journal of hydrology...wellbore skin effects may contribute prominently to slug re-sponse (faust and...

Journal of Hydrology 403 (2011) 66–82

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Information content of slug tests for estimating hydraulic propertiesin realistic, high-conductivity aquifer scenarios

Michael Cardiff a,⇑, Warren Barrash a, Michael Thoma a, Bwalya Malama b

a Boise State University, Center for Geophysical Investigation of the Shallow Subsurface (CGISS), Department of Geosciences, 1910 University Drive, MS 1536, Boise, ID 83725-1536, USAb Montana Tech of the Univ. of Montana, Dept. of Geological Engineering, 1300 West Park Street, Butte, MT 59701, USA

a r t i c l e i n f o s u m m a r y

Article history:Received 12 August 2010Received in revised form 30 January 2011Accepted 24 March 2011Available online 2 April 2011

This manuscript was handled by P. Baveye,Editor-in-Chief

Keywords:Slug testAquifer characterizationWellbore skinIdentifiabilityKozeny–Carman

0022-1694/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jhydrol.2011.03.044

⇑ Corresponding author. Tel.: +1 208 426 4678; faxE-mail addresses: [email protected]

boisestate.edu (W. Barrash), [email protected] (B. Malama).

A recently developed unified model for partially-penetrating slug tests in unconfined aquifers (Malamaet al., in press) provides a semi-analytical solution for aquifer response at the wellbore in the presenceof inertial effects and wellbore skin, and is able to model the full range of responses from over-damped/monotonic to underdamped/oscillatory. While the model provides a unifying framework forrealistically analyzing slug tests in aquifers (with the ultimate goal of determining aquifer propertiessuch as hydraulic conductivity K and specific storage Ss), it is currently unclear whether parameters ofthis model can be well-identified without significant prior information and, thus, what degree of infor-mation content can be expected from such slug tests. In this paper, we examine the information contentof slug tests in realistic field scenarios with respect to estimating aquifer properties, through analysis ofboth numerical experiments and field datasets.

First, through numerical experiments using Markov Chain Monte Carlo methods for gauging parameteruncertainty and identifiability, we find that: (1) as noted by previous researchers, estimation of aquiferstorage parameters using slug test data is highly unreliable and subject to significant uncertainty; (2)joint estimation of aquifer and skin parameters contributes to significant uncertainty in both unless priorknowledge is available; and (3) similarly, without prior information joint estimation of both aquiferradial and vertical conductivity may be unreliable. These results have significant implications for thetypes of information that must be collected prior to slug test analysis in order to obtain reliable aquiferparameter estimates. For example, plausible estimates of aquifer anisotropy ratios and bounds on well-bore skin K should be obtained, if possible, a priori.

Secondly, through analysis of field data – consisting of over 2500 records from partially-penetratingslug tests in a heterogeneous, highly conductive aquifer, we present some general findings that haveapplicability to slug testing. In particular, we find that aquifer hydraulic conductivity estimates obtainedfrom larger slug heights tend to be lower on average (presumably due to non-linear wellbore losses) andtend to be less variable (presumably due to averaging over larger support volumes), supporting thenotion that using the smallest slug heights possible to produce measurable water level changes is animportant strategy when mapping aquifer heterogeneity.

Finally, we present results specific to characterization of the aquifer at the Boise HydrogeophysicalResearch Site. Specifically, we note that (1) K estimates obtained using a range of different slug heightsgive similar results, generally within ±20%; (2) correlations between estimated K profiles with depth atclosely-spaced wells suggest that K values obtained from slug tests are representative of actual aquiferheterogeneity and not overly affected by near-well media disturbance (i.e., ‘‘skin’’); (3) geostatisticalanalysis of K values obtained indicates reasonable correlation lengths for sediments of this type; and(4) overall, K values obtained do not appear to correlate well with porosity data from previous studies.

� 2011 Elsevier B.V. All rights reserved.

ll rights reserved.

: +1 208 426 3888.(M. Cardiff), wbarrash@

e.edu (M. Thoma), bmalama@

1. Introduction

Slug tests have become a primary method for analyzing aquifertransmissivity due to their relative speed and simplicity as com-pared with more labor-intensive tests such as pumping tests orhydraulic tomography. Likewise, slug tests have proven beneficialat contaminated sites since they do not produce water during a test

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Fig. 1. Slug test setup diagram, showing packer and port system utilized. Beforeeach test, the water column was pressurized through a manifold (1) using acompressed gas source. Water level in the unscreened column (2) is depressed dueto outflow in the screened interval (3). At time to, excess air pressure is releasedthrough a valve in the manifold, resulting in flow into the well at the interval thathas been isolated with packers (3). Water level in column is measured as itequilibrates with surrounding head, using a transducer (4) connected to a dataacquisition system (DAQ) at the surface through a pressure-tight fitting. Dimen-sions are as explained in the text.

M. Cardiff et al. / Journal of Hydrology 403 (2011) 66–82 67

and thus may reduce characterization costs. In addition, partially-penetrating slug tests such as those that can be performed withinpacked-off intervals in a borehole are a beneficial source of infor-mation about depth variations in hydraulic conductivity, which isnot generally obtainable with traditional fully-penetrating pump-ing tests. In order to obtain aquifer parameter estimates from slugtest records, curve fitting is generally carried out using one of avariety of analytic or semi-analytic models which assume homoge-neous aquifer properties within the volume interrogated by slugtest measurements.

Depending on the type of aquifer being investigated (confined/unconfined), the type of slug response observed (overdamped/underdamped), the location of the test (shallow/deep), the typeof slug test performed (fully-penetrating/partially-penetrating),and the existence of near-well disturbance (skin/no skin) a varietyof models may be used to analyze slug test data. Relatively simplerslug-test models may be used for analysis when the response ob-served is non-oscillatory or ‘‘overdamped’’ – as is generally thecase in very shallow or low-conductivity aquifers – and where ver-tical flow in the aquifer is deemed insignificant (Hvorslev, 1951;Cooper et al., 1967; Bouwer and Rice, 1976). In the overdamped re-sponse case where partial penetration produces significant verticalflow, more complex models may be used (e.g., Hyder et al., 1994,which also incorporates wellbore skin).

While partially-penetrating slug tests present a potentiallyquick and information-rich source of data for estimating depth-dependent aquifer heterogeneity, several difficulties are associ-ated with their implementation in highly conductive aquifers.Firstly, in highly conductive aquifers, slug response may be so fastthat inertial effects within the wellbore become important, result-ing in oscillatory well water level responses. In order to duplicatesuch responses in numerical or analytical models, both the headresponse in the aquifer and inertial balances in the wellbore mustbe modeled, resulting in more computationally complex solutions(Bredehoeft et al., 1966; Van Der Kamp, 1976; Kipp, 1985; Spring-er and Gelhar, 1991; Hyder et al., 1994; Zlotnik and McGuire,1998; Butler and Zhan, 2004). In the case of extremely fast-mov-ing wellbore water, even turbulent energy loss may contribute toslug response, resulting in non-linear responses (see, e.g., McEl-wee and Zenner, 1998). Secondly, the fast response of highly con-ductive systems means that, simply due to finite measurementfrequency, it may be difficult to exactly define both the initialslug height and the time at which the test began (see, e.g., Butler,1996, 1998), both of which are required as parameters for slugtest modeling. Thirdly, since most slug test setups perform bothslug injection/extraction and measurement at the same well,wellbore skin effects may contribute prominently to slug re-sponse (Faust and Mercer, 1984; Hyder et al., 1994; Malamaet al., in press), meaning analysis of wellbore skin parameters(such as radial extent and hydraulic conductivity Ksk within theskin) may be crucial to determining accurate aquifer conductivi-ties. Finally, typical models used for slug test data analysis gener-ally assume homogeneity within the ’’region of influence’’ of theslug test, which may impact K estimates obtained depending onthe scale of the slug used and the extent of heterogeneity withinthe aquifer being analyzed (see, e.g., analyses in Butler et al.(1994) and Beckie and Harvey (2002)).

