Molz, FJ, Rajaram, H. y Lu, S. (2004). Modelos de heterogeneidad basados en fractales estocásticos en hidrología del.pdf

STOCHASTIC FRACTAL-BASED MODELS OF
HETEROGENEITY IN SUBSURFACE HYDROLOGY:
ORIGINS, APPLICATIONS, LIMITATIONS,
AND FUTURE RESEARCH QUESTIONS
Received 12 February 2003; revised 30 September 2003; accepted 28 October 2003; published 6 March 2004.
[1] Modern measurement techniques have shown that
property distributions in natural porous and fractured media
appear highly irregular and nonstationary in a spatial
statistical sense. This implies that direct statistical analyses
of the property distributions are not appropriate, because
the statistical measures developed will be dependent on
position and therefore will be nonunique. An alternative,
which has been explored to an increasing degree during the
past 20 years, is to consider the class of functions known
as nonstationary stochastic processes with spatially
stationary increments. When such increment distributions
are described by probability density functions (PDFs) of
the Gaussian, Levy, or gamma class or PDFs that converge
to one of these classes under additions, then one is also
dealing with a so-called stochastic fractal, the mathematical
theory of which was developed during the first half of the
last century. The scaling property associated with such
fractals is called self-affinity, which is more general that
geometric self-similarity. Herein we review the application
of Gaussian and Levy stochastic fractals and multifractals
in subsurface hydrology, mainly to porosity, hydraulic
conductivity, and fracture roughness, along with the
characteristics of flow and transport in such fields.
Included are the development and application of fractal
and multifractal concepts; a review of the measurement
techniques, such as the borehole flowmeter and gas
minipermeameter, that are motivating the use of fractal-
based theories; the idea of a spatial weighting function
associated with a measuring instrument; how fractal fields
are generated; and descriptions of the topography and
aperture distributions of self-affine fractures. In a
somewhat different vein the last part of the review deals
with fractal- and fragmentation-based descriptions of
fracture networks and the implications for transport in
such networks. Broad conclusions include the implication
that models based on increment distributions, while more
realistic, are inherently less predictive than models based
directly on stationary stochastic processes; that there is
presently an unresolved ambiguity when a measurement is
attempted in a medium that exhibits property variations on
all scales; the strong possibility that log(property)
increment distributions that appear to be described by the
Levy PDF are actually superpositions of several PDFs of
finite variance, one for each facies; that there are apparent
similarities in the transport behavior of heterogeneous
porous media and fractured rock at the field scale that
appear to be related to the existence of a few preferential
flow paths in both types of media; and finally, that
additional carefully collected data sets are needed to clarify
and advance the fractal-based theories, particularly in the
case of three-dimensional fracture networks where few data
are available. Further refinement is needed also in the
understanding of instrument spatial weighting functions
in heterogeneous media and how measurements in
media exhibiting variations on all scales should be
interpreted. INDEX TERMS: 1829 Hydrology: Groundwater
hydrology; 1832 Hydrology: Groundwater transport; 1869
Hydrology: Stochastic processes; 1894 Hydrology: Instruments
and techniques; 3250 Mathematical Geophysics: Fractals and
multifractals; KEYWORDS: subsurface hydrology, stochastic
fractals, heterogeneity, transport, self-affine fractures, fracture
networks.
Citation: Molz, F. J., H. Rajaram, and S. Lu (2004), Stochastic fractal-based models of heterogeneity in subsurface hydrology:
Origins, applications, limitations, and future research questions, Rev. Geophys., 42, RG1002, doi:10.1029/2003RG000126.
1. INTRODUCTION
[2] For individuals not familiar with the discipline of
subsurface hydrology, it is probably worth reviewing the
subject matter and basic motivations, both practical and
intellectual, which drive the field. From a hydrologic
viewpoint, nations have traditionally been interested in the
subsurface because it contains a huge supply of fresh water,
second only to that contained in ice caps and glaciers. Thus,
during most of the twentieth century, water supply problems
Fred J. Molz
Environmental Engineering
and Science Department
Clemson University
Clemson, South Carolina, USA
Harihar Rajaram
Department of Civil,
Environmental and
Architectural Engineering
University of Colorado
Boulder, Colorado, USA
Silong Lu
Tetra Tech, Inc.
Atlanta, Georgia, USA
Copyright 2004 by the American Geophysical Union.
8755-1209/04/2003RG000126
Review of Geophysics, 42, RG1002 / 2004
1 of 42
Paper number 2003RG000126
RG1002

dominated the interests of subsurface hydrologists, and
water supply is still the major interest in many parts of
the world. Evaluating the gross water supply capability of
an aquifer (body of rock that both stores and transmits water
in economical quantities) is based mainly on large-scale
measures of water transmission properties, such as hydrau-
lic conductivities resulting from pumping tests. When con-
structing a well for a water supply purpose, relatively little
concern is given to whether most of the water enters the top
half of the well, the bottom half, or some other location. The
success or failure of the construction will depend mainly on
the total amount of water that the well delivers over an
extended period of time. This is not to say that water supply
problems are insignificant; they certainly are not. However,
their solution requires information that is different from
much of the material reviewed herein.
[3] Starting about 35 years ago with the beginning of the
modern environmental movement, the concern of various
governments and regulatory agencies began to shift, to an
increasing extent, toward groundwater quality problems.
Because groundwater is such a valuable natural resource,
it became important to protect it from a myriad of point and
nonpoint pollution sources. It was also deemed necessary to
identify and to hold responsible for their actions those
individuals, groups of individuals, companies, and govern-
ment agencies causing pollution. In order to answer ques-
tions related to water quality and pollution problems, a
much more detailed understanding of water and solute
movement in the subsurface is required. If groundwater
pollution was discovered at a particular location, questions
often arose as to where the pollutants came from, how long
it took them to get from their source(s) to the location where
they were detected, if and how rapidly were they decaying,
and where they would travel in the future. To answer such
questions, a much more detailed understanding of the
groundwater flow pattern is needed than that required for
water supply purposes. Subsurface hydrologists began talk-
ing about distributions of hydraulic conductivity (K) (the
measured quantity relating groundwater flux (Darcy veloc-
ity) to hydraulic head gradient), particularly the distribution
as a function of vertical position (along a well screen) in an
aquifer and also actual point values. As increasingly so-
phisticated measurements were attempted, it was realized
that much more heterogeneity was present in natural K and
other property distributions than expected initially. In this
way, subsurface hydrologists were led into the detailed
study of property distributions, such as K, porosity, and
other geophysical variables in natural, heterogeneous sedi-
ments, motivated by the complex mix of governmental and
private groundwater quality concerns. This review is de-
voted to these types of detailed measurements and one of
their most recent interpretations.
[4] From an intellectual viewpoint, which obviously is
very important to many scientists cited in this review, the
modern story of subsurface hydrology is the classical
scientific process of searching for order. Where is the order
in the hydraulic properties of natural heterogeneous sedi-
mentary materials, upon which an understanding can be
built? Early efforts at understanding used deterministic
concepts built around assumed homogeneity within geo-
logic units or else smooth, gradual variation. This was
followed by the rise of stochastic theory in subsurface
hydrology, based initially on treating heterogeneous property
distributions as stationary, correlated, random processes.
Here the order was assumed to be in the statistical charac-
terization (probability density functions (PDFs), means,
variances, etc.) of the properties themselves or the loga-
rithms of the properties. This approach has had limited
success because property distributions based on these sta-
tistically homogeneous concepts were too regular when
compared to those that occur in natural systems. The next
step in the search for order was to consider statistically
heterogeneous concepts that conceptualize heterogeneity in
terms of nonstationary stochastic processes with stationary
increments, the mathematical basis for stochastic fractals.
The hoped-for order then lies in the property or log(prop-
erty) increment statistics and scaling properties displayed by
the increment PDFs. Thus deterministic and homogeneous,
deterministic and heterogeneous, stochastic and statistically
homogeneous, and stochastic but statistically heterogeneous
define the evolving intellectual pathway that has been
followed in the attempt to describe our evolving concepts
of pervasive natural heterogeneity. There is certainly no
universal agreement on where we are or where we should be
in this hierarchy. Numerous viewpoints are valid, especially
when one considers the difficulty of measuring property
distributions in porous media and the approximations needed
to perform practical analyses and simulations of contempo-
rary problems. So the question naturally arises, Are the
emerging fractal-based theories the final step in understand-
ing natural heterogeneity? The present authors feel that the
answer is probably no, and regardless of the final result, we
view the stochastic fractal-based model as a hypothesis
undergoing refinement at the present time but hopefully
leading in the right direction. However, as the present
review will suggest, the emerging statistically heteroge-
neous concepts offer a new perspective and capture many
of the properties of subsurface heterogeneity in a natural
way. They have also motivated new attempts at property
measurement on a wide variety of scales.
[5] Most subsurface hydrologists and hydrogeologists
would agree that the central unsolved problem preventing
a satisfactory understanding of flow and transport processes
in natural geologic media revolves around property mea-
surement and how to deal with the pervasive heterogeneity
that is typical of such domains. The problem is multifaceted:
How should one define relevant properties, how can such
properties be measured, and how should the measurements
be used in (hopefully) predictive models? Related problems
exist in other areas of geophysics, for example, the atmo-
sphere and the oceans, where physical properties are not as
variable but fluid flow processes are turbulent. In all three
areas, whether one can ultimately produce classical deter-
ministic predictions is an open question, especially since the
development and understanding of the physical/mathemat-
ical basis for deterministic chaos [Lighthill, 1994; Turcotte,
RG1002 Molz et al.: FRACTALS IN SUBSURFACE HYDROLOGY
2 of 42
RG1002

