Transmissivity estimation for highly heterogeneous

Transmissivity estimation for highly heterogeneous aquifers:

comparison of three methods applied

to the Edwards Aquifer, Texas, USA

Scott L. Painter & Allan D. Woodbury & Yefang Jiang

Abstract Obtaining reliable hydrological input parame-ters is a key challenge in groundwater modeling. Althoughmany quantitative characterization techniques exist, expe-rience applying these techniques to highly heterogeneousreal-world aquifers is limited. Three geostatistical charac-terization techniques are applied to the Edwards Aquifer, alimestone aquifer in south-central Texas, USA, for thepurposes of quantifying the performance in an 88,000-cellgroundwater model. The first method is a simple krigingof existing hydraulic conductivity data developed primar-ily from single-well tests. The second method involvesnumerical upscaling to the grid-block scale, followed bycokriging the grid-block conductivity. In the third method,the results of the second method are used to establish theprior distribution for a Bayesian updating calculation.Results of kriging alone are biased towards low values andfail to reproduce hydraulic heads or spring flows. Theupscaling/cokriging approach removes most of the sys-tematic bias. The Bayesian update reduced the meanresidual by more than a factor of 10, to 6 m, approxi-mately 2.5% of the total head variation in the aquifer. Thisagreement demonstrates the utility of automatic calibra-tion techniques based on formal statistical approaches andlends further support for using the Bayesian updatingapproach for highly heterogeneous aquifers.

Résumé Disposer de paramètres hydrogéologiques d’en-trée fiables est une des clés de la modélisation hydro-géologique. Bien que de nombreuses techniques decaractérisation quantitative existent, leur application àdes aquifères réels fortement hétérogènes est limitée.Trois techniques géostatistiques de caractérisation sontappliquées à l’aquifère Edwards, un aquifère calcaire situéau sud du centre du Texas, USA, afin d’évaluer leurperformance dans un modèle hydrogéologique de 88 000cellules. La première méthode est un krigeage simple desdonnées disponibles de conductivité hydraulique obtenuesà partir d’essais de puits. La seconde méthode faitintervenir une mise à l’échelle au niveau de celle dumaillage puis un co-krigeage de la conductivité à cetteéchelle bloc. Dans la troisième méthode, les résultatsobtenus à la deuxième méthode sont utilisés commedistribution préalable à une approche bayésienne. Lesrésultats du krigeage seul sont biaisés pour les faiblesvaleurs et ne permettent pas de reproduire les niveauxpiézomètriques ou les écoulements des sources. L’ap-proche changement d’échelle/co-krigeage supprime unegrande partie du biais systématique. L’approche bayési-enne a réduit le résidu moyen d’un facteur supérieur à 10,jusqu’à 6 m, environ 2.5% de la variation piézomètriquetotale dans l’aquifère. Ceci démontre l’utilité des techni-ques de calibration automatique basées sur des approchesstatistiques formelles et accrédite davantage l’utilisationde l’approche bayésienne dans le cas d’aquifères forte-ment hétérogènes.

Resumen La obtención de parámetros hidrológicos deentrada confiables es un desafío clave en la elaboración demodelos de aguas subterráneas. Aunque existen muchastécnicas cuantitativas de caracterización se cuenta conexperiencia limitada para aplicar estas técnicas a acuíferosdel mundo real altamente heterogéneos. Se han aplicadotres técnicas de caracterización geoestadística al acuíferoEdwards, un acuífero de calizas en la parte sur central deTexas, Estados Unidos, con el propósito de cuantificar eldesempeño en un modelo de agua subterránea de 88,000celdas. El primer método es una interpolación krigingsimple de los datos existentes de conductividad hidráulicaobtenidos principalmente de pruebas en un solo pozo. Elsegundo método involucra escaleo ascendente numérico ala escala de bloques de la malla seguida por cokriging enla conductividad de los bloques de la malla. En el tercer

Received: 30 December 2005 /Accepted: 26 May 2006Published online: 29 July 2006

© Springer-Verlag 2006

S. L. Painter ())Geosciences and Engineering Division,Southwest Research Institute,6220 Culebra Rd, P. O. Drawer 28510,San Antonio, TX 78228-0510, USAe-mail: [email protected]

A. D. WoodburyDepartment of Civil Engineering,University of Manitoba,15 Gillson St., Winnipeg, Manitoba R3T 3V5, Canada

Y. JiangWater Management Division,Department of Environment and Energy,11 Kent Street, P. O. Box 2000, CharlottetownPrince Edward Island, C1A 7N8, Canada

Hydrogeology Journal (2007) 15: 315–331 DOI 10.1007/s10040-006-0071-y

método, se utilizan los resultados del segundo métodopara establecer la distribución anterior para un cálculo deactualización Bayesiano. Los resultados únicos de krigingestán sesgados hacia valores bajos y fallan en reproducirpresiones hidráulicas o flujos de manantial. El enfoque deescaleo superior/cokriging remueve casi todo el sesgosistemático. La actualización Bayesiana reduce el residualmedio en más de un factor de 10, a 6m, aproximadamente2.5 por ciento de la variación total de presión en elacuífero. Esta consistencia demuestra la utilidad de lastécnicas de calibración automática basadas en enfoquesestadísticos formales y confiere apoyo adicional para eluso del enfoque de actualización Bayesiano en acuíferosaltamente heterogéneos.

Keywords Inverse modeling . Geostatistics . Numericalmodeling . Edwards Aquifer . Bayesian methods

Introduction

Obtaining reliable hydrological input parameters is one ofthe key challenges in groundwater modeling applications.Hydraulic conductivity often varies dramatically andexhibits spatial fluctuations at a wide range of spatialscales (e.g., Painter 2003), and it is rarely feasible toobtain enough measurements to fully characterize anaquifer. Further, hydraulic conductivity in heterogeneousaquifers depends on the spatial scale, and the spatial scaleinterrogated by typical aquifer tests is often much smallerthan that of the numerical grid cells. These two aspects ofquantitative aquifer characterization, sparse data and scaleinconsistency, exist independently of issues of data qualityand are not fully eliminated by improving the quality offield measurements.

The determination of model parameters in incompletelycharacterized aquifers may be aided by indirect informationsuch as hydraulic heads or springflows, either in a “trial anderror” calibration or in formal parameter identificationapproaches. The latter approach requires the solution to aninverse problem (e.g., Neuman and Yakowitz 1979), whichis ill posed and generally plagued by instability and non-uniqueness. The groundwater inverse problem has attractedconsiderable research efforts, and several effectiveapproaches have been developed (see reviews by Yeh1986; Ginn and Cushman 1990; McLaughlin and Townley1996; Kitanidis 1996). The majority of these approachesrequire the aquifer to be subdivided into a relatively smallnumber of constant-property zones, a conceptualization thatis at odds with the spatial variability observed in manyaquifers.