Recently, a semi-analytical solution was developed by Malamaet al. (in press) that models partially-penetrating slug test responsein unconfined aquifers, in the presence of both wellbore inertial ef-fects and wellbore skin. As such, this ‘‘unified’’ model is a highlyflexible platform for slug test analysis and is able to reproducethe full range of possible slug responses, from overdamped/mono-tonic to underdamped/oscillatory. Using this unified model, it be-comes possible to gain insight into the utility of slug tests inhighly conductive unconfined aquifers, where fast slug response

and wellbore skin may contribute significantly to uncertainty inaquifer hydraulic parameters.

In this work, we undertake a study of slug test response inhighly conductive aquifers under realistic field conditions (i.e., tak-ing into account wellbore skin as well as timing issues that may bepresent in field data). In the first section of this work, we performnumerical experiments with the goal of understanding how esti-mates of aquifer properties (and their associated uncertainty) areaffected by such conditions. We utilize the above-mentioned uni-fied model in order to determine the sensitivity of estimated aqui-fer parameters and their uncertainty (as determined via inversionof synthetic data) under several different types of ‘‘prior’’ informa-tion. Then, in the second section of this work, we analyze a largeset of slug test field data from the Boise HydrogeophysicalResearch Site (BHRS) using the unified model with prior informa-tion from studies at the site (Barrash et al., 2006).

2. Mathematical model

Building on the work of Hyder et al. (1994) and Butler and Zhan(2004) among others, Malama et al. (in press) developed a semi-analytical model for slug test response in unconfined aquifers thattakes into account partial penetration as well as skin effects at thesource well and inertial effects within the borehole. The solutionpresented by Malama et al. (in press) satisfies the governing partialdifferential equations (PDEs), boundary conditions, and initial con-ditions described below.

68 M. Cardiff et al. / Journal of Hydrology 403 (2011) 66–82

We consider a system for performing partially-penetrating slugtests in an unconfined aquifer, such as the pressurized-gas systemwith geometry as described in Fig. 1. Within the aquifer, the satu-rated flow equations apply assuming a small displacement of thewater table:

Ss;aq@saq

@t¼ Kr;aq

r@

@rr@saq

@r

� �þ Kz;aq

@2saq

@z2 ð1Þ

saqðr; z; t ¼ 0Þ ¼ 0saqðr; z ¼ 0; tÞ ¼ 0@saq

@z

��z¼B

¼ 0

limr!1

saqðr; z; tÞ ¼ 0

where saq represents the change in head from static conditions inthe aquifer [L], r represents radial distance from the wellbore center[L], z represents depth below the water table [L], t represents time[T], B represents the aquifer saturated thickness [L], Ss,aq is specificstorage of the aquifer [L�1], and Kr,aq and Kz,aq represent radial andvertical hydraulic conductivities within the aquifer [L T�1],respectively.

Within the wellbore skin, similar saturated flow equationsapply:

Ss;sk@ssk

@t¼ Kr;sk

r@

@rr@ssk

@r

� �þ Kz;sk

@2ssk

@z2 ð2Þ

sskðr; z; t ¼ 0Þ ¼ 0sskðr; z ¼ 0; tÞ ¼ 0@ssk

@z

��z¼B

¼ 0

where ssk represents change in head from static conditions in thewellbore skin [L], and Ss,sk [L�1], Kr,sk, and Kz,sk [L T�1] represent stor-age and hydraulic conductivity parameters for the wellbore skin.

Between the aquifer and wellbore skin, continuity of head anddischarge are enforced:

saqðrsk; z; tÞ ¼ sskðrsk; z; tÞ ð3Þ

Kr;aq@saq

@r

��r¼rsk

¼ Kr;sk@ssk

@r

��r¼rsk

where rsk is the radius of the wellbore skin [L].

Table 1Definitions of physical parameters required for performing simulation of slug tests. Those p(or errors in their measurement) is not considered here.

Model parameter (Units) Definition

Kr,aq (L/T) Hydraulic conductivity in the radial direction within theKz,aq (L/T) Hydraulic conductivity in the vertical direction within tSs,aq (L�1) Specific storage within the main aquiferKr,sk (L/T) Hydraulic conductivity in the radial direction within theKz,sk (L/T) Hydraulic conductivity in the vertical direction within tSs,sk (L�1) Specific storage within the wellbore skinrw (L) Radius of wellrc (L) Radius of slug water columnrsk (L) Radius of the wellbore skin from the center of the welldT (L) Depth to the top of the test intervaldB (L) Depth to the bottom of the test intervalB (L) Aquifer saturated thicknessm (L2 T�1) Kinematic viscosity of waterg (L2 T�1) Acceleration due to gravityH0 (L) Initial water level displacementHoffset (L) Correction factor for imperfect choice of static water levHscale (L) Correction factor for imperfect estimation of initial slugtoffset (T) Correction factor for imperfect choice of test start time

Finally, a linearized inertial balance is applied to the open waterwithin the wellbore (see Butler and Zhan, 2004), resulting in thefollowing PDE and initial conditions:

d2H

dt2 þ8mLr2

c Le

dHdtþ g

LeH ¼ g

bLe

Z dT

dB

sskðrw; z; tÞdz ð4Þ

Hðt ¼ 0Þ ¼ Ho

dHdt

��t¼0¼ H0o

where H is the displacement of the wellbore water level from staticconditions [L], v is kinematic viscosity [L2 T�1], dT and dB representthe z coordinates of the top and bottom of the test interval [L], rw isthe radius of the well [L], and rc is the radius of the water column.L and Le are length parameters related to the flow geometry in

the well, with L ¼ dT þ ðdT�dBÞ2

rcrw

� �4(Butler, 2002), and Le ¼ dTþ

ðdT�dBÞ2

rcrw

� �2(Kipp, 1985).

The water level changes in the well are linked to head changesin the wellbore skin, lastly, through a mass balance:

2pðdT � dBÞKr;sk r@ssk

@r

� ��r¼rw

¼ pr2c

dHdt 8z 2 ½dT ;dB�

0 elsewhere

(ð5Þ

A summary of all physical parameters of the model is given inTable 1. While it is recognized that all of the physical modelparameters are subject to measurement errors or uncertainties, afew of the parameters can be very well constrained by simple fieldmeasurements and are thus considered constant and known forthe remainder of this work, as marked with Xs in Table 1.

The solution to these coupled equations (the ‘‘unified solution’’),as detailed in Malama et al. (in press) and references therein, is ob-tained through a combination of a Laplace transform in time and afinite Fourier transform in z, resulting in a solution in the Laplacedomain that involves an infinite series of Bessel functions. We thusrefer to this solution as semi-analytical since a finite approxima-tion of the infinite series is necessary to obtain a result and sincethe inverse Laplace transform used to obtain the solution must,in general, be carried out numerically.

