Prepared in cooperation with the

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

Arkansas Soil and Water Conservation Commission

CONJUNCTIVE-USE OPTIMIZATION MODEL OF THE 

MISSISSIPPI RIVER VALLEY ALLUVIAL AQUIFER OF 

SOUTHEASTERN ARKANSAS 

Water-Resources Investigations Report 03-4233

U.S. Department of the Interior

U.S. Geological Survey
Cover: Irrigation of cotton and rice fields in southeastern Arkansas represent two of the major uses of 

ground water from the Mississippi River Valley alluvial aquifer. Photographs by John B. Czarnecki, U.S. 

Geological Survey
CONJUNCTIVE-USE OPTIMIZATION MODEL OF THE 

MISSISSIPPI RIVER VALLEY ALLUVIAL AQUIFER OF 

SOUTHEASTERN ARKANSAS

By John B. Czarnecki, Brian R. Clark, and Gregory P. Stanton 

U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 03-4233

Prepared in cooperation with the

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

Arkansas Soil and Water Conservation Commission

Little Rock, Arkansas

2003
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

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

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

Hydrogeology............................................................................................................................................ 5

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

Flow Model ............................................................................................................................................... 5

Optimization Model .................................................................................................................................. 7

Problem Formulation ....................................................................................................................... 9

Objective Function................................................................................................................10

Hydraulic-Head Constraints .................................................................................................10

Streamflow Constraints.........................................................................................................11

Ground-Water Withdrawal Limits ........................................................................................11

Surface-Water Withdrawal Limits ........................................................................................11

Wells Used in Optimization Model.......................................................................................13

Streamflow............................................................................................................................13

Optimization Results......................................................................................................................15

Sustainable Yield ..................................................................................................................15

Unmet Demand .....................................................................................................................20

Optimal Simulated Hydraulic-Head Altitude .......................................................................20

Nonlinear Effects ..................................................................................................................23

Limitations ............................................................................................................................23

Summary ..........................................................................................................................................................24

References Cited ..............................................................................................................................................25

ILLUSTRATIONS

Figures 1-3. Maps showing:

1. Location of study and modeled area..............................................................................................3

2. Hydrogeologic sections showing lithology and variation in thickness through the model area.... 6

3. Potentiometric surface within the alluvial aquifer, spring 1998................................................8

4. Flow chart of optimization modeling process......................................................................................10

5. Map showing location of hydraulic-head constraint points and thickness of aquifer below

hydraulic-head constraint...............................................................................................................12

6-11. Maps showing:

6. Location of streams within model showing cells and rates at which water could be withdrawn

and still meet constraints within optimization model.............................................................14

7. Ratio of optimal ground-water withdrawal calculated by the optimization model to the

amount withdrawn in 1997 for withdrawal limits at each well set to 100 percent of the

1997 withdrawal rate ..............................................................................................................16
IV Contents

8. Ratio of optimal ground-water withdrawal calculated by the optimization model to the

amount withdrawn in 1997 for withdrawal limits at each well set to 150 percent of the

1997 withdrawal rate.............................................................................................................. 18

 9. Ratio of optimal ground-water withdrawal calculated by the optimization model to the

amount withdrawn in 1997 for withdrawal limits at each well set to 200 percent of the

1997 withdrawal rate.............................................................................................................. 19

10. Simulated hydraulic head at steady state using (A) 1997 withdrawal rates; and

(B) sustainable yield............................................................................................................... 21

11. Difference between simulated hydraulic head and altitude of half the aquifer thickness........... 22

12. Graph showing cumulative percentage of hydraulic-head constraint points less than or equal

to the difference between simulated hydraulic head and the altitude corresponding to

half the aquifer thickness ....................................................................................................... 23

TABLES

Table 1. Characteristics of the flow model ............................................................................................................7

2. Rivers, streamflows, and streamflow constraints.................................................................................. 13

3. Sustainable yield and unmet demand by county or parish for different upper limits on withdrawals

and different demand rates................................................................................................................ 17

4. Optimized streamflow withdrawal........................................................................................................ 20

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, ABBREVIATIONS, AND VERTICAL DATUM

Multiply By To obtain

inch (in.) 2.54 centimeter (cm)

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

million cubic feet per day (Mft3/d) 7.481 million gallons per day (Mgal/d)

square foot (ft2) 0.0929 square meter (m2)

square mile (mi2) 2.590 square kilometer (km2)

cubic foot (ft3) 7.481 gallon (gal)

cubic foot per day (ft3/d) 0.0283 cubic meter per day (m3/d)
Abstract 1

Conjunctive-Use Optimization Model of the Mississippi 

River Valley Alluvial Aquifer of Southeastern Arkansas

By John B. Czarnecki, Brian R. Clark, and Gregory P. Stanton

ABSTRACT

The Mississippi River Valley alluvial aquifer is a 

water-bearing assemblage of gravels and sands that 

underlies about 32,000 square miles of Missouri, Kentucky, Tennessee, Mississippi, Louisiana, and Arkansas. Because of the heavy demands placed on the 

aquifer, several large cones of depression have formed 

in the potentiometric surface, resulting in lower well 

yields and degraded water quality in some areas. A 

ground-water flow model of the alluvial aquifer was 

previously developed for an area covering 3,826 square 

miles, extending south from the Arkansas River into 

the southeastern corner of Arkansas, parts of northeastern Louisiana, and western Mississippi. The flowmodel results indicated that continued ground-water 

withdrawals at rates commensurate with those of 1997 

could not be sustained indefinitely without causing 

water levels to decline below half the original saturated 

thickness of the aquifer. 

Conjunctive-use optimization modeling was 

applied to the flow model of the alluvial aquifer to 

develop withdrawal rates that could be sustained relative to the constraints of critical ground-water area designation. These withdrawal rates form the basis for 

estimates of sustainable yield from the alluvial aquifer 

and from rivers specified within the alluvial aquifer 

model. A management problem was formulated as one 

of maximizing the sustainable yield from all groundwater and surface-water withdrawal cells within limits 

imposed by plausible withdrawal rates, and within 

specified constraints involving hydraulic head and 

streamflow. Steady-state conditions were selected 

because the maximized withdrawals are intended to 

represent sustainable yield of the system (a rate that can 

be maintained indefinitely).

One point along the Arkansas River and one 

point along Bayou Bartholomew were specified for 

obtaining surface-water sustainable-yield estimates 

within the optimization model. Streamflow constraints 

were specified at two river cells based on average 7-day 

low flows with 10-year recurrence intervals.

Sustainable-yield estimates were affected by the 

allowable upper limit on withdrawals from wells specified in the optimization model. Withdrawal rates were 

allowed to increase to 200 percent of the withdrawal 

rate in 1997. As the overall upper limit is increased, the 

sustainable yield generally increases. Tests with the 

optimization model show that without limits on pumping, wells adjacent to sources of water, such as large 

rivers, would have optimal withdrawal rates that were 

orders of magnitude larger than rates corresponding to 

those of 1997. Specifying an upper withdrawal limit of 

100 percent of the 1997 withdrawal rate, the sustainable yield from ground water for the entire study area 

is 70.3 million cubic feet per day, which is about 96 

percent of the amount withdrawn in 1997 (73.5 million 

cubic feet per day). If the upper withdrawal limit is 

increased to 150 percent of the 1997 withdrawal rate, 

the sustainable yield from ground water for the entire 

study area is 80.6 million cubic feet per day, which is 

about 110 percent of the amount withdrawn in 1997. If 

the upper withdrawal limit is increased to 200 percent 

of the 1997 withdrawal rate, the sustainable yield from 

ground water for the entire study area is 110.2 million 

cubic feet per day, which is about 150 percent of the 

amount withdrawn in 1997. Total sustainable yield 

from the Arkansas River and Bayou Bartholomew is 

about 4,900 million cubic feet per day, or about 6,700 

percent of the amount of ground-water withdrawn in 

1997. The large, sustainable yields from surface water 

represent a potential source of water that could supplement ground water and meet the total water demand.

