improved characterization of small ‘‘u’’ for jacob pumping ...for jacob pumping test...

Improved Characterization of Small ‘‘u’’for Jacob Pumping Test Analysis Methodsby Scott C. Alexander1 and Martin O. Saar2

AbstractNumerous refinements have been proposed to traditional pumping test analyses, yet many hydrogeologists

continue to use the Jacob method due to its simplicity. Recent research favors hydraulic tomography and inversenumerical modeling of pumping test data. However, at sites with few wells, or relatively short screens, the datarequirements of these methods may be impractical within physical and fiscal constraints. Alternatively, an improvedunderstanding of the assumptions and limitations of Theis and, due to their widespread usage, Jacob analyses,leads to improved interpretations in data-poor environments. A fundamental requirement of Jacob is a “small”value of u = f (r2/t), with radial distance, r , and pumping time, t . However, selection of a too stringent (i.e., toolow) maximum permissible u-value, umax, results in rejection of usable data from wells beyond a maximum radius,rmax. Conversely, data from small radii, less than rmin, where turbulent- and vertical-flow components arise, canresult in acceptance of inappropriate data. Usage of drawdown data from wells too close to the pumping well,and exclusion of data from wells deemed too far, can cause unrealistic aquifer transmissivity, permeability, andstorativity determinations. Here, data from an extensive well field in a glacial-outwash aquifer in north-centralMinnesota, USA, are used to develop a new estimate for umax. Traditionally quoted values for umax range from0.01 to 0.05. Our proposed value for Jacob distance-drawdown analyses is significantly higher with umax up to 0.2,resulting in larger allowable rmax-values and a higher likelihood of inclusion of additional wells in such pumpingtest analyses.

IntroductionPumping tests are an important method for in situ

studies of hydraulic aquifer properties at the meso-scale.Stressing the aquifer, by pumping water, and monitoringresultant water table drawdown in unconfined aquifers orthe potentiometric surface in confined aquifers, importanthydrogeologic parameters can be obtained at the scale

1Corresponding author: Department of Earth Sciences, NewtonHorace Winchell School of Earth Sciences, University of Minnesota,310 Pillsbury Dr. SE, Minneapolis, MN 55455, (651)626-4164;fax: (651)625-3819; [email protected]

2Department of Earth Sciences, Newton Horace WinchellSchool of Earth Sciences, University of Minnesota, 310 PillsburyDr. SE, Minneapolis, MN 55455.

Received October 2010, accepted June 2011.© 2011, The Author(s)Ground Water © 2011, National Ground Water Association.doi: 10.1111/j.1745-6584.2011.00839.x

of water usage. These parameters include transmissivityand storativity as well as derived quantities such ashorizontal and vertical permeability in some cases. Atsmaller scales, Darcy-type tests may be conducted, forexample, core analyses (Darcy 1856; Wenzel 1942;Saar and Manga 1999) and slug tests (Hvorslev 1951,Butler et al. 1996). At larger scales, numerical modelsare frequently constructed to constrain hydrogeologicparameters (Forster and Smith 1988; Anderson andWoessner 1992; Saar and Manga, 2003, 2004; Anderson2005; Walsh and Saar 2010; Saar 2011).

Recent research efforts favor interpretation ofpumping test data by employing hydraulic tomo-graphy and inverse numerical modeling (Yeh and Lee2007), although a discussion of the limitations ofthese methods has begun (Bohling and Butler 2010). Atsites with limited numbers of wells, and/or wells with rel-atively short well screens, the data required for hydraulic

256 Vol. 50, No. 2–GROUND WATER–March-April 2012 (pages 256–265) NGWA.org

tomography may be impractical to gather and may bebeyond the scope of the investigation. Although analyti-cal models that account for unconfined conditions, partialpenetration, and so on are available, classic Theis curvematching (Theis 1935) and, in particular, Jacob distance-drawdown (Cooper and Jacob 1946) relationships are stillwidely used for many field applications (Todd and Mays2005).

Consequently, an improved understanding of the sim-ple, classic Theis and Jacob pumping test interpretationmethods can make a significant contribution to improvingthe quality of inferred aquifer property data in a muchlarger number of investigations than would be the casewith improvements regarding more sophisticated meth-ods. Here, we focus our investigation on Jacob time anddistance-drawdown analyses as these methods are par-ticularly widely used (Todd and Mays 2005) and haveproduced useful results over more than five decadesthroughout the hydrogeologic and environmental consult-ing industries (Sterrett 2007).

