# ASTM D4105D4105M-15e1

Designation: D4105/D4105M − 15´1Standard Test Method for(Analytical Procedure) for Determining Transmissivity andStorage Coefficient of Nonleaky Confined Aquifers by theModified Theis Nonequilibrium Method1This standard is issued under the fixed designation D4105/D4105M; the number immediately following the designation indicates theyear of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of lastreapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.ε1NOTE—Editorially corrected designation to match the units of measurement statement in September 2015.1. Scope*1.1 This test method covers an analytical procedure fordetermining transmissivity and storage coefficient of a non-leaky confined aquifer under conditions of radial flow to a fullypenetrating well of constant flux. This test method is a shortcutprocedure used to apply the Theis nonequilibrium method. TheTheis method is described in Test Method D4106.1.2 This test method, along with others, is used in conjunc-tion with the field procedure given in Test Method D4050.1.3 Limitations—The limitations of this test method areprimarily related to the correspondence between the fieldsituation and the simplifying assumptions of this test method(see 5.1). Furthermore, application is valid only for values ofu less than 0.01 (u is defined in Eq 2,in8.6).1.4 All observed and calculated values shall conform to theguidelines for significant digits and rounding established inPractice D6026.1.4.1 The procedures used to specify how data are collected/recorded or calculated, in this standard are regarded as theindustry standard. In addition, they are representative of thesignificant digits that generally should be retained. The proce-dures used do not consider material variation, purpose forobtaining the data, special purpose studies, or any consider-ations for the user’s objectives; and it is common practice toincrease or reduce significant digits of reported data to becommensurate with these considerations. It is beyond the scopeof this standard to consider significant digits used in analyticalmethods for engineering design.1.5 Units—The values stated in either SI Units or inch-pound units are to be regarded separately as standard. Thevalues in each system may not be exact equivalents; thereforeeach system shall be used independently of the other. Combin-ing values from the two systems may result in non-conformance with the standard. Reporting of test results inunits other than SI shall not be regarded as nonconformancewith this test method.1.6 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2D653 Terminology Relating to Soil, Rock, and ContainedFluidsD3740 Practice for Minimum Requirements for AgenciesEngaged in Testing and/or Inspection of Soil and Rock asUsed in Engineering Design and ConstructionD4043 Guide for Selection of Aquifer Test Method inDetermining Hydraulic Properties by Well TechniquesD4050 Test Method for (Field Procedure) for Withdrawaland Injection Well Testing for Determining HydraulicProperties of Aquifer SystemsD4106 Test Method for (Analytical Procedure) for Deter-mining Transmissivity and Storage Coefficient of Non-leaky Confined Aquifers by the Theis NonequilibriumMethodD6026 Practice for Using Significant Digits in GeotechnicalData3. Terminology3.1 Definitions:3.1.1 For common definitions of terms in this standard, referto Terminology D653.1This test method is under the jurisdiction ofASTM Committee D18 on Soil andRock and is the direct responsibility of Subcommittee D18.21 on Groundwater andVadose Zone Investigations.Current edition approved April 15, 2015. Published May 2015. Originallyapproved in 1991. Last previous edition approved in 2008 as D4105 – 96 (2008).DOI: 10.1520/D4105_D4105M-15E01.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.*A Summary of Changes section appears at the end of this standardCopyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States13.2 Symbols and Dimensions:3.2.1 K [LT−1]—hydraulic conductivity.3.2.2 Kxy—hydraulic conductivity in the horizontal direc-tion.3.2.3 Kz—hydraulic conductivity in the vertical direction.3.2.4 T [L2T−1]—transmissivity.3.2.5 S—dimensionless storage coefficient.3.2.6 Ss [L−1]—specific storage.3.2.7 s [L]—drawdown.3.2.8 Q [L3T−1]—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 Method4.1 This test method describes an analytical procedure foranalyzing data collected during a withdrawal or injection welltest. The field procedure (see Test Method D4050) involvespumping a control well at a constant rate and measuring thewater level response in one or more observation wells orpiezometers. The water-level response in the aquifer is afunction of the transmissivity and coefficient of storage of