Unmet demand (defined as the difference 

between the optimized withdrawal rate or sustainable 

yield, and the anticipated demand) was calculated 
2 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

using different demand rates based on multiples of the 

1997-withdrawal rate. Assuming that demand is the 

1997 withdrawal rate, and that sustainable-yield estimates are those obtained using upper limits of withdrawal rates of 100-, 150-, and 200-percent of 1997 

withdrawal rates, then the resulting unmet demand for 

the entire model area is 3.3, -7.1, and -36.6 million 

cubic feet per day, respectively. Whereas, if the demand 

is specified as 100-, 150-, and 200-percent of the 1997 

withdrawal rate, and the sustainable-yield estimates 

remain the same, then the resulting unmet demand for 

the entire model area is 3.3, 29.7, and 36.9 million 

cubic feet per day. 

INTRODUCTION

The Mississippi River Valley alluvial aquifer, 

often termed simply the “alluvial aquifer,” is a waterbearing assemblage of gravels and sands that underlies 

about 32,000 mi2

 of Missouri, Kentucky, Tennessee, 

Mississippi, Louisiana, and Arkansas (fig. 1). In 

Arkansas, the alluvial aquifer occurs in an area generally 50 to 125 mi wide and about 250 mi long adjacent 

to the Mississippi River. The alluvial aquifer is the 

uppermost aquifer in this area, and generally ranges in 

thickness between 50 and 150 ft. The alluvial aquifer is 

under both confined and unconfined conditions 

depending on location (Czarnecki and others, 2002). 

The Mississippi River Valley alluvial aquifer supplies 

large volumes of water for agriculture in Arkansas. 

Withdrawal of ground water from the alluvial aquifer 

for agriculture started in the early 1900’s in the Grand 

Prairie area for irrigation of rice, and to a lesser extent, 

soybeans. Water-level declines in the alluvial aquifer 

were first documented in 1927 in the Grand Prairie 

(Engler and others, 1963, p. 21). From 1965 to 2000, 

water use from the alluvial aquifer in eastern Arkansas 

increased 637 percent. In 1997, 635.6 Mft3/d of water 

were pumped from the aquifer, primarily for irrigation 

and fish farming. Most of the water use from the alluvial aquifer occurred north of the Arkansas River (outside the area of investigation of this report). In 2000, 97 

percent of the ground water obtained in eastern Arkansas came from wells completed in the alluvial aquifer 

(Terry Holland, U.S. Geological Survey, written commun., 2002). 

Because of heavy demands placed on the aquifer, 

several large cones of depression have formed in the 

potentiometric surface, resulting in lower well yields 

and degraded water quality from salinity increases in 

some areas. Several counties in southeastern Arkansas, 

which are within the extent of the alluvial aquifer, have 

been designated Critical Ground-Water Areas by the 

Arkansas Soil and Water Conservation Commission 

(ASWCC). One criterion associated with the designation of a Critical Ground-Water Area applies when 

water levels drop below half the original saturated 

thickness of the formation. 

Ground-water flow models of the alluvial aquifer 

show that continued ground-water withdrawals at rates 

equal to those of 1997 could not be sustained indefinitely without causing water levels to decline below 

half the original saturated thickness of the formation 

(Stanton and Clark, 2003). To develop estimates of 

withdrawal rates that could be sustained in compliance 

with the constraints of critical ground-water area designation, the U.S. Geological Survey (USGS), in cooperation with the ASWCC and the Vicksburg District of 

the U.S. Army Corps of Engineers (USACE) applied 

conjunctive-use optimization modeling to an existing 

model of ground-water flow for the alluvial aquifer in 

southeastern Arkansas (fig. 1). Conjunctive use 

involves the withdrawal of both ground water and surface water. Conjunctive-use optimization modeling is 

a technique that can be used to determine maximum 

withdrawal rates from both surface water and ground 

water while meeting constraints with respect to water 

levels and streamflow. These withdrawal rates would 

form the basis for estimates of sustainable yield from 

the alluvial aquifer and from rivers specified within the 

alluvial aquifer model. 

Purpose and Scope

The purpose of this report is to describe the 

application and evaluation of a conjunctive-use optimization model (hereafter referred to as the optimization 

model) of the Mississippi River Valley alluvial aquifer 

of southeastern Arkansas. The objective of this study is 

to provide estimates of sustainable yield from ground 

water and surface water using the optimization model. 

The optimization model was formulated as a linear programming problem, and utilized a ground-water model 

developed for the study area (Stanton and Clark, 2003) 

as a basis for evaluation. The purpose of the optimization model was to: (1) determine maximum withdrawal 

rates from model cells at which ground-water withdrawals occurred in 1997 and (2) determine maximum 

withdrawal rates from model cells at stream locations 

while maintaining ground-water levels at or 
Introduction 3

Figure 1. Location of study and modeled area. 

River

Monticello Ridge

Bayou Macon

Boeuf

River

Big Bayou

Drew

Ashley

Desha

Chicot

Lincoln

Bradley

Jefferson

Arkansas

Union

Union

Morehouse

Bolivar

Washington

Phillips

West Carroll East Carroll

Issaquena

Active Model

Area Boundary

92° 91°

33°

34°

0 10 20 Miles 5

0 10 5 20 Kilometers

Arkansas

Louisiana

Mississippi

Arkansas

River

Mississippi

River

Saline River

Ouachita

Bayou Bartholomew

Louisiana

Arkansas

Mississippi

Base from U.S. Geological Survey digital data, 1:100,000
4 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

above specified levels and streamflow at or above specified rates. The report describes the amount of projected total water demand that can be met by the 

alluvial aquifer and by available surface water while 

maintaining specified constraints. In this report, sustainable yield is defined as the amount of water that can 

be withdrawn indefinitely from ground water and from 

surface water without violating specified hydraulichead or streamflow constraints. If an anticipated 

demand for water is known, an unmet demand may be 

calculated by subtracting the sustainable yield from the 

anticipated demand. Sustainable yield from ground 

water will be compared to anticipated demand for various withdrawal rates, because of concerns about 

water-level declines in the alluvial aquifer. The results 

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

to assist in the management of the ground-water 

resources of the alluvial aquifer in southeastern Arkansas in a sustainable manner. 

Previous Studies

Many investigators have described the underlying sediments of the Mississippi River Alluvial Plain. 

One of the earliest reports describing subsurface geology and ground-water resources in southern Arkansas 

and northern Louisiana was written by Veatch (1906). 

Ground-water resources of northeastern Arkansas were 

described and a detailed inventory was provided by 

Stephenson and Crider (1916). Fisk (1944) reported on 

extensive geologic investigations along the Mississippi 

River Valley made by the U.S. Army Corps of Engineers between 1941 and 1944. Krinitzsky and Wire 

(1964) expanded on the hydrogeologic work of Fisk 

with a comprehensive look at ground-water conditions. 

Cushing and others (1964) and Boswell and others 

(1968) provided an overview of the alluvial aquifer in 

their discussions of Quaternary-age aquifers of the 

Mississippi Embayment. Boswell and others (1968) 

first referred to the water-yielding sediments underlying the alluvial plain as the Mississippi River Valley 

alluvial aquifer.

The MODFLOW (McDonald and Harbaugh, 

1988; Harbaugh and McDonald, 1996) ground-water 

flow model (hereafter referred to as the flow model) 

used in the optimization modeling of this report is 

based on a recalibrated and extended model of Mahon 

and Poynter (1993), which included hydraulic-head 

observations for the years 1992 and 1998 (Stanton and 

Clark, 2003). Many researchers have applied conjunctive-use optimization models to the management of 

ground-water systems. Reichard (1995) provides a 

thorough review of many of these studies. Nishikawa 

(1998) used MODMAN 3.0 (Greenwald, 1998) (the 

precursor program to the one used for optimization in 

this report) to minimize the cost of supplying water 

during a design drought in Santa Barbara, California, 

by optimizing delivery of surface water and operation 

of the city’s reservoirs.

The first effort to estimate optimized groundwater withdrawals from the alluvial aquifer was done 

by Peralta and others (1985) who estimated future 

ground-water availability in the Grand Prairie area by 

using a flow model coupled to an optimization routine. 

Their analysis focused on a small subset of the alluvial 

aquifer north of the Arkansas River, and did not couple 

the conjunctive use of ground water and surface water. 

