Appendix Part 4. 221

Appendix Part 4. Conjunctive-Management 

Models for the Eastern Pawcatuck and Lower 

Wood River Model Areas

Figures

A4–1. Map showing locations of the lower Wood and eastern Pawcatuck River conjunctivemanagement model areas in the Pawcatuck River Basin, southwestern 

Rhode Island..............................................................................................................................225

A4–2. Boxplot showing the distribution of all estimated daily mean streamflows (daily) and 

the minimum monthly streamflow of each year (mins.) during the 1960–2004 period 

from the HSPF model, at the streamflow-constraint site on the Wood River (WOOD6, 

RCHRES 65) in the Pawcatuck River Basin, southwestern Rhode Island.......................230

A4–3. Graphs showing monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during which one-day 

monthly-minimum streamflows equaled or exceeded a selected streamflow value at 

conjunctive-management model constraint site BEAVM on the Beaver River, in the 

Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004 ....................232

A4–4. Graphs showing monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during which one-day 

monthly-minimum streamflows equaled or exceeded a selected streamflow value at 

conjunctive-management model constraint site QUEENM on the Usquepaug-Queen 

River, in the Pawcatuck River Basin in southwestern Rhode Island, during 

1960–2004 ...................................................................................................................................233

A4–5. Graphs showing monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during which one-day 

monthly-minimum streamflows equaled or exceeded a selected streamflow value at 

conjunctive-management model constraint site PAWCD on the Pawcatuck River, in 

the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004..............234

A4–6. Graphs showing monthly flow-duration curves (data from the HSPF model of 

conditions without withdrawals) showing the percentage of months during which 

one-day monthly-minimum streamflows equaled or exceeded a selected streamflow 

value at conjunctive-management model constraint site MEAD2 (RCHRES 48) on 

Meadow Brook, in the Pawcatuck River Basin in southwestern Rhode Island, 

during 1960–2004.......................................................................................................................235

A4–7. Graphs showing monthly flow-duration curves (data from the HSPF model of 

conditions without withdrawals) showing the percentage of months during which 

one-day monthly-minimum streamflows equaled or exceeded a selected streamflow 

value at conjunctive-management model constraint site WOOD5 (RCHRES 63) on the 

Wood River, in the Pawcatuck River Basin in southwestern Rhode Island, 

during 1960–2004.......................................................................................................................236

A4–8. Graphs showing monthly flow-duration curves (data from the HSPF model of 

conditions without withdrawals) showing the percentage of months during which 

one-day monthly-minimum streamflows equaled or exceeded a selected streamflow 

value at conjunctive-management model constraint site WOOD6 (RCHRES 65) on the 

Wood River, in the Pawcatuck River Basin in southwestern Rhode Island, 

during 1960–2004.......................................................................................................................237
222 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

A4–9. Graphs showing monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during which one-day 

monthly-minimum streamflows equaled or exceeded a selected streamflow value at 

conjunctive-management model constraint site PAWC4 (RCHRES 50) on the 

Pawcatuck River, in the Pawcatuck River Basin in southwestern Rhode Island, during 

1960–2004 ...................................................................................................................................238

A4–10. Graph showing hypothetical streamflow depletion caused by two wells pumping 

independently for 180 days at 1.0 million gallons per day..................................................239

A4–11. Boxplot showing the distribution of monthly withdrawals as a fraction of total annual 

withdrawals during the period 1995–2004 from four water-supply systems that withdraw primarily from groundwater sources in the Pawcatuck River Basin, 

southwestern Rhode Island ....................................................................................................241

A4–12. Boxplots showing the distribution of monthly irrigation withdrawals as a fraction of 

total annual withdrawals during the period January 1960 through September 2004 

estimated by use of logistic-regression equations developed for (A) turf-farm irrigation, 

(B) golf-course irrigation from surface-water withdrawals, and (C) golf-course 

irrigation from groundwater withdrawals, southwestern Rhode Island.........................242

A4–13. Boxplot showing the distribution of annual irrigation withdrawals as estimated from 

the logistic-regression equations for irrigation withdrawals for the eastern Pawcatuck 

River conjunctive-management model (EPRCMM) area and the lower Wood River 

conjunctive-management model (LWRCMM) area in the Pawcatuck River Basin, 

southwestern Rhode Island, 1960–2004................................................................................243

A4–14. Graphs showing selected examples of simulated response coefficients for the 

(A) Beaver, (B) Usquepaug-Queen, and (C) Pawcatuck Rivers, in the Eastern Pawcatuck River Conjunctive-Management Model (EPRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for three hypothetical wells at different distances 

between the rivers....................................................................................................................246

A4–15. Graphs showing selected examples of simulated response coefficients for the 

(A) Beaver, (B) Usquepaug-Queen, and (C) Pawcatuck Rivers, in the Eastern Pawcatuck River Conjunctive-Management Model (EPRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for four hypothetical wells at different distances 

between the rivers....................................................................................................................247

A4–16. Graphs showing selected examples of simulated response coefficients for the 

(A) Beaver, (B) Usquepaug-Queen, and (C) Pawcatuck Rivers, in the Eastern Pawcatuck River Conjunctive-Management Model (EPRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for four hypothetical wells at different distances 

between the rivers....................................................................................................................248

A4–17. Graphs showing selected examples of simulated response coefficients for the 

(A) Beaver, (B) Usquepaug-Queen, and (C) Pawcatuck Rivers, in the Eastern Pawcatuck River Conjunctive-Management Model (EPRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for four hypothetical wells at different distances 

between the rivers....................................................................................................................249

A4–18. Graphs showing selected examples of simulated response coefficients for the 

(A) Beaver, (B) Usquepaug-Queen, and (C) Pawcatuck Rivers, in the Eastern Pawcatuck River Conjunctive-Management Model (EPRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for four hypothetical wells at different distances 

between the rivers....................................................................................................................250

A4–19. Graphs showing selected examples of simulated response coefficients for 

(A,B) Wood River, (C) Meadow Brook, and (D) Pawcatuck River, in the Lower Wood 

River Conjunctive-Management Model (LWRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for three hypothetical wells at different distances 

between the rivers....................................................................................................................251
Appendix Part 4. 223

A4–20. Graphs showing selected examples of simulated response coefficients for 

(A,B) Wood River, (C) Meadow Brook, and (D) Pawcatuck River, in the Lower Wood 

River Conjunctive-Management Model (LWRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for three hypothetical wells at different distances 

between the rivers....................................................................................................................252

A4–21. Graphs showing selected examples of simulated response coefficients for 

(A,B) Wood River, (C) Meadow Brook, and (D) Pawcatuck River, in the Lower Wood 

River Conjunctive-Management Model (LWRCMM) area in the Pawcatuck River 

Basin southwestern Rhode Island, for three hypothetical wells at different distances 

between the rivers....................................................................................................................253

Tables

A4–1. Characteristics of existing and potential water-withdrawal sites in the eastern 

Pawcatuck and lower Wood River conjunctive-management model areas, 

Pawcatuck River Basin, southwestern Rhode Island ........................................................226

A4–2. Characteristics of selected streamflow constraint sites in the eastern Pawcatuck and 

lower Wood River conjunctive-management model areas, Pawcatuck River Basin, 

southwestern Rhode Island ....................................................................................................227

A4–3. Irrigation management-model scenarios for existing and potential water-withdrawal 

sites in the eastern Pawcatuck River conjunctive-management model (EPRCMM) 

area, Pawcatuck River Basin, southwestern Rhode Island..............................................228
224 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

Introduction

Conjunctive-management models were developed for 

selected areas in the Pawcatuck River Basin to evaluate 

relations between streamflow-depletion criteria and 

water-withdrawal volumes for different withdrawal-well 

networks. These conjunctive-management models are based 

on statistical analysis of water-use data, simulations with 

the transient numerical groundwater-flow (MODFLOW) 

models developed for the study areas, and simulations with 

the basinwide surface-water hydrology model to formulate 

linear-optimization models for water-resource management. 

