Technical Commentary/

On the Utility of Calculating and InterpretingApparent Storativityby Robert Shaver

IntroductionThe hydrogeology of glaciated terrains is highly com-

plex. Buried glaciofluvial deposits that form many of themost productive aquifers are characterized by complexboundary conditions (Shaver and Pusc 1992). For manyproject areas within glaciated settings, initial hydrogeolog-ical data are often sparse. As a result, the hydrogeologistis faced with developing conceptual hydrogeologic mod-els with limited data. It is not uncommon for existing datato include only pumping-test data from a production welland a driller’s log of that production well. Transmissivityis often calculated using the method of Cooper and Jacob(1946) applied to production well time vs. drawdown data.If a nearby barrier boundary exists, the assumption of aninfinite aquifer is violated and calculation of aquifer trans-missivity by the Cooper and Jacob (1946) method is notvalid. A barrier boundary can easily remain undetectedwith only production well water-level data to evaluate.Depending on the hydraulic properties of the aquifer andthe distance to the barrier boundary from the produc-tion well, analysis of early time water-level data may berequired to calculate a valid transmissivity. Unfortunately,early time water levels often are not systematically mea-sured or if measured, may not provide valid data points foranalysis because the pumping rate was not kept constantduring the early stage of the test.

Owing to well loss and the affects of partialpenetration, true storativity cannot be calculated usingproduction well time vs. drawdown data. As a result,calculation of true storativity generally is omitted.

Water Appropriations Division, North Dakota State WaterCommission, 900 E. Blvd. Bismarck, ND 58505; bshaver@nd.gov

Received January 2012, accepted January 2012.© 2012, The Author(s)Ground Water © 2012, National Ground Water Association.doi: 10.1111/j.1745-6584.2012.00919.x

The purpose of this paper is to demonstrate that eval-uation of apparent storativity calculated using the methodof Cooper and Jacob (1946) can be used to validate cal-culated transmissivity and to detect the existence of anearby barrier boundary using time vs. drawdown datafrom a pumping well thereby improving the conceptualhydrogeologic model. The analytical procedure is appliedto a theoretical non-leaky, confined buried aquifer forwhich synthesized time vs. drawdown data were generatedfor two different barrier configurations and a Pleistoceneburied aquifer in the Bottineau aquifer complex in northcentral North Dakota where two short-term pumping testswere conducted.

Analytical Procedure—Theoretical ConfinedBuried Aquifer

Case #1—Linear Barrier Boundary Located 700 Feet(213.4 m) from a Pumping Well

A production well is completed in a buried aquiferand is located 700 feet (213.4 m) from the near flankof the aquifer (barrier boundary; Figure 1). The oppositeflank of the buried aquifer is located at a large distancesuch that the affects of the far barrier boundary are notmanifested in the synthesized drawdown data. All otherTheis (1935) assumptions are valid. The barrier boundaryis assumed to be vertical. The aquifer is 20 feet (6.1 m)thick and the hydraulic conductivity is 75 ft/d (22.9 m/d).Aquifer transmissivity is 1500 ft2/d (139.4 m2/d), stora-tivity is 2.0 × 10−4, and specific storage is 1.0 × 10−5ft−1

(3.28 × 10−5m−1). The fully penetrating, 100% efficient,12-inch (30.48 cm) diameter production well is pumpedcontinuously for 10,000 min at a constant rate of 175 gal-lons per minute (661.5 L/min).

A semilogarithmic plot of drawdown vs. timesince pumping began for the above example is shownin Figure 2. Applying the method of Cooper and

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Figure 1. Schematic diagram showing an idealized Pleis-tocene buried valley aquifer.

Jacob (1946), aquifer transmissivity is calculated usingEquation 1 and storativity is calculated using Equation 2.

T = 2.3Q

4π�s(1)

S = 2.25T t0

r2w

(2)

where T is the transmissivity, (L2/T); Q is the wellpumping rate, (L3/T); �s is the slope of line connectingdrawdown points over one complete log time cycle, (L);t0 is the point of zero drawdown or static water level, (T);rw is the production well radius, (L); S is the storativity,dimensionless.

Figure 2. Effect of a nearby barrier boundary on theanalysis of a semilogarithmic plot of drawdown vs. time sincepumping began for a 100 and an 80% efficient well.

The line connecting the early time-drawdown pointsin Figure 2 is defined by Equation 3.

−y = 19.206 + 4.135 log(x) (3)

Solving for y = 0, Equation 3 becomes: log (x) =−19.206/4.135 = 2.3 × 10−5 min which is the valueof t0. Applying Equations 1 and 2 to the early time-drawdown data unaffected by the barrier boundary yields atransmissivity of 1500 ft2/d (139.4 m2/d) and a storativityof 2 × 10−4. The u < 0.01 validity criterion is met before1 min. Applying the above analytical approach to the latertime vs. drawdown data affected by the barrier boundaryyields an apparent transmissivity of 750 ft2/d (69.7 m2/d)and an apparent storativity of 0.49. The barrier boundaryincreases the slope of the time vs. drawdown plotsby a factor of two thereby halving the transmissivitybecause transmissivity is inversely proportional to �s.The apparent storativity calculated using barrier-affecteddrawdown is about three orders of magnitude larger thanthe actual value because the barrier boundary shifts thet0 value to the right, increasing t0 which is directlyproportional to S (Equation 2). The t0 value is shiftedfurther to the right as the distance to the barrier boundaryincreases.

