Abstract
Longitudinal dispersion coefficients in given realizations of the transport computed by two currently used approximations of the first‐order in velocity variance are compared with accurate global random walk simulations. The comparisons are performed for the same ensemble of realizations of the Darcy velocity field, approximated by a quasiperiodic random field, for lognormal hydraulic conductivity with small variance and finite correlation lengths. The results show that at finite times of about one dispersion timescale, the mean coefficient is underestimated by ≈20%, and the fluctuations are overestimated by ≈80%. At larger times the errors decrease monotonously, and the first‐order approximations yield fairly good predictions for the mean and the fluctuations of the dispersion coefficient.
Authors
N. Suciu
Institute of Applied Mathematics, Friedrich-Alexander University of Erlangen-Nuremberg, Erlangen, Germany
C. Vamoş
‘T. Popoviciu Institute of Numerical Analysis, Romanian Academy,
J. Eberhard
Simulation in Technology, University of Heidelberg, Germany
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N. Suciu, C. Vamoş, J. Eberhard (2006), Evaluation of the first-order approximations for transport in heterogeneous media, Water Resour. Res., W11504,
doi: 10.1029/2005wr004714
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Evaluation of the first-order approximations for
transport in heterogeneous media
N. Suciu,
1
C. Vamos¸,
2
and J. Eberhard
3
Received 7 November 2005; revised 5 July 2006; accepted 2 August 2006; published 17 November 2006.
[1] Longitudinal dispersion coefficients in given realizations of the transport computed
by two currently used approximations of the first-order in velocity variance are compared
with accurate global random walk simulations. The comparisons are performed for the
same ensemble of realizations of the Darcy velocity field, approximated by a quasiperiodic
random field, for lognormal hydraulic conductivity with small variance and finite
correlation lengths. The results show that at finite times of about one dispersion timescale,
the mean coefficient is underestimated by 20%, and the fluctuations are overestimated
by 80%. At larger times the errors decrease monotonously, and the first-order
approximations yield fairly good predictions for the mean and the fluctuations of the
dispersion coefficient.
Citation: Suciu, N., C. Vamos¸, and J. Eberhard (2006), Evaluation of the first-order approximations for transport in heterogeneous
media, Water Resour. Res., 42, W11504, doi:10.1029/2005WR004714.
1. Introduction
[2] In this technical note, by ‘‘first-order approxima-
tions’’ we denote asymptotic expansions of the solutions
of the transport equations, for given velocity realizations,
which approximate second-order moments of the concen-
tration or dispersion coefficients at the same order of
magnitude as the velocity variance. These first-order
approaches to transport in saturated porous formations can
be classified as ‘‘Eulerian perturbation methods’’ (EPM)
and ‘‘Langevin iteration methods’’ (LIM).
[3] The EPM method considered here derives the approx-
imations by a perturbation approach applied to the solution
of the partial differential equation for the concentration field
c(x,t)[Bouchaud and Georges, 1990; Dentz et al., 2003;
Eberhard, 2004]. For constant porosity and nonreactive
transport, c obeys the advection-dispersion equation
@
t
c þ Vrc ¼ D
0
r
2
c: ð1Þ
For the sake of comparisons, we consider in (1) a constant
local dispersion coefficient D
0
and a stationary velocity
field V which is a realization of a statistically homogeneous
random space function. The unperturbed problem in EPM
methods is obtained by replacing V in (1) by the constant
mean of the velocity field. Systematic EPM expansions are
obtained by iterations of an integral equation for Fourier
transformed concentration equivalent to (1) [Bouchaud and
Georges, 1990, section 4.2.2]. Here we consider the ap-
proximation of the dispersion coefficients for a single
realization of the velocity field given by the first EPM
iteration of Eberhard [2004].
[4] LIM methods use the Langevin equation for the
trajectories of the transport process which is equivalent to
the Eulerian equation (1) [see, e.g., Bouchaud and Georges,
1990, section 2.3]. In the following we consider a point-like
instantaneous source located at the origin of the coordinate
system. For a given realization of the velocity field, the
equivalent description is given by the ensemble of all
trajectories starting from X(0) = 0, which solve the integral
form of the Langevin equation
X
i
t ðÞ¼
Z
t
0
V
i
X t
0
ðÞ ð Þdt
0
þ
Z
t
0
dW
i
t
0
ðÞ; ð2Þ
where i = 1,..,d and d is the space dimension. The second
integral in (2) is a Wiener process with zero mean and
variance 2D
0
t.
