• TABLE OF CONTENTS
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 Title Page
 Executive summary
 Table of Contents
 Introduction
 Review of current literature
 Ground water simulation models
 Water supply needs and sources
 Ground water optimization...
 Summary and conclusions
 Appendices
 References














Title: Water resource allocation and quality optimization modeling: final report
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Publication Date: 1995
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Table of Contents
    Title Page
        Title Page
    Executive summary
        Page i
        Page ii
        Page iii
        Page iv
        Page v
        Page vi
        Page vii
        Page viii
    Table of Contents
        Page ix
        Page x
        Page xi
        Page xii
        Page xiii
        Page xiv
        Page xv
        Page xvi
        Page xvii
        Page xviii
        Page xix
        Page xx
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
    Review of current literature
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
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    Ground water simulation models
        Page 44
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    Water supply needs and sources
        Page 55
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    Ground water optimization models
        Page 69
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    Summary and conclusions
        Page 177
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    Appendices
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    References
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Full Text








Special Publication SJ96-SP5


WATER RESOURCE ALLOCATION AND QUALITY OPTIMIZATION MODELING


FINAL REPORT


July 5, 1995





Prepared for

Saint Johns River Water Management District
P.O. Box 1429
Palatka, Florida 32178-1429








Patrick C. Burger Graduate Research Assistant
Mark E. Holland Graduate Research Assistant


Kirk Hatfield, Ph.D. Principal Investigator
Donald W. Hearn, Ph.D. Co-Principal Investigator
Wendy Graham, Ph.D. Co-Principal Investigator


Department of Civil Engineering
University of Florida
Gainesville, Florida 32611














EXECUTIVE SUMMARY


PROJECT SCOPE

Due to the increasing demands placed on Florida's water resources, the state of

Florida adopted legislation in 1990 to improve water resource management and to direct

future growth through planning programs. This legislation requires each Water

Management District to completely evaluate their water needs and sources through the

year 2010 and delineate critical areas identified as water resource problems. Once

completed, Districts were expected to develop possible alternative water supply scenarios

which avoid adverse or otherwise unacceptable changes in the environment or the

availability of water.

This report presents systematic modeling methods for determining optimum water

supply strategies that satisfy various environmental and hydrological requirements. Five

site specific water resource allocation optimization models were developed for Volusia

County, Florida and were executed to investigate a variety of management objectives.

These optimization models incorporate both water quantity and quality aspects of water

resource management to determine optimum ground water allocation strategies which

satisfy future water service demands and minimize adverse environmental impacts at

specified areas. These areas include sensitive wetlands where projected water table

declines are predicted to induce a high level of vegetative harm and well fields where

excessive withdrawal is predicted to cause a degradation in water quality due to salt-

water intrusion or upcoming.









The five optimization models were designed to elucidate water resource allocation

strategies for water service areas that would:

1) Satisfy the water demands of both municipal and agricultural water demands.

2) Explore development of both existing and proposed ground water supply

areas.

3) Select wastewater effluent as a feasible supply to supplement agricultural

demands.

Water quality aspects were incorporated in two models by constraining chloride

concentrations changes at wells while simultaneously minimizing the maximum

drawdown at sensitive wetland areas in one model and by minimizing relative chloride

concentration increases at wells in a second model.

The optimization models were a product of a research project that sought to

accomplish two goals. The first was to obtain a broad understanding of numerical

optimization modeling and its recent applications to ground water resource management.

The second goal was to demonstrate optimum resource allocation modeling at a selected

site that would satisfy specified environmental and hydrological requirements. In order

to achieve the established two goals and fulfill the scope of the project, the following

three objectives were accomplished:

1) Review of current literature of numerical optimization modeling with respect

to ground water resource management.

2) Construct site specific optimization models capable of generating water

resource allocation scenarios which satisfy projected demands and environmental

constraints for the year 2010.









3) Generate and summarize various optimum water resource allocation scenarios

to meet specific objectives.

LITERATURE REVIEW

To meet the first objective a literature review was conducted to investigate

applications of ground water resource management where tools of optimization were

applied to demonstrate optimum scenarios of resource allocation. Chapter 2 presents a

review of various water management models; a number of existing ground water

optimization models were summarized in tabular form. Based on knowledge acquired

through the literature review and the size and complexity of the problem, it was

determined that the most efficient combined simulation/optimization model would be one

developed using of the unit "response matrix" technique. This method consists

incorporating a matrix of coefficients which represent the response of ground water to

a specified change in withdrawal rate.

PROJECT SIMULATION MODELS

Chapter 3 presents details on the three-dimensional ground water flow and solute

transport models used to generate the steady-state ground water system responses (i.e.,

predicted head values, solute concentrations, etc.) to various stresses (i.e., specified

pumping and recharge rates) in the study area; these system responses are generated as

needed to create response matrices that are later used to construct the larger optimization

models. The study area includes most of Volusia County where in order to alleviate the

problem of salt-water intrusion and satisfy water service demands, the general trend since

the 1950's has been to locate additional wells to the west toward central Volusia County.

The flow simulation model as originally acquired from the SJRWMD simulates the









aquifer systems of Volusia County using MODFLOW (McDonald and Harbaugh 1988).

This regional simulation model was revised to simulate flow in the study area. The

transport model for the same study area was also developed by the District using the

program DSTRAM (Huyakorn and Panday 1991).

WATER SUPPLY NEEDS AND SOURCES

Chapter 4 presents details on water supply needs and sources as currently viewed

by SJRWMD under the District Water Management Plan. Projections of future water

use for the year 2010 are presented that were based on historical trends, local

government comprehensive plans, and direct communication with both public and private

public supply utilities (SJRWMD, 1992). Major components of the water uses categories

include municipal and agricultural service area demands. Some of the larger projected

municipal demand increases include an increase in the Port Orange service area in excess

of 70 percent of the current use, an increase in Daytona Beach service area of 62

percent, and an increase of 400 percent over current uses in the Smyrna and Samsula

areas. The overall demand in the Volusia County area is projected to increased by

approximately 75 percent from 1988 to 2010. Also included in Chapter 4 are maps and

tables that depict wastewater treatment plant information used to incorporate water reuse

in the development of the ground water management models.

GROUND WATER OPTIMIZATION MODELS

Chapter 5 presents the five aquifer optimization models developed to investigate

optimal allocation of ground water to meet year 2010 demand in Volusia County project

area. These models were formulated to investigate future water allocation strategies

assuming feasible withdrawal scenarios must meet or exceed projected water service









areas demands and not exceed available water supplies. It was assumed with these

models that adverse environmental effects could be minimized at specific locations by

constraining pressure head changes (i.e., drawdown) to meet specified environmental

goals or standards. These models are essentially numerical optimization models

comprised of an objective function, decision variables, and constraints. Of the five

optimization models, three models dealt strictly with hydraulic constraints and two

models incorporated both hydraulic and water quality constraints. Objective functions

included the following: 1) minimizing the maximum drawdown value, 2) maximizing

the minimum pressure head, 3) minimizing average drawdown over all sensitive wetland

areas, 4) minimizing the maximum drawdown while limiting chloride concentrations, and

5) minimizing relative chloride concentration increases.

Data supplied by SJRWMD and gathered during the project were used to

formulate the allocation models (i.e., in the development of an objective function and

management constraints). This included well data specifying location, capacity,

withdrawal rates, corresponding service areas, and additional data identifying potential

well locations. Other data included information concerning effluent rates, capacities, and

locations of wastewater treatment plants that could supplement the demand of agricultural

areas. Water supply demand data consisted of municipal and agricultural water volumes

used in year 1988 and projected water quantity requirements for the year 2010.

The District also supplied environmental impact data specifying locations where

drawdown was expected to induce adverse effects on wetland vegetation. These target

locations were used to constrain the optimization model to identify resource allocation

scenarios that would achieve minimum environmental impacts.









Aquifer system response constraints were developed from steady-state unit

response matrices containing influence coefficients generated from ground water flow and

water quality simulation models described in Chapter 3. These steady-state unit response

matrices represent steady-state changes in pressure head and water chloride

concentrations with respect to pumpage. A ground water flow simulation model was

executed once for each existing and potential well location to determine the head

response at specific locations. A similar approach was taken with the solute transport

simulation model to determine chloride changes at wells due to pumpage. Water level

and water quality unit response matrices were developed for both municipal and

agricultural wells. These response matrices summarize the influence of each well upon

itself, upon all other wells, and also upon specified control points throughout Volusia

County.

The GAMS program (Brooke, Kendrick, and Meeraus 1989) was used to

formulate each allocation model into a linear program optimization model. A model

specific objective function and various constraints (i.e., water supply, demand, and

environmental constraints) constitute the basic framework of each resource allocation

model. All the models contain specific constraints that define allowable connections

between water supplies and demand areas. Three of the models incorporate data from

a unit response matrix created for various hydraulic management constraints, while two

models incorporates two response matrices needed to construct both hydraulic and water

quality management constraints. Once formulated the optimization models were executed

using the linear program solution algorithm available in the GAMS software.

Year 2010 water resource allocation strategies were examined with the









optimization models. Results of each model revealed where future well fields should be

developed and to what extent future and existing wells should be utilized. Aquifer

responses induced by the optimized allocation strategies were compared to those induced

by the projected year 2010 allocation strategy. Predicted pumping strategies identified

from each optimization model were then reviewed and tabulated. To verify optimal

resource allocation strategies identified by the five models the ground water flow and

transport simulation models were used to simulate the optimal pumping scenarios

generated. Predicted pressure heads, drawdowns, and chloride concentrations from the

simulation models were then compared to optimization model results.

SUMMARY AND CONCLUSIONS

Chapter 6 summarizes the overall research effort and presents several salient

conclusions. This work endeavor demonstrated the use of optimization modeling as a

valuable tool for the management of water resources. Optimization modeling elucidated

the potential for an abatement of adverse environmental impacts associated with declining

water table elevations and this was verified when the optimized allocation strategies were

simulated with ground water flow and transport models. It was also shown that the

minimum discharge constraint is an important determinant of the identified optimum

allocations. This constraint requires a minimum flow of at least 50 percent of the 1988

ground water pumping rate from all active wells. Several of the optimal allocation

scenarios incorporate new wells in areas of Port Orange West well field, Daytona Beach

West South Daytona water treatment plant well field, and Ormond Beach State Road

40 / Hudson well fields. In general, each model identified a scenario that decreased

ground water pumpage at the Daytona Beach West water service area and expanded









pumpage in areas east and west where the native vegetation is less sensitive to ground

water withdrawals.

Sensitivity analyses were performed with the optimization models to determine

where management strategies could be changed to induce improvements in ground water

levels. These analyses identified areas were the balance between supplies and demands

could be changed to decrease the environmental impacts of ground water withdrawals.

For example, it was shown that a ten percent decrease in the demand at the DBS

(Daytona Beach West South TP) water service area would induce a water level

improvement of 17 percent. These types of analyses give the water resource manager

valuable information on where conservation efforts and new well developments should

be pursued and let the manager quickly determine how the resource allocation scenario

given by each optimization model responds to changes in projected demands and

withdrawal limits.















TABLE OF CONTENTS

EXECUTIVE SUMMARY .................................. i

LIST OF TABLES .................... ................. xi

LIST OF FIGURES .................... ............... xiii

LIST OF VARIABLES ................... .............. xvii

1.0 INTRODUCTION ................... ................ 1
1.1 STUDY AREA BACKGROUND ...................... 4
1.2 TECHNICAL APPROACH .......................... 8

2.0 REVIEW OF CURRENT LITERATURE ................... 11
2.1 COMBINED SIMULATION/OPTIMIZATION MODELS ..... 11
2.1.1 Embedding Method ........................ 13
2.1.2 Unit Response Matrix Method ................. 19
2.2 SALT-WATER INTRUSION MODELS ................. 32
2.3 INTERFACE PROGRAMS ........................ 36
2.4 SUMMARY ................... ............. 38

3.0 GROUND WATER FLOW SIMULATION MODEL ............. 44
3.1 REGIONAL FLOW SIMULATION MODEL ............. 47
3.2 FLOW SIMULATION MODEL FOR PROJECT AREA ...... 48
3.3 SOLUTE TRANSPORTATION MODEL ............... 54

4.0 WATER SUPPLY NEEDS AND SOURCES .................. 55

5.0 GROUND WATER OPTIMIZATION MODELS ............... 69
5.1 OPTIMIZATION MODEL DECISION VARIABLES ........ 70
5.2 OPTIMIZATION MODEL OBJECTIVE FUNCTIONS ....... 71
5.3 OPTIMIZATION MODEL CONSTRAINTS .............. 73
5.3.1 Aquifer Response Constraints .................. 73
5.3.2 Management Constraints ................. ... 77
5.3.3 Chloride Concentration Constraints ............. 80
5.3.4 Nonnegativity Constraints ................. ... 82
5.4 OPTIMIZATION MODEL FORMULATION ............. 83
5.4.1 General Optimization Model Formulation ......... 83
5.4.2 Specific Optimization Model Formulations ........ 86
5.5 PROCEDURE FOR IDENTIFYING STRATEGIES ......... 86









5.6 RESULTS AND DISCUSSION .... ................. 90
5.6.1 M odel 1 ... ....... ...... ....... ...... 90
5.6.2 Model 2 ....... ...................... 104
5.6.3 M odel3 .................. ...... ....... 118
5.6.4 Model 4 ....................... ....... 124
5.6.5 M odel 5 ............................... 138
5.6.6 General Observations ....................... 150
5.6.7 Sensitiviy Analysis ........................ 167

6.0 SUMMARY AND CONCLUSIONS ......................... 177

7.0 APPENDICES ..................... ....... .......... 181
7.1 PROGRAM CODES FOR DETERMINING RESPONSE
MATRICES ........... ....................... 182
7.2 OPTIMIZATION MODEL FORMULATION USING GAMS ... 190
7.2.1 Sets ............. .................... 190
7.2.2 Tables ..................... .......... 191
7.2.2.1 Source to Demand Link Tables ........ 191
7.2.2.2 Influence Coefficient Tables ............. 192
7.2.3 Parameters ........ ............ ......... 195
7.2.4 Variables ............ .... ............. 195
7.2.5 Equations ..... .......... ............... 196
7.2.6 GAMS Optimization Modeling ................ 196
7.3 GAMS OPTIMIZATION MODEL INPUT FILE ........... 197
8.0 REFERENCES ................... ................. 208














LIST OF TABLES


Tables page

2.1 Ground water optimization models .................... 41

4.1 Volusia County municipal well data for project area ............ 56

4.2 Volusia County agricultural well data for project area ............ 61

4.3 Year 1988 and projected year 2010 demand rates for municipal
water service areas ................... ............. 65

4.4 Year 1988 and projected Year 2010 demand rates for agricultural
water service areas ................... .............. 65

4.5 Volusia County wastewater treatment plant data for project area . 67

5.1 Specific optimization model formulations . . . ..... 86

5.2 Municipal water supply grid cell locations and withdrawal rates . 91

5.3 Agricultural water supply grid cell locations and withdrawal rates .. 95

5.4 Calculated pressure head and drawdown values while
minimizing maximum drawdown at sensitive wetland control
points (Results from initial formulation) . . . ...... ...... 97

5.5 Calculated pressure head and drawdown values while
minimizing maximum drawdown at sensitive wetland control
points (Results from revised model 1) ................ .... 101

5.6 Calculated pressure head and drawdown values while
maximizing minimum head at sensitive wetland control
points (Results from revised model 2) ..................... 119

5.7 Calculated pressure head and drawdown values while
minimizing average head at sensitive wetland control
points (Results from revised model 3) ................ .... 125









Tables page

5.8 Calculated pressure head and drawdown values while
maximizing minimum head at sensitive wetland control
points (Results from revised model 4) ..................... 130

5.9 Calculated chloride concentrations at municipal well grid cells
while minimizing maximum drawdown at wetland control points . 134

5.10 Calculated pressure head and drawdown values while
maximizing minimum head at sensitive wetland control
points (Results from revised model 4) ..................... 142

5.11 Calculated chloride concentrations at municipal well grid cells
while minimizing maximum drawdown at wetland control points .. 145

5.12 Pressure head, drawdown, and difference between projected and
optimized values at sensitive wetland control points ............. 154

5.13 Shadow prices for municipal well grid cells .................. 168

5.14 Shadow prices for agricultural well grid cells ................. 174

5.15 Shadow prices for wastewater treatment plants ................ 174

5.16 Shadow prices for municipal water service areas ............... 174

5.17 Shadow prices for agricultural water service areas .............. 175














LIST OF FIGURES


Figure page

1.1 Location of Volusia County, Florida within the St. Johns River Water
Management District .................................... 5

