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 OF THE 

MISSISSIPPI RIVER VALLEY ALLUVIAL AQUIFER OF 

NORTHEASTERN ARKANSAS

Water-Resources Investigations Report 03-4230

U.S. Department of the Interior

U.S. Geological Survey
Cover: Center pivot irrigation system in eastern Pulaski County, Arkansas. Photograph by John B. 

Czarnecki, U.S. Geological Survey.
CONJUNCTIVE-USE OPTIMIZATION MODEL OF THE 

MISSISSIPPI RIVER VALLEY ALLUVIAL AQUIFER OF 

NORTHEASTERN ARKANSAS

By John B. Czarnecki, Brian R. Clark, and Thomas B. Reed

U.S. GEOLOGICAL SURVEY

Water-Resources Investigations Report 03-4230

Prepared in cooperation with the

U.S. Army Corps of Engineers, Memphis 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

Irrigation Project Areas............................................................................................................................. 4

Conjunctive-Use Optimization........................................................................................................................... 6

Flow Model ............................................................................................................................................... 6

Optimization Model .................................................................................................................................. 8

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

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

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

Streamflow Constraints.........................................................................................................10

Ground-Water Withdrawal Limits ........................................................................................10

Surface-Water Withdrawal Limits ........................................................................................12

Predevelopment Recharge From Non-River Sources ...........................................................12

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

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

Optimization Results......................................................................................................................13

Sustainable Yield ..................................................................................................................13

Unmet Demand .....................................................................................................................21

Optimal Simulated Hydraulic-Head Altitude .......................................................................23

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

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

Summary ..........................................................................................................................................................27

References Cited ..............................................................................................................................................29

ILLUSTRATIONS

Figures 1-3. Maps showing:

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

2. Irrigation project areas within the model area ........................................................................... 5

3. Potentiometric surface within the alluvial aquifer Spring, 1998 ...............................................7

4. Flow chart of optimization modeling process ...................................................................................8

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

hydraulic-head constraint...........................................................................................................11

6. Graph showing hypothetical variation in recharge to the alluvial aquifer from non-river sources

as a function of time ..................................................................................................................12

7-11. Maps showing:

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

and still meet constraints within optimization model.........................................................14
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 100 percent of the

1997 withdrawal rate.............................................................................................................. 16

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 150 percent of the

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

10. 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

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

(B) sustainable yield .............................................................................................................. 24

12. Graph showing cumulative percentage of model cells with values less than or equal to the

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

aquifer thickness .................................................................................................................... 25

13. Map showing difference between simulated hydraulic head and altitude of half the aquifer

thickness ................................................................................................................................ 26

TABLES

Table 1. Characteristics of the flow model ............................................................................................................ 6

2. Rivers, streamflows, and streamflow constraints................................................................................... 15

3. Sustainable yield and unmet demand for different upper limits on withdrawals and different demand

rates................................................................................................................................................... 17

4. Optimized total streamflow withdrawals from the optimization model ................................................ 20

5. Comparison of sustainable yield obtained for hydraulic-head constraints everywhere in the model

area and only in the Bayou Meto and Grand Prairie irrigation project areas.................................... 21

6. Sustainable yield from ground water and unmet demand relative to 1997 withdrawal rates in six

irrigation Project areas and the remainder of the model area............................................................ 22
Contents V

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 Northeastern Arkansas

By John B. Czarnecki, Brian R. Clark, and T.B. Reed

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 over 100 feet 

deep 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 

14,104 square miles, extending northeast from the 

Arkansas River into the northeast corner of Arkansas 

and parts of southeastern Missouri. 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 aquifer. 

To develop estimates of withdrawal rates that 

could be sustained in compliance with the constraints 

of critical ground-water area designation, conjunctiveuse optimization modeling was applied to the flow 

model of the alluvial aquifer in northeastern Arkansas. 

Ground-water 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 ground-water and 

surface-water withdrawal cells within limits imposed 

by plausible withdrawal rates, and within specified 

constraints involving hydraulic head and streamflow. 

Steady-state flow conditions were selected because the 

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

Within the optimization model, 11 rivers are 

specified. Surface-water diversion rates that occurred 

in 2000 were subtracted from specified overland flow 

at the appropriate river cells. Included in these diversions were the planned diversions of 63,339,248 ft3

/d 

for the Bayou Meto project area and 55,078,367 ft3/d 

for the Grand Prairie project area, which factor in an 

additional 30 and 40 percent transmission loss, respectively. Streamflow constraints were specified at all 

1,165 river cells based on average 7-day minimum 

flows for 10 years. Sustainable yield for all rivers 

ranged from 0 (Current, Little Red, and Bayou Meto 

Rivers) to almost 5 billion cubic feet per day for the 

Arkansas River. Total sustainable yield from all rivers 

combined was 12.8 billion cubic feet per day, which 

represents a substantial source for supplementing 

ground water to meet the total water demand. 

Sustainable-yield estimates are affected by the 

allowable upper limit on withdrawals from wells specified in the optimization model. Ground-water withdrawal rates were allowed to vary as much as 200 

percent of the withdrawal rate in 1997. As the overall 

upper limit on withdrawals 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 

rates corresponding to those of 1997. The sustainable 

yield from ground water for the entire study area while 

setting the maximum upper limit as the amount withdrawn in 1997 is 360 million cubic feet per day, which 

is only about 57 percent of the amount withdrawn in 

1997 (635.6 million cubic feet per day). Optimal sustainable yields from within the Bayou Meto irrigation 

project area and within the Grand Prairie irrigation 

project area are 18.1 and 9.1 million cubic feet per day, 

respectively, assuming a maximum allowable withdrawal rate equal to 1997 rates. These values of sustainable yield represent 35 and 30 percent respectively of 

the amount pumped from these project areas in 1997. 