In this work, we utilize a slightly modified version of the unifiedsolution in order to model slug tests from the BHRS. A modifiedcode has been implemented in MATLAB and takes advantage ofvectorization for most coding loops, resulting in a forward model

arameters marked with Xs are considered known, and uncertainty in these parameters

Assumed known/constantthroughout?

main aquiferhe main aquifer

wellbore skinhe wellbore skin

XXXXXXXX

els (shifts time/head change graph up/down)height (scales solution head values by a constant)

(shifts time/head change graph left/right)


that can simulate roughly 20–30 temporal observations per secondon a standard laptop CPU. The input parameters for this model aresummarized in Table 1. In addition to the physical parameters uti-lized in the governing equations, our revised model includes threenew parameters used to allow for errors in field data collection anddata pre-processing. These errors may include misjudgement ofstatic water levels, misjudgement of initial slug height, and mis-estimation of time to when slug test initiation occurs. Given modelresults that calculate H(t), the water level displacement in theborehole at time t, a revised estimate Hr(t) can be calculated takinginto account these errors using the formula

HrðtÞ ¼ Hscale � Hðt � toffsetÞ þ Hoffset ð6Þ

where Hscale, Hoffset, and toffset can be used to correct the slug height,static water levels, and time zeroing, respectively. During data-fit-ting these parameters may be modified within reasonable rangesin order to improve curve matching.

3. Analysis of parameter identifiability

The unified model presented above provides a method for real-istically simulating slug tests in unconfined, high-conductivityaquifers. In general, though, the more important feature of suchmodels is that it allows estimation of important hydrologic param-eters (primarily, hydraulic conductivity in the aquifer) given a setof field data, through inverse modeling. However, as is made evi-dent by the above mathematical formulation, slug test responsewill also be dependent on factors such as wellbore skin hydraulicconductivity and on storage parameters in both the aquifer andskin. For this reason, as discussed by prior researchers (Butler,1996, 1998; Faust and Mercer, 1984), care must be taken in evalu-ating the information content of slug tests for identifying aquiferparameters.

In this section, we investigate the issue of parameter identifi-ability under a number of realistic scenarios. For example, if slugtest data are available from a wellbore which may have a wellboreskin, can we uniquely identify the hydrologic characteristics ofboth the wellbore skin and the aquifer from the data? Alternately,are there a variety of different possibilities for combinations ofaquifer and skin parameters that fit the data well, such that accu-rate joint estimation of these parameters is, in effect, impossible?

To investigate the issue of parameter identifiability, we take thefollowing approach. First, a set of field data is synthetically gener-ated using the unified model, and a small amount of noise is addedto the data in order to simulate measurement error and/or concep-tual error in our model. For comparability with the field data anal-ysis presented, the ‘‘true’’ parameter values used are similar tothose expected from the BHRS aquifer, and the degree of measure-ment error added is comparable to that seen in real field data fromthe field site (as discussed in later sections). We then treat thehydrologic parameters of the unified model as unknown andsearch for parameter sets that result in good fits to the data.

Table 2Unknowns and prior knowledge assumed in three sample cases investigated using MCMC

Model parameter (Units) Case 1 Case 2

True value Assumed knowledge True v

Kr,aq (m/s) 3.70e�03 <1 3.70eKz,aq (m/s) 2.70e�03 <1 3.70eSs,aq (m�1) 5.00e�05 =5e�5 5.00e�Kr,sk (m/s) 2.00e�04 =2e�4 2.00e�Kz,sk (m/s) 2.00e�04 =2e�4 2.00e�Ss,sk (m�1) 5.00e�05 =5e�5 5.00e�

More formally, given a set of field data and a set of estimatedhydrologic parameters, the likelihood of the parameters given thedata is, within a constant:

LðmjdÞ / exp �12ðd� GðmÞÞT R�1ðd� GðmÞÞ

� �

where m is an (n � 1) vector of unknown model parameters, d is an(m � 1) vector of datapoints, G() is the forward model operator,Rn ! Rm, which converts from a given set of parameter values tothe equivalent simulated field data, R is an (m �m) matrix repre-senting the covariance of measurement errors, and where the stan-dard assumption of Gaussian measurement errors has beeninvoked. To evaluate the distribution L(mjd), (i.e., to determinewhich parameter values are ‘‘likely’’) we perform Metropolis–Has-tings sampling (Metropolis et al., 1953; Hastings, 1970), a MarkovChain Monte Carlo (MCMC) sampling method which results in a ser-ies of equally likely parameter sets given the data. While computa-tionally intensive, MCMC methods are more likely to focussampling and detail on high probability density areas of the likeli-hood function (which are of the most interest), in comparison to na-ive strategies such as grid search.

To implement the Metropolis–Hastings algorithm, we beginfrom an initial parameter set and implement the following stepsiteratively:

1. Starting from a current realization of the parameter set m, cal-culate the negative log-likelihoodNLLðmÞ ¼ 1

2 ðd� GðmÞÞT R�1ðd� GðmÞÞ.2. Take a random step from the current parameter set m to a new

set m0. Calculate the negative log-likelihood at the new location,NLL(m0).

3. Select a random variable u from a uniform distribution on [0, 1].4. If ln(u) > NLL(m0) � NLL(m), accept m0 as the new current reali-

zation, otherwise re-accept m as the current realization. Returnto step 1.

To arrive at a good representative sample of conditional realiza-tions, the above algorithm should first go through a ‘‘burn-in’’ per-iod where new realizations are accepted, but not stored inmemory. In practice, the ‘‘burn-in’’ period is utilized so that theMarkov Chain is independent of its initial state (i.e., the initialparameter set supplied by the user), and so that the initial guessdoes not show a prominent signature in the final obtained distribu-tion. After the set number of iterations in the burn-in period, allrealizations of the variables are stored. Ideally, it can be shown thatconditional realizations from the algorithm above will effectivelybe drawn from the true probability distribution as the number ofaccepted realizations approaches infinity.

In the following sections, we present a few specific sample casesthat simplify the problem of parameter estimation in order to focuson identifiability of a few key parameters. We focus on the issues ofwhether aquifer anisotropy, storage coefficients, and wellbore skinparameter can be effectively estimated given realistic, noisy fielddata. In all cases presented below, we utilized synthetic slug tests

sampling to determine parameter identifiability.

Case 3

alue Assumed knowledge True value Assumed knowledge

�03 <1 3.70e�03 <1�03 =Kr,aq 3.70e�03 =Kr,aq

05 >1e�11 5.00e�05 =5e�504 =2e�4 2.00e�04 <104 =2e�4 2.00e�04 =Kr,sk

05 =Ss,aq 5.00e�05 =5e�5

Fig. 2. Realizations of Kr,aq and Kz,aq accepted by the Metropolis–Hastings algorithm, with true values and maximum likelihood estimate highlighted (top). Non-linear trade-off between Kr,aq and Kz,aq results in a wide range of reasonable parameter values that fit the data well. Selected fits for three randomly selected parameter realizations areplotted beneath, emphasizing that good data fits can be obtained with quite different parameter sets.


that generated underdamped responses, though a similar analysiscould be carried out for synthetic cases in which the responseswere overdamped. The true parameter values used in each case,and types of prior knowledge assumed are detailed in Table 2.The sets of prior information used in each case are ‘‘strict’’ in thesense that we assume some parameters of the model are knownperfectly a priori in each scenario, but only upper bounds are de-fined for the parameters that are allowed to vary. Thus these sam-ple cases focus on the relationship between parameters that areallowed to vary in each sample case, which are essentially assumedto have a uniform or ’’non-informative’’ prior distribution. Similaranalysis may be carried out if other types of prior informationare available (e.g. prior mean and variance estimates) by perform-ing Metropolis–Hastings sampling on the posterior probabilitydensity function of the parameters given the data.

