U.S. Department of the Interior

U.S. Geological Survey

WTAQ—A Computer Program for 

Calculating Drawdowns and Estimating 

Hydraulic Properties for Confined and 

Water-Table Aquifers

By PAUL M. BARLOW and ALLEN F. MOENCH 

Water-Resources Investigations Report 99-4225

A product of the Ground-Water Resources Program

Northborough, Massachusetts

1999 
U.S. DEPARTMENT OF THE INTERIOR

BRUCE BABBITT, Secretary

U.S. GEOLOGICAL SURVEY

Charles G. Groat, Director

The use of trade or product names in this report is for identification 

purposes only and does not constitute endorsement by the 

U.S. Geological Survey.

For additional information write to: Copies of this report can be purchased 

Chief, Massachusetts-Rhode Island District U.S. Geological Survey

U.S. Geological Survey Information Services

Water Resources Division Box 25286

10 Bearfoot Road Denver, CO 80225-0286

Northborough, MA 01532
PREFACE

This report presents a computer program for calculating drawdowns and estimating 

hydraulic properties for confined and water-table aquifers. The program has been successfully 

tested for a variety of applications. Users are requested to notify the USGS if errors are found in 

this report or the program.

Although this program has been used by the USGS, no warranty, expressed or implied, is 

made by the USGS or the United States Government as to the accuracy and functioning of the 

program and related materials. Nor shall the fact of distribution constitute any such warranty, 

and no responsibility is assumed by the USGS in connection therewith.

Con

CONTENTS

Abstract .............................................................................................................................................................................

Introduction .......................................................................................................................................................................

Axial-Symmetric Flow in Confined and Water-Table Aquifers ........................................................................................

Program WTAQ..................................................................................................................................................................

Simplifying Assumptions ........................................................................................................................................

Description of Selected Program Options and Variables ........................................................................................

Data-Input Files.......................................................................................................................................................

Type-Curve Format .......................................................................................................................................

Dimensional Format......................................................................................................................................

Result and Plot Files................................................................................................................................................

Sample Problems...............................................................................................................................................................

Sample Problem 1—Estimation of Horizontal and Vertical Hydraulic Conductivity and 

Specific Yield Using Type-Curve Format..................................................................................................

Sample Problem 2—Estimation of Specific Storage Using Dimensional Format ..................................................

Sample Problem 3—Automatic Parameter Estimation Using Dimensional Format...............................................

Summary ...........................................................................................................................................................................

References .........................................................................................................................................................................

Attachment 1: Narrative for Program WTAQ ....................................................................................................................

Attachment 2: Sample Input Files for use of WTAQ with UCODE ..................................................................................

FIGURES

1–3. Schematic sections showing:

1. One-dimensional, axial-symmetric flow in a confined aquifer with a fully penetrating

pumped well ..............................................................................................................................................

2. Two-dimensional, axial-symmetric flow in a confined aquifer with a partially penetrating

pumped well ..............................................................................................................................................

3. Two-dimensional, axial-symmetric flow in a water-table aquifer with a partially penetrating

pumped well ..............................................................................................................................................

4. Graph showing drawdown at an observation well 3.2 meters from a fully penetrating pumped

well in hypothetical confined and water-table aquifers .....................................................................................

5. Schematic section showing finite-diameter, partially penetrating pumped well in a water-table aquifer .........

6, 7. Representations showing:

6. Example data-input file using type-curve format......................................................................................

7. Example data-input file using dimensional format ...................................................................................

8. Schematic section showing hypothetical, homogeneous water-table aquifer with a partially penetrating 

pumped well of finite diameter and the locations of the open interval of four observation piezometers..........

9. Representation showing data-input file using type-curve format for sample problem 1...................................

10–12. Graphs showing:

10. Composite plots of measured drawdown and calculated dimensionless drawdown for (A) KD=0.1 and 

(B) KD=0.5 for the hypothetical water-table aquifer.................................................................................

11. Drawdown versus time for the hypothetical water-table aquifer...............................................................

12. Drawdown versus time at the four observation piezometers for the hypothetical water-table aquifer. ....

13–16. Representations showing:

13. Example result file using type-curve format .............................................................................................

14. Example plot file using type-curve format................................................................................................

15. Example result file using dimensional format...........................................................................................

16. Example plot file using dimensional format .............................................................................................
VI Contents

TABLES

1, 2. Input data for:

1. Type-curve format ......................................................................................................................................

2. Dimensional format ....................................................................................................................................

3. Parameters of the hypothetical water-table aquifer and pumped well ................................................................

4. Locations of observation piezometers in hypothetical water-table aquifer ........................................................

5. Drawdown versus time for the pumped well and four observation piezometers in the hypothetical 

water-table aquifer ..............................................................................................................................................

6. Hydraulic properties of the hypothetical water-table aquifer determined using parameter estimation ..............

CONVERSION FACTORS AND DEFINITION OF SYMBOLS

CONVERSION FACTORS

DEFINITION OF SYMBOLS

[L, length; T, time; --, dimensionless]

Multiply By To Obtain

cubic meter per second (m3/s) 35.31 cubic foot per second

meter (m) 3.281 foot

meter per second (m/s) 3.281 foot per second

square meter per second (m2/s) 10.76 square foot per second

Symbol Dimension Definition

L Thickness of confined aquifer or saturated thickness of watertable aquifer

L Thickness of the well-bore skin

L Modified Hvorslev (1951) observation well shape factor

L Head in aquifer

L Model calculated drawdown

-- Dimensionless drawdown

L Initial head (or potentiometric surface) in aquifer

L Measured drawdown

-- Dimensionless ratio of vertical to horizontal hydraulic 

conductivity

L/T Hydraulic conductivity of well-bore skin

L/T Horizontal and vertical hydraulic conductivity of aquifer, 

respectively

L Length of the screened interval of observation well

L3/T Pumping rate of well

L Radial distance from axis of pumped well

L Inside radius of the pumped well in the interval where water 

levels are changing during pumping

b

ds

F′

h

hc

hD

hi

hm

KD

Ks

Kr, Kz

L

Q

r

rc
Cont

-- Dimensionless radial distance to observation well or 

piezometer from axis of pumped well

L Inside radius of the observation well in the interval where 

water levels are changing during pumping

L Radius of the screened interval of the pumped well

-- Storativity (storage coefficient) of aquifer

1/L Specific storage of aquifer

-- Well-bore skin

-- Specific yield of aquifer

T Time

-- Dimensionless time

-- Dimensionless time with respect to specific yield

L2/T Transmissivity of aquifer

-- Dimensionless well-bore storage

-- Dimensionless delayed response factor

L Depth below top of aquifer or initial water table

L Depth below top of aquifer or initial water table to the top of 

screened interval of observation well

L Depth below top of aquifer or initial water table to the bottom 

of screened interval of observation well

L Depth below top of aquifer or initial water table to center of 

piezometer

L Depth below top of aquifer or initial water table to the top of 

screened interval of pumped well

L Depth below top of aquifer or initial water table to the bottom 

of screened interval of pumped well

1/T Ith empirical drainage constant

-- Dimensionless product of and the square of dimensionless 

radial distance to observation well or piezometer

--

Dimensionless product of anisotropic ratio of vertical to 

horizontal hydraulic conductivity and the square of 

dimensionless radius of screened interval of pumped well

-- Ith dimensionless empirical drainage constant

-- Pi (3.141592654)

-- Dimensionless ratio of aquifer storativity to aquifer specific 

yield

-- Infinity

Symbol Dimension Definition

rD

rp

rw

S

Ss

Sw

Sy

t

t

D

tDy

T

WD

W'D

z

z1

z2

zp

zpd

zpl

αi

β βw

βw

γ i

π

σ

∞

Introd

WTAQ—A Computer Program for 

Calculating Drawdowns and Estimating 

Hydraulic Properties for Confined and 

Water-Table Aquifers

By Paul M. Barlow and Allen F. Moench

Abstract

The computer program WTAQ calculates hydraulic-head drawdowns in a confined or water-

aquifer that result from pumping at a well of finite or infinitesimal diameter. The program is based 

analytical model of axial-symmetric ground-water flow in a homogeneous and anisotropic aquifer

program allows for well-bore storage and well-bore skin at the pumped well and for delayed draw

response at an observation well; by including these factors, it is possible to accurately evaluate the 

specific storage of a water-table aquifer from early-time drawdown data in observation wells and 

piezometers. For water-table aquifers, the program allows for either delayed or instantaneous drain

from the unsaturated zone. WTAQ calculates dimensionless or dimensional theoretical drawdowns 

can be used with measured drawdowns at observation points to estimate the hydraulic properties o

confined and water-table aquifers. Three sample problems illustrate use of WTAQ for estimating 

horizontal and vertical hydraulic conductivity, specific storage, and specific yield of a water-table a

by type-curve methods and by an automatic parameter-estimation method.

INTRODUCTION

Determination of aquifer hydraulic properties is a basic component of most ground-water supply and 

contaminant-transport investigations. A frequently used method for estimating hydraulic properties is graph

type-curve analysis of aquifer tests, in which dimensionless type curves derived from an assumed analytica

of ground-water flow to a pumped well are used to analyze time-drawdown measurements of hydraulic hea

observation wells and piezometers. These analyses are done to estimate the transmissivity and storativity o

confined aquifers or the hydraulic conductivity and specific yield of water-table (unconfined) aquifers. An 

alternative approach to dimensionless type-curve analysis is to generate dimensional time-drawdown curve

the analytical model that are compared directly to the measured values. In this approach, the hydraulic prope

the model are adjusted in a series of model simulations until the model-calculated drawdowns closely matc

measured values. This procedure is called model calibration and can be done graphically, as in the dimensi

type-curve approach, or automatically by use of a parameter-estimation technique.
2 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aquife

Many analytical models have been developed for evaluation of axial-symmetric flow to a well that pu

from a confined or water-table aquifer. The ability of these models to represent realistic field conditions suc

well-bore storage and skin, partial penetration of wells, and, in the case of a water-table aquifer, drainage fr

unsaturated zone, has steadily improved since the pioneering work of Theis (1935), who presented a transie

analytical model of flow to a fully penetrating well in a confined aquifer. Recently, Moench (1997) develop

analytical model of flow to a partially penetrating, finite-diameter well in a homogeneous, anisotropic wate

aquifer. The model accounts not only for well-bore storage and skin at the pumped well, but also for delaye

drawdown response of an observation well. By including these factors, it is possible to accurately evaluate 

specific storage of a water-table aquifer from early-time drawdown data in observation wells and piezomete

also theoretically possible to use the model to interpret pumped-well data. For confined aquifers, the mode

expands upon the work of Dougherty and Babu (1984) and allows for anisotropic hydraulic conductivity. F

unconfined aquifers, the model expands upon the early work of Boulton (1954, 1963) and of Neuman (1972

and allows for well-bore storage and skin, delayed piezometer response, and delayed drainage from the unsa

zone.

This report describes and provides documentation for the computer program WTAQ, which is based o

Moench’s (1997) analytical model for axial-symmetric flow in a confined or water-table aquifer. WTAQ cal

dimensionless or dimensional drawdowns that can be used with measured drawdowns at observation points 

estimate hydraulic properties of confined and water-table aquifers. Three sample problems illustrate use of 

for estimating horizontal and vertical hydraulic conductivity, specific storage, and specific yield of a water-

aquifer by graphical, type-curve methods and by an automatic parameter-estimation method. The report als

includes instructions for constructing the necessary data-input files for program execution.

The authors thank Thomas Reilly (U.S. Geological Survey) for providing the results of a numerical 

simulation of the hypothetical aquifer described in the sample problems.

AXIAL-SYMMETRIC FLOW IN CONFINED AND WATER-TABLE AQUIFERS

When a well is pumped at a constant rate , ground-water levels around the well decline, and a hyd

gradient is established that causes ground water to flow toward the well. These water-level declines are refe

as drawdowns, and are largest in the immediate vicinity of the well (fig. 1). When the aquifer from which t

pumps is homogeneous, of infinite lateral extent, horizontal, and of uniform thickness, drawdowns will be 

symmetric about the axis of the well, and the resulting flow toward the well will have axial symmetry. Conce

and mathematically, the simplest type of axial-symmetric flow occurs in an aquifer that is overlain and unde

impermeable geologic materials that prevent vertical flow into or out of the aquifer (confined aquifer). If th

pumped well fully penetrates the thickness of a horizontal confined aquifer, then the flow caused by pu

will be horizontal (that is, one-dimensional, or radial), and drawdown in the aquifer will be a function of th

distance from the axis of the pumped well and time (fig. 1). If, however, the pumped well only par

penetrates the thickness of the aquifer, flow will consist of both horizontal and vertical components (that is

dimensional flow), and drawdown will be a function of the vertical position in the aquifer as well as th

distance from the pumped well and time (fig. 2).

Theis (1935) developed an analytical model for one-dimensional, radial flow in a confined aquifer in 

the hydraulic conductivity of the aquifer is assumed to be isotropic (that is, , where and a

horizontal and vertical hydraulic conductivity of the aquifer, respectively). His solution provides a means t

determine the transmissivity and storativity (or storage coefficient) of the aquifer, w

is specific storage. Later, Hantush (1961) developed an analytical model for two-dimensional, axial-symme

flow in confined aquifers that accounts for the effects of partial penetration of the pumped well and anisotr

the aquifer material .

Q

( ) b

( )r ( )t

( )z

Kr = Kz Kr Kz

T Kr ( ) = b S Ss ( ) = b 

Kr Kz ( ) ≠
Axial-Symmetric Flow in Confined and Water-Table Aq

EXPLANATION

Pumped well Observation

well

POTENTIOMETRIC

SURFACE IN

AQUIFERÐDashed

portion indicates

initial potentiometric

surface

SCREENED INTERVAL

OF WELL

hi

h(r,t)

r

b

Q

Drawdown

AQUIFER

CONFINING LAYER

IMPERMEABLE

(NO-FLOW)

BOUNDARY

DIRECTION OF

GROUND-WATER

FLOW

hi

h(r,z,t)

r

z

z

zpl

pd 

zp

Q

Drawdown

Pumped well Obse

Observation w

piezometer

EXPLANATION

AQUIFER

CONFINING LAYER

IMPERMEABLE

(NO-FLOW)

BOUNDARY

DIRECTION OF

GROUND-WATER

FLOW

POTENTIOME

SURFACE IN

AQUIFER–Da

portion indica

initial potentio

surface

SCREENED I

OF WELL

Two-dimensional, axial-symmetric flow in a water-table aquifer is more complex than that in a confi

aquifer because the water table, which is the upper boundary of the aquifer, is free to move in response to p

(fig. 3). When the water table is lowered, the saturated thickness of the aquifer is reduced; this reduction in 

thickness contrasts with the response of a confined aquifer in which the thickness of the aquifer remains co

during pumping. Furthermore, as the water table is lowered by pumping, water drains from pores at the wat

and from the overlying unsaturated zone and flows vertically into the aquifer. The specific yield ( ) of a w

table aquifer is defined as the volume of pore water that is released from storage per unit surface area of aqu

unit decline in the water table (Freeze and Cherry, 1979). Specific yields of water-table aquifers are much l

than the storativities of confined aquifers because specific yields reflect an actual drainage of the pores in th

above the water table. In a confined aquifer, on the other hand, water is released from storage only by comp

of the aquifer matrix and expansion of the water.

A comparison between drawdowns at observation wells in hypothetical confined and water-table aqu

illustrated in figure 4 for the condition of a fully penetrating pumped well. The aquifers are equivalent in al

respects with the exception of the upper boundary condition of each aquifer. Because most of the water pum

from the water-table aquifer is supplied by dewatering of pores, drawdown at the observation well in the w

Sy

Figure 1. One-dimensional, axial-symmetric flow in a 

confined aquifer with a fully penetrating pumped well. 

[b, aquifer thickness; h(r,t), head in aquifer at radial distance 

r from axis of pumped well at time t ; hi, initial head in 

aquifer; Q, pumping rate of well.]

Figure 2. Two-dimensional, axial-symmetric flow in 

confined aquifer with a partially penetrating pumped 

[b, aquifer thickness; h(r,z,t), head in aquifer at radi

distance r from axis of pumped well and depth z b

of aquifer at time t ; hi, initial head in aquifer; Q, pum

rate of well; zpd, zpl, depth below top of aquifer to the 

bottom, respectively, of the screened interval of the 

well; z1, z2, depth below top of aquifer to the top and 

respectively, of the screened interval of the observa

zp, depth below top of aquifer to center of piezomet
4 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aquife

hi

h(r,z,t)

b

z

zpl

pd z

1

2

z

zp

=

Q

Drawdown

Pumped well

Observation 

well Observation 

piezometer

Unsaturated zone

EXPLANATION

WATER TABLE–Dashed

portion indicates

initial water table

SCREENED INTERVAL 

OF WELL

r z

AQUIFER

IMPERMEABLE

(NO-FLOW)

BOUNDARY

DIRECTION OF

GROUND-WATER

FLOW

table aquifer is always less and is slower than drawdown in the confined aquifer. A characteristic feature of 

drawdown curve that is frequently seen in log-log plots for the water-table aquifer is the inflection in the cu

during the intermediate time (between about 1 to 3,000 seconds after the start of pumping in fig. 4). This in

is caused by vertical drainage from the zone above the water table, which slows the drawdown response in 

aquifer. Following the inflection there is an apparent (due to log-scale compression of time) increase in the 

drawdown as the drainage from the zone above the water table keeps pace with the decline in the free surfa

“...the time-drawdown curve merges with the Theis non-equilibrium curve associated with the coefficient of 

for water-table conditions” (that is, specific yield) (Boulton, 1963, p. 470).

Two conceptual approaches have been used to analytically model the release of water from the zone a

the water table. In the first approach, developed primarily by Boulton (1954, 1963), drainage from the zone 

the water table is assumed to occur gradually in a manner that varies exponentially with time in response to 

decline in the elevation of the water table. As part of his mathematical treatment of the problem, Boulton intr

an empirical drainage constant. In the second approach, developed by Neuman (1972, 1974), water is assum

released instantaneously from the zone above the water table in response to lowering of the water table. Ne

justified this assumption, in part, based upon numerical models that appeared to show that effects of flow fr

unsaturated zone upon drawdown in the saturated zone were negligible.

Figure 3. Two-dimensional, axial-symmetric flow in a watertable aquifer with a partially penetrating pumped well.

[b, initial saturated thickness of aquifer; h(r,z,t), head in 

aquifer at radial distance r from axis of pumped well and 

depth z below initial water table at time t ; hi, initial head in 

aquifer; Q, pumping rate of well; zpd, zpl, depth below initial 

water table to the top and bottom, respectively, of the 

screened interval of the pumped well; z1, z2, depth below 

initial water table to the top and bottom, respectively, of the 

screened interval of the observation well; zp, depth below 

initial water table to center of piezometer.]
Axial-Symmetric Flow in Confined and Water-Table Aq

Moench (1995) developed an analytical model of axial-symmetric flow in water-table aquifers that co

the Boulton and Neuman approaches. His model allows for either gradual (delayed) or instantaneous releas

water from the zone above the water table, and also accounts for vertical flow in the aquifer. Moench show

inclusion of the delayed drainage process in the analytical model improved the match between theoretically 

drawdowns and those measured in shallow, water-table observation piezometers. He also infers that the use 

multiple empirical drainage constants, as opposed to a single constant, may better represent the delayed dra

process, but did not explicitly incorporate such constants into his model.

