aquifer operator-scaling and the eﬁect on solute mixing...

WATER RESOURCES RESEARCH, VOL. ???, XXXX, DOI:10.1029/,

Aquifer Operator-Scaling and the Effect on Solute Mixing andDispersion

David A. BensonDesert Research Institute, Reno, Nevada, USA(now at Colorado School of Mines, Golden, CO, USA)

Mark M. MeerschaertUniversity of Otago, Dunedin, New Zealand

Boris BaeumerUniversity of Otago, Dunedin, New Zealand

Hans-Peter SchefflerUniversity of Nevada, Reno, Nevada, USA

Abstract.The creation of representative aquifer facsimiles is a fundamental building block of Monte

Carlo stochastic hydrology. This process hinges on the identification and re–creation ofthe statistical properties (distribution and dependence structure) of aquifer character-istics such as hydraulic conductivity (K). Since aquifer parameters may have dependencestructures, such as correlation, that are present across a huge range of scales, the con-cepts of fractional Brownian motion (fBm) have been explored previously in both an-alytic and numerical stochastic settings. Most previous models have used isotropic scal-ing, in which the self-similarity properties do not vary with direction and are describedby a scalar Hurst coefficient. Any real-world anisotropy in the degree of statistical de-pendence has been handled by an elliptical stretching of the scalar random K field. Wedefine an extension of fBm to d−dimensions in which the fractional-order integration infBm may take on different orders in the d primary scaling directions (which do not haveto be orthogonal) and the degree of connectivity and long-range dependence is freely as-signed according to a discrete or continuous measure on the unit sphere. This approachaccounts for the different scaling found in the vertical versus horizontal in real granu-lar aquifers, and allows very general degrees of continuity of K in certain directions. Takento an end member, it also allows for the direct representation of fracture networks ina continuum setting: the eigenvectors of the scaling matrix describe the primary frac-ture scaling directions, and discrete weights of fractional integration represent fracturecontinuity that may be limited to a small number of directions. In a numerical exper-iment simulating a granular aquifer, the motion of solutes through these “operator-fractional”Gaussian fields depends very much on both the longitudinal and the transverse scaling(Hurst) coefficients. Lower integration order in the transverse direction (in the range offractional Gaussian noise) engenders greater mixing and a transition to a Fickian regime,contrary to previous stochastic analytic results. Higher orders of integration, in the rangeof fractional Brownian motion, are associated with thicker layering of aquifer sedimentsand more preferential, non-mixing flow and transport. Therefore, the methods in thispaper, which account for the (possibly) unique scaling properties of an aquifer in all di-rections, are important for realistic Monte Carlo simulation of aquifers with long-rangestatistical dependence.

1. Introduction

Predicting the movement of dissolved chemicals throughreal-world aquifer material has been a priority of hydrogeol-ogists for decades. In the 1970s, an understanding evolvedthat no aquifer could be completely characterized [Freeze,

Copyright 2005 by the American Geophysical Union.0043-1397 /05/$9.00

1975; Gelhar et al., 1979; Smith and Schwartz, 1980]. Un-certain aquifer properties, at every unsampled point, mustbe represented by some statistical model. Two interre-lated methods evolved to model the effect of the uncer-tain aquifer, using either analytic or numerical methods.The analytic method replaces the deterministic advection–dispersion equation with a stochastic version and solves forthe moments of the head and concentration fields [see theextensive review by Gelhar, 1993]. The numerical approachuses classical Monte Carlo methods (e.g., Tompson and Gel-har, 1990), and more recently, Bayesian and other condition-ing methods [Woodbury and Ulrych, 2000; Feyen et al., 2002;

1

X - 2 BENSON ET AL.: OPERATOR-FRACTIONAL FIELDS

Morse et al., 2003; Huang et al., 2004], to simulate a largenumber of aquifer realizations (some more likely than oth-ers), and either analyzes the ensemble statistics of the headand concentration fields or solves the moment equationsthemselves [Graham and Mclaughlin, 1989; Li and Mclaugh-lin, 1991; Li et al., 2003, 2004; Guadangnini and Neuman,1999a,b; 2001; Ye et al., 2004]. A number of techniquesare widely used to generate scalar random field replicas ofaquifers for use in the Monte Carlo studies [Tompson et al.,1989; Deutsch and Journel, 1992; Dietrich and Newsam,1993; Robin et al., 1993; Carle and Fogg, 1997; Bruininget al., 1997; Hassan et al., 1997; Ruan and McGlaughlin,1998; Painter, 2001; Lu et al., 2003].

Both the analytic and numerical methods have enjoyedgreat success for various subsets of the variety of aquifertypes. The strength of the analytic method lies in thesimplicity of representing the aquifer statistical characteris-tics, but is burdened with assumptions needed to gain well-posedness (closure) of the stochastic equations (see a recentdiscussion by Wood et al. [2003]). The strength of the nu-merical method lies in the robustness of each deterministicsolution, the ability to condition with data from specific lo-cations (e.g., Dietrich and Newsam [1996]), and the abilityto incorporate any number of uncertain parameters [Burret al., 1994; Cushman et al., 1995; Hu et al., 1995; Huangand Hu, 2001]. Furthermore, the rapid and predictable ad-vancement of computer speed and addressable memory al-lows the representation of aquifer features from many scales.The downside of the numerical method, aside from the com-putational time required, is that the aquifer facsimiles mustbe created according to some defined statistical model thatis typically an oversimplification. The chosen model shouldadhere to the real characteristics of the aquifer, includingthe statistical dependence structures that might be presentacross all scales [Neuman, 1995; Molz and Boman, 1993;Molz et al., 1997].

One early statistical model that parsimoniously accountsfor evolving heterogeneity at all scales is a fractional Brow-nian motion (fBm), since this stochastic process has theproperty of self-similarity [Mandelbrot and van Ness, 1968].However, true multidimensional fBm is an overly restrictivemodel for several reasons. First, non–Gaussian incrementsof K or ln(K) may be the rule, not the exception. Thedistribution of K or ln(K) is typically heavier tailed thana Gaussian [Eggleston et al., 1996; Painter and Paterson,1994; Painter, 1996, 2001; Rasmussen et al., 1993; Liu andMolz, 1996, 1997; Benson et al., 2001; Hyun and Neuman,2003; Meerschaert et al., 2004; Aban et al., 2005]. Sincesimple fBm can replicate the long-range K dependence thatcan account for continual faster-than-Fickian growth rate ofplumes [e.g., Kemblowski and Wen, 1993; Neuman, 1995;Rajaram and Gelhar, 1995; Zhan and Wheatcraft, 1996;Bellin et al., 1996; Cushman, 1997; Hassan et al., 1997; DiFederico and Neuman, 1998; Painter and Mahinthakumar,1999], it is still the basis of several simulation techniquesthat have relaxed the restrictive assumption of Gaussian in-crements [Painter, 2001; Meerschaert et al., 2004]. Second,a more serious drawback of simple fBm is its functional, orscaling isotropy, in which the self-similarity parameter is thesame in all directions [Bonami and Estrade, 2003]. There isno physical reason that a sedimentary aquifer should havethe same scaling properties in the vertical and horizontaldirections. Evidence for different scaling in different direc-tions at several sites is presented by Hewitt [1986], Molz andBoman [1993], Liu et al. [1997], Deshpande et al., [1997],Tennekoon et al. [2003], and Castle et al. [2004], amongothers. In this paper we provide a brief re-analysis of thedata collected by the latter researchers. Techniques havebeen developed that can handle this anisotropy in the scal-ing or the Hurst coefficients [Dobrushin, 1978; Hudson andMason, 1982; Schertzer and Lovejoy, 1985, 1987; Kumar and

Foufoula-Georgiou, 1993; Mason and Xiao, 2001; Bonamiand Estrade, 2003] yet the mathematical character of nu-merical implementations is still under investigation, sincemany of the properties of these fields have only recentlybeen worked out in one-dimension (1-D) [Pipiras and Taqqu,2000, 2003].

A third complication is that aquifers may possess strongdirectionality in the K structure. Continuity within high-Kunits may be restricted to a small subset of the unit cir-cle in 2-D or the unit sphere in 3-D. For example, braidedstreams deposit gravel in continuous channels with distinctpreferential directions associated with the mean transportdirection (Fig. 1). One can easily calculate the propor-tion of the channels that lie in any angular interval dθ andconstruct a histogram of the stream channel directions (Fig.2). If the high-K portions of an aquifer are deposited withinthese channels, then we presume that the continuity of theaquifer K should follow the same directional control. Forreasons that will become clear in subsequent derivations, wedescribe the proportion of the stream channel lying in anydirection by the so-called mixing measure, or weight func-tion, M(θ). The histogram of directions (Fig. 2) shows thestrong preference for flow, hence deposition of similar ma-terial, in the direction parallel to the bottom edge of thephotograph (0◦/180◦) and almost none in the ±90◦ direc-tions. The measure of directional dependence M(θ) is spe-cific to any given site and should be freely assigned within

Figure 1. Braided stream channels from above.