Production wells commonly are partially penetratingand rarely, if ever, 100% efficient. Partial penetration andwell inefficiency cause a downward displacement inthe time vs. drawdown data and the t0 intercept isshifted to the left yielding a smaller t0 value and asmaller calculated storativity value (Figure 2). This isdemonstrated by comparing time vs. drawdown data from100 and 80% efficient production wells (Figure 2). Theslope of time vs. drawdown line remains the same andtherefore the calculated transmissivity from Equation 1 isthe correct value. Due to the effects of partial penetrationand well inefficiency, true storativity is underestimatedwhen calculated using production well time vs. drawdowndata. However, calculated values of apparent storativityand apparent specific storage can be used to validatetransmissivity and infer the existence of nearby barrierboundaries that commonly occur in non-leaky, confinedaquifers. This is demonstrated by the following example(Case #2).

Case #2—Linear Barrier Boundary Located 100 Feet(30.5 m) from a Pumping Well

The variables described in Case #1 are the samefor Case #2 except that the barrier boundary is located100 feet (30.5 m) from the pumping well rather than700 feet (213.4 m) from the pumping well. A semilog-arithmic plot of time vs. drawdown data for Case #2is shown in Figure 3. Early time vs. drawdown datadoes not plot on a straight line but rather displays a“roll-off” pattern because the criterion that u < 0.01 isviolated for the image well located 200 feet (61.0 m) fromthe pumping well. Applying the method of Cooper andJacob (1946) to the later time vs. drawdown data yieldsan apparent transmissivity of 750 ft2/d (69.7 m2/d) and

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Figure 3. Semilogarithmic plot of drawdown vs. time sincepumping began with a barrier boundary located 100 feetfrom the pumping well.

an apparent storativity of 0.04. As expected, calculatedtransmissivity is one-half the true transmissivity andcalculated apparent storativity is about two orders ofmagnitude larger than true storativity. Given the above,the method of Cooper and Jacob (1946) cannot be appliedto any segment of the time vs. drawdown data to cal-culate transmissivity. However, calculation of the muchlarger than actual apparent storativity using later time vs.drawdown data provides an indicator that calculated trans-missivity is invalid and a nearby barrier boundary exists.

Reference Values of Storativities and Specific Storagein Non-Leaky, Confined, Buried Aquifers of GlaciofluvialOrigin in North Dakota

As part of the North Dakota County Ground-Water Studies program carried out from the 1950sthrough the 1980s, numerous aquifer tests were conductedby the North Dakota State Water Commission in Pleis-tocene non-leaky, confined buried aquifers comprised ofsand and/or gravel (North Dakota State Water Commis-sion Open-File Aquifer Test Reports, various dates). Waterlevels in two or more observation wells were measuredduring the pumping and recovery periods. Standard Theis(1935) and Cooper and Jacob (1946) analytical proceduresusing time vs. drawdown and distance vs. drawdown datawere used to calculate aquifer hydraulic properties. Basedon the analyses of 15 aquifer tests, mean storativities cal-culated from observation wells ranged from 1.0 × 10−4

to 6.3 × 10−4 and mean specific storage ranged from8.4 × 10−6ft−1 (2.8 × 10−5m−1) to 1.2 × 10−5 (3.9 ×10−5m−1).

In Pleistocene buried aquifers, calculated apparentstorativity and apparent specific storage using the methodof Cooper and Jacob (1946) for time vs. drawdown data

measured in a pumping well that are two to three ordersof magnitude larger than the above reference values likelyindicates the existence of a nearby barrier boundary andthat calculated transmissivity likely is invalid.

Example—The Bottineau Buried Aquifer ComplexIn 2001 to 2002, the North Dakota State Water

Commission conducted a hydrogeologic investigation toevaluate the municipal ground water supply for the City ofBottineau, located in north central North Dakota (Shaver2002). Test drilling, water-level, and pumping-test data inthe Bottineau study area indicate at least five and possiblysix sand and gravel channels that are hydraulicallydiscrete and appear, for the most part, to occupy differentstratigraphic positions (Shaver 2002). The non-leaky,confined sand and gravel channels commonly are overlainand underlain by a clay till.

The City of Bottineau municipal well #2 is completedin one of the above buried sand and gravel channels.Municipal well #2 was installed in January 1958. Thedriller’s log of the pilot hole is shown in Table 1.

The 12-inch (30.48 cm) diameter well is completedwith 12 feet (3.7 m) of 8-inch (20.32 cm) diameter, #60slot screen set from 68 feet (20.7 m) to 80 feet (24.4 m)below land surface. The screen was gravel packed.