Most of the early analytical models of axial-symmetric flow in confined and water-table aquifers wer

developed with the assumption that the pumped well is infinitesimal in diameter, such as shown schematica

figures 1–3. This assumption is made to simplify the mathematics, but it is unrealistic because it does not al

water stored in the well bore (fig. 5). Also unrealistic is the lack of accounting for the resistance to flow, or 

effect,” that can occur at the interface between the aquifer and well bore. This resistance is caused by restric

the well screen, bridging by sand particles across screen openings, and damage to the aquifer caused by dril

a result of these factors, a skin of material of lower hydraulic conductivity than that of the aquifer can form 

the well-bore casing (fig. 5). Well-bore storage and well-bore skin at the pumped well can affect drawdown

aquifer and at the pumped well itself. In particular, it has been shown (Narasimhan and Zhu, 1993; Moench

that such effects can mask early-time drawdown data that are required for accurate determination of the spe

storage of a water-table aquifer. Well-bore storage effects were included in analytical models developed by 

Papadopulos and Cooper (1967) and Papadopulos (1967); well-bore storage and skin effects combined hav

considered by Agarwal and others (1970), Dougherty and Babu (1984), and Moench (1997).

Additionally, most analytical models have been developed under the assumption that observation we

piezometers are infinitesimal in diameter, are in good hydraulic connection with the aquifer material, and r

instantaneously to aquifer head changes. Because of stored water in the observation well or piezometer or b

of a poor hydraulic connection with the aquifer material, however, a time delay often is observed between t

drawdown in the aquifer and that in the observation installation (Hvorslev, 1951). The magnitude of this de

function of the geometry of the observation well or piezometer, the permeability and specific storage of the 

and the integrity of the hydraulic connection between the observation installation and aquifer. Delayed draw

response in an observation well or piezometer can be important during the early period of an aquifer test in 

0.01 0.1 1 10 100 1,000 10,000 100,000

0.01

2

0.02

0.03

0.04

0.05

0.07

0.1

0.2

0.3

0.4

0.5

0.7

1

TIME SINCE START OF PUMPING, IN SECONDS

DRAWDOWN, IN METERS

RESPONSE IN CONFINED

AQUIFER

RESPONSE IN WATER-TABLE

AQUIFER

Figure 4. Drawdown at an observation well 3.2 meters from a 

fully penetrating pumped well in hypothetical confined and watertable aquifers.
6 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aquife

ds

hi b

z

1

2

z

z pd

=

r

r

z r pl

p

w

c

Q

h(r,z,t)

Drawdown

Pumped well Observation 

well

Unsaturated zone

EXPLANATION

AQUIFER

WELL-BORE SKIN

IMPERMEABLE

(NO-FLOW)

BOUNDARY

WATER TABLE–Dashed

portion indicates

initial water table

SCREENED INTERVAL 

OF WELL

r z

heads are changing rapidly, and, in addition to well-bore storage and skin at the pumped well, is an importan

to consider in the accurate evaluation of specific storage. Moench (1997) has shown that the effects of this d

response are most important in observation wells closest to the pumped well where drawdown is most rapi

PROGRAM WTAQ

Program WTAQ implements the Laplace-transform solutions for drawdown at a pumped well, observ

well, or observation piezometer that are presented in equations 18–27 and 30–31 in Moench (1997). The L

transform solutions are numerically inverted to the time domain by means of the Stehfest (1970) algorithm 

same manner as was done in Moench (1996, 1997). Some improvements were made to the numerical inver

algorithm as a result of the work of Gieske (1999). The program calculates dimensionless or dimensional 

drawdowns for a given set of input conditions that are specified by the user in a data-input file. Instructions 

Figure 5. Finite-diameter, partially penetrating pumped 

well in a water-table aquifer. [b, initial saturated thickness 

of aquifer; ds, thickness of the well-bore skin; h(r,z,t), head 

in aquifer at radial distance r from axis of pumped well and 

depth z below initial water table at time t ; hi, initial head in 

aquifer; Q, pumping rate of well; rc, rp, inside radius of the 

pumped well and observation well, respectively, in the 

interval where water levels are changing during pumping; 

rw, radius of the screened interval of the pumped well; 

zpd, zpl, depth below initial water table to the top and 

bottom, respectively, of the screened interval of the 

pumped well; z1, z2, depth below initial water table to the 

top and bottom, respectively, of the screened interval of the 

observation well.]
Program 

preparing the data-input files are provided in this section, as are a summary of the simplifying assumptions 

the program, a description of selected program options and variables, and a description of the result and plo

generated by the program. A narrative description of the computer code is provided in Attachment 1.

Simplifying Assumptions

Several simplifying assumptions were required in the development of the analytical model on which p

WTAQ is based. Discussions pertaining to the adequacy and limitations of these assumptions can be found 

Boulton (1954, 1963), Neuman (1972, 1974), Black and Kipp (1977), Dougherty and Babu (1984), and Mo

(1995, 1997). The assumptions are as follows:

Assumptions for both confined and water-table aquifers (no well-bore storage or skin effects)—

1. The aquifer is homogeneous, of infinite lateral extent, horizontal, and of uniform thickness.

2. The aquifer can be anisotropic provided that the principal directions of the hydraulic-conductivity ten

parallel to the coordinate axes.

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

impermeable).

4. A well discharges at a constant rate from a specified zone below an initially horizontal potentiometric 

(for a confined aquifer) or water table.

5. The porous medium and fluid are slightly compressible and have constant physical properties.

6. The pumped well and observation wells or piezometers are infinitesimal in diameter.

Additional assumption for confined aquifers—

7. Vertical flow across the upper boundary of the aquifer is negligible (that is, the upper boundary is 

impermeable).

Additional assumptions for water-table aquifers—

8. The upper boundary of the aquifer is the water table. Vertical flow of water into the aquifer from the 

unsaturated zone can be instantaneous in response to a decline in the elevation of the water table (N

1972, 1974), or can vary exponentially with time in response to a step decline in the elevation of th

table (Boulton, 1954, 1963; Moench, 1995).

9. The change in saturated thickness of the aquifer due to pumping is small compared with the initial sat

thickness.

Additional assumptions for the condition of a finite-diameter pumped well (Dougherty and Babu, 1984)—

10. The head within the pumped well does not vary spatially.

11. The radial flow from the aquifer to the pumped well does not vary in magnitude along the length of th

screened section.

12. Vertical flow from the aquifer through the base of the pumped well is negligible.

13. A thin skin of low-permeability material having no storage capacity may be present at the interface be

the pumped well screen and the aquifer.

Additional assumptions for the condition of delayed drawdown response in a finite-diameter observation w

(Black and Kipp, 1977)—

14. The hydraulic head in the observation well changes at a rate that is proportional to the head difference 

between the observation well and the adjacent aquifer.

r z,
8 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aquife

Description of Selected Program Options and Variables

Several of the program options and variables that are used in WTAQ are described below. Program va

are shown in upper-case text.

Analysis formats.--Two analysis formats are provided in the program: type curve and dimensional. S

input files are required for each of the analysis formats. The character variable FORMAT is used to specify 

the TYPE CURVE or DIMENSIONAL format. Type curves are the theoretical drawdown responses that re

from pumping at a constant rate. For the type-curve format, WTAQ generates dimensionless drawdowns at 

observation wells and/or piezometers (but not at the pumped well) that can be used for graphical matching 

measured time-drawdown data. Procedures for estimating aquifer hydraulic properties by type-curve match

be found in many journal articles and texts on ground-water hydrology and aquifer-test analysis, such as th

Freeze and Cherry (1979), Marsily (1986), Driscoll (1986), Walton (1970, 1987), and Kruseman and de Ri

(1991).

For the dimensional format, WTAQ generates dimensional drawdowns for the pumped well or observ

wells and/or piezometers. Using the dimensional format, the user typically adjusts the hydraulic parameter

analytical model in a series of simulations until drawdowns calculated by WTAQ closely match those meas

during the aquifer test. This approach for estimating aquifer hydraulic properties by calibration of the analy

model often is referred to as the trial-and-error approach. The dimensional format also can be used with aut

parameter-estimation techniques for model calibration.

Example applications of these two formats are described in the sample problems.

Aquifer type and hydraulic properties.--WTAQ simulates either confined or water-table aquifer typ

characteristics that differentiate the two aquifer types are the condition of the upper boundary and the thick

the aquifer. For confined aquifers, the upper boundary is impermeable, and there is no vertical flow into the 

the thickness of the aquifer remains constant during pumping. For water-table aquifers, the upper boundary 

water table, water flows vertically into the aquifer from the overlying unsaturated zone in response to a dec

the elevation of the water table caused by pumping, and the saturated thickness of the aquifer decreases as 

elevation of the water table falls.

Aquifer type is specified by use of character variable AQTYPE, which is either CONFINED or WAT

TABLE. The thickness of the aquifer, or, for water-table aquifers, the saturated thickness of the aquifer at t

beginning of the simulation, is specified by use of variable BB. For the type-curve format, the user specifie

ratio of the vertical to horizontal hydraulic conductivities using variable XKD. For the dimensional format, b

horizontal (variable HKR) and vertical (variable HKZ) hydraulic conductivity must be specified. Transmiss

the aquifer is calculated by the program for the dimensional format and written to the result file.

Aquifer storage properties are specified by use of variables SIGMA (the ratio of aquifer storativity to 

yield, ), SS (specific storage), and/or SY (specific yield). SIGMA is used for the type-curve format. It is s

to zero for a confined aquifer and to for a water-table aquifer. Variables SS and SY are used for th

dimensional format. SS is specified for both confined and water-table aquifers; SY is specified for water-ta

aquifers only. Aquifer storativity is calculated by the program (variable SS multiplied by variable BB) for t

dimensional format and written to the result file.

σ

S

s

b Sy ⁄
Program 

Drainage at water table.--For water-table aquifers, WTAQ simulates either instantaneous or delayed 

drainage into the aquifer from the unsaturated zone. The type of drainage is specified by variable IDRA, w

set to 0 for instantaneous drainage and to 1 for delayed drainage. (For a confined aquifer, IDRA = 0). For d

drainage, up to five empirical drainage constants can be included. The number of drainage constants is spec

variable NGAMMA for the type-curve format and by variable NALPHA for the dimensional format. In Mo

(1995, 1996, 1997), only a single value of NGAMMA or NALPHA is indicated. Multiple empirical draina

constants have been used in an analysis of aquifer-test data in a sand and gravel water-table aquifer, Cape C

Massachusetts (A.F. Moench, U.S. Geological Survey, written commun., 1999). The analysis shows that thr

constants are adequate to account for deviations between measured drawdown and drawdown simulated by 

under the assumption of instantaneous drainage.

Dimensionless drainage constants are specified for the type-curve format using the array GAMMA. T

constants are calculated by the user independently of the program from

 ,

where

is the ith dimensionless empirical drainage constant;

is the ith empirical drainage constant (units of inverse time);

is the initial saturated thickness of the aquifer (units of length);

is the specific yield of the aquifer (dimensionless); and

is the vertical hydraulic conductivity of the aquifer (units of length per time).

For the dimensional format, the empirical drainage constants are specified using the array ALPHA; W

calculates the dimensionless values of GAMMA internally. For instantaneous drainage at the water table 

(IDRA = 0), the user enters a single value of 1.0D09 (an essentially infinite value) for either GAMMA(1) o

ALPHA(1), depending on the analysis format that is being used. (It is recommended that instantaneous drai

assumed for initial estimates of aquifer parameters.)

Time information.--For the type-curve format, dimensionless drawdowns are calculated for a series 

equally-spaced, dimensionless, and logarithmically-scaled times. These times are calculated from values sp

for variables TDLAST, NLC, and NOX. Variable TDLAST is the largest value of dimensionless time that i

desired; variable NLC is the total number of logarithmic cycles on the time scale for which drawdown will 

calculated; and variable NOX is the number of equally spaced times per logarithmic cycle for which drawd

will be calculated. The total number of time steps for which drawdown will be calculated is equal to 

, and the length of each time step equals .

Two options are provided for specifying the times of drawdown calculations in the dimensional form

the first option (ITS = 0), dimensional drawdown is calculated for a series of equally-spaced, dimensional, 

logarithmically-scaled times. These times are calculated from values specified for variables TLAST, NLC, 

NOX. Variable TLAST is the largest value of dimensional time that is desired; variables NLC and NOX are 

above. In the second option (ITS = 1), the user enters a series of specific times at which drawdown will be 

calculated for the pumped well or each of the observation wells and piezometers. The number of specific ti

GAMMA I( )

αi

bSy

Kz

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

GAMMA I( )

αi

b

Sy

Kz

( ) NLC ( ) NOX + 10 1 1.0 ⁄ ( ) NOX
10 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

which drawdown will be calculated is set using variable NTSPW for the pumped well and variable NTSOB 

each of the observation wells and piezometers. The actual times are read using arrays TIMEPW for the pum

well and TIMEOB for observation wells and piezometers.

Program solution variables.--Five program-solution variables (RERRNR, RERRSUM, NMAX, NT

and NS) are used in the numerical-inversion algorithms of WTAQ. Suggested values are provided for each o

these variables and the user should not have to make substantial changes to these values. Variables RERRS

NMAX, and NS are used for confined aquifers and variables RERRNR, NTMS, and NS are used for water

aquifers. Variable RERRSUM is the relative error in the finite summations of the analytical solutions for 

drawdown in a confined aquifer. Decreasing the value of RERRSUM will increase solution precision and t

A value between 1.0D-07 to 1.0D-08 is suggested. Variable NMAX is the maximum number of terms perm

in the finite summations of the analytical solutions for drawdown in a confined aquifer. The suggested value 

If the relative error criterion (RERRSUM) has not been met after NMAX terms have been summed, an erro

message is printed to the result file and the program continues to the next time. The user should ensure that 

sufficient number of terms is being used in the summations by making multiple runs in which NMAX is 

increased from one simulation to the next, and should continue these runs until simulation results do not va

substantially when NMAX is increased.

Variable NS is the number of terms used in the Stehfest algorithm. NS must be an even number, the v

of which depends upon computer precision. If the computer holds 16 significant figures in double precision

NS = 8 to 12. A value of 8 is often sufficient. If calculated drawdowns are unstable, NS can be reduced to 6 (

4). Precision will be reduced, however, and results should be checked for accuracy. The user can compare 

simulation results using alternative values of NS (such as 6, 8, and 12) to determine if numerically stable re

have been attained.

Variable RERRNR is the relative error for Newton-Raphson iteration used in the finite summations o

analytical solutions for drawdown in a water-table aquifer. Smaller values of RERRNR will increase soluti

precision and time. A value of 1.0D-10 is suggested. If the value of RERRNR is exceeded after 100 Newto

Raphson iterations, a message is printed to the result file and the program is stopped. Should this happen, t

can increase the value of RERRNR and re-run the simulation. Variable NTMS is a factor that is used to det

the number of terms in the finite summations of the analytical solutions for drawdown in a water-table aqui

Suggested values are 20 or 30. The user should ensure that a sufficient number of terms is being used in the 

summations by making multiple runs in which NTMS is increased from one simulation to the next (for exa

doubled), and should continue until simulation results do not vary substantially when NTMS is increased. 

Characteristics of the pumped well.--The simulated pumped well can either partially or fully penet

aquifer and can be either finite or infinitesimal in diameter. These options are provided for both confined an

table aquifer types. For all options, a nonzero value of the radius of the pumped well screen (fig. 5) must be 

specified for variable RW. The screened interval of the pumped well is specified using variables ZPD and Z

which are the depths below the top of the aquifer or initial water table to the top (ZPD) and bottom (ZPL) o

screened interval of the well (fig. 5). For the dimensional format, the pumping rate of the well must be spec

use of variable QQ.
Program W

Well-bore storage effects are simulated when the finite-diameter pumped-well option is used (IPWD 

Simulation of well-bore storage requires specification of additional information on the pumped well. For th

curve format, this information is the dimensionless well-bore storage parameter (variable WD), which is cal

independently of the program from

 ,

where

is the inside radius of the pumped well in the interval where water levels are changing during pum

(units of length);

is the radius of the pumped well screen (units of length);

is the specific storage of the aquifer (units of inverse length);

is the depth below top of aquifer or initial water table to the bottom of the screened interval of the p

well (units of length); and 

is the depth below top of aquifer or initial water table to the top of the screened interval of the pum

well (units of length). [Note that is the length of the screened interval of the pumped w

For the dimensional format, the user must specify values of and .

Well-bore skin can be specified for either a finite-diameter or infinitesimal-diameter pumped well. Th

of the well-bore skin parameter (variable SW) is calculated independently of the program from

 ,

where

is the hydraulic conductivity of the well-bore skin (units of length per time); and

is the thickness of the well-bore skin (units of length);

and other parameters are defined previously. In the absence of well-bore skin, variable SW is set to 0.0D0.

Well-bore skin can be simulated with or without well-bore storage at the pumped well. When a well-

skin of low permeability relative to the aquifer material is simulated but well-bore storage is not, drawdown

aquifer (and, thus, at observation wells and piezometers) will not be affected by the skin effect at the pumpe

Drawdowns at the pumped well, however, will be larger than they would be in the absence of the skin. Whe

well-bore skin and well-bore storage are simulated (the usual situation), early-time drawdowns at observatio

and piezometers will be delayed relative to those that occur in the absence of the skin, and drawdown at the 

pumped well will be larger than it would be in the absence of well-bore storage (Moench, 1985, fig. 7).

Characteristics of observation wells and piezometers.--WTAQ can simulate drawdowns at up to 2

locations in an aquifer; the number of observation locations is specified with variable NOBWC. The three t

observation wells or piezometers that can be simulated are a partially penetrating observation well (IOWS 

fully penetrating observation well (IOWS = 1), or an observation piezometer (IOWS = 2). For both partiall

fully penetrating observation wells, the screened interval of the well must be specified using variables Z1 a

which are the depths below the top of the aquifer or initial water table to the top (Z1) and bottom (Z2) of th

screened interval of the well, respectively (fig. 3). For an observation piezometer, the depth below the top o

aquifer or initial water table to the center of the piezometer is specified with variable ZP (fig. 3). In all case

radial distance from the axis of the pumped well to the observation well or piezometer is specified with var

WD

πrc

2

2πrw

2

Ss z pl z pd [ ] ( ) –

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

rc

rw

Ss

z pl

z pd

z pl z pd –

rc rw Ss z pl ,,, z pd

SW

Krds

Ksrw

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

Ks

ds
12 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Delayed drawdown response in an observation well can be simulated with or without well-bore storag

pumped well. The option is activated by setting variable IDPR = 1. For the type-curve format, the dimensio

delayed response factor, , is specified for each observation well. This factor (variable WDP in the prog

must be specified by the user, and is determined from

 ,

where is the inside radius of the observation well in the interval where water levels are changing during 

ing (units of length) and is a shape factor (units of length). For a well in a uniform aquifer medium, 

defined by (Hvorslev, 1951, case 8)

 ,

where

 , ,

and is the length of the screened interval of the observation well ( ). For the dimensional format, t

need only specify the inside radius of the observation well over the range of changing water levels (variabl

and the length of the screened interval of the observation well (variable XLL); the program will calculate W

internally. Note that delayed drawdown response in an observation piezometer as defined here cannot be sim

using WTAQ because of the dependence of the delayed response on the screened length of the observation w

do so requires treating the zero length piezometer as a finite-length observation well and specifying and 

Measured drawdown data.--Measurements of drawdown made at specific times at either the pump

or observation wells and piezometers may be specified using the dimensional format. Measured drawdown 

used for model calibration and the user may want to include these data in the input file for comparison to m

calculated drawdowns (see section on “Result and Plot Files”). When measured data are specified, variable 

is set to 1 (IMEAS is set to 0 when no data are specified). The measured data are read using arrays XMEAS

the pumped well and XMEASOB for the observation wells and piezometers. (The times associated with the

are read using arrays TIMEPW and TIMEOB.) If IMEAS=1, drawdown measurements must be specified a

time. An option also is provided to suppress drawdown calculations during a simulation for the pumped wel

of the observation wells and piezometers. This option allows the user to specify time-drawdown data in the 

file without having that data used in all simulations. The option is activated by setting IRUN = 0; hence, IRU

probably most often be set to 1.

