Observation

piezometer

Pumped well

Saturated zone

Unsaturated zone

Land surface

Base of aquifer

Initial water table

Moisture

 content

Water table

Ground-water flow

Q

Estimation of Hydraulic Parameters 

from an Unconfined Aquifer Test 

Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachusetts 

U.S. Department of the Interior

U.S. Geological Survey

Professional Paper 1629
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Estimation of Hydraulic Parameters from an 

Unconfined Aquifer Test Conducted in a 

Glacial Outwash Deposit, 

Cape Cod, Massachusetts

By Allen F. Moench, Stephen P. Garabedian, and Denis R. LeBlanc

Professional Paper 1629
Library of Congress Cataloging-in-Publications Data

Moench, A. F.

Estimation of hydraulic parameters from an unconfined aquifer test conducted in a glacial outwash deposit, Cape Cod, Massachusetts / by Allen F. Moench, Stephen P. Garabedian, and 

Denis R. LeBlanc.

p. cm. – (Professional paper ; 1629)

Includes bibliographical references (p. ).

ISBN 0-607-95032-3

1. Aquifers–Massachusetts–Cape Cod–Testing. I. Garabedian, Stephen P. II. LeBlanc, Denis 

R. III. Geological Survey (U.S.) IV. Title. V. U.S. Geological Survey professional paper ; 1629.

QE75 .P9 v. 1629

[GB1199.3.M]

557.3 s–dc21

[551.49’09744’92]

00-060051

Any use of trade, product, or firm names in this report is for

identification purposes only and does not constitute endorsement 

by the U.S. Government.

For sale by U.S. Geological Survey, Information Services,

Box 25286, Federal Center,

Denver, CO 80225

Reston, Virginia 2001

U.S. DEPARTMENT OF THE INTERIOR

Gale A. Norton, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director
Contents III

CONTENTS

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

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Purpose and Scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Response of an Idealized Unconfined Aquifer to Pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Hydrogeology of the Aquifer-Test Site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

 Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Boundary-Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Dimensionless Boundary-Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Laplace Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Aquifer-Test Design and Operation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Analyses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Analysis by Nonlinear Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Evaluation of Sy, Kr, Kz Using Late-Time Data (Step 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Evaluation of Sy, b, Kr, Kz Using Late-Time Data (Step 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Evaluation of Sw Using Late-Time Pumped-Well Data (Step 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Evaluation of Ss Using Early-Time Data (Step 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Evaluation of Ss, Sy, b, Kr, Kz, and αm Using Data for Entire Time Range (Step 5) . . . . . . . . . . . . . . . . . . . . . 20

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Effect of Heterogeneity on Aquifer Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Parameter-Estimation Experiments with Measured Drawdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Experiments with Pumped-Well Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Experiments with Limited Piezometer Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Experiments with Reduced Length of Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Appendix I—Derivation of the Laplace Transform Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Appendix II—Drawdown Data Plots for Well Test and Data Selected for Parameter Estimation . . . . . . . . . . . . . . . . . . . . . 59
IV Contents

Figures

1. A, Schematic diagram of a pumping well and observation piezometer in an idealized, anisotropic unconfined 

aquifer with a hypothetical moisture distribution indicated for the unsaturated zone. B, A typical double-logarithmic 

plot of drawdown in an observation piezometer versus time that defines the approximate ranges of early-,

intermediate-, and late-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Schematic diagram of a finite-diameter pumped well, observation well, and observation piezometer in a 

homogeneous, anisotropic water-table aquifer of infinite lateral extent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3. Regional location and local plan view showing the positions of the pumped well (F507-080) and 

observation wells and piezometers in the study area. The reference piezometer (F343-036) is not used to 

measure drawdown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4. Vertical cross section of the aquifer at the study site showing the lengths and positions of the piezometers 

and observation wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5. Measured and simulated drawdowns for the pumped well (F507-080), for the model parameters

shown in table 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, piezometers F505-080 and F504-080. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

7. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, wells F505-059 and F504-060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, piezometers F505-032 and F504-032. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, piezometers F377-037 and F347-031. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, (A) piezometers F383-061 and F383-032, and

(B) piezometers F383-082 and F383-129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, (A) piezometers F384-033 and F385-032, and 

(B) piezometers F381-056 and F376-037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

12. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone, (A) wells F434-060 and F450-061, and 

(B) wells F476-061 and F478-061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

13. Measured drawdowns and drawdowns simulated for piezometer F377-037 using (A) the assumption of instantaneous 

drainage, (B) gradual drainage using a single empirical parameter, (C) gradual drainage using

two empirical parameters, and (D) gradual drainage using three empirical parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

14. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release of 

water from the unsaturated zone, piezometers F505-080 and F504-080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

15. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release 

of water from the unsaturated zone, wells F505-059 and F504-060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

16. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release 

of water from the unsaturated zone, piezometers F505-032 and F504-032. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

17. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release 

of water from the unsaturated zone, piezometers F377-037 and F347-031. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

18. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release 

of water from the unsaturated zone, (A) piezometers F383-061 and F383-032, and 

(B) piezometers F383-082 and F383-129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

19. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release 

of water from the unsaturated zone, (A) piezometers F384-033 and F385-032, and 

(B) piezometers F381-056 and F376-037 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Contents V

20. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release 

of water from the unsaturated zone, (A) wells F434-060 and F450-061, and 

(B) wells F476-061 and F478-061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

21. Measured and simulated drawdowns for the pumped well (F507-080) for the model parameters

shown in table 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

22. Measured drawdowns compared with drawdowns simulated under the assumption of instantaneous 

release of water from the unsaturated zone for selected deep-seated piezometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

23. Measured drawdowns compared with drawdowns simulated under the assumption of 

gradual drainage of water from the unsaturated zone for selected deep-seated piezometers . . . . . . . . . . . . . . . . . . . . . . . . . 47

Tables

1. Dimensionless expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2. Locations of observation piezometers, number of PEST values and measurement numbers . . . . . . . . . . . . . . . . . . . . . . . 15

3. Parameters obtained from preliminary analysis of hand-measured drawdown data, where 

Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in the horizontal direction, 

and Kz equals hydraulic conductivity in the vertical direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4. Parameters obtained from late-time data exclusively using PEST with b equals 160 feet, 

where Sy equals specific yield, Kr equals hydraulic conductivity in the horizontal direction, and 

Kz equals hydraulic conductivity in the vertical direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5. Parameters obtained from late-time data exclusively using PEST with b as an estimated parameter, 

where Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in the horizontal 

direction, and Kz equals hydraulic conductivity in the vertical direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6. Parameters estimated from early and late-time data exclusively, using PEST, where Ss equals specific 

storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in the horizontal 

direction, and Kz equals hydraulic conductivity in the vertical direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7. Parameters estimated from the complete data set using PEST, where Ss equals specific storage, 

Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in the horizontal direction, 

Kz equals hydraulic conductivity in the vertical direction, and α1 , α2 , and α3 are empirical constants for 

gradual drainage from the unsaturated zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

8. Correlation coefficient matrix for table 7, where Ss equals specific storage, Sy equals specific yield, 

b equals saturated thickness, Kr equals hydraulic conductivity in the horizontal direction, Kz equals hydraulic

conductivity in the vertical direction, and α1 , α2 , and α3 are empirical constants for gradual drainage from 

the unsaturated zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9. Parameters estimated from the complete data set using PEST with alternative initial values, 

where Ss equals specific storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic 

conductivity in the horizontal direction, Kz equals hydraulic conductivity in the vertical direction, and 

α1 , α2 , and α3 are empirical constants for gradual drainage from the unsaturated zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10A. Estimates of the variance and standard deviation of head in the aquifer for three-dimensional and 

one-dimensional flow using equations 27 and 28, where αf

2 equals variance of ln Kr, J equals horizontal hydraulic 

gradient, λ1 equals horizontal log hydraulic conductivity correlation scale, and λ2 equals vertical log 

hydraulic conductivity correlation scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

10B. Estimates of the variance and standard deviation of head in the aquifer for three-dimensional and 

one-dimensional flow using equations 27 and 28, where αf

2 equals variance of ln Kr, J equals horizontal hydraulic 

gradient, λ1 equals horizontal log hydraulic conductivity correlation scale, and λ2 equals vertical log 

hydraulic conductivity correlation scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

11. Column headings for tables 12–15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

12. Analysis of selected piezometer groups assuming gradual drainage and adjustable saturated thickness, 

where Ss equals specific storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic 

conductivity in the horizontal direction, Kz equals hydraulic conductivity in the vertical direction, and α1, α2, 

and α3 are empirical constants for gradual drainage from the unsaturated zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
VI Contents

13. Analysis of selected piezometer groups assuming delayed drainage and fixed saturated thickness, 

where Ss equals specific storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic 

conductivity in the horizontal direction, Kz equals hydraulic conductivity in the vertical direction, and α1, α2, 

and α3 are empirical constants for gradual drainage from the unsaturated zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14. Analysis of selected piezometer groups assuming instantaneous drainage for times greater than 430 

minutes and adjustable saturated thickness, where Ss equals specific storage, Sy equals specific yield, b equals 

saturated thickness, Kr equals hydraulic conductivity in the horizontal direction, and Kz equals hydraulic 

conductivity in the vertical direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

15. Analysis of selected piezometer groups assuming instantaneous drainage for times greater than 430

minutes with fixed saturated thickness, where Ss equals specific storage, Sy equals specific yield, b equals saturated

thickness, Kr equals hydraulic conductivity in the horizontal direction, Kz equals hydraulic conductivity in the 

vertical direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

16A. Analysis of time-limited tests for all piezometers and deep-seated piezometers, where Ss equals 

specific storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in 

the horizontal direction, Kz equals hydraulic conductivity in the vertical direction, and α1, α2, and α3 are 

empirical constants for gradual drainage from the unsaturated zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

16B. Analysis of time-limited tests for long-screened piezometers and piezometer clusters, where Ss

equals specific storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in 

the horizontal direction, Kz equals hydraulic conductivity in the vertical direction, and α1, α2, and α3 are 

empirical constants for gradual drainage from the unsaturated zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

17. Various data analyses for piezometers F505-059, F505-080, F504-080, and F383-129, where Ss

equals specific storage, Sy equals specific yield, b equals saturated thickness, Kr equals hydraulic conductivity in 

the horizontal direction, Kz equals hydraulic conductivity in the vertical direction, and α1, α2, and α3 are 

empirical constants for gradual drainage from the unsaturated zone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Contents VII

CONVERSION FACTORS

Multiply By To obtain

Length

inch (in) 2.54 centimeter

inch (in) 25.4 millimeter

foot (ft) 0.3048 meter

Volume

gallon (gal) 3.785 liter

Flow rate

foot per minute (ft/min) 0.3048 meter per minute

gallon per minute (gal/min) 0.06309 liter per second

Hydraulic conductivity

foot per day (ft/d) 0.3048 meter per day

Sea level: In this report, “sea level” refers to the National Geodetic Vertical Datum of 1929—a geodetc datum derived from a general 

adjustment of the first-order level nets of the United States and Canada, formerly called Sea Level Datum of 1929.
VIII Contents
ABSTRACT 1

Estimation of Hydraulic Parameters from an Unconfined 

Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachusetts

By Allen F. Moench, Stephen P. Garabedian, and Denis R. LeBlanc

ABSTRACT

An aquifer test conducted in a sand and gravel, 

glacial outwash deposit on Cape Cod, Massachusetts 

was analyzed by means of a model for flow to a partially penetrating well in a homogeneous, anisotropic 

unconfined aquifer. The model is designed to account 

for all significant mechanisms expected to influence 

drawdown in observation piezometers and in the 

pumped well. In addition to the usual fluid-flow and 

storage processes, additional processes include effects 

of storage in the pumped well, storage in observation 

piezometers, effects of skin at the pumped-well screen, 

and effects of drainage from the zone above the water 

table. The aquifer was pumped at a rate of 320 gallons 

per minute for 72-hours and drawdown measurements 

were made in the pumped well and in 20 piezometers 

located at various distances from the pumped well and 

various depths below the land surface. To facilitate the 

analysis, an automatic parameter estimation algorithm 

was used to obtain relevant unconfined aquifer parameters, including the saturated thickness and a set of 

empirical parameters that relate to gradual drainage 

from the unsaturated zone. 

Drainage from the unsaturated zone is treated in 

this paper as a finite series of exponential terms, each 

of which contains one empirical parameter that is to be 

determined. It was found necessary to account for 

effects of gradual drainage from the unsaturated zone 

in order to obtain satisfactory agreement between measured and simulated drawdown, particularly in piezometers located near the water table. The commonly used 

assumption of instantaneous drainage from the unsaturated zone gives rise to large discrepancies between 

measured and predicted drawdown in the intermediatetime range and can result in inaccurate estimates of 

aquifer parameters especially when automatic parameter estimation procedures are used. 

The values of the estimated hydraulic parameters 

are consistent with estimates from prior studies and 

from what is known about the aquifer at the site. 

Effects of heterogeneity at the site were small as measured drawdowns in all piezometers and wells were 

very close to the simulated values for a homogeneous 

porous medium. The estimated values are: specific 

yield, 0.26; saturated thickness, 170 feet; horizontal 

hydraulic conductivity, 0.23 feet per minute; vertical 

hydraulic conductivity, 0.14 feet per minute; and specific storage, 1.3x10–5 per foot.

It was found that drawdown in only a few piezometers strategically located at depth near the pumped 

well yielded parameter estimates close to the estimates 

obtained for the entire data set analyzed simultaneously. If the influence of gradual drainage from the 

unsaturated zone is not taken into account, specific 

yield is significantly underestimated even in these 

deep-seated piezometers. This helps to explain the low 

values of specific yield often reported for granular 

aquifers in the literature. If either the entire data set or 

only the drawdown in selected deep-seated piezometers was used, it was found unnecessary to conduct the 

test for the full 72-hours to obtain accurate estimates of 

the hydraulic parameters. For some piezometer groups, 

practically identical results would be obtained for an 

aquifer test conducted for only 8-hours. Drawdowns 

measured in the pumped well and piezometers at distant locations were diagnostic only of aquifer transmissivity.

INTRODUCTION

Proper management of ground-water resources 

requires an accurate evaluation of the parameters 

(hydraulic properties) that control the movement and 

storage of water. Aquifer tests, performed by pumping 
2 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

a well at a constant rate and observing the resulting 

changes in hydraulic head in the aquifer, are the most 

commonly used method for determination of aquifer 

hydraulic properties. Unconfined aquifers, also known 

as water-table aquifers, are of particular interest to 

hydrogeologists and to the general public not only 

because of their accessibility as a water supply but also 

because of their vulnerability to contamination from 

activities at the land surface. Unconfined aquifers have 

special features that set them apart from other aquifer 

types and make analyses of tests conducted in them 

more difficult. The primary added complication has to 

do with the existence of the free surface (or water table) 

and the overlying unsaturated zone.

Background

Hydraulic parameters that control an unconfined 

aquifer's capacity to transmit and store water are generally obtained by aquifer-test analysis using one of several classical analytical models, the most popular of 

which are those of Boulton (1954, 1963), Dagan 

(1967), and Neuman (1972, 1974). The model of Boulton (1954) was the first to provide a plausible explanation for changes in hydraulic head observed in 

unconfined aquifers in response to pumping from a 

well. The model takes into account gradual drainage of 

water from the zone above the water table, a feature 

that is now, belatedly, becoming recognized as being 

important for unconfined aquifers. The Boulton model 

has the drawback, however, that it does not account for 

vertical components of flow in the aquifer and, consequently, cannot be used to evaluate vertical hydraulic 

conductivity: the model is strictly valid only at large 

distances from the pumped well where the flow might 

be assumed to be horizontal. Also, because of the horizontal flow assumption, the model cannot account for 

effects of partial penetration by the pumped well. (This 

limitation was subsequently eliminated in a paper by 

Boulton and Streltsova, 1975.)

The Dagan (1967) and Neuman (1972, 1974) 

models both account for vertical components of flow in 

the aquifer and, hence, for effects of partial penetration 

by the pumped well, but neither consider effects of 

gradual drainage from the zone above the water table to 

be an important consideration. Both models assume 

drainage from the unsaturated zone to occur instantaneously in response to lowering of the water table. The 

Dagan model, in contrast with both the Boulton and 

Neuman models, has the additional limitation that it 

does not account for compressive characteristics of the 

aquifer and therefore cannot be used to evaluate aquifer 

specific storage. The Neuman (1972, 1974) model has 

come to be accepted by many hydrogeologists as the 

preferred model ostensibly because it appears to make 

the fewest simplifying assumptions and because of the 

perception that neglecting the effects of gradual drainage from the zone above the water table is reasonable 

for purposes of aquifer parameter estimation.

While both the Boulton and Neuman models 

account for aquifer compressive characteristics, they 

make the mathematical simplifying assumption that the 

pumped well is a line-sink (that is, the pumped well is 

assumed to be infinitesimal in diameter). Thus, it is 

impossible to account for effects of wellbore storage. 

This limits the usefulness of the models for accurate 

evaluation of specific storage. The line-sink assumption in these models requires that observation piezometers be located at large distances from the pumped 

well to reduce the influence of wellbore storage. Unfortunately, this last requirement makes it difficult to make 

accurate early-time measurements because of small 

drawdowns at large distances.

Use of the Boulton and Neuman models for analysis of early-time data from piezometers located near 

the pumped well may result in values of specific storage that are overestimated by as much as one or two 

orders of magnitude (Moench, 1997), depending on 

aquifer compressibility. Moench (1997) extended the 

range of validity of the Neuman (1974) model by 

accounting for the finite diameter of the pumped well. 

This greatly improves upon the accuracy of specific 

storage estimates made by using drawdown data from 

piezometers located near the pumped well. It also 

makes it theoretically possible to evaluate other unconfined-aquifer parameters from pumped-well data. 

Unfortunately, however, effects of well-bore skin, turbulence, and other non-ideal flow conditions make the 

use of pumped well data difficult and frequently impossible for parameter estimation.

As mentioned above, the Boulton (1963) model 

differs from the Neuman (1972) model in that the latter 

assumes instantaneous drainage of water from the 

unsaturated zone and the former assumes the drainage 

occurs gradually in response to a lowering of the water 

table. Boulton (1954, 1963) approximates drainage 

from the zone above the water table by using an exponential relation containing an empirical parameter or 

“delay index.” Neuman (1975) found that the delay 

index as used by Boulton (1963) is not a characteristic 
INTRODUCTION 3

property of the aquifer. (He found it to be a function of 

radial distance from the pumped well.) Also, based on 

numerical modeling, Neuman (1972) found that effects 

of gradual drainage from the unsaturated zone could be 

neglected in aquifer tests. It has since been found, however, that there may exist a significant difference 

between measured field data and theoretical drawdowns in observation piezometers, particularly those 

located near the water table (see Moench, 1995). These 

differences exist independently of whether the Boulton 

model, which does not account for vertical components 

of flow in the saturated zone, or the Neuman model, 

which does not account for gradual drainage, is used.

