use of time subsidence data during pumping to characterize specific storage and hydraulic...

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Use of time–subsidence data during pumping to characterize

specic storage and hydraulic conductivity of semi-conning unitsT.J. Burbey *

Department of Geological Sciences, Virginia Tech, 3053 Derring Hall, Blacksburg, VA 24061, USA

Received 17 April 2002; accepted 1 May 2003

Abstract

A new graphical technique is developed that takes advantage of time–subsidence data collected from either traditionalextensometer installations or from newer technologies such as xed-station global positioning systems or interferometricsynthetic aperture radar imagery, to accurately estimate storage properties of the aquifer and vertical hydraulic conductivity of semi-conning units. Semi-log plots of time–compaction data are highly diagnostic with the straight-line portion of the plotreecting the specic storage of the semi-conning unit. Calculation of compaction during one-log cycle of time from theseplots can be used in a simple analytical expression based on the Cooper– Jacob technique to accurately calculate specic storageof the semi-conning units. In addition, these semi-log plots can be used to identify when the pressure transient has migratedthrough the conning layer into the unpumped aquifer, precluding the need for additional piezometers within the unpumpedaquifer or within the semi-conning units as is necessary in the Neuman and Witherspoon method. Numerical simulations areused to evaluate the accuracy of the new technique. The technique was applied to time–drawdown and time–compaction datacollected near Franklin Virginia, within the Potomac aquifers of the Coastal Plain, and shows that the method can be easilyapplied to estimate the inelastic skeletal specic storage of this aquifer system.q 2003 Elsevier B.V. All rights reserved.

Keywords: Subsidence; Aquifer compaction; Well hydraulics

1. Introduction

Aquifer tests are the most fundamental means bywhich groundwater practitioners can quantitativelyestimate the transmissivity and storage properties of the aquifer and semi-conning unit. The pioneeringwork of Meinzer (1928), Theis (1935), Jacob (1940),and later Hantush (1956, 1959, 1960, 1967) for theanalysis of conned and leaky aquifers still remains

the benchmark by which other techniques andmodications have been based. Extensions andmodications to these earlier works are too numerousto list here. However, several key works have beenpivotal in describing the nature of drawdown duringpumping and are still widely used today to character-ize aquifer properties under a wide variety of settings(e.g. Cooper and Jacob, 1946; Neuman and With-erspoon, 1968; 1969a– c, 1972; Walton, 1970;Moench, 1985; Sridharan et al., 1987 ). Many of these techniques for estimating the storage andhydraulic conductivity properties of the aquifer

0022-1694/03/$ - see front matter q 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0022-1694(03)00197-5

Journal of Hydrology 281 (2003) 3–22www.elsevier.com/locate/jhydrol

* Tel.: þ 1-5402316696; fax: þ 1-5402313386.E-mail address: [email protected] (T.J. Burbey).

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system have been incorporated into softwarepackages, developed for rapid analysis of aquifertest data.

Each of these techniques makes a number of assumptions in order to obtain an analytic solution oran approximation to the exact solution that allows themethod to be used to estimate the transmissivity andstorativity of the aquifer. In addition, several of themethods allow for estimation of the conning unit

diffusivity (e.g. Neuman and Witherspoon, 1972 ). Theeld data collected during the aquifer test andrequired to estimate these parameters generallyinvolves only the collection of hydraulic head andtime data for various hydrologic units. Semi-conningunits are known to contribute large quantities of water to the pumped aquifer (Poland and Davis, 1969)as a result of the large compressibilities of clay-richunits. However, in order to evaluate the specicstorage and vertical hydraulic conductivity of thesemi-conning unit, head data (or time-lag data)within the clay unit are required along with laboratorydata on the compressibility of the clay ( Neuman and

Witherspoon, 1972 ). Because only head data arecollected during aquifer testing, further renementand characterization of the semi-conning unitscannot be made without piezometers located in eachhydrogeologic unit.

The implementation of extensometer data ( Pope,2002; Harmon, 2002 ) and the recent advent of radartechniques such as interferometric synthetic apertureradar (InSAR) and high precision global positioningsystems (GPS) in hydrogeologic applications ( Zebkeret al., 1994; Ikehara, 1994; Thom et al., 1995; Zebkeret al., 1997; Fielding et al., 1998; Galloway et al.,1998; Amelung et al., 1999; Davies and Blewitt,2000; Hoffmann et al., 2001 ) have allowed for the real-time determination of subsidence or total compactionoccurring as the result of pumping. The application of these techniques during an aquifer test of moderatelength of 10 days or longer, could provide valuabledata regarding the compressible units in the aquifersystem that would allow further parameterization of the ne-grained units that cannot be made withhydrograph information alone.

This paper presents a straightforward analyticaland graphical methodology based on the Cooper–Jacob method ( Cooper and Jacob, 1946 ) for estimat-ing the storage coefcient of conned aquifers and

the specic storage and vertical hydraulic conduc-tivity of the semi-conning units in leaky aquifersystems under transient pumping conditions. Numeri-cal simulations are used to evaluate the validity andaccuracy of the methodology described below. Thetechnique has advantages over traditional straight-linemethods in that it is not necessary to know the exacttime of the start of pumping.