A conjunctive-management model consists of a mathematical 

formulation (statement) of the groundwater-development goals 

for the model area and a set of constraints that limit those 

goals (Barlow and Dickerman 2001a,b; Granato and Barlow, 

2005). The term “conjunctive-management model” commonly 

is used in the hydrologic literature to refer to the combined 

use of numerical simulation and optimization techniques to 

evaluate alternative strategies for balancing water withdrawals 

with aquatic-habitat protection goals (Barlow and Dickerman 

2001a,b; Granato and Barlow, 2005). This appendix describes 

the mathematical formulation of the models and the responsematrix technique used to solve them.

Two conjunctive-management model areas, described 

herein as the eastern Pawcatuck River conjunctive-management model area (EPRCMM area) and the lower-Wood 

conjunctive-management model area (LWRCMM area) were 

selected to demonstrate concepts that may be useful for waterresource planning for the Pawcatuck River Basin and similar 

basins in the northeastern United States. These two areas were 

selected from the larger area simulated with the groundwaterflow (MODFLOW) models within the Pawcatuck River Basin 

(fig. A4–1) to demonstrate the application of conjunctivemanagement models for irrigation and municipal watersupply development (table A4–1) (Vicky Drew, U.S. Natural 

Resource Conservation Service, and Juan Marsical, Rhode 

Island Water Resources Board, oral commun., March 2006). 

The EPRCMM area has three streamflow-constraint sites 

(BEAVM, QUEENM, and PAWCD) shown on figure 4–1 and 

the constraints are defined in table A4–2. The 20 existing and 

potential water-withdrawal sites in this area are shown on figure 4–1 and described in table A4–1. The LWRCMM area has 

four streamflow-constraint sites (MEAD2, WOOD5, WOOD6, 

and PAWC4) shown on figure 4–2 and defined in table A4–2. 

The 12 existing and potential water-withdrawal sites in this 

area are shown on figure 4–2 and defined in table A4–1.

Formulation of the Conjunctive-Management 

Model

The formulation of each conjunctive-management model 

consists of defining a set of decision variables, an objective 

function, and a set of constraints. The decision variables are 

the monthly withdrawal rates calculated by the model for each 

of the simulated water-withdrawal sites. Mathematically, the 

decision variables were expressed as Qwi,t, which represents 

the withdrawal rate at water-withdrawal site i in month t. 

The subscript t ranges from t=1 for January through t=12 for 

December. The decision variables are the withdrawal rate at 

each water-withdrawal site for each of the 12 months in a year, 

during which the site is active. The set of water-withdrawal 

sites used for specific formulations of the models in each area 

(table A4–1) differed from one model application to the next. 

For example, some groundwater-withdrawal sites were not 

included in the models that were used to evaluate the potential 

effects of withdrawals at community irrigation wells in the 

EPRCMM area. Also, it was assumed that municipal waterwithdrawal sites will be active for all 12 months and irrigation 

sites will be active only during the growing-season months 

defined by the analysis of water-use data.

Management-Model Objective Function

The objective function of the model was formulated to 

maximize total annual groundwater withdrawals from the 

aquifer, and was written mathematically as

max , imize NDt Qwi t t

NM

i

NW

× =

∑ =∑

1 1 , (1)

where 

NW is the number of water-withdrawal sites, 

NM is the number of months (12), and 

NDt is the number of days in month t. 

Management-model constraints were formulated in terms of 

streamflow depletion and water-withdrawal rates for each 

month, but the objective function of the model is calculated 

as the total volume of water by weighting each monthly value 

by the number of days in each month. Values of the objective 

function were calculated in cubic feet of water withdrawn 

from the basin during the 12-month period. Total annual withdrawals were converted to units of million gallons because this 

Appendix Part 4. Conjunctive-Management Models for the Eastern 

Pawcatuck and Lower Wood River Model Areas

By Gregory E. Granato and Donald A. Walter
Management-Model Objective Function 225

71°44' 71°36'

41°28'

41°36'

Base from U.S. Geological Survey digital data, 1:24,000,

Rhode Island stateplane projection. 1983 North American datum

0

0 1

1 2 MILES

2 KILOMETERS

EXPLANATION

Inactive model areas 

(shown in shades of grey)

Hydraulic cells

Ponds

Lower Wood River

model area

Eastern Pawcatuck 

River model area

Lower Wood River model area

Eastern Pawcatuck River model area

Conjunctive-management

model area

Pawcatuck River

Basin boundary

Constant head boundary

Specified flux

Head-dependent flux

boundary

Pawcatuck River Basin

South Coastal Basin

Lower Wood River 

conjunctivemanagement model 

area

Wood River

Meadow Brook

Beaver River

Eastern Pawcatuck 

River conjunctivemanagement model 

Paw

area catuck River

Usquepaug River

Chickasheen Brook

Queen River

Chipuxet River

Figure A4–1. Locations of the lower Wood and eastern Pawcatuck River conjunctive-management model areas in the 

Pawcatuck River Basin, southwestern Rhode Island.
226 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

Table A4–1. Characteristics of existing and potential water-withdrawal sites in the eastern Pawcatuck and 

lower Wood River conjunctive-management model areas, Pawcatuck River Basin, southwestern Rhode Island.

[Index, conjunctive-management model index number; ID, identifier. Site type: SW, surface water; GW, groundwater; GW–P, 

groundwater-pond. Water use: IG, irrigation golf; MS, potential municipal supply; IT, irrigation turf; IV, irrigation vegetable. 

Response coefficient information: the number of the appendix, which shows a graph of the response coefficients for the withdrawal site (response coefficient information equal to 1 indicates a surface-water site where streamflow depletion occurs simultaneously with withdrawals)]

Index

Site ID

(site locations 

shown in 

figure 4–1)

Site type Water use MODFLOW 

row

MODFLOW 

column

Response coefficient 

information

Eastern Pawcatuck River conjunctive-management model (EPRCMM) area

1 AUQ6A SW IT 166 82 1

2 PR–AUQ8A GW IT 175 91 A4–15

3 PR–AUQ9A GW–P IT 185 80 A4–15

4 PR–AUQ11A GW–P IT 236 74 A4–15

5 PR–AB8A GW–P IT 231 69 A4–15

6 PR–GB1A GW IG 193 48 A4–18

7 RIW–336A GW MS 205 70 A4–18

8 RIW–385 GW MS 239 56 A4–14

9 PR–AUQ7A GW IT 178 88 A4–18

10 PR–AUQ10A GW IT 215 69 A4–17

11 PR–AUQ10B GW IT 221 83 A4–17

12 PR–AUQ10C GW IT 220 76 A4–17

13 PR–AB1A GW IT 207 67 A4–16

14 PR–AB2A GW IT 213 67 A4–16

15 PR–AB2B GW IT 224 68 A4–16

16 PR–AB3A GW IT 216 69 A4–16

17 PR–AB3B GW IT 225 54 A4–14

18 PR–AB4A GW IT 219 69 A4–14

19 PR–AB5A GW IV 231 54 A4–17

20 PR–GUQ2AB GW IG 204 82 A4–18

Index

Site ID

(site locations 

shown in 

figure 4–1)

Site type Water use MODFLOW 

row

MODFLOW 

column

Response coefficient 

information

Lower Wood River conjunctive-management model (LWRCMM) area

1 AW2A SW IT 165 138 1

2 AP1A SW IT 196 220 1

3 GW1A GW–P IG 22 111 A4–20

4 GW2A SW IG 80 82 1

5 GW3A GW–P IG 100 138 A4–20

6 GW4A GW IG 183 120 A4–20

7 GW4B GW IG 185 111 A4–21

8 GW4C GW IG 181 91 A4–21

9 RIW–458 GW MS 158 164 A4–19

10 RIW–458A GW MS 162 152 A4–19

11 RIW–458B GW MS 165 141 A4–19

12 PR–AW1A GW IT 129 155 A4–21
Management-Model Objective Function 227

Table A4–2. Characteristics of selected streamflow constraint sites in the eastern Pawcatuck and lower Wood River conjunctive-management model areas, Pawcatuck River Basin, southwestern Rhode Island.[Site locations for EPRCMM shown in figure 4–1 and site locations for LWRCMM shown in figure 4–2 and described in table A2–4. Estimates of the minimum monthly one-day streamflow values without withdrawals are derived from the output of the basinwide surface-water model. mi2, square miles; *, HSPF subbasin outlets]

Managementmodel site 

designation

River

MODFLOW 

row

MODFLOW 

column

Drainage 

area 

(mi2)

Description

Minimum monthly one-day streamflow without withdrawals during 1960–2004, in cubic feet per second 

Jan. Feb. Mar. Apr. May June July Aug. Sept Oct. Nov. Dec.

