nonlinear spectral analysis of flow through multifractal porous media

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Nonlinear spectral analysis of flow through multifractal porous media Daniele Veneziano * , Albert K. Essiam Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Room 1-348, Cambridge, MA 02139, USA Abstract In previous work, we have considered flow in a saturated D-dimensional aquifer when the hydraulic conductivity K is an isotropic lognormal multifractal field. We found that the resulting hydraulic gradient rH and specific flow q are also multifractal fields and obtained their scaling properties in analytical form. Here we use these scaling properties to derive the spectral density tensors of rH and q and the macrodispersivities. The analysis is nonlinear and accounts for the local random rotation of the rH and q fields as the resolution to which the lnðKÞ field is developed increases. The spectra show that rH and q are anisotropic at large scales, but become increasingly isotropic at smaller scales. The rate at which isotropy is approached depends on the space dimension D and the multifractal characteristics of K. Con- ventional first- and second-order methods do not capture this important feature and further give incorrect amplitudes and power decays of the spectral density tensors. At short distances, the macrodispersivities from nonlinear theory are significantly larger than those from linear perturbation analysis. Ó 2003 Elsevier Ltd. All rights reserved. 1. Introduction The present paper continues the flow analysis of Veneziano and Essiam [1]. That earlier study derived the scaling properties of the hydraulic gradient rH and specific discharge q when the hydraulic conductivity K is an isotropic lognormal multifractal field in D-dimensional space. Under these conditions, the spectral density of the log-conductivity F ¼ lnðKÞ has the form S F ð kÞ ¼ 2 S D C K k D ; k 0 6 k 6 rk 0 ¼ 0; otherwise ð1Þ where k is the length of the wavenumber vector k,0 < C K < D is the so-called co-dimension parameter of K, r > 1 is a resolution parameter, k 0 and rk 0 are low and high wavenumber cutoffs that determine the limits of multifractal scaling, and S D is the surface area of the unit ball in R D ; hence S 1 ¼ 2, S 2 ¼ 2p and S 3 ¼ 4p. Veneziano and Essiam [1] found that the resulting fields rH and q are homogeneous multifractal and derived analytically their scaling properties. The analysis of [1] holds for a cubic region X of unit side length under k 0 1 and C K D. The condition on k 0 ensures that the spatial average of the log-conductivity field in X is close to the ensemble mean and may be viewed as a requirement of ergodicity of K inside X. The condition on C K is theoretically needed for rH and q to have multifractal scale invariance. However, we have found through simulation that multifractality of rH and q is very closely satisfied also for large values of C K [2]. * Corresponding author. Tel.: +1-617-253-7199; fax: +1-617-253-6044. E-mail address: [email protected] (D. Veneziano). 0960-0779/04/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00043-2 Chaos, Solitons and Fractals 19 (2004) 293–307 www.elsevier.com/locate/chaos

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Page 1: Nonlinear spectral analysis of flow through multifractal porous media

Chaos, Solitons and Fractals 19 (2004) 293–307

www.elsevier.com/locate/chaos

Nonlinear spectral analysis of flow throughmultifractal porous media

Daniele Veneziano *, Albert K. Essiam

Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Room 1-348, Cambridge, MA 02139, USA

Abstract

In previous work, we have considered flow in a saturated D-dimensional aquifer when the hydraulic conductivity Kis an isotropic lognormal multifractal field. We found that the resulting hydraulic gradient rH and specific flow q are

also multifractal fields and obtained their scaling properties in analytical form. Here we use these scaling properties to

derive the spectral density tensors of rH and q and the macrodispersivities. The analysis is nonlinear and accounts for

the local random rotation of the rH and q fields as the resolution to which the lnðKÞ field is developed increases. The

spectra show that rH and q are anisotropic at large scales, but become increasingly isotropic at smaller scales. The rate

at which isotropy is approached depends on the space dimension D and the multifractal characteristics of K. Con-ventional first- and second-order methods do not capture this important feature and further give incorrect amplitudes

and power decays of the spectral density tensors. At short distances, the macrodispersivities from nonlinear theory are

significantly larger than those from linear perturbation analysis.

� 2003 Elsevier Ltd. All rights reserved.

1. Introduction

The present paper continues the flow analysis of Veneziano and Essiam [1]. That earlier study derived the scaling

properties of the hydraulic gradient rH and specific discharge q when the hydraulic conductivity K is an isotropic

lognormal multifractal field in D-dimensional space. Under these conditions, the spectral density of the log-conductivity

F ¼ lnðKÞ has the form

* Co

E-m

0960-0

doi:10.

SF ðkÞ ¼ 2

SDCKk�D; k0 6 k6 rk0

¼ 0; otherwise

ð1Þ

where k is the length of the wavenumber vector k, 0 < CK < D is the so-called co-dimension parameter of K, r > 1 is a

resolution parameter, k0 and rk0 are low and high wavenumber cutoffs that determine the limits of multifractal scaling,

and SD is the surface area of the unit ball in RD; hence S1 ¼ 2, S2 ¼ 2p and S3 ¼ 4p. Veneziano and Essiam [1] found that

the resulting fields rH and q are homogeneous multifractal and derived analytically their scaling properties.

The analysis of [1] holds for a cubic region X of unit side length under k0 � 1 and CK � D. The condition on k0ensures that the spatial average of the log-conductivity field in X is close to the ensemble mean and may be viewed as a

requirement of ergodicity of K inside X. The condition on CK is theoretically needed for rH and q to have multifractal

scale invariance. However, we have found through simulation that multifractality of rH and q is very closely satisfied

also for large values of CK [2].

rresponding author. Tel.: +1-617-253-7199; fax: +1-617-253-6044.

ail address: [email protected] (D. Veneziano).

779/04/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

1016/S0960-0779(03)00043-2

Page 2: Nonlinear spectral analysis of flow through multifractal porous media

294 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

Quantities derived using the spectrum in Eq. (1) are said to be ‘‘at resolution r’’. Whenever necessary for clarity, rwill be used as a subscript. The main result of Veneziano and Essiam [1] is that the hydraulic gradient and flow fields at

different resolutions r1 and rr1 ðr; r1 P 1Þ satisfy the multifractal scaling relations

rHrr1ðxÞ ¼d JrRrrHr1ðrRT

r xÞ ð2aÞ

qrr1ðxÞ ¼d BrRrqr1ðrR

Tr xÞ ð2bÞ

where ¼d denotes equality of all finite dimensional distributions, Jr and Br are random variables and Rr is a random

rotation matrix. The distributions of Jr, Br and Rr depend on D and CK and are given in Section 4 of [1]. These results

were obtained using a nonlinear analysis method in which interactions from large-scale to small-scale fluctuations of K,rH and q are considered. The scaling relations in Eq. (2) are examples of what is sometimes called generalized scale

invariance or gsi; see Lovejoy and Schertzer [3] for a general introduction to gsi and Section 2 of [1] for application to

subsurface flow.