Barlow and others (2003) developed conjunctive-use 

management models for estimating sustainable yield 

from surface water and ground water within an alluvial-valley stream-aquifer system in Rhode Island.

Acknowledgments

The conjunctive-use optimization routine used in 

this report is an adaptation of enhancements to MODMAN (Greenwald, 1998) made by Brian Wagner (U.S. 

Geological Survey). Wesley Danskin (U.S. Geological 

Survey) provided guidance on the use of MODMAN 

and the large-scale optimization solver MINOS (Murtaugh and Saunders, 1998). Streamflow constraints 

used in the optimization model were compiled by Steve 

Loop (Arkansas Soil and Water Conservation Commission). Elton Porter (U.S. Geological Survey) computed 

mean annual flow rates and runoff estimates for the 

model streams used in this report.

Study Area

The study area (fig. 1), which is the same as the 

model area, is 3,826 mi2

, and includes all or part of six 

counties south of the Arkansas River in Arkansas, part 

of four parishes in northeastern Louisiana, and part of 

one county in Mississippi. The active cells of the model 

encompassed the area south of the Arkansas River, 

mostly west of the Mississippi River, and north of an 

arbitrary east-west line about 10 mi south of the Arkan-
Conjunctive-Use Optimization 5

sas/Louisiana State line encompassing a small part of 

northeastern Louisiana (fig. 1).

Hydrogeology

Deposition of sediment from the Mississippi and 

Arkansas Rivers during Pleistocene and Holocene time 

(herein referred to as Mississippi River alluvium) has 

produced a sequence of sands, silts, and clays that constitute the alluvial aquifer and semiconfining units in 

eastern Arkansas. From a regional perspective this collection of sediment can be divided into two units. The 

lower unit, which contains the alluvial aquifer, is composed of coarse sand and gravels that grade upward to 

fine sand. The upper unit consisting of clay, silts, and 

fine sand confines the alluvial aquifer. It is this regional 

alluvial aquifer and overlying confining unit, along 

with its flow system, that has been defined and investigated previously (Broom and Lyford, 1981; Broom and 

Reed, 1973; Ackerman, 1989a, 1989b, 1990; Mahon 

and Ludwig, 1990; Mahon and Poynter, 1993).

The geology of the model area typically is characterized by clay and sand beds of the Mississippi 

River alluvium of Pleistocene and Holocene time overlying the silt and clay sequence of the Tertiary-age 

Jackson Group in the study area (Ackerman, 1989a). 

Channel fill, point bar, and backswamp deposits, associated with present or former channels of the major rivers, locally can produce abrupt differences in lithology 

of alluvial deposits, that result in spatial variations in 

the hydraulic properties of both the aquifer and confining unit within small distances. On a local scale, alluvial deposits are dominated by the complex 

heterogeneity of small, discontinuous sand and silt 

beds dispersed laterally and vertically, which represent 

local features of the aquifer and flow system. Laterally 

to the west, the contact between alluvial deposits and 

older strata is masked by terrace deposits (Ackerman, 

1989a), which form a slight topographic high, locally 

referred to as the Monticello Ridge. During Tertiary 

time, the last marine inundation of the Mississippi 

Embayment occurred depositing the clays and silts of 

the Jackson Group. A truncated erosional surface of the 

Jackson Group underlies the Mississippi River alluvium in the study area forming an effective underlying 

hydraulic confining unit (Cushing and others, 1964). 

The Mississippi River alluvium consists of two 

distinct but gradational lithologies; clays and silts overlie coarse sands and gravels forming a fining upward 

sequence (Ackerman, 1989a). These two lithologies 

are used as a basis for subdividing the alluvium into the 

basal alluvial aquifer and the overlying confining unit. 

Figure 2 shows two hydrogeologic sections of the aquifer through the model area. The overlying confining 

unit ranges in thickness from 0 to 60 feet within the 

study area (Gonthier and Mahon, 1993). The sands and 

gravels that underlie the overlying confining unit and 

comprise the alluvial aquifer range from 50 to 100 feet 

thick.

CONJUNCTIVE-USE OPTIMIZATION

The following sections describe the development 

and application of the conjunctive-use optimization 

approach applied to the model area, beginning with a 

review of the flow model. The optimization model is 

described and results evaluated.

Flow Model

The flow model discussed in this report is based 

on a recalibrated and extended model of Mahon and 

Poynter (1993), which included hydraulic-head observations for the years 1992 and 1998 (Stanton and Clark, 

2003). Characteristics for the flow model are listed in 

table 1. The flow model was developed using MODFLOW-2000 and incorporates river, no-flow, drain, and 

areally distributed recharge boundary conditions. The 

combinations of boundary conditions result in flow 

occurring from recharge sources to areas with extensive 

ground-water withdrawals and a consequent widespread lowering of the water table. The potentiometric 

surface within the alluvial aquifer for spring 1998 is 

shown in figure 3, which shows a substantial cone of 

depression near the boundary of Desha and Chicot 

Counties resulting from sustained pumping. Model 

parameters were estimated in part using MODFLOW2000 (Hill and others, 2000) to assist model calibration 

to observed values of hydraulic head in 1972, 1982, 

1992, and 1998. Values of the mean, mean absolute, 

and root-mean-square difference between observed and 

simulated hydraulic head for all observations for all 

periods simulated by the model are listed in table 1. The 

mean difference (-0.33 ft) is a sum of the differences, 

both positive and negative, divided by the total number 

of observations; for an unbiased model this value 

would be zero. The mean absolute difference (5.1 ft) is 

the sum of the magnitudes of the difference at each 

observation, divided by the total number of observa-
6 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 2. Hydrogeologic sections showing lithology and variation in thickness through the model area.

EXPLANATION

MISSISSIPPI RIVER VALLEY CONFINING UNIT

MISSISSIPPI RIVER VALLEY ALLUVIAL AQUIFER

DEPOSITS OF TERTIARY AGE AND OLDER

TRACE OF HYDROGEOLOGIC SECTION--Points

on the trace are location of wells used for control

WELL USED FOR CONTROL

A A’ B - B' Chicot County Desha County

VERTICAL SCALE GREATLY EXAGGERATED MODIFIED FROM GONTHIER AND MAHON, 1993

200

150

100

50

NGVD of 1929

250

200

150

100

50

NGVD of 1929

FEET

250

SOUTH NORTH

B B’

NGVD of 1929

FEET

100

50

150

Ashley County

Chicot County

A - A'

VERTICAL SCALE

GREATLY EXAGGERATED

MODIFIED FROM GONTHIER AND

MAHON, 1993

NGVD of 1929

100

50

150

WEST EAST

0 25 50 MILES

0 25 50 KILOMETERS

B

B’

A

Lincoln A’

Arkansas

Desha

Drew

Ashley

Chicot

Arkansas

Mississippi
Conjunctive-Use Optimization 7

Table 1. Characteristics of the flow model (Stanton and Clark, 2003)

[mi2, square mile; ft3/d, cubic foot per day; ft/d, foot per day; ft-1, inverse foot; ft, foot]

Characteristic Value

Model area 3,826 mi2

Cells with wells corresponding to 1997 withdrawals 1,841

Total pumpage in 1997 617,000 ft3/d

Average hydraulic conductivity 250 - 450 ft/d

Specific yield 0.27 - 0.30

Specific storage 3x10-5 - 9x10-4 ft-1

River cells 470

Hydraulic-head observations 521

Hydraulic-head observation periods 1972, 1982, 1992, 1998

Range in observed hydraulic head values in 1998

 (feet above National Geodetic Vertical Datum of 1929) 61.85 - 186.01 ft

Mean difference between observed and simulated hydraulic head, all four periods -0.33 ft

Mean absolute difference between observed and simulated hydraulic head, all four periods 5.1 ft

Root-mean-square difference between observed and simulated hydraulic head, all four periods 6.5 ft

tions; a value of zero is preferable. Assuming a normal 

distribution, approximately 67 percent of the residual 

values (that is, the difference between observed and 

simulated water-level altitudes) would lie within positive or negative values of the root-mean-square difference (that is, +/-6.5 ft). Given the difference in the 

range in observed hydraulic-head values (220 ft), these 

differences are small and indicate a good fit to the 

observed values. 