Eastern Pawcatuck River conjunctive-management model (EPRCMM) area

BEAVM Beaver River 261 50 12.47 Model cell at confluence of the 

Beaver and Pawcatuck Rivers.

7.02 9.74 13.93 10.68 9.84 6.18 2.93 2.30 1.99 2.20 2.20 3.14

QUEENM UsquepaugQueen River 240 87 36.57 Model cell at confluence of the Usquepaug-Queen and Pawcatuck 

Rivers.

19.6 26.1 36.8 28.1 26.5 17.0 9.13 7.10 6.09 6.80 7.00 9.03

PAWCD Pawcatuck River 261 49 90.91 Model cell downstream of the 

confluence of the UsquepaugQueen and Pawcatuck Rivers.

56.5 79.1 106 87.1 81.2 54.4 30.3 23.0 19.5 23.7 27.6 34.7

Lower Wood River conjunctive-management model (LWRCMM) area

MEAD2* Meadow Brook

(RCHRES 48)

241 186 7.21 Model cell at confluence of the 

Meadow Brook and the Pawcatuck River.

3.00 4.40 6.50 5.10 5.30 3.40 2.00 1.60 1.40 1.50 1.40 1.70

WOOD5* Wood River

(RCHRES 63)

144 118 85.13 Model cell downstream of the 

confluence of Canonchet Brook and the Wood River.

41.9 55.1 79.5 60.9 58.2 36.3 17.5 12.9 10.8 13.1 12.1 18.6

WOOD6* Wood River

(RCHRES 65)

232 111 88.31 Model cell downstream of the 

confluence of an unnamed tributary and the Wood River.

46.9 62.0 89.3 68.3 65.3 40.9 20.0 14.9 12.6 13.8 13.9 21.1

PAWC4* Pawcatuck River

(RCHRES 50)

281 98 204.33 Model cell on the Pawcatuck River 

at the outlet of the MODFLOW model below the confluence with the Wood River.

118 160 223 177 170 112 60.8 45.8 39.0 44.8 46.9 65.6
228 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

unit is commonly used for evaluating water-supply scenarios 

in Rhode Island (Barlow and Dickerman, 2001a,b; Granato 

and Barlow, 2005).

The value of the objective function was limited by a 

set of specified constraints that included combinations of 

maximum rates of streamflow depletion at the streamflowconstraint sites, minimum and maximum withdrawal rates at 

each water-withdrawal site, and seasonal patterns of monthly 

water demands. The set of constraints differed from one model 

application to the next.

Streamflow-Depletion Criteria

Streamflow depletions were required to be less than or 

equal to maximum specified streamflow-depletion rates for 

each month. These streamflow-depletion criteria were applied 

at selected streamflow-constraint sites that corresponded to 

selected points of interest in the EPRCMM area (fig. 4–1; 

table A4–2) and the LWRCMM area (fig. 4–2; table A4–2). 

These constraints were written as

Qsd j t Qsd , ,j t max

≤ 

  , (2)

where 

Qsdj,t is the streamflow depletion at streamflowconstraint site j in month t, and 

(Qsdj,t)max is the maximum rate of streamflow depletion 

allowed at site j in month t. 

Maximum rates of streamflow depletion for each of the 12 

months were specified for each of the three constraint sites in 

the EPRCMM area and each of the four constraint sites in the 

LWRCMM area for a total of 36 and 48 streamflow-depletion 

constraints in each model, respectively.

Table A4–3. Irrigation management-model scenarios for existing and potential water-withdrawal sites in the 

eastern Pawcatuck River conjunctive-management model (EPRCMM) area, Pawcatuck River Basin, southwestern 

Rhode Island.

[Site locations shown in figure 4–1 and described in table 4–1. Index, conjunctive-management model index number; ID, identifier]

Index Site ID

Scenarios

Individual irrigation wells Community-well option A Community-well option B

1 AUQ6A X X X

2 PR–AUQ8A X X X

3 PR–AUQ9A X X X

4 PR–AUQ11A X X X

5 PR–AB8A X X X

6 PR–GB1A X X X

7 RIW–336A

8 RIW–385 

9 PR–AUQ7A X X X

10 PR–AUQ10A X

11 PR–AUQ10B X X X

12 PR–AUQ10C X X X

13 PR–AB1A X X

14 PR–AB2A X

15 PR–AB2B X

16 PR–AB3A X

17 PR–AB3B X X X

18 PR–AB4A X X X

19 PR–AB5A X X X

20 PR–GUQ2AB X X X
Streamflow-Depletion Criteria 229

Granato and Barlow (2005) evaluated 11 commonly 

referenced instream-flow criteria and determined that criteria 

based on the allowable monthly streamflow-depletion rate 

in the month with the lowest streamflows would effectively 

limit total annual groundwater withdrawals because of the 

long response times (about 3–12 months) for such withdrawals. Granato and Barlow (2005) also demonstrated that many 

aquatic-baseflow criteria, originally developed under the 

assumption of the availability of a substantial amount of water 

in storage in an actively managed upstream surface-water 

reservoir, may not be protective for groundwater withdrawals from a limited stream-aquifer system. This is especially 

true for municipal water suppliers with a limited ability to 

enforce dry-year water conservation efforts that may result 

in a 10- to 20-percent reduction in summer water withdrawals. For this reason, Granato and Barlow (2005) developed an 

alternative instream-flow paradigm based on the maintenance 

of some nonzero streamflow at a defined return period to limit 

or eliminate zero-flow days. Granato and Barlow (2005) used 

an estimated streamflow record for the period 1961–2000 to 

evaluate allowable depletion criteria based on the one-day 

monthly minimum streamflows during this period. In that 

study, postoptimization analysis indicated that the application 

of such criteria may prevent zero-flow events.

The calibrated basinwide HSPF model was used to 

estimate streamflows without any anthropogenic water withdrawals at selected sites during the study period January 1960 

through September 2004. Data available from streamflow gaging stations in the area reflect an unknown amount of existing 

streamflow depletion because there is no long-term record of 

all existing municipal and irrigation withdrawals available for 

the 1960–2004 period. A long-term simulated record without 

withdrawals was used because the record would represent the 

range of hydrologic conditions that occurred during the period, 

including the drought of record, and thus would provide 

statistics representative of variations in hydrologic conditions in Rhode Island. The HSPF model-generated estimates 

of streamflows without withdrawals were used as the basis 

for calculating values of allowable streamflow depletion that 

represent various risks for inducing a zero-flow condition as 

an effect of water withdrawals. 

In two cases (BEAVM and QUEENM), output from the 

calibrated HSPF model was adjusted to address differences in 

surface-water and groundwater contributing areas to the HSPF 

HRUs. In the EPRCMM area, MODFLOW model results 

indicated that withdrawals from some of the wells considered 

in the conjunctive-management model analysis may cause 

streamflow depletion below the outlets of the HRUs in the 

HSPF model for the Beaver and Usquepaug-Queen Rivers. 

