Prediction uncertainty for tracer migration in random heterogeneities
with multifractal character
S. Painter
a,*, G. Mahinthakumar
baAustralian Petroleum Cooperative Research Centre, Commonwealth Scienti®c and Industrial Research Organization, Glen Waverley, Vic. 3150,
Australia
bCenter 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 sta-tistical 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 approx-imations 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 eective 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 eects such as deviations from Gaussian behavior. This sensitivity illustrates the need for careful aquifer characterization and conditional numerical simulation in practical situations re-quiring 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 dicult 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 theo-retical work [8,5] and numerical experiments.
The vast majority of studies addressing transport in random heterogeneity rely on a stationary Gaussian
random ®eld with exponentially decaying covariance function hY xrY xi r2
Yexpÿjrj=IY as a model for the log conductivity Y x lnK x. HereK xis the local hydraulic conductivity, r2
Y the variance in the stationaryY ®eld, andrthe lag distance. This particular model predates the large datasets of hydraulic conduc-tivities 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 sub-surface data [9,3,27,17,23,20] have produced clear evi-dence for long-range depenevi-dence or fractal structure, which is consistent with a hierarchy of heterogeneity scales instead of the single characteristic scale of heter-ogeneityIY 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 het-erogeneity [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 experi-ments, the statistics of non-reacting tracer migration in random heterogeneities that deviate from the classical assumptions. The goal is to assess the sensitivity of
*Corresponding author. Address: Center for Nuclear Waste
Regu-latory Analyses, Southwest Research Institue, 6220 Culebra Rd., San Antonio, TX 78238-5166, USA.
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 re-cently proposed [23,19,20] fractal model based on frac-tional Levy motion [26]. We then trace the movement of non-reacting tracers through a large number of realiza-tions 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 re-quired for a plume centroid to move through imper-fectly known heterogeneities. The spreading of the plume about its centroid due to diusion is not ad-dressed here. However it is noted that the spreading of a ®nite-sized plume in fractal media and its quanti®cation through an eective dispersion coecient are also of considerable practical interest and have been addressed [1,6].
The need to simulate multiple realizations of rela-tively 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 distribu-tions, ®nite-size scalings, and other non-ideal features that are dicult 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 ground-water 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 ap-proach to modeling fractal noises is to specify a two-point covariance function that is power-law C jrj / jrjÿb for large separation distances jrj. The non-sta-tionary fractal motions, on the other hand, do not have a well-de®ned C jrj and are usually characterized by their power-law variogram:
c jrj DY xr ÿY x2E/ jrj2H; 1 whereH is the Hurst coecient (0<H61). In the sit-uationH>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
H1=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 frac-tional 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 het-erogeneity 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 distri-bution in situations where this is appropriate. The main feature of this class of models is the distribution of in-crements. We model the increments as having a sym-metric Levy distribution
ProbfY x ÿY xr6zg L z;a;0;C0jrjH; 2
where L z;a;l;C is the symmetric Levy distribution centered at l with width parameter C and Levy index
Here H2 0;1 is the Hurst parameter and C0 is the
Levy width parameter at unit lag. The lag distancejrjis in units of lattice spacing. The Levy index measures the degree of deviation from the Gaussian distribution for the increments; a2 corresponds to the Gaussian dis-tribution and the fBm model. The deviation from the Gaussian distribution increases with decreasinga. In the particular construction [11] of fLm used here, the role of the Hurst parameterH is similar to its role in the fBm model: H<1=2 implies anti-persistence and H>1=2 implies persistence. However, the situation H1=2 does not necessarily imply independent increments ex-cept 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 C2
0 is the
semi-variogram at unit lag in the special case of fBm. A complete description of our fractal modeling ap-proach and details of its implementation in the LevySim computer code can be found elsewhere [22]. The con-ditional simulation algorithm relies on the generic geo-statistical method known as sequential simulation (see Ref. [10], for example). In this technique, each simula-tion node is visited in turn. At each node the probability distribution conditional on any data and on the previ-ously simulated values is constructed and then sampled. Within the bfLm model, the lower-dimensional condi-tional probability distribution, which is necessary for application of sequential simulation techniques, is not available in closed form. An ecient numerical proce-dure [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 to-ward the more isolated regions of the simulation do-main. 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 station-arity 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 situa-tion, 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
jY x
h r ÿY xjqi / jrjp q 3 with multifractal spectrum p q that deviates from the monofractal resultp q qH.
