Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue1.1999:

Advances in Water Resources 23 (1999) 49±57

Prediction uncertainty for tracer migration in random heterogeneities
with multifractal character
S. Painter
a

a,*

, G. Mahinthakumar

b

Australian Petroleum Cooperative Research Centre, Commonwealth Scienti®c and Industrial Research Organization, Glen Waverley, Vic. 3150,
Australia
b
Center for Computational Sciences, Bldg 4500N, MS 6203, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
Received 12 August 1997; received in revised form 23 December 1998; accepted 29 December 1998

Abstract
Travel-time statistics for non-reacting tracers in fractal and multifractal media are addressed through numerical simulations. The

logarithm of hydraulic conductivity is modeled using fractional Brownian motion (fBm) and more recently developed multifractal
model based on bounded fractional Levy motion (bfLm). These models have been shown previously to accurately reproduce statistical properties of large conductivity datasets. The ensemble-mean travel time increases nearly linearly with travel distance and the
variance in the travel time increases nearly parabolically with travel distance. This is consistent with near-®eld analytical approximations developed for non-fractal media and suggests that these analytical results may have some degree of robustness to non-ideal
features in the random-®eld models. The magnitudes of the travel-time moments are dependent on the system size. For fBm media,
this size dependence can be explained using an e€ective variance that increases with increasing size of the ¯ow system. However, the
magnitudes of the travel-time moments are also sensitive to other non-ideal e€ects such as deviations from Gaussian behavior. This
sensitivity illustrates the need for careful aquifer characterization and conditional numerical simulation in practical situations requiring accurate estimates of uncertainty in the plume position. Ó 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction
Uncertainty created by incomplete knowledge of
subsurface heterogeneity is a widespread issue facing
geoscientists concerned with predicting the movement of
contaminant plumes. Managing this uncertainty in a
systematic and quantitative manner is a dicult task
due to the great sensitivity of ¯ow and transport to
spatial variability and to the complex, highly variable
nature of subsurface formations. Although numerical
methods combining conditional stochastic simulation
[10] with ¯ow and transport simulations can be used for
this problem, it is convenient in some situations to have

approximate analytical models as alternatives to these
more elaborate numerical calculations. This need for
analytical methods devoted to transport in random
heterogeneities has motivated a large amount of theoretical work [8,5] and numerical experiments.
The vast majority of studies addressing transport in
random heterogeneity rely on a stationary Gaussian
*
Corresponding author. Address: Center for Nuclear Waste Regulatory Analyses, Southwest Research Institue, 6220 Culebra Rd., San
Antonio, TX 78238-5166, USA.

random ®eld with exponentially decaying covariance
function hY …x ‡ r†Y …x†i ˆ r2Y exp ‰ÿjrj=IY Š as a model
for the log conductivity Y …x† ˆ ln ‰K…x†Š. Here K…x† is
the local hydraulic conductivity, r2Y the variance in the
stationary Y ®eld, and r the lag distance. This particular
model predates the large datasets of hydraulic conductivities that are now available for analysis, and in our
opinion, should be regarded as a rule-of-thumb rather
than an absolute fact. Indeed, direct analyses of subsurface data [9,3,27,17,23,20] have produced clear evidence for long-range dependence or fractal structure,
which is consistent with a hierarchy of heterogeneity
scales instead of the single characteristic scale of heterogeneity IY implicit in the standard model. In addition,

the scale-dependence in ®eld-derived longitudinal
dispersivities [8] is known to be consistent with fractal
correlations and inconsistent with a single scale of heterogeneity [18]. Further, the assumptions of stationarity
and Gaussian behavior of the log-conductivity ®eld have
also been questioned [19,20].
This paper addresses, through numerical experiments, the statistics of non-reacting tracer migration in
random heterogeneities that deviate from the classical
assumptions. The goal is to assess the sensitivity of

0309-1708/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 0 4 - 4

50

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

transport predictions to the choice of random ®eld
models and to provide some initial statistics on travel
times to guide the development of analytical models. We
focus on non-stationary random ®elds with stationary

increments and long-range dependence (fractal and
multifractal motions), as this class of models has been
shown to reproduce observed statistical features of ®eld
measurements of hydraulic conductivity. Speci®cally, we
model the logarithm of hydraulic conductivity using
fractional Brownian motion (fBm) [16] and a more recently proposed [23,19,20] fractal model based on fractional Levy motion [26]. We then trace the movement of
non-reacting tracers through a large number of realizations generated from these models and collect statistics
on the travel times.
For each of our Monte Carlo simulations, we trace a
single particle along a streamline and collect statistics
on the time required to pass through a set of control
planes perpendicular to the macroscopic ¯ow direction.
The issue addressed then is uncertainty in the time required for a plume centroid to move through imperfectly known heterogeneities. The spreading of the
plume about its centroid due to di€usion is not addressed here. However it is noted that the spreading of a
®nite-sized plume in fractal media and its quanti®cation
through an e€ective dispersion coecient are also of
considerable practical interest and have been addressed
[1,6].
The need to simulate multiple realizations of relatively large three-dimensional systems places large
computational burdens on Monte Carlo studies like