1.2 Hydrologic cross sections of Volusia County, Florida ............. 7

3.1 Map of Volusia County, Florida ......................... 49

3.2 Volusia County regional and subregional numerical model grid
discretizations ..................................... 50

3.3 Project Flow Model Cell Location Map ................... 52

4.1 Location of municipal water service areas within Project area ...... 60

4.2 Location of agricultural water service areas and wastewater treatment
plants with respect to numerical model ................. .... 63

4.3 Areas of potential impact to plant communities resulting from projected
changes in ground water withdrawals between 1988 and 2010 ....... 68

5.1 Example of nonlinear aquifer response with increased discharge rate at a
surficial aquifer control point ........................... 88

5.2 Correlation between simulation model results and optimization model 1
initial formulation results a) Pressure heads for year 2010;
b) Drawdowns from year 1988 to 2010 . . . ...... ...... 99

5.3 Correlation between simulation model results and revised optimization
model 1 results a) Pressure heads for year 2010;
b) Drawdowns from year 1988 to 2010 . . . ...... ....... 103

5.4 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Port Orange East water service area . 105

5.5 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Port Orange West water service area . 106









Figure


5.6 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Daytona Beach water service area . ... 107

5.7 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Spruce Creek water service area . ... 108

5.8 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Holly Hill water service area . . .. 109

5.9 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for New Smyrna Beach water service area . 110

5.10 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Ormond Beach water service area ........ 111

5.11 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for Tymber Creek & The Trails water service areas 112

5.12 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for agricultural water service areas 1, 3, and 9 .113

5.13 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for agricultural water service areas 2, 4, and 5 .114

5.14 Year 1988, projected year 2010, and model 1 optimized year 2010
withdrawal strategies for agricultural water service areas 6, 7, and 8 .115

5.15 Flow simulation model pressure head results in surficial aquifer
due to optimization model 1 withdrawal strategy for year 2010 . 116

5.16 Flow simulation model drawdown results in surficial aquifer
due to optimization model 1 withdrawal strategy for year 2010 . 117

5.17 Correlation between simulation model results and revised optimization
model 2 results a) Pressure heads for year 2010;
b) Drawdowns from year 1988 to 2010 .................... 121

5.18 Flow simulation model pressure head results in surficial aquifer
due to optimization model 2 withdrawal strategy for year 2010 . 122


page











5.19 Flow simulation model drawdown results in surficial aquifer
due to optimization model 2 withdrawal strategy for year 2010 ...... 123

5.20 Correlation between simulation model results and revised optimization
model 3 results a) Pressure heads for year 2010;
b) Drawdowns from year 1988 to 2010 .................... 127

5.21 Flow simulation model pressure head results in surficial aquifer
due to optimization model 3 withdrawal strategy for year 2010 . 128

5.22 Flow simulation model drawdown results in surficial aquifer
due to optimization model 3 withdrawal strategy for year 2010 . 129

5.23 Correlation between simulation model results and revised optimization
model 4 results a) Pressure heads for year 2010;
b) Drawdowns from year 1988 to 2010 .................... 133

5.24 Chloride concentration correlation between simulation model and
optimization model 4 ................... ............. 138

5.25 Flow simulation model pressure head results in surficial aquifer
due to optimization model 4 withdrawal strategy for year 2010 . 139

5.26 Flow simulation model drawdown results in surficial aquifer
due to optimization model 4 withdrawal strategy for year 2010 . 140

5.27 Chloride concentrations in upper Floridan aquifer due to optimization
model 4 water allocation strategy for year 2010 . . ..... 141

5.28 Correlation between simulation model results and revised optimization
model 5 results a) Pressure heads for year 2010;
b) Drawdowns from year 1988 to 2010. . . ........ 149

5.29 Chloride concentration correlation between simulation model and
optimization model 5 ................................ 150

5.30 Flow simulation model pressure head results in surficial aquifer
due to optimization model 5 withdrawal strategy for year 2010 . 151

5.31 Flow simulation model drawdown results in surficial aquifer
due to optimization model 5 withdrawal strategy for year 2010 . 152


page


Figure









5.32 Chloride concentrations in upper Floridan aquifer due to optimization
model 5 water allocation strategy for year 2010 . . ..... 153

5.33 Flow simulation model pressure head results in surficial aquifer
due to projected year 2010 withdrawal strategy . . ..... 158

5.34 Flow simulation model drawdown results in surficial aquifer
due to projected year 2010 withdrawal strategy . . ..... 159

5.35 Chloride concentrations in upper Floridan aquifer due to the
projected year 2010 water allocation strategy . . . ..... 160

5.36 Difference in induced pressure head between the projected year 2010
strategy and the strategy identified by model 1 (projected-optimized) .162

5.37 Difference in induced pressure head between the projected year 2010
strategy and the strategy identified by model 2 (projected-optimized) .163

5.38 Difference in induced pressure head between the projected year 2010
strategy and the strategy identified by model 3 (projected-optimized) .164

5.39 Difference in induced pressure head between the projected year 2010
strategy and the strategy identified by model 4 (projected-optimized) .165

5.40 Difference in induced pressure head between the projected year 2010
strategy and the strategy identified by model 5 (projected-optimized) .166









LIST OF VARIABLES


ca, is the aquifer influence coefficient defining pressure head change at

each sensitive wetland area control point due to a change in discharge

rate at each municipal well grid cell i.

3,j is the aquifer influence coefficient defining pressure head change at

each sensitive wetland area control point due to a change in discharge

rate at each agricultural well n.

BOTELEVh is the bottom elevation of the surficial aquifer in feet from mean

sea level at each well grid cell control point h.

CA, is the capacity rate in cfd of each agricultural well grid cell i.

CCh is the chloride concentration in ten-thousandths of a milligrams per liter

(mg/1) at each well grid cell control point h.

CCOh is the initial chloride concentration in mg/1 at well grid cell control

points h.

CIh is the increase in chloride concentration in ten-thousandths of a

milligrams per liter (mg/1) at each well grid cell control point h.

CLh is the maximum chloride concentration limit in mg/1 of each well grid

cell control point h.

CM, is the capacity rate in cfd of each municipal well grid cell i.

CW. is the capacity rate in cfd of each wastewater treatment plant i.

DA, is the demand rate in cfd of each agricultural water service area o.

DDj is the drawdown at sensitive wetland control point.

xvii














DDPRIVj is the drawdown in feet at each sensitive wetland area control point

due to private wells not incorporated in the optimization process.

DDW, is the drawdown in millionths of a foot at each well grid cell h.

DMk is the demand rate in cfd of each municipal water service area k.

Yi,h is the aquifer influence coefficient defining pressure head change at well

grid cell control point h due to a change in discharge rate at each

municipal well grid cell i.

h defines all the well grid points incorporated in the optimization model.

HDj is the pressure head in millionths of a foot at each sensitive wetland

area control point.

HDWh is the pressure head in millionths of a foot at each well grid cell h.

HOj is the initial pressure head in feet at each sensitive wetland area control

point.

HWOh is the initial pressure head in feet at each well grid cell h.

i defines the 120 municipal well grid cells in the optimization model.

j defines the 100 grid cell points in the optimization model where there is

a high potential of harm to vegetation in sensitive wetland areas.

k defines the 14 Municipal water service areas in the optimization model.

m defines the five wastewater treatment plants in optimization model.

n defines the 28 agricultural well grid cells in the optimization model.

xviii














o defines the 9 different agricultural water service areas in the

optimization model.

40n,h is the aquifer influence coefficient defining chloride concentration

change at each well grid cell control point h due to a change in

discharge rate at each agricultural well n.

QAn,o is the discharge rate in cfd of each agricultural well grid cell n which

supplies each agricultural water service area o.

QAO, is the initial discharge rate in cfd of each agricultural well grid cell n.

QAT, is the total discharge rate in cfd of each agricultural well grid cell n.

QMik is the discharge rate in cubic feet per day (cfd) of each municipal well

grid cell i which supplies each municipal water service area k.

QMO, is the initial discharge rate in cfd of each municipal well grid cell i.

QMT, is the total discharge rate in cfd at each municipal well grid cell i.

QW,o is the effluent reuse rate in cfd of each wastewater treatment plant m

which supplements each agricultural water service area o.

QWTm is the total effluent reuse rate in cfd of each wastewater treatment

plant m.

RCIh is the relative chloride concentration increase in ten-thousandths of a

miligram per liter (mg/1) at each well grid cell control point h.

SERVEi,k designates which municipal well grid cells i supply which municipal

xix












water service areas k.

SERVE2,,o designates which agricultural well grid cells n supply which

agricultural water service areas o.

SERVE3,,o designates which wastewater treatment plants m can supplement the

demand of which agricultural water service areas o.

TDDj is the drawdown in feet at each sensitive wetland area control point.

TDDWh is the drawdown in feet at each well grid cell h.

THDj is the total pressure head in feet at each sensitive wetland area control

point j.

THDWh is the total pressure head in feet at each well grid cell h.

On,h is the aquifer influence coefficient defining pressure head change at

well grid cell control point h due to a change in discharge rate at each

agricultural well grid cell n.

TCIh is the total increased chloride concentration, respectively, in mg/l at

each well grid cell control point h.

TCCh is the total chloride concentration in mg/1 at each well grid cell control

point h.

4i,h is the aquifer influence coefficient defining chloride concentration

change at each well grid cell control point h due to a change in

discharge rate at each municipal well grid cell i.


XX














CHAPTER 1
1.0 INTRODUCTION


In recent years, population growth, expanding agricultural and industrial activities,

and rapid urban development have significantly increased the demand for clean, fresh

water in the state of Florida. Within the boundaries of the St. Johns River Water

Management District (SJRWMD), demand for public water has increased 66% from 1975

to 1990 and is predicted to increase another 120% by the year 2010 (SJRWMD 1992).

This increased demand presents an ominous threat to ground water sources in terms of

both water quantity and quality. Along coastal areas of the District, increased

development has generated water quality problems related to high chloride concentrations

due to salt-water intrusion and localized upcoming. Sensitive wetland areas have also

been adversely affected by recent declines in the water table.

Due to the increasing demands placed on Florida's water resources, the state of

Florida adopted legislation in 1990 to improve water resource management and to direct

future growth through planning programs. This legislation requires each Water

Management District to completely evaluate their water needs and sources through the

year 2010 and delineate critical areas identified as water resource problems. Districts

are then expected to develop possible alternative water supply scenarios which avoid

adverse or otherwise unacceptable changes in the environment or the availability of

water.








2

A study of regional water supply needs and sources, could identify many possible

resource allocation plans that meet the needs of the District. The distribution of available

supplies to service areas is an allocation problem and can have several feasible solutions.

It is also likely that the number of plans will be too extensive to permit detailed

examination of each and every scenario. In order to identify a concise subset of water

allocation plans which best meet environmental and developmental goals of the District,

optimization modeling can be incorporated into the decision or plan elucidation process.

This project involves the development of a systematic method of determining

optimum water supply strategies that satisfy various environmental and hydrological

requirements. The purpose of this type of water supply strategy is to optimize the

pattern of water supply development and usage to meet projected needs. This resource

management problem requires the use of optimization modeling to identify desirable

scenarios of resource allocation; otherwise, resources may not be used in the most

effective and efficient manner. When environmental impacts are also incorporated, the

allocation problem expands to include identifying feasible scenarios that must also satisfy

environmental constraints (i.e., ground water quality standards, minimum water levels).

To balance projected needs against available sources, it is possible that the management

problem may become one of balancing projected development against adverse

environmental impacts.

The objective of this project is to accomplish two goals. The first is to obtain a

broad understanding of numerical optimization modeling and its recent applications to

ground water resource management. The second goal is to demonstrate optimum

resource allocation modeling at a selected site that would satisfy specified environmental








3

and hydrological requirements. The demonstration will give the SJRWMD essential

knowledge and experience to incorporate optimization technology into the District's

decision-making framework. In order to achieve the established two goals and fulfill the

scope of the project, the following three objectives were accomplished:

1) Review current literature of numerical optimization modeling with respect

to ground water resource management.

2) Construct site specific optimization models capable of generating water

resource allocation scenarios which satisfy projected demands and environmental

constraints for the year 2010.

3) Generate and summarize various optimum water resource allocation

scenarios which satisfy specific objectives.


To meet the first objective the literature review was limited to applications of

ground water resource management and only where tools of optimization were applied

to demonstrate optimum scenarios of resource allocation. Formulations of various water

management models were reviewed and a number of existing ground water optimization

models were summarized in tabular form. Based on knowledge acquired through the

literature review and the size and complexity of the problem, it was determined that the

most efficient simulation/optimization model would be one developed using the unit

"response matrix" technique. This method consists of incorporating a matrix of

coefficients which represent the response of ground water to a specified change in

withdrawal rate. To achieve the second and third objectives it was necessary to select

a study site within the district boundaries.










1.1 STUDY AREA BACKGROUND

Based on the extensive ground water flow and contaminant transport modeling

effort completed by the SJRWMD, four distinct regions within the district were identified

as candidates for project study. These areas included the Volusia County region, the

Wekiva River Basin region, the East-Central Florida region, and the Geneva

Area/Seminole County region. The regional models and areas were evaluated with

respect to model accuracy and complexity, computer requirements and efficiency, data

availability, and applicability. Based on this review, A study area within Volusia County

Florida was selected as the site to apply the numerical optimization modeling techniques

of this project, shown in Figure 1.1.

The study area includes most of Volusia County, thus a general geographical and

hydrological characterization of this county is in order. Volusia County alone covers an

area of approximately 1,200 square miles in the east-central region of Florida. As shown

in Figure 1.1, the county is bounded by the Atlantic Ocean to the east, the St. Johns

River to the West, Flagler and Putnam Counties to the north, and Brevard County to the

south. Although the county consists of approximately 120 lakes larger than 5 acres in

size, ground water supplies are currently the sole means of meeting public water demands

(Knochenmus and Beard 1971, Kimrey 1990).

Most of the population of Volusia County occupies the region in and near the

coastal cities of Daytona Beach, New Smyrna Beach, and Ormond Beach. Extensive

development of these coastal areas has resulted in increased demands for ground water

and an encroachment of the fresh-water/salt-water interface. In order to alleviate the

problem of salt-water intrusion and satisfy water service demands, the general trend since



























PUTNAM
-FLAER\ Project Area



MARION


VOLUSIA
LAKE

SEMINOLE
BREVARD
















10 miles
0 5 kilometers




Figure 1.1

Location of Project in Volusia County, Florida within the
St. Johns River Water Management District


I








6

the 1950's has been to locate additional wells to the west towards central Volusia County.

The hydrogeologic system of Volusia County is made up of two aquifer systems,

the surficial and the Floridan aquifers, separated by an intermediate confining layer (See

Figure 1.2). The surficial aquifer is the uppermost formation, consists of silts, clays,

cemented shell, and quartz sands, and is considered to be unconfined. The water table

is usually at or near the surface in lowland and flatland regions and is generally a

suppressed image of the ground level in highland regions. Precipitation, lakes, wetlands,

and irrigation are the main sources of recharge into the surficial aquifer, which ranges

in thickness between 50 and 100 feet. Via the intermediate confining layer, leakage can

occur in and out of the aquifer depending on the difference in potentiometric head

between the surficial and Floridan aquifer systems (Tibbals 1990).

The Floridan aquifer system contains the Upper Floridan and Lower Floridan

aquifers and is divided by a middle confining unit. This confining layer of low

permeability is located completely within the Avon Park limestone geologic formation.

The lower portion of the Avon Park Formation along with the Oldsmar Formation

comprise the Lower Floridan aquifer, which consists mainly of saline water. The Upper

Floridan aquifer consists of the Ocala Limestone Formation and the upper portion of the

Avon Park Formation (Miller 1986, Tibbals 1990). Although the Upper Floridan aquifer

contains brackish water in the St. Johns River Valley, near the Atlantic coast, and north

of Volusia County line, this ground water supply is the main means of meeting public

water demands (Kimrey 1990). The entire thickness of the Floridan aquifer system

ranges from approximately 1,800 to 2,300 feet in Volusia County.

Separating the surficial aquifer from the Floridan aquifer throughout most of the












Section A-A
100-


50.


MSL
50-






-100-


-150-


-200


Feet


Feet


KEY


LSURFICIAL AQUIFER

16ONFINING UNIT

OFLORIDAN AQUIFER


Figure 1.2 Hydrologic cross sections of Volusia County, Florida.
(source: McKenzie-Arenberg, 1989)








8

county is the intermediate or upper confining layer. This layer consists of clay or silty

sand of the Miocene to Pleistocene age and has a thickness range of 0 to 60 feet. The

confining unit in the western portion of the county is thinner than the eastern portion and

is sometimes totally absent. Although the low permeability layer is leaky, it is capable

of confining the pressurized water in the Floridan aquifer system (Phelps 1990).