[5] Following Eberhard [2004], the LIM method can be
formulated as an approximation of (2) given by
X
0 ðÞ
i
t ðÞ¼ Ut ð Þd
i1
þ
Z
t
0
dW
i
t
0
ðÞ; ð3Þ
X
1 ðÞ
i
t ðÞ¼ Ut ð Þd
i1
þ
Z
t
0
u
i
X
0 ðÞ
t
0
ðÞ
dt
0
þ
Z
t
0
dW
i
t
0
ðÞ; ð4Þ
where U is the constant mean velocity, u
i
= V
i
Ud
i1
are the
random velocity fluctuations and the 1 axis of the
coordinate system is oriented in the direction of the mean
flow. It is obvious that the first iteration of the Langevin
equation (4) is the approximation obtained ‘‘by leaving the
‘Brownian motion’ component as part of the zeroth-order
term’’ which gives the particle displacement covariance
(3.14) of Dagan [1987]. This formula is a key tool for
investigations in the spirit of Dagan’s theory of transport
1
Institute of Applied Mathematics, Friedrich-Alexander University of
Erlangen-Nuremberg, Erlangen, Germany.
2
‘‘T. Popoviciu’’ Institute of Numerical Analysis, Romanian Academy,
Cluj-Napoca, Romania.
3
Simulation in Technology, University of Heidelberg, Heidelberg,
Germany.
Copyright 2006 by the American Geophysical Union.
0043-1397/06/2005WR004714
W11504
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[Fiori and Dagan, 2000]. It is also noteworthy to remark
that (3) and (4) represent the first step in the Picard method
of successive approximations, as used for instance in
numerical schemes for solving the Langevin equation
[Carvalho et al., 2005] or in analytical approximations for
the Lagrangian equation of motion in random fields
[Phythian, 1975].
[6] The Langevin equation (3) describes a diffusion in the
mean velocity field U and is strictly equivalent to the
unperturbed problem of the EPM method. Both EPM and
LIM approximate ensemble averaged dispersion coeffi-
cients up to the order of the velocity variance. It is expected
therefore that EPM and LIM, having in common the
reference to the same unperturbed problem yield equivalent
first-order approximations.
[7] The validity of the first-order stochastic theories of
flow and transport in porous media has mainly been
investigated in the case of vanishing local dispersion [see,
e.g., Chin and Wang, 1992]. For nonvanishing local disper-
sion, the existing comparisons between EPM solution for
transport in Gaussian velocity fields and particle tracking
simulations, using approximations by quasiperiodic random
functions of the Gaussian field, show large disagreement,
mostly on the longitudinal dispersion coefficient [Dentz et al.,
2003]. What has not yet been done is an evaluation of the
predictions provided by the first-order theories for the
fluctuations of the dispersion coefficients. The latter is
crucial in assessing the reliability of first-order theory for
investigations on the self-averaging properties of the trans-
port. Here, we disjoint the transport and flow problems.
Using the same ensemble of velocity realizations, we
evaluate the EPM and LIM approximations for the mean
and the fluctuations of the longitudinal dispersion coeffi-
cient by comparisons with a numerical method which was
independently validated.
2. Dispersion Quantities
[8] In the following by X we denote the longitudinal
trajectory component described by (2). The longitudinal
dispersion of the solute in a given realization of the
transport is described by the second-order central moment
St ðÞ¼h Xt ðÞhXt ð Þi
w
2
i
w
; ð5Þ
where the subscript w denotes the average over the
realizations of the Wiener process. The dispersion coeffi-
cients D =
1
2
S/t and D
1
=
1
2
dS/dt have the meaning of a
diffusion constant when the transport in heterogeneous
media has ‘‘normal’’ (or ‘‘Fickian’’) diffusion behavior at
large times, i.e., D = D
1
and S =2tD [Bouchaud and
Georges, 1990]. Other dispersion quantities are provided by
the second central moment of the ensemble averaged
concentration
S t ðÞ¼h Xt ðÞhXt ð Þi
wv
2
i
wv
; ð6Þ
where the subscript v stands for an additional ensemble
average, and by the corresponding ‘‘average’’ dispersion
coefficients D
av
=
1
2
S/t and D
1
av
=
1
2
dS/dt. We note that the
difference between (6) and the ensemble average of (5)
gives the variance of the center of mass of the plume [Suciu
et al., 2006, equation (8)]. The coefficient D
av
characterizes
the envelope of the diffusion fronts in given realizations and
is in general larger than the ensemble average hDi
v
of the
single realization dispersion coefficient [see, e.g., Bouchaud
and Georges, 1990, section 2.1, and references therein].
The ensemble average predictor of the dispersive flux J =
Vc D
0
rc corresponding to processes described by (1) is
nonlocal in space-time and non-Fickian. Even if localized
forms can be derived under some restrictive conditions, the
localized dispersion coefficients still depend on space and
time [Morales-Casique et al., 2006a]. Therefore, at finite
times the dispersion coefficients can merely be used as
convenient representations of the second moments (5) and (6).
