ASTM D4105/D4105M-20

Standard Practice for (Analytical Procedure) for Determining Transmissivity and Storage Coefficient of Nonleaky Confined Aquifers by the Modified Theis Nonequilibrium Method

SIGNIFICANCE AND USE
5.1 Assumptions:  
5.1.1 Well discharges at a constant rate, Q.  
5.1.2 Well is of infinitesimal diameter and fully penetrates the aquifer, that is, the well is open to the full thickness of the aquifer.  
5.1.3 The nonleaky aquifer is homogeneous, isotropic, and areally extensive. A nonleaky aquifer receives insignificant contribution of water from confining beds.  
5.1.4 Discharge from the well is derived exclusively from storage in the aquifer.  
5.1.5 The geometry of the assumed aquifer and well conditions are shown in Fig. 1.    
5.2.3 Application of Theis Nonequilibrium Method to Unconfined Aquifers:  
5.2.3.1 Although the assumptions are applicable to confined conditions, the Theis solution may be applied to unconfined aquifers if drawdown is small compared with the saturated thickness of the aquifer or if the drawdown is corrected for reduction in thickness of the aquifer and the effects of delayed gravity yield are small.
5.2.3.2 Reduction in Aquifer Thickness—In an unconfined aquifer, dewatering occurs when the water levels decline in the vicinity of a pumping well. Corrections in drawdown need to be made when the drawdown is a significant fraction of the aquifer thickness as shown by Jacob (8). The drawdown, s, needs to be replaced by s′, the drawdown that would occur in an equivalent confined aquifer, where:
5.2.3.3 Gravity Yield Effects—In unconfined aquifers, delayed gravity yield effects may invalidate measurements of drawdown during the early part of the test for application to the Theis method. Effects of delayed gravity yield are negligible in partially penetrating observation wells at a distance, r, from the control well, where:
after the time, t, as given in the following equation from Neuman (9):
     where:
  Sy  =  the specific yield.  
For fully penetrating observation wells, the effects of delayed yield are negligible at the distance, r, in Eq 11 after one tenth of the time given in the Eq 12.
Note ...
SCOPE
1.1 This practice covers an analytical procedure for determining transmissivity and storage coefficient of a nonleaky confined aquifer under conditions of radial flow to a fully penetrating well of constant flux. This practice is a shortcut procedure used to apply the Theis nonequilibrium method. The Theis method is described in Practice D4106.  
1.2 This practice, along with others, is used in conjunction with the field procedure given in Test Method D4050.  
1.3 Limitations—The limitations of this practice are primarily related to the correspondence between the field situation and the simplifying assumptions of this practice (see 5.1). Furthermore, application is valid only for values of u less than 0.01 (u is defined in Eq 2, in 8.6).  
1.4 All observed and calculated values shall conform to the guidelines for significant digits and rounding established in Practice D6026.  
1.4.1 The procedures used to specify how data are collected/recorded or calculated, in this standard are regarded as the industry standard. In addition, they are representative of the significant digits that generally should be retained. The procedures used do not consider material variation, purpose for obtaining the data, special purpose studies, or any considerations for the user’s objectives; and it is common practice to increase or reduce significant digits of reported data to be commensurate with these considerations. It is beyond the scope of this standard to consider significant digits used in analytical methods for engineering design.  
1.5 Units—The values stated in either SI Units or inch-pound units are to be regarded separately as standard. The values in each system may not be exact equivalents; therefore each system shall be used independently of the other. Combining values from the two systems may result in non-conformance with the standard. Reporting of results in units other than SI shall not be regarded as nonconformance with...

General Information

• Status: Published
• Publication Date: 14-May-2020
• ICS: 93.160 - Hydraulic construction
• Technical Committee: D18 - Soil and Rock
• Drafting Committee: D18.21 - Groundwater and Vadose Zone Investigations

Relations

• Refers: ASTM D3740-23 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Nov-2023
• Refers: ASTM D4050-20 - Standard Test Method for (Field Procedure) for Withdrawal and Injection Well Testing for Determining Hydraulic Properties of Aquifer Systems
• Effective Date: 01-Jan-2020
• Refers: ASTM D3740-19 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Oct-2019
• Refers: ASTM D4043-17 - Standard Guide for Selection of Aquifer Test Method in Determining Hydraulic Properties by Well Techniques
• Effective Date: 01-Jan-2017
• Refers: ASTM D4106-15 - Standard Test Method for (Analytical Procedure) for Determining Transmissivity and Storage Coefficient of Nonleaky Confined Aquifers by the Theis Nonequilibrium Method
• Effective Date: 15-Apr-2015
• Refers: ASTM D653-14 - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• Effective Date: 01-Aug-2014
• Refers: ASTM D4050-14 - Standard Test Method for (Field Procedure) for Withdrawal and Injection Well Testing for Determining Hydraulic Properties of Aquifer Systems
• Effective Date: 01-Jun-2014
• Refers: ASTM D3740-12a - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-May-2012
• Refers: ASTM D3740-12 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Mar-2012
• Refers: ASTM D3740-11 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Sep-2011
• Refers: ASTM D653-11 - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• Effective Date: 01-Sep-2011
• Refers: ASTM D4043-96(2010)e1 - Standard Guide for Selection of Aquifer Test Method in Determining Hydraulic Properties by Well Techniques
• Effective Date: 01-Aug-2010
• Refers: ASTM D3740-10 - Standard Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in Engineering Design and Construction
• Effective Date: 01-Mar-2010
• Refers: ASTM D653-09 - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• Effective Date: 01-Jan-2009
• Refers: ASTM D653-08a - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• Effective Date: 01-Dec-2008