1.2 TECHNICAL APPROACH

Once the specific study site was selected, it was possible to initiate development

of the water resource allocation models. These models were essentially numerical

optimization models comprised of an objective function, decision variables, and

constraints. A total of five optimization models were formulated to examine water

resource allocations for year 2010, these included three models with hydraulic constraints

and two models that incorporate both hydraulic and water quality constraints. The five

model specific objective functions included: 1) minimizing the maximum drawdown

value, 2) maximizing the minimum pressure head, 3) minimizing average drawdown over

all sensitive wetland areas, 4) minimizing the maximum drawdown while limiting

chloride concentrations, 5) minimizing the maximum chloride concentration, and 6)

minimizing the maximum relative chloride concentration increase. From the literature

review, it was determined that the formulation of these water resource allocation models

would be predicated on the unit response matrix approach as described in the Literature

Review.

Data supplied by SJRWMD and gathered during the project were used to develop

the allocation models consisting of an objective function and management constraints.








9

The data supplied included well data specifying location, capacity, withdrawal rates,

corresponding service areas, and additional data identifying potential well locations.

Other data included information concerning effluent rates, capacities, and locations of

wastewater treatment plants that could supplement the demand of agricultural areas.

Water supply demand data consisted of municipal and agricultural water volumes used

in year 1988 and projected water quantity requirements for the year 2010.

The District also supplied environmental impact data specifying locations where

drawdown was expected to induce adverse effects on wetland vegetation. These target

locations were used to constrain the optimization models. Thus, various ground water

allocation scenarios could be examined that involve minimizing environmental impacts.

Aquifer system response constraints were incorporated in each optimization

model. These constraints were developed from steady-state unit response matrices

containing influence coefficients generated from ground water flow and solute transport

simulation models. These steady-state unit response matrices represent steady-state

changes in pressure head and water chloride concentrations with respect to pumpage.

A ground water flow simulation model was executed once for each existing and potential

well location to determine the head response at specific locations. A similar approach

was taken with the solute transport simulation model to determine chloride changes at

wells due to pumpage. Water level and water quality unit response matrices were

developed for both municipal and agricultural wells. These response matrices summarize

the influence of each well upon itself, upon all other wells, and also upon specified

control points throughout Volusia County.

Pumping strategies identified from each optimization model were reviewed and








10

tabulated. To verify these optimum water allocation strategies, the ground water flow

and the solute transport simulation models were used to simulate these pumping

strategies. Predicted pressure head, drawdown, and water quality changes generated from

the simulation models were then compared to predictions given by the five optimization

models to determine how well the optimization models emulated the simulation models.

When simulated and optimized model values for drawdown, pressure head, and chloride

concentration differed severely, response matrices were recreated using initial conditions

that better approximate the optimum pumping scenarios identified by each of the original

optimization models.














CHAPTER 2
2.0 REVIEW OF CURRENT LITERATURE


Over the last three decades, ground water management models have been

formulated using a variety of computational techniques to simulate and optimize the

management of aquifer systems. These models have been produced by applying

economic theory, heuristic or intuitive procedures, optimization algorithms, hydraulic

flow equations, and complex combined simulation/optimization algorithms. Although

each of these methods has its own advantages and disadvantages, the combined

simulation/optimization approach is found to be the most powerful since it is capable of

incorporating economic, physical, and policy considerations within a single model

environment. The literature review that follows is limited to only the combined

simulation/optimization approach and discusses the various methods and applications of

this modeling technique. Also reviewed are applications where optimization modeling

has been combined with salt-water intrusion models and computer programs which

facilitate model formulation and act as an interface between the simulation and

optimization models.


2.1 COMBINED SIMULATION/OPTIMIZATION MODELS

Models developed using the simulation/optimization approach contain both

simulation equations and operations research style optimization algorithms. The

simulation equations assure that the management model correctly emulates the aquifer







12

responses to external and internal fluxes. The optimization algorithms allow the water

management objective and restrictions to be specified as algebraic equations. The

combined models are then capable of identifying ground water withdrawal scenarios

which optimize given objective functions (Peralta and Willardson, 1992).

Ground water flow simulation models alone are only able to compute pressure

heads and fluxes resulting from specified boundary conditions and pumping rates. Using

the simulation model alone to determine an optimal pumping scenario can be tedious and

error prone process; the simulation model must be executed many times with different

pumping strategies in order to determine the best scenario. Unless all possible

combinations of pumping strategies are evaluated, the optimum pumping strategy is not

likely to be identified. The simulation/optimization models, however, directly compute

the optimum strategies under identified management objectives subject to specified system

constraints (i.e. assuring the heads and fluxes are constrained within desired limits).

Optimization methods which have been applied in the field of ground water

management include linear programming, nonlinear programming, mixed-integer linear

programming, quadratic programming, dynamic programming, differential dynamic

programming, and goal programming. While these applications differ greatly in their

mathematical basis, they may be grouped into general categories according to the method

by which ground water flow equations are incorporated into the problem formulation.

Ground water flow equations have been combined with mathematical optimization

algorithms through the use of various types of approaches, methods, or techniques.

Historically, most simulation/optimization models represent the relationship between

aquifer system response and stimuli by incorporating the "embedding" method or the








13

"unit response matrix" approach. For ground water management models, pressure heads

or fluid fluxes represent the system response and withdrawal rates represent the stimuli.

The following is a review of current literature on the embedding and unit response matrix

approaches.

2.1.1 Embedding Method

Models incorporating the "embedding" technique use numerical methods such as

finite-differences or finite-elements to transform the partial differential ground water flow

equations into a set of linear algebraic equations. These numerical methods both involve

the use of a mesh or grid to discretize the aquifer system, with each grid cell specified

as having homogenous properties. Pressure heads and discharge (and/or recharge) rates

are then depicted by flow equations formulated at each cell node in terms of control or

decision variables. These equations are then incorporated into the optimization model

as constraints.

The embedding technique with respect to ground water management was first

proposed by Aguado and Remson (1974). Their initial work demonstrated that linear

programming could be used in conjunction with finite-difference approximations to study

the physical response of ground water systems under conditions which define optimal

resource management. The optimization models were developed to determine optimum

well production scenarios which produce a maximized sum total of hydraulic heads over

the area of interest. The method was demonstrated for both confined and unconfined

aquifer systems under one-dimensional, two-dimensional, steady-state, and transient flow

conditions. For the transient flow case, equations were developed for each time step

and then solved simultaneously. For the unconfined aquifer system, the nonlinear







14

equations were approximated by a succession of systems of linear difference equations.

Although the authors conclude that nonlinear flow equations and irregular

boundary or initial conditions can be modeled adequately, they concede that the resulting

linear programs are typically very large.

A site-specific application of the embedding technique was presented by Aguado

et al. (1974) to determine an optimum aquifer dewatering strategy for a proposed dry

dock. A linear programming model incorporating steady-state, finite-difference

approximations of a two-dimensional unconfined aquifer was developed in order to

predict the optimal number of wells, well locations, and corresponding withdrawal rates

required to produce and maintain desired ground water levels. The optimization model

formulation permits the identification of scenarios that minimize total pumping without

restricting the number or location of utilized wells. Model execution led to a solution

in which a maximum number of wells were positioned as close to the dewatering region

as possible. To verify the accuracy of the optimization model, aquifer response results

were compared to those predicted by unconstrained numerical models and also electrical

analog ground water models (Remson, Aguado, and Remson, 1974). The authors

concluded that a finer grid spacing in the management model would produce improved

accuracy with respect to aquifer response, thus an improved optimum pumping strategy.

Alley et al. (1976) demonstrated the embedding technique by using a finite-

difference approximation of two-dimensional transient flow. The optimization model

determined optimum withdrawal strategies depicting well location and pumping rates for

a 20 day time period. Since an existing withdrawal strategy was specified as an initial

condition, the optimization model predicted the rate of increased withdrawal (as opposed







15

to the absolute withdrawal rate). To optimize the transient condition, submodels were

created which divided the time period into four equal time steps. These models were

then executed sequentially to produce initial conditions for the subsequent time interval.

This stepwise approach was found to reduce size dimensions of the model, and thus

computation time, when compared to the lumped transient approach, which solves all

time steps simultaneously. However, this method has been criticized due to conflict

between long-range management goals and short-term, sequential optimization (Gorelick

1983). The availability of future water resources is affected by decisions made in earlier

time periods.

The Galerkin-finite element method was incorporated by Willis and Newman

(1977) to simulate a hypothetical aquifer consisting of heterogeneous anisotropic porous

media. The two-dimensional flow equations were embedded into an optimal control

ground water problem to determine well locations and pumping rates which would satisfy

water demands at minimum cost over a number of planning periods. Since the objective

function is non-linear, the algorithm was developed to solve a sequence of linear

programming submodels until no improvement in the objective function could be

obtained. Additional constraints and "penalty" costs were added to the management

model to minimize deviations from the desired final system state. This modification

enabled the model to find a more equally distributed pumping pattern, thus minimizing

environmental impacts due to drawdown.

Yazicigil and Rasheeduddin (1987) extended the embedding approach to include

multi-aquifer systems. Both a transient, single-objective approach and a steady-state,

multiple-objective approach were presented. For both cases, a hypothetical, leaky








16

confined two-layer aquifer system was discretized to represent the physical system.

Quasi-three-dimensional finite- difference approximations were formulated for each active

grid cell and embedded as constraints in the linear programming model.

For the transient case, the water management objective was specified to determine

optimal well locations and corresponding withdrawal rates which would minimize

drawdowns in the two aquifer system over a four season time period. Equations for all

four time steps were effectively solved simultaneously, but an attempt to sequentially

solve the four season problem generated an infeasible solution for the fourth time step.

These results supported earlier discoveries by Alley et al. (1976) which suggested that

the use of the embedding technique with sequential linear programming may result in

unacceptable long range management decisions.

To further demonstrate the management model, the authors applied the embedding

technique to steady-state, multi-objective analysis by replacing equations representing

transient responses and adding a second objective function to the original problem.

Constraint and weighing methods were then used to solve the resultant linear

programming problem and to develop graphical trade-off curves between total withdrawal

rate and total hydraulic head.

Jones et al. (1987) applied a modified embedding approach by incorporating a

differential dynamic programming (DDP) algorithm to obtain optimal control of an

unsteady, unconfined aquifer. DDP is a successive approximation technique for solving

optimal control problems. Equations were incorporated which 1) represented the

simulation model of the aquifer system, 2) constrained the hydraulic head and pumping

parameters such as well capacity, drawdown, and supply demands, and 3) optimized







17

objective functions such as maximize the sum of heads or minimize yield. The method

was found to reduce the dimensionality problems associated with embedding hydraulic

equations, linearize the exponential growth in computer time with respect to stages and

time periods, and cause to solution to converge quadratically. The authors concluded

DDP to be a powerful tool for management of transient ground water hydraulics.

The embedding technique has not only been applied to ground water quantity, but

also to ground water quality. Willis (1979) demonstrated the use of ground water quality

constraints in linear programming by developing an embedded algorithm for contaminant

transport equations. The technique was applied to a ground water system utilized as both

a water supply resource and storage reservoir for waste water residuals. The Galerkin

method was used to transform the flow and transport differential equations, and the

resulting equations were embedded as constraints into the management model to simulate

the aquifer system. The model was developed to determine optimum pumping and waste

injection rates which satisfy water service demands while maintaining ground water

quality as mandated by current ground water standards. The analytical solutions to the

transport equations are nonlinear with respect to the decision variables (pumping and

injection rates). Therefore, the equations could not be embedded directly into a linear

programming model. In order to approximate this nonlinear response and to determine

optimal strategy, the management model was redefined to consist of two interdependent

linear submodels. Optimal pumping and injection strategies were first determined by a

flow submodel, then these results were input into a ground water quality submodel to

predict the maximum waste injection concentrations for each operational cycle.

The embedding technique was again applied to ground water quality management








18

by Shamir et al. (1984) for a coastal aquifer in Isreal. Aquifer response was simulated

by two sets of finite-difference equations which represent the flow of ground water and

the movement of the fresh-water/salt-water interface. These equations were included as

constraints in the optimization model. However, in order to represent and constrain the

location of the fresh-water/salt-water interface, a linearized model of salt-water intrusion

was incorporated to approximate the nonlinear response. Multi-objective linear

programming was incorporated to determine optimal annual operation of the coastal

aquifer.

Multi-objective optimization involved performing single objective linear

programming optimization and developing trade-off functions for conflicting objectives

which are then utilized to obtain a final compromise solution. The management model

contained four objective functions which minimized 1) changes in water levels due to

pumping, 2) intrusion of the fresh-water/salt-water interface, 3) chloride concentrations,

and 4) energy costs. Chloride mass balance equations and maximum upper concentration

limits were also specified for each aquifer cell location.

Gorelick et al. (1979) applied a numerical finite-difference approximation method to

solute transport convective-dispersive equations to determine chloride concentrations of

a transient pollutant source. Numerical approximation was utilized to generate a system

of linear equations which were then incorporated in a water quality optimization model

as part of the formulated constraint set. Water quality standards were specified in the

management model, and the model was executed to determine the maximum permissible

single source pollutant concentration. The method demonstrated the feasibility of

balancing water supply and waste disposal needs while satisfying water quality









requirements over long time frames.

As discussed, the embedding technique has been successfully applied to a wide

range of ground water management problems. The approach is capable of handling both

linear and nonlinear ground water hydraulics; steady-state and transient problems with

single or multipurpose objective functions. The primary advantage of the method is the

ability to emulate the physical processes of an aquifer system at a detailed level by

including ground water flow and/or solute transport equations for each grid cell directly

as constraints in the optimization model.

A disadvantage of the method is the dimensionality associated with large scale

management problems. The degree of dimensionality is a function of the number of

decision variables, aquifer grid cells, and time steps. Therefore, spatially large or multi-

time step management models could potentially have hundreds or even thousands of

constraint equations. Attempts by Alley et al. (1976) and Yazicigil and Rasheedudd

(1987) to redefine multi-time step embedded problems into a sequential set of smaller

problems have resulted in management decisions biased towards the short-term. To

avoid these difficulties, the embedded technique should be limited to local aquifer

systems and steady-state management problems.

2.1.2 Unit Response Matrix Method

The "unit response matrix" method is utilized by simulation/optimization models

by incorporating the use of influence coefficients, superposition, and linear systems

theory. The simulation model is used initially to compute the system response to a unit

stimuli. For ground water management models, the response matrix approach is based

on the concept that the influence of discharging or recharging a single well on aquifer








20

drawdown at selected locations may be expressed as simple algebraic functions. The

individual influence functions are then combined using the principle of superposition to

obtain the aquifer response due to multiple wells. Therefore, a "response matrix" can

be developed which may be expressed algebraically as
m
dj = E [ aiJqi ] = [ aiJq + a(i1)Jq(i+) ".. amjqm
i=1

where
dj = drawdown at control point j
qi = pumpage at well i
aj = unit response function of well i on point j
i = 1,2,..., m
j = 1,2,..., n
m = number of wells
n = number of control points


The response matrix of influence coefficients, aj's, are generally determined from

analytical or numerical ground water simulation models. Since the response equations

need only be developed for selected points of interest, it is not necessary for equations

to be developed for each grid cell within the aquifer system. This characteristic

significantly reduces the dimensionality of the management problem when compared to

other simulation/optimization techniques.

One of the earliest pioneers to combine the response matrix approach with ground

water management, Deininger (1970) developed a management model which maximized

well field production by optimally selecting well locations and withdrawal rates. A

response matrix representing aquifer drawdown was developed from an analytical

solution to the Theis nonequilibrium equation and incorporated in the linear programming

model. Drawdown, pumping, and well characteristics were all specified as constraints








21

in the optimization model. Maximum drawdown limits were specified at well field

boundaries, and head versus discharge curves were approximated by piecewise

linearization to incorporate drawdown constraints at wells. Although head losses in well

casings were also linearized using a chordal approximation of the Manning formula, well

screen head losses were assumed to be constant. Deininger demonstrated the formulated

problem was easily solved by linear programming techniques.

Rosenwald and Green (1974) used the response matrix method in combination

with branch and bound mixed-integer linear programming to identify optimal well sites

in an "underground reservoir". Pressure coefficients, which represent potentiometric

drawdowns at wells due to changes in discharge rate, were determined by executing a

numerical simulation model. A set of constraint equations were then developed which

combined the pressure response matrix with maximum allowable drawdown limits. A

mixed-integer programming model was employed which assumed constant well flow rates

and involved a binary switching variable allowing each potential well location to be either

active or inactive.