We apply results from a pumping test conducted at theUniversity of Minnesota Hydrogeology Field Site in July2009 to improve our understanding of the assumptions andlimitations of Jacob analyses, allowing better applicationin data-poor environments. Specifically, we reevaluateselection criteria for a maximum u-cutoff value, umax,which affects the maximum usable radial distance ofmonitoring wells from the pumping well, rmax, for agiven pumping time, in the Jacob distance-drawdownmethod. We also discuss the effects of turbulent andnon-horizontal groundwater flow at monitoring wells thatare located close to the pumping well. Combined, thesetwo limitations define an allowable intermediate dis-tance interval for permissible monitoring well distancesfrom the pumping well. The larger this intermediatedistance interval is, the higher is the likelihood thatexisting monitoring wells can be included in simpledistance-drawdown Jacob-type pumping tests. Largerpermissible distances also allow greater flexibility indesigning well fields for pumping tests and increasethe volume of aquifer material represented in thesetests.

BackgroundTheis, in collaboration with C.I. Lubin at the

University of Cincinnati, USA, found in 1935, thatthe drawdown, s, in a monitoring well between initial,unperturbed water table or potentiometric surface andthose values at some time, t , after pumping began and ata radial distance from the pumping well, r , is given by:

s = Q

4πT

∫ ∞

u

e−u

udu, (1)

where

u = r2S

4T t, (2)

Q is the constant volumetric pumping rate or discharge,T is transmissivity, and S is the dimensionless storativity.In step-drawdown pumping tests, Q is not constant,as shown in the later discussion, and thus adjustmentshave to be made to t for all pumping rates butthe first one, to account for previous pumping steps’influence on a given pumping step drawdown. Equation 1produces the drawdown of the water table in confinedaquifers based on the assumption of uniform pumpingin “a homogeneous aquifer of constant thickness andinfinite areal extent” (Theis 1935) but can be readilyexpanded to unconfined aquifers. For a given pumpingrate and measured drawdown at radii, r , and at times, t ,Equation 1 can be solved semi-analytically or via a curvematching procedure to yield transmissivity and storativityas explained in standard hydrogeology textbooks (Freezeand Cherry 1979; Fetter 2001; Schwartz and Zhang2003). If aquifer thickness, b, is known, then hydraulicconductivity, K = T /b, can be determined and from itpermeability.

The integral in Equation 1 is known as the wellfunction, W(u), and can be approximated by a Taylor-series expansion (Theis 1935):

s = Q

4πT

(−0.577216 − ln(u) + u − u2

2·2!

+ u3

3·3!− u4

4·4!+ u5

5·5!· · ·

), (3)

where higher order terms may be neglected for smallvalues of u. In fact, Cooper and Jacob (1946) suggestneglecting all terms in the expansion beyond the loga-rithmic term, ln(u), to allow plotting of drawdown, s, vs.time, t , or vs. distance, r , data on a semi-log plot (withdrawdown on a linear scale) and thus more straightfor-ward linear fitting of data, as opposed to fitting of curveson superimposed log-log plots as in the Theis method.In addition, if the distance-drawdown Jacob method isemployed, where drawdowns in multiple monitoring wellsare used at the same instance in time (late during thepumping test), rather than the time-drawdown method fora single well, then only one (late-time) data point of draw-down in each monitoring well is sufficient. This simplifiesdata collection and facilitates better results as inaccuraciesin drawdown measurements over time are less critical.However, it should be noted that a heightened dependenceon obtaining accurate drawdown measurements in eachmonitoring well is introduced.

Frequently, Jacob-type pumping test analyses are thenfocused on ensuring small values of u in Equation 2, tojustify omission of the higher order terms beyond thelogarithmic term in the Taylor-series expansion of the wellfunction integral (the terms in parentheses in Equation 3).The question then arises, what the maximum allowablevalue of u, that is, umax, should be. Cooper and Jacob(1946) assert that “The approximation will be tolerablewhere u is less than about 0.02” but note four paragraphslater that “in most cases little, if any, of the data will

NGWA.org S.C. Alexander and M.O. Saar GROUND WATER 50, no. 2: 256–265 257

fall off the straight line.” Schwartz and Zhang (2003)and Todd and Mays (2005) suggest that a maximumvalue of umax = 0.01 is acceptable, whereas Fetter (2001)states that umax = 0.05 is sufficient. However, as we showin the following, even this latter, higher umax valueappears to be almost an order of magnitude smallerthan necessary, potentially causing omission of valuabledata collected in distal monitoring wells during pumpingtests. Furthermore, this focus on a small value of u canlead to an overemphasis of data collected in monitoringwells that are located close to the pumping well andthus potentially result in neglect of some of the otherassumptions, especially those of laminar and horizontalflow.