45

90

135

180

225

270

315

0.02 0.04 0.06 0.080.12

0.1

Figure 2. Approximate measure M(θ) of the directionalproportion of the wet portions in Fig. 1. Angles are indegrees.

BENSON ET AL.: OPERATOR-FRACTIONAL FIELDS X - 3

a statistical model of that site. In this paper we constructrandom fields based on a convolutional formula that allowsa general degree of statistical dependence in any given direc-tion. In this example, only the direction of the channel wasmeasured; therefore, the measure in Figure 2 is symmetricwith the addition of 180◦. When measuring actual K values,the dependence may be non-symmetric with respect to thedownstream direction, since a disturbance in the water flowwould be expected to deposit a fining-downstream sequenceof gravel and sand. For supercritical flow, no random distur-bance should be reflected in the sediments in the upstreamdirection at all. We investigate the influence of directional“causality” in a subsequent section.

Aside from the deposition of sediment grains of varioussizes, aquifers may gain permeability structure through de-formation. This is particularly important in low permeabil-ity rock, which may acquire fractures that convey essen-tially all of the groundwater flow. This scenario takes onincreased importance, since most of the nations that pro-duce high-level nuclear waste plan to store it within frac-tured, low permeability host rock (hence the numerous stud-ies of Yucca Mountain in the United States [Bodvarsson andTsang, 1999; Bodvarsson, 2003], and the Aspo repository inSweden [SKB, 2001]). Studies of the suitability of these sites[e.g., Outters and Shuttle, 2000; Stigsson et al., 2000; BSC2004a,b] have endeavored to map the locations of all impor-tant fractures and faults that intersect tunnels and drifts.But the final buildout of the repositories will uncover mul-titudes of new, as yet unknown fractures, and the regions ofrock between the repository and potential receptors will al-ways remain uncharacterized. Some sort of statistical repre-sentation of the aquifer (fracture) properties is still needed tosupplement the mapping information [SKB, 2001, Chapter8]. Real fracture networks are almost universally observed topossess fractal structure (see the extensive review by Bonnetet al. [2001]). Permeability measurements in fractured rockalso show scale effects and may support fractal models [Hyunet al., 2002; Neuman and Di Federico, 2003; Martinez-Landaand Carrera, 2005]. The specifics of the more subtle fracturestatistics have a large bearing on the transport characteris-tics of the networks [Bour and Davy, 1998; Bour et al., 2003;Darcel et al., 2003a,b; deDreuzy et al., 2004].

When simulating flow through fracture networks, sev-eral approaches can be adopted (see the recent review byNeuman, 2005a). The first treats the network, or a subsetof the network (e.g., BSC, 2004a; Carrera and Martinez-Landa, 2000), as an equivalent porous medium (EPM), andderives or measures an effective upscaled K tensor based onthe contributions of “subgrid” fractures. The EPM is jus-tified in highly fractured media where the fracture spacingis small compared to some region of interest, say a numer-ical grid block or the distance between observation points[Snow, 1969; Long et al., 1982; Endo et al., 1984; Hsiehand Neuman, 1985; Hsieh et al., 1985; Neuman et al., 1984;Neuman, 2005b]. However, when the fracture spacing islarge compared to the region of interest, then an effectiveparameter does not reasonably represent the subgrid fea-tures, and individual fractures (and perhaps as importantlythe intervening impermeable rock) may need to be explicitlyrepresented. This can be done using discrete fracture net-works [Adler and Thovert, 1999; Benke and Painter, 2003;Cvetkovic et al., 2004], or by accepting as an approximationa continuous random field, preferably with conditioning onthe locations of fractures. One must accept that a continu-ous random field might assign some regions of impermeablerock a positive-definite K tensor, which assumes sufficientlydense subgrid fractures. In the case of fractal networks,there may not be a convenient separation of scales such thatindividual fractures can be ignored in favor of an equiva-lent porous medium [Long, et al., 1982]. In this paper, weshow that a random field, based on an extension of fBm, can

represent fractal fracture permeability and preserve fractureorientations and unique scaling properties in different direc-tions. As a result, no separation of scales is invoked todivide the domain into large mappable fractures and sub-grid features that constitute an equivalent porous medium.Furthermore, if fractures orientations are concentrated in afew directions, the generation of a continuous random fieldshould be able to restrict the continuity of high-K areasto those directions, as opposed to the more isotropic priordefinitions of fBm.

To expand the capabilities of current generators of con-tinuous random fields including classical fBm, we intro-duce operator-scaling scalar random field generators basedon multidimensional fractional integration. The randomfields have “operator scaling” that is described by a ma-trix of values—the linear operator—instead of a single scalarHurst coefficient. The matrix-valued, or operator rescal-ing properties mean that the scaling indices (for example,Hurst coefficients for Gaussian fields) can be different ineach coordinate. For maximal generality, we choose a frac-tional integration based on the inverse of the recently definedanisotropic fractional Laplacian [Meerschaert et al., 2001].The fields we describe in this paper allow all of the following:1) self-similar structure at all scales, particularly when thescaling rates are unique in different directions, 2) arbitrarystrength of statistical dependence along discrete directions,and 3) generality in choosing the statistical distribution ofthe scalar random field, including non-stationarity. Finally,in Section 3, we investigate the influence of simple operator-scaling K structure in synthetic aquifers on the transportof solutes and compare the results to previous stochasticanalytic predictions.

2. Mathematical Background

The first d–dimensional fBm BH(x) was defined as anisotropic special case of the moving average

Bϕ(x) =

Z[ϕ(x− y)− ϕ(−y)] B(dy) (1)

with B(dy) the increment of a Brownian random field,ϕ(y) = ‖y‖H−d/2, and Hurst index 0 < H < 1 [Samorod-nitsky and Taqqu, 1994; Cushman, 1997]. The first termin the integral is the convolution kernel, which mathemat-ically spreads the random value at each point over space.The second term centers the motion so that Bϕ(0) = 0.The stochastic integral I(f) =

Rf(y)B(dy) is the distribu-

tional limit of approximating sumsP

j f(yj)B(∆yj) where∆yj is a small d-dimensional rectangle, yj is a point insidethat rectangle, B(∆yj) is a normal random variable withmean zero and variance equal to the volume of that rectan-gle, and these normal random variables are independent fordisjoint rectangles. A finite sum gives a faithful approxima-tion so long as the area covered spreads over entire spaceas the mesh of the partition tends to zero. The integralI(f) is a mean zero normal random variable with varianceproportional to

Rf(y)2dy, and the integral exists so long

asR

f(y)2dy < ∞ [e.g., Yaglom, 1987]. In this case wesay that the function f(y) is square integrable. Since thevolume of c∆yj is cd times the volume of ∆yj , and usingthe formula VAR(aZ) = a2VAR(Z) for random variables Z,B(c dy) = cd/2B(dy) in distribution for any scale c > 0, andthen it follows immediately from (1) that an isotropic frac-tional Brownian field is self-similar: BH(cx) = cHBH(x) indistribution. The restriction 0 < H < 1 comes from therequirement that the integrand in (1) is square-integrable.Fractional Gaussian noise is a stationary discrete param-eter process obtained by taking differences of a fractional


Brownian field, GH(j) = BH(j)−BH(j− 1) in 1-dimensionor the corresponding analogue in d-dimensions: GH(i, j) =BH(i, j)− BH(i, j − 1)− BH(i− 1, j) + BH(i− 1, j − 1) in2-dimensions and so forth. The noise process inherits thesame scaling: GH(cx) = cHGH(x).