One of the two cases used to demonstrate the technique involved the selection of

well locations in a hypothetical ground water aquifer. Enumeration of the alternatives

and comparison with simulation results verified that the response matrix approach based

upon linear superposition was applicable to ground water hydraulics. In the second case,

the authors applied the response matrix method to select well sites in a gas storage

reservoir. The response matrix initially developed erroneously estimated pressure heads

due to the nonlinear behavior of gas flow. Using various corrective techniques, attempts

were made to improve the pressure response equations. Although the results of the







22

optimization model were only slightly improved, the authors concluded that corrective

procedures offer some potential for improving the optimization of nonlinear systems with

response matrices.

Maddock (1972) discussed the use of transient algebraic technological functions

(response functions) for aquifers whose transmissivities vary with drawdown. These

algebraic functions were developed to relate seasonal pumping rates to drawdown levels

at specified wells, and the relationship was derived from an analytical solution of the

two-dimensional transient ground water flow system. A hypothetical example is used to

demonstrate the method of combining response functions and quadratic programming to

determine optimal semiannual well withdrawal strategies.

Algebraic response functions were again developed by Maddock (1974) to

incorporate the optimization technique in nonlinear aquifer systems. Boussinesq's

equation for unsteady flow due to pumping in an unconfined aquifer was approximated

by an infinite power series. The total aquifer drawdown response function was then

expressed as the finite sum of an infinite power series in pumping values. The

approximation assumed fully penetrating wells, no vertical flow, and constant pumpage

over a single time horizon. The number of terms required for an accurate water

elevation estimate is dependent upon the ratio of drawdown to saturated thickness. To

demonstrate the methodology, the author formulated and executed the nonlinear

programming problem to determine least cost pumping distributions.

One of the first site specific applications of the response matrix technique was

presented by Heidari (1982) and involved determining optimum ground water allocation

strategies in Kansas. A two-dimensional ground water simulation model developed for








23

the region was combined with the algebraic influence function proposed by Maddock

(1972,1974) to develop the response matrix. For simplicity, sixty-one hypothetical well

fields were created to represent total withdrawal from actual wells in the study area.

Once the response matrix was incorporated into the management model, constraint

equations were specified which limited drawdowns at each well to a fraction of the

saturated aquifer thickness.

The optimization model was formulated to maximize total pumpage under two

policy scenarios. With respect to the net appropriation (difference between appropriation

and recharge), maximum limits were specified for the first scenario but no limits were

specified for the second. The models were executed for five and ten year time periods

and each contained five time steps. Results revealed that removing the net appropriation

constraint created barely a noticeable increase in pumpage when the drawdown fraction

was allowed to be equal to or greater than 20%. The model also indicated that under

optimal conditions, only about 50% of the net appropriation could be satisfied over the

ten year time period.

Willis and Liu (1984) incorporated response functions to develop a bi-objective

optimization model to allocate ground water to competing irrigation demands of the Yun

Lin basin in southwestern Taiwan. Objectives of the model were to maximize the sum

of hydraulic head and to minimize the total water deficit in the basin. The simulation

model, which was used to develop the response matrices, was developed using the

Galerkin finite-element approximation method for ground water flow and was validated

using field data from over 350 monitoring wells. The response matrices were

incorporated as constraints and the model was initially executed as a steady-state linear








24

programming problem. Comparison of results from this initial steady-state optimization

to existing allocation strategies revealed that the total water deficit could be decreased

substantially without decreasing the sum total of hydraulic heads. By assigning weights

to the objective functions, solutions to the optimization problem were depicted in the

form of trade-off curves to express the relationship between hydraulic head and total

water deficit.

The authors reformulated the optimization model to perform a transient analysis.

Piecewise linearization was used to incorporate time dependent boundary conditions for

the development of the response functions. Using the same objectives specified for the

steady-state condition, the model was executed and trade-off curves were again developed

to depict the relationship between hydraulic head and water deficit. Results of the

transient analysis indicated that the steady-state formulation overestimated the reduction

in water deficit. However, the authors concluded that a significant water savings could

still be achieved with the use of the optimization model.

Danskin and Gorelick (1985) were some of the first to implement response

functions in the management of multi-layer aquifer systems. The study area consisted

of a multi-layer aquifer system connected to a surface water system, and the model was

formulated to evaluate the efficiency of flow between the two systems. Critical factors

controlling basin management decisions were also identified through execution of the

mixed-integer linear programming model. The response matrix was developed using a

quasi-three-dimensional finite-difference simulation model, and the influence coefficients

for the upper unconfined aquifer was linearized using an iterative approach. The

response functions were used to develop water elevation constraints and additional








25

limitations were specified with respect to surface water flow, vertical leakance, and water

service demands. Mass balance constraints were also incorporated to regulate the

concentration of dissolved solids.

Evaluation of historical basin management practices revealed that "the cost of

operation during the study period was twice that of optimal basin management."

However, the largest inefficiencies were localized and most activities were within 20%

of optimal. Sensitivity analyses indicated that the controlling factor in basin operations

involved surface water and ground water relationships.

Willis and Finney (1985) developed response equations using finite-difference

methods, quasi-linearization, and matrix calculations. Nonlinear optimization for an

unconfined aquifer system was performed by incorporating a quasi-linearization

optimization algorithm and projected Langrangean methods. The model was structured

as a discrete optimal control problem and determined the optimal pumping pattern while

satisfying water demands. Quasi-linearization and MINOS (Modular In-Core Nonlinear

Optimization System) algorithms were both found to be efficient for solving moderately

sized nonlinear ground water management models. However, Willis and Finney

concluded that large management problems could be solved more efficiently by applying

quasi-linearization optimization.

Herrling and Heckele (1986) used both the embedding and response matrix

techniques to couple a finite-element simulation model with an linear simplex

optimization model. Management of the nonlinear ground water system was performed

by optimizing well locations along with pumping and infiltration rates while satisfying

ecological constraints. These constraints consisted of sustaining ground water levels and








26

meeting contamination standards. The advantages of both coupling techniques were

incorporated. The embedding method involved the implicit consideration of the flow

model, and the influence function method reduced the amount of computer storage

required.

The response matrix technique was combined with a stochastic approach by Tung

(1986) for the management of ground water resources. The "chance constrained"

management model was formulated using linear systems theory to determine optimal

withdrawal rates in a well field subject to specified reliability constraints. Transient,

nonleaky drawdown response functions were developed from an analytical solution to the

Cooper-Jacob equation while transmissivity and storativity values were treated as

independent random variables. First-order analysis was employed to estimate the

statistical characteristics of drawdown at each control point. Stochastic drawdown

response functions implicitly incorporating parameter uncertainty were then developed

as linear constraints. Drawdown limitations were stated at each control point which

required drawdowns to be less than a specified value times a reliability factor.

The management model was applied to a hypothetical confined aquifer where

optimal withdrawal rates were determined for three potential wells over three time

periods. While maximizing total well production, a sensitivity analysis was performed

on the model to investigate the effect of parameter uncertainty and of varying reliability

factors. Tung discovered that model results were insensitive to changes in storativity but

quite sensitive to changes in transmissivity and the specified reliability factor. A

postoptimal analysis indicated that the linear programming model produced acceptable

results in terms of complying with reliability requirements, but only when the









transmissivity uncertainty was small.

Lindner et al. (1988) applied the response matrix technique to a two-aquifer

system to determine the optimum ground water withdrawal while meeting established

environmental criteria. The two-aquifer system consisted of an upper unconfined and a

lower confined aquifer with a leaky intermediate layer. The environmental criteria

included meeting specified ecological conditions and ground water levels in the upper

aquifer. Though ground water was withdrawn from the confined aquifer only, the

unconfined aquifer was also affected due to leakage from the separating layer.

Galeati and Gambolati (1988) employed the response matrix technique to solve

a three-dimensional aquifer dewatering problem. The water management model was

formulated to identify the optimal spatial distribution and corresponding well rates

required to maintain desired water levels during two planning periods. A three-

dimensional finite-element ground water flow model was developed and repeatedly

executed to develop a steady-state response for each abstraction well. The individual

influence coefficients were then combined using the principle of superposition, and

hydraulic head limits were specified to create a set of water elevation constraints. The

model was formulated to minimize total withdrawal while still dewatering the study area.

The authors assumed linear response since the dewatering area responds as a

hydrodynamically closed system having essentially no influence on the surrounding

unconfined aquifer.

Once the management model was executed and an optimal dewatering strategy

was prescribed, it was discovered that some wells which were deemed active during the

first planning period were deemed inactive during the second period, and vice versa.







28

Therefore, in order to avoid large installation costs and withdrawal rate nonuniformity

associated with the solution, the model was reformulated as a mixed-integer linear

programming model. Mixed-integer linear programming involved designating wells with

a withdrawal rate of either zero or some constant value. Additional constraints were

added which required wells be utilized in both time periods if they are utilized at all.

Although the solution to this reformulated problem reduced the number of wells by 36%,

the total pumping rate increased slightly. After performing yet another reformulation and

execution with respect to minimizing costs, the authors concluded that the intermediate

mixed-integer programming solution represented the best compromise between

withdrawal and installation costs.

Similar research was performed by Lall and Santini (1989) to extend the response

matrix approach to a nonlinear, multilayered, unconfined aquifer system. The Grinski

potential concept was utilized to develop a linear approximation of the nonlinear aquifer

system and was described as being analogous to the velocity potential for a confined

aquifer system. For this case, hydraulic head is directly related to the vertical depth and

hydraulic conductivity of each layer. The authors demonstrated that the steady-state

continuity equation is a linear function of Grinski potential when hydraulic heads were

converted to Grinski potentials. By incorporating the method of finite elements, the

aquifer responses were then modeled as linear functions of the Grinski potential. The

superposition principle was employed to sum the individual responses and obtain a

response matrix in terms of the Grinski potential. The use of the Grinski potential for

linear approximation was also shown to be applicable to transient ground water

hydraulics under certain conditions. The method was also demonstrated using three









different variations of the original dewatering problem.

Response functions and mixed integer programming were applied by Chau (1989) to

analyze pressure relief systems. Optimal well sites and discharge schedules were

determined while discharge was minimized and hydraulic heads were maintained with

respect to soil stability and ground water flow. Response functions were determined

from a simulation model and represented the drawdown as induced by discharge from

another well and time period. The effects of varying performance parameters which

represent aquifer characteristics, hydraulic head limits, well capacities, and discharge

elevation at well locations were evaluated. Two hypothetical examples were analyzed

to demonstrate the trade-offs between system parameters and system performance.

Yazicigil (1990) incorporated the response function approach to determine optimal

planning and operating policies of a multiaquifer system in Eastern Saudi Arabia.

MODFLOW, a three-dimensional finite-difference ground water flow model developed

by McDonald and Harbaugh (1988), was executed under transient conditions and used

to generate response coefficients for each well field. Since the flow model was

formulated to simulate an eight year planning period with monthly time steps, over

20,000 individual influence coefficients were developed. These response functions were

then combined with both linear and quadratic programming to determine optimal water

management strategies for the 52 well fields in the basin. Three different objective

functions were formulated which maximized withdrawals, minimized drawdowns, and

minimized pumping costs. Trade-off costs which related withdrawal to drawdown,

aquifer dewatering, and pumping costs were also determined from results of the

optimization model.

The response matrix technique has not only been applied to management models







30

with respect to ground water quantity, but also ground water quality. Colarullo et al.

(1984) demonstrated this approach using a hypothetical unconfined aquifer which had

historically been used for surface waste disposal, but was soon to be developed as a

fresh-water supply. A two-dimensional ground water flow model was incorporated to

develop linear response equations which represented the aquifer response due to a change

in pumpage rate. Using this same method, response equations were also defined for

pumpage induced velocities and incorporated in the model as constraints to limit

contaminant flow. The optimization model was formulated to determine what quantity

of water could be removed to supplement water service demands and how interception

wells should be operated to avoid contamination of fresh-water supplies. A nonlinear

optimization algorithm was utilized to identify optimal well discharges for supply and

interception wells. The authors verified previous assumptions of linearity for the

response matrices by executing the simulation model with the prescribed strategy.

Influence coefficients derived from an approximation of a solute transport model

were applied by Datta and Peralta (1985) to revise a quantity optimization model to

include quality. The authors applied the approach of Peralta and Killian (1985) to

optimize the potentiometric surface and identify the water use strategy required to

maintain the surface. Steady-state hydraulic stresses were determined from the

simulation model, and steady-state ground water concentrations were determined from

the solute transport model. The influence coefficients were determined based on the

hydraulic head levels required to meet quality limits, and then used in establishing new

hydraulic head constraints. The modified optimization model was created through the

use of constrained derivatives of the quadratic optimization model. The methodology







31

was found useful in determining concentrations in a subsystem of a larger model and/or

the influence of water quantity changes on water quality.

Similar to the effort presented by Tung (1986), Wagner and Gorelick (1987)

combined the response matrix approach with a stochastic optimization model for ground

water quality management. A two-dimensional finite element flow and solute transport

model was utilized to develop the head and concentration response, respectively, as a

function of pumping and aquifer parameters. Simulation and multiple regression were

used to develop parameter estimates and minimize the differences between simulated and

observed values. In order to incorporate parameter uncertainty, the response equations

were transformed from deterministic constraints to probabilistic constraints through first-

order, first- and second-moment analysis. The chance constrained nonlinear optimization

model was formulated to identify well locations and withdrawal rates for aquifer

remediation under specified parameter uncertainty.

The authors applied the management model to both steady-state and transient

conditions to determine which withdrawal strategy would satisfy water quality standards.

Results revealed that the prescribed location, number, and pumping rate of wells were

very sensitive to reliability level. Monte Carlo simulations verified that the true mean

concentration and the concentration predicted by first-order analyses were nearly identical

and normally distributed.

Finney et al. (1992) incorporated response equations into a management model

for the control of salt-water intrusion in a multiaquifer system in Jakarta, Indonesia.

Hydraulic response equations were also used by Willis and Finney (1988) in the

development of a simulation/optimization model for the management of seawater







32

intrusion in Yun-Lin ground water basin of southwestern Taiwan. These efforts along

with others are discussed in the following literature review of salt-water intrusion models.


2.2 SALT-WATER INTRUSION MODELS

In many coastal areas, excessive abstraction of ground water has resulted in a

decline of fresh-water potentiometric heads and therefore an intrusion of salt water.

When this encroachment may have adverse effects on water supply quality,

simulation/optimization models are applied to these coastal areas in order to minimize

the effect of salt-water intrusion on fresh-water supplies. Simulation models which

predict the location of the fresh-water/salt-water interface are combined with optimization

models to minimize encroachment of the interface while meeting pumping demands.

Some of the initial research involving the optimum exploitations of ground water

reserves located in coastal aquifers was performed by Cummings (1971) and Youngs

(1971). Cummings created a ground water management optimization model which dealt

with the economic impacts of aquifer exploitation and salt-water intrusion with respect

to time. The model focused on the interrelated problems between annual ground water

pumping rate and the annual cost of pumping. Although Cummings mentions how

hydrological simulation models and optimization models complement each other, most

of his research involved the economic side of ground water mining.

Youngs' (1971) research involved determining the optimum well conditions in

order to maximize fresh-water output from coastal aquifers. By performing an analysis

of the horizontal seepage, Youngs determined that the maximum pumping rate of fresh

water could be calculated precisely and that the optimum conditions for well installation







33

can be found which will produce the maximum continuous pumping rate. Knowing that

pore water pressure is reduced near an operating well which causes an upcoming of saline

water, the analysis identifies the pumping rate required to raise the interface to the

bottom of the well. Youngs then determined that maximum withdrawal is obtained when

wells are positioned to a elevation equal to sea level and are pumped to maximum

capacity. The analysis is presented using two hypothetical examples.

The dynamic programming approach was applied by Nutbrown et al. (1975) in

a digital simulation/optimization model in order to effectively and efficiently manage a

coastal aquifer near Brighton, England. The goal of their research was to predict the

maximum yield of the aquifer while considering the limiting factor of salt-water

intrusion. The simulation model was created to describe the transient effects of natural

infiltration and abstraction of both fresh and saline ground water. The dynamic

programming method was combined with the simulation model to generate abstraction

regimes which would maximize the final storage of fresh water in the aquifer.

The approach was based on the methodology that at regular intervals of time, the

spatial abstraction distribution for succeeding intervals could be calculated on the basis

of existing water levels. The concept that pumping rate was proportional to the

magnitude of local ground water flow was assumed. The model was then executed under

average infiltration conditions until cyclic equilibrium was attained. Once validity had

been determined, the model was applied to various drought and recharge conditions to

determine the various optimum scenarios.