Data-Input Files

Each of the two analysis formats requires a separate data-input file for execution of WTAQ. Most of t

and other information that is required for the data-input files is described in detail in the preceding section. 

input data are read using free-format style, which means that data values in each line of input do not have t

specific columns; however, data values must be separated by one or more blank spaces. A consistent set of 

and time units must be used throughout the input file, for example, meters and seconds or feet and days. Al

valued variables are double-precision format in WTAQ; consequently, double-precision format should be us

real-valued variables in the input file.

WD'

WDP

πr

p

2

2πr

w

2

S

s [ ] F′

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

r

p F′ F′

F′ L

x 1 x

2 ln( ) + +

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

x

mL

2r

p

= --------- m

K

r

K

z

= ------

L z1 z2 –

z1 
Program W

Type-Curve Format

Line-by-line instructions for creating a data-input file for the type-curve format follow. Variable nam

are used in the input file and computer program are shown in upper-case text.

Part I: Aquifer Characteristics and General Information

Line 1:

TITLE--Title of simulation, which can be up to 70 characters in length. Leave this line blank if no title is sp

Line 2:

FORMAT--Analysis format. Enter TYPE CURVE. 

Line 3:

AQTYPE--Type of aquifer being simulated. Two options are provided:

AQTYPE = CONFINED or

AQTYPE = WATER TABLE.

Line 4:

BB--Thickness or saturated thickness of aquifer at beginning of simulation, in units of length.

XKD--Ratio of vertical to horizontal hydraulic conductivity of aquifer, dimensionless.

SIGMA--Ratio of storativity to specific yield of aquifer, dimensionless. Enter 0.0D0 if AQTYPE = CONFI

Line 5:

IDRA--Type of drainage at water table. Enter 0 if AQTYPE = CONFINED. Two options are provided:

IDRA = 0: Instantaneous drainage.

IDRA = 1: Delayed drainage.

NGAMMA--Number of dimensionless drainage constants. Enter 0 if IDRA = 0. NGAMMA must be less t

equal to 5.

Line 6:

GAMMA(I)--A single row of NGAMMA dimensionless drainage constants. Successive values must be sep

by at least 1 space. Maximum of 5 values is allowed. Enter 1.0D09 if IDRA = 0.

Line 7:

TDLAST--Largest value of dimensionless time.

NLC--Number of logarithmic cycles on the time scale for which drawdown will be calculated.

NOX--Number of equally spaced times per logarithmic cycle for which drawdown will be calculated.

Line 8: Program-solution variables. See “Description of Selected Program Options and Variables” for infor

on these variables.

RERRNR--Relative error for Newton-Raphson iteration and finite summations of drawdown for water-tabl

aquifers. A value of 1.0D-10 is suggested. Enter 0.0D0 for AQTYPE = CONFINED.

RERRSUM--Relative error for finite summations of drawdown for confined aquifers. Suggested value is 1

to 1.0D-08. Enter 0.0D0 if AQTYPE = WATER TABLE.

NMAX--Maximum number of terms permitted in the finite summations of drawdown for confined aquifers

Suggested value is 200. Enter 0 if AQTYPE = WATER TABLE.

NTMS--Factor used to determine number of terms in the finite summations for drawdown for water-table a

Suggested values are 20 or 30. Enter 0 if AQTYPE = CONFINED.

NS--Number of terms used in the Stehfest algorithm. This must be an even integer, the value of which depe

upon computer precision. If the computer holds 16 significant figures in double precision, let NS = 8 to 

value of 8 is often sufficient.
14 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Part II: Pumped-Well Information

Line 9:

IPWS--Type of pumped well:

IPWS = 0: Partially penetrating pumped well.

IPWS = 1: Fully penetrating pumped well.

IPWD--Type of diameter of pumped well:

IPWD = 0: Infinitesimal diameter (line-source theory).

IPWD = 1: Finite diameter.

Line 10:

RW--Radius of pumped well screen, in units of length.

ZPD--Depth below top of aquifer or initial water table to the top of the screened interval of the pumped we

units of length.

ZPL--Depth below top of aquifer or initial water table to the bottom of the screened interval of the pumped 

units of length.

WD--Well-bore storage parameter, dimensionless. Enter 0.0D0 if IPWD = 0.

SW--Well-bore skin parameter, dimensionless.

Part III: Observation-Well Information

Line 11:

NOBWC--Number of observation wells or piezometers for which type curves will be calculated. Must be le

or equal to 25.

Lines 12 and 13: Repeat lines 12 and 13 for each of the NOBWC observation wells or piezometers.

Line 12:

IOWS--Type of observation well or piezometer:

IOWS = 0: Partially penetrating observation well.

IOWS = 1: Fully penetrating observation well.

IOWS = 2: Observation piezometer.

IDPR--Option for delayed response of observation well. Valid only for IOWS = 0 or 1:

IDPR = 0: No delayed response.

IDPR = 1: Delayed response.

Line 13:

R--Radial distance from axis of pumped well to observation well or piezometer, in units of length.

Z1--Depth below top of aquifer or initial water table to the top of screened interval of observation well, in u

length. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2.

Z2--Depth below top of aquifer or initial water table to the bottom of screened interval of observation well, 

of length. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2.

ZP--Depth below top of aquifer or initial water table to center of piezometer, in units of length. Use for IOW

Enter 0.0D0 if IOWS = 0 or 1.

WDP--Delayed response factor, dimensionless. Enter 0.0D0 if IDPR = 0.

A summary of the data-input requirements for the type-curve format is shown in table 1. An example 

input file that is based on the hypothetical water-table aquifer described in the sample problems is shown in 

figure 6. Variable names are shown for convenience on the right side of each line of the example data-input 

(fig. 6).
Program W

Table 1. Input data for type-curve format—Continued

Line Variable

name Type Explanation

PART I: AQUIFER CHARACTERISTICS AND GENERAL INFORMATION

1 TITLE Character Title of simulation (up to 70 characters)

2 FORMAT Character Analysis format. Enter TYPE CURVE

3 AQTYPE Character Aquifer type: 

AQTYPE = CONFINED or

AQTYPE = WATER TABLE 

4 BB Real Thickness or saturated thickness of aquifer

XKD Real Ratio of vertical to horizontal hydraulic conductivity of aquifer

SIGMA Real Ratio of storativity to specific yield of aquifer. Enter 0.0D0 if AQTYPE = CONFINED

5 IDRA Integer Type of drainage at water table. Enter 0 if AQTYPE = CONFINED.

IDRA = 0 : instantaneous drainage

IDRA = 1 : delayed drainage

NGAMMA Integer Number of dimensionless drainage constants. Enter 0 if IDRA = 0.

NGAMMA must be less than or equal to 5

6 GAMMA(I) Real A single row of NGAMMA dimensionless drainage constants, successive values sepa

at least 1 space. Enter 1.0D09 if IDRA = 0

7 TDLAST Real Largest value of dimensionless time

NLC Integer Number of logarithmic cycles on time scale

NOX Integer Number of equally spaced times per logarithmic cycle for which drawdown will be ca

8 RERRNR Real Relative error for Newton-Raphson iteration and finite summations for water-table aq

Suggested value is 1.D-10. Use for AQTYPE = WATER TABLE. Enter 0.0D0 if AQ

CONFINED

RERRSUM Real Relative error for finite summations for confined aquifers. Suggested value is 1.D-07 to 

Enter 0.0D0 if AQTYPE = WATER TABLE

NMAX Integer Maximum number of terms permitted in the finite summations for confined aquifers. 

gested value is 200. Enter 0 if AQTYPE = WATER TABLE 

NTMS Integer Factor used to determine number of terms in the finite summations for water-table aqu

Suggested values are 20 or 30. Enter 0 if AQTYPE = CONFINED

NS Integer Number of Stehfest terms. Must be an even integer. Eight terms are usually sufficient

PART II: PUMPED-WELL INFORMATION

9 IPWS Integer Type of pumped well:

IPWS = 0: Partially penetrating pumped well

IPWS = 1: Fully penetrating pumped well

IPWD Integer Type of diameter of pumped well:

IPWD = 0 : Infinitesimal diameter (line-source theory)

IPWD = 1 : Finite diameter

10 RW Real Radius of pumped well screen

ZPD Real Depth below top of aquifer or initial water table to the top of the screened interval of 

pumped well 

ZPL Real Depth below top of aquifer or initial water table to the bottom of the screened interval 

pumped well

WD Real Well-bore storage parameter. Enter 0.0D0 if IPWD = 0

SW Real Well-bore skin parameter

Table 1. Input data for type-curve format

[Free-format input style—each variable in a line must be separated by at least one space]
16 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

PART III: OBSERVATION-WELL INFORMATION

11 NOBWC Integer Number of observation wells or piezometers for which type curves will be calculated. 

less than or equal to 25

Repeat lines 12 and 13 for each observation well or piezometer

12 IOWS Integer Type of observation well or piezometer:

IOWS = 0: Partially penetrating observation well

IOWS = 1: Fully penetrating observation well

IOWS = 2: Observation piezometer

IDPR Integer Option for delayed response of observation well. Valid only for

IOWS = 0 or 1

IDPR = 0 : No delayed response

IDPR = 1 : Delayed response

13 R Real Radial distance from axis of pumped well to observation well or piezometer

Z1 Real Depth below top of aquifer or initial water table to top of screened interval of observat

Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2

Z2 Real Depth below top of aquifer or initial water table to bottom of screened interval of obs

well. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2

ZP Real Depth below top of aquifer or initial water table to center of piezometer. Use for IOW

Enter 0.0D0 for IOWS = 0 or 1

WDP Real Delayed response factor. Enter 0.0D0 if IDPR = 0

Table 1. Input data for type-curve format—Continued

Line Variable

name Type Explanation

Example input file using type-curve format.

TYPE CURVE FORMAT 

WATER TABLE AQTYPE

 10.0D0 0.5D0 1.0D-3 BB XKD SIGMA 

 0 0 IDRA NGAMMA

 1.0D09 GAMMA(I)

 1.0D12 11 5 TDLAST NLC NOX

 1.0D-10 0.0D0 0 30 8 RERRNR RERRSUM NMAX NTMS NS 

 0 1 IPWS IPWD--PUMPED WELL

 0.10D0 5.0D0 10.0D0 5.0D03 0.0D0 RW ZPD ZPL WD SW 

 4 NOBWC

 2 0 IOWS IDPR--OBPIEZ PD1

 3.16D0 0.0D0 0.0D0 7.5D0 0.0D0 R Z1 Z2 ZP WDP

 2 0 IOWS IDPR--OBPIEZ PD2 

 31.6D0 0.0D0 0.0D0 7.5D0 0.0D0 R Z1 Z2 ZP WDP

 2 0 IOWS IDPR--OBPIEZ PS1

 3.16D0 0.0D0 0.0D0 1.0D0 0.0D0 R Z1 Z2 ZP WDP

 2 0 IOWS IDPR--OBPIEZ PS2

 31.6D0 0.0D0 0.0D0 1.0D0 0.0D0 R Z1 Z2 ZP WDP

Figure 6. Example data-input file using type-curve format.
Program W

Dimensional Format

Line-by-line instructions for creating a data-input file for the dimensional format follow. Variable nam

are used in the input file and computer program are shown in upper-case text.

Part I: Aquifer Characteristics and General Information

Line 1:

TITLE--Title of simulation, which can be up to 70 characters in length. Leave this line blank if no title is sp

Line 2:

FORMAT--Analysis format. Enter DIMENSIONAL. 

Line 3:

AQTYPE--Type of aquifer being simulated. Two options are provided:

AQTYPE = CONFINED or

AQTYPE = WATER TABLE.

Line 4:

BB--Thickness or saturated thickness of aquifer at beginning of simulation, in units of length.

HKR--Horizontal hydraulic conductivity of aquifer, in units of length per time.

HKZ--Vertical hydraulic conductivity of aquifer, in units of length per time.

SS--Specific storage of aquifer, in units of inverse length.

SY--Specific yield of aquifer, dimensionless. Enter 0.0D0 if AQTYPE = CONFINED.

Line 5:

IDRA--Type of drainage at water table. Enter 0 if AQTYPE = CONFINED. Two options are provided:

IDRA = 0: Instantaneous drainage.

IDRA = 1: Delayed drainage.

NALPHA--Number of drainage constants. Enter 0 if IDRA = 0. NALPHA must be less than or equal to 5.

Line 6:

ALPHA(I)--A single row of NALPHA drainage constants, in units of inverse time. Successive values must 

separated by at least 1 space. Maximum of 5 values is allowed. Enter 1.0D09 if IDRA = 0.

Line 7:

ITS--Time specification:

ITS = 0: Log-cycle time steps (use when dimensional theoretical curves are desired).

ITS = 1: User-specified times.

IMEAS--Specification of measured drawdown data. Enter 0 if ITS = 0. Options for ITS = 1:

IMEAS = 0: Measured drawdown data not specified for each time.

IMEAS = 1: Measured drawdown data specified for each time.

Line 8:

TLAST--Largest value of time. Enter 0.0D0 if ITS = 1.

NLC--Number of logarithmic cycles on the time scale for which drawdown will be calculated. Enter 0 if IT

NOX--Number of equally spaced times per logarithmic cycle for which drawdown will be calculated. Ente

ITS = 1.
18 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Line 9: Program-solution variables. See “Description of Selected Program Options and Variables” for infor

on these variables.

RERRNR--Relative error for Newton-Raphson iteration and finite summations of drawdown for water-tabl

aquifers. A value of 1.0D-10 is suggested. Enter 0.0D0 for AQTYPE = CONFINED.

RERRSUM--Relative error for finite summations of drawdown for confined aquifers. Suggested value is 1

to 1.0D-08. Enter 0.0D0 if AQTYPE = WATER TABLE.

NMAX--Maximum number of terms permitted in the finite summations of drawdown for confined aquifers

Suggested value is 200. Enter 0 if AQTYPE = WATER TABLE.

NTMS--Factor used to determine number of terms in the finite summations for drawdown for water-table a

Suggested values are 20 or 30. Enter 0 if AQTYPE = CONFINED.

NS--Number of terms used in the Stehfest algorithm. This must be an even integer, the value of which depe

upon computer precision. If the computer holds 16 significant figures in double precision, let NS = 8 to 

value of 8 is often sufficient.

Part II: Pumped-Well Information

Line 10:

IPWS--Type of pumped well:

IPWS = 0: Partially penetrating pumped well.

IPWS = 1: Fully penetrating pumped well.

IPWD--Type of diameter of pumped well:

IPWD = 0: Infinitesimal diameter (line-source theory).

IPWD = 1: Finite diameter.

Line 11:

QQ--Pumping rate of well, in units of cubic length per time.

RW--Radius of pumped well screen, in units of length.

RC--Inside radius of pumped well in the interval where water levels are changing during pumping, in units 

length. Enter 0.0D0 if IPWD = 0.

ZPD--Depth below top of aquifer or initial water table to the top of the screened interval of the pumped we

units of length.

ZPL--Depth below top of aquifer or initial water table to the bottom of the screened interval of the pumped 

units of length.

SW--Well-bore skin parameter, dimensionless.

Lines 12 and 13: Information on user-specified times for which drawdowns will be calculated for the pum

well. Skip to line 14 if ITS = 0.

Line 12:

NTSPW--Number of user-specified times for which drawdown at the pumped well will be calculated. If NT

0, no drawdowns are calculated for the pumped well.

IRUN--Option to suppress drawdown calculations for the pumped well. Allows user to specify time-drawdo

(Line 13), but those data are ignored during the simulation. Options are:

IRUN = 0: Drawdowns not calculated.

IRUN = 1: Drawdowns calculated.

Line 13: Time and measured drawdown data for pumped well. Repeat this line of input data NTSPW time

NTSPW = 0, skip to line 14):

TIMEPW(I)--Time I.

XMEASPW(I)--Measured drawdown at pumped well for time I, in units of length. Leave blank if IMEAS 
Program W

Part III: Observation-Well Information

Line 14:

NOBWC--Number of observation wells or piezometers for which drawdown curves will be calculated. Mu

less than or equal to 25.

Lines 15 through 18: Repeat lines 15 through 18 for each of the NOBWC observation wells or piezomete

Line 15:

IOWS--Type of observation well or piezometer:

IOWS = 0: Partially penetrating observation well.

IOWS = 1: Fully penetrating observation well.

IOWS = 2: Observation piezometer.

IDPR--Option for delayed response of observation well. Valid only for IOWS = 0 or 1:

IDPR = 0: No delayed response.

IDPR = 1: Delayed response.

Line 16:

R--Radial distance from axis of pumped well to observation well or piezometer, in units of length.

Z1--Depth below top of aquifer or initial water table to the top of screened interval of observation well, in u

length. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2.

Z2--Depth below top of aquifer or initial water table to the bottom of screened interval of observation well, 

of length. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2.

ZP--Depth below top of aquifer or initial water table to center of piezometer, in units of length. Use for IOW

Enter 0.0D0 if IOWS = 0 or 1.

RP--Inside radius of the observation well standpipe in the interval over which water levels are changing du

pumping, in units of length. Enter 0.0D0 if IDPR = 0.

XLL--Length of screened interval of observation well, in units of length. Enter 0.0D0 if IDPR = 0.

Lines 17 and 18: Information on user-specified times for which drawdowns will be calculated for the obse

well or piezometer. Skip these lines if ITS = 0.

Line 17:

NTSOB--Number of user-specified times for which drawdown at the observation well or piezometer will b

calculated. If NTSOB = 0, no drawdowns are calculated for the observation well or piezometer.

IRUN--Option to suppress drawdown calculations for the observation well or piezometer. Allows user to sp

time-drawdown data (Line 18), but those data are ignored during the simulation. Options are:

IRUN = 0: Drawdowns not calculated.

IRUN = 1: Drawdowns calculated.

Line 18: Time and measured drawdown data for observation well or piezometer. Repeat this line of input d

NTSOB times (if NTSOB = 0, skip to next observation point):

TIMEOB(I)--Time I.

XMEASOB(I)--Measured drawdown at observation well or piezometer for time I, in units of length. Leave 

IMEAS = 0.