Approaches based on geostatistics have also beendeveloped (e.g., Dagan 1985; Rubin and Gomez-Hernandez1992; Kitanidis 1996). These approaches do not requirezonation and can thus better honor natural variability.Geostatistical approaches are typically based on cokriging,which requires knowledge of the spatial correlationbetween observed heads and unknown hydraulic conduc-tivities. They are usually based on small-variance approx-

imations and cannot take full advantage of hydraulic headinformation in highly heterogeneous situations (Yeh et al.1996; Zhang and Yeh 1997). Moreover, the analyticalexpressions for cross-covariance are based on simple flowgeometries. How such approaches would be applied to flowconfigurations with complex geometry and arbitraryarrangements of sources and sinks is not clear.

New stochastic inverse modeling approaches based onBayesian statistics have recently been developed (Woodburyand Ulyrich 1998, 2000). As in the geostatistical approach,the Bayesian approach explicitly acknowledges the non-unique nature of the inverse problem and seeks aprobability distribution for the hydraulic conductivity ineach cell. It was recently demonstrated (Jiang 2002; Jianget al. 2004) that the approach can be successfully applied tohighly heterogeneous aquifers with complex flow geome-tries and arbitrary arrangements of sources and sinks. In apreliminary application (Jiang et al. 2004) to the EdwardsAquifer in south central Texas, the Bayesian updatingmethod was used to estimate about 88,000 transmissivityvalues. Steady-state hydraulic head data were successfullyreproduced, but the approach was less successful atmatching spring flows.

Stochastic inverse approaches, and indeed all inversemethods, benefit greatly from prior information about thetransmissivity field. Ideally, such prior information shouldcome from multi-day interference tests that interrogatelarge aquifer volumes. Unfortunately, most well tests aresingle-well specific-capacity tests, which interrogate rela-tively small volumes and are of uncertain reliability.Nevertheless, it is not unusual to have large numbers ofthese specific capacity tests; when taken in aggregate theymay contain useful information about the aquifer trans-missivity field, provided that methods are available forhandling the scale inconsistency.

The difficulty in addressing aquifer characterizationissues, and the degree to which they are important,depends critically on the level of heterogeneity in theaquifer. In mildly heterogeneous granular aquifers, scaledependence may be relatively weak, sparse measurementsof hydraulic conductivity may be sufficient, and it may bepossible to calibrate to hydraulic head data using a varietyof approaches. Quantitative characterization of highlyheterogeneous aquifers is much more difficult, and fewexamples of advanced techniques applied to highlyheterogeneous aquifers exist. Karstic limestone aquifersare particularly challenging. They are often highlyheterogeneous with flow dominated by extremely trans-missive zones corresponding to dissolution featuresaligned with faults, fractures, stratigraphic features, orbedding plane partings. Moreover, the multiple porositynature of these aquifers, with flow through porouslimestone, fractures, or dissolution features, makes thescale-dependence in hydraulic conductivity particularlypronounced (e.g., Halihan et al. 2000).

In this study, the performance of three quantitativecharacterization approaches applied to a highly heteroge-neous karstic limestone aquifer, the Edwards Aquifer insouth-central Texas, is systematically evaluated. The

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evaluation starts with a straightforward geostatisticalinterpolation, then combines geostatistical interpolationwith numerical upscaling, and finally considers a combi-nation of geostatistics, upscaling and Bayesian updating.The main contributions of this work are as follows. First,it is demonstrated how geostatistical interpolation andnumerical upscaling can be combined to create usefulblock-scale transmissivity models from widely availablespecific capacity tests. The combination of numericalupscaling and interpolation avoids the small-variancelimitations of previous geostatistical approaches and isapplicable to complex flow geometry, thus extending theapplicability and utility of geostatisical characterization.

Second, it is shown how the data consistency and datasparseness challenges can be simultaneously addressed bycombining numerical upscaling, geostatistical interpola-tion, and Bayesian updating. Bayesian updating has onlyrecently been tested in highly heterogeneous situations;sensitivity of the method to choice of a prior transmissiv-ity model is explored, and the performance is improvedover the previous preliminary application to the EdwardsAquifer. Finally, a successful demonstration is provided ofadvanced statistical characterization techniques applied toa large, highly heterogeneous aquifer with complexboundaries and source/sink configurations. Using mea-sured hydraulic heads, spring flows, and known scaledependence in the Edwards Aquifer as performancemetrics, the degree to which the performance of a large

groundwater model is improved by systematic applicationof statistical characterization is quantified.

Edwards aquifer

Study areaThe Edwards Aquifer located in south-central Texas(Fig. 1), covers an area of more than 105 km2 extendingfrom Austin to a groundwater divide about 280 km to thesouthwest. The aquifer has high economic, social, andenvironmental importance. It supplies agricultural anddrinking water for more than 1.5 million people, includingalmost all of the city of San Antonio. Major springsflowing from the Edwards Aquifer are crucial habitats forseveral endangered species.

The complex faulted aquifer lies within and near theBalcones fault zone and consists of thinly to massivelybedded limestone and dolomites. Well-developed solutionconduits have formed in association with former erosionalsurfaces within the carbonate rocks and along faults andinclined fractures. The aquifer is recharged mainly bystream flow losses and through sinkholes in an outcropregion along the northern edge. Discharge is primarilyfrom major springs located between San Antonio andAustin and from municipal water supply wells near SanAntonio. Smaller springs and pumping wells existthroughout the region.

Fig. 1 Location map and model region for the Edwards Aquifer, Texas. The numerical grid is 700 cells by 370 cells. The lightly shadedarea represents the outcrop region, the unshaded area represents the confined zone. The darker shaded area is inactive in the model

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Hydraulic conductivity dataThe starting point in the analysis is the transmissivity/hydraulic conductivity dataset compiled by Mace andcoworkers (Mace 1997; Mace and Hovorka 2000;Hovorka et al. 1998). Although a few hydraulic conduc-tivity data are from traditional interference tests, most ofthese data are from single borehole tests and are onlyappropriate for the local scale (a few meters). Transmis-sivity estimates for the single-well tests were obtainedfrom measured specific capacity using a relationshipbetween transmissivity and specific capacity that wasdetermined empirically for the Edwards Aquifer (Mace1997). Vertically averaged hydraulic conductivity was

then obtained from transmissivity by dividing by thescreened interval of the well. The only manipulation of thedata performed for the present study was to geometricallyaverage values when multiple values (representing differ-ent tests) exist in the same well. After this averaging, thedataset contains 653 values of hydraulic conductivity inthe confined zone and 108 values in the outcrop zone.Data locations are shown in Fig. 2. The color scale islogarithmic in hydraulic conductivity with warm colorsrepresenting large values.