The drainage-area-ratio method (Stedinger and others, 1993) 

was used to estimate streamflow values at the mouth of 

each river to include the contributing area to each well 

(table A4–2).

The one-day monthly minimum streamflows differ from 

one another over a large range of values. The boxplots in 

figure A4–2 indicate the variability from month to month and 

year to year in the populations of all estimated daily-mean 

streamflows and the one-day monthly minimum streamflows 

for each month of each year during the 1960–2004 period at 

streamflow-constraint site WOOD6 (defined in table A4–2). 

The daily mean streamflows ranged over one or two orders of 

magnitude, and one-day monthly minimum streamflows by 

a factor of three or more during each month. The maximum 

one-day monthly minimum streamflow value in each month 

exceeds the all of medians of the daily-mean values for the 

same month and exceeds 75 percent of all the daily-mean 

values in the months of March, July, August, November, 

and December.

Potential allowable streamflow-depletion rates were 

selected by identifying the one-day monthly minimum 

streamflow values for each station in each month of each year 

from the HSPF-generated streamflow record for 1960–2004. 

To select streamflow-depletion rates from the population of 

values, the one-day monthly minimum streamflow values were 

ranked and the percentage of the values that would equal or 

exceed each value was estimated by use of its Weibull plotting-position value (Helsel and Hirsch, 2002). Each Weibull 

plotting-position percentile value (WPj) is calculated as

WPj

Rj

N = ⋅ +

100 1

, (3)

where

Rj is the rank of value j, and 

N is the number of one-day monthly-minimum 

streamflow values (45 values for each 

month for January through September 

and 44 values for each month for October, 

November, and December).

Flow-duration curves commonly are expressed as the 

percentage of flows that may equal or exceed a given value. In 

setting minimum-flow criteria, however, the complementary 

probability percentile (100-WPj) is of concern. For example, 

the low-flow value with a 20-percent chance of occurring 

in any given year is the 80-percent flow duration, and the 

low-flow value with a 10-percent chance of occurring in any 

given year is the 90-percent flow duration. The minimum flow 

for each month in this 45-year data set has about a 2-percent 

chance (98-percent flow duration) of occurring in any given 

year. The 80-, 90-, 95-, and 98-percent flow durations were 

used as maximum rates of streamflow depletion (Qsdj,t)max in 

each management model. The management models with these 

streamflow constraints indicate the probability that the watersupply-development plan calculated by use of the management 

model may result in zero flows in one or more streams in each 

model area. 

Management models also are developed from the 

minimum daily mean streamflow for each month of the 

year. Four percentages—75, 60, 50, and 25 percent—of the 

minimum daily-mean streamflow values estimated for the 

45-year period were used as maximum rates of streamflow 
230 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

1

10

100

0.1

STATISTICS BY MONTH

Jan. daily

Jan. mins.

Feb. daily

Mar. daily

Apr. daily

May daily

June daily

July daily

Aug. daily

Sept. daily

Oct. daily

Nov. daily

Dec. daily

July mins.

June mins.

May mins.

Apr. mins.

Mar. mins.

Feb. mins.

Oct. mins.

Sept. mins.

Aug. mins.

Nov. mins.

Dec. mins.

STREAMFLOW, IN CUBIC FEET PER SECOND PER SQUARE MILE Maximum 99th percentile 75th percentile Mean Median

25th percentile1st percentileMinimum

Figure A4–2. The distribution of all estimated daily mean streamflows (daily) and the minimum monthly streamflow of each year (mins.) during the 1960–2004 period from the HSPF model, at the streamflow-constraint site on the Wood River (WOOD6, RCHRES 65) in the Pawcatuck River Basin, southwestern Rhode Island. (Site location shown in figure 4–1 and described in table 4–2.)
Water-Withdrawal Criteria 231

depletion (Qsdj,t)max in each management model. These 

streamflow-depletion constraints were used to evaluate waterwithdrawal plans that may be sustainable for dry periods when 

agricultural, golf, and municipal water demands are driven by 

irrigation needs. These allowable-depletion criteria would not 

result in a condition of zero flow on the day of lowest flow 

on record because these values are a only a fraction of the 

minimum streamflow value.

The percentages of September months during the study 

period which have one-day monthly minimum streamflows 

that are equaled or exceeded a given value are shown for each 

model area in figure A4–5. In each model area, the minimum 

of the one-day minimum September streamflow values is 

about 0.18 ft3/s/mi2. Flow-duration curves for the one-day 

monthly minimum streamflows (in ft3/s) during each 

month at each streamflow-constraint location are shown in 

figures A4–3–A4–9.

Use of the one-day monthly minimum streamflows 

to estimate the maximum streamflow-depletion constraint 

(Qsdj,t)max does not preclude the potential for surface-water 

withdrawals to exceed the instantaneous streamflow during 

the hours when the surface-water withdrawals are active. The 

potential effects of the differences in the time scales between 

variable hourly irrigation withdrawals with instantaneous 

depletions from surface-water sites and the monthly average 

rates calculated by the management model are minimized 

as surface-water withdrawals are converted to groundwater 

withdrawals. This is because the effects of variations in 

withdrawals at wells are damped by the long streamflowdepletion response times for groundwater withdrawals. For 

example, Barlow (1997) calculated the hypothetical effects 

on streamflow depletion from a 180-day-long period of well 

withdrawals at 1.0 Mgal/d. This example shows that the 

effect of changes in withdrawals in wells at various distances 

from the stream would not be fully realized in the amount of 

streamflow depletion for one or more months (fig. A4–10). 

Water-Withdrawal Criteria

Constraints on minimum and maximum withdrawal rates 

at each potential water-withdrawal site were written as

Qwi t Qwi t Qw , min , ,i t max ( ) ≤ ≤ ( ) , (4)

where

(Qwi,t)min and (Qwi,t)max are the minimum and maximum

withdrawal rates, respectively, at

water-withdrawal site i in month t;

and

 Qwi,t is the withdrawal rate at site i in

month t. 

The minimum withdrawal rate at each water-withdrawal site, 

which was zero, did not need to be specified explicitly in the 

model. The maximum water-withdrawal rates were selected 

to evaluate relations between the total maximum withdrawals 

estimated by use of optimization methods and the associated 

streamflow depletions. The maximum withdrawal rates at 

each water-withdrawal site for each month were 1.0, 1.4, and 

2.0 Mgal/d in each set of models.

Many of the water-withdrawal sites in each management 

model area are hypothetical wells, and thus, lack long-term 

well-yield information. For this reason, potential maximum 

groundwater-withdrawal rates were selected on the basis of 

generalized information about the extent and properties of the 

aquifer in each management-model area. These rates, however, should not be viewed as estimates of actual production 

capacities. Furthermore, the locations of selected sites should 

be viewed as hypothetical. The sites were chosen to demonstrate concepts rather than to produce a specified water yield; 

aquifer tests would be needed to explore the potential of each 

site to produce a desired water yield. Similarly, maximum 

withdrawal rates at the surface-water-withdrawal sites depend 

on streamflow availability, local channel geometry, and pump 

rating. Instantaneous surface-water withdrawals may greatly 

exceed monthly average values calculated by the management models. 
232 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

1

10

100

1

10

100

1

10

100

1

10

100

Figure A4–3.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site BEAVM on the Beaver River, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–1 and described in table 4–2.)
Water-Withdrawal Criteria 233

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

10

100

1,000

10

100

1,000

1

10

100

2

10

200100

PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

Figure A4–4.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site QUEENM on the Usquepaug-Queen River, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–1 and described in table 4–2.)
234 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1

PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

10

100

1,000

10

100

1,000

10

100

1,000

10

100

1,000

Figure A4–5.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site PAWCD on the Pawcatuck River, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–1 and described in table 4–2.)
Water-Withdrawal Criteria 235