For all of the sample cases investigated below, we utilized asinitial guess the ‘‘best estimate’’ obtained from deterministic opti-mization. A burn-in period of 5000 and acceptance total of 150,000realizations were utilized for each case. While proper burn-in per-iod length and accepted Markov Chain length are both the subjectof some debate (see, e.g., discussion in Liu et al., 2010), the choicesutilized were found to perform well on similar but faster-runningtest problems.

3.1. Case 1 – Identifiability of aquifer anisotropy

In Case 1, a synthetic slug test data set was generated for aslightly anisotropic aquifer (Kr,aq = 3.7e�3 m/s, Kz,aq = 2.7e�3 m/s)using the unified model and parameters as given in Table 2. In

order to specifically investigate the identifiability of aquifer anisot-ropy during parameter estimation, we assumed that wellbore skinparameters (Ss,sk, Kr,sk, Kz,sk) and aquifer specific storage (Ss,aq) wereknown. Additionally, an upper bound of 1 m/s was set for both Kr,aq

and Kz,aq. The set of accepted parameter values for Kr,aq and Kz,aq

found using Metropolis–Hastings sampling are shown in Fig. 2,along with a few examples of the fit of simulated curves to the syn-thetic data.

We note the following interesting features of the obtainedparameter distribution. Firstly, the maximum likelihood estimateof the parameters is not equal to the true parameter values, sug-gesting that even moderate amounts of noise in the data may biasestimates of aquifer anisotropy. Also obvious is the interestingnon-linear trade-off between Kr,aq and Kz,aq. A wide range of bothKr,aq and Kz,aq values were found that fit the data well, and thusjoint identifiability of these parameters may be difficult withoutprior information. However, if prior information about one of theseparameters is available (e.g. Kz,aq � 3e�3 m/s), or if prior informa-tion is available relating the two parameters (e.g. Kr,aq � 1.3Kz,aq),then conditional distributions will contain only very narrow uncer-tainty bounds, owing to the thinness of the given likelihood distri-bution. Of course, prior information must also be used with caresince assuming inaccurate information (e.g. Kr,aq � 2Kz,aq) can leadto quite inaccurate parameter estimates.

3.2. Case 2 – Identifiability of storage coefficient

In Case 2, we generated synthetic slug test data from an isotro-pic aquifer and assumed that both aquifer conductivity and storage

Fig. 3. Realizations of Kr,aq and Ss accepted by the Metropolis–Hastings algorithm, with true values and maximum likelihood estimate highlighted (top). Several orders ofmagnitude in Ss are spanned, suggesting poor identifiability of this parameter. Uncertainty in Kr,aq remains low, even with unknown Ss. Selected fits for three randomlyselected parameter realizations are plotted beneath, emphasizing that good data fits can be obtained with quite different parameter sets.


parameters were unknown. That is, Kr,aq and Ss,aq were to be esti-mated assuming knowledge of skin parameters and also assumingknowledge that Kr,aq = Kz,aq. In addition, a lower bound of1e�11 m�1 was assumed for Ss,aq. The true values used for Kr,aq

and Ss,aq are 3.7e�3 m/s and 5e�5 m�1, respectively.The distribution of likely parameters in Case 2 provides in-

sights into joint estimation of these parameters (see Fig. 3). First,we note that the marginal distribution of K is quite narrow, onlyspanning a range of a few percent around the true value. In con-trast, the marginal distribution of Ss is extremely wide, suggesting– as other researchers have noted – that estimation of storageparameters using slug test data will be prone to large errors.The relative lack of correlation between K and Ss estimates ac-cepted indicates, practically, that if a reasonable value of Ss is as-sumed, it will not have a large impact on estimation of K. Forexample, if one assumed in this particular case thatSs = 5e�6 m�1, the conditional distribution of K obtained from thisestimation of Ss will still give values close to the true K. Thisagrees well with the analyses of Beckie and Harvey (2002), whichfound that transmissivity estimates from slug tests were notstrongly influenced by storage properties, but that estimates ofstorage properties obtained from slug tests had dubious value.

3.3. Case 3 – Identifiability of skin conductivity

In Case 3, we assume an isotropic aquifer with known storageparameters and seek to estimate both aquifer and skin hydraulicconductivity (Kr,aq and Kr,sk). Synthetic data were generated with

Kr,aq = 3.7e�3 m/s and Kr,sk = 2e�4 m/s, i.e. the slug response is af-fected by both aquifer conductivity and a low-conductivity or ‘‘po-sitive’’ skin. In order to bound the problem, a reasonable upperbound of 1 m/s was assumed for both parameters.

Similarly to Case 1, the distribution of Kr,aq and Kr,sk shows aninteresting non-linear trade-off between accepted values of thetwo parameters (see Fig. 4). This suggests that, as noted by otherresearchers, joint estimation of both aquifer and skin conductivitymay be difficult without prior information (Faust and Mercer,1984; Hyder et al., 1994). As before, if good prior information isavailable about either one of the parameters, then the resultingconditional uncertainty in the other parameter will generally below due to the thinness of the overall likelihood distribution. Thedistribution of accepted parameter sets also indicates that it maybe possible to obtain good lower bounds on skin conductivity val-ues without prior information. Note, as shown by the acceptedparameter estimates, that as one assumes progressively lowerand lower skin K estimates, drastic increases in the aquifer K valueare required in order to fit the data (see also Malama et al., in press).Since we know that aquifer K estimates should be reasonable (forexample, below 1 m/s), this information can be used to place rea-sonable lower bounds on the skin K values supported by the data.

4. Field data analysis

In the following sections, we apply the lessons learned from ourinvestigation of parameter identifiability to a large set of slug testdata collected at the Boise Hydrogeophysical Research Site (BHRS).

Fig. 4. Realizations of Kr,sk and Kr,aq accepted by the Metropolis–Hastings algorithm, with true values and maximum likelihood estimate highlighted (top). Non-linear trade-off between Kr,sk and Kr,aq results in a wide range of parameter values that fit the data well. Selected fits for three randomly selected parameter realizations are plottedbeneath, emphasizing that good data fits can be obtained with quite different parameter sets.


4.1. Field site and data collection

The BHRS is a research wellfield in an uncontaminated fluvialaquifer established in a gravel bar of the Boise River (see Fig. 5),developed by the Center for Geophysical Investigation of the Shal-low Subsurface (CGISS) at Boise State University. The purpose ofthe site is to provide a field-scale control volume for the develop-ment and testing of hydrologic and geophysical methods used foraquifer characterization (Barrash and Knoll, 1998). The site setupconsists of 18 wells – arranged in a series of roughly concentricrings around a central well (designated A1) – which were drilledthrough the roughly 18m thick aquifer and completed into theunderlying clay aquitard. The wells themselves are small-diameter(10 cm inner diameter) PVC pipe and are fully screened throughoutthe aquifer depth with the exception of blank segments, roughly0.3 m in length and spaced 3 m apart, representing locations wheresections of pipe are threaded together. Disturbance of the naturalmaterial near the wellbores during drilling is thought to be mini-mal based on the drilling and well-finishing techniques employed(Barrash et al., 2006; Morin et al., 1988).

