nonlinear spectral analysis of flow through multifractal porous media
TRANSCRIPT
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
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
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
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
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 .
-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
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
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.
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
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.
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.
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
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.
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.
D. Veneziano, A.K. Essiam / Chaos, Solitons and Fractals 19 (2004) 293–307 307
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