Overview

ASTM D4105/D4105M-20 establishes a standardized analytical practice for determining the transmissivity and storage coefficient of nonleaky confined aquifers using the Modified Theis Nonequilibrium Method. Issued by ASTM International, this standard provides a practical, shortcut approach for aquifer evaluation based on data from well tests conducted under constant discharge or controlled injection rates. The standard is widely applied in hydrogeology to assess groundwater resources, manage aquifer systems, and support engineering and environmental projects.

Key Topics

• Analytical Procedure: Outlines a step-by-step method for analyzing water level changes in observation wells in response to pumping or injection in a control well.
• Core Assumptions:
  • The aquifer is homogeneous, isotropic, and areally extensive.
  • The control well is fully penetrating and operates at a constant discharge rate.
  • The aquifer is nonleaky, with minimal contribution from confining layers.
  • Flow is radial, and drawdown responses are measured in strategically placed observation wells.
• Modified Theis Nonequilibrium Method: Applies the Theis solution to data from confined aquifers, with potential corrections for unconfined settings when drawdown is minimal relative to the aquifer thickness.
• Data Requirements and Calculations:
  • Drawdown is measured over time and at varying distances from the pumping well.
  • Graphical and mathematical procedures are used to estimate transmissivity and storage coefficient.
  • Calculations are valid only under specified conditions (e.g., for a dimensionless parameter "u" less than 0.01).
• Limitation and Considerations:
  • Results rely on how closely the real-world site matches the assumed conditions.
  • Corrections for aquifer thickness and gravity yield may be needed for unconfined aquifers.
  • Data reporting must follow industry practices for significant digits and unit consistency (SI or inch-pound units).

Applications

• Groundwater Resource Assessment: Fundamental for estimating the hydraulic properties of aquifers, enabling water supply planning and sustainable groundwater management.
• Engineering and Environmental Studies: Used by geotechnical engineers, hydrologists, and environmental consultants in site investigations, including contamination assessment, dewatering studies, and groundwater remediation.
• Regulatory Compliance and Permitting: Supports permitting processes for water well construction, withdrawal, and injection projects by providing industry-accepted hydraulic property data.
• Field Procedures Integration: Often used alongside ASTM D4050 for conducting the field component (well pumping or injection and observation) and ASTM D4106 for alternative analytical approaches.

Related Standards

• ASTM D653 - Standard Terminology Relating to Soil, Rock, and Contained Fluids
• ASTM D3740 - Practice for Agencies Engaged in Testing of Soil and Rock
• ASTM D4043 - Guide for Selection of Aquifer Test Method in Determining Hydraulic Properties by Well Techniques
• ASTM D4050 - Test Method for Field Procedures for Withdrawal and Injection Well Testing
• ASTM D4106 - Practice for Analytical Procedure for Determining Transmissivity and Storage Coefficient by the Theis Nonequilibrium Method
• ASTM D6026 - Practice for Using Significant Digits in Geotechnical Data

Practical Value

Implementing ASTM D4105/D4105M-20 ensures the accurate and consistent determination of key hydraulic properties critical for effective groundwater management. By following standardized methods, professionals can:

• Minimize data interpretation errors.
• Achieve regulatory acceptance of aquifer test results.
• Support robust hydrogeologic modeling.
• Enhance the design and implementation of infrastructure and remediation systems.