those considered here. High-performance massively
parallel supercomputers were used for the random-®eld
generation and for the numerical solution of Laplace's
equation for the hydraulic head. This combination of
improved random-®eld models and high-performance
computers allow us to address non-Gaussian distributions, ®nite-size scalings, and other non-ideal features
that are dicult to treat theoretically.

2. Fractal models for subsurface heterogeneities
A large number of studies of subsurface properties
have produced evidence for long-range dependence or
fractal structure. For recent examples in the groundwater literature see Refs. [17,20].
When dealing with fractal models it is important to
make the distinction between stationary fractal noises
and the non-stationary fractal motions. A typical approach to modeling fractal noises is to specify a twopoint covariance function that is power-law C…jrj† /
ÿb
jrj for large separation distances jrj. The non-stationary fractal motions, on the other hand, do not have
a well-de®ned C…jrj† and are usually characterized by
their power-law variogram:

D
E
c…jrj† ˆ ‰Y …x ‡ r† ÿ Y …x†Š2 / jrj2H ;

…1†

where H is the Hurst coecient (0 < H 6 1). In the situation H > 1=2 positive increments tend to be followed
by positive increments (persistence), while the opposite
is true for H < 1=2 (anti-persistence). The situation
H ˆ 1=2 corresponds to independent increments. The
prototype fractal models are fBm and its increment
process, fractional Gaussian noise (fGn) [16,15].
There are also classes of fractal models [26] for which
the two-point correlation and the variogram are not
de®ned. This class of models is based on the Levy-stable
distributions [12,7,28] which have diverging theoretical
moments. The correlation function and the variogram
can be replaced by analogous measures of the two-point
dependence in this case [24]. The main appeal of fractional Levy motion (fLm) as an alternative to the fBm
model for subsurface properties is that fLm captures

abrupt changes in subsurface properties associated with
geological strati®cation [23].

3. Bounded fractional motion models and simulation
algorithm
Proposed fractal models for subsurface heterogeneity
include anti-persistent fractal motions and the persistent
fractal noises. We believe the failure of the geoscience
community to settle on one class of models is likely due
to subsurface formations displaying aspects of both
models. It has been pointed out [19] that incremental
values at short separation distances display much
greater degree of consistency in their histograms as
compared to the property values themselves, which is
compelling evidence for the use of a non-stationary
fractal motion with stationary increments. At large
scales, a strict fractal motion model would predict large
¯uctuations in the property of interest. Ultimately this is
inconsistent with subsurface properties, which always
have a limited dynamic range.

The approach used here is to model subsurface heterogeneity using fractional Levy motion coupled with
explicit bounds on the simulated variables. This
bounded fractional Levy motion (bfLm) model includes
fBm and bounded fBm as special cases, and can also
model signi®cant deviations from the Gaussian distribution in situations where this is appropriate. The main
feature of this class of models is the distribution of increments. We model the increments as having a symmetric Levy distribution
H

ProbfY …x† ÿ Y …x ‡ r† 6 zg ˆ L…z; a; 0; C0 jrj †;

…2†

where L…z; a; l; C† is the symmetric Levy distribution
centered at l with width parameter C and Levy index
a 2 …0; 2Š. The special case L…z; 2; l; C† is the Gaussian
(Normal) distribution with mean l and variance 2C 2 .

51

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

Here H 2 …0; 1† is the Hurst parameter and C0 is the
Levy width parameter at unit lag. The lag distance jrj is
in units of lattice spacing. The Levy index measures the
degree of deviation from the Gaussian distribution for
the increments; a ˆ 2 corresponds to the Gaussian distribution and the fBm model. The deviation from the
Gaussian distribution increases with decreasing a. In the
particular construction [11] of fLm used here, the role of
the Hurst parameter H is similar to its role in the fBm
model: H < 1=2 implies anti-persistence and H > 1=2
implies persistence. However, the situation H ˆ 1=2
does not necessarily imply independent increments except in the fBm case. This is because the particular
choice of the multivariate Levy distribution used in the
simulation algorithm does not include the situation of
independent variables. The parameter C02 is the semivariogram at unit lag in the special case of fBm.
A complete description of our fractal modeling approach and details of its implementation in the LevySim
computer code can be found elsewhere [22]. The conditional simulation algorithm relies on the generic geostatistical method known as sequential simulation (see
Ref. [10], for example). In this technique, each simulation node is visited in turn. At each node the probability
distribution conditional on any data and on the previously simulated values is constructed and then sampled.
Within the bfLm model, the lower-dimensional conditional probability distribution, which is necessary for