This paper uses Eq. (2) to derive the spectral density tensors of rHr and qrand examine transport properties of

media with lognormal multifractal K. Section 2 provides background on the spectral properties of multifractal fields,

including vector fields with gsi renormalization properties of the type in Eq. (2). The main results of Section 2 are

proved in Appendix A. Section 3 uses the properties in Section 2 to derive the spectral tensors of rHr and qr, while

Section 4 examines the implications of these tensors on macrodispersivities. We find important differences with results

from conventional first-order perturbation analysis. The spectral density tensors from nonlinear theory have scale-

dependent (k-dependent) anisotropy and further differ in both amplitude and decay rate from small-perturbation re-

sults. Due to these differences, at small distances our longitudinal and transversal macrodispersivities are much larger

than those predicted by linear theory. The macrodispersivities from nonlinear theory further exhibit near-isotropy at

small travel distances and become anisotropic at large travel distances, whereas according to linear theory the mac-

rodispersivities have the same degree of anisotropy at all scales. Conclusions and future developments are discussed in

Section 5.

2. Spectral relations for homogeneous multifractal measures

We present some key results on the spectral densities of homogeneous multifractal measures, which will be needed in

subsequent sections. First, we derive the spectral density functions of scalar random measures and their logarithms

under isotropic multifractality (these conditions are satisfied by our model of the hydraulic conductivity K). In par-

ticular, we obtain Eq. (1) for lnðKÞ. Then we analyze the spectral density tensors of homogeneous random vector

measures under gsi conditions of the type in Eq. (2). These results will be used in Section 3 to derive the spectral density

tensors of rH and q.

2.1. Isotropically multifractal measures and their logarithms

Let KðSÞ, S � RD, be a homogeneous random measure in RD and denote by KðSÞ ¼ KðSÞ=jSj the associated measure

density. Suppose that K has isotropic multifractality, meaning that it satisfies the scale invariance condition [1,4]

KðSÞ ¼d ArKðrSÞ; rP 1 ð3Þ

where Ar is a non-negative random variable independent of KðrSÞ. The function WKðsÞ ¼ logrfE½Asr�g is called the

moment scaling function of K. For example, if Ar has lognormal distribution (in this case we say that K is a lognormal

multifractal measure), then

WKðsÞ ¼ CKðs2 � sÞ ð4Þ

and CK ¼ 12Var½logrðArÞ� has values between 0 and D.

It is shown in Appendix A that, if the spectral density functions SK and SlnðKÞ are finite (this is the case if K is a

lognormal multifractal measure), then these functions scale as

SKðkeÞ ¼ k�DþWK ð2ÞSKðeÞ ð5aÞ

Page 3: Nonlinear spectral analysis of flow through multifractal porous media

D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 295

SlnðKÞðkeÞ ¼ k�DSlnðKÞðeÞ ð5bÞ

where e is any given unit D-dimensional vector. In the lognormal case, Eq. (4) gives WKð2Þ ¼ 2CK ¼ Var½logrðArÞ�.Eq. (5) shows that both SK and SlnðKÞ behave like power laws along any given direction in Fourier space, although

they have different decay exponents. In practice, the parameter CK might range from 0.1 to 0.4 or higher depending on

how erratic the K field is. Hence the two exponents might differ by 0.2–0.8 or more. This is an important result, since

according to first-order perturbation analysis the decay exponents of SK and SlnðKÞ are both equal to �D. For applicationto our flow problem, we are interested in isotropic lognormal K fields. For that case, it is shown in Appendix A that

SlnðKÞ has the form in Eq. (1).

2.2. Spectral energy tensors under generalized scale invariance

For the analysis of rH and q, one needs to know the spectral density tensors of homogeneous vector fields, say

V rðxÞ, that satisfy a renormalization condition of the type in Eq. (2). Here we write that renormalization condition as

V rr1ðxÞ ¼d Ar½RrV r1ðrR

Tr xÞ� ð6Þ

where Ar is a random variable and Rr is a random rotation matrix, independent of Ar. The expression in brackets in Eq.

(6) is obtained from V r1ðxÞ through space contraction (by r) and space and field rotation (by Rr). We denote this

transformed vector field by V 0r1 ;r

ðxÞ ¼ RrV r1ðrRTr xÞ. For any given rotation matrix Rr, the spectral density tensor of V 0

r1 ;r,

SV 0r1 ;rðkÞ, can be obtained from the spectral density tensor of V r1 as

SV 0r1 ;rðkÞ ¼ r�D RrSV r1

1

rRTr k

� �RTr

� �ð7Þ

Eq. (7) holds for given Rr. Since in our case Rr is random, one must take expectation of the right-hand side of Eq. (7)

with respect to Rr. Further using Eq. (6) and E½A2r � ¼ rW ð2Þ, one obtains that the spectral density tensors of V r1ðxÞ and

V rr1ðxÞ are related as

SV rr1ðkÞ ¼ r�DþW ð2ÞE

Rr

RrSV r1

1

rRTr k

� �RTr

� �ð8Þ

3. Spectral analysis of $H and q

The objective of this section is to obtain the spectral density tensors of rH and q. We start by noting the limitations

of first-order (FO) and second-order (SO) analyses and then derive nonlinear results using the scaling properties in Eqs.

(2) and (8).

3.1. First-order and second-order results

Many authors have solved flow problems with stochastic K by approximate first-order methods. A general outline of

the methodology was presented by Beran [5] and later extended by many others (see e.g. [6–14]). The main difference

among alternative approaches is the nature of the perturbation expansion. Here we present the first-order method as

developed by Gelhar and Axness [13] and point out its limitations. Then, in Sections 3.2 and 3.3, we develop an al-

ternative nonlinear approach.