A summary of the data-input requirements for the dimensional format is shown in table 2. An examp

input file that is based on the hypothetical water-table aquifer described in the sample problems is shown in 

figure 7. Variable names are shown for convenience on the right side of each line of the example data-input 

(fig. 7).
20 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Table 2. Input data for dimensional format—Continued

Line Variable

name Type Explanation

PART I: AQUIFER CHARACTERISTICS AND GENERAL INFORMATION

1 TITLE Character Title of simulation (up to 70 characters)

2 FORMAT Character Analysis format. Enter DIMENSIONAL

3 AQTYPE Character Aquifer type: 

AQTYPE = CONFINED or

AQTYPE = WATER TABLE 

4 BB Real Thickness or saturated thickness of aquifer

HKR Real Horizontal hydraulic conductivity of aquifer

HKZ Real Vertical hydraulic conductivity of aquifer

SS Real Specific storage of aquifer

SY Real Specific yield of aquifer. Enter 0.0D0 if AQTYPE = CONFINED

5 IDRA Integer Type of drainage at water table. Enter 0 if AQTYPE = CONFINED.

IDRA = 0 : instantaneous drainage

IDRA = 1 : delayed drainage

NALPHA Integer Number of drainage constants. Enter 0 if IDRA = 0. NALPHA must be less than o

to 5

6 ALPHA(I) Real A single row of NALPHA drainage constants, successive values separated by at le

space. Enter 1.0D09 if IDRA = 0

7 ITS Integer Time specification:

ITS = 0: Log-cycle times

ITS = 1: User-specified times

IMEAS Integer Measured drawdown data. Enter 0 if ITS = 0. Options for ITS = 1:

IMEAS = 0: Measured data not specified for each time

IMEAS = 1: Measured data specified for each time

8 TLAST Real Largest value of time. Enter 0.0D0 if ITS = 1

NLC Integer Number of logarithmic cycles on time scale. Enter 0 if ITS = 1

NOX Integer Number of equally spaced times per logarithmic cycle for which drawdown will be 

calculated. Enter 0 if ITS = 1

9 RERRNR Real Relative error for Newton-Raphson iteration and finite summations for water-table 

Suggested value is 1.D-10. Use for AQTYPE = WATER TABLE. Enter 0.0D0 if 

AQTYPE = CONFINED

RERRSUM Real Relative error for finite summations for confined aquifers. Suggested value is 1.D-

1.D-08. Enter 0.0D0 if AQTYPE = WATER TABLE

NMAX Integer Maximum number of terms permitted in the finite summations for confined aquife

Suggested value is 200. Enter 0 if AQTYPE = WATER TABLE 

NTMS Integer Factor used to determine number of terms in the finite summations for water-table 

Suggested values are 20 or 30. Enter 0 if AQTYPE = CONFINED

NS Integer Number of Stehfest terms. Must be an even integer. Eight terms are usually sufficie

Table 2. Input data for dimensional format

[Free-format input style—each variable in a line must be separated by at least one space]
Program W

PART II: PUMPED-WELL INFORMATION

10 IPWS Integer Type of pumped well:

IPWS = 0: Partially penetrating pumped well

IPWS = 1: Fully penetrating pumped well

IPWD Integer Type of diameter of pumped well:

IPWD = 0 : Infinitesimal diameter (line-source theory)

IPWD = 1 : Finite diameter

11 QQ Real Pumping rate of well

RW Real Radius of pumped well screen

RC Real Inside radius of pumped well in the interval where water levels are changing durin

pumping. Enter 0.0D0 if IPWD = 0

ZPD Real Depth below top of aquifer or initial water table to the top of the screened interval 

pumped well

ZPL Real Depth below top of aquifer or initial water table to the bottom of the screened interv

pumped well

SW Real Well-bore skin parameter

User-specified times and drawdowns. Skip lines 12 and 13 if ITS = 0

12 NTSPW Integer Number of user-specified times at which drawdown will be calculated (if NTSPW 

drawdowns are calculated for the pumped well)

IRUN Integer Option to suppress drawdown calculations for the pumped well:

IRUN = 0: Drawdowns not calculated

IRUN = 1: Drawdowns calculated

13 Time and measured drawdown data for pumped well. Repeat this line NTSPW tim

NTSPW = 0, skip to line 14):

TIMEPW(I) Real Time I

XMEASPW(I) Real Measured drawdown for time I. Leave blank if IMEAS = 0

PART III: OBSERVATION-WELL INFORMATION

14 NOBWC Integer Number of observation wells or piezometers for which drawdown curves will be ca

Must be less than or equal to 25

Repeat lines 15 - 18 for each observation well or piezometer

15 IOWS Integer Type of observation well or piezometer:

IOWS = 0: Partially penetrating observation well

IOWS = 1: Fully penetrating observation well

IOWS = 2: Observation piezometer

IDPR Integer Option for delayed response of observation well. Valid only for

IOWS = 0 or 1

IDPR = 0 : No delayed response

IDPR = 1 : Delayed response

16 R Real Radial distance from axis of pumped well to observation well or piezometer

Z1 Real Depth below top of aquifer or initial water table to top of screened interval of obse

well. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2

Z2 Real Depth below top of aquifer or initial water table to bottom of screened interval of o

tion well. Use for IOWS = 0 or 1. Enter 0.0D0 if IOWS = 2

ZP Real Depth below top of aquifer or initial water table to center of piezometer. Use for IO

Enter 0.0D0 for IOWS = 0 or 1

Table 2. Input data for dimensional format—Continued

Line Variable

name Type Explanation
22 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

PART III: OBSERVATION-WELL INFORMATION—Continued

RP Real Inside radius of the observation well standpipe in the interval over which water lev

changing during pumping. Enter 0.0D0 if IDPR = 0

XLL Real Length of screened interval of observation well. Enter 0.0D0 if IDPR= 0

User-specified times and drawdowns. Skip lines 17 and 18 if ITS = 0

17 NTSOB Integer Number of user-specified times at which drawdown will be calculated (if NTSOB 

drawdowns are calculated for the observation well or piezometer)

IRUN Integer Option to suppress drawdown calculations for the observation well or piezometer:

IRUN = 0: Drawdowns not calculated

IRUN = 1: Drawdowns calculated

18 Time and measured drawdown data for observation well or piezometer. Repeat thi

NTSOB times (if NTSOB = 0, skip to next observation point):

TIMEOB(I) Real Time I

XMEASOB(I) Real Measured drawdown for time I. Leave blank if IMEAS = 0

Table 2. Input data for dimensional format—Continued

Line Variable

name Type Explanation

Figure 7. Example data-input file using dimensional format.

Sample problem 2, using dimensional format.

DIMENSIONAL FORMAT 

WATER TABLE AQTYPE

 10.0D0 1.0D-4 0.5D-4 2.0D-5 2.0D-01 BB HKR HKZ SS SY 

 0 0 IDRA NALPHA

 1.0D09 ALPHA(I)

 1 1 ITS IMEAS

 0.0D0 0 0 TLAST NLC NOX

 1.D-10 0.0D0 0 30 8 RERRNR RERRSUM NMAX NTMS NS

 0 1 IPWS IPWD--P.WELL

 2.0D-3 0.10D0 0.10D0 5.0D0 10.0D0 0.0D0 QQ RW RC ZPD ZPL SW 

 14 1 NTSPW IRUN

 9.28D00 0.51 TIMEPW(ID), XMEASPW(ID)

 2.00D01 0.97

 4.31D01 1.64

 9.28D01 2.31

 2.00D02 2.65

 4.31D02 2.71

 9.28D02 2.71

 2.00D03 2.72

 4.31D03 2.74

 9.28D03 2.76

 2.00D04 2.81

 4.31D04 2.89

 9.28D04 2.98

 2.00D05 3.09
Program W

 4 NOBWC

 2 0 IOWS IDPR--OBPIEZ PD1

 3.16D0 0.0D0 0.0D0 7.5D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN

 9.28D00 0.090 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.204

 4.31D01 0.384

 9.28D01 0.568

 2.00D02 0.662

 4.31D02 0.680

 9.28D02 0.683

 2.00D03 0.690

 4.31D03 0.703

 9.28D03 0.729

 2.00D04 0.775

 4.31D04 0.847

 9.28D04 0.942

 2.00D05 1.052

 2 0 IOWS IDPR--OBPIEZ PD2

 31.6D0 0.0D0 0.0D0 7.5D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN

 9.28D00 0.0000 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.0005

 4.31D01 0.0028

 9.28D01 0.0071

 2.00D02 0.0099

 4.31D02 0.0105

 9.28D02 0.0110

 2.00D03 0.0119

 4.31D03 0.0140

 9.28D03 0.0186

 2.00D04 0.0292

 4.31D04 0.0524

 9.28D04 0.0983

 2.00D05 0.1722

 2 0 IOWS IDPR--OBPIEZ PS1

 3.16D0 0.0D0 0.0D0 1.0D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN 

 9.28D00 0.007 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.019

 4.31D01 0.039

 9.28D01 0.060

 2.00D02 0.072

 4.31D02 0.077

 9.28D02 0.084

 2.00D03 0.098

 4.31D03 0.126

 9.28D03 0.177

 2.00D04 0.258

Figure 7. Example data-input file using dimensional format—Continued.
24 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Result and Plot Files

WTAQ generates a result file and, optionally, a plot file for each simulation. The contents of these file

between the type-curve and dimensional formats. Example result and plot files generated using the type-cu

format are shown in figures 13 and 14 (at back of report), respectively, and are based on the data-input file 

for the water-table aquifer example in figure 6. The result file consists of three main sections. The first inclu

program banner, analysis format, aquifer type and hydraulic properties, and program time specification and 

solution variables. The second section lists the characteristics and calculated drawdowns for the pumped w

the third section lists the characteristics and calculated drawdowns for observation wells and piezometers. T

dimensionless variables, BETAW and BETA, which are calculated by WTAQ from information provided in 

data-input file, are listed in the second and third sections. These variables are based on the dimensionless 

parameters and , and are defined as (Moench, 1997)

 ,

and

.

βw β

βw

Kz

Kr

----- rw

b

-----     2

=

β

Kz

Kr

----- r

b

--     2

=

 4.31D04 0.365

 9.28D04 0.487

 2.00D05 0.612

 2 0 IOWS IDPR--OBPIEZ PS2

 31.6D0 0.0D0 0.0D0 1.0D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN

 9.28D00 0.0000 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.0001

 4.31D01 0.0005

 9.28D01 0.0012

 2.00D02 0.0017

 4.31D02 0.0019

 9.28D02 0.0022

 2.00D03 0.0028

 4.31D03 0.0042

 9.28D03 0.0075

 2.00D04 0.0160

 4.31D04 0.038

 9.28D04 0.084

 2.00D05 0.162

Figure 7. Example data-input file using dimensional format—Continued.
Program W

In the type-curve format, WTAQ calculates dimensionless time-drawdown values for the pumped we

observation wells and piezometers. Dimensionless time ( ) is divided by the dimensionless radial distanc

each well or piezometer from the axis of the pumped well ( ) for use in composite time-drawdow

(see sample problem 1). The resulting dimensionless time variable TDRDSQ is written to the result file, an

defined as

 . 

(For the pumped well, .) In the remainder of this report, the dimensionless time divided by the di

sionless radial distance of each well or piezometer will be referred to simply as dimensionless time.

Dimensionless drawdown ( ) is defined as

 , 

where

is the initial hydraulic head at the pumped well, observation well, or observation piezometer (units 

length), and

is the hydraulic head at the pumped well, observation well, or observation piezometer at time sin

start of pumping (units of length). [Note that is drawdown at the pumped well, observa

well, or observation piezometer (units of length).]

Other parameters in equation 2 were defined previously. Calculated values of dimensionless drawdown les

0.01 are set to zero.

The plot file for the type-curve format provides a tabulation of all dimensionless time-drawdown calcu

for a particular simulation. Several variables shown in the plot file are equivalent in definition to those in th

file. These include dimensionless time (TDRDSQ), which is defined by equation 1; dimensionless drawdow

the pumped well (HDPW), which are defined by equation 2; and dimensionless drawdowns at the observatio

and piezometers (HDOB1, HDOB2, and so forth), which also are defined by equation 2. Three additional v

are listed in the plot file that are not written to the result file. The first of these is HDT, which is the dimens

drawdown at dimensionless time TDRDSQ that results from strictly one-dimensional, radial flow in a confi

aquifer with storativity equal to (that is, Theis-type flow). The second and third variables, TDYRDSQ 

HDTY, are the dimensionless time and dimensionless drawdown, respectively, that result from strictly onedimensional, radial flow in a confined aquifer with storativity equal to (that is, Theis-type flow with resp

specific yield). This second dimensionless time— —is defined as

, 

and is related to by 

or

. 

The alternative dimensionless times and drawdowns are provided in columns 1-4 of the plot file for the con

ience of the user. The choice of dimensionless time (equation 1 or 3) is at the discretion of the user.

tD

rD r rw = ⁄

tD rD

2 ⁄

Krt

r

2

S

s

= ----------

rD = 1.0

hD

hD

4πKrb

Q ---------------- hi = ( ) – h

hi

h t

hi ( ) – h

S

s

b

Sy

tDy rD

2 ⁄

tDy rD

2 ⁄

Krbt

r

2

Sy

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

tD rD

2 ⁄ σ

tDy rD

2 ( )σ ⁄ tD rD2 = ( ) ⁄

tDy σt = D
26 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Example result and plot files generated using the dimensional format are shown in figures 15 and 16 (

of report), respectively, and are based on the data-input file shown in figure 7. The result file consists of thre

sections that are similar in scope to those generated by the type-curve format. The primary difference betw

results generated for the dimensional format and those generated for the type-curve format is that time-draw

values for the dimensional format are expressed in units of time and length, respectively. Also, when measu

drawdowns are specified in the input file, WTAQ calculates a relative error between the measured and calcu

drawdowns at each time. This error is defined as

 , 

where

is the measured drawdown at time at the pumped well, observation well, or observation piezome

(units of length); and

is the calculated drawdown at time at the pumped well, observation well, or observation piezom

(units of length).

To avoid division by zero in equation 6, WTAQ automatically sets the relative error equal to zero when the 

sured drawdown equals zero, and writes asterisks in the relative error column of the result and plot files.

There are two formats for writing results to the plot file for the dimensional format. In the first, which 

when log-cycle time steps are selected (ITS = 0), time-drawdown calculations are written for all wells and 

piezometers in tabular form, as is done for the type-curve format. In the second, which is used when user-sp

times are selected (ITS = 1), time-drawdown calculations are written separately for each well or observatio

piezometer, as shown in figure 16. This is done because the user-specified times may not be the same for th

different wells and piezometers.

SAMPLE PROBLEMS

Three sample problems demonstrate application of WTAQ to the estimation of hydraulic properties o

hypothetical, homogeneous water-table aquifer. A cross section illustrating the aquifer and the locations of 

observation piezometers is shown in figure 8. It is assumed that the effects of delayed drawdown response 

observation piezometers have been effectively eliminated with hydraulic packers. Aquifer parameters and 

characteristics of the pumped well are given in table 3, and observation-piezometer locations are given in t

Water is assumed to drain instantaneously from the unsaturated zone [variables GAMMA(1) or ALPHA(1) e

1.0 x 109 and variable IDRA = 0] and well-bore skin is assumed to be negligible (variable SW = 0). As see

figure 8, the pumped well partially penetrates the aquifer. These partial-penetration effects are accounted fo

each of the sample problems.

A finite-difference numerical model also was used to simulate the hypothetical water-table aquifer. T

pose of using a numerical model to simulate the aquifer was to generate a set of synthetic drawdowns at th

pumped well and observation piezometers that could be used to demonstrate application of WTAQ. The US

MODFLOW model (McDonald and Harbaugh, 1988) was used in conjunction with a radial-flow preproces

(Reilly and Harbaugh, 1993a,b) for the simulations. The numerical model was developed by T.E. Reilly (U

Geological Survey) and is described in detail by Moench (1997). Drawdowns calculated by the numerical m

differ from those reported by Moench who used a value of equal to 2 x 10-6 m-1. The value used here 

(2 x 10-5 m-1) is an order of magnitude larger and is considered to be a more realistic value of the specific s

of a shallow, water-table aquifer.

hm hc ( ) –

hm

--------------------- × 100

hm t

hc t

S

s
Sample Prob

EXPLANATION

AQUIFER

IMPERMEABLE

(NO-FLOW)

BOUNDARY

SCREENED INTERVAL 

OF PUMPED WELL

PIEZOMETER AND

IDENTIFIER

PS1

Pumped well

Initial water table

Unsaturated zone

Saturated zone

PS1

PD1

PS2

PD2

Land surface

Drawdowns calculated for the hypothe

aquifer using the data-input file shown in figu

and table 4 by WTAQ and those calculated by 

MODFLOW are summarized in table 5. Draw

calculated by MODFLOW differ slightly from 

calculated for the same conditions by WTAQ

Possible explanations for the discrepancies b

the analytical (WTAQ) and numerical results 

given by Moench (1997). Drawdowns calcul

MODFLOW will be referred to as measured 

drawdowns for the remainder of this section.

Sample Problem 1—Estimation of 

Horizontal and Vertical Hydraulic 

Conductivity and Specific Yield Us

Type-Curve Format

In this first sample problem, WTAQ is u

generate dimensionless type curves for the pu

of estimating horizontal hydraulic conductivi

( ), vertical hydraulic conductivity ( ), a

specific yield ( ) of the hypothetical aquife

Moench (1994) provides a detailed descriptio

the methods that are followed here. The data

file for a type-curve analysis requires specific

of three dimensionless parameters ( , a

). These dimensionless parameters are in

unknown, because they depend upon the as-y

undetermined values of , and . F

the determination of , and , the effe

of , which occur at early time, can be igno

( will be reconsidered in Sample Problem 

Kr Kz

Sy

KD, σ

WD

Kr Kz Ss , , Sy

Kr Kz , Sy

Ss

Ss

Figure 8. Hypothetical, homogeneous water-table 

aquifer with a partially penetrating pumped well of finite 

diameter and the locations of the open interval of the four 

observation piezometers. (Piezometers extend to land 

surface, as shown in fig. 3.)

Table 3. Parameters of the hypothetical water-table aquifer 

and pumped well

[m, meter; m/s, meter per second, m3/s, cubic meter per second]

Parameter Value

Horizontal hydraulic conductivity (Kr) 1.0 x 10-4 m/s

Vertical hydraulic conductivity (Kz) 0.5 x 10-4 m/s

Specific storage (Ss) 2.0 x 10-5 m-1

Specific yield (Sy) 0.2

Saturated thickness (b) 10 m

Radius of the screened interval of the

pumped well (rw) 0.1 m

Inside radius of the pumped well (rc) 0.1 m

Pumping rate of well (Q) 2 x 10-3 m3/s

Depth below initial water table to top of 

screened interval of pumped well (zpd) 5 m

Depth below initial water table to bottom of 

screened interval of pumped well (zpl) 10 m

Well-bore skin (Sw) 0

Empirical drainage constant (α1) 1.0 x 109

Table 4. Locations of observation piezometers in hyp

water-table aquifer

[distances in meters]

Piezometer Radial distance from 

axis of pumped well

Depth below 

water table to 

of piezome

PD1 3.16 7.5

PD2 31.6 7.5

PS1 3.16 1.0

PS2 31.6 1.0
AQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aquifers

calculated by MODFLOW]

Time

Pumped well PD1 PD2 PS1 PS2

Drawdown Drawdown Drawdown Drawdown Drawdown

A N t/r2 A N t/r2 A N t/r2 A N t/r2 A N

9.28×100 0.521 0.508 9.28×10-1 0.091 0.090 9.28×10-3 0.0000 0.0000 9.28×10-1 0.006 0.007 9.28×10-3 0.0000 0.0000

2.00×101 .996 .971 2.00×100 .211 .204 2.00×10-2 .0000 .0005 2.00×100 .018 .019 2.00×10-2 .0000 .0001

4.31×101 1.70 1.64 4.31×100 .402 .384 4.31×10-2 .0029 .0028 4.31×100 .039 .039 4.31×10-2 .0000 .0005

9.28×101 2.39 2.31 9.28×100 .594 .568 9.28×10-2 .0076 .0071 9.28×100 .060 .060 9.28×10-2 .0000 .0012

2.00×102 2.70 2.65 2.00×101 .682 .662 2.00×10-1 .0101 .0099 2.00×101 .071 .072 2.00×10-1 .0018 .0017

4.31×102 2.74 2.71 4.31×101 .693 .680 4.31×10-1 .0106 .0105 4.31×101 .075 .077 4.31×10-1 .0019 .0019

9.28×102 2.74 2.71 9.28×101 .695 .683 9.28×10-1 .0110 .0110 9.28×101 .082 .084 9.28×10-1 .0022 .0022

2.00×103 2.74 2.72 2.00×102 .702 .690 2.00×100 .0119 .0119 2.00×102 .095 .098 2.00×100 .0028 .0028

4.31×103 2.76 2.74 4.31×102 .715 .703 4.31×100 .0140 .0140 4.31×102 .123 .126 4.31×100 .0042 .0042

9.28×103 2.79 2.76 9.28×102 .741 .729 9.28×100 .019 .019 9.28×102 .174 .177 9.28×100 .0074 .0075

2.00×104 2.83 2.81 2.00×103 .788 .775 2.00×101 .029 .029 2.00×103 .256 .258 2.00×101 .016 .016

4.31×104 2.91 2.88 4.31×103 .861 .847 4.31×101 .053 .052 4.31×103 .365 .365 4.31×101 .037 .038

9.28×104 3.01 2.98 9.28×103 .957 .942 9.28×101 .099 .098 9.28×103 .488 .487 9.28×101 .086 .084

2.00×105 3.12 3.09 2.00×104 1.068 1.052 2.00×102 .175 .172 2.00×104 .613 .612 2.00×102 .166 .162
Sample Prob

This is done by setting to some small value (such as 10-9). Also, well-bore storage should be ignored (tha

equal to zero) because it does not substantially affect the intermediate- and late-time drawdown data that ar

to estimate , and . Well-bore storage effects are removed from the analysis by setting variables IPW

and WD = 0.0D0 in the data-input file. The only remaining dimensionless parameter that must be specified

therefore, is . For this analysis, an initial estimate of equal to 0.1 was assigned. The data-input file 

generate dimensionless type curves for this initial simulation is shown in figure 9. 