Professionals are encouraged to apply this standard in conjunction with other ASTM procedures and to consult the official ASTM document for detailed test protocols and supporting information.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: D4105/D4105M −20
Standard Practice for
(Analytical Procedure) for Determining Transmissivity and
Storage Coefficient of Nonleaky Confined Aquifers by the
Modified Theis Nonequilibrium Method
This standard is issued under the fixed designation D4105/D4105M; the number immediately following the designation indicates the
year of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last
reapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope* conformance with the standard. Reporting of results in units
other than SI shall not be regarded as nonconformance with
1.1 This practice covers an analytical procedure for deter-
this practice.
mining transmissivity and storage coefficient of a nonleaky
confined aquifer under conditions of radial flow to a fully 1.6 This practice offers a set of instructions for performing
penetrating well of constant flux. This practice is a shortcut one or more specific operations. This document cannot replace
procedureusedtoapplytheTheisnonequilibriummethod.The education or experience and should be used in conjunction
Theis method is described in Practice D4106. with professional judgment. Not all aspects of the practice may
be applicable in all circumstances. This ASTM standard is not
1.2 This practice, along with others, is used in conjunction
intended to represent or replace the standard of care by which
with the field procedure given in Test Method D4050.
the adequacy of a given professional service must be judged,
1.3 Limitations—The limitations of this practice are primar-
nor should this document be applied without the consideration
ily related to the correspondence between the field situation
of a project’s many unique aspects. The word “Standard” in
and the simplifying assumptions of this practice (see 5.1).
the title of this document means only that the document has
Furthermore, application is valid only for values of uless than
been approved through the ASTM consensus process.
0.01 (uis defined in Eq 2,in 8.6).
1.7 This standard does not purport to address all of the
1.4 All observed and calculated values shall conform to the
safety concerns, if any, associated with its use. It is the
guidelines for significant digits and rounding established in
responsibility of the user of this standard to establish appro-
Practice D6026.
priate safety, health, and environmental practices and deter-
1.4.1 Theproceduresusedtospecifyhowdataarecollected/
mine the applicability of regulatory limitations prior to use.
recorded or calculated, in this standard are regarded as the
1.8 This international standard was developed in accor-
industry standard. In addition, they are representative of the
dance with internationally recognized principles on standard-
significant digits that generally should be retained. The proce-
ization established in the Decision on Principles for the
dures used do not consider material variation, purpose for
Development of International Standards, Guides and Recom-
obtaining the data, special purpose studies, or any consider-
mendations issued by the World Trade Organization Technical
ations for the user’s objectives; and it is common practice to
Barriers to Trade (TBT) Committee.
increase or reduce significant digits of reported data to be
commensuratewiththeseconsiderations.Itisbeyondthescope
2. Referenced Documents
of this standard to consider significant digits used in analytical
2.1 ASTM Standards:
methods for engineering design.
D653Terminology Relating to Soil, Rock, and Contained
1.5 Units—The values stated in either SI Units or inch-
Fluids
pound units are to be regarded separately as standard. The
D3740Practice for Minimum Requirements for Agencies
values in each system may not be exact equivalents; therefore
Engaged in Testing and/or Inspection of Soil and Rock as
each system shall be used independently of the other. Combin-
Used in Engineering Design and Construction
ing values from the two systems may result in non-
D4043Guide for Selection of Aquifer Test Method in
Determining Hydraulic Properties by Well Techniques
This practice is under the jurisdiction of ASTM Committee D18 on Soil and
Rock and is the direct responsibility of Subcommittee D18.21 on Groundwater and
Vadose Zone Investigations. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved May 15, 2020. Published June 2020. Originally contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
approved in 1991. Last previous edition approved in 2015 as D4105–15. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/D4105_D4105M-20. the ASTM website.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4105/D4105M − 20
D4050Test Method for (Field Procedure) for Withdrawal and:
and Injection Well Testing for Determining Hydraulic 2y
` e
dy 5 W u 520.577216 2 log u (3)
Properties of Aquifer Systems * ~ !
e
u y
D4106Practice for (Analytical Procedure) for Determining
2 3 4
Transmissivity and Storage Coefficient of Nonleaky Con-
u u u
1u 2 1 2 1…
fined Aquifers by the Theis Nonequilibrium Method 2!2 3!3 4!4
D6026Practice for Using Significant Digits in Geotechnical
4.3 The sum of the terms to the right of log uin the series
e
Data
of Eq 3 is not significant when ubecomes small.
3. Terminology
NOTE 1—The errors for small values of u, from Kruseman and
DeRidder (1) are as follows:
3.1 Definitions:
Error less than, %: 1 2 5 10
3.1.1 For definitions of common technical terms in this
