission to be sustainable as determined 

in a previous study. The optimized solutions maintained hydraulic heads at or above the top of the 

Sparta Sand (except in the outcrop areas where 

unconfined conditions occur) and streamflow 

within the outcrop areas was maintained at or 

above minimum levels. Scenario 3 used limits of 

100, 150, and 200 percent of baseline 1990-1997 

withdrawal rates for the upper specified limit on 

1,119 withdrawal decision variables (managed 

wells) resulting in estimated sustainable yields 

ranging from 11.6 to 13.2 Mft3

/d in Arkansas and 

0.3 to 0.5 Mft3/d in Louisiana. Assuming the total 
2 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

water demand is equal to the baseline 1990-1997 

withdrawal rates, the sustainable yields estimated 

from the three scenarios only provide 52 to 59 percent of the total ground-water demand for Arkansas; the remainder is defined as unmet demand that 

could be obtained from large, sustainable surfacewater withdrawals.

INTRODUCTION

The Sparta aquifer is a confined aquifer of 

regional importance within the Mississippi Embayment 

aquifer system. It consists of varying amounts of unconsolidated sand, interstratified with silt and clay lenses 

within the Sparta Sand of the Claiborne Group. It 

extends through eastern and southeastern regions of 

Arkansas, northern Louisiana, and portions of Texas, 

Mississippi, and Tennessee (fig. 1). The Sparta Sand 

ranges in total thickness from 100 to 1,000 feet (ft) and 

is stratigraphically positioned and confined between the 

overlying Cook Mountain Formation and the underlying Cane River Formation (Payne, 1968).

The Sparta aquifer of southeastern Arkansas and 

north-central Louisiana covers more than 32,500 mi2

and is a major source of water for municipal, industrial, 

and agricultural uses. Since the early 1900’s, the Sparta 

aquifer has provided abundant water of high quality primarily for industry and public supply. Total groundwater withdrawals from the Sparta aquifer in Arkansas 

have increased from 14.2 Mft3

/d in 1965 (Halberg and 

Stephens, 1966) to an estimated 35.4 Mft3

/d in 2000 

(T.W. Holland, U.S. Geological Survey, written commun., 2002)—an increase of 150 percent. Through the 

years, the growing water demands of industries and 

municipalities have resulted in ground-water withdrawals from the Sparta aquifer that substantially exceed 

recharge to the aquifer. In addition, where ground-water 

level declines in the Mississippi River Valley alluvial 

aquifer are decreasing well yields, the Sparta aquifer is 

increasingly used for crop irrigation. These large withdrawals have caused considerable drawdown in the 

potentiometric surface resulting in the development of 

large cones of depression over 175 feet deep. Hydraulic 

heads (often used interchangeably with water-level altitude or potentiometric surface) are now below the top of 

the Sparta Sand in some parts of Union and Columbia 

Counties, Arkansas, and in several parishes of Louisiana (T.P. Schrader, U.S. Geological Survey, written 

commun., 2003). 

Several Arkansas counties within the extent of the 

Sparta aquifer have been designated Critical GroundWater Areas (CGWA) by the Arkansas Soil and Water 

Conservation Commission (ASWCC) because hydraulic heads in the Sparta aquifer are below the top of the 

Sparta Sand (fig. 1). This indicates that withdrawal rates 

have already exceeded those that can be sustained indefinitely. A recently calibrated flow model of the Sparta 

aquifer (McKee and Clark, 2003) simulates this condition and indicates these areas will expand if withdrawal 

rates equivalent to those representing the period from 

1990 to 1997 continue.

To estimate withdrawal rates that could be sustained relative to the constraints of critical ground-water 

designation, the U.S. Geological Survey (USGS), in 

cooperation with the Memphis District of the U.S. 

Army Corps of Engineers (MCOE) and the ASWCC, 

applied conjunctive-use optimization modeling to the 

existing Sparta aquifer flow model during 2000-2002. 

Optimized withdrawal rates form the basis for estimating sustainable yield from the Sparta aquifer. Additionally, the conjunctive-use optimization model determines 

the maximum available withdrawals from major 

streams for supplementing ground water to meet the 

total water demand. The Sparta aquifer flow model was 

modified and recalibrated for the purpose of evaluating 

potential pumping scenarios and optimizing withdrawal 

rates to determine sustainable yield for the Sparta aquifer. Documentation of the development, calibration, and 

application of the Sparta aquifer flow model is given by 

McKee and Clark (2003). 

Purpose and Scope

This report describes the application and evaluation of a conjunctive-use optimization model that was 

developed from a linear programming routine utilizing 

the calibrated Sparta aquifer flow model (McKee and 

Clark, 2003). The objective of the optimization modeling was to simulate maximum ground-water withdrawals from the Sparta aquifer and surface-water 

withdrawals from major streams in the Sparta outcrop/

subcrop area while maintaining desirable hydraulic 

heads in the aquifer and streamflow in the outcrop. In 

essence, the objective was to estimate the “sustainable 

yield”—indefinite ground-water withdrawals without 

compromising the integrity of the Sparta aquifer in 

Arkansas and Louisiana or streamflow in seven streams. 

Although the numerical flow model includes western 

Mississippi for hydrologic boundary purposes, no 
Introduction 3

Figure 1. Extent of Sparta aquifer and location of model area with Arkansas Soil and Water Conservation Commission Critical 

Ground-Water Areas shaded.

Active Model

Area Boundary

Hodge

Ruston

Monroe

Belzoni

Bastrop

Magnolia

Magnolia

Cleveland

El Dorado

Springhill

Pine

Bluff

West

Monroe

Stuttgart

94° 93° 92° 91° 90° 89° 88° 87°

30°

31°

32°

33°

34°

35°

36°

37°

0 60 120 Miles

0 60 120 Kilometers

Mississippi Embayment Boundary

Sparta Aquifer Boundary

(Modified from Arthur and Taylor, 1990)

Southern extent of aquifer with

water having dissolved-solids

concentrations less than 10,000

milligrams per liter

Camden

Red

Saline

Cache

Arkansas

Mississippi

Mississippi

Bayou

White

Dorcheat

River

Ouachita

Bodcau

Creek

River

River

River

River

River

River

River

Grand

Prairie

Fall Line

Arkansas

Mississippi

Alabama

Tennessee

Missouri

Texas Louisiana Oklahoma

Kentucky

Critical GroundWater Areas
4 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

hydraulic-head data from Mississippi were used for 

calibration; therefore, sustainable-yield estimations for 

Mississippi are not included in this report. The results 

of the optimization modeling can provide water managers and policy makers with information that can be used 

to assist in management of the ground-water resources 

of the Sparta aquifer.

Previous Studies

Optimization management of ground-water systems has been explored and successfully applied by 

many researchers. Reichard (1995) offers a thorough 

review of these studies discussing an optimization 

approach to ground-water/surface-water management. 

For the Sparta aquifer, sustainable yield was estimated 

in Union County using one control point of specified 

minimum acceptable hydraulic head and varying 

Union County pumpage within the ground-water 

model to achieve the target hydraulic head (Hays, 

2000). Results from Hays (2000) determined that 28 

percent of the 1997 withdrawal rate in Union County 

could be sustained while maintaining acceptable 

hydraulic heads above the top of the Sparta Sand Formation at one control location centered in the deepest 

cone of depression. This rate has since been recognized 

by ASWCC as a goal to limit pumping from the Sparta 

aquifer in Union County through conservation and 

alternate water supplies. The study described in this 

report, however, is the first to explore and describe conjunctive-use optimization management for the Sparta 

aquifer, although similar investigations have been conducted in the overlying Mississippi River Valley alluvial aquifer by Peralta and others (1985) for the Grand 

Prairie region and most recently by Czarnecki and others (2003) for the alluvial aquifer in northeastern 

Arkansas. 

Study Area Description

The study area comprises terrain in southeastern 

Arkansas, north-central Louisiana, and northwestern 

Mississippi (fig. 1). Land-surface altitudes range from 

more than 500 ft along the western boundary and outcrop recharge zones to less than 100 ft along the Mississippi River. The principal drainages are the 

Mississippi, Arkansas, Saline, Ouachita, and Red Rivers, Bayou Dorcheat, and Bodcau Creek. Mean annual 

precipitation is approximately 50 inches (Freiwald, 

1985). Water use from the Sparta aquifer is for municipal supply, manufacturing of forest, chemical, and 

other industrial products, agriculture, and aquaculture. 