Univariate statistical distributions of hydraulic conduc-tivity data are shown in Fig. 3 for the confined andoutcrop sections of the Edwards Aquifer. Both distribu-

Fig. 2 Spatial location of hydraulic conductivity measurements in the Edwards Aquifer

Fig. 3 Cumulative distribution for log K for both the confined and outcrop (unconfined) zones

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tions are reasonably well approximated as lognormal,although the hydraulic conductivity distribution for theconfined section does have a lower tail that is enhancedrelative to the lognormal distribution. The mean andvariance for the confined and outcrop sections aresignificantly different, which is the main motivation fortreating them as separate populations. The geometric meanis 5.7 m/day (18.8 ft/day) for the confined sectioncompared to 0.4 m/day (1.3 ft/day) for the outcrop zone.The log-variance is 6.4 and 9.7 for the confined andoutcrop sections. Because it is vertically averaged hydrau-lic conductivity that is analyzed and not transmissivity, thedifferences are not due to differences in aquifer thicknessbetween the confined and outcrop zones. The cause of thesmaller hydraulic conductivity in the outcrop zone is notclear, but it is noted that the uppermost subunits of theEdwards group are not present in the outcrop region;lower values of hydraulic conductivity in the outcropregion would result if the most permeable subunits of theEdwards group were located near the top of the formation.

The data from the single well tests have three principallimitations that may affect performance of the parameterassignment to numerical models. First, specific capacitytests are not direct measurements of transmissivity.Although the empirical correlation between transmissivityand specific capacity is strong, it is imperfect (Mace1997). For a given specific capacity value, there may be asmuch as an order of magnitude spread in thecorresponding transmissivity values. Thus, the data form-ing the starting point of the study have significantuncertainty. This uncertainty has two consequences: itenhances the spread in the univariate distribution ascompared with the true hydraulic conductivity, and tendsto mask spatial correlation. Given that Mace and Hovorka(2000) found that transmissivity values, as inferred fromdrawdown and recovery data, range over five orders ofmagnitude, the additional spreading introduced by anorder-of-magnitude spread in the specific capacity versustransmissivity relationship is relatively unimportant. Thesecond effect, the masking of spatial correlation, is likelyto be more important. This effect is manifest as anincreased “nugget effect” in the spatial correlation, withthe result being to increase uncertainty in interpolations atpoints near to the actual measurements.

Second, the location data are imprecise for some of thehydraulic conductivity data. This imprecision can be seenby visual examination of the data locations in Fig. 2. Notethat some of the measurements are arranged into horizon-tal lines. This arrangement into horizontal lines is likelycaused by imprecise recording of well locations in driller’srecords. The net effect of this location imprecision is tomask spatial correlation.

Finally, nearly 15% of the single-well tests havedrawdown that is less than the limit of detection andrecorded as zero. These tests clearly represent largehydraulic conductivity values and to discard them wouldintroduce significant bias in the result. In the Macedataset, these single-well tests have been assigned adrawdown of 0.3 m (1 ft), which is thought to be near

the limit of detection for drawdown (Mace and Hovorka2000). Hydraulic conductivities based on these wellsrepresent a lower bound on their true hydraulic conduc-tivity. Thus, the data are biased toward low values.

Forward groundwater modelA steady-state groundwater flow model is required to testthe performance of the three methods for mappinghydraulic conductivity. In addition, the Bayesian updatingmethod also requires a groundwater flow model because itworks directly on a linear system resulting from a finite-difference or finite-element discretization. The EdwardsAquifer Authority (EAA) and US Geological Survey(USGS) recently constructed a new groundwater modelthat will be used for groundwater management purposes(Lindgren et al. 2005). The two-dimensional MODFLOW(Harbaugh and McDonald 1996) grid is composed ofabout 88,000 grid blocks of size 402×402 m (1/4×1/4 mi)and spans the region shown in Fig. 1. The computationalgrid, pumping data, recharge data, and boundary con-ditions from a preliminary version of the Edwards AquiferManagement Model were used in this study. However, afinite-element model was used instead of the MODFLOWfinite-difference model because previously developedcomputer codes (Jiang et al. 2004) for the Bayesianupdating calculation were based on the finite-elementmethod. This choice was purely one of convenience; theentire calculation could have been performed with a finite-difference model.

The resulting finite-element model has no-flow bound-aries on all sides except for a short edge on the northeasternedge that corresponds to the Colorado River, which is set asa constant head boundary. Recharge is specified across theoutcrop region, which occupies a narrow strip along thenorthern edge of the model region. Discharge is frompumping wells with specified removal rates, springs that areset as constant head nodes, and the Colorado River.Although the finite-element grid is based on the EdwardsAquifer Management Model grid, a few simplificationswere made in the interest of computational expediency:

1. The model is confined throughout the region asopposed to the Edwards Aquifer Management Model,which uses the confined/unconfined mode of MODFLOW. The aquifer thickness in the Edwards Aquiferrecharge zone was approximated as one-half theaquifer thickness. Because the inversion is based onsteady-state conditions, this approximation has littleconsequences for estimation of transmissivity. Ifvertically averaged hydraulic conductivity is needed,then the effects of this assumption can be removed as apost-processing step by using independent informationabout saturated thickness when converting the updatedtransmissivity map to hydraulic conductivity.

2. Inflow along the northern boundary from the TrinityAquifer was neglected. Specifically, the northernboundary of the Edwards Aquifer is set as a no-flowboundary. Flow from the Trinity Aquifer is estimated to

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be less than 10% of the total recharge (Lindgren et al.2005) and this approximation is thus not expected tosignificantly affect the estimated transmissivity.

3. Springs and the Colorado River were set as constanthydraulic head nodes. This is a slightly differentcondition compared with the drainage and river pack-ages employed in MODFLOW. However, given thelarge spring conductances for the major springs in theregion, the differences between the two approaches areexpected to be minor.