1997; Faybishenko, 1999]. Unfortunately, we now realize
that increased understanding and increased predictability do
not necessarily coincide.
[6] If one must deal with irregular (heterogeneous) prop-
erty distributions in natural porous and fractured geologic
media, it is logical to attempt to employ mathematical
representations that are also irregular. An infinite variety
of functions fall under the broad classification of irregular,
so within the present review we will limit our consideration
to continuous functions with discontinuous first derivatives.
(In what follows, this is what will be meant by the term
irregular function.) By definition, such functions are not
smooth, so the traditional theoretical basis for interpolation
and extrapolation fails for this class of functions. The
problem is illustrated in Figure 1, where it is shown
schematically that two measurements a short distance apart
do not in any obvious geometrical sense bound an interme-
diate measurement. Thus the classical process of interpola-
tion between measurements simply acts to smooth the data
in an unrealistic manner. In order to maintain the natural
irregularity of data measurements, researchers have found it
necessary to employ a stochastic approach, wherein prop-
erty values between measurements are chosen based on a
PDF rather than some smoothing interpolation scheme.
Thus irregular functions form the logical basis for stochastic
subsurface hydrology, and stochastic fractal-based func-
tions, the central topic of this review, form a subset of this
class.
[7] As mentioned previously, the field of stochastic
subsurface hydrology developed initially based on the
properties of what are called stationary stochastic processes,
with and without autocorrelation. When autocorrelation was
present, the so-called autocorrelation length was usually
assumed to be finite and relatively short. Often such models
of heterogeneity are called ‘‘statistically homogeneous,’’
meaning loosely that for a sufficiently large averaging
volume the mean property value is independent of position.
(Much of this terminology falls within the discipline of
geostatistics [Journel, 1989].) The study of these types of
irregular functions in subsurface hydrology, which we will
refer to as traditional stochastic subsurface hydrology, may
be traced largely to a paper by Freeze [1975], with several
notable contributions following [Bakr et al., 1978; Dagan,
1979; Gelhar and Axness, 1983; Neuman et al., 1987] that
initiated a decades-long study. The stationary stochastic
process field may now be viewed as fairly mature, with
results summarized in several books [Dagan, 1989; Gelhar,
1993; Dagan and Neuman, 1997]. Figure 2 illustrates one-
dimensional examples (realizations) of hypothetical K dis-
tributions derived from stationary stochastic processes, with
and without positive autocorrelation. When compared with
the nonstationary K distribution shown in Figure 3, the
rather featureless nature of the stationary realizations is
evident. Also, data collected throughout the 1980s increas-
ingly supported the irreducible and nonasymptotic scale
dependency of mechanical dispersivity, an observation that
was inconsistent with the common existence of statistically
homogeneous K fields in natural porous media. (For a data
summary, see Gelhar et al. [1985, 1992] and Neuman
[1990].)
[8] The concern with so-called scaling and scale-depen-
dent properties was evident in several overview papers
published in the August 1986 supplement of Water Resources
Research (22(9)) entitled ‘‘Trends and Directions in
Hydrology.’’ Dagan [1986] dealt with transport processes
in groundwater and discussed, among other things, different
measurement or study scales such as the pore scale, labo-
ratory scale, formation scale, and regional scale. Gelhar
[1986], using the significantly different terminology of core
scale, fluvial aquifer scale, alluvial basin scale, and inter-
basin aquifer scale, discussed the idea of variograms with
scale-dependent correlation lengths. The hope was that
within each scale the property measured would be statisti-
cally homogeneous with a finite correlation length, and so
stationary stochastic theory would apply. With hindsight
this appears to be a somewhat inadequate concept. First of
all, the scales that were selected appear at least somewhat
arbitrary. Where does one end and the other begin? When
dealing with natural aquifers, scaling features appear much
more related to the geologic concept of facies than to
Figure 1. Illustration of the difference between (top) a
smooth heterogeneous property distribution and (bottom) an
irregular heterogeneous distribution. For an irregular
distribution with variation on all scales such as a stochastic
fractal-like distribution, it is not precisely clear what an
instrument measures.
3 of 42
RG1002

regional hydrologic scales [Goggin, 1988; Davis et al.,
1993, 1997; Koltermann and Gorelick, 1996; Allen-King
et al., 1998; Lu et al., 2002b]. Within a given facies, there is
more orderly behavior, with the possibility of larger jumps
in property values between facies. Second, there are no
obvious changes in the scale-dependent plots of apparent
dispersivity [Neuman, 1990] that can be related to some
type of universal scales such as formation scale or regional
scale. Again, with hindsight it appears that a more general
theory is needed, and that need was a major motivation for
development of the present theory of nonstationary stochas-
tic processes with stationary increments or the stochastic
fractal theory.
[9] It is noteworthy that within the same supplemental
issue of Water Resources Research, Rodriguez-Iturbe
[1986] discussed, in part, rainfall scaling issues within the
context of nonstationary stochastic processes with stationary
increments. So the time was drawing near for similar
applications in subsurface hydrology. In fact, a symposium
held at the 1988 Fall Annual Meeting of the American
Geophysical Union (Hierarchy in Subsurface Transport,
Session H22B), and a subsequent book-length proceedings
[Cushman, 1990] were probably the earliest collection of
experimental and theoretical work in subsurface hydrology
that was moving unambiguously into the realm of stochastic
fractal concepts. In the meantime, the role of fractures in
flow and transport through rock masses became increasingly
appreciated, and the study of this role developed largely in
parallel with the developments related to flow and transport
in heterogeneous porous media. The roughness on fracture
surfaces was observed to possess a stochastic character, and
a stochastic fractal model of natural surface roughness was
proposed by Brown and Scholz [1985]. In the late 1980s and
throughout the 1990s, additional observations of natural
rock fracture surfaces supported the validity of the stochas-
tic fractal model, and these have motivated attempts to
relate flow and transport properties at the single fracture
scale to the fractal surface properties. At around the same
time the notion of stochastic network models of fracture
systems was proposed in a variety of contexts in the fracture
mechanics and hydrology literature. The importance of
crystalline rock formations as potential sites for waste
repositories greatly motivated the development of models
for fracture networks, which offer fast pathways for flow
and transport in these otherwise low-permeability environ-
ments. As these stochastic network models [e.g., Long et
al., 1982; Andersson and Dverstrop, 1987; Chiles, 1988]
became increasing popular, so did the application of fractal
geometric concepts to describe fracture networks [e.g.,
Barton and Larsen, 1985]. Although the initial application
of fractal geometric concepts to fracture networks focused
more on estimating fractal dimensions, fractal fracture
network models have since been developed to realistically
incorporate scale-dependent fracture density and power law
length distributions for fractures that are characteristic of
fractal networks. These models are beginning to find
applications in explaining scale dependence of flow and
transport processes in fractured rock formations.
[10] Throughout the 1990s much fractal-based research
dealing with the unsaturated zone took place, with applica-
tions mainly to agronomy and soil science. This work was
quite productive and has been reviewed more extensively to
Figure 2. Plots showing the rather featureless variability
inherent in the stationary stochastic processes: (top)
uncorrelated Gaussian noise and (bottom) positively
correlated fractional Gaussian noise.
Figure 3. Plots of fractional Brownian motion (log(K))
with (top) negatively correlated increments and (bottom) the
exponentiated form (K) of the same function. The nonsta-
tionary functions immediately remind one of the types of
distributions often encountered in natural sediments.
4 of 42
RG1002

date than the more geohydrologic-type applications consid-
ered herein. Therefore the unsaturated zone application area
will not be included in the present review. The interested
reader may find several recent and excellent reviews,
including those of Gimenez et al. [1997], Anderson et
al. [1998], Baveye et al. [1998], and Pachepsky et al.
[2000a].
2. CONCEPTUAL ORIGIN OF STOCHASTIC
FRACTALS
[11] As mentioned in section 1, stochastic fractals had
their origin in the mathematics of nonstationary stochastic
processes with stationary increments. For this case the focus
falls naturally on the increments of a property, the difference
between the property values measured at two points a
known distance apart, rather than simply the property value
itself at each point. Thus, if a property is measured at n
points with a constant separation h, one will have n 1
increments associated with the separation h, also called the
lag. These increment distributions may then be studied
using statistical techniques. For example, an obvious first
step would be to attempt to fit a PDF or a cumulative
distribution function (CDF) to the increment distributions.
Thus, with irregular functions having stationary increments,
one is led to study the statistical properties of increment
distributions. Such properties would be expected to vary in
some way with the lag size, which is one way to define
quantitatively the ‘‘scale’’ associated with a set of local
measurements. Since irregular functions appear irregular on
all scales of measurement, mathematical representations
must reproduce this property, with the possibility of doing
so in some orderly manner that (hopefully) agrees with
experiment. It turns out that mathematicians were working
on such representations throughout the first half of the
twentieth century, with that early work collected in the
classical texts of Feller [1968, 1971].
[12] As discussed by Feller [1971], the most fundamental
property of PDFs capable of representing increment distri-
butions in an orderly manner is that such distributions be
infinitely divisible. A probability distribution (PD) is said to
be infinitely divisible if and only if for any number n, it can
be represented as the sum of n independent random varia-
bles with a common distribution. Mathematically, this may
be written as
PDn ¼ PD1;n þ PD2;n þ . . . þ PDn;n: ð1Þ
All so-called stable distributions, which include the Levy
distributions and the Gaussian special case, are infinitely
divisible [Feller, 1971]. This will be obvious when the basic
properties of such distributions are summarized in section 2.
However, it turns out that the most inclusive and
fundamental requirement for a particular distribution type
to be infinitely divisible is that the family be closed under
convolutions [Feller, 1971]. This will be the case if the
convolution of one family member with a second family
member always produces a third member of the family. The
well-known gamma distribution displays this property and
is therefore infinitely divisible. Two independent random
variables (x1 and x2) that follow the gamma distribution will
satisfy
G x1 þ x2
ð Þ ¼ G x1
ð Þ * G x2
ð Þ; ð2Þ
with the asterisk representing the convolution operation. To
the authors’ knowledge, this property of the gamma
distribution may not be exploited fully in subsurface
hydrology to represent irregular property distributions. It
is mentioned here mainly to indicate that there are
candidates other than the stable distributions that have the
property of infinite divisibility.
[13] The important aspects of stable distributions, includ-
ing the Gaussian, as PDFs for property increments of a
given lag will be introduced using the Levy stable nomen-
clature. In general, such distributions in one dimension
may be defined conveniently as the inverse Fourier trans-
form of their characteristic functions, since this inverse
transform in general does not have an analytical expression
[Samorodnitsky and Taqqu, 1994]. Thus the general Levy
stable PDF (LPDF(x)) with zero median may be written as
LPDFðxÞ ¼
1
p
Z1
0
exp sk
j ja
ð Þ cosðkxÞdk; ð3Þ
where s is the width parameter (sa
is analogous to the
variance of Gaussian distributions), k is the Fourier variable,
and a (0 a 2) is the Levy index. Such a distribution has
an undefined (infinite) variance for a 2 and an undefined
mean for a 1. For 1 a 2 the zero mean and median
are identical. When a = 2, equation (3) reduces to the
Gaussian or normal distribution (NPDF(x)), which has the
well-known analytical expression given by [Samorodnitsky
and Taqqu, 1994]:
NPDFðxÞ ¼
1
p
Z1
0
exp sk
j j2

cosðkxÞdk ¼
1
s
ffiffiffiffiffiffi
2p
p exp
x2
2s2

:
ð4Þ
Four stable distributions with different a values are shown
in Figure 4.
[14] The fundamental property of stable distributions
upon which infinite divisibility is based may be stated as
follows: The PDF of n independent random variables,
characterized by stable distributions with fixed index a, is
still a stable distribution with index a and the following
width parameter
sa
¼
X
n
i¼1
sa
i ) s2
¼
X
n
i¼1
s2
i ; ð5Þ
where si is the width parameter for the ith independent
random variable; the second equation is the well-known
variance rule for Gaussian PDFs.
5 of 42
RG1002