Hydrogeology of the Sparta Aquifer

The Sparta aquifer is composed of a sequence of 

unconsolidated sand, silt, and clay units within the 

Sparta Sand of the Mississippi Embayment aquifer system and extends from south Texas, northward into Louisiana, Arkansas, and Tennessee, and eastward into 

Mississippi (fig. 1). A detailed description of the 

hydrogeology of the Sparta aquifer is discussed in 

numerous reports including Payne (1968), Hosman and 

others (1968), and Petersen and others (1985). A large 

part of the Sparta aquifer is overlain by Quaternary-age 

alluvial and terrace deposits that form the Mississippi 

River Valley alluvial aquifer. The Sparta Sand ranges in 

total thickness from 100 to 1,000 ft and is stratigraphically positioned between the overlying Cook Mountain 

Formation and underlying Cane River Formation. 

These two formations are low-permeability, finegrained, clay-rich units that confine flow within the 

more permeable Sparta Sand. The Sparta Sand outcrops on both the west and east sides of the Mississippi 

Embayment. North of 35 degrees latitude, the Sparta 

Sand becomes part of the Memphis Sand as the Cane 

River Formation undergoes a facies change where 

deposits change from marine clays to sand (Hosman 

and others, 1968).

The Sparta aquifer is unconfined in the outcrop 

area and becomes confined as it dips toward the central 

axis of the Mississippi Embayment (generally corresponding with the Mississippi River) and southward 

toward the Gulf of Mexico where it is deeply buried in 

the subsurface (Hosman and others, 1968). The total 

sand thickness within the aquifer generally ranges from 

200 to 600 ft. Sources of recharge to the Sparta aquifer 

are infiltration of precipitation on the outcrop, leakage 

from streams in the outcrop, leakage from the alluvium 

where the Sparta subcrops, and leakage from adjacent 

aquifers through confining layers where the vertical 

hydraulic gradient is towards the Sparta aquifer. Most 

natural discharge is to streams in the outcrop and to 

adjacent units with lower potentiometric surfaces. Generalized ground-water flow in the Sparta aquifer is 

from the outcrop areas to the axis of the Mississippi 

Embayment except where heavy withdrawals in some 

areas have caused a reversal of flow towards the resultant cones of depression (Hosman and others, 1968). 
Conjunctive-Use Optimization Model 5

The horizontal hydraulic conductivity of the Sparta 

aquifer generally ranges from 10 to 200 ft/d with an 

average of about 70 ft/d across the extent of the Mississippi Embayment. The average specific storage for the 

Sparta aquifer is about 6 x 10-7/ft and storativity is 

3x10-4 based on an average thickness of 500 ft (Hosman and others, 1968). 

Sparta Aquifer Development and 

Associated Effects

The earliest known withdrawals from the Sparta 

aquifer in the study area began in 1898 in Pine Bluff, 

Arkansas, for industrial use. It was not until about 1920 

that substantial withdrawals from the aquifer began for 

rice irrigation in the Grand Prairie area. Since that time, 

the aquifer has been heavily used for industrial and 

municipal supply, providing water of excellent quality. 

The principal areas of ground-water withdrawal in the 

study area are in Jefferson, Union, and Arkansas Counties in Arkansas, Ouachita Parish in Louisiana, and 

Bolivar County in Mississippi (all counties and parishes shown on figure 3). Currently, more than 1,760 

wells are screened in the Sparta aquifer in Arkansas 

and Louisiana, and more than 35.4 Mft3

/d was withdrawn from the aquifer in Arkansas in 2000 from more 

than 900 wells (T.W. Holland, U.S. Geological Survey, 

written commun., 2003). Individual wells completed in 

the Sparta aquifer generally yield 100 to 500 gal/min, 

with less common rates up to 1,200 gal/min. Continued 

ground-water withdrawal through time has caused the 

potentiometric surface to decline more than 175 ft in 

some areas in the aquifer (Joseph, 1998; T.P. Schrader, 

U.S. Geological Survey, written commun., 2003), 

which also has affected natural flow directions. 

Acknowledgements

The conjunctive-use optimization model routine 

used in this report is an adaptation of enhancements 

made by Brian Wagner (U.S. Geological Survey) to 

MODMAN 3.0. Wesley Danskin (U.S. Geological Survey) provided guidance on the use of MODMAN and 

the large-scale optimization solver MINOS. Streamflow constraints for streams used in the optimization 

model were compiled by Steve Loop (Arkansas Soil 

and Water Conservation Commission) for six streams 

in Arkansas and by Paul Ensminger (U.S. Geological 

Survey) for one stream in Louisiana.

CONJUNCTIVE-USE OPTIMIZATION 

MODEL

A conjunctive-use optimization model, hereafter 

referred to as the “optimization model,” was developed 

for the Sparta aquifer using three modeling packages 

known as MODFLOW-96, MODMAN, and MINOS 

(Harbaugh and McDonald, 1996; Greenwald, 1998; 

Murtaugh and Sanders, 1998). The optimization model 

determined maximum total surface-water and groundwater withdrawals from the system while maintaining 

desirable hydraulic heads in the aquifer and streamflows in the outcrop.

Ground-Water Flow Model

The Sparta aquifer ground-water flow model, 

hereafter referred to as the “ground-water model,” used 

in the optimization model was developed, calibrated, 

and documented by McKee and Clark (2003). Characteristics of the ground-water model are listed in table 1. 

The ground-water model simulates flow in the Sparta 

aquifer covering 38,220 mi2

 in southeastern Arkansas, 

north-central Louisiana, and western Mississippi (fig. 

1). The ground-water model is a two-layer, transient, 

three dimensional (3D) numerical model with a uniform grid spacing of 1 mi2 developed using MODFLOW-2000 (Harbaugh and others, 2000). The quasi3D confining unit between the two layers represents a 

mixture of continuous and discontinuous clay lenses in 

much of south-central Arkansas and north-central Louisiana separating the upper and lower water-bearing 

zones of the Sparta aquifer. Calibration of the groundwater model involved a simulation period from 1898 to 

1997 and used non-linear regression methods (Hill and 

others, 2000) to match a total of 795 hydraulic-head 

observations from 316 wells over four time periods—1970, 1985, 1990, and 1997. Field measurements 

were not available to compare with simulated fluxes in 

and out of the Sparta outcrop streams. Thorough analysis of the calibration indicated that the model could 

simulate observed conditions in the Sparta aquifer 

within acceptable error. The calibrated flow model has 

a root mean square error (RMSE) of 17.99 ft for all 

observations, and 18.93 ft in 1997 (McKee and Clark, 

2003). Calibration statistics such as the mean, mean 
6 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

absolute, and root-mean-square difference between 

observed and simulated hydraulic head for all observations are listed in table 1.

 Ground-water flow in the Sparta aquifer was 

simulated from 1898 through 1997. Steady-state conditions were simulated in the ground-water model for the 

year 1898 and represent aquifer predevelopment conditions. The ground-water model then simulates 100 

years of system response to stress by dividing the 

model into 27 stress periods (28 total, including steadystate predevelopment) with each stress period representing a period of time for which ground-water withdrawals were relatively unchanged. The length of stress 

periods ranges from 1 to 20 years; therefore, seasonal 

effects are not simulated. The most recent stress period 

(1990-1997) is 8 years. 

Boundary conditions include no flow on the bottom, western edge, and a portion of the eastern edge of 

the model, as well as head-dependent flow boundaries 

on much of the top, northern, southern, and portions of 

the eastern edge of the model. Specified flow boundaries exist along the western edge of the model in what 

is known as the outcrop-subcrop area of the Sparta 

aquifer. In addition, 15 streams in Arkansas and Louisiana were simulated in the Sparta aquifer outcrop area 

using head-dependent flow boundaries (McKee and 

Clark, 2003).

Hydraulic properties of the ground-water system 

are represented in the Sparta aquifer flow model by the 

distribution of faults, hydraulic conductivity, storage, 

and vertical leakance parameters throughout the model 

domain. The final parameter estimates of the model are 

considered reasonable estimates for the type of material and conditions found in the Sparta aquifer. Horizontal hydraulic conductivity values ranged from 2.5 to 

47.9 ft/d. Horizontal hydraulic conductivity of horizontal flow boundaries, representing faults, range from 5.0 

x 10-6 to 1.0 ft/d. Specific yield values were fixed at 

0.2. Specific storage values range from 3.54 x 10-6 to 

1.5 x 10-4ft-1. The hydraulic conductivity value used in 

the calculation of riverbed conductance representing 

leakance through overlying units, ranged from 3.1 x

10-5 to 9.3 x 10-3. Recharge ranges from 0.11 to 1.10 

inches per year, with the highest recharge applied to the 

northern portion of the outcrop-subcrop area. 

