Flow in multiscale log conductivity fields with truncated power

WATER

RESOURCES

RESEARCH,

VOL. 34, NO. 5, PAGES 975-987, MAY 1998

Flow in multiscale log conductivity fields with truncated power variograms Vittorio

Di Federico

Dipartimentodi Ingegneriadelle Strutture,dei Transporti,delle Acque, del Rilevamento,del Territorio (D.I.S.T.A.R.T.), Universit• di Bologna,Bologna,Italy

Shlomo

P. Neuman

Department of Hydrologyand Water Resources,Universityof Arizona, Tucson

Abstract. In a previouspaper we offered an interpretationfor the observationthat the log hydraulicconductivityof geologicmedia often appearsto be statisticallyhomogeneous but with varianceand integral scalewhich growwith the size of the region (window)being sampled.We did so by demonstratingthat suchbehavioris typicalof any randomfield with a truncatedpower (semi)variogramand that this field can be viewedas a truncated hierarchyof mutuallyuncorrelatedhomogeneous fieldswith either exponentialor Gaussianspatialautocovariancestructures.The low- and high-frequencycutoff scalesX• and Xu are related to the length scalesof the samplingwindowand data support, respectively.We showedhow this allowsthe use of truncatedpowervariogramsto bridge information about a multiscale random field acrosswindows of different sizes, either at a

givenlocale or betweendifferent locales.In this paper we investigatemean uniform steady state groundwaterflow in unboundeddomainswhere the log hydraulicconductivityforms a truncatedmultiscalehierarchyof Gaussianfields,each associatedwith an exponential autocovariance. We start by derivingan expressionfor effectivehydraulicconductivity,as a function of the Hurst coefficient H and the cutoff scalesin one-, two-, and three-

dimensionaldomainswhich is qualitativelyconsistentwith experimentaldata. We then developleading-orderanalyticalexpressions for two- and three-dimensional autocovarianceand cross-covariance functionsof hydraulichead,velocity,and log hydraulicconductivityversusH, X• and Xu; examinetheir behavior;and comparethem with thosecorrespondingto an exponentiallog hydraulicconductivityautocovariance.Our resultssuggestthat it shouldbe possibleto bridge informationabout hydraulicheadsand groundwatervelocitiesacrosswindowsof disparatescales.In particular,when X• >> Xu,

thevariance of headisinfinitein twodimensions andgrowsin proportion to X•+2Hin three dimensions,while the varianceand longitudinalintegral scaleof velocitygrow in

proportion to X•H andX•,respectively, in bothcases. 1.

Introduction

It hasbecomecommonin subsurfacehydrologyto interpret spatiallyvaryingdata geostatistically and to analyzeflow and transportin geologicmedia stochastically. One importantaspectof geostatistical analysisis the inferencefrom suchdata of a (semi)variogram 3/(s),wheres is a vectordefiningthe spatial separationbetweenanytwo points.Natural log hydraulicconductivity(Y = In K) and transmissivity (Y = In T) data often appearto fit variogrammodelsassociated with a constantvari-

ance0-2andintegral(spatialautocorrelation) scaleX. Thisis commonlytaken to implythat the data are representativeof a statistically homogeneous randomfield Y(x) where x is a vector of spatial coordinates.Hence much of the stochastic groundwaterliterature is devotedto the analysisof flow and transportin homogeneous Y(x) fields IDagan, 1989; Gelhar, 1993;Dagan and Neuman, 1997]. There is, however,evidence[Gelhar, 1993, Table 6.1; Neu-

man,1994]that X, andto a lesserextent0-2,increase consis-

tentlywith the sizeof the region(window)beingsampled.This suggests that homogeneitymay not be a true propertyof Y(x) but rather an artifactof the scaleof investigationand method of inference.Indeed, when samplevariogramsare plotted on logarithmicrather than on arithmeticpaper (as hasbeen the standardprocedureuntil quite recently),the majority of the data often lie closeto a straightline, which is not consistent with the assumptionof homogeneity.Suchbehavioris being observedat an increasingnumber of siteson distancescales rangingfrom a few meters to 100 km [Neuman, 1995]. The straightline is indicativeof a nonstationaryfield with homogeneousspatialincrements.If the field is statisticallyisotropic, a line with slope 2H representsa power variogram•/(s) =

Cos2H,wheres isseparation distance (themagnitude of s),CO is a constant, and H is the Hurst coefficient. Since the vario-

gramscales as•/(rs) = r2H'y(s)the fieldis self-affine [Neuman, 1990] and, within the range 0 < H < 1, constitutesa random

fractal with dimension

D

= E

+

1 -

H where E is

Euclidean(topologic)dimension[Voss,1985]. If the field is additionallyGaussian,it constitutesfractional Brownian motion (fBm) [Mandelbrotand Van Ness,1968]. The Hurst coefficientof Y(x) is not the sameat each site,

Copyright1998by the American GeophysicalUnion. Paper number98WR00220. 0043-1397/98/98 WR-00220509.00 975