The spectral representation of the Brownian random field(1) is given by

Bϕ(x) =

Z heik·x − 1

iϕ(k)B(dk) (2)

where ϕ(k) = (2π)−d/2R

e−ik·yϕ(y)dy. In the isotropiccase, ϕ(k) = CH‖k‖−H−d/2 and CH > 0 is a constantdepending only on H and d. The stochastic integralR

g(k)B(dk) is the distributional limit of approximating

sumsP

` g(k`)B(∆k`) where ∆k` is a small d-dimensionalrectangle in wave space, and the complex-valued randommeasure B(∆k`) = m1(∆k`) + im2(∆k`). The real andcomplex parts m1(∆k`) and m2(∆k`) are independent nor-mal random variables with mean zero and variance equalto half the volume of the rectangle ∆k`, independent fordisjoint rectangles, except that m1(∆k`) = m1(−∆k`) andm2(∆k`) = −m2(−∆k`) [Samorodnitsky and Taqqu, 1994].The random measure B is the Fourier transform of the ran-dom measure B in the sense that

B(∆k`) ≈ 1√N

Xj

e−ik`·yj B(∆yj) (3)

where N is the total number of grid points, the distributionalapproximation becoming exact as the mesh of the partitiontends to zero. The conditions on m1 and m2 follow from(3), since this formula implies that the complex conjugateof B(−∆k`) equals B(∆k`). Then both the integrand andthe increment process in (2) are Fourier transforms of therespective terms in (1). The restriction 0 < H < 1 is alsoshown in the requirement that the integrand in (2) is square-integrable, and the distributional equality of the two forms(1) and (2) is a result of the Parseval identity (compareProposition 7.2.7 in Samorodnitsky and Taqqu, [1994]).

The fractional Brownian field (1) can be simulated usingdiscrete Fourier transforms. Using the spectral representa-tion (2) we approximate Bϕ(x) ≈ Bϕ(x)− Bϕ(0) where

Bϕ(x) =X

eik`·xϕ(k`)B(∆k`) (4)

which is evidently the inverse discrete Fourier transform ofthe product ϕ(k`)B(∆k`). Then in order to simulate Bϕ wecan begin by simulating B(∆yj) as iid random variables ona regular grid yj , take discrete Fourier transforms to obtainB(∆k`) on a regular grid k`, multiply by the Fourier filterϕ(k`), and invert the discrete Fourier transform. Then wesubtract the result at grid point x = yj from the result atgrid point x = 0 to obtain Bϕ(x) ≈ Bϕ(x) − Bϕ(0). Thediscrete Fourier (and inverse Fourier) transforms can be ef-ficiently computed using fast Fourier transforms. While wehave described this simulation procedure for Gaussian ran-dom fields, a variety of stochastic integrals can be used,allowing a flexible and realistic tool for simulating K fields.For example, ln(K) increments at several field sites arewell modelled by a Laplace distribution [Meerschaert et al.,2004], which is essentially a double-sided exponential. Adifferent method for simulating only Gaussian processes,starting in Fourier space [Voss, 1989; Peitgen and Saupe,1988; Gelhar, 1993; Ruan and McGlaughlin, 1998; Diekerand Mandjes, 2003] will not be pursued here.

2.1. Fractional Integration

Although the stochastic (convolution) integral

Bϕ(x) =

Zϕ(x− y)B(dy) (5)

does not exist because the function ϕ(x−y) is not square in-tegrable, we can heuristically write Bϕ(x) = Bϕ(x)−Bϕ(0),the convergent difference of two divergent stochastic inte-grals. Since the form (5) represents a stationary process,the theory of linear filters can be applied, and it turns outthat these filters correspond to certain fractional integrals.For a simple example, consider fractional Brownian motion(fBm) in one dimension. Mandelbrot and van Ness [1968] de-fined a one dimensional fractional Brownian motion (fBm)using

BH(x) =1

Γ(H + 1/2)

Z x

−∞(x− y)H−1/2B(dy). (6)

In this formulation, the stochastic integral depends only onthe values of B(dy) for y < x (the past, if x represents time)and hence is termed “causal.” When x represents a spatialvariable, this restriction may not be needed, and Mandel-brot [1982] introduced a symmetric, bilateral version of fBmusing

BH(x) =1

Γ(H + 1/2)

Z ∞

−∞|x− y|H−1/2B(dy). (7)

The convolutions in (6) and (7) take values of B(dy) andspread them according to the respective power-law kernels.From the point of view of fractional calculus (see Samko, etal., 1993 for a comprehensive exposition), these integralscan be viewed as fractional-order integrals of the “whitenoise” B(dy) [Meyer et al., 1999, Pipiras and Taqqu, 2000;Chechkin and Gonchar, 2001]. Fractional derivatives andintegrals are natural extensions of their integer-order coun-terparts, easily understood in terms of their Fourier trans-forms. Given a function f(x) with Fourier transform f(k)and compact support (i.e., f(x) = 0 for |x| large), it is wellknown that the nth derivative of f(x) has Fourier trans-form (ik)nf(k), and hence the n-fold integral has Fouriertransform (ik)−nf(k). Fractional derivatives and integralsreplace the integer exponent n with a rational numberα. A computation using generalized functions shows thatΓ(−α)(ik)α is the Fourier transform of the function x−1−α

on x > 0 [Samko et al., 1993]. Hence the α-order fractionalderivative of f(x) is the convolution of f(x) with a powerlaw:

dαf(x)

dxα=

1

Γ(−α)

Z x

−∞(x− y)−α−1f(y)dy (8)

where this integral is to be understood in terms of gener-alized functions. If α < 0 this formula defines a fractionalintegral of order −α. Now the one-sided fBm formula (6)can be viewed as an A = H + 1/2 fractional integral ofwhite noise. One also defines a negative–direction fractionalderivative

dαf(x)

d(−x)α=

1

Γ(−α)

Z ∞

x

(y − x)−α−1f(y)dy (9)

and then the two-sided fBm formula (7) is seen as a combina-tion of positive– and negative–direction fractional integralsof white noise with the same order A = H + 1/2. Since0 < H < 1, this means that heuristically fBm can be writ-ten BH(x) = BH(x)− BH(0) where BH(x) is a white noisefractional integral of order 1/2 < A < 3/2. One-sided fBmcan be simulated using the Fourier filter ϕ(k) = (ik)−A in(4). Two-sided fBm results from using the equally weightedforward and backward fractional integrals, hence the sym-metric filter ϕ(k) = (ik)−A + (−ik)−A = 2 cos(Aπ/2)|k|−A


where again A = H + 1/2 is the order of fractional integra-tion. Using the filter ϕ(k) = |k|−A multiplies the resultingfBm by a constant.

We note here that the causal function (6) was originallyintended for time series, since the present cannot influencethe past. A similar treatment, in multiple dimensions, mustbe considered for the deposition of sediment in supercriti-cal flow, since a disturbance in the flow is not propagatedupstream. Therefore, a convolution kernel that mathemat-ically transmits fluctuations in the K field should not nec-essarily have equal (or any) weight in certain directions.In 1–dimension, a simple extension of the symmetric ker-nel in formula (7) incorporates a weight function M(θ),where θ = ±1: ϕ(k) = M(1)(ik)−A+M(−1)(−ik)−A, whereagain, A = H + 1/2 is the order of fractional integration.This concept of weights on the unit circle extends to 2–and 3–dimensions (below). It is also worth noting here thatthe one–sided and symmetric fBm are models of differentprocesses and have noticeably different traces, even thoughthe correlation structure of the increments (i.e., the spectraldensity functions) are identical.

Fractional Gaussian noise (fGn) GH(x) = BH(x) −BH(x − 1) is the increment process of fractional Brow-nian motion, hence we can simulate one-sided fGn usingthe Fourier filter ϕ(k) = [1 − e−ik](ik)−H−1/2 in (4). Us-ing a Taylor series approximation we can simplify this toϕ(k) = (ik)−H+1/2. Similarly we can simulate two-sidedfGn using the filter ϕ(k) = |k|−A where now A = H − 1/2.This extends the simulation procedure through the interval−1/2 < A < 1/2, where again A indicates the order of frac-tional integration. For simplicity, we will continue to usethis notation, since using H becomes somewhat confusingwhen referring to both fGn and fBm.