Unmet demand (defined as the difference 

between the optimized withdrawal rate or sustainable 

yield, and the anticipated demand) was calculated 

using different demand rates based on multiples of the 
2 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

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 275.5, 190.9, and 110 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 275.5, 508.8, and 745.8 million 

cubic feet per day, respectively. These unmet demands 

for ground water could be obtained from large sustainable surface-water withdrawals.

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. 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). 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 (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 million 

cubic feet per day (Mft3/d) of water were pumped from 

the aquifer, primarily for irrigation and fish farming. 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 over 100 ft deep have 

formed in the potentiometric surface, resulting in lower 

well yields and degraded water quality in some areas. 

Several counties in the Grand Prairie area, 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 

(Reed, 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 Memphis 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 (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 northeastern Arkansas. The optimization model was 

formulated as a linear program, and utilized a groundwater model developed for the study area by Reed 

(2003) as a basis for evaluation. The purpose of the 

optimization model was to: (1) determine maximum 

withdrawal rates from model cells at which groundwater withdrawals occurred in 1997 and (2) determine 

maximum withdrawal rates from model cells at stream 

locations while maintaining ground-water levels at or 

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 
Introduction 3

Figure 1. Location of study and modeled area. 

Tennessee

Mississippi

Missouri

Arkansas

LEE

WHITE

CLAY

SHELBY

IZARD

LONOKE

ARKANSAS

RIPLEY

SHARP

CROSS

PANOLA

OREGON

PRAIRIE

PHILLIPS

DESHA

TATE

MISSISSIPPI

JEFFERSON

POINSETT

STONE

MONROE

GREENE

JACKSON

BUTLER

LINCOLN

TUNICA

PULASKI

DUNKLIN

CRAIGHEAD

BOLIVAR

RANDOLPH

COAHOMA

DE SOTO

CLEBURNE

ST FRANCIS

TIPTON

LAWRENCE

WOODRUFF

CRITTENDEN

PEMISCOT

INDEPENDENCE

GRANT

NEW MADRID

CLEVELAND

FAULKNER

DYER

STODDARD

LAUDERDALE

SALINE

Mississippi

River

St. Francis

River

L'Anguille

River

River

Cache

White

River

Arkansas

River

92° 90°

34°

36°

0 10 20 MILES

0 10 20 KILOMETERS

EXPLANATION

Boundary of active model

cells

Mississippi River Valley

alluvial aquifer

Crowleys Ridge

Condsolidated formations

of Mesozoic and Paleozoic

age

Jackson Group

Base from U.S. Geological Survey digital data, 1:100,000 and 1:2,000,000

Digital geological data adapted from Haley and others (1976) and Bicker (1969)

GRAND PRAIRIE

AREA

MISSISSIPPI

ALLUVIAL

PLAIN

MISSISSIPPI

ALLUVIAL

PLAIN

MODEL

AREA

Missouri

Arkansas

Tennessee

Mississippi

FALL

LINE

FALL

LINE
4 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

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 northeastern 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 on 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) ground-water flow model (hereafter referred to 

as the flow model) used in the optimization modeling 

of this report is based on the work of Reed (2003), who 

recalibrated and extended the model of Mahon and 

Poynter (1993) to include hydraulic-head observations 

for the years 1992 and 1998. 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 simulate scenarios 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 optimize ground-water 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. This analysis 

focused on a small subset of the area contained in this 

report, 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). Paul Barlow (U.S. Geological Survey) provided a comprehensive and extensive 

review of the report. Streamflow constraints used in the 

optimization model were compiled by Steve Loop 

(Arkansas Soil and Water Conservation Commission).

Study Area

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

model area, is 14,104 mi2, and includes all or part of 23 

counties north of the Arkansas River in Arkansas and 

all or part of 5 counties in southeastern Missouri. The 

active cells of the model cover the area north of the 

Arkansas River, west of the Mississippi River, east of 

the surficial exposure of Paleozoic-age formations (or 

Fall Line), and south of about 8 mi north of the Arkansas/Missouri State line encompassing a small part of 

southeastern Missouri (fig. 1).
Introduction 5

Figure 2. Irrigation project areas within the model area.

92° 91° 90°

34°

35°

36° Crowleys Ridge

Irrigation project areas

within the model area

Bayou Meto

L'Anguille

Grand Prairie

North Prairie

Bayou DeView

Little Red

Non-project area

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

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

EXPLANATION

Data from U.S. Army Corps of Engineers

and U.S. Fish and Wildlife Service

0 10 20 MILES

0 10 20 KILOMETERS
6 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

Irrigation Project Areas

Six irrigation project areas (fig. 2) have been 

established by various agencies for managing water 

resources within the alluvial aquifer (Ken Bright, U.S. 

Army Corps of Engineers, written commun., 2002; 

Jason Phillips, U.S. Fish and Wildlife Service, written 

commun., 2002). To meet the water needs of these 

areas, delivery of supplemental water from surfacewater sources is being considered by agencies such as 

the U.S. Army Corps of Engineers and the ASWCC. 

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

Although ground-water flow models had been 

developed previously for the Mississippi River Valley 

alluvial aquifer in northeastern Arkansas, these models 

were either at a scale that was too large to analyze the 

effects of projected ground-water withdrawals within 

the study area or were too limited in their areal extent. 

The flow model discussed in this report is based on the 

work of Reed (2003), which improved on previous 

models by simulating transient conditions to include 

observed hydraulic head from 1972 to 1998 (the previous model only included hydraulic-head observations 

to 1982). Characteristics for the flow model are listed 

in table 1. The flow model incorporates river, general 

head, no-flow, and areally distributed recharge boundary conditions. Flow from sources located outside the 

model area (such as from the underlying Sparta aquifer, 

or from the Interior Highlands to the west) is simulated 

by either general-head or areally-distributed-recharge 

boundary conditions. The distribution of boundary 

conditions specified in the model results in groundwater flow from recharge sources to areas with extensive ground-water pumping, and consequent widespread lowering of the water table. Water-level 

altitudes within the aquifer in spring 1998 are shown in 

figure 3, which shows substantial cones of depression 

resulting from sustained ground-water withdrawals. 