2. Methodology and results

2.1. Conned aquifers

Typical aquifer tests employ the use of time-dependent drawdown data for making estimates of aquifer storativity and transmissivity. Semi-logstraight-line methods such as the Cooper–Jacobtime–drawdown method are commonly used becausethey allow one to readily estimate these parametersprovided that the aquifer test is sufciently long andthe assumptions inherent in the technique are not

grossly violated. Furthermore, straight-line methodsprovide visual indicators of the presence of bound-aries or signicant leakage from adjacent semi-conning units because the eld data depart fromthe perfectly conned straight-line semi-log plot.Estimation of aquifer transmissivities tends to be quiteaccurate using this approach even for leaky aquifers aslong as r ; the radial distance from the pumping well tothe observation well, is sufciently small. Theaccuracy of this technique for estimating transmissiv-ity is attributed to the fact that transmissivity is highlysensitive to drawdown during pumping. The Cooper–Jacob equation can be written as

s ¼2:303Q

4p T log

2:25Tt r 2S ð1Þ

where s is the drawdown, Q is the constant pumpingrate, T is the aquifer transmissivity, and S is theaquifer storativity. Fig. 1 shows the semi-log plot of the numerically produced time–drawdown data at anobservation well 50 m from a constantly pumped wellin a perfectly conned 100 m thick aquifer (a Theis-type aquifer) with a hydraulic conductivity of 5 m/d.Plots are shown for storage coefcients spanning threeorders of magnitude. The slope of each plot in Fig. 1

T.J. Burbey / Journal of Hydrology 281 (2003) 3–224


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can be quantied from Eq. (1) asD s

logðt 2 = t 1Þ¼

2:303Q4p T

ð2Þ

where D s is the change in drawdown over the timeinterval t 2 2 t 1: The slopes are constant because thetransmissivity remains the same for all three storagevalues simulated.

The storage coefcient of the aquifer is esti-mated from these data plots by extending thestraight-line portion of the plot to the zero draw-down axis. The estimated time, t 0; at which thedrawdown is zero is used in Eq. (1) to yield

0 ¼ log 2:25Tt 0

r 2S ð3Þor, by solving for S

S ¼2:25Tt 0

r 2 ð4Þ

This is the graphical approach for estimatingstorativity described by Cooper and Jacob (1946) .Unfortunately, storage properties tend to be onlymoderately sensitive to drawdown in many settings(Neuman, 1979; Anderson and Woessner, 1992 ).Furthermore, estimating t

0 can potentially lead to

large errors, particularly in settings in which thestorage coefcient tends to be large. It is notuncommon for estimated storage errors to beupwards of one order of magnitude using thisapproach. In addition, this approach requires one toknow the exact time since the start of pumping.Hydrogeologists often correlate the insensitivity of the storage with the lack of importance for a goodestimate. This can be troublesome when trying to

preserve groundwater supplies through propermanagement.A more diagnostic method for accurately estimat-

ing storage is to plot total compaction (subsidenceexpressed at the land surface) with time during theaquifer test. This approach is preferred becausecompaction is directly related to the compressibilityand is therefore directly related to the storage of theaquifer. Although this method does not account forthe release of water resulting from the expansion of water, in compressible systems where this techniquewould be applied, the matrix compressibility istypically several orders of magnitude greater thanwater compressibility. Therefore, the contributionfrom water expansion is considered negligible. Fig. 2shows the semi-log straight-line plots of the time–compaction record for the same conned aquifer

Fig. 1. Simulated drawdown as a function of time during pumpingfor three different values of storage derived from a perfectlyconned 100 m-thick homogeneous and isotropic aquifer.

Fig. 2. Simulated subsidence as a function of time during pumpingfor three different values of storage derived from a perfectlyconned 100 m-thick homogeneous and isotropic aquifer.

T.J. Burbey / Journal of Hydrology 281 (2003) 3–22 5


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using the same three storage coefcients as illustratedin Fig. 1. Note that the slope is vastly different foreach of the three values of storage coefcient.Calculation of storage can be obtained from this plotusing the slope in much the same way the transmis-sivity was estimated from Eq. (2) and Fig. 1. Thisyields a potentially more accurate calculation of thestorage coefcient than the traditional use of time–drawdown plots. Furthermore, this method no longer

depends on the estimation of time. That is, the time of the start of pumping is not required.This is particularlybenecial in settings where compaction data has notbeen measured until sometime after the start of pumping. If accurate extensometer, GPS or InSARvalues can be obtained during some portion of theaquifer test or period of pumping, then this techniqueshould be favored over the traditional techniqueswhere only time– drawdown plots are used.

For one-dimensional consolidation the storativityof the subsiding system can be expressed simply as(Burbey, 2001 )

S ¼D bD h ð5Þ

where D b is the change in thickness or measuredcompaction measured within the aquifer systemduring D h; the change in head or total drawdownduring pumping. The storage expressed in Eq. (5)assumes that the contribution of water expansion issmall. In Eq. (2) D h ¼ D s; that is, the change in headis equivalent to the change in drawdown. If the changein drawdown and change in compaction are measuredover one log cycle of time, the denominator in the leftside of Eq. (2) becomes equal to 1. Substituting Eq. (5)into Eq. (2) and solving for S gives

S ¼D b4p T 2:303Q

¼5:46D bT

Q ; ð6Þ

where D b is the straight-line measured compactionoccurring over one log cycle of time. In thisexpression, the more diagnostic slope is used on thetime–compaction plot as opposed to the timeintercept at s ¼ 0 on the time–drawdown plot. Forthe hypothetical aquifer described above (here it isassumed that the aquifer represents the compressibleunit of interest), time–subsidence data were producedusing the IBS3 package ( Leake and Prudic, 1991 ) withModow ( McDonald and Harbaugh, 1988 ). These

simulated data are plotted in Fig. 3 along with thecalculated D b expressed over one log cycle of time.Transmissivity was estimated using the classicalTheis method on the time–drawdown data, with thedrawdown values calculated using Modow. Thesedata are then incorporated into Eq. (6) to yield anestimated S of 1.0 £ 10

2 4 , which is identical to thenumerical value used in the simulation. Additionalanalyses using numerical models resulted in predictedstorage coefcients that were always within 10percent of the actual value.

