Designation: D5850 − 18Standard Test Method for (Analytical Procedure)Determining Transmissivity, Storage Coefficient, andAnisotropy Ratio from a Network of Partially PenetratingWells1This standard is issued under the fixed designation D5850; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope*1.1 This test method covers an analytical procedure fordetermining the transmissivity, storage coefficient, and ratio ofvertical to horizontal hydraulic conductivity of a confinedaquifer using observation well drawdown measurements froma constant-rate pumping test. This test method uses data froma minimum of four partially penetrating, recommended to bepositioned observation wells around a partially penetratingcontrol well.1.2 The analytical procedure is used in conjunction with thefield procedure in Test Method D4050.1.3 Limitations—The limitations of the technique for deter-mination of the horizontal and vertical hydraulic conductivityof aquifers are primarily related to the correspondence betweenthe field situation and the simplifying assumption of this testmethod.1.4 Units—The values stated in inch-pound units are to beregarded as the standard. The SI units given in parentheses aremathematical conversions, which are provided for informationpurposes only and are not considered standard.1.5 All observed and calculated values shall conform to theguidelines for significant digits and rounding established inPractice D6026, unless superseded by this standard.1.6 The procedures used to specify how data are collected/recorded or calculated in this standard are regarded as theindustry standard.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 objective; and it is common practice toincrease or reduce the significant digits of reported data to becommensurate with these considerations. It is beyond the scopeof this standard to consider significant digits used in analysismethod or engineering design.1.7 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, health, and environmental practices and deter-mine the applicability of regulatory limitations prior to use.1.8 This international standard was developed in accor-dance with internationally recognized principles on standard-ization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.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 ConstructionD4050 Test Method for (Field Procedure) for Withdrawaland Injection Well Testing for Determining HydraulicProperties of Aquifer SystemsD5473/D5473M Test Method for (Analytical Procedure for)Analyzing the Effects of Partial Penetration of ControlWell and Determining the Horizontal andVertical Hydrau-lic Conductivity in a Nonleaky Confined AquiferD6026 Practice for Using Significant Digits in GeotechnicalData3. Terminology3.1 Definitions:3.1.1 For definitions of common technical terms in thisstandard, see Terminology D653.3.2 The following definitions from Terminology D653 are1This 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 Jan. 1, 2018. Published February 2018. Originallyapproved in 1995. Last previous edition approved in 2012 as D5850 – 95 (2012).DOI: 10.1520/D5850-18.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] 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 StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.1used in this standard and are presented for the convenience ofthe user.3.2.1 anisotropy—having different properties in differentdirections.3.2.2 confined aquifer—in hydrogeology, an aquiferbounded above and below by confining beds and in which thestatic head is above the top of the aquifer.3.2.3 control well—in aquifer testing, well by which theaquifer is stressed, for example, by pumping, injection, orchange of head.3.2.4 drawdown [L]—in field aquifer tests, vertical distancethe free water elevation is lowered or the pressure head isreduced due to the removal of free water.3.2.5 hydraulic conductivity—in field aquifer tests, the vol-ume of water at the existing kinematic viscosity that will movein a unit time under a unit hydraulic gradient through a unitarea measured at right angles to the direction of flow.3.2.6 monitoring well (observation well), n—inhydrogeology, a well installed, usually of small diameter, formeasuring water levels, collecting water samples, or determin-ing other groundwater characteristics.3.2.6.1 Discussion—The well may be cased or uncased, butif cased the casing should have openings to allow flow ofgroundwater into or out of the casing, such as a well screen.3.2.7 storage coeffıcient—in aquifers, the volume of waterthat an aquifer releases from or takes into storage per unitsurface area of the aquifer per unit change in head. For aconfined aquifer, the storage coefficient is equal to the productof the specific storage and aquifer thickness. For an unconfinedaquifer, the storage coefficient is approximately equal to thespecific yield.3.2.8 transmissivity—in aquifers, the volume of water at theexisting kinematic viscosity that will move in a unit time undera unit hydraulic gradient through a unit width of the aquifer.3.2.8.1 Discussion—It is equal to an integration of thehydraulic conductivities across the saturated part of the aquiferperpendicular to the flow paths.3.3 Symbols and Dimensions:3.3.1 A—Kz/Kr, anisotropy ratio [nd].3.3.2 b—thickness of aquifer [L].3.3.3 Cf—drawdown correction factor, equal to the ratio ofthe drawdown for a fully penetrating well network to thedrawdown for a partially penetrating well network (W(u)/(W(u) + fs)).3.3.4 d—distance from top of aquifer to top of screenedinterval of control well [L].3.3.5 d —distance from top of aquifer to top of screenedinterval of observation well [L].3.3.6 fs—incremental dimensionless drawdown componentresulting from partial penetration [ nd].3.3.7 K—hydraulic conductivity [LT−1].3.3.7.1 Discussion—The use of symbol K for the termhydraulic conductivity is the predominant usage in groundwa-ter literature by hydrogeologists, whereas the symbol k iscommonly used for this term in the rock and soil mechanicsliterature.3.3.8 Ko—modified Bessel function of the second kind andzero order.3.3.9 Kr—hydraulic conductivity in the plane of the aquifer,radially from the control well (horizontal hydraulic conductiv-ity) [LT−1].3.3.10 Kz—hydraulic conductivity normal to the plane of theaquifer (vertical hydraulic conductivity) [LT−1].3.3.11 l—distance from top of aquifer to bottom of screenedinterval of control well [L].3.3.12 