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thoughit hasbeen foundto lie near the midrangeof 0 < H < 0.5 in severalrecentstudiesquotedby Neuman [1995]and in

the studiesof Molz andBoman[1995],Liu andMolz [1996], and Guzman and Neuman [1996]. Similar values of H were

reportedby Painter[1996]in the contextof fractionalLevy

CONDUCTIVITY

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nally proposedby Neuman [1990],and other generalizedvariogramparametersderivedin their paper. In this paper we use the aboveideasto analyzemean uniform steady state groundwaterflow in unboundeddomains within whichthe log hydraulicconductivityformsa truncated multiscalehierarchyof Gaussianfields,eachassociated with an exponentialautocovariance. We startby derivingan expression for effectivehydraulicconductivity,as a functionof the Hurst

motion(fLm), a generalization of fBm whichdoesnot require Gaussianity.Within this range of Hurst coefficientsthe incrementsare negativelycorrelatedand relativelynoisy,exhibiting antipersistent behavior.Nevertheless, whenonejuxtaposes ap- coefficient and cutoff scales in one-, two-, and threedomains.We then developleading-orderanalytiparentvaluesof o'2andh frommanydifferentsites,inferred dimensional for two- and three-dimensionalautocovariance from Y data by assumingthat the underlyingfield is homoge- cal expressions and cross-covariance functionsof hydraulichead,velocity,and neous,one findsthat they fit a generalizedpower variogram with H •- 0.25 [Neuman,1994].Suchgeneralizedbehaviorhas log hydraulicconductivityversusHurst coefficientsand cutoff been deducedearlier by Neuman [1990] from the observed scales;examinetheir behavior;and comparethem with those to an exponentialloghydraulicconductivity auscale dependenceof juxtaposedapparent dispersivitiesre- corresponding tocovariance. We concludeby explaininghow our resultsallow ported for a large number of tracer studiesworldwide. Neuman [1990] has shownthat any random field with ho- informationto be bridgedabouthydraulicheadsand groundmogeneousincrementscan be viewed as an infinite hierarchy water velocitiesacrosswindowsof disparatescales. of mutuallyuncorrelatedhomogeneous fields(modes)characterized by exponentialautocovariancefunctionsand variances that increaseas a power of scale.He noted that accounting deterministically for large-scalespatialvariabilityis equivalent to filtering out low-frequencymodesfrom this hierarchy.Di Federicoand Neuman[1997]extendedtheseideasby demonstrating that both the power variogram and the associated spectraof a randomfield with homogeneous isotropicincrements can be constructedas weightedintegralsfrom zero to infinity (an infinite hierarchy)of either exponentialor Gaussian variogramsand spectraof uncorrelatedhomogeneous iso-

tropicmodes.They analyzedthe effectof filteringout (truncating) high- and low-frequency modes from this infinite hierarchyin the real andspectraldomainsandshowedthat a low-frequencycutoff rendersthe truncatedhierarchystatisticallyhomogeneous, with a spatialautocovariance functionthat variesmonotonicallywith separationdistancein a mannernot too dissimilarfrom that of its constituentmodes.The integral scalesof the lowest-andhighest-frequency modes(cutoffs)are relatedto the lengthscalesof the samplingwindowandsample

(datasupport) volume,respectively. Takingthisrelationship to be one of proportionalityrenderedtheir expressions for integral scaleandvariancedependenton windowsizein a manner consistent

with

observations.

Di

Federico

and Neuman

2. Autocovarianceof Log Hydraulic Conductivity Di Federicoand Neuman[1997]consideredan infinitehierarchyof mutuallyuncorrelated,statistically homogeneous, and isotropicrandomfields(modes)of naturallog hydraulicconductivityY(x), eachof whichis associated with an exponential variogram

yes, x) =

drift from the data.

The traditionalapproachhasbeen one of truncatingpower spectraldensitiesrather than the hierarchyof spectraassociated with exponentialor Gaussianautocovariance functionsas proposedbyDi FedericoandNeuman[1997].Di Federicoand Neuman showedthat truncatedpower spectrayield autocovariancefunctionswhich oscillateabout zero with vanishing integralscalein one dimension,They concludedtheir paperby describinghowtheir hierarchicaltheoryallowsbridgingacross scales.They showedthat this canbe done at a givenlocaleby calibratinga truncatedvariogrammodelto data observedon a

(1)

and a variance C

cry(n)= 2u

(2)

whereX is integralscale,n = 1/X is modenumber(a measure

of spatialfrequency), C is a constant (dimensions [L-2H]) so that the variancedecreasesas a power of the mode number, andH is the Hurst coefficient.Theyshowedthat integrationof (1) over all modesbetweena lower cutoffnl = l/hi and an upper cutoffn, = 1/X,, weightedby 1/n, leadsfor 0 < H
o,) as functionsof the dimensionless parallel to mean flow in two and three dimensions,respecdistances/Iy.and the angleof inclinationfromx•. To explorethe tively. Its behavior is similar in principle to that of C•.h in influenceof H, we showresultscorresponding to H = 0.10,0.25, Figure 2. The cross covarianceis negative, more so in two dimensions than in three. It exhibits a minimum which diminand 0.40. Sincethe integralscaleIv = (2H)/[(1 + 2H)nl] andthevariance rr• = C/(2Hn••) dependonH, wevaryC ishesin absolutevalue and is shiftedtoward larger lagsas one

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2.00

2.00

3-D

7h

7h

.......H=0.10

1.50

.........

....... H=0.25 ...... H=0.40 ..... expo n.

..'"'

1.50

.........,-"•J

.......

H=0.10

.....

H=0.25

•

expon.

H=0.40

.." ..-C•

1.00

1.00

0.50

0.50

0.00

0.00

2.0

0.0

4.0

6.0

2.0

0.0

8.0 S/Iy 10.0

4.0

6.0

8.0 S/Iy 10.0

Figure 3. Dimensionlesshead variogramas in Figure 2.

movesfrom the exponentialmode toward hierarchiesof such modeswith progressivelylower Hurst coefficients.As H diminishes,so doesthe decayrate toward the minimum,while the distance over which cross correlation persistsbecomes larger. The log-conductivity-headcrosscovarianceis zero in a direction

normal

to mean flow in all cases. Cross-correlations

betweentransversevelocityand log conductivityor head (not shown)exhibitbehaviorssimilarto thoseillustratedby Zhang and Neuman [1992,Figureslb and ld] for the caseof a single exponentialmode but with lower peaksand longer tails. Figure 6 depictsthe autocovarianceCa of longitudinal

modeswith progressively lowerHurst coefficients; the opposite is true, to a lesserdegree, at large lags.