One might argue that this technique is of limitedvalue because subsidence occurs from conning unitsand not aquifers. However, in many settings in whichsubsidence occurs, the ‘aquifer system’ is used torepresent the aquifer containing compressible lensesof clays and silty-clays. Hence, this technique wouldproduce an average value for the aquifer system. Thecase study used for the application of this techniquerepresents such a system.

2.2. Leaky aquifers

The approach of incorporating time–compactiondata can be extended beyond perfectly conned

Fig. 3. Simulated time– subsidence plot for a perfectly connedaquifer with calculated compaction evaluated over one log-cycle of time. This calculated compaction is used in Eq. (6) to determine thestorage coefcient of the compacting unit.



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aquifers to the more commonly encountered leakyaquifer systems in which storage within the conningunit is considered important. In these settings thecompressibilities of the conning unit are typicallyone to several orders of magnitude greater than that of the aquifer unit. Hence, the compaction data largelyreect the compressibility of the semi-conning unit,whereas the time–drawdown data largely reect theaquifer properties where the head data are being

measured. Field data that depart from the Theis-typecurve for the aquifer being pumped are related to (1)the distance from the observation well where the dataare being acquired to pumping well, (2) the thicknessand vertical hydraulic conductivity of the boundingsemi-conning units, and (3) the head in theunpumped aquifer if one exists. Because we assumehere full penetration of the pumping well with a smallannulus and homogeneous and isotropic aquiferconditions, deviations in the time– drawdownresponse due to a departure from these conditionsare not considered.

Two hydrogeologic conditions will be evaluated:

(1) Two aquifers separated by a single conning unitwhere no drawdown occurs in the unpumpedaquifer that overlies the conning unit ( Fig. 4).This condition is valid where the overlyingaquifer is unconned with a sufciently largestorage such that downward leakage induced bypumping of the lower aquifer would notappreciably affect the water level of thisunconned aquifer.

(2) Multiple aquifer and conning-unit system(Fig. 5) in which one aquifer is pumped andheads in the overlying aquifer can be affected bypumping from within the pumped aquifer.

2.2.1. Condition 1: one semi-conning unit and no drawdown in unpumped aquifer

The method for evaluating the rst hydrogeologiccondition was rst described analytically by Hantush(1960) . The solution for drawdown in the pumpedaquifer of this well known governing equation that

mathematically describes the leaky aquifer conditionpresented in Fig. 4 has been estimated for both earlyand late times and is expressed as

sðr ; t Þ ¼Q

4p T ð1

u

e2 y

y erfc

b u1 = 2

yð y 2 uÞ1 = 2 !d y ð7Þwhere b is known as the Hantush leakage factor forleaky aquifers, and u is the well function expressed as

u ¼r 2S 4Tt

;

and erfc is the complementary error function.According to Hantush (1960) the validity of Eq. (7)

Fig. 4. Conceptual model of the one-conning-unit aquifer systemused to evaluate specic storage and hydraulic conductivity of thesemi-conning unit.

Fig. 5. Conceptual model of the multiple-conning unit aquifersystem used to evaluate an average specic storage and hydraulicconductivity of the semi-conning units.



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is limited to the range of time given by theexpression

t #0:1b0S 0

K 0 ð8Þ

where the primes on b; K and S indicate thethickness, vertical hydraulic conductivity and stor-ativity of the semi-conning unit, respectively.Neuman and Witherspoon (1969c) determined thatEq. (8) is far too conservative and that its validitycan be expanded to a much larger range of timethan suggested by Eq. (8). They concluded thatEq. (7) is valid while the time–drawdown curve issteep. That is, the solution expression is valid if it isnot affected by the constant head condition imposedby the overlying aquifer. We will show that thiscondition can be evaluated graphically using atime–subsidence plot obtained from the aquifertest. In Eq. (7) the dimensionless leakage factor isevaluated from type curves and is expressedmathematically as

b ¼r 4

K 0

Kbb 0S 0

S 1 = 2

: ð9Þ

To evaluate the usefulness of the methodologydescribed in the previous section on connedaquifers for the leaky-aquifer condition, a simulationwas performed in which 10 model layers were usedto simulate the semi-conning unit and one modellayer was used for each the unconned and connedaquifers. The 10 layers were implemented so that amore accurate and realistic head distribution throughtime could be simulated through the semi-conningunit as the underlying aquifer was pumped ( Fig. 4).Modow ( McDonald and Harbaugh, 1988 ) was usedto simulate hydraulic heads and the IBS3 package(Leake and Prudic, 1991 ) was used to simulatecompaction and total subsidence. Table 1 and Fig. 4provide the parameters and aquifer system geometryused in the simulation. The lower conned aquiferwas pumped at a rate of 2000 m 3 /d for 30 days.Three assumptions are made regarding the hydro-geologic conditions that lead to the simulationresults discussed below:

(1) Horizontal strain is negligible. Burbey (2001)indicates that horizontal strain may not be

negligible in many settings in which aquifersare pumped from unconsolidated sediments.However, we are making evaluations near thepumped well where horizontal deformation islimited as a result of the zero-radial straincondition imposed by the well casing. In thisanalysis we assume that all strain is vertical.The IBS3 package does not account forhorizontal strain.