In order to reduce the magnitude of this discrepancy and still account for vertical flow in the aquifer's 

saturated zone, Moench (1995) substituted Boulton's 

(1963) convolution integral for Neuman's (1972, 1974) 

boundary condition for the free surface and solved the 

revised boundary-value problem. By so doing, the 

redefined delay index becomes, for the particular test, 

a property of the homogeneous aquifer and associated 

homogeneous unsaturated zone, albeit not a very accurate one. No physical basis was assigned to the revised 

delay index other than that it could be considered the 

inverse of a “characteristic drainage time” for the particular medium. Nevertheless, the discrepancy between 

measured and theoretical drawdowns was diminished 

(over a limited time range) as seen in piezometers 

located near the water table (Moench, 1995). The reason that the discrepancy is not completely eliminated is 

likely due to Boulton's convolution integral not accurately describing the drainage process (see, Narasimhan and Zhu, 1993). Boulton's approach is based on the 

incorrect but plausible assumption that drainage from 

the unsaturated zone follows an exponential decline in 

response to a step decline in the elevation of the water 

table. Boulton and Pontin (1971) recognized deficiencies in Boulton's (1954, 1963) original theory; that is, 

the exponential relation did not accurately reflect physical reality and that it was necessary to account for vertical components of flow in the aquifer. To improve 

upon the single-parameter exponential relation, they 

used two exponential terms containing four adjustable 

parameters (two delay indices and two delayed yield 

parameters that when summed and added to a third 

parameter called “instantaneous yield,” form the total 

specific yield). To account for vertical components of 

flow, they adapted the model of Dagan (1967) to meet 

their needs. Unfortunately, their model (like Dagan's) 

does not account for aquifer compressibility. Also, the 

treatment of specific yield (as the sum of three components rather than as a characteristic constant) is rather 

cumbersome for purposes of analysis.

In 1990, the U.S. Geological Survey (USGS) 

conducted an aquifer test at its Cape Cod Toxic Substances Research Site in Falmouth, MA in order to 

evaluate the hydraulic parameters of the unconfined 

aquifer. Moench and others (1996) provide a preliminary analysis of this test using hand-measured drawdown data, the Neuman (1974) model, and traditional 

type-curve matching methods. The traditional 

approach to evaluation of the hydraulic parameters is 

by visual trial-and-error matching of field data with 

dimensionless type curves (for recommended procedures see, for example, Prickett, 1965; Kruseman and 

de Ridder, 1990; Moench, 1994; and Batu, 1998). A 

single match point was found that yielded excellent 

late-time matches between theoretical and measured 

drawdowns in the 16 piezometers used for the analysis. 

Thus, a single set of hydraulic parameters (vertical 

hydraulic conductivity, horizontal hydraulic conductivity, and specific yield) was obtained. The agreement 

suggested a remarkable degree of homogeneity in 

hydraulic conductivity to ground water flow at the scale 

of the test. Early-time drawdowns, however, recorded 

with the help of transducers located in seven of the piezometers and in the pumped well, and intermediatetime drawdowns recorded in all piezometers, but especially those located near the water table, could not be 

interpreted satisfactorily with the Neuman model.

Purpose and Scope

It is the purpose of this report to provide a thorough interpretation of the aquifer test that was conducted in the summer of 1990 at the USGS Cape Cod 

Toxic Substances Research Site in Falmouth, Mass. It 

is intended to expand upon the preliminary analysis of 

Moench and others (1996) by using a modification of a 

model developed by Moench (1997) and a method of 

automatic parameter estimation. The model modification is designed to permit an accurate representation of 

the process of gradual drainage from the zone above 

the water table.

The Moench (1997) model in its unmodified 

form allowed for evaluation of specific storage, and the 

other unconfined aquifer parameters, but did not fully 

account for discrepancies observed between measured 

and theoretical drawdowns calculated by the Neuman 

(1974) model in the intermediate-time range. Discrep-
4 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

ancies that exist between theoretical and measured 

drawdowns in the intermediate-time range are found to 

be a consequence of neglecting gradual drainage from 

the unsaturated zone. These discrepancies are only partially eliminated with a model that assumes exponentially declining drainage of water from the unsaturated 

zone in response to a step decline in the elevation of the 

water table (Moench, 1995, 1997). This is corroborated 

indirectly by the results of laboratory (Vachaud, 1968), 

field (Nwankwor and others, 1992), and numerical 

experiments (Narasimhan and Zhu, 1993). In this 

report, a necessary additional modification is made in 

the water-table boundary condition by including a 

series of exponential terms to represent drainage from 

the unsaturated zone caused by a decline in the elevation of the water table. The drainage in this instance is 

controlled by a finite number of empirical parameters 

αM. It is not intended in this report that the empirical 

parameters αM be ascribed a physical basis. However, 

it is possible to set up and solve a simple boundaryvalue problem demonstrating that the parameters can 

be used in combination to approximate the rate of flow 

across the water table in response to a step change in its 

elevation (R.L. Cooley, USGS, written commun., 

2000).

In addition to the improved interpretation of the 

aquifer test, resulting from the model used, apparent 

homogeneity of the aquifer, and extent and quality of 

the data set, a number of secondary analyses are performed so that some general recommendations about 

unconfined aquifer tests can be made concerning: 1. the 

number of piezometers needed, 2. the placement of piezometers, 3. the timing and frequency of data sampling, 

and 4. the minimum test duration required to obtain satisfactory results. Computer simulations are also conducted to examine the consequences of specific model 

assumptions.

The scope of this report is limited to analyses of 

the aquifer test by means of an analytical model 

assisted by automated parameter estimation using nonlinear least squares. Because of the apparent homogeneity of the aquifer, the lack of interference from 

recharge and/or evapotranspiration, and the apparent 

validity of the model assumptions, this aquifer test 

might be deemed a benchmark test and a good candidate for illustration of a broad range of unconfined 

aquifer phenomena.

Response of an Idealized Unconfined Aquifer to 

Pumping

Figure 1A is a schematic diagram showing a typical well/aquifer configuration and depicting the 

response of an idealized granular, homogeneous and 

anisotropic unconfined aquifer to pumping. The vertical (Kz) and horizontal (Kr) hydraulic conductivity vectors, the relative magnitude of which is indicated by the 

length of the arrows in the inset, indicate the anisotropic character of the aquifer. Flow to the finite-diameter 

pumped well is axisymmetric and three-dimensional.

Drawdowns (changes in hydraulic head due to 

pumping) in the pumped well may be greater than that 

in the aquifer adjacent to the well because of resistance 

to flow (wellbore skin) at the well screen. Vertical components of flow in the aquifer near the pumped well are 

enhanced if the length of the pumped-well screen is 

less than the full saturated thickness of the aquifer (that 

is, the pumped well partially penetrates the aquifer). 

Because of vertical components of flow, drawdowns 

observed in piezometers located near the pumped well 

cannot be assumed to accurately indicate the position 

of the falling water table. In addition, the response of 

finite-diameter piezometers to rapid changes in hydraulic head in the aquifer may be delayed due to wellbore 

storage in the piezometer.

With regard to the unsaturated zone in the idealized unconfined aquifer: 1. Water held by adsorption 

and surface tension in the unsaturated zone is in direct 

hydraulic connection with the falling water table. 2. 

The equilibrium moisture distribution in the unsaturated zone decreases monotonically with an increase in 

elevation (zu ) as depicted in figure 1A, where θ is the 

moisture content and θs is the moisture content at saturation. 3. The zone of near saturation immediately 

above the water table is referred to as the capillary 

fringe and will vary in thickness depending on the soil 

texture.

Figure 1B depicts a typical plot of drawdowns 

versus time (using double-logarithmic coordinates) 

and defines what is described in this report as “early 

time,” “intermediate time,” and “late time.” The time 

ranges are approximate and would vary depending 

upon the aquifer parameters.

Hydrogeology of the Aquifer-Test Site

The aquifer at the study site is composed of 

unconsolidated glacial outwash sediments that were 
INTRODUCTION 5

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Early time Intermediate time Late time

B.

Observation

piezometer

Pumped well

Q

0 θs

θ

Saturated zone

Unsaturated zone

Kz

zu

Kr

Land surface

Base of aquifer

Initial water table

Moisture

 content

Water table

Ground-water flow

A.

Q

θs

θ

Kz

zu

Kr

Pumping rate

Vertical distance above water table

Moisture content of unsaturated zone

Moisture content above the water 

 table at saturation

Vertical hydraulic conductivity—Length 

 of arrow indicates relative magnitude

Horizontal hydraulic conductivty—Length 

 of arrow indicates relative magnitude

EXPLANATION

Figure 1. A. Schematic diagram of a pumping well and observation piezometer in an idealized, anisotropic 

unconfined aquifer with a hypothetical moisture distribution indicated for the unsaturated zone. B. A typical doublelogarithmic plot of drawdown in an observation piezometer versus time that defines the approximate ranges of early-

, intermediate-, and late-time.
6 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

deposited during the recession, 14,000 to 15,000 years 

before present, of the late Wisconsinan continental ice 

sheet that had previously covered New England. 

Although the unconsolidated sediments in the test area 

overlie crystalline bedrock at a depth of approximately 

300 ft, detailed lithologic studies indicate that clean, 

medium to coarse-grained, high-permeability glacial 

outwash deposits overlie fine-grained, relatively lowpermeability material at a depth of about 160 ft below 

the water table (LeBlanc, 1984; LeBlanc, and others, 

1986; Masterson, and others, 1997). The horizontal 

hydraulic conductivity of the upper material in western 

Cape Cod generally, ranges from 150 to 350 ft/d with a 

ratio of horizontal to vertical hydraulic conductivity of 

3:1 to 10:1. The horizontal hydraulic conductivity of 

the material below the transition ranges from 10 to 70 

ft/d with a ratio of horizontal to vertical hydraulic conductivity of 30:1 to 100:1. The estimate of the saturated 

thickness is corroborated hydraulically by the preliminary analysis of the aquifer test (Moench and others, 

1996).

 Acknowledgments

The USGS Toxic Substances Hydrology Program provided funding for the aquifer test. The logistical and field support from the Massachusetts District 

office is gratefully acknowledged. In particular, thanks 

are given to Kathryn M. Hess and Richard D. Quadri 

for their assistance during the field experiment. It is 

because of the careful work of all participants that the 

test successfully yielded the high quality data needed 

for the analysis. The analysis could not have been 

accomplished otherwise. The detailed and insightful 

reviews of the manuscript by Paul M. Barlow and Richard L. Cooley greatly enhanced the content and presentation of the report.

MATHEMATICAL MODEL

In this section the mathematical model of 

Moench (1997) is presented in a slightly modified 

form. The modification is to allow for improved representation of drainage from the zone above the water 

table. The bulk of the material in this section derives 

from Moench (1997). It is presented here for the convenience of the reader and to incorporate the necessary 

modifications.

Assumptions

As with all mathematical models, several simplifying assumptions are required. Most of the assumptions are identical to those of Neuman (1974). Those 

that are identical are as follows:

1. The aquifer is homogeneous, infinite in lateral 

extent, horizontal, and of uniform thickness.

2. The aquifer can be anisotropic provided that the 

principal directions of the hydraulic conductivity 

tensor are parallel to the coordinate axes.

3. Vertical flow across the lower boundary of the aquifer is negligible.

4. A well discharges at a constant rate from a specified 

zone below the water table.

5. The change in saturated thickness of the aquifer due 

to pumping is small compared with the initial saturated thickness.

6. The porous medium and fluid are slightly compressible and have physical properties that do not vary 

in space or time.

7. The initial hydraulic head is the same everywhere. 

The Neuman (1972, 1974) model assumes that 

water in the unsaturated zone is released instantaneously as the water table declines. It is pointed out in 

the introduction that, under this assumption, there may 

exist a significant difference between measured and 

theoretical drawdowns in piezometers located near the 

water table. The introduction of Boulton's (1963) convolution integral by Moench (1995) into the boundary 

condition for the free surface reduces this discrepancy. 

Moench (1995) assumed, as did Boulton (1954, 1963), 

that the vertical flux of water into the aquifer occurs in 

a manner that varies exponentially with time in 

response to a step decline in the elevation of the water 

table. The rate of exponential decline is controlled by 

an empirical constant α1 (see Moench, 1995). [The 

subscript on α is included to avoid confusion with 

Boulton's reciprocal of “delay index” (Boulton, 1963), 

which has a meaning that is slightly different from the 

α1 used by Moench (1995). The difference in meaning 

is due to the fact that Boulton worked with vertically 

averaged heads (no vertical components of flow) in the 

aquifer and included the term containing α in the governing partial differential equation rather than as a 

boundary condition for the water-table.]

It is known, however, that the assumption of an 

exponential decline provides only a crude approximation of the actual drainage process (see, Vachaud, 1968; 

Narasimhan and Zhu, 1993). In this report, the representation of the actual drainage process can be made as 
MATHEMATICAL MODEL 7

precise as desired by extending the single, empiricalparameter approximation to multiple empirical parameters. It should be pointed out that the same set of 

parameters should not be assumed to accurately represent the reverse process (i.e., absorption of water during recovery) due to effects of hysteresis or entrapped 

air. Nor should it be assumed to represent the drainage 

process at a different time or location. This is because 

conditions in the unsaturated zone may differ from 

time to time and place to place. Although, in this report, 

the empirical parameters are given little physical meaning individually, as a group they can be used to quantify 

drainage from the unsaturated zone in response to a 

driving force such as the lowering of the water table.

The assumptions pertaining to the finite-diameter pumped well are identical to those of Dougherty 

and Babu (1984) and are listed here as follows:

1. The head within the well does not vary vertically.

2. The radial flux from the aquifer to the well does not 

vary along the length of the screened section. 

3. Vertical flux from the aquifer through the base of the 

well is negligible.

4. A thin skin of homogeneous porous material having 

no significant storage capacity may be present at 

the interface between the well screen and the 

aquifer. The hydraulic conductivity of this material may be less than or greater than that of the 

aquifer, and is assumed to be constant during the 

course of the aquifer test. (Low hydraulic conductivity skin may be present for a number of reasons 

as, for example, flow constrictions due to the well 

screen itself, bridging by sand particles across 

screen openings, or damage to the aquifer caused 

by drilling. High hydraulic conductivity skin may 

be due to well development or to the presence of 

a gravel pack installed to increase well productivity.)

The influence of a delayed response of the observation piezometers is often overlooked in the analysis 

of aquifer tests. The effect is treated approximately in 

this report (following Black and Kipp, 1977) by assuming the hydraulic head in an observation piezometer 

changes with a rate that is proportional to the head difference between the piezometer and the adjacent aquifer material. Delayed piezometer response is most 

important at early time when the head changes are most 

rapid, and if not taken into account, the estimate of specific storage may be exaggerated.

Figure 2 is a diagrammatic cross-section through 

a part of an idealized unconfined aquifer with a finitediameter partially penetrating pumped well, an observation piezometer, and an observation well (or longscreened piezometer). The figure illustrates the parameters used to define the well radii, the location of the 

screens, the location of the observation piezometer, and 

the saturated thickness of the aquifer. Also shown is the 

location of the origin of the coordinate system. Symbols used in the mathematical development are defined 

in the Notation section.

Boundary-Value Problem

The governing equation in the domain rw ≤ r ≤ ∞

and 0 ≤ z ≤ b for axisymmetric flow to a pumped well 

in a slightly compressible, anisotropic, unconfined 

aquifer may be written as

(1)

The initial condition in the domain of equation 1 

is

(2)

where hi

 is the initial hydraulic head. The outer boundary condition at r = ∞ is

(3)

The inner boundary condition at r=rw requires 

a wellbore balance equation for a partially penetrating 

well. Following Dougherty and Babu (1984), this condition is 

(4a)

where l – d is the length of the screen, Q is the pumping 

rate, C is the wellbore storage (assumed constant), and 

hw is the average head in the wellbore. Ramey and 

Agarwal (1972) point out that effects of wellbore storage can occur as a result of changing liquid level in the 

∂

2

h

∂r

2 -------- 1

r

--∂h

∂r

----- Kz

Kr

----- ∂

2

h

∂z

2 -------- Ss

Kr

----- ∂h

∂t

+ + = -----

hi – hrz ( ) , , 0 = 0

hi – h( ) ∞, ,z t = 0

2πrw( ) l – d Kr

∂h

∂r

-----

r rw =

Q C+

∂hw

∂t = --------- b – l ≤ ≤zbd –
8 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

z2

rw

zp

z1

2rp

2rc

z

r

d

b

Observation

piezometer

Observation

well

Pumped well

Saturated zone

Unsaturated zone

Land surface

Base of aquifer

Initial

water table

Q

l

rw

zp

z1, z2

2rp

2rc

z

r

d

b

Q Pumping rate

Effective diameter of pumped well in the interval 

 where water levels are changing

Diameter of observation piezometer in the interval 

 where water levels are changing

Outside radius of the pumped well screen

Vertical distance from initial water table to top of 

 pumped well screen

Vertical distance from initial water table to bottom 

 of pumped well screen

Initial saturated thickness of aquifer

Vertical distance above base of aquifer

Vertical distance above base of aquifer to the bottom 

 and top, respectively, of the observation well screen

Vertical distance above base of aquifer to the center 

 of the piezometer screen

Radial distance from axis of pumped well

EXPLANATION

l

Figure 2. Schematic diagram of a finite-diameter pumped well, 

observation well, and observation piezometer in a homogeneous, 

anisotropic water-table aquifer of infinite lateral extent.
MATHEMATICAL MODEL 9

well or, for confined and leaky aquifers, by virtue of 

wellbore-fluid compressibility in a pressurized test. 

Effects of wellbore storage are greatest when due to 

changing water level. In this case, C is the cross-sectional area of the free surface in the well. In this report, 

for convenience, C =π rc

2 where rc is the effective 

radius of the well in the interval where water levels are 

changing. The term “effective radius” is used here to 

allow for the presence of a column pipe or other tubing 

that might reduce the cross-sectional area of the 

pumped well in the vicinity of changing water levels. 

The radial flow through the screen from the aquifer to the well, expressed by the left-hand-side of equation 4a, is assumed to be independent of z and to vary 

only with time. Ruud and Kabala (1997) found that 

flow variations along the well screen can be significant, 

especially for wells with short screen lengths in thick 

aquifers; however, the effect upon drawdowns in the 

wellbore was found to be insignificant.

The vertical average of the head in the wellbore, 

hw, is related to the average head in the aquifer adjacent to the pumped-well screen, h*, by 

(4b)

where Ks is the hydraulic conductivity of the wellbore 

skin, ds is the skin thickness, and h* is defined by

(4c)

Equation 4b derives from the heat-flow literature 

(Carslaw and Jaeger, 1959, p. 20). For simplicity, the 

storage capacity of the skin is assumed negligible. One 

additional equation is required for completion of the 

boundary condition at the interface between the 

pumped well and the aquifer. Namely,

(4d)

This condition implies that a well casing of constant 

external radius rw extends from the top of the screened 

section to the water table, and that no radial flow occurs 

across an imaginary cylinder that extends from the bottom of the screened section of the well to the base of 

the aquifer. 

The condition along the base of the aquifer (z =

0) for r ≥ rw is that of a no-flow boundary and is written 

as

(5)

The condition used by Moench (1997) at the 

water table (z=b) for r ≥ rw, which approximates the 

rate of drainage per unit area from the unsaturated 

zone, is written as 

(6)

Moench (1995) provides details pertaining to the theoretical justification for the use of equation 6, which 

derives from the work of Boulton (1954). As α1→ ∞, 

equation 6 approaches the boundary condition used by 

Neuman (1972, 1974). Also, it can be seen by inspection that if α1= 0, a no-flow condition is obtained for 

equation 6 and the boundary-value problem reduces to 

that solved by Dougherty and Babu (1984) for a confined, single-porosity aquifer. 

It was stated at the beginning of this section that 

a single-parameter exponential relation does not represent drainage from the unsaturated zone very well. In 

this report, drainage from the unsaturated zone is represented by a modification to equation 6 such that, 

instead of the single exponential relation suggested by 

Boulton (1954, 1963), a finite series of M exponential 

terms is used, each with a different empirical parameter 

αm: 

(7)

The summation term in equation 7, in systems terminology, is an input response function or kernel of the 

convolution integral. It represents an average of a series 

of exponential terms. If M = 1, equation 7 reverts to 

equation 6.