Figure8 shows thedimensionless cross covariance Cu•u2= Cu2u• between longitudinal andtransverse velocities, in a direction inclined at 45ø to that of mean flow, in two and three

dimensions (whereCu•uk = Cuku• andk = 2 and3).Thecross covarianceis positive,more soin two dimensions than in three. It vanishesboth at zero lag and at infinity.The crosscovariance exhibits a maximum

which diminishes

and is shifted toward

smallerlagsas o.nemovesfrom the exponentialmode toward hierarchiesof suchmodeswith progressively lower Hurst covelocities, normalized with respect toU2tr•;•arallel tomeanefficients.As H diminishes,so doesthe decayrate of the tail, flow in two and three dimensions. Its behavior is similar in while the distanceover which crosscorrelation persistsbe-

principle to thatof Cu•vin Figure4. Thedimensionless auto- comes larger.Asin theexponential case[Rubin,1990a],Cu•u2 covarianceis positiveand decaysmonotonicallytoward zero. is antisymmetric(it changessignwith a changein signof the Its rate of decay at small lags is slowestfor the exponential inclinationangle) and vanishesin both the longitudinaland mode and increases as one moves toward

hierarchies

of such

transverse directions.

It is important to emphasizethe difference between the modeswith progressively lowerHurst coefficients; the opposite is true, to a lesserdegree,at large lags. present resultsfor multiscalefields and single-scaleresults. Figure 7 illustratesthe dimensionless transversevelocityau- Even if our graphicalresultslook quite similar to published tocovariance Cu2u2 (equalto Cu3tt 3in threedimensions), par- graphs,they differ from them fundamentallyin their formal allelto meanflowin twoandthreedimensions. UnlikeCu•u•, dependenceon Hurst coefficientH supportand domainscales. Cu2u2 is a hole-type autocovariance witha zerointegralscale; We quoteampleempiricalevidencethat multiscalebehaviorof otherwise, itsbehavior issimilar, in principle, to thatof Cu•u •. subsurfaceflow and transportis ubiquitousand thereforethe Its rate of decay at small lags is slowestfor the exponential rule rather than the exception.[Di FedericoandNeuman,1997; mode and increases as one moves toward hierarchies of such this study, section 1]. We therefore concludethat when in

0.70

0.70

CulY

0.60

0.60

....... 0.50 0.40 0.30

0.20

3-D

CulY

2-D

H=0.10

0.50

H=0.25

••,

H=0.40

0.40

expon•.

i• ....... H=0.10 •!• H=0.40 ......

H=0.25

, \.\

expon.

0,30

:...,

',

0.20

0.10

0.10

i

0.0

I

2.0

i

I

4.0

i

I

6.0

i

i •"•'•1• -•

8.0 S/Iy 10.0

'•' ',.,;' !

0.00 0.0

i

2.0

4.0

i

6.0

8.0 S/Iy 10.0

Figure 4. Dimensionlesscrosscovariancebetween longitudinalvelocity and log hydraulic conductivity versusdimensionless distances/Iv parallel to mean flow for variousvaluesof H without a high-frequency cutoff and for exponentiallog conductivityautocovariance.

DI FEDERICO

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2.0

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8.0 S/Iy 10.0

0.0

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2.0

4.0

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6.0

981

8.0 S/Iy 10.0

Figure 5. Dimensionlesscrosscovariancebetweenlongitudinalvelocityand head as in Figure 4. doubt,one shouldprefer a multiscalerather than a single-scale stochasticmodel.Di Federicoand Neurnan[1997] make specific recommendationsas to how one might discriminateformally betweenalternativemodelsof spatialvariability. 6.

Conclusions

We draw the followingmajor conclusions from this paper: 1. The earlierwork of Di Federicoand Neuman[1997]has establisheda geostatistical frameworkwhich allowsthe applicationof standardmethodsof stochasticanalysisto problems of flow in random media with continuouslyevolvingscalesof heterogeneity betweentwo limits:a lowerlimit proportionalto measurementor supportscaleand an upperlimit proportional to samplingdomain or window size. We have demonstrated this analyticalcapabilityin this paper by using a standard method of perturbation to solve the stochasticproblem of mean uniform steadystate groundwaterflow in an infinite, two- or three-dimensional multiscalehierarchyof randomlog hydraulicconductivityfields.Preciselybecausethe methodof analysisis standard,it is easyto implementand leadsto familiar-lookingresultswhich are, nevertheless,novel in their explicit dependenceon Hurst coefficientH supportand window scales.It shouldbe possibleto derive other solutionsto stochasticgroundwaterflow problemsby perturbationor other meanswithin the samegeostatistical frameworksoasto render them dependenton thesescales. 2. Within the geostatistical frameworkof Di Federicoand