A number of numerical tests were performed to ensure that the simulated random ®elds have the desired char-acteristics. As an example we compute the semi-vario-gram for an ensemble of 16 one-dimensional realizations of fBm withH0:25 (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 andL´20´ 20 lattice units withLin the range 50±600. Each realization in our Monte Carlo ex-periments involves the following steps:
(1) The LevySim multifractal simulation code is used to generate a randomY map. Because the model used is 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 vario-gram model characteristic of fBm. The Hurst parameterHin this test is 0.25, which is in the range for antipersistent motion.
0 1 2 3 4 5 6
multifractal (bounded fLm)
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 K0expY, where K0 is the geometric mean of the
permeability.
(2) The steady state saturated groundwater ¯ow equation is solved numerically using theKmap obtained from Step 1. Porosity variations are neglected and we set the porosity equal to unity for convenience. The boun-dary conditions are constant heads on the x0 and
xLfaces 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 paralleli-zed for distributed memory machines using a two-di-mensional domain decomposition strategy. For the linear system solution ecient multigrid and conjugate gradient solvers have been implemented [14,25]. Excel-lent 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 condi-tioning datum used in the generation of theY 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 diusion and fo-cusing on the uncertainty in the position of a plume centroid. A tri-linear interpolation is used in the calcu-lation 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 u0JK0 where J is the
macroscopic head gradient andK0 the geometric mean
of theK-®eld.
The above steps were repeated for a large number of realizations (512±2000). We generated the independent realizations of theY ®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 par-ticle 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 independentY ®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 indexa C0 H bl bu
fBm6 200´40´40 2 0.05 0.25 ÿ4 4
fBm7 200´40´40 2 0.15 0.25 ÿ4 4
fBm8 50´40´40 2 0.25 0.25 ÿ4 4
fBm9 200´40´40 2 0.05 0.35 ÿ4 4
fBm10 400´40´40 2 0.15 0.25 ÿ4 4
fBm11 50´40´40 2 0.15 0.25 ÿ4 4
fBm12 400´40´40 2 0.10 0.25 ÿ4 4
fBm21 100´20´20 2 0.20 0.35 ÿ4 4
fBm22 100´20´20 2 0.15 0.35 ÿ4 4
fBm23 100´20´20 2 0.10 0.25 ÿ4 4
fBm24 50´20´20 2 0.30 0.25 ÿ4 4
fBm25 50´20´20 2 0.20 0.25 ÿ4 4
fBm27 100´20´20 2 0.25 0.25 ÿ4 4
fBm28 200´20´20 2 0.10 0.35 ÿ4 4
fBm29 50´20´20 2 0.25 0.15 ÿ4 4
bfLm1 600´40´40 1.5 0.25 0.25 ÿ4 4
bfLm2 400´40´40 1.5 0.25 0.25 ÿ4 4
bfLm3 300´40´40 1.5 0.25 0.25 ÿ4 4
bfLm4 200´40´40 1.5 0.25 0.25 ÿ4 4
bfLm5 100´40´40 1.5 0.25 0.25 ÿ4 4
bfLm6 200´40´40 1.5 0.25 0.25 ÿ8 8
bfLm7 100´40´40 1.5 0.25 0.25 ÿ8 8
bfLm8 200´40´40 1.5 0.05 0.25 ÿ6 6
bfLm9 200´40´40 1.5 0.05 0.25 ÿ8 8
bfLm11 200´40´40 1.5 0.15 0.25 ÿ4 4
bfLm12 200´40´40 1.5 0.15 0.25 ÿ6 6
bfLm13 200´40´40 1.5 0.15 0.25 ÿ8 8
bfLm14 100´40´40 1.5 0.15 0.25 ÿ4 4
aThe parameterHis the Hurst exponent,C2
Dierent combinations of model parameters and domain sizes were considered. These are shown in Ta-ble 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 x1, 50 and 390. Heresis the time required for the particle to travel the distance x. These histograms are reasonably well approximated by a Gaussian probability density func-tion.