The parameters used in equation 7 might be 

found to differ from one aquifer test to the next, even at 

the same location, but are assumed to be unique for a 

given test. Due to the assumption of homogeneity, the 

Ks

h

* hw –

ds

----------------- Kr

∂h

∂r

----- r rw =

=

h

* 1

( ) l – d -------------- h rw ( ) , ,z t dz

b – l

b – d

∫ =

∂h

∂r

-----

r rw =

= 0 z b < – l

zbd > –

∂h

∂z

-----( ) r, , 0 t = 0

Kz

∂h

∂z

-----( ) rbt , ,

α1 – S

y

∂h

∂t′ ------

0

t

∫ = () α rbt , , 1 exp[– ( )] t t – ' dt'

Kz

∂h

∂z

-----( ) rbt , ,

Sy

∂h

∂t′ ------( ) rbt , , αm

M

------ αm exp[ ] – ( ) t t – '

m= 1

M

∑ dt'

0

t

∫ = –
10 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

same set of αM parameters should apply irrespective of 

the location of the individual piezometers. This set of 

empirical parameters will be subject to change, however, as antecedent conditions in the unsaturated zone 

change. This arrangement allows the hydrogeologist to 

obtain not only better agreement between measured 

and predicted drawdowns over the entire intermediate 

time range of a time-drawdown curve but also allows 

for improved parameter estimation, as will be shown. 

The representation of the drainage process suggested 

by Boulton and Pontin (1971) makes use of two empirical α parameters and a combination of instantaneous, 

short-term, and long-term “delayed-yield” parameters 

summed together to represent specific yield. In this 

report, a single value of specific yield is used in order 

to be consistent with the hypothesis that specific yield 

is a characteristic property of the aquifer and does not 

vary with time. 

Dimensionless Boundary-Value Problem

By substituting the dimensionless parameters 

listed in table 1 into the above equations, one obtains 

the following dimensionless boundary-value problem:

(8)

(9)

(10)

(11a)

(11b)

 (11c)

(11d)

(12)

(13)

Laplace Transform Solution

Use of Laplace transforms and Fourier cosine 

series leads to a Laplace transform solution to the 

above dimensionless boundary-value problem. The 

derivation is provided in Appendix I. The Laplace 

transform solution for dimensionless drawdown in the 

pumped well is

(14)

And the solution for the dimensionless drawdown in 

the aquifer is

(15)

where

(16)

∂

2

hD

∂rD

2 ----------- 1

rD

-----∂hD

∂zD

--------- βw

∂

2

hD

∂zD

2 ----------- ∂hD

∂tD

+ + = ---------

hD rD zD ( ) , , 0 = 0

hD ∞ zD tD ( ) , , = 0

∂hD

∂rD

--------- 2

(l D dD – ) ---------------------- WD

∂hwD

∂tD

– = – ------------

1 lD zD 1 dD – ≤ ≤ –

rD=1

∂hD

∂rD

---------= 0

zD 1 l

D zD 1 dD < – ; > –

rD= 1

hwD hD

* Sw

∂hD

∂rD

= 1 – --------- l

D zD 1 dD – ≤ ≤ –

hD

* 1

l

D dD ( ) – --------------------- hD 1 zD tD ( ) , , zD d

1 l

D –

1 dD –

∫ =

∂hD

∂zD

--------- rD 0 tD ( ) , , = 0

∂hD

∂zD

--------- rD 1 tD ( ) , , ∂hD

∂τ --------- rD 1 tD ( ) , , 0

tD

∫ = –

γ m

M

----- γ mσβw tD exp[– ( – τ )]

m = 1

M

• ∑ dτ

hwD( ) p

2 A S + w ( )

p lD dD ( ) – 1 WD pA S + w [ ] + ( ) = --------------------------------------------------------------------------

hD rD zD ( ) , , p

2E

p lD dD ( ) – 1 WD pA S + w [ ] + ( ) = --------------------------------------------------------------------------

A 2

lD dD – ---------------- Koqn

n = 0

∞

= ∑

εn 1 dD – ) ] εn 1 l

D ( –sin[ ( )]} – 2 { sin[

εnqnK1 qn ( )εn 0.5 2εn [ + sin( ) ] ------------------------------------------------------------------------------------------- •
MATHEMATICAL MODEL 11

(17)

(18)

(19)

and εn, where n = 0, 1, 2, ..., are the roots of

(20)

K0 and K1 are the modified Bessel functions of the second kind and of zero and first order, respectively.

For long screened piezometers (observation 

wells) it is assumed that the measured drawdown is the 

average drawdown over the screened interval zD2–zD1

as determined by

(21)

The Laplace transform solution for dimensionless 

drawdown in a long-screened piezometer in the aquifer 

then becomes

(22)

where

(23)

E 2 Ko qnrD ( )ε( nzD cos )

n = 0

∞

= ∑

εn 1 dD – ) ] εn 1 l

D sin[ ( –sin[ ( )] –

qnK1 qn ( )εn 0.5 2εn [ + sin( ) ] •---------------------------------------------------------------------------------

qn ε

2

nβw ( ) + p

1 2⁄ =

qnrD ε

2

nβ rD

2 ( ) +

1 2⁄ =

εn εn ( ) p

M

---- 1

σβw p γ m ( ) + ⁄ ----------------------------------

n = 1

M

tan = ∑

hD rD zD1 zD2 ( ) ,,, p

1

zD2 zD1 – ---------------------- hD rD zD ( ) , , p zD d

zD1

zD2

∫ =

hD rD zD1 zD2 ( ) ,,, p

2E'

p l

D dD ( ) – 1 WD pA S + w [ ] + ( ) = --------------------------------------------------------------------------

E 2

K0 qnrD ()εn 1 dD ( )] ε – n { sin[ – sin[ ( )]} 1– lD

εnqnK1 qn ( )εn 0.5 2εn [ ] + sin( ) --------------------------------------------------------------------------------------------------------------

n = 0

∞

= ∑

εnzD2 ) εnzD1 [sin( –sin( ) ]

zD2 zD1 ( ) – • ---------------------------------------------------------------

Table 1. Dimensionless expressions.

dD d/b

hD 4π Kr b(hi – h)/Q

hmD 4π Kr b(hi –hm)/Q

hwD 4πKr b(hi –hw)/Q

KD Kz/Kr

l D l /b

rD r/rw

rwD rw /b

Sw Kr ds/Ksrw

tD Krt/rw

2Ss

tDy Kr bt /rw

2Sy

WD π rc

2/[2π rw2Ss(l –d )]

WD' π rc

2/(2π rw2SsF')

zD z/b

zD1 z1/b

zD2 z2/b

βw KDrwD

2

β βw rD

2

α Ss b/Sy

γm αm bSy /Kz

τ Krt'/rw

2Ss
12 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

Equations 15 and 22 are solutions for head 

changes in the aquifer. Observation wells or piezometers used to measure hydraulic head variations in an 

aquifer are often open holes containing a significant 

quantity of stored water. With the start of pumping, 

rapid changes in head in the aquifer may not be accurately reflected by measurements in the piezometer 

because of the finite time it takes to dissipate stored 

water and come into equilibrium with the hydraulic 

head in the aquifer. This delayed piezometer response 

is important for accurate evaluation of specific storage 

and is treated analytically in the manner described in 

detail by Moench (1997). By choosing an appropriate 

shape factor F' = F/2π, where F is defined by Hvorslev 

(1951) for various geometrical configurations, and, 

assuming good hydraulic connection between the piezometer and the aquifer so that screen clogging is not a 

factor, it is possible to account for delayed piezometer 

response by use of the following equation:

(24)

where hmD is the Laplace transform solution for the 

piezometer response and W'D is the dimensionless piezometer storage parameter defined in table 1. In the 

event that screen clogging is a factor, it is possible to 

estimate F for use in (24) by slug testing the piezometer 

and following the procedure indicated by Black and 

Kipp (1977).

Dimensionless or dimensional drawdowns are 

obtained by numerical inversion of equations 14, 15, or 

22. The Stehfest (1970) algorithm is particularly useful 

in this regard because of its computational efficiency. 

The FORTRAN code WTAQ3 (see Moench, 1997) was 

modified to enhance speed of computation and to 

include the summation in equation 20. The modified 

version of WTAQ3 used in this report can be sent to 

interested readers upon request from the first author. 

Also available for downloading from the World Wide 

Web is a fully documented computer program WTAQ, 

described by Barlow and Moench (1999), that includes 

all the physical processes available in WTAQ3. WTAQ 

can be used for both type-curve analysis and automated 

parameter estimation for both confined and unconfined 

aquifers.

hmD

hD

1 W'

D + p = ----------------------
AQUIFER-TEST DESIGN AND OPERATION 13

AQUIFER-TEST DESIGN AND OPERATION

As part of an effort to quantify the hydraulic 

properties of the unconfined, sand and gravel, glacialoutwash aquifer at the Cape Cod Toxic-Substances 

Hydrology Research site near Falmouth, Massachusetts, a three-day aquifer test was carried out from 

August 28–31, 1990. The test was conducted by pumping a well at a constant rate and by observing the resulting changes in hydraulic head at locations that differ in 

distance and azimuth from the pumped well and in 

depth below the water table. 

Figure 3 shows the location of the study area and 

the locations in plan view, within an abandoned gravel 

pit, of the pumped well and points of observation. The 

reference piezometer (F343-036) is included in figure 

3 as its position is the location of the origin of coordinates of a magnetic north-oriented grid that was overlaid on the site and used to locate the positions of points 

of observation for this study and others (LeBlanc, and 

others, 1991). The last three digits of the well numbers 

represent the approximate depth in feet below land surface to the bottom of the well or piezometer. Figure 4 

illustrates the positions of the observation well screens 

in vertical section and is drawn roughly to scale. The 

pumped well was drilled in July 1990 by cable tool 

methods to a depth of 80 ft below land surface. An 8-

inch, inside-diameter (i.d.) polyvinyl chloride (PVC) 

casing was installed with an 8-inch i.d. PVC screen 

along the bottom 47 ft of the well. Backfill consisted of 

natural collapse material and cuttings from the hole. 

The top and bottom of the screen were located 13.2 and 

60 ft, respectively, below the initial water table, which 

was approximately 19 ft below land surface. Immediately prior to the test, the elevation of the water table at 

the pumped well was 46.80 ft above mean sea level. 

The regional temporal trend of the water table was 

determined by measuring the elevation of the water 

table, in areas unaffected by pumping, before, during, 

and after the aquifer test. The water table was found to 

have declined at a rate of 9.26x10–6 ft/min over this 

time period, requiring a slight correction (diminution) 

in the late-time drawdown data. Observation piezometers and wells were constructed by auguring to prescribed depths and installing 2-inch i.d., PVC casings 

with 2-ft-, 9-ft-, or 39-ft-long PVC screens. Details 

with regard to the exact radial and vertical positions 

and lengths of the well screens are given in table 2. 

F376-037

F385-032

F478-061

F381-056

Pumped well

F507-080

Reference piezometer

F343-036

Lat. 41°38'15"

Long. 70°32'35" True 

north

Magnetic

north (-14°)

F504-032

F504-060

F504-080

F377-037

F450-061

F476-061

F434-060

F383-129

F383-061

F383-032

F383-082

F505-059

F505-032

F505-080 F347-031

F384-033

0 100 FEET

0 25 METERS

Study

Area

Boston

Cape

Cod

N M A

R I

Providence

Figure 3. Regional location and local plan view showing the positions 

of the pumped well (F507-080) and observation wells and 

piezometers in the study area. The reference piezometer (F343-036) is 

not used to measure drawdown.
14 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

F381 F385 F376 F347 F384 F383

F434 F476 F450 F377 F478 Pumped well

F507-080

Approximate scale

Screened

interval

F505 F504 Land surface

Water table

Saturated zone

F384 Well identification number

Observation well

Screened interval(s) of observation 

 wells and piezometers

EXPLANATION

Henceforth, for convenience, the various observation 

piezometers and wells will be referred to as piezometers regardless of the length of the screen.

Well F507-080 was pumped at a rate of approximately 320 gal/min (42.8 ft3

/min) for 72 hours. Discharge water was diverted through a fire hose to a 

Massachusetts Military Reservation sewage-infiltration 

bed located about 500 to 600 ft up gradient (north) of 

the test area. The rate of well discharge was monitored 

(1) by a manometer and orifice at the discharge point in 

the infiltration bed and (2) by noting the time required 

to fill a 55-gal drum at the end of the fire hose. Adjustments to the wellhead valve were made as necessary to 

maintain a constant flow rate. 

Water levels in all piezometers were measured 

manually using a steel tape. In addition, drawdown data 

were collected in the pumped well and in seven piezometers with pressure transducers connected to data 

loggers. An electric tape was used for manual collection of drawdown data in the pumped well. Recovery 

measurements also were made in selected piezometers 

and in the pumped well. 

Appendix II (fig. A–K) shows plots of the drawdown data used for analysis, which have been corrected 

for the regional decline of the water table, mentioned 

above. Where transducer data are available, the plots 

show hand-measured values at late time (solid circles) 

continuing beyond the transducer values (dots). This is 

because, after about 300 min of pumping, the transducer values of drawdown are discarded since, with one 

exception (F377-037), they appear to drift apart from 

the hand-measured values. The latter are considered 

more accurate than the transducer values in the latetime range. Values of drawdown (open diamonds, 

squares, and circles) selected for use in the automatic 

parameter estimation algorithm are also shown. 

Numerical values for parameter estimation are listed in 

Appendix II table A. Recovery data are not analyzed in 

this report, but were measured in three piezometers 

with transducers (F505-032, F504-032, and F377-037). 

Figure 4. Vertical cross section of the aquifer at the study site 

showing the lengths and positions of the piezometers and 

observation wells.
AQUIFER-TEST DESIGN AND OPERATION 15

Table 2. Locations of observation wells and piezometers, number of observation values for PEST, and measurement 

numbers for PEST.

Location 

number

Well number 

(feet)

Radial 

distance1

(feet)

Depth2 (feet) Screen length

(feet)

Number of 

observation 

values for 

PEST

Measurement numbers 

for PEST

1 F507-080 0.333 13.2 47 24 8–313

2 F505-032 23.9 10.7 2 32 32–63

3 F505-059 19.5 30.6 9 32 64–95

4 F505-080 21.6 58.4 2 33 96–128

5 F504-032 46.6 9.6 2 31 129–159

6 F504-060 49.8 30.0 9 33 160–192

7 F504-080 53.1 57.5 2 32 193–224

8 F377-037 85.1 13.3 2 30 225–254

9 F383-032 93.0 12.1 2 18 255–272

10 F383-061 92.9 39.9 2 24 273–296

11 F383-082 94.8 61.8 2 18 297–314

12 F383-129 96.7 107.8 2 17 315–331

13 F384-033 137.3 15.8 2 23 332–354

14 F381-056 159.8 20.0 2 20 355–374

15 F347-031 225.7 14.8 2 18 375–392

16 F434-060 38.6 2.0 39 15 393–407

17 F450-061 66.3 1.7 39 16 408–423

18 F476-061 65.6 2.2 39 16 424–439

19 F478-061 101.3 2.2 39 16 440–455

20 F385-032 224.6 10.0 2 17 456–472

21 F376-037 227.6 13.2 2 20 473–492

1. Distance from center of pumped well.

2. Depth below the initial water table to the top of the screen.

3. First seven values eliminated (see text).
16 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

All measured data (both drawdowns and recovery) are 

available from the USGS in electronic form. 

ANALYSES

Preliminary Analysis

The preliminary analysis of Moench and others 

(1996) involved traditional type-curve matching procedures using Neuman's (1974) model and late-time 

drawdown data to obtain the aquifer parameters Kr, Kz, 

and Sy. Type curves were generated with WTAQ1 

(Moench, 1993) for the particular well screen/aquifer 

configuration, taking partial penetration of the pumped 

well into consideration and assuming the saturated 

thickness to be a known quantity. This preliminary 

analysis involved use of the hand-measured data only, 

because the rapidly changing early-time data, required 

for evaluation of Ss , was not needed. The preliminary 

analysis involved the use of composite plots of drawdowns versus time divided by the square of the distance 

between the observation point and the pumped well 

(double logarithmic plots of h vs. t/r2

). The procedure 

is essential for accurate evaluation of the aquifer 

hydraulic properties by analytical methods (see 

Moench, 1994) as it allows the hydrogeologist to 

obtain a single match point and, hence, a single set of 

hydraulic parameters for data obtained from a number 

of observation points simultaneously. 

Initially, based on limited local well-log information, a saturated thickness of 80 ft was assumed (see 

Hess and others, 1992). This resulted in three problems: (1) complete inability to match theoretical type 

curves with composite plots of drawdown data from 

piezometers located near the pumped well, (2) significant differences between the estimated parameters and 

estimates based on prior studies (for example, the value 

of Kr obtained by the Jacob method was twice the 

expected value), and (3) having to account for head 

variations in a piezometer located 28 ft below the 

assumed base of the aquifer. A second analysis was 

then performed after doubling the saturated thickness 

to 160 ft. The justification for this change was not just 

based on the three problems listed but also on regional 

geological studies (see LeBlanc, 1984, fig. 5) that indicate the presence of a sharp transition from very coarse 

to very fine-grained sediments with increased depth at 

about 150 ft. Upon changing the saturated thickness to 

160 ft, an estimate of Kr was obtained that was consistent with prior studies at the site, and excellent matches 

were obtained for composite plots of all late-time 

drawdowns leading to a single match point (see 

Moench and others, 1996). The values of vertical and 

horizontal hydraulic conductivity and specific yield 

calculated from the match point are given in table 3. 

Also shown in table 3 is the value of saturated thickness 

b that was derived, in part, from the preliminary analysis. The fact that a doubling of the saturated thickness 

could make such a difference in the theoretical 

responses was an indication that an aquifer test conducted in a homogeneous aquifer might be used to 

obtain an estimate of saturated thickness (in addition to 

the primary unconfined aquifer parameters). The 

agreement between the hand-measured drawdowns and 

the theoretical late-time responses, obtained with a single set of hydraulic parameters, provide support for the 

primary assumptions in the Neuman model that control 

late-time piezometer responses. The preliminary analysis also shows that the use of an incorrect estimate of 

the saturated thickness could lead one to conclude that 

the lack of agreement between measured and theoretical responses is due to aquifer heterogeneity.

Analysis by Nonlinear Least Squares

In this report, various analyses are performed by 

automatic nonlinear parameter estimation using the 

model WTAQ3 (Moench, 1997). In this approach, differences between simulated drawdowns based on estimated hydraulic parameters and measured (observed) 

drawdowns are minimized using a weighted sum of 

squared errors objective function. The parameter-estimation code used in this report, PEST1

 (Doherty, 1994) 

and the upgraded version PEST2000, runs WTAQ3 

repeatedly while automatically varying the hydraulic 

parameters in a systematic manner from one run to the 

next until the objective function is minimized. Statistics are provided showing the precision of the estimated 

parameters (for example, the 95 percent confidence 

limits, and correlation coefficients). The literature on 

automatic parameter estimation is extensive and is not 

reviewed in this report. Papers by Poeter and Hill 

(1997) and Hill (1998) provide excellent discussions of 

1

The use of this product does not imply endorsement by the 

U.S. Government.
ANALYSES 17

the methodology. The manual and code documentation 

that comes with PEST software provides additional 

helpful information regarding parameter estimation 

methodology. 

Proper use of PEST, or any parameter estimation 

algorithm, often requires a certain amount of fine-tuning accomplished only by trial and error. Such adjustments involve, for example, the proper selection of 

initial parameter estimates, whether or not to use logarithmic parameter transformations, or other adjustments necessary to optimize PEST's performance. 

Probably because of the relatively high quality of the 

test data and the scale of the aquifer test relative to the 

correlation length of the heterogeneity (discussed at 

greater length later in the report), the necessary trial 

and error adjustments were minimal for the analysis of 

the Cape Cod data reported here.