As mentioned earlier in the Embedding Method section of this literature review,

Shamir et al. (1984) developed an annual operating plan for a coastal aquifer in Israel.







34

A linearized model of salt-water intrusion was combined with a flow simulation model

to represent the movement of the salt-water interface. This simulation model was

incorporated into a multi-objective linear programming model to produce the management

model.

A planning model was developed by Willis and Finney (1988) to control salt-

water intrusion and declining ground water levels in the Yun Lin ground water basin of

southwestern Taiwan. The simulation/optimization model applied hydraulic response

equations to relate the location and magnitude of ground water pumping and recharge to

the movement of the salt-water interface. Finite-difference methods were used in the

simulation model to approximate the aquifer's response to various management strategies.

The model was based on the following hydraulic assumptions: 1) Hydrodynamic

dispersion is negligible so the sharp interface theory is valid; 2) The Dupuit

approximation is valid throughout the aquifer system; 3) The aquifer base is

impermeable; 4) The hydraulic conductivity and storage coefficients are invariant with

depth; and 5) Leakage to or from the aquifer is at steady-state.

The solution algorithms were based on the influence-coefficient method combined

with quadratic programming and also the reduced-gradient method combined with a

quasi-Newton algorithm. The influence-coefficient algorithm is based on hydraulic

response equations of the ground water system, and the coefficients are applied to the

quadratic programming optimization problem of the management model. The quadratic

program determines the optimal direction vector which is then used to revise the current

solution using a gradient-based algorithm. The reduced-gradient/quasi-Newton algorithm

can also be used to determine optimal solutions of the planning model. This algorithm







35

is implemented using MINOS and is able to determine a feasible decent direction vector

at any iteration. The simulation/optimization model was created using historical ground

water data for the basin in order to develop optimal pumping and recharge schedules.

The authors made the following conclusions based on the results of the

optimization analysis: 1) Both algorithms produced stable and reliable solutions to the

salt-water management problem; 2) Greatly differing pumping and recharge schedules

produced essentially the same objective function values for the applied basin; 3) The

management problem is characterized by local optimality problems; and 4) Different

starting solutions for the algorithms produce different optimal schedules.

Finney et al. (1992) incorporated response equations into a management model

for the control of salt-water intrusion in a multiaquifer system in Jakarta, Indonesia.

Response functions relating pumpage to the location and movement of the salt-water

interface were developed from a finite-difference simulation model. Within the model,

multiple aquifer systems were linked through their recharge terms. The response

equations were then included as part of a nonlinear optimization model which sought to

minimize the total squared volume of salt water in each aquifer. Initial attempts to reach

a solution with the MINOS programming package resulted in solutions that were not truly

local optimum values. To correct the problem, Box's sequential search algorithm was

used. This resulted in a 20% improvement over the MINOS generated values. Finney

et al. concluded that the combined use of simulation and optimization was able to reduce

the magnitude of salt-water intrusion in the Jakarta basin by 6%.









2.3 INTERFACE PROGRAMS

Due to the increasing popularity of the U.S. Geological Survey Modular Three-

Dimensional Finite-Difference Ground water Flow Model (MODFLOW) and of ground

water optimization models, recent efforts have been made by various groups to combine

these models via interface computer programs. AQMAN3D and MODMAN are two

such programs which simplify the interaction and iteration process between MODFLOW

and particular standard optimization programs (GAMS, MINOS, etc.).

Puig et al. (1992) developed AQMAN3D by modifying the two-dimensional code

AQMAN originally developed by Lefkoff and Gorelick (1987). The revised version is

a mathematical programming system data set generator for aquifer management using

MODFLOW as its ground water flow simulation subroutine. The FORTRAN-77

computer code can be used to formulate a variety of aquifer management models;

depending on the chosen objective function to be optimized and the constraints imposed

on hydraulic conditions. The program creates input files to be used by any standard

optimization program using Mathematical Programming System (MPS) input format.

When AQMAN3D is used with an optimization program, the optimum pumping

and/or recharge strategy can be determined while ground water hydraulic conditions are

maintained within specified limits. The applied management function may be linear or

nonlinear and restrictions can be applied to ground water heads, gradients, and/or

velocities. The program is limited to confined or quasi-confined aquifer systems and to

nonlinear sources/sinks in the ground water flow model.

The AQMAN3D aquifer management modeling process is initiated by

qualitatively and quantitatively conceptualizing the aquifer system by using MODFLOW.







37

The simulation flow model is then calibrated to steady-state or transient conditions and

then executed to simulate particular unmanaged aquifer conditions. The optimization

portion of the model is then initiated by developing a objective function and a system of

constraints which reflect the aquifer management condition. The AQMAN3D program

is now executed to interact with MODFLOW to produce the MPS input file containing

the objective function, constraints, and response coefficient matrix. The MPS file is

applied to the standard optimization model to determine the optimal scenario for the

specified management conditions. This optimal scenario should then be applied back into

the MODFLOW model to observe and verify the response of the aquifer system.

Another model which has been developed to perform the interface operations

between MODFLOW and a standard optimization program is recognized as MODMAN

(MODFLOW Management An Optimization Module for MODFLOW). GeoTrans, Inc.

(1990) developed MODMAN for use by the Southwest Florida Water Management

District to assist in determining optimum pumping scenarios in their region. MODMAN

is analogous to AQMAN3D in that it is a modified extension of AQMAN (Lefkoff and

Gorelick 1987) and accommodates three-dimensional problems. However, unlike

AQMAN3D, MODMAN is actually a linked module or subroutine of a revised version

of MODFLOW and can be executed in two modes. The MODFLOW code was modified

by the creators of MODMAN in order to facilitate the process of determining the

response coefficient matrix.

When used together with standard optimization software, MODMAN determines

the optimum strategy depicting the location and rate of extraction and/or injection wells

while satisfying user-specified constraints. One of many objective functions may be







38

maximized or minimized depending on the goals of the ground water manager. Once the

optimal pumping/injection strategy has been determined, MODMAN is incorporated

again by acting as a postprocessor of the optimization output data. MODMAN

automatically inserts the optimal well rates into the MODFLOW input file, executes a

simulation based on the optimal well strategy, indicates which constraints are binding in

the optimization model, and warns the user if nonlinearities have significantly affected

the optimization process.


2.4 SUMMARY

The approach of combining simulation and optimization models was first used in

the development of management models in the late 1960's and early 1970's. This type

of an approach offers the water resource manager a means of considering physical,

policy, and environmental factors simultaneously. Generally, the combined approaches

can be divided into two separate categories according to the method in which ground

water flow equations are incorporated into mathematical optimization models.

In the "embedding" technique, numerical or analytical solutions to ground water

flow and/or transport equations are written for each control node and are subsequently

included as constraints in the optimization model. This method has been successfully

applied to a large number of ground water management problems and has been

demonstrated to handle both steady-state and transient conditions. This approach has also

been used with problems involving linear and nonlinear ground water hydraulics and

single and multipurpose objective functions. Although the ability to accurately simulate

physical processes of an aquifer system is a major advantage, there is a dimensionality








39

problem associated with the embedding approach when applied to large scale management

problems. Therefore, this technique is generally only applicable to small aquifer systems

and steady-state management problems.

The "response matrix" approach has also been demonstrated as a technique to

combine simulation and optimization models for ground water management. As opposed

to the embedding technique however, this method is capable of handling large scale

management problems. The response matrix is composed of coefficients which represent

the influence of specified wells on the aquifer system. The aquifer response in terms of

change in hydraulic head, or drawdown, is usually derived from a simulation model.

The response is then expressed as simple algebraic functions in the mathematical model

using the principle of superposition. Constraints are incorporated into the algebraic

functions to limit drawdown at specified control points. Because equations are developed

only for the selected control points, use of the response matrix method generally results

in a reduced problem.

Additional ground water optimization methods including differential dynamic

programming, multi-objective programming, and quadratic programming were reviewed.

Techniques such as combining embedding and response matrix approaches for a

simulation/optimization model and revising an existing quantity optimization model to

incorporate quality were also discussed.

An assortment of optimization models were reviewed which incorporated varying

methods of managing coastal aquifers while minimizing the adverse effects of salt-water

intrusion. Many of the models were combined with simulation models which applied

finite-difference methods to determine the aquifer's response to varying pumping








40

strategies. All of the models assumed that the effect of hydraulic dispersion between

fresh and salt water was negligible so that the "sharp interface" concept was acceptable.

In order to simplify the interaction process between the ground water simulation

model and the numerical programming software, interface programs have been developed

which link the ground water flow model MODFLOW with a standard optimization

program. Two such models which were reviewed are AQMAN3D and MODMAN.

Both models are extended modifications of AQMAN (a previously developed two-

dimensional interface model), incorporate the response matrix approach, and

accommodate three-dimensional problems. MODMAN differs from AQMAN3D in that

it is actually a module of MODFLOW, it can be applied in a second mode as a

postprocessor, and it is more "user-friendly".

Existing simulation/optimization models pertaining to ground water resource

allocation and management are displayed in tabular form in Table 2.1. The model name,

authorss, type, aquifer condition, optimization technique, objective function, and

constraints are shown for easy comparison.












Table 2.1 Ground water optimization models.


Author Type Equation / Solution Objective Function Constraints Optimization Model
[Year] [Dimension] Aquifer Condition Technique [Liner / Nonlinear] [Linear / Nonlinear] Technique Name


Ahlfeld Quantity / Saturated Confined Linear / Finite Minimize Total Pumping Rates, Linear VCON
[1988] Quality Steady / Unsteady Element Pumping Hydraulic Head, Programming
[2-D] [Linear] Magnitude & Direction
of Groundwater
Velocity [Linear]

Ahlfeld & Quantity / Saturated Confined Linear / Finite Minimize Total Minimum Nonlinear GW2SEN
Pinder Quality Steady / Unsteady Element Volume of Pumping Concentrations at All Programming
[1988] [1 & 2-D] [Linear] Nodes at Future Times
[Non-linear]

Alley Quantity Unsteady Confined Linear / Finite Maximize Well Flow Rates, Linear AQMG
et al. [2-D] Saturated Difference Hydraulic Heads Hydraulic Heads Programming
[1976] [Linear] [Linear]

Aly & Quantity Steady / Unsteady Theis Analytical Minimize Extraction, Hydraulic Gradients, Linear, Quadratic, or US/WELL
Peralta [2-D] Confined / [Well Function] Injection Extraction or Injection Nonlinear
Unconfined Deterministic / [Linear / Nonlinear] Rates, Hydraulic Head Programming
Stochastic [Linear / Nonlinear]

Aquado & Quantity Steady Unconfined Linearized / Finite Minimize Total Hydraulic Heads, Linear OPAQD
Remson [2-D] Saturated Difference Pumping Pumping Rates Programming
[1974] [Linear] [Linear]

Deninger Quantity Unsteady,Unconfined Analytical Maximize Well Drawdown, Linear SAWSS
[1970] Radial Saturated [Well Formula] Production Well Facility Programming
[Linear] [Linear]

Elango & Quantity Steady Confined Linear / Finite Maximize Total Flow Equilibrium, Linear
Rouve [2-D] Saturated Element Pumpages Piezometric Heads, Programming
[1980] [Linear] Pumping Capacities
[Linear]












Table 2.1 continued


Author Type Equation / Solution Objective Function Constraints Optimization Model
[Year] [Dimension] Aquifer Condition Technique [Liner / Nonlinear] [Linear / Nonlinear] Technique Name


Gorelick Quality Steady State Linear / Finite Maximize Waste Water Quality Linear OLMWDF
& Remson [2-D] Flow and Transport Difference Disposal [Linear] Programming
[1982] Confined Saturated [Linear]

Haimes & Quantity Aquifer / Stream Linearized / Maximize User's Water Requirements, Quadratic MGWSW
Dretzen [2-D] System, Unsteady- Cell Model Net Benefit Lift & Pumping limits, Programming
[1977] Unconfined [Quadratic] Capacity of Recharge
Saturated Facility [Linear]

Heidari Quantity Unsteady,Unconfined Linearized / Finite Maximize Pumping Pumping Demands, Linear LSTLP
[1982] [2-D] Saturated Difference Rates over Time Drawdown Programming
[Linear] [Linear]

Koltermann Quantity Unsteady Confined Linear / Finite Maximize Hydraulic Groundwater Flow, Linear
[1983] [2-D] Saturated Difference Head and Minimize Piezometric Heads, Programming
Water Transfer and Pumping Capacities
Recharge [Linear] [Linear]

Larson Quantity Steady Unconfined Linearized / Finite Maximize Steady Pumping Rates, Mixed Integer OTGWD
et al. [2-D] Difference State Pumping Number of Wells, Linear
[1977] [Linear] Drawdown [Linear] Programming

Molz & Quantity Steady Confined Linear / Finite Minimize Total Hydraulic Heads, Linear HGCAQ
Bell [2-D] Saturated Difference Pumping Head Gradients Programming
[1977] [Linear] [Linear]

Peralta Quantity / Confined Saturated Linearized Boussinesq, Minimize Changes in Steady/Unsteady Flow Linear Goal MODCON
et al. Quality Steady / Unsteady Linearized & Nonlinear Piezometric Head and Transport Water Programming
[1987] [2-D] Transport/ Method for [Linear] Quality [Linear]
Characteristics

Peralta Quantity Steady / Unsteady Linear / Finite Minimize Withdrawal Head, Head Gradient, Linear US/REMAX
et al. [3-D] Confined / Unconfined Difference and Recharge Rates Flow Velocities, Programming
[1992] Saturated/Unsaturated [Linear] Demand, Capacity &
Pumping [Linear]












Table 2.1 continued


Author Type Equation / Solution Objective Function Constraints Optimization Model
[Year] [Dimension] Aquifer Condition Technique [Liner / Nonlinear] [Linear / Nonlinear] Technique Name


Rosenwald Quantity Unsteady-Unconfined Linearized / Finite Minimize Total Production Demand, Mixed Integer OLWR
& Green [2-D] Saturated Difference Pumping Number of Wells Linear
[1974] [Linear] [Linear] Programming

Shamir Quantity Unsteady-Unconfined Linearized / Finite Minimize Energy Pumping Demands, Linear OPOCAQ
et al. [2-D] Saturated Difference Demands for Import/Export Fluxes, Programming
[1983] Pumping / Recharge Drawdown Position,
[Linear] Water Quality
Limits [Linear]

Takahashi Quantity Confined/Unconfined Linear / Finite Maximize Extraction Potentiomentric Head, Linear USUGWM
& Peralta [Quasi 3-D] Steady Difference [Linear] Pumping Rate Programming
[1991] [Linear]

Willis Quantity / Confined Saturated Linear / Finite Maximize Lowest of Water Target, Linear PMMGWQ
[1979] Quality Unsteady Transport Element Waste Concentration Pumping & Injection, Programming
[2-D] Quasi-Steady Flow [Linear] Water Quality [Linear]

Willis Quantity Unsteady Confined/ Boussinesq Equation/ Maximize Sum of Agricultural Demand, Linear UARGWM
[1983] [2-D] Unconfined Finite Element Heads, Minimze deficit Heads, Well Capacity Programming
Analytical [Linear] [Linear]

Yazicigil Quantity Unsteady Confined Linear / Finite Minimize Total Water Demands, Linear OPMRA
et al. [2-D] Saturated Difference Pumping Maximum & Minimum Programming
[1987] [Linear] Pumping Rates,
Drawdowns [Linear]

Yazicigill & Quantity Saturated Confined Linear / Finite Maximize Total Water Demands, Linear OPMGW
Rashee- [3-D] Steady / Unsteady Difference Hydraulic Head Hydraulic head bounds, Programming
duddin [Linear] Maximize Pump Rates
[1987] [Linear]

Reference: El-Kadi et al. (1991)














CHAPTER 3
3.0 GROUND WATER SIMULATION MODELS


From the literature review, it was determined that the most efficient general

formulation of the allocation models would be one containing aquifer response constraints

developed from steady-state unit response matrices created with ground water flow and

transport simulation models. Two- and three-dimensional finite-difference ground water

simulation models have been employed extensively for the last twenty years by the

United States Geological Survey and other ground water professionals. These numerical

simulation models apply particular boundary conditions to a spatially discretized aquifer

system in order to predict potentiometric heads, fluxes, and solute concentrations

throughout a specified region. These boundary conditions include the size, shape,

hydrogeologic framework, hydraulic characteristics, and particular known fluxes affecting

the aquifer system. Although there are various ground water flow simulation programs

available, MODFLOW was used in this project, because a simulation model had already

been developed for the project area. Similarly the solute transport model program

DSTRAM was incorporated into the project because it had been successfully implemented

in the project area.