For u smaller than: 0.03 0.05 0.1 0.15
standard, refer to Terminology D653.
The value of udecreases with increasing time, t, and
3.2 Symbols and Dimensions:
decreases as the radial distance, r, decreases. Therefore, for
−1
3.2.1 K [LT ]—hydraulic conductivity.
large values of tand reasonably small values of r, the terms to
the right of log uin Eq 3 may be neglected as recognized by
3.2.2 K —hydraulic conductivity in the horizontal direc-
e
xy
tion. Theis (2)andJacob (3).TheTheisequationcanthenbewritten
as follows:
3.2.3 K —hydraulic conductivity in the vertical direction.
z
2 −1
Q S
3.2.4 T [L T ]—transmissivity.
s 5 20.577216 2 ln r (4)
F S DG
4πT 4Tt
3.2.5 S—dimensionless storage coefficient.
−1
from which it has been shown by Lohman (4) that
3.2.6 Ss [L ]—specific storage.
2.3Q
3.2.7 s [L]—drawdown.
T 5 (5)
4π∆s/∆log t
3 −1
3.2.8 Q [L T ]—discharge.
and:
3.2.9 r [L]—radial distance from control well.
2.3Q
3.2.10 t [T]—time.
T52 (6)
2π∆s/∆log r
3.2.11 b [L]—thickness of the aquifer.
where:
3.2.12 u—dimensionless time parameter.
∆s/∆log t = the drawdown (measured or projected) over
4. Summary of Practice
one log cycle of time, and
∆s/∆log r = the drawdown (measured or projected) over
4.1 This practice describes an analytical procedure for 10
one log cycle of radial distance from the
analyzing data collected during a withdrawal or injection well
control well.
test. The field procedure (see Test Method D4050) involves
pumping a control well at a constant rate and measuring the
5. Significance and Use
water level response in one or more observation wells or
piezometers. The water-level response in the aquifer is a
5.1 Assumptions:
function of the transmissivity and coefficient of storage of the
5.1.1 Well discharges at a constant rate, Q.
aquifer. Alternatively, the test can be performed by injecting
5.1.2 Well is of infinitesimal diameter and fully penetrates
water at a constant rate into the aquifer through the control
the aquifer, that is, the well is open to the full thickness of the
well.Analysisofbuildupofwaterlevelinresponsetoinjection
aquifer.
issimilartoanalysisofdrawdownofwaterlevelinresponseto
5.1.3 The nonleaky aquifer is homogeneous, isotropic, and
withdrawal in a confined aquifer. Drawdown of water level is
areally extensive. A nonleaky aquifer receives insignificant
analyzed by plotting drawdown against factors incorporating
contribution of water from confining beds.
either time or distance from the control well, or both, and
5.1.4 Discharge from the well is derived exclusively from
matching the drawdown response with a straight line.
storage in the aquifer.
5.1.5 The geometry of the assumed aquifer and well condi-
4.2 Solution—The solution given by Theis (1) can be
tions are shown in Fig. 1.
expressed as follows:
2y
5.2 Implications of Assumptions:
Q ` e
s 5 dy (1)
*
u 5.2.1 Implicitintheassumptionsaretheconditionsofradial
4πT y
flow. Vertical flow components are induced by a control well
where:
that partially penetrates the aquifer, that is, not open to the
r S
aquifer through its full thickness. If the control well does not
u 5 (2)
4Tt
fully penetrate the aquifer, the nearest piezometer or partially
penetrating observation well should be located at a distance, r,
beyond which vertical flow components are negligible, where
The boldface numbers in parentheses refer to a list of references at the end of
this standard. according to Reed (5)
D4105/D4105M − 20
aquifer thickness as shown by Jacob (8). The drawdown, s,
needs to be replaced by s', the drawdown that would occur in
an equivalent confined aquifer, where:
s
s' 5 s 2 (10)
2b
5.2.3.3 Gravity Yield Effects—In unconfined aquifers, de-
layed gravity yield effects may invalidate measurements of
drawdownduringtheearlypartofthetestforapplicationtothe
Theismethod.Effectsofdelayedgravityyieldarenegligiblein
partiallypenetratingobservationwellsatadistance, r,fromthe
control well, where:
b
r 5 (11)
FIG. 1 Cross Section Through a Discharging Well in a Nonleaky
K
z
Confined Aquifer
Œ
K
xy
after the time, t, as given in the following equation from
1.5b
Neuman (9):
r 5 (7)
K
r
z
Πt 5 10S (12)
y
K T
xy
This section applies to distance-drawdown calculations of where:
transmissivity and storage coefficient and time-drawdown cal-
S = the specific yield.
y
culations of storage coefficient. If possible, compute transmis-
For fully penetrating observation wells, the effects of de-
sivity from time-drawdown data from wells located within a
layed yield are negligible at the distance, r,in Eq 11 after one
distance, r, of the pumped well using data measured after the
tenth of the time given in the Eq 12.
effectsofpartialpenetrationhavebecomeconstant.Thetimeat
which this occurs is given by Hantush (6) by: NOTE 2—The quality of the result produced by this standard is
dependent on the competence of the personnel performing it, and the
t 5 b s/2T ~K /K ! (8)
z r
suitability of the equipment and facilities used. Agencies that meet the
criteria of Practice D3740 are generally considered capable of competent
Fully penetrating observation wells may be placed at less
and objective testing/sampling/inspection/etc. Users of this standard are
than distance r from the control well. Observation wells may
cautioned that compliance with Practice D3740 does not in itself assure
beonthesameoronvariousradiallinesfromthecontrolwell.
reliable results. Reliable results depend on many factors; Practice D3740
provides a means of evaluating some of those factors.
5.2.2 The Theis method assumes the control well is of
NOTE 3—The injection of water into an aquifer may be regulated or
