Designation: D5850 − 95 (Reapproved 2012)Standard 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. Scope1.1 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, properly positionedobservation wells around a partially penetrating control 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 The values stated in inch-pound units are to be regardedas the standard. The SI units given in parentheses are forinformation only.1.5 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 ContainedFluidsD4050 Test Method for (Field Procedure) for Withdrawaland Injection Well Testing for Determining HydraulicProperties of Aquifer SystemsD5473 Test Method for (Analytical Procedure for) Analyz-ing the Effects of Partial Penetration of Control Well andDetermining the Horizontal and Vertical Hydraulic Con-ductivity in a Nonleaky Confined Aquifer3. Terminology3.1 Definitions:3.1.1 aquifer, confined—an aquifer bounded above and be-low by confining beds and in which the static head is above thetop of the aquifer.3.1.2 confining bed—a hydrogeologic unit of less permeablematerial bounding one or more aquifers.3.1.3 control well—well by which the head and flow in theaquifer is changed, for example, by pumping, injection, orimposing a constant change of head.3.1.4 drawdown—vertical distance the static head is low-ered due to the removal of water.3.1.5 hydraulic conductivity—(field aquifer test) the volumeof water at the existing kinematic viscosity that will move in aunit time under a unit hydraulic gradient through a unit areameasured at right angles to the direction of flow.3.1.6 observation well—a well open to all or part of anaquifer.3.1.7 piezometer—a device so constructed and sealed as tomeasure hydraulic head at a point in the subsurface.3.1.8 storage coeffıcient—the volume of water an aquiferreleases from or takes into storage per unit surface area of theaquifer per unit change in head.3.1.9 transmissivity—the volume of water at the existingkinematic viscosity that will move in a unit time under a unithydraulic gradient through a unit width of the aquifer.3.1.10 For definitions of other terms used in this testmethod, see Terminology D653.3.2 Symbols and Dimensions:3.2.1 A—Kz/Kr, anisotropy ratio [nd].3.2.2 b—thickness of aquifer [L].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 May 1, 2012. Published December 2012. Originallyapproved in 1995. Last previous edition approved in 2006 as D5850 – 95 (2006).DOI: 10.1520/D5850-95R12.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

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United States13.2.3 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)).3.2.4 d—distance from top of aquifer to top of screenedinterval of control well [L].3.2.5 d —distance from top of aquifer to top of screenedinterval of observation well [L].3.2.6 fs—incremental dimensionless drawdown componentresulting from partial penetration [ nd].3.2.7 K—hydraulic conductivity [LT−1].3.2.7.1 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.3.2.8 Ko—modified Bessel function of the second kind andzero order.3.2.9 Kr—hydraulic conductivity in the plane of the aquifer,radially from the control well (horizontal hydraulic conductiv-ity) [LT−1].3.2.10 Kz—hydraulic conductivity normal to the plane of theaquifer (vertical hydraulic conductivity) [LT−1].3.2.11 l—distance from top of aquifer to bottom of screenedinterval of control well [L].3.2.12 l —distance from top of aquifer to bottom of screenedinterval of observation well [L].3.2.13 Q—discharge [L3T−1].3.2.14 r—radial distance from control well [L].3.2.15 S—storage coefficient [nd].3.2.16 s—drawdown observed in partially penetrating wellnetwork [L].3.2.17 sf—drawdown observed in fully penetrating wellnetwork [L].3.2.18 T—transmissivity [L2T−1].3.2.19 t—time since pumping began [T].3.2.20 u—(r2S)/(4Tt)[nd ].3.2.21 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:sf5Q4πTW~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, whentb2S/(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 possible 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. This3The boldface numbers given in parentheses refer to a list of references at theend of the text.D5850 − 95 (2012)2process 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, all of the calculationsdescribed above must be performed for several differentassumed anisotropy ratios. The assumed anisotropy value thatleads to the best solution, that is, best straight line fit or bestcurve match, is deemed 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), that relies ondistance-drawdown data, and Way and McKee (6), that utilizestime-drawdown data. An important restriction of the Weeksdistance-drawdown method is that the observation wells musthave identical construction (screened intervals) and two ormore of the observation wells must be located at a distancefrom the pumped well beyond the effects of partial penetration.The procedure described in this test method general distance-drawdown method, in that it works in theory for any observa-tion well configuration incorporating three or more wells,provided some of the wells are within the zone where flow isaffected 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 is not readily tractable by hand. Because of this,it is assumed the practitioner using this test method will haveavailable a computerized procedure for evaluating the functionfs. This can be accomplished using commercially availablemathematical software including some spreadsheetapplications, or by writing programs in languages such asFortran or C.6. Apparatus6.1 Apparatus for withdrawal tests is given in Test MethodD4050. The apparatus described below are those componentsof the apparatus that require special attributes for this specifictest.6.2 Construction of the Control Well—Screen the controlwell through only part of the vertical extent of the aquifer to betested. The exact distances from the top of the aquifer to the topand bottom of the pumped well screen interval must be known.6.3 Construction and Placement of Observation Wells—Theprocedure will work for arbitrary positioning of observationwells and placement of their screens, as long as three or moreobservation wells are used and some of the observation wellsfall inside the zone where flow is affected by partialpenetration, that is, the area where significant vertical flowcomponents exists. However, strategic selection of the numberand location of observation wells will maximize the quality ofthe data set and improve the reliability of the interpretation.6.3.1 Optimum results will be obtained by using a minimumof four observation wells incorporating two pairs of observa-tion wells located at two different distances from the pumpedwell, both within the zone where flow is affected by partialpenetration. Each well pair should consist of a shallow welland a deep well, that span vertically the area in which verticalanisotropy is sought. For each well pair, one observation wellscreen should be at the same elevation as the screen in thepumped well, whereas the other observation well screen shouldbe at a different elevation than the screen in the pumped well.6.3.2 This test method relies on choosing several arbitraryanisotropy ratios, correcting the observed drawdowns forpartial penetration, and evaluating the results. If all observationwells are screened at the same elevation, the quality of the datatrace produced by correcting the observed drawdown measure-ments is not sensitive to the choice of anisotropy, making itdifficult to determine this parameter accurately. If, however,observation well screens are located both within the pumpedzone (where drawdown is greater than the fully penetratingcase) and the unpumped zone (where drawdown is less than thefully penetrating case), the quality of the corrected data issensitive to the choice of anisotropy ratio, making it easier toquantify this parameter.7. Procedure7.1 Pre-test preparations, pumping test guidelines, and post-test procedures associated with the pumping test itself aredescribed in Test Method D4050.7.2 Verify the quality of the data set. Review the record ofmeasured flow rates to make sure the rate was held constantduring the test. Check to see that hand measurements ofdrawdown agree well with electronically measured values.Finally, check the background water-level fluctuations ob-served prior to or following the pumping test to see ifadjustments must be made to the observed drawdown values toaccount for background fluctuations. If appropriate, adjust theobserved drawdown values accordingly.7.3 Analysis of the field data is described in Section 8.8. Calculation and Interpretation of Results8.1 Initial Estimates of Transmissivity and StorageCoeffıcient—This test method requires that initial estimates ofT and S be obtained. These estimates can be made using a widevariety of procedures, including time-drawdown analysis,