[15] In order to make equation (5) more concrete, let us
note specifically that we will be dealing with the increments
of a property P over a lag h, hP. Then the width
parameter of the increments for a given h to the a power is
h(hP)a
i, where the brackets denote the expected value. In
order to change from a set of increments of lag h to a set of
lag 2h one simply adds each neighboring pair of increments
together; that is, for each i, Pi+2 Pi = (Pi+2 Pi+1) +
(Pi+1 Pi). Combining this with equation (5) yields the
scaling relationship:
h 2hP
ð Þa
i ¼ h hP
ð Þa
i þ h hP
ð Þa
i ¼ 2h hP
ð Þa
i; ð6Þ
and for a change from lag h to an arbitrary lag of rh,
equation (6) obviously generalizes to
h rhP
ð Þa
i ¼ rh hP
ð Þa
i; ð7Þ
with a = 2 for the Gaussian case.
[16] As written, the scaling relationship implied by equa-
tion (7) applies to stable distributions with independent
(uncorrelated) increments, which are the increments of Levy
flights (a 2) or Brownian motion (a = 2). Mandelbrot
and Van Ness [1968] generalized the (a = 2) case of
equation (7), and this may be extended to the general case
[Taqqu, 1987] to read
h rhP
ð Þa
i ¼ raH
h hP
ð Þa
i; 0 H 1: ð8Þ
This generalization is consistent with the existence of long-
range positive correlation (H 1/a) or negative correlation
(H 1/a) in the property increments, with the classical case
of independent increments given by H = 1/a. H is called the
Hurst coefficient, and when correlation is present, the
increments constitute fractional Levy noise (a 2) or
fractional Gaussian noise (a = 2). The sums of the
increments are known as fractional Levy motion or
fractional Brownian motion, respectively. Convenient
abbreviations for fractional Levy noise, fractional Levy
motion, fractional Gaussian noise, and fractional Brownian
motion are fLn, fLm, fGn, and fBm, respectively. For a
detailed review and development of the resulting properties
of these stochastic processes within the context of subsur-
face hydrology, see Painter [1995, 1996a, 1996b] and Molz
et al. [1997].
[17] In principle, the manner in which the characteristics
of stable distributions are used to represent an irregular
property may now be stated clearly and simply. In a
mathematical sense one is dealing with an infinite set of
property increment distributions (random variables), each
defined by a different lag h. Such a family of increment
distributions could potentially be represented by an a family
of stable PDFs, because such PDFs display the property of
infinite divisibility. If in reality an a family of stable
distributions does represent an infinite set of property
increment distributions, then for each lag the increments
must be distributed according to the a family member for
that lag, and the expected value of the increments to the a
power (variance in the Gaussian case) must, in general,
satisfy the scaling relationship defined by equation (8). To
determine experimentally if an a family of stable PDFs does
represent an irregular property distribution, one must mea-
sure the property at many locations for a number of lags h,
2h, 3h, etc. For each lag the property increments should be
distributed according to a fixed a stable distribution, the
expected value of the measured increments to the a power
should change with increment size according to equation (8),
and the measured autocovariance function should yield a
fixed H value, also consistent with equation (8) [Molz et al.,
1997].
[18] The scaling relationship represented by equation (8)
relates only to the a moment of the underlying increment
PDF (a-order structure function and the variogram in the
Gaussian case). A natural generalization would be to
consider the extension of this relationship to include all
qth order structure functions, i.e.,
h hP
ð Þq
i ¼ AðqÞhxðqÞ
) rhP
ð Þq
h i ¼ rxðqÞ
h hP
ð Þa
i: ð9Þ
For the q scaling represented by the right-hand side of
equation (9) to hold, A and x must be functions only of q.
Equation (9) gives rise to the concept of a multifractal. The
function x(q) is sometimes called the multifractal spectrum,
anditsvariabilitywithqimpliesanonuniquefractaldimension
[Falconer, 1990; Harte, 2001]. This concept is potentially
more general than the Levy or Gaussian monofractal case
given by x(q) = aH, and early applications in geophysics were
to atmospheric processes [Schertzer and Lovejoy, 1987;
Wilson et al., 1991]. Later applications followed in subsur-
face hydrology; these will be reviewed in section 3.2 [Liu
and Molz, 1997c; Painter and Mahinthakumar, 1999;
Boufadel et al., 2000; Essiam, 2001; Tennekoon et al.,
2003; Veneziano and Essiam, 2003].
[19] As implied by the discussion in this section, and
supported by material in section 4, checking the validity of
Figure 4. Plots of stable probability density distributions
for various values of the Levy index a, with 0 a 2. As
a gets smaller, the probability density function (PDF)
displays increased peaking around the mean and a more
slowly decaying tail that leads to the divergence of all
statistical moments of order a. The Gaussian distribution
results when a = 2.
6 of 42
RG1002

the various types of stochastic fractal models in subsurface
hydrology has been based mainly on making small-scale
property measurements, calculating increment PDFs, and
then checking for the PDF type and required scaling
properties. Definite indications of stochastic fractal behavior
have resulted from these efforts, but very few data sets are
sufficiently well defined and sufficiently large to support a
definitive conclusion, especially when one considers the
complicating aspect of geologic facies to be discussed in
section 4. Motivation for further study would certainly be
enhanced if a physical theory were developed, offering a
potential explanation for why one might expect stochastic
fractals to describe natural heterogeneity. To the authors’
knowledge, to date, such a theory has not been developed.
However, we note in passing that fractals are associated
with chaotic processes, such as many body problems in
Newtonian gravitational physics, weather phenomena, tur-
bulence, etc. [Lighthill, 1994; Turcotte, 1997]. The likely
connection to turbulence is particularly intriguing since
turbulent processes involving air and water are entwined
with the deposition and construction of sediments. What
type of property distribution would a potentially chaotic
process create? Maybe it is no coincidence that the fractal-
based theories now used to describe natural heterogeneity
were previously used to describe the velocity fluctuations of
turbulence, although in this endeavor such models have not
yet succeeded in detail [Molz, 2003].
3. FRACTALS AND MULTIFRACTALS
IN SUBSURFACE HYDROLOGY
3.1. Monofractals
[20] Like many new ideas, the stochastic fractal concept
entered subsurface hydrology through a number of parallel
pathways, with the various contributions motivated by
different but related reasons. Some of the earliest papers
[Ross, 1986; Wheatcraft and Tyler, 1988; Wheatcraft et al.,
1990] were motivated by a desire to offer a concrete reason
for observed scale-dependent dispersion; Ross [1986] dealt
with fracture networks, and Wheatcraft and Tyler [1988]
dealt with porous media. While not based upon the use of
stochastic fractals to represent irregular property distribu-
tions, the main topic of this review, these papers were able
to show that if solute particles followed self-similar fractal
paths, a concentration distribution standard deviation
(width) could grow in proportion to the travel distance
rather than the square root of the travel distance, the case
for diffusion-like (dispersion) processes. Trying to fit a
dispersion coefficient to such a spreading process would
cause the dispersion coefficient to be scale-dependent. In a
somewhat similar vein, Cushman [1991] published the
beginnings of his nonlocal theories of diffusion and disper-
sion in fractal porous media [Cushman and Ginn, 1993a,
1993b; Hassan et al., 1997].
[21] Otherearlycontributions[Neuman,1990;Kemblowksi
and Wen, 1993; Molz and Boman, 1993; Grindrod and Impey,
1993; Painter and Paterson, 1994] were related in varying
degrees to a very influential paper by Hewett [1986] entitled,
‘‘Fractal distributions of reservoir heterogeneity and their
influence on fluid transport’’ [Arya et al., 1988; Fayers and
Hewett, 1992]. Part of the motivation for introducing the
stochastic fractals fGn and fBm to hydrology [Mandelbrot
andVanNess,1968;MandelbrotandWallis,1968,1969a]was
the ability of such models to offer a potential explanation for
the Hurst effect [Hurst, 1951] in terms of stochastic models
with (theoretically) infinite memory (autocorrelation of incre-
ments through time). As such, these models were applied to
time series of various hydrologic and geophysical quantities
such as annual river flows or tree ring thicknesses. Hewett
[1986] changed the independent variable from time to space
and applied the analogous (spatial) concepts to a series of
porosity measurements collected along a well bore. Using
rescaled range analysis [Mandelbrot and Wallis, 1969b], as
well as related spectral techniques, Hewett [1986] calculated a
Hurst coefficient H of 0.86, assuming that the vertical distri-
bution of porosity was best represented by fGn, a statistically
stationary stochastic process. This implied that the porosity
values themselves were positively correlated. In generating
horizontal porosity distributions, Hewett [1986] assumed that
inthatdirection porosity wasrepresented bythenonstationary
function fBm, with the same Hurst coefficient (0.86) that was
derived from the vertical data. In the horizontal direction
therefore the porosity value increments would be positively
correlated for H = 0.86. This approach became known as the
anisotropic fractal model. It was applied by Molz and Boman
[1993] to the natural logs of hydraulic conductivity (lnK)
obtained using borehole flowmeter data [Molz et al., 1989,
1990], resulting in a Hurst coefficient of 0.82, very similar to
Hewett’s porosity-derived value. However, Neuman’s [1990]
estimate of H (his w) based on using lnK fBm to explain the
observed scale dependence of apparent dispersivity was 0.25.
When the Molz and Boman [1993] data were reanalyzed
assuming that the vertical variation of ln(K) was represented
by the nonstationary function fBm [Molz and Boman, 1995],
the resulting Hurst coefficient was around 0.3, implying
negative correlation of the lnK increments, reasonably con-
sistent with the Neuman [1990] lnK result and the porosity
result(H=0.34)ofGrindrodandImpey[1993].Earlyworkby
KemblowskiandChang[1993]usingrelativelysmalldatasets
resulted in fBm-based H values in the vertical and horizontal
directions of 0.18 and 0.53, respectively. Most of the subse-
quentworkinsubsurfacehydrologywasbasedontheassump-
tion that vertical lnK is best represented by a nonstationary
fBm. However, additional data and analyses supporting the
Hewett[1986]concepthavebeenpublishedalso[Tubmanand
Crane,1995].Sothequestion isnot settled fully,and it may be
that both concepts have potential applications. As detailed by
LiuandMolz[1996],withshortdatasetsitiseasytoconfusean
fBm of H 0.25 with an fGn of H (1 0.25).
[22] Drawingonseveralsourcesofpreviousworkrelatedto
fractal geometry [Mandelbrot and Wallis, 1968; Mandelbrot,
1982; Burrough, 1983a, 1983b; Voss, 1985; Arya et al., 1988;
Wheatcraft and Tyler, 1988; Cushman, 1990] and motivated
by a desire for an improved understanding of the scale
dependence of longitudinal dispersivity, Neuman [1990]
published his first journal article dealing with what he
7 of 42
RG1002