In this report, a systematic (5-step) approach to 

data interpretation was taken in order to gradually gain 

confidence in the validity of the parameters obtained by 

the parameter estimator. 

1. The parameters Sy , Kr, and Kz were estimated using 

very late-time data and the instantaneous drainage 

assumption. This allowed for comparison with 

estimates reported in the preliminary analysis 

(Moench and others, 1996). 

2. The parameters Sy , b, Kr, and Kz were estimated 

using very late-time data and the instantaneous 

drainage assumption. This again allowed for comparison with parameter estimates reported in the 

preliminary analysis. 

3. Based on the previously estimated aquifer parameters, the late-time pumped-well data were used 

alone to obtain the wellbore skin factor (Sw). 

4. The parameter Ss was estimated with the exclusive 

use of very early-time transducer data, the previously obtained aquifer parameters, the wellbore 

skin factor, and by accounting for delayed piezometer response. 

5. Finally, under the gradual drainage assumption, the 

complete data set with 461 drawdown values in 

20 piezometers was analyzed with PEST to obtain 

all relevant aquifer parameters simultaneously 

(Ss, Sy , b, Kr, Kz, and three empirical αm parameters). Table 2 provides the numbers used by the 

parameter estimator to identify the measured 

observations.

Evaluation of Sy , Kr , Kz Using Late-Time Data (Step 1)

An analysis was performed on drawdowns measured in all piezometers at times greater than one day 

(about three values for each piezometer at approximately 2,000, 3,000, and 4,300 minutes; see Appendix 

II figures B–K). The analysis assumes instantaneous 

drainage from the unsaturated zone. With the chosen 

saturated thickness of 160 ft, used in the preliminary 

analysis by Moench and others (1996), the parameterestimation code produced the parameter values listed in 

table 4. Table 4 also shows the upper and lower values 

of the 95 percent confidence limits and initial values 

used in the simulation. The 60 observation values used 

in the simulation were given equal weights. The parameter values in table 4 agree reasonably well with the 

values shown in table 3, although Sy in table 4 is about 

25 percent greater than the value in table 3. This difference is apparently a consequence of the reduced time 

span of the selected data and the unbiased treatment 

performed by parameter-estimation code as compared 

with visual type-curve matching. An experiment conducted by using a reduction in the initial time span 

(2,000–4,300 min), to 3,000–4,300 min, gave rise to an 

increase in the value of Sy from 0.287 to 0.304, which 

is consistent with what one might expect as a consequence of gradual drainage from the unsaturated zone. 

In the reverse sense, if the time span is expanded to 

include values of time closer to the start of pumping, 

the estimated values of Sy decrease, evidently as a consequence of the assumption of instantaneous drainage. 

Such was the case in the analysis of the hand-measured 

data from this aquifer test performed by Heidari and 

Moench (1997, table 5) whereby only the first 10 minutes of data were excluded from the analysis and a 

value of Sy= 0.183 was obtained. In general, attempts 

to incorporate as much data as possible from the interTable 3. Parameters obtained from preliminary analysis of 

hand-measured drawdown data, where Sy=specific yield, 

b = saturated thickness, Kr = hydraulic conductivity in the 

horizontal direction, and Kz = hydraulic conductivity in the 

vertical direction.

Parameter Estimated value

Sy 0.23

b (feet) 160

Kr (feet per minute) .24

Kz (feet per minute) .12
18 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

mediate-time range without placing emphasis on the 

late-time data can explain the relatively small values of 

specific yield often obtained by both visual type-curve 

matching and by methods of automatic parameter estimation.

Evaluation of Sy , b, Kr , Kz Using Late-Time Data (Step 2)

Results of a PEST simulation with the same data 

set but allowing the saturated thickness to be one of the 

estimated parameters is shown in table 5. Interestingly, 

the estimated saturated thickness is close to the value of 

160 ft chosen for use in the preliminary analysis. The 

finding suggests that late-time drawdown measurements in piezometers properly located in the aquifer 

are sensitive to the effective base of the saturated zone. 

Of course, other factors may be at play, such as possible leakage from the fine-grained material underlying 

the aquifer and/or flow paths diverted by heterogeneous materials or lithologic structures located in the 

aquifer at depth. These factors might lead to an estimated saturated thickness (based on the hydraulic data) 

that is, by coincidence, close to the value obtained from 

the geologic cross section. Similar results from additional, carefully designed aquifer tests conducted in 

homogeneous unconfined aquifers are therefore 

needed before hydraulic tests can be considered an 

effective method to identify aquifer thickness. 

Evaluation of Sw Using Late-Time Pumped-Well Data 

(Step 3)

In addition to the aquifer parameters evaluated 

from the late-time piezometer data, it was necessary to 

use late-time pumped-well data (Appendix II, fig. A) to 

evaluate the wellbore skin factor (Sw). The latter was 

needed for an improved estimate of specific storage 

(Ss), as shown in the next section. If the ability of a skin 

to transmit water is less than that of the aquifer, drawdowns in the pumped well are enhanced, and there is an 

apparent increase in wellbore storage, the consequence 

of which is to reduce drawdowns in the aquifer at early 

time over the value that would have been obtained with 

no skin (see, Moench, 1985, figure 7b). If the hydraulic 

conductivity of the skin is enhanced over that of the 

aquifer (as by gravel pack or well development), drawdowns in the pumped well are decreased and there is an 

apparent decrease in wellbore storage, which increase 

drawdowns in the aquifer at early time. The decrease in 

wellbore storage in this instance is due to an effective 

increase in wellbore radius rw as a consequence of the 

gravel pack or well development (see WD in table 1). 

Because an accurate evaluation of specific storage is 

dependent on proper analysis of early-time data, it is 

important to obtain an estimate of Sw (or an effective 

rw). The factor Sw is easily evaluated by a trial and error 

comparison of measured late-time pumped-well data 

with drawdowns predicted for the pumped well at late 

time using known values of the aquifer parameters (see 

table 5) and known pumped-well dimensions. This can 

be accomplished by visual data matching or by automated procedures. 

Using the parameters listed in table 5, a negligibly small value of specific storage, and the drawdown 

data in Appendix II (table A) for the pumped well 

(F507-080), the PEST algorithm yields Sw=1.375 with 

a 95 percent confidence range of 1.301 to 1.454. Figure 

5 shows a plot of measured and simulated drawdowns 

using the parameters in table 5 and Sw=1.4. The curves 

in figure 5 for Sw=1.2 and 1.6 demonstrate the sensitivity of drawdowns in the pumped well to variation in 

wellbore skin. Note that the larger value of Sw in figure 

5 produces greater theoretical drawdowns in the 

Table 4. Parameters obtained from late-time data exclusively, using PEST, with b =160 feet, where Sy=specific yield, Kr=

hydraulic conductivity in the horizontal direction, and Kz =hydraulic conductivity in the vertical direction.

Parameter Estimated value

95-percent confidence limits

Initial value

Lower limit Upper limit

Sy 0.2868 0.2790 0.2947 0.1

Kr (feet per minute) .2318 .2299 .2337 .01

Kz (feet per minute) .1325 .1277 .1375 .01
ANALYSES 19

0

1

2

3

4

5

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Measured drawdown

Simulated drawdown, Sw = 1.6

Simulated drawdown, Sw = 1.2

Simulated drawdown, Sw = 1.4

pumped well. Additional discussion and analysis of the 

pumped well data can be found in the discussion section.

Evaluation of Ss Using Early-Time Data (Step 4)

With estimates of the aquifer parameters (Sy , Kr, 

and Kz) and the wellbore skin factor (Sw) obtained 

from late-time data, it was possible to estimate the specific storage (Ss) of the aquifer. The additional parameters needed for the computations are the wellbore radii 

for the pumped well (rc and rw), and the radii (rp ) and 

screen lengths (L) of the observation piezometers. 

Inclusion of delayed piezometer response can be 

accomplished either by slug testing each piezometer, 

which is the recommended approach if piezometers 

respond slowly to slug testing, or by use of a theoretical 

formula derived by Hvorslev (1951). The latter was 

used in this report and was obtained from the following 

formula (Moench, 1997):

(25)

where x = kL/2rp, k = (Kr/Kz)1/2, and L=z2 – z1. The 

screen length L is given in table 2, rp=1 inch for all piezometers, and Kr/Kz has been determined from the lateF' L

1n x 1 x

2

)

1 2⁄

[ + ( + ]

= -------------------------------------------------

Table 5. Parameters obtained from late-time data exclusively, using PEST, with b as an estimated parameter, where Sy=

specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal direction, and Kz =hydraulic conductivity in 

the vertical direction.

Parameter Estimated value

95-percent confidence limits

Initial value

Lower limit Upper limit

Sy 0.2536 0.2356 0.2730 0.1

b (feet) 171.3 165.3 177.4 100

Kr (feet per minute) .2289 .2265 .2313 .01

Kz (feet per minute) .1369 .1316 .1424 .01

Figure 5. Measured and simulated drawdowns for the pumped well (F507-

080), for the model parameters shown in table 5.
20 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

time data (table 5). From Appendix II (fig. B–E, which 

show drawdowns in the piezometers that have transducers), it is evident that early time data exist only for 

a few tens of seconds after the onset of pumping.

In this section, for estimation of Ss, the analysis 

was performed using only the first 6 values of drawdown selected for PEST (see fig. B–E, Appendix II) 

from piezometers F505-080 and F504-080, the first 5 

values from piezometers F504-060, F505-032, F504-

032, and F377-037, and the first 4 values from piezometer F505-059. The values of Sy , Kr, Kz and b used 

were those in table 5, and the value of Sw (1.4) was that 

determined in the previous section. From these earlytime values of drawdown the estimated value of Ss was 

determined to be 1.26x10–5

 ft–1.

It is of interest to note that if delayed piezometer 

response is not included in the analysis, the value of Ss

is increased by 50 percent to 1.97x10–5

 ft–1, demonstrating that an accurate estimate of specific storage 

requires inclusion of this effect. It is also of interest to 

determine the value of Ss obtained if effects of wellbore 

skin are not included in the analysis. Ignoring the 

effects of both wellbore skin at the pumped well and 

delayed piezometer response results in a value of Ss

equal to 2.33x10–5 ft–1. If a line-source model is used 

a value of Ss equal to 3.6x10–5

 ft–1 is obtained. 

Having estimated the primary aquifer parameters from the late-time data, and the specific storage 

from the early-time data, it should be noted that the 

parameters can also be estimated simultaneously, using 

PEST, by weighting the early-time data and late-time 

data equally and giving the intermediate-time data zero 

weight. The results of such an analysis are shown in 

table 6, with small differences in parameter values 

from those in table 5. Figures 6–12 show the comparison of measured and simulated drawdowns, using the 

parameters in table 6 and assuming instantaneous 

release of water from the unsaturated zone. 

Evaluation of Ss , Sy , b, Kr , Kz, and αm Using Data for Entire 

Time Range (Step 5)

Drawdowns in the shallow piezometers that have 

transducers clearly show the early-, intermediate-, and 

late-time ranges defined in the Introduction (see the 

responses for piezometers F505-032, F504-032, and 

F377-037 illustrated in fig. 8 and 9). It is evident upon 

close examination of figures 6–12 that the difference 

between measured and simulated drawdowns in the 

intermediate-time range is most apparent in piezometers closest to the water table and that the difference 

appears to increase with distance from the pumped 

well. Any attempt to estimate the aquifer parameters by 

visual or automated methods without accounting for 

this difference is likely to yield inaccurate results. 

Accuracy can be improved by avoiding the use of shallow piezometers as discussed by Moench (1994), or by 

restricting the analysis to very late-time data and earlytime data as was done to obtain the parameter estimates 

in table 6.

As discussed previously, it is possible to account 

for effects of gradual drainage by insertion of a modified form of the Boulton (1963) integral, which 

includes more than one empirical parameter, into the 

boundary condition for the free surface (see equation 

7). To demonstrate that three empirical parameters are 

necessary and apparently sufficient for the model to 

Table 6. Parameters estimated from early- and late-time data exclusively, using PEST, where Ss=specific storage, Sy=

specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal direction, and Kz =hydraulic conductivity in 

the vertical direction.

Parameter Estimated value

95-percent confidence limits

Initial value

Lower limit Upper limit

Ss (feet–1) 1.299E-05 1.239E-05 1.362E-05 1.E-06

Sy .2645 .2402 .2913 .1

b (feet) 168.3 160.6 176.3 100

Kr (feet per minute) .2303 .2272 .2334 .01

Kz (feet per minute) .1295 .1235 .1357 .01
ANALYSES 21

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Simulated drawdown

Measured drawdown

Figure 7. Measured drawdowns compared with drawdowns simulated 

under the assumption of instantaneous release of water from the 

unsaturated zone, wells F505-059 and F504-060.

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-080

F504-080

Simulated drawdown

Measured drawdown

Figure 6. Measured drawdowns compared with drawdowns simulated 

under the assumption of instantaneous release of water from the 

unsaturated zone, piezometers F505-080 and F504-080.
22 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-032

F504-032

Simulated drawdown

Measured drawdown

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F377-037

F347-031

Simulated drawdown

Measured drawdown—F377-037

Measured drawdown—F347-031

Figure 8. Measured drawdowns compared with drawdowns simulated 

under the assumption of instantaneous release of water from the 

unsaturated zone, piezometers F505-032 and F504-032.

Figure 9. Measured drawdowns compared with drawdowns simulated 

under the assumption of instantaneous release of water from the 

unsaturated zone, piezometers F377-037 and F347-031.
ANALYSES 23

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F383-061

F383-032

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F383-082

F383-129

Simulated drawdown

Measured drawdown—F383-061

Measured drawdown—F383-032

Simulated drawdown

Measured drawdown—F383-082

Measured drawdown—F383-129

A.

B.

Figure 10. Measured drawdowns compared with drawdowns simulated under 

the assumption of instantaneous release of water from the unsaturated zone, 

(A) piezometers F383-061 and F383-032, and (B) piezometers F383-082 and F383-

129.
24 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F384-033

F385-032

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F381-056

F376-037

Simulated drawdown

Measured drawdown—F384-033

Measured drawdown—F385-032

Simulated drawdown

Measured drawdown—F381-056

Measured drawdown—F376-037

A.

B.

Figure 11. Measured drawdowns compared with drawdowns simulated under 

the assumption of instantaneous release of water from the unsaturated zone, (A) 

piezometers F384-033 and F385-032, and (B) piezometers F381-056 and F376-

037.
ANALYSES 25

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F434-060

F450-061

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F476-061

F478-061

Simulated drawdown

Measured drawdown—F434-060

Measured drawdown—F450-061

Simulated drawdown

Measured drawdown—F476-061

Measured drawdown—F478-061

A.

B.

Figure 12. Measured drawdowns compared with drawdowns simulated under 

the assumption of instantaneous release of water from the unsaturated zone, (A) 

wells F434-060 and F450-061, and (B) wells F476-061 and F478-061.
26 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

TIME, IN MINUTES

F377-037

Simulated drawdown

Measured drawdown

A.

Instantaneous drainage

10–2 10–1 100 101 102 103 104

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

TIME, IN MINUTES

Simulated drawdown

Measured drawdown

B.

F377-037

Delayed drainage: 

 α1 = 0.0417 min–1

10–2 10–1 100 101 102 103 104

simulate actual drawdowns in the intermediate-time 

range, parameter-estimation runs were made by fixing 

the primary hydraulic parameters with the values given 

in table 6 and allowing the algorithm to estimate the 

best-fit empirical parameters using data from piezometer F377-037 (figure E, Appendix II). Figure 13 compares measured data with the simulated responses for 

instantaneous drainage (fig. 13A), and gradual drainage 

using one (fig. 13B), two (fig. 13C), or three (fig. 13D) 

parameters. Experimentation led to the conclusion that 

three parameters are adequate for purposes of parameter estimation for this aquifer test. Because of local 

head variations caused by aquifer heterogeneity, the 

use of additional empirical drainage parameters did not 

appear justified. 

Table 7 shows the values of the 5 characteristic 

aquifer parameters and 3 empirical parameters estimated simultaneously using the PEST algorithm. The 

complete set of evenly weighted drawdown data from 

all 20 piezometers was used (see Appendix II, figures 

Figure 13. Measured drawdowns and drawdowns simulated for piezometer F377-037 using (A) the 

assumption of instantaneous drainage, (B) gradual drainage using a single empirical parameter.
ANALYSES 27

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F377-037

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Simulated drawdown

Measured drawdown

Simulated drawdown

Measured drawdown

C.

D.

F377-037

Delayed drainage: 

 α1 = 0.000364 min–1

 α2 = 0.0181 min–1

 α3 = 0.412 min–1

Delayed drainage: 

 α1 = 0.00117 min–1

 α2 = 0.117 min–1

B–K and table A). A unit weight was applied to each 

data point. Experiments with different weighting 

schemes resulted in little or no change in the estimated 

primary water-table parameters. For example, when 

weights were made proportional to the ratio of maximum drawdown in a given piezometer to the value at 

the time of interest, the estimated value of Sy decreased 

about 1 percent, Kr and Kz each increased about 2 percent, the value of b increased by about 4 percent, and 

the value of estimated Ss increased from 1.3x10–5

 ft–1

to 2.3x10–5

 ft–1. The value of wellbore skin (Sw=1.4) 

required for the computations had been determined 

independently, as discussed previously.

The correlation coefficient matrix provided by 

PEST for the results presented in table 7 is shown in 

table 8. The table suggests that the various estimated 

parameters should be independent of one another; that 

is, the estimated value of one parameter does not significantly influence the estimated value of any other. 

However, because of the method used by PEST to calFigure 13—Continued. Measured drawdowns and drawdowns simulated for piezometer F377-

037 using (C) gradual drainage using two empirical parameters, and (D) gradual drainage using 

three empirical parameters.
28 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

culate the correlation coefficients, the reader is cautioned against placing a great deal of reliance on values 

that differ significantly from unity (Hill, 1998). A recommended approach to examine the uniqueness of the 

estimated parameters is to rerun PEST using different 

initial parameter estimates. The results of one rerun are 

shown in table 9, and are offered as evidence of the 

reproducibility of the results in table 7. Differences in 

the estimated values do not have a noticeable effect 

upon simulated drawdown responses. The reproducibility in tables 7 and 9 is due in large measure to the 

quality and quantity of the data set. It is shown in the 

discussion section of this report that the use of a limited 

number of piezometers may not always provide consistent and unique results. 

Comparisons of observed and simulated drawdowns based on the parameters in table 7 are shown in 

figures 14–20. The degree of agreement between 

observed and simulated drawdowns demonstrates that: 

(1) the relevant physical processes occurring in the 

course of the aquifer test appear to not only include the 

horizontal and vertical flow and storage of water in the 

aquifer, but also the time-dependent drainage of water 

from the unsaturated zone across the water table and 

(2) the scale of the aquifer test (that is, the pumping rate 

and the size of the drawdown cone) was large enough 

that the known aquifer heterogeneity at this site (Hess 

and others, 1992) had little influence on the properties 

estimated from the aquifer test analysis. (The interbedded nature of the sand and gravel, glaciofluvial 

deposits is revealed in a photograph published by LeBlanc and others (1991, fig. 3) and described by Hess 

and others (1992, p. 2,012).) The estimated parameters 

(Ss, Sy , b, Kr, and Kz) are entirely reasonable for this 

type of aquifer and can be supported, in part, by independent investigations at the site (Hess and others, 

1992; Springer, 1991). 