To ensure compatibility between the Edwards AquiferGroundwater Management Model and the new finite-element model, the mesh was designed so that each nodeis located at the center of an active MODFLOW cell. Thismesh was optimally renumbered so that the half-band widthof the global stiffness matrix was limited to as small as 196.The total node number of this finite-element mesh, which isequal to the number of active cells in the MODFLOW grid,is 87,890. This compatibility between the two models allowsus to transfer data between the two numerical grids.

Hydraulic head and spring flow dataThe USGS has compiled a database of steady-statehydraulic head measurements in the Edwards Aquifer(Lindgren et al. 2005). The locations of the head measure-ments are plotted in Fig. 4; the color scale represents thehydraulic head value, and the x-y locations are in terms ofcell number on the numerical grid. Of the original 153measurements, 26 values located in regions of extremelylarge gradient of hydraulic head were culled from thetarget set. These regions of large gradient are situatedmostly along the contact between the outcrop andconfined zones of the aquifer. They may represent trueflow barriers or they may alternatively reflect errors in

well location. At least some of the culled data are thoughtto have inaccurate well location information.

Although the 26 culled data were not used directly inthe inversion, they were retained for comparison with thecalculated heads in the evaluation of the performance ofthe aquifer characterization methods. Spring flows weretreated the same way. Springs were treated as constant-head nodes in the inversion, an approach that offers nodirect control over spring discharges. In this paper,calculated spring discharges are compared with observedor estimated spring discharges.

Hydraulic conductivity based on kriging

The procedure begins by constructing a map of verticallyaveraged hydraulic conductivity using kriging. No effortwas made to adjust for the scale-of-support differencesbetween the single-well and model grid cells. Thisupscaling issue is addressed in the following section.Because hydraulic conductivities in the outcrop andconfined zones form different statistical populations, eachwas kriged independently to obtain the hydraulic conduc-tivity map. In the confined zone, an isotropic, nestedexponential correlation of the form

� hð Þ ¼ p1exp � hk k�1

� �þ p2exp � hk k

�2

� �ð1Þ

was used to model the spatial correlation in Y=1nK. Here3λ1 and 3λ2 are the practical correlation ranges, h is thelag, p1 and p2 quantify the relative contributions of thetwo correlation structures, and (1-p1-p2) is the relativenugget value. Values for the parameters were determinedby geostatistical analysis as described in the Appendix.

Fig. 4 Measured hydraulic heads across the Edwards Aquifer used as conditioning data in the Bayesian inversion. Locations for the citiesof Austin and San Antonio are also indicated

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The fitted practical correlation ranges are 2.1–3.6 km forthe short correlation and 15–21 km for the longcorrelation, depending on the cutoff. The data from theoutcrop zone are too few to support a nested model and aspherical semi-variogram model with a single practicalrange of 15 km provides a reasonable fit. Note that it wasnot possible to fit directional variograms or correlogramsbecause of too few data and the long, thin aspect of theEdwards Aquifer. Thus, the possibility of higher correla-tion along preferred directions coinciding with northeasttrending faults cannot be ruled out. Results of the krigingare shown in Fig. 5.

Hydraulic conductivity based on upscalingand cokriging

As was noted in the section Introduction, approaches toquantitative aquifer characterization for the purposes ofaquifer model construction need to address the datainterpolation and scale consistency issues. Data interpola-tion is necessary because the grid blocks used in agroundwater model do not necessarily correspond to welllocations where measurements of hydraulic conductivityare available. Scale consistency is an issue because

hydraulic conductivity in heterogeneous formations isdependent on the scale on which it is defined. If local-scale (a few meters) hydraulic conductivity derived fromwell tests is applied unaltered to the larger grid blocks of agroundwater model, a systematic bias towards lowerconductivity values may be introduced.

Rubin and Gomez-Hernandez (1992), and Dagan (1985)developed a method for simultaneous treatment of the scaleconsistency and data interpolation problem. They suggestcokriging block scale transmissivity (or, equivalently,vertically averaged hydraulic conductivity) based on localscale measurements. This approach requires the cross-covariance between the local scale hydraulic conductivityK, and the grid block scale hydraulic conductivity K bespecified. Dagan (1985) and Rubin and Gomez-Hernandez(1992) develop analytical relationships between K and Kbased on the first-order perturbation approach. A similarapproach is taken here, but without reliance on analyticalsolutions for the cross-covariance between K and K that arerequired for the cokriging. Such analytical forms aredeveloped for situations when the variance in Y=1nK isless than unity. In the Edwards aquifer, the log-variance is6.8 in the confined zone and approximately 9 in the outcropzone. Because of this large variance, a purely numericalapproach was used to develop the necessary cross

Fig. 5 a Expected log-K and b log-K variance obtained by kriging without upscaling

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covariance. Specifically, unconditional stochastic simula-tion was performed at the local scale, each realization wasnumerically upscaled to the block scale, and then geo-statistical models were fitted to the upscaled result to obtainthe necessary K model and the K and Kcross covariance.Please refer to Fig. 6 for a schematic representation of theentire upscaling procedure.

Upscaling in this context means calculating or estimatingthe univariate distribution of the block-scale hydraulicconductivity, K, and a spatial cross-correlation betweenY ¼ 1nK and Y=1nK (the block-to-point cross correlation).Note that the block-to-block correlation is not neededbecause there are no direct measurements of K to match inthe interpolation.

The first step in the upscaling procedure is uncondi-tional stochastic simulation at the local scale. Local-scalegeostatistical models are needed for this step. For theconfined zone, a non-parametric indicator model with fivecategories was constructed (see Appendix). This local-scale geostatistical model was then used in unconditionalstochastic simulations of hydraulic conductivity based onthe sequential indicator simulation method. The SISIMmodule of the GSLIB (Version 2.1) system (Deutsch andJournel 1998) was used. This indicator approach does notpresume a multi-Gaussian form, and is able to betterrepresent random fields with connected regions of high orlow K. Twenty realizations with 20×2,000 cells in eachrealization were generated. Each cell has dimensions of20×20 m. The geostatistical model fit to the Y data is

presumed valid at this scale, because the original well testdata support scale (the local scale), from which thegeostatistical model was derived, is of comparable size.

To obtain realizations of the K, 20×20 cell subregions(corresponding to the 402×402 m (1/4×1/4 mi) grid blocksof the USGS MANAGEMENT model) were successivelyremoved from the simulated hydraulic conductivity fields.A head gradient was applied across each subregion in the xdirection with no flow conditions on the other two sides,and the resulting head calculated using a finite differencecode. Because the fitted geostatistical model is isotropic,numerical flow experiments are required for only onedirection. Once the head solution was obtained, the totalflux through the subregion was calculated, and thenconverted to an effective hydraulic conductivity by dividingby the magnitude of the applied gradient. This procedurewas repeated for each subregion in each realization. Theresult is simulated realizations of the K field.