called universal scaling of hydraulic conductivities and
dispersivities in geologic media, which immediately gen-
erated comment [Anderson, 1991; Neuman, 1991, 1993b;
Gelhar et al., 1993]. This terminology meant that by
assuming that lnK is a nonstationary stochastic process
with stationary (or homogeneous) increments, which is
equivalent to assuming that lnK is represented by a fBm,
one can provide a rough explanation for the observed scale
dependence of longitudinal dispersivity (aL) if the Hurst
coefficient H is selected as 0.25. (As shown in Figure 5,
which is a plot of apparent correlation length associated
with measured hydraulic conductivities, Neuman’s [1990]
approximate scaling rule is supported by hydraulic tests,
not only dispersivity plots.) ‘‘Travel distance’’ and ‘‘size’’
are placed in quotes to indicate that these quantities are
often somewhat ill defined in a mathematical/physical
sense, as are the ‘‘experiments’’ used to do the measuring.
For this reason, Neuman [1990] used the term ‘‘apparent
length scale.’’ Nevertheless, such plots cluster along a
straight line for a variety of media over apparent length
scales ranging from roughly 10 cm to several kilometers
[Gelhar et al., 1992; Gelhar, 1993; Neuman, 1990, 1994].
Neuman [1990] differed from previous work in porous
media in that scale dependence was tied to a lnK incre-
ment distribution of a distinct stochastic fractal class, fBm,
rather than the arbitrary assumption of a fractal velocity
field. Later, however, Dagan [1994], Neuman [1995], and
Rajaram and Gelhar [1995] showed that an fBm lnK
increment field did give rise to an fBm velocity field. The
initial work of Neuman [1990, 1994, 1995] was followed
by a series of elaborations that explicitly accounted for the
finite domains within which fieldwork is actually done [Di
Federico and Neuman, 1997, 1998a, 1998b; Di Federico
et al., 1999]. Dagan [1994] and Rajaram and Gelhar
[1995] suggested that dispersion in media with fBm or fGn
lnK fields will depend strongly on the source dimensions,
as in Richardson’s [1926] theory of relative dispersion in
turbulent flows, further complicating interpretation of the
data presented by Gelhar et al. [1992] using a fractal
model. Related explanations of the scale effect were also
published [Sposito, 1996]. Specifically, the importance of
the source dimensions as another characteristic length
scale, potentially even more fundamental than ‘‘travel
distance’’ in fBm lnK fields, suggests that inference of
fractal character or dispersivity scaling from data sets, such
as Figure 5, may not be altogether appropriate.
[23] A second line of inquiry that developed during the
early 1990s was not motivated primarily by a desire to
explain scale-dependent dispersivity. Rather, the objective
was a better understanding of the fundamental spatial
variability of natural heterogeneity. This second line was
based mainly on smaller scale, and hopefully better defined,
K measurements [Kemblowski and Chang, 1993; Molz and
Boman, 1993, 1995], porosity measurements [Grindrod
and Impey, 1993], and acoustic transit time logs [Painter
and Paterson, 1994]. The work by Painter and Paterson
[1994] was unique in that they based their property incre-
ment distributions on the Levy probability density function
(LPDF) [Taqqu, 1987] rather than the Gaussian distribution.
Later, Painter [1995, 1996a, 1996b] extended his analysis
to lnK increments.
[24] By the middle 1990s, and within the limitations of
contemporary measurement techniques, the fractal-like
scaling (also called pseudofractal scaling [Molz et al.,
1998]) of physical properties in natural porous media
was reasonably well established, but what this meant was
not clear. If nothing else, the scaling range was limited by
natural boundaries, and there still are insufficient data to
test the scaling range carefully even within the geometrical
limitations. (Similar concerns have been expressed in other
areas of physics [Avnir et al., 1998].) Evidence presented
for the Levy model versus the Gaussian model was
persuasive [Painter and Paterson, 1994; Painter, 1995,
1996a, 1997; Liu and Molz, 1997a, 1997c], but the Levy
model has its own internal inconsistencies associated with
the divergence of all statistical moments of K when the lnK
increments are assumed to follow fLm. This required
truncation of the LPDF used to generate Levy monofrac-
tals or lnK multifractals [Painter, 1996b, 1998; Liu and
Molz, 1997c; Painter and Mahinthakumar, 1999; Boufadel
et al., 2000]. A detailed study of k data from eolian
sandstone, based on gas permeability measurements, and
K data from fluvial sediments, based on borehole flowme-
ter measurements [Lu and Molz, 2001], verified the non-
Levy nature of the increment PDF tails. Yet as shown in
Figures 6 and 7, the strong non-Gaussian shape of the
increment frequency distributions for a broad variety of
data is obvious.
[25] In response to such observations, Painter [2001]
proposed a ‘‘flexible scaling model’’ that considered the
Figure 5. Plot of the apparent correlation length l as a
function of the size (field length) of the experiment used to
infer l (taken from Neuman [1994] and based on data from
Gelhar [1993]). As indicated on Figure 5, the data are from
a variety of formations, and they support the fact that K is
not in general a stationary process. This and similar plots of
apparent dispersivity motivated Neuman [1990, 1994] to
propose his stochastic fractal-based scaling hypothesis.
8 of 42
RG1002

increment PDFs to be a superposition of Gaussian distribu-
tions with random variances (also called variance subordi-
nation). Different variance PDFs resulted in increment PDFs
that could be ‘‘tuned’’ between the full Gaussian and full
Levy cases. As shown convincingly by Painter [2001], this
model can be made to fit field data quite nicely (Figure 7)
and, in general, will have a finite variance. However, what
might be the physical basis for such a model?
[26] When geologists classify and map sedimentary
deposits, they usually do so in terms of facies. ‘‘Sedimentary
facies commonly are defined on the basis of distinct textural,
structural, and/or lithologic features that reflect changes in
sediment transport or deposition mechanisms, including
changes in flow competence, capacity, and/or variability’’
[Allen-King et al., 1998, p. 385]. The key property from our
perspective is that a facies be a rock body that reflects the
unique combination of processes through which it was
created. Thus the K distribution in a given sediment will
reflect variation between facies and variation within facies
(Figure 8). During the development of fractal scaling con-
cepts in subsurface hydrology, facies have sometimes been
mentioned [Anderson, 1991; Bellin et al., 1996] but not dealt
with in detail. However, apparently realistic transport models
have also been developed recently based solely on hydraulic
property variation between facies [Carle and Fogg, 1996;
Fogg et al., 1998; Weissmann et al., 1999], indicating the
often dominant role played by these structures. Recent
additional analysis of the Macro-Dispersion Experiment
(MADE) experiments have moved toward bringing facies
structure into the analysis [Julian et al., 2001]
[27] In a multifaceted work aimed mainly at developing a
more rigorous methodology for applying the surface gas
minipermeameter to perform small-scale k measurements,
Figures 6. Empirical frequency distribution for the ln(K)
increments obtained from the fluvial sediments that were
studied during the MADE experiment [after Liu and Molz,
1997a]. (The data were obtained using a borehole flowmeter
[Boggs et al., 1992, 1993]). A non-Gaussian, Levy-like
appearance is evident with increased peaking around the
mean and heavy tails.
Figure 7. Empirical frequency distributions for increment logs of intrinsic permeability (Perm) and
electrical resistivity, mostly from sandstones [after Painter, 2001]. Once again, Levy-like PDFs are
evident that are well fit (solid curves) by Painter’s [2001] flexible scaling model.
9 of 42
RG1002

Goggin [1988] developed a k data set based on vertical
cores from an eolian sandstone that was a mix of three
facies: grain flow, wind ripple, and interdune [Goggin et al.,
1992]. As shown in the analysis by Lu and Molz [2001], the
histogram for the entire ln(k) data set is distinctly non-
Gaussian and, as usual, resembles a Levy distribution.
However, Lu et al. [2002b] present an analysis of data from
the interdune facies as well as a preliminary analysis of
more recent data collected from a bioturbated sandstone
facies in Utah that was collected using the new drill hole gas
minipermeameter [Dinwiddie et al., 2003]. The cumulative
distribution functions from both single-facies data sets
appear more Gaussian. This observation motivated in part
the so-called fractal/facies model [Lu et al., 2002b] that
appears consistent with the theory and observations of
Painter [1995, 1996a, 2001], Liu and Molz [1997a], and
Lu and Molz [2001], namely, that the observed lnK incre-
ment PDFs are not Levy but a superposition of a finite
number of Gaussian distributions with different variances.
(However, on the basis of analyses done to date, one cannot
rule out with certainty non-Gaussian distributions with finite
variances.) How this might come about is illustrated in
Figure 9 that is taken from Lu et al. [2002b] and based on a
data set from the four facies of an alluvial fan channel,
levee, floodplain, and debris flow [Fogg et al., 1998]. If the
fractal/facies model, or some future variation still based on
facies architecture, turns out to be valid, then it may not
make sense to mix data from distinct facies and perform a
single statistical analysis. A K realization combining sto-
chastic fractal and facies concepts is shown in Figure 10.
[28] If rainfall and sediment permeability both display
fractal-like structure, one would suspect that the flow
velocity through sediments and the resulting discharge to
streams might display fractal structure also. Recently,
Kirchner et al. [2000] have reported such an observation.
3.2. Multifractals
[29] In principle, the scaling relationship represented by
equation (9) does not imply a unique stochastic function. To
date, the most popular method for generating functions that
scale according to equation (9) is to use so-called multipli-
cative random cascades [Schertzer and Lovejoy, 1987;
Gupta and Waymire, 1990, 1993; Wilson et al., 1991;
Holley and Waymire, 1992; Harte, 2001]. In subsurface
hydrology, there are two ways in which multifractal theory
has been applied to K distributions. Similar to the mono-
Figure 8. Illustration of the different types of geologic sedimentary processes that give rise to different
types of facies [after Anderson, 1991]. Given the very different physical/chemical origin of different
facies, it probably does not make sense to do statistical analyses based on combined data from multiple
facies. This point was made by Anderson [1991] in her comment on Neuman’s [1990] universal scaling
hypothesis, which did not allow for multimodal increment ln(K) distributions characteristic of multiple
facies.
10 of 42
RG1002