(2) The hydraulic conductivity of the conningunit is at least two orders of magnitude lessthan that of the aquifers. This essentially leadsto the condition of vertical ow through theconning unit. Neuman and Witherspoon(1969a–c) indicate through nite-elementmodeling that when this condition occurs theerror introduced by this assumption is lessthan 5%.

(3) The compressibility of the conning unit ismore than one order of magnitude greater thanthe compressibility of the aquifers. This isgenerally true for conning units rich in clayand undergoing nonrecoverable or virgincompression. If preconsolidation heads havebeen lowered signicantly within the conningunit prior to the aquifer test such thatpermanent deformation of the clays hasalready occurred, then the deformation tendsto be elastic and the active compressibilities of the conning unit are closer to that of theaquifer. In this scenario the approach would beto obtain an aquifer system (combined aquiferand semi-conning unit average value) specicstorage through use of the previously describedmethod for conned aquifers.

Table 1Simulated and estimated aquifer and conning unit parameters usedfor the rst hydrogeologic condition with one conning unit and nodrawdown occurs in the unpumped aquifer

Aquifer and conningunit parameters

Simulationvalue

Estimated valuefrom new method

T (m 2 /d) 125.0 132.4S (dimensionless) 5.0 £ 10

2 5 5.2 £ 102 5

K 0v (m/d) 0.005 0.003

S 0s (1/m) 8.0 £ 102 5

5.6 £ 102 5



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Fig. 6a shows the numerically derived time–drawdown plot for this simulation at an observationwell 8.6 m from the pumping well. The resultsindicate that after 30 days of pumping the pumpedaquifer has reached equilibrium with the overlyingunconned aquifer as the time rate of drawdown has

asymptotically approached zero. The Hantush typecurve displayed in Fig. 6b is obtained from Eq. (7)with a best t b of 0.01. The value of b is small(representing a curve that closely approaches theTheis curve) because the extent to which leakageaffects drawdown in pumped aquifers is a function of the radial distance from the pumping well ( Neumanand Witherspoon, 1972 ). The Hantush method is usedin favor of the Theis method here because an estimate

of b is desired for the technique used here and alsoprovides a better estimate of T and S for the aquifer.The curve match yields a transmissivity and storati-vity for the aquifer that are nearly identical to theactual values used in the simulation ( Table 1 ). The b value can be used in Eq. (9) but this still leaves theproduct K 0S 0 that cannot be further reduced with thetraditional time–drawdown data plot alone. Neumanand Witherspoon (1972) indicate that the only way toevaluate the properties of the semi-conning unit is tohave an observation wells located in this clay unit andto either measure the head relative to that of thepumped aquifer, or to evaluate the lag time for the

pressure transient to propagate from the aquifer to theopening of the piezometer in the conning unit.

Time–compaction data collected during the aqui-fer test can be used to provide accurate estimates of S 0

and K 0: This data set provides important informationthat precludes the need for piezometers in the semi-conning unit, or laboratory estimates of compressi-bility, information that was necessary when only headdata were available during aquifer testing. As GPSand InSAR techniques become further rened sub-millimeter resolution will be possible. Currently,properly designed extensometers can provide nearmicron resolution and xed-station GPS using choke-ring technology can provide near millimeter verticalresolution (Blewitt, NBMG, oral commun., 2002).The magnitude of deformation that can be expected inthese limited duration aquifer tests is typically of theorder of 1 cm.

Fig. 7a is the semi-log time–compaction plot forthis aquifer test. The curve is extremely diagnostic forindicating the source of compaction and the unitsresponsible for the release of water from storage. Thecurve can be divided into three segments. Early-timedata (labeled as segment 1 on the curve) do not fall ona straight line using semi-log graph paper because theinitial release of water and subsequent compaction

Fig. 6. (a) Time–drawdown curve at an observation well 8.6 mfrom the pumping well for the conceptual model depicted in Fig.4; (b) Hantush type-curve match for this time– drawdown plotshowing leakage factor, transmissivity and storage coefcient forthe type-curve match.



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occurs from the less compressible pumped aquifer.Fig. 8a is a histogram of the simulated compaction foreach model layer. Note that during this early timevirtually all the water released from storage occursfrom within the pumped aquifer. Water does not begin

to be released from the semi-conning unit until theinection point on the curve is reached at a time inexcess of 0.3 days. From this time until approximately4 days into the test (segment 2), the subsidence datafall on a straight line using semi-log paper. During this

Fig. 7. (a) Time–subsidence curve at an observation well 8.6 mfrom the pumping well for the conceptual model depicted in Fig.4; (b) calculated compaction over one log cycle of time used inEq. (6) to estimate the storage and specic storage of the semi-conning unit.

Fig. 8. Histograms showing the distribution of compaction by layer(a) after 0.3 days of pumping and (b) after 40 days of pumping.Layer 1 is the overlying unconned aquifer, layers 2–11 representthe conning unit, and layer 12 is the pumped conned aquifer.