Neuman [1997], log hydraulicconductivityis viewed as a statistically nonhomogeneousrandom field with homogeneous isotropicspatialincrements.The field thus forms a self-affine random fractal characterizedby a power (semi)variogram. Boththe powervariogramandassociated spectraareviewedas an infinite hierarchyof either exponentialor Gaussianvariograms and spectraof mutually uncorrelated,statisticallyhomogeneous, andisotropicrandomfields(modes).Filteringout (truncating)low-frequency(large-scale)modesfrom this infinite hierarchyrendersthe truncatedhierarchystatisticallyhomogeneous with a positivespatialautocovariance functionthat decaysmonotonicallywith separationdistancein a mannernot too dissimilarfrom that of its constituent(exponentialor Gaussian)modes.The integral scaleX• of the low-frequency cutoffmodeis relatedto the length-scaleof a samplingwindow definedby the regionunder investigation.If a high-frequency (small-scale)cutoff is present, then its integral scale Xu is related to the length-scaleof data support(volume of measurement).Taking eachrelationshipto be one of proportionality rendersthe integralscaleandvarianceof a truncatedfield dependenton windowand supportscalesin a mannerconsistent with observations. We have shownin this paper how the latter translatesinto a dependenceof effectivehydraulicconductivity,andsecondensemblemomentsof hydraulicheadand velocity, on window and support scalesin unboundedflow domainsunder uniform mean steadystate flow. 3. We developedan expressionfor effectivehydrauliccon-

0.60

0.60

2-D

Culul

0.50

0.50

0.40

.......

H=0.10

.....

H=0.25

....

H=0.40

I....... H:0.10 [..... H:0.25

0.40

] ....

0.30

0.30

0.20

0.20

0.10

0.10

2.0

4.0

6.0

8.0 S/Iy 10.0

H:0.40

I--- expon.

",., ':•,,,•,

...... 7."..* ......

0.00

0.00 0.0

3-D

Culul

0.0

2.0

4.0

6.0

:.•'",.•::.-• ::-'. 8.0 S/Iy 10.0

Figure 6. Dimensionless autocovariance of longitudinalvelocityversusdimensionless distances/Iy parallel to mean flow for variousvaluesof H without a high-frequencycutoff and for exponentiallog conductivity autocovariance.

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0.15

CONDUCTIVITY

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0.15

Cu2u2 .

2-D

0'10[{/ II ..... ....... H=0'10 1 I:,\ H=O.25I 0.05

0.10

expon.

' 0.0

• 2.0

'

' 4.0

.......

.....

H=O. 10

H=0.25

H=0.40

'"' I.....

-0.05

3-D

Cu2u 2

0,05

,•,•• "•,,•.• •-.c=ex..;._. IX)::__ n.:... i.._...._...._ :._.,._,, .............. •_ '

• 6.0

'

•

'

i

-0.05

8.0 S/Iy lO.O

0.0

i

2.0

i

i

i

I

4.0

i

i

6.0

i

8.0 S/Iy 10.0

Figure 7. Dimensionlessautocovarianceof transversevelocityas in Figure 6.

ductivityin one-, two-, and three-dimensionalunboundedflow

and crosscovariancefunctionsincreasewith a diminishing Hurst coefficient.Regardlessof how large or small these diftiscale log hydraulic conductivityfield is Gaussianwith an ferencesmay be, our multiscaleresultshavean advantageover exponentialautocovariance.Our expressionis closelyrelated correspondingsingle-moderesultsin that they dependexplicto those derived earlier by Neuman [1994] for the effective itly on Hurst coefficientH windowand supportscales. 5. When Xl >> Xu, the varianceof hydraulichead is inficonductivities of finite domainsof lengthscaleL without consideringtruncation.It suggeststhat log effectiveconductivity nitein twodimensions andgrowsin proportion to X]+2•/in increasesasa powerof L in three dimensions,decreasesin one three dimensions.The varianceand longitudinalintegralscale dimension,and showsno systematicvariation with L in two of velocity growin proportion to X]•/and Xl, respectively, in dimensions.This appearsto be qualitativelyconsistentwith both cases.The latter growth rate of the velocityvarianceis with an earlier analysisbyNeuman [1995,equations publishedlaboratoryand field data. It differsfrom an earlier consistent analysisbyAbabouand Gelhar[1990]for a Hurst coefficientof (5a) and (5b)], accordingto which the varianceof velocities zero in which suchbehavioris exhibitedby the effectivecon- experiencedby a travelingparticlegrowsas a power2H of its ductivityrather than by its logarithm. meantravel distance,i.e., of the lengthof path (window)sam4. We developedleading-orderexpressionsfor two- and pled by the particle. three-dimensionalhydraulichead variograms,head-log con6. Our hierarchicalflow theory thus allows,in principle, ductivitycrosscovariance,velocityautocovariance,and veloc- bridgingacrossscalesby predictingthe effect of viewingmulity-log conductivityas well as velocity-headcrosscovariance tiscalerandom hydraulichead and velocityfields,definedand functions for the case where each mode of the truncated mulmeasuredon a given supportscale,througha larger window tiscalelog hydraulicconductivityhierarchyis Gaussianwith an definedby the regionunderinvestigation. At a givenlocaleone exponentialautocovariance.All of thesefunctionsdependon can start by calibratinga truncatedvariogram model of log the Hurst coefficientand cutoff scales.Qualitatively,all mul- hydraulicconductivitiesto sampleconductivitydata observed tiscale autocovariance and cross covariance functions behave on a givensupportscalewithin one window,then predictthe asthosecorresponding to a singlemodewith the sameintegral autocovariance structureof the corresponding log conductivity scale,but the multiscalefunctionstend to decaymore rapidly field within windowsthat are either smalleror larger, as proat short separationdistancesand to exhibit longer tails. Dif- posedby Di Federicoand Neuman [1997]. One can then evalferencesbetween multiscaleand single-modeautocovariance uate (to a leadingorder of approximation)the associated vardomains

for the case where

each mode of the truncated

mul-

O.O5

0.05

Culu2

3-D

Culu2

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0.00

.......