fractal case, Liu and Molz [1997c] considered ln(K) to be a
nonstationary process with stationary increments. They then
used the structure function exponent for a so-called contin-
uous random cascade for lnK based on the Levy PDF
[Wilson et al., 1991; Lavallee et al., 1993; Schmitt et al.,
1995], which is given by
xðqÞ ¼ qH
C
a 1
qa
q
ð Þ; a 6¼ 1; ð10Þ
where H, C, and a are constants (a is the Levy index). For
C ! 0, equation (10) yields the monofractal relationship
given by x(q) = qH, and for this case only, H is identified
as the Hurst coefficient. Parameter values defining
equation (10) were extracted from the MADE, Cape Cod,
Mobile, and Borden (Canada) data sets [Boggs et al., 1993;
Hess et al., 1992; Molz et al., 1990; Sudicky, 1986]. Results
were mixed, with one ln(K) data set displaying monofractal
behavior, others displaying weak to strong multifractal
behavior, and another inconsistent with either type of
behavior. An interesting observation was that two data sets,
Cape Cod and vertical Borden, displayed lnK variations
consistent with multifractal noise, the increments of a
multifractal rather than the multifractal itself. The effect was
rather striking, with the significance still not understood
fully. Just as with the Levy monofractal case, however,
when lnK is exponentiated, the resulting field displays
statistical moments that diverge for all orders of q, so a
truncation process is needed.
[30] A potential advantage of the continuous cascade
(Levy based) multifractal approach is that it can be applied
to K directly rather than only to lnK, and this was done
recently by Boufadel et al. [2000]. They used two intrinsic
permeability data sets, one an eolian sandstone from north-
ern Arizona [Goggin, 1988] and the other a near-shoreface
sandstone from the Coalinga Formation in California. Both
data sets appear to display multifractality, with the fit being
reasonably good. Recently, the multifractal K model has
been further elaborated and extended [Essiam, 2001;
Tennekoon et al., 2003; Veneziano and Essiam, 2003, 2004].
[31] So where do we stand with multifractal K distribu-
tions? Painter and Mahinthakumar [1999] noted that a
bounded fBm and fLm model of lnK, when exponentiated,
Figure 9. Diagram showing (top) how the superposition of
four Gaussian distributions of the same mean but different
variances s can result in a Levy-like distribution when they
are added together and (bottom) when they are renormalized
with respect to area under the curve [after Lu et al., 2002b,
Figure 7]. Other finite-variance distributions can produce
similar results. #Springer-Verlag Berlin Heidelberg 2002.
Figure 10. Illustration of a K realization that preserves facies structure (channel, debris flow, levee, and
floodplain deposits) while displaying a different type of stochastic fractal structure within each of the four
facies [after Lu et al., 2002b, Figure 11]. If K data from such a structure are mixed, the increment log(K)
frequency distribution will be Levy-like, and the K frequency distribution will be multimodal.
#Springer-Verlag Berlin Heidelberg 2002.
11 of 42
RG1002

produced a multifractal-like K distribution. It was multi-
fractal in the sense that the structure function exponent was
nonlinear. Painter [2001] noted also that lnK increment
distributions do appear stationary and ‘‘have a consistent
shape that is remarkably reproducible from formation to
formation.’’ This observation does not hold for K distribu-
tions, although it did appear to hold for the K increment
distributions studied by Lu and Molz [2001]. One would
also expect the fractal/facies concept discussed in section 4
to apply to the multifractal case, so some of the statistical
parameters may be ill-defined because of the mixing of
properties measured in different facies. Clearly, more work
is needed, and our understanding is still limited by property
measurement capability, the meaning of such measurements
in heterogeneous natural material, and the large amount of
labor required to collect large data sets. Much of the
development of fractal-based concepts in subsurface hy-
drology went hand-in-hand with improved measurement
techniques and the development of larger data sets. There-
fore section 4 will be devoted to recent progress in mea-
surement technology, applications to the measurement of K
(or k) distributions, and an improved understanding of the
averaging volume of an instrument.
4. PERMEABILITY MEASUREMENT AND
INTERPRETATION
[32] The collection and study of innovative data are vital
to scientific advances, and two different types of data sets
were behind the development of fractal concepts (nonsta-
tionary stochastic processes with stationary increments) in
subsurface hydrology. One was the log/log plot of apparent
longitudinal dispersivity versus scale of measurement
[Gelhar et al., 1985, 1992; Neuman, 1990, 1994], which
resulted mainly from larger-scale tracer tests. The other data
type resulted from direct K measurements made on a small
scale (1 m3
averaging volume) to an extremely small scale
(1 cm3
averaging volume) using various types of borehole
flowmeters [Schimschal, 1981; Morin et al., 1988; Rehfeldt
et al., 1989; Molz et al., 1989, 1990; Guven et al., 1992;
Hess et al., 1992; Wolf et al., 1992; Molz and Young, 1993;
Ruud and Kabala, 1996; Ruud et al., 1999], laboratory
measurements on small cores using water or gas [Bakr,
1976; Sudicky, 1986], or gas minipermeameter measure-
ments [Goggin, 1988; Goggin et al., 1992; Desbarats and
Bachu, 1994; Boult et al., 1995; Liu et al., 1996]. Also of
interest are tomographic-based methods, wherein the poten-
tial exists for large-scale tracer tests to yield small-scale
permeability data [Zhan and Yortsos, 2001]. In addition,
promising results have been obtained recently using numer-
ical inversions of pneumatic cross-hole flow tests [Vesselinov
et al., 2001], the dipole flow test [Kabala, 1993; Zlotnik and
Ledder, 1996; Zlotnik and Zurbuchen, 1998; Zlotnik et al.,
2001], and direct push methods [Butler et al., 2002].
[33] Precise testing of fractal-like scaling theories requires
large numbers of saturated K or k measurements (preferably
thousands, tens of thousands, or more), and probably the
most practical way to do this at present is to measure
intrinsic permeability using core plugs or various forms of
the gas minipermeameter [Goggin et al., 1988; Goggin,
1993; Hurst and Goggin, 1995; Suboor and Heller, 1995].
Initially, most applications of the gas minipermeameter were
made in the area of petroleum engineering and geology for
the purpose of reservoir characterization. More recently,
however, instruments and approaches oriented to ground-
water applications have appeared [Sharp et al., 1994; Davis
et al., 1994]. Since that time, instrumentation has been
improved, and increasingly sophisticated data sets have been
obtained [Tidwell and Wilson, 1997, 1999a, 1999b, 2000,
2001]. Mainly because of surface seal integrity considera-
tions, the most precise data sets, to the authors’ knowledge,
have been obtained in the laboratory by Tidwell and Wilson
[1997] on blocks of carefully cut porous rocks. A sophisti-
cated probe (Figure 11) and computer-controlled positioning
system [Tidwell and Wilson, 1997] enables these researchers
to obtain large data sets (more than 10,000 measurements per
block in some cases) with excellent repeatability. By varying
the radius of the circular tip seal, and thus the averaging
volume (sample support) of the probe, permeability upscal-
ing may be studied experimentally [Tidwell and Wilson,
1999a, 1999b, 2000, 2001]. The result was very fundamental
and interesting data, with the full implications still being
elaborated. Among other things the apparent range of the
experimental k variograms increased continuously with the
sample support volume, as one would expect in a multiscale
system. It seems also that such data would have implications
for the degree of validity of the representative elementary
volume concept in porous media.
[34] While the surface-applied and mechanically con-
trolled minipermeameter probe enables one to make precise
k measurements in a laboratory setting, it is much more
difficult to get precise and representative data in the field.
This is due to surface roughness, which inhibits a good seal,
the difficulty of manually maintaining a steady and repeat-
able sealing pressure, and the fact that an unprepared
surface is often weathered and not representative of the
interior rock properties. These considerations motivated
Dinwiddie et al. [2003] to develop and analyze the small
drill hole minipermeameter probe (Figure 12). This probe is
well suited for field use in drillable rock, because drilling
produces a smooth and nonweathered surface, and the
normal force applied by the annular seal to the walls of
the drill hole is robust and easily controlled by a mechanical
torque wheel. In a recent application to near-shoreface
sandstone in southern Utah, Castle et al. [2004] were able
to obtain measurements in triplicate at 515 points at the rate
of about 30 points per day (three measurements per point),
not considering set up and takedown time, when all instru-
mentation was working properly.
[35] Another indirect, and mostly unexpected, benefit of
developing the gas minipermeameter is that use of the
instrument focused attention on the following questions
[Tidwell et al., 1999]: What is the sample support or sample
volume associated with the instrument? How does the
instrument spatially weight the heterogeneities composing
the sample support? The same questions could be applied to
12 of 42
RG1002

conventional permeability measurements on cores, but the
common assumption of one-dimensional (1-D) flow in such
samples obscures such questions, even though the flow is
actually 3-D. With the obviously diverging flow field
associated with a minipermeameter probe, the questions
have meaning even in homogeneous media. This led
Tidwell et al. [1999] to define and attempt to calculate the
‘‘spatial weighting function’’ (SWF) (x, y, z) associated with
the surface-applied gas minipermeameter probe applied to a
homogeneous medium, and a full 3-D version was pre-
sented by Wilson and Aronson [1999] and was calculated
numerically from the results of an adjoint state sensitivity
analysis [Aronson, 1999]. Tartakovsky et al. [2000]
approached the same problem using a streamline analysis.
[36] A spatial weighting function associated with an
instrument flow field-based measurement in a homogeneous
system may be defined implicitly by
kI ¼
Z Z
v
Z
k SWF x; y; z
ð Þ
½ dxdydz; ð11Þ
where kI is the permeability value resulting from application
of the instrument and k is the (assumed constant)
permeability of the medium. In other words, even in a
homogeneous system, the analysis applied to the diverging
flow field set up by the permeameter probe does not weight
all volumes of the medium equally, so some spatial
weighting function does exist. A previous experimental
attempt aimed at determining rough spatial weighting
function characteristics for the gas minipermeameter
involved measuring the distribution of gas bubbles on the
surface of a sandstone sample that had been saturated with
water [Goggin, 1993]. The quantity and size of air bubbles,
a surrogate for gas flux, decreased rapidly with radial
distance from the tip seal, suggesting a very local sample
support. Similar conclusions were obtained by injecting
colored dyes through a gas permeameter tip seal and
observing the dye distribution [Garrison et al., 1996].
[37] At a recent fall meeting (2000) of the American
Geophysical Union it was realized that the sensitivity
analysis by Knight [1992] and Knight et al. [1997] applied
to water content measurements using the time domain
Figure 11. Diagram of an automated apparatus described by Tidwell and Wilson [1997] to make precise
gas intrinsic permeability measurements on large blocks of cut rock. Using this apparatus, it was feasible
to make tens of thousands of measurements on a single block.
13 of 42
RG1002

reflectometer was analogous to the spatial weighting func-
tion concept outlined above, with voltage being analogous
to hydraulic head [Molz et al., 2000]. Molz et al. [2000,
2002] brought the two concepts together by deriving the
rather general result that spatial weighting functions for
steady state potential flow systems in homogeneous media
are given physically by the distribution of the potential
energy dissipation rate per unit volume of the medium
divided by the total energy dissipation rate in the flow
domain. For a system like the gas minipermeameter that is
driven approximately by the energy gradient of a pseudo-
potential [Goggin, 1988; Dinwiddie et al., 2003], the
applicable spatial weighting function is given by
SWFðx; y; zÞ ¼
qm
j j r
r
j j
R
v
qm
j j r
r
j jdv
) kI ¼
Z
v
k
qm
j j r
r
j j
R
v
qm
j j r
r
j jdv
dv;
ð12Þ
where qm is the mass flux vector, r
r is the pseudopotential
gradient, and n is the volume of the flow domain set up by
the minipermeameter probe. Once again, it is important to
keep in mind that even though kI and k are equal in a
homogeneous system, SWF represents how the instrument
and analytical procedure used to calculate kI weights the
different volumes of the flow domain. Thus portions of the
domain where SWF is high, such as around the tip seals, are
critical to an accurate measurement, while portions where
SWF is low contribute little [Molz et al., 2000, 2002;
Tartakovsky et al., 2000; Dinwiddie, 2001]. A weighting
function distribution for the small drill hole minipermea-
meter probe, based on equation (12), is shown in Figure 13
[Molz et al., 2002].
[38] Obviously, the future challenge to a full understand-
ing of what an instrument measures in natural sedimentary
systems involves a potential extension of the concepts
developed so far to heterogeneous and anisotropic media.
Until this is done, the ‘‘scale’’ of a measurement will be
ambiguous, since the measurement process and the local
heterogeneities interact in an, as yet, unknown manner.
Steps in these directions have already been taken by Knight
[1992] for mild heterogeneity and by Aronson [1999] for
layered systems [Wilson and Aronson, 1999]. Such further
understanding is essential in order to refine the testing of
various theories describing the structure of natural hetero-
geneity in a rigorous manner [Molz, 2003].
5. DETECTING AND GENERATING FRACTAL
STRUCTURE
[39] There are two obvious computational processes
associated with various applications for nonstationary sto-
chastic processes with stationary increments. One is detect-
ing fractal scaling properties in data sets, and the other is
Figure 12. Diagram of the new small drill hole gas minipermeameter probe [after Dinwiddie et al.,
2003]. This probe is well suited for field applications and was used recently to obtain increment ln(k) data
sets from two well-defined facies. The data are not yet fully analyzed.
14 of 42
RG1002