These simulated K fields were then analyzed statisti-cally. The geometric mean of K is increased by 65% andthe log-variance is decreased by 32% compared with thelocal scale values. The increase in geometric mean withincreasing scale is consistent with previous studies ofhydraulic conductivity in the Edwards Aquifer (Halihan,et al. 2000). This scale-dependency is explored furtherlater in this paper. A cross-covariance model was alsofitted to model the joint variability between the Y and Yvariables, which is needed in the cokriging step. The crosscovariance (not shown) is well approximated by models of

Fig. 6 Graphical representation of upscaling and cokriging process. Note that one quater of a mile is 402 m

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the form of Eq. 1, with practical correlation ranges3 λ1=2,700 m and 3 λ2=15,000 m.

For the outcrop region, unconditional simulations ofhydraulic conductivity were obtained using the sequentialGaussian simulation method as implemented in theSGSIM module of GSLIB (Deutsch and Journel 1998).The remaining procedure was the same as for the confinedregion. The upscaling procedure increased the geometricmean hydraulic conductivity for the outcrop zone by 74%and decreased the log-variance by 29%. As with theconfined zone, a block-to-point cross covariance modelwas fitted.

The next step is to estimate the grid-scale conductivitybased on the local-scale conductivity measurements. Forthis step, the log-transformed variables are used. Cokrigingprovides the best estimate of a spatially distributed variable(Y in this case) based on some correlated secondarymeasurements (Y data from the Mace dataset). The resultY� �

is the best estimate of Y at each grid block in themanagement model. The result also provides the krigingvariance �2

K , which quantifies the uncertainty in the Y� �

estimate. These two kriging results are shown in Fig. 7a,b.In constructing these maps, the cokriging estimates for theconfined and outcrop zones were merged using maps ofthe outcrop region. Gray areas are outside the model

region. In general, Y� �

is smaller in the outcrop zone ascompared to the confined zone because of differences inthe underlying Y distributions. The kriging variance issmallest near well locations, but is larger than zero even ifthe estimation point corresponds to a well location,because the well-scale and block-scale values are imper-fectly correlated. The kriging variance is much larger inthe outcrop region, indicating large uncertainty in theoutcrop region. This increased uncertainty is due primarilyto undersampling in the outcrop region.

The final step is to convert the Y� �

to the best estimateof block-scale hydraulic conductivity. Since the krigingsystem was constructed in terms of the Y variable, theresulting kriging result provides Y

� �, which is different

from the natural log of the best estimate of K. The two arerelated through the kriging variance, if one assumes thatthe posterior distribution is log normal (e.g., Journel andHuijbregts 1978).

Best estimates of the K field in the Edwards Aquiferare shown in Fig. 8a. The color scale is on a base-10logarithmic scale. The largest red regions in the K

� �map

correspond approximately to the high permeability zoneunderneath San Antonio. In general, the K

� �is slightly

smaller in the outcrop region compared with the confinedregion, but the difference is smaller than for Y

� �, as the

Fig. 7 a Is the upscaled block ln (K) in m/day, and b is the ln (K) variance. Both result from the cokriging and upscaling exercise and areused to specify the prior hydraulic conductivity model for use in the Bayesian updating

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outcrop region has a larger kriging variance whichpartially compensates the lower Y

� �.

Bayesian updating using hydraulic headmeasurements

The hydraulic conductivity model shown in Fig. 7 wasbased only on well-test data and did not utilize thehydraulic-head data existing over the Edwards Aquiferregion. Hydraulic head data carry considerable informa-tion about the underlying hydraulic conductivity distribu-tion. As an example, for the same boundary and sourceconditions, regions of smaller hydraulic gradient implylarger hydraulic conductivity. However, quantitative infer-ence of hydraulic conductivity from hydraulic head datais, in general, difficult. Specifically, this inference requiresthe solution to an inverse problem, which is nonunique.

The map in Fig. 8a was taken as a starting point andthen modified to be more consistent with measuredhydraulic head data. Specifically, a recently developedBayesian updating procedure (Woodbury and Ulrych1998, 2000) was used to update the hydraulic conductivitymodel. In this approach, the nonunique nature of theinverse problem is explicitly acknowledged and the resultsare given in terms of probability distributions for thehydraulic conductivity in each cell. In addition, theBayesian method allows prior information of varioustypes to be incorporated into the inversion procedure.This feature allowed the results on upscaled hydraulicconductivity to be retained and used in the inversion.

Details of the Bayesian updating procedure arereported elsewhere (Woodbury and Ulrych 2000; Jiang etal. 2004). Here, a brief description is provided and somefeatures described that are pertinent to the followingdiscussion. The objective of the inversion is to determine

Fig. 8 Expected value for hydraulic conductivity in each grid cellas produced by a upscaling and cokriging, and b Bayesianupdating. The hydraulic conductivity from a was used to set the

prior distribution in the Bayesian updating, as described in the text.Locations of major springs are indicated. Note that the two plotshave different color scales

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the expected transmissivity field of the Edwards Aquiferconditioned on the hydraulic head measurements and theupscaled 1nT field. Here T denotes upscaled transmissiv-ity of the aquifer. The solution is not unique because thereare fewer data (heads) than unknown model parameters.To obtain a unique solution to the inverse problem, amethod for singling out precisely one from the infinitenumber of solutions is required. In a probabilisticapproach, it is assumed that the model is composed of avery large, but finite number of model parameters. Themodel parameters are assumed to be random and theinversion is approached from the viewpoint of probabilitytheory (see Ulrych et al. 2001). Bayesian solutions can besought for this problem. The solution is a model that fitsthe observed data and in some sense minimizes thedeviation away from a preconceived model.

Before proceeding to the groundwater inversion problemit is useful to review results from Bayesian linear inversiontheory. The objective is to find a solution m to a linearinverse problem d*=Gm+ν where d* is an observed datavector, m is the parameter vector to be estimated, G is akernel that transforms parameters into observables, and ν isa noise vector that represents measurement errors. Thenoise is presumed to have zero mean; that is, the measure-ments may have some uncertainty, but are unbiased.