In the first-order analysis of Gelhar and Axness [13], one starts by expressing the log hydraulic conductivity

F ¼ lnðKÞ and the hydraulic head H as F ¼ F þ f and H ¼ H þ h, where F ¼ E½F �, H ¼ E½H � and f and h are devi-

ations from the mean values. It is customary to put J 0 ¼ �rHðxÞ and assume J 0 ¼ J0e1, where e1 is the unit vector inthe direction of x1. Since H and q are proportional to J0 and q is proportional to E½K�, one may further put J0 ¼ 1 and

K ¼ E½K� ¼ 1, as was done in [1].

In terms of log deviation f , the hydraulic conductivity K may be written as KðxÞ ¼ K0f1þ f ðxÞ þ f 2ðxÞ=2þ . . .gwhere K0 ¼ expfF g is the geometric mean of K. Under the condition that f , its derivatives oh=oxi and the derivatives of

h, oh=oxi are small so that higher order terms may be neglected, the mean value of q is K0J 0 ¼ K0e1 and the mean-

corrected specific discharge q0 ¼ q� E½q� is given by

q0 ¼ �Krh ¼ K0½e1f �rh� ð9Þ

Page 4: Nonlinear spectral analysis of flow through multifractal porous media

296 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

Also the flow equation, r2H þ 1K ðrK � rHÞ ¼ 0, may be expressed in terms of F ;H and the perturbations f and h.

When higher order terms in f and h are neglected, this gives

r2h ¼ e1 � rf ¼ ofox1

ð10Þ

Gelhar and Axness [13] used Eqs. (9) and (10) to derive the spectral density S and spectral tensor S of various quantities

from the power spectrum of the log conductivity, SF ðkÞ. They found

S0KðkÞ ¼ K2

0SF ðkÞ ð11aÞ

S0hðkÞ ¼ e21k

�2SF ðkÞ ð11bÞ

S0rH ðkÞ ¼ e21ðeeTÞSF ðkÞ ð11cÞ

S0qðkÞ ¼ K2

0

ðdi1 � e1eiÞðdj1 � e1ejÞi; j ¼ 1; . . . ;D

� �SF ðkÞ ð11dÞ

where k is the length of k, e ¼ k=k is the unit vector in the direction of k, and the prime sign denotes first-order ap-

proximation. There is some controversy on whether K20 is the appropriate prefactor in Eq. (11d) [12,15,16].

For K multifractal down to infinitesimal scales, the variance of K diverges and the geometric mean conductivity

K0 ! 0. Therefore, the spectral densities in Eqs. (11a) and (11d) are nonzero only if one limits the scaling range of K. As

we shall see later in Section 3.3, this vanishing behavior of Sq is qualitatively correct, but that of SK is not.

A second problem with first-order analysis is that all the spectral densities in Eq. (11) have incorrect decay expo-

nents. This is easily seen in the case of Eq. (11a), because when K is an isotropic lognormal multifractal field in RD the

exact spectral densities of f and K have the form Sf ðkÞ / k�D and SKðkÞ / k�Dþ2CK , respectively; see Eqs. (1) and (5).

The term 2CK in the expression for SKðkÞ is a positive constant, which in practice might be rather large. Therefore, the

error in the decay exponent for SK may be significant. As we shall show later, the exponents of Eqs. (11b)–(11d) contain

similar errors.

A third problem with the first-order spectra S0rH and S 0

q is that they have scale-invariant anisotropy (their contour

sets have all the same shape). By contrast, we shall find in Sections 3.2 and 3.3 that the spectral tensors of rH and qfrom nonlinear analysis are anisotropic for small k and gradually become isotropic for large k.

Dagan [17] made a second-order spectral analysis of the hydraulic head fluctuation h. His results are analyzed in

Appendix B for the case when the log-hydraulic conductivity has a spectral density of the type Sf ðkÞ / k�a. It is found

that, in the case of multifractal K (when a ¼ D), the second-order correction to S0hðkÞ diverges and is therefore not

useful.

A case that is important for our spectral analysis and for which first-order theory is exact is when r in Eq. (1) is

infinitesimally close to 1. In this case the fluctuations f of F and h of H are infinitesimal and the geometric mean K0

satisfies K0 ¼ K ¼ 1. Further denoting by k0 any wavenumber vector of length k0 and using SF ðk0Þ ¼ 2SDCKk�D

0 from Eq.

(1), one obtains from Eqs. (11b) and (11d) that

Shðk0Þ ¼2

SDCKk�D�2

0

� �e21 ð12aÞ

Sqðk0Þ ¼2

SDCKk�D

0

� �ðdi1 � e1eiÞðdj1 � e1ejÞ

i; j ¼ 1; . . . ;D

� �ð12bÞ

where the prime signs have been omitted because these results are exact.

3.2. Nonlinear spectral analysis of rH based on Eq. (2a)

Our analysis of rH is based on multifractal scaling (Eq. (2a)), the spectral density tensor transformation in Eq. (8),

and the result in Eq. (12a). It follows from Eqs. (8) and (2a) and the independence of Jr and Rr that

SrHrr1ðkÞ ¼ r�DE½J 2

r �ERr

RrSrHr1ðRT

r k=rÞRTr

h ið13Þ

As r ! 1, rHr tends to a non-degenerate random measure rH and SrHrðkÞ in Eq. (13) converges to a spectral tensor

SrH ðkÞ that satisfies

Page 5: Nonlinear spectral analysis of flow through multifractal porous media

D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 297

SrH ðkÞ ¼ r�Dþ2ðCK=DÞERr

RrSrH ðRTr k=rÞRT

r

� �ð14Þ

where we have used the relation E½J 2r � ¼ r2ðCK=DÞ. This relation follows from properties of the distribution of Jr; see Eq.