Several authors have discussed the importance of including drawdown data from all available observ

points simultaneously in the analysis of aquifer-test results (Stallman, 1971; Weeks, 1977; van der Kamp, 1

Moench, 1994). This is accomplished by creating a single composite plot of all measured drawdown data p

against , where is the time of each drawdown measurement and is the square of the radial distanc

observation point from the axis of the pumped well. This approach was taken for this analysis, and the resu

values of for the four hypothetical observation piezometers are shown with the drawdown data in tab

double-logarithmic composite plot using data from all four observation points was made, and is shown in b

figure 10A and 10B. 

WTAQ calculates two values of dimensionless time for the type-curve format ( and )

dimensionless times are defined in equations 1 and 3, and are identified in the plot file by variables TDRDS

TDYRDSQ, respectively. A double-logarithmic graph is made of either TDRDSQ or TDYRDSQ plotted ag

each of the dimensionless drawdowns (shown in figure 14 as HDOB1, HDOB2, and so forth). For matchin

purposes this graph must be at the same size and have the same number of log cycles that were used for the 

composite plot of the measured data. For this analysis, dimensionless-time values TDYRDSQ were used (fi

because of the emphasis on late-time data.

σ

Kr Kz , Sy

KD KD

t r

2 ⁄ t r2

t r

2 ⁄

tD rD

2 ⁄ tDy rD2 ⁄

Sample problem 1, using infinitesimal diameter pumped well.

TYPE CURVE FORMAT 

WATER TABLE AQTYPE

 10.0D0 0.1D0 1.0D-9 BB XKD SIGMA 

 0 0 IDRA NGAMMA

 1.0D09 GAMMA(I)

 1.0D12 11 5 TDLAST NLC NOX

 1.0D-10 0.0D0 0 30 8 RERRNR RERRSUM NMAX NTMS NS 

 0 0 IPWS IPWD--PUMPED WELL

 0.10D0 5.0D0 10.0D0 0.0D00 0.0D0 RW ZPD ZPL WD SW 

 4 NOBWC

 2 0 IOWS IDPR--OBPIEZ PD1

 3.16D0 0.0D0 0.0D0 7.5D0 0.0D0 R Z1 Z2 ZP WDP

 2 0 IOWS IDPR--OBPIEZ PD2 

 31.6D0 0.0D0 0.0D0 7.5D0 0.0D0 R Z1 Z2 ZP WDP

 2 0 IOWS IDPR--OBPIEZ PS1

 3.16D0 0.0D0 0.0D0 1.0D0 0.0D0 R Z1 Z2 ZP WDP

 2 0 IOWS IDPR--OBPIEZ PS2

 31.6D0 0.0D0 0.0D0 1.0D0 0.0D0 R Z1 Z2 ZP WDP

Figure 9. Data-input file using type-curve format for sample problem 1.
30 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

+

Match point

A.

B.

DIMENSIONLESS DRAWDOWN (hD )

t/r2, IN SECONDS PER SQUARE METER

DIMENSIONLESS TIME (t

Dy /rD

2

)

DRAWDOWN (hi - h), IN METERS

DIMENSIONLESS DRAWDOWN (hD )

t/r2, IN SECONDS PER SQUARE METER

DIMENSIONLESS TIME (t

Dy /rD

2

)

DRAWDOWN (hi - h), IN METERS

100

100

100

100

100

100

100

100

101

101

101

102

102

103

103

104 105

10-1

10-1

10-2

10-2

10-3

101

10-1

10-2

10-3

101

101

102

102

103 104 105

10-1

10-1 101

10-2

10-2

10-3

101

10-1

10-2

10-3

10-3

EXPLANATION

Observation Measured Calcu

 piezometer data type 

PD1

PD2

PS1

PS2

Figure 10. Composite plots of measured drawdown (hi - h) and calculated dimensionless drawdown (type curves) for 

(A) KD = 0.1 and (B) KD = 0.5 for the hypothetical water-table aquifer. (KD, ratio of horizontal to vertical hydraulic condu

t/r2, time divided by the square of the radial distance of the piezometer from the axis of pumped well.)
Sample Prob

The composite plot of the measured data is superimposed onto the theoretical type curves, while the 

coordinate axes of each graph are kept parallel, such that the late-time drawdown data for a given piezomet

overlie as much of the late-time type curve for the same piezometer as possible (fig. 10A). The large deviat

between measured and theoretical data in figure 10A is due to the erroneous first guess of 0.1 for . Beca

measured data cannot be overlain onto the type curves in figure 10A, the value of must be adjusted in t

input file and new type curves generated. In this analysis, was increased from 0.1 to 0.5 because, with 

exception of the type curve for piezometer PS1, type-curve drawdowns were larger than the measured draw

data for (see, for example, the curve for piezometer PD1 in fig. 10A). The increase in effect

decreases the calculated drawdowns by increasing vertical flow in the hypothetical aquifer, and, as seen in fi

resulted in an improvement in the match between the intermediate- and late-time portions of the measured 

drawdown data with the intermediate- and late-time portions of the type curves. The deviation at early time 

piezometers PS1 and PD1 in figure 10B is due to the assumption that . Therefore, the of the aqu

taken to be 0.5, which is equal to the ratio of the values of vertical and horizontal hydraulic conductivity gi

table 3.

A single, global match point is now selected on the superimposed graphs (fig. 10B) from which simul

values of and , and of and can be determined (the values of these four variable

match point are identified by asterisks). This global match point is chosen arbitrarily, and for this problem 

chosen to be and , at which and 

(fig. 10B).

Transmissivity of the aquifer is calculated from

 . 

Using the values of and determined in figure 10B and the pumping rate given in table 3,

.

Horizontal hydraulic conductivity of the aquifer is calculated from the relation , where 

(table 3): 

. 

This calculated value equals that for the aquifer given in table 3. Vertical hydraulic conductivity is then calc

from the definition of and the calculated value of ; that is,

.

KD

KD

KD

KD = 0.1 KD

Ss = 0 KD

tDy rD

2 ⁄ hD t r2 ⁄ hi ( ) – h

tDy rD

2 ( ) ⁄ * 0.1 = hD* 1.0 = t r2 ( ) ⁄ * 20.0 s m

2 = ⁄ hi ( ) – h * 0.16 = m

T Q

4π

------ hD

*

hi ( ) – h * ---------------------     =

hD

* hi ( ) – h *

T 2.0x10–3

m

3 ( ) ⁄ s

4π --------------------------------------- 1.00.16m --------------     1.0x10–3

m

2 = = ⁄ s

T Kr = b b = 10m

Kr T b⁄ 1.0x10–3

m

2 ⁄ s

10m ---------------------------------- 1.0x10–4 == = m s⁄

KD Kr

Kz KDKr ( ) 0.5 1.0x10–4 ( ) m s⁄ 0.5x10–4 == = m s⁄
32 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Specific yield is calculated using either

if values of are used in the analysis, or

if values of are used. Because values of were used in this problem, equation 9 gives

,

which equals the value of specific yield given in table 3.

Sample Problem 2—Estimation of Specific Storage Using Dimensional Format

A type-curve analysis was done in sample problem 1 to estimate the horizontal and vertical hydrauli

conductivity and specific yield of the hypothetical water-table aquifer. The advantage of the type-curve appr

analysis of aquifer tests conducted in water-table aquifers is that only one parameter, , needs adjusting 

evaluate , , and . Of the four hydraulic properties of the aquifer that must be estimated ( 

), only remains unknown. To estimate , the dimensional format of WTAQ is now used to generate 

theoretical, dimensional time-drawdown responses that are compared graphically to the measured time-dra

data. The value of is varied in a series of simulations until the measured data overlie the theoretical resp

Because well-bore storage at the pumped well affects the estimation of , the finite diameter of the pumpe

must now be taken into consideration. Well-bore storage is accounted for using the dimensional format by 

variable IPWD = 1, specifying a value for the inside radius of the pumped well (variable RC, which equals 0

this example; table 3), and specifying a value of (variable SS). An example data-input file for this probl

shown in figure 7.

Dimensional time-drawdown graphs that show the measured data overlain onto theoretical time-draw

curves calculated by WTAQ are given in figure 11. Three separate simulations were done with WTAQ using 

of equal to 2.0 x 10-6 m-1 (simulation I), 2.0 x 10-5 m-1 (simulation II), and 2.0 x 10-4 m-1 (simulation I

best match between the measured and theoretical responses corresponded to a value of equal to 2.0 x 10

which is the value given for the hypothetical aquifer in table 3.

The theoretical time-drawdown curves demonstrate several aspects of the drawdown response of a w

table aquifer to variations in specific storage. First, calculated drawdowns for the pumped well are not noti

changed by variations in the value of . Second, as noted by Moench (1997), reductions in the value of sp

storage below 2.0 x 10-6 m-1 bring about no substantial change in the calculated time-drawdown responses. 

because hydraulic head at the observation piezometers in the hypothetical aquifer respond almost instantan

to changing drawdown in the pumped well when is less than about 10-6 m-1. Under this condition, the re

of the water-table aquifer approaches that of a rigid aquifer in which specific storage equals zero. Finally, fi

also shows that as the value of specific storage is increased from 2.0 x 10-6 m-1 to 2.0 x 10-4 m-1, there is a

increased delay in the early-time drawdowns at the observation piezometers.

Sy

T

σ

--- t r

2 ( ) ⁄ *

tD rD

2 ( ) ⁄ *

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

tD rD

2 ⁄

Sy T t r

2 ( ) ⁄ *

tDy rD

2 ( ) ⁄ *

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

tDy rD

2 ⁄ tDy rD2 ⁄

Sy 1.0x10–3

m

2 ⁄ s

20.0 s m

2 ( ) ⁄

0.1 = = ---------------------------- 0.2

KD

Kr Kz Sy Kr Kz Ss , ,

Sy Ss Ss

Ss

Ss

Ss

Ss

Ss

Ss

Ss
Sample Prob

Sample Problem 3—Automatic Parameter Estimation Using Dimensional Format

In the previous two problems, trial-and-error approaches were used to estimate , and o

hypothetical water-table aquifer. In both approaches, the user adjusts the hydraulic parameters of the analy

model in a series of simulations until the measured time-drawdown data can be overlain graphically onto ei

dimensionless or dimensional theoretical time-drawdown curves. The process of finding a set of hydraulic 

parameters that causes the theoretical curves to match the measured data closely is called model calibration

alternative to this graphical approach is the use of automatic parameter estimation based on nonlinear-regre

techniques. In that approach, a computer program is used to find a set of hydraulic parameters of the mode

minimizes the sum of squared errors between measured and model-calculated drawdowns at each observat

Kr Kz Ss , , Sy

0.001

10

0.002

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

2

5

1 1,000,000 10 100 1,000 10,000 100,000

1 1,000,000 10 100 1,000 10,000 100,000

0.001

10

0.002

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

2

5

TIME, IN SECONDS

DRAWDOWN (hi - h), IN METERS

TIME, IN SECONDS

DRAWDOWN (hi - h), IN METERS

A.

B.

I, II, III

I

II

III

I II III

I

II

III I

II III

PW

PD1

PD2

PS1

PS2

EXPLANATION

Location Measured Calculated

data curve

PD1

PD2

PS1

PS2

PW

Figure 11. Drawdown versus time for the hypothetical water-table aquifer. Calculated curves use values of spec

storage equal to 2.0 x 10-6 m-1 (I), 2.0 x 10-5 m-1 (II), and 2.0 x 10-4 m-1 (III). (A) Wells PW, PD1, and PD2. (B) We

PS1 and PS2.
34 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

location and time. The set of model parameters that minimizes these errors is said to be the optimal set of pa

values. The use of automatic parameter-estimation techniques for estimating aquifer hydraulic properties fr

aquifer-test data is described by Sayed (1990), Johns and others (1992), and Heidari and Moench (1997), as 

in several references listed in those citations.

The purpose of this sample problem is to demonstrate how aquifer hydraulic properties can be estim

linking WTAQ with an automatic parameter-estimation program. The computer program UCODE (Poeter a

1998) was chosen for the parameter estimation, although other parameter-estimation programs may have w

equally well. The objective of the analysis was to estimate all four hydraulic properties of the hypothetical 

table aquifer ( , and ) simultaneously. It is necessary to use the dimensional format in the WTA

simulations so that calculated drawdowns can be compared directly with the measured data.

The parameter-estimation program requires that measured drawdowns at observation locations be sp

In this analysis, measured data from all four of the observation piezometers were used. The parameter-estim

program also requires specification of initial estimates of each of the parameters that are to be determined; 

used in this analysis are shown in table 6. The values of these initial estimates should be varied in a series o

parameter-estimation runs to ensure that the same set of optimal parameter values is determined regardless 

particular values selected for the initial estimates. Data-input files used by UCODE for this problem are giv

Attachment 2.

As shown in table 6, the estimated values determined by UCODE for each of the hydraulic propertie

the water-table aquifer are close to the true values. The relatively large discrepancy between the true value 

(2.00 x 10-5 m-1) and the estimated value (2.53 x 10-5 m-1) of is most likely because of differences in 

drawdowns calculated at the four piezometer locations by WTAQ (the theoretical drawdowns) and by MOD

(the ‘measured’ drawdowns) at early times. Also shown in table 6 are the lower and upper 95-percent confi

intervals for each of the estimated values. These optimal estimates were obtained after only eight iterations 

parameter-estimation process, in which WTAQ was called by UCODE to simulate eight separate combinati

hydraulic properties. As shown by the time-drawdown curves on figure 12, drawdowns calculated at each o

piezometer locations using the estimated values of the hydraulic properties are nearly identical to those cal

by MODFLOW.

Kr Kz Ss , , Sy

Ss

Table 6. Hydraulic properties of the hypothetical water-table aquifer determined using parameter estimation 

[m, meter; m/s, meter per second]

Parameter

(units) True value Initial estimate Estimated value

95-percent confidence interv

of estimated parameter

Lower Uppe

Kr (m/s) 1.00x10-4 1.00x10-5 1.01x10-4 1.01x10-4 1.02x

Kz (m/s) 0.50x10-4 1.00x10-5 0.52x10-4 0.52x10-4 0.53x

Ss (m-1) 2.00x10-5 1.00x10-2 2.53x10-5 2.35x10-5 2.71x

Sy 2.00x10-1 1.00x10-1 2.02x10-1 1.97x10-1 2.06x
Sum

0.001

10

0.002

0.005

0.01

0.02

0.05

0.1

0.2

0.5

1

2

5

1 1,000,000 10 100 1,000 10,000 100,000

TIME, IN SECONDS

DRAWDOWN (hi - h), IN METERS

EXPLANATION

PD1

PD2

PS1

PS2

Observation

piezometer

Measured data

(MODFLOW)

Calculated

by WTAQ

SUMMARY

This report describes the computer program WTAQ, which was developed to implement a previously 

analytical solution for axial-symmetric flow to a partially penetrating, finite-diameter well in a homogeneo

anisotropic water-table aquifer. The solution accounts for well-bore storage and well-bore skin at the pumpe

delayed response at an observation piezometer, and delayed or instantaneous drainage from the unsaturated 

For the particular case of zero drainage from the unsaturated zone, the solution simplifies to that of axial-sym

flow in a confined aquifer.

WTAQ calculates dimensionless or dimensional drawdowns that can be used in conjunction with mea

drawdowns at observation points to estimate hydraulic properties of confined and water-table aquifers. Thr

sample problems illustrate use of WTAQ for estimating horizontal and vertical hydraulic conductivity, spec

storage, and specific yield of a hypothetical water-table aquifer by type-curve methods and by use of an au

parameter-estimation method. In the first sample problem, dimensionless type-curve analysis is used to est

the horizontal and vertical hydraulic conductivity and specific yield of the aquifer. This is done by ignoring 

storage of the aquifer and well-bore storage, which do not affect the intermediate- and late-time drawdown 

that are used to estimate , and . In the second sample problem, specific storage of the aquifer is es

by using WTAQ to generate dimensional time-drawdown curves that include well-bore storage effects caus

the finite diameter of the pumped well. These theoretical, model-derived time-drawdown curves are then co

directly to the measured time-drawdown data to estimate . A third sample problem illustrates use of WTA

conjunction with an automatic parameter-estimation program that is based on nonlinear regression to estim

four hydraulic properties of the water-table aquifer simultaneously. In all three sample problems, the estim

techniques yield parameter values that are equal or nearly equal to those specified for the hypothetical aqui

The report includes a summary of the simplifying assumptions used in the development of the analyt

model, descriptions of the program variables, and instructions for constructing the necessary data-input file

program execution.

Kr Kz , Sy

Ss

Figure 12. Drawdown versus time at the four observation piezometers for the hypothetical water-table aquifer. 

Drawdowns calculated by WTAQ using estimated values of hydraulic properties (lines) and by MODFLOW

using true values of hydraulic properties (symbols).
36 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

REFERENCES

Agarwal, R.G., Al-Hussainy, R., and Ramey, H.J., Jr., 1970, An investigation of wellbore storage and skin effect in un

liquid flow, 1. analytical treatment: Transactions of the Society of Petroleum Engineering, AIME, v. 249, p. 279–

Black, J.H., and Kipp, K.L., Jr., 1977, Observation well response time and its effect upon aquifer test results: Journal 

Hydrology, v. 34, p. 297–306.

Boulton, N.S., 1954, Unsteady radial flow to a pumped well allowing for delayed yield from storage: International Ass

of Scientific Hydrology, Rome, Publication 37, p. 472–477.

_____1963, Analysis of data from non-equilibrium pumping tests allowing for delayed yield from storage: Proceedin

Institute of Civil Engineers, v. 26, p. 469–482.

Dougherty, D.E., and Babu, D.K., 1984, Flow to a partially penetrating well in a double-porosity reservoir: Water Res

Research, v. 20, no. 8, p. 1116–1122.

Driscoll, F.G., 1986, Groundwater and wells (2d ed.): St. Paul, Minn., Johnson Filtration Systems, 1089 p.

Freeze, R.A., and Cherry, J.A., 1979, Groundwater: Englewood Cliffs, N.J., Prentice-Hall, 604 p.

Gieske, A., 1999, Flow to a well in a water table aquifer—speeding up the computation of type curves for low beta va

Ground Water, v. 37, no. 2, p. 171–174.

Hantush, M.S., 1961, Drawdown around a partially penetrating well: Proceedings of the American Society of Civil En

Journal of the Hydraulics Division, v. 87, p. 83–98.