As T and S are assumed to be constant for analysis bythe Theis method, the only contributions to the value of u

in Equation 2 that can be varied are radial distance of themonitoring well from the pumping well, r , measurementtime after pumping began, t , and discharge (i.e., pump-ing rate), Q, which is included in the calculation of T .Thus, to minimize u, r should be small and t should belarge, with a much more significant dependence on r , asit is squared in Equation 2. Consequently, finances andtime permitting, the pumping time, t , may be extendedand monitoring wells may be located relatively close tothe pumping well to reduce r . However, and especially athigh pumping rates, for monitoring wells located too closeto the pumping well, wells that are not fully screened, orwhere low efficiency within the pumping well has reducedthe effective screen length, non-horizontal and/or turbulentgroundwater flow components may become significant.However, integration over larger subsurface regions, wheninferring hydrogeologic parameters, is often desired whichrequires larger spacing of monitoring wells and thus largervalues of r . Similarly, pre-existing monitoring wells maybe employed at whatever radial distances available due tobudgetary and/or time constraints. Similarly, data pointsfrom more distal monitoring wells may be required forvarious reasons such as lack of a sufficient number ofmonitoring wells close to the pumping well or an interestin investigating heterogeneity by interpreting pumping testresults utilizing monitoring wells at different distances andin various directions from the pumping well.

Field SiteThe University of Minnesota Hydrogeology Field Site

(Hydro Site) is located in north-central Minnesota nearthe town of Akeley. The 20 acre (8.1 hectare) field siteis surrounded by the much larger United States Geo-logical Survey (USGS) Shingobee Headwaters AquaticEcosystem Project site which includes Williams and Crys-tal Lakes (Rosenberry et al. 1997). This site was selectedto be representative of a typical glacial outwash systemand assumed to have a relatively isotropic and homo-geneous distribution of sediments based on pre-existingUSGS well logs. The original pumping well (PW-1) andfour monitoring wells were installed in 1995 as part of aNational Science Foundation grant. Since then, one or

more wells have been added each summer to the mon-itoring network, in cooperation with the USGS, as part ofa Hydrogeology Field Camp taught by the Universityof Minnesota (www.geo.umn.edu/orgs/camp/hydrocamp).The well field currently contains two pumping wells, 18monitoring wells, and four additional USGS WilliamsLake wells, the latter at large radii. Figure 1 shows thelocation of these wells superimposed on an air-photo ofthe field site and the surrounding region. Both pumpingwells have 15.2 cm diameter custom-made screens. Allmonitoring wells have 5.1 cm diameter and 0.025 mmscreens. Table 1 lists the radial distance from PumpingWell 1 (PW-1), the only well used for pumping in thisanalysis, the depth of completion, and the screen lengthfor each well.

The University of Minnesota Hydro Site is locatedbetween Crystal and Williams Lakes on an isthmus ofglacial outwash sands extending between two ice blocksthat melted to form the present-day lakes. The outwashsands originated from the Itasca Moraine immedi-ately to the Northeast and the St. Croix Moraineimmediately to the Southeast (Mooers and Norton 1997).The water table aquifer system is contained within a 32-mthick section of outwash sediments from the Wadena,Itasca, and St. Croix moraines and is underlain by adense, low-permeability basal till of over 30-m thickness(Figure 2) from the Wadena Lobe. The aquifer has about12 m saturated thickness and about 20 m of unsaturatedoutwash sediments above it. Water from the pumping testis discharged to a till-lined ice-block depression located80 m northeast of pumping well PW-1.

The aquifer is typical of outwash systems withrelatively homogeneous horizontal properties but strongvertical heterogeneities, based on interpretations of a con-tinuous rotosonic core recovered from drilling pumpingwell PW-1 and multiple split-spoon samples from instal-lation of the various monitoring wells since 1995. Thehighest-permeability sediments are found in the bottom5 m of the aquifer. These gray sediments are dominated

Figure 1. Location and layout of the University of MinnesotaHydro Site in north-central Minnesota, USA, with 0.5 mcontour lines indicating water table elevation. Pumpingwell PW-1, at the center of the circles, is located at(46◦56′52.86′′N, 94◦39′47.39′′W).

258 S.C. Alexander and M.O. Saar GROUND WATER 50, no. 2: 256–265 NGWA.org

Table 1Wells at the University of Minnesota Hydro Site

Well IDRadius toPW-1 (m)

WellDepth (m)

Well ScreenLength (m)

PW-1 — 30.5 12.2MW-03 2.13 25.9 1.3MW-05 2.99 25.9 1.5MW-04 4.39 19.8 2.4MW-01 6.48 26.5 2.0MW-02 13.22 25.9 1.5MW-07 13.25 21.0 2.4MW-09 13.28 28.7 3.7MW-08 30.09 24.8 1.5MW-06 30.22 20.7 1.8MW-10 30.27 32.0 3.7MW-11 30.51 22.7 2.4MW-14 60.33 21.2 2.4MW-12 60.62 21.2 2.4MW-15 74.65 32.9 3.0PW-2 78.28 32.0 4.6MW-17 89.30 22.9 1.5WL-16 99.72 21.2 1.5MW-18 133.3 22.7 1.5WL-25 179.1 14.5 1.5SW-1 306.7 22.0 1.5WL-2 306.9 25.0 1.5WL-15 306.9 16.9 1.5

by moderately well-sorted medium sands to medium grav-els related to the Wadena Lobe. The upper 7 m of theaquifer are dominated by poorly sorted fine sands inter-spersed with thin, typically less than 20-cm thick muddytills. The muddy tills are interpreted as “outburst” eventsfrom the nearby Itasca moraine which was located lessthan 2 km from the field site.