DISCUSSION

Many hydrogeologists would likely agree that 

the aquifer test analyzed in this report is atypical. Not 

only is the scale of the test large enough to average the 

effects of smaller-scale heterogeneity in the aquifer, but 

also there appear to be few or no interfering effects (for 

example, effects of recharge or discharge due to precipitation or evapotranspiration, extraneous pumping, lateral boundaries, and so forth) over the duration of the 

test. In addition, the test was designed and executed 

with a minimum of problems: the pumping rate varied 

only slightly over the course of the test; the pumping 

rate was set at a rate that was sufficiently great to obtain 

accurate drawdown measurements, but not so great as 

Table 7. Parameters estimated from the complete data set, using PEST, where Ss=specific storage, Sy=specific yield, b =

saturated thickness, Kr=hydraulic conductivity in the horizontal direction, Kz=hydraulic conductivity in the vertical direction, 

and α1, α2 , and α3 are empirical constants for gradual drainage from the unsaturated zone.

Parameter Estimated value

95-percent confidence limits

Initial value

Lower limit Upper limit

Ss (feet–1) 1.305E-05 1.205E-05 1.412E-05 1.E-06

Sy .2660 .2525 .2802 .1

b (feet) 168.9 162.5 175.4 200

Kr (feet per minute) .2331 .2299 .2362 .01

Kz (feet per minute) .1418 .1365 .1474 .01

α1 (minutes–1) 2.78E-04 1.50E-04 5.14E-04 1.E-03

α2 (minutes–1) 1.68E-02 1.27E-02 2.22E-02 1.E-02

α3 (minutes–1) .416 .318 .545 1.E-01
DISCUSSION 29

Table 9. Parameters estimated from the complete data set, using PEST with alternative initial values, where Ss =specific 

storage, Sy =specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal direction, Kz = hydraulic 

conductivity in the vertical direction, and α1, α2 , and α3 are empirical constants for gradual drainage from the unsaturated 

zone.

Parameter Estimated value

95-percent confidence limits

Initial value

Lower limit Upper limit

Ss (feet–1) 1.306E-05 1.206E-05 1.415E-05 1.E-07

Sy .2623 .2500 .2752 .05

b (feet) 172.8 165.3 180.7 80

Kr (feet per minute) .2336 .2304 .2368 .005

Kz (feet per minute) .1409 .1356 .1465 .005

α1 (minutes–1) 1.92E-04 9.94E-05 3.72E-04 1.E-04

α2 (minutes–1) 1.79E-02 1.37E-02 2.33E-02 1.E-02

α3 (minutes–1) .441 .333 .584 1.E00

Table 8. Correlation coefficient matrix for table 7, where Ss = specific storage, Sy = specific yield, b =saturated thickness, 

Kr =hydraulic conductivity in the horizontal direction, Kz =hydraulic conductivity in the vertical direction, and α1, α2, and α3

are empirical constants for gradual drainage from the unsaturated zone.

Ss Sy b Kr Kz α1 α2 α3

Ss 1

Sy .006 1

b –.058 –.195 1

Kr .213 –.217 –.406 1

Kz –.290 –.002 .173 –.703 1

α1 –.035 –.380 –.627 .025 .065 1

α2 .012 –.797 .142 .103 –.070 .378 1

α3 –.093 –.180 –.020 .294 –.539 .092 .160 1
30 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-080

F504-080

Simulated drawdown

Measured drawdown

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-059

F504-060

Simulated drawdown

Measured drawdown

Figure 15. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release of water from the unsaturated zone,

wells F505-059 and F504-060.

Figure 14. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release of water from the unsaturated zone,

piezometers F505-080 and F504-080.
DISCUSSION 31

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-032

F504-032

Simulated drawdown

Measured drawdown

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F377-037

F347-031

Simulated drawdown

Measured drawdown—F377-037

Measured drawdown—F347-031

Figure 16. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release of water from the unsaturated zone,

piezometers F505-032 and F504-032.

Figure 17. Measured drawdowns compared with drawdowns simulated under the assumption of gradual release of water from the unsaturated zone,

piezometers F377-037 and F347-031.
32 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F383-061 F383-032

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F383-082

F383-129

Simulated drawdown

Measured drawdown—F383-061

Measured drawdown—F383-032

Simulated drawdown

Measured drawdown—F383-082

Measured drawdown—F383-129

A.

B.

Figure 18. Measured drawdowns compared with drawdowns simulated under

the assumption of gradual release of water from the unsaturated zone, (A) piezometers F383-061 and F383-032, and (B) piezometers F383-082 and F383-129.
DISCUSSION 33

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F384-033

F385-032

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F381-056

F376-037

Simulated drawdown

Measured drawdown—F384-033

Measured drawdown—F385-032

Simulated drawdown

Measured drawdown—F381-056

Measured drawdown—F376-037

A.

B.

Figure 19. Measured drawdowns compared with drawdowns simulated under

the assumption of gradual release of water from the unsaturated zone, (A) piezometers F384-033 and F385-032, and (B) piezometers F381-056 and F376-037.
34 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F434-060 F450-061

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F476-061

F478-061

Simulated drawdown

Measured drawdown—F434-060

Measured drawdown—F450-061

Simulated drawdown

Measured drawdown—F476-061

Measured drawdown—F478-061

A.

B.

Figure 20. Measured drawdowns compared with drawdowns simulated under the

assumption of gradual release of water from the unsaturated zone, (A) wells F434-

060 and F450-061, and (B) wells F476-061 and F478-061.
DISCUSSION 35

to cause significant changes in saturated thickness near 

the pumped well; the well-discharge disposal site was 

sufficiently remote that infiltrating water had a negligible effect upon evaluation of aquifer parameters or prediction of aquifer response; and, the piezometers were 

positioned at such depths and distances that the aquifer 

parameters could be accurately diagnosed from the 

transducer and hand-measured drawdown data. 

Although the aquifer test described in this report is 

ideal in several respects, the approach used here should 

be applicable elsewhere given data of sufficient quality 

and quantity.

Parameter values in table 5 show that Neuman's 

model, which assumes instantaneous drainage from the 

unsaturated zone and a line source for the pumped well, 

yields reasonable estimates of the primary unconfined 

aquifer parameters Sy , Kr, and Kz when using late-time 

test data. It also appears that by using late-time data 

Neuman's model produces an estimate of the saturated 

thickness of the aquifer that is consistent with that 

which is known from the regional geology. As mentioned previously, leakage from the “impermeable”

base of the aquifer and, perhaps, undetected heterogeneity at depth, casts some doubt on whether an aquifer 

test analysis can be used in general for the estimation 

of aquifer thickness. To obtain accurate estimates of 

these four parameters with the Neuman (1974) model, 

the aquifer test should be run without interference from 

extraneous influences for an extended period of time 

(three days, in the test examined here), the analysis 

must be weighted heavily toward the late-time data 

(times greater than 2,000 minutes), the piezometers 

should be located near a partially penetrating pumped 

well at various depths and distances such as illustrated 

in figures 3 and 4, and the model assumptions must be 

reasonably met.

Results illustrated in figures 6–12 demonstrate 

that the assumption of instantaneous drainage from the 

unsaturated zone does not provide a satisfactory 

description of flow processes in the time range of about 

1 to 1,000 minutes. All piezometers, with three exceptions, show significant differences between measured 

and theoretical responses in this time range. The three 

exceptions (F505-059, F505-080, and F504-080) are 

relatively deep piezometers located in close proximity 

to the pumped well where effects of drainage from the 

unsaturated zone, although present, do not manifest 

themselves to the same extent as in the other piezometers. The differences between measured and theoretical 

drawdowns in the logarithmic plots shown in figures 6–

12 are greatest in piezometers located closest to the 

water table, and appear to increase with distance from 

the pumped well. It is of interest to note that the shortscreened piezometers located closest to the water table 

are still at least 9 ft below the water table. Thus, one 

might expect to see even greater differences in piezometers located closer to the water table. In addition, the 

aquifer is coarse-grained and highly permeable, so one 

might expect to find still greater differences in aquifers 

that are less permeable. 

The above findings, demonstrated in figures 6–

12, contradict statements by Neuman (1972, 1974, 

1975, 1979, 1987) that drawdowns in unconfined aquifers are not significantly affected by drainage from the 

unsaturated zone. It is only in the three deep-seated piezometers located close to the pumped well (F505-059, 

F505-080, and F504-080), mentioned above, that Neuman's assumption of instantaneous drainage might 

appear to be satisfactory. This point is examined in 

greater detail toward the end of this discussion. In general, if automated methods are used for parameter estimation with models that assume instantaneous 

drainage from the unsaturated zone, without judicious 

weighting of the data to eliminate intermediate-time 

data (and early-time data if line-source models are 

used), then one should be skeptical of the validity of the 

estimated parameters. On the other hand, if type-curve 

methods are used for parameter estimation with models 

using the instantaneous drainage assumption, an experienced practitioner can accomplish the judicious 

weighting visually and the results should be satisfactory. 

The results shown in figures 14–20 demonstrate 

that the differences (seen in fig. 6–12) between theoretical and measured drawdowns can be almost entirely 

eliminated by use of a single set of three empirical 

parameters, estimated by PEST, designed to account 

for the effects of gradual drainage of water from the 

unsaturated zone across the water table. The fact that 

the same set of empirical parameters essentially eliminates the intermediate-time discrepancies in all piezometers simultaneously shows that they can be 

considered “characteristic” constants for this particular 

aquifer test. As discussed in the section that follows, 

minor deviations between measured and simulated 

drawdowns as seen in one plot relative to another in figures 14–20 appear to be randomly distributed and can 

be attributed to local variations in aquifer properties 

(heterogeneity). Two possible exceptions to this statement can be seen in figure 19A and 19B for piezometers 
36 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

F385-032 and F376-037, which are located about 225 

ft north of the pumped well (see fig. 3). Close examination of figure 19 reveals that the measured late-time 

drawdowns in these piezometers is less than the 

expected drawdowns based on the simulated responses. 

A likely explanation for these diminished drawdowns, 

which are in the range of 0.01–0.03 ft after about 1,000 

minutes of pumping, is the influence of recharge from 

the disposal of the well discharge water in the infiltration pit some 300–400 feet further north.

Effect of Heterogeneity on Aquifer Test Results

Although the fit of the gradual drainage model to 

drawdown measurements is quite good, there are minor 

differences noted between simulated and measured 

drawdowns (see fig. 14–20). The analysis approach 

used in this report was to fit all of the observations of 

drawdown as a group, and in so doing, produce an estimate of the average aquifer properties. It was expected 

that there would be differences (errors) between the 

simulated and measured drawdowns caused by local 

variations in aquifer properties (that is, aquifer heterogeneity, primarily in hydraulic conductivity), and that 

these deviations would be small and random in distribution across the area of the aquifer test. In general, 

these differences can be characterized as minor fluctuations in an otherwise uniformly varying drawdown 

distribution around the pumped well. The output from 

the PEST algorithm for the results in table 7 indicate 

the sum of squared residuals between simulated and 

measured drawdowns to be 0.0848 ft2

, from which the 

variance and standard deviation are calculated to be 

1.75x10–4

 ft2 and 0.013 ft, respectively. Deviations 

from the mean head distribution are expected at the test 

site because the aquifer is mildly heterogeneous (Hess 

and others, 1992), and the variations in aquifer properties (particularly hydraulic conductivity) produce perturbations in the head field relative to the mean 

distribution.

A comparison of aquifer characteristics developed as a result of this test to those reported by Hess 

and others (1992) for the same aquifer help put the 

effects of heterogeneity on the test results reported here 

in some context. The average values of horizontal and 

vertical hydraulic conductivity found by Hess and others (1992) were 0.123 cm/s (0.242 ft/min) and 0.099 

cm/s (0.195 ft/min), respectively. These compare 

closely with the values reported for this test (Kr= 0.233 

ft/min and Kz= 0.142 ft/min, see table 7) considering 

the comparatively large radial and vertical extent of the 

aquifer test. Measurements of horizontal hydraulic 

conductivity reported by Hess and others (1992) were 

made using flowmeter tests conducted in the upper 25 

ft of the saturated zone. Estimates of vertical hydraulic 

conductivity were made using geostatistical analyses, 

as was the spatial correlation structure of the aquifer. 

The flowmeter tests are small-scale single-well tests 

conducted in wells located within the cone of depression of the aquifer test approximately 30–50 ft northeast of the pumped well. The flowmeter tests produced 

hundreds of small-scale measurements of hydraulic 

conductivity in the upper aquifer, and the above values 

are geometric means of those measurements. The close 

comparison between the results of these two independent sets of experiments indicates that not only are the 

results of the aquifer test good estimates of average 

conditions, but also that the average hydraulic conductivity values don't vary much at the scale of the aquifer 

test. This latter conclusion supports the assumption of 

homogeneity made for the analysis reported here, if 

homogeneity is viewed in an average sense. That is, 

although the aquifer is known to be heterogeneous, the 

statistical variability of the aquifer properties is constant across the area of the test, and therefore, the average aquifer properties remain constant over the area of 

the test.

Hess and others (1992) also estimated a variance 

of 0.24, a horizontal correlation scale of 3.5–8 m 

(11.5–26.2 ft), and a vertical correlation scale of 0.19–

0.38 m (0.62–1.25 ft) for log hydraulic conductivity (ln

Kr) at the site. These estimates indicate that the sand 

and gravel sediments at the site are mildly heterogeneous and that the correlation scales are about the same 

size as the lenses and layers that compose the aquifer 

(see photograph in LeBlanc and others, 1991, fig. 3; 

and description by Hess and others, 1992, p. 2,012). A 

comparison of the correlation scales to the size of the 

cone of depression formed during the aquifer test, with 

measurable drawdowns over a radial distance of over 

200 ft (60m) and depth below the water table of up to 

110 ft, also indicate that the size of the aquifer test was 

many times larger than of the aquifer correlation scales. 

This is additional evidence that the aquifer test very 

likely integrated the effect of many different values of 

hydraulic conductivity, and that the analysis of the test 

measurements should result in a good estimate of the 

average values of the aquifer properties. 

The value of the variance of ln Kr can be used to 

estimate the expected anisotropy of aquifer hydraulic 
DISCUSSION 37

conductivity (the ratio of horizontal to vertical hydraulic conductivity) by use of an equation modified from 

the results of Gelhar and Axness (1983, equation 59):

(26)

where αf

2

 is the variance of ln Kr. Using the value of ln 

Kr variance given above (0.24), this equation produces 

an aquifer anisotropy ratio of 1.27, which is reasonably 

close to the value that can be calculated from Kr/Kz in 

table 7 (1.64). These values are also very similar to the 

value reported by Hess and others (1992) (1.24). This 

provides another indication that the aquifer test results 

are comparable to the earlier results of Hess and others, 

and are reasonable estimates of average aquifer conditions.

An estimate of the expected variance of head in 

the aquifer can be made using an equation developed 

by Naff and Vecchia (1986) for three-dimensional flow 

in a horizontally stratified aquifer:

(27)

where α3Dh

2

 is the variance of head for three-dimensional flow, J is the horizontal hydraulic gradient, λ1 is 

the ln Kr correlation scale in the horizontal plane, and 

λ3 is the ln Kz correlation scale in the vertical plane. A 

one-dimensional version of the equation can also be 

used (Gelhar, 1993, p. 143):

(28)

where α1Dh

2

 is the variance of head for one-dimensional flow. A uniform gradient was assumed in the derivation of equations 27 and 28.

In order to calculate the head variance using the 

above equations, an estimate of the hydraulic gradient 

is needed. This could be obtained using the gradual 

drainage equation, but a simpler approximation can 

also be applied by taking the spatial derivative of the 

Jacob (1950) straight-line equation:

(29)

where T is the aquifer transmissivity and S is the storage coefficient, or storativity. The derivative of drawdowns with radial distance for the above equation is:

(30)

If the values for the aquifer test are used (Q = 42.8 ft3

/

min and T = 39.1 ft2

/min), hydraulic gradients can be 

calculated; these vary from about 0.01 (ft/ft) at a distance of 17.4 ft to 0.001 (ft/ft) at a distance of 174 ft 

from the pumped well. (For purposes of comparison, 

the measured horizontal hydraulic gradient prior to the 

test was approximately 0.0015 ft/ft and just prior to the 

end of the test near the pumped well, between piezometers F505-059 and F504-060, was approximately 

0.014 ft/ft). 

Tables 10A and 10B provide values of the variance and standard deviation in metric and English 

units, respectively, for the indicated values of αf

2

, J, 

λ1 , and λ3. The tables indicate that the standard deviation of the aquifer head variations (columns 6 and 8) 

are likely small, and if the normal range of these head 

variations are approximately four (that is, ± 2) standard 

deviations, then this range is on the order of a few centimeters (0.1 ft) or less. This magnitude of head variation around the mean compares well with those 

differences seen between the simulated and measured 

drawdowns for the aquifer test (standard deviation 

0.014 ft and a range of 0.06 ft).

There are limitations to the above analysis that 

should be noted. First, the analysis is strictly valid only 

for a stationary hydraulic gradient, that is, one that is 

constant in space. This is certainly not true for the case 

of radially convergent flow around a pumped well; 

however, it would be unlikely that the head variations 

would be larger than those for the one-dimensional 

case, as the standard deviations are about an order of 

magnitude larger than those for three-dimensional 

flow, and represent an extreme case where flow is 

forced to travel within each nonuniformity in the aquifer. The reader is referred to Gelhar (1993, chapter 4.1) 

for a discussion of this assumption and the effect of 

dimensionality on the head variance. In addition, equations 27 and 28 are based on an assumption of a small 

value of ln Kr variance, which appears to be satisfied 

here because the value of 0.24 is much lower than the 

often-observed value of approximately 1.0 for the ln Kr

variance (see Gelhar, 1993, p. 103). Finally, the 

assumption of steady flow conditions for these equations isn't strictly satisfied, but the small variation over 

Kr Kz ⁄ 1 αf

2 ( ) + ( ⁄ 2 1 αf2 = ⁄ ( ) – ⁄ 2

α3Dh = παf

2 J2 2λ1λ3 ⁄ 8

α1Dh 5.34αf J

2

λ1

2 2 =

2

h Q ⁄ 4πT')1n(2.25 Tt r

2 = ( ⁄ S )

dh dr ⁄ = –Q ⁄ ( ) 2πTr
38 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

time that would be necessary for the Jacob straight-line 

method could be applied. Here, the value of the dimensionless parameter u (= r

2

S/ 4Tt ) would need to be less 

than 0.05, and with this assumption the hydraulic gradient would be changing very slowly over time, satisfying the basic assumption of steady flow conditions.

One can conclude that given the assumptions and 

limitations noted above, the effects of heterogeneity on 

the results of the aquifer test site are small, and in general it was observed that the measured drawdowns at 

piezometers and wells were very close to the simulated 

values for a homogeneous porous medium. In addition, 

the calculated aquifer property values from the aquifer 

test results are very similar to geometric mean values 

calculated from many small-scale measurements made 

in the same aquifer, supporting the contention that the 

aquifer test values are average results for the aquifer in 

the area of the test.

Parameter-Estimation Experiments with 

Measured Drawdown

Because of the quality and quantity of the measured drawdown data, the scale of the test relative to the 

aquifer heterogeneity, and the availability of the necessary information, some general recommendations can 

be made about the planning, execution, and analysis of 

unconfined aquifer tests. Addressed are the consequences of (1) having only pumped-well data to analyze, (2) having a limited number and distribution of 

piezometers and assuming (a) gradual drainage from 

the unsaturated zone, and (b) instantaneous drainage 

from the unsaturated zone, and (3) having an aquifer 

test of limited duration.

Experiments with Pumped-Well Data 

Because observation piezometer data are often 

unavailable, hydrogeologists will commonly attempt to 

Table 10A. Estimates of the variance and standard deviation of head in the aquifer for three-dimensional and onedimensional flow using equations 27 and 28, where αf

2= variance of ln Kr , J =horizontal hydraulic gradient, λ1=

horizontal log hydraulic conductivity correlation scale, and λ2=vertical log hydraulic conductivity correlation scale. Metric 

units are used.