application of sequential simulation techniques, is not
available in closed form. An ecient numerical procedure [22] is used to construct the lower-dimensional
conditional probability distribution, which is then
sampled by a rejection method. The trajectory through
the simulation nodes, which is usually a purely random
one in the sequential simulation method, is biased toward the more isolated regions of the simulation domain. This helps reproduce the large-scale ¯uctuation
characteristics of the fractal motions [21].
We also impose explicit bounds (bu and bl ) on the
simulated variables. These enforce a return to stationarity at large scales which is potentially important in
both the non-Gaussian and Gaussian situations. In the
non-Gaussian situation, the bounds also eliminate the
diverging theoretical moments characteristic of Levy
models and force a return to Gaussian with increasing
separation distances, a feature which Liu and Molz [13]
have observed in the MADE dataset [2]. In this situation, the distribution of incremental values depends on
the lag separation, which is consistent with multifractal
rather than monofractal properties. By multifractal, we
mean that the media is characterized by a generalized
variogram of the form
q

hjY …x ‡ r† ÿ Y …x†j i / jrj

p…q†

…3†

with multifractal spectrum p…q† that deviates from the
monofractal result p…q† ˆ qH .

Fig. 1. The LevySim random-®eld generator used in this study can
produce realizations of a random ®eld with properties very similar to
fractional Brownian motion. Shown is the experimental variogram
versus lag distance in lattice units compared to the power-law variogram model characteristic of fBm. The Hurst parameter H in this test
is 0.25, which is in the range for antipersistent motion.

1.4

p(q)

1.2

monofractal (fBm)
p(q) = qH

1
0.8
0.6

x

x

x

0.4

x

x

x

x

x

multifractal
(bounded fLm)

x
x

0.2

x

α=1.5

x

0
0

1

2

3

4

5

6

q
Fig. 2. The LevySim random-®eld generator can also produce realizations of non-Gaussian random ®elds with multifractal properties.
Shown are the multifractal spectra p…q† obtained by ®tting the generalized variogram Eq. (3) for the fBm and bounded fLm models. The
solid line is the theoretical result for a fBm monofractal.

A number of numerical tests were performed to ensure
that the simulated random ®elds have the desired characteristics. As an example we compute the semi-variogram for an ensemble of 16 one-dimensional realizations
of fBm with H ˆ 0:25 (Fig. 1) and compare this with the
power-law variogram (1). In Fig. 2 we also show the
multifractal spectrum p…q†.

4. Numerical simulations of tracer migration
We consider three-dimensional domains of size
L ´ 40 ´ 40 and L ´ 20 ´ 20 lattice units with L in the
range 50±600. Each realization in our Monte Carlo experiments involves the following steps:
(1) The LevySim multifractal simulation code is used
to generate a random Y map. Because the model used is

52

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

a non-stationary one, it is necessary to condition this to
at least one measurement. We specify a value of zero for
Y at the center of the upstream face of the domain. A
conductivity map is obtained from the Y map via
K ˆ K0 exp Y , where K0 is the geometric mean of the
permeability.
(2) The steady state saturated groundwater ¯ow
equation is solved numerically using the K map obtained
from Step 1. Porosity variations are neglected and we set
the porosity equal to unity for convenience. The boundary conditions are constant heads on the x ˆ 0 and
x ˆ L faces and no-¯ow conditions on the other (longer)
faces. The solution of this equation is obtained using a
Galerkin ®nite-element discretization with eight-node
linear brick elements. The ¯ow code has been parallelized for distributed memory machines using a two-dimensional domain decomposition strategy. For the
linear system solution ecient multigrid and conjugate
gradient solvers have been implemented [14,25]. Excellent scalability of the ¯ow code has been demonstrated
on up to 1024 parallel processors of the Intel Paragon
XP/S 150. Mass balance tests indicate that the errors in
the nodal mass ¯ux balance are less than 0.1% for the
conditions considered here.
(3) A single particle is released from the center of the
upstream face, which is also the location of the conditioning datum used in the generation of the Y map. This