(15) of [1]. Further using SrH ðkÞ ¼ ðkkTÞShðkÞ, the term in brackets in Eq. (14) may be written askrkT

r

� �ShðRT

r k=rÞ, whereall rotation matrices Rr except the one in the argument of Sh have disappeared. This simplification results from the fact

that rH is a potential field (is the gradient of a scalar field). Substitution into Eq. (14) gives

SrH ðkÞ ¼ r�Dþ2ðCK=DÞ krkT

r

� �ERr

ShðRTr k=rÞ

� �ð15Þ

Eq. (16) relates the spectral tensor of rH at some wavenumber vector k to the spectral density of the head fluctuation hat smaller rotated wavenumber vectors RT

r k=r. In order to obtain an explicit expression for SrH , we assume that, for

k ¼ k0 such that jk0j ¼ k0 (at the low-wavenumber end of the scaling range), the spectral density of h is evaluated

correctly by Eq. (12a). Then, setting k ¼ rk0, Eq. (15) becomes

SrH ðrk0Þ ¼ r�Dþð2=DÞCK ðk0kT0 Þ2

SDCKk�D�2

0

� �E½e2r1 � ð16Þ

where ei is the ith component of the unit vector e ¼ k0=k0 and er1 is the first component of the vector RTr e. The spectral

density of h ¼ H � H follows directly from Eq. (16) and is given by

Shðrk0Þ ¼ r�D�2þð2=DÞCK2

SDCKk�D�2

0

� �E½e2r1 � ð17Þ

In particular, in the case of planar flow (D ¼ 2) Eqs. (16) and (17) become

SrH ðrk0Þ ¼1

pk�20 CK

� �r�2þCK ½bre21 þ ð1� brÞe22�ðeeTÞ

Shðrk0Þ ¼1

pk�40 CK

� �r�4þCK ½bre21 þ ð1� brÞe22�

8>>><>>>: ð18Þ

where br ¼ E½cos2ðarÞ� and the angle ar has normal distribution with mean zero and variance 2DðDþ2ÞCK lnðrÞ; see Section

4 of [1]. The spectral results in Eqs. (16)–(18) have some interesting features:

(i) The exponents of r in Eqs. (16) and (17) give the asymptotic high-frequency decay of the spectral densities. From

first-order theory (insert Eq. (1) into Eq. (11)), these exponents are �D for rH and �ðDþ 2Þ for h. The nonlinearexponents differ due to the term ð2=DÞCK , which depends on the space dimension D and the parameter CK of the Kfield. In order for K to exist, CK must be between 0 and D [18,19]. Hence the exponent of r in Eq. (17) is between �Dand �ðDþ 2Þ, which is the range of spectral exponents for fractional Brownian surfaces (fBs) [20]. Notice however

that h is not a Gaussian field and therefore differs from fBs.

(ii) The term E½e2r1 � in Eqs. (16) and (17) and the terms in square brackets in Eq. (18) are anisotropic factors, which vary

with the wavenumber factor r as well as the direction in Fourier space. Consider for example Eq. (18). For r ¼ 1 (at

very low frequencies), ar ¼ 0 and br ¼ 1; hence the term in brackets equals e21 as in first-order theory. On the other

hand, for r ! 1 (at very high frequencies) the variance of ar diverges, br ! 0:5, and the term in brackets! 1. This

transition of rH and h from anisotropy at large scales to isotropy at small scales is not predicted by first-order

theory, according to which the fluctuations of H and rH have the same anisotropy at all scales; see Eq. (11).

Fig. 1 shows numerical comparisons between the linear and nonlinear spectral densities of h, the former from Eqs.

(1) and (11) and the latter from Eq. (17). The cases considered are CK ¼ 0:1 and 0.3, both for D ¼ 2. Due to the nearly

power behavior of the functions Sh and S0h, we have transformed logarithmically both the functions and their arguments.

Hence a point k0 ¼ ½k01; k02� in Fig. 1 corresponds to the wavenumber vector k ¼ k0

k0 10k0 , where k0 P 0 is the length of k0.

The innermost contour levels have a log-spectral level of )4 and subsequent contour levels have a log spacing of )4.Hence the outermost contours shown in the figures have a log level of )20. Two important differences between the

linear and nonlinear results may be observed: (1) the spectral density from linear theory decays at a faster rate (the

decay exponent is )4 for the linear spectra and �4þ CK for the nonlinear spectra) and (2) the contour lines from linear

theory have the same non-circular shape at all scales, whereas those from nonlinear theory exhibit anisotropy at large

scales and isotropy at small scales. The transition towards isotropy is faster for larger values of CK .

Page 6: Nonlinear spectral analysis of flow through multifractal porous media

-4 -2 0 2 4

-4-2

0

2

4

-4 -2 0 2 4

-4-2

0

2

4

-4 -2 0 2 4

-4-2

02

4

02

4

-4 -2 0 2 4

-4-2

Log(k/ko)Log(k/ko)

Log(k/ko)Log(k/ko)

Log

(k/k

o)L

og(k

/ko)

Linear Theory Nonlinear Theory CK =0.1

Linear Theory Nonlinear Theory CK =0.3

Fig. 1. Linear and nonlinear power spectra of the hydraulic head fluctuation for CK ¼ 0:1 and 0.3.

298 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

3.3. Nonlinear spectral analysis of q based on Eq. (2b)

The spectral analysis of the specific flow q follows closely that of rH , with two main differences. One is that rH is

conservative (its mean value does not depend on the resolution r to which K is developed) whereas q is non-conservative(its mean value depends on r and approaches zero as r ! 1). Also the spectral density of q vanishes as r ! 1; hence

the spectral density tensors that we obtain here for q are for finite r. The other difference is thatrH is a potential field (it

is the gradient of H ), whereas q is a solenoidal field (due to conservation of flow, q has zero divergence). The fact that qis not a potential field introduces some algebraic complications in the spectral analysis.

We start by relating the spectral densities of qr1and q

rr1. Using Eqs. (2b) and (8), we obtain

Sqrr1

ðkÞ ¼ r�DE½B2r �ERr

RrSqr1

ðRTr k=rÞRT

r

� �ð19Þ

where, from [1, Eq. (17)], E½B2r � ¼ r2ððD�1Þ=DÞCK r�ð4=DÞCK . Eq. (19) is analogous to Eq. (13) for rH . The factor r�ð4=DÞCK in

the expression of E½B2r � is the square of jE½qr�j ¼ r�ð2=DÞCK (see [1, Eq. (22)]) and reflects the non-conservative property of

Page 7: Nonlinear spectral analysis of flow through multifractal porous media

D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 299

q. This factor is also responsible for the vanishing behavior of Sqras r ! 1. To obtain a non-degenerate limit, we define

the conservative normalized flow q0rðxÞ ¼ r2CK=Dq

rðxÞ. The spectral density tensor of the normalized flow is given by

Sq0rðkÞ ¼ r4CK=DSq

rðkÞ and, from Eq. (19), satisfies

Sq0rr1

ðkÞ ¼ r�Dþ2CK ððD�1Þ=DÞERr

RrSq0r1

ðRTr k=rÞRT

r

� �ð20Þ

It follows that Sq0 ðkÞ ¼ limr!1 Sq0rðkÞ scales as

Sq0 ðkÞ ¼ r�Dþ2CK ððD�1Þ=DÞERr

RrSq0 ðRTr k=rÞRT

r

h ið21Þ

Assuming that at the low-wavenumber end of the scaling range (for k ¼ k0 with amplitude k0) Sq0 ðk0Þ is given by Eq.