Symmetric fBm in one dimension (1-D) is a special caseof an isotropic Brownian random field. The d-dimensionalisotropic version can be simulated using the Fourier filterϕ(k) = ‖k‖−H−d/2 in (4), which corresponds to (1) withϕ(x) = C‖x‖H−d/2 for some constant C depending only onH and the dimension d of the variable x. A stretching of therandom field, resulting in greater strength or weight of corre-lation of the increments in some direction, but still decayingwith the same power law (i.e., the same Hurst coefficient inall directions), is achieved by a scalar multiplication of theorthogonal vector components in the Fourier filter [Rajaramand Gelhar, 1995; Zhan and Wheatcraft, 1996; Di Federicoet al., 1999]:

ϕ(k) = |λ · k|−A = ((λ1k1)2+(λ2k2)

2+(λ3k3)2)−A/2. (10)

This approach does not address the possibility that the de-cay of statistical dependence in any direction may fall offaccording to a different power law. Hence, in contrast to pre-vious researchers, we call this stretched version with scalarH “functionally isotropic,” since the power law is the sameexcept for a prefactor, in all directions. Real aquifers mayshow different fractal scaling properties in the horizontal di-rection than the vertical [Hewitt, 1986; Molz and Boman,1993; Liu and Molz, 1996]. Other aquifer characteristicsthat may serve as surrogates to K, such as sonic velocity,may also show differences in the scaling exponent in the di-rection of strike versus dip [Deshpande et al., 1997]. Molz etal. [1997] suggest a scheme to change the order of integra-tion in one direction via an expression similar to:

ϕ(k) = ((λ1k1)2 + (λ2k2)

2 + |λ3k3|2β)−A/2. (11)

In a subsequent section, we investigate the scaling proper-ties of the random field generated by this and more generalanisotropically scaling filters.

Finally, in many depositional environments, the degreeof dependence and connectivity of high-K units may havestrong directionality and/or causality. If the stretched

isotropic filter (10) is represented in polar coordinatesϕ(r, θ) = M(θ)r−A, then the set of weights is a smoothellipse with axes lengths given by λ−A

i , clearly dissimilarto the shape measured in Figure 2. Even more compellingexamples are given by fractured aquifers, which may havefractures restricted to a small number of preferred orienta-tions in 3-D. In the directions transverse to fracture sets,there may or may not be any correlation in the fracturespacing or point K values. In the fracture planes, however,the connectivity and long-range dependence of K values willbe much larger. Only upon upscaling of sufficiently densefractures [Snow, 1964, 1969; Long et al., 1982] can an equiv-alent porous medium with a K tensor be defined. In thecase of fractal fracture networks, it is unclear whether aseparation of scales exists that allows upscaling to an equiv-alent porous medium with the typical K tensor. Upon anyupscaling, the flow may be dominated by just a few majorfractures brought into the observation window.

The isotropically scaling filter ϕ(k) = ‖k‖−A cor-responds to multivariable fractional integration of or-der A, since multiplication by ‖k‖A in Fourier space isequivalent to taking the fractional Laplacian ∆A/2 =�∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2

�A/2in real space [Bojdecki and

Gorostiza, 1999]. Similarly, the filter (10) corresponds to the

inverse of the operator�λ2

1∂2/∂x2 + λ2

2∂2/∂y2 + λ2

3∂2/∂z2

�A/2

and (11) to the inverse of�λ2

1∂2/∂x2 + λ2

2∂2/∂y2 + λ2

3∂β/∂|z|β�A/2

.These are three special cases of the general multivariablefractional derivative operator recently introduced by Meer-schaert, et al. [2001] to model anomalous diffusion, andapplied to solute transport by Schumer, et al. [2003a]. Thegeneral form of this operator allows the order of derivative,as well as the weight M(θ) that determines the strength, orprefactor, of the derivative, to vary with coordinate.

3. Analytic Predictions of Solute Transportin Monofractal K Fields

Kemblowski and Wen [1993], Neuman [1995], Rajaramand Gelhar [1995], Zhan and Wheatcraft [1996], Bellin et al.[1996], and Di Federico and Neuman [1998] predict the rateof plume spreading through a fractal (typically fBm) K fieldbased on a single value of H in the direction of transport.Our numerical experiments (Section 5) are most closelyaligned with the assumptions of Neuman [1995], Rajaramand Gelhar [1995], and Di Federico and Neuman [1998], sowe focus here on their results. Neuman [1995] and Di Fed-erico and Neuman [1998] assume purely advective flow sothat each particle stays within a streamtube whose velocityautocorrelation function then becomes a scaled copy of thethe K correlation function in the direction of transport. Thisassumption is based on low variance of the K field, since theflow will bypass low K zones and focus through high-K zonesto a greater degree as the variance grows, thereby allowingthe solute particles to subsample the random K field.

Making three additional and distinct assumptions, DiFederico and Neuman [1998] predict that the plume willgrow either in an early pre-asymptotic (pre-ergodic), asymp-totic (ergodic), or permanently pre-asympototic state.Their results depend primarily on the average travel dis-tance of a plume relative to a large-scale (low-frequency)fractal cutoff in either the K field or the plume itself. Anearly preasymptotic plume may follow totally “unmixed”ballistic motion, similar to the stratified model described byMercado [1967]. This earliest stage is characterized by lineargrowth of the ensemble particle trajectory standard devia-tion and macrodispersivity versus travel distance (aL ∝ X).An asymptotic plume occurs when the plume has travelled


a greater distance than the scale of the largest heterogene-ity present. If the particles stop experiencing new scales ofheterogeneity (either through truncation of the K field het-erogeneity or by subsampling do to advective mixing) theplume will transition to an asymptotic, or Fickian, growthrate (aL = constant). Finally, Di Federico and Neuman[1998] reason that a plume travelling in fractal material iscontinually sampling larger scales of heterogeneity that are,on average, the same size as the growing mean travel dis-tance. Such a plume is permanently pre-asymptotic and hasa longitudinal macrodispersivity (aL) that grows accordingto [Di Federico and Neuman, 1998]:

aL ≡ 1

2

d

dXVAR(X) ∝ X1+2H , (12)

where VAR(X) is the variance of the particle travel dis-tance, X is the mean travel distance, and H refers to theHurst coefficient in the direction of travel.

If one assumes that particles are free to roam a domainwith a fixed large-scale fractal cutoff (for example, a fi-nite domain), then at early time the mean travel distancemay be much smaller than both the possible particle tra-jectories (due to the large heterogeneity of the flowpathvelocities), and the large-scale K cutoff. Furthermore, astime progresses, and a plume grows around its mean po-sition, the mean distance becomes smaller relative to theincreasing proportion of large excursions, which dominatethe plume variance. Hence the mean position may be muchsmaller than either the plume or K large-scale cutoff and theplume may be approximately “early preasymptotic”. In anycase (early pre-asymptotic, permanently pre-asymptotic, orasymptotic), Di Federico and Neuman [1998] predict theplume dynamics based only on the Hurst coefficient in thedirection of travel.

Rajaram and Gelhar [1995] use a two-particle, relativedispersion approach to calculate the macrodispersivity ina plume-specific (not ensemble) manner that correspondsmost closely to Di Federico and Neuman’s [1998] perma-nently pre-asymptotic results. Rajaram and Gelhar [1995]compute the macrodispersivity in an fBm K field with asingle value of H as:

aL ∝ XH (13)

The common feature of the analytic results on transportthrough fractal porous media is the isotropic scaling accord-ing to a single scaling parameter H. Liu and Molz [1996]posit that a different H in the direction transverse to flowmay influence transport, since lateral mixing is determinedby the layered structure of the aquifer material. Larger val-ues of A or H correspond to greater persistence of the ran-dom increments, hence thicker layering of similar material.The thicker, more extensive layers should contribute to lessmixing and more Mercado-type [1967] (ballistic) dispersion.Less extensive layering, characteristic of smaller values ofA or H, should promote more mixing and perhaps moreFickian-type dispersion. In Section 5 we investigate Liu andMolz’ [1996] forecast that knowledge of H in the directionof transport alone is inadequate to predict plume growth.

4. Operator-Scaling Random Fields

A functionally isotropic fractional Brownian random fieldBH(x) in d-dimensions results from applying the Fourier fil-ter ϕ(k) = ‖k‖−A to a Gaussian white noise in the spectralrepresentation (2), where the scalar exponent A = H + d/2is used. Using the scaling relations ϕ(ck) = c−Aϕ(k) and

B(c dk) = cd/2B(dk) we get

BH(cx) =

Z heick·x − 1

iϕ(k)B(dk)

=

Z heik·x − 1

iϕ(c−1k)B(c−1dk)

=

Z heik·x − 1

icAϕ(k)c−d/2B(dk)

and the last integral equals cHBH(x) since H = A − d/2.Hence the scaling property BH(cx) = cHBH(x) also followsdirectly from the scaling of the Fourier filter. The stretchedFourier filter in (10) also satisfies ϕ(ck) = c−Aϕ(k), hencethe process (2) with this filter has the same scaling propertyBϕ(cx) = cHBϕ(x) as a fractional Brownian field.