Optimization Model

For optimization-modeling described in this 

report, modifications were made to MODMAN 4.0 to 

(1) incorporate stream withdrawal cells as decision 

variables, (2) allow specification of streamflow constraints, and (3) account for streamflow water budgeting. Modifications to the MODMAN code were 

initially provided by Brian Wagner (U.S. Geological 

Survey) in a modification of MODMAN 3.0, and 

adapted by John Czarnecki (U.S. Geological Survey) to 

MODMAN 4.0. In addition, the ability to aggregate 

wells within a subarea of the model and to treat an 

aggregate-well pumping rate as a single decision variable was added to MODMAN 4.0. However, that ability was not utilized; instead, 1,152 ground-waterwithdrawal decision variables and 7 surface-waterwithdrawal decision variables were specified.

Table 1. Characteristics of the Sparta aquifer ground-water 

flow model

[RMSE, root mean square error; Mft3/d, million cubic feet per day]

Characteristic Value

Area (square miles) 38,220

Model layers 2

Calibration period 1898 to 1997

Stress periods 28

Stress period length, range (years) 1 to 20

Cells with wells representing 1996

distribution of withdrawals 1,152

Baseline 1990-1997 withdrawals

(Mft3/d) 32

Hydraulic conductivity (feet per

day) 2.5 to 47.9

Specific yield 0.20

Specific storage (feet-1) 3.54x10-6 to 1.5x10-4

River cells (defines streams included in 

the model) 488

Hydraulic head observation periods 1990, 1985, 1990, 1997

Hydraulic head observations, all four

periods 795

Range in observed hydraulic head values

in 1997 (altitude, in feet relative to

NGVD of 1929)

-224 to +306

Mean difference between observed and

simulated hydraulic head, all four

periods (feet)

0.36

Mean absolute difference between

observed and simulated hydraulic head,

all four periods (feet)

13.78

Root mean square error (RMSE) between

observed and simulated hydraulic head,

all four periods (feet)

17.99
Conjunctive-Use Optimization Model 7

The optimization modeling process (fig. 2) 

begins with the calibration and adaptation of a MODFLOW-based ground-water flow model to be compatible with the optimization modeling software 

(MODMAN 4.0). Adaptation entailed the conversion 

of the flow model from MODFLOW 2000 (Hill and 

others, 2000) to MODFLOW 96 (McDonald and Harbaugh, 1988; Harbaugh and McDonald, 1996), and verifying that the results were the same. Steady-state 

conditions were selected (rather than transient conditions) because the maximized withdrawals are intended 

to represent sustainable yield of the system (a rate that 

can be maintained indefinitely).

A management problem is formulated to maximize a parameter, such as water production from 

ground water and surface water, within selected constraints, such as maintaining hydraulic heads in the 

aquifer above a minimum altitude or maintaining a 

minimum amount of streamflow. An input objective 

function is prepared from the component of the management problem for a conjunctive-use version of 

MODMAN 4.0 to generate response coefficients for 

each specified withdrawal cell in the model. The 

response-coefficient matrix consists of changes in 

hydraulic head or streamflow at each constraint location that occur in response to pumping at a single well 

or river cell at a unit rate (Greenwald, 1998; Ahlfeld 

and Mulligan, 2000). 

After all the response coefficients are calculated, 

they are combined to form a data-input set along with 

hydraulic-head and streamflow constraints, and are formulated as a linear program in mathematical programming system (MPS) format. The linear optimization 

program is run under MINOS. If a feasible solution 

Figure 2. Flow chart of optimization modeling process (modified from Greenwald, 1998)

Develop and Calibrate Site Specific

Ground-Water Flow Model

Formulate Management

Problem

Input Objective Function and

Constraints

Generate Response Coefficient Matrix

Transform Management

Problem into a Linear

Program in Mathematical

Programming System Format

Solve Linear Optimization

Problem

;

;

;

;

;
8 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

exists, MINOS will provide estimates of optimal (maximum) values of ground-water and surface-water withdrawals. MINOS also identifies points in the model 

where hydraulic-head or streamflow constraints have 

been reached.

Optimal ground-water withdrawal rates calculated using the optimization model were evaluated by 

applying them in the flow model, to compare the resulting simulated hydraulic head and streamflow against 

the specified hydraulic-head and streamflow constraints. In a strictly linear model, such as one for a confined aquifer, ground-water flow is a function of 

hydraulic head through only the hydraulic-gradient 

term in Darcy’s law:

(1)

where Q is ground-water flow (in cubic feet per day);

K is the hydraulic conductivity (in feet per 

day);

is the hydraulic gradient, dimensionless;

h is hydraulic head, (in feet);

l is a distance over which the gradient is measured (in feet); and

A is the cross-sectional area through which 

flow occurs (in square feet).

For unconfined conditions, A also is a function of 

hydraulic head. If changes in hydraulic head are small 

relative to the total saturated thickness, then A will 

remain about the same. However, if substantial change 

in saturated thickness occurs, A can change appreciably, because

(2)

where b is the saturated thickness (in feet), which

varies with hydraulic head; and

w is the width through which flow occurs (in 

feet).

Problem Formulation

The optimization model was formulated as a linear programming problem with the objective of maximizing water production from wells and from streams 

subject to: (1) maintaining streamflow at or above minimum specified rates; (2) maintaining ground-water 

levels at or above specified levels; and (3) limiting 

ground-water withdrawals to a maximum of 200 percent of the baseline 1990-1997 withdrawal rate. 

Steady-state conditions were selected (rather than transient conditions) because the maximized withdrawals 

are intended to represent sustainable yield of the system (a rate that can be maintained indefinitely). In this 

model, the decision variables (a term used in optimization modeling to identify variables that can be part of a 

management scheme) are the withdrawal rates at 1,152 

model cells corresponding to well locations and at 7 

model cells corresponding to stream reaches on the outcrop/subcrop area of the Sparta aquifer.

Constraints (specified limits) on hydraulic head 

and streamflow are designated at specific cells called 

control points within the ground-water model grid (fig. 

3). These control points and the constraints assigned 

must be satisfied for a feasible solution of the objective 

function.

Objective Function

The objective of the optimization model is to 

maximize water production from ground-water and 

surface-water sources. The objective function of the 

optimization model has the form:

maximize z = (3)

where z is the total managed water withdrawal, in million cubic feet per day; 

 is the sum of ground-water withdrawal 

rates from all managed wells, in million 

cubic feet per day; and

is the sum of surface-water withdrawal 

rates from all managed stream reaches, in 

million cubic feet per day.

Streamflow Constraints

Streamflow is regulated in Arkansas by ASWCC 

for purposes of maintaining water quality, navigation, 

and species habitat. Streamflow constraints are specified as the minimum amount of flow required at individual river model cells. The equation governing the 

relation between streamflow constraints and flow into 

and out of a stream is

(4)

Q K– dh

dl = -----A

dh

dl

-----

A bw =

q ∑ well q + ∑ river

q ∑ well

q ∑ river

qhead

R

qoverland

R

qground water

R

qdiversions

R

qriver

R

qminimum

R

∑ – ∑ ≥

+ ∑ ±∑ –
Conjunctive-Use Optimization Model 9

where is the flow rate into the head of stream 

reach R, in cubic feet per day; 

 is the sum of all overland and tributary flow to stream reach R, in cubic feet 

per day;

 is the net sum of all groundwater flow to or from stream reach R, in 

cubic feet per day;

 is the sum of all surface-water 

diversions from stream reach R, in cubic 

feet per day;

 is the sum of all potential withdrawals, not including diversions, from stream 

reach R, in cubic feet per day; and

is the minimum permissible surface-water flow rate for stream reach R, in 

cubic feet per day.

Optimal surface-water withdrawals were calculated at seven points along seven streams within the 

Sparta aquifer outcrop/subcrop area for which streamflow constraints were specified. These streams include 

the Arkansas River, Cache River, Bayou Dorcheat, 

Ouachita River, Red River, Saline River, and White 

River (fig. 3), all of which have mean annual flows 

greater than 1,000 ft3

/s within the Sparta outcrop/subcrop area. Streamflow constraint data were determined 

using the median August mean daily flow values for the 

period of record for each stream (Steve Loop, Arkansas 

Soil and Water Conservation Commission, written 

commun., 2002; Paul Ensminger, U.S. Geological Survey, written commun., 2002). All streamflow data were 

based on historical streamflow data obtained from 

USGS streamflow gaging stations located on nearby 

reaches of these seven streams. Streamflow constraints 

were specified in the most downstream point of each 

stream segment (seven total). Flow into the most 

upstream point of each stream was specified based on 

mean annual flow. Overland flow was distributed 

equally at river model cells within a stream reach based 

on the difference in long-term mean streamflow 

between neighboring streamflow-gaging stations for a 

specific stream reach; or if such data were unavailable, 

areal estimates of runoff based on drainage areas were 

used (Elton Porter, U.S. Geological Survey, written 

commun., 2001). Surface-water diversion rates that 

occurred in 2000 (T.W. Holland, U.S. Geological Survey, written commun., 2002) were subtracted from 

specified overland flow at the appropriate river cells. 