(12b) and (21) gives

Sq0 ðrk0Þ ¼2

SDCKk�D

0

� �r�Dþ2ððD�1Þ=DÞCK E

Rr

Rrðdi1 � er1eriÞðdj1 � er1erj Þ

i; j ¼ 1; . . . ;D

� �RTr

ð22Þ

where eri is the ith component of the vector er ¼ RTr e and e is the unit vector in the direction of k0. If K is multifractal

down to resolution rmax and no further (meaning that the spectral density of F ¼ lnðKÞ is zero beyond rmaxk0), thespectral density tensor of q is SqðkÞ ¼ r�4CK=D

max Sq0 ðkÞ for k < rmaxk0 and SqðkÞ ¼ 0 otherwise.

To obtain a more explicit form of Sq0 ðkÞ, one must evaluate the expectation term in Eq. (22). For D ¼ 1, this term is

1. For D ¼ 2, we obtain in Appendix C that

Sq0 ðrk0Þ ¼1

pk�20 CK

� �r�2þCK ðð1� brÞe21 þ bre22Þ

e22 �e1e2�e1e2 e21

� �ð23Þ

where br ¼ E½cos2ðarÞ� and ar has normal distribution with mean zero and variance 14CK lnðrÞ. Notice that

(i) Since q0 has zero divergence, its spectral density tensor must satisfyP

i ki½Sq0 ðkÞ�ij ¼ 0 for each j andPj kj½Sq0 ðkÞ�ij ¼ 0 for each i [21, Eq. (9.4)]. Eq. (23) satisfies these conditions.

(ii) For k in the direction of k1 (for e1 ¼ 1 and e2 ¼ 0), Sq01q01ðkÞ and Sq0

1q02ðkÞ vanish and Sq0

2q02ðkÞ becomes

Sq02q02

rk00

� �� �¼ ð1� brÞ

1

pk�20 CK

� �r�2þCK ð24Þ

The term ð1� brÞ in Eq. (24) is close to zero for small r (at low frequencies) and approaches 0.5 as r ! 1.

Contrary to linear theory, the spectral density Sq02q02ðkÞ does not vanish identically along k1. The reason for this non-

vanishing behavior is the random rotation of the average flow vector at small scales.

(iii) Like the hydraulic gradient rH , the specific flow q is anisotropic at large scales but tends to isotropy at small

scales. The scale below which q may be considered isotropic is controlled by the term br and hence by CK , which

appears in the variance of ar. For example, if one considers isotropy to be effectively realized when br ¼ 0:6, thenthis requires Var½ar� ¼ 0:9 and a resolution riso ¼ e3:6=CK , e.g. riso ¼ 8103 for CK ¼ 0:4. If riso exceeds the multifractal

scaling range rmax of the hydraulic conductivity, then near-isotropy is not observed, even at the smallest scales.

The spectral components Sq01q01ðkÞ, Sq0

2q02ðkÞ and Sq0

1q02ðkÞ are contour plotted in Figs. 2–4, using a representation similar

to Fig. 1. In each figure, the nonlinear analysis results in Eq. (23) are compared with the linear analysis results in Eq.

(11) for CK ¼ 0:1 and CK ¼ 0:3. The contours in Figs. 2–4 have a log-spacing of )2, starting from an innermost log-level

of )2. As in the case of the hydraulic head fluctuations, the main differences between the linear and nonlinear spectra

are that the latter are flatter and consistent with the fact that, at small scales, q approaches isotropy (at large wave-

numbers, Sq01q01ðkÞ and Sq0

2q02ðkÞ are identical except for a 90� rotation and Sq0

1q02ðkÞ is symmetrical). Both features (slow

decay of the spectrum relative to the linear case and high-wavenumber isotropy) are more pronounced for higher CK .

4. Implications on macrodispersivity

We now examine the implications of the previous flow results on the transport of solutes. How a solute spreads with

mean travel time or mean travel distance from the point of injection is usually described by the dispersion tensor fDijg

Page 8: Nonlinear spectral analysis of flow through multifractal porous media

Log(k/ko)Log(k/ko)

Log(k/ko) og(k/ko)

Log

(k/k

o)L

og(k

/ko)

Linear Theory Nonlinear Theory CK =0.1

Linear Theory Nonlinear Theory CK =0.3

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

L

Fig. 2. Linear and nonlinear power spectrum Sq1q1 of the longitudinal specific discharge for CK ¼ 0:1 and 0.3.

300 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

and macrodispersivity tensor fAijg, respectively. The dispersion coefficients Dij give the growth rate in time of the

second spatial moments of the solute concentration Mij, whereas the macrodispersivity coefficients Aij give the growth

rate of Mij with mean travel distance.

When the mean flow has the direction of x1, the growth rate of Mij with mean travel distance hx1i is accurately

described by the following expression, which has been presented in equivalent forms in [22–25]:

Aijðhx1iÞ ¼1

2

dMij

dhx1i¼ n2

�qq2

Z hx1i

0

Z 1

�1eðik1�ak2ÞnSqiqjðkÞdkdn ð25Þ

where MijðtÞ ¼ nm

R1�1ðxi � hxiðtÞiÞðxj � hxjðtÞiÞCðx; tÞdx are the ensemble second spatial moments, hxi is the center of

mass of the ensemble average concentration, Cðx; tÞ the ensemble average concentration, n the porosity, assumed

spatially constant, m the total mass of injected solute, �qq the uniform mean flow in the x1 direction, a the pore scale

dispersion coefficient and SqiqjðkÞ is the specific discharge spectrum.

Eq. (25) estimates the second moment growth of the solute plume relative to the ensemble centroid.

Page 9: Nonlinear spectral analysis of flow through multifractal porous media

Log(k/ko)Log(k/ko)

Log(k/ko)Log(k/ko)

Log

(k/k

o)L

og(k

/ko)

Linear Theory Nonlinear Theory CK =0.1

Linear Theory Nonlinear Theory CK =0.3

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

Fig. 3. Linear and nonlinear power spectrum Sq2q2 of the transverse specific discharge for CK ¼ 0:1 and 0.3.