αf

2 J

λ1

(meters)

λ3

(meters)

α3Dh

2

(meters2)

α3Dh

(meters)

α1Dh

2

(meters2)

α1Dh

(meters)

 0.24 0.01 3.5 0.19 6.3x10–6 0.0024 1.5x10–3 0.039

 .24 .01 8 .38 2.9x10–5 .0054 8.0x10–3 .089

 .24 .001 3.5 .19 6.3x10–8 2.4x10–4 1.5x10–5 3.9x10–3

 .24 .001 8 .38 2.9x10–7 5.4x10–4 8.0x10–5 8.9x10–3

Table 10B. Estimates of the variance and standard deviation of head in the aquifer for three-dimensional and onedimensional flow using equations 27 and 28, where αf

2= variance of ln Kr

, J =horizontal hydraulic gradient, λ1=

horizontal log hydraulic conductivity correlation scale, and λ2=vertical log hydraulic conductivity correlation scale. English 

units are used.

αf

2 J

λ1

(feet)

λ3

(feet)

α3Dh

2

(feet2)

α3Dh

(feet)

α1Dh

2

(feet2)

α1Dh

(feet)

 0.24 0.01 11.5 0.62 6.8x10–5 0.0079 1.6x10–2 0.13

 .24 .01 26.2 1.25 3.1x10–4 .018 8.6x10–2 .29

 .24 .001 11.5 .62 6.8x10–7 7.9x10–4 1.6x10–4 1.3x10–2

 .24 .001 26.2 1.25 3.1x10–6 1.8x10–3 8.6x10–4 2.9x10–2
DISCUSSION 39

3

3.5

4

4.5

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Measured drawdown

Simulated drawdown

F507-080

estimate aquifer parameters from pumped-well data 

alone. It is shown in this section that the pumped-well 

data obtained during this aquifer test, although of reasonably good quality, do not lend themselves to accurate evaluation of the aquifer parameters. In general, 

efforts to obtain reliable estimates of aquifer parameters other than transmissivity with pumped-well data 

are unlikely to be successful. To do so would require 

that the test be designed with that in mind and special 

precautions would have to be taken. Such precautions 

might involve the use of hydraulic packers to reduce 

effects of wellbore storage and/or involve the use of 

special devices to monitor and control the flow rate 

accurately. Although special drilling methods were 

used in this investigation to avoid the introduction of 

foreign material, effects of wellbore skin were nonetheless apparent. This was likely because of turbulence 

in the well bore or flow constrictions caused by the well 

screen.

Figure 21 is a semi-logarithmic plot similar to 

figure 5 but with an expanded drawdown scale that 

amplifies the fluctuations in drawdown as recorded by 

a transducer. The fluctuations appear to be a consequence of minor variations in discharge. A plot of theoretical drawdowns, simulated by using the parameters 

listed in table 6 and with Sw= 1.375 (as determined previously), is superimposed on the transducer data. With 

the exception of the first 0.3 min, and a portion of the 

intermediate-time data between 10 and 100 min, the 

simulated response fits the measured data reasonably 

well. The slope of the straight line in the late-time portion of the simulated drawdown curve is 0.21/log cycle 

and, by the method of Cooper and Jacob (1946), gives 

rise to the expected value of transmissivity (38 ft2

/min) 

corresponding to the product of Kr and b in table 6. 

Thus, by judicious placement of a straight edge on the 

measured late-time data, or by means of a least squares 

polynomial fit, it should be possible to obtain a reasonable estimate of the transmissivity. It is evident from 

the data, however, that the choice of slope is open to 

interpretation. Unfortunately, for several reasons, 

transmissivity is about the only parameter that can be 

estimated to any degree of reliability from pumped 

well data. This is due, in large part, to enhanced drawdowns in the pumped well because of wellbore skin, 

fluctuations in drawdowns due to discharge variations, 

and insensitivity of drawdowns in the pumped well to 

vertical components of flow in the aquifer. 

An attempt (using PEST) to estimate the aquifer 

parameters Sy, Kr, and Kz, from the pumped well data 

(figure A, Appendix II) using the “known” value of b

(170 ft) and assuming, for lack of contrary information, 

that Sw equal zero, gave rise to the values Sy= 0.094, 

Kr= 0.21 ft/min, and Kz= 0.018 ft/min. Because of 

variations in measured drawdowns at early time, the 

first 0.5 min of data (see figure A, Appendix II) were 

eliminated from the analysis. The results were not significantly different when the analysis was restricted to 

Figure 21. Measured and simulated drawdowns for the pumped well (F507-

080) for the model parameters shown in table 6.
40 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

drawdowns at late time (times greater than 400 min). 

These estimated aquifer property values are clearly at 

odds with the parameters obtained by using the 20 

observation piezometers. Even with known wellbore 

skin parameters (which could conceivably be estimated 

by means of a step-drawdown test) analyses of the 

pumped-well data did not yield improved estimation 

results. Repetition of the above tests with Sw=1.375 

gave rise to the values Sy= 0.056, Kr= 0.26 ft/min, and 

Kz= 0.032 ft/min.

Based on these results, it is recommended that 

analysis of pumped-well data without supporting aquifer piezometer data be approached with extreme caution, if at all.

Experiments with Limited Piezometer Distribution

In this section, the results of a number of computer runs performed on selected piezometer groups, 

listed in table 11, are presented in an effort to determine 

the effectiveness of one location or distribution of piezometers compared with another. Because of the 

reduced data sets, it was not always possible to find the 

global optimum using the original initial values (shown 

in table 6) for the characteristic aquifer parameters (Ss, 

Sy , b, Kr, and Kz). These instances involved two groups 

of piezometers located near the water table. The alternative initial values were for the parameters Sy , Kr, and 

Kz and are 0.2, 0.1 ft/min, and 0.1 ft/min, respectively. 

In all instances, the initial values of the empirical 

parameters α1, α2, and α3 are, respectively, 10–4

, 10–2, 

and 10–1

. Letters A–H in table 11 correspond to columns of estimated parameters in tables 12–15. Columns A–D in tables 12–15 contain results of analyses 

conducted on groups of piezometers located at similar 

depths, column E on the group of long-screened wells, 

and columns F–H on groups of piezometers located at 

different distances (i.e., the three piezometer clusters).

Gradual drainage. Table 12 shows the results of 

PEST simulations using the selected piezometer 

groups with the assumption of gradual drainage from 

the unsaturated zone and adjustable saturated thickness. The values of the characteristic aquifer parameters (Ss, Sy , b, Kr, and Kz) should be compared with 

the values determined using all piezometers (table 7), 

which are shown in table 12 (and 13) for convenience. 

Results in table 12 (and 13), columns A, B, and E–H, 

were obtained using initial values for the characteristic 

aquifer parameters that are the same as those in table 6. 

Results in table 12 (and 13), columns C and D required 

the use of initial values for Sy , Kr, and Kz that were 

Table 11. Column headings for tables 12–15.

Column Piezometers Description

A F505-080 F504-080 

F383-082 F383-129

Deep-seated piezometers

B F505-059 F504-060 

F383-061

Mid-depth piezometer and wells

C F505-032 F504-032 

F377-037 F383-032

Shallow, close-in piezometers

D F384-033 F381-056 

F347-031 F385-032 

F376-037

Shallow, distant piezometers

E F434-060 F450-061 

F476-061 F478-061

Long-screened wells

F F505-032 F505-059 

F505-080

Piezometer cluster F505 (20–24 feet from 

pumped well)

G F504-032 F504-060 

F504-080

Piezometer cluster F504 (47–53 feet from 

pumped well)

H F383-032 F383-061 

F383-082 F383-129

Piezometer cluster F383 (93–97 feet from 

pumped well)
DISCUSSION 41

close to the expected values as indicated in the paragraph above. 

The estimated characteristic aquifer parameters 

in table 12 columns A and B, for deep-seated and middepth piezometers, are reasonably close to the values 

given in table 7 and are reproducible using different initial values. The differences found in the empirical 

parameters between the values given in table 7 and the 

values in columns A and B are not surprising and are 

indicative of the sensitivity of these parameters to measurement error and aquifer heterogeneity. They do not 

noticeably change the match between theoretical and 

measured drawdown seen in figures 14, 15, and 18. The 

results in columns C and D for the shallow, close-in 

piezometers and shallow, distant piezometers do not 

appear to be unique. They appear to require the use of 

initial values for Kr, Kz, and Sy that are close to the 

expected values (namely, 0.2, 0.1, and 0.1, respectively, 

as indicated above). In the case of column C, however, 

through further experimentation, it was found that the 

need to revise the initial values of Kr, Kz, and Sy , was 

a consequence of the attempt to estimate specific storage. By fixing the value of specific storage at the value 

given in table 7 it was found unnecessary to change the 

initial values of any parameters. Also, the use of an 

additional empirical parameter α4 improved the estimate of specific storage (1.6x10–5

 ft–1) but did not significantly change the other parameters. In the case of 

column D, the reason for the need to revise the initial 

values of Kr, Kz, and Sy may be partially explained by 

inaccurate hand measurements of small drawdowns 

early in the test (see figures E–I in Appendix II), and 

the fact that the flow regime at the larger distances is 

essentially horizontal and would likely not be sensitive 

to the aquifer saturated thickness or vertical hydraulic 

conductivity. The results in both columns C and D suggest that analyses performed using only shallow piezometers may not always be trusted, and that deepseated piezometers may be required to obtain satisfactory parameter estimates.

The results presented in columns E–H of table 12 

are, with one exception, reasonably consistent with the 

results in table 7. The exception is the simulation using 

the piezometer cluster F383 (column H), whose results 

depart significantly from the others. Note the near 

equality of Kr and Kz and the large values of Sy and b. 

(The choice of initial values close to the values in table 

7 did not change the final results shown in column H.) 

Further experimentation with this piezometer cluster, 

wherein drawdowns at times less than 100 min were 

eliminated from the estimation process, yielded a 

totally unrealistic estimate of specific yield (Sy =

0.739). These results suggest that the locations of the 

piezometers in the F383 cluster are such that the aquifer response there is insensitive to the exact value of the 

hydraulic parameters. This is apparently due to essentially horizontal flow at that location, which is at a distance of approximately 95 ft from the pumped well.

Because it is the usual procedure to specify the 

saturated thickness as a known quantity, the runs in 

Table 12. Analysis of selected piezometer groups assuming gradual drainage and adjustable saturated thickness, where 

Ss = specific storage, Sy =specific yield, b =saturated thickness, Kr=hydraulic conductivity in the horizontal direction, Kz =

hydraulic conductivity in the vertical direction, and α1, α2, and α2 are empirical constants for gradual drainage from the 

unsaturated zone; ft, feet; min, minute.

Parameter Table 7 A B C D E F G H

Ss(ft–1) 1.3E-05 1.4E-05 1.3E-05 1.9E-05 1.3E-05* 1.3E-05* 1.2E-05 1.8E-05 1.3E-05*

Sy .266 .249 .248 .280 .241 .278 .217 .240 .360

b(ft) 169 176 157 198 217 178 159 202 253

Kr(ft/min) .233 .243 .248 .230 .203 .226 .249 .227 .157

Kz(ft/min) .142 .135 .123 .110 .104 .144 .135 .134 .152

α1(min1) 2.8E-04 2.8E-05 1.7E-04 4.9E-05 8.2E-04 2.3E-04 8.7E-05 1.7E-06 8.0E-04

α2(min–1) 1.7E-02 2.8E-02 2.6E-02 1.2E-02 2.2E-02 2.4E-02 1.8E-02 1.9E-02 8.8E-03

α3(min–1) .42 1.4 3.2 .65 6.6E-02 21 .60 .32 6.6E-02

*, fixed value
42 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

table 12 were repeated by using a fixed value of saturated thickness (170 ft). The results of these runs are 

presented in table 13 and, with one or two exceptions, 

appear to be reasonably consistent with the values in 

table 7. The primary exception is the value of Kz in column D for the shallow distant piezometers. This result 

again suggests a lack of sensitivity of the data from the 

shallow, distant piezometers to aquifer thickness (b) 

and vertical hydraulic conductivity (Kz). Specification 

of the saturated thickness clearly results in an improvement in the estimated hydraulic conductivities in column H for the most distant piezometer cluster as 

compared with column H in table 12.

Instantaneous drainage. As mentioned repeatedly, many conventional analyses of aquifer test data 

make the assumption that drainage from the unsaturated zone occurs instantaneously in response to a 

decline in elevation of the water table. This assumption 

was made to obtain the results shown in table 5, but to 

do so with the PEST algorithm it was necessary to utilize only the very late-time data (times greater than 

2,000 minutes) measured in each piezometer. Tables 14 

and 15 show results obtained under the assumption of 

instantaneous drainage without and with, respectively, 

an assumed known saturated thickness (b). Because 

there are significant discrepancies between measured 

and simulated drawdowns during the intermediate time 

range for most piezometers (see fig. 6–12), it would be 

counterproductive to use PEST with the model 

assumption of instantaneous drainage over the entire 

time range without appropriate weighting considerations. For the results presented in tables 14 and 15, the 

analyses are limited to data occurring over the final log 

cycle of time (generally the last 7 values chosen for 

analysis by PEST, or time greater than 430 min; (see 

figures B–K in Appendix II). Over this time period, and 

for most piezometers the discrepancies mentioned 

above (see fig. 6–12) are not large and one might expect 

to get reasonable estimates of the aquifer parameters 

either by type-curve analysis or by application of the 

PEST algorithm. That this may not always be the case 

is evident from tables 14 and 15.

Inspection of the estimated parameters in each of 

the selected piezometer groups of table 14 reveals that 

there is at least one parameter that is significantly different from the parameters listed in table 5. Table 15 

shows that specification of the saturated thickness 

reduces some of the variability that is evident from one 

piezometer group to the next. In all columns of table 15 

the specific yield is less than the values obtained under 

the model assumption of gradual drainage from the 

unsaturated zone (table 13). The low values of Sy are a 

consequence, in large part, of the extended time range 

used in these analyses compared with the significantly 

shorter time range used to obtain the parameters in 

table 5. The values of hydraulic conductivity in table 15 

show greater variability than the values in table 13 even 

though early-time data, and most of the intermediatetime data, were eliminated from the analysis. The 

results suggest the significant influence of gradual 

Table 13. Analysis of selected piezometer groups assuming delayed drainage and fixed saturated thickness, where Ss=

specific storage, Sy=specific yield, b =saturated thickness, Kr=hydraulic conductivity in the horizontal direction, Kz=

hydraulic conductivity in the vertical direction, and α1, α2, and α2 are empirical constants for gradual drainage from the 

unsaturated zone; ft, feet; min, minute.

Parameter Table 7 A B C D E F G H

Ss(ft–1) 1.3E-05 1.4E-05 1.3E-05 2.0E-05 1.3E-05* 1.3E-05* 1.1E-05 2.0E-05 1.3E-05*

Sy .266 .260 .235 .277 .252 .275 .217 .206 .331

b(ft) 169 170* 170* 170* 170* 170* 170* 170* 170*

Kr(ft/min) .233 .243 .247 .230 .256 .227 .246 .252 .211

Kz(ft/min) .142 .132 .124 .135 .0693 .147 .137 .127 .129

α1(min–1) 2.8E-04 4.6E-05 3.8E-05 2.1E-04 7.3E-04 4.1E-04 1.1E-05 1.5E-05 6.3E-04

α2(min–1) 1.7E-02 2.6E-02 3.0E-02 1.2E-02 3.2E-02 2.7E-02 2.1E-02 2.4E-02 1.2E-02

α3(min–1) .42 4.1 3.2 .30 .35 15 .61 .43 .17

*, fixed value
DISCUSSION 43

drainage processes (even for times greater than 430 

minutes) and the importance of using a model that 

takes account of this process.

Experiments with Reduced Length of Test

Tables 16A and 16B show the results of PEST 

simulations for different piezometer groups with the 

model assumption of gradual drainage from the unsaturated zone for aquifer tests of different duration. Some 

important findings emerge from examination of these 

tables: tables 16A and 16B demonstrate (1) that it may 

not have been necessary to run this aquifer test for as 

long as it was and (2) only a few piezometers located in 

proximity to the pumped well at appropriate depths 

may be needed to define the aquifer characteristics. By 

use of either the complete set of piezometers or just the 

deep-seated piezometers (see table 16A for both), it 

appears that the results for an 8-hour test are as valid as 

the results for a 72-hour test. This finding indicates that 

the drawdown data obtained in the first 8 hours of this 

aquifer test are adequate to define the primary aquifer 

characteristics as well as the three empirical parameters that account for gradual drainage. Of course, without measurements made in distant piezometers it 

would not be possible to judge the aerial extent to 

which the evaluated aquifer characteristics apply.

Results obtained for the long-screened piezometers and the combination of three piezometer clusters 

shown in table 16B show that the 16-hour and 8-hour 

tests do not agree as well with the results of the 72-hour 

test as do the results of the 24-hour test. The parameters 

with the greatest differences are the specific yield and 

the saturated thickness. The differences are not great, 

however, and may be satisfactory for most applications. 

Examination of figures 6–12 reveals that the piezometers whose measured responses deviate the least 

from simulated responses (based on the assumption of 

instantaneous drainage) are the deep-seated, close-in 

piezometers (F505-059, F505-080, F504-080, and 

F383-129). This set of piezometers is the same as the 

set of deep-seated piezometers in table 16A except that 

F383-082 is replaced by F505-059. The results of analyses of the drawdown data from these piezometers 

based on the assumptions of gradual drainage on the 

one hand and instantaneous drainage on the other, are 

shown in table 17. The primary difference in the estimated parameters is in the estimate of specific yield. 

Table 14. Analysis of selected piezometer groups assuming instantaneous drainage for times greater than 430 minutes and 

adjustable saturated thickness, where Sy =specific yield, b = saturated thickness, Kr =hydraulic conductivity in the horizontal 

direction, Kz = hydraulic conductivity in the vertical direction; ft, feet; min, minute.

Parameter Table 5 All A B C D E F G H

Sy 0.254 0.175 .152 0.164 0.264 0.181 0.190 0.251 0.241 0.310

b(ft) 171 198 188 172 260 243 228 229 266 403

Kr(ft/min) .229 .227 .238 .285 .222 .232 .218 .209 .208 .124

Kz(ft/min) .137 .137 .140 .0591 .0712 .0682 .123 .154 .101 .186

Table 15. Analysis of selected piezometer groups assuming instantaneous drainage for times greater than 430 minutes and 

adjustable saturated thickness, where Sy =specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal 

direction, Kz =hydraulic conductivity in the vertical direction; ft, feet; min, minute.

Parameter Table 5 All A B C D E F G H

Sy 0.254 0.208 0.171 0.169 0.230 0.193 0.211 0.124 0.148 0.197

b(ft) 171 170* 170* 170* 170* 170* 170* 170* 170* 170*

Kr(ft/min) .229 .237 .247 .286 .228 .335 .252 .250 .258 .238

Kz(ft/min) .137 .125 .121 .0584 .132 .274 .0923 .131 .0963 .142

*, fixed value
44 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

Under the assumption of instantaneous drainage, the 

specific yield is a little more than half of that obtained 

under the assumption of gradual drainage from the 

unsaturated zone. In each instance it does not matter 

whether the test was run for 8-hours or 72-hours. 

Based on the parameters in table 17 for the 8-

hour test, simulated and measured drawdown 

responses are compared for the case of instantaneous 

drainage in figure 22 and for the case of gradual drainage in figure 23. The improvement seen in figure 23, 

although marginal, is evident in the intermediate time 

range. There are no noticeable differences, however, in 

the simulated responses at late time (last log cycle of 

time). The explanation for the significantly larger value 

of Sy obtained by including gradual drainage lies in the 

fact that a larger value of Sy is needed to compensate 

for the fact that water is retained in the unsaturated 

zone and gives rise to only small increased drawdowns 

in the deep-seated observation piezometers.