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1

PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

1

10

100

1

10

100

1

10

100

1

10

100

Figure A4–6.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site MEAD2 (RCHRES 48) on Meadow Brook, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–2 and described in table 4–2.)
236 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1

PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

10

100

1,000

10

100

1,000

10

100

1,000

10

100

1,000

Figure A4–7.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site WOOD5 (RCHRES 63) on the Wood River, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–2 and described in table 4–2.)
Water-Withdrawal Criteria 237

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1

PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

10

100

1,000

10

100

1,000

10

100

1,000

10

100

1,000

Figure A4–8.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site WOOD6 (RCHRES 65) on the Wood River, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–2 

and described in table 4–2.)
238 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

ONE-DAY MINIMUM-MONTHLY STREAMFLOW, IN CUBIC FEET PER SECOND

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

1 10 5 2 50 40 30 20 99 98 95 90 80 70 60 1 10 5 2 50 40 30 20 99 98 95 90 80 70 60

PERCENTAGE OF MONTHS DURING WHICH ONE-DAY MONTHLY-MINIMUM STREAMFLOWS WERE EQUALED OR EXCEEDED

JanuaryFebruaryMarch

AprilMayJune

JulyAugustSeptember

OctoberNovemberDecember

10

100

1,000

10

100

1,000

10

100

1,000

10

100

1,000

Figure A4–9.

Monthly flow-duration curves (data from the HSPF model of conditions without withdrawals) showing the percentage of months during 

which one-day monthly-minimum streamflows equaled or exceeded a selected streamflow value at conjunctive-management model constraint site PAWC4 (RCHRES 50) on the Pawcatuck River, in the Pawcatuck River Basin in southwestern Rhode Island, during 1960–2004. (Site location shown in figure 4–2 and described in table 4–2.)
Seasonal Water-Demand Criteria 239

Seasonal Water-Demand Criteria

The conjunctive-management models developed for both 

areas include seasonal water-demand patterns for municipal 

water use, and irrigation for agriculture and golf courses. The 

timing of water demands in Rhode Island is a critical factor 

because evapotranspiration reduces available streamflows 

during the summer months when monthly water demands 

commonly peak. Factors that influence the annual distribution of monthly demands for municipal water-supply systems 

include increased lawn and garden irrigation, implementation 

of water-use restrictions, and increased summer populations in 

recreational areas (L.K. Barlow and E.C. Wild, U.S. Geological Survey, written commun., September 2003). The logisticregression equations developed to predict water-use patterns 

(see Part 1) indicate that the factors influencing the annual 

distribution of monthly demands for agricultural and golfcourse irrigation are precipitation and potential evapotranspiration. Furthermore, municipal and agricultural water demands 

commonly increase during dry years, which may exacerbate 

the effect of precipitation deficits on instream flow.

Granato and Barlow (2005) demonstrated that management-models with the same streamflow constraints and 

no prespecified monthly demand-pattern constraints could 

produce substantially more water withdrawals than management models with the monthly demand-pattern constraint. 

The actual implementation of a management model without 

such constraints, however, requires the availability of a large 

reservoir as an alternate or supplemental summer supply or 

as a storage facility to retain excess fall, winter, and spring 

withdrawals to meet peak summer demands. Water suppliers 

typically have distribution reservoirs (small surface-water reservoirs, standpipes, or tanks) that are used to meet fluctuations 

in daily demand, maintain pressure in the system, and provide 

water for emergencies. The amount of storage provided for 

these objectives, however, is commonly sufficient for only 

several days, not months or seasons (Viessman and Hammer, 

1985). For example, the Kent County Water Authority currently has a storage capacity for a 1- to 3-day supply in water 

tanks connected to the distribution network (Timothy Brown, 

Kent County Water Authority, written commun., 2003).

0

0

30 60 90 120 150 180 210 240 270 300 330 360

1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

TIME, IN DAYS

STREAMFLOW DEPLETION, IN MILLION GALLONS PER DAY

Well A 

Well B 

Figure A4–10. Hypothetical streamflow depletion caused by two wells pumping independently for 180 days at 1.0 million 

gallons per day. Well A is 250 feet from the streambank and well B is 1,000 feet from the streambank (Modified from 

Barlow, 1997).
240 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

The monthly demand pattern constraint was specified 

as a set of 11 constraints that control the relation among total 

withdrawals from one month to the next for each withdrawal 

point as

Qwi t i

NW

Qwi t i

NW

, , =1 1 1 2

∑ = × =∑

, (5)

where

NW is the number of active withdrawal sites in 

each model (table A4–1); and 

Qwi,t are the withdrawal rates at water-withdrawal 

point i in months t

1

 and t

2

, respectively; 

and 

α1,2 is the ratio of the percentage of total demands 

in month t

1

 to total demands in month t

2

(adjusted for the number of days in each 

month). 

All calculations of α1,2 were normalized to the July demand 

for each type of withdrawal and to the number of days in each 

month because irrigation demands do not occur over the entire 

year and the July value is the maximum of all 12 monthly 

values for municipal and irrigation withdrawals. 

Monthly water-withdrawal patterns for four water-supply 

systems (Kingston, Richmond, United Water, and Westerly) 

that are largely dependent on groundwater (fig. A4–11) were 

used to estimate the monthly distribution of withdrawals from 

the hypothetical wells included in the management models. 

These data, which were collected as part of the current study 

and for USGS water-use studies in Rhode Island (Wild and 

Nimiroski, 2004), were provided by water suppliers and 

include the period from October 1995 through September 

2004. The boxplots in figure A4–11 indicate the month-tomonth and year-to-year variability in water withdrawals for 

each month. The thick line in figure A4–11 connects the average monthly fractions, which add up to one. The fractions are 

printed along the top of the graph. The boxplots indicate both 

the higher fractions of water demand and higher within-month 

variability during the warm months (May through October 

for municipal wells). This indicates that the highest municipal water-supply demands occur during the irrigation season 

when streamflows are lowest. The boxplots also indicate 

lower demand and less variability during the rest of the year 

(November through April for municipal wells). These patterns 

are similar to the monthly distribution of water-supply withdrawals defined by Granato and Barlow (2005) as the Rhode 

Island groundwater-demand pattern for six groundwater-based 

water-supply systems throughout Rhode Island for the period 

1995–1999. The monthly water-supply factor (α1,2) was calculated by dividing the fraction of annual withdrawals for each 

month, August through June, by the July fraction shown in 

figure A4–11.

Three different monthly water-withdrawal patterns for 

irrigation were used in the management-model formulations 

(fig. A4–12). The records of precipitation and potential 

evapotranspiration during the 1960–2004 period were used 

with the logistic-regression equations and daily water-use 

data to estimate irrigation withdrawals during this period. 

The pattern for agricultural irrigation was based on irrigation 

data from turf farms (fig. A4–12). It was assumed that the 

vegetable farms follow a similar irrigation pattern. Results of 

the logistic-regression analysis indicate that there was a sharp 

peak in the fraction of agricultural irrigation patterns in July. 

The results also indicate that small amounts of agricultural 

irrigation may have occurred in the months of April and 

October in only few years during the 45-year simulation 

period. The pattern of surface-water golf-course irrigation 

(fig. A4–12B) was similar to the pattern for agricultural 

irrigation (fig. A4–12A). However, golf courses on surfacewater irrigation systems (fig. A4–12B) followed different 

irrigation patterns than golf courses on groundwater irrigation 

systems (fig. A4–12C). The data indicated that golf courses 

irrigating with groundwater withdrawals used automated 

watering systems, with a regular irrigation pattern that was 

fairly consistent from month to month during the irrigation 

season (fig. A4–12C) and daily peak-irrigation rates occurring 

in the middle of the night. Furthermore, the logistic-regression 

equation for golf courses irrigating with groundwater 

withdrawals indicates that the irrigation season for these 

systems includes parts of March and November (although it 

should be noted that the equation indicates the potential for 

withdrawals during March in only a few years during the 

1960–2004 period).