MODFLOW is a modular three-dimensional finite-difference ground water flow

program created by the United States Geological Survey (McDonald and Harbaugh

1988). The model is based on the following well known governing equations describing

44










the movement of an incompressible fluid through porous material:

a a h a h a ah ah
(K ) + (K -) + (K ) W= S (3.1)
x ax aY ay az zzaz sat

where K,, Ky, and K,, are values of hydraulic conductivity along the x, y, and z

coordinate axes, which are assumed to be parallel to the primary axes of hydraulic

conductivity (Lt'). H is the potentiometric head (L), W is a volumetric flux per unit

volume and represents sources and/or sinks of water (t'), S, is the specific storage of the

porous material (L'), and t is time (t).

Equation 3.1 approximates flow under non-equilibrium conditions in a

heterogeneous and anisotropic medium, assuming the principal axes of hydraulic

conductivity are aligned with the coordinate directions. Since analytical solutions to this

equation are rarely possible, the finite-difference numerical method is implemented to

procure approximate solutions. In this method, the continuous system described by the

partial-differential equation is replaced by a finite set of discretized equations in time and

space. Continuous partial derivatives are replaced by discrete algebraic functions

describing the change in pressure head at the distinct points. A system of linear

algebraic difference equations results from this methodology, and its solution produces

values of pressure head at specific points and time.

MODFLOW is able to simulate aquifer systems with layers that are confined,

unconfined, or a combination of both. Each cell of every layer is specified as being

inactive (no flow), active (variable head), or having a constant head. Boundary

conditions such as specific flux, specific head, or a head-dependant flux can be applied









46

in the model. The model is capable of simulating external flows such as discharge and

injection wells, rivers and streams, evapotranspiration, precipitation, and agricultural

drains. Because of its diversity and effectiveness, MODFLOW has become one of the

most popular ground water flow simulation models amongst hydrogeologic professionals.

DSTRAM developed by HydroGeologic Inc. is a three-dimensional numerical

finite element program that simulates fluid flow and solute transport saturated porous

media. The code is capable of performing several types of analysis. These include

ground water flow analyses, trace concentration solute transport analyses, and density

dependent coupled flow and transport analyses. The code is based on the following

governing equation for three-dimensional density-dependent flow and transport in an

aquifer system:


a I p gej) p i,j = 1,2,3 (3.2)
axi aPxi at

where p is fluid pressure, klj is the intrinsic permeability tensor, p is the fluid density,

1A is the dynamic viscosity, g is the gravitational acceleration, ej is the unit vector in the

upward vertical direction, and 4 is the porosity of the porous medium.

DSTRAM analysis can be performed in an areal plane, a vertical cross-section,

an axisymmetric configuration, or a fully three-dimensional mode. Because of its special

design features, DSTRAM is capable of handling a wide range of complex three-

dimensional, steady-state or transient, field problems and producing values of solute

concentration at specific points and times (Huyakorn and Panday, 1991).











3.1 REGIONAL FLOW SIMULATION MODEL

A regional model covering the Volusia County study area was acquired from

SJRWMD. The model was originally developed by Geraghty & Miller, Inc. (1991), but

several modifications were incorporated by SJRWMD to increase simulation accuracy

(Williams, 1993). The model as received is capable of simulating three different steady-

state stress conditions with respect to time, redevelopment, year 1988, and year 2010.

The finite-difference grid of the regional model area consists of 86 columns, 91

rows, and 5 layers for a total of 39,130 cells. The model covers an area of

approximately 1,850 square miles and consists of planar cell spacings varying from 0.25

to 2.0 miles. The model is bounded by the St. John's river on the west and extends

approximately seven miles off the Atlantic Coast on the east.

The 5 layers of the model aquifer system consist of the surficial aquifer in layer

1, the Upper Floridan aquifer in layer 2, the middle semi-confining unit of the Floridan

aquifer in layer 3, and the Lower Floridan aquifer in layers 4 and 5. The upper

confining unit which separates the surficial and Upper Floridan aquifers are

interconnected through the use of leakance coefficients which represent the hydraulic

connection between the two layers. The surficial aquifer is modeled as unconfined and

the Floridan aquifer system is modeled as confined.

The boundary conditions used in the development of the regional model included

constant flux, constant head, and head-dependant flux boundaries. Constant head

boundary conditions were applied to regions where impacts of pumping were assumed

to be negligible. These areas include the Atlantic Ocean, Halifax River, St. Johns River,









48

Blue Springs vicinity, and miscellaneous lakes within the county (See Figure 3.1).

Constant heads were also applied along the edge of the model near Blue Springs in the

surficial aquifer to allow flow from Lake County. Constant flux boundary conditions

were used in the model to represent the various discharge and recharge areas. The

discharge flow rates for each cell were calculated by summing the pumping rates from

all wells located within that cell for a specific layer.

Head-dependant flux boundaries are often known as mixed-type boundary

conditions since they are essentially a combination of specified head and constant flux

conditions. They are applied to represent an unknown flux which is dependant on a

specified, hydraulically connected, and externally-located head. Blue Spring, Ponce De

Leon Springs, Gemini Springs, and a variety of small creeks were modeled using head-

dependant flux boundaries. The western and northern boundaries of the Floridan Aquifer

system were defined with a type of head-dependant boundary called general head

boundaries in MODFLOW. These conditions provide adequate inflow to the model

while minimizing boundary effects due to varying pumping rates.



3.2 FLOW SIMULATION MODEL FOR PROJECT AREA

The five optimization models produced through the efforts of this project were

applied to the region defined by SJRWMD (See Figure 3.2). This project area covers

a smaller subarea of the above described regional flow simulation model. To facilitate

the development of this model, the Volusia County regional flow simulation model was

modified to create a MODFLOW subregional flow simulation model for the project area.













































I RvA.rCa


EXPLANATION

= SURFACE WATER

***..** MAJOR DRAINAGE
-- -- MIN DRAINAGE


BODIES


DMDE
nn fn


0
\
Jk


(from Rutledge, 1985o. fig 22)
09 SPRING


0 5 10 KILOMETERS


Figure 3.1

Map of Volusia County, Florida.














Fagler


Putnam


Lake


REGIONAL MODEL


FHI


PROJECT MODEL


volusia





0 5 miles
o0 5 kilomes I



Figure 3.2
Volusia County regional and project
numerical model grid discretizations.









51

The process of developing the project flow simulation model was initiated by

essentially cropping the northern, western, and southern portions of the regional model.

(The eastern border coincides with the larger regional model). This process involved

modifying all the regional model input files so they would correspond with the project

geometry. Computer programs written in FORTRAN code were developed to facilitate

this process. Removed from the regional model were 19 columns at the western border

and 20 and 11 rows from the northern and southern borders respectively. The location

of the project model with respect to the regional model is depicted in Figure 3.2. Figure

3.3 provides a reference for flow model cell locations.

The next step in the modification process involved applying the appropriate

boundary conditions to the project model in order to simulate the regional model as

closely as possible. Boundary conditions were applied using the general head boundary

option on the three boundaries of the model which do not coincide with regional

boundaries. This was performed by applying a constant external head value and a value

which represents the conductance between the external location and the boundary.

Effective conductance values between the regional and project borders were

calculated for each layer. Initially, the conductance of each cell located outside the

model was calculated in the direction perpendicular to the border. The effective

conductances were then calculated between the regional and project borders using the

individual cell conductances, or






























SAmOH


Figure 3.3
Project Flow Model
Cell Location Reference Map


1 L[TITITl


0


..... :.. .


^ ^ ^ ^ ^ ^ ^ll IN 1111 1 11 1











1 1 1 1. (3.2)
C C1 C2 C.

where C is conductance, K is hydraulic conductivity, A is cross sectional area

perpendicular to flow, L is length of flow path, and n is the number of cells.

FORTRAN programs were developed to facilitate the process of gathering and

manipulating the geometric and hydraulic data for these calculations. Results from the

execution of the regional model were also used to determine the conductance and external

head values for the general head boundary conditions. These boundary conditions varied

depending on the simulated condition of the regional model redevelopmentn, year 1988,

or year 2010). For layers 2 through 5 of the regional model, general head boundary

conditions were applied to the northern and western borders. Therefore, external head

values used for the regional model were applied to the project model, and conductance

values applied to the project model were determined by combining the conductance

values used for the regional model with the conductance between the borders of the two

models. To determine the external head values to be applied at the southern border of

the project model, the regional model was executed for specific conditions to determine

head values along its southern border. This process assures that the influences of the

regional models on the smaller project model is incorporated in the simulation.

For the surficial aquifer (layer 1), there are often many bodies of water between

the regional and project borders which are modeled as constant heads (especially on the

western border). In these cases, the constant head values which were closest and

external to project border were used as the external head values for the project model.









54

The effective conductances were calculated in these cases between the project border and

the location of the external head. In cases where constant heads did not exist between

the regional and project borders, effective conductances were calculated between the two

borders as described previously. External head values for the project model were

determined by using the corresponding head values along the regional border.



3.3 SOLUTE TRANSPORT SIMULATION MODEL

In order to incorporate water quality constraints into several of the optimization

models a solute transport simulation model was needed. The DSTRAM program was

used for the project area was chosen for this application because SJRWMD had already

developed a model for another project which covers area common to the subregion

modeled with MODFLOW. The DSTRAM model incorporates an uniform grid of 30

columns by 51 row over the project area. This model uses 15 layers to simulate the

Upper Floridan aquifer, Lower Floridan aquifer, and the confining units in the system.

The majority of the information needed to create the DSTRAM project model was taken

from the regional MODFLOW model (Williams, 1994). This information includes

boundary conditions, the modeled region size and shape, hydrogeologic framework,

hydraulic characteristics, and known fluxes affecting the aquifer system. DSTRAM

simulation runs for the project were made by SJRWMD as needed during the

optimization model development.














CHAPTER 4
4.0 WATER SUPPLY NEEDS AND SOURCES


Water supply needs and sources are currently being evaluated by SJRWMD under

the District Water Management Plan. The needs and sources of Volusia County have

been recorded and projected by SJRWMD to the year 2010 for several use categories.

Projections of future water use are based on historical trends, local government

comprehensive plans, and direct communication with both public and private public

supply utilities (SJRWMD, 1992). Computer data files depicting municipal, agricultural,

and miscellaneous well information and water use needs have been obtained from the

District. Maps and tables have also been obtained which depict wastewater treatment

plant information. This data was used to incorporate water reuse in the development of

the ground water management model.

Municipal well data include the municipality and water treatment plant name, the

water treatment plant permitted capacity, existing and proposed wells which service these

areas, location of these wells in state-planar and numerical-grid coordinates, pumping

rates for years 1988 and 1990, and the predicted pumping rate for year 2010. See Table

4.1. Within the project area, there are 97 existing and 74 proposed wells in 17 separate

well zones which supply water to 8 different municipalities or water service areas. Nine

different water treatment plants supply these service areas. These municipalities or water

service areas include Holly Hill, Port Orange, Spruce Creek, Daytona Beach, Ormond

Beach, Tymber Creek Utilities, The Trails, Inc., and a section of New Smyrna Beach,

55
















Table 4.1 Volusia County municipal well data for Project area.


Project 2010
State Planar Flow Model Well 1988 1990 Projected Water Permitted
Municipal Coord. Location Cell Location Grid Cell Pumpage Pumpage Pumpage Treatment Capacity
Water Service Area X Y row col. Name (cu ft/mo) (cu ftmo) (cu ft/mo) Plant Name (mgd)
*PORT ORANGE *** Port Orange 6.21


WESTERN WELLFIELD


POW Proposed







POW Proposed











POW Proposed


457956
457862
457594
457505
457592
457946
456001
455911
455999
456175
453961
454491
454046
457948
454222
453874
453874
453874
456084
456084
456084
422871
420211
425531
422877
420218
425537
422865
420205
425525
457946
457946
458355
458355
458355
458672
458672
458672


1736744
1732805
1730684
1730381
1729068
1728664
1734120
1733009
1731999
1730888
1734022
1732708
1731597
1729674
1730486
1735638
1735638
1735638
1729070
1729070
1729070
1756899
1756905
1756894
1759929
1759935
1759924
1753869
1753874
1753864
1729674
1732805
1729068
1729674
1732805
1729068
1729674
1732805


MWELLI
MWELL9
MWELL5
MWELL5
MWELL4
MWELLIO
MWELL3
MWELL6
MWELL7
MWELLII
MWELL12
MWELL8
MWELL8
MWELL13
MWELL2
MWELL14
MWELL14
MWELL14
MWELL15
MWELL15
MWELLI5
MWELL16
MWELL17
MWELL18
MWELL19
MWELL20
MWELL21
MWELL22
MWELL23
MWELL24
MWELL25
MWELL26
MWELL27
MWELL28
MWELL29
MWELL30
MWELL31
MWELL32


1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040
1079040


1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841
1184841


961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
961230
0
0
0
0
0
0
0
0
0
961230
961230
961230
961230
961230
961230
961230
961230


EASTERN WELLFIELD 491132 1745711 49 46 MWELL34 124503 136712 221822
491132 1746216 49 46 MWELL34 124503 136712 221822
491132 1746721 49 46 MWELL34 124503 136712 221822
491132 1746014 49 46 MWELL34 124503 136712 221822
490600 1745610 49 45 MWELL35 124503 136712 221822
491133 1748236 47 46 MWELL38 124503 136712 221822
491133 1748741 47 47 MWELL36 124503 136712 221822
491133 1749246 47 47 MWELL36 124503 136712 221822
490690 1749448 47 47 MWELL36 124503 136712 221822
490335 1749448 46 46 MWELL33 124503 136712 221822
489891 1749448 46 46 MWELL33 124503 136712 221822
489093 1749448 46 45 MWELL37 124503 136712 221822
488916 1749448 46 45 MWELL37 124503 136712 221822














Table 4.1 continued


Project 2010
State Planar Flow Model Well 1988 1990 Projected Water Permitted
Municipal Coord. Location Cell Location Grid Cell Pumpage Pumpage Pumpage Treatment Capacity
Water Service Area X Y row col. Name (cu fl/mo) (cu ft/mo) (cu ft/mo) Plant Name (mgd)
** DAYTONA BCH *** Daytona Beach 35.17
EASTERN WELLFIELD 478108 1765916 32 43 none 0 0 Marion Street
(7 wells inactive) 477221 1765513 32 42 none 0 0
476512 1765109 32 42 none 0 0
475182 1765110 32 41 none 0 0
474473 1764606 32 40 none 0 0
473852 1764303 32 40 none 0 0
472522 1763799 32 38 MWELL39 0 0
471990 1763496 32 38 MWELL39 2388368 2508913 1246680
471192 1763093 32 37 MWELL42 2388368 2508913 1246680
469862 1762589 32 36 MWELIA0 2388368 2508913 1246680
464897 1761482 31 32 MWEL41A 2388368 2508913 1246680
464188 1760978 31 32 MWELIAL 2388368 2508913 1246680
WESTERN WELLFIELD 462769 1760272 31 31 MWELL46 3796082 2092352 3110003 Ralph Brennan
461794 1761081 30 30 MWELLA7 3796082 2092352 3110003
458589 1747752 39 26 MWELL44 3796082 2092352 3110003
458145 1747551 39 25 MWELL52 3796082 2092352 3110003
457169 1747148 39 25 MWELL52 3796082 2092352 3110003
456371 1746846 39 24 MWELL43 3796082 2092352 3110003
455306 1746342 39 23 MWELIA5 3796082 2092352 3110003
454507 1746040 39 22 MWELLA8 3796082 2092352 3110003
454242 1746848 38 23 MWELLA9 3796082 2092352 3110003
453800 1747960 37 23 MWELL50 3796082 2092352 3110003
453269 1748769 36 22 MWELL51 3796082 2092352 3110003
WESTERN WELLFIELD 451854 1751800 34 22 MWELL53 0 2092352 3110003 South Daytona
(1988 construction) 451146 1752407 33 22 MWELL54 0 2092352 3110003
450969 1753115 33 22 MWELL54 0 2092352 3110003
451148 1753922 32 23 MWELL55 0 2092352 3110003
450972 1754933 31 23 MWELLS6 0 2092352 3110003
451062 1756044 31 23 MWELL56 0 2092352 3110003
449910 1756449 30 23 MWELL57 0 2092352 3110003
453265 1745638 38 21 MWELL58 0 2092352 3110003
452917 1750587 35 23 MWELL59 0 2092352 3110003
452297 1751093 34 23 MWELL60 0 2092352 3110003
DBW Proposed 451242 1757861 29 24 MWELL61 0 0 3110003
450977 1758872 29 24 MWELL61 0 0 3110003
450712 1759882 28 24 MWELL62 0 0 3110003
450448 1760893 27 24 MWELL63 0 0 3110003
450183 1761903 26 24 MWELL64 0 0 3110003