The local aquifer system links surface waters flowingfrom Crystal Lake to Williams Lake, that is, from A’to A in Figure 1, two typical ice-block lakes. Low-permeability boundaries are created near the lakes bydraped tills, as shown in Figure 2. The draped tills, re-lated to the formation of the ice-block lakes, result in arelatively flat water table surface in between the lakes,and underneath the field site, with steep changes in thewater table elevation immediately adjacent to the lakesand particularly at Williams Lake.

For purposes of pumping test analysis, the well fieldcan be collapsed into a cross-section of radial distancesas shown in Figure 3, where only monitoring wells areincluded that are located within 15 m of the pumpingwell. Monitoring wells MW-3, MW-4, and MW-5, inparticular, are all located within 5 m of the pumping well.MW-3 and MW-5 are screened in the vertical middle ofthe aquifer, whereas MW-4 is screened across the watertable surface. Monitoring wells MW-2, MW-7, and MW-9are three individual wells completed at the same radialdistance, 13.2 m, but at varying depths, roughly northwestof pumping well PW-1. Note that although PW-1 has afully penetrating screen, none of the monitoring wells arefully penetrating. Fully penetrating screens would greatlyincrease the expense of the monitoring wells. Furthermore,the hollow-stem auger drilling rig, provided mostly by theUSGS, has a maximum depth of 26 m. MW-9 was drilledby a commercial mud rotary rig.

Results and DiscussionDrawdown, s, vs. time, t , data recorded in two

monitoring wells during the 2009 pumping test at theUniversity of Minnesota Hydro Site are presented inFigure 4. The single-pumping-well, multiple-monitoring-wells test was conducted as a step-drawdown test that

Figure 2. Vertically exaggerated stratigraphy of the University of Minnesota Hydro Site and surrounding region along crosssection A-A’ in Figure 1 (msl = meters above mean sea level, bgs = meters below ground surface).


Figure 3. Cross section showing the radial distance ofmonitoring wells from pumping well PW-1, well depths andscreen lengths, as well as the stratigraphy at the Universityof Minnesota Hydro Site to a maximum radius of 15 m(msl = meters above mean sea level, bgs = meters belowground surface).

Figure 4. Four-step drawdown pumping test at a linear timescale from the beginning of pumping, where curves forMW-1 and MW-2 represent drawdown, s , while the curvefor Q indicates discharge or pumping rate. Total pumpingtest time was just over 6 d.

included four discharge or pumping rate steps, Q1 throughQ4.

The pumping rate was measured employing aBernoulli tube (Sterrett 2007) and a 5-min running aver-age filter was applied to smooth out short-term pressurefluctuations within the tube that are caused by flow tur-bulence and do not reflect variations in pumping rate.Druck® pressure transducers were used to monitor waterlevels in the Bernoulli tube and in the wells and the datawere recorded on a Campbell® CR3000 data-logger at1-s intervals.

Time DrawdownTimes for the step-drawdown analysis are corrected

as outlined by Birsoy and Summers (1980) to accountfor reduced drawdown values due to lower discharge or

Figure 5. Jacob time-drawdown semi-log plot showingcorrected alpha time, α(t), vs. discharge- or pumping-rate-corrected drawdown, s/Qn, for the four pumping ratesteps, n = 1, 2, 3, 4.

pumping rates during previous pumping steps. Figure 5 isa semi-log plot of corrected time, α(t), as pumping beganagainst the observed drawdown, s, in two monitoringwells, MW-1 and MW-3, divided by the pumping rate,Qn, at each of the four pumping rate steps.

At early times, especially less than 10,000 s for thispumping test, the data plot above the respective Jacob fitlines (Figure 5). This deviation is due to the differencebetween the straight-line Jacob approximation and theTheis well function, where u is not small enough, dueto pumping time, t , being too small (see also Equation 2).At 10,000 < t < 200,000 s, the data fall on the straightline given by the Jacob method, whereas at t > 200,000 s,the rate of drawdown with time begins to drop, plottingunderneath the Jacob line. This latter deviation from a linesuggests effects due to semi-confined aquifer conditions,a higher permeability zone at some distance from thepumping well, and/or simply leakage of the dischargedwater back to the aquifer.

On the basis of Figure 5, transmissivity, T , andstorativity, S, can be calculated as outlined in Cooperand Jacob (1946) for a time-distance analysis, withmodifications as proposed by Birsoy and Summers (1980)replacing �s by �(s/Qn) (while dropping Q from thenumerator) and t0 by β0, respectively, resulting in

T = 2.3

4π�(s/Qn)(4)

and

S = 2.25Tβ0

r2, (5)

where T in Equation 5 is given by Equation 4. The resultsare summarized in Table 2.