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period all the measured compaction originates fromthe semi-conning unit as signicant quantities of water are being released from this unit through thedissipation of excess pore pressure. At the nextinection point (at approximately 6 days) the pressuretransient has made its way to the overlying unconnedaquifer as the subsidence asymptotically approaches aterminal value (segment 3). The time betweeninection points should be close to the time constant

of the semi-conning unit for a singly drainingcontinuous layer. This time constant derived byRiley (1969) is expressed as

t ¼S 0sb

2

K 0vð10Þ

and results in the time required for 93% of the porewater to dissipate from the unit. The time constantfrom Eq. (10) is very close to the time between thetwo inection points on time–subsidence plot shownin Fig. 7b. Fig. 8b is the histogram indicating the nalcompaction, by model layer, at the end of the aquifer

test. The total compaction from the semi-conninglayer represents 92.5 % of the total subsidencemeasured at the land surface. It should be noted thatthe distance from the pumped well at whichobservations are made does slightly affect the slopeof the time–subsidence curve at distances greater thanabout 2.5 times the aquifer thickness. At distancesequaling four times the aquifer thickness the decreasein slope results in about 10 percent less estimatedcompaction over one log cycle of time. This wouldtranslate into a smaller value of estimated specicstorage for the conning unit. However, because thelength of the aquifer test results in small quantities of total compaction, one would in practice want to makethe observations near the pumping well where thesubsidence is greatest, thus alleviating this source of error.

The next step is similar to the method described byCooper and Jacob (1946) and discussed previously forperfectly conned aquifers where we use a straight-line approximation through segment 2, which shouldnearly always approximate a straight line on semi-logpaper (Fig. 7b). In other words, the dissipation of porepressure from the semi-conning unit and its relatedcompaction, as expressed through the poroelasticstress– strain constitutive relation, responds like

a conned aquifer under a pumping stress. Jorgensen(1980) presents an expression for compaction withinthe semi-conning unit that is logarithmicallyproportional to the effective stress at the boundaryof the unit, assuming no signicant change in porosity(void ratio) during compression. Using the straight-line approach through segment 2, the compactionoccurring over one log cycle of time is 3.9 mm. TheHantush method was used to estimate T and S of the

aquifer because the head changes used to estimatethese parameters were obtained from the aquifer withmodication resulting from leakage from the over-lying semi-conning unit. It is proposed that Eq. (6)can be used to calculate the storativity of the semi-conning unit because the compaction, which is usedto estimate storativity, is occurring largely within thismore highly compressible unit ( Fig. 8b). Once theconning-unit storativity is known (and specicstorage if b0 is known) it can be used to solve for thevertical hydraulic conductivity by rearranging Eq. (9)

K 0 ¼16b 2Kbb 0S

r 2S

0 : ð11Þ

Table 1 lists the nal estimated values and shows howthey compare with the simulated values. Closeagreement occurs for each parameter and only onepiezometer was required. It is assumed here that theaquifer and conning unit thicknesses are knownthrough driller’s logs or other available data.

It should be noted that it does not matter if thepressure transient has propagated through the conn-ing unit (segment 3 effectively would be absent). Aslong as a straight-line can be drawn through the datarepresenting compression of the semi-conning unit(segment 2); then accurate values of specic storageand hydraulic conductivity can be obtained with thistechnique.

2.2.2. Condition 2: multiple semi-conning unitswith drawdown in the unpumped aquifer

The second hydrogeologic condition to be eval-uated involves the more general case of multipleaquifers and intervening semi-conning units. Theassumptions presented in the rst condition areapplied here except that the overlying unconnedaquifer has a variable head. That is, drawdown canoccur in this overlying aquifer unit. Fig. 5 representsthe conditions used for the numerical simulation to



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obtain data sets for drawdown and subsidence as afunction of time. The values for the semi-conningunits and aquifers are the same as those listed inTable 1 . That is, each conning unit and each aquiferhave identical hydraulic properties. This distinction isnot required, but as we shall see, without thisassumption explicit values of specic storage andhydraulic conductivity for each semi-conning unitcan not be made without head date for each unpumped

aquifer unit.Neuman and Witherspoon (1969a– c, 1972)present a complex general solution for the conditionshown in Fig. 5 such that each aquifer depends on vedimensionless parameters, one of which is Hantush’sleakage factor expressed in Eq. (9). A tractablesolution is obtained by making predictions at earlytimes or alternatively late times and thus reducing thenumber of parameters, or by setting dimensionlessparameters equal and thus effectively making hydrau-lic properties within individual units equal. Toexplicitly calculate an estimate of the specic storageof the semi-conning units a piezometer must be

installed within these low permeable units at a knownvertical distance above or below the pumped aquifer.In other words, to characterize the semi-conningunits, hydraulic head data or transient pressure datawithin the low permeable units are required. In manyeld applications, however, an observation pointwithin the semi-conning unit is not available andin practice is avoided.

Fortunately, the methodology described earlier canbe used in this more complex setting without the moredata intensive, and mathematically complex, solutionof Neuman and Witherspoon (1969a–c) . As beforethe key is to plot the semi-log time–subsidencedata, which is diagnostic for evaluating (1) whencompaction is largely occurring within the semi-conning units, and (2) when signicant drawdownbegins within the unpumped aquifer such that thedrawdown curves within the pumped aquifer would beaffected by the contribution from the overlyingaquifer. Fig. 9 represents the 30-day time–subsidenceplot for the aquifer test described by Fig. 5, withaquifer properties listed in Table 1 . The straight-linedrawn through the linear portion of the curverepresents, as before, the compression within thesemi-conning units, whose slope is representative of the specic storage of the most compressible units.