Geologically, the shallow unconfined aquifer at the BHRS con-sists of a heterogeneous mixture of unconsolidated, unaltered sandand gravel deposits with ages from Pleistocene to Holocene. Grainsize analyses from cores show varying distributions with depth,from primarily sand-dominated units to bimodal sand-gravel mix-tures to cobble-dominated units with sand in the interstices of theframework (Reboulet and Barrash, 2003; Barrash and Reboulet,2004). Similarly, neutron porosity logs show significant variationsin porosity with depth, with mean values for individual strati-

graphic units ranging from 0.172 to 0.425 in addition to significantchanges in porosity variance between units (Barrash and Clemo,2002).

Earlier analyses of datasets to obtain hydraulic conductivity (K)estimates at the site have noted differences in fitted K values whenusing analytic models on a well-by-well basis (Fox, 2006; Barrashet al., 2006), suggesting lateral heterogeneity. Similarly, inversemodeling of dipole pumping tests has likewise suggested variabil-ity in depth-integrated K throughout the site (Cardiff et al., 2009),providing further evidence for lateral heterogeneity. In addition tolateral heterogeneity affecting head responses at wells, wellboreskin has been found to contribute to differing responses at pump-ing/testing vs. observation wells. Drawdown curves show consis-tent, systematic evidence for positive wellbore skin (Barrashet al., 2006) at the BHRS, in terms of larger than expected draw-downs at pumping wells given observation wells’ responses.

A series of slug tests was performed at the BHRS with the goal ofcontributing to the understanding of vertical variability in hydrau-lic conductivity at the BHRS in the vicinity of the wells and to pro-vide more information about the 3D geologic structuresinfluencing groundwater flow and solute transport.The slug testdata analyzed in this paper were collected using a partially-pene-trating, pressurized-gas slug test system (Leap, 1984; Levy et al.,1993) as diagrammed in Fig. 1. For all tests performed, the follow-ing parameters of the system geometry were constant:rc = 1.905 cm, rw = 5.08 cm, rsk = 5.715 cm, and dB � dT = 30.48 cm.Thickness of the aquifer B varied depending on water levels duringtesting conditions and minor variations in the depth of the clayaquitard at the site. The depth to the top of the slug interval dT


varied based on the interval of the aquifer being evaluated. Logis-tically, each slug test proceeded as follows: before the test, the air

Fig. 5. Location of BHRS with respect to Boise River and upstream

Fig. 6. Normalized sample slug test records, showing overall independence of response ffirst measured datapoint.

above the water column was pressurized and water in the watercolumn was allowed to come to equilibrium with the surrounding

Diversion Dam. Inset shows arrangement of central wells.

rom initial slug height. Slug test records have been normalized by the height of the

Fig. 7. Samples of field data and optimized simulated slug test curves for both analysis cases. Curve fits obtained with and without wellbore skin are near-identical in mostcases (e.g., Samples 1–3), but produce different aquifer K estimates. In a few cases (e.g., Sample 4), field data analyzed using the model with skin could not be fit withreasonable aquifer K estimates. Note aquifer K estimates were constrained to be below 1 m/s (log10(K) = 0).

Table 3Slug test data collected during BHRS field campaigns. Tests were performed at each 0.3 m (1 ft) interval, except at intervals where well casing is blank (i.e., unscreened). Slugheights represent approximate equivalent head changes in height of water based on column gas pressurization.

Well name Highest elevation(m AMSL)

Lowest elevation(m AMSL)

Data collection date Slug height 1(cm H2O)

Slug height 2(cm H2O)



Al 846.93 332.60 7/1/2008–7/2/2008 30 25 20 –

Bl 847.70 831.24 6/16/2009–6/17/2009 5 13 20 5B2 847.29 832.05 6/17/2008–6/19/2008 30 25 20 –B3 847.21 831.67 6/24/2008–6/26/2008 30 25 20 –B4 847.38 832.14 5/23/2008–6/03/2008 30 25 20 –B5 847.21 832.16 6/30/2009–6/30/2009 5 13 20 5B6 847.53 832.59 6/30/2009–7/1/2009 5 13 20 5

Cl 847.16 832.22 7/14/2009 5 13 5 –C2 847.19 831.65 3/5/2008–8/6/2008 30 25 20 –C3 846.84 831.90 7/9/2008–7/11/2008 30 25 20 –C4 847.22 831.18 6/4/2008–6/6/2008 30 25 20 –C5 847.30 837.54 7/6/2008 5 13 20 –C5 836.93 832.06 7/7/2008 5 13 5 –C6 847.23 831.69 7/7/2009–7/8/2009 5 13 5 –

XI 847.26 831.72 7/20/2009 5 13 5 –X2 847.31 831.46 7/15/2009 5 13 5 –X3 847.33 831.18 7/21/2009 5 13 5 –X4 847.53 823.79 6/8/2009–6/10/2009 5 13 20 5X5 847.23 829.30 7/22/2009 5 13 5 –



aquifer. Then, at time t0, the excess air pressure was near-instanta-neously released, resulting in a loss of equivalent head in the watercolumn and, hence, flow into the well (i.e., a ‘‘slug out’’ test).

Fig. 8. Well C3 profiles, showing consistency of K estimates across slug heights. Spread oAnalysis Case 2.

Fig. 9. Well C5 profiles, showing consistency of K estimates across slug heights. Note usuggesting that 2e�4 m/s represents a lower bound for reasonable skin K values.

In accordance with practical guidance from prior research stud-ies (see, e.g., Butler, 1996, 1998; McElwee, 2002), a series of at leastthree different slug heights (i.e., air pressurizations to produce

f estimates is <20% for 98% of intervals in Analysis Case 1, and for 89% of intervals in

nreasonable aquifer K estimate obtained near 839 m in Analysis Case 2 (�1.0 m/s),


three different Ho starting values) were performed for each depthinterval. These repetitions allow for testing whether slug responseis independent of slug height, which can be used to validatewhether the assumption of linearity invoked by most analytic the-ories (i.e., the linearity of Eq. (4)) applies. In many cases, like thoseshown in Fig. 6, normalized slug response was largely repeatableand independent of initial slug height, suggesting that the data todo not grossly violate the assumptions of linearity. However, insome slug test records a trend of increased damping with largerslug heights was observed, suggesting that tests using larger slugheights resulted in slight flow non-linearities within the wellbore.

In many of the wells, multiple repetitions of the smallest slugheight were collected for a given depth interval (see Table 3).The repetition of a given slug height allows testing of whether welldevelopment is taking place during the performance of slug tests.Given prior well development activities undertaken at the BHRSand the variety of prior hydraulic tests performed at the site, welldevelopment during slug tests was not expected. In cases wheremultiple replicates of the same slug height were performed, visualinspection of slug test data showed little to no difference betweenthese records.

The field data utilized in this work were collected throughoutthe summers of 2008 and 2009. Slug tests were performed at0.3m depth increments for all intervals in each well, except forthose intervals in the well that are unscreened due to pipe thread-ing. Intervals were progressively isolated using a down-boreholepacker system consisting of 1 m long packers with a 0.3 m isolatedscreen interval in between. The tests were performed at all 18wells of the BHRS, resulting in a total of over 2500 response curvesto be analyzed. Due to the highly conductive nature of the aquiferand the quick response time, a full set of slug tests for a well couldbe collected quite quickly, and routinely took only 1–2 days. Theparticular slug heights used for each well varied and, as discussedearlier, in more than half the zones, tests using the smallest slugheight were repeated multiple times.