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 lin-early with travel distance. Exploring this further, en-semble mean travel-times s are plotted versus travel distance in Fig. 4 for fBm10 and fBm11 runs (L400 and L50). The nearly straight lines on this double logarithmic plot indicate that s has a near power-law dependence on travel distance x,sA1xa1. The values
forA1 anda1 obtained by linear regression of logson
logx are summarized in Table 2. In all cases a11,
indicating a near linear increase with travel distance. Power-law scaling was also obtained for the variance of travel times r2
s, except that, in this situation, the
variance increases nearly quadratically with travel dis-tance (r2
s A2xa2 witha2near 2). These results for the
fBm runs are shown in Fig. 5 and the best-®t values of
A2 anda2 are shown in Table 2. Note also that A2
in-creases with increasing system size.
In order to assess the relative uncertainty in the ®tted values ofA1 and A2, we performed a simple numerical
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 ex-perimental standard deviation was calculated from the set of eight ®tted values. Making use of the inverse de-pendence 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 realiza-tions. For the fBm21 situation, we estimate a coecient of variation (standard deviation divided by the mean) of 0.014 forA1 and 0.061 forA2. 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 r2
s and s when Y is stationary and multiGaussian
-1 -0.5 0 0.5 1
0 0.02 0.04 0.06 0.08
Log [τ/x]
10
Probability Density
Fig. 3. The distributions of travel times in random fBm heterogeneity are approximately log-normal independent of travel distancex. Shown are the probability densities for travel times for several dierent values of the travel distancex. The travel times have been normalized by the mean ¯ow velocity and the distance traveled.
Fig. 4. The mean travel time s increases nearly linearly with travel distance (sA1xa1 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 1.02 1.00 0.0259 2.05
fBm7 1.12 1.00 0.304 2.04
fBm8 1.10 1.01 0.277 2.16
fBm9 1.05 1.00 0.0659 2.05
fBm10 1.27 0.999 0.537 2.03
fBm11 1.05 1.00 0.0923 2.11
fBm12 1.12 0.999 0.192 2.02
fBm13 1.30 0.998 0.914 2.02
fBm21 1.22 0.997 0.481 2.03
fBm22 1.34 0.998 0.925 2.07
fBm23 1.11 0.999 0.216 2.04
fBm24 1.30 0.997 0.829 2.07
fBm25 1.15 0.999 0.242 2.08
fBm27 1.43 0.999 1.39 2.04
fBm28 1.23 0.997 0.495 2.01
fBm29 1.08 1.00 0.121 2.08
aThe parameters are obtained as ®ts to the meansand variancer2
sof
travel time versus travel distancex. Speci®cally,A1anda1are ®tting
parameters in sA1xa1, and A2 and a2 are ®tting parameters in
r2
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 approxi-mations, which are valid when the integral scale of heterogeneity is comparable to the travel distance. For example, in 3-D isotropic media,
These were derived in a manner completely analogous to the 2-D ones in Ref. [4], but make use of 3-D relation-ships 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, respec-tively. 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 increasingLis also consistent qualitatively with the near-®eld results. In the non-stationary fractal media considered here, there is no position-independent r2
Y. Instead the variance increases with increasing distance from the conditioning data. For the particular con®gu-ration studied, we also expect the variance in the lon-gitudinal velocity to increase with increasing system size leading to an increase in A1 andA2. Exploring this
sys-tem-size dependence further, we de®ne an eective variance that incorporates the size dependence
r2eff andV the volume of the simulation domainD. For fBm media and a single conditioning datum located at the origin,
whereDis the transverse dimension of the system. The eective variance r2
eff 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 A1exp 8=30r2eff
system size. Shown isA1for fBm runs versus an eective variance
in-corporating the system-size dependence (Eq. (7)). The solid lineA1
exp 8=30r2
effobtained from Eq. (4) overestimates the numerical
re-sults slightly but captures the general trend.
0.2
0.4
0.6
0.8
1
system size. Shown isA2for fBm runs versus an eective variance
in-corporating the system-size dependence (Eq. (7)). The solid lineA2
exp 8=15r2
obtained from Eq. (5) captures the general shape.