H=0.10

.....

H=0.25

....

H=0.40

--

expon.

0.00 0.0

2.0

4.0

6.0

8.0 S/Iy 10.0

0.0

2.0

4.0

6.0

8.0 S/Iy 10.0

Figure 8. Dimensionlesscrosscovariancebetweenlongitudinaland transversevelocitiesversusdimensionlessdistances/Iv inclinedat 45øto meanflow directionfor variousvaluesof H without a high-frequencycutoff and for exponentiallog conductivityautocovariance.

DI FEDERICO AND NEUMAN: FLOW IN MULTISCALE CONDUCTIVITY

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983

iograms and/orautocovariance andcrosscovariance functions of headandvelocityby meansof the formulaein thispaper. P(s) - 2(1+ H)(3+ 2H)n• Alternatively,one may use theseformulaeto calibratethe multiscaleflow modelnot only againstconductivity data but 2/-](3+ 2/•- 2/-](1+ 2I•e -r alsoagainst measurements of headandvelocity(asobtained, ß 8H(i+H)r say,from pointdilutionor othertracertests).One mayalso venture(we suspect with lesserpredictive power)to transfer (A6) suchpredictions fromonelocaleto another,withoutcalibration,by meansof parameters corresponding to a generalized whereEi(x) is the exponential-integral function log conductivity variogramas proposed by Di Federico and respectively,

1 8H(i+H) rr l e+2Hre -r-r2e -r+r2+2•F(1 - 2H, r)}

and E = 0.57721 is the Euler constant [Abramowitzand Neuman [1997]. 1972,pp.228-229].Thesecond of equations (22)yields 7. The multiscaleresultsin this paper are unconditional, Stegun,

but the geostatistical frameworkof Di FededcoandNeuman for [1997],andanystochastic flowanalysis baseduponit including the one in thispaper,are amenableto conditioning on mea-

two and three dimensions

surementsin the samemannerasone doesconditioningin the

standard single-mode case.However,onecannotdoconditioningwithoutfirstinferringthe unconditional statistics of the underlying randomfield, andwe proposethat thisbe done usingthe abovemultiscalegeostatistical framework.Even thoughit is highlydesireable to conditionstochastic flowso-

aQ(s) fos --=ds

s

aQ(s)

s'P(s') ds'

(A7)

s2 s'2p(s ') ds'

ds

(A8)

lutionson data,unconditionalsolutionssuchasthosewe offer

respectively. Substituting (A5) and (A6) into (A7) and (A8) in this paperservea very importantpurposein stochasticrespectively, and integratinggives[Gradshteyn and Rhyzik, subsurface hydrologythat has been recentlyarticulatedby 1994,equations (2.322.1)-(2.322.4), p. 112,equation(2.723.1),

Neuman [1997].

p. 248]

AppendixA: Evaluationof P and Q Functions

dQ(s) ds

The first of equations(22) yields for two- and threedimensional

cases

8(1 + H)2(2 + H)nl3

ß{2H(2 +H)[2(1 - E)+H(3 - 2E)]r - 12H(1 +H) 21 F

a?(s) ds

a?(s) ds

s'Cr(s') ds' l0s

(A1)

loS

(A2)

•

+ 12H(1 + H)2e-r+ F

s2 s'2Cr(s ') ds'

+ 2Hr2e-r- r3e-r + 4H(1 + H)(2 + H)r[Ei(-r)

ds

112] a?(s)

0'2y [ 1 e-r +2He -r-re -r+rl+2•F(1 - 2H, r)] dP(s) rr2v-4H ? -•1+4He -r+4He,2 -r +2He -r-re -r+rl+2•F(1 - 2H, r)1 ds

(3 + 2H)nl

(A9)

cry, 6(1 + H)(3 + 2H)(5 + 2H)nl3

1

ß -2H(3 + 2H)(5+ 2H)r- 48H(1+ H)(3+ 2H)r2

2(l+H)nl -2H-+2H-F

- In (r)]

+r3+2"r(1 - 2H, r)}

respectively. Substituting (10)-(11) with rn = l into these equations andintegrating (bypartsfor thesecond term)gives [Gradshteyn andRyzhik,1994,equations (2.322.1)-(2.322.2), p. dQ(s)

ds

12H(1 + H)2e-r- 2H(1 + 2H)re -r

F

(A3)

+ 48H(1+ H)(3+ 2H)•-+ 48H(1+ H)(3+ 2H) r + 12H(1 +H)(5+2H)

t

(A4)

+ 12H(1 +H)(1

+2H)e -r-6H

ß(1+2H)re -r+6Hr2e -r-3r3e -r+3r3+2•F(1 - 2H, r)} (AlO)

respectively, wherer = nlS.Subsequent integration yieldsfor

thesecond derivative of twoandthreedimensions [Gradshteyn andRhyzik,1994,equa- From(A9)-(A10)it iseasyto develop Q(s) whichis requiredto computethe headvariogram. tions(2.322.1),(2.322.2),and(2.325.1),pp. 112-113],

P(s)= 4(1+H)2n• {2H[2(1 +H)(1- E)- 1]

AppendixB: Log Conductivity-HeadCross

- 2H(1 + 2H)e -r + 2Hre-r- r2e-r + 4H(1 + H)

Covariance and Head Variogram Thelog-conductivity-head crosscovariance andheadvario-

ß[Ei(-r) - In (r)] + r2+2•F(1- 2H, r)}

gramare givenin two dimensions by

(A5)

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e-r

2Jjrj [-2H? 1+2He.2 -• Crh(S) H)rtl .= =2(1+cry, ;•1 e-r

+ 2H---

+ 12H(1+ H)(13+ 10H)•T+ 12H(!+ H)

e-r+r2nF(1 - 2H, r)1

O•,

(B1)

-6(1 +H)r2"F(1-2H, r)l}

2 2

')/h(S) =8(1+H)2(2 +H)n• • • JiJj

(B4)

Hereo-} = o-}(n•),r = n•s,ri = n•si,ri = n•si,•o isthe

ß

t=l

ß(1+2H)•2--12H(l+H) • +6(l+H)e -•

j=l

ß{15ij[ 4H(2 +H)(1 +H)(ln (r)- Ei(-r)) +2H(2 +H) 1

Kroneckerdelta, Ei(x) is the exponential-integral function, and E - 0.57721 is the Euler constant[Abramowitzand Stegun,1972,pp. 228-229].