generating realizations of these irregular functions in one,
two, and three dimensions, which may or may not be
conditioned on data. (Conditioning means generating a
realization that takes on prescribed values at a set of fixed
locations.) During the past decade, advances were made in
all of these areas.
[40] Traditional methods for detecting fractal scaling in a
3-D data set, such as ln(K) data, would begin by selecting
1-D data transects along the three coordinate directions,
which ideally would also be the principal directions for any
anisotropy that may be present. This makes the detection
problem analogous to a set of time series analyses. If
potentially dealing with the Gaussian fractals fGn or fBm,
the next task is to decide which model, stationary or
nonstationary, best represents the data. The answer to this
question may be obvious based on the physics of the
application, but if not, Liu and Molz [1996] describe a
straightforward procedure for discriminating between fGn
and fBm based on range analyses [Mandelbrot and Wallis,
1969b]. The basic idea is that for the stationary process fGn
both the range of the data set and the rescaled range, which
involves dividing the range by the increment variance,
should have the same slope on a log/log plot against the
lag h. However, since fBm is nonstationary, with the
increment variance a power function of h, log/log plots of
the range and rescaled range will have significantly different
slopes. This property may be used to discriminate between
the two irregular functions.
[41] In most applications the basic stochastic process that
one is dealing with will be fBm or fLm. By basic definition
the increments of both irregular functions will be stationary,
with width parameter, or variance in the Gaussian case,
displaying the scaling property represented by equation (8).
Testing for so-called fractal structure is built around various
methods for determining whether equation (8) is satisfied
and if so what the correct values of H and a are. Since data
set size is usually limiting, the best methods have the
greatest tolerance for this main limitation. Direct plotting
of relation (8), which is variogram analysis in the Gaussian
case, is acceptable for the smaller lags but fails rapidly
because of diminishing data for the larger lags [Journel and
Huijbregts, 1978]. This observation was a major motivation
for the development of rescaled range analysis [Mandelbrot
and Wallis, 1969b], and rescaled range (R/S) analysis
became more or less standard for calculating the Hurst
coefficient for applications in subsurface hydrology, since
it works well theoretically for large lags [Hewett, 1986;
Molz and Boman, 1993; Liu and Molz, 1996; Molz et al.,
1997]. The spectral density functions of fBm and fGn also
have power law behavior, with slopes of the log/log plots
being known functions of H [Molz et al., 1997]. In their
early lnK analyses, Kemblowski and Chang [1993] used a
combination of variogram and spectral analysis, while Molz
and Boman [1993] used R/S analysis. In their analysis of
porosity data, Grindrod and Impey [1993] used variogram
analysis.
[42] When approaching a data set from the more general
perspective of fLm, it is necessary to calculate the Levy
index a, as well as H. This was done initially [Painter and
Paterson, 1994; Painter, 1996a; Liu and Molz, 1997a] using
a histogram-based estimator developed by Fama and Roll
[1972]. Recently, several improved techniques have been
developed for estimating a based on analyses of LPDF tail
behavior and/or higher moments [Meerschaert and Scheffler,
1998; Lu and Molz, 2001]. On the basis of equation (8), H
can be determined using the Levy analog of variogram
analysis (width parameter analysis), and this was the tech-
nique employed by Painter [1996a]. In his new flexible
scaling model for lnK using variance subordination of fBm,
Painter [2001] used first and second statistical moment
estimators in order to extract model parameter values from
data.
[43] For a Gaussian or Levy PDF family to serve as the
basis for a nonstationary stochastic process with stationary
increments, the a parameter (2 for the Gaussian case)
should be independent of lag. Liu and Molz [1997a]
reported that for the MADE lnK increment data set the a
Figure 13. Isopleths of the spatial weighting function for
the small drill hole minipermeameter probe as they would
appear in a homogeneous medium. The weighting at
contour 1 is about 750 times less (0.0013) than that at
contour 14, which shows that the instrument tends to
‘‘sense’’ the medium very close to the drill hole and
particularly in the vicinity of the tip seal. An interesting
future question is how the spatial weighting distribution is
modified by natural heterogeneity and anisotropy.
15 of 42
RG1002

value based on the Fama and Roll [1972] estimation
procedure trended toward 2, for Gaussian behavior, as the
lag increased. Painter [1997] initially took this calculation
at face value but later became concerned when he found out
through a personal communication that the ‘‘below detec-
tion’’ K values were dropped from the data set [Painter,
2001]. According to Painter [2001] this would have the
effect of accelerating, but not removing, the trend toward
Gaussian behavior. Most of the data points dropped by Liu
and Molz [1997a] were at the top and bottom of the various
flowmeter logs, so the motivation was to better define the
actual aquifer that was being studied at the MADE site.
However, Liu and Molz do not disagree with Painter’s
[2001] concerns. The present authors believe, however, that
there is a more fundamental issue. Lu and Molz [2001]
reanalyzed the MADE K data using a ‘‘higher moments’’
technique that looks primarily at the power law PDF tail
behavior. (This differs fundamentally from the Fama and
Roll [1972] a estimation procedure that considers the form
of the entire PDF.) They found an a value of 3.6, which is
typical of a Pareto distribution with a finite variance. Any
Levy cumulative distribution function displays the property
Prob(jxj s) / sa
, as s ! 1, and it is this tail behavior,
when a 2, that leads to the Levy infinite variance. Thus
we view an a value determined from tail behavior as more
fundamental than an a determined by fitting to an entire
distribution. Both Painter’s [2001] flexible scaling model
and the fractal/facies concept of Lu et al. [2002b] are
consistent with the lnK increment distributions not being
Levy and the possibility of the Fama and Roll [1972]
estimator, when applied to such distributions, leading to
ambiguous a values. Once again, however, clear answers to
such concerns will require further analysis of high-quality
data.
[44] In practical applications of fractal concepts, both
fBm and fLm, it is often necessary to generate multidi-
mensional realizations of such functions, with the realiza-
tions sometimes conditioned on data. A variety of methods
exist, including successive random additions (SRA) [Voss,
1988; Saupe, 1988], Fourier filtering methods [Saupe,
1988], the modified turning band method [Yin, 1996],
and the technique known as fractional integration
[Mandelbrot, 1982; Tatom, 1995]. However, many tech-
niques that grew out of the above method classes are not
developed fully in program form for 3-D applications.
Maeder [1995] outlined an algorithm based on the use of
the Mathematica Software Package to simulate 3-D fBm.
This might be a practical option for individuals who are
Mathematica users. Gaynor et al. [2000] used a version of
a code called LevySim that was developed previously
by Painter [1996b], but the code is not readily available
(S. Painter, personal communication, 2002). (The Gaynor
et al. [2000] application, involving 3.9 million permeabil-
ity values generated on a complex, faulted, 3-D grid,
which are then upscaled for a multiphase reservoir simu-
lator, is one of the largest to date.) More recently, Painter
[2001] described a code that is available for research
purposes from the Southwest Research Institute (S. Painter,
personal communication, 2002). This is also based on
previous work [Painter, 1998] and is a 3-D version of
sequential simulation that can be used to generate fBm,
fLm, or Painter’s variance-subordinated scaling model.
[45] As another computationally efficient alternative, Lu
et al. [2003] published a 3-D code based on a version of
SRA that allows for conditioning and the generation of
anisotropic fractals, both fBm and fLm. An advantage
of this code is its simplicity. Lu et al. [2003] plan to update
the code as improvements to the basic SRA procedure are
identified.
[46] Recently, several publications appeared in the
literature that revisit the traditional stochastic fractal genera-
tion and scaling tests, with the objective of identifying
shortcomings and developing improved methodology
[Bassingthwaighte and Raymond, 1995; Caccia et al.,
1997; Cannon et al., 1997; Eke et al., 2000; McGaughey
and Aitken, 2000]. This series of publications is built around
a new method for generating 1-D fGn or fBm using a so-
called fractional Gaussian process (fGp) [Davies and Harte,
1987; Caccia et al., 1997] and a new method for determin-
ing the Hurst coefficient called dispersional analysis. An
fGp is essentially an exact fGn that is generated directly.
Exact means that the series is truly stationary and produces
the correct autocorrelation function. Summing the series
then produces a precise fBm. Caccia et al. [1997] generated
1-D time series in this way and showed that the procedure
was superior to spectral methods and SRA. They therefore
advocated that these methods be abandoned. Using the
exact fGn series with known H, Caccia et al. then showed
that dispersion analysis for H determination was distinctly
superior to R/S analysis, especially for H 0.4. For this
reason, Lu et al. [2003] used dispersional analysis rather
than R/S analysis, and a code for that algorithm is avail-
able also from the Computers and Geosciences web site.
Numerical experiments supported the conclusion of Caccia
et al. [1997] that dispersional analysis is superior to R/S
analysis as a means to extract Hurst coefficients from data
sets. However, to the authors’ knowledge, multidimensional
fGp methods have not been implemented. Accomplishing
this should be a high priority for future research.
6. FLOW AND TRANSPORT IN FRACTAL K FIELDS
[47] A major motivation for developing stochastic models
of permeability heterogeneity is that they provide a frame-
work for representing the ‘‘effective’’ or ‘‘upscaled’’ behav-
ior of flow and transport processes. These effective
parameters, such as the effective conductivity or dispersiv-
ity, are derived based on stochastic partial differential
equations (PDEs) describing flow and transport in random
porous media. The lnK field is involved as a parameter in
these stochastic PDEs. The theoretical framework for
deriving these effective parameters has evolved beginning
from the later half of the 1970s and is summarized by
Dagan [1989], Gelhar [1993], and Zhang [2002]. The PDEs
for flow and solute transport in porous media implicitly
invoke the notion of a representative elementary volume
16 of 42
RG1002