SUMMARY

A model for flow to a well in an unconfined aquifer was applied to the analysis of a 72-hour aquifer test 

conducted in a sand and gravel, glacial-outwash 

deposit in Cape Cod, Massachusetts. In addition to vertical components of flow in the saturated zone and 

effects of partial penetration, this model allows for 

gradual drainage from the unsaturated zone, wellbore 

storage and skin at the pumped well, and delayed piezometer response. An automated parameter estimation 

algorithm was used to obtain all relevant unconfined 

aquifer parameters, including the saturated thickness. 

The detailed analysis supports the results of a preliminary type-curve analysis, reported by Moench and others (1996), which made use of an analysis method of 

Neuman (1974) to evaluate horizontal and vertical 

hydraulic conductivity and specific yield. Although the 

preliminary analysis showed good agreement between 

simulated and measured drawdowns at late time, there 

were found to be significant discrepancies between 

simulated responses and the intermediate-time data. 

The analysis presented in this report was extended 

using a modified version of the Moench (1997) model, 

with the result that simulated responses based on the 

estimated aquifer parameters compare well with measured drawdowns in all piezometers at all times. Due to 

the high quality and quantity of the data and the small 

variability in the aquifer parameters it was possible to 

provide some guidelines for the design and execution 

of unconfined aquifer tests of the type found at this site.

The model modification involves the substitution 

of multiple empirical parameters (as coefficients in 

exponential relations) for the single empirical parameter used previously to describe drainage from the zone 

above the water table. The single empirical parameter, 

which assumes that the vertical flux of water at the free 

surface varies exponentially in response to a step 

change in the elevation of the water table, was found to 

provide only moderate improvement over the assumption of instantaneous drainage. The introduction of a 

finite series of terms, each with an additional empirical 

parameter, effectively eliminated discrepancies 

between measured and computed drawdowns. Three 

such terms appeared to provide an adequate representation of the drainage process that occurred during this 

test.

The values of the estimated hydraulic parameters 

are consistent with estimates from prior studies and 

from what is known about the aquifer at the site. The 

estimated values are: specific yield, 0.26; saturated 

thickness, 170 feet; horizontal hydraulic conductivity, 

0.23 feet per minute; vertical hydraulic conductivity, 

0.14 feet per minute; and specific storage, 1.3x10−5 per 

foot.

Apart from the aquifer parameters, the principal 

findings that result from the analysis and the additional 

parameter-estimation experiments are the following:

1. Pumped-well data by themselves are not useful for 

estimating most unconfined aquifer parameters. 

The primary information obtained using these 

data is an approximate value of aquifer transmissivity from the late-time drawdown values. 

Pumped-well data are needed, however, in order 

to estimate the wellbore skin factor that is used to 

improve estimates of specific storage.

2. An analysis of early-time data from the piezometers 

with transducers can yield reasonable estimates of 

specific storage (Ss) provided wellbore skin, wellbore storage, and delayed piezometer responses 

are included in the analysis.

3. Under the assumption of instantaneous drainage, 

estimates of specific yield (Sy ), saturated thickness (b), horizontal hydraulic conductivity (Kr) 

and vertical hydraulic conductivity (Kz) can be 

obtained using late-time drawdown data, given a 

number of piezometers strategically located at 

depth in the vicinity of the pumped well and given 
SUMMARY 45

Table 16A. Analysis of time-limited tests for all piezometers and deep-seated piezometers, where Ss =specific storage, 

Sy = specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal direction, Kz =hydraulic 

conductivity in the vertical direction, and α1, α2, and α3 are empirical constants for gradual drainage from the unsaturated 

zone; ft, feet; min, minutes.

Parameter

All piezometers

Deep piezometers

F505-080 F505-080 F383-082 F383-129

72 hour# 24 hour 16 hour 8 hour 72 hour* 24 hour 16 hour 8 hour

Ss (ft–1) 1.3E-05 1.3E-05 1.3E-05 1.3E-05 1.4E-05 1.3E-05 1.4E-05 1.4E-05

Sy .263 .255 .258 .239 .249 .261 .266 .270

b (ft) 174 179 177 186 176 169 166 163

Kr (ft/min) .233 .236 .237 .240 .243 .242 .242 .245

Kz (ft/min) .141 .140 .140 .139 .135 .134 .132 .130

α1 (min–1) 1.7E-04 6.6E-05 3.1E-05 9.2E-05 2.8E-05 6.1E-05 5.2E-05 6.5E-05

α2 (min–1) 1.7E-02 1.8E-02 1.7E-02 2.0E-02 2.8E-02 2.8E-02 2.8E-02 2.4E-02

α2 (min–1) .42 .45 .42 .43 1.4 4.4 3.7 8.8

#Values slightly different from those in tables 7 and 9 because of different initial values (consistent with others used in generating this 

table).

 *Values taken from table 12, column A.

Table 16B. Analysis of time-limited tests for long-screened piezometers and piezometer clusters, where Ss =specific 

storage, Sy = specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal direction, Kz =hydraulic 

conductivity in the vertical direction, and α1, α2, and α3 are empirical constants for gradual drainage from the unsaturated 

zone; ft, feet; min, minutes; —, no data.

Parameter

Long-screened piezometers Piezometer clusters

F434-060 F450-061 F476-061 F478-061 F505 F504 F383

72 hour* 24 hour 16 hour 8 hour 72 hour 24 hour 16 hour 8 hour

Ss (ft–1) ———— 1.3E-05 1.3E-05 1.3E-05 1.3E-05

Sy 0.278 0.278 .294 0.294 0.235 .241 .234 .215

b (ft) 178 167 146 132 183 178 181 199

Kr (ft/min) .226 .233 .235 .248 .235 .239 .240 .244

Kz (ft/min) .144 .149 .152 .150 .139 .139 .137 .136

α1 (min–1) 2.3E-04 8.0E-05 2.5E-04 2.4E-05 1.4E-05 2.1E-05 3.1E-05 1.3E-04

α2 (min–1) 2.4E-02 2.2E-02 1.8E-02 1.8E-02 2.2E-02 1.7E-02 1.9E-02 2.1E-02

α2 (min–1) 21 6.1 5.2 22 .44 .45 .47 .52

 *Values taken from table 12, column E.
46 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

that the duration of the aquifer test is sufficient 

(72 hours in this instance). 

4. The assumption of instantaneous drainage from the 

unsaturated zone does not always lead to an adequate simulation of drawdowns in the intermediate time range, which in this test are times from 

approximately 1 to 1,000 min. Even in those situations where analysis using the assumption of 

instantaneous drainage might be deemed appropriate (for example, with deep-seated piezometers 

located relatively close to the pumped well), estimates of specific yield will likely be low relative 

to values obtained using an assumption of gradual 

drainage or relative to values obtained from latetime data alone.

5. Drawdown data from the shallow piezometers 

located at large distances from the pumped well 

where flow is essentially horizontal appear to be 

diagnostic of only horizontal hydraulic conductivity and specific yield. Data from the shallow 

piezometers located near the pumped well, even 

though in a part of the aquifer that is strongly 

influenced by vertical components of flow, do not 

necessarily yield a unique set of aquifer parameters. It is possible that this is a consequence of the 

variably distributed drainage from the unsaturated 

zone due to heterogeneity in the material overlying the piezometers.

6. Data from mid-depth and deep-seated piezometers 

located near the pumped well, treated as an independent group, appear to provide parameter estimates that are consistent with results from the 

data set as a whole and would be the recommended locations given limited resources.

7. Tests performed with the modified model on the 

complete set of piezometers and on selected piezometer groups, independent of one another, indicate that it is not necessary to run the aquifer test 

for as long as 72-hours. It appears that for some 

piezometer groups a test as short as 8-hours may 

be adequate. For other piezometer groups, a 16-

hour test may be adequate.

It may be concluded from the analysis provided 

here that accurate estimates of unconfined aquifer 

parameters using automatic parameter estimation techniques require a model that accounts for all the physical 

processes that influence the measured drawdowns. If 

the relevant processes are not taken into account, the 

parameter estimation algorithm treats differences 

between measured and simulated drawdowns as errors 

in measurement with a subsequent degradation in the 

validity of the estimated parameters.

It has been mentioned repeatedly that the Cape 

Cod aquifer test is exceptional: the data are of unusually good quality, there are a sufficient number of 

Table 17. Various data analyses for piezometers F505-059, F505-080, F504-080, and F383-129, where Ss =specific 

storage, Sy =specific yield, b =saturated thickness, Kr =hydraulic conductivity in the horizontal direction, Kz =hydraulic 

conductivity in the vertical direction, and α1, α2, and α3 are empirical constants for gradual drainage from the 

unsaturated zone; —, not applicable.

Parameter

Delayed drainage Instantaneous drainage

72 hour 8 hour 72 hour 8 hour

Ss (feet–1) 1.2E-05 1.2E-05 1.3E-05 1.3E-05

Sy .298 .297 .163 .160

b (feet) 170 174 180 178

Kr (feet per minute) .233 .236 .243 .246

Kz (feet per minute) .136 .134 .124 .121

α1 (minute–1) 1.5E-04 1.9E-05 — —

α2 (minute–1) 2.8E-02 2.3E-02 — —

α3 (minute–1) 21 44 — —
SUMMARY 47

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-059

F505-080

F504-080

F583-129

Simulated drawdown

Measured drawdown

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F505-059

F505-080

F504-080

F583-129

Simulated drawdown

Measured drawdown

Figure 22. Measured drawdowns compared with drawdowns simulated 

under the assumption of instantaneous release of water from the 

unsaturated zone for selected deep-seated piezometers.

Figure 23. Measured drawdowns compared with drawdowns simulated 

under the assumption of gradual drainage of water from the unsaturated 

zone for selected deep-seated piezometers.
48 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

observation piezometers, the scale of heterogeneity is 

small in comparison with the scale of the aquifer test 

resulting in estimated parameters that are average values for the aquifer in the area of the test, the boundary 

conditions of the physical system appear to conform 

satisfactorily with the mathematical model, and interference from extraneous sources is minimal. Field conditions do not often reach this ideal: the aquifer may 

have large-scale heterogeneity in hydraulic conductivity or saturated thickness; the aquifer may be bounded 

laterally within the cone of depression; piezometers 

may be too few in number, improperly located, or not 

in good hydraulic connection with the aquifer; there 

may be interference from one or more extraneous 

sources; and relevant pumped-well and observationpiezometer characteristics may not be known. Because 

of the nearly ideal conditions and high data quality it is 

suggested that this aquifer test be used as a benchmark 

case or standard against which other tests, conducted 

under less than ideal conditions, are compared.
NOTATION 49

NOTATION

b initial saturated thickness of aquifer, L

C wellbore storage, L2

d vertical distance from initial water table to top 

of pumped well screen, L

ds thickness of the wellbore skin, L

F ' modified Hvorslev shape factor, L

F Hvorslev shape factor, L

h hydraulic head, L

h* vertical average of hydraulic head in the aquifer adjacent to and over the length of the 

pumped-well screen, L

hi initial hydraulic head, L

hm measured hydraulic head in a piezometer, L

hw average hydraulic head in the pumped well, L

J hydraulic gradient in the horizontal direction

k square root of ratio Kr/Kz

Kz hydraulic conductivity in the vertical direction, 

LT–1

Kr hydraulic conductivity in the horizontal direction, LT–1

Ks hydraulic conductivity of the wellbore skin, 

LT–1

l vertical distance from initial water table to bottom of pumped-well screen, L

L length of a piezometer screen, L

M number of empirical constants for gradual 

drainage from the unsaturated zone

Q pumping rate, L3

T–1

p Laplace transform variable

r radial distance from axis of pumping well, L

rc effective radius of the pumped well in the 

interval where water levels are changing, L

rp radius of observation piezometer in the interval 

where water levels are changing, L

rw outside radius of the pumped-well screen, L

S storativity

Ss specific storage, L–1

Sw wellbore skin factor

Sy specific yield

T transmissivity, L2

T−1

t time since start of pumping, T

t ' variable of integration in equation 6

u dimensionless parameter defined as r

2

S/4Tt

z vertical distance above bottom of aquifer, L

zp elevation of the midpoint of an observationpiezometer screen above the base of the aquifer, L

zu vertical distance above the water table, L

z1 elevation of the bottom of an observation-well 

screen above the base of the aquifer, L

z2 elevation of the top of an observation-well 

screen above the base of the aquifer, L

α Boulton's (1963) reciprocal “delay index,” T–1

αm m–th empirical constant for gradual drainage 

from the unsaturated zone, T–1

αf

2 variance of ln Kr

α1Dh

2 variance of head for one-dimensional flow, L2

α3Dh

2 variance of head for three-dimensional flow, L2

λ1 horizontal log hydraulic conductivity correlation scale, L

λ3 vertical log hydraulic conductivity correlation 

scale, L

θs moisture content above the water table at saturation

θ(z) moisture content of unsaturated zone

τ variable of integration in equation 13
50 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

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Grenoble, France, University of Grenoble, Ph.D. thesis, 

159 p.
52 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts
Appendix I Derivation of the Laplace Transform Solution 53

APPENDIX I

DERIVATION OF THE LAPLACE 

TRANSFORM SOLUTION
54 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

APPENDIX I

In this section, a derivation of the Laplace transform solution is provided. Application of the method of 

Laplace transformation to equations 8–13 leads to the following subsidiary boundary-value problems:

(A1)

(A2)

(A3a)

(A3b)

(A3c)

(A3d)

(A4)

(A5)

The solution to the above problem is obtained with the help of Fourier cosine series in the manner followed 

by Dougherty and Babu (1984) for well tests in confined double-porosity aquifers and by Moench (1997, 1998).

A solution to equation A1 that satisfies equations A4 and A5 is 

(A6)

where n = 0, 1, 2,..., and εn are the roots of

∂

2

hD

∂rD

2 ----------- 1

rD

----- ∂hD

∂rD

--------- βw

∂

2

hD

∂rD

2

+ + -----------= phD

hD ∞ zD ( ) , , p = 0

∂hD

∂rD

--------- 2

p l

D dD ( ) – – ------------------------- WD phwD

1 l

D zD 1 dD – ≤ ≤ –

rD= 1 = +

∂hD

∂rD

--------- 0 = zD 1 l

D zD 1 dD < – ; > –

rD = 1

hwD hD

* Sw

∂hD

∂rD

= 1 – --------- l

D zD 1 dD – ≤ ≤ –

hD

* 1

l

D dD ( ) – --------------------- hD 1 zD ( ) , , p zD d

1 l

D –

1 dD –

∫ =

∂hD

∂zD

--------- rD ( ) , , 0 p = 0

∂hD

∂zD

--------- rD ( ) , , 1 p hD rD ( ) , , 1 p p

M

---- 1

σβω p γ m ( ) + ⁄ ----------------------------------

m = 1

M

= – ∑

hD f n rD ( )ε , p nzD cos( )

n = 0

∞

= ∑
APPENDIX I 55

(A7)

Substitution of equation A6 into A1 yields

(A8)

Hence, fn must satisfy

(A9)

the general solution of which can be written as

(A10)

where qn = (ε2

nβw+ p)

1/2. K0 and I0 are the zero order modified Bessel functions of the second and first kind, 

respectively, and un and vn are coefficients to be determined.

Because of equation A2, vn(p)= 0 and,consequently

(A11)

Substitution of equation A11 into A6 yields

(A12)

The coefficients un(p) are determined from equations A3a and A3b. First substitute equation A12 into A3a

and let rD = 1 to get

(A13)

where

(A14)

Likewise, use equation A3b to get

εn εn tan( ) p

M

---- 1

σβω p γ m ( ) + ⁄ -----------------------------------

m = 1

M

= ∑

f ′′n 1

rD

----- f 'n εn

2

βw ( ) + p f + – n εnzD cos( )= 0

m = 0

M

∑

f ′′n 1

rD

----- f 'n εn

2

βw ( ) + p f + – n = 0

f n = un( ) p K0 qnrD ( ) vn( ) p I0 qnrD + ( )

f n un( ) p K0 qnrD = ( )

hD un( ) p K0 qnrD ( )εnzD cos( )

n = 0

∞

=∑

un( ) p qnK1 qn () εnzD cos( )

n = 0

∞

– ∑ qs 1 l

D zD 1 dD = – ≤ ≤ –

qs

2

p l

D dD ( ) – – ------------------------- WD = + phwD
56 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

(A15)

Multiplying equation A13 through by cos (εmzD), where m is an integer, and integrating over the indicated 

intervel one obtains

(A16)

Multiplying equation A15 by cos(εmzD) and integrating over the interval below the screen one obtains

(A17)

Also, by performing the same operation over the interval above the screen one obtains

(A18)

Adding equations A17 and A18 to the left-hand side of equation A16 one obtains

(A19)

It can be shown quite simply by use of fundamental trigonometric indenties, equation A7, and direct integration that all terms in the sum on the left-hand side of equation A19 are zero except those for which n=m. Thus, the 

set cos (εnzD)cos (εmzD) is orthogonal over the interval 0,1and equation A19 becomes

(A20)

Performing the integration and rearranging terms, one obtains

(A21)

un( ) p qnK1 qn () εnzD ( ) 0

zD 1 l

D < –

zD 1 dD > – cos =

n = 0

∞

– ∑

un( ) p qnK1 qn ( )

n = 0

∞

– ∑ εnzD cos( ) εmzD cos( ) zD d

1 l

D –

1 dD –

∫ qs εmzD cos( ) zD d

1 l

D –

1 dD –

∫ =

un( ) p qnK1 qn ( )

n = 0

∞

–∑ εnzD cos( ) εmzD cos( ) zD d =0

0

1 l

D –

∫

un( ) p qnK1 qn ( )

n = 0

∞

–∑ εnzD cos( ) εmzD cos( ) zD d =0

1 l

D –

1

∫

un( ) p qnK1

n = 0

∞

–∑ qn () εnzD cos( ) εmzD cos( )dzD

0

1

∫ qs εmzD cos( ) dzD

1 l D –

1 dD –

∫ =

un( ) p qnK1 qn () εnzD ( ) zD d 2cos qs εnzD cos( ) zD d

1 l D –

1 dD –

∫ =

0

1

∫ –

un( ) p

2qs εn 1 dD ( )] ε – n 1 l

D { sin[ – sin[ ( )]} –

qnK1 qn ( )εn 0.5 2εn [ + sin( )] = –--------------------------------------------------------------------------------------------------
APPENDIX I 57

Thus, the solution (equation A12) becomes

(A22)

where E is defined by equation 17.

The Laplace transform of the hydraulic head in the pumped well is obtained by inserting equation A22 into 

A3d. Thus, at rD =1, one obtains

(A23)

where A is defined by equation 16. From equations A3c and A23

(A24)

Substitution of equation A14 into A24 yields the solution (equation 14) for the Laplace transform of the dimensionless head in the pumped well.

Substitution of equation A24 into A14 and combining the result with A22 yields the solution (equation 15) 

for the Laplace transform of the dimensionless head in the aquifer.

hD rD zD ( ) , , p qs = – E

h*

D q = – s A

hwD q – s A Swqs = –
58 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts
Appendix II DRAWDOWN DATA PLOTS FOR WELL TEST AND DATA SELECTED FOR PARAMETER ESTIMATION 59

APPENDIX II

DRAWDOWN DATA PLOTS FOR WELL TEST AND

DATA SELECTED FOR PARAMETER ESTIMATION
60 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0

1

3

4

2

5

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Drawdown values selected for parameter estimation

Drawdown—Pressure transducer measurements

F507-080

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Drawdown values selected for parameter estimation

Drawdown—Pressure transducer measurements

Drawdown—Manual measurements

F505-080

F504-080

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Drawdown values selected for parameter estimation

Drawdown—Pressure transducer measurements

Drawdown—Manual measurements

F505-059

F504-060

Figure A. Drawdown data for the pumped well F507-080.