4.2. Parameter estimation

Incorporating the lessons learned from our synthetic investiga-tions, we apply parameter estimation to the full set of over 2500slug test records available from the BHRS. Useful prior information

Table 4Trends in mean and variance of K estimates per well (Analysis Case 1) due to slug height usaveraged K estimates. Similarly, in all but two cases (bold), use of larger slug heights is as

Well Slug height (cm) K mean (m/s)

Slug 1 Slug 2 Slug 3 Slug 4 Slug 1 Slug 2 Sl

A1 30 25 20 – 9.57e�04 9.86e�04 1.

B1 5 13 20 5 1.25e�03 1.17e�03 1.B2 30 25 20 – 8.74e�04 8.99e�04 9.B3 30 25 20 – 8.58e�04 8.81e�04 9.B4 30 25 20 – 1.09e�03 1.14e�03 1.B5 5 13 20 5 8.99e�04 8.37e�04 8.B6 5 13 20 5 1.10e�03 1.02e�03 9.

C1 5 13 5 – 1.16e�03 1.07e�03 1.C2 30 25 20 – 6.66e�04 6.88e�04 6.C3 30 25 20 – 7.13e�04 7.51e�04 7.C4 30 25 20 – 5.61e�04 5.82e�04 6.C5 5 13 20 – 1.00e�03 9.63e�04 9.C5 5 13 5 – 1.05e�03 1.00e�03 1.C6 5 13 5 – 1.21e�03 1.16e�03 1.

X1 5 13 5 – 1.61e�03 1.48e�03 1.X2 5 13 5 – 7.64e�04 7.63e�04 7.X3 5 13 5 – 8.30e�04 7.90e�04 8.X4 5 13 20 5 1.20e�03 1.13e�03 1.X5 5 13 5 – 1.21e�03 1.12e�03 1.

from earlier studies at the site is used to constrain model parame-ters that otherwise would contribute to significant uncertainty inthe estimation of aquifer hydraulic conductivity.

Since identification of storage parameters from slug test dataappears unreliable, we have chosen a reasonable value of specificstorage Ss that applies to both the aquifer material and the well-bore skin. Prior studies at the BHRS (e.g. Barrash et al., 2006) haveobtained estimates of Ss between 3 � 10�5 m�1 and 1 � 10�4 m�1.For the analysis of slug test data, we have thus chosen to utilizea constant Ss = 5 � 10�5 m�1. We note that, based on our earliersynthetic analyses, estimation of aquifer K values should not beparticularly dependent on this choice. Estimation of aquifer anisot-ropy from slug test data likewise appears unreliable based on oursynthetic analyses, and we thus resort to utilizing prior informa-tion available from earlier analyses. Based on analyses fromfully-penetrating pumping tests, Barrash et al. (2006) found littleevidence for significant aquifer anisotropy. We therefore assumeduring our analyses of slug test data that Kr,aq = Kz,aq.

With regards to wellbore skin, Barrash et al. (2006) found evi-dence of positive (i.e., lower conductivity) skin through analysesof drawdown curves at both pumping and observation wells. How-ever, since the exact conductivity values for wellbore skin may bedifficult to estimate, and since pumping tests may be affected bymechanisms such as strongly-convergent radial flow which resultin ‘‘pseudo-skin’’ behaviors (see, e.g. Desbarats, 1992; Neumanand Orr, 1993; Rovey and Niemann, 2001), we have analyzed slugtest records using three different analysis cases. In Analysis Case 1,we have assumed that there is no wellbore skin present, i.e. by set-ting Kr,aq = Kz,aq = Kr,sk = Kz,sk. If positive wellbore skin is present inany magnitude, this assumption results in decreased estimates ofaquifer K, meaning that Analysis Case 1 provides a lower boundon the aquifer K values from the site. In Analysis Case 2, we assumeisotropic wellbore skin with magnitude Kr,sk = Kz,sk = 2 � 10�4 m/s,which is somewhat higher conductivity than the skin value ob-tained by Barrash et al. (2006). Finally, in Analysis Case 3 we utilizethe skin value obtained by Barrash et al. (2006), withKr,sk = Kz,sk = 2 � 10�5 m/s.

In order to perform parameter estimation, we utilized an auto-matic routine based on MATLAB’s built-in fmincon (constrained,gradient-based optimization) and fminsearch (simplex search)optimization routines in order to minimize data-fitting residuals.

ed. In all except one case (bold), larger slug heights are associated with smaller depth-sociated with decreased variance in K estimates obtained.

K variance (m2/s2)

ug 3 Slug 4 Slug 1 Slug 2 Slug 3 Slug 4

05e�03 1.91e�07 1.98e�07 3.30e�07

11e�03 1.25e�03 6.02e�07 4.81e�07 3.69e�07 6.14e�0736e�04 1.15e�07 1.21e�07 1.31e�0707e�04 2.56e�07 2.76e�07 2.74e�0716e�03 3.64e�07 4.06e�07 4.14e�0715e�04 8.83e�04 4.73e�07 3.34e�07 2.39e�07 4.04e�0782e�04 1.08e�03 5.80e�07 4.07e�07 3.14e�07 5.15e�07

16e�03 3.94e�07 3.45e�07 4.02e�0793e�04 9.73e�08 1.06e�07 1.11e�0763e�04 3.09e�07 3.58e�07 3.29e�0708e�04 7.10e�08 7.70e�08 8.23e�0819e�04 7.82e�07 5.76e�07 4.45e�0708e�03 7.27e�07 6.66e�07 8.88e�0722e�03 1.48e�06 1.42e�06 1.58e�06

58e�03 1.79e�06 1.41e�06 1.71e�0646e�04 3.76e�07 3.52e�07 3.42e�0733e�04 1.79e�07 1.58e�07 1.80e�0710e�03 1.23e�03 4.74e�07 3.95e�07 3.25e�07 4.97e�0716e�03 5.76e�07 3.88e�07 4.81e�07

Table 5Trends in mean and variance of K estimates per well (Analysis Case 2) due to slug height used. In all except four cases (bold), larger slug heights are associated with smaller depth-averaged K estimates. Similarly, in all but six cases (bold), use of larger slug heights is associated with decreased variance in K estimates obtained.

Well Slug height (cm) K mean (m/s) K variance (m2/s2)

Slug 1 Slug 2 Slug 3 Slug 4 Slug 1 Slug 2 Slug 3 Slug 4 Slug 1 Slug 2 Slug 3 Slug 4

A1 30 25 20 – 1.62e�03 1.65e�03 3.05e�03 3.62e�06 2.95e�06 9.77e�05

B1 5 13 20 5 4.56e�03 2.95e�03 2.10e�03 4.40e�03 1.38e�04 2.98e�05 5.22e�06 1.02e�04B2 30 25 20 – 1.23e�03 1.28e�03 1.34e�03 3.79e�07 4.20e�07 4.63e�07B3 30 25 20 – 1.38e�03 1.46e�03 1.56e�03 2.00e�06 2.39e�06 3.00e�06B4 30 25 20 – 2.83e�03 2.15e�02 2.44e�02 3.52e�05 1.66e�02 2.17e�02B5 5 13 20 5 2.66e�03 1.90e�03 1.29e�03 2.07e�03 5.12e�05 1.95e�05 2.25e�06 1.75e�05B6 5 13 20 5 1.37e�02 7.00e�03 2.77e�03 1.67e�02 3.32e�03 1.09e�03 6.23e�05 8.54e�03