Fig. 5. The variance in the travel timer2
sincreases nearly parabolically
with travel distance asr2
sA2xa2witha2near 2. That is, in logarithmic
scale log r2
s log A2 2 log x. The magnitudeA2is dependent on
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 eective 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).
6. Travel-time distribution and scaling in bfLm media
Power-law scaling was also found forr2
s andsin the
bfLm situation. The scaling exponentsa1anda2are 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 decreasingawith everything else ®xed. A1 and A2 also increase with
in-creasing 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 eect of increasingA1andA2.
The dierence 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 eects caused by non-Gaussian behavior in the bfLm ®elds. We now attempt to separate these eects by comparing the
various runs on an equivalent basis de®ned in terms of the eective variance. First note that although the Levy distributions underlying the bfLm model have diverging moments (in®nite variance), the bounds on the simu-lations truncate these distributions and render the variance ®nite. The variance of a truncated Levy dis-tribution can be calculated numerically and then used to de®ne an eective variance. Proceeding as in the fBm case, Eq. (6) becomes
r2 function (pdf) centered at 0 with width parameter C. The parameter Cc u C0jujH is the Levy width
pa-rameter for theY ®eld at the spatial pointu, conditional on the data. The factor g is required to normalize the truncated pdf. Speci®cally, gRbu
bl ` n; 0;Ccdn. The
eective variance can be calculated numerically, pro-vided 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 prefactorA2in the scaling r2 contrast with the fBm results shown in Fig. 7, the bfLm media have a range of ®ttedA2values for a givenr2eff. In
addition, theA2's for bfLm are larger than those for fBm
Table 3
Scaling parameters for ensemble-mean and variance of travel times in bfLm mediaa
Run A1 a1 A2 a2
bfLm1 2.08 0.984 5.99 1.95
bfLm2 1.99 0.982 5.42 1.98
bfLm3 1.53 0.995 1.95 2.04
bfLm4 1.42 0.974 4.39 1.86
bfLm5 1.35 0.996 1.03 2.04
bfLm6 2.46 0.978 21.82 1.88
bfLm7 1.76 0.994 5.17 1.99
bfLm8 1.04 0.998 0.10 2.00
bfLm9 1.04 1.000 0.14 2.03
bfLm11 1.30 0.995 1.15 1.99
bfLm12 1.53 0.994 10.38 1.94
bfLm13 1.87 0.964 41.72 1.83
bfLm14 1.22 0.991 1.47 1.85
aThe parameters are obtained as ®ts to the meansand variancer2
sof
travel time versus travel distancex. Speci®cally,A1anda1are ®tting
parameters in sA1xa1, and A2 and a2 are ®tting parameter in
degree of non-Gaussianity in the random ®eld. Shown isA2for fBm
runs versus an eective variance incorporating the system-size depen-dence (Eq. (9)). TheA2for the bfLm runs are consistently larger than
the corresponding fBm runs with the same reff. The solid line is
A2exp 8=15r2eff
at the same eective variance. Similar, but less dramatic, results were found for the ®tted prefactors A1 (not
shown). These results suggest that the dierences in the fBm and bfLm results are not due purely to dierences in the eective variance. The dierences can be attrib-uted to dierence caused by the non-multiGaussian nature of the bfLm ®elds.
7. Discussion and conclusions
Analytical theories based on short-range heteroge-neities 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 para-bolically with x at small x. Similarly, the mean travel time is expected to change from one linear scaling to another linear scaling with dierent 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 heteroge-neity ¯uctuations regardless of system size or travel distance.
The practical implications of this are clear: extrapo-lations 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 underes-timated signi®cantly.
Although the scaling exponents were nearly the same for all cases considered, the magnitudes of the travel-time 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 re-placing theYvariance with a (conditional) eective 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 be-havior in the log conductivity ®eld, suggesting that re-sults applicable to GaussianY®elds should be used with care. This sensitivity underscores the need for aquifer
characterization combining ®eld measurements, model selection/validation studies, conditional stochastic sim-ulations, and numerical ¯ow/transport modeling.
Acknowledgements
This work was partially supported by a travel grant from the Bilaterial Collaboration Program of the Aus-tralian Department of Industry, Science and Tourism. We also acknowledge the support by the Center of Computational Sciences at Oak Ridge National Labo-ratory for the use of Intel Paragon XP/S 150 super-computer.
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