The flow model was used to simulate groundwater flow for the period from 1918 through 2049, and 

to evaluate effects from the demand for ground water 

from the alluvial aquifer, which has increased steadily 

for the last 40 years (Stanton and Clark, 2003). The 

flow model results indicated that continued groundwater withdrawals at rates commensurate with those of 

1997 could not be sustained indefinitely without causing water levels to decline below half the original saturated thickness of the formation. Based on measured 

water levels, the saturated thickness of the alluvial 

aquifer has been greatly reduced in some areas 

(Schrader, 2001; Czarnecki and others, 2002). This has 

resulted in degraded water quality, decreasing water 

availability, increased pumping costs, and lower well 

yields. 

Optimization Model

For the optimization model 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 to MODMAN 3.0, and 

adapted 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, 9,979 groundwater-withdrawal decision variables and 1,165 surfacewater-withdrawal decision variables were specified. 

The optimization modeling process (fig. 4) 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). 
8 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 3. Potentiometric surface within the alluvial aquifer, spring 1998 (Stanton and Clark, 2003). Boeuf R ver i Big Bayou

80

90

140

100

130

160

120

110

170

180

190

100

Drew

Ashley

Desha

Chicot

Arkansas

Jefferson

Bradley

Lincoln

Cleveland

Union

Union

Morehouse

Bolivar

Washington

Phillips

West Carroll Issaquena East Carroll Mississippi

150

100

Bayou Macon

90

Ouachita River

Saline River

Arkansas River

M ssissippi i

River

Arkansas

Louisiana

92° 91°

33°

34°

Active Model

Area Boundary

Modified from Joseph, (1999) 0 10 20 Miles 5

0 10 20 Kilometers 5

EXPLANATION

POTENTIOMETRIC CONTOUR--

Shows altitude at which water level

would have stood in tightly cased

wells. Dashed where approximately

located. Contour interval 10 feet.

Datum is National Geodetic Vertical

Datum of 1929

OBSERVATION WELL COMPLETED

IN ALLUVIAL AQUIFER

20
Conjunctive-Use Optimization 9

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 (as opposed to 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. The conjunctive-use 

version of MODMAN 4.0 was used 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). The unit rate was specified at 10,000 ft3

/d. To accurately represent the 

response of the flow model to a unit rate of pumping 

under unconfined conditions, selection of starting 

hydraulic heads should be similar to those that would 

result when optimal withdrawal rates are applied. Starting hydraulic head values were selected as those simulated for 1997 from the model of Reed (2003). Because 

some dry cells occurred in that model at that simulated 

point in time, hydraulic-head values at dry cells were 

assigned similar values as adjacent model cells that 

were not dry. 

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 optimization program in mathematical programming system (MPS) format. The linear 

program is run under MINOS. If a feasible solution 

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 against the specified 

hydraulic-head constraints. Non-linear flow model 

behavior was expected for this model because of the 

unconfined condition of the aquifer and head-dependent flow boundary conditions at the rivers. For this 

reason, starting values of hydraulic head were specified 

as those simulated for 1997. 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 feet squared.

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, which varies with 

hydraulic head, in feet; and 

w is the width through which flow occurs, in 

feet. 

This is an important consideration in selecting starting 

values of hydraulic head for the flow model to produce 

a more efficient solution to the ground-water flow 

equation (McDonald and Harbaugh, 1988). 

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 ground-water levels at or 

above specified levels; (2) maintaining streamflow at or 

above minimum specified rates; and (3) limiting

Q K– dh

dl = -----A

dh

dl

-----

A bw =
10 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 4. Flow chart of optimization modeling process (modified from Greenwald, 1998). 

Develop and Calibrate Site-Specific GroundWater 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 

ground-water withdrawals to a maximum of either 100, 

150, or 200 percent of the rate pumped in 1997. Steadystate 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 1,841 

model cells corresponding to well locations and at 2 

river cells. 

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 

cubic feet per day; 

 is the sum of ground-water withdrawal 

rates from all managed wells, in cubic feet 

per day; and

 is the sum of surface-water withdrawal rates from all managed river reaches, 

in cubic feet per day. 

Hydraulic-Head Constraints

Equation 3 is computed such that the following 

constraints are maintained:

(4)

where is the hydraulic head (water-level altitude) at 

constraint location c, in feet; and 

 is the water-level altitude at half the 

thickness of the aquifer, in feet. q ∑ well q ∑ river +

q ∑ well

q ∑ river

hc hminimum ≥

hc

hminimum
Conjunctive-Use Optimization 11

To accommodate the ASWCC Critical Ground-Water 

Area criteria that water levels within the alluvial aquifer should remain above half the original saturated 

thickness of the aquifer, hydraulic-head constraints 

were specified at 2,804 model cells. For a few cells 

where the original saturated thickness of the aquifer is 

less than 60 ft but at least 30 ft, the hydraulic head constraint was specified as 30 ft, a minimum thickness 

considered necessary for the aquifer to remain viable in 

those areas. The spatial distribution of constraint points 

represents approximately every fifth model cell (fig. 5). 

If water levels were to drop everywhere to the level of 

the head constraint, then the resulting saturated thickness of the alluvial aquifer would range from 30 to 100 

ft, and generally be thinnest in the Grand Prairie area 

(fig. 5).

Streamflow Constraints

Streamflow is regulated in Arkansas by ASWCC 

for purposes of maintaining water quality, navigation, 

and species habitat. Streamflow constraints for several 

rivers specified in the optimization model are based on 

7-day, 10-year-recurrence low-flow data (7Q10). 

Streamflow constraints are specified as the minimum 

amount of flow required at individual river cells. The 

equation governing the relation between streamflow 

constraints and flow into and out of a stream is 

 (5)

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.

Ground-Water Withdrawal Limits

The proximity of managed wells to model flow 

boundaries was taken into account to properly formulate the management objective. If no limit is imposed 

on the potential amount of water that can be pumped at 

each managed well, then those wells nearest model 

sources of water, such as rivers or general head-boundaries, will be the first to be supplied water, thus capturing flow that would otherwise reach wells further from 

the sources. Test simulations done with the optimization model show that without limits on pumping, wells 

adjacent to sources of water would have optimized 

withdrawal rates that were orders of magnitude larger 

than rates corresponding to those of 1997. Not only is 

it physically unlikely that individual wells could pump 

that much more water, but construction of sufficient 

additional wells in the one-square mile cells also is 

unlikely. The phenomenon of wells near rivers capturing induced recharge from the rivers and preventing 

sufficient water from flowing to interior wells is, however, consistent with current conditions (Czarnecki and 

others, 2002). 

Test simulations using 1997 withdrawal rates 

applied to steady-state conditions yielded large areas 

with dry cells in the flow model. Therefore, groundwater demand limits were specified at each cell as a 

multiple of the amount pumped in 1997, such that 

(6)

where, is the optimal ground-water withdrawal 

for well i, in cubic feet per day; 

 is a multiplier between 1 and 2; and

 is the total amount withdrawn in

 1997 from all wells, in cubic feet per day. 

Surface-Water Withdrawal Limits

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

water available (greater than 7Q10) at a given point in 

a given river. This specification permitted analysis of 

where water could be produced and the maximum 

amount available. Withdrawals were allowed only at 

the two points where river constraints were specified on 

both the Arkansas River and Bayou Bartholomew.

qhead

R

qoverland

R

qground water

R

qdiversions

R

qriver

R

qminimum

R

∑ – ∑ ≥

+ ∑ ±∑ –

qhead

R

qoverland

R

∑

qground water

R

∑

qdiversions

R

∑

qriver

R

∑

qminimum

R

0 qwelli Mqwell 1997≤ ≤

q welli

M

qwell 1997
12 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 5. Location of hydraulic-head constraint points and thickness of aquifer below hydraulic-head constraint.