The smallest value of u, where the data starts tofollow a linear trend in the semi-log plot of Figure 5, withincreasing pumping time, can then be found. For MW-1,which is located at a radial distance of r = 6.5 m from


Table 2Time-Drawdown Results from Step Pumping Test

Well r (m)�(s/Qn)(s/m2) β0 (s) T (m2/s) S

MW-1 6.5 11.6 80 0.0158 0.065MW-3 2.1 11.8 1.3 0.0155 0.010

Note: Variables are defined in the main text.

the pumping well, the data begin to follow a linear trendafter approximately 3000 s (50 min). Using Equation 2with r , T , and S as given in Table 2, results in u = 0.01.Similarly, for MW-3, where r = 2.1 m, data begin tofollow a linear trend at about 2400 s (40 min), resulting inu = 0.0003. For both monitoring wells, a linear trend isnot achieved in the semi-log plot before u is less thanabout 0.01 which is in agreement with quoted estimatesfor what constitutes a small value of u.

Distance DrawdownTransmissivity, T , and storativity, S, can also

be determined employing the Cooper-Jacob distance-drawdown method (Cooper and Jacob 1946), using pointsalong the linear segments of the semi-log plot depicted inFigure 6 with T and S now given by:

T = 2.303

4π�s(6)

and

S = 2.25T tc

r2Ø

, (7)

where �s is the change in drawdown per one logarithmicdecade, rØ is the calculated x-axis intercept, where

Figure 6. Regression of distance-drawdown data for threedifferent values of umax, for Pumping Step 3 with Q =0.00576 m3/s and for a corrected common pumping time of75,000 s. Error envelopes of the regression lines are givenas 3σ errors for illustration purposes while the calculationsdiscussed in the main text use 1σ errors.

drawdown would be zero for the Jacob line, and tc isthe “common” time late in each pumping step at whichdrawdowns in all monitoring wells are measured. Notethat tc is a corrected time, α(t), accounting for lowerpumping rates during previous steps, as discussed beforefor the time-drawdown analysis. T in Equation 7 is givenby Equation 6.

We perform a series of linear regressions using datapoints meeting the requirements of small u, that is,u below some maximum allowable value, umax, withumax ranging from 0.01 to 0.2. During this process,we ignore data taken in monitoring wells located atradial distances, r , from the pumping well that are largerthan the maximum allowable radial distance, rmax, givenfor each respective umax value. In this linear regressionanalysis, the three monitoring wells at very small radialdistances (<5 m) are also ignored because the drawdowndata for these wells indicate that they are significantlyinfluenced by turbulent and/or non-horizontal groundwaterflow components. At these small radial distances, less thansome minimum radius, rmin, the conditions required forthe Theis, and thus also the Jacob, analyses are violated.The remaining allowable data align on a semi-log plot ofdrawdown vs. radial distance from the pumping well asshown in the Cooper-Jacob-type distance-drawdown plotin Figure 6. The regression analysis, therefore, is of theform:

s = A + B(log10 r), (8)

where, as before, s is drawdown and r is the radialdistance between each monitoring well and the pumpingwell. Standard errors of the slope and x-axis (i.e., radius-axis) intercept for s = 0 of the regression line, in thesemi-log plot, can be calculated as described in Taylor(1982) by

σ 2A = σ 2

y

(�x2

i

)/� (9)

and

σ 2B = Nσ 2

y /�, (10)

where σ 2y = (N − 2)−1�(yi − A − Bxi)

2,� = N(�x2i )

− (�xi)2, and N is the number of data points. Calculation

of the 1σ errors for the A and B coefficients allowsdetermination of an error envelope around the best-fit line.These error envelopes have a parabolic form where theuncertainties grow larger with increasing distance fromthe centroid of the data. For visualization purposes, 3σ

error envelopes are plotted in Figure 6, instead of the 1σ

error envelopes used in the actual calculations, to helpdifferentiate the error envelopes from the fitted regressionlines.

Figure 6 is an example of this analysis for thethird pumping step. If a restrictively low value of umax

is used, for example umax = 0.01, then, as Equation 2indicates, after calculation of temporary values for T

and S, only wells less than a maximum radial distance


from the pumping well of rmax = 28 m can be includedin the analysis. For this time step, only four monitoringwells result in both r < rmax = 28 m and r > rmin = 5 m,leading to a regression line based on only four data points.Not surprisingly, this produces a regression line with arelatively large error envelope, as defined by the dashedred lines. If we accept a larger umax value, for exampleumax = 0.1 or umax = 0.2, then the maximum allowableradial distances of monitoring wells that can be includedin the analysis increase to rmax = 80 m and rmax = 100 m,respectively. We can thus include a larger range andnumber of data points in the Jacob pumping test analysis(Figure 6). By including more data points, and regressingover a greater range of radii, the error envelopes of theregression line in Figure 6 shrink significantly. However,at yet larger values of umax, and correspondingly largermaximum allowable radial distances, rmax, points areadded to the regression analysis that are clearly no longerfollowing a linear trend. For this data set, umax = 0.5,resulting in rmax = 160 m, would lead to inclusion of suchnonlinear data points.