The boxplots in figure A4–12 indicate that the fraction 

of irrigation withdrawals within each month varied substantially with differences in weather patterns. Average monthly 

fractions were used to formulate the conjunctive-management 

models because the effect of such variation cannot be predicted with great certainty for any future year. The thick line 

in each panel of figure A4–12 connects the mean monthly 

fractions, which are printed along the top of the graph and 

add up to one. As with the municipal withdrawal pattern, the 

monthly water-supply factors (α1,2) for irrigation withdrawals 

are calculated by dividing the fraction of annual withdrawals 

for each month in the irrigation season by the July fraction 

shown in figure A4–12.

Because the objective of the example management models is to maximize water withdrawals for a given set of streamflow criteria, a management model that is not constrained 

by a monthly water-withdrawal pattern will indicate that the 

optimal solution for maximizing irrigation withdrawals would 

be to make such withdrawals during months when the ensuing streamflow depletion would occur during the high-flow 

months of the year (for example, fig. A4–2). The resulting 

withdrawal pattern would not be useful for irrigation needs, 

however, because the withdrawals would be made in the late 

fall, winter, and early spring when irrigation is unnecessary. 

This example illustrates why monthly withdrawal-pattern constraints are necessary to formulate the optimization models. 

It should be noted, however, that use of the mean withdrawal 

α1,2
Seasonal Water-Demand Criteria 241

JAN. FEB. MAR. APRIL MAY JUNE JULY AUG. SEPT. OCT. NOV. DEC.

0.00

0.05

0.10

0.15

0.20

0.25

0.096 0114 0.104 0.087 0.079 0.072 0.072 0.073 0.066 0.073 0.073 0.091

Mean monthly proportion of annual withdrawals

MONTHLY WATER SUPPLY WITHDRAWALS, AS A FRACTION OF ANNUAL WITHDRAWALS

MONTH

Maximum

95th percentile

25th percentile

Median

Mean

75th percentile

5th percentile

Minimum

Figure A4–11. The distribution of monthly withdrawals as a fraction of total annual withdrawals during the period 

1995–2004 from four water-supply systems that withdraw primarily from groundwater sources in the Pawcatuck River 

Basin, southwestern Rhode Island.
242 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

fractions may not produce optimal irrigation patterns for every 

year because irrigation patterns vary with climate conditions 

from year to year. Further refinement of withdrawal patterns 

or the generation of multiple optimization models based on 

irrigation patterns for the 45-year record would not be useful 

because it is impossible to produce accurate estimates of daily 

precipitation and potential evaporation at the beginning of the 

irrigation season. For example, results of the logistic-regression equations indicate that total annual irrigation withdrawals 

would range from 147 to 445 Mgal/yr in the EPRCMM area 

and from 102 to 279 Mgal/yr in the LWRCMM area under 

1960–2004 the climatic conditions (fig. A4–13).

Response-Matrix Technique for Solution of the 

Conjunctive-Management Model

The optimization method used to solve the conjunctivemanagement model is based on a widely applied technique for 

solving many types of groundwater-management problems. 

Called the response-matrix technique, the method is based on 

the assumption that the rate of streamflow depletion at each 

streamflow-constraint site is a linear function of the rate of 

groundwater withdrawal at each production well. By assuming 

linearity, it is possible to estimate total streamflow depletion 

at a constraint site by summation of the individual streamflow 

depletions caused by each well. Detailed descriptions of 

the response-matrix technique are given by Gorelick and 

others (1993) and Ahlfeld and Mulligan (2000). Specific 

applications of the technique to problems in stream-aquifer 

management in Massachusetts and Rhode Island are given by 

Male and Mueller (1992), Mueller and Male (1993), Barlow 

(1997), Barlow and Dickerman (2001a,b), DeSimone and 

others (2002), Barlow and others (2003), DeSimone (2004), 

Eggleston (2004), and Granato and Barlow (2005). 

The response-matrix technique is developed using 

a transient dynamic-equilibrium groundwater modeling 

technique. The monthly hydrologic inputs for an average 

year are used repeatedly a transient groundwater model over 

a number of simulation years so the system will reach a 

dynamic-equilibrium from year-to-year to quantify the effect 

of one monthly withdrawal on streamflow depletions in the 

months following each withdrawal period (Barlow, 1997; 

Barlow and Dickerman, 2001a,b; DeSimone and others, 2002; 

Barlow and others, 2003; DeSimone, 2004; Eggleston, 2004; 

Granato and Barlow, 2005). The technique is valid as long as 

the saturated thickness and transmissivity of the aquifer do not 

vary substantially with changes in withdrawal rates, and other 

nonlinear effects simulated by the transient model, such as 

head-dependent boundary conditions, are not large. 

To employ this technique, the monthly inputs for the 

year 2000 in the calibrated transient MODFLOW models 

developed for each of the conjunctive-management model 

area were used in a 5-year dynamic-equilibrium simulation 

for each model area. Eight MODFLOW simulations were run 

to calculate a characteristic streamflow-depletion response in 

Maximum

95th percentile

25th percentile

Median

Mean

75th percentile

5th percentile

Minimum

J F M A M J J A S O N D

MONTH

A Estimated turf-farm irrigation

0.005 0.082 0.224 0.317 0.231 0.134 0.007

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

MONTHLY IRRIGATION WITHDRAWALS, AS A FRACTION OF ANNUAL WITHDRAWALS

0.00

0.05

0.10

0.15

0.20

0.25

0.001 0.034

C Estimated golf-course irrigation from groundwater withdrawals

0.077 0.148 0.148 0.154 0.154 0.148 0.136

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

B Estimated golf-course irrigation from surface-water withdrawals

0.005 0.110 0.219 0.284 0.226 0.142 0.014

Mean monthly 

fraction of 

annual withdrawals

Figure A4–12. The distribution of monthly irrigation withdrawals as a 

fraction of total annual withdrawals during the period January 1960 through 

September 2004 estimated by use of logistic-regression equations developed 

for (A) turf-farm irrigation, (B) golf-course irrigation from surface-water 

withdrawals, and (C) golf-course irrigation from groundwater withdrawals in 

southwestern Rhode Island.
Response-Matrix Technique for Solution of the Conjunctive-Management Model 243

100

200

300

400

500

600

700

800

900

1,000

IRRIGATION

WITHDRAWALS, I

N

MILLIONS OF GALLONS PER YEAR

CONJUNCTIVE-MANAGEMENT MODEL AREA

EPRCMM LWRCMM

Maximum

Mean

95th percentile

75th percentile

50th percentile

Minimum

5th percentile

25th percentile

EXPLANATION

Figure A4–13. The distribution of annual irrigation withdrawals as estimated from the logistic-regression 

equations for irrigation withdrawals for the eastern Pawcatuck River conjunctive-management model (EPRCMM) 

area and the lower Wood River conjunctive-management model (LWRCMM) area in the Pawcatuck River Basin, 

southwestern Rhode Island, 1960–2004.
244 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

response coefficients for all potential groundwater withdrawal 

sites that affect each streamflow constraint site in each conjunctive-management model area. Factors that affect the values of the response coefficients are the relative positions of the 

withdrawal wells and streamflow-constraint sites (including 

the vertical positions of the screened intervals of the wells), 

the geometry and hydraulic properties of the aquifer, the 

streambed conductance, and other physical characteristics of 

the streams as simulated with the groundwater models. Graphs 

of the response coefficients indicate substantial variability in 

the quantity and timing of streamflow-depletion responses to 

the simulated unit withdrawals.