particle is tracked through the velocity ®eld obtained
from Step 2 until it reaches the downstream face of the
domain. A single particle was tracked instead of a group
of particles because we are neglecting di€usion and focusing on the uncertainty in the position of a plume
centroid. A tri-linear interpolation is used in the calculation of the local velocity from the elemental velocities.
The times at which the particle reaches a set of reference
surfaces perpendicular to the macroscopic ¯ow direction
are recorded. All travel times are normalized by the
macroscopic ¯ow velocity u0 ˆ JK0 where J is the
macroscopic head gradient and K0 the geometric mean
of the K-®eld.
The above steps were repeated for a large number of
realizations (512±2000). We generated the independent
realizations of the Y ®eld in parallel with one realization
per computational node on an Intel Paragon XP/S 150.
These multiple realizations were then input one at a time
into the ¯ow code which was distributed across several
(typically 64±128) computational nodes. The computed
velocity ®elds from this step were then used in the particle tracking transport code. This step was also done in
parallel, with one realization per computational node of
the XP/S 150. This setup is justi®ed since the memory
requirements for the generation of independent Y ®elds
and particle tracking are much smaller than the ¯ow
calculations and are intrinsically parallel.

Table 1
Model parameters for fBm and bfLm media considered in this studya
Run

Size

Levy index a

C0

H

bl

bu

fBm6
fBm7
fBm8
fBm9
fBm10
fBm11
fBm12
fBm21
fBm22
fBm23
fBm24
fBm25
fBm27
fBm28
fBm29
bfLm1
bfLm2
bfLm3
bfLm4
bfLm5
bfLm6
bfLm7
bfLm8
bfLm9
bfLm11
bfLm12
bfLm13
bfLm14

200 ´ 40 ´ 40
200 ´ 40 ´ 40
50 ´ 40 ´ 40
200 ´ 40 ´ 40
400 ´ 40 ´ 40
50 ´ 40 ´ 40
400 ´ 40 ´ 40
100 ´ 20 ´ 20
100 ´ 20 ´ 20
100 ´ 20 ´ 20
50 ´ 20 ´ 20
50 ´ 20 ´ 20
100 ´ 20 ´ 20
200 ´ 20 ´ 20
50 ´ 20 ´ 20
600 ´ 40 ´ 40
400 ´ 40 ´ 40
300 ´ 40 ´ 40
200 ´ 40 ´ 40
100 ´ 40 ´ 40
200 ´ 40 ´ 40
100 ´ 40 ´ 40
200 ´ 40 ´ 40
200 ´ 40 ´ 40
200 ´ 40 ´ 40
200 ´ 40 ´ 40
200 ´ 40 ´ 40
100 ´ 40 ´ 40

2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5

0.05
0.15
0.25
0.05
0.15
0.15
0.10
0.20
0.15
0.10
0.30
0.20
0.25
0.10
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.05
0.05
0.15
0.15
0.15
0.15

0.25
0.25
0.25
0.35
0.25
0.25
0.25
0.35
0.35
0.25
0.25
0.25
0.25
0.35
0.15
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25

ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ4
ÿ8
ÿ8
ÿ6
ÿ8
ÿ4
ÿ6
ÿ8
ÿ4

4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
8
8
6
8
4
6
8
4

a

The parameter H is the Hurst exponent, C02 is the semi-variogram at unit lag, and bl and bu are upper and lower bounds on the Y ®eld.

53

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

Di€erent combinations of model parameters and
domain sizes were considered. These are shown in Table 1.

5. Travel-time distribution and scaling in fBm media
Histograms of normalized travel times log10 …s=x†
from run fBm1 are shown in Fig. 3 for x ˆ 1, 50 and
390. Here s is the time required for the particle to travel
the distance x. These histograms are reasonably well
approximated by a Gaussian probability density function.
The three histograms of normalized travel time
overlay each other in Fig. 3. This indicates that the
shape of the travel-time distribution is independent of
the travel distance. It also suggests that the mean and
standard deviation of the log travel-times increase linearly with travel distance. Exploring this further, ensemble mean travel-times s are plotted versus travel
distance in Fig. 4 for fBm10 and fBm11 runs (L ˆ 400
and L ˆ 50). The nearly straight lines on this double
logarithmic plot indicate that s has a near power-law
dependence on travel distance x, s ˆ A1 xa1 . The values
for A1 and a1 obtained by linear regression of logs on
log x are summarized in Table 2. In all cases a1  1,
indicating a near linear increase with travel distance.
Power-law scaling was also obtained for the variance
of travel times r2s , except that, in this situation, the
variance increases nearly quadratically with travel distance (r2s ˆ A2 xa2 with a2 near 2). These results for the
fBm runs are shown in Fig. 5 and the best-®t values of
A2 and a2 are shown in Table 2. Note also that A2 increases with increasing system size.
In order to assess the relative uncertainty in the ®tted
values of A1 and A2 , we performed a simple numerical

Fig. 4. The mean travel time s increases nearly linearly with travel
distance (s ˆ A1 xa1 with a1 near 1). That is, in logarithmic scale
log …s†  log …A1 † ‡ log…x†. The magnitude A1 is dependent on the
system size.