92° 91°

33°

34°

0 10 20 Miles 5

0 10 5 20 Kilometers

Arkansas River

Mississippi River

EXPLANATION

Thickness of aquifer

below the hydraulichead constraint,

in feet

71 - 80

61 - 70

51 - 60

41 - 50

30 - 40

Constraint locations

Base from U.S. Geological Survey digital data, 1:100,000
Conjunctive-Use Optimization 13

Wells Used in Optimization Model

For optimization modeling, 1,841 one-square 

mile cells were used to represent pumping in 1997. In 

Louisiana, pumping data were available only by parish, 

and were divided equally over all model cells corresponding to a given parish. Multiple wells can exist 

within any model cell, but the total pumping from all 

wells in a cell is the value specified. In 1997, the annual 

pumping rate for all wells was 73.5 million cubic feet 

per day. For the sustainable-yield analysis, the optimized rate was allowed to vary between a rate of zero 

to a maximum rate equal to twice that which was withdrawn in 1997. The latter condition was specified 

because no limit on withdrawal rates led to unrealistic 

optimal withdrawal rates from wells adjacent to rivers. 

Model cells used to denote ground-water withdrawals 

correspond to the distribution of wells in 1997. No 

additional wells other than those existing in 1997 are 

assumed. Because withdrawal rates in Louisiana are 

reported by parish rather than for individual wells as in 

Arkansas, withdrawal rates in Louisiana were evenly 

distributed by parish at every model cell within the flow 

model. This distribution gives the impression of a 

denser well distribution in Louisiana than probably 

exists.

Wells are optimized as individual wells, and 

therefore, have individual rates associated with each 

cell. For each optimization model run, the multiplier M 

is specified as a uniform value that applies to all 1,841 

ground-water withdrawal cells. 

Streamflow

To allow for both the optimal conjunctive use of 

surface water and ground water within the optimization 

model, streamflows from the two largest rivers (Arkansas River and Bayou Bartholomew) are specified for 

estimating optimal withdrawals (table 2). Streamflow 

constraints are specified at one cell from each river (fig. 

6) based on 7-day low flows with 10-year recurrence 

intervals (7Q10) (Steve Loop, Arkansas Soil and Water 

Conservation Commission, written commun., 2001), 

which are derived from historical streamflow for the 

rivers. Streamflow constraints were unavailable for the 

other streams within the model. By specifying a minimum flow constraint based on 7Q10 values, available 

streamflow within the optimization model would be 

limited all year long to the amount of streamflow 

greater than the 7Q10; 7Q10 historically is expected to 

occur only once every 10 years, and then for only 7 

consecutive days. Flow into the most upstream cell of 

each river contained within the model was specified 

based on mean annual flow, as were the cells at which 

tributaries connect. Because stream gages are not 

located at the start of the rivers simulated in the model, 

mean annual flow was prorated based on the drainage 

area upstream from that point. Overland flow (that is, 

surface-water runoff that would enter river cells from 

minor tributaries or sheet flow that was not explicitly 

represented in the model) was distributed equally at 

river cells within a river reach based on the difference 

in long-term average streamflow for a specific river 

Table 2. Rivers, streamflows, and streamflow constraints

[ASWCC, Arkansas Soil and Water Conservation Commission. Flow constraint based on an annual minimum 7-consecutiveday average flow with a recurrence interval of 10 years]

River name

Number of

 model cells

Flow into

 uppermost

river cell

of model

(million cubic

feet per day)

Overland flow

per river cell

(million cubic

 feet per day)

Flow constraint

(million cubic

 feet per day)

Source for

 value of

 constraint

Arkansas 83 3,500 18 100 ASWCC

Bayou Bartholomew 139 6.65 0.8 8.38 ASWCC
14 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 6. Location of streams within model showing cells and rates at which water could be withdrawn and still meet 

constraints within optimization model. 

92° 91°

33°

34°

EXPLANATION

Model Rivers

Arkansas River

Bayou Bartholomew

Other rivers

38,921,000

Arkansas River

Mississippi

Bayou Bartholomew

38,921,000

4,879,400,000

Optimized river

withdrawal location

and rate, cubic feet

per day

River

0 10 5 20 Miles

0 10 5 20 Kilometers

Base from U.S. Geological Survey digital data, 1:100,000
Conjunctive-Use Optimization 15

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., 2000). 

Surface-water diversion rates that occurred in 2000 

were subtracted from specified overland flow at the 

appropriate river cells. Diversions totaling 1,787,103 

ft3

/d were subtracted from Bayou Bartholomew, 

whereas no diversions from the Arkansas River were 

accounted for in this model. 

 Optimization Results

Sustainable Yield 

The ultimate objective of the optimization model 

is to provide estimates of sustainable yield from both 

ground water and surface water. Sustainable yield is 

defined as a withdrawal rate from the aquifer or from a 

stream that can be maintained indefinitely (that is, to 

steady-state conditions) without causing violation of 

either hydraulic-head or streamflow constraints. 

Streamflow constraints (the 7Q10 streamflow) are 

specified in table 2. 

Because sustainable yield from ground water is a 

function of the withdrawal limit specified for each 

managed well, multiples of the rate withdrawn in 1997 

were used to set the maximum withdrawal limit. The 

distribution of optimal withdrawal rates using upper 

limits specified as 100-, 150-, and 200-percent multiples of the total 1997 ground-water withdrawal rates 

(scenarios 1, 2, and 3, respectively) are shown in figures 8-10. These limits, while arbitrary, are specified to 

examine the distribution of optimal withdrawals that 

would result for each scenario. The distribution of optimal withdrawal rates for each of these scenarios is such 

that most wells are withdrawing water at a rate equal to 

the upper limit or are not withdrawing water. This is 

convenient from a management standpoint because 

wells are generally on or off. As the withdrawal rate 

limit is increased, the total number of wells that can 

pump decreases (although the total amount withdrawn 

increases), with those wells withdrawing water being 

nearest to sources of water (major rivers) within the 

model. Test runs with the optimization model show that 

if no limits are placed on ground-water withdrawals, all 

of the withdrawals would come from wells adjacent to 

model sources of water, at rates that are orders of magnitude greater than were pumped in 1997. Although 

overall optimized withdrawal would be largest for such 

a scenario, the distribution of withdrawal cells would 

be unacceptable from a management standpoint 

because virtually all of the water production would 

come from wells adjacent to rivers, at the expense of 

the remaining interior wells being unable to pump at 

all. Not only is it physically unlikely that individual 

wells could pump that much more water, but construction of sufficient additional wells in the one-square 

mile cells also is unlikely. 

Specifying an upper withdrawal limit of 100 percent of the 1997 withdrawal rate (scenario 1; fig. 7), the 

sustainable yield from ground water for the entire 

model area is 70.3 Mft3

/d (table 3), which is about 96 

percent of the amount withdrawn in 1997 (73.5 Mft3/

d). If the upper withdrawal limit is increased to 150 percent of the 1997 withdrawal rate (scenario 2; fig. 8), the 

sustainable yield from ground water for the entire 

model area is 80.6 Mft3

/d (table 3), which is about 110 

percent of the amount withdrawn in 1997. If the upper 

withdrawal limit is increased to 200 percent of the 1997 

withdrawal rate (scenario 3; fig. 9), the sustainable 

yield from ground water for the entire model area is 

110.2 Mft3/d (table 3), which is about 150 percent of 

the amount withdrawn in 1997. 

Sustainable yields from the Arkansas River and 

Bayou Bartholomew are much larger than those from 

ground water. Total sustainable yield from the Arkansas River and Bayou Bartholomew is about 4,900 Mft3/

d for scenarios 1, 2, and 3 (table 4) or about 6,700 percent of the amount of ground-water withdrawn in 1997. 

These large sustainable yields represent a potential 

source of water that could supplement ground water 

and meet total water demand, but to do so will require 

the construction of withdrawal and distribution facilities, which will have legal, political, economic, and 

social consequences.
16 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 7. Ratio of optimal ground-water withdrawal calculated by the optimization model to the amount withdrawn in 1997 for 

withdrawal limits at each well set to 100 percent of the 1997 withdrawal rate.