Because of the assumed linearity of the system, the total 

streamflow depletion Qsdj,t at each constraint site j and for 

each month t can be calculated with the response coefficients 

by summation of the individual streamflow depletions caused 

by withdrawals at each well in each month. This summation is 

written as

Qsd j t r

j i k k

Qwi k i

NW

, , , , = =

∑ × ′ =∑

1

12

1 , (7)

where

 { k ́ = t – k + 1 for t – k + 1 > 0

 k ́ = 12 + (t – k + 1), for t – k + 1 ≤ 0

The two-part definition of k ́ (the monthly withdrawal 

index number) is required as a consequence of the annual 

cycle of withdrawals. For example, streamflow depletions in 

January (t=1, k ́=1) can be affected by withdrawals in 

December (t=12, k ́=2). For example, to calculate depletions 

in the Beaver River at BEAVM from withdrawals at well 

RIW-385 a response coefficient of about 0.06 would be used 

for December withdrawals and a response coefficient of about 

0.93 would be used for January withdrawals (fig. A4–14). 

Although the summation includes 12 terms for each well/

streamflow-constraint-site pair, many of the terms may equal 

zero if many of the response coefficients equal zero. Implementation of equation 7 provides the information necessary 

to estimate the cumulative effect of multiple withdrawals on 

streamflow depletion from one or more withdrawal sites on a 

selected streamflow-depletion site in the area of interest.

The response coefficients are the link between the 

groundwater flow model and the conjunctive-management 

models in the Pawcatuck River Basin, and are incorporated 

into the water-resource-management model by replacing the 

definition of Qsdj,t in the expression defining the streamflowdepletion constraints (equation 2) by the right-hand side of 

equation 7. The constraints are then written as 

r

j i k k

Qwi k i

NW

Qsd = , , , j t, max

∑ × ′ =∑ ≤ 

  1

12

1 . (8)

each month to a unit withdrawal at each potential groundwater-withdrawal site in each model area. Different simulations 

for the four withdrawal rates 0.1, 0.5, 1.0, and 2.0 Mgal/d in 

two different months (January and June) were run to calculate 

response coefficients for different withdrawal rates and different seasons. Each groundwater withdrawal site was simulated 

independently to isolate the effects of withdrawals at that site 

on flows in neighboring streams. In each simulation, the withdrawal rate for one of the wells was specified in the groundwater model for a duration of one month; at the end of the month, 

the withdrawal rate at the well was returned to zero. The 

single-month increase in withdrawals is referred to as the unit 

withdrawal Qwi* at well i. The amount of streamflow depletion 

resulting from the unit withdrawal was determined by subtracting streamflow rates calculated by the model with the unit 

withdrawal active from those calculated by the model without 

any withdrawals. Streamflow-depletion responses to the unit 

withdrawals are defined as Qsd*j,i,t, in which the subscripts 

indicate that the streamflow depletion occurs at site j in month 

t in response to a withdrawal at well i. Streamflow-depletion 

response coefficients (rj,i,t) are then defined as

rj i t

Qsd j i t

Qw i , ,

* , ,

* = . (6)

The response coefficients are dimensionless and can 

range from 0.0 to 1.0. A response of 1.0 in the first month of 

withdrawals indicates that all of the water removed by the 

well can be accounted for as streamflow depletion in the first 

month. Similarly, surface-water withdrawals, which are taken 

directly from a site on the stream, are assigned a response of 

1.0 in the first month of withdrawals. Response coefficients for 

groundwater withdrawals may represent depletions induced 

directly from the stream (by lowering groundwater levels 

below the local stream stage) and depletions caused by intercepting groundwater that would otherwise have discharged to 

the stream in the current and subsequent months (Granato and 

Barlow, 2005).

If a well causes depletions in only one stream, the sum 

of the response coefficients for that stream would be to almost 

1.0; the remainder would be attributable to minor reductions 

in riparian evapotranspiration under dynamic equilibrium. 

Similarly, if a well affects streamflow in more than one stream, 

the sum of the response coefficients from the affected streams 

would be about 1.0. Response coefficients for both areas were 

adjusted to a value of 1.0 because the groundwater models did 

not simulate variable riparian-evapotranspiration rates or variable water-withdrawal rates. In addition, the use of response 

coefficients that add up to 1.0 provides a small margin of 

safety for the effects of withdrawals on streamflow depletion. 

Response coefficients for each potential groundwater 

withdrawal site were determined by interpretation of transient groundwater model results for the streamflow-constraint 

sites in each area. Figures 4A–14–A4–21 show the nonzero 
Response-Matrix Technique for Solution of the Conjunctive-Management Model 245

Equation 8 replaces equation 2 in the conjunctive-management 

model.

The use of the response-matrix technique is limited 

because relations between groundwater head and groundwater 

discharge to streamflow in unconfined aquifers commonly are 

weakly nonlinear (Barlow and Dickerman 2001a,b; Granato 

and Barlow, 2005). These nonlinearities commonly are the 

result of two factors under normal conditions and an additional 

factor under extreme drought conditions. First, the saturated 

thickness and transmissivity in an unconfined aquifer change 

as recharge, evapotranspiration, and withdrawal rates at the 

wells change. These nonlinearities were not characterized by 

the transient groundwater models because the models were 

developed with the fixed-transmissivity option. The fixedtransmissivity option was used to address numerical-stability 

issues in areas where relatively steep-slope low-transmissivity 

till areas contact high-transmissivity aquifer areas. Second, 

streamflow leakages were simulated as piecewise-linear functions of calculated groundwater heads. Under extreme drought 

conditions (or with water withdrawals that exceed available 

streamflow), a stream reach may go dry. In this case, the 

stream is no longer a source of water to the well, and aquifer 

storage becomes the sole source of water to the well. Theoretically, the amount of water taken from aquifer storage will 

appear as streamflow depletion in subsequent months because 

the well will capture water that would otherwise discharge to 

the stream until the deficit in aquifer storage is replaced by 

subsequent increases in recharge. Because of these nonlinearities, the response coefficients for each well/streamflowconstraint-site pair can change as withdrawal rates change, 

and such changes can affect the solution of the conjunctivemanagement model.

Conjunctive-management models for both areas are 

affected by an additional source of potential nonlinearity. In 

each are the watersheds consist of large relatively low-relief 

sand and gravel aquifers that span the groundwater divide 

between two streams. In the EPRCMM area, 10 of the existing and potential groundwater withdrawal sites are between 

the Beaver and the Usquepaug-Queen Rivers, and some sites 

cause streamflow depletion in the main stem of the Pawcatuck 

River (fig. 4–1). In the LWRCMM area, five of the existing and potential groundwater withdrawal sites are between 

the Wood River and Meadow Brook (fig. 4–2). Variations in 

recharge and groundwater withdrawal rates near the natural 

groundwater divides may alter the fraction of withdrawals that 

cause depletions in each stream leading to the potential for 

variations in the response coefficients for different hydrologic conditions. 

These types of nonlinearities have been addressed 

in groundwater-management problems by sequential (or 

iterative) linearization of the nonlinear problem (Danskin 

and Gorelick, 1985; Danskin and Freckleton, 1989; Gorelick 

and others, 1993, p. 206–208; Barlow, 1997; Ahlfeld and 

Mulligan, 2000, p. 160–163). Barlow and Dickerman (2001) 

indicated that such sequential-linearization approaches were 

not necessary for conjunctive-management models of the 

unconfined Hunt-Annaquatucket-Pettasquamscutt streamaquifer system because the methods were computationally 

intensive and because simulations with transient groundwater 

models indicated that the response coefficients changed very 

little with the simulated withdrawal conditions. Granato and 

Barlow (2005) obtained similar results for the unconfined sand 

and gravel aquifer in the Big River Area.

Another complicating factor in the use of the responsematrix technique is that the lengths of the stress periods in the 

transient model were not constant, but ranged from 28 to 

31 days. This difference violates the assumption of equallength stress periods. Equal-length stress periods divide the 

response over the same number of days. Barlow and Dickerman (2001) indicated that violation of this assumption should 

not have a substantial effect on the model solution because 

the lengths of the stress periods used in the model do not 

differ substantially. This factor may affect the calculation of 

response coefficients, but does not directly affect the solution of the management model because the components are 

weighted by month length. 