Heidari, Manoutchehr, and Moench, Allen, 1997, Evaluation of unconfined-aquifer parameters from pumping test dat

nonlinear least squares: Journal of Hydrology, v. 192, p. 300–313.

Hvorslev, M.J., 1951, Time lag and soil permeability in ground-water observations: Vicksburg, Miss., U.S. Army Corp

Engineers, Waterways Experiment Station, Bulletin 36, 50 p.

Johns, R.A., Semprini, L., and Roberts, P.V., 1992, Estimating aquifer properties by nonlinear least-squares analysis o

test response: Ground Water, v. 30, no. 1, p. 68–77.

Kruseman, G.P., and de Ridder, N.A., 1991, Analysis and evaluation of pumping test data (2d ed.): Publication 47, 

International Institute for Land Reclamation and Improvement, The Netherlands, 377 p.

Marsily, Ghislain de, 1986, Quantitative hydrogeology: Orlando, Fla, Academic Press, 440 p.

McDonald, M.G., and Harbaugh, A.W., 1988, A modular three-dimensional finite-difference ground-water-flow mode

Geological Survey Techniques of Water-Resources Investigations, book 6, chap. A1, 586 p.

Moench, A.F., 1985, Transient flow to a large-diameter well in an aquifer with storative semiconfining layers: Water R

Research, v. 21, no. 8, p. 1121–1131.

_____1994, Specific yield as determined by type-curve analysis of aquifer-test data: Ground Water, v. 32, no. 6, p. 94

_____1995, Combining the Neuman and Boulton models for flow to a well in an unconfined aquifer: Ground Water, v

no. 3, p. 378–384.

_____1996, Flow to a well in a water-table aquifer—an improved Laplace transform solution: Ground Water, v. 34, n

593–596.

_____1997, Flow to a well of finite diameter in a homogeneous, anisotropic water table aquifer: Water Resources Res

v. 33, no. 6, p. 1397–1407.
Refere

Narasimhan, T.N., and Zhu, Ming, 1993, Transient flow of water to a well in an unconfined aquifer—applicability of 

conceptual models: Water Resources Research, v. 29, no. 1, p. 179–191.

Neuman, S.P., 1972, Theory of flow in unconfined aquifers considering delayed response of the water table: Water Re

Research, v. 8, no. 4, p. 1031–1044.

_____1974, Effects of partial penetration on flow in unconfined aquifers considering delayed aquifer response: Water 

Resources Research, v. 10, no. 2, p. 303–312.

Papadopulos, I.S., 1967, Drawdown distribution around a large-diameter well: San Francisco, Cal., Proceedings of th

National Symposium on Groundwater Hydrology, American Water Resources Association, p. 157–168.

Papadopulos, I.S., and Cooper, H.H., Jr., 1967, Drawdown in a well of large diameter: Water Resources Research, v. 3

p. 241–244.

Poeter, E.P., and Hill, M.C., 1998, Documentation of UCODE, a computer code for universal inverse modeling: U.S. 

Geological Survey Water-Resources Investigations Report 98-4080, 116 p.

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computing (2d ed.): New York, Cambridge University Press, 963 p.

Reilly, T.E., and Harbaugh, A.W., 1993a, Simulation of cylindrical flow to a well using the U.S. Geological Survey M

Finite-Difference Ground-Water Flow Model: Ground Water, v. 31, no. 3, p. 489–494.

_____1993b, Source code for the computer program and sample data set for the simulation of cylindrical flow to a we

the U.S. Geological Survey Modular Finite-Difference Ground-Water Flow Model: U.S. Geological Survey Ope

Report 92-659, 7 p.

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using ground-water storage: EOS, Transactions of the American Geophysical Union, v. 16, p. 519–524.

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service of man: Cambridge, Memoirs of the 18th Congress of the International Association of Hydrogeologists, 

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Walton, W.C., 1970, Groundwater resource evaluation: New York, McGraw-Hill Book Company, 664 p.

_____1987, Groundwater pumping tests—design and analysis: Chelsea, Mich., Lewis Publishers, 201 p.

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Symposium, October 19–21, p. 14–26.

FIGURES 13–16

Figu

 *****************************************************

 * *

 * **** U.S. GEOLOGICAL SURVEY **** *

 * *

 * **** WTAQ: PROGRAM OUTPUT **** *

 * *

 * COMPUTER PROGRAM FOR CALCULATING DRAWDOWN *

 * *

 * IN A CONFINED OR WATER-TABLE AQUIFER WITH *

 * *

 * AXIAL-SYMMETRIC FLOW TO A FINITE- OR *

 * *

 * INFINITESIMAL-DIAMETER PUMPED WELL *

 * *

 * VERSION CURRENT AS OF 10/01/99 *

 * *

 *****************************************************

 Example input file using type-curve format. 

 TYPE-CURVE FORMAT WATER-TABLE AQUIFER 

 *** AQUIFER HYDRAULIC PROPERTIES ***

 SATURATED THICKNESS (BB): 0.100D+02 (units of length)

 RATIO OF VERTICAL TO HORIZONTAL

 HYDRAULIC CONDUCTIVITY (XKD): 0.500D+00 (dimensionless)

 RATIO OF STORATIVITY TO

 SPECIFIC YIELD (SIGMA): 0.100D-02 (dimensionless)

 DRAINAGE AT WATER TABLE (IDRA): 0 (instantaneous)

 *** PROGRAM SOLUTION VARIABLES ***

 LARGEST VALUE NUMBER OF DRAWDOWN CALCULATIONS

 OF TIME (TDLAST) LOG CYCLES (NLC) PER LOG CYCLE (NOX)

 ------------------------------------------------------------

 0.100D+13 11 5

 RERRNR RERRSUM NMAX NTMS NS

 --------------------------------------------------

 0.100D-09 0.000D+00 0 30 8

Figure 13. Example result file using type-curve format.
42 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 *** PUMPED-WELL CHARACTERISTICS AND CALCULATED DRAWDOWN ***

 WELL-DIAMETER TYPE (IPWD): 1 (finite diameter)

 SCREENED INTERVAL (IPWS): 0 (partially penetrating)

 SCREENED INTERVAL WELL BORE WELL BORE

 WELL RADIUS ZPD ZPL STORAGE SKIN

 ----------------------------------------------------------

 0.10D+00 0.50D+01 0.10D+02 0.50D+04 0.00D+00

 BETAW = 0.500D-04

 DIMENSIONLESS TIME DIMENSIONLESS DRAWDOWN

 (TDRDSQ) (HD)

 ------------------ ----------------------

 0.1000D+02 0.0000D+00

 0.1585D+02 0.1267D-01

 0.2512D+02 0.2006D-01

 0.3981D+02 0.3178D-01

 0.6310D+02 0.5031D-01

 0.1000D+03 0.7962D-01

 0.1585D+03 0.1259D+00

 0.2512D+03 0.1989D+00

 0.3981D+03 0.3138D+00

 0.6310D+03 0.4937D+00

 0.1000D+04 0.7742D+00

 0.1585D+04 0.1208D+01

 0.2512D+04 0.1868D+01

 0.3981D+04 0.2856D+01

 0.6310D+04 0.4286D+01

 0.1000D+05 0.6260D+01

 0.1585D+05 0.8784D+01

 0.2512D+05 0.1164D+02

 0.3981D+05 0.1428D+02

 0.6310D+05 0.1612D+02

 0.1000D+06 0.1697D+02

 0.1585D+06 0.1718D+02

 0.2512D+06 0.1720D+02

 0.3981D+06 0.1720D+02

 0.6310D+06 0.1721D+02

 0.1000D+07 0.1724D+02

 0.1585D+07 0.1729D+02

 0.2512D+07 0.1736D+02

 0.3981D+07 0.1746D+02

 0.6310D+07 0.1761D+02

 0.1000D+08 0.1781D+02

 0.1585D+08 0.1807D+02

Figure 13. Example result file using type-curve format—Continued.
Figu

 0.2512D+08 0.1840D+02

 0.3981D+08 0.1877D+02

 0.6310D+08 0.1917D+02

 0.1000D+09 0.1960D+02

 0.1585D+09 0.2004D+02

 0.2512D+09 0.2049D+02

 0.3981D+09 0.2094D+02

 0.6310D+09 0.2140D+02

 0.1000D+10 0.2186D+02

 0.1585D+10 0.2231D+02

 0.2512D+10 0.2277D+02

 0.3981D+10 0.2323D+02

 0.6310D+10 0.2369D+02

 0.1000D+11 0.2415D+02

 0.1585D+11 0.2461D+02

 0.2512D+11 0.2507D+02

 0.3981D+11 0.2553D+02

 0.6310D+11 0.2600D+02

 0.1000D+12 0.2646D+02

 0.1585D+12 0.2692D+02

 0.2512D+12 0.2738D+02

 0.3981D+12 0.2784D+02

 0.6310D+12 0.2830D+02

 0.1000D+13 0.2876D+02

 *** OBSERVATION-WELL CHARACTERISTICS AND CALCULATED DRAWDOWN ***

   OBSERVATION WELL OR PIEZOMETER 1 ** **

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+01 0.750D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D-01

Figure 13. Example result file using type-curve format—Continued.
44 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 DIMENSIONLESS TIME DIMENSIONLESS DRAWDOWN

 (TDRDSQ) (HD)

 ------------------ ----------------------

 0.1000D+02 0.1321D+01

 0.1585D+02 0.2001D+01

 0.2512D+02 0.2788D+01

 0.3981D+02 0.3527D+01

 0.6310D+02 0.4043D+01

 0.1000D+03 0.4282D+01

 0.1585D+03 0.4346D+01

 0.2512D+03 0.4355D+01

 0.3981D+03 0.4363D+01

 0.6310D+03 0.4380D+01

 0.1000D+04 0.4408D+01

 0.1585D+04 0.4451D+01

 0.2512D+04 0.4516D+01

 0.3981D+04 0.4612D+01

 0.6310D+04 0.4753D+01

 0.1000D+05 0.4947D+01

 0.1585D+05 0.5202D+01

 0.2512D+05 0.5518D+01

 0.3981D+05 0.5883D+01

 0.6310D+05 0.6284D+01

 0.1000D+06 0.6709D+01

 0.1585D+06 0.7148D+01

 0.2512D+06 0.7596D+01

 0.3981D+06 0.8048D+01

 0.6310D+06 0.8503D+01

 0.1000D+07 0.8961D+01

 0.1585D+07 0.9419D+01

 0.2512D+07 0.9878D+01

 0.3981D+07 0.1034D+02

 0.6310D+07 0.1080D+02

 0.1000D+08 0.1126D+02

 0.1585D+08 0.1172D+02

 0.2512D+08 0.1218D+02

 0.3981D+08 0.1264D+02

 0.6310D+08 0.1310D+02

 0.1000D+09 0.1356D+02

 0.1585D+09 0.1402D+02

 0.2512D+09 0.1448D+02

 0.3981D+09 0.1494D+02

 0.6310D+09 0.1540D+02

 0.1000D+10 0.1586D+02

 0.1585D+10 0.1632D+02

 0.2512D+10 0.1678D+02

 0.3981D+10 0.1724D+02

 0.6310D+10 0.1770D+02

Figure 13. Example result file using type-curve format—Continued.
Figu

 0.1000D+11 0.1817D+02

 0.1585D+11 0.1863D+02

 0.2512D+11 0.1909D+02

 0.3981D+11 0.1955D+02

 0.6310D+11 0.2001D+02

 0.1000D+12 0.2047D+02

 0.1585D+12 0.2093D+02

 0.2512D+12 0.2139D+02

 0.3981D+12 0.2185D+02

 0.6310D+12 0.2231D+02

 0.1000D+13 0.2277D+02

   OBSERVATION WELL OR PIEZOMETER 2 ** **

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+02 0.750D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D+01

 DIMENSIONLESS TIME DIMENSIONLESS DRAWDOWN

 (TDRDSQ) (HD)

 ------------------ ----------------------

 0.1000D+02 0.7498D-01

 0.1585D+02 0.8151D-01

 0.2512D+02 0.9203D-01

 0.3981D+02 0.1091D+00

 0.6310D+02 0.1369D+00

 0.1000D+03 0.1824D+00

 0.1585D+03 0.2565D+00

 0.2512D+03 0.3741D+00

 0.3981D+03 0.5502D+00

 0.6310D+03 0.7928D+00

 0.1000D+04 0.1099D+01

 0.1585D+04 0.1455D+01

 0.2512D+04 0.1848D+01

 0.3981D+04 0.2265D+01

 0.6310D+04 0.2698D+01

 0.1000D+05 0.3141D+01

 0.1585D+05 0.3591D+01

Figure 13. Example result file using type-curve format—Continued.
46 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 0.2512D+05 0.4044D+01

 0.3981D+05 0.4500D+01

 0.6310D+05 0.4958D+01

 0.1000D+06 0.5417D+01

 0.1585D+06 0.5876D+01

 0.2512D+06 0.6336D+01

 0.3981D+06 0.6796D+01

 0.6310D+06 0.7256D+01

 0.1000D+07 0.7717D+01

 0.1585D+07 0.8177D+01

 0.2512D+07 0.8638D+01

 0.3981D+07 0.9098D+01

 0.6310D+07 0.9559D+01

 0.1000D+08 0.1002D+02

 0.1585D+08 0.1048D+02

 0.2512D+08 0.1094D+02

 0.3981D+08 0.1140D+02

 0.6310D+08 0.1186D+02

 0.1000D+09 0.1232D+02

 0.1585D+09 0.1278D+02

 0.2512D+09 0.1324D+02

 0.3981D+09 0.1370D+02

 0.6310D+09 0.1416D+02

 0.1000D+10 0.1462D+02

 0.1585D+10 0.1508D+02

 0.2512D+10 0.1555D+02

 0.3981D+10 0.1601D+02

 0.6310D+10 0.1647D+02

 0.1000D+11 0.1693D+02

 0.1585D+11 0.1739D+02

 0.2512D+11 0.1785D+02

 0.3981D+11 0.1831D+02

 0.6310D+11 0.1877D+02

 0.1000D+12 0.1923D+02

 0.1585D+12 0.1969D+02

 0.2512D+12 0.2015D+02

 0.3981D+12 0.2061D+02

 0.6310D+12 0.2107D+02

 0.1000D+13 0.2153D+02

Figure 13. Example result file using type-curve format—Continued.
Figu

   OBSERVATION WELL OR PIEZOMETER 3 ** **

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+01 0.100D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D-01

 DIMENSIONLESS TIME DIMENSIONLESS DRAWDOWN

 (TDRDSQ) (HD)

 ------------------ ----------------------

 0.1000D+02 0.1144D+00

 0.1585D+02 0.1860D+00

 0.2512D+02 0.2711D+00

 0.3981D+02 0.3529D+00

 0.6310D+02 0.4122D+00

 0.1000D+03 0.4441D+00

 0.1585D+03 0.4605D+00

 0.2512D+03 0.4766D+00

 0.3981D+03 0.5011D+00

 0.6310D+03 0.5397D+00

 0.1000D+04 0.5993D+00

 0.1585D+04 0.6895D+00

 0.2512D+04 0.8230D+00

 0.3981D+04 0.1014D+01

 0.6310D+04 0.1274D+01

 0.1000D+05 0.1606D+01

 0.1585D+05 0.2003D+01

 0.2512D+05 0.2444D+01

 0.3981D+05 0.2907D+01

 0.6310D+05 0.3378D+01

 0.1000D+06 0.3847D+01

 0.1585D+06 0.4314D+01

 0.2512D+06 0.4779D+01

 0.3981D+06 0.5243D+01

 0.6310D+06 0.5705D+01

 0.1000D+07 0.6167D+01

 0.1585D+07 0.6628D+01

 0.2512D+07 0.7089D+01

 0.3981D+07 0.7550D+01

 0.6310D+07 0.8011D+01

 0.1000D+08 0.8471D+01

Figure 13. Example result file using type-curve format—Continued.
48 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 0.1585D+08 0.8932D+01

 0.2512D+08 0.9392D+01

 0.3981D+08 0.9853D+01

 0.6310D+08 0.1031D+02

 0.1000D+09 0.1077D+02

 0.1585D+09 0.1123D+02

 0.2512D+09 0.1170D+02

 0.3981D+09 0.1216D+02

 0.6310D+09 0.1262D+02

 0.1000D+10 0.1308D+02

 0.1585D+10 0.1354D+02

 0.2512D+10 0.1400D+02

 0.3981D+10 0.1446D+02

 0.6310D+10 0.1492D+02

 0.1000D+11 0.1538D+02

 0.1585D+11 0.1584D+02

 0.2512D+11 0.1630D+02

 0.3981D+11 0.1676D+02

 0.6310D+11 0.1722D+02

 0.1000D+12 0.1768D+02

 0.1585D+12 0.1814D+02

 0.2512D+12 0.1860D+02

 0.3981D+12 0.1906D+02

 0.6310D+12 0.1952D+02

 0.1000D+13 0.1998D+02

   OBSERVATION WELL OR PIEZOMETER 4 ** **

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+02 0.100D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D+01

 DIMENSIONLESS TIME DIMENSIONLESS DRAWDOWN

 (TDRDSQ) (HD)

 ------------------ ----------------------

 0.1000D+02 0.1751D-01

 0.1585D+02 0.2176D-01

 0.2512D+02 0.2884D-01

 0.3981D+02 0.4088D-01

 0.6310D+02 0.6179D-01

 0.1000D+03 0.9879D-01

Figure 13. Example result file using type-curve format—Continued.
Figu

 0.1585D+03 0.1646D+00

 0.2512D+03 0.2783D+00

 0.3981D+03 0.4592D+00

 0.6310D+03 0.7156D+00

 0.1000D+04 0.1040D+01

 0.1585D+04 0.1413D+01

 0.2512D+04 0.1819D+01

 0.3981D+04 0.2246D+01

 0.6310D+04 0.2685D+01

 0.1000D+05 0.3132D+01

 0.1585D+05 0.3584D+01

 0.2512D+05 0.4040D+01

 0.3981D+05 0.4497D+01

 0.6310D+05 0.4955D+01

 0.1000D+06 0.5414D+01

 0.1585D+06 0.5874D+01

 0.2512D+06 0.6334D+01

 0.3981D+06 0.6794D+01

 0.6310D+06 0.7254D+01

 0.1000D+07 0.7715D+01

 0.1585D+07 0.8175D+01

 0.2512D+07 0.8636D+01

 0.3981D+07 0.9096D+01

 0.6310D+07 0.9557D+01

 0.1000D+08 0.1002D+02

 0.1585D+08 0.1048D+02

 0.2512D+08 0.1094D+02

 0.3981D+08 0.1140D+02

 0.6310D+08 0.1186D+02

 0.1000D+09 0.1232D+02

 0.1585D+09 0.1278D+02

 0.2512D+09 0.1324D+02

 0.3981D+09 0.1370D+02

 0.6310D+09 0.1416D+02

 0.1000D+10 0.1462D+02

 0.1585D+10 0.1508D+02

 0.2512D+10 0.1554D+02

 0.3981D+10 0.1600D+02

 0.6310D+10 0.1646D+02

 0.1000D+11 0.1692D+02

 0.1585D+11 0.1739D+02

 0.2512D+11 0.1785D+02

 0.3981D+11 0.1831D+02

 0.6310D+11 0.1877D+02

 0.1000D+12 0.1923D+02

 0.1585D+12 0.1969D+02

 0.2512D+12 0.2015D+02

 0.3981D+12 0.2061D+02

 0.6310D+12 0.2107D+02

 0.1000D+13 0.2153D+02

Figure 13. Example result file using type-curve format—Continued.
AQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aquifers