92° 91°

33°

34°

0 10 20 Miles 5

0 10 20 Kilometers 5

Arkansas River

EXPLANATION

Ratio of optimal

ground-water

withdrawal to the

1997 withdrawal rate

0

0.21

0.75

1

Head at lower limit

Mississippi

Bayou Bartholomew

Scenario 1

No wells

River

Base from U.S. Geological Survey digital data, 1:100,000
Conjunctive-Use Optimization 17

Table 3. Sustainable yield and unmet demand by county or parish for different upper limits on withdrawals and different demand rate[Negative unmet demand values indicate surplus water availability; Mft3/d, million cubic feet per day]

County

or

parish State

1997

withdrawal

rate

(Mft3/d)

Sustainable yield (Mft3/d) based on

 an upper withdrawal limit of:

Unmet demand (Mft3/d) based

on 1997 demand and a

sustainable yield from:

Unme

 susta

 1,2, 

100 percent

of 1997

withdrawal

rate

(scenario 1)

150 percent

 of 1997

withdrawal

rate

(scenario 2)

200 percent

 of 1997

withdrawal

 rate

(scenario 3) Scenario 1 Scenario 2 Scenario 3

100 percen

of 1997

 withdrawa

 rate

Ashley Arkansas 10.1 8.7 12.1 15.5 1.5 -2.0 -5.4 1.5 Chicot Arkansas 7.9 7.9 11.8 15.7 0.0 -3.9 -7.9 0.0 Desha Arkansas 20.9 19.6 22.2 22.1 1.3 -1.3 -1.2 1.3 Drew Arkansas 6.8 6.8 9.7 11.9 0.0 -2.9 -5.1 0.0 Jefferson Arkansas 8.6 8.2 12.3 16.5 0.3 -3.8 -7.9 0.3 Lincoln Arkansas 16.1 15.9 7.7 22.1 0.2 8.4 -6.0 0.2 East Carroll Louisiana 0.7 0.7 1.1 1.5 0.0 -0.4 -0.7 0.0 Morehouse Louisiana 1.8 1.8 2.7 3.6 0.0 -0.9 -1.8 0.0 Union Louisiana 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0 West Carroll Louisiana 0.6 0.6 0.9 1.2 0.0 -0.3 -0.6 0.0 Issaquena Mississippi 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0 Total 73.5 70.3 80.6 110.2 3.3 -7.1 -36.6 3.3 
18 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 8. Ratio of optimal ground-water withdrawal calculated by the optimization model to the amount withdrawn in 1997 for 

withdrawal limits at each well set to 150 percent of the 1997 withdrawal rate.

92° 91°

33°

34°

0 10 5 20 Miles

0 10 5 20 Kilometers

Arkansas River

EXPLANATION

Ratio of optimal

ground-water

withdrawal to the

1997 withdrawal rate

0

0.42 - 0.74

0.75 - 0.99

1.00 - 1.49

1.5

Head at lower limit

Mississippi River

Bayou Bartholomew

Scenario 2

No wells

Base from U.S. Geological Survey digital data, 1:100,000
Conjunctive-Use Optimization 19

Figure 9. Ratio of optimal ground-water withdrawal calculated by the optimization model to the amount withdrawn in 1997 for 

withdrawal limits at each well set to 200 percent of the 1997 withdrawal rate.

92° 91°

33°

34°

Arkansas River

EXPLANATION

Ratio of optimal

ground-water

withdrawal to the

1997 withdrawal rate

0

0.16 - 0.25

0.25 - 1.00

1.01 - 1.25

1.25 - 1.50

1.55 - 1.90

2

Head at lower limit

Mississippi

Scenario 3

No wells

0 10 5 20 Miles

0 10 5 20 Kilometers

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

River
20 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Unmet Demand

Unmet demand is defined in this report as the difference between the sustainable yield of ground water 

(or optimized withdrawal rate), and the anticipated 

demand: 

(7)

where U is the unmet demand, in cubic feet per day;

D is the demand, in cubic feet per day; and 

S is the sustainable yield, in cubic feet per day.

For example, if the demand is 73.5 Mft3/d (the amount 

withdrawn in 1997), and the sustainable yield is calculated to be 70 Mft3/d, the unmet demand is the difference of these two values, or 3.5 Mft3

/d. Because, unmet 

demand is not solely a function of the sustainable yield, 

unmet demand calculations are provided for different 

values of anticipated demand. Anticipated demand is 

specified as being either 100-, 150-, or 200-percent of 

the 1997 withdrawal rate for the entire model area 

(table 3).

Unmet demand was calculated using different 

demand rates based on multiples of the 1997 withdrawal rate. From a management perspective, it is useful to calculate the unmet demand under different 

anticipated demands and sustainable-yield estimates 

based on different limits on withdrawals. If the demand 

is specified as the 1997 withdrawal rate for the entire 

model area, and the sustainable-yield estimates for scenarios 1, 2, and 3 are used to calculate unmet demand, 

then the resulting unmet demand for the entire model 

area is 3.3, -7.1, and -36.6 Mft3/d, respectively. Note 

that a negative value of unmet demand indicates surplus water availability. Whereas, if the demand is specified as 100-, 150-, and 200-percent of the 1997 

withdrawal rate, and the sustainable-yield estimates for 

scenarios 1, 2, and 3 are used to calculate unmet 

demand, then the resulting unmet demand for the entire 

model area is 3.3, 29.7, and 36.9 Mft3

/d, respectively. 

Hydraulic-head constraints restrict where and 

how much ground water and surface water can be 

extracted. The red triangles in figures 7-9 show the 

locations where the simulated value of hydraulic head, 

derived from the optimization model, reached the lower 

limits of the hydraulic-head constraint (known as a 

binding constraint). The ring of active pumping cells 

surrounding the binding constraint lower the potentiometric surface. These pumping cells are effective in 

reducing the flow toward the center of the model, and 

dropping the potentiometric surface as a result. If the 

hydraulic head at the binding constraint is relaxed or 

lowered, then more withdrawal cells would become 

active.

Pumping cells in the north-central part of the 

model, for which an optimized rate of zero was calculated, lie mostly between the Arkansas River and 

Bayou Bartholomew (figs. 7-9). An additional set of 

zero-pumping cells occurs to the west of Bayou Bartholomew. Both sets of zero-pumping cells are adjacent 

to points at which the hydraulic-head constraint (red 

triangles in figs. 7-9) prevented further withdrawals for 

all three scenarios.

Optimal Simulated Hydraulic-Head Altitude

Substantial differences occur between simulated 

hydraulic-head altitudes (or water-level altitudes) for 

steady-state flow-model simulations using 1997 withdrawal rates and for steady-state simulations using 

optimal withdrawal rates (sustainable yield) from scenario 1 (fig. 10). Because the 1997 withdrawal rates are 

unsustainable, some cells of the model are dry, particularly in the area between the Arkansas River and Bayou 

Bartholomew. In contrast, the optimal hydraulic-head 

altitude using sustainable yield from scenario 1 shows 

no dry cells.

The spatial distribution of the difference between 

simulated hydraulic-head altitude using optimized 

withdrawal rates from scenario 1 and the altitude corresponding to the half-aquifer-thickness hydraulic-head 

constraint is shown in figure 11. Over the vast majority 

of the model area, simulated hydraulic head is at or 

above the altitude of the constraint. 

Table 4. Optimized streamflow withdrawal

River name

Optimized streamflow withdrawal

(million cubic feet per day)

100 percent

of 1997

withdrawal

rate

(scenario 1)

150 percent

 of 1997

withdrawal

rate

(scenario 2)

200 percent

 of 1997

withdrawal

 rate

(scenario 3)

Arkansas River 4,879.4 4,879.8 4,871.4

Bayou Bartholomew 38.9 40.7 30.7

Total 4,918.3 4,920.5 4,902.1

U DS = –
Conjunctive-Use Optimization 21Figure 10. Simulated hydraulic-head altitude at steady state using (A) 1997 withdrawal rates and (B) sustainable yield.

92° 91°

33°34°

Arkansas River

Mississippi

EXPLANATION

Hydraulic head altitudeafter steady statepumping at 1997withdrawal rates, in feet.Datum is National GeodeticVertical Datum of 1929

181 - 200161 - 180141 - 160121 - 140101 - 12081 - 10061 - 8051 - 60Dry cell

A

Bayou Bartholomew

0 10 5 20 Miles

0 10 5 20 Kilometers

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

River

92° 

33°34°

Arkansas Rive

Bayou Bartholomew

EXPLANATION

Hydraulic head altitudeafter steady state pumpingat optimal withdrawal rates,in feet. Datum is NationalGeodetic Vertical Datumof 1929

181 - 200161 - 180141 - 160121 - 140101 - 12081 - 10061 - 8051 - 60

Base from U.S. Geological Survey digital data, 1:100,000
22 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

Figure 11. Difference between simulated hydraulic-head altitude using optimized withdrawal rates and altitude of half-theaquifer-thickness hydraulic-head constraint. 