The Fourier filter (11) proposed by Molz et al. [1997] hasa more complicated scaling ϕ(ck1, ck2, c

1/βk3) = c−Aϕ(k)most conveniently handled using matrix powers [see, e.g.,Section 2.2 of Meerschaert and Scheffler, 2001]. The matrixpower cQ is defined by a Taylor series

cQ = exp(Q ln c) = I + Q ln c +(Q ln c)2

2!+

(Q ln c)3

3!+ · · ·

where I is the d × d identity matrix. Using this nota-tion, the scaling for the Fourier filter (11) can be writtenas ϕ(cQk) = c−Aϕ(k) where Q−1 = diag(1, 1, β). In thisrespect, we see that the matrix Q denotes deviations fromthe overall order of integration A. Then the random field ofMolz et al. [1997] satisfies

Bϕ(cQx) =

Z heicQk·x − 1

iϕ(k)B(dk)

=

Z heik·x − 1

iϕ(c−Qk)B(c−Qdk)

=

Z heik·x − 1

icAϕ(k)|c−Q| 12 B(dk)

(14)

where |c−Q| = c−1−1−1/β is the determinant of the matrixc−Q. Hence this random field satisfies the scaling relation

Bϕ(cQx) = cA−1− 1

2β Bϕ(x). Recalling that A = H +3/2 fora 3-dimensional field, we can also write this in the form

Bϕ(cx1, cx2, c1/βx3) = c

H+ 12− 1

2β Bϕ(x)

which reduces to the familiar scaling for a fractional Brow-nian field if we take β = 1.

A wide variety of multi-scaling random fields can beobtained in this manner (using eq. (2)), so long asthe integrand in (14) is square-integrable. Another suit-able choice of Fourier filter yields a random field withBϕ(c1/H1x1, c

1/H2x2, c1/H3x3) = cBϕ(x) so that a differ-

ent Hurst index 0 < Hi < 1 applies to each coordinate.For example, the random field of Molz et al. [1997] hasHurst coefficient Hi = H + 1

2− 1

2βfor coordinates i = 1, 2

and H3 = β(H + 12) − 1

2. A more general form of the fil-

ter comes from the theory of multivariable fractional dif-ferentiation where the generalized fractional derivative ofa suitable function f(x) is defined as the inverse Fouriertransform of ψ(k)f(k) and exp[−ψ(k)] is the Fourier trans-form of an operator stable density function [Meerschaert,et al., 2001]. Using an example from section 2, the un-streched, isotropic Laplacian operator has the Fourier sym-bol ψ(k) = ‖k‖2 corresponding to the Fourier transform of azero mean multiGaussian density exp[−‖k‖2]. The generalform of the Fourier symbol ψ(k), given by the Levy repre-sentation [Meerschaert and Scheffler, 2001; Schumer et al.,2003a] is somewhat complex, but for our purposes it sufficesto note that this function always satisfies a matrix scalingrelation cψ(k) = ψ(cQk). Then the filter corresponding to a


fractional integration ϕ(k) = ψ(k)−A, where A = H + d/2,scales according to cAϕ(k) = ϕ(c−Qk). Once again a changeto polar coordinates splits the function into the weight func-tion M(θ) and a scaling function rQ; however, since Q mayspecify a different power law in any direction, a modified po-lar coordinate system is needed. A procedure for calculatingthis filter is listed in the appendix.

4.1. Examples

To illustrate the effect of the weight function alone, wegenerated several 2-D random fields (Fig. 5) with identical“random” Gaussian input fields but with four different func-tions M(θ). The fields are all isotropic with respect to thescaling coefficient H = 0.3. These realizations are typicalfor this scaling parameter. The first field uses the measureestimated by the directions of the braided stream in Figure1 without respect to flow direction (Fig. 5a). We then as-sume that the K values can only be influenced by upstreamperturbations by zeroing upstream weights and adding themto those in the downstream direction (Fig. 5b). Finally, weapproximate the measure using a classical elliptical stretch-ing (Fig. 5d) and a simple “cross” of four weights on the x–and y–axes (Fig. 5c). The plots clearly show the effect ofgeneralizing the description of weights. The “downstream”measure develops much more continuity in the x–directionthan any other measure, while the classical elliptical measureshows the least. The latter is due to the unavoidable andpossibly unrealistic placement of convolution weight trans-verse to the prevailing stream flow direction. The simple4-point “cross” measure develops a random field (Fig. 5c)that bears a closer resemblance to the “downstream” mea-sure. In section 5, we use this simple “cross” measure to con-struct fields with operator scaling (H different in the trans-verse and longitudinal directions) to investigate the impactof transverse H values on plume migration.

In general, the eigenvalues of the scaling matrix Q, andthe overall order of integration A, define the Hurst indexin each coordinate. The eigenvectors of Q specify thosecoordinates, and the weight function M(θ) indicates the im-portance of each direction in determining the dependencestructure. The primary scaling directions (eigenvectors ofthe matrix Q) need not be orthogonal, and the various di-rections can be given unequal weight of overall correlation.

x1

0

1

2

3

−4 −3 −2 −1 0 1 2 3 4

10

4.6

2.2

c =

1

x2

−2

−1

3

Q =

0.6 0

0 1.4cQ

=

c0.6

0

0 c1.4

−

0.46

Figure 3. Ellipses of the operator rescaling of 2-D spacecQx for various rescaling values c and Q = diag(0.6, 1.4).The dashed line corresponds to the remapping of thepoint on the 45◦ angle for c = 1 (heavy circle) onto otherellipses.

Because of the multiple scaling directions, these fields havethe property of “generalized scale invariance” [Schertzer andLovejoy, 1989], but do not have the multiple scaling dimen-sions associated with multiplicative cascades [Mandelbrot,1974; Frisch and Parisi, 1983; Benzi et al., 1984; Lovejoyand Schertzer, 1985; Gupta and Waymire, 1990; Boufadel etal., 2000; Veneziano and Essiam, 2003]. However, we noteone serious consequence of the operator scaling relationship:In general, the random process is distributionally self-similaralong curves in physical space (Fig. 3) unless one samplesalong the eigenvectors of the scaling matrix Q. Samplingthe process along straight lines between the eigenvectors willnot follow a single power law (for any measure), and theobserved scaling (Hurst) coefficient will change. For somemeasures used to distinguish multiplicative multifractals [seeSchertzer and Lovejoy, 1987; Boufadel et al., 2000; Tenekoonet al., 2003], such as the structure function and spectralmethods, the perceived fractal dimension will change un-less care is first taken to find the eigenvectors of the scalingprocess.

A closed form expression for ψ(k) or ϕ(k) can only bewritten in certain instances. For a simple example of a fil-ter that can be written in closed form, specify an order ofintegration A, and let the deviation matrix Q be diagonalin 2–D with eigenvalues Q1 and Q2. Furthermore, let therebe 4 weights on the unit circle of 0.45 on the ±x-axis and0.05 along the ±y-axis. Then the orders of integration in thetwo coordinates are A1 = A/Q1 and A2 = A/Q2, and thekernel can be built from the Fourier symbol of the fractionalderivative:

ψ(k) = 0.45C1

�(ik1)

1/Q1 + (−ik1)1/Q1)

�+

0.05C2

�(ik2)

1/Q2 + (−ik2)1/Q2)

�, (15)

where Ci = Γ(1 − 1/Qi). A plot of ϕ(k) = (ψ(k))−A—theconvolution kernel—with A1 = 1.2 and A2 = 0.2 (Fig. 4)shows the different decay rates in the two directions and thediscrete weights that transfer correlation in the chosen di-rections. In Section 5, we use (15) to fractionally integrateGaussian noise and create a series of random fields, eachwith an identical A1 but a different value of the parameter

x8

160

-16-8

y

816

0-8

x8

160

-16-8

816

0

-16-8

y

816

0-8

816

0-8

Figure 4. One possible convolution kernel leading to anoperator–self–similar random process: 2–D kernel repre-senting Ax = 1.2, Ay = 0.5 with total weights concen-trated on the x– and y–axes of 0.9 and 0.1, respectively.Note both the slower decay due to larger order of inte-gration, and the greater convolution weight, in the x–direction.


-50 50

-50

0

50

-50 50

-50

0

50

-50 50

-50

0

50

-50 50

-50

0

50

Braided Downstream

Cross Elliptical

0.24

0.12

0.24

0.12

0.24

0.12

0.24

0.12

(a) (b)

(d)(c)

Figure 5. Effect of different weight functions M(θ), discretized at intervals of π/10 (polar plots), onisotropically scaling fBm (shaded upper plots) a) measure estimated from braided stream directions,b) downstream effects only, c) simplified 4–point measure, and d) elliptical approximation. All weightfunctions are normalized to sum over θ to unity.