Table 2
Scaling parameters for ensemble-mean and variance of travel times in
fBm mediaa
Run

A1

a1

A2

a2

fBm6
fBm7
fBm8
fBm9
fBm10
fBm11
fBm12
fBm13
fBm21
fBm22
fBm23
fBm24
fBm25
fBm27
fBm28
fBm29

1.02
1.12
1.10
1.05
1.27
1.05
1.12
1.30
1.22
1.34
1.11
1.30
1.15
1.43
1.23
1.08

1.00
1.00
1.01
1.00
0.999
1.00
0.999
0.998
0.997
0.998
0.999
0.997
0.999
0.999
0.997
1.00

0.0259
0.304
0.277
0.0659
0.537
0.0923
0.192
0.914
0.481
0.925
0.216
0.829
0.242
1.39
0.495
0.121

2.05
2.04
2.16
2.05
2.03
2.11
2.02
2.02
2.03
2.07
2.04
2.07
2.08
2.04
2.01
2.08

a

The parameters are obtained as ®ts to the mean s and variance r2s of
travel time versus travel distance x. Speci®cally, A1 and a1 are ®tting
parameters in s ˆ A1 xa1 , and A2 and a2 are ®tting parameters in
r2s ˆ A2 xa2 .

Probability Density

0.08
0.06
0.04
0.02
0
-1

-0.5

0
Log [τ/x]

0.5

1

10

Fig. 3. The distributions of travel times in random fBm heterogeneity
are approximately log-normal independent of travel distance x. Shown
are the probability densities for travel times for several di€erent values
of the travel distance x. The travel times have been normalized by the
mean ¯ow velocity and the distance traveled.

experiment. The 2000 realizations for the fbm21 case
were divided into eight batches of 250 realizations. The
®tting procedure was applied to each batch. The experimental standard deviation was calculated from the
set of eight ®tted values. Making use of the inverse dependence of the variance on the number of realizations,
we then extrapolated these standard deviations based on
the 250 realizations to the one based on 2000 realizations. For the fBm21 situation, we estimate a coecient
of variation (standard deviation divided by the mean) of
0.014 for A1 and 0.061 for A2 . Thus we have con®dence
that the trends identi®ed in this study are systematic
trends and not due to random ¯uctuations in the results.
Cvetkovic et al. [4] have developed analytical models
for r2s and s when Y is stationary and multiGaussian

54

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

where r2c …u† is the variance at u conditional on the data,
and V the volume of the simulation domain D. For fBm
media and a single conditioning datum located at the
origin,
r2eff

2C0
ˆ 2
DL

Z
Z D=2
ZL D=2

H

…x2 ‡ y 2 ‡ z2 † dx dy dz;

…7†

0 ÿD=2ÿD=2

Fig. 5. The variance in the travel time r2s increases nearly parabolically
with travel distance as r2s ˆ A2 xa2 with a2 near 2. That is, in logarithmic
scale log …r2s †  log …A2 † ‡ 2 log…x†. The magnitude A2 is dependent on
the system size.

with exponentially decaying correlations. Although the
conditions in the numerical simulation deviate markedly
from this standard model, the qualitative features from
the numerical experiments can be explained using this
model. Of speci®c interest are the near-®eld approximations, which are valid when the integral scale of
heterogeneity is comparable to the travel distance. For
example, in 3-D isotropic media,


8 2 x
s ˆ exp
rY
…4†
30
u0
and





x2
8 2
8 2
2
r
r ÿ1 :
exp
…5†
rs ˆ 2 exp
u0
15 Y
15 Y
These were derived in a manner completely analogous to
the 2-D ones in Ref. [4], but make use of 3-D relationships between velocity and Y ¯uctuations [5]. The key
feature here is the scaling with travel distance x. The
exponents a1 and a2 ®tted to the numerical results for
fBm are close to the near-®eld values 1 and 2, respectively. This means that, as far as fractal media are
concerned, the particles can always be considered in the
near-®eld of the heterogeneity ¯uctations, regardless of
the system size or travel distance.
The increase in the prefactors A1 and A2 in Table 2
with increasing L is also consistent qualitatively with the
near-®eld results. In the non-stationary fractal media
considered here, there is no position-independent r2Y .
Instead the variance increases with increasing distance
from the conditioning data. For the particular con®guration studied, we also expect the variance in the longitudinal velocity to increase with increasing system size
leading to an increase in A1 and A2 . Exploring this system-size dependence further, we de®ne an e€ective
variance that incorporates the size dependence
Z
1
2
r2c …u† du;
…6†
reff ˆ
V
D

where D is the transverse dimension of the system.
The e€ective variance r2eff was calculated for each
combination of fBm model parameters in Table 1. The
prefactor A1 in the ®tted simulation results is plotted
versus this reff in Fig. 6. Also shown as a continuous
line is the theoretical expression A1 ˆ exp …8=30 r2eff †