Both confined and unconfined conditions are simulated 

in the model. The model was divided into two layers 

of equal thickness with the lower layer having a larger 

hydraulic conductivity than the upper layer. This designation is consistent with observations of coarser, more 

transmissive sediments in the lower part of the aquifer. 

Model parameters were estimated in part using MODFLOW-2000 (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 (-0.46 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 

(4.9 ft) is the sum of the magnitudes of the difference at 

each observation, divided by the total number of observations; a value close to 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-meansquare difference (that is, +/-6.4 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 hydraulic-head values.

Table 1. Characteristics of the flow model (modified from 

Reed, 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 14,104 mi2

Cells with wells corresponding to 1997 withdrawals 9,979

Total pumpage in 1997 635,600,000 ft3/d

Average hydraulic conductivity 230 - 480 ft/d

Specific yield 0.30

Specific storage 1x10-6 ft-1

River cells 1,165

Hydraulic-head observations 1,698

Hydraulic-head observation periods 1972, 1982,

1992, 1998

Range in observed hydraulic head

values in 1998(feet above National

Geodetic Vertical Datum of 1929)

78 - 298 ft

Mean difference between observed and

simulated hydraulic head, all four periods 

-0.46 ft

Mean absolute difference between observed

and simulated hydraulic head, all four periods 

4.9 ft

Root-mean-square difference between observed and

simulated hydraulic head, all four periods 

6.4 ft
Conjunctive-Use Optimization 7

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

280

270

260

250

240

230

220

210

200

190

140

150

150

160

170

180

270

260

250 240

220

230

240

250

220

210

200

190

180

170

150

190

160

14

130

0

170

180

160

170

180

190

80

90

100

110

120

130

140

150

160

100

200

190

180

170

160

150

140

130

120

110

Tennessee

Mississippi

Missouri

Arkansas

92° 90°

34°

36°

EXPLANATION

Boundary of active model cells

POTENTIOMETRIC CONTOUR--

Shows altitude at which water

level would have stood in tightly

cased wells. Contour interval 10

feet. Datun is National Geodetic

Vertical Datum of 1929

180

0 10 20 MILES

0 10 20 KILOMETERS

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

Crowleys

Ridge
8 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

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

to evaluate the demand for ground water from the alluvial aquifer, which has increased steadily for the last 40 

years (Reed, 2003). The flow model showed that water 

is being withdrawn from the aquifer at rates that are 

much greater than what can be sustained for the long 

term. 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). 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

Figure 4. Flow chart of optimization modeling process.

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 
Conjunctive-Use Optimization 9

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. A 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 is ground-water flow, in cubic feet per day;

 is hydraulic conductivity, in feet per day;

 is the hydraulic gradient, dimensionless;

 is hydraulic head, in feet;

 is a distance over which the gradient is 

measured, in feet; and

 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 

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 9,979 

model cells corresponding to well locations and at 

1,165 river cells. 

Q K– dh

dl = -----A

Q

K

dh

dl

-----

h

l

A

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

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

constraint location c, in feet; and

hminimum is the water-level altitude at half the 

thickness of the aquifer, in feet. 

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 surfacewater 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 

q ∑ well q ∑ river +

q ∑ well

q ∑ river

hc hminimum ≥

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
Conjunctive-Use Optimization 11

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

92° 91° 90°

34°

35°

36°

0 10 20 MILES

0 10 20 KILOMETERS

Thickness of aquifer

below the hydraulichead constraint, in feet

30

31 - 40

41 - 50

51 - 60

61 - 70

71 - 80

81 - 90

91 - 100

Constraint location

Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

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

EXPLANATION
12 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

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. 

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 9,979 

ground-water withdrawal cells. 

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 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 at all river cells. 

Predevelopment Recharge From Non-River Sources

Optimal pumping rates are affected by the rate of 

recharge from non-river sources explicitly specified in 

the flow model because these non-river sources can be 

a source of water to wells instead of the rivers. 

Recharge from non-river sources corresponds to 

recharge from underlying hydrogeologic units such as 

the Sparta aquifer or from the Interior Highlands west 

and north of the modeled area. Recharge to the aquifer 

from non-river sources will vary with time as groundwater withdrawals reduce the hydraulic head in the 

alluvial aquifer (fig. 6). Initially, flow is induced from 

underlying and adjacent units. As the hydraulic head 

within the aquifer is lowered because of pumping, 

increased flow from underlying and adjacent units 

0 qwelli Mqwell 1997≤ ≤

qwelli

M

qwell 1997

Figure 6. Hypothetical variation in recharge to the alluvial aquifer from non-river sources as a function of time.

Predevelopment 

Induced flow 

from underlying 

and adjacent 

units begins 

Water released from storage 

from underlying and adjacent 

units begins to decline; 

hydraulic gradient declines 

until equilibrium is reached 

Recharge 

From 

Non-River 

Sources 

Equilibrium 

reached 

Time 
Conjunctive-Use Optimization 13

occurs as water is released from storage. A point is 

reached when the gradient between the aquifer and the 

non-river sources equilibrates, and the change in storage is zero. At this point, recharge to the system is at 

predevelopment rates, which can be attributed to the 

long-term recharge from precipitation from distant 

sources. This is important because the recharge from 

non-river sources is specified explicitly in the model 

and is allowed to vary with time. Therefore, specifying 

recharge from non-river sources associated with predevelopment conditions for steady-state simulations is 

appropriate for obtaining estimates of sustainable 

yield.

Wells Used in Optimization Model

For optimization, 9,979 one-square mile cells 

were used to represent pumping from 35,043 wells in 

1997. Each cell was specified as a managed well (that 

is, a decision variable) within MODMAN. In 1997, the 

annual pumping rate for all wells was 635.6 Mft3

/d. 