92° 91°

33°

34°

Arkansas River

Mississippi

EXPLANATION

Difference between

simulated hydraulic

head and altitude

corresponding to

half the aquifer

thickness, in feet.

Datum is National

Geodetic Vertical

Datum of 1929

101 - 120

81 - 100

61 - 80

41 - 60

21 - 40

1 - 20

-19 - 0

River

0 10 5 20 Miles

0 10 5 20 Kilometers

Base from U.S. Geological Survey digital data, 1:100,000
Conjunctive-Use Optimization 23

Nonlinear Effects

Because sustainable yield is obtained with the 

assumption that the model behaves linearly (that is, the 

change in hydraulic head is a constant multiple of the 

change in withdrawal rate, regardless of the withdrawal 

rate), it is important to compare the resulting simulated 

hydraulic-head altitudes from the flow model derived 

using sustainable yield to the altitudes corresponding to 

the hydraulic-head constraints specified in the optimization model. This is done by using the sustainable 

yield from scenario 1 (upper limit of withdrawals set to 

100 percent of the 1997 rates) in the flow model and 

simulating hydraulic-head altitudes at steady-state conditions. Less than 1.4 percent of the 3,826 hydraulichead constraint points (that is, all of the active model 

cells) had flow-model derived values of hydraulic head 

that was below the hydraulic head constraint (fig.12). 

The optimized pumping distribution is a good approximation of sustainable yield, despite nonlinear behavior 

inherent in the model. 

The values of sustainable yield should be considered maximum rates, in that hydraulic-head constraints 

are violated in some areas because of nonlinear 

responses in hydraulic head to incremental changes in 

withdrawal rates within the flow model. When the sustainable yield rates are used in the flow model, a few 

cells have hydraulic heads at steady state that are below 

the hydraulic-head constraints, which could have been 

corrected by reducing withdrawal rates further. This 

was not done, however, because of the few points 

affected. 

Limitations

Sustainable yield results from the optimization 

model should be used cautiously, mindful that the 

model represents a simplification of a complex system. 

The assumption that the flow system behaves linearly 

is likely the largest discrepancy from actual conditions. 

Nonetheless, the optimization model does provide estimates of sustainable yield from both the ground-water 

and surface-water sources that result in hydraulic-head 

values remaining at or above an altitude corresponding 

to half the thickness of the aquifer throughout the bulk 

of the model area, and maintaining streamflows at or 

above specified minimum amounts. 

Figure 12. Cumulative percentage of hydraulic-head constraint points less than or equal to the difference between simulated 

hydraulic head and the altitude corresponding to half the aquifer thickness. 
24 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

SUMMARY

The Mississippi River Valley alluvial aquifer is a 

water-bearing assemblage of gravels and sands that 

underlies about 32,000 mi2

 of Missouri, Kentucky, 

Tennessee, Mississippi, Louisiana, and Arkansas. The 

Mississippi River Valley alluvial aquifer supplies large 

volumes of water for agriculture in Arkansas. Because 

of the heavy demands placed on the aquifer, several 

large cones of depression have formed in the potentiometric surface, resulting in lower well yields and 

degraded water quality in some areas. Several counties, 

which are within the extent of the alluvial aquifer, have 

been designated Critical Ground-Water Areas by the 

Arkansas Soil and Water Conservation Commission 

(ASWCC). These criteria state that if water levels drop 

below half the original saturated thickness of the formation, then a “critical ground-water area” may be designated. 

A ground-water flow model of the alluvial aquifer was developed for an area covering 3,826 mi2, 

extending south from the Arkansas River into the 

southeastern corner of Arkansas, northeast Louisiana, 

and Mississippi. The flow model showed that continued ground-water withdrawals at rates commensurate 

with those of 1997 could not be sustained indefinitely 

without causing water levels to decline below half the 

original saturated thickness of the formation. To 

develop estimates of withdrawal rates that could be sustained relative to the constraints of critical groundwater area designation, the U.S. Geological Survey, in 

cooperation with the Arkansas Soil and Water Conservation Commission, and Vicksburg District of the U.S. 

Army Corps of Engineers, applied conjunctive-use 

optimization modeling to the flow model of the Mississippi River Valley alluvial aquifer in southeastern 

Arkansas. Conjunctive-use optimization modeling is a 

technique that can be used to simulate maximum withdrawal rates from both surface water and ground water 

while honoring constraints with respect to water levels 

and streamflow. These withdrawal rates form the basis 

for estimates of sustainable yield from the alluvial 

aquifer and from rivers specified within the alluvial 

aquifer model. 

The purpose of this report is to describe the 

application and evaluation of a conjunctive-use optimization model of the Mississippi River Valley alluvial 

aquifer. A management problem was formulated as one 

of maximizing the sustainable yield from all groundwater and surface-water withdrawal cells within limits 

imposed by plausible withdrawal rates, and within 

specified constraints involving hydraulic head and 

streamflow. Steady-state conditions were selected 

because the maximized withdrawals are intended to 

represent sustainable yield of the system. The optimization model was used to generate response coefficients for each specified withdrawal cell in the model. 

After all the response coefficients were calculated, they 

were combined to form a data-input set along with 

hydraulic-head and streamflow constraints, and formulated as a linear program. Optimal sustainable yield 

values were obtained by running the linear program 

under MINOS.

Optimal sustainable-yield values are affected by 

the rate of recharge and limits to potential withdrawals 

assigned within the optimization model. Limits were 

assigned to ground-water withdrawals at individual 

model cells at multiples of 100, 150, and 200 percent of 

1997 withdrawal rates. If no limit is placed on withdrawals, the majority of water production will be from 

wells proximal to model water sources such as rivers, 

depriving wells of water that are distant from these 

sources of water. No limit on withdrawals led to unrealistic optimal withdrawal from wells adjacent to rivers 

and most interior withdrawal cells had no withdrawal.

One point along the Arkansas River and one 

point along Bayou Bartholomew were specified for 

obtaining surface-water sustainable-yield estimates 

within the optimization model. Streamflow constraints 

were specified at two river cells based on average 7-day 

low flows with 10-year recurrence intervals. 

Surface-water diversion rates that occurred in 

2000 were subtracted from specified overland flow at 

the appropriate river cells. Diversions totaling 

1,787,103 ft3

/d were subtracted from Bayou Bartholomew, whereas no diversions from the Arkansas River 

were accounted for in this model. 

Sustainable-yield estimates for ground water are 

affected by the allowable upper limit on withdrawals 

from wells specified in the optimization model. 

Ground-water withdrawal rates were allowed to 

increase to 200 percent of the withdrawal rate in 1997. 

As the overall upper limit is increased, the sustainable 

yield generally increases. Tests with the optimization 

model show that without limits on pumping, wells adjacent to sources of water would have optimized withdrawal rates that were orders of magnitude larger than 

1997 withdrawal rates. Not only is it physically 

unlikely that individual wells could pump that much 

more water, but construction of sufficient additional 

wells in the one-square mile cells also is unlikely. 
References Cited 25

Specifying an upper withdrawal limit of 100 percent of the 1997 withdrawal rate, the sustainable yield 

from ground water for the entire study area is 70.3 

Mft3

/d, which is 96 percent of the amount withdrawn in 

1997 (73.5 Mft3/d). If the upper withdrawal limit is 

increased to 150 percent of the 1997 withdrawal rate, 

the sustainable yield from ground water for the entire 

study area is 80.6 Mft3

/d, which is about 110 percent of 

the amount withdrawn in 1997. If the upper withdrawal 

limit is increased to 200 percent of the 1997 withdrawal 

rate, the sustainable yield from ground water for the 

entire study area is 110.2 Mft3/d, which is about 150 

percent of the amount withdrawn in 1997. 