 0.251E+02 0.404E+01 0.251E-01 0.000E+00 0.201E-01 0.279E+01 0.920E-01 0.271E+00 0.288E-01

 0.398E+02 0.450E+01 0.398E-01 0.000E+00 0.318E-01 0.353E+01 0.109E+00 0.353E+00 0.409E-01

 0.631E+02 0.496E+01 0.631E-01 0.000E+00 0.503E-01 0.404E+01 0.137E+00 0.412E+00 0.618E-01

 0.100E+03 0.542E+01 0.100E+00 0.248E-01 0.796E-01 0.428E+01 0.182E+00 0.444E+00 0.988E-01

 0.158E+03 0.588E+01 0.158E+00 0.899E-01 0.126E+00 0.435E+01 0.257E+00 0.461E+00 0.165E+00

 0.251E+03 0.634E+01 0.251E+00 0.222E+00 0.199E+00 0.435E+01 0.374E+00 0.477E+00 0.278E+00

 0.398E+03 0.680E+01 0.398E+00 0.430E+00 0.314E+00 0.436E+01 0.550E+00 0.501E+00 0.459E+00

 0.631E+03 0.726E+01 0.631E+00 0.709E+00 0.494E+00 0.438E+01 0.793E+00 0.540E+00 0.716E+00

 0.100E+04 0.772E+01 0.100E+01 0.104E+01 0.774E+00 0.441E+01 0.110E+01 0.599E+00 0.104E+01

 0.158E+04 0.818E+01 0.158E+01 0.142E+01 0.121E+01 0.445E+01 0.146E+01 0.690E+00 0.141E+01

 0.251E+04 0.864E+01 0.251E+01 0.183E+01 0.187E+01 0.452E+01 0.185E+01 0.823E+00 0.182E+01

 0.398E+04 0.910E+01 0.398E+01 0.225E+01 0.286E+01 0.461E+01 0.227E+01 0.101E+01 0.225E+01

 0.631E+04 0.956E+01 0.631E+01 0.269E+01 0.429E+01 0.475E+01 0.270E+01 0.127E+01 0.269E+01

 0.100E+05 0.100E+02 0.100E+02 0.314E+01 0.626E+01 0.495E+01 0.314E+01 0.161E+01 0.313E+01

 0.158E+05 0.105E+02 0.158E+02 0.359E+01 0.878E+01 0.520E+01 0.359E+01 0.200E+01 0.358E+01

 0.251E+05 0.109E+02 0.251E+02 0.404E+01 0.116E+02 0.552E+01 0.404E+01 0.244E+01 0.404E+01

 0.398E+05 0.114E+02 0.398E+02 0.450E+01 0.143E+02 0.588E+01 0.450E+01 0.291E+01 0.450E+01

 0.631E+05 0.119E+02 0.631E+02 0.496E+01 0.161E+02 0.628E+01 0.496E+01 0.338E+01 0.496E+01

 0.100E+06 0.123E+02 0.100E+03 0.542E+01 0.170E+02 0.671E+01 0.542E+01 0.385E+01 0.541E+01

 0.158E+06 0.128E+02 0.158E+03 0.588E+01 0.172E+02 0.715E+01 0.588E+01 0.431E+01 0.587E+01

 0.251E+06 0.132E+02 0.251E+03 0.634E+01 0.172E+02 0.760E+01 0.634E+01 0.478E+01 0.633E+01

 0.398E+06 0.137E+02 0.398E+03 0.680E+01 0.172E+02 0.805E+01 0.680E+01 0.524E+01 0.679E+01

 0.631E+06 0.142E+02 0.631E+03 0.726E+01 0.172E+02 0.850E+01 0.726E+01 0.571E+01 0.725E+01

 0.100E+07 0.146E+02 0.100E+04 0.772E+01 0.172E+02 0.896E+01 0.772E+01 0.617E+01 0.771E+01

 0.158E+07 0.151E+02 0.158E+04 0.818E+01 0.173E+02 0.942E+01 0.818E+01 0.663E+01 0.818E+01

 0.251E+07 0.155E+02 0.251E+04 0.864E+01 0.174E+02 0.988E+01 0.864E+01 0.709E+01 0.864E+01

 0.398E+07 0.160E+02 0.398E+04 0.910E+01 0.175E+02 0.103E+02 0.910E+01 0.755E+01 0.910E+01

 0.631E+07 0.165E+02 0.631E+04 0.956E+01 0.176E+02 0.108E+02 0.956E+01 0.801E+01 0.956E+01

 0.100E+08 0.169E+02 0.100E+05 0.100E+02 0.178E+02 0.113E+02 0.100E+02 0.847E+01 0.100E+02

 0.158E+08 0.174E+02 0.158E+05 0.105E+02 0.181E+02 0.117E+02 0.105E+02 0.893E+01 0.105E+02

 0.251E+08 0.178E+02 0.251E+05 0.109E+02 0.184E+02 0.122E+02 0.109E+02 0.939E+01 0.109E+02

Figure 14. Example plot file using type-curve format.
Figure 14 51

 0.158E+09 0.197E+02 0.158E+06 0.128E+02 0.200E+02 0.140E+02 0.128E+02 0.112E+02 0.128E+02

 0.251E+09 0.202E+02 0.251E+06 0.132E+02 0.205E+02 0.145E+02 0.132E+02 0.117E+02 0.132E+02

 0.398E+09 0.206E+02 0.398E+06 0.137E+02 0.209E+02 0.149E+02 0.137E+02 0.122E+02 0.137E+02

 0.631E+09 0.211E+02 0.631E+06 0.142E+02 0.214E+02 0.154E+02 0.142E+02 0.126E+02 0.142E+02

 0.100E+10 0.215E+02 0.100E+07 0.146E+02 0.219E+02 0.159E+02 0.146E+02 0.131E+02 0.146E+02

 0.158E+10 0.220E+02 0.158E+07 0.151E+02 0.223E+02 0.163E+02 0.151E+02 0.135E+02 0.151E+02

 0.251E+10 0.225E+02 0.251E+07 0.155E+02 0.228E+02 0.168E+02 0.155E+02 0.140E+02 0.155E+02

 0.398E+10 0.229E+02 0.398E+07 0.160E+02 0.232E+02 0.172E+02 0.160E+02 0.145E+02 0.160E+02

 0.631E+10 0.234E+02 0.631E+07 0.165E+02 0.237E+02 0.177E+02 0.165E+02 0.149E+02 0.165E+02

 0.100E+11 0.238E+02 0.100E+08 0.169E+02 0.242E+02 0.182E+02 0.169E+02 0.154E+02 0.169E+02

 0.158E+11 0.243E+02 0.158E+08 0.174E+02 0.246E+02 0.186E+02 0.174E+02 0.158E+02 0.174E+02

 0.251E+11 0.248E+02 0.251E+08 0.178E+02 0.251E+02 0.191E+02 0.178E+02 0.163E+02 0.178E+02

 0.398E+11 0.252E+02 0.398E+08 0.183E+02 0.255E+02 0.195E+02 0.183E+02 0.168E+02 0.183E+02

 0.631E+11 0.257E+02 0.631E+08 0.188E+02 0.260E+02 0.200E+02 0.188E+02 0.172E+02 0.188E+02

 0.100E+12 0.261E+02 0.100E+09 0.192E+02 0.265E+02 0.205E+02 0.192E+02 0.177E+02 0.192E+02

 0.158E+12 0.266E+02 0.158E+09 0.197E+02 0.269E+02 0.209E+02 0.197E+02 0.181E+02 0.197E+02

 0.251E+12 0.271E+02 0.251E+09 0.202E+02 0.274E+02 0.214E+02 0.202E+02 0.186E+02 0.201E+02

 0.398E+12 0.275E+02 0.398E+09 0.206E+02 0.278E+02 0.218E+02 0.206E+02 0.191E+02 0.206E+02

 0.631E+12 0.280E+02 0.631E+09 0.211E+02 0.283E+02 0.223E+02 0.211E+02 0.195E+02 0.211E+02

 0.100E+13 0.284E+02 0.100E+10 0.215E+02 0.288E+02 0.228E+02 0.215E+02 0.200E+02 0.215E+02

Figure 14. Example plot file using type-curve format—Continued.
52 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 *****************************************************

 * *

 * **** U.S. GEOLOGICAL SURVEY **** *

 * *

 * **** WTAQ: PROGRAM OUTPUT **** *

 * *

 * COMPUTER PROGRAM FOR CALCULATING DRAWDOWN *

 * *

 * IN A CONFINED OR WATER-TABLE AQUIFER WITH *

 * *

 * AXIAL-SYMMETRIC FLOW TO A FINITE- OR *

 * *

 * INFINITESIMAL-DIAMETER PUMPED WELL *

 * *

 * VERSION CURRENT AS OF 10/01/99 *

 * *

 *****************************************************

 Sample problem 2, using dimensional format. 

 DIMENSIONAL FORMAT WATER-TABLE AQUIFER 

 *** AQUIFER HYDRAULIC PROPERTIES ***

 SATURATED THICKNESS (BB): 0.100D+02 (units of length)

 HORIZONTAL HYDRAULIC

 CONDUCTIVITY (HKR): 0.100D-03 (units of length per time

 VERTICAL HYDRAULIC

 CONDUCTIVITY (HKZ): 0.500D-04 (units of length per time

 RATIO OF VERTICAL TO HORIZONTAL

 HYDRAULIC CONDUCTIVITY (XKD): 0.500D+00 (dimensionless)

 CALCULATED TRANSMISSIVITY: 0.100D-02 (units of length squared

 per time)

 SPECIFIC STORAGE (SS): 0.200D-04 (units of inverse length)

 SPECIFIC YIELD (SY): 0.200D+00 (dimensionless)

 CALCULATED STORATIVITY: 0.200D-03 (dimensionless)

 RATIO OF STORATIVITY TO

 SPECIFIC YIELD (SIGMA): 0.100D-02 (dimensionless)

 DRAINAGE AT WATER TABLE (IDRA): 0 (instantaneous)

Figure 15. Example result file using dimensional format.
Figu

 *** PROGRAM SOLUTION VARIABLES ***

 USER-SPECIFIED TIMES; MEASURED DRAWDOWN DATA SPECIFIED

 RERRNR RERRSUM NMAX NTMS NS

 --------------------------------------------------

 0.100D-09 0.000D+00 0 30 8

 *** PUMPED-WELL CHARACTERISTICS AND CALCULATED DRAWDOWN ***

 WELL-DIAMETER TYPE (IPWD): 1 (finite diameter)

 SCREENED INTERVAL (IPWS): 0 (partially penetrating)

 PUMPING RATE OF WELL (QQ): 0.200D-02 (cubic length per time)

 SCREENED INTERVAL WELL BORE WELL BORE

 WELL RADIUS ZPD ZPL STORAGE SKIN

 ----------------------------------------------------------

 0.10D+00 0.50D+01 0.10D+02 0.50D+04 0.00D+00

 BETAW = 0.500D-04

 RELATIVE

 MEASURED CALCULATED ERROR

 TIME DRAWDOWN DRAWDOWN (PERCENT)

 ---- ---------- ---------- ---------- 

 0.9280D+01 0.5100D+00 0.5214D+00 -0.2244D+01

 0.2000D+02 0.9700D+00 0.9963D+00 -0.2713D+01

 0.4310D+02 0.1640D+01 0.1699D+01 -0.3626D+01

 0.9280D+02 0.2310D+01 0.2388D+01 -0.3394D+01

 0.2000D+03 0.2650D+01 0.2700D+01 -0.1895D+01

 0.4310D+03 0.2710D+01 0.2737D+01 -0.9962D+00

 0.9280D+03 0.2710D+01 0.2738D+01 -0.1016D+01

 0.2000D+04 0.2720D+01 0.2744D+01 -0.8975D+00

 0.4310D+04 0.2740D+01 0.2759D+01 -0.6761D+00

 0.9280D+04 0.2760D+01 0.2786D+01 -0.9340D+00

 0.2000D+05 0.2810D+01 0.2834D+01 -0.8686D+00

 0.4310D+05 0.2890D+01 0.2910D+01 -0.6791D+00

 0.9280D+05 0.2980D+01 0.3008D+01 -0.9310D+00

 0.2000D+06 0.3090D+01 0.3119D+01 -0.9524D+00

Figure 15. Example result file using dimensional format—Continued.
54 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 *** OBSERVATION-WELL CHARACTERISTICS AND CALCULATED DRAWDOWN ***

   OBSERVATION WELL OR PIEZOMETER 1 ** **

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+01 0.750D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D-01

 RELATIVE

 MEASURED CALCULATED ERROR

 TIME DRAWDOWN DRAWDOWN (PERCENT)

 ---- ---------- ---------- ---------- 

 0.9280D+01 0.9000D-01 0.9071D-01 -0.7844D+00

 0.2000D+02 0.2040D+00 0.2106D+00 -0.3236D+01

 0.4310D+02 0.3840D+00 0.4018D+00 -0.4629D+01

 0.9280D+02 0.5680D+00 0.5939D+00 -0.4567D+01

 0.2000D+03 0.6620D+00 0.6816D+00 -0.2954D+01

 0.4310D+03 0.6800D+00 0.6928D+00 -0.1881D+01

 0.9280D+03 0.6830D+00 0.6951D+00 -0.1769D+01

 0.2000D+04 0.6900D+00 0.7015D+00 -0.1669D+01

 0.4310D+04 0.7030D+00 0.7148D+00 -0.1676D+01

 0.9280D+04 0.7290D+00 0.7407D+00 -0.1609D+01

 0.2000D+05 0.7750D+00 0.7875D+00 -0.1608D+01

 0.4310D+05 0.8470D+00 0.8606D+00 -0.1610D+01

 0.9280D+05 0.9420D+00 0.9572D+00 -0.1615D+01

 0.2000D+06 0.1052D+01 0.1068D+01 -0.1526D+01

Figure 15. Example result file using dimensional format—Continued.
Figu

   OBSERVATION WELL OR PIEZOMETER 2 ** **

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+02 0.750D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D+01

 RELATIVE

 MEASURED CALCULATED ERROR

 TIME DRAWDOWN DRAWDOWN (PERCENT)

 ---- ---------- ---------- ---------- 

 0.9280D+01 0.0000D+00 0.0000D+00  

 0.2000D+02 0.5000D-03 0.0000D+00 0.1000D+03

 0.4310D+02 0.2800D-02 0.2890D-02 -0.3217D+01

 0.9280D+02 0.7100D-02 0.7606D-02 -0.7127D+01

 0.2000D+03 0.9900D-02 0.1012D-01 -0.2258D+01

 0.4310D+03 0.1050D-01 0.1059D-01 -0.8321D+00

 0.9280D+03 0.1100D-01 0.1100D-01 0.3762D-01

 0.2000D+04 0.1190D-01 0.1194D-01 -0.3075D+00

 0.4310D+04 0.1400D-01 0.1400D-01 -0.2910D-01

 0.9280D+04 0.1860D-01 0.1861D-01 -0.4377D-01

 0.2000D+05 0.2920D-01 0.2905D-01 0.4981D+00

 0.4310D+05 0.5240D-01 0.5245D-01 -0.8710D-01

 0.9280D+05 0.9830D-01 0.9934D-01 -0.1057D+01

 0.2000D+06 0.1722D+00 0.1750D+00 -0.1641D+01

Figure 15. Example result file using dimensional format—Continued.
56 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 ** ** OBSERVATION WELL OR PIEZOMETER 3  

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+01 0.100D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D-01

 RELATIVE

 MEASURED CALCULATED ERROR

 TIME DRAWDOWN DRAWDOWN (PERCENT)

 ---- ---------- ---------- ---------- 

 0.9280D+01 0.7000D-02 0.6376D-02 0.8912D+01

 0.2000D+02 0.1900D-01 0.1824D-01 0.4023D+01

 0.4310D+02 0.3900D-01 0.3858D-01 0.1066D+01

 0.9280D+02 0.6000D-01 0.5984D-01 0.2670D+00

 0.2000D+03 0.7200D-01 0.7069D-01 0.1825D+01

 0.4310D+03 0.7700D-01 0.7491D-01 0.2714D+01

 0.9280D+03 0.8400D-01 0.8152D-01 0.2949D+01

 0.2000D+04 0.9800D-01 0.9542D-01 0.2633D+01

 0.4310D+04 0.1260D+00 0.1231D+00 0.2311D+01

 0.9280D+04 0.1770D+00 0.1739D+00 0.1729D+01

 0.2000D+05 0.2580D+00 0.2558D+00 0.8476D+00

 0.4310D+05 0.3650D+00 0.3652D+00 -0.6745D-01

 0.9280D+05 0.4870D+00 0.4878D+00 -0.1672D+00

 0.2000D+06 0.6120D+00 0.6126D+00 -0.9086D-01

Figure 15. Example result file using dimensional format—Continued.
Figu

 ** ** OBSERVATION WELL OR PIEZOMETER 4  

 OBSERVATION PIEZOMETER

 DISTANCE FROM

 CENTER OF DELAYED RESPONSE

 PUMPED WELL ZP FACTOR

 -------------------------------------------

 0.316D+02 0.100D+01 0.000D+00

 DRAWDOWN CALCULATED FOR BETA = 0.499D+01

 RELATIVE

 MEASURED CALCULATED ERROR

 TIME DRAWDOWN DRAWDOWN (PERCENT)

 ---- ---------- ---------- ---------- 

 0.9280D+01 0.0000D+00 0.0000D+00 ** **

 0.2000D+02 0.1000D-03 0.0000D+00 0.1000D+03

 0.4310D+02 0.5000D-03 0.0000D+00 0.1000D+03

 0.9280D+02 0.1200D-02 0.0000D+00 0.1000D+03

 0.2000D+03 0.1700D-02 0.1762D-02 -0.3626D+01

 0.4310D+03 0.1900D-02 0.1929D-02 -0.1507D+01

 0.9280D+03 0.2200D-02 0.2193D-02 0.3342D+00

 0.2000D+04 0.2800D-02 0.2788D-02 0.4134D+00

 0.4310D+04 0.4200D-02 0.4152D-02 0.1136D+01

 0.9280D+04 0.7500D-02 0.7423D-02 0.1024D+01

 0.2000D+05 0.1600D-01 0.1575D-01 0.1575D+01

 0.4310D+05 0.3800D-01 0.3728D-01 0.1898D+01

 0.9280D+05 0.8400D-01 0.8546D-01 -0.1738D+01

 0.2000D+06 0.1620D+00 0.1656D+00 -0.2233D+01

Figure 15. Example result file using dimensional format—Continued.
58 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