Fig. 10a and b indicate the proportion of compactionoccurring at early time (primarily aquifer) and latertime (primarily aquitards) during the aquifer test forthe multiple aquifer system where the aquifer isrepresented by model layer 12. Drawdown within theunpumped aquifer is not signicant based on the slopeof this plot. If drawdown was signicant, the slope of the time–subsidence plot would undergo an abruptchange indicating that the pressure gradient across theconning unit had become smaller as the transient

pressure wave propagated to the unpumped aquifer.The simulated drawdown within the unpumpedaquifer at a distance of 8.6 m from the pumped wellwas only 0.25 m after 6 days and 1.0 m after 40 days;only a small fraction of the drawdown in the pumpedaquifer that reached 12 m at this same horizontaldistance at the conclusion of the aquifer test. Thissmall amount of drawdown apparently is notsignicant enough to lower the slope of thecompaction curve, which would occur if the headgradient across this unit were reduced as a result of signicant contribution from the overlying aquifer.Fig. 10b indicates that the compaction through

Fig. 9. Time–subsidence curve at an observation well 8.6 m fromthe pumping well for the conceptual model depicted in Fig. 5. Thecalculated compaction is measured over one log cycle of time andused in Eqs. (6) and (13) to estimate the average specic storage of

the semi-conning units.



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the upper conning unit is greatly reduced at the topof the unit, conrming this conclusion (see Eq. (6)).As a result, it can be concluded that the Hantushmethod that assumes no drawdown in the overlyingaquifer can be used with condence without any

appreciable error. This assessment can be made withthe time–subsidence plot and without an observationwell in the overlying aquifer.

The Hantush (1960) method is more tractable thanthe Neuman and Witherspoon (1969a– c) method andrequires only a single leakage factor expressed by

b ¼r 4

K 0 = b0

T S 0

S 1 = 2

þK 00 = b00

T S 00

S 1 = 2( ); ð12Þ

where the double primes refer to the secondunderlying semi-conning unit shown in Fig. 5.

As before, the storage coefcient and transmissiv-ity of the aquifer can be evaluated by matching thedrawdown curve of the pumped aquifer with theHantush type curve as was done previously. Fig. 11represents the time–drawdown plot for this aquifertest with the matching Hantush type curve that yieldsa dimensionless leakage factor (Eq. 12) of 0.028. Thecurve match yields a hydraulic conductivity andstorage coefcient very close to the simulated valuesused (Fig. 11 and Table 2 ).

The information necessary to determine the storageproperties of the semi-conning units is now avail-able. The estimated aquifer transmissivity of 135.7 m 2 /d from the Hantush type curve and thesemi-log subsidence value (subsidence occurring overone log cycle of time in Fig. 9) of 0.0084 m can nowbe used in Eq. (6) to estimate the storage coefcient of the semi-conning units. This yields a storagecoefcient of 3.11 £ 10

2 3 . This estimated valuerepresents the average storage of the more highlycompressible semi-conning units. To obtain aspecic storage for each conning unit, a cumulativethickness of all compressible units undergoing

compression is required. In this case each semi-conning unit was simulated to be 20 m thick. Thus,dividing the estimated storage by 40 m yields aspecic storage value nearly identical to the simulatedvalue (Table 2 ). The general expression for estimatingthe specic storage from Eq. (6) for multiple semi-conning units is

S 0save ¼ S 0ave

1b1

þ1b2 ð13Þ

where the thickness of the bounding semi-conningunits b1 and b2 is required to estimate the averagespecic storage.

Fig. 10. Histogramsshowing the distribution of compaction by layer(a) after 0.3 days of pumping and (b) after 40 days of pumping.Layer 1 is the overlying unconned aquifer, layers 2–11 representthe overlying conning unit, and layer 12 is the pumped connedaquifer, and layers 13–22 represents the lower conning unit.



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If the semi-conning units all reside above thepumped aquifer and are separated by more highlypermeable aquifer units, then the storage valueobtained by Eq. (6) is dependent upon both thethickness of the unit and the head change across eachsemi-conning unit. It should be noted, however, thatif conning units above the unpumped aquifer are

compressing, then the head within the unpumpedaquifer is being affected by pumping within thepumped aquifer. In this case the Hantush method isnot valid and the method described above by Neumanand Witherspoon (1969a–c) would be needed toobtain a legitimate type curve for the pumped aquifer.Nonetheless, Eq. (6) is still valid if piezometers arelocated within each aquifer so that a head gradientacross each semi-conning unit can be measured, thena specic storage can be estimated as well as anestimate of the contribution of compaction from eachsemi-conning unit if it is assumed that the compres-sibilities of each unit are the same.

3. Case study: Franklin Virginia

Groundwater withdrawals from the lower connedaquifers of the Virginia Coastal Plain have increaseddramatically over the past century, particularly nearthe city of Frankin ( Fig. 12), where withdrawalsassociated with paper milling have resulted in a large

cone of depression. Water levels currently are more

Fig. 11. Time–drawdown curve at an observation well 8.6 m from the pumping well for the conceptual model depicted in Fig. 5. A Hantushtype-curve match for this time–drawdown plot shows the leakage factor, transmissivity and storage coefcient for the type-curve match.

Table 2Simulated and estimated aquifer and semi-conning unit parametersused for the second hydrogeologic condition with two conningunits and variable drawdown in the unpumped aquifer

Aquifer and conningunit parameters

Simulationvalue

Estimated valuefrom new method

T (m 2 /d) 125.0 135.7S (dimensionless) 5.0 £ 10

2 5 5.1 £ 102 5

K 0v (m/d) 0.005 0.0038S 0s (1/m) 8.0 £ 10

2 5 7.8 £ 102 5



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than 50 m lower than predevelopment conditions.Subsidence in this region was rst identied in the1970s through a high-precision re-leveling survey(Holdahl and Morrison, 1974 ). In an attempt to furthercharacterize subsidence trends, the US GeologicalSurvey installed an extensometer in close proximity tothe center of the cone of depression in Franklin.Sixteen years of compaction data from 1979 to 1995

were collected before the extensometer was removedfrom service.