ß(2E(1 + H) - (3H + 2)) + 12H(1 + H) 2-1.2

Appendix C: Log Conductivity, Velocity, and

- 12H(1+ H)2-•-- 12H(1+ H)2--

Crosscovarianceand autocovariance expressions of log conductivity,velocity,and head are givenin two dimensionsby

Head

Cross Covariance

and Autocovariance

tb 2(1+H) cr •,K•J 1 {-2H?+2H72-+2H-1 e-• e-• +2H(1 +2H)e -r-2Hre -r+r2e -r-r2H+2I'(1 - 2H, r)1 Cu,r(s)= + (1 + 2H)e -r __(1 -{-2H)r2U-2F(1 - 2H, r) +fir j -24H(1 +H)2 •-• +H)2e-r 1+24H(1 + • [4H -•- - 4H•r) e-•- 4H•-e-•- 2H•-e-r+ 2Hr2"2F(•-2H, /.4

e -r

1

+ 24H(1+ H)27x-+4H(2+ H)(1+ H) r2

(c•)

cr•.K•J rlr2[ 1 e-r e-r

+4H(1+2H)(1+H)•--4H(1 +H) r

Cu2r(S)d) 2(1+H) 4H•-4H74--4H .3 !

+2(1 +H)e -•-2(1 +H)r2UF(12H, r)]}

(B2)

e-r

J

(c2)

- 2H-•T+ 2Hr2U-2F(1 - 2H,r)

and in three dimensionsby

Culh(S ) --

3JjrjI-4H•I +4He.3-• Crh(S) (3+cry. 2H)n• .=

4(2 + H)(1 + H)n,

!

+ 4H • + 4H--

1

e-r+r2•rI'(1 --2H, r)] (B3)

O'•

e-•

e-•

ß -36H(1+ H) •4+ 36H(1+ H) 7r + 36H(1+ H) r3 1

e -r

e -r

+2H(2+H)•+2H(7+8H)•-+2H(l+2H)

r

- (1 + 2H)e-• + (1 + 2H)r2UF(1 - 2H, r)

3 3

•/•(s) - 6(1+H)(3+2H)(5 +2H)n• • • J•Jj i=1

•2[

([

1

e-r

1

1

e -r

e -r

ß 30 2H(3+ 2H)(5+ 2H)+ 48H(1+ H)(3+ 2H)1'3

- 4H(2+ H) •- 4H(4+ 5H)V

- 48H(1+ H)(3+ 2H)-•-- 48H(1+ H)(3+ 2H) r2

+2Hr22Hr2n2r(12H, r)l}

1

e-•

+ r 48H(1+ H) •- 48H(1+ H) 7g-- 48H(1+ H) rS

j=l

4H2r3

(C3)

e -r

- 12H(1 + H) (5 + 2H)

Cu•(S) =

12H(1 + H)(1 + 2H)

4(2 + H)(1 + H)n,

1

e-•

e-•

+6H(1 +2/•e -•- 6Hre -•+3r2e -•- 3r 2u+ 2F(1 - 2H, r)1

ß -12H(1+ H) • + 12H(1+ H) •-+ 12H(1+ H) r3

+ ri•)- 144H(1 + H)(3+ 2/• • + 144H(1 + H)(3+ 2/• -•-

+ 2H(2+ H) ? + 2H(1+ 2H)72-- 2H-

e -r

1

+ 144H(1+ H)(3+ 2H)-•-+ 12H(1+ H)(5+ 2H) .3

q[- e-r

-r2U(1-2H, r)+rl 248H(1 +H)•-48H(1 +H)e/.6-r 1

DI FEDERICO

NEUMAN:

1

e-r

- 48H(1+ H) •-e -r

AND

FLOW

IN MULTISCALE

e-r

4H(2+H)•-4H(4+5H) r4

e -r

CONDUCTIVITY

tr2vK3I 2

985

rlr2

Cu,u,(S)= +2 4(2+H)(1 +H)

2Hr2H-2F(1-2H, r)l} (C4) ß 144H(1+H)•-

-4H 2•+2H•-r-

FIELDS

144H(l+H)•1

CUlul(S)-'--

4>2 4(2 + H)(1 + H)

1

e-r

1

(1- 2H,r) '•-r- 2H(1+ 2H)•- + 2H(1+ 2H)r2•-2F

e -r

-- 2H(2+ H) • + 2H(11+ 10H)•-+ 2H(5+ 4H) r + [4H(2 + H) + 3]e-r- [4H(2 + H) + 3]r2"F(1 - 2H, r)

+r•2 -288H(l+H)•+288H(l+H)•+288H(l+H)