(REV), and it is only above the scale of the REV that these
continuum PDEs are meaningful. So porous medium
properties such as K or lnK are also meaningfully defined
only at scales larger than the REV.
[48] The effective properties typically represent mean
or averaged behavior across an ensemble of possible
realizations of the small-scale or unresolved heterogene-
ity. The issue of when such effective properties are
meaningful and provide useful representations of upscaled
behavior requires important consideration. Unless the
maximum scale of variability that is represented stochas-
tically is much smaller than the scale associated with an
upscaled or effective property, the resulting effective
properties will vary significantly from realization to
realization. Under these conditions the effective property
will have a large degree of uncertainty associated with it.
On the other hand, if the largest scale of variability that
is represented stochastically is of the order of 1/10–1/5
of the scale at which predictions are required (e.g., the
grid block scale), the corresponding effective properties
may provide robust representations of larger-scale flow
and transport. This is the classical requirement of ergo-
dicity in stochastic flow and transport theories [e.g.,
Dagan, 1989; Gelhar, 1993]. If permeability variations
are modeled as a stationary random field, robust effective
properties result when the scale of averaging is several times
the integral scale.
[49] Development of effective representations for flow
and transport in fractal permeability fields thus poses prob-
lems because fractal fields are by definition nonstationary
and furthermore nondifferentiable. The integral scale and
variance of a truly fractal permeability field are both infinite.
For instance, in the case of self-similar or monofractal fBm
fields discussed in section 3.1, the spectrum of log hydraulic
conductivity (lnK) variations behaves as a power function,
with one-dimensional spectra of the form C/w2H+1
, where C
is a constant, w is the wave number, and H (0 H 1) is the
Hurst exponent. The corresponding variograms grow as
power functions x2H
, with separation x. In case fractal
behavior is manifested only below a low-wave number
cutoff (1/L), the field is effectively stationary at scales much
larger than L. The variance is then given by
s2
ln K ¼
C
2H
L2H
: ð13Þ
The integral scale associated with such a field is propor-
tional to and of the order of the cutoff scale L. The issue of
differentiability arises because effective parameters are
derived from stochastic PDEs, where lnK is involved as a
parameter and is differentiated. Monofractal fields are not
differentiable, as is well known (the one-dimensional
spectrum should decay faster than 1/w3
at large w for
differentiability). However, because permeability itself is a
well-defined quantity only at scales larger than the REV
scale, stochastic continuum theories implicitly invoke a
high-wave number cutoff that renders the lnK field
differentiable. The high-wave number cutoff does not
significantly influence the effective properties obtained for
self-similar fields, as long as there is a sufficient scale
separation (2 orders magnitude) between the low- and
high-wave number cutoffs. Di Federico and Neuman [1997]
discuss in detail the properties of permeability fields with
truncated power function variograms.
[50] The low-wave number cutoff scale L used in
equation (13) may be regarded as a natural cutoff scale
indicated by lnK data, a filtering scale (i.e., variability at
scales larger than L are filtered out and represented
deterministically), or a cutoff scale associated with a
physical process (e.g., plume scale). Most results in the
literature related to flow and transport in fractal permeability
fields have been derived by either implicitly or explicitly
assuming that a low-wave number cutoff exists. The
resulting effective parameters will depend on the cutoff
scale. Thus, in media with long-range correlations, effective
properties are rendered scale-dependent as the cutoff scale is
varied. In the remainder of this section we will discuss the
significant developments reported in the literature on flow
and transport in fractal permeability fields and other types of
fields that exhibit long-range correlations.
[51] An important hydraulic property in the context of
flow in a random permeability field is the effective conduc-
tivity. The effective conductivity provides a relationship
between the mean hydraulic gradient applied across the
medium and the resulting mean flux in a macroscopic Darcy
equation. The basic theoretical results for the effective
conductivity in the simplest case of steady uniform mean
flow in infinite domains are discussed in detail in books on
stochastic subsurface hydrology [e.g., Dagan, 1989; Gelhar,
1993]. Extensions to nonuniform mean flow, transient
conditions, and finite domains have also been accomplished
in the recent literature, and some of these developments are
summarized in a more recent book [Zhang, 2002].
[52] A classical result for the effective conductivity tensor
in the case of steady uniform mean flow through a lognor-
mal permeability field with statistically isotropic correlation
structure, attributed to Matheron [1967], is
Keff;ij ¼ KG exp s2
ln K
1
2

1
d

dij; ð14Þ
where KG is the geometric mean conductivity, d is the
dimensionality of the flow system, and dij is the Kronecker
delta function. Approximations to equation (14) have been
derived using various perturbation approaches [e.g., Bakr et
al., 1978; Dagan, 1979]. The exponential functional form in
equation (14) was, in fact, obtained by a conjecture that the
terms obtained in the perturbation approximations are part
of a series expansion of the exponential function. For
isotropic media, equation (14) has been demonstrated to be
robust even for large values of slnK
2
[e.g., Ababou et al.,
1988; Dykaar and Kitanidis, 1992; Neuman and Orr, 1993].
At first glance, it becomes evident that application of
equation (14) to quantify the effective conductivity for a
fractal medium poses problems: For a truly fractal medium
17 of 42
RG1002

the variance is infinite. To get around this problem, cutoffs
may be introduced to represent fractal behavior in a
truncated range of wave number space, as noted above.
When a low-wave number cutoff (1/L) is introduced, the
variance is given by equation (13), and correspondingly, the
effective conductivity may be expressed as
Keff;ij ¼ KG exp
C
2H
L2H 1
2

1
d

dij: ð15Þ
Note that the effective conductivity in equation (15)
depends on the cutoff scale L for d 6¼ 2, while it is
independent of the cutoff scale for d = 2. Ababou and
Gelhar [1990] considered the case of three-dimensional 1/f
noise, for which case, slnK
2
is proportional to lnL. The
resulting effective conductivity given by equation (15)
increases linearly with the cutoff scale L. Di Federico and
Neuman [1998a] quantified the effective conductivity for
the general three-dimensional case and suggested that lnKeff
exhibits a power function dependence on L, with exponent
2H. Ababou and Gelhar [1990] interpreted the increase in
effective conductivity with scale in equation (15) for d = 3
as a manifestation of the increasing dominance of flow by
connected high-conductivity regions. Di Federico and
Neuman [1998a] used equation (13) to interpret conductiv-
ity data from crystalline rocks at various sites [Clauser,
1992] that exhibit an increase with support scale at smaller
scales and do not exhibit scale dependence at the regional
scale. They suggested that equation (13) with d = 2 is
consistent with Clauser’s data at large scales, perhaps
because flow at regional scales is approximately two-
dimensional. For the one-dimensional flow case, which is
largely of academic interest, equation (15) predicts an
effective conductivity decreasing with L as a power
function. By extension of Ababou and Gelhar’s [1990]
argument this feature may be interpreted as follows: The
smallest conductivity value encountered within a region of
size L decreases as L increases; Keff decreases with L
because it is controlled by the smallest K values for d = 1.
[53] Exactly at what scales equation (15) provides a
meaningful representation of the effective conductivity is
a subtle point. Consider the application of equation (15) to
the classic problem of defining a grid block-scale effective
conductivity for a cubic grid block in a numerical model,
with dimensions (B B B), and assume for argument’s
sake that the permeability field is isotropic. In order to use
equation (15), estimates of H, KG, and L are required. The
exponent H may be determined using data from a wide
range of scales if monofractal behavior is evident across
those scales. The geometric mean conductivity KG may be
estimated using a local geometric average of K values from
within the grid block. Reliable estimation of KG within the
grid block requires that the cutoff scale L be B/5, for
example. In other words, the integral scale associated with
the filtered conductivity field, which is proportional to L,
should be significantly smaller than L. Under these
conditions, equation (15) may potentially provide robust
estimates of the effective conductivity. By ‘‘robust’’ we
mean here that Keff predicted without a detailed knowledge
of the permeability variations within the grid block provides
an accurate estimate of the flux resulting from the
application of a given hydraulic gradient, or in other words,
the variance of Keff is relatively small. Di Federico et al.
[1999] obtained a rigorous expression for the variance of
ln(Keff) obtained from equation (15) for the case of lnK
fields with truncated power function variograms. They
showed that the variance of ln(Keff) induced by variability at
scales larger than the grid block scale (i.e., cutoff scale
larger than the grid block scale) is identical to that of the
grid block-scale mean log conductivity, which grows
approximately as the cutoff scale raised to the power of
2H. It should also be noted in this connection that in
deriving equation (15) or its approximate forms, the mean
hydraulic gradient is assumed to be constant; that is, the
large-scale flow is assumed to be uniform. This assumption
can be restrictive in the case of fractal permeability fields,
where large-scale variations can produce nonuniform
‘‘mean’’ flows at the ‘‘grid block’’ scale.
[54] For anisotropic fBm (e.g., in the sense of Rajaram
and Gelhar [1995] or Di Federico et al. [1999], where the
Hurst exponent is the same in all directions) the effective
conductivity can be obtained, in principle, from the
generalization of equation (14) to anisotropic correlation
structures [e.g., Gelhar and Axness, 1983]:
Keff;ij ¼ KG exp s2
ln K
1
2
gij

: ð16Þ
In equation (16) the term gij depends on the statistical
anisotropy of the lnK field. However, the range of s2
lnK
within which the exponential generalization of the
perturbation approximation is valid for equation (16) has
not been investigated in detail, even for the case of
stationary lnK fields. Di Federico et al. [1999] also present
evaluations of a generalized expression for effective
conductivity in the case of multiscale anisotropic fields
with truncated power function variograms (analogous to
anisotropic fBm with cutoffs) that accounts additionally for
a limited domain size [Tartakovsky and Neuman, 1998].
The effective conductivity in a finite domain is given by an
expression similar to equations (14) and (16), with 1/d or
gij replaced by a function that varies within the domain.
The equivalent conductivity obtained by averaging the
effective conductivity over the domain was shown to
depend on H, the anisotropy ratios, and a characteristic
length scale (related to the domain size) raised to the power
of 2H. Recently, Hyun et al. [2002] attempted to apply the
scaling relationships of Di Federico et al. [1999] to explain
observations of air permeability at the Apache Leap
research site in Arizona. While their analysis is not
conclusive, there is some qualitative evidence for a scale
effect in permeability. Results analogous to equation (15)
for general anisotropic monofractal (in the sense of Molz et
al. [1997]) fields, with different values of H corresponding
to one-dimensional spectra in different directions, are yet to
be derived.
18 of 42
RG1002