A1 1.6
1.5
1.4
1.3
1.2
1.1
0.2
0.4
0.6
0.8
2
Effective Variance σeff

1

Fig. 6. The pre-factor A1 in the ®t s ˆ A1 xa1 increases with increasing
system size. Shown is A1 for fBm runs versus an e€ective variance incorporating the system-size dependence (Eq. (7)). The solid line A1 ˆ
exp…8=30r2eff † obtained from Eq. (4) overestimates the numerical results slightly but captures the general trend.

A2
10
1
0.1

0.2

0.4

0.6

0.8

1

2
Effective Variance σeff
Fig. 7. The pre-factor A2 in the ®t r2s ˆ A2 xa2 increases with increasing
system size. Shown is A2 for fBm runs versus an e€ective variance incorporating
the
dependence
(Eq. (7)). The solid line A2 ˆ
ÿ

 system-size
ÿ



exp 8=15 r2eff exp 8=15 r2eff ÿ 1 obtained from Eq. (5) captures
the general shape.

55

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

obtained from Eq. (4). The results of the simulations
are slightly greater than this model but otherwise track
the model reasonably well. A similar good ®t was also
obtained for the prefactor A2 in the ®tted travel-time
variance (Fig. 7). Thus our approach of utilizing an
e€ective variance in existing near-®eld theories for
travel-time statistics captures the complex dependence
on the various model parameters, including the system
size. However, we emphasize that this heuristic model
is proposed here as a way of understanding the results
from the large number of travel-time simulations. The
validity conditions for Eqs. (4) and (5) are clearly
violated in our simulations. The comparison should
not be considered a check on the accuracy of Eqs. (4)
and (5).

various runs on an equivalent basis de®ned in terms of
the e€ective variance. First note that although the Levy
distributions underlying the bfLm model have diverging
moments (in®nite variance), the bounds on the simulations truncate these distributions and render the
variance ®nite. The variance of a truncated Levy distribution can be calculated numerically and then used to
de®ne an e€ective variance. Proceeding as in the fBm
case, Eq. (6) becomes
r2eff

1
ˆ
V

Z

1
g…u†

Zbu

D

1
ˆ
V

Z

Power-law scaling was also found for r2s and s in the
bfLm situation. The scaling exponents a1 and a2 are near
1 and 2, respectively, similar to the fBm situation.
The magnitudes A1 and A2 for bfLm in Table 3 are
larger than the corresponding values for fBm. This is
due in part to increasing variance with decreasing a with
everything else ®xed. A1 and A2 also increase with increasing system size for the same reason. By comparing
bfLm11 with bfLm12 and bfLm13, it is possible to see
that increasing the distance between the upper and lower
bounds has the e€ect of increasing A1 and A2 .
The di€erence between the fBm and bfLm results and
the sensitivity of the bfLm results to the upper and lower
bounds have two possible causes: the change in the
overall variance just mentioned, and possible e€ects
caused by non-Gaussian behavior in the bfLm ®elds. We
now attempt to separate these e€ects by comparing the
Table 3
Scaling parameters for ensemble-mean and variance of travel times in
bfLm mediaa
Run

A1

a1

A2

a2

bfLm1
bfLm2
bfLm3
bfLm4
bfLm5
bfLm6
bfLm7
bfLm8
bfLm9
bfLm11
bfLm12
bfLm13
bfLm14

2.08
1.99
1.53
1.42
1.35
2.46
1.76
1.04
1.04
1.30
1.53
1.87
1.22

0.984
0.982
0.995
0.974
0.996
0.978
0.994
0.998
1.000
0.995
0.994
0.964
0.991

5.99
5.42
1.95
4.39
1.03
21.82
5.17
0.10
0.14
1.15
10.38
41.72
1.47

1.95
1.98
2.04
1.86
2.04
1.88
1.99
2.00
2.03
1.99
1.94
1.83
1.85

a

The parameters are obtained as ®ts to the mean s and variance r2s of
travel time versus travel distance x. Speci®cally, A1 and a1 are ®tting
parameters in s ˆ A1 xa1 , and A2 and a2 are ®tting parameter in
r2s ˆ A2 xa2 .