D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 301

Alternatively, one can calculate the second moments of the plume relative to the actual centroid �xxðtÞ. The latter

moments are RijðtÞ ¼ E½SijðtÞ� where

SijðtÞ ¼nm

Z þ1

�1ðxi � �xxiðtÞÞðxj � �xxjðtÞÞCðx; tÞdx ð26Þ

In general, the ensemble moments Mij are larger than the relative moments Rij; in fact the ensemble moments include

two sources of plume dispersion: the random location of the plume centroid �xxðtÞ and the expected plume size Rij.

Following Rajaram and Gelhar [16], the macrodispersivities relative to the plume centroid, Arij, can be evaluated as

Arijð�xx1Þ ¼

1

2

dRij

d�xx1¼ n2

�qq2

Z �xx1

0

Z 1

�1eðik1�ak2Þn 1

n� e�kikjRijð�xx1Þ

oSqiqj ðkÞdkdn ð27Þ

Here we limit consideration to the ensemble macrodispersivities. Specifically, we compare the dispersion coefficients Aij

in Eq. (25) when Arij in Eq. (25) when SqiqjðkÞ is taken to be the spectrum in Eq. (11d) from linear theory or the spectrum

in Eq. (22) from nonlinear theory. It should be noted that here ‘‘linear’’ and ‘‘nonlinear’’ refers to the method used to

Page 10: Nonlinear spectral analysis of flow through multifractal porous media

Log(k/ko)Log(k/ko)

Log(k/ko)Log(k/ko)

Log

(k/k

o)L

og(k

/ko)

Linear Theory Nonlinear Theory CK =0.1

Linear Theory Nonlinear Theory CK =0.3

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

02

4

-4 -2 0 2 4

-4-2

0

2

4

Fig. 4. Linear and nonlinear cross power spectrum Sq1q2 of the longitudinal and transversal specific discharge for CK ¼ 0:1 and 0.3.

302 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

determine the spectral tensor of the flow q. In both cases we use Eqs. (25) and (27), which were derived from linear

theory.

Fig. 5 shows plots of the longitudinal and transversal macrodispersivities, A11 and A22, against the dimensionless

mean travel distance �xx1k0 or hx1ik0 for the planar case (D ¼ 2), zero pore scale dispersion a, and codimension parameter

CK ¼ 0:3. As a consequence of local isotropy of the flow field (here multifractality extends to very high wavenumbers),

the longitudinal and transversal macrodispersivities from nonlinear theory are practically identical at short travel

distances. By contrast, the macrodispersivity from linear theory remains anisotropic at all scales. Another difference is

that nonlinear theory gives larger macrodispersivities at small distances; this is due to the increased high-frequency

content of the flow predicted by the nonlinear analysis. Note that for small travel distances Aijð�xx1Þ / �xxc1, where c ¼ 1

according to linear theory and c ¼ 1� CK according to nonlinear theory.

Qualitatively similar differences between linear and nonlinear analysis are found for macrodispersivities Ar11 and Ar

22.

In particular, like the ensemble macrodispersivities in Fig. 5, the relative macrodispersivities from nonlinear theory are

higher than those from linear theory and are identical at short distances due to local isotropy of the flow field.

Page 11: Nonlinear spectral analysis of flow through multifractal porous media

Fig. 5. Comparison between linear and nonlinear ensemble macrodispersivities for CK ¼ 0:3.

D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 303

5. Summary and discussion

In this and a previous paper [1], we have analyzed the flow in a D-dimensional porous medium under the condition

that the hydraulic conductivity K is an isotropic lognormal field with multifractal scale invariance. Isotropic multi-

fractality is a scale invariance property whereby, after the support of K is isotropically contracted by a given factor and

the amplitude of K is multiplied by a suitable independent random variable, the transformed field is statistically

identical to the original field. There is evidence that this modeling hypothesis is satisfied by at least some natural

aquifers.

In [1] we have shown that, when K is multifractal, also the hydraulic gradient rH and specific flow vector q satisfy

multifractal scaling conditions. These conditions involve random rotations of the vector fields; hence the multifractality

of rH and q is of the type often referred to as generalized scale invariance (gsi) [3]. In [1], we derived the distribution of

the random amplitude scaling factor and random rotation matrix that characterize this gsi property. To the authors�knowledge, this is the first time that gsi has been derived from first principles. Then, using the scaling properties of rHand q, we have obtained the marginal distributions of these two vector fields. In particular, we have found that rH and

q have lognormal amplitudes and that their orientation has the distribution of Brownian motion on the unit sphere. As

a by-product of this result, we have obtained an expression for the effective conductivity under ergodic conditions. The

expression corresponds to an early conjecture by Matheron [6].

The present paper has continued the flow analysis by deriving consequences of multifractality of rH and q in the

frequency domain. First we have obtained the spectral density tensors of these vector fields. In contrast with traditional

first-order (FO) analysis, which neglects nonlinearities under the assumption that fluctuations are small, our analysis is

nonlinear. We find that FO results are inaccurate in various respects. The FO spectral tensors of rH and q have er-

roneous (too high) decay exponents along any given direction in Fourier space and incorrectly characterize the an-

isotropy ofrH and q as being the same at all scales. Our nonlinear analysis shows thatrH and q are indeed anisotropic

at large scales, but become progressively isotropic at small scales. This scale-dependent behavior is important for the

evaluation of the macrodispersivities, which are themselves anisotropic at large scales and nearly isotropic locally, as

generally observed from field experiments.

Beyond these specific results, we have shown how the problem of flow through random porous media can be an-

alyzed using novel nonlinear methods. The reason why traditional perturbation methods are inadequate is that the

fluctuations encountered in flow analysis are typically large and higher-order terms tend to be significant.

Our treatment has been limited to the case when K is isotropic lognormal and multifractal, which means that the log

conductivity F ¼ lnðKÞ has a power spectral density of the type SF ðkÞ / k�D, where D is the space dimension. One

sometimes observes power spectra of F that decay like k�a with a smaller or larger than D [16]. In this case the hydraulic

conductivity is not multifractal, although K may share the multiplicative structure of multifractal fields. We are cur-

rently extending the present theory to this more general case and in fact to the case when K is any isotropic lognormal

field, not necessarily with a spectrum of the power type.