C1 5 13 5 – 3.49e�03 1.04e�02 4.41e�03 4.39e�05 2.97e�03 1.68e�04C2 30 25 20 – 8.74e�04 9.25e�04 9.71e�04 5.29e�07 6.57e�07 8.36e�07C3 30 25 20 – 1.22e�03 1.59e�03 1.46e�03 2.73e�06 1.08e�05 4.85e�06C4 30 25 20 – 6.82e�04 7.19e�04 7.57e�04 1.68e�07 1.88e�07 2.06e�07C5 5 13 20 – 2.27e�02 3.06e�03 1.88e�03 1.22e�02 5.93e�05 7.73e�06C5 5 13 5 – 2.95e�03 2.89e�03 3.10e�03 2.68e�05 2.82e�05 3.24e�05C6 5 13 5 – 5.80e�02 4.58e�02 6.01e�02 4.55e�02 4.23e�02 4.92e�02

X1 5 13 5 – 9.01e�02 6.18e�02 8.18e�02 6.35e�02 4.74e�02 5.43e�02X2 5 13 5 – 1.67e�03 1.49e�03 1.92e�03 1.24e�05 7.39e�06 2.53e�05X3 5 13 5 – 1.20e�03 1.11e�03 1.21e�03 1.01e�06 8.55e�07 9.73e�07X4 5 13 20 5 2.71e�03 2.39e�03 1.95e�03 3.12e�03 1.68e�05 1.80e�05 4.55e�06 3.03e�05X5 5 13 5 – 1.66e�02 2.24e�03 2.64e�03 1.03e�02 7.10e�06 1.14e�05


In all three Analysis Cases, log10(Kr,aq)was treated as a variablebounded on the interval [�1, 0], in addition to the scaling and tim-

Fig. 10. Example of similarity of estimated log10(K) profiles at closely-spaced wells (bothlikewise suggesting horizontal continuity of individual layers or lenses. Similar slug heigC1 = 4.67 m, B2–C1 = 5.08 m.

ing parameters mentioned earlier (Hscale, Hoffset, and toffset). Afteroptimization, all data fits were visually examined and a subset of

analysis cases), suggesting that K values are representative of aquifer properties andhts chosen for comparison. Inter-well distances are as follows: B1–B2 = 3.33 m, B1–

Fig. 11. Visualization of anisotropy ellipse for log10(K) variogram for Analysis Case 1 (no skin), estimated using RML methodology. Dimensions of the ellipse representoptimized correlation length L of exponential variogram, and are compared with dimensions of BHRS well field (black lines representing wells onsite). Axes of ellipserepresent optimized principle anisotropy directions.

Fig. 12. Visualization of anisotropy ellipse for log10(K)variogram for Analysis Case 2 (skin K = 2e�4 m/s), estimated using RML methodology. Dimensions of the ellipserepresent optimized correlation length L of exponential variogram and are compared with dimensions of BHRS well field (black lines representing wells onsite). Axes of ellipserepresent optimized principle anisotropy directions.



records with poor fits (due to convergence to local minima) werere-run. In cases where optimization had converged to unrealisticlocal minima, modified starting parameter estimates were trieduntil acceptable convergence was obtained.

Examples of slug test data fits for a range of responses – usingAnalysis Cases 1 and 2 – are shown in Fig. 7. In Analysis Case 3,which is not pictured, all simulated slug responses were over-damped even when extremely high aquifer K values up to 1 m/swere utilized. Given that a large number of records from the fielddata exhibit underdamped behavior, this suggests that the skinconductivities obtained by Barrash et al. (2006) are too low, possi-bly due to ‘‘pseudo-skin’’ effects from strongly convergent flownear the pumping well. For this reason, Analysis Case 3 is not con-sidered further.

In Analysis Cases 1 and 2, very consistent estimates of aquifer Kwere obtained for the majority of records across the full set of slugheights utilized. Estimated K profiles for wells C3 and C5 are shownin Figs. 8 and 9, respectively. For the majority of depths, K estimatevariability across slug heights was within ±20% of the average K ob-tained. However, it should be noted that at a few intervals (e.g.,around 839 m AMSL in well C5, Fig. 9), Analysis Case 2 resultedin aquifer K estimates that were unrealistically high. This suggests,as discussed in the synthetic analyses, that the skin value of2 � 10�4 m/s used represents a lower bound on the skin K valueand, thus, aquifer K estimates obtained using Analysis Case 2 rep-resent an upper bound on actual aquifer K. Given our previous rea-soning that Analysis Case 1 represents a reasonable lower boundon actual aquifer K values, then, it is likely that true aquifer K val-ues are somewhere between those found in Analysis Case 1 andAnalysis Case 2.

Fig. 13. Comparison between neutron log-derived porosity (line) and estimated log10(K)similar K trends. Note some regions in which porosity and K appear positively correlated awells).

4.3. Analysis of K estimates

4.3.1. Dependence on initial slug heightDifferences between aquifer K estimates obtained using differ-

ent slug heights is slight in most cases when compared to overallK variability throughout the BHRS aquifer, as shown in Figs. 8and 9. However, subtle trends are visible in overall K estimatesdependent on slug height when viewed on a well-by-well basis.In Tables 4 and 5, we note two predominant trends, both of whichsuggest that K estimates obtained from small slug heights may bethe most useful for future analysis. The first trend we note is that,analyzed on a well-by-well basis, average K estimates tend to de-crease as slug height increases. While such behavior could bedue to a variety of factors, one of the most likely is perhaps theexistence of higher in-well flow velocities and, thus, some non-lin-ear energy losses within the well (McElwee and Zenner, 1998;McElwee, 2002), which incurs extra damping and thus lower-Kparameter estimates. The second trend made evident in Tables 4and 5 is that, analyzed on a well-by-well basis, the variance of Kestimates obtained tends to decrease as slug height increases. Inthis case, a likely cause for this behavior is that, at larger slugheights, a larger region of influence is interrogated resulting in Kestimates that average over a larger aquifer volume, i.e. producea measurable water level disturbance in a larger portion aroundthe slug region.

Because of these trends, and their likely causes, K estimates ob-tained when using smaller initial slug heights are thought to bemore representative of fine-scale depth-dependent aquifer hetero-geneity. The analyses performed in the following sections thus fo-

from slug tests (points) for wells B1, B2, and C1, which are adjacent wells showingnd others in which negative correlation is evident (e.g., near 838–844 m AMSL in all


cus on K estimation results for each well using data from stimula-tions with the smallest slug height.

4.3.2. K autocorrelation analysisBecause of slug tests’ high sensitivity to wellbore skin conduc-

tivity values, depth variability in aquifer hydraulic conductivityestimates could easily be attributed to depth variability of wellboreskin – a complication with slug tests that has not received muchattention. However, if consistency in profiles of K with depth areseen at several wells, this suggests that aquifer K estimates ob-tained are actually related to true aquifer K values near thepumped interval, and not just to random variations in wellboreskin conductivity. A high degree of consistency is seen between Kprofiles obtained from wells that are located near each other atthe BHRS, as exhibited in Fig. 10.