Using the range of slopes for the plus and minus1σ error envelopes, we can calculate a 1σ range for �s.Substituting the range of �s into Equation 6, we can thencalculate the corresponding range of transmissivities, T ,implied by the regression line and related error estimates.Similarly, we can also calculate the range of rØ, or x-axis intercepts, of the 1σ error envelope of the regressionline to determine a range of corresponding storativities,S, using Equation 7, where the input value of T is alsoa range, calculated from Equation 6, as just discussed.Consequently, error propagation of two ranges, for rØ andfor T , is required when calculating S, in Equation (7),resulting in a larger uncertainty for S than for T . Theseresults are summarized in Table 3.

Figure 7 is a distance-drawdown plot, where datapoints are color-coded by the pumping rate employedduring the step pumping test. Heavy dark lines representbest Jacob-fit lines. Superimposed lighter curves are theTheis well functions, calculated to the sixth factorial termsin the Taylor-series expansion (Equation 3), using T andS values derived from the corresponding Jacob lines.The inset figure shows how the interval of convergenceexpands to include larger radii as additional terms areadded to the Taylor-series expansion. The Theis functionsshow a slowly increasing deviation from the Jacoblines at large radial distances, r , illustrating breakdownof the small u assumption as monitoring wells arelocated too far (i.e., beyond rmax) from the pumpingwell.

The horizontally stratified aquifer at this field site canbe approximated as equivalent to an isotropic, homoge-neous aquifer where the horizontal hydraulic conductivity,Kx , is the arithmetic mean of the individual horizontalhydraulic conductivities, Kxi , of each horizontal layer, i,with thicknesses, bi , given as

Kx = (�Kxibi)/b, (11)

Figure 7. Semi-log plot of radial distance, r, from moni-toring wells from the pumping well vs. drawdown, s , forfour different pumping rates, Qn, from the example pump-ing test. The Cooper-Jacob lines are shown in darkercolors, whereas the Theis curves are plotted in lighter col-ors and employ the Taylor-series expansion approximation(Equation 3) of the integral in the well function to the sixthfactorial term as shown in the figure inset. The 2.5 mmdrawdown line in the figure inset marks the approximateresolution of the water level measurement tape. The firstthree monitoring wells, at r < 5 m (gray-shaded region), donot follow the Theis solution as the assumptions of laminarand/or horizontal flow break down near the pumping well.

where b is the sum of all layer thicknesses (Maasland1957). The arithmetic mean is dominated by the highest-permeability layer. This horizontally stratified aquifer,which is typical of many glacial outwash systems,produces the clear Jacob fits of Figure 7.

The three wells at the smallest radial distances, lessthan 5 m, show an increasing deviation from the fittedJacob lines with higher pumping rates (Figure 7). Thesethree wells were installed prior to conduction of the firstpumping test and before the stratigraphy of the site wasfully defined. MW-3 and MW-5 were installed at themaximum depth range of the drill rig and, due to limitedrig time, MW-4 was constructed as shallow as possible.

At small radial distance, where r < rmin, two of thefundamental pumping test assumptions become tenuousand increasingly so as pumping rate increases. First, theassumption of horizontal flow begins to be violated asdistance from the pumping well decreases and drawdownsin the cone of (hydraulic head) depression, which ismore pronounced at higher pumping rates, become deeper.MW-4, at a radial distance of r = 4.39 m, is screenedonly at the very top of the aquifer (Figure 3) whereflow is constrained to an upper, higher-permeability sandabove a muddy till unit. Therefore, drawdown values forMW-4 are not the result of averaging over the wholeaquifer thickness. In contrast, wells in the (vertical)middle of the aquifer show an accentuated vertical-flowcomponent, creating a larger-than-predicted drawdown asseen in MW-3 and MW-5, located at radial distances fromthe pumping well of only r = 2.13 m and r = 2.99 m,respectively (Figure 3). Both these wells are screened


Table 3From Transmissivity, T, Hydraulic Conductivity, K = T/b, and Permeability, k = K μ/ρg), Are Calculated,