Note that in the model of Reed (2003), dry cells 

occurred causing pumping wells at the dry cells to 

become inactive, reducing total pumping to 631

Mft3/d. For the sustainable-yield analysis, the optimized rate at each of the 9,979 cells was allowed to 

vary between a rate of zero to a maximum rate equal to 

a multiple between 1 and 2 to that which was pumped 

in 1997. An upper limit was specified because no limit 

on pumping led to unrealistic optimal withdrawal from 

wells adjacent to rivers.

Streamflow

To allow for both the optimal conjunctive-use of 

surface water and ground water within the optimization 

model, 11 rivers were specified (table 2). Of the 11 rivers specified, 7 have streamflow constraints specified at 

each river cell based on 7-day, 10-year-recurrence low 

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

Conservation Commission, written commun., 2001), 

which are derived from historical streamflow for the 

rivers. Where a constraint was not provided by 

ASWCC, an arbitrary value of zero was specified, 

except in the case of the Mississippi River where a 

value of 50 billion cubic ft per day was specified. By 

specifying a minimum flow constraint based on 7Q10 

data, available streamflow within the optimization 

model would be limited all year long to an amount 

equal to or greater than 7Q10, although 7Q10 data 

reflect a statistically low flow that occurs only once 

every 10 years, and then for only 7 consecutive days. 

Streamflow constraints were specified at every river 

cell (1,165 total) to facilitate the calculation of optimized streamflow withdrawals (fig. 7). It should be 

noted that flow constraints based on 7Q10 flows are 

only one criterion that could be selected, the results 

from which reflect one specific application of the optimization model. 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 up stream from that point. Overland flow (that is, 

surface-water runoff that would enter river cells from 

minor tributaries or sheet flow that were 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 

reach, as measured between the upstream and downstream ends of the 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. Included in 

these diversions were the planned diversions of 

63,339,248 ft3

/d for the Bayou Meto project area and 

55,078,367 ft3

/d for the Grand Prairie project, which 

factor in an additional 30 and 40 percent transmission 

loss area for deliveries to the project areas, respectively.

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. For 

this model, ground-water levels were not allowed to 

drop below half the thickness of the aquifer or 30 ft 

above the bottom of the aquifer, whichever resulted in 

the higher ground-water level. Streamflow was not 

allowed to drop below a minimum amount specified in 

table 2. The optimization model was used to obtain 

estimates of sustainable yield at 9,979 ground-water 

and 1,165 streamflow withdrawal cells. 
14 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

Figure 7. Location of streams within model showing cells and rates at which water could be withdrawn and still meet 

constraints within optimization model. The upper limit for well withdrawals was set at 100 percent of the 1997 withdrawal rate.

92° 91° 90°

34°

35°

36°

Optimized streamflow withdrawal,

cubic feet per day

0 - 999,999

Greater than 0 - 999,999

1,000,000 - 9,999,999

10,000,000 - 99,999,999

100,000,000 - 999,999,999

1,000,000,000 - 4,896,900,000 Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

0 10 20 MILES

0 10 20 KILOMETERS

Current

River

Black

River Cache

River

St. Francis

River

Right

Hand

Chute of

Little River

Little Red

River

White

River

Bayou

Meto

L'Anguille

River

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

EXPLANATION
Conjunctive-Use Optimization 15

Table 2. Rivers, streamflows, and streamflow constraints

[ASWCC, Arkansas Soil and Water Conservation Commission; flow constraint from ASWCC based on an annual minimum 7-consecutive-day

average flow with a recurrence interval of 10 years; ft3/d, cubic foot per day]

River name

Number of

model cells

Flow into

uppermost

river cell

of model

(ft3/d)

Overland flow

per river cell

(ft3/d)

Overland flow

per river reach

(ft3/d)

Flow

constraint

(ft3/d)

Source for

value of

constraint

Arkansas 97 4,903,200,000 1,000,000 97,000,000 100,224,000 ASWCC

Bayou Meto 77 17,020,800 1,000,000 77,000,000 605,000 ASWCC

Black 88 148,996,800 8,000,000 704,000,000 27,302,400 ASWCC

Cache 105 50,328,000 2,000,000 210,000,000 950,400 ASWCC

Current 31 280,886,400 2,000,000 62,000,000 0 Arbitrary

L’Anguille 54 21,556,800 2,000,000 108,000,000 3,974,400 ASWCC

Little Red 15 247,017,600 1,400,000 21,000,000 0 Arbitrary

Mississippi 305 50,185,440,000 3,000,000 915,000,000 50,000,000,000 Arbitrary

Right Hand Chute 74 244,944,000 1,000,000 74,000,000 0 Arbitrary

St. Francis 169 231,552,000 12,000,000 2,028,000,000 7,257,600 ASWCC

White 150 1,248,480,000 25,000,000 3,750,000,000 665,000,000 ASWCC

Total 1,165 57,579,422,400 -- 8,046,000,0001

1Summation assumes that overland flow is applied at river cell.

50,805,313,800 --

Because sustainable yield from ground water is a 

function of the pumping limit specified for each managed well, multiples of the rate withdrawn in 1997 were 

used to set the upper limit of pumping. The distribution of optimal withdrawal rates using upper limits 

specified as 100-, 150-, and 200-percent multiples of 

the 1997 withdrawal rates are shown in figures 8 

through 10. The distribution of withdrawal rates for 

each of these scenarios is such that most wells are 

either withdrawing water at a rate equal to the upper 

limit or at a rate of zero. 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 actually decreases 

(although the total amount withdrawn increases), with 

those capable of withdrawing water being nearest to 

sources of water within the model (that is, major rivers). 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 

sources of water within the model and at rates that are 

orders of magnitude higher than were pumped in 1997. 

Although overall optimized withdrawal would be largest for such a scenario, the distribution of wells would 

be unacceptable from a management standpoint 

because nearly all of the water production would come 

from wells that are adjacent to rivers, with the remaining interior wells being unable to pump at all. 