[9] With the transformations X =
^
X l, t =
^
t l/U, V = U +
euˆU and W =
^
W (D
0
l/U)
1/2
, the nondimensional form of (2)
for the longitudinal displacement reads
^
X
^
t ðÞ¼
^
t þ
Z
^t
0
^u
^
X
^
t
0
ðÞ
d
^
t
0
þ Pe
1=2
Z
^t
0
d
^
W
^
t
0
ðÞ;
where Pe = Ul/D
0
is the Pe´clet number. The small
parameter which describes the velocity fluctuations can
be estimated by the relative standard deviation of the
longitudinal velocity ’ s
u
/U. Since for the typical
advection dominated transport investigated here = 0.19
and Pe
1/2
= 0.1 (see next section), we have the
following order of magnitude relation = O(Pe
1/2
). In
a consistent expansion of
^
X for ! 0, the leading term
cannot contain contributions of order higher than
0
. Thus
(3) has to be replaced by X
(0)
(t)= Ut and the first
iteration (4) yields a diffusion process in a deterministic
time-dependent velocity field U + u(Ut), with variance
S(t)= h[X
(1)
(t) hX
(1)
(t)i
w
]
2
i
w
=2D
0
t. It follows that for
all realizations D =
1
2
S/t = D
0
. However, the consistent
first order approximation captures the enhanced dispersion
caused by velocity fluctuations through the moment (6) of
the ensemble averaged concentration. Owing to the
independence between dispersive and advective displace-
ments, the corresponding average longitudinal coefficient
D
1
av
=
1
2
dS/dt is readily shown to be
D
av
1
t ðÞ¼ D
0
þ
Z
t
0
r Ut
0
ð Þdt
0
; ð7Þ
where r(Ut)= hu(0)u(Ut)i
v
is the Eulerian correlation of
the longitudinal velocity component. The coefficient (7) is
the output of the earlier ‘‘first-order’’ approximation of
Dagan [1984], investigated numerically (for D
0
= 0) by
Chin and Wang [1992].
3. Evaluation Method
[10] Our approach is based on an approximation of Darcy
flow by quasiperiodic random fields, which has the advan-
tage to result in explicit EPM and LIM expressions for the
effective coefficients in single realizations of the transport
and to allow large-scale numerical simulations at reasonable
computational costs. This approximation, in the form of the
Kraichnan routine, was already successfully used in numer-
ical investigations on large-scale behavior of the dispersion
coefficients [Dentz et al., 2003; Eberhard, 2004]. The
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incompressible Darcy flow for log hydraulic conductivity
with small variance s
2
and Gauss-shaped isotropic correla-
tion with correlation length l is approximated in the order s
by a superposition of N
p
periodic modes
V
i
x ð Þ¼U d
i1
þ U s
ffiffiffiffiffi
2
N
p
s
X
Np
l¼1
p
i
q
l
ð Þ cos q
l
x þ a
l
ð Þ: ð8Þ
The wave vectors q
l
are independently normally distributed
random variables with zero mean and variance l
2
, and the
phases a
l
are uniformly distributed in the interval [0, 2p].
The functions p
l
are projectors which ensure the incompres-
sibility of the flow. For N
p
!1, V
i
tends to a Gaussian
velocity field [Kraichnan, 1970].
[11] For the evaluation of the first-order approximations
we consider the two-dimensional transport problem for
fixed local dispersion coefficient, D
0
= 0.01 m
2
/d, and an
ensemble of 512 velocity realizations characterized by the
parameters s
2
= 0.1, l = 1 m, U = 1 m/d, and N
p
= 6400.
The estimated standard deviation of the longitudinal
velocity component (8) is s
u
= 0.19 m/d. For the small
value s
2
= 0.1 the Darcy velocity field can be accurately
described by a Gaussian distribution (as indicated, for exam-
ple, by Figure 20 of Morales-Casique et al. [2006b]) and the
results based on the Kraichnan procedure can be relevant for
real cases. It should also be noted that accuracy for larger
s
2
and/or Pe´clet numbers demands higher-order approxi-
mations, as plentifully demonstrated by recent investiga-
tions on the iterative expansions of the exact nonlocal
equations for the ensemble average and the variance of
the concentration [Morales-Casique et al., 2006a, 2006b].
[12] The evaluation is performed by comparisons for the
same ensemble of transport realizations between first-order
approximations and numerical simulations by the ‘‘global
random walk’’ (GRW) algorithm, implemented in the so-
called ‘‘reduced fluctuations’’ version, see [Vamos¸ et al.,
2003]. Statistically stable simulations of transport in each
realization of the velocity field are obtained by releasing