Another needed extension is to treat cases with anisotropic hydraulic conductivity, for example K fields that display

different correlation decay on the horizontal plane and in the vertical direction. Although multifractality is compatible

with anisotropy, this extension is expected to pose significant technical problems.

Page 12: Nonlinear spectral analysis of flow through multifractal porous media

304 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

Acknowledgements

Support for this research has been provided by the Italian National Research Council, under a Cooperative

Agreement for the Study of Climatic Changes and Hydrogeologic Risks, and by the National Science Foundation

under Grant CMS-9612531. Partial funding for Albert Essiam has been provided by the Office of Naval Research

(ONR) and the Historically Black Engineering Colleges (HBEC) fellowship and by the Broken Hill Propriety (BHP)

Minerals Exploration Department. The authors are grateful to Lynn Gelhar for many useful discussions.

Appendix A. Proof of spectral results in Section 2.1

We derive Eqs. (1) and (5) and other scaling relations in Fourier space for isotropic multifractal measures KðSÞ. Key

to these results is the multifractal renormalization property of the spectral measure of K, which we derive first and show

to be dual of the renormalization property of K in physical space.

A.1. Isotropic multifractality in Fourier space

For KðSÞ a homogeneous random measure in RD, the average measure density KðSÞ ¼ KðSÞ=jSj has the Fourier

representation

KðSÞ ¼ 1

jSj

ZRD

bIISðkÞbKK ðdkÞ ðA:1Þ

where jSj is the volume of S, bIISðkÞ is the D-dimensional Fourier transform of the indicator function of S, and bKK ðdkÞ is acomplex measure in Fourier space, whose properties are given in [26]. Since bIIrSðkÞ ¼ rDbIISðrkÞ and rS has volume rDjSj,substitution of Eq. (A.1) into Eq. (3) gives

1

jSj

ZRD

bIISðkÞbKK ðdkÞ ¼d rD1

rDjSjAr

ZRD

bIISðrkÞbKK ðdkÞ ¼d Ar1

jSj

ZRD

bIISðk0ÞbKK ðdk0=rÞ ðA:2Þ

where we have used k0 ¼ rDk. Eq. (A.2) must hold for any region S. Therefore, the spectral measure bKK ðdkÞ must satisfy

bKK ðdkÞ ¼d ArbKK ðdk=rÞ; rP 1 ðA:3Þ

Eq. (A.3) is the dual in Fourier space of the renormalization property in Eq. (3). An important difference between the

two scaling conditions is that K is multifractal under contraction, whereas bKK is multifractal under dilation; see [4].

Another difference is that K is statistically homogeneous, whereas bKK is non-homogeneous. Next we use Eq. (A.3) to

derive Eqs. (5a) and (5b).

A.2. Derivation of Eqs. (5a) and (5b)

Suppose that the spectral density SKðkÞ ¼ E½jbKK ðdkÞj2 �dk exists. Then, from Eq. (A.3),

SKðkÞ ¼ r�DE½A2r �SKðk=rÞ ðA:4Þ

Substitution of E½A2r � ¼ rKð2Þ produces Eq. (5a).

To determine the spectral density of lnK, we work with the sequence of low-passed fields KrðxÞ obtained by elim-

inating all Fourier components of lnðKÞ outside the range k0 6 jkj6 rk0, where k0 is a positive constant. Having

eliminated the high-wavenumber components, the point values KrðxÞ exist. The average value of KrðxÞ in S is denoted by

KrðSÞ.Suppose that KðSÞ ¼ limr!1 KrðSÞ satisfies Eq. (3). Then, at least for large r1 and any rP 1

Krr1ðxÞ¼d ArKr1ðrxÞ ðA:5Þ

or

lnðKrr1ðxÞÞ¼dlnðArÞ þ lnðKr1ðrxÞÞ ðA:6Þ

It follows that the spectral densities of lnðKrr1 Þ and lnðKr1Þ must satisfy SlnðKrr1 ÞðkÞ ¼ rDSlnðKr1 Þðk=rÞ. Hence, if SlnðKrÞconverges as j ! 1 to a finite limit SlnðKÞ, then this limit must satisfy

Page 13: Nonlinear spectral analysis of flow through multifractal porous media

D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 305

SlnðKÞðkeÞ / k�D ðA:7Þ

where e is any given unit wavenumber vector. Eq. (5b) is a restatement of Eq. (A.7).

A.3. Derivation of Eq. (1)

In the isotropic case, Eq. (A.7) gives

SlnðKÞðkÞ ¼ ck�D ðA:8Þ

for some c and k ¼ jkj. We want to show that, when Ar in Eq. (3) has lognormal distribution with unit mean,

c ¼ 2CK=SD and thus Eq. (1) applies.

We notice that Ar and Kr1ðrxÞ in Eq. (A.5) are independent. Then, from Eq. (A.6),

Var½lnðKrr1ðxÞÞ� ¼ Var½lnðArÞ� þ Var½lnðKr1ðrxÞÞ� ðA:9Þ

The spectral density of Kr1 is the same as SlnðKÞ in Eq. (A.8) in the interval k0 6 k < r1k0 and is zero otherwise. Therefore,

using Eq. (A.9),

Var½lnðArÞ� ¼ Var½lnðKrr1ðxÞÞ� � Var½lnðKr1ðrxÞÞ� ¼ cZr1k0 6 k<rr1k0

k�Ddk ¼ cSD lnðrÞ ðA:10Þ

where SD is the same constant as in Eq. (1). We conclude that c ¼ 1SD

Var½lnðArÞ�lnðrÞ . From the review of multifractal fields in

Section 2 of [1], Var½lnðArÞ�lnðrÞ ¼ Var½lnðAeÞ� is independent of r. Moreover, for Ae lognormal with mean value 1,

Var½lnðAeÞ� ¼ lnE½A2e � ¼ WKð2Þ ¼ 2CK . Hence c ¼ 2CK=SD and Eq. (1) follows from Eq. (A.8).