Overland flow, streamflow constraints, and optimum 

withdrawal rates for each stream are listed in table 2. 

qhead

R

qoverland

R

∑

qground water

R

∑

qdiversions

R

∑

qriver

R

∑

qminimum

R

Table 2. Flow and optimized surface-water withdrawals for major streams within the Sparta aquifer outcrop area specified 

within the conjunctive-use optimization model for all three scenarios (3A-3C)

[Mft3/d, million cubic feet per day; August mean is the median August mean daily flow for period of record]

Stream name

Number of

groundwater

model

cells

Flow in 

stream

entering

outcrop/

subcrop

area

(Mft3/d)

Net

overland

flow per

river cell

(Mft3/d)

Flow in

stream

leaving

outcrop/

subcrop

 area

(Mft3/d)

Streamflow

constraint

(Mft3/d)

Rationale

for value

of

constraint

Period of

record

Optimized

surfacewater

withdrawal

(Mft3/d)

Arkansas 13 3,751 -0.006 3,751 1,492 August mean 1970-1997 2,259

Bayou Dorcheat 26 67 1.2 98 2 August mean 1929-1932,

 1937-1979

96

Cache 50 109 0.2 119 36 August mean 1988-2000 83

Ouachita 39 340 9.6 713 171 August mean 1929-1997 542

Red 55 1,700 2.4 1,832 473 August mean 1945-1997 1,359

Saline 14 74 0.7 85 16 August mean 1941-1999 69

White 11 2,316 4.6 2,366 1,378 August mean 1981-1995 988
10 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Figure 3. Control point locations for Sparta aquifer hydraulic-head and streamflow constraints.

Texas

Arkansas

Louisiana

Tennessee

Mississippi

Winn

White

Union

Rapides

Lee

Drew

Union

Yazoo

Ashley

Bolivar Desha

Miller

Grant

Saline

Bossier

Grant

Pulaski

Arkansas

Lonoke

Chicot

Dallas

Cross

Prairie

Holmes

Phillips

Natchitoches

Jefferson

Bienville

Poinsett

Avoyelles

Leflore

Tensas

Bradley Ouachita

La Salle

Lincoln

Monroe

Columbia

Nevada

Claiborne

Franklin

Jackson

Madison

Calhoun

Webster

Catahoula

Concordia

Morehouse

Sunflower

Tunica

Ouachita

Jackson

Lincoln

Woodruff

Richland

Washington

Caldwell

Crittenden

Coahoma

St. Francis

Cleveland

Lafayette

Hot Spring

Sharkey

Quitman

East Issaquena

Carroll

Humphreys

West

Carroll

White

River

Cache

River

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop Active Model

Area Boundary

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

EXPLANATION

Simulated rivers

Hydraulic head constraints

(top of the Sparta Sand)

Hydraulic head constraints

(50 percent saturated thickness)

Streamflow constraints (Table 2,

all located at down-stream

most point in outcrop streams)

Red

River

Bayou

Dorcheat

Ouachita

River

Saline

River

Arkansas

River
Sustainable-Yield Estimation 11

Hydraulic-Head Constraints

Equation 3 is computed such that the following 

constraints are maintained:

(5)

where hc is the hydraulic-head altitude at constraint 

location c; and

hminimum is the hydraulic-head altitude at the 

top of the Sparta Sand. 

To accommodate the ASWCC Critical Ground-Water 

Area criterion that hydraulic heads within the Sparta 

aquifer should remain above the top of the Sparta Sand, 

hydraulic-head constraints were specified at 2,549 

model cells (fig. 3).

A uniform distribution of model cells with an 

approximate 5-mile spacing was selected as the control 

points for which hydraulic heads are maintained at or 

above the minimum specified head during the optimization process. Specification of hydraulic-head constraints at every model cell is numerically difficult and 

was found to be unneeded for obtaining an acceptable 

solution for ground-water withdrawals that yield 

hydraulic-head values at or above the specified constraint. In the simulation/optimization model of Czarnecki and others (2003), the differences between 

simulated values of hydraulic head and associated 

hydraulic-head constraints were compared to differences at all model cells and resulted in nearly identical 

distribution for the two data sets. Based on the results 

of Czarnecki and others (2003), the assumption was 

made that hydraulic-head control points specified at an 

approximate 5-mile spacing provided an adequate distribution of control points for the optimization model.

The Sparta aquifer is not confined everywhere in 

the model, such as in the outcrop/subcrop area or in a 

small area in northeastern Louisiana (fig. 3). For this 

reason, hydraulic-head constraints were set at 50 percent of predevelopment saturated thickness in these 

areas to prevent an unrealistic optimized solution or 

hydraulic heads from dropping below the bottom of the 

Sparta Sand. Everywhere else, hydraulic-head constraints were set at the top of the Sparta Sand.

Specified Limits on Ground-Water Withdrawals

The spatial distribution of the withdrawal decision variable (managed wells) was specified as those 

wells “that represent withdrawal rates” of the Sparta 

aquifer flow model from 1990-1997. The withdrawal 

rates from these wells were based on 1995 and 1996 

water-use data from Louisiana and Arkansas that were 

adjusted to adequately represent the time period from 

1990-1997 in the last stress period of the Sparta aquifer 

flow model (ground-water model stress period 28; 

McKee and Clark, 2003), hereafter referred to as “baseline 1990-1997 withdrawal rates.” For optimization, 

1,152 one-square mile cells were used to represent 

managed wells from more than 1,760 individual wells 

and aggregated water-use data in 1995-1996. All managed wells were assigned to layer 2 of the ground-water 

model; 398 in Arkansas, 725 in Louisiana, and 29 in 

Mississippi. The baseline 1990-1997 withdrawal rates 

for all managed wells total 32 Mft3

/d.

Test simulations using the baseline 1990-1997 

withdrawal rates applied to steady-state conditions 

yielded areas with dry cells. Therefore, ground-water 

withdrawal limits were specified at each well in the 

model as a multiple of the amount pumped in the baseline 1990-1997 withdrawal rate, such that 

(6)

where is the optimal ground-water withdrawal 

for well i, in cubic feet per day;

M is a multiplier between 1 and 2; and

 is the total amount withdrawn using the baseline 1990-1997 withdrawal rate from well i, in cubic feet per 

day.

This equation holds true for all withdrawals except 

those in Union County, which were held at 28 percent 

of the Hays (2000) 1997 withdrawal rate, discussed further in the Scenario 3 section.

Specified Limits on Surface-Water Withdrawals

No limits were imposed on optimized withdrawals from streams such that the range in optimal withdrawal was between zero and the maximum amount of 

water available at the most downstream point of each 

stream segment where constraints were specified. This 

specification permitted analysis of where water could 

be produced and the maximum amount available.

SUSTAINABLE-YIELD ESTIMATION

Sustainable yield was estimated by using the 

optimization model. In this section, sustainable yield 

hc hminimum ≥

0 qwelli M qwelli( ) 1990 1997 – ≤ ≤

qwelli

qwelli( ) 1990 1997 –
12 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

and unmet demand are defined and the development 

and results of scenarios are described.

Determination of Sustainable Yield and 

Unmet Demand

The optimization model provides estimates of 

sustainable yield from both ground water and surface 

water. The primary concern of State planners and 

water-resource managers for the Sparta aquifer is sustainability—the development and use of ground water 

for an indefinite time without causing unacceptable 

environmental, economic, or social consequences 

(Alley and others, 1999). The term “sustainable yield” 

is used to define that quantity of water that can be withdrawn indefinitely by reaching a system equilibrium 

(steady state) without compromising the integrity of 

the aquifer or with respect to agreed upon criteria. In 

Arkansas, the ASWCC has established critical criteria 

for a confined aquifer, such as the Sparta aquifer. Criteria include (1) hydraulic head in wells must be above 

the top of the aquifer formation and (2) the rate of 

decline of hydraulic head in wells must not be more 

than 1 foot per year over a 5-year period. If these criteria are violated, the ASWCC may designate an area to 

be a Critical Ground-Water Area. 

Unmet demand is defined as the difference 

between the optimized ground-water withdrawal rate 

(sustainable yield) and a desired withdrawal rate 

(demand that is not met by ground water). For example, 

if the desired withdrawal rate is 9.8 Mft3

/d, but the optimized ground-water withdrawal rate is calculated to be 

only 6.7 Mft3/d, the unmet demand is 3.1 Mft3

/d (the 

difference between these two values). Note that determining the optimized withdrawal rate is an independent 

process that is not affected by the desired withdrawal 

rate. Therefore, unmet demand will vary as the desired 

withdrawal rate changes.