A thorough evaluation of the relationship betweenpumping and subsidence in this complex aquifersystem was conducted by Pope (2002) , who was ableto accurately match historical leveling and extenso-meter data through simulation using the one-dimen-sional compaction model COMPAC developed by

Fig. 12. Location of study area in the Virginia Coastal Plain (modied from Pope, 2002 ).



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Helm (1975) . In addition, Pope evaluated thecontribution of compaction attributed to each aquiferand conning unit within the Coastal Plain system thatconsists of 10 aquifers and intervening conningunits. Fig. 13 shows the lithologic log and therepresentative hydrogeologic and modeling unitsand their thicknesses at the Franklin site. Resultsfrom Pope (2002) suggest that much of the permanent

(inelastic) compaction occurred during the mostintensive pumping and subsequent water-leveldeclines between 1940 and 1966. The vast majorityof pumping occurred within the Potomoc group(Fig. 13), which represents the lowest unit of thesedimentary wedge that makes up the Coastal Plainand represents the thickest most prolic aquifers(Meng and Harsh, 1988; Hamilton and Larson, 1988 ).

Fig. 13. Lithology and hydrogeologic and modeled units at the Franklin site. Model unit numbers correspond to the unit numbers in Fig. 13b(from Pope, 2002 ).



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Additionally, prior to 1980, only the lower and middlePotomac aquifers were pumped. Fig. 14a and brepresent the measured and reconstructed water-leveland compaction data for the Franklin site ( Pope,2002). The units shown in Fig. 14b correspond to theunits shown in Fig. 13. Pope determined thecumulative compaction as a function of timeassociated with the lower and middle Potomacaquifers and lower Potomac conning unit. These

totals were used in this investigation.Aquifer properties for the lower and middlePotomac aquifers and the lower Potomac conningunit separating these aquifers have been estimatedfrom hydraulic testing and numerical modeling. Theaquifer transmissivities reported in the literature rangefrom 400 to 1500 m 2 /d (Harsh and Lacnziak, 1990;Hamilton and Larson, 1988 ). The vertical hydraulicconductivity of the conning unit has been estimatedfrom laboratory analyses to be as low as5.8 £ 10

2 7 m/d (Harsh and Lacnziak, 1990 ), whilevalues estimated from aquifer tests and modelingrange from 2.6 £ 10

2 4 to 1.3 £ 102 5 m/d (Harsh and

Lacnziak, 1990 ).Storage properties of the Potomac aquifers have

varied widely. Numerical models and aquifer testingindicate that the estimated storage coefcients forthese aquifers range from 1.6 £ 10

2 6 to 9.3 £ 102 3

(Hamilton and Larson, 1988 ). No storage value for theconning unit has been estimated.

While the range of estimated transmissivities of the aquifers differ by less than a factor of four, theestimated range of storage coefcients differ byclose to four orders of magnitude, indicating theneed for techniques to better quantify the storageproperties of aquifer systems. Pope (2002) used thesubsidence data and the COMPAC model of Helm(1975) to obtain an inelastic storage coefcient of 1.5 £ 10

2 5 for the lower two aquifers (whichincludes the conning unit as part of a bulk storage value). This estimate is probably moreaccurate than previous estimates because itaccounts not only for the head response due topumping, but also takes into account the compac-tion record as well as the role of the storage withinthe conning units, which previous numericalmodels did not account for.

Fig. 14a and b were used to test the graphicalmethod presented here. The application of the method

developed here does not require an entire record fromthe outset of pumping. Although the entire record wasplotted on semi-log paper, only the portion of therecord between 1941 and 1966 was evaluated fortransmissivity and storage. The reason for this istwofold: Firstly, the pumping rates were moreconsistent during this time period. Much smallerrates were used prior to 1941 much larger rates wereapplied shortly after 1966. After 1990 additional

aquifers above the lower Potomac were also pumped.Secondly, water levels were declining past their pastmaximum levels during this period indicating thatinelastic or permanent compaction was occurringallowing for a more consistent evaluation of theinelastic storage coefcient of the aquifer during thistime period. Fig. 15a and b represent the semi-logplots of the time–drawdown and time–compactionrecords, respectively. Although the log time scalesappear to be different, the portion of the slopeapproximated with a straight line represents thesame period of record. The time axis values aredifferent because of the length of record differences as

shown in Fig. 14.The Cooper–Jacob straight-line method was used

to estimate the transmissivity from Fig. 15a assuminga thickness of 175 m (from Fig. 13) for a connednon-leaky aquifer system. A non-leaky system isassumed because the thickness of the aquifer is muchgreater than the overlying conning unit thickness andthe bottom of the system is bounded by crystallinebedrock. The average pumping rate for the portion of the record of concern is 1.54 £ 105 m3 /d. The early-time data represent the earlier portion of the recordwhere lower pumping rates were used. In addition, thedata are responding to the fact that the entire recordfrom the start of pumping was not used. The estimatedtransmissivity using Eq. (2) is 430 m 2 /d, which isconsidered to be at the low end of the range of estimated values. Although this estimate is in therange of reported values, it is near the low end of values in the literature. The reason is likely due to thefact that most reported values are from modelingestimates that do not take into account the signicantleakage from conning units. Thus, in order tocalibrate the model to measured head values, largertransmissivities of the aquifer units are requiredbecause of the source of water not accounted forfrom these ne-grained units. Consequently,



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Fig. 14. Water level and measured and simulated compaction records for the Franklin site. Unit numbers correspond to model unit numbers inFig. 12 (from Pope, 2002 ).