The model estimation problem can now be approachedfrom the point of view of probability theory. Given anestimate of the probability density function for the model,f(m|I), subject to new information in the form of a sample(data, d*), one can obtain an estimate of the model bycomputing expected values. If one assumes the observa-tions have errors as described by a matrix Cd and if theerrors in the data and the prior information on the modelparameters are adequately described by the Gaussianhypothesis (with covariance Cm) then the posteriorprobability in the model space is also Gaussian. The firsttwo moments of this probability distribution are given byTarantola (1987, Eq.1.93)

mh i ¼ s þ CpGT GCpG

T þ Cd

� ��1d� � Gsð Þ ð2Þ

Cq ¼ Cp � CpGT GCpG

T þ Cd

� ��1GCp ð3Þ

where mh i and Cq are the expected value and covarianceof the posterior distribution, respectively. These results arewell known (Woodbury 1989).

The vector s can be thought of as the prior mean for m,and the quantity mh i given by Eq. (2) is the Bayesianupdate. As shown, the conditional variance Cq can also bedetermined, but is not addressed here.

Groundwater flow equations, in their usual form, are notin the required framework for the linear inversion theory. Ifone associates upscaled transmissivity with m, then thegroundwater flow equations are linear in the parametersas required, but m is not adequately approximated asmultivariate Gaussian (e.g., lnT may be approximated asmultivariate Gaussian, making T lognormal). Conversely,

if one associates 1nT with m, then m may be adequatelyapproximated as multivariate Gaussian, but then theparameter estimation problem is nonlinear and Eq. (2) isnot applicable. Woodbury and Ulrych (2000) addressedthis issue by employing a perturbation approach to theusual groundwater flow equation. The resulting partialdifferential equation is linear in 1nT, and in the form of anadvection-dispersion equation, which is linear as requiredfor application of Eq. (2). The advection-dispersionequation can be readily discretized by standard finite-difference or finite-element techniques. The result of thisdiscretization provides the kernel G for use in Eq. (2). Itshould be recognized that this perturbation approach isstrictly applicable only when the variance of lnT is lessthan unity, a condition not met by our upscaled lnT model.However, perturbation approaches often provide accept-able results outside the range of applicability. Theperformance of the Woodbury and Ulrych (2000) full-Bayesian updating procedure has been studied (Jiang2002) using synthetic examples with lnT variances aslarge as 6.8. The root-mean-square error between thehidden-truth head and the results of the Bayesian updatewas less than 5% of the maximum head difference in thesimulation. The findings from these synthetic examplessuggest that the Bayesian methodology can be applied tothe automatic calibration of two-dimensional steady-stategroundwater flow model with large variance of lnT.

The results of the cokriging (Fig. 8a) were used to setthe prior distribution in the inversion process. Specifically,maps of the aquifer thickness were used in combinationwith the expected values of hydraulic conductivity to setthe prior lnT. This prior was modified in the areasimmediately surrounding Comal and San Marcos Springs.The borehole data are too sparse to constrain thetransmissivity in those areas, and the presence of large-discharge springs suggests that the transmissivity must belarge there. The hydraulic conductivity in those areas wasset to 6,706 m/day, roughly one third of the largest valuein the hydraulic conductivity dataset.

A nested model with integral scales of 1,200 and5,000 m (practical correlation scales of 3,600 and15,000 m) and a 50% relative nugget effect was used forthe spatial covariance, consistent with the geostatisticalanalyses of the upscaled K. For simplicity, this model wasassumed to apply across the outcrop and confined zoneseven though it was developed only for the confined zone.However, the variance in the outcrop zone, as shown inFig. 7b, was used unmodified in Eq. (1).

Results are shown in Fig. 8b. The color scalerepresents the logarithm of vertically averaged hydraulicconductivity with warmer colors representing largervalues. Locations of major springs are also indicated. Ingeneral, the inversion increased hydraulic conductivitycompared with the prior K.

Calculated heads using the updated hydraulic conduc-tivity are shown in Fig. 9. The average residual betweenthe calculated and observed heads is 0.15 m, indicating nosignificant bias in the result. The root-mean-squareresidual and the average residual are 7.0 and 4.6 m,

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respectively. Given that the difference between themaximum and minimum observed head is about 215 m,this is a good agreement. When the 26 culled heads arereturned to the target set, the agreement is still good, aswill be discussed in the following section.

Performance of the hydraulic conductivity models

The performance of the three hydraulic conductivitymodels is evaluated in this section. Specifically, eachblock scale hydraulic conductivity map is converted to atransmissivity and input into the forward finite element

model. The success at matching observed hydraulic headsand spring flows is evaluated. In addition, the results arecompared with empirical trends relating average hydraulicconductivity to scale of investigation. This section isintended to quantify the improvement obtained in thehydraulic conductivity model over a basic spatial interpo-lation of the Mace (1997) dataset.

Scale dependenceOne of the key goals of this study was to address thedependence of hydraulic conductivity on spatial scale ofinvestigation. It has been noted previously (Halihan et al.

Fig. 10 Geometric mean K versus scale of the investigation from various studies

Fig. 9 Computed hydraulic heads based on effective K values from inversion

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2000) that effective hydraulic conductivity in the EdwardsAquifer tends to increase with increasing spatial scale ofsupport. Specifically, the geometric means from coremeasurements, single well tests, and previous calibratedgroundwater models form a nearly straight line whenplotted against spatial scale on a double logarithmic plot.Neglecting this scale dependence would have introducedsignificant bias in the resulting hydraulic conductivitymap. The geometric means of the three methods forestimating hydraulic conductivity model are shown on thesame type plot in Fig. 10. The upscaling procedureincreased the geometric mean hydraulic conductivity by67% over that of the Mace (1997) data, as the spatial scaleis increased by a factor of 20. This increase withincreasing scale is consistent with the previously identifiedtrend, but is below the trend line. The Bayesian updatingprocedure increases the geometric mean to about 46 m/day,which corresponds to a factor of 7 increase over that of theMace (1997) data. This value of 46 m/day for thegeometric mean is close to the previously identified trendline, which provides confidence in the result of this study.

It is emphasized that this increase was accomplishedwithout resorting to an ad-hoc adjustment of the hydraulicconductivity distribution; the only exception to this wasaround the Comal and San Marcos Springs as discussedearlier.

Match to hydraulic heads and spring flowsThe three hydraulic conductivity fields were input into thefinite-element groundwater flow model described, and theresulting hydraulic heads and spring flow rates werecompared with observed values. The target hydraulic headdataset included 153 measurements (i.e., the 26 that wereremoved for the Bayesian inversion are reintroduced inthis validation exercise). Springs were modeled by fixingthe hydraulic head at the spring locations and calculatingthe discharges. The hydraulic head in the vicinity of SanAntonio Springs is lower than the spring elevation in allthe models; i.e., the spring is dry in the simulations.