infinitesimal diameter. Also, it assumes that the water level in
require regulatory approvals. Withdrawal of contaminated waters may
the control well is the same as in the aquifer contiguous to the
require that the removed water be properly treated prior to discharge.
well. In practice these assumptions
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
´1
Designation: D4105/D4105M − 15 D4105/D4105M − 20
Standard Test Method Practice for
(Analytical Procedure) for Determining Transmissivity and
Storage Coefficient of Nonleaky Confined Aquifers by the
Modified Theis Nonequilibrium Method
This standard is issued under the fixed designation D4105/D4105M; the number immediately following the designation indicates the
year of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last
reapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Editorially corrected designation to match the units of measurement statement in September 2015.
1. Scope*
1.1 This test method covers an analytical procedure for determining transmissivity and storage coefficient of a nonleaky
confined aquifer under conditions of radial flow to a fully penetrating well of constant flux. This test method is a shortcut procedure
used to apply the Theis nonequilibrium method. The Theis method is described in Test Method D4106.
1.2 This test method, along with others, is used in conjunction with the field procedure given in Test Method D4050.
1.3 Limitations—The limitations of this test method are primarily related to the correspondence between the field situation and
the simplifying assumptions of this test method (see 5.1). Furthermore, application is valid only for values of u less than 0.01 (u is
defined in Eq 2, in 8.6).
1.4 All observed and calculated values shall conform to the guidelines for significant digits and rounding established in Practice
D6026.
1.4.1 The procedures used to specify how data are collected/recorded or calculated, in this standard are regarded as the industry
standard. In addition, they are representative of the significant digits that generally should be retained. The procedures used do not
consider material variation, purpose for obtaining the data, special purpose studies, or any considerations for the user’s objectives;
and it is common practice to increase or reduce significant digits of reported data to be commensurate with these considerations.
It is beyond the scope of this standard to consider significant digits used in analytical methods for engineering design.
1.5 Units—The values stated in either SI Units or inch-pound units are to be regarded separately as standard. The values in each
system may not be exact equivalents; therefore each system shall be used independently of the other. Combining values from the
two systems may result in non-conformance with the standard. Reporting of test results in units other than SI shall not be regarded
as nonconformance with this test method.
1.6 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory
limitations prior to use.
2. Referenced Documents
2.1 ASTM Standards:
D653 Terminology Relating to Soil, Rock, and Contained Fluids
D3740 Practice for Minimum Requirements for Agencies Engaged in Testing and/or Inspection of Soil and Rock as Used in
Engineering Design and Construction
D4043 Guide for Selection of Aquifer Test Method in Determining Hydraulic Properties by Well Techniques
D4050 Test Method for (Field Procedure) for Withdrawal and Injection Well Testing for Determining Hydraulic Properties of
Aquifer Systems
D4106 Practice for (Analytical Procedure) for Determining Transmissivity and Storage Coefficient of Nonleaky Confined
This test method practice is under the jurisdiction of ASTM Committee D18 on Soil and Rock and is the direct responsibility of Subcommittee D18.21 on Groundwater
and Vadose Zone Investigations.
Current edition approved April 15, 2015May 15, 2020. Published May 2015June 2020. Originally approved in 1991. Last previous edition approved in 20082015 as
D4105 – 96D4105 – 15. (2008). DOI: 10.1520/D4105_D4105M-15E01.10.1520/D4105_D4105M-20.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
D4105/D4105M − 20
Aquifers by the Theis Nonequilibrium Method
D6026 Practice for Using Significant Digits in Geotechnical Data
3. Terminology
3.1 Definitions:
3.1.1 For common definitions of terms in this standard, refer to Terminology D653.
3.2 Symbols and Dimensions:
−1
3.2.1 K [LT ]—hydraulic conductivity.
3.2.2 K —hydraulic conductivity in the horizontal direction.
xy
3.2.3 K —hydraulic conductivity in the vertical direction.
z
2 −1
3.2.4 T [L T ]—transmissivity.
3.2.5 S—dimensionless storage coefficient.
−1
3.2.6 Ss [L ]—specific storage.
3.2.7 s [L]—drawdown.
3 −1
3.2.8 Q [L T ]—discharge.
3.2.9 r [L]—radial distance from control well.
3.2.10 t [T]—time.
3.2.11 b [L]—thickness of the aquifer.
3.2.12 u—dimensionless time parameter.
4. Summary of Test Method
4.1 This test method describes an analytical procedure for analyzing data collected during a withdrawal or injection well test.
The field procedure (see Test Method D4050) involves pumping a control well at a constant rate and measuring the water level
response in one or more observation wells or piezometers. The water-level response in the aquifer is a function of the transmissivity
and coefficient of storage of the aquifer. Alternatively, the test can be performed by injecting water at a constant rate into the aquifer
through the control well. Analysis of buildup of water level in response to injection is similar to analysis of drawdown of water
level in response to withdrawal in a confined aquifer. Drawdown of water level is analyzed by plotting drawdown against factors