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the model-estimated transmissivities are inated. Thelower transmissivity estimated in this analysis islikely closer to the actual value of the aquifer units.This lower estimate of transmissivity was used toestimate the storage using Eq. (6). The straight-lineestimate ( Fig. 15b ) was made for the period of record

as described above. Late time data do not t becausethe aquifer began to respond elastically at these latertimes as heads began to equilibrate. The early-timedata do not t the curve ( Fig. 15b ) for the same reasonthe transmissivity data did not-much lower pumpingrates than the average rate used for the period of

Fig. 15. Semi-log time–drawdown and time–compaction plots for the Franklin data. Straight-line approximations are made through the datafrom 1941 to 1966.



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record and because the semi-log data was not initiatedwith the start of pumping. The amount of compactionoccurring over one log cycle of time through theperiod of record was measured to be 0.173 m. Theestimated inelastic storage from the slope method (Eq.6) is calculated to be 1.5 £ 10

2 5 , which is theidentical value estimated by Pope (2002) by meansof one-dimensional compaction modeling that incor-porated the extensometer data, past pumping history,

previous levelings, and water-level data.As radar techniques progress allowing for real-timecompaction data to be obtained during pumping, thegraphical technique presented here will provide aquick and accurate approach to obtaining a legitimatestorage value for the aquifer or conning unit of question. Accurate storage estimates are necessary inorder for water managers to accurately predict thequantity of potable water supplies available to them.

4. Summary and conclusions

Traditional semi-log straight-line methods forestimating storage require that the time at s ¼ 0 beevaluated. Not only can this estimate lead tosubstantial errors, but it also requires the estimationof the exact time since the start of pumping. A newgraphical technique based on the Cooper–Jacobmethod uses a highly diagnostic plot of compactionor subsidence as a function of time to estimate thestorage coefcient (skeletal storage is nearly equal tothe storage in highly compressible aquifer systems).This method does not require that one explicitly knowthe time since the start of pumping. Time subsidencedata can be obtained from extensometer data, or fromnewer radar techniques such as GPS or InSARimagery. The vertical deformation signatures pro-vided from these techniques provide valuable infor-mation about the aquifer system compressibility, ormore precisely about the most highly compressibleunits within the aquifer system during pumping. Theclay-rich layers that typically are one or two orders of magnitude more compressible than the aquiferdeposits represent the units contributing most of thesubsidence in an aquifer system. Plotted on semi-logpaper, the slope of the time–subsidence plotoccurring over one log cycle of time can be used toaccurately calculate the specic storage of the aquifer

system, or specically of the semi-conning unitswithin the aquifer system.

Previously developed analytical solutions byHantush (1960) and Neuman and Witherspoon(1969a–c) for leaky aquifer systems depend exclu-sively on hydraulic head data to characterize theaquifer properties. Without additional head or press-ure data in the unpumped units, including the semi-conning units, only limited information can be

drawn from the hydrograph data regarding the semi-conning units. The method presented here does notrequire additional sets of head data and the estimatedspecic storage and vertical hydraulic conductivityvalues obtained with this technique are consistentlyclose to the actual values of the system duringnumerical tests.

In leaky aquifer systems, the time–subsidence plotis highly diagnostic with the semi-log plot represent-ing three distinct periods of deformation. During theearly stages of the aquifer test compaction is almostentirely occurring within the pumped aquifer and theslope of the time– drawdown curve is very small.

During the second stage of deformation the curvequickly steepens and forms a straight line representingthe period when compression is occurring almostexclusively within the more highly compressiblesemi-conning units. If the head in the unpumpedaquifer is constant, the third portion of the time–drawdown curve will asymptotically approach a nalsubsidence value. However, if the head within theunpumped aquifer is variable, the third segment of thecurve will change noticeably to a smaller slopeindicating that drawdown within the unpumpedaquifer has occurred and a smaller pressure gradientis developing across the conning unit.

By measuring the amount of subsidence occur-ring within one log cycle of time during thestraight-line portion of the time– compaction plot,a highly accurate estimate of specic storage of thecompacting unit can be readily obtained. Theleakage factor for the aquifer and adjacent semi-conning units can then be used to estimate thevertical hydraulic conductivity of the semi-conningunits. Estimates of vertical hydraulic conductivityusing this technique are consistently close to thenumerical values used. This technique has beensuccessfully applied to systems with multipleconning units as well.



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The power of the method is not merely theaccurate values of specic storage and hydraulicconductivity obtained, but also the minimumamount of eld data that needs to be obtained.Extensive and expensive piezometer networks thatinclude observation points within each aquifer andeach semi-conning unit are not required. Only thetime–drawdown data of the pumped aquifer andthe time–subsidence data obtained from radar

techniques at the land surface are needed to usethis technique.

The method was applied to the Coastal Plain nearFranklin in southeastern Virginia where long-termpumping associated with the milling of paper hascreated an extensive cone of depression. Furthermore,compaction has been monitored at the site and recentcompaction modeling data have provided a data setthat allows the application of the method presentedherein. Earlier estimates of storage for this aquifersystem varied by nearly four orders of magnitude. Theapplication of the graphical method described hereinproduced a storage value that matched the storageobtained from extensive numerical modeling using thehydraulic head and compaction records from the site.

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