A2. The resulting operator–fractional Gaussian fields are

then used to investigate the effect of the order of aquifer

scaling, transverse to mean flow, on contaminant migration.

The possibility of non-orthogonal eigenvectors makes an

operator-fractional field a good candidate for simulating

physical properties of structurally deformed rock, since thefracture sets formed by stress relief are rarely orthogonal(e.g., Davis, 1984). Presently, geostatistical simulation isused primarily to simulate undeformed sedimentary rock.Notable exceptions are Ando et al.’s [2003] non-parametricsimulation of the densely fractured Fanay-AugPres site,


�

��

�

x x

yy

Figure 6. Operator–fractional Brownian fields (unconditional) for non-orthogonal scaling directions andweight functions. High K values are depicted by darker colors. The Hurst coefficient is isotropic with avalue of H = 0.3. The dots on the lower polar plots illustrate the weights used in the upper simulations,i.e., the right field has three times more weight along the π/8 radian direction.

and Tsang et al.’s [1996] sequential simulation and addi-tion of fracture sets as an equivalent porous medium. Inthese studies, the scaling properties of the simulated fieldswere not fractal. In the latter study, individual fields weregenerated for each of the correlation structures and re-combined, potentially altering the overall statistical prop-erties of the fields. Using operator-fractional random fields,realistic self-similar permeability structures can be gener-ated in one pass by varying the d principal scaling direc-tions in d–dimensional space and their corresponding Hurstcoefficients, as well as their relative weights.

To illustrate, we chose two fracture sets (scaling eigen-vectors) oriented ±π/8 radians from the x–axis. The weightfunction M(θ) is zero everywhere except for the π/8 and7π/8 directions (polar plots, Fig. 6). Along those direc-tions, the weights were made equal along both fracture sets(Fig. 6, left side), and three times greater on one of the frac-ture sets (Fig. 6, right side). The input uncorrelated noise(with no conditioning) was chosen to be Gaussian. The out-put field was normalized to have a standard deviation of0.9 and exponentiated. The darker areas shown in Fig. 6are those with higher value of K. Note the fracture–likesparse and linear regions of connected high permeability.The positions of known fractures can be placed in the in-put noise, and the variability of effective K in individualfractures (or zones) is explicitly represented. Flow throughsuch a limited “network” would strongly depend on boththe scale of simulation and the specifics of both the weightfunction M(θ) and the fractal dimensions (power laws of theconvolution) in various fracture directions. Such behavior isa hallmark of DFN simulations (e.g., Bour and Davy, 1998;

Bour et al., 2003; Darcel et al., 2003a,b; deDreuzy et al.,2004). In a future study, we will investigate the correspon-dence of flow and transport through these random fields toDFN. In the next section, we investigate transport throughoperator-fractional granular aquifers and the applicability ofmono-fractal analytic theories.

To motivate the numerical experiment in the next sec-tion, we re–examine the gas mini–permeameter data thatwas collected near Escalante, Utah, and presented by Castleet al. [2004]. Point permeability data were collected alongseveral horizontal and vertical transects in 1) an undisturbedupper shoreface cross–bedded sandstone and 2) an underly-ing bioturbated (disturbed) lower shoreface sandstone. Theresearchers calculated Hurst coefficients for a mixture of allhorizontal transects (two in the disturbed and one in theundisturbed facies) and four vertical transects that spanboth facies. We calculated the rescaled range statistic (R/S)for each horizontal line (with 110, 132, and 135 data pointsspaced 15 cm apart) and found that the lower two in the bio-turbated facies do not show fractal structure. We assumethat the extensive reworking of the sediments by burrow-ing animals altered any fractal structure that might havebeen present. Therefore, we used only data from the uppercross–bedded unit, which left one horizontal and the upperportion of two vertical transects (with only 16 and 18 datapoints with spacing of 15 cm). While the vertical transectsare severely shortened, the R/S statistics show reasonablyfractal structure indicative of fBm with H ≈ 0.33 over asmall range (Fig. 7a).

On the other hand, the horizontal R/S plot shows frac-tal structure indicative of fBm with H ≈ 0.45 over a wider


range of scales (Fig. 7a). The R/S statistic is only valid forlarger lags [Molz et al., 1997]. Dispersional analysis of thedeviations of means taken over different partitions [Caccia etal., 1997] is a less biased measure than the R/S or variogramfor fGn, and indicates slightly different structure—fBm withH ≈ 0.6—at smaller partitions (Fig. 7b). This points to ei-ther a multifractal structure, or that the measurement tran-sects were not aligned with the principal scaling directions.One of the principal directions may, in fact, coincide withthe cross–beds [e.g., Ritzi et al., 2004], which dip at low an-gle [Castle et al., 2004]. Given the wide confidence intervalsassociated with the R/S statistic, and all other measures in-cluding the semivariogram and structure functions of variousorder [see, e.g., Tennekoon et al., 2004], a number of differentdependence models cannot be ruled out. For the purposesof our numerical experiment, we simply acknowledge thatthe statistics for this site support a model of ln(K) that isan operator-fBm with horizontal H > 0.5 and a lower H inthe vertical. We have not attempted to find the principaldirections of the scaling process.

Numerous K measurements, like those from the Escalantesite, very seldom are taken in the horizontal direction. Oneexception is Desbarats and Bachu’s [1994] large-scale (up to100 km) analysis of aquifer transmissivity, which is consis-tent with an fBm with H ≈ 0.26. Further evidence for frac-tal K structure in the horizontal direction comes indirectly

1 10 1000.1

1

10

Hvert = 0.33

1 10 1000.01

0.1

1

Hhoriz = 0.6

Hhoriz = 0.45

(a)

(b)

Partition Size (number of data)

Partition Size (number of data)

Dis

per

sio

n S

tati

stic

Res

cale

d R

ang

e S

tati

stic

UndisturbedBioturbated (1)Bioturbated (2)All Combined

Figure 7. a) Calculated R/S statistic from horizon-tal transect and portions of vertical transects within theundisturbed cross–bedded sandstone unit [Castle et al.,2004]. Dashed lined indicate fBm in the transects withHhoriz ≈ 0.45 and Hvert ≈ 0.33. Thin lines indicateR/S statistic plus or minus one standard deviation. Notethat for small lags, the R/S statistic is biased low, sinceit equals zero for lag 1. b) Dispersion statistic (standarddeviation of means) versus partition size for the horizon-tal transects.

by inference from tracer data [Neuman, 1995; Benson, 1998].Using the asymptotic assumption and (12) applied to appar-ent dispersivity data from a large number of sites, Neuman[1995] argues that Hhoriz should be on the order of 0.25 foran fBm.

More extensive vertical K data from boreholes have beenanalyzed to estimate the vertical Hurst coefficient [Molz andBoman, 1993; Liu et al., 1997]. Very often, surrogate dataare used to infer long-range dependence, such as gammaray, spontaneous potential, or porosity logs from boreholegeophysical surveys. Reported values of vertical H froma variety of these surrogates measured in sediments fromvarious depositional environments span a very large rangethat includes both fGn and fBm [Hewett, 1986; Tubmanand Crane, 1995; Pelletier and Turcotte, 1996; Deshpandeet al., 1997]. The latter study also directly looked at theanisotropy of fractal structure of fluvial and deltaic sedi-ments on the multi-kilometer scale using well logs and seis-mic reflection data. Using reflection amplitude from twoplanes extracted from distinct sandy horizons, they ana-lyzed N-S and E-W transects for fractal structure. Theyconcluded that the fluvial sediments (with high amplitudechannels trending north-northwest) in one horizontal planewere consistent with fBm with H ≈ 0.4 to 0.65 in roughlythe direction of channels (along dip) and H ≈ 0.05 trans-verse to the channels (along strike). Deshpande et al. [1997]found less striking evidence of horizontal anisotropy in thedeltaic sediments, with different H values of 0.26 versus 0.4.All of their vertical data was in the fBm range, with H ≈0.3 to 0.5. To sum up the K and surrogate data from manystudies using different data types and different scales, ver-tical measurement may show a large range of the order ofintegration, consistent with both fGn and fBm, while hori-zontal sediments typically are consistent with an fBm withH ≈ 0.25 to 0.65. Also note that these studies do not al-ways test the assumption that the increments are normallydistributed. Many studies have shown that the K and/orln K data from a particular site are distinctly nonGaussian.