incorporating either time or distance from the control well, or both, and matching the drawdown response with a straight line.
4.2 Solution—The solution given by Theis (1) can be expressed as follows:
2y
Q ` e
s 5 * dy (1)
u
4πT y
where:
r S
u 5 (2)
4Tt
and:
2y
` e
dy 5 W u 520.5772162log u (3)
* ~ !
e
u y
2 3 4
u u u
1u 2 1 2 1…
2!2 3!3 4!4
4.3 The sum of the terms to the right of log u in the series of Eq 3 is not significant when u becomes small.
e
NOTE 1—The errors for small values of u, from Kruseman and DeRidder (1) are as follows:
Error less than, %: 1 2 5 10
For u smaller than: 0.03 0.05 0.1 0.15
The value of u decreases with increasing time, t, and decreases as the radial distance, r, decreases. Therefore, for large values
of t and reasonably small values of r, the terms to the right of log u in Eq 3 may be neglected as recognized by Theis (2) and Jacob
e
(3). The Theis equation can then be written as follows:
Q S
s 5 20.577216 2 ln r (4)
F S DG
4πT 4Tt
The boldface numbers in parentheses refer to a list of references at the end of this standard.
D4105/D4105M − 20
from which it has been shown by Lohman (4) that
2.3Q
T 5 (5)
4πΔs/Δlog t
and:
2.3Q
T 52 (6)
2πΔs/Δlog r
where:
Δs/Δlog t = the drawdown (measured or projected) over one log cycle of time, and
Δs/Δlog r = the drawdown (measured or projected) over one log cycle of radial distance from the control well.
5. Significance and Use
5.1 Assumptions:
5.1.1 Well discharges at a constant rate, Q.
5.1.2 Well is of infinitesimal diameter and fully penetrates the aquifer, that is, the well is open to the full thickness of the aquifer.
5.1.3 The nonleaky aquifer is homogeneous, isotropic, and areally extensive. A nonleaky aquifer receives insignificant
contribution of water from confining beds.
5.1.4 Discharge from the well is derived exclusively from storage in the aquifer.
5.1.5 The geometry of the assumed aquifer and well conditions are shown in Fig. 1.
5.2 Implications of Assumptions:
5.2.1 Implicit in the assumptions are the conditions of radial flow. Vertical flow components are induced by a control well that
partially penetrates the aquifer, that is, not open to the aquifer through its full thickness. If the control well does not fully penetrate
the aquifer, the nearest piezometer or partially penetrating observation well should be located at a distance, r, beyond which vertical
flow components are negligible, where according to Reed (5)
1.5b
r 5 (7)
K
z
Œ
K
xy
This section applies to distance-drawdown calculations of transmissivity and storage coefficient and time-drawdown calculations
of storage coefficient. If possible, compute transmissivity from time-drawdown data from wells located within a distance, r, of the
pumped well using data measured after the effects of partial penetration have become constant. The time at which this occurs is
given by Hantush (6) by:
t 5 b s/2T K /K (8)
~ !
z r
Fully penetrating observation wells may be placed at less than distance r from the control well. Observation wells may be on
the same or on various radial lines from the control well.
5.2.2 The Theis method assumes the control well is of infinitesimal diameter. Also, it assumes that the water level in the control
well is the same as in the aquifer contiguous to the well. In practice these assumptions may cause a difference between the
theoretical drawdown and field measurements of drawdown in the early part of the test and in and near the control well. Control
well storage is negligible after a time, t, given by the following equation after weeks (7).
25 r
c
t 5 (9)
T
FIG. 1 Cross Section Through a Discharging Well in a Nonleaky Confined Aquifer
D4105/D4105M − 20
where:
r = the radius of the control well in the interval that includes the water level changes.
c
5.2.3 Application of Theis Nonequilibrium Method to Unconfined Aquifers:
5.2.3.1 Although the assumptions are applicable to confined conditions, the Theis solution may be applied to unconfined
aquifers if drawdown is small compared with the saturated thickness of the aquifer or if the drawdown is corrected for reduction
in thickness of the aquifer and the effects of delayed gravity yield are small.
5.2.3.2 Reduction in Aquifer Thickness—In an unconfined aquifer, dewatering occurs when the water levels decline in the
vicinity of a pumping well. Corrections in drawdown need to be made when the drawdown is a significant fraction of the aquifer
thickness as shown by Jacob (8). The drawdown, s, needs to be replaced by s', the drawdown that would occur in an equivalent
confined aquifer, where:
s
s'5 s 2 (10)
2b
5.2.3.3 Gravity Yield Effects—In unconfined aquifers, delayed gravity yield effects may invalidate measurements of drawdown
during the early part of the test for application to the Theis method. Effects of delayed gravity yield are negligible in partially
penetrating observation wells at a distance, r, from the control well, where:
b
r 5 (11)
K
z
Œ
K
xy
after the time, t, as given in the following equation from Neuman (9):
r
t 5 10S (12)
y
T
where:
S = the specific yield.
y
For fully penetrating observation wells, the effects of delayed yield are negligible at the distance, r, in Eq 11 after one tenth of
the time given in the Eq 12.
NOTE 2—The quality of the result produced by this standard is dependent on the competence of the personnel performing it, and the suitability of the
equipment and facilities used. Agencies that meet the criteria of Practice D3740 are generally considered capable of competent and objective