Where b Is Aquifer Thickness, μ Is Dynamic Water Viscosity (1.390 × 10−3 Pa s at 8 ◦C), ρ Is WaterDensity (999.8 kg/m3 at 8 ◦C), and g Is Earth’s (Vertical) Gravitational Acceleration (9.81 m/s2)

umax rmax (m) K (m/s) k (×10−12 m2) T (m2/s) T +σ (m2/s) T −σ (m2/s) S S +σ S −σ

Q1 = 0.00306 m3/s, time = 7200 s

0.01 11 No usable wells0.02 16 0.00137 ± 0.00001 194 ± 1 0.0164 0.0163 0.0165 0.037 0.036 0.0390.05 18 0.0012 ± 0.0004 170 ± 50 0.0143 0.0115 0.0191 0.062 0.011 0.2110.10 25 no change from u < 0.050.2 31 0.00100 ± 0.00008 156 ± 11 0.0132 0.0123 0.0141 0.077 0.049 0.1160.5 61 0.00127 ± 0.00011 181 ± 15 0.0153 0.0142 0.0166 0.058 0.031 0.101

Q2 = 0.00458 m3/s, corrected alpha time α(t) = 11,500 s

0.01 11 no usable wells0.02 15 0.0012 ± 0.0002 180 ± 30 0.0151 0.0131 0.0179 0.059 0.020 0.1400.05 23 No change from u < 0.050.10 32 0.00121 ± 0.00006 172 ± 8 0.0146 0.0140 0.0152 0.066 0.049 0.0890.2 45 No change from u < 0.100.5 87 0.00136 ± 0.00005 193 ± 7 0.0164 0.0158 0.0169 0.050 0.037 0.066


0.01 28 0.00127 ± 0.00010 179 ± 6 0.0152 0.0141 0.0165 0.058 0.029 0.1080.02 28 No change from u < 0.010.05 44 0.00113 ± 0.00004 160 ± 6 0.0135 0.0131 0.0141 0.108 0.078 0.1460.10 70 0.00118 ± 0.00004 167 ± 5 0.0142 0.0138 0.0146 0.092 0.069 0.1220.2 100 0.00118 ± 0.00003 170 ± 3 0.0144 0.0142 0.0147 0.086 0.070 0.1040.5 160 No change from u < 0.2


0.01 39 0.00117 ± 0.00004 167 ± 6 0.0141 0.0137 0.0145 0.188 0.138 0.2500.02 66 0.00125 ± 0.00004 177 ± 6 0.0149 0.0145 0.0154 0.139 0.098 0.1920.05 105 0.00125 ± 0.00003 178 ± 4 0.0150 0.0147 0.0153 0.138 0.107 0.1750.10 153 0.00127 ± 0.00003 180 ± 4 0.0153 0.0150 0.0165 0.130 0.102 0.1650.2 220 No change from u < 0.100.5 370 0.00130 ± 0.00003 184 ± 4 0.0157 0.0153 0.0160 0.117 0.085 0.158

For this field site the unconfined aquifer thickness is the height of the saturated zone, i.e., b = 12 m (from Figures 2 and 3).

in the midsection of the aquifer which is the lowest-permeability portion (Figure 3). The variation may be dueto the difference between the average vertical permeabil-ity and the horizontal permeability. Vertical permeabilityacross a horizontally layered formation is controlled by theharmonic mean (Maasland 1957) instead of the arithmeticmean (Equation 11), that is, by

kz = b/(�bi/kzi). (12)

The harmonic mean is dominated by the lowest-permeability unit which, in this case, is a leaky, siltysand in the (vertical) middle of the aquifer (Figure 3).Where the low-permeability units dominate, the apparentpermeability is low, leading to larger observed draw-downs. Additional discussion of aquifer layering andnon-uniform gradient distributions along a well screen

can be found in Hemker (1999) and in Perina and Lee(2006).

The second assumption potentially broken near thepumping well is that of laminar flow. The Reynoldsnumber,

Re = qρd10

φμ, (13)

can be used to estimate if flow is turbulent (e.g., atapproximately Re > 1 for fluid flow around a singlesphere, here, for simplicity, assumed to be applicableto flow around multiple grains in sediments). For ourdilute, fresh water system with temperature T = 8.0 ◦C,the density is ρ = 999.8 kg/m3 and the dynamic viscosityis μ = 1.390 × 10−3 Pa s. In Equation 13, the seepagevelocity is given by the term q/φ, where q is the Darcy


velocity (or specific discharge) and φ is the pore fraction.The intergranular flow diameter is assumed to be theeffective grain size, d10 (Todd and Mays 2005). Thissuggests that turbulent flow may be expected at radialdistances of r < 0.5 m for the pumping rates consideredin this study.