For the optimization run in which an upper withdrawal limit of 100 percent of the 1997 withdrawal rate 

was specified (scenario 1; fig. 8), the sustainable yield 

from ground water for the entire study area is 360

Mft3/d (table 3), which is only about 57 percent of the 

amount withdrawn in 1997 (635.6 Mft3

/d). If the upper 

withdrawal limit is increased to 150 percent of the 1997 

withdrawal rate (scenario 2; fig. 9), the sustainable 

yield from ground water for the entire study area is 445 

Mft3

/d (table 3), which is about 70 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. 10), the sustainable yield from 

ground water for the entire study area is 526 Mft3

/d 

(table 3), which is about 83 percent of the amount withdrawn in 1997.
16 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

Figure 8. Location 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° 90°

35°

36°

34°

0 10 20 MILES

0 10 20 KILOMETERS

Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

White

River

Ratio of optimal groundwater withdrawal to the

amount withdrawn in

1997

0

0.5

0.75

1

Hydraulic head at

half the aquifer

thickness

No wells in cell

GRAND PRAIRIE

EXPLANATION
Conjunctive-Use Optimization 17

Table 3. Sustainable yield and unmet demand for different upper limits on withdrawals and different demand rates[Negative unmet demand values indicate surplus water availability. All values are in million cubic feet per day]

County

1997

withdrawal

rate

(Mft3/d)

Sustainable yield based on

 an upper withdrawal limit of:

Unmet ground-water demand (baseline

 rate minus sustainable yield) based on

1997 demand and a sustainable yield from:

Unmet ground-water de

 rate minus sustainable 

 sustainable yields from 

and 3, respectively, and 

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 percent

of 1997

 withdrawal

 rate

150 percen

of 1997

 withdrawa

 rate

Arkansas 51.4 24.0 27.0 29.6 27.5 24.4 21.9 27.5 50.1 Butler 4.4 4.3 6.5 8.6 0.1 -2.1 -4.2 0.1 0.1 Clay 31.4 31.0 43.4 54.9 0.3 -12.0 -23.5 0.3 3.7 Coahoma 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Craighead 49.4 26.3 30.4 34.4 23.2 19.1 15.1 23.2 43.8 Crittenden 17.3 15.1 16.9 18.4 2.2 0.4 -1.1 2.2 9.1 Cross 32.2 19.2 22.6 24.2 13.0 9.6 8.0 13.0 25.7 Desha 1.4 1.4 2.0 2.7 0.0 -0.7 -1.4 0.0 0.0 Dunklin 5.1 5.0 7.5 10.0 0.1 -2.4 -4.9 0.1 0.2 Dyer 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Greene 20.3 14.9 14.0 16.4 5.3 6.3 3.8 5.3 16.4 Independence 3.3 2.4 3.3 4.3 1.0 0.0 -1.0 1.0 1.7 Jackson 37.2 2.5 3.8 5.1 34.7 33.4 32.2 34.7 52.1 Jefferson 34.6 24.4 32.6 39.3 10.1 1.9 -4.7 10.1 19.2 Lake 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Lawrence 40.1 24.2 30.4 35.4 15.9 9.7 4.7 15.9 29.8 Lee 22.5 10.3 12.2 10.4 12.2 10.3 12.1 12.2 21.6 Lincoln 0.7 0.7 1.0 1.4 0.0 -0.3 -0.6 0.0 0.1 Lonoke 42.1 17.7 17.3 16.7 24.4 24.8 25.4 24.4 45.9 Mississippi 17.5 17.5 25.1 31.5 0.0 -7.5 -13.9 0.0 1.3 Monroe 28.3 20.8 22.0 30.0 7.5 6.3 -1.7 7.5 20.5 New Madrid 4.8 4.8 7.2 9.5 0.1 -2.3 -4.7 0.1 0.1 Pemiscot 3.9 3.9 5.8 7.7 0.1 -1.8 -3.8 0.1 0.1 Phillips 19.6 3.5 4.3 4.8 16.1 15.3 14.8 16.1 25.1 Poinsett 57.4 29.0 41.3 48.9 28.4 16.1 8.5 28.4 44.9 Prairie 27.3 13.8 16.2 18.9 13.5 11.1 8.4 13.5 24.7 Pulaski 3.1 3.1 4.7 6.3 0.0 -1.6 -3.1 0.0 0.0 Randolph 8.7 7.9 10.7 12.6 0.8 -2.0 -4.0 0.8 2.3 Ripley 0.2 0.2 0.3 0.4 0.0 -0.1 -0.2 0.0 0.0 Shelby 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.0 0.0 St. Francis 27.2 9.7 9.9 11.5 17.4 17.2 15.6 17.4 30.8 Tipton 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 White 8.5 5.8 7.0 9.5 2.7 1.5 -0.9 2.7 5.8 Woodruff 35.5 16.7 19.4 22.2 18.8 16.1 13.3 18.8 33.9 Total 635.7 360.3 444.9 525.8 275.5 190.9 110.0 275.5 508.8 
18 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

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 150 percent of the 1997 withdrawal rate.

92° 91° 90°

34°

35°

36° Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

0 10 20 MILES

0 10 20 KILOMETERS

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

Ratio of optimal groundwater withdrawal to the

amount withdrawn in

1997

0

0.5

0.75

1

1.5

Hydraulic head at

half the aquifer

thickness

No well in cell

White

River

EXPLANATION

GRAND PRAIRIE
Conjunctive-Use Optimization 19

Figure 10. 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° 90°

34°

35°

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

0 10 20 MILES

0 10 20 KILOMETERS

Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

36°

Ratio of optimal groundwater withdrawal to the

amount withdrawn in

1997

0

0.5

0.75

1

1.5

2

Hydraulic head at

half the aquifer

thickness

No well in cell

White

River

EXPLANATION

GRAND PRAIRIE
20 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

Total sustainable yield from streamflow is 12.8, 

12.7, and 12.6 billion ft3/d for scenarios 1, 2, and 3, 

respectively (table 4), which is on the order of 20 times 

larger than the sustainable yield from ground water. 