The development of an optimization model scenario to achieve feasible optimized solutions of estimated sustainable yield from the Sparta aquifer was a 

step-wise process involving several scenario formulations. The general process involved formulating a scenario by assigning initial conditions, upper and lower 

specified limits on withdrawals from managed wells, 

and hydraulic-head and streamflow constraints at control points. An infeasible solution occurs when optimization is not possible for the given scenario formulation 

without violating constraints. Despite the difficulties of 

investigating the cause of infeasible solutions, much is 

learned about the system behavior to direct and guide 

scenario development. For the purpose of discussion, 

the progression of scenario development is separated 

and described in three scenarios of which the results of 

scenario 1 lead to scenario 2 which lead to scenario 3.

For all scenarios, MODMAN was used to estimate optimized ground-water withdrawal rates from all 

1,152 managed wells in the optimization model. The 

lower and upper specified limits on withdrawals from 

these managed wells were a percentage from baseline 

1990-1997 withdrawal rates (100 percent). The distribution and rates of withdrawal for the baseline 1990-

1997 wells managed in the optimization model are 

shown in figure 4. For feasible scenario solutions, the 

sustainable yield was estimated for the Sparta aquifer 

in Arkansas and Louisiana independently (table 3). 

Sustainable yield estimations for ground water are 

listed in table 4 by county and parish for each feasible 

scenario, as well as examples of unmet demand calculated using hypothetical ground-water demand rates. 

Optimized withdrawals from streams were the same 

through each feasible scenario (3A-3C).

Total optimized withdrawals of surface water 

from all streams for which low-flow constraints were 

specified were about 5,300 and 100 Mft3

/d in Arkansas 

and Louisiana, respectively. The sustainable yield from 

surface water is much larger than from ground water, 

and these large sustainable yields represent a potential 

source of water that could supplement ground water 

and meet total water demand.

The following sections describe three scenarios 

that were tested with the optimization model. Scenario 

1 looks at the viability of using 1997 simulated hydraulic heads as initial conditions in the optimization 

model. Scenario 2 tests the feasibility of stressing all 

pumping wells at the continued withdrawal rate of 

1990-1997. As described in the following sections, scenarios 1 and 2 are infeasible. Scenario 3 provides estimates of sustainable yield using variations on 

withdrawals throughout the model area while recognizing predefined sustainable yield in Union County from 

Hays (2000). To accomplish this, withdrawals in Union 

County were treated as managed wells for ease of postprocessing. Union County managed wells were not 

allowed to decrease below 28 percent of the Hays 

(2000) 1997 withdrawal rate, discussed further in the 

Scenario 3 section. In all cases, withdrawals in Union 

County were optimized at the lower specified limit of 

28 percent of the Hays (2000) 1997 withdrawal rate.
Sustainable-Yield Estimation 13

Figure 4. Baseline 1990-1997 ground-water withdrawal rates used in conjunctive-use optimization model.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boundary

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

500,000 - 1,410,000

200,000 - 499,999

100,000 - 199,999

75,000 - 99,999

50,000 - 74,999

25,000 - 49,999

0 - 24,999

Baseline 1990-1997

withdrawal rate, cubic

feet per day

EXPLANATION
14 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Table 3. Summary of optimization scenarios and associated constraints including sustainable yield estimates and total 1997 

withdrawals for Arkansas and Louisiana

[Mft3/d, million cubic feet per day; Sum of sustainable yield and unmet demand may not equal baseline withdrawal rate because of rounding; NA, not applicable]

Study

area

(managed

wells1)

1

Does not include 29 wells in Mississippi.

Scenario

Number

of

managed

wells with

 non-zero

 flow

Baseline

1990-1997

withdrawal

rate

(Mft3/d)

Lower

specified

limit

withdrawal

rate2

2

Withdrawal rate in percent of baseline 1990-1997 withdrawals; 22.4, 8.2, and 1.3 Mft3/d for Arkansas, Louisiana, and Mississippi, respectively.

Upper

specified

limit

withdrawal

rate2

Sustainable

groundwater

yield

(Mft3d)

Unmet

demand3

(Mft3/d)

3

Unmet demand assuming a desired demand equal to baseline 1990-1997 withdrawals.

Optimized

withdrawal

from

streams4

(Mft3/d)

4

Not considered in unmet demand calculation; available streamflow to be considered as source of water for areas where current demand is 

unmet.

Arkansas (398) 1 NA 22.4 0 100 infeasible NA NA

2 NA 22.4 100 100 infeasible NA NA

3A 193 22.4 0 100 11.6 10.9 5,300

3B 146 22.4 0 150 12.7 9.8 5,300

3C 135 22.4 0 200 13.2 9.2 5,300

Louisiana (725) 1 NA 8.2 0 100 infeasible NA NA

2 NA 8.2 100 100 infeasible NA NA

3A 110 8.2 0 100 0.3 7.9 96

3B 73 8.2 0 150 0.4 7.8 96

3C 69 8.2 0 200 0.5 7.7 96
Sustainable-Yield Estimation 15

Table 4. Estimated sustainable yield and unmet demand by county and parish for each scenario in Arkansas and Louisiana[All values are in million cubic feet per day. Negative values indicate no unmet demand]

County

name

Baseline

1990-1997

withdrawal

rate

Sustainable yield based on an

upper withdrawal limit of:

Unmet demand based on a demand of

the baseline rate and a sustainable yield from:

Unmet demand ba

 scenarios 3a, 3b, and 

100 percent

of baseline

rate

(scenario 3a)

150 percent

of baseline

rate

(scenario 3b)

200 percent

of baseline

rate

(scenario 3c) Scenario 3a Scenario 3b Scenario 3c

100 percent

of baseline

rate

(scenario 3a)

Arkansas 4.73 0.84 0.02 0.02 3.89 4.72 4.71 3.89 Bradley 0.12 0.00 0.00 0.00 0.12 0.12 0.12 0.12 Calhoun 0.05 0.03 0.00 0.00 0.02 0.05 0.05 0.02 Chicot 0.03 0.03 0.05 0.06 0.00 -0.02 -0.03 0.00 Cleveland 0.04 0.04 0.05 0.07 0.00 -0.01 -0.03 0.00 Columbia 0.74 0.00 0.00 0.00 0.74 0.74 0.74 0.74 Cross 0.15 0.15 0.22 0.29 0.00 -0.07 -0.15 0.00 Dallas 0.08 0.08 0.00 0.00 0.00 0.08 0.08 0.00 Desha 1.04 1.04 1.56 2.08 0.00 -0.52 -1.04 0.00 Drew 0.22 0.22 0.08 0.07 0.00 0.14 0.15 0.00 Grant 0.15 0.15 0.22 0.29 0.00 -0.07 -0.15 0.00 Jackson 0.10 0.10 0.15 0.20 0.00 -0.05 -0.10 0.00 Jefferson 5.58 5.58 6.80 6.27 0.00 -1.22 -0.69 0.00 Lafayette 0.15 0.00 0.00 0.00 0.15 0.15 0.15 0.15 Lee 0.16 0.16 0.00 0.00 0.00 0.16 0.16 0.00 Lincoln 0.25 0.25 0.37 0.14 0.00 -0.12 0.10 0.00 Lonoke 1.79 0.13 0.02 0.03 1.65 1.76 1.76 1.65 Miller 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 Monroe 0.05 0.00 0.00 0.00 0.05 0.05 0.05 0.05 Ouachita 0.28 0.16 0.00 0.00 0.12 0.28 0.28 0.12 Phillips 0.57 0.57 0.70 0.94 0.01 -0.13 -0.36 0.01 Poinsett 0.60 0.60 0.77 0.80 0.00 -0.17 -0.20 0.00 Prairie 2.82 0.14 0.11 0.15 2.67 2.70 2.66 2.67 
16 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Pulaski 0.02 0.02 0.03 0.04 0.00 -0.01 -0.02 0.00 Saline 0.02 0.02 0.03 0.05 0.00 -0.01 -0.02 0.00 St. Francis 0.07 0.06 0.08 0.11 0.01 -0.01 -0.04 0.01 Union 2.20 0.78 0.78 0.78 1.42 1.42 1.42 1.42 Woodruff 0.43 0.40 0.60 0.80 0.02 -0.17 -0.37 0.02 LouisianaBienville 1.17 0.00 0.00 0.00 1.17 1.17 1.17 1.17 Bossier 0.01 0.00 0.01 0.01 0.00 0.00 0.00 0.00 Caddo 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Caldwell 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Claiborne 0.29 0.00 0.00 0.00 0.29 0.29 0.29 0.29 Jackson 0.84 0.00 0.00 0.00 0.84 0.84 0.84 0.84 La Salle 0.01 0.01 0.02 0.02 0.00 -0.01 -0.01 0.00 Lincoln 0.98 0.00 0.00 0.00 0.98 0.98 0.98 0.98 Morehouse 0.75 0.03 0.04 0.05 0.72 0.71 0.70 0.72 Natchitoches 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 Ouachita 2.70 0.00 0.00 0.00 2.70 2.70 2.70 2.70 Richland 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 Union 0.59 0.00 0.00 0.00 0.59 0.59 0.59 0.59 Webster 0.60 0.01 0.02 0.03 0.58 0.58 0.57 0.58 Winn 0.26 0.23 0.28 0.37 0.03 -0.02 -0.11 0.03 