…8†

Ccÿ1 n2 `…Ccÿ1 n ; 0; 1† dn du;

…9†

bl

1
g…u†

Zbu

D

6. Travel-time distribution and scaling in bfLm media

n2 `…n ; 0; Cc † dn du

bl

where `…n ; 0; C† is the Levy-stable probability density
function (pdf) centered at 0 with width parameter C.
H
The parameter Cc …u† ˆ C0 juj is the Levy width parameter for the Y ®eld at the spatial point u, conditional
on the data. The factor g is required
to normalize the
Rb
truncated pdf. Speci®cally, g ˆ blu `…n ; 0; Cc † dn. The
e€ective variance can be calculated numerically, provided care is taken when calculating the Levy-stable pdf.
A procedure [22] based on the inverse fast fourier
transform was used.
In Fig. 8 we show the ®tted prefactor A2 in the scaling
r2s ˆ A2 x2 versus e€ective variance de®ned in Eq. (9)
for ÿ the bfLm
The
solid
line is A2 ˆ
ÿ




 media.
exp 8=15 r2eff exp 8=15 r2eff ÿ 1 (see Eq. (5)). In
contrast with the fBm results shown in Fig. 7, the bfLm
media have a range of ®tted A2 values for a given r2eff . In
addition, the A2 's for bfLm are larger than those for fBm

A2
10
1
0.1

0.2

0.4

0.6

0.8

1

Effective Variance σ 2
eff
Fig. 8. The pre-factor A2 in the ®t r2s ˆ A2 xa2 also depends on the
degree of non-Gaussianity in the random ®eld. Shown is A2 for fBm
runs versus an e€ective variance incorporating the system-size dependence (Eq. (9)). The A2 for the bfLm runs are consistently larger than
the corresponding

ÿruns with
the  same reff . The solid line is
ÿ

fBm
A2 ˆ exp 8=15 r2eff exp 8=15 r2eff ÿ 1 , obtained from Eq. (5).

56

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57

at the same e€ective variance. Similar, but less dramatic,
results were found for the ®tted prefactors A1 (not
shown). These results suggest that the di€erences in the
fBm and bfLm results are not due purely to di€erences
in the e€ective variance. The di€erences can be attributed to di€erence caused by the non-multiGaussian
nature of the bfLm ®elds.

7. Discussion and conclusions
Analytical theories based on short-range heterogeneities predict non-linear transitions in the travel-time
moments versus travel distance. For example, these
theories predict the travel-time uncertainty to increase
linearly with travel distance x for large x and parabolically with x at small x. Similarly, the mean travel
time is expected to change from one linear scaling to
another linear scaling with di€erent magnitude as the
travel distance increases. This non-linear transition
from near-®eld to far-®eld behavior was not observed in
any of the simulations. In all the cases studied, the
ensemble-mean travel time increased nearly linearly
while the variance in the travel times increased nearly
parabolically with travel distance, consistent with near®eld theories. This means that, as far as the travel-time
distributions are concerned, particles can always be
treated as if they are in the near-®eld of the heterogeneity ¯uctuations regardless of system size or travel
distance.
The practical implications of this are clear: extrapolations of ®eld observations using theoretical results
appropriate for ergodic conditions may result in signi®cant errors in the predicted travel-times moments in
situations where a fractal model is more appropriate.
This is particularly true for the uncertainty in the plume
centroid (travel-time variance), which may be underestimated signi®cantly.
Although the scaling exponents were nearly the same
for all cases considered, the magnitudes of the traveltime moments were dependent on the details of the
particular situation. The prediction uncertainty is, for
example, strongly dependent on the system size, the
deviations from Gaussian behavior in the Y ®eld, and
the bounds on the Y ®eld. For media that are well
represented by an fBm model, a heuristic modi®cation
of existing near-®eld models can be used to make
quantitative predictions. This modi®cation involves replacing the Y variance with a (conditional) e€ective one
de®ned as a volume average. However, the applicability
of this procedure for more complicated arrangements of
conditioning data needs to be studied. Finally, we note a
strong dependence on the degree of non-Gaussian behavior in the log conductivity ®eld, suggesting that results applicable to Gaussian Y ®elds should be used with
care. This sensitivity underscores the need for aquifer

characterization combining ®eld measurements, model
selection/validation studies, conditional stochastic simulations, and numerical ¯ow/transport modeling.

Acknowledgements
This work was partially supported by a travel grant
from the Bilaterial Collaboration Program of the Australian Department of Industry, Science and Tourism.
We also acknowledge the support by the Center of
Computational Sciences at Oak Ridge National Laboratory for the use of Intel Paragon XP/S 150 supercomputer.