testing/sampling/inspection/etc. Users of this standard are cautioned that compliance with Practice D3740 does not in itself assure reliable results. Reliable
results depend on many factors; Practice D3740 provides a means of evaluating some of those factors.
NOTE 3—The injection of water into an aquifer may be regulated or require regulatory approvals. Withdrawal of contaminated waters may require that
the removed water be properly treated prior to discharge.
6. Apparatus
6.1 Analysis of data from the field procedure (see Test Method D4050) by this test method requires that the control well and
observation wells meet the requirements specified in 6.2 – 6.4.
6.2 Control Well—Screen the control well in the aquifer and equip with a pump capable of discharging water from the well at
a constant rate for the duration of the test. Preferably, screen the control well throughout the full thickness of the aquifer. If the
control well partially penetrates the aquifer, take special precaution in the placement or design of observation wells (see 5.2.1).
6.3 Observation Wells—Construct one or more observation wells or piezometers at a distance from the control well. Observation
wells may be partially open or fully open throughout the thickness of the aquifer.
6.4 Location of Observation Wells—Locate observation wells at various distances from the control well within the area of
influence of pumping. However, if vertical flow components are significant and if partially penetrating observation wells are used,
locate them at a distance beyond the effect of vertical flow components (see 5.2.1). If the aquifer is unconfined, constraints are
imposed on the distance to partially penetrating observation wells and the validity of early time measurements (see 5.2.3).
7. Procedure
7.1 The overall procedure consists of conducting the field procedure for withdrawal or injection well tests described in Test
Method D4050 and analysis of the field data as addressed in this test method.
7.2 Use a graphical procedure to solve for transmissivity and coefficient of storage as described in 8.2.
8. Calculation
8.1 Plot drawdown, s, at a specified distance on the arithmetic scale and time, t, on the logarithmic scale.
8.2 Plot drawdown, s, for several observation wells at a specified time on the arithmetic scale and distance on the logarithmic
scale.
D4105/D4105M − 20
8.3 For convenience in calculations, by choosing drawdown, Δ s , as that which occurs over one log cycle of time:
t
t
Δ log t 5 log 5 1 (13)
S D
10 10
t
and, similarly for convenience in calculations, by choosing the drawdown, Δs , as that which occurs over one log cycle of
r
distance,
r
Δ log r 5 log 5 1 (14)
S D
10 10
r
8.4 Calculate transmissivity using the semilog plot of drawdown versus time by the following equation derived from Eq 5:
t 5 2.3Q/2πΔs (15)
r
or calculate transmissivity using the semilog plot of drawdown versus radial distance from control well by the following
equation derived from Eq 6:
2.3Q
T 52 (16)
2πΔs
r
8.5 Determine the coefficient of storage from these semilog plots of drawdown versus time or distance by a method proposed
by Jacob (2) where:
2.3Q 2.25Tt
s 5 log (17)
S D
4πT r S
Taking s = 0 at the zero-drawdown intercept of the straight-line semilog plot of time or distance versus drawdown,
2.25Tt
S 5 (18)
r
where:
eitherrort = the value at the zero-drawdown intercept.
8.6 To apply the modified Theis nonequilibrium method to thin unconfined aquifers, where the drawdown is a significant
fraction of the initial saturated thickness, apply a correction to the drawdown in solving for T and S (see 5.2.3.2).
8.7 This test method is applicable only for values of u < 0.01, that is:
r S
u 5 ,0.01 (19)
4Tt
It is seen from Eq 19 that u decreases as time increases, other things being equal. Because S is in the numerator, the value of
−5 −3
u is much smaller for a confined aquifer, whose storage coefficient may range from only about 10 to 10 , than for an unconfined
aquifer, whose specific yield may be from 0.1 to 0.3. To compensate for this, t must be greater by several orders of magnitude in
testing an unconfined aquifer than testing a confined aquifer.
8.7.1 In a drawdown-time test (s versus log t or log t/r ), data points for any particular distance will begin to fall on a straight
10 10
line only after the time is sufficiently long to satisfy the above criteria. In a drawdown-distance test (s versus log r), the well must
be pumped long enough that the data for the most distant observation well satisfy the requirements; then only the drawdowns at
or after this value of t may be analyzed on a semilogarithmic plot for one particular value of t.
NOTE 4—The analyst may also find it useful to analyze the data using the Theis nonequilibrium procedure (see Test Method D4106).
NOTE 5—Commercially available software is available from several sources that can perform the calculation and plotting.
9. Report: Test Data Sheets/Forms
9.1 Report as a minimum the information described below. The report of the analytical procedure will include information from
the report on test method selection (see Guide D4043) and the field testing procedure (see Test Method D4050).
9.1.1 Introduction—The introductory section is intended to present the scope and purpose of the recovery method for
determining transmissivity and storativity in a nonleaky confined aquifer. Summarize the field hydrogeologic conditions and the
field equipment and instrumentation including the construction of the control well and observation wells and piezometers, the
method of measurement of discharge and water levels, and the duration of the test and pumping rate. Discuss rationale for selecting