5. Solute Transport in Granular Aquifers

To examine the effect of the difference of the Hurst co-efficients in different directions, we conducted a numericalexperiment. K fields were created that are identical in everyway, except for the Hurst coefficient in the direction trans-verse to the mean flow. The fields were created using thesame input Gaussian noise, so only the Fourier filter ϕ(k)was different. The 2-dimensional Gaussian noise consistedof a 1024×1024 array of independent and identically dis-tributed standard normal random variables. The noise wasfast Fourier transformed, multiplied by the Fourier filter,and inverse fast Fourier transformed. The middle 1/4th ofthe field (now 512×512) was removed for flow and transportsimulation.

Based on the limited data in the horizontal direction, wespecified an fBm with Hurst index of H1 = 0.7 in the hori-zontal direction of flow, and varied the scaling index in thetransverse direction, using orders of integration from Avert

= 0.1 to 1.5, signifying fGn with H = 0.6 up to an fBm withH = 1.0. Four of these fields are shown in Figure 8. Theweights in the horizontal are equal left to right for a total of0.9, versus similarly symmetric weight in the vertical of 0.1.For reference, when the vertical order of integration Avert

= Ahoriz = 1.2, the field is similar to previous isotropicand stretched models of fBm [Rajaram and Gelhar, 1995;Zhan and Wheatcraft, 1996]. The only difference is our useof four discrete, rather than continuous (elliptical) weights.The output fields were adjusted to have zero mean and astandard deviation of 1.5, then exponentiated, so that each


(a) (b)

(d)(c)

Figure 8. Shaded plots of four of the ln(K) fields. The fields are identical except for the order of verticalintegration: a) 0.1, b) 0.5, c) 0.9, and d) 1.3. The fields are fBm in the horizontal direction with H =0.7, or an order of integration Ahoriz = 1.2. For particle tracking, 10,000 particles were released fromthe right side between nodes 128 and 384 (brackets).

field is a lognormal operator-fractional field. The fields weretested for scaling properties in the two principal scaling di-rections by sampling all rows and columns from the field andapplying both rescaled range and disperional analyses [Liuand Molz, 1996; Caccia et al., 1997]. These plots are notshown for brevity.

After solving the head field using MODFLOW (assumingan average hydraulic gradient of 0.01 and no-flow bound-aries on the top and bottom), particles were placed on thehigh head side and tracked using LaBolle et al.’s [1996] par-ticle tracking code. To make the simulations representativeof real-world conditions, we included a small local dispersiv-ity (equal in longitudinal and transverse directions) of 0.1times the constant grid size of 1 foot. Since this does notcorrespond exactly to Neuman’s [1995] and Di Federico andNeuman’s [1998] analytic results, we performed additionalsimulations with no local dispersion and found no notice-

able differences. Cushman [1997] and Hassan et al. [1998]more fully discuss the influence of local and nonlocal fluxeson the accuracy of analytic methods in monofractal media.

Increased orders of vertical integration (Avert) have adistinct effect on both the simulated aquifer structure andthe plume statistics. Lower values have less persistence ofaquifer layers and allow 1) more advective mixing, 2) lessspreading from persistent preferential flow, and 3) a transi-tion from Mercado (ballistic) to Fickian flow (Fig. 9). Incontrast, as aquifer gains more vertical structure and lay-ering from the increased order of integration, the plumespreads at a nearly Mercado-like (ballistic) rate over theentirety of the simulation. Even thought the mean and vari-ance of the overall K distribution are exactly the same in allsimulations, greater orders of transverse integration (Avert)lead to thicker layers of high and low K material, hence more


mass in the leading edge and trailing edges of the plume(Fig. 10). The duration of breakthrough is greater by overan order-of-magnitude for the most, versus least, layeredaquifer. The contribution of early arrivals and late tailingare roughly equal. The lowest value of Avert = 0.1 showsa near-Gaussian plume after 10,000 days (Fig. 11a). Theplumes become progressively less Gaussian and multi-modalwith greater transverse Avert.

Our numerical K fields have a low wave number cutoffequal to twice the domain size, since we subsampled theoriginal 1024×1024 field. The Nyquist high wave numbercutoff corresponds to the size of an individual cell. There-

Ap

par

ent

dis

per

siv

ity

(ft

)

Mean distance (ft)

0.1

1

10

100

0.1 1 10 100 1000

Fickian

R&

G

Avert

= 0.1

Avert

= 0.5

Avert

= 0.9

Avert

= 1.3

D-F

&N

(per

man

entl

y p

re-a

sym

pto

tic)

Mer

cado

Figure 9. (a) Apparent dispersivity versus mean par-ticle distance for various orders of integration (Avert)in the vertical direction. Solid and dash-dot lines in-dicate permanently pre-asymptotic and early preasymp-totic (i.e. Mercado) solutions for monofractals [Di Fed-erico and Neuman, 1998]. Dotted line indicates two-particle (single-plume) slope from Rajaram and Gelhar[1995]. Dashed line indicates Fickian motion.

10-4

10-3

10-2

10-1

1

103

104

105

106

Avert = 0.1

Avert = 0.5

Avert = 0.9

Avert = 1.3

Avert = 0.1

Avert = 0.5

Avert = 0.9

Avert = 1.3

Time (d)

No

rmal

ized

Co

nce

ntr

atio

n

Figure 10. Breakthrough of particles at the boundaryx = 512 ft.

fore, the mean positions of our numerical plumes are be-tween the large and small cutoffs. According to the criteriaof Di Federico and Neuman [1998], the plumes should bein a permanently pre-asymptotic state, where the appar-ent dipersivity should spread supralinearly (denoted by thesteepest, solid line in Fig. 9). The K fields should not fosterasymptotic (Fickian) growth, if the assumptions inherent tothe analytic theories are applicable. Since the eliminationof local dispersion had negligible effect, we conclude thatthe transition to a Fickian regime by plumes in the aquiferswith the least persistent layering is due primarily to advec-tive mixing. The particles do not experience the exact Kcorrelation structure in the longitudinal direction prescribedby the value of Hhoriz, contrary to the assumption inherentto the analytic solutions [Neuman, 1995; Di Federico andNeuman, 1995; Rajaram and Gelhar, 1998]. The particlesare free to “shortcut” between high-K zones and avoid thelow-K zones, making the Lagrangian correlation structuredifferent from the Eulerian K structure.

Only the fields that are nearly mono-fractal, with Avert =0.9 and 1.3, agree qualitatively with the analytic predictionsof Rajaram and Gelhar (Fig. 9). In these cases the particlesexperience roughly the same Hurst coefficient along a tra-jectory no matter how much lateral mixing is taking place.Since these are single, not ensemble plumes, it is not sur-prising that Rajaram and Gelhar’s two-particle predictionsare more accurate in the nearly mono-fractal cases. How-ever, these plumes are far from Gaussian in shape (Figs. 11cand d). The plumes in the aquifers with the least persistentlayering—with lower values of Avert = 0.1 and 0.5—showconvergence to a Fickian growth rate and an approximatelyGaussian plume shape (Figs. 10, 11a and b), since we spec-ified a finite-variance lognormal K distribution. However,all of the plumes have significant fractions that travel wellahead of the Gaussian best fit (Fig. 11). Because of thelow ln(K) variance, none of the plumes show as much rapidtransport in the leading edge as an α-stable plume under-going fractional-order dispersion [Benson et al., 2001]. Onecan expect that mono-fractally scaling fields with heavy-tailed, α-stable K increments can create even more rapidtransport and α-stable plumes [Herrick et al., 2002; Grabas-njak, 2003]; however, Trefry et al. [2003] show that lognor-mally distributed aquifers with exponential correlation de-cay may also engender highly channelized flow and α-stablegrowth behavior when both the integral scale and the vari-ance of ln(K) are large.

On the other end of the transport spectrum, the non-Fickian plumes (Fig. 10) do not display the anomalous(power-law) late-time tailing associated with fractal mo-bile/immobile and typical continuous time random walkformulations [Berkowitz and Scher, 1997; Berkowitz et al.,2000; Schumer et al., 2003b; Dentz and Berkowitz, 2003].Since our VAR(ln K) was only 2.25 with a geometric meanof 1 ft/d, the minimum K values were on the order of 10−3

ft/d. This value is not low enough to make diffusion a dom-inant flux in our low-K zones [Haggerty et al., 2000]. Hadthe variance of ln(K) been on the order of 4.0 or more, thepresence of a fractal distribution of low-K zones, with fluxdominated by diffusion, should induce this type of fractaldelayed transport [Haggerty et al., 2000].