On January 9, 1958, a pumping test was conducted onmunicipal well #2. The well was pumped at a relativelyconstant pumping rate, which averaged about 148 gallonsper minute (559.4 L/min) for 1444 min. A semilogarith-mic plot of time since pumping began vs. drawdownis shown in Figure 4. The data indicate minor scatterfrom a straight-line trend likely due to minor fluctua-tions in pumping rate. Linear regression analysis yieldsa slope (�s) of 8.22 feet (2.5 m). Using the analyticalmethod of Cooper and Jacob (1946), a transmissivity of635 ft2/d (59.0 m2/d) is calculated (Equation 1). Appar-ent storativity is calculated at 0.24. Based on an estimatedaquifer thickness of 21 feet (6.4 m), apparent specific stor-age is 6.3 × 10−3ft−1 (2.1 × 10−2m−1). These storativityand specific storage values are two to three orders ofmagnitude larger than the buried aquifer reference val-ues previously provided. This suggests that the drawdowndata are affected by a nearby barrier boundary and that thecalculated transmissivity could be up to about one-halfthe actual transmissivity.

Table 1Driller’s Log of Municipal Well #2

Description Depth (ft bls) (m. bls)

From ToTopsoil 0 0.5 (0.15 m)Gray clay 0.5 (0.15 m) 4 (1.2 m)Yellow clay 4 (1.2 m) 11 (3.4 m)Gray clay, rocks 11 (3.4 m) 68 (20.7 m)Sand and coarse gravel 68 (20.7 m) 73 (22.3 m)Very clayey sand,

becoming finer73 (22.3 m) 100 (30.5 m)

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Figure 4. Plot of log of time vs. arithmetic pumping level inmunicipal well #2 (pumping test—1958).

On October 3, 2001, the North Dakota State WaterCommission conducted a pumping test on municipal well#2 (Shaver 2002). The well was pumped continuouslyat a rate of 65 gallons per minute (245.7 L/min) for350 min. A semilogarithmic plot of time since pumpingbegan vs. pumping level in municipal well #2 is shownin Figure 5. Water levels were corrected to account forrecovery that occurred during previous pumping periods.The pumping water levels plot on a straight line forabout the first 12 min of pumping. The slope (�s) of thisline is 1.57 feet (0.48 m). Using Equation 1 and a �s of1.57 feet (0.48 m), an aquifer transmissivity of 1460 ft2/d(135.7 m2/d) is calculated. Although the pump columndiameter was not known, the effects of casing storage withno pump in the well would not affect the well drawdownafter 4 min of pumping at a rate of 65 gallons per minute(245.7 L/min). Extending the line connecting the first12 min of pumping-level points to determine t0 and usingEquation 2 yields an apparent storativity of 1.7 × 10−6.This value of storativity is about two orders of magnitudesmaller than the previously reported reference values. Thisis to be expected as the drawdown is displaced downwardand the t0 value is shifted to the left due likely to theeffects of partial penetration and well loss. It is importantto note that low storativity and specific storage valuescan also be an indication of leakage that would cause theslope of the time vs. drawdown line to decrease. However,leakage is not considered significant because the aquiferis confined above and below by a clay till.

The �s of the early time data of 1.57 ft (0.48 m)in Figure 5 is about one-half that of the later-time data(2.95 ft [0.90 m]). Thus, using the early time-drawdowndata yields a transmissivity of 1460 ft2/d (135.7 m2/d)that is about double the transmissivity calculated usinglater time-drawdown data (775 ft2/d [72.0 m2/d]). Thedoubling of the slope (�s) after about 12 min of pumpingsupports the existence of a nearby barrier boundary.

Figure 5. Plot of log of time vs. arithmetic pumping level inmunicipal well #2.

ConclusionsMost production wells are rarely, if ever, 100%

efficient and are often partially penetrating. Therefore, itis not possible to calculate true aquifer storativity andspecific storage using time vs. drawdown data obtainedfrom a pumping well. As a result, calculation of storativityis often omitted. For situations where pumping-test dataonly is available from a production well, and early time vs.drawdown data is sparse or unusable, apparent storativityshould always be calculated using the method of Cooperand Jacob (1946) as it can provide an indicator as to thevalidity of calculated transmissivity and the existence ofa nearby barrier boundary. If possible, reference values ofstorativity and specific storage should be compiled fromother sources that are derived from similar hydrogeologicsystems. Apparent storativity should be compared withthese reference values.

ReferencesCooper, H.H., and C.E. Jacob. 1946. A generalized graphical

method for evaluating formation constants and summarizingwell field history. American Geophysical Union Transac-tions 27: 526–534.

North Dakota State Water Commission Open-File Aquifer TestReports. Various dates.

Shaver, R.B. 2002. A hydrogeologic analysis to determine thesustained yield of the Bottineau Municipal Well Fieldand All Seasons Rural Water Systems I and II, BottineauCounty, ND. North Dakota State Water CommissionGround-Water Studies No. 109, 196.

Shaver, R.B., and S.W. Pusc. 1992. Hydraulic barriers inPleistocene buried-valley aquifers. Ground Water 30, no.1: 21–28.

Theis, C.V. 1935. The relation between the lowering ofthe piezometric surface and the rate and duration ofdischarge of a well using ground-water storage. AmericanGeophysical Union Transactions 16, 519–524.

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