For the linearity approximation to be valid for solution 

of the management models, the values of the response coefficients for each well/streamflow-constraint-site pair must 

be approximately constant for all simulated withdrawal and 

hydrologic conditions. The assumption of linearity was tested 

in the eastern Pawcatuck and lower Wood River groundwater 

models by calculating response coefficients with four different pumping rates (0.1, 0.5, 1.0, and 2.0 Mgal/d) for different 

seasons of the year (January and June). The differences in 

calculated response coefficients were within the precision of 

the numerical solution for most of the stressed and unstressed 

conditions tested. The primary reason the response coefficients 

change very little for different simulated withdrawal conditions is that the aquifer is assumed to be highly transmissive 

near many of the wells. As a consequence, drawdowns caused 

by different conditions do not cause substantial changes in 

the saturated thickness or transmissivity of the aquifer beyond 

the immediate area of the well, unless the local stream goes 

dry. The median response coefficients from among different 

groundwater-model runs were selected to obtain the most representative response coefficients for different conditions.

The modified conjunctive-management model defined by 

equations 1, 3–5, and 8 constitutes a linear program based on 

the assumption of the linearity of the streamflow responses to 

groundwater withdrawals. The LINDO linear-programming 

computer software (LINDO Systems, 1996; Schrage, 1997) 

was used to solve each specific application of the conjunctivemanagement models for each area. The program mathematically searches for the monthly withdrawal rates at each well 

that maximize the yield of the aquifer subject to the set of constraints. Output from the LINDO program includes the total 

withdrawal rate, withdrawal rates for each well, and information about binding criteria. 
246 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

Figure A4–14. Selected examples of simulated 

response coefficients for the (A) Beaver, (B)

Usquepaug-Queen, and (C) Pawcatuck Rivers, 

in the Eastern Pawcatuck River ConjunctiveManagement Model (EPRCMM) area in the 

Pawcatuck River Basin southwestern Rhode 

Island, for three hypothetical wells at different 

distances between the rivers. (Site locations 

shown in figure 4–1 and described in tables 4–1 

and 4–2.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

 PR-AB4A

 PR-AB3B

RIW-385

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pawcatuck River at PAWCD

Beaver River at BEAVM Usquepaug River at QUEENM

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12

A

C

B

END OF MONTH
Response-Matrix Technique for Solution of the Conjunctive-Management Model 247

Figure A4–15. Selected examples of simulated 

response coefficients for the (A) Beaver, (B)

Usquepaug-Queen, and (C) Pawcatuck Rivers, 

in the Eastern Pawcatuck River ConjunctiveManagement Model (EPRCMM) area in the 

Pawcatuck River Basin southwestern Rhode 

Island, for four hypothetical wells at different 

distances between the rivers. (Site locations 

shown in figure 4–1 and described in tables 4–1 

and 4–2.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pawcatuck River at PAWCD

Beaver River at BEAVM Usquepaug River at QUEENM

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

 PR-AB8A

 PR-AUQ8A

 PR-AUQ9A

 PR-AUQ11A

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12

A

C

B

END OF MONTH
248 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

 PR-AB1A

 PR-AB2A

 PR-AB3A

 PR-AB2B

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pawcatuck River at PAWCD

Beaver River at BEAVM Usquepaug River at QUEENM

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12

A

C

B

Figure A4–16. Selected examples of simulated 

response coefficients for the (A) Beaver, (B)

Usquepaug-Queen, and (C) Pawcatuck Rivers, 

in the Eastern Pawcatuck River ConjunctiveManagement Model (EPRCMM) area in the 

Pawcatuck River Basin southwestern Rhode 

Island, for four hypothetical wells at different 

distances between the rivers. (Site locations 

shown in figure 4–1 and described in tables 4–1 

and 4–2.)
Response-Matrix Technique for Solution of the Conjunctive-Management Model 249

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

 PR-AB5A

 PR-AUQ10A

 PR-AUQ10B

 PR-AUQ10C

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pawcatuck River at PAWCD

Beaver River at BEAVM Usquepaug River at QUEENM

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12

A

C

B

END OF MONTH

Figure A4–17. Selected examples of simulated 

response coefficients for the (A) Beaver, (B)

Usquepaug-Queen, and (C) Pawcatuck Rivers, 

in the Eastern Pawcatuck River ConjunctiveManagement Model (EPRCMM) area in the 

Pawcatuck River Basin southwestern Rhode 

Island, for four hypothetical wells at different 

distances between the rivers. (Site locations 

shown in figure 4–1 and described in tables 4–1 

and 4–2.)
250 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

PR-AUQ7A

PR-GB1A

PR-GUQ2AB

RIW336A

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pawcatuck River at PAWCD

Beaver River at BEAVM Usquepaug River at QUEENM

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12

A

C

B

END OF MONTH

Figure A4–18. Selected examples of simulated 

response coefficients for the (A) Beaver, (B)

Usquepaug-Queen, and (C) Pawcatuck Rivers, 

in the Eastern Pawcatuck River ConjunctiveManagement Model (EPRCMM) area in the 

Pawcatuck River Basin southwestern Rhode 

Island, for four hypothetical wells at different 

distances between the rivers. (Site locations 

shown in figure 4–1 and described in tables 4–1 

and 4–2.)
Response-Matrix Technique for Solution of the Conjunctive-Management Model 251

Figure A4–19. Selected examples of simulated response coefficients for (A,B) Wood River, (C) Meadow Brook, and (D) Pawcatuck 

River, in the Lower Wood River Conjunctive-Management Model (LWRCMM) area in the Pawcatuck River Basin southwestern 

Rhode Island, for three hypothetical wells at different distances between the rivers. (Site locations shown in figure 4–2 and 

described in tables 4–1 and 4–2.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

RIW-458

RIW-458A

RIW-458B

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Meadow Brook at MEAD2 (RCHRES 48) Pawcatuck River at PAWC4 (RCHRES 50)

Wood River at WOOD5 (RCHRES 63) Wood River at WOOD6 (RCHRES 65)

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

A

C D

B
252 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

Figure A4–20. Selected examples of simulated response coefficients for (A,B) Wood River, (C) Meadow Brook, and (D) Pawcatuck 

River, in the Lower Wood River Conjunctive-Management Model (LWRCMM) area in the Pawcatuck River Basin southwestern 

Rhode Island, for three hypothetical wells at different distances between the rivers. (Site locations shown in figure 4–2 and 

described in tables 4–1 and 4–2.)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

GW1A

GW3A

GW4A

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Meadow Brook at MEAD2 (RCHRES 48) Pawcatuck River at PAWC4 (RCHRES 50)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Wood River at WOOD5 (RCHRES 63) Wood River at WOOD6 (RCHRES 65)

RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

A

C D

B
Response-Matrix Technique for Solution of the Conjunctive-Management Model 253

Figure A4–21. Selected examples of simulated response coefficients for (A,B) Wood River, (C) Meadow Brook, and (D) Pawcatuck 

River, in the Lower Wood River Conjunctive-Management Model (LWRCMM) area in the Pawcatuck River Basin southwestern 

Rhode Island, for three hypothetical wells at different distances between the rivers. (Site locations shown in figure 4–2 and 

described in tables 4–1 and 4–2.) RESP

ONSE C

OEFFICIENT (CHAN

GE IN STREAMFL

OW FOR UNIT WITHDRAWAL RATE AT WELL), DIMENSI

ONLESS

END OF MONTH

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

A

C D

B

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

GW4B

GW4C

 PR-AW1A

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Meadow Brook at MEAD2 (RCHRES 48) Pawcatuck River at PAWC4 (RCHRES 50)

Wood River at WOOD5 (RCHRES 63) Wood River at WOOD6 (RCHRES 65)
254 Appendix Part 4. Conjunctive-Management Models for the Eastern Pawcatuck and Lower Wood River Model Areas

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