A final test of the inferred values is to comparethe results to published ranges of permeability, k =Kμ/(ρg), where K = T/b, and storativity, S. Calculatedvalues of T for this site range from 0.013 to 0.016 m2/s(Table 3) leading to a best estimate for permeability ofk = (178 ± 4) × 10−12 m2. This value of k agrees withthe mid-range values for clean sands (Freeze and Cherry1979). Storativity, S, for most aquifer systems shouldfall into two categories, values for confined and uncon-fined aquifers. In the confined setting, storativity is definedby the aquifer and fluid compressibility yielding specificstorage, Ss , times the aquifer thickness, b, resultingin Sconfined = Ssb. Under normal pressure variations theexpected storativity range is 0.00005 < S < 0.005. Inunconfined systems, storativity, S, is the sum of Ssb andthe specific yield, Sy , resulting in Sunconfined = Ssb + Sy ,where Sy is limited by the porosity, φ, of the formationwith 0.10 < φ < 0.30 for glacial outwash sands. Asgravity drainage of water from sands is almost 100%,that is, very little water is typically retained in sandsby specific retention, Sr , and as φ = Sy + Sr , we obtainSy ≈ φ. Typically φ � Ssb in many cases, including ourfield site and thus Sunconfined ≈ φ. The results from thelargest and longest pumping rate, Q4, should producethe most representative results for Sunconfined. At this site,we find Sunconfined = 0.13 ± 0.3 (Table 3) for PumpingStep 4, which is also typical of clean outwash sands.

ConclusionsTraditional Jacob distance-drawdown analyses of

pumping test data are frequently employed to date,however, practitioners may apply too stringent (i.e., toolow) an estimate of a small maximum allowable value ofu, that is, too small umax. This can lead to the discardingof entirely useful data measured in monitoring wells thatare located at greater distances, r , from the pumping welland simultaneous inclusion of potentially poor data tooclose to the pumping well. Inclusion of data too close tothe pumping well and discarding of data points deemedtoo far from the pumping well both lead to unrealisticor statistically poor regression lines in the Jacob analysisproducing transmissivity, T , hydraulic conductivity, K ,permeability, k, and/or storativity, S, values outside ofexpected ranges.

Data from all wells that fall along a linear segmentshould be used in Jacob distance-drawdown analyses. Inthe example from the University of Minnesota HydroSite, inclusion of monitoring wells with radial distances,r , from the pumping well that result in u-values upto umax = 0.2 produce linear regression fits within thedrawdown, s, measurement error to a Jacob line. At lowpumping rates the effect is larger, as shown in Table 2,

where estimates for permeability improve from (170 ±50) × 10−12 m2 (umax = 0.05) to (160 ± 15) × 10−12 m2

(umax = 0.2). At higher pumping rates and longer timesthe improvement is smaller but still significant. For thefourth pumping rate, the permeability estimate impro-ves from (167 ± 6) × 10−12 m2 to (180 ± 4) ×10−12 m2. umax = 0.2 is significantly larger than themaximum value cited in various textbooks (umax =0.05), allowing inclusion of drawdown data from moremonitoring wells that are located farther away from thepumping well than previously assumed thereby improvingestimates of hydraulic parameters. It is also important tonote that for umax = 0.01, we would not have any usabledata for the first two pumping rates.

Conversely, Jacob time-drawdown analyses should berun to sufficiently large times to produce a linear segmentwhere u is significantly less than 0.01. The Jacob time-drawdown method relies on the extension of time datawell beyond some small u value.

By improving the applicability of Jacob style analy-ses, simple pumping tests can be designed to use existingwells, where appropriate, and avoid the technical and eco-nomic demands of more advanced hydraulic tomographymethods. Improved application of the Jacob method hasan important role to play in water supply and groundwatersustainability studies where the primary interests are oftenthe overall, large-scale, or effective, hydraulic parametersof an aquifer.

AcknowledgmentsWe thank Hans-Olaf Pfannkuch, Mark Person, and

E. Calvin Alexander, Jr., for initiating the first Universityof Minnesota Hydrogeology Field Camp (Hydrocamp)in 1995. We also thank Steffan Fay for selecting thefield site and laying out the original well field. GeoffDelin, Tom Winters, and Don Rosenberry from the U.S.Geological Survey (USGS) are thanked for establishingthe USGS SHAEP site in which the Hydrocamp site islocated and for maintaining active research programs therefor many years as well as for supporting Hydrocampby drilling a new monitoring well almost every year atthe Hydrocamp site since 1995. We are also grateful to theover 360 students who have attended Hydrocamp andhave helped install new wells and conduct pumping testsat the site almost every year since 1995. The NationalScience Foundation (NSF) is gratefully acknowledged forproviding initial funding to establish the Hydrocamp siteand for supporting the particular research project leadingto this paper through Grant No. EAR-0838541. Anyopinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors anddo not necessarily reflect the views of the NSF. M.O.S.also thanks the George and Orpha Gibson as well asthe McKnight Land-Grant Professorship endowments fortheir generous support of the Hydrogeology and Geofluidsresearch group. We also thank reviewers Tomas Perina,Todd Schweisinger, and an anonymous reviewer for


their insightful comments and suggestions that greatlyimproved the paper.

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