theaquifer. Alternatively, the test can be performed by injectingwater at a constant rate into the aquifer through the controlwell.Analysis of buildup of water level in response to injectionis similar to analysis of drawdown of water level in response towithdrawal in a confined aquifer. Drawdown of water level isanalyzed by plotting drawdown against factors incorporatingeither time or distance from the control well, or both, andmatching the drawdown response with a straight line.4.2 Solution—The solution given by Theis (1)3can beexpressed as follows:s 5Q4πT*u` e2yydy (1)where:u 5r2S4Tt(2)and:*u` e2yydy 5 W~u! 520.577216 2 logeu (3)1u 2u22!21u33!32u44!41…4.3 The sum of the terms to the right of logeu in the seriesof Eq 3 is not significant when u becomes small.NOTE 1—The errors for small values of u, from Kruseman andDeRidder (1) are as follows:Error less than, %: 1 2 5 10For u smaller than: 0.03 0.05 0.1 0.15The value of u decreases with increasing time, t, anddecreases as the radial distance, r, decreases. Therefore, forlarge values of t and reasonably small values of r, the terms tothe right of logeu in Eq 3 may be neglected as recognized byTheis (2) and Jacob (3). The Theis equation can then be writtenas follows:s 5Q4πTF20.577216 2 lnSr2S4TtDG(4)from which it has been shown by Lohman (4) thatT 52.3Q4π∆s/∆log10t(5)and:T 522.3Q2π∆s/∆log10r(6)where:∆s/∆log10t = the drawdown (measured or projected) overone log cycle of time, and∆s/∆log10r = the drawdown (measured or projected) overone log cycle of radial distance from thecontrol well.5. Significance and Use5.1 Assumptions:5.1.1 Well discharges at a constant rate, Q.5.1.2 Well is of infinitesimal diameter and fully penetratesthe aquifer, that is, the well is open to the full thickness of theaquifer.5.1.3 The nonleaky aquifer is homogeneous, isotropic, andareally extensive. A nonleaky aquifer receives insignificantcontribution of water from confining beds.5.1.4 Discharge from the well is derived exclusively fromstorage in the aquifer.5.1.5 The geometry of the assumed aquifer and well condi-tions are shown in Fig. 1.5.2 Implications of Assumptions:5.2.1 Implicit in the assumptions are the conditions of radialflow. Vertical flow components are induced by a control wellthat partially penetrates the aquifer, that is, not open to theaquifer through its full thickness. If the control well does not3The boldface numbers in parentheses refer to a list of references at the end ofthis standard.FIG. 1 Cross Section Through a Discharging Well in a NonleakyConfined AquiferD4105/D4105M − 15´12fully penetrate the aquifer, the nearest piezometer or partiallypenetrating observation well should be located at a distance, r,beyond which vertical flow components are negligible, whereaccording to Reed (5)r 51.5bŒKzKxy(7)This section applies to distance-drawdown calculations oftransmissivity and storage coefficient and time-drawdown cal-culations of storage coefficient. If possible, compute transmis-sivity from time-drawdown data from wells located within adistance, r, of the pumped well using data measured after theeffects of partial penetration have become constant. The time atwhich this occurs is given by Hantush (6) by:t 5 b2s/2T ~Kz/Kr! (8)Fully penetrating observation wells may be placed at lessthan distance r from the control well. Observation wells maybe on the same or on various radial lines from the control well.5.2.2 The Theis method assumes the control well is ofinfinitesimal diameter. Also, it assumes that the water level inthe control well is the same as in the aquifer contiguous to thewell. In practice these assumptions may cause a differencebetween the theoretical drawdown and field measurements ofdrawdown in the early part of the test and in and near thecontrol well. Control well storage is negligible after a time, t,given by the following equation after weeks (7).t 525 rc2T(9)where:rc= the radius of the control well in the interval that includesthe water level changes.5.2.3 Application of Theis Nonequilibrium Method to Un-confined Aquifers:5.2.3.1 Although the assumptions are applicable to confinedconditions, the Theis solution may be applied to unconfinedaquifers if drawdown is small compared with the saturatedthickness of the aquifer or if the drawdown is corrected forreduction in thickness of the aquifer and the effects of delayedgravity yield are small.5.2.3.2 Reduction in Aquifer Thickness—In an unconfinedaquifer, dewatering occurs when the water levels decline in thevicinity of a pumping well. Corrections in drawdown need tobe made when the drawdown is a significant fraction of theaquifer thickness as shown by Jacob (8). The drawdown, s,needs to be replaced by s , the drawdown that would occur inan equivalent