[55] It should also be emphasized that equation (15) is
evidently exact only for lognormal permeability distribu-
tions. For cases where the permeability distribution is not
lognormal, perturbation approximations to equation (15)
may still be valid; however, their validity is restricted by a
requirement of small slnK
2
. The behavior of the effective
conductivity in cases where lnK is a fractional Levy motion
[e.g., Painter, 1996a, 1996b, 1998] or a multifractal process
[e.g., Liu and Molz, 1997c] has not been analyzed in detail
to date. Evaluation of the scaling behavior of the effective
conductivity in these fields appears to be a promising
avenue for further research. Besides perturbation methods
and effective medium approximations, techniques from
continuum percolation theory and critical path analysis
appear to hold promise in this context [Friedman and
Seaton, 1998; Hunt, 1998, 2003; Mukhopadhyay and
Sahimi, 2000]. The latter approaches are not constrained
by the requirement of small variance, and they also
provide a framework for interpretation of nonergodic
behavior of the effective conductivity. At scales much
smaller than the cutoff scale, the effective conductivity is
perhaps strongly controlled by connectivity of high-
conductivity regions.
[56] Another important quantity of interest in the context
of flow in random permeability fields is the head variance
sh
2
. The head variance represents the mean square deviation
of the actual head field from the mean head field, which is
given by a linear head variation in the case of uniform mean
flow or an appropriately smooth surface obtained from
numerical modeling on a coarse large-scale grid. The head
field is typically smoother than the lnK field, since it
represents the integrated effect of lnK variations. Gutjahr
and Gelhar [1981] analyzed this feature extensively using
intrinsic random function theory and identified the condi-
tions when the head field is stationary or is a nonstationary
field with stationary increments. Dagan [1989] and Gelhar
[1993] discuss the primary theoretical results for sh
2
from
perturbation-based stochastic theories (valid for small
values of slnK
2
). In three-dimensional flows through
stationary lnK fields, sh
2
slnK
2
l2
, where l is the integral
scale and the head field is stationary [Gutjahr and Gelhar,
1981]. In two-dimensional flows, Mizell et al. [1982]
showed that even with a stationary lnK field, the head field
is a zero-order intrinsic random function (i.e., it is
nonstationary, although its increments are stationary and
the head variogram depends only on separation). Di
Federico and Neuman [1998a] analyzed the head variance
in lnK fields with truncated power function variograms and
obtained analogous results: In three dimensions, sh
2
L2H+2
(where the exponent 2H results from slnK
2
and the additional
2 results from the square of the integral scale), while sh
2
is
infinite in two-dimensional flows. The increase in head
variance as a power function of the cutoff scale suggests
that the model error variance in a regional-scale numerical
model will increase as a power function of the grid block
scale. This feature reflects the fact that the ‘‘intrinsic
variability’’ of unresolved heterogeneity (the product of the
variance and integral scale) increases with the grid block
size in a fractal medium, leading to an increase in the degree
of prediction uncertainty.
[57] The flux variations in a random permeability field
control the behavior of advective-dispersive transport. Most
of the results for the flux statistics that have been used in the
context of fractal models of lnK are based on small-
perturbation approximations and have been derived assum-
ing a uniform mean flow and a constant porosity. These
results on the statistical structure of flux variations in
random porous media are summarized by Dagan [1989],
Gelhar [1993], and Zhang [2002]. Although these results are
based on the small-perturbation approximations (slnK 1),
computational results suggest that they are valid for slnK
values even larger than 1.0. Under uniform mean flow
conditions the flux field is a vector random field whose
statistics are controlled by the lnK spectrum and the
requirement of zero divergence (i.e., continuity). In
stationary lnK fields, sustained deviations of particle
trajectories away from the mean flow direction are precluded
by the zero divergence condition, which results in
asymptotic transverse macrodispersivities controlled by
local dispersion. Rajaram and Gelhar [1995] presented
expressions for the flux spectra for the case of fGn and fBm
lnK fields. In each case the flux spectra also exhibit power
function behavior with the same Hurst exponents as the
corresponding lnK fields, and the flux components have
variances proportional to the lnK variance. The implications
of power function behavior of flux spectra for dispersion are
discussed further in section 7. Di Federico and Neuman
[1998a, 1998b] discuss the statistical structure of flux
variations in lnK fields with truncated power function
variograms. With a low-wave number cutoff (1/L) in the lnK
spectrum the flux components have variances proportional
to L2H
. The integral scale along the mean flow direction is
proportional to L for the longitudinal flux component, while
the transverse components have a zero integral scale. The
latter is an important consequence of the zero divergence
condition. The introduction of cutoffs essentially renders the
flux field stationary at scales much larger than the cutoff
scale.
[58] Transport in random velocity fields has been exten-
sively analyzed in the context of random porous media and
turbulent flows, dating back to the pioneering studies of
Taylor [1921] and Richardson [1926] in the turbulence
literature. Taylor’s [1921] classical result for the dispersion
coefficient in a random velocity field forms the basis for
most theoretical treatments of macrodispersion in hetero-
geneous porous media as well and may be expressed in the
following form:
DijðtÞ ¼
Zt
0
RL
ijðt tÞdt; ð17Þ
where Dij(t) is the time-dependent dispersion coefficient
with t denoting time and Rij
L
is the Lagrangian velocity
covariance function. Taylor’s result strictly applies for a
stationary random velocity field, in which case the mean
velocity can be meaningfully defined and the velocity
covariance function exists. Although Taylor’s result,
19 of 42
RG1002

equation (17), is exact, it is impossible to compute the
dispersion coefficient exactly because the Lagrangian
velocity covariance function can only be represented
approximately. Nevertheless, it forms the basis for most
theoretical results for the dispersion coefficient in hetero-
geneous porous media. Stochastic flow theories describe the
Eulerian velocity covariance structure and employ approx-
imate Eulerian-Lagrangian transformations to represent the
Lagrangian velocity covariance function in equation (17).
For instance, the particle displacement during a time
interval may be approximated by its mean displacement
for purposes of approximating the Lagrangian correlation
function [e.g., Dagan, 1994]. The resulting expressions for
the macrodispersivity tensor Aij (which is the dispersion
coefficient divided by the mean velocity) in three-dimen-
sional flows may be expressed as
AijðsÞ ¼
1
^
u2
Zs
0
Ruiuj
ðx; 0; 0Þdx; ð18Þ
where s represents the mean displacement, ^
u is the mean
velocity, and Rui
uj is the Eulerian velocity covariance
function, evaluated using a spatial lag along the mean flow
direction. Without loss of generality the direction of the
mean flow and mean displacement can be approximated
using the ‘‘1’’ index. The limiting behavior predicted by
equation (18) at small and large displacements provides
important insights on transport in random velocity fields. At
small displacement (s ! 0), equation (19) may be evaluated
by using a Taylor’s series expansion around s = 0 to yield
Aijðs ! 0Þ ¼
suiuj
^
u2
s: ð19aÞ
In equation (19a), suiuj
represents the cross-variance of the i
and j components of the velocity vector (which is the cross-
covariance function at zero lag). In stationary random
velocity fields with well-defined integral scales the large
displacement limit (s ! 1) yields
A11ðs ! 1Þ ¼ s2
ln Kl;
A22ðs ! 1Þ ¼ 0;
A33ðs ! 1Þ ¼ 0:
ð19bÞ
In equation (19b), l is the integral scale of lnK in the flow
direction. In a stationary random velocity field the
macrodispersivity components grow linearly at small
displacements and approach constant asymptotic values at
large displacements (typically about 20 l). A constant value
for the macrodispersivity indicates a Fickian regime of
transport, while a macrodispersivity varying with displace-
ment (or equivalently time) indicates non-Fickian transport.
Thus transport in a random velocity field cannot be
described using a Fickian model until a plume traverses a
displacement of the order of 20 integral scales of lnK.
[59] In the case of transport in a fractal lnK field, where
there is variability across a wide range of scales, several
issues arise in the context of applying equation (18) for
quantifying the dispersivity. First and foremost is the issue
of whether a mean velocity may be meaningfully defined.
Richardson [1926] raised this issue with his famous quip,
‘‘Does the wind have a mean velocity?’’ In effect, if there
are velocity variations at several scales, we should expect
that the ‘‘mean’’ velocity experienced by a finite-size cloud
of solute particles would also evolve with time (or
displacement). Second, there is the issue of whether the
Eulerian velocity covariance function exists. Third, even if a
dispersion coefficient can be defined, it only quantifies the
rate of growth of the displacement variance of a tracer
particle (or equivalently the rate of second-moment growth
associated with the ensemble average concentration field).
Strictly speaking, the ensemble average concentration field
is described by a nonlocal transport equation [e.g., Koch
and Brady, 1988; Cushman and Ginn, 1993b; Neuman,
1993a]. However, the dispersivity still provides a useful
basis for representing scale dependence of transport in the
case of a uniform mean flow.
[60] Attempts to describe dispersion in fractal lnK fields
or other lnK fields with long-range correlation have resorted
to various ways of overcoming the aforementioned issues.
The issue of the mean flow is glossed over in essentially all
these studies, which simply assume a uniform mean flow at
the large scale (or equivalently, in an ensemble average
sense). One class of studies deals with cases where the lnK
field is stationary, so that a covariance function (or velocity
covariance function) exists, although the field has an infinite
(or very large) integral scale. Philip [1986] considered
transport in a velocity field where the Lagrangian velocity
covariance function decays as a power function for long
travel times and showed that the corresponding dispersion
coefficient in equation (16) grows as a power function of
time. Koch and Brady [1988] developed a nonlocal theory
for the ensemble average concentration for transport in a
lnK field with a covariance function that decays as x2H2
at
large separation. The corresponding longitudinal dispersiv-
ity A11 was shown to grow with displacement s as s2H1
for
1/2 H 1. Glimm and Sharp [1991] and Dagan [1994]
obtained similar results, showing that a Fickian transport
regime results for H 1/2, while a power function growth of
dispersivity corresponding to non-Fickian transport results
for 1/2 H 1 [Glimm et al., 1993]. Rajaram and Gelhar
[1995] noted that the foregoing cases are similar to
fractional Gaussian models for lnK and obtained analogous
results for three-dimensional isotropic media. Their results
indicated that both longitudinal and transverse dispersivities
grow with displacement as s2H1
. Fractional Gaussian
behavior has been hypothesized for horizontal lnK variation
at the Borden site [Robin et al., 1991], and it was also
observed for horizontal porosity variations in the early work
of Hewett [1986]. The appealing feature of the results for
the case 1/2 H 1 is that there is a sustained increase in
dispersivity with displacement, which can potentially
explain the phenomenon of ‘‘scale-dependent dispersion.’’
However, fractional Gaussian media are essentially sta-
tionary and are not suitable as models for nonstationary lnK
20 of 42
RG1002

Molz, FJ, Rajaram, H. y Lu, S. (2004). Modelos de heterogeneidad basados en fractales estocásticos en hidrología del.pdf

Molz, FJ, Rajaram, H. y Lu, S. (2004). Modelos de heterogeneidad basados en fractales estocásticos en hidrología del.pdf

Recommended

Recommended

More Related Content

Similar to Molz, FJ, Rajaram, H. y Lu, S. (2004). Modelos de heterogeneidad basados en fractales estocásticos en hidrología del.pdf

Similar to Molz, FJ, Rajaram, H. y Lu, S. (2004). Modelos de heterogeneidad basados en fractales estocásticos en hidrología del.pdf (20)

More from SandroSnchezZamora

More from SandroSnchezZamora (8)

Recently uploaded

Recently uploaded (20)