Appendix B. Second-order spectral density corrections for multifractal K

Dagan [17] has made a second-order spectral analysis of the hydraulic head fluctuation hðxÞ. He has found that the

second-order spectra S00fhðkÞ and S00

h ðkÞ are obtained by multiplying the first-order spectra by ½1þ LðkÞ� and ½1þ 2LðkÞ�,respectively, where LðkÞ is given by

LðkÞ ¼ � 1

ð2pÞD=2ZRD

½k1 � ðk1 þ kÞ�ðk � k1Þðk21 þ k2Þk2 1

�þ k � k1

k21

�Sf ðk1Þdk1 ðB:1Þ

To examine the behavior of the integral in Eq. (B.1) near the origin and at infinity, we set k ¼ ke and k1 ¼ k1e1, where eand e1 are unit vectors. Then the factor that multiplies Sf ðk1Þ becomes

ðk21kÞ½e1 � ðk1e1 þ keÞ�ðe � e1Þðk21 þ k2Þk2 1

�þ k1k

k21ðe � e1Þ

�¼ k21

k21 þ k2k1k

�þ ðe � e1Þ

�ðe � e1Þ 1

�þ 1

ðk1=kÞðe � e1Þ

�ðB:2Þ

For isotropic K fields, Sf ðk1Þ has rotational symmetry. Therefore, what matters in Eq. (B.1) is the average of the term in

Eq. (B.2) for e1 a vector on the unit spherical surface in RD. Terms in ðe � e1Þnwith n odd make no contribution to this

average. Neglecting these terms, the expression in Eq. (B.2) reduces to

2k21

k21 þ k2ðe � e1Þ

2 ðB:3Þ

Notice that ðe � e1Þ2 ¼ ðcos aÞ2 where a is the angle between e and e1. From the Appendix of [1], the expected value of

ðcos aÞ2 is 1=D. Hence one may replace the integrand in Eq. (B.1) with

2

Dk21

k21 þ k2Sf ðk1Þ ðB:4Þ

As ðk1=kÞ ! 1, the ratio k21=ðk21 þ k2Þ ! 1, implying divergence of LðkÞ for Sf ðkÞ / k�a with a6D. The value a ¼ D,which corresponds to a multifractal K field, is included in this condition for divergence. The case a < D corresponds to

fractional Gaussian noise (fGn) models of logK.

Page 14: Nonlinear spectral analysis of flow through multifractal porous media

306 D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307

For ðk1=kÞ ! 0, the ratio k21=ðk21 þ k2Þ behaves like k21 . Therefore, low-frequency divergence of LðkÞ occurs only for

aPDþ 2, which does not include the multifractal case. We conclude that, for multifractal K, the second-order cor-

rection factors to S0fhðkÞ and S0

hðkÞ have high-frequency divergence. Also notice that k21=ðk21 þ k2ÞP 0. Therefore, in the

pre-multifractal case when the scaling range extends to a large but finite wavenumber, LðkÞ is large negative and the

second-order spectrum S00h ðkÞ ¼ ½1þ 2LðkÞ�S0

hðkÞ is large negative. We conclude that second-order spectral analysis is not

useful for multifractal hydraulic conductivity fields.

Appendix C. The expectation term in Eq. (22) for D5 2 and D5 3

We analyze the expected value

ERr

Rrðdi1 � er1eriÞðdj1 � er1erjÞ

i; j ¼ 1; . . . ;D

� �RTr

ðC:1Þ

where Rr is the random rotation matrix in Eq. (2), eri is the ith component of the vector er ¼ RTr e, and e is the unit vector

in the direction of k0. The term in Eq. (C.1) appears in the spectral density tensor of the specific flow q; see Eq. (22). We

obtain an explicit expression for the matrix in Eq. (C.1) for the case D ¼ 2 and a more manageable but not completely

explicit expression for D ¼ 3.

D ¼ 2

Using ð1� e2r1Þ ¼ e2r2 , the matrix in brackets in Eq. (C.1) may be written as

ðdi1 � er1eriÞðdj1 � er1erjÞi; j ¼ 1; 2

� �¼ 1 0

0 0

� �� 2e2r1 er1er2

er1er2 0

� �þ e2r1

e2r1 er1er2er1er2 e2r2

� �

¼ ð1� e2r1Þ1 0

0 0

� �þ e2r2

0 0

0 1

� �� ere

Tr þ e2r1ere

Tr ¼ e2r2ðI � ere

Tr Þ ðC:2Þ

Therefore, the expectation in Eq. (C.1) becomes

ERr

Rrðdi1 � er1eriÞðdj1 � er1erjÞ

i; j ¼ 1; 2

� �RTr

¼ E

Rr

½e2r2 �ðRrRTr � Rrere

Tr R

Tr Þ ¼ E

Rr

½e2r2 �ðI � eeTÞ ¼ ERr

½e2r2 �e22 e1e2e1e2 e21

� �ðC:3Þ

Using er2 ¼ � sinðarÞe1 þ cosðarÞe2 where ar has normal distribution with mean zero and variance 14CK lnðrÞ [1,

Section 3.2] and the fact that E½sinðarÞ cosðarÞ� ¼ 0 due to symmetry of the distribution of ar, we obtain ERr

½e2r2 � ¼ð1� brÞe21 þ bre22 where br ¼ E½cos2ðarÞ�. Hence, in the two-dimensional case, the matrix in Eq. (C.1) may be written

explicitly as

ERr

Rrðdi1 � er1eriÞðdj1 � er1erjÞ

i; j ¼ 1; 2

� �RTr

¼ ðð1� brÞe21 þ bre22Þ

e22 �e1e2�e1e2 e21

� �ðC:4Þ

D ¼ 3

Results in the 3-D case are less explicit, although Eq. (C.1) can be simplified somewhat. For D ¼ 3, the matrix in

brackets in Eq. (C.1) may be written as

ðdi1 � er1eriÞðdj1 � er1erjÞi; j ¼ 1; . . . ; 3

� �¼

1 0 0

0 0 0

0 0 0

24

35�

2e2r1 er1er2 er1er3er1er2 0 0

er1er3 0 0

24

35þ e2r1ere

Tr

¼ ð1� 2e2r1Þ1 0 0

0 0 0

0 0 0

24

35�

0 er1er2 er1er3er1er2 0 0

er1er3 0 0

24

35þ e2r1ere

Tr ðC:5Þ

Then, using Eq. (C.5), the expectation in Eq. (C.1) becomes

ERr

Rr

ðdi1 � er1eriÞðdj1 � er1erjÞi; j ¼ 1; . . . ; 3

� �RTr

¼ E½ð1� e2r1ÞRr1R

Tr1� � E½er1er2ðRr1R

Tr2þ Rr2R

Tr1Þ�

� E½er1er3ðRr1RTr3þ Rr3R

Tr1Þ� þ E½e2r1 �ee

T ðC:6Þ

where Rri is the ith column of Rr.

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D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 307

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