*SPRUCE CREEK *** 483674 1724907 56 34 MWELL66 658423 493538 326649 Spruce Creek 7.73
484295 1725513 56 35 MWELL65 658423 493538 326649
SC- Proposed 484118 1725311 56 34 MWELL66 0 0 326649
483496 1724705 56 34 MWELL66 0 0 326649
484739 1725815 56 35 MWELL65 0 0 326649

0** HOLLY HILL *** I I I Holly Hill 1.43
EASTERN WELLFIELD 484940 1784396 22 53 MWELL68 69192 66195 79694
485117 1784093 22 54 MWELL67 69192 66195 79694
485294 1783689 22 53 MWELL68 69192 66195 79694
485472 1784800 21 54 MWELL69 69192 66195 79694
484409 1784700 21 53 MWELL70 69192 66195 79694
482991 1784094 21 52 MWELL71 69192 66195 79694
WESTERN WELLFIELD 461629 1773101 22 34 MWELL72 593074 567314 597705
461896 1773606 22 34 MWELL72 593074 567314 597705
462251 1774009 22 35 MWELL73 593074 567314 597705
462694 1774211 22 35 MWELL73 593074 567314 597705
461632 1775424 21 35 MWELL74 593074 567314 597705
462163 1774918 21 35 MWELL74 593074 567314 597705
463402 1773705 22 35 MWELL73 593074 567314 597705
HHW Proposed 462783 1775019 21 35 MWELL74 0 0 597705














Table 4.1 continued
Project 2010
State Planar Flow Model Well 1988 1990 Projected Water Permitted
Municipal Coord. Location Cell Location Grid Cell Pumpage Pumpage Pumpage Treatment Capacity
Water Service Area X Y row col. Name (cu ft/mo) (cu fl/mo) (cu fl/mo) Plant Name (mgd)
*NEW SMYRNA BCH New Smyrna 3.60
SAMSULA WELLFIELD 471322 1701078 60 20 MWELL75 1207921 1290193 1521308 Beach
470701 1701483 60 20 MWELL75 1207921 1290193 1521308
SR44 WF Proposed 461913 1702702 59 16 MWELL76 0 0 1521308
461380 1702804 59 15 MWELL77 0 0 1521308
462358 1704318 58 16 MWELL78 0 0 1521308
461470 1704117 58 16 MWELL78 0 0 1521308
461915 1704722 58 16 MWELL78 0 0 1521308
461560 1705127 58 16 1MWELL78 0 0 1521308
ORMOND BCH *** Ormond Beach 8.00
DIVISION AVE 479276 1795409 12 53 MWELL80 917447 645276 659368
WELLFIELD 479365 1795308 12 53 MWELL8O 917447 645276 659368
478833 1793692 13 52 MWELL81 917447 645276 659368
478479 1794197 13 52 MWELL81 917447 645276 659368
478302 1794399 13 52 MWELL81 917447 645276 659368
478124 1794096 13 52 MWELL81 917447 645276 659368
476176 1793794 12 50 MWELL79 917447 645276 659368
475644 1793290 13 50 MWELL83 917447 645276 659368
475024 1793088 13 49 MWELL84 917447 645276 659368
478480 1796116 12 53 MWELL80 917447 645276 659368
478125 1795914 12 52 MWELL82 917447 645276 659368
475822 1794704 12 50 MWELL79 917447 645276 659368
SR 40 WELLFIELD 467319 1793397 10 44 MWELL86 2582442 645276 659368
464038 1789966 11 41 MWELL87 2582442 645276 659368
467940 1794912 9 45 MWELL88 0 645276 659368
468294 1794305 10 45 MWELL89 0 645276 659368
462089 1789564 11 39 MWELL85 2582442 645276 659368
HUDSON WELLFIELD 450488 1791092 6 32 MWELL90 0 645276 659368
450489 1792002 6 32 MWELL90 0 645276 659368
450490 1792911 5 32 MWELL91 0 645276 659368
450491 1793719 4 33 MWELL92 0 645276 659368
450492 1794628 4 33 MWELL92 0 645276 659368
450494 1795537 3 33 MWELL93 0 645276 659368
450495 1796446 3 33 MWELL93 0 645276 659368
451553 1792909 5 33 MWELL94 0 645276 659368
451997 1793818 5 34 MWELL95 0 645276 659368
451998 1794727 4 34 MWELL96 0 645276 659368
452001 1797151 3 35 MWELL97 0 645276 659368
449516 1792912 5 32 MWELL91 0 645276 659368
448719 1792913 4 31 MWELL98 0 645276 659368
OB- Proposed 419697 1764986 15 6 MWELL99 0 0 659368
CENTRAL RECHARGE 420758 1764075 16 7 MWELL100 0 0 659368
WELLFIELD 421643 1763265 16 7 MWELL100 0 0 659368
421995 1761951 17 7 MWELL101 0 0 659368
422080 1760436 18 7 MWELL102 0 0 659368
422166 1758820 20 6 MWELL103 0 0 659368
422163 1757708 20 6 MWELL103 0 0 659368
422160 1756193 21 5 MWELL104 0 0 659368
422159 1755385 22 5 MWELL105 0 0 659368
423769 1762452 18 8 MWELL106 0 0 659368
424833 1762652 18 9 MWELL107 0 0 659368
425719 1762549 18 10 MWELL108 0 0 659368
426872 1762749 18 11 MWELL109 0 0 659368
425544 1763762 17 10 MWELL110 0 0 659368
425280 1764671 17 10 MWELLI10 0 0 659368
420039 1759430 19 5 MWELLI1 0 0 659368
419152 1759129 18 4 MWELL112 0 0 659368
418000 1759333 18 3 MWELL113 0 0 659368
417820 1757920 19 3 MWELL114 0 0 659368
417639 1756405 20 2 MWELL115 0 0 659368
417548 1755395 21 2 MWELL116 1 0 0 659368
OB-Proposed 441343 1777571 13 23 MWELL117 0 0 659368
RIMARIDGE 441168 1778784 12 24 MWELL118 0 0 659368
**TYMBER CREEK UTIL *** 459436 1792900 8 39 MWELL119 372103 409982 728944 Tymber Creek 160000
THE TRAILS INC. '** 453145 1790887 7 34 MWELLI20 1299484 1299484 2355224 ???? ????
Total 100230505 105037423 174947036









(Figure 4.1).

Agricultural water service data include both agricultural and golf course well data.

This data depicts well locations in state-planar and numerical-grid coordinates, water

application types, and pumping rates for year 1990. See Table 4.2. These rates have

been predicted to remain approximately constant through the year 2010 (Geraghty &

Miller 1992). According to data obtained from SJRWMD, there are 80 agricultural and

golf course wells throughout the Volusia County subregion. The ground water

management model was formulated to supply a portion of the ground water needs with

effluent from wastewater treatment plants. The agricultural areas and wastewater

treatment plants are shown in Figure 4.2.

The miscellaneous well data includes the well description, the well location in

state-planar and numerical-grid coordinates, and the 1988 and 1990 pumping rates. The

Volusia County subregion consists of 9 miscellaneous wells which supply the Florida

Mining and Materials Department, the Florida Department of Education, and the Tomoka

Correctional Facility. Additional data depicting location and discharge rates of private

wells within the region for years 1988 and 2010 are also included. Data were supplied

by SJRWMD in a form to facilitate execution of the simulation model and subsequent

incorporation into the optimization model. However, these non-municipal wells were not

optimized during this study.

In order for the simulation/optimization model to predict a feasible water

management strategy, water service demands must be incorporated into the model.

Municipal and agricultural area water supply demands for the year 2010 were calculated

by taking the sum of all projected well discharge rates which supplied a specific water






















HOLLY HILLS


TYMBER CREEK
& THE TRAILS


DAYTONA BEACH


NEW SMYRNA BEACH


o 5mile
0 5kitmeras


SPrivate areas not managed


Figure 4.1
Location of Municipal Water Service Areas
within Project Area.


F'I













Table 4.2 Volusia County agricultural well data project area.


Project 1990 2010
State Planar Flow Model Estimated Projected
Agricultural Application Coord. Location Cell Location Well Grid Pumpage Pumpage
Service Area Number X Y row col Cell Name (cu ft/mo) (cu ft/mo)

AGAREA1 2-127-0396AN 495388 1736822 54 46 AGWELL2 55355 55355
2-127-0396AN 496541 1736519 54 46 AGWELL2 55355 55355
2-127-0396AN 495388 1737024 54 46 AGWELL2 55355 55355
2-127-0396AN 496452 1737327 54 47 AGWELL1 55355 55355
2-127-0396AN 496452 1736923 54 47 AGWELL1 55355 55355

AGAREA2 2-127-0269AN 479539 1789550 16 51 AGWELL4 191139 191139
2-127-0269AN 481045 1789953 17 53 AGWELL5 191139 191139
2-127-0269AN 482639 1790256 17 54 AGWELL3 191139 191139

AGAREA3 2-127-0279AU 505496 1758941 45 58 AGWELL6 14255 14255
2-127-0279AU 505674 1757931 45 58 AGWELL6 14255 14255
2-127-0279AU 506117 1758032 45 58 AGWELL6 14255 14255
2-127-0279AU 505674 1758638 45 58 AGWELL6 14255 14255
2-127-0279AU 505496 1758941 45 58 AGWELL6 14255 14255
2-127-0279AU 505851 1758436 45 58 AGWELL6 14255 14255
2-127-0279AU 505762 1758739 45 58 AGWELL6 14255 14255
2-127-0279AU 505940 1758234 45 58 AGWELL6 14255 14255
2-127-0279AU 505851 1758436 45 58 AGWELL6 14255 14255
2-127-0279AU 505585 1758840 45 58 AGWELL6 14255 14255
2-127-0279AU 505674 1758739 45 58 AGWELL6 14255 14255
2-127-0279AU 505585 1757931 45 58 AGWELL6 14255 14255
2-127-0279AU 506738 1756921 46 58 AGWELL7 14255 14255
2-127-0279AU 506206 1757830 46 58 AGWELL7 14255 14255
2-127-0279AU 506294 1757729 46 58 AGWELL7 14255 14255
2-127-0279AU 506383 1757527 46 58 AGWELL7 14255 14255
2-127-0279AU 506206 1756719 46 58 AGWELL7 14255 14255
2-127-0279AU 506472 1757325 46 58 AGWELL7 14255 14255
2-127-0279AU 506826 1757022 46 58 AGWELL7 14255 14255
2-127-0279AU 505940 1756820 46 58 AGWELL7 14255 14255
2-127-0279AU 506560 1757123 46 58 AGWELL7 14255 14255
2-127-0279AU 506383 1757628 46 58 AGWELL7 14255 14255
2-127-0279AU 506206 1757224 46 58 AGWELL7 14255 14255
2-127-0279AU 506383 1757325 46 58 AGWELL7 14255 14255
2-127-0279AU 506117 1756618 46 58 AGWELL7 14255 14255

AGAREA4 2-127-0565ANV 484418 1806921 6 58 AGWELL12 92010 92010
2-127-0565ANV 484506 1807022 6 59 AGWELL11 92010 92010
2-127-0565ANV 485480 1805708 7 59 AGWELL10 92010 92010
2-127-0565ANV 485745 1804799 8 59 AGWELL8 92010 92010
2-127-0565ANV 485833 1804395 8 59 AGWELL8 92010 92010
2-127-0565ANV 485834 1804698 8 59 AGWELL8 92010 92010
2-127-0565ANV 485833 1803183 9 58 AGWELL9 92010 92010

AGAREA5 2-127-0647AUS 469188 1804304 3 49 AGWELL16 193220 193220
2-127-0647AUS 468566 1802688 4 48 AGWELL14 193220 193220
2-127-0647AUS 470692 1803596 4 50 AGWELL13 193220 193220
2-127-0647AUS 470071 1802182 5 49 AGWELL15 193220 193220












Table 4.2 continued


Project 1990 2010
State Planar Flow Model Estimated Projected
Agricultural Application Coord. Location Cell Location Well Grid Pumpage Pumpage
Service Area Number X Y row col Cell Name (cu ft/mo) (cu ft/mo)

AGAREA6 2-127-0147AU 483584 1722786 56 33 AGWELL17 302712 302712
2-127-0147AU 485536 1723189 56 35 AGWELL18 302712 302712
2-127-0147AU 485536 1723189 56 35 AGWELL18 302712 302712
2-127-0147AU 484118 1726321 56 35 AGWELL18 302712 302712

AGAREA7 2-127-0236AN 465995 1701284 59 17 AGWELL19 857472 857472
2-127-0236AN 468304 1701787 59 18 AGWELL20 857472 857472
2-127-0236AN 466614 1698456 60 17 AGWELL21 857472 857472

AGAREA8 2-127-0237AN 487399 1719754 57 35 AGWELL22 141258 141258
2-127-0237AN 488907 1717128 58 35 AGWELL24 141258 141258
2-127-0237AN 488907 1717128 58 35 AGWELL24 141258 141258
2-127-0237AN 488907 1717128 58 35 AGWELL24 141258 141258
2-127-0237AN 488907 1717128 58 35 AGWELL24 141258 141258
2-127-0237AN 488907 1717128 58 35 AGWELL24 141258 141258
2-127-0237AN 488907 1717128 58 35 AGWELL24 141258 141258
2-127-0237AN 490416 1719652 58 37 AGWELL23 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258
2-127-0237AN 490681 1715511 59 36 AGWELL25 141258 141258

AGAREA9 2-127-0085AN 483427 1768641 32 47 AGWELL26 24536 24536
2-127-0085AN 483782 1768843 32 48 AGWELL28 24536 24536
2-127-0085AN 483516 1768742 32 48 AGWELL28 24536 24536
2-127-0085AN 484047 1768943 32 48 AGWELL28 24536 24536
2-127-0085AN 483427 1768944 32 48 AGWELL28 24536 24536
2-127-0085AN 483604 1769247 32 48 AGWELL28 24536 24536
2-127-0085AN 484668 1770155 31 49 AGWELL27 24536 24536

Total 9686206 9686206


















Area 4


ea \
Area 5 1'


I--


-, \
Daytona Beach Bethune Pt. WIP
Area 9


Il \Area 3


F1 /
/ Port Orange WrP
// Areal
/- olusia Co.- Spruce Cr. WIP
SArea 6 (r 7
L -j
Area 8

Area 7 r





0 ,5 miles
0 5 kilanem rs




Figure 4.2
Agricultural Areas and Wastewater Treatment Plants
with respect to Numerical Model Grid Location


Hd


\
X







64

service area. Municipal and agricultural service area demands are summarized in Tables

4.3 and 4.4, respectively, for years 1988 and 2010.

Municipal water service areas were generally defined by individual well fields

and/or water treatment plants which supply ground water to the municipalities. Proposed

well fields which were not in service before 1991 were not considered as individual water

service areas. They were incorporated in the optimization model as possible well sites

and their year 2010 withdrawal rates contributed to projected demand. However, these

locations were not specified as demand areas. The Central Recharge and Rima Ridge

well fields are two such proposed areas which will supply the Ormond Beach

municipality by the year 2010. Therefore, the year 2010 projected demand of these areas

were equally divided between the State Road 40 and Hudson well fields. These well

fields were selected because of the proximity to the proposed Central and Rima well

fields.

Defining water service areas as individual well fields is only one method of

designating demand areas. The definition of a water service area within the optimization

model can easily be revised depending on the objectives of the water resource manager.

This includes defining each municipality, or the entire Volusia County region for that

matter, as a single water service area.

Effluent from wastewater treatment plants can often be used to replace ground

water currently being used to irrigate agricultural areas and golf courses. The

wastewater treatment plant data includes tables depicting facility names and locations,

current permitted capacities and mean flow rates, and predicted permitted capacities and

mean flow rates for the year 2010. There are 9 wastewater treatment facilities in the











Table 4.3 Year 1988 and projected year 2010 demand rates for
municipal water service areas.