   PUMPED WELL ** **

 TIME MEASDD CALCDD RELERR

 0.9280E+01 0.5100E+00 0.5214E+00 -0.2244E+01

 0.2000E+02 0.9700E+00 0.9963E+00 -0.2713E+01

 0.4310E+02 0.1640E+01 0.1699E+01 -0.3626E+01

 0.9280E+02 0.2310E+01 0.2388E+01 -0.3394E+01

 0.2000E+03 0.2650E+01 0.2700E+01 -0.1895E+01

 0.4310E+03 0.2710E+01 0.2737E+01 -0.9962E+00

 0.9280E+03 0.2710E+01 0.2738E+01 -0.1016E+01

 0.2000E+04 0.2720E+01 0.2744E+01 -0.8975E+00

 0.4310E+04 0.2740E+01 0.2759E+01 -0.6761E+00

 0.9280E+04 0.2760E+01 0.2786E+01 -0.9340E+00

 0.2000E+05 0.2810E+01 0.2834E+01 -0.8686E+00

 0.4310E+05 0.2890E+01 0.2910E+01 -0.6791E+00

 0.9280E+05 0.2980E+01 0.3008E+01 -0.9310E+00

 0.2000E+06 0.3090E+01 0.3119E+01 -0.9524E+00

   OBSERVATION WELL OR PIEZOMETER 1 ** **

 TIME MEASDD CALCDD RELERR

 0.9280E+01 0.9000E-01 0.9071E-01 -0.7844E+00

 0.2000E+02 0.2040E+00 0.2106E+00 -0.3236E+01

 0.4310E+02 0.3840E+00 0.4018E+00 -0.4629E+01

 0.9280E+02 0.5680E+00 0.5939E+00 -0.4567E+01

 0.2000E+03 0.6620E+00 0.6816E+00 -0.2954E+01

 0.4310E+03 0.6800E+00 0.6928E+00 -0.1881E+01

 0.9280E+03 0.6830E+00 0.6951E+00 -0.1769E+01

 0.2000E+04 0.6900E+00 0.7015E+00 -0.1669E+01

 0.4310E+04 0.7030E+00 0.7148E+00 -0.1676E+01

 0.9280E+04 0.7290E+00 0.7407E+00 -0.1609E+01

 0.2000E+05 0.7750E+00 0.7875E+00 -0.1608E+01

 0.4310E+05 0.8470E+00 0.8606E+00 -0.1610E+01

 0.9280E+05 0.9420E+00 0.9572E+00 -0.1615E+01

 0.2000E+06 0.1052E+01 0.1068E+01 -0.1526E+01

   OBSERVATION WELL OR PIEZOMETER 2 ** **

 TIME MEASDD CALCDD RELERR

 0.9280E+01 0.0000E+00 0.0000E+00  

 0.2000E+02 0.5000E-03 0.0000E+00 0.1000E+03

 0.4310E+02 0.2800E-02 0.2890E-02 -0.3217E+01

 0.9280E+02 0.7100E-02 0.7606E-02 -0.7127E+01

 0.2000E+03 0.9900E-02 0.1012E-01 -0.2258E+01

Figure 16. Example plot file using dimensional format.
Figu

 0.4310E+03 0.1050E-01 0.1059E-01 -0.8321E+00

 0.9280E+03 0.1100E-01 0.1100E-01 0.3762E-01

 0.2000E+04 0.1190E-01 0.1194E-01 -0.3075E+00

 0.4310E+04 0.1400E-01 0.1400E-01 -0.2910E-01

 0.9280E+04 0.1860E-01 0.1861E-01 -0.4377E-01

 0.2000E+05 0.2920E-01 0.2905E-01 0.4981E+00

 0.4310E+05 0.5240E-01 0.5245E-01 -0.8710E-01

 0.9280E+05 0.9830E-01 0.9934E-01 -0.1057E+01

 0.2000E+06 0.1722E+00 0.1750E+00 -0.1641E+01

 ** ** OBSERVATION WELL OR PIEZOMETER 3  

 TIME MEASDD CALCDD RELERR

 0.9280E+01 0.7000E-02 0.6376E-02 0.8912E+01

 0.2000E+02 0.1900E-01 0.1824E-01 0.4023E+01

 0.4310E+02 0.3900E-01 0.3858E-01 0.1066E+01

 0.9280E+02 0.6000E-01 0.5984E-01 0.2670E+00

 0.2000E+03 0.7200E-01 0.7069E-01 0.1825E+01

 0.4310E+03 0.7700E-01 0.7491E-01 0.2714E+01

 0.9280E+03 0.8400E-01 0.8152E-01 0.2949E+01

 0.2000E+04 0.9800E-01 0.9542E-01 0.2633E+01

 0.4310E+04 0.1260E+00 0.1231E+00 0.2311E+01

 0.9280E+04 0.1770E+00 0.1739E+00 0.1729E+01

 0.2000E+05 0.2580E+00 0.2558E+00 0.8476E+00

 0.4310E+05 0.3650E+00 0.3652E+00 -0.6745E-01

 0.9280E+05 0.4870E+00 0.4878E+00 -0.1672E+00

 0.2000E+06 0.6120E+00 0.6126E+00 -0.9086E-01

 ** ** OBSERVATION WELL OR PIEZOMETER 4  

 TIME MEASDD CALCDD RELERR

 0.9280E+01 0.0000E+00 0.0000E+00 ** **

 0.2000E+02 0.1000E-03 0.0000E+00 0.1000E+03

 0.4310E+02 0.5000E-03 0.0000E+00 0.1000E+03

 0.9280E+02 0.1200E-02 0.0000E+00 0.1000E+03

 0.2000E+03 0.1700E-02 0.1762E-02 -0.3626E+01

 0.4310E+03 0.1900E-02 0.1929E-02 -0.1507E+01

 0.9280E+03 0.2200E-02 0.2193E-02 0.3342E+00

 0.2000E+04 0.2800E-02 0.2788E-02 0.4134E+00

 0.4310E+04 0.4200E-02 0.4152E-02 0.1136E+01

 0.9280E+04 0.7500E-02 0.7423E-02 0.1024E+01

 0.2000E+05 0.1600E-01 0.1575E-01 0.1575E+01

 0.4310E+05 0.3800E-01 0.3728E-01 0.1898E+01

 0.9280E+05 0.8400E-01 0.8546E-01 -0.1738E+01

 0.2000E+06 0.1620E+00 0.1656E+00 -0.2233E+01

Figure 16. Example plot file using dimensional format.

ATTACHMENT 1.—NARRATIVE FOR

PROGRAM WTAQ

Attachm

Program WTAQ calculates ground-water-level drawdowns in confined and water-table aquifers in resp

withdrawals at a pumped well. WTAQ is written in the FORTRAN-77 computer language. A description of 

numbered steps in the code follows:

Main program:

1. Call subroutine OFILE to open input (variable IN), result (variable IO), and plot (variable IP) files.

2. Call subroutine DATAPREP to read most input data from input file, write input data to result file, and 

input data for use in main program.

3. Define selected program parameters.

4. Call subroutine LINVST to calculate coefficients used for the Stehfest (1970) algorithm.

5. Calculate time-step parameters for log-cycle formats.

6. Calculate required number of time-drawdown curves.

7. Calculate time-drawdown curves for each well:

a. Calculate dimensionless variables to pass to Laplace transform solution subroutines.

b. Define dimensionless time (TD) and iteration variables.

c. Calculate dimensionless drawdown (HD) for each dimensionless time (TD). If aquifer is confined

subroutine LTST2; if water-table aquifer, call subroutine LTST3.

d. Write time-drawdown calculations to result file and, for dimensional data, to plot file.

8. Calculate Theis drawdowns and write all time-drawdown calculations to plot file for type-curve forma

(including dimensional):

a. Calculate Theis drawdowns using subroutine LTST1.

b. Write dimensionless time-drawdown calculations to plot file.

c. Write dimensional time-drawdown calculations to plot file.

9. Close input, result, and plot files; stop program execution; end.

Subroutine OFILE:

10. Subroutine OFILE opens input, result, and plot files.

Function LENCHR:

11. Function LENCHR calculates the length of a character string (code from R. S. Regan, USGS, written 

commun., 1997).

Subroutine DATAPREP:

12. Subroutine DATAPREP reads most input data from the input file, writes this data to the result file, and 

prepares input data for use in main program.

Subroutine BANNER:

13. Subroutine BANNER writes a program banner to the result file.

Subroutine RDERR:

14. If an error is found in the input data, subroutine RDERR writes a message to the result file indicating th

line on which the error was found and then stops program execution.

Subroutine LINVST:

15. Subroutine LINVST calculates coefficients used for the Stehfest (1970) algorithm.
64 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

Subroutine LTST1:

16. Subroutine LTST1 calculates the Laplace-transform solution for drawdown for flow to a fully penetrati

of infinitesimal diameter in a confined aquifer (Theis solution). A value of TD (dimensionless time) 

passed to LTST1 from the Main program and a value of HDT (dimensionless drawdown) is returne

Main program from LTST1.

Subroutine LTST2:

17. Subroutine LTST2 calculates the Laplace-transform solution for drawdown for flow to a finite diamet

partially penetrating well in a confined aquifer (modified solution of Dougherty and Babu, 1984). A 

of TD (dimensionless time) is passed to LTST2 from the Main program and a value of HD (dimens

drawdown) is returned to the Main program from LTST2.

Subroutine LTST3:

18. Subroutine LTST3 calculates the Laplace-transform solution for drawdown for flow to a finite diamet

partially penetrating well in a water-table aquifer with delayed drawdown response at observation w

value of TD (dimensionless time) is passed to LTST3 from the Main program and a value of HD 

(dimensionless drawdown) is returned to the Main program from LTST3.

Function BESSK0:

19. Function BESSK0 calculates the zero-order modified Bessel function of the second kind. Mathematic

of the function is given in Press and others (1992).

Function BESSI0:

20. Function BESSI0 calculates the zero-order modified Bessel function of the first kind. Mathematical b

the function is given in Press and others (1992).

Function BESSK1:

21. Function BESSK1 calculates the first-order modified Bessel function of the second kind. Mathematic

of the function is given in Press and others (1992).

Function BESSI1:

22. Function BESSI1 calculates the first-order modified Bessel function of the first kind. Mathematical ba

the function is given in Press and others (1992).
ATTACHMENT 2.—SAMPLE INPUT

FILES FOR USE OFWTAQ WITH UCODE

Attachm

Several input files that were used by WTAQ and the parameter-estimation program UCODE for samp

problem 3 are listed in this Attachment. The files are provided for reference. Detailed information on prepa

input files for UCODE are not given here; that information is provided by Poeter and Hill (1998). Four UC

input files are shown, the universal (file prob3.uni), prepare (file prob3.pre), template (file prob3.tpl), and e

(file prob3.ext) files. In addition, two short files that are used to run WTAQ in batch mode (which is require

UCODE) on a UNIX-language computer also are given. These files (prob3.bat and prob3.in) are at the end 

Attachment.

Universal file (prob3.uni):

3 #Phase: 1=test files; 3=perform regression

###Sensitivity and regression control information

1 #Forward differencing

0.01 #TOL

0.0 #SOSC

0 #NOPT

10 #MAX-ITER

2.0 #MAX-CHANGE

###Inversion algorithm

/export/home/pbarlow/ucode1.08/bin/mrdrive

###Application model

1 #Number of application models

prob3.bat #Name of batch file to run WTAQ executable

###Output control information

3 #SCALE-SENSITIVITIES

1 #PRINT-INTERMEDIATE

1 #GRAPH

4 #NUMBER-RESIDUAL-SETS

#OBS-NAME OBS-VALUE STATISTIC STAT-FLAG PLOT-SYMBOL

PD1-1 0.090 1.0 0 1

PD1-2 0.204 1.0 0 1

PD1-3 0.384 1.0 0 1

PD1-4 0.568 1.0 0 1

PD1-5 0.662 1.0 0 1

PD1-6 0.680 1.0 0 1

PD1-7 0.683 1.0 0 1

PD1-8 0.690 1.0 0 1

PD1-9 0.703 1.0 0 1

PD1-10 0.729 1.0 0 1

PD1-11 0.775 1.0 0 1

PD1-12 0.847 1.0 0 1

PD1-13 0.942 1.0 0 1

PD1-14 1.052 1.0 0 1

PD2-1 0.0000 1.0 0 1

PD2-2 0.0005 1.0 0 1

PD2-3 0.0028 1.0 0 1

PD2-4 0.0071 1.0 0 1

PD2-5 0.0099 1.0 0 1

PD2-6 0.0105 1.0 0 1

PD2-7 0.0110 1.0 0 1

PD2-8 0.0119 1.0 0 1

PD2-9 0.0140 1.0 0 1
68 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

PD2-10 0.0186 1.0 0 1

PD2-11 0.0292 1.0 0 1

PD2-12 0.0524 1.0 0 1

PD2-13 0.0983 1.0 0 1

PD2-14 0.1722 1.0 0 1

PS1-1 0.007 1.0 0 1

PS1-2 0.019 1.0 0 1

PS1-3 0.039 1.0 0 1

PS1-4 0.060 1.0 0 1

PS1-5 0.072 1.0 0 1

PS1-6 0.077 1.0 0 1

PS1-7 0.084 1.0 0 1

PS1-8 0.098 1.0 0 1

PS1-9 0.126 1.0 0 1

PS1-10 0.177 1.0 0 1

PS1-11 0.258 1.0 0 1

PS1-12 0.365 1.0 0 1

PS1-13 0.487 1.0 0 1

PS1-14 0.612 1.0 0 1

PS2-1 0.0000 1.0 0 1

PS2-2 0.0001 1.0 0 1

PS2-3 0.0005 1.0 0 1

PS2-4 0.0012 1.0 0 1

PS2-5 0.0017 1.0 0 1

PS2-6 0.0019 1.0 0 1

PS2-7 0.0022 1.0 0 1

PS2-8 0.0028 1.0 0 1

PS2-9 0.0042 1.0 0 1

PS2-10 0.0075 1.0 0 1

PS2-11 0.0160 1.0 0 1

PS2-12 0.0375 1.0 0 1

PS2-13 0.0844 1.0 0 1

PS2-14 0.1624 1.0 0 1

END

Prepare file (prob3.pre):

#Do not use a function file

F no

#Names of template and substitution files

<prob3.tpl

>prob3.sub

#Parameters to estimate

#NAME START MIN MAX PERT FORM LOG EST?

/!Kr,,,,! 1.00e-5 1.00e-6 1.00e-2 0.01 %7.2e 1 1

/!Kz,,,,! 1.00e-5 1.00e-6 1.00e-2 0.01 %7.2e 1 1

/!Ss,,,,! 1.00e-2 1.00e-7 1.00e-2 0.01 %7.2e 1 1

/!Sy,,,,! 1.00e-1 1.00e-2 1.00e+0 0.01 %7.2e 1 1

END
Attachm

Template file (prob3.tpl):

Sample problem 3, parameter estimation using dimensional format.

DIMENSIONAL FORMAT 

WATER TABLE AQTYPE

 10.0D0 !Kr,,,,! !Kz,,,,! !Ss,,,,! !Sy,,,,! BB HKR HKZ SS SY 

 0 0 IDRA NALPHA

 1.0D09 ALPHA(I)

 1 1 ITS IMEAS

 0.0D0 0 0 TLAST NLC NOX

 1.D-10 0.0D0 0 30 8 RERRNR RERRSUM NMAX NTMS NS

 0 1 IPWS IPWD--P.WELL

 2.0D-3 0.10D0 0.10D0 5.0D0 10.0D0 0.0D0 QQ RW RC ZPD ZPL SW 

 14 0 NTSPW IRUN

 9.28D00 0.51 TIMEPW(ID), XMEASPW(ID)

 2.00D01 0.97

 4.31D01 1.64

 9.28D01 2.31

 2.00D02 2.65

 4.31D02 2.71

 9.28D02 2.71

 2.00D03 2.72

 4.31D03 2.74

 9.28D03 2.76

 2.00D04 2.81

 4.31D04 2.89

 9.28D04 2.98

 2.00D05 3.09

 4 NOBWC

 2 0 IOWS IDPR--OBPIEZ PD1

 3.16D0 0.0D0 0.0D0 7.5D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN

 9.28D00 0.090 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.204

 4.31D01 0.384

 9.28D01 0.568

 2.00D02 0.662

 4.31D02 0.680

 9.28D02 0.683

 2.00D03 0.690

 4.31D03 0.703

 9.28D03 0.729

 2.00D04 0.775

 4.31D04 0.847

 9.28D04 0.942

 2.00D05 1.052

 2 0 IOWS IDPR--OBPIEZ PD2

 31.6D0 0.0D0 0.0D0 7.5D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN

 9.28D00 0.0000 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.0005

 4.31D01 0.0028

 9.28D01 0.0071
70 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

 2.00D02 0.0099

 4.31D02 0.0105

 9.28D02 0.0110

 2.00D03 0.0119

 4.31D03 0.0140

 9.28D03 0.0186

 2.00D04 0.0292

 4.31D04 0.0524

 9.28D04 0.0983

 2.00D05 0.1722

 2 0 IOWS IDPR--OBPIEZ PS1

 3.16D0 0.0D0 0.0D0 1.0D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN 

 9.28D00 0.007 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.019

 4.31D01 0.039

 9.28D01 0.060

 2.00D02 0.072

 4.31D02 0.077

 9.28D02 0.084

 2.00D03 0.098

 4.31D03 0.126

 9.28D03 0.177

 2.00D04 0.258

 4.31D04 0.365

 9.28D04 0.487

 2.00D05 0.612

 2 0 IOWS IDPR--OBPIEZ PS2

 31.6D0 0.0D0 0.0D0 1.0D0 0.0D0 0.0D0 R Z1 Z2 ZP RP XLL

 14 1 NTSOB IRUN

 9.28D00 0.0000 TIMEOB(ID) XMEASOB(ID)

 2.00D01 0.0001

 4.31D01 0.0005

 9.28D01 0.0012

 2.00D02 0.0017

 4.31D02 0.0019

 9.28D02 0.0022

 2.00D03 0.0028

 4.31D03 0.0042

 9.28D03 0.0075

 2.00D04 0.0160

 4.31D04 0.0375

 9.28D04 0.0844

 2.00D05 0.1624
Attachm

Extract file (prob3.ext):

# Open file wtaq.plt and read simulated drawdowns

<wtaq.plt

# Read simulated value for observation PD1-1

o PD1-1

+6

c33_42

# Read next simulated value (PD1-2)

o PD1-2

+1

c33_42

# And so forth....

o PD1-3

+1

c33_42

o PD1-4

+1

c33_42

o PD1-5

+1

c33_42

o PD1-6

+1

c33_42

o PD1-7

+1

c33_42

o PD1-8

+1

c33_42

o PD1-9

+1

c33_42

o PD1-10

+1

c33_42

o PD1-11

+1

c33_42

o PD1-12

+1

c33_42

o PD1-13

+1

c33_42

o PD1-14

+1

c33_42

# Now, read simulated values for PD2

o PD2-1

+6

c33_42
72 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

o PD2-2

+1

c33_42

o PD2-3

+1

c33_42

o PD2-4

+1

c33_42

o PD2-5

+1

c33_42

o PD2-6

+1

c33_42

o PD2-7

+1

c33_42

o PD2-8

+1

c33_42

o PD2-9

+1

c33_42

o PD2-10

+1

c33_42

o PD2-11

+1

c33_42

o PD2-12

+1

c33_42

o PD2-13

+1

c33_42

o PD2-14

+1

c33_42

# Now, read simulated values for PS1

o PS1-1

+6

c33_42

o PS1-2

+1

c33_42

o PS1-3

+1

c33_42

o PS1-4

+1

c33_42

o PS1-5

+1
Attachm

c33_42

o PS1-6

+1

c33_42

o PS1-7

+1

c33_42

o PS1-8

+1

c33_42

o PS1-9

+1

c33_42

o PS1-10

+1

c33_42

o PS1-11

+1

c33_42

o PS1-12

+1

c33_42

o PS1-13

+1

c33_42

o PS1-14

+1

c33_42

# Now, read simulated values for PS2

o PS2-1

+6

c33_42

o PS2-2

+1

c33_42

o PS2-3

+1

c33_42

o PS2-4

+1

c33_42

o PS2-5

+1

c33_42

o PS2-6

+1

c33_42

o PS2-7

+1

c33_42

o PS2-8

+1

c33_42

o PS2-9
74 WTAQ—A Program for Calculating Drawdowns and Estimating Hydraulic Properties for Confined and Water-Table Aqui

+1

c33_42

o PS2-10

+1

c33_42

o PS2-11

+1

c33_42

o PS2-12

+1

c33_42

o PS2-13

+1

c33_42

o PS2-14

+1

c33_42

END

Batch file to run WTAQ (prob3.bat):

rm wtaq.scratch

rm wtaq.out

rm wtaq.plt

wtaq.exe <prob3.in >wtaq.scratch

File containing names of input, result, and plot files used or generated by WTAQ (prob3.in):

prob3.sub

wtaq.out

wtaq.plt