References
[1] Bellin A, Pannone M, Fiori A, Rinaldo A. On transport in porous
formations characterized by heterogeneity of evolving scales.
Water Resour Res 1996;32(12):3485±496.
[2] Boggs JM, Young SC, Beard LM, Gelhar LW, Rehfeldt KR,
Adams EE. A ®eld study of dispersion in a heterogeneous aquifer,
1: Overview and site description. Water Resour Res 1992;28:3281±
292.
[3] Crane SD, Tubman KM. Reservoir variability and modeling with
fractals. In: Barton CC, Pointe' PRL, editors. Fractals in
Petroleum Geology and Earth Processes, ch. 13. New York:
Plenum Press, 1995.
[4] Cvetkovic V, Cheng H, Wen X-H. Analysis of nonlinear e€ects on
tracer migration in heterogeneous aquifers using lagrangian travel
time statistcs. Water Resour Res 1996;32(6):1671±680.
[5] Dagan G. Flow and Transport in Porous Formations. New York:
Springer, 1989.
[6] Dagan G. The signi®cance of heterogeneity of evolving scales and
of anomalous di€usion to transport in porous formations. Water
Resour Res 1994;30(12):3327±336.
[7] Feller W. An Introduction to Probability Theory and its Applications, vol. 2. New York: Wiley, 1971.
[8] Gelhar LW. Stochastic subsurface hydrology from theory to
applications. Water Resour Res 1986;22(9):135S±145S.
[9] Hewett T. Fractal distributions of reservoir heterogeneity and
their in¯uence on ¯uid transport. In: Proceedings of Society of
Petroleum Engineers Annual Technical Meeting, 1986.
[10] Journel AG, Huijbregts CJ. Mining Geostatistics. New York:
Academic Press, 1978.
[11] Kumar P, Foufoula-Georgiou E. A multicomponent decomposition of spatial rainfall ®elds 2. self-similarity in ¯uctuations. Water
Resour Res 1993;29:2533±544.
[12] Levy P. Theorie de l'Addition des Variables Aleatoires. GauthierVillars, Paris, 1937.
[13] Liu HH, Molz FJ. Comment on `evidence for non-gaussian
scaling behavior in heterogeneous sedimentary formations'. Water
Resour Res 1997;33(4):907±908.
[14] Mahinthakumar G, Saied F. Distributed memory implementation
of multigrid methods for groundwater ¯ow problems with rough
coecients. In: Tentner AM, editor. Proceedings of the Simulation Multiconference: High Performance Computing 1996:51±57.
San Diego, 1996. Simulation Councils, Inc.
[15] Mandelbrot B. The Fractal Geometry of Nature. New York:
Freeman, 1982.
[16] Mandelbrot B, Ness JWV. Fractional brownian motions, fractional noises and applications. SIAM Review 1968;10:422±37.

S. Painter, G. Mahinthakumar / Advances in Water Resources 23 (1999) 49±57
[17] Molz FJ, Boman GK. Further evidence of fractal structure in
hydraulic conductivity distributions. Water Resour Res
1995;22(18):2545±548.
[18] Neuman SP. Universal scaling of hydraulic conductivities and
dispersivities in geologic media. Water Resour Res 1990;26(8):
1749±758.
[19] Painter S. Random fractal models of heterogeneity: the levy-stable
approach. Math Geol 1995;27:813±30.
[20] Painter S. Evidence for non-gaussian scaling behavior in heterogeneous sedimentary formations. Water Resour Res
1996;32(5):1183±195.
[21] Painter S. Stochastic interpolation of aquifer properties using
fractional levy motion. Water Resour Res 1996;32(5):1323±332.
[22] Painter S. Numerical method for conditional simulation of levy
random ®elds. Math Geol 1998;30(2):163±79.

57

[23] Painter S, Paterson L. Fractional levy motion as a model for
spatial variability in sedimentary rock. Geophys Res Letters
1994;21:2857±860.
[24] Press SJ. Estimation in univariate and multivariate stable distributions. J Am Stat Assoc 1972;67:842±46.
[25] Saied F, Mahinthakumar G. Ecient parallel multigrid based
solvers for large scale groundwater ¯ow simulations. Comput
Math Appl 1998;35(7):45±54.
[26] Taqqu MS. Random processes with long-range dependence and
high variability. J Geophys Res 1987;92:9683±686.
[27] Todoeschuck JP, Jensen OG, Labonte S. Scaling geology:
evidence from well logs. Geophysics 1990;55:480±84.
[28] Zolotarev VM. One-dimensional Stable Distributions. American
Mathematical Society, Providence Rhode Island, 1986.