Table 4. Estimated sustainable yield and unmet demand by county and parish for each scenario in Arkansas and Louisiana--Continu[All values are in million cubic feet per day. Negative values indicate no unmet demand]

County

name

Baseline

1990-1997

withdrawal

rate

Sustainable yield based on an

upper withdrawal limit of:

Unmet demand based on a demand of

the baseline rate and a sustainable yield from:

Unmet demand ba

 scenarios 3a, 3b, and 

100 percent

of baseline

rate

(scenario 3a)

150 percent

of baseline

rate

(scenario 3b)

200 percent

of baseline

rate

(scenario 3c) Scenario 3a Scenario 3b Scenario 3c

100 percent

of baseline

rate

(scenario 3a)
Sustainable-Yield Estimation 17

Scenario 1

The first scenario formulated for the optimization model attempted to use 1997 simulated hydraulic 

heads from the ground-water model as the initial heads. 

However, continued heavy withdrawals from the Sparta 

aquifer caused hydraulic heads at the end of 1997 to be 

below the top of the Sparta Sand in many areas within 

the aquifer (fig. 5). Therefore, in the optimization 

model, some initial heads were below the constraint; 

thus, the solution was infeasible from the outset. To 

resolve this discrepancy, steady-state simulations of 

predevelopment conditions were used for initial conditions in scenarios 2 and 3 because hydraulic heads were 

above the top of the Sparta Sand prior to aquifer development. However, an area of approximately 100 mi2

 in 

northeastern Webster and western Claiborne Parishes 

in Louisiana (fig. 3) had predevelopment heads below 

the top of the Sparta Sand. Lithologic data were 

unavailable to dispute or confirm this condition. 

Unconfined conditions are most likely the result of cell 

geometry misrepresenting the actual aquifer geometry. 

To resolve this problem in scenarios 2 and 3, this area 

was treated in similar fashion as the outcrop/subcrop 

areas with regard to assigning hydraulic-head constraints at 50 percent of the predevelopment saturated 

thickness. 

Scenario 2

The second scenario transitioned from the first, 

after resolving issues with initial heads, by attempting 

to find an optimized solution using 100 percent of the 

baseline 1990-1997 withdrawal rates as the lower specified limit on withdrawals from managed wells. A feasible solution for this scenario would indicate this 

ground-water demand could be sustainable indefinitely. Unfortunately, this solution is infeasible, 

because the scenario 1 discussion already indicated 

observed and simulated 1997 hydraulic heads below 

the top of the Sparta Sand as a result of continued, 

heavy withdrawals. The demand of baseline 1990-1997 

withdrawal rates cannot be met; therefore, the lower 

specified limit on withdrawals from managed wells 

must be lowered to something less than 100 percent, 

which will create an unmet demand.

Scenario 3

In previous work by Hays (2000), sustainable 

yield estimates were calculated for the Sparta aquifer in 

Union County. In these estimates, simulated withdrawals were varied iteratively to achieve the specified minimum hydraulic head at the top of the Sparta Sand at 

one constraint location centered in the deepest cone of 

depression formed beneath the city of El Dorado. 

Hays’s study resulted in an estimated sustainable yield 

of 28 percent of the 1997 withdrawal rate in Union 

County. This result, which was recognized by 

ASWCC, was used as a goal to reduce withdrawals 

from the Sparta aquifer in Union County. Since that 

time, local industry, the city of El Dorado, and Union 

County have been initiating conservation measures to 

reduce withdrawals from the Sparta aquifer through 

water reuse and alternative water sources for industry. 

Therefore, in all subsets of scenario 3, withdrawals in 

Union County are not allowed to decrease below 28 

percent of the Hays (2000) 1997 withdrawal rate, 

which is equivalent to approximately 35.6 percent of 

the baseline 1990-1997 withdrawal rate specified for 

Union County in the current optimization model. 

Scenario 3 transitioned from scenario 2 by allowing withdrawals from managed wells other than those 

in Union County to vary between 0 (lower specified 

limit) and some percent of the baseline 1990-1997 

withdrawals (upper specified limit). Upper specified 

limits of 100, 150, and 200 percent were used in three 

subsets (A-C) that make up scenario 3 to find optimized 

solutions and determine any unmet demand assuming a 

desired demand equal to baseline withdrawals. Lower 

specified limits of 0 percent were used for all three scenarios for wells other than those in Union County.

All subsets of scenario 3 initially resulted in 

infeasible solutions. However, only three constraint 

locations in the southwest portion of the model (fig. 

6A) caused the optimization to be infeasible. The head 

constraint (as discussed previously in the HydraulicHeads Constraint section) is set equal to the Sparta top 

in areas where the aquifer is confined and to 50 percent 

of the original saturated thickness where the aquifer is 

unconfined (outcrop area). The three constraint locations that caused the infeasible solutions are within or 

close to the defined Sparta Sand outcrop area (fig. 6). 

Because of uncertainty in the boundary between confined and unconfined in that transition area, and 

because there is some inherent uncertainty in the interpolated top of the Sparta Sand, it was reasonable to 

adjust the constraint at each of the three infeasible
18 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Figure 5. Difference between Sparta aquifer 1997 simulated heads and the top of the Sparta Sand.

Arkansas

Louisiana

Texas

Tennessee

Mississippi

Active Model

Area Boundary

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

0 40 80 20 Miles

0 40 80 20 Kilometers

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

-300 to -200

-199 to -100

-99 to -50

-49 to 0

1 to 50

51 to 100

101 to 200

201 to 300

301 to 500

501 to 1,000

1001 to 2,000

2001 to 4,050

EXPLANATION

1997 Simulated

hydraulic heads

minus top of Sparta

Sand (feet)

Base from U.S. Geological Survey digital data, 1:100,000
Sustainable-Yield Estimation 19

locations. A relaxation of 10 feet in the constraint at 

these three locations resulted in feasible optimization 

solutions.

For scenario 3A, lower and upper specified limits 

on withdrawals from managed wells other than those in 

Union County were set at zero and 100 percent of baseline 1990-1997 withdrawal rates, respectively. These 

specified limits on managed wells, coupled with the 

hydraulic-head constraint over much of the Sparta 

aquifer results in an optimized solution. However, 

withdrawals from many managed wells were optimized 

at the lower specified limit of zero in the Grand Prairie 

area and Columbia County, and in much of the already 

stressed areas in north-central Louisiana (fig. 6A). 

Managed wells with non-zero flow totaled 193 out of 

398 possible wells in Arkansas and 110 out of 725 

wells in Louisiana (table 3). Other than Union County, 

withdrawals from these managed wells with non-zero 

flow were optimized at the upper specified limit (100 

percent of the baseline 1990-1997 withdrawal rate). 

The distribution of optimal withdrawals was restricted 

when the hydraulic head at any point in the optimization model reached the minimum-head constraint. This 

occurred in two locations in scenario 3A—the first near 

the border of Lonoke and Prairie Counties in Arkansas 

and the second in Claiborne Parish in Louisiana (fig. 

6A). Sustainable yield for scenario 3A was calculated 

to be 11.6 Mft3

/d in Arkansas and 0.3 Mft3/d in Louisiana. Using baseline 1990-1997 withdrawal rates as 

the desired demand, unmet demand was calculated to 

be 10.9 Mft3/d and 7.9 Mft3/d in Arkansas and Louisiana, respectively (table 3).

For scenario 3B, the withdrawals from managed 

wells other than those in Union County were allowed to 

vary between zero and 150 percent of baseline 1990-

1997 withdrawal rates. This scenario produced an optimized solution that was similar to the scenario 3A solution because most withdrawals from managed wells in 

the stressed areas of the Grand Prairie area and northcentral Louisiana were optimized at the lower specified 

limit of zero. Managed wells with non-zero flow as 

compared to Scenario 3A were fewer in Arkansas with 

146 out of 398 possible wells and fewer in Louisiana 

with 73 out of 725 possible wells (table 3). As in scenario 3A, withdrawals from these managed wells with 

non-zero flow were optimized at the upper specified 

limit. Hydraulic-head constraints were reached at the 

same two locations as for scenario 3A, along with an 

additional point located in the northwestern corner of 

the model area (fig. 6B). Sustainable yield for scenario 

3B was calculated to be 12.7 Mft3

/d in Arkansas and 

0.4 Mft3

/d in Louisiana. Using baseline 1990-1997 

withdrawal rates as the desired demand, unmet demand 

was calculated to be 9.8 Mft3/d and 7.8 Mft3/d in 

Arkansas and Louisiana, respectively (table 3), which 

could be obtained from the optimized surface-water 

withdrawals.