Calculated versus observed heads are plotted inFig. 11. Quantitative measures of the agreement between

Fig. 11 a–c These are the computed versus observed hydraulic heads for various K models of the aquifer

327


the calculated and observed heads are provided in Table 1.The improvement obtained by the combination of upscal-ing, cokriging, and Bayesian updating can be seen bycomparing Fig. 11a–c. The hydraulic conductivity modelobtained by kriging alone produces significant error in thepredicted hydraulic heads. In particular, there is a largespread and a significant bias in the heads produced by thekriged model. The upscaling/cokriging approach removesmost of the systematic bias towards low values, but stillhas significant spread around the trend line. Theseresiduals are reduced greatly by Bayesian updating; themean residual is reduced by more than a factor of 10, andthe mean absolute residual is reduced by a factor of 3compared with kriging. The mean absolute residual is 6 m,which is roughly 2.5% of the total head drop across theEdwards region, representing good agreement. Thisagreement between calculated and observed heads clearlydemonstrates the utility of the automatic calibrationtechniques applied in this work.

Spring flow information was not used as a formalconstraint in the Bayesian updating. Nevertheless, theBayesian updated hydraulic conductivity field still pro-duced significant improvements in the spring flows ascompared with the kriged model (Table 2). Specifically,the spring discharges in the down-gradient regions areroughly consistent with the calibration set, with the mainexception being San Antonio Spring, which is dry in the

simulations. Flow at Leona Spring is also significantlylarger than the calibration target of 0.425 m3/s. However,this discharge rate is potentially underestimating the actuallosses from the Edwards aquifer by a large amountbecause it ignores flow into the highly permeable LeonaGravel formation (Green et al. 2004). For this reason, noattempt was made to match discharge at Leona Spring.

Discussion and conclusions

In summary, three quantitative methods for estimatinghydraulic conductivity were applied to the EdwardsAquifer and tested in a forward groundwater model. Eachrefinement in the hydraulic conductivity field producedimprovement in the ability of the groundwater model toreproduce steady-state hydraulic heads and spring flows.The final result was a calibrated groundwater model withabout 88,000 computational elements. The ability toachieve a calibrated model of this size and the observedsuccessive improvements in the model performanceclearly demonstrate the utility of adopting a formalstatistical approach to parameter estimation. Such anapproach minimizes subjective input and manual adjust-ment of the hydraulic conductivity model.

The simplest method based on a simple interpolationwithout adjusting for scale resulted in significant bias in theestimated hydraulic conductivity field and poor performancein the forward groundwater model. The systematic biastoward lower hydraulic conductivity was removed throughthe use of a numerical upscaling and cokriging technique.The method uses stochastic simulation and numericalupscaling to develop an upscaled geostatistical model, thusavoiding the small variance approximation inherent incokriging approaches that require an analytical cross-covariance model. Using this approach, the mean-absoluteresidual in the hydraulic head was about 5% of the total headdrop across the region. This level of residual is likely to beacceptable for some applications, thus demonstrating thathydraulic conductivity data derived from specific capacitytests (e.g., Mace 1997) can be combined with numericalupscaling and interpolation to develop useful transmissivityinputs for groundwater models. In the final revision to thehydraulic conductivity field, the result of the cokriging stepwas used in a Bayesian updating, which further improvedthe performance of the forward groundwater model.

These results provide some insights into the relativevalue of different types of data. Specifically, the largechange in the hydraulic conductivity field and thedramatic improvement in performance of the forwardgroundwater model achieved by the Bayesian updating,underscore the degree to which hydraulic head measure-ments carry information about the underlying arealdistribution of hydraulic conductivity. Additional effortsto improve and expand the database of water levelmeasurements, through compilation of existing data andongoing water level monitoring, may be the most cost-effective approach to improving the hydraulic conductiv-ity model of the Edwards Aquifer.

Table 2 Calculated spring flows obtained from three methods forassigning transmissivity values. Calibration targets are also shown.Values are spring discharges in cubic meters per second

Spring Kriging withno upscaling

Upscaledandcokriged

Bayesianupdate

Calibrationtarget

SanMarcos

0.584 0.478 2.24 4.22

Comal 6.91 6.55 7.62 9.34SanAntonio

0a 0* 0a 0.283

San Pedro 6.24 3.68 0.325 0.181Leona 4.06 3.40 4.76 0.425b

aHydraulic head at node corresponding to spring is less than springelevation. Spring is not flowingbDoes not include losses into Leona Gravel Formation. Maybe significantly less than true discharge from Edwards AquiferNo attempt was made to match this value

Table 1 Hydraulic head residual for three methods for assigningtransmissivity values. The total head drop across the region is about250 m. Values are in meters

Kriging withno upscaling

Upscaled andcokriged

Bayesianupdate

Mean residual –12.9 –3.9 1.2Root meansquareresidual

26.1 17.2 8.2

Mean absoluteresidual

17.5 12.5 6.0

Maximumresidual

84.2 45.9 48

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The Bayesian updating is based on a perturbationapproach that, strictly speaking, requires a small varianceof log conductivity. The fact that the method performedwell for log K variances as high as six suggests that it maybe more robust and with a much wider applicability thanthe small-variance requirement would suggest. Jiang et al.(2004) reached similar conclusions. However, theBayesian updating method did perform poorly when theprior distribution was biased to low values. This under-scores the need for a good method for setting the priordistribution. The numerical upscaling/cokriging approachprovides a practical alternative to existing analyticalmodels for setting the prior distribution.

The Bayesian updating method also performed poorly inareas of large head gradients. Large hydraulic gradients canoccur near pumping wells and in regions of low hydraulicconductivity. It was found empirically that good results areproduced if the head difference between two adjacentcomputational cells is sufficiently small. This condition isnot met over all regions of the model grid, but acceptableresults could still be achieved by simply removing hydraulichead data associated with very high gradients. This dataculling has the effect of spreading in space what wouldotherwise be narrow flow barriers. When viewed atsufficiently large scales, the resulting smearing of the lowhydraulic conductivity features does not appear to signifi-cantly affect the flow. However, it does introduce potentialerrors in the calculated water levels near the features. Thisinaccuracy in regions of high gradient is not a fundamentallimitation and could be eliminated by simply refining thecomputational grid in those regions.