Figure B. Drawdown data for piezometers F505-080 and F504-080.

Figure C. Drawdown data for wells F505-059 and F504-060.
Appendix II DRAWDOWN DATA PLOTS FOR WELL TEST AND DATA SELECTED FOR PARAMETER ESTIMATION 61

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Drawdown values selected for parameter estimation

Drawdown—Pressure transducer measurements

Drawdown—Manual measurements

F505-032

F504-032

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

Drawdown values selected for parameter estimation

Drawdown—Pressure transducer measurements

Drawdown—Manual measurements from F377-037

Drawdown—Manual measurements from F347-031

F377-037

F347-031

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F383-061

F383-032

Drawdown values selected for parameter estimation—F383-061

Drawdown values selected for parameter estimation—F383-032

Drawdown—Manual measurements from F383-061

Drawdown—Manual measurements from F383-032

Figure D. Drawdown data for piezometers F505-032 and F504-032.

Figure E. Drawdown data for piezometers F377-037 and F347-031.

Figure F. Drawdown data for piezometers F383-061 and F383-032. 
62 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F383-082

F383-129

Drawdown values selected for parameter estimation—F383-082

Drawdown values selected for parameter estimation—F383-129

Drawdown—Manual measurements from F383-082

Drawdown—Manual measurements from F383-129

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F384-033

F385-032

Drawdown values selected for parameter estimation—F384-033

Drawdown values selected for parameter estimation—F385-032

Drawdown—Manual measurements from F384-033

Drawdown—Manual measurements from F385-032

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F381-056

F376-037

Drawdown values selected for parameter estimation—F381-056

Drawdown values selected for parameter estimation—F376-037

Drawdown—Manual measurements from F381-056

Drawdown—Manual measurements from F376-037

Figure G. Drawdown data for piezometers F383-082 and F383-129. 

Figure H. Drawdown data for piezometers F384-033 and F385-032. 

Figure I. Drawdown data for piezometers F381-056 and F376-037. 
Appendix II DRAWDOWN DATA PLOTS FOR WELL TEST AND DATA SELECTED FOR PARAMETER ESTIMATION 63

0.001

0.1

0.01

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F476-061

F478-061

Drawdown values selected for parameter estimation—F476-061

Drawdown values selected for parameter estimation—F478-061

Drawdown—Manual measurements from F476-061

Drawdown—Manual measurements from F478-061

0.001

0.01

0.1

1

DRAWDOWN, IN FEET

10–2 10–1 100 101 102 103 104

TIME, IN MINUTES

F434-060

F450-061

Drawdown values selected for parameter estimation—F434-060

Drawdown values selected for parameter estimation—F450-061

Drawdown—Manual measurements from F434-060

Drawdown—Manual measurements from F450-061

Figure J. Drawdown data for wells F434-060 and F450-061. 

Figure K. Drawdown data for wells F476-061 and F478-061.
64 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

 * Values not used

Table A. Data selected for parameter estimation, with PEST observation numbers; t, time, in minutes; h, drawdown, in feet; 

Obs. #, observation number.

F507-080 PEST

Obs. #

F505-032 PEST

Obs. #

F505-059 PEST

 Obs. #

F505-080 PEST

Obs. #

t (min) h(ft) t (min) h(ft) t (min) h(ft) t (min) h(ft)

0.0337 2.560 1 * 0.0337 0.0250 32 0.0337 0.1190 64 0.0170 0.0220 96

0.0504 3.530 2 * 0.0504 0.0530 33 0.0504 0.2370 65 0.0337 0.0440 97

0.0670 3.680 3 * 0.0670 0.0810 34 0.0670 0.3350 66 0.0504 0.0910 98

0.1000 3.750 4 * 0.1000 0.1320 35 0.1000 0.4550 67 0.0670 0.1310 99

0.1500 3.870 5 * 0.1500 0.1770 36 0.1500 0.5160 68 0.1000 0.2160 100

0.2170 3.790 6 * 0.2170 0.2040 37 0.2170 0.5310 69 0.1500 0.3130 101

0.3170 3.760 7 * 0.3170 0.2140 38 0.3170 0.5350 70 0.2170 0.3560 102

0.4670 3.760 8 0.4670 0.2250 39 0.4670 0.5450 71 0.3170 0.3720 103

0.6840 3.740 9 0.6840 0.2310 40 0.6840 0.5500 72 0.4670 0.3820 104

1.0000 3.760 10 1.0000 0.2440 41 1.0000 0.5590 73 0.6840 0.3900 105

1.4700 3.750 11 1.4700 0.2570 42 1.4700 0.5670 74 1.0200 0.3980 106

2.1500 3.750 12 2.1500 0.2750 43 2.1500 0.5760 75 1.5000 0.4020 107

3.1700 3.780 13 3.1700 0.2960 44 3.1700 0.5890 76 2.1500 0.4070 108

4.5800 3.790 14 4.7500 0.3210 45 4.5800 0.6020 77 3.1500 0.4150 109

6.7500 3.850 15 6.7500 0.3450 46 6.7500 0.6140 78 4.5800 0.4230 110

10.100 3.840 16 10.100 0.3770 47 10.100 0.6340 79 6.7500 0.4330 111

14.900 3.900 17 14.900 0.4150 48 14.900 0.6560 80 10.100 0.4460 112

21.900 3.920 18 21.900 0.4510 49 21.900 0.6740 81 14.900 0.4590 113

46.900 3.940 19 31.900 0.4820 50 31.900 0.6930 82 21.900 0.4730 114

67.900 3.960 20 46.900 0.5160 51 46.900 0.7200 83 31.900 0.4860 115

101.00 3.990 21 67.900 0.5450 52 67.900 0.7460 84 46.900 0.5020 116

151.00 4.010 22 99.900 0.5820 53 101.00 0.7810 85 68.900 0.5230 117

221.00 4.020 23 151.00 0.6130 54 151.00 0.8190 86 101.00 0.5470 118

321.00 4.090 24 221.00 0.6540 55 221.00 0.8580 87 151.00 0.5780 119

461.00 4.120 25 325.00 0.7200 56 312.00 0.9000 88 211.00 0.6020 120

681.00 4.230 26 492.00 0.7800 57 488.00 0.9600 89 321.00 0.6600 121

1000.0 4.210 27 675.00 0.8200 58 672.00 0.9900 90 491.00 0.7000 122

1470.0 4.240 28 1050.0 0.8600 59 1050.0 1.0200 91 674.00 0.7300 123

2150.0 4.290 29 1470.0 0.8900 60 1470.0 1.0500 92 1050.0 0.7700 124

3160.0 4.260 30 2190.0 0.9200 61 2010.0 1.0700 93 1470.0 0.7900 125

4360.0 4.350 31 3100.0 0.9500 62 3090.0 1.1000 94 2190.0 0.8100 126

4330.0 0.9700 63 4350.0 1.1400 95 3090.0 0.8500 127

4330.0 0.8700 128
Appendix II DRAWDOWN DATA PLOTS FOR WELL TEST AND DATA SELECTED FOR PARAMETER ESTIMATION 65

Table A—Continued. Data selected for parameter estimation, with PEST observation numbers; t, time, in minutes; 

h, drawdown, in feet; Obs. #, observation number.

F504-032 PEST

Obs. #

F504-060 PEST

Obs. #

F504-080 PEST

 Obs. #

F377-037 PEST

Obs. #

t (min) h(ft) t (min) h(ft) t (min) h(ft) t (min) h(ft)

0.0500 0.0050 129 0.0333 0.0050 160 0.0667 0.0030 193 0.0720 0.0030 225

0.0667 0.0080 130 0.0500 0.0090 161 0.0833 0.0110 194 0.1050 0.0060 226

0.1000 0.0250 131 0.0667 0.0230 162 0.1000 0.0210 195 0.1550 0.0120 227

0.1500 0.0520 132 0.1000 0.0720 163 0.1330 0.0460 196 0.2220 0.0200 228

0.2170 0.0740 133 0.1500 0.1310 164 0.1670 0.0750 197 0.3220 0.0260 229

0.3170 0.0860 134 0.2170 0.1590 165 0.2170 0.1060 198 0.4720 0.0300 230

0.4670 0.0930 135 0.3170 0.1740 166 0.3170 0.1330 199 0.6880 0.0340 231

0.6830 0.1000 136 0.4670 0.1820 167 0.4670 0.1480 200 1.0050 0.0370 232

1.0000 0.1060 137 0.6830 0.1890 168 0.6830 0.1570 201 1.4720 0.0410 233

1.4700 0.1150 138 1.0000 0.1940 169 1.0000 0.1630 202 2.2380 0.0470 234

2.1200 0.1250 139 1.4700 0.1990 170 1.4700 0.1680 203 3.2380 0.0520 235

3.1200 0.1410 140 2.1200 0.2070 171 2.1200 0.1740 204 4.7380 0.0620 236

4.6200 0.1580 141 3.1200 0.2180 172 3.1200 0.1810 205 6.7380 0.0720 237

6.7800 0.1790 142 4.6200 0.2280 173 4.6200 0.1890 206 10.072 0.0870 238

10.100 0.2040 143 6.7800 0.2390 174 6.7800 0.1980 207 15.905 0.1070 239

14.900 0.2310 144 10.100 0.2570 175 10.100 0.2080 208 21.905 0.1160 240

21.900 0.2560 145 14.900 0.2760 176 15.900 0.2240 209 31.905 0.1290 241

31.900 0.2810 146 21.900 0.2930 177 21.900 0.2320 210 46.905 0.1490 242

46.900 0.3130 147 31.900 0.3080 178 31.900 0.2420 211 67.905 0.1710 243

67.900 0.3470 148 46.900 0.3300 179 46.900 0.2580 212 100.91 0.2030 244

101.00 0.3880 149 67.900 0.3560 180 67.900 0.2740 213 150.91 0.2390 245

151.00 0.4290 150 101.00 0.3880 181 101.00 0.2980 214 220.91 0.2760 246

211.00 0.4620 151 151.00 0.4260 182 151.00 0.3280 215 351.90 0.3300 247

356.00 0.5500 152 211.00 0.4550 183 211.00 0.3530 216 500.90 0.3600 248

497.00 0.5700 153 301.00 0.4870 184 353.00 0.4200 217 687.90 0.3900 249

682.00 0.6000 154 354.00 0.5000 185 499.00 0.4400 218 1001.9 0.4300 250

1060.0 0.6500 155 498.00 0.5600 186 686.00 0.4800 219 1481.9 0.4600 251

1480.0 0.6800 156 684.00 0.5900 187 1000.0 0.5000 220 2202.9 0.4900 252

2200.0 0.7100 157 1000.0 0.6100 188 1480.0 0.5300 221 3103.9 0.5100 253

3100.0 0.7300 158 1600.0 0.6400 189 2200.0 0.5600 222 4336.9 0.5400 254

4330.0 0.7600 159 2200.0 0.6700 190 3100.0 0.5800 223

3100.0 0.6900 191 4330.0 0.6100 224

4340.0 0.7200 192
66 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

Table A—Continued. Data selected for parameter estimation, with PEST observation numbers; t, time, in minutes; 

h, drawdown, in feet; Obs. #, observation number.

F383-032 PEST

Obs. #

F383-061 PEST

Obs. #

F383-082 PEST

 Obs. #

F383-129 PEST

Obs. #

t (min) h(ft) t (min) h(ft) t (min) h(ft) t (min) h(ft)

3.3200 0.0500 255 0.3170 0.0300 273 4.7000 0.1000 297 8.2300 0.0800 315

10.000 0.0800 256 0.4330 0.0500 274 11.100 0.1300 298 18.500 0.0900 316

15.600 0.0800 257 0.7000 0.0500 275 16.800 0.1300 299 29.500 0.1000 317

27.000 0.1000 258 0.8670 0.0500 276 28.500 0.1400 300 36.000 0.1000 318

34.500 0.1100 259 1.5000 0.0700 277 35.300 0.1400 301 53.600 0.1100 319

51.800 0.1300 260 2.1700 0.0700 278 52.800 0.1500 302 64.900 0.1200 320

86.700 0.1600 261 5.7800 0.1000 279 64.100 0.1600 303 88.700 0.1400 321

119.00 0.1800 262 8.7300 0.1000 280 87.700 0.1800 304 151.00 0.1600 322

147.00 0.2000 263 14.500 0.1200 281 149.00 0.2100 305 228.00 0.1800 323

226.00 0.2400 264 22.900 0.1300 282 227.00 0.2400 306 333.00 0.2100 324

329.00 0.2800 265 33.700 0.1400 283 331.00 0.2700 307 501.00 0.2400 325

506.00 0.2900 266 51.100 0.1500 284 505.00 0.3100 308 703.00 0.2700 326

706.00 0.3600 267 85.600 0.1800 285 708.00 0.3300 309 1010.0 0.3000 327

1000.0 0.3900 268 118.00 0.1900 286 1000.0 0.3600 310 1490.0 0.3200 328

1490.0 0.4200 269 146.00 0.2100 287 1490.0 0.4000 311 2390.0 0.3600 329

2210.0 0.4600 270 224.00 0.2500 288 2210.0 0.4200 312 3300.0 0.3900 330

3120.0 0.4800 271 328.00 0.2900 289 3300.0 0.4600 313 4350.0 0.4200 331

4350.0 0.5200 272 507.00 0.3300 290 4350.0 0.4900 314

773.00 0.3700 291

1010.0 0.3900 292

1490.0 0.4300 293

2210.0 0.4500 294

3120.0 0.4800 295

4350.0 0.5100 296
Appendix II DRAWDOWN DATA PLOTS FOR WELL TEST AND DATA SELECTED FOR PARAMETER ESTIMATION 67

Table A—Continued. Data selected for parameter estimation, with PEST observation numbers; 

t, time, in minutes; h, drawdown, in feet; Obs. #, observation number.

F384-033 PEST

Obs. #

F381-056 PEST

Obs. #

F347-031 PEST

Obs. #

t (min) h (ft) t (min) h (ft) t (min) h (ft)

0.2330 0.0100 332 1.0800 0.0300 355 6.9000 0.0100 375

0.9000 0.0200 333 2.3200 0.0200 356 10.2000 0.0100 376

1.6200 0.0200 334 3.2200 0.0300 357 15.4000 0.0200 377

2.3000 0.0300 335 4.7300 0.0300 358 22.0000 0.0200 378

3.5700 0.0300 336 6.7300 0.0400 359 32.6000 0.0200 379

4.9000 0.0400 337 10.9000 0.0400 360 50.3000 0.0300 380

6.5200 0.0400 338 15.9000 0.0400 361 70.6000 0.0300 381

10.3000 0.0400 339 22.9000 0.0500 362 106.0000 0.0500 382

15.2000 0.0500 340 32.9000 0.0600 363 157.0000 0.0600 383

22.6000 0.0500 341 47.9000 0.0700 364 227.0000 0.0800 384

32.1000 0.0600 342 70.9000 0.0800 365 324.0000 0.1000 385

46.3000 0.0700 343 101.0000 0.0900 366 475.0000 0.1300 386

68.8000 0.0800 344 151.0000 0.1200 367 694.0000 0.1500 387

104.0000 0.1000 345 231.0000 0.1600 368 1010.0000 0.1900 388

154.0000 0.1200 346 332.0000 0.1800 369 1490.0000 0.2200 389

223.0000 0.1500 346 483.0000 0.2000 370 2210.0000 0.2500 390

321.0000 0.2000 348 699.0000 0.2300 371 3290.0000 0.2800 391

474.0000 0.2400 349 1010.0000 0.2700 372 4340.0000 0.3000 392

695.0000 0.2600 350 2220.0000 0.3200 373

1010.0000 0.3000 351 3850.0000 0.3700 374

2210.0000 0.3600 352

3300.0000 0.4000 353

4340.0000 0.4200 354
68 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts

Table A—Continued. Data selected for parameter estimation, with PEST observation numbers; t, time, in minutes; 

h, drawdown, in feet; Obs. #, observation number.

F434-060 PEST

Obs. #

F450-061 PEST

Obs. #

F476-061 PEST

 Obs. #

F478-061 PEST

Obs. #

t (min) h(ft) t (min) h(ft) t (min) h(ft) t (min) h(ft)

1.97 0.1500 393 2.88 0.0900 408 3.62 0.0900 424 4.37 .0400 440

9.92 0.2400 394 13.2 0.1400 409 10.6 0.1300 425 11.8 .0700 441

20.00 0.2900 395 22.4 0.1700 410 20.7 0.1700 426 21.5 .0900 442

29.00 0.3200 396 33.3 0.1900 411 30.0 0.1900 427 30.8 .0900 443

54.90 0.3700 397 64.2 0.2300 412 55.7 0.2200 428 57.8 .1300 444

100.00 0.4400 398 96.0 0.2700 413 97.9 0.2700 429 96.9 .1600 445

149.00 0.4800 399 152.0 0.3000 414 150.0 0.3100 430 151.0 .1900 446

240.00 0.5200 400 243.0 0.3500 415 242.0 0.3500 431 243.0 .2300 447

337.00 0.5900 401 350.0 0.4100 416 343.0 0.4400 432 346.0 .2800 448

545.00 0.6500 402 496.0 0.4400 417 494.0 0.4400 433 495.0 .3000 449

765.00 0.6900 403 681.0 0.4600 418 678.0 0.4800 434 680.0 .3300 450

1150.00 0.7100 404 1060.0 0.5100 419 1060.0 0.5100 435 1060.0 .3800 451

2200.00 0.7600 405 1480.0 0.5300 420 1470.0 0.5300 436 1480.0 .3900 452

3280.00 0.7900 406 2200.0 0.5600 421 2200.0 0.5600 437 2200.0 .4200 453

4330.00 0.8200 407 3100.0 0.5800 422 3280.0 0.5900 438 3280.0 .4500 454

4330.0 0.6100 423 4330.0 0.6200 439 4330.0 .4700 455
Appendix II DRAWDOWN DATA PLOTS FOR WELL TEST AND DATA SELECTED FOR PARAMETER ESTIMATION 69

Table A—Continued. Data selected for parameter estimation, 

with PEST observation numbers; t, time, in minutes; 

h, drawdown, in feet; Obs. #, observation number.

F385-032 PEST

Obs. #

F376-037 PEST

Obs. #

t (min) h(ft) t (min) h(ft)

10.400 0.0200 456 1.9000 0.0100 473

15.700 0.0200 457 3.3000 0.0200 474

22.400 0.0300 458 5.5700 0.0200 475

32.400 0.0300 459 8.5500 0.0200 476

51.300 0.0300 460 12.300 0.0200 477

61.200 0.0300 461 19.000 0.0300 478

101.00 0.0400 462 23.600 0.0300 479

151.00 0.0600 463 32.200 0.0400 480

221.00 0.0800 464 45.700 0.0400 481

326.00 0.1100 465 68.100 0.0400 482

477.00 0.1300 466 111.00 0.0500 483

692.00 0.1500 467 156.00 0.0600 484

1020.0 0.1700 468 231.00 0.0800 485

1480.0 0.2100 469 327.00 0.1300 486

2210.0 0.2300 470 478.00 0.1500 487

3290.0 0.2600 471 690.00 0.1600 488

4340.0 0.2800 472 1020.0 0.1800 489

1480.0 0.2000 490

3290.0 0.2500 491

4340.0 0.2700 492
70 Estimation of Hydraulic Parameters from an Unconfined Aquifer Test Conducted in a Glacial Outwash Deposit, 

Cape Cod, Massachuesetts
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