Sustainable yield for all rivers ranged from 0 (Current 

and Little Red Rivers, and Bayou Meto) to almost 5 billion ft3/d for the Arkansas River. These rates decrease 

as ground-water discharge to these streams decrease as 

the upper limit of ground-water withdrawal increases. 

Nonetheless, these large sustainable yields represent a 

potential source of water to supplement ground water 

and meet overall water demand, but to do so will 

require the construction of withdrawal and distribution 

facilities, which will have legal, political, economic, 

and social consequences.

Hydraulic-head constraints restrict where and 

how much ground water and, to a lesser extent, surface 

water can be extracted. The red triangles in figure 8 

through 10 show the locations where the simulated 

value of hydraulic head, derived from the optimization 

model, reached the lower limits. In the Grand Prairie 

area, the red triangles delimit the boundary between 

cells that can produce water and those that cannot 

resulting from the limits imposed by the hydraulichead constraints. Between Crowleys Ridge and the 

White River, a few points constrain withdrawals over 

large areas. 

Wells in the western half of the model area, for 

which an optimized rate of zero was calculated, lie 

partly between the Arkansas and White Rivers (fig. 8). 

The Grand Prairie area of the model, which lies 

between these two rivers, contains parts of two U.S. 

Army Corps of Engineers irrigation project areas: 

Bayou Meto and Grand Prairie (fig. 2). The locations of 

these project areas are consistent with current and 

anticipated water needs, and the inability of the aquifer 

to supply water sustainably as demonstrated by the 

optimization-model results (figs. 8 through 10). Optimal sustainable yields from within the Bayou Meto irrigation project area and within the Grand Prairie 

irrigation project area are 18.1 and 9.1 Mft3/d, respectively, assuming a maximum allowable withdrawal rate 

equal to 1997 rates. These values of sustainable yield 

represent 35 and 30 percent respectively of the amount 

pumped from these project areas in 1997.

Table 4. Optimized total streamflow withdrawals from the optimization model

River name

Sum of optimized total streamflow withdrawal

(cubic foot per day)

100 percent of

baseline rate

(scenario 1)

150 percent of

baseline rate

(scenario 2)

200 percent of

baseline rate

(scenario 3)

Arkansas 4,949,237,300 4,939,029,200 4,930,806,700

Bayou Meto 0 0 0

Black 1,401,763,000 1,386,626,000 1,370,421,000

Cache 225,434,500 222,633,400 213,770,000

Current 0 0 0

L’Anguille 123,381,780 122,834,070 123,086,270

Little Red 0 0 0

Mississippi 1,110,849,200 1,102,986,600 1,096,987,400

Right Hand Chute 323,995,740 314,608,571 306,391,523

St. Francis 2,036,571,700 2,014,040,840 2,001,651,420

White 2,635,151,400 2,627,994,800 2,613,921,400

Total 12,806,384,620 12,730,753,481 12,657,035,713

Difference from scenario 1 -- -75,631,139 -149,348,907
Conjunctive-Use Optimization 21

The distribution of hydraulic-head constraint 

points has a substantial effect on the distribution and 

amount of optimal pumping. Test simulations with the 

model were conducted to compare the sustainable yield 

derived from a model simulation with hydraulic-head 

constraints only in place in the Bayou Meto and Grand 

Prairie irrigation project areas. The reason that this 

configuration of constraints may be of interest is that 

imposition of pumping restrictions within a Critical 

Ground-Water Area is possible only if an alternate 

source of water is available, which would be the case in 

the Bayou Meto and Grand Prairie irrigation project 

areas. Table 5 shows a comparison of sustainable yields 

obtained using that constraint configuration with one 

that has constraints assigned throughout the model 

area. Removal of constraints outside the project areas 

allows substantially more water to be pumped in those 

areas without constraints. However, doing so allows 

water levels outside the project areas to drop below half 

the thickness of the aquifer, violating Critical GroundWater Area requirements. 

Additional testing with the optimization model 

showed that specification of non-zero lower limits on 

withdrawals at every pumping well in the model area 

led to an infeasible solution, regardless of the value of 

the lower limit that was specified. This was done to test 

if at least some water could be produced in those areas 

for which an optimal withdrawal rate was zero. 

Attempts at further subdividing areas, where some of 

the cells were specified with a non-zero lower limit on 

withdrawals (for example, applying a lower limit of 10 

percent of the rate pumped in 1997 only to cells in the 

Bayou Meto or Grand Prairie project areas), also 

resulted in infeasible solutions. However, an exhaustive 

application of this approach to areas with optimized 

pumping rates of zero (figs. 8 through 10) was not 

done.

A test was done to evaluate the effect of optimizing ground-water withdrawals without optimizing 

withdrawals from streams, while still maintaining the 

minimum streamflow constraints. Sustainable yield 

from ground water (with a maximum limit of groundwater withdrawals set at 100-percent of the 1997 rate) 

for all wells increased by 13.6 percent to 409 Mft3

/d.

Unmet Demand

Unmet demand is defined 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 anticipated demand, in cubic feet per 

day; and 

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

For example, if the anticipated demand is 635 Mft3/d 

(the amount withdrawn in 1997), and the sustainable 

yield is calculated to be 360 Mft3/d, the unmet demand 

is the difference of these two values, or 275 Mft3

/d. 

Note that the sustainable yield is an independent calculation based on model results that is not affected by the 

demand. Therefore, unmet demand is not solely a function of the sustainable yield.

Although none of the three scenarios that were 

considered provided a sustainable yield for the entire 

model area that met the 1997 demand, sustainable yield 

for some counties did meet or exceed the demand (table 

3). These counties (Coahoma, Desha, Dyer, Lake, Mississippi, Pulaski, Shelby, and Tipton) tended to have the 

lowest withdrawal rates specified in the flow model and 

also are located near a large river. Mississippi County 

had the largest 1997-withdrawal rate of those counties 

whose demand rate was met and exceeded by sustainable yield. Arkansas, Jackson, Lonoke, and Poinsett 

Counties consistently have the largest unmet demand 

for the different scenarios considered, although the 

ranking for these four counties changes depending on 

the scenario. 