Sustainable yields from the Arkansas River and 

Bayou Bartholomew are much larger than that from 

ground water. Total sustainable yield from the Arkansas River and Bayou Bartholomew is about 4,900

Mft3/d or about 6,700 percent of the amount of ground 

water withdrawn in 1997. These large sustainable 

yields represent a potential source of water that could 

supplement ground water and meet the total water 

demand, but to do so will require the construction of 

withdrawal and distribution facilities, which will have 

legal, political, economic, and social consequences.

Unmet demand (defined as the difference 

between the optimized ground-water withdrawal rate 

or sustainable yield, and the anticipated demand) was 

calculated using different demand rates based on multiples of the 1997-withdrawal rate. Assuming that 

demand is the 1997 withdrawal rate, and that sustainable-yield estimates are those obtained using upper 

limits of withdrawal rates of 100-, 150-, and 200-percent of 1997 withdrawal rates, then the resulting unmet 

demand for the entire model area is 3.3, -7.1, and -36.6 

Mft3/d, respectively. Whereas, if the demand is specified as 100-, 150-, and 200-percent of the 1997 withdrawal rate, and the sustainable-yield estimates remain 

the same, then the resulting unmet demand for the 

entire model area is 3.3, 29.7, and 36.9 Mft3

/d. A negative value of unmet demand indicates surplus water 

availability. 

A check of sustainable-yield rates was performed by applying these rates in the flow model run to 

steady state and comparing simulated hydraulic-head 

altitude to hydraulic-head constraints. Application of 

the sustainable-yield rates resulted in hydraulic-head 

values that were above 98.6 percent of the constraint 

values. At those points where hydraulic head is below 

the constraint, deviation from linear model response 

(that is, a unit incremental change in withdrawal rate 

results in a unit incremental change in hydraulic head) 

is suspected. 

REFERENCES CITED

Ackerman, D.J., 1989a, Hydrology of the Mississippi River 

Valley alluvial aquifer, south-central United States--A 

preliminary assessment of the regional flow system: 

U.S. Geological Survey Water-Resources Investigations Report 88-4028,74 p.

———1989b, Potentiometric surfaces of the Mississippi 

River Valley alluvial aquifer in eastern Arkansas, spring 

1972 and 1980: U.S. Geological Survey WaterResources Investigations Report 88-4075, 1 sheet.

———1990, Hydrology of the Mississippi River Valley 

alluvial aquifer, south-central United States: U.S. Geological Survey Open-File Report 90-358, 228 p.

Ahlfeld, D.P., and Mulligan, A.E., 2000, Optimal management of flow in groundwater systems: San Diego, 

Calif., Academic Press, 185 p.

Barlow, P.M., Ahlfeld, D.P., and Dickerman, D.C., 2003, 

Conjunctive-management models for sustained yield of 

stream-aquifer systems: Journal of Water Resources 

Planning and Management, v. 129, no. 1, p. 35-48.

Broom, M.E., and Lyford, F.P., 1981, Alluvial aquifer of the 

Cache and St. Francis River basins, northeastern Arkansas: U.S. Geological Survey Open-File Report 81-476, 

48 p.

Broom, M.E., and Reed, J.E., 1973, Hydrology of the Bayou 

Bartholomew alluvial-aquifer stream system, Arkansas: U.S. Geological Survey Open-File Report, 91 p.

Boswell, E.H., Cushing, E.M., Hosman, R.L., 1968, Quaternary aquifers in the Mississippi embayment with a discussion of Quality of the water by H.G. Jeffery: U.S. 

Geological Survey Professional Paper 448-E, 15 p.

Cushing, E.M., Boswell, B.H., and Hosman, R.L., 1964, 

General geology of the Mississippi embayment: U.S. 

Geological Survey Professional Paper 448-B, 28 p.

Czarnecki, J.B., McKee, P.W., and Hays, P.D., 2002, The 

Mississippi River Valley alluvial aquifer in Arkansas: A 

sustainable water resource?: U.S. Geological Survey 

Fact Sheet FS-041-02, 4 p.

Engler, Kyle, Bayley, F.H., III, and Sniegocki, R.T., 1963, 

Studies of artificial recharge in the Grand Prairie 

region, Arkansas; environment and history: U.S. Geological Survey Water-Supply Paper 1615-A, 32 p.

Fisk, H.N., 1944, Geological investigation of the alluvial 

aquifer of the lower Mississippi River: U.S. Department 

of the Army, Mississippi River Commission, 78 p.

Gonthier, G.J., and Mahon, G.L., 1993, Thickness of the 

Mississippi River Valley confining unit, eastern Arkan-
26 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Southeastern Arkansas

sas: U.S. Geological Survey Water-Resources Investigations Report 92-4121, 4 sheets.

Greenwald, R. M., 1998, Documentation and user’s guide: 

MODMAN, An optimization module for MODFLOW 

version 4.0: Freehold, New Jersey, HSI GeoTrans, 112 

p.

Harbaugh, A.W., and McDonald, M.G., 1996, User’s documentation for MODFLOW-96, an update to the U.S. 

Geological Survey modular finite-difference groundwater flow model: U.S. Geological Survey Open-File 

Report 96-485, 56 p.

Hill, M.C., Banta, E.R., Harbaugh, A.W., and Anderman, 

E.R., 2000, MODFLOW-2000, The U.S. Geological 

Survey modular ground-water model user guide to the 

observation, sensitivity, and parameter-estimation processes and three post-processing programs: U.S. Geological Survey Open-File Report 00-184, 209 p.

Krinitzsky, E.L., and Wire, J.C., 1964, Ground water in alluvium of lower Mississippi Valley (upper and central 

areas): U.S. Department of the Army, Corps of Engineers, Waterways Experiment Station Technical Report 

3-658, v. 1 and 2, 400 p.

Mahon, G.L. and Ludwig, A.H., 1990, Simulation of 

ground-water flow in the Mississippi River Valley alluvial aquifer in eastern Arkansas: U.S. Geological Survey Water-Resources Investigations Report 89-4145, 26 

p.

Mahon, Gary L., and Poynter, David T., 1993, Development, 

calibration, and testing of ground-water flow models 

for the Mississippi River Valley alluvial aquifer in eastern Arkansas using one-square-mile cells: U.S. Geological Survey Water- Resources Investigations Report 92-

4106, 33 p., 11 plates.

McDonald, M.G., and Harbaugh, A.W., 1988, A modular 

three-dimensional finite-difference ground-water flow 

model: U.S. Geological Survey Techniques of WaterResources Investigations Book 6, Chapter Al.

Murtaugh, B.A., and Saunders, M.A., 1998, MINOS 5.5 

user’s guide: Systems Optimization Laboratory Technical Report SOL 83-20R, Stanford, California, Stanford 

University, 150 p.

Nishikawa, Tracy, 1998, A simulation-optimization model 

for water-resources management, Santa Barbara, California: U.S. Geological Survey Water-Resources Investigations Report 97-4246, 99 p.

Peralta, R.C., Yazdanian, Amin, Killian, P.J., and Shulstad, 

R.N., 1985, Future Quaternary groundwater accessibility in the Grand Prairie--1993: University of Arkansas, 

Agricultural Experiment Station Bulletin no. 877, 37 p.

Reichard, E.G., 1995, Groundwater-surface water management with stochastic surface water supplies: A simulation-optimization approach: Water Resources 

Research, v. 31, no.11, p. 2845-2865.

Schrader, T.P., 2001, Status of water levels and selected 

water-quality conditions in the Mississippi River Valley 

alluvial aquifer in eastern Arkansas: U.S. Geological 

Survey Water-Resources Investigations Report 01-

4124, 52 p.

Stanton, G.P., and Clark, B.R., 2003, Recalibration of a 

ground-water flow model of the Mississippi River Valley alluvial aquifer in southeastern Arkansas, 1918-

1998, with simulations of hydraulic heads caused by 

projected ground-water withdrawals through 2049: 

U.S. Geological Survey Water-Resources Investigations Report 03-4232, __ p.

Stephenson, L.W., and Crider, A.F., 1916, Geology and 

ground water of northeastern Arkansas with a discussion of the chemical character of the waters by R.B. 

Dole: U.S. Geological Survey Water-Supply Paper 399, 

315 p.

Veatch, A.C., 1906, Geology and underground water 

resources of northern Louisiana and southern Arkansas: 

U.S. Geological Survey Professional Paper 46, 422 p.