Projected
Year 1988 Year 2010
Well Demand Demand
ID Water Service Area Name Rate (cfd) Rate (cfd)
POW Port Orange West Wellfield 532127 916458
POE Port Orange East Wellfield 53211 94807
DBM Daytona Beach East Marion TP 392609 204934
DBB Daytona Bch West Brennan TP 1372832 1124716
DBS Daytona Beach West South TP 0 1533702
SCK Spruce Creek 43294 53695
HHE Holly Hill East Wellfield 13650 15720
HHW Holly Hill West Wellfield 136489 157205
NSB Smyrna Beach / Samsula 79425 400124
OBD Ormond Beach Division Ave WF 361952 260135
OB4 Ormond Beach State Rd. 40 WF 254706 346848
OBH Ormond Beach Hudson Wellfield 0 541950
TCU Tymber Creek Utilities 12234 24298
TTI The Trails, Inc. 42723 78507

Total 3295252 5753099


Table 4.4 Year 1988 and projected year 2010 demand rates for
agricultural water service areas.

Projected
Year 1988 Year 2010
Well Demand Demand
ID Agricultural Application Number Rate (cfd) Rate (cfd)
AGAREA1 AG-0396AN 9100 9100
AGAREA2 AG-0269AN 18852 18852
AGAREA3 AG-0279AU 11717 11717
AGAREA4 AG-0565ANV 21175 21175
AGAREAS AG-0647AUS 25408 25408
AGAREA6 AG-0147AU 39808 39808
AGAREA7 AG-0236AN 84573 84573
AGAREA8 AG-0237AN 102170 102170
AGAREA9 AG-0085AN 5645 5645

Total 318448 318448








66

Volusia County subregion with permitted capacities of 0.1 mgd or greater. The

wastewater and agricultural data was reviewed and compared to determine which

agricultural areas could supplement their ground water demand with wastewater plant

effluent. From this review and based on proximity to agricultural areas, it was

determined that 5 of the 9 treatment plants could feasibly supplement the agricultural

demands. Constraints for the ground water optimization model were then formulated

based on the results of the review and comparison. See Table 4.5.

The needs and sources evaluation of Volusia County by SJRWMD includes a

water resources impact assessment. This assessment involves identifying potential

problem areas when the year 2010 projected allocation strategy is implemented. One

such problem area involves the possibility of hydrophytes being harmed as a result of

declines in water table. Based on the evaluation, several areas of Volusia County have

a high potential for vegetative harm if proposed strategies were incorporated.

In order to reduce the possibility of harm to wetland vegetation, areas depicted

as having high potential for harm were incorporated into the optimization model as

control points. These control points are locations where pressure head and/or drawdown

values were constrained or optimized. To reduce the number of control points and

therefore the size of the optimization model, 100 out of approximately 500 points were

selected throughout the high potential harm areas. Figure 4.3 displays the control point

locations along with areas of high, medium, and low potential harm.



















Table 4.5 Volusia County subregional model wastewater treatment plant data.


Coordinate Numerical Grid
Location Cell Location Permit Capacity
Supply ID Plant Name latitude longitude Row Col. (mgd) (cfd) Wastewater Reuse Service Area
WASTE1 Port Orange WWTP 290812 805949 51 53 12.00 1604400 AGAREAl and/or AGAREA3
WASTE2 Holly Hill WWTP 291426 810240 21 54 2.40 320880 AGAREA2
WASTE3 Daytona Beach-Bethune Pt. WWTP 291205 810031 31 55 12.00 1604400 AGAREA3 and/or AGAREA9
WASTE4 Ormond Beach WWTP 291720 810426 6 52 6.00 802200 AGAREA4 and/or AGAREA5
WASTES Volusia Co. Spruce Creek WWTP 290443 810318 55 36 0.35 46795 AGAREA6 and/or AGAREA8









-- I


I -



I --,


Daytona Beach WSA
I "- A I
-I


Port Orange WSA

-- 1%
I ,l .-


- Low Harm
SModerate Harm
SHigh Harm
SControl Point


5 mits
I I l I


I I I I I I
0 5 kemdeas

Figure 4.3
Areas of potential impacts to plant communities resulting from
projected changes in ground water withdrawals between 1988 and 2010.


4














CHAPTER 5
5.0 GROUND WATER OPTIMIZATION MODELS



Five aquifer optimization models were developed to investigate optimal allocation

of ground water to meet year 2010 demand in the project area. These models were

developed to investigate future water allocation strategies assuming that feasible

withdrawal scenarios meet or exceed projected water service area demands and do not

exceed available water resource supplies. It was assumed with these models that adverse

environmental effects could be minimized at specific locations by constraining pressure

head changes (i.e. drawdown) to meet specified environmental goals or standards. For

example, one hundred control point locations were chosen at which ground water levels

changes were constrained. These points were in areas where native vegetation could be

harmed by declines in the surficial aquifer due to pumping. In addition to the ground

water level constraints two optimization models developed that incorporate a set of

constraints to allocate water in a manner that preserves ground water quality.

It is important to note the optimization models were developed using data

generated from both numerical flow and transport simulation models (e.g., information

describing aquifer responses to changing stresses such as pumping). The modeling grid

from the flow simulation model was incorporated in the formulations of all the

optimization models (see Chapter 3). Information on needs and sources was also








70

included in the formulation of each the optimization model (see Chapter 4). For

example, elemental discharge rates and pressure heads given by each optimization model

correspond to elemental cumulative discharges (from wells located in a grid cell) and

elemental average pressure heads in associated cells defined in the flow simulation

model. In addition, the term "well grid cells" refers to numerical model cells where one

or more wells are located, and is used herein to reflect the fact that the optimization

model identifies the cumulative well flows in each grid cell (in contrast to individual well

flows).



5.1 OPTIMIZATION MODEL DECISION VARIABLES

Each model is comprised of combinations of several groups of decision variables.

One group defines steady-state drawdowns and pressure heads at specified control points.

These are DDj, the drawdown at sensitive wetland control point; DDWh, the drawdown

at each well grid cell h; HDj, the pressure head at sensitive wetland control point; and

HDWh, the pressure head at each well grid cell h.

Another group defines cumulative and elemental flows from well grid cells used

to meet service area demands. These are QM,k, the discharge rate of each municipal

well grid cell i which supplies each municipal water service area k; QA,o, the discharge

rate of each agricultural well grid cell n which supplies each agricultural water service

area o; QW,,,, the effluent reuse rate each wastewater treatment plant m which

supplements each agricultural water service area o; QMT1, the total discharge rate at each

municipal well grid cell i; QAT,, the total discharge rate of each agricultural well grid










cell n; and QWT,, the total effluent reuse rate of each wastewater treatment plant m.

Additional decision variables used in one optimization model that incorporate

water quality constraints define steady-state chloride concentrations and changes in

concentrations at well points. These are CIh, the increase in chloride concentration at

each well grid cell control point h and CCh, the chloride concentration at each well grid

cell control point h.



5.2 OPTIMIZATION MODEL OBJECTIVE FUNCTIONS

Five optimization model formulations were identified under the assumption that

it was desirable to determine feasible ground water allocation in which minimum aquifer

system responses are maximized or maximum aquifer system responses are minimized.

For example, when the objective is to minimize the maximum drawdown then the value

of the objective function must be less than or equal to all drawdowns in the management

area. However, if the objective is to maximize a minimum pressure head, the objective

function must have a value greater than or equal to all the pressure heads in the

management area. In the optimization models presented, objective functions will appear

as statements specifying that the value of the objective function S is to be maximize or

minimize.

The objective function of the first model is to minimize the maximum drawdown

at all sensitive wetland control points (i.e. minimize S). For this model, the following

constraint was used to define S:











S DD, for all j (5.1)

where DDi is the drawdown at sensitive wetland control point.

The objective of the second model is to maximize the minimum pressure head at

all sensitive control points (i.e. maximize S). The appertain constraint associated with

this objective is:


S HDj for all j (5.2)

where HDj is the pressure head at sensitive wetland control point.

The objective of the third model is to minimize the average drawdown of all

sensitive wetland control points (i.e. minimize S). This model objective, which is equal

to maximizing average pressure heads, requires the following constraint to define S:


P
SDD (5.3)
S t -J-- for all j
P

where p is the number of sensitive wetland control points.

The objective of the fourth model is to minimize the maximum drawdown while

constraining concentration levels. The objective function of this model is formulated the

same as the first model, equation 5.1. The difference between model one and model four

is the addition of water quality constraints to model four. These additional constraints

function to elucidate pumping strategies which satisfying specified water quality

standards.

Finally, the objective of the fifth model is to minimize the maximum relative










chloride concentration increase. This objective function is written as follows:


S RCLh (5.4)

where RCLh is the relative chloride concentration increase at all well grid cells h.



5.3 OPTIMIZATION MODEL CONSTRAINTS

Other than model specific objective functions and their appurtenant constraint

equations (i.e., Equations 5.1, 5.2, 5.3, and 5.4) the six optimization model share a large

common block of constraint equations. This block of equations includes aquifer response

constraints, management constraints, and nonnegativity constraints. The common block

and water quality constraints were used in different combinations to formulate the five

different optimization models with unique goals to identify optimal allocation strategies.

The following sections discuss these constraint in greater detail.

5.3.1 Aquifer Response Constraints

The optimization models were developed to allow pressure head and/or drawdown

to be constrained or optimized. Drawdown constraints at the specified control points

were developed using influence coefficients that describe pressure head changes at each

control point created by ground water pumpage at each well grid cell. The following

general drawdown constraint for a control point includes a linear combination of aquifer

responses to the municipal, agricultural, and private wells.











DD, = aj (QMT QMOi) + E P, (QAT QA0)
n (5.5)
+ DDPRIVW (10'6) for all i, j, and n

where DDj is the drawdown at sensitive wetland control point j, aj is the aquifer

influence coefficient defining pressure head change at each sensitive wetland area control

point due to a change in discharge rate at each municipal well grid cell i, QMTi is the

total discharge rate in cfd at each municipal well grid cell i, QMO, is the initial discharge

rate in cfd of each municipal well grid cell i, f,n is the aquifer influence coefficient

defining pressure head change at each sensitive wetland area control point j due to a

change in discharge rate at each agricultural well n, QATn is the total discharge rate in

cfd of each agricultural well grid cell n, QAO, is the initial discharge rate in cfd of each

agricultural well grid cell n, and DDPRIVj is the drawdown in feet at each sensitive

wetland area control point j due to private wells not incorporated in the optimization

process.

QMT, values were calculated by summing the discharge rates at a specific well

grid cell over all municipal service areas for which it supplies. QATn values were

calculated by summing the discharge rates at a specific well grid cell over all agricultural

service areas for which it supplies. QMO, and QAO, values were determined using the

year 1988 water allocation strategy. QMO, and QAO, were used also in calculating the

drawdown from years 1988 to 2010 at the control points. DDPRIVj is the drawdown

from year 1988 to 2010 under steady-state conditions. These values were calculated by

increasing the discharge of private wells alone in the simulation model and were used in

calculating the total drawdown at the control points. To facilitate the linear programming








75

solver, the original influence coefficients were multiplied by 10+6 to give resulting

coefficients values on the order of one. As a result, calculated DDj drawdown values

were divided by 10+6 to produce the decision variable TDDj, which has units of feet:


DD.
TDD, for all j (5.6)
10+6

where TDDj is the total drawdown at each sensitive wetland area control point.

Pressure head at the sensitive wetland control points, HD,, are determined by a

constraint equation subtracting the drawdown at control point j from the initial pressure

head:


HDj = HOj (10+6) DDi for all j (5.7)

where HDj is the pressure head in millionths of a foot at each sensitive wetland area

control point and HOj is the initial pressure head in feet at each sensitive wetland area

control point.

Initial pressure heads at sensitive wetland area control points, HOj, were

determined by executing the simulation model at year 1988 conditions. Again, a second

decision variable was created to describe pressure head in feet. The total pressure head,

THDj, values were calculated by dividing HDj by 10+6:


HD.
THD. J (5.8)
D 10+6

where THDj is the total pressure head in feet at each sensitive wetland area control point

j.

Similar to the aquifer influence coefficient matrices use to create constraint








76

equations 5.6 and 5.8 for sensitive wetland control points, influence coefficient matrices

were also developed for constraints expressing the aquifer response at each well grid

locations due to pumpage within a well grid cell and at every other well grid cell. The

following constraint equations define the pressure head at well grid cell h:


DDWh = Y, (QMTi QMO,) + E O, (QAT, QAOn) (5.9)
i n

HDWh = HWOh (106) DDWh for all h (5.10)

where DDWh is the drawdown in millionths of a foot at each well grid cell h, 7,h* is the

aquifer influence coefficient defining pressure head change at well grid cell control point

h due to a change in discharge rate at each municipal well grid cell i, Oh is the aquifer

influence coefficient defining pressure head change at well grid cell control point h due

to a change in discharge rate at each agricultural well grid cell n, HDWh is the pressure

head in millionths of a foot at each well grid cell h, and HWOh is the initial pressure

head in feet at each well grid cell h.

Initial pressure heads at well grid cells, HWOh, were determined by executing the

simulation model at year 1988 conditions and were used in calculating the year 2010

pressure head at the well grid cells. Again, i,h and On,h were multiplied by one million

to obtain values on the order of one. THDWh and TDDWh values were calculated by

dividing HDWh and DDWh by 10+6:











TDDWh = DDWh (5.11)
10+6

HDW
THDWh = h for all h (5.12)
10+6

where TDDWh is the drawdown in feet at each well grid cell h and THDWh is the total

pressure head in feet at each well grid cell h.

An additional constraint was incorporated into the optimization model to preclude

pumpage that would dewater the surficial aquifer. The following constraint was

implemented to ensure water table elevations do not decrease below a level of one foot

above the bottom of the surficial aquifer. This constraint enables the pressure heads to

be constrained in order to avoid drying out of well grid cells:


HDWh > BOTELEVh + 1.0 for all h (5.13)


where BOTELEVh is the bottom elevation of the surficial aquifer in feet from mean sea

level at each well grid cell control point h. BOTELEVh values were obtained from the

input file of the simulation model.

5.3.2 Management Constraints

Management constraints used in these optimization models define the capacity

of available resources, the demand for available resources, and the source to demand

links. The first set of constraints specify limits on capacities associated with the

production of water from aquifer systems. These following constraints were incorporated

in the model to limit the maximum withdrawal rates at the well grid cells:











QMT, = QMi CM, (5.14)
k

QAT, = QAn,o CAn (5.15)
0

QWT = E QWr,,o CWm for all i, k, m, n, and o (5.16)

where QM,k is the discharge rate in cubic feet per day (cfd) of each municipal well grid

cell i which supplies each municipal water service area k, CM, is the total capacity rate

in cfd of each municipal well grid cell i, QAn,o is the discharge rate in cfd of each

agricultural well grid cell n which supplies each agricultural water service area o, CA,

is the total capacity rate in cfd of each agricultural well grid cell n, QWT, is the total

effluent reuse rate in cfd of each wastewater treatment plant m, QW,,o is the effluent

reuse rate in cfd of each wastewater treatment plant m which supplements each

agricultural water service area o, and CWm is the capacity rate in cfd of each wastewater

treatment plant i.

Municipal well grid cell capacities, CM, (for all i), were set at 600,000 cfd. CA,

values for each agricultural well grid cell were set at the service area demand for which

the cell supplied. CWm limits reflect the available wastewater effluent which could be

used to supplement the agricultural demand.

Minimum withdrawal rates on municipal well grid cell were also incorporated into

the optimization model to prevent the shutting off of existing wells. This is achieved by

placing lower discharge limits on the well grid cells. These constraints were formulated

to require minimum flows that equal a percentage of the 1988 withdrawal rates. From

a review of the projected year 2010 water allocation strategy, discharges at existing well








79

grid cells were allowed to decrease to approximately 50 percent of the 1988 distribution

discharge rate estimates. Thus, the following constraints incorporated in the optimization

models, which requires year 2010 discharges at municipal well grid cells to be greater

than or equal to half of the year 1988 rates:


QMT, = QMi 0.50 (QMO) (5.17)
k
In order to be a feasible water allocation strategy, the strategy must meet or exceed the

demands of the service areas. The following demand constraints ensure that water needs

of municipal and agricultural service areas are satisfied:


SQMj ; DMk (5.18)


SQAn,o + QWmo DAo for all i, k, m, n, and o (5.19)
n m
where DMk is the demand rate in cfd of each municipal water service area k and DA, is

the demand rate in cfd of each agricultural water service area o.

Municipal demand, DMk, and agricultural demand, DAo, were calculated using

projected year 2010 discharge rates (See Chapter 4). These demands ensure that the

model identifies discharge from well grid cells that meet future demands of the water

service areas. It may be seen from constraint equations 5.19 that agricultural well grid

cells are not constrained to lower limits as long as demand can be satisfied with

wastewater effluent.

Also, since every water supply area does not (and can not) supply every water

service area, the optimization model was constructed to link specific water sources with

specific demand areas. The following constraints were formulated to specify which well




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