the modified Theis method.
9.1.2 Hydrogeologic Setting—Review the information available on the hydrogeology of the site; interpret and describe the
hydrogeology of the site as it pertains to the selection of this method for conducting and analyzing an aquifer test. Compare the
hydrogeologic characteristics of the site as it conforms and differs from the assumptions in the solution to the aquifer test method.
9.1.3 Equipment—Report the field installation and equipment for the aquifer test, including the construction, diameter, depth of
screened interval, and location of control well and pumping equipment, and the construction, diameter, depth, and screened interval
of observation wells.
D4105/D4105M − 20
9.1.4 Describe the methods of observing water levels, pumping rate, barometric changes, and other environmental conditions
pertinent to the test. Include a list of measuring devices used during the test, the manufacturers name, model number, and basic
specifications for each major item, and the name and date and method of the last calibration, if applicable.
9.1.5 Testing Procedures—State the steps taken in conducting pre-test, drawdown, and recovery phases of the test. Include the
date, clock time, and time since pumping started or stopped for measurements of discharge rate, water levels, and other
environmental data recorded during the testing procedure.
9.1.6 Presentation and Interpretation of Test Results:
9.1.6.1 Data—Present tables of data collected during the test. Show methods of adjusting water levels for barometric changes
and calculation of drawdown and residual drawdown.
9.1.6.2 Data Plots—Present data plots used in analysis of the data.
9.1.6.3 Evaluate qualitatively the determinations of transmissivity and coefficient of storage on the basis of the adequacy of
instrumentation, observations of stress and response, and the conformance of the hydrogeologic conditions, and the performance
of the test to the assumptions of the method.
10. Precision and Bias
10.1 Precision—Test data on precision is not presented due to the nature of the material (groundwater) tested by this test
method. It is either not feasible or too costly at this time to have ten or more laboratories participated in a round-robin testing
program. It is not practicable to specify the precision of this test method because the response of aquifer systems during aquifer
tests is dependent upon ambient system stresses.
10.2 Bias—There is no accepted reference value for this test method, therefore bias cannot be determined. No statement can be
made about bias because no true reference values exist.
11. Keywords
11.1 aquifer tests; aquifers; confined aquifers; control wells; groundwater; hydraulic properties; observation wells; storage
coefficient; transmissivity; unconfined aquifers
REFERENCES
(1) Kruseman, G. P., and DeRidder, N. A., “Analysis and Evaluation of Pumping Test Data,” ILRI Publication 47, 1990, p. 377.
(2) Theis, C. V., “The Relation Between the Lowering of the Piezometric Surface and the Rate and Duration of Discharge of a Well Using Ground-Water
Storage,” American Geophysical Union Transactions, Vol 16, Part 2, 1935, pp. 519–524.
(3) Jacob, C. E., “Flow of Ground Water,” in Engineering Hydraulics, Proceedings of the Fourth Hydraulics Conference, June 12–15, 1949, New York,
John Wiley and Sons, Inc., 1950, pp. 321–386.
(4) Lohman, S. W., “Ground-Water Hydraulics,” U.S. Geological Survey Professional Paper 708, 1972.
(5) Reed, J. E., “Type Curves for Selected Problems of Flow to Wells in Confined Aquifers,” U.S. Geological Survey Techniques of Water-Resources
Investigations, Book 3, Chapter B3, 1980.
(6) Hantush, M. S., and Jacob, C. E., “Non-Steady Radial Flow in an Infinite Leaky Aquifer,” American Geophysical Union Transactions, Vol 36, No.
1, 1955, pp. 95–100.
(7) Papadopulos, S. S., and Cooper, H. H., Jr., “Drawdown in a Well of Large Diameter,” Water Resources Research, Vol 1, 1967, pp. 241–244.
(8) Jacob, C. E., “Determining Permeability of Water-Table Aquifers,” in Bentall, Ray, compiler, Methods of Determining Permeability, Transmissibility,
and Drawdown, U.S. Geological Survey Water-Supply Paper 1536-I, 1963, pp. 272–292.
(9) Neuman, S. P., “Effect of Partial Penetration on Flow in Unconfined Aquifers Considering Delayed Gravity Response,” Water Resources Research
, Vol 10, No. 2, 1974, pp. 303–312.
D4105/D4105M − 20
SUMMARY OF CHANGES
In accordance with Committee D18 policy, this section identifies the location of changes to this standard since
the last edition (1996 (Reapproved 2008)) that may impact the use of this standard.
(1) Deleted terminology that is already in D653.
(2) Added D653 and D6026 to list of referenced documents.
(3) Added SI units notes, D3740 notes.
(4) Revised Precision and Bias to current format.
(5) Edits made throughout to comply with the D18 Procedures Preparation Manual.
(6) Added new note on commercially available software for calculations and plotting.
ASTM International takes no position respecting the validity of any patent rights asserted in connection with any item mentioned
in this standard. Users of this standard are expressly advised that determination of the validity of any such patent rights, and the risk
of infringement of such rights, are entirely their own responsibility.
This standard is subject to revision at any time by the responsible technical committee and must be reviewed every five years and
if not revised, either reapproved or withdrawn. Your comments are invited either for revision of this standard or for additional standards
and should be addressed to ASTM Intern
...