For scenario 3C, withdrawals from managed 

wells other than those in Union County were allowed to 

vary between zero and 200 percent of the baseline 

1990-1997 withdrawal rate. This scenario produced an 

optimized solution that was similar to the scenario 3B 

solution. Managed wells with non-zero flow as compared to scenario 3B were fewer in Arkansas with 135 

out of 398 possible wells and fewer in Louisiana with 

69 out of 725 possible wells (table 3). As with both scenarios 3A and 3B, withdrawals from these managed 

wells with non-zero flow were optimized at the upper 

specified limit. Hydraulic-head constraints also were 

reached at the same three locations in the model as that 

of scenario 3B (figs. 6B, 6C). Sustainable yield for scenario 3C was calculated to be 13.2 Mft3

/d in Arkansas 

and 0.5 Mft3/d in Louisiana. Using baseline 1990-1997 

withdrawal rates as the desired demand, unmet demand 

was calculated to be 9.2 Mft3

/d and 7.7 Mft3/d in 

Arkansas and Louisiana, respectively (table 3).

Analysis of Scenario Sustainable Yield 

Estimation

Sustainable yield is a function of managed well 

distributions and lower and upper specified limits on 

withdrawals from managed wells. However, in each of 

the scenarios, the managed well distributions are fixed. 

For the scenarios with feasible solutions, sustainableyield estimations increase as the upper specified limit 

on withdrawals from managed wells increase (fig. 7). 

However, continuing to increase withdrawals at managed well locations beyond 200 percent of the baseline 

rate may be unrealistic.

The source of water for the increased sustainable 

yields with increased ground-water withdrawals is 

leakage from surface water in the outcrop/subcrop area 

and through the overlying confining unit. The amount 

of water required by simulations with maximum allowable ground-water withdrawals greater than 200 percent of the 1990-1997 baseline rate is considered 

unrealistic because the wells physically would not be 

able to pump that much water, and the distribution of 

pumping wells would need to be changed to satisfy 
20 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Figure 6. Ratio of optimal ground-water withdrawal to baseline 1990-1997 withdrawals (in percent) for (A) scenario 3A,

(B) scenario 3B, and (C) scenario 3C including locations where hydraulic-head constraints are reached.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Ratio of optimal withdrawal rate

to baseline 1990-97 withdrawal

rate (in percent)

Adjusted constraints

Minimum head constraint

reached

0

36

100

EXPLANATION

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boundary

A

Base from U.S. Geological Survey digital data, 1:100,000
Sustainable-Yield Estimation 21

Figure 6. Ratio of optimal ground-water withdrawal to baseline 1990-1997 withdrawals (in percent) for (A) scenario 3A,

(B) scenario 3B, and (C) scenario 3C including locations where hydraulic-head constraints are reached.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Ratio of optimal withdrawal rate

to baseline 1990-97 withdrawal B

rate (in percent)

Adjusted constraints

Minimum head constraint

reached

0

36

150

EXPLANATION

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boundary

Base from U.S. Geological Survey digital data, 1:100,000
22 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Figure 6. Ratio of optimal ground-water withdrawal to baseline 1990-1997 withdrawals (in percent) for (A) scenario 3A,

(B) scenario 3B, and (C) scenario 3C including locations where hydraulic-head constraints are reached.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

C

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Ratio of optimal withdrawal rate

to baseline 1990-97 withdrawal

rate (in percent)

0

36

200

Adjusted constraints

Minimum head constraint

reached

EXPLANATION

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boundary

Base from U.S. Geological Survey digital data, 1:100,000
Sustainable-Yield Estimation 23

model constraints for managed wells with no flow. In 

addition, the distribution of wells likely would not be 

suitable for ground-water users.

Sustainable yield may be increased by strategic 

placement of managed wells in the model (other than 

the present fixed well locations) and optimizing withdrawal rates. This strategy could be used to determine 

the location of future wells to be drilled into the Sparta 

aquifer. Results of the optimization modeling indicate 

that substantial increases in maximum allowable 

ground-water withdrawals produce small incremental 

increases in sustainable yield (fig. 7). Assuming the 

total water demand is equal to the baseline 1990-1997 

withdrawal rates, the sustainable yield estimated from 

scenario 3A-C only provides 52 to 59 percent of the 

total demand for Arkansas; the remainder is unmet 

demand that could be obtained from surface water.

Infeasible Scenarios

Several other scenarios were investigated. In 

many, the optimization problem was formulated with 

the lower specified limit on withdrawals from managed 

wells set at something greater than zero. For these scenarios, there were no feasible solutions. This means 

that some areas of the Sparta aquifer cannot be pumped 

and maintain hydraulic head above the top of the Sparta 

Sand while other areas are allowed to pump at rates 

equal to or greater than baseline 1990-1997 withdrawal 

rates.

Other scenarios were formulated whereby all 

hydraulic-head constraints were relaxed to the Sparta 

Sand bottom in Louisiana because currently there is no 

designated CGWA, as there is in Arkansas. In these 

scenario formulations, estimated sustainable yield 

appears to be much larger. However, non-linear model 

behavior, resulting from large changes in saturated 

Figure 7. Graph showing scenario results of estimated sustainable yield as a function of the maximum allowable ground-water 

withdrawal.

6

8

10

12

14

16

80 100 120 140 160 180 200 220

MAXIMUM ALLOWABLE GROUND-WATER WITHDRAWAL

(percent of baseline 1990-1997 withdrawals)

ESTIMATED SUSTAINABLE YIELD, IN MILLION

CUBIC FEET PER DAY

Scenario 3AScenario 3BScenario 3C

0

2

4

0.4 0.5 0.3

Louisiana 13.2

12.7

11.6

Arkansas
24 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

thickness and transmissivity, was too large to render 

viable solutions with these scenario formulations. Nonlinear optimization would be required to adequately 

determine the sustainable yield. These techniques are 

more complicated than linear or nearly linear boundary 

conditions (Ahlfeld and Mulligan, 2000). However, 

these types of scenarios could explore the effect of nonuniform water-management strategies between adjacent states. Therefore, two reasons emerged for continuing to maintain hydraulic-head constraints at the 

top of the Sparta Sand in Louisiana: (1) to maintain 

consistency with Arkansas criteria in lieu of any established criteria in Louisiana; and (2) to minimize any 

non-linear effects that could cause optimization problems as the Sparta aquifer becomes unconfined. 

Verification of Optimized Withdrawals

Where unconfined conditions occur, large 

changes in saturated thickness and transmissivity can 

cause non-linear model response. Optimized withdrawal rates must be checked for errors associated with 

non-linear response of the ground-water model. Verification was done using the optimized withdrawals from 

each of the three scenarios as input to the Sparta aquifer 

flow model (MODFLOW-2000 version) to evaluate 

whether simulated steady-state hydraulic heads were 

within the specified constraints as computed by the 

optimization model (fig. 8). With the exception of the 

Sparta aquifer outcrop areas where hydraulic-head constraints were specified as the aquifer bottom, the optimized withdrawal rates produced hydraulic heads at or 

above the top of the Sparta Sand in most of the model 

area. In areas near the outcrop, hydraulic heads were 

below the top of the Sparta Sand by 10 to 50 ft, but 

maintained greater than 50 percent saturated thickness, 

as did all hydraulic heads in the outcrop area. 

Model Assumptions and Limitations

To effectively utilize flow model results, an 

understanding of model assumptions and limitations is 

essential. The accuracy of ground-water models is limited by simplification of the conceptual model of a 

complex flow system, by space and time discretization 

effects, and by assumptions made in the formulation of 

the governing flow equations. Models also are limited 

by cell size, number of layers, boundary conditions, 

accuracy and availability of hydraulic property values, 

accuracy of calibration, accuracy of ground-water 

withdrawal estimates, historical data for simulations, 

and parameter sensitivity. Models are limited by the 

availability of data and by the interpolations and 

extrapolations that are inherent in using data in a 

model. Although a model is calibrated, the calibration 

parameter values may not be unique in yielding the specific simulations of hydraulic head.

No additional managed wells other than those 

that were used to represent the baseline 1990-1997 

withdrawal rates were assumed to be available for 

pumping in the optimization scenarios; only existing 

wells were used. Therefore, additional wells strategically located might have resulted in more yield. Also, 

the optimization scenarios focused on adjusting maximum allowable withdrawal rates based on the percentage of baseline 1990-1997 withdrawal rates. Some 

areas represented in the model likely could pump at 

rates much higher (>200 percent) than baseline 1990-

1997 withdrawal rates. Both of these changes might 

increase yield. Another method of formulating the scenarios could be based on withdrawal rate magnitudes 

rather than percentages of a base rate. 