To summarize the results of the numerical experiment,we confirmed the fact that higher orders of integrationin the vertical direction (corresponding to fBm with highH values) describe increased persistence of random values,hence greater layering or stratification. Plumes released inoperator-fractional aquifers will spread at different rates,even with the same value of H in the direction of flow andidentical ln(K) mean and variance. Those plumes in nearlymonofractal aquifers plumes agree best with the monofrac-tal analytic solution of Rajaram and Gelhar [1995], perhaps


1

10

100

1000

0 100 200 300 400 500 600

1

10

100

1000

0 100 200 300 400 500 600

1

10

100

1000

0 100 200 300 400 500 6001

10

100

1000

0 100 200 300 400 500 600

Longitudinal Distance (ft)

Inte

gra

ted

Co

nce

ntr

atio

n


Inte

gra

ted

Co

nce

ntr

atio

n


Inte

gra

ted

Co

nce

ntr

atio

n


Inte

gra

ted

Co

nce

ntr

atio

n

Avert

= 0.5

(fBm, H = 0.0)

t = 1,000 d

t = 10,000 d

t = 1,000 d

t = 10,000 d

t = 1,000 d

t = 10,000 d

t = 1,000 d

t = 10,000 d

(a) (b)

(c) (d)

Avert

= 0.9

(fBm, H = 0.4)

Avert

= 1.3

(fBm, H = 0.7)

Avert

= 0.1

(fGn, H = 0.6)

Figure 11. Semi-log plots of vertically integrated concentration (plumes) versus longitudinal distancefor the fields shown in Fig. 8 with Avert = [0.1, 0.5, 0.9, 1.3]. Best-fit Gaussian curves underlie points.

because their theory is developed for single, not ensembleplumes. A more extensive analysis of ensembles of operator-fractional plumes would be needed to confirm this. Thebreakthrough of plumes is drastically affected by the trans-verse scaling properties of the aquifer. The most stratifiedaquifer (fBm in the vertical with H=0.8) had particles arrivenearly an order of magnitude before and after the least strat-ified aquifer. We used a simplistic measure of directional Kdependence in our approximation of a granular aquifer. Ourmeasure of the directionality within the surface expressionof a braided stream environment may show more complexstructure and different plume mixing effects. We have notyet examined the effect of more complex directional mea-sures M(θ), whether they represent granular (Fig. 5) orfractured media (Fig. 6).

6. Conclusions

1) A multi-dimensional operator-fractional Gaussiannoise/Brownian motion, or combination of the two, can becreated via Fourier transform by convolving an uncorrelatedGaussian noise with a function that has the correct matrixscaling relationship.

2) Many other distributions can be chosen for the uncor-related noise in the numerical procedure, yielding a processwith operator-self-similar increments. Furthermore, the in-put noise may be given non-stationary statistics to representsequences of geologic material.

3) In d–dimensions, the d primary scaling directions ofthe self-similar random field need not be orthogonal. Fur-

thermore, the weights assigned to the directional fractional

integrals, i.e., the convolution kernel, are freely assigned to

the d–dimensional unit sphere.

4) The longitudinal moments of a moving plume depend

largely on the scaling (e.g., Hurst) coefficients in the di-

rections transverse to flow. Prior analytic results do not

account for this.

5) Smaller orders of integration, representing noise or mo-

tion with small Hurst coefficients, lead to less persistent

aquifer layering and greater advective mixing of a plume. In

contrast to previous analytic predictions, plumes in material

with small orders of integration (i.e., noisy or less persistent

layering) in the transverse direction may converge relatively

quickly to Fickian-type transport, regardless of the K struc-

ture in the direction of transport.


7. Notation

aL – longitudinal dispersivity [L].A – order of fractional integration.B(dx) – uncorrelated (white) Gaussian noise.BH(x) – isotropic fractional Brownian motion.Bϕ(x) – (operator) fractional random field.fBm – fractional Brownian motion.d – number of dimensions.H – Hurst coefficient.I – identity matrix.k – wave vector [L−1].K – hydraulic conductivity [LT−1].M(θ) – measure of directional weight within ϕ(x).Q – deviations from isotropy matrix.VAR(Y ) – variance of random variable Y .X – mean particle longitudinal travel distance.θ – unit vector on the d−dimensional unit sphere.ϕ(x) – scaling (convolution) kernel.ψ(k) – log-characteristic function of operator-stable density.

Appendix A: Calculation of ϕ(k)

Recall that the Fourier filter ϕ(k) = ψ(k)−A, whereA = H + d/2 and ψ(k) is the log-characteristic function ofsome operator stable law with exponent Q. It follows (e.g.,Jurek and Mason, 1993) that ψ(k) has the representation

ψ(k) =

Z‖θ‖=1

Z ∞

0

�e−ik·(rQθ) − 1− ik · (rQθ)

�dr

r2λ(dθ)

(A1)if the real parts of the eigenvalues of Q are in the interval(1/2, 1), whereas

ψ(k) =

Z‖θ‖=1

Z ∞

0

�e−ik·(rQθ) − 1

�dr

r2λ(dθ) (A2)

if the real parts of the eigenvalues of Q are greater thanone. In the following we will only consider the latter case.The first case can be dealt with similarly. The measure λ onthe unit sphere is called the mixing measure or the spectralmeasure of ψ.

Let 1 < a1 ≤ a2 ≤ · · · ≤ ad denote the real parts ofthe eigenvalues of Q. Then Bϕ(x) defined by (2) exists, if(d/2)(ap − 1) < H < (d/2)(a1 − 1) + a1 which is only pos-sible if ad < (1 + 2/d)a1, putting some restriction on Q. Itfollows that

Bϕ(cQx) = cA−trace(Q)/2Bϕ(x) (A3)

in distribution, where trace(Q) is the sum of the eigenvaluesof Q. Furthermore, the variance V (x) = VAR(Bϕ(x)) hasthe scaling relation V (cQx) = c2A−trace(Q)V (x). Note thatin the isotropic fractional Brownian field case, correspondingto ψ(k) = ‖k‖, we have a1 = ad = 1, so our condition in thisspecial case reduces to the standard 0 < H < 1 conditionon H and (A3) is just Bϕ(cx) = cHBϕ(x).

In order to compute ψ(k) we use a discrete mixing mea-sure λ(dθ) =

PM(θi)δθi , a sum of point masses δθi with

user-specified weights M(θi) (indicating the strength of thedirectional dependence). For a continuous measure λ(dθ)we use a discrete approximation.

An important special case, generalizing the example in eq.(15) can be obtained in the following way (we only describethe case d = 2, higher dimensional examples are obtainedsimilarly): Fix any two non-parallel vectors θ1 and θ2 on

the unit sphere and any two numbers 1 < a1 < a2 < 2a1.Pick any weights M(±θi) and define the spectral measureλ(dθ) = M(θ1)δθ1 +M(−θ1)δ−θ1 +M(θ2)δθ2 +M(−θ2)δ−θ2 .Let Q be a 2 × 2-matrix with eigenvectors θ1 and θ2 andcorresponding eigenvalues a1 and a2, respectively, so thatQθi = aiθi. Then, by some calculations we get from (A2)for some constants C1, C2 > 0 (only depending on a1 anda2) that

ψ(k) = C1

�M(θ1)(ik · θ1)

1/a1 + M(−θ1)(−ik · θ1)1/a1

�+ C2

�M(θ2)(ik · θ2)

1/a2 + M(−θ2)(−ik · θ2)1/a2

�which corresponds to a weighted fractional derivative of or-der 1/a1 in direction ±θ1 and of order 1/a2 in direction ±θ2.If we let ϕ(k) = ψ(k)−H−1 for some a2 − 1 < H < 2a1 − 1then Bϕ(x) defined by (2) exists and scales as Bϕ(cQx) =cH+1−(a1+a2)/2Bϕ(x).

Acknowledgments. This work is supported by NSF grantsDMS-0417869, DMS-0417972, DMS–0139927, DMS–0139943,EAR–9980484, EAR–9980489, and US DOE grant DE–FG03–98ER14885. We thank the two reviewers, Shlomo Neuman andBrian Wood, and the associate editor, John Selker, for excellentreview comments.

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David A. Benson, Desert Research Institute, 2215 RaggioParkway, Reno, NV 89512, USA. ([email protected])

Mark M. Meerschaert, Department of Mathematics &Statistics, University of Otago, Dunedin, New Zealand.([email protected])

Boris Baeumer, Department of Mathematics & Statistics, Uni-versity of Otago, Dunedin, New Zealand. ([email protected])

Hans-Peter Scheffler, Department of Mathematics & Statistics,University of Nevada, Reno, NV, 89512, USA. ([email protected])

aquifer operator-scaling and the eﬁect on solute mixing...

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