l —distance from top of aquifer to bottom of screenedinterval of observation well [L].3.3.13 Q—discharge [L3T−1].3.3.14 r—radial distance from control well [L].3.3.15 S—storage coefficient [nd].3.3.16 s—drawdown observed in partially penetrating wellnetwork [L].3.3.17 sf—drawdown observed in fully penetrating wellnetwork [L].3.3.18 T—transmissivity [L2T−1].3.3.19 t—time since pumping began [T].3.3.20 u—(r2S)/(4Tt)[nd ].3.3.21 W(u)—an exponential integral known in hydrologyas the Theis well function of u[nd].4. Summary of Test Method4.1 This test method makes use of the deviations in draw-down near a partially penetrating control well from those thatwould occur near a control well fully penetrating the aquifer. Ingeneral, drawdown within the screened horizon of a partiallypenetrating control well tends to be greater than that whichwould have been observed near a fully penetrating well,whereas the drawdown above or below the screened horizon ofthe partially penetrating control well tends to be less than thecorresponding fully penetrating case. Drawdown deviationsdue to partial penetration are amplified when the verticalhydraulic conductivity is less than the horizontal hydraulicconductivity. The effects of partial penetration diminish withincreasing distance from the pumped well, becoming negli-gible at a distance of about 1.5b/(Kz/Kr)1/2. This test methodrelies on obtaining drawdown measurements at a minimum oftwo locations within this distance of the pumped well and ateach location obtaining data from observation wells completedto two different depths. At each location, one observation wellshould be screened at about the same elevation as the screen inthe pumped well, while the other observation well should bescreened in sediments not screened by the pumped well.4.2 According to Theis (1),3the drawdown around a fullypenetrating control well pumped at a constant rate and tappinga homogeneous, confined aquifer is as follows:3The boldface numbers given in parentheses refer to a list of references at theend of the text.D5850 − 182sf5Q4πTW~u! (1)where:W~u! 5 *u` e2xxdx (2)4.2.1 Drawdown near a partially penetrating control wellpumped at a constant rate and tapping a homogeneous,anisotropic, confined aquifer is presented by Hantush (2, 3, 4):s 5Q4πT~W~u!1fs! (3)According to Hantush (2, 3, 4), at late pumping times, whent>b2S/(2TA), fscan be expressed as follows:fs54b2π2~l 2 d!~l 2d !(n51`S1n2DKoSnπr=Kz/KrbD (4)FsinSnπibD2 sinSnπdbDGFsinSnπl bD2 sinSnπd bDG4.2.2 For a given observed drawdown, it is practicable tocompute a correction factor, Cf, defined as the ratio of thedrawdown for a fully penetrating well to the drawdown for apartially penetrating well:Cf5W~u!W~u!1fs(5)The observed drawdown for each observation well may becorrected to the fully penetrating equivalent drawdown bymultiplying by the correction factor:sf5 Cfs (6)The drawdown values corresponding to the fully penetratingcase may then be analyzed by conventional distance-drawdownmethods to compute transmissivity and storage coefficient.4.2.3 The correction factors are a function of both transmis-sivity and storage coefficient, that are the parameters beingsought. Because of this, the test method relies on an iterativeprocedure in which an initial estimate of T and S are made fromwhich initial correction factors are computed. Using thesecorrection factors, fully penetrating drawdown values arecomputed and analyzed using distance-drawdown methods todetermine revised values for T and S. The revised T and Svalues are used to compute revised correction factors, Cf. Thisprocess is repeated until the calculated T and S values changeonly slightly from those obtained in the previous iteration.4.2.4 The correction factors are also a function of theanisotropy ratio, A. For this reason, the calculations describedabove must be performed for several different assumed anisot-ropy ratios. The assumed anisotropy value that leads to the bestsolution, that is, best straight line fit or best curve match, isdeemed to be the actual anisotropy ratio.5. Significance and Use5.1 This test method is one of several available for deter-mining vertical anisotropy ratio. Among other available meth-ods are Weeks ((5); see Test Method D5473/D5473M), thatrelies on distance-drawdown data, and Way and McKee (6),that utilizes time-drawdown data. An important restriction ofthe Weeks distance-drawdown method is that the observationwells need to have identical construction (screened intervals)and two or more of the observation wells need to be located ata distance from the pumped well beyond the effects of partialpenetration. The procedure described in this test methodgeneral distance-drawdown method, in that it works in theoryfor most observation well configurations incorporating three ormore wells, provided some of the wells are within the zonewhere flow is affected by partial penetration.5.2 Assumptions:5.2.1 Control well discharges at a constant rate, Q.5.2.2 Control well is of infinitesimal diameter and partiallypenetrates the aquifer.5.2.3 Data are obtained from a number of partially penetrat-ing observation wells, some screened at elevations similar tothat in the pumped well and some screened at differentelevations.5.2.4 The aquifer is confined, homogeneous and areallyextensive. The aquifer may be anisotropic, and, if so, thedirections of maximum and minimum hydraulic conductivityare horizontal and vertical, respectively.5.2.5 Discharge from the well is derived exclusively fromstorage in the aquifer.5.3 Calculation Requirements—Application of this methodis computationally intensive. The function, fs, shown in (Eq 4)must be evaluated numerous times using arbitrary input pa-rameters. It is not practical to use existing, somewhat limited,tables of values for fsand, because this equation is ratherformidable, it may not be easily tractable by hand. Because ofthis, it is assumed the practitioner using this test method willhave available a computerized procedure for evaluating thefunction fs. This can be accomplished using commerciallyavailable mathematical software including some spreadsheetapplications, or by writing programs. (7)NOTE 1—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 2—Most fractured (unconfined) aquifers, even noncarbonates,will have some form of convergent flow to master fissures or channels(Worthington et al., 2016). A relationship is known to occur in carbonatesw