Unmet demand was tallied for the six irrigation 

project areas (table 6) based on different upper limits 

Table 5. Comparison of sustainable yield obtained for

hydraulic-head constraints everywhere in the model area

and only in the Bayou Meto and Grand Prairie irrigation 

project areas

[Limit on maximum pumping for both cases was set at 1997 pumping 

rates]

Irrigation project

area

Sustainable yield

(million cubic feet per day)

Hydraulic-head

 constraints

everywhere

Hydraulic-head

constraints only

in project areas

Bayou Meto 18.1 17.9

Grand Prairie 9.1 5.0

Outside of Bayou Meto and

and Grand Prairie project

areas

 333 434 

U = D – S
22 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas Table 6. Sustainable yield and unmet demand relative to 1997 withdrawal rates in six irrigation project areas and the remainder of th

[mi2, square mile; ft3/d, cubic foot per day] Project area Area (mi2) Number of model pumping cells Baseline 1997 withdrawal rate (ft3/d) Sustainable yield based on an upper withdrawal limit of: Unmet ground-water demand based on 1997 demand and a sustainable yield from: 

100 percent of baseline rate (scenario 1) 150 percent of baseline rate (scenario 2) 200 percent of baseline rate (scenario 3) Scenario 1 Scenario 2 Scenario 3 

Bayou Meto 661 601 52,014,938 18,098,444 17,244,950 15,847,078 33,916,494 34,769,988 36,167,860 Grand Prairie 570 432 30,196,068 9,120,136 10,219,743 10,740,002 21,075,932 19,976,325 19,456,066 North Prairie 176 137 9,507,588 5,379,767 5,208,230 6,849,813 4,127,821 4,299,358 2,657,775 Little Red River 129 103 5,433,200 2,819,435 2,866,419 3,231,044 2,613,765 2,566,781 2,202,156 L’Anguille River 241 215 26,778,165 10,366,204 12,377,300 13,032,644 16,411,961 14,400,865 13,745,521 Bayou Deview 188 169 21,062,884 1,677,939 928,016 1,237,360 19,384,945 20,134,868 19,825,524 Non-project areas 12,139 8,322 490,788,396 312,808,687 396,044,423 474,880,308 177,979,709 94,743,973 15,908,088 Total 14,104 9,979 635,781,239 360,270,612 444,889,081 525,818,247 275,510,627 190,892,158 109,962,992 
Conjunctive-Use Optimization 23

on pumping from scenarios 1, 2, and 3, and different 

demand rates. A comparison of the sustainable yield 

and the 1997 withdrawal rates within the six irrigation 

project areas is listed in table 6. Sustainable yield is 

substantially less in all project areas than the rate withdrawn in 1997, resulting in unmet demand ranging 

from 2.2 to 88.1 Mft3

/d for all the project areas. Bayou 

Meto project area consistently has the largest unmet 

demand of all the project areas, followed by the Grand 

Prairie project area. However, unmet demand for nonproject areas was considerably larger for scenarios 1 

and 2. 

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 resulting in sustainable yield 

(fig. 11). Because the 1997 withdrawal rates are unsustainable, large areas of the model area are dry, particularly in the area between the Arkansas and White 

Rivers (the area that includes the Bayou Meto and 

Grand Prairie irrigation project areas). In contrast, the 

optimal hydraulic-head altitude using sustainable yield 

shows a gradual decline from the rivers to the troughs 

caused by ground-water withdrawals.

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 values from the flow model derived 

using sustainable yield, to the altitudes corresponding 

to the hydraulic-head constraints specified in the optimization model. Such a comparison is provided in figure 12, which shows the cumulative percentage of 

model cells with values less than or equal to the difference between simulated hydraulic head and the altitude 

corresponding to half the aquifer thickness. Values to 

the left of zero represent cells with hydraulic head 

below half the aquifer thickness and represent less than 

5 percent of the total. The comparison was made for 

data sets consisting of the 2,804 hydraulic-head constraint points and for all of the active model cells. There 

is no appreciable difference. The comparison indicates 

that (1) additional hydraulic-head constraint locations 

would have had little effect on the simulation results as 

a whole; and (2) the optimized pumping distribution is 

a good approximation of sustainable yield, despite nonlinear behavior inherent in the model. 

Limitations

The values of sustainable yield should be considered maximum rates, in that head constraints are violated in some areas because of non-linear 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 where this 

occurred. From a management standpoint, however, the 

values might be considered to be conservative because 

they apply to steady-state conditions that will not be 

reached for possibly hundreds of years. 

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. 

The spatial distribution of the difference between 

simulated hydraulic head and the altitude corresponding to half the aquifer thickness is shown in figure 13. 

Over the vast majority of the model area, simulated 

hydraulic head is at or above the constraint. Those areas 

where this is not the case occur where large changes in 

saturated thickness have occurred when compared to 

predevelopment conditions, which has caused the largest change in aquifer transmissivity. This is particularly 

true in the Grand Prairie area. 
24 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

Figure 11. Simulated hydraulic head at steady state using (A) 1997 withdrawal rates; and (B) sustainable yield.

92° 91° 90°

4°5°6°

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

0 10 20 MILES

0 10 20 KILOMETERS

47 - 6061 - 8081 - 100101 - 120121 - 140141 - 160161 - 180181 - 200201 - 220221 - 240241 - 260261 - 280281 - 300301 - 320

Dry cell

Simulated hydraulichead, in feet aboveNGVD of 1929

Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

EXPLANATION

White

River

A

92° 91° 

34°35°36°

47 - 6061 - 8081 - 100101 - 120121 - 140141 - 160161 - 180181 - 200201 - 220221 - 240241 - 260261 - 280281 - 300301 - 320

Dry cell

Simulated hydraulichead, in feet aboveNGVD of 1929

Crowl

FALL

LINE

FALL

LINE

Arkansas

River

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

White

River

EXPLANATION
Conjunctive-Use Optimization 25

Figure 12. Cumulative percentage of model cells with values less than or equal to the difference between simulated hydraulic 

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

Figure 13. Difference between simulated hydraulic head and altitude of half the aquifer thickness.