+r•2 288H(1 +H)•-288H(1 +H)•-g--288H(1 1

1

e-r

F7

e -r

e-r

e-r e-r

e -r

e -r

4H(2+ H) V- 4H(16+ 17H)•-- 4H(4+ 5H)

r5

ß -36H(1+ H) • + 36H(1+ H) •-+ 36H(1+ H) •

!44H(1+H)

e -r

16H(2+ H) • + 16H(7+ 8H)• + 16H(1+ 2H)

+ H)

ßrS 4H(1- H) •- + 4H(1- H)r2•-4F (1- 2H,r)

e -r

r5 8H(2+ H) r4 8H(!6+ !7H) r4 8H(4+ 5H)

(c7) and in three dimensionsby

e-r

4H(!+ 2H)• + 4H(1+ 2H)r2•-2F(1 - 2H,r)

rr2vK/ 1

Cgtl½8) --'•

+ r•4 -288H(1+H) ? + 288H(1 +H)•- + 288H(1 +H) e -r

1

3+ 2•

1 e-r e-r e-r

ß -4H•+4H•+4H•+2H--+2(1

e -r

r?+ 16H(2+ H) • + 16H(7+ 8H)•-+ 16H(1+2H)

+H)e-r

+ 2(1 + H)r2m-2F(1- 2H, r)

e-r

1

4H(1- H) •- + 4H(1- H)r2•-4r (1- 2H,r)

e-r

e-r e-r

+r2•12H•- 12H•- 12H•--6H .3 1

(C5) Cu,u,(S)=

{r•K•J 2

(C8)

- 2H•2-+ 2Hr2m-2F (1- 2H,r)

I

qb 2 4(2 + H)(1 + H)

e-r

ck 3+2H •--r - 12H rn tr2rK/ r•rk I12H•1- 12He - 6H•-•-+ - 2H,r)] e-r 2He -r 2Hr2H-2F(1

Cu½S) =

1

e-r

e-r

ß -12H(1+ H) • + 12H(1+ H) •-+ 12H(1+ H) r3 1

e -r

(C9)

e -r

+ 2H(2+ H) • + 2H(1+ 2H)•-- 2H r

Culh(S ) --

+ e-r- r2•q['(1- 2H, r) + (r2•+ r•)

I

1

e-r

ß 48H(l+H)•-48H(l+H)7a--48H(l+H) 1

e -r

er5-r

rl

4>

(3 + 2H)(5 + 2H)nl

ß -72H(3+2H)•+72H(3+2H)•-+72H(3 +2H) e -r

e -r

1

e -r

r4 + 2H(5+ 2H)•j + 2H(49+ 34H)r3

-- 4H(2+ H) •- 4H(4+ 5H)•-- 4H2r-•-

e-r

•r•Kd•'

I 22

+ 2H(13+ 10H)•-+ 4H(1+ H) -

+ 2H-•-- 2Hr2•-2F(1 - 2H,r) + rlr2

2(1+ H)e-r

+ 2(1 + H)r2mF(1- 2H, r) + r•2

ß -288H(1+ H) ? + 288H(1+ H) • + 288H(1+ H) e-r

1

ß 120H(3 + 2H)?- 120H(3 + 2H)•-•-- 120H(3

e -r

--+ 16H(2+ H) ?+16H(7+ 8H)• + 16H(1+ 2H)

e-r

F6

e-r

4H(1- H) •- + 4H(1- H)r2"-4r(1 - 2H,r) (C6)

1

+ 2H)

e -r

6H(5+ 2H)ff- 6H(25+ 18H)•-- 2H(15+ 14H)

e-r

F4

4H2• + 2H•2-- 2Hr2n-2F (1- 2H,r)

(C10)

986

DI FEDERICO

2

C.•n(s) =

2

cr•,KoJ

AND

NEUMAN:

FLOW

IN MULTISCALE

CONDUCTIVITY

-r

1

r,

e

qb (3 + 2H)(5 + 2H)nl

1

r4

e-r

ß -24H(3+ 2H)• + 24H(3+ 2H)•-+ 24H(3+ 2H) e -•

1

e-r

e-r

+ e-•- r2"F(1- 2/4,r) + (d + d)

r2

ß 120H(3 + 2/• •- 120H(3 + 2/•-•-e -•

+ 2H(1+ 2H)-•-- 2H r

+ e-r-- r2nF(1- 2H, r) + r•2

1

1

e -•

2H(15 + 14H)

r6 6H(5+ 2/• •- 6H(25 + 18/-/)• e-r

e -•

e-r

e-r

r4 4H2 •- + 2H-•--

r4 4H••- + 2H• --

2Hr2"-2F(1 - 2H, r)]}

(c•)

2Hr2"'r(1 - 2H, r)]+(•d)

1

e -•

ß-•-•-+30H(5+ 2H)• + 30H(37+ 26H)r7 e-r

Cu•u•(S) =

2H(15 + 14H)

ß -840H(3 + 2/• ? + 840H(3 + 2/•-•-+ 840H(3 + 2/4) e-•

e-r

120H(3 + 2/•

r6 6H(5+ 2/• • - 6H(25 + 18/-/) -•- e-r

ß 120H(3 + 2H)? - 120H(3 + 2H)• - 120H(3 + 2H)

e-r

e -r

2H(S+ 2H)• + 2H(•3+ •0H)•-+ 2H(•+ 2H)

e -•

r•- + 2H(5+ 2H)• + 2H(13+ 10H)r3

e -•

FIELDS

e-r

e-r

+ 10H(27 + 22/-/) •- + 10H(3 + 4H)-•- - 4H(1- H) r4 (k2

(3 + 2H)(5 + 2H)