As expected from geostatistical theory, correlation between Kprofiles in wells is high when distances between wells are rela-tively small (as in Fig. 10), and correlation decreases with increas-ing distance. A geostatistical analysis of the K values obtained in all

Table 6Correlation between porosity term of Kozeny–Carman equation and estimated K values frohave statistically significant P-values. Negative correlations occur often and represent alm

Well name Correlation coefficient

A1 0.439

B1 0.209B2 0.287B3 0.508B4 �0.200B5 �0.127B6 �0.279

C1 0.239C2 0.135C3 0.063C4 0.117C5 �0.076C6 0.153

X1 0.576X2 �0.014X3 �0.329X4 �0.286X5 0.041

Table 7Correlation between porosity term of Kozeny–Carman equation and estimated K values(highlighted) have statistically significant positive correlations. For all other wells, correla

Well name Correlation coefficient

A1 0.167

B1 0.050B2 0.268B3 0.531B4 0.027B5 �0.065B6 �0.040

C1 0.180C2 0.018C3 0.164C4 0.075C5 �0.064C6 �0.102

X1 0.644X2 �0.011X3 �0.158X4 �0.081X5 �0.036

wells (which are treated as point-wise information) was performedin order to gain more insight into the correlation structure. In ouranalysis, we utilized the Restricted Maximum Likelihood (RML)method of Kitanidis (1987), and analyzed the aquifer K estimatesfrom both Analysis Cases. We assume an anisotropic exponentialvariogram in which the sill of the variogram, the correlationlengths along the principle anisotropy directions, and the direc-tions of the principle anisotropy vectors were all optimized. Sincethe optimization of these parameters is non-linear, a number ofdifferent initial guesses were chosen for the geostatistical parame-ters. Several of the optimization cases converged to the sameparameter estimates, and in the other cases local minima were ob-tained, all of which had lower likelihood values. The maximumlikelihood geostatistical parameters obtained from Analysis Cases1 and 2 are shown visually in Figs. 11 and 12, respectively, andare quite similar. For Analysis Case 1, i.e. for the K estimates ob-tained when no wellbore skin was assumed, a minimum correla-tion length of about 1.5 m was obtained along a principle axisthat is only slightly tilted relative to vertical (�8�), and longer cor-

m field data (Analysis Case 1) computed on a well-by-well basis. Highlighted entriesost half of statistically significant correlations.

Correlation type P-value

Positive 0.4%

Positive 15.5%Positive 5.3%Positive 0.0%Negative 18.3%Negative 41.0%Negative 8.1%

Positive 11.4%Positive 37.3%Positive 68.0%Positive 42.4%Negative 61.7%Positive 30.9%

Positive 0.0%Negative 92.8%Negative 2.1%Negative 3.4%Positive 77.1%

from field data (Analysis Case 2) computed on a well-by-well basis. Only two wellstion is weakly positive or negative.

Correlation type P-value

Positive 29.1%

Positive 73.7%Positive 7.2%Positive 0.0%Positive 85.9%Negative 67.7%Negative 80.6%

Positive 23.7%Positive 90.7%Positive 28.2%Positive 60.8%Negative 67.1%Negative 50.0%

Positive 0.0%Negative 94.5%Negative 27.8%Negative 55.5%Negative 79.8%


relation lengths of 6 m and 10 m are observed along the other twoprinciple axes, which are roughly parallel and perpendicular to theBoise River, respectively. In Analysis Case 2, where a skin conduc-tivity of Ksk = 2e�4 m/s was assumed, only slight differences in theestimated geostatistical parameters are observed. The minimumcorrelation length in this case is about 1.2 m, and is tilted 18� rel-ative to vertical. As before, the longer correlation lengths are ori-ented roughly parallel and perpendicular to the Boise River again,with correlation length values equal to 4 m and 8 m, respectively.Regardless of which Analysis Case is considered, the geostatisticalresults obtained support the notion that the BHRS is a predomi-nantly layered system, as has been suggested in numerous prioranalyses of both hydrologic and geophysical data from the site(e.g., Barrash and Clemo, 2002; Tronicke et al., 2004; Irving et al.,2007; Bradford et al., 2009).

4.3.3. K-porosity correlation analysisFinally, one very interesting feature of the K distribution esti-

mated using slug testing is its overall lack of correlation withporosity data. Porosity values at the BHRS wells have been esti-mated using neutron well logs (Barrash and Clemo, 2002), andtrends in porosity observed at wellbores have been verified usingnumerous geophysical methods (Tronicke et al., 2004; Clementand Knoll, 2006; Irving et al., 2007; Bradford et al., 2009). Theoriessuch as the Kozeny–Carman equation have been used extensivelyin hydrologic and geophysical research to make inferences aboutaquifer K given either direct or indirect measurements of porosity/. For example, the Kozeny–Carman Bear model suggests that Kshould be proportional to a function of the porosity K / /3/(1 � /)2 (Domenico and Schwartz, 1998). However, as shown for selectedwells in Fig. 13, a strong correlation between porosity and K is notapparent. Correlations such as those suggested by the oft-usedKozeny–Carmen equation do not appear to apply throughout theBHRS aquifer. While positive correlation between K and porosityexists for some intervals, there are equally many intervals in whichno correlation is apparent or even in which negative correlationsappear to occur. As suggested by both Fig. 13 and shown morecompletely, for both Analysis Cases 1 and 2 in Tables 6 and 7,respectively, porosity data alone (whether collected through direct,hydrologic, or geophysical methods), may be insufficient for esti-mating aquifer hydraulic conductivity in sediments similar tothose present at the BHRS. Investigation into possible sedimentaryand post-depositional causes for this discrepancy at the BHRS arecurrently in progress.

5. Conclusions

Partially-penetrating slug tests are an information-rich and rel-atively easy-to-collect source of data for estimating vertical vari-ability of hydraulic conductivity, even in highly conductiveaquifers such as the aquifer at the BHRS. However, in interpretingdata from such tests, care must be taken in order to be realisticabout the level of information that can be obtained from analyzingsuch tests. For example, estimation of aquifer anisotropy and aqui-fer storage properties using slug test data may be highlyinaccurate.

Since often both the stimulation (i.e., the slug injection/extrac-tion) and measurement take place at the same location, slug testdata may be highly affected by wellbore skin, and in highly con-ductive aquifers a key concern is the existence of ‘‘positive’’, orlow-conductivity skin. However, by analyzing slug test data undera variety of different scenarios, it may be generally possible toplace bounds on aquifer K. By making the assumption of no well-bore skin, aquifer K estimates obtained will represent a lowerbound. Similarly, by analyzing data using a model with wellbore

skin (as indicated by prior information in the case of the BHRS),and by steadily decreasing the skin conductivity values, one can ar-rive at an approximate upper bound of aquifer K.

Even after depth-dependent aquifer K has been estimated byslug test data analysis, we believe these values must be validatedthrough other methods such as comparison against other datasources and autocorrelation (geostatistical) analyses. Due to slugtests’ high sensitivity to low-conductivity skins, it is possible thataquifer K estimates obtained may simply represent depth-variabil-ity in a low-conductivity skin. By comparing K profiles obtained atclosely-spaced wells, though, we increase our confidence that thevolume being interrogated by the slug test response extends be-yond the wellbore skin and represents true aquifer K variability.When slug test data are treated carefully and when K estimates ob-tained are validated, the information content present in this datasource can help to provide new insights into aquifer heterogeneity,and can likewise provide validation for hydrologic and geophysicalmodels.

Acknowledgments

The authors wish to thank Brady Johnson, who participated inthe collection of the data analyzed in this work, and Drs. James But-ler Jr. and Geoff Bohling, who provided numerous helpful sugges-tions for field operation and model development. In addition, theauthors would like to thank the helpful comments of Walter Illmanand one anonymous reviewer, whose input helped improve thispublication. Support for this research was provided by NSF GrantsEAR-0710949 and DMS-0934680, and is gratefully acknowledged.

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