From an applications perspective, an obvious extensionto this study would be to refine the computational grid inthe areas around Comal and San Marcos Springs, and nearthe contact between outcrop and confined regions.Another extension would be to refine boundary conditionsand recharge values. To expedite the model development,preliminary data on recharge and simplified boundaryconditions were used. Updated recharge values, morerealistic boundary conditions, and refined grid in criticalregions could result in improved models for hydraulicconductivity in the Edwards Aquifer.

One of the key products of this work is the develop-ment of a general quantitative framework combiningnumerical upscaling, cokriging, and Bayesian updating.It is possible to combine various types of data within thisframework. Using the approach, data on aquifer thickness,boundary conditions, recharge, pumping rates, observedhydraulic heads, and well-derived transmissivities werecombined in a self-consistent manner that honors thephysics of groundwater flow. Spring flow informationcould also be incorporated. For the present application, arefined grid would be required near the major springs ifthe spring flows were to be included as constraints.

The Bayesian methodology also has the flexibility toincorporate some types of geological information. Forexample, one could include expert judgment on thelocation of highly conductive zones by enhancing priorconductivity in areas corresponding to mapped conduits.

Indeed, a preliminary step was taken down this path byenhancing the prior hydraulic conductivity in the areasaround Comal and San Marcos Springs, where geologicalunderstanding indicated that the geostatistically derivedconductivity was too low. This flexibility for merging softgeological information with more quantitative constraintsis an important advantage of Bayesian approach and apotentially productive area for future research.

Acknowledgements The authors are grateful to the EdwardsAquifer Authority for supporting this study. The authors also thankDr. Ron Green and two anonymous reviewers for reviewing themanuscript.

Appendix

Geostatistical analysisGeostatistical analyses of the local scale hydraulicconductivity are needed to establish models for the spatialcorrelation. The confined and outcrop sections of theaquifer were treated as separate populations. A moredetailed geostatistical analysis was undertaken for theconfined region to better model spatial correlation in theextreme values of hydraulic conductivity.

There are several methods for establishing the two-point spatial correlation, each with its advantages anddisadvantages in a given situation (e.g., Journel andHuijbregts 1978; Deutsch and Journel 1998). The mostfamiliar approach is the sample semi-variogram, which isthe sample semi-variance as a function of lag (separation)distance. An alternative approach is to calculate thesample nonergodic covariance

* hð Þ ¼ 1

2N hð ÞXN hð Þ

i¼1

Z uið ÞZ ui þ hð Þ � m2h

or the closely related nonergodic correlogram

� hð Þ ¼ * hð Þ��2h

where Z(ui) is a sample value (of the random field Z) atlocation ui, and h is the lag (separation) distance. Thequantities mh and �2

h are the sample mean and samplevariance of the N(h) data pairs, which need not be the sameas the mean and variance of the entire population. Thecorrelogram ranges between 0 and 1 with the value 1corresponding to perfect correlation; the value zerocorresponds to the uncorrelated situation. The semi-vario-gram is the most widely used measure of spatial correla-tion. However, it is particularly sensitive to two statisticalfeatures that often occur in combination in hydrologicaldata: heteroscedasticity and clustering of extreme values.The term heteroscedasticity refers to the situation when thelocal variability of the data is related to the local mean, ormore generally, changes over the study area. When this

329


occurs in combination with clustering of data, spatialcorrelation, as measured by the semi-variogram, is oftenmasked. The nonergodic correlogram is preferred in suchsituations, as it is known to be robust against thesestatistical artifacts (Deutsch and Journel 1998).

It is advantageous to apply these measures of spatialcorrelation not to the original K data, but to sometransforms of K. For example, it is common to calculatethe semi-variogram or correlogram for the logarithmictransforms. Another powerful technique is to apply thesemi-variogram or correlogram to indicator transforms.The indicator transform at a given cutoff value is obtainedby replacing those values above the cutoff with a 1 andthose below with a 0. By repeating this process for severalcutoff values and calculating the variogram or correlogramfor each, it is possible to develop a nonparametric modelfor the spatial correlation. Such an indicator modelcontains much more information than a traditional semi-variogram model, and is preferred if enough data areavailable. In particular, an indicator model is better in

reproducing spatial correlation in the tails of the distribu-tion, which controls upscaling.

Fig. 13 Semi-variogram for the 9th decile. Unlike the non-ergodiccorrelogram of Fig. 12, the semi-variogram reveals no significantspatial correlation

Fig. 12 Results of the geostatistical analysis. The solid curves are fitted correlogram models

330


Omnidirectional indicator correlograms for the confinedregion K are shown in Fig. 12 for five cutoff valuescorresponding to the 1st, 2nd, 5th, 8th and 9th decile. Thefact that these correlograms are nonzero for nonzero lagdistances means that significant spatial correlation exists forall cutoff values. For these data, the indicator correlogramsfor the 1st decile are larger than the corresponding ones forthe median or upper cutoffs. This suggests that low values ofhydraulic conductivity are better correlated spatially than thelarge values. The fact that all of these correlograms aresignificantly less than 1 as lags approach 0 is a manifestationof the nugget effect. A nonzero nugget value means thathydraulic conductivity measured in two closely spaced wellswould be imperfectly correlated due to the effects of verysmall-scale variability or measurement errors.

The solid curves in Fig. 12 are fitted correlogrammodels. The indicator correlogram for each cutoff is wellfitted by a nested exponential model of the form Eq. (1).For each cutoff, the two correlation structures contributeroughly equally to the composite correlation structure. Theshort correlation range is 2.1–3.6 km, depending on thecutoff value, and the long correlation range is 15 kmexcept for the first decile, which has a practical correlationrange of 21 km.

The indicator semi-variogram for the 9th decile isshown for comparison purposes in Fig. 13. The semi-variogram is constant for all lags except for randomfluctuations. This suggests no spatial correlation in theupper 10% of the hydraulic conductivity distribution, asopposed to the correlogram in Fig. 12, which showssignificant spatial correlation. Careful review of the datasuggests that this apparent discrepancy is caused by mh

and �2h being different for small and large lags, a statistical

feature that is known to mask spatial correlation. There aretwo possible causes for mh and �2

h being different for smalland large lags: heteroscedasticity and sampling bias.Sampling bias is likely in this dataset, as a landownerthat drills an unproductive well is likely to try again on thesame property. This type of biased sampling may alterthe statistics of closely spaced wells compared with thegeneral population. Whatever the cause, the correlogramis robust to this statistical feature and picks up spatialcorrelation that would have been overlooked using thetraditional semi-variogram.

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Transmissivity estimation for highly heterogeneous