Nonlinear effects are negligible except in the 

outcrop areas of the Sparta Sand. Because hydraulichead constraints are set at the top of the Sparta Sand, 

the ground-water system must maintain confined conditions, which prevents the nonlinearity associated with 

unconfined systems. When the Sparta aquifer is confined, the flow equation is linear, but becomes nonlinear for unconfined conditions and, in this case, the 

linear solution from the optimization model begins to 

deviate from the non-linear solution. The system is 

unconfined in the outcrop-subcrop areas where hydraulic-head constraints were set to 50 percent saturated 

thickness of the aquifer, and some nonlinearity effects 

cause the estimated hydraulic heads from the optimized 

solutions to differ from simulated hydraulic heads from 

a flow ground-water flow model run using optimized 

withdrawal rates.

Hydraulic-head constraints near topographic 

high points in the top of the Sparta Sand cause the optimization model to reduce withdrawals over large areas 

to maintain hydraulic heads above the topographic high 

head constraint. The reduced withdrawals result in 

hydraulic heads more than 100 feet above the top of the 

Sparta Sand (fig. 8A-C) over much of the Sparta aquifer. Future alternative scenario runs might include discrete assignment of the hydraulic-head constraints that 

accommodate the highs and lows inherent in the top of 
Sustainable-Yield Estimation 25

Figure 8. Difference between steady-state simulated hydraulic heads and the top of the Sparta Sand for (A) scenario 3A, (B) 

scenario 3B, and (C) scenario 3C using optimized withdrawals in the Sparta aquifer flow model for verification.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

A

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Scenario 3 simulated heads

minus the top of the Sparta

Sand (feet)

Adjusted constraints

Minimum head constraint

reached

-129 to -50

-49 to -25

-24 to 0

1 to 25

26 to 50

51 to 100

101 to 500

> 500

EXPLANATION

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

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boundary
26 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

Figure 8. Difference between steady-state simulated hydraulic heads and the top of the Sparta Sand for (A) scenario 3A, (B) 

scenario 3B, and (C) scenario 3C using optimized withdrawals in the Sparta aquifer flow model for verification.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

Scenario 3 simulated heads

minus the top of the Sparta B

Sand (feet)

Adjusted constraints

Minimum head constraint

reached

-129 to -50

-49 to -25

-24 to 0

1 to 25

26 to 50

51 to 100

101 to 500

> 500

EXPLANATION

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boundary

Base from U.S. Geological Survey digital data, 1:100,000
Sustainable-Yield Estimation 27

Figure 8. Difference between steady-state simulated hydraulic heads and the top of the Sparta Sand for (A) scenario 3A, (B) 

scenario 3B, and (C) scenario 3C using optimized withdrawals in the Sparta aquifer flow model for verification.

Arkansas

Louisiana

Mississippi

Texas

Tennessee

94° 93° 92° 91° 90°

31°

32°

33°

34°

35°

36°

0 40 80 20 Miles

0 40 80 20 Kilometers

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

Scenario 3 simulated heads

minus the top of the Sparta

Sand (feet)

C

Adjusted constraints

Minimum head constraint

reached

-129 to -50

-49 to -25

-24 to 0

1 to 25

26 to 50

51 to 100

101 to 500

> 500

EXPLANATION

Eastern Extent of the Sparta

Aquifer Outcrop-Subcrop

Active Model

Area Boudary
28 Conjunctive-Use Optimization Model and Sustainable-Yield Estimation for the Sparta Aquifer of Southeastern Arkansas and 

North-Central Louisiana

the Sparta Sand. These scenarios also may result in 

higher sustainable yields for the system because of 

relaxed constraints in areas where the hydraulic heads 

drop below the top of the Sparta Sand early in the simulation.

SUMMARY

Conjunctive-use optimization modeling was 

done to assist water managers and planners by estimating the maximum amount of ground water that hypothetically could be withdrawn indefinitely from wells 

within the Sparta aquifer without violating hydraulichead or streamflow constraints. The Sparta aquifer is 

largely a confined aquifer of regional importance that 

comprises a sequence of unconsolidated sand units that 

are contained within the Sparta Sand. In 2000, more 

than 35.4 Mft3

/d of water were withdrawn from the 

aquifer in Arkansas by more than 900 wells, primarily 

for industry, municipal supply, and crop irrigation. Several large cones of depression have formed in the 

potentiometric surface in the Grand Prairie area and 

near the cities of Pine Bluff and El Dorado in Arkansas 

and Monroe in Louisiana. Hydraulic heads in the 

Sparta aquifer are now below the top of the Sparta Sand 

in parts of Union and Columbia Counties and several 

areas in north-central Louisiana. Problems related to 

overdraft in the Sparta aquifer can result in increased 

drilling and pumping costs, reduced well yields, and 

degraded water quality in areas of large drawdown. 

A finite-difference ground-water flow model was 

developed for the Sparta aquifer using MODFLOW, in 

eastern and southeastern Arkansas and north-central 

Louisiana. Observed aquifer conditions in 1997 supported by model simulations indicate that continued 

pumping at withdrawal rates representative of 1990-

1997 rates cannot be sustained indefinitely and maintain hydraulic heads above the top of the Sparta Sand; 

a criteria used to designated Arkansas's Critical 

Ground-Water Areas. This report describes a steadystate conjunctive-use optimization model that was 

developed to simulate optimized surface-water and 

ground-water withdrawals while maintaining hydraulic-head and streamflow constraints, thus determining 

the “sustainable yield” for the aquifer.

Attempts to estimate sustainable yield initially 

resulted in infeasible solutions. However, continued 

scenario development produced a scenario with feasible solutions. A feasible solution returns estimated sustainable yield—optimized ground-water withdrawals 

that can be maintained indefinitely without lowering 

hydraulic heads below the top of the Sparta Sand 

(except in the outcrop areas where unconfined conditions occur) or decreasing streamflows below minimum levels.

Scenarios 1 and 2 resulted in infeasible solutions, 

but were important findings in the progression of scenario development. From scenario 1, it was determined 

that steady-state simulations of predevelopment 

hydraulic heads should be used as the initial conditions 

for the optimization model to ensure that all hydraulic 

heads start out above the top of the Sparta Sand. For 

this reason, the solution was infeasible using the 1997 

simulated hydraulic heads because hydraulic heads in 

many areas already were below the top of the Sparta 

Sand. 

Scenario 2 transitioned from scenario 1 by 

attempting to find an optimized solution using 100 percent of the baseline 1990-1997 withdrawals as the 

lower specified limit on withdrawals from managed 

wells. This infeasible solution proves that the baseline 

1990-1997 withdrawals cannot be sustained indefinitely.

For scenarios 3A-C, withdrawals from managed 

wells varied between zero and rates equal to 100, 150, 

and 200 percent of the baseline 1990-1997 withdrawal 

rates except for wells in Union County, which were 

held at 35.6 percent of the baseline 1990-1997 withdrawal rate. Estimated sustainable yields for scenario 3 

ranged from 11.6 to 13.2 Mft3

/d in Arkansas and 0.3 to 

0.5 Mft3/d in Louisiana. Assuming the total water 

demand is equal to the baseline 1990-1997 withdrawal 

rates, the sustainable yield estimated from scenario 3AC only provides 52 to 59 percent of the total demand for 

Arkansas; the remainder is unmet demand that could be 

obtained from surface water.

Sustainable yield is a function of managed well 

distributions and lower and upper specified limits on 

withdrawals from managed wells. For this report, 1,152 

managed wells were used to represent the distribution 

of existing wells in the Sparta aquifer. In addition, 

lower and upper specified limits on withdrawals from 

managed wells were set as a percentage of the baseline 

1990-1997 withdrawal rates. Hydraulic-head constraints have the largest effect on estimations of sustainable yield. Therefore, with no established criteria 

for maintaining hydraulic heads above the top of the 

Sparta Sand in Louisiana, decreasing the hydraulichead constraint in Louisiana would increase the 

amount of estimated sustainable yield for both Arkan-
Selected References 29

sas and Louisiana. However, non-linear model behavior was too great to render a viable solution for 

quantifying this increase. Sustainable yield also may be 

increased by strategic placement of managed wells in 

the model and optimizing withdrawal rates. This strategy could be used to determine the location of future 

wells to be drilled into the Sparta aquifer.

The results of this optimization model must be 

viewed within the limitations of the quality of data 

available for the Sparta aquifer system and representation of the system with a numerical simulation model. 

Results of model application scenarios demonstrate the 

usefulness of coupling numerical simulation and optimization for regional-scale evaluation of waterresource management strategies. 

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