confined aquifer, where:s 5 s 2s22b(10)5.2.3.3 Gravity Yield Effects—In unconfined aquifers, de-layed gravity yield effects may invalidate measurements ofdrawdown during the early part of the test for application to theTheis method. Effects of delayed gravity yield are negligible inpartially penetrating observation wells at a distance, r, from thecontrol well, where:r 5bŒKzKxy(11)after the time, t, as given in the following equation fromNeuman (9):t 5 10Syr2T(12)where:Sy= the specific yield.For fully penetrating observation wells, the effects of de-layed yield are negligible at the distance, r,inEq 11 after onetenth of the time given in the Eq 12.NOTE 2—The quality of the result produced by this standard isdependent on the competence of the personnel performing it, and thesuitability of the equipment and facilities used. Agencies that meet thecriteria of Practice D3740 are generally considered capable of competentand objective testing/sampling/inspection/etc. Users of this standard arecautioned that compliance with Practice D3740 does not in itself assurereliable results. Reliable results depend on many factors; Practice D3740provides a means of evaluating some of those factors.NOTE 3—The injection of water into an aquifer may be regulated orrequire regulatory approvals. Withdrawal of contaminated waters mayrequire that the removed water be properly treated prior to discharge.6. Apparatus6.1 Analysis of data from the field procedure (see TestMethod D4050) by this test method requires that the controlwell and observation wells meet the requirements specified in6.2 – 6.4.6.2 Control Well—Screen the control well in the aquifer andequip with a pump capable of discharging water from the wellat a constant rate for the duration of the test. Preferably, screenthe control well throughout the full thickness of the aquifer. Ifthe control well partially penetrates the aquifer, take specialprecaution in the placement or design of observation wells (see5.2.1).6.3 Observation Wells—Construct one or more observationwells or piezometers at a distance from the control well.Observation wells may be partially open or fully open through-out the thickness of the aquifer.6.4 Location of Observation Wells—Locate observationwells at various distances from the control well within the areaof influence of pumping. However, if vertical flow componentsare significant and if partially penetrating observation wells areused, locate them at a distance beyond the effect of verticalflow components (see 5.2.1). If the aquifer is unconfined,constraints are imposed on the distance to partially penetratingobservation wells and the validity of early time measurements(see 5.2.3).7. Procedure7.1 The overall procedure consists of conducting the fieldprocedure for withdrawal or injection well tests described inTest Method D4050 and analysis of the field data as addressedin this test method.7.2 Use a graphical procedure to solve for transmissivityand coefficient of storage as described in 8.2.D4105/D4105M − 15´138. Calculation8.1 Plot drawdown, s, at a specified distance on the arith-metic scale and time, t, on the logarithmic scale.8.2 Plot drawdown, s, for several observation wells at aspecified time on the arithmetic scale and distance on thelogarithmic scale.8.3 For convenience in calculations, by choosingdrawdown, ∆ st, as that which occurs over one log cycle oftime:∆ log10t 5 log10St2t1D5 1 (13)and, similarly for convenience in calculations, by choosingthe drawdown, ∆sr, as that which occurs over one log cycle ofdistance,∆ log10r 5 log10Sr2r1D5 1 (14)8.4 Calculate transmissivity using the semilog plot of draw-down versus time by the following equation derived from Eq 5:t 5 2.3Q/2π∆sr(15)or calculate transmissivity using the semilog plot of draw-down versus radial distance from control well by the followingequation derived from Eq 6:T 522.3Q2π∆sr(16)8.5 Determine the coefficient of storage from these semilogplots of drawdown versus time or distance by a methodproposed by Jacob (2) where:s 52.3Q4πTlog10S2.25Ttr2SD(17)Taking s = 0 at the zero-drawdown intercept of the straight-line semilog plot of time or distance versus drawdown,S 52.25Ttr2(18)where:eitherrort = the value at the zero-drawdown intercept.8.6 To apply the modified Theis nonequilibrium method tothin unconfined aquifers, where the drawdown is a significantfraction of the initial saturated thickness, apply a correction tothe 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:u 5r2S4Tt,0.01 (19)It is seen from Eq 19 that u decreases as time increases, otherthings being equal. Because S is in the numerator, the value ofu is much smaller for a confined aquifer, whose storagecoefficient may range from only about 10−5to 10−3, than f