ß -36H(3+ 2H)• + 36H(3+ 2H)•-+ 36H(3+ 2H) e -•

1

e -•

+4H(1 - H)r2H-4F(1 - 2H, r)]} o'•.K3] 2

Cu,u•(S)=

r4 H(5+ 2H)•j + H(59+ 38H)r•

(c•3)

2Hrlrl•

(k2 (3 + 2H)(5 + 2H) e-r

+ H(23+14H)•-+ H(7+4H) 7 + 2(I + H)(2+H)e-r

]

- 2(1 + H)(2 + H)r2nF(1 - 2H, r) + r•2

1

360H(3 + 2H)

e -r

- 3(5+ 4H)p 2(1+m)? + 2(•+m)r2"'•r(• - 2H,r)

e -r

6H(5+ 2/• • - 6H(85 + 58H)-•- - 6H(25 + 18H) e -r

e -r

-- 3(5+ 2H)•- 3(85+ 58H)7S--3(25+ 18H)r4

ß 360H(3 + 2H)•- 360H(3 + 2H)-•-e-r

ß 180(3+ 2H)?- 180(3+ 2H)-•-- 180(3+ 2H)

e -r

e -r

+r•2 -420(3+2H)?+420(3+2H)•+420(3+2H) e -r

r4 6H(5+ 4H)-•x--4H(1+ H) r2

1

e -•

r8+ 15(5+ 2H)• + 15(37+ 26H)r7

+4H(1 +H)r2U-2F(1 - 2H, r)1+r•

+ 5(27+ 22H)• + 5(3+ 4H)-•-- 2(1-H) r4

ß -420H(3+ 2H)• + 420H(3+ 2H) r9

+2(1-H)r2n-4F(1-2H, r)l}

e -r

e -r

e -r

(C14)

ßr•-+ 2H(1- H)r2n-4F (1- 2H,r)

cr}K•J 2

CU2U3(S) --

2Hr2r3

(k2 (3+2H)(5+2H)

e -r

'•-W+ 5H(27+ 22H)• + 5H(3+ 4H)-•-- 2H(1-H)

Cu•u•(S)=

e -r

1

+ 420H(3 + 2/• •-+ 15H(5 + 2/• ? + 15H(37 + 26H) e -r

e -r

(C12)

(k2 (3 + 2H)(5 + 2H)

ß -24H(3+ 2H)• + 24H(3+ 2H)-•-+ 24H(3+ 2H)

e-r

ß 60(3+ 2H)•- 60(3+ 2H)-•-- 60(3+ 2H) 1'6 1

e -r

e-r

- 3(5+ 2H)•- 3(25+ 18H)-•-- 3(15+ 14H)r4 r2n-2F(1- 2H, r) + r•2

ß -420(3+ 2H)?+ 420(3+ 2H)• + 420(3+ 2H)

DI FEDERICO AND NEUMAN: FLOW IN MULTISCALE CONDUCTIVITY

e-r

1

½-r

½-r

ßr•- + 5(3+ 4H)-•-- 2(1-H) r4

+2(1-H)r2H-4F(1-2H,

987

Liu, H. H., and F. J. Molz, Discriminationof fractionalBrownian movementand fractionalGaussiannoisestructuresin permeability andrelatedpropertydistribution with rangeanalysis, WaterResour. Res.,32(8), 2601-2605,1996. Mandelbrot,B. B., and J. W. Van Ness,FractionalBrownianmotions, fractionalnoisesand applications, SIAM Rev.,10, 422-437, 1968. Matheron, G., ElementsPour une TheoriedesMilieuxPoreux,166 pp.,

e-r

r8+ 15(5 + 2/•? + 15(37 + 26H)-?-+ 5(27 + 22/• ½-r

FIELDS

Masson et Cie, Paris, 1967.

(C15)

Molz, F. J., and G. K. Boman,Furtherevidenceof fractalstructurein

hydraulicconductivity distribution,Geophys. Res.Lett., 22(18), 2545-2548, 1995.

where k = 2 and 3.

Neuman,S. P., Universalscalingof hydraulicconductivities and dis-

persivities in geologic media,WaterResour. Res.,26(8), 1749-1758, 1990.

Acknowledgments. This work was conducted,in part, duringa sojournof thefirstauthorasFulbrightScholarin the Departmentof Hydrologyand Water Resources at The Universityof Arizonain Tucson.He thanksUSIA-CIES and the Fulbright ScholarshipPro-

Neuman,S. P., Generalizedscalingof permeabilities: Validation and effectof supportscale,Geophys. Res.Lett.,21(5), 349-352,1994. Neuman,S. P., On advectivetransportin fractalvelocityand perme-

abilityfields,WaterResour. Res.,31(6), 1455-1460,1995. gramfor partialsupportin theformof a travelgrant.Theworkwas Neuman,S. P., Stochastic approachto subsurface flowandtransport: alsosupported by the U.S. NuclearRegulatoryCommission under A viewto the future,in Subsurface Flowand Transport: A Stochastic contractNRC-04-95-038andby the ItalianMinisterodell'Universith e Approach,editedby G. Daganand S. P. Neuman,pp. 231-241, dellaRicercaScientifica e Tecnologica (MURST) 40% "Scalee definizioni dei parametrinello studiodellefalde."

CambridgeUniv. Press,New York, 1997. Neuman,S. P., and S. Orr, Predictionof steadystateflow in nonuni-

form geologicmediaby conditional moments: Exactnonlocalformalism,effectiveconductivities, andweak approximation,WaterResour.Res.,29(2), 341-364, 1993. Painter,S., Evidencefor non-Gaussianscalingbehaviorin heterogeneoussedimentary formations, WaterResour. Res.,32(5),1183-1195,

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