92° 91° 90°

34°

35°

36° Crowleys Ridge

FALL

LINE

FALL

LINE

Mississippi

River

Arkansas

River

0 10 20 MILES

0 10 20 KILOMETERS

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

100 - 120

81 - 100

61 - 80

41 - 60

21 - 40

1 - 20

-4 - 0

-9 - -5

-14 - -10

-19 - -15

-40 - -20

Dry cell

Difference between

simulated hydraulic

head and altitude of

half the aquifer

thickness, in feet

EXPLANATION
Summary 27

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 over 100 ft deep have formed 

in the potentiometric surface, resulting in lower well 

yields and degraded water quality in some areas. Several counties in the Grand Prairie area, 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 14,104 mi2, 

extending northeast from the Arkansas River into the 

northeast corner of Arkansas and parts of southeast 

Missouri. 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 ground-water area 

designation, the U.S. Geological Survey, in cooperation with the Arkansas Soil and Water Conservation 

Commission, and Memphis 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 northeastern Arkansas. 

Conjunctive-use optimization modeling is a technique 

that simulates 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 the optimization model described 

in this report is to: (1) obtain maximum withdrawal 

rates from model cells at which ground-water withdrawals would have occurred in 1997; (2) obtain maximum withdrawal rates from model cells at stream 

locations; (3) maintain ground-water levels at or above 

specified levels; and (4) maintain streamflow at or 

above specified rates. A management problem was formulated as one of maximizing the sustainable yield 

from all ground-water 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 (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). 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 hydraulichead and streamflow constraints, and formulated as a 

linear program in mathematical programming system 

format. 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. For obtaining 

estimates of sustainable yield, specified recharge was 

assumed to be the same as for predevelopment conditions. Optimal sustainable yield is a function of limits 

assigned to ground-water withdrawals, which in this 

report were set to multiples of 100, 150, and 200 percent of 1997 rates. Some areas represented in the model 

likely could pump at rates higher than those in 1997. 

However, if no limit is placed on withdrawals, the 

majority of water production will be from wells near 

model water sources such as rivers or general-head 

boundaries, depriving wells of water that are distant 

from these sources of water. The maximum potential 

withdrawal from wells was limited to twice the amount 

withdrawn in 1997 because even at the 1997 rate, many 

areas within the model went dry. No limit on withdrawals led to unrealistic optimal withdrawal from wells 

adjacent to rivers and most interior withdrawal cells 

having no withdrawal.

Within the optimization model, 11 rivers are 

specified. Surface-water diversion rates that occurred 

in 2000 were subtracted from specified overland flow 

at the appropriate river cells. Included in these diversions were the planned diversions of 63,339,248 ft3/d 

for the Bayou Meto project area and 55,078,367 ft3/d 

for the Grand Prairie project area, which factor in an 

additional 30 and 40 percent transmission loss, respectively. Streamflow constraints were specified at all 

1,165 river cells based on average 7-day minimum 

flows for 10 years. Sustainable yield for all rivers 

ranged from 0 (Current, Little Red, and Bayou Meto 
28 Conjunctive-Use Optimization Model of the Mississippi River Valley Alluvial Aquifer of Northeastern Arkansas

Rivers) to almost 5 billion ft3/d for the Arkansas River. 

Total sustainable yield from all rivers combined was 

12.8 billion ft3

/d. Nonetheless, these large sustainable 

yields represent a potential source of water to supplement ground water and meet overall water demand, but 

to do so will require the construction of withdrawal and 

distribution facilities, which will have legal, political, 

economic, and social consequences.

Sustainable-yield estimates are affected by the 

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

allowed to vary up to 200 percent of the withdrawal rate 

in 1997. As the overall upper limit is increased, the sustainable yield generally increases because wells closer 

to water sources can produce more water. 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 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 is also unlikely. The sustainable 

yield from ground water for the entire study area with 

the maximum upper limit set as the amount withdrawn 

in 1997 is 360 Mft3

/d, which is only about 57 percent 

of the amount withdrawn in 1997 (635.6 Mft3

/d). Optimal sustainable yields from within the Bayou Meto irrigation project area and within the Grand Prairie 

irrigation project area are 18.1 and 9.1 Mft3/d, respectively, assuming a maximum allowable withdrawal rate 

equal to 1997 rates. These values of sustainable yield 

represent 35 and 30 percent respectively of the amount 

pumped from these project areas in 1997.

Unmet demand (defined as the difference 

between the optimized withdrawal rate, or sustainable 

yield, and the anticipated demand) was calculated 

using different demand rates based on multiples of the 

1997-withdrawal rate. Sustainable-yield values were 

based on upper limits of ground-water withdrawals set 

at 100, 150, and 200 percent of the 1997 withdrawal 

rate. 

Hydraulic-head constraints have the largest 

effect on sustainable yield estimates in the areas in 

which the hydraulic-head constraints are applied. 

Removal of head constraint outside the Bayou Meto 

and Grand Prairie irrigation project areas had a much 

smaller effect on those areas than in the areas distant 

from them. 

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 

values to hydraulic-head constraints. In 95 percent of 

the model area, application of the sustainable-yield 

rates resulted in hydraulic heads that were above the 

constraint values. In those areas where this was not the 

case, 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, largely because of changes in transmissivity 

resulting from substantial change in saturated thickness 

between the starting hydraulic-head distribution used 

in the optimization model and hydraulic head for 

steady-state conditions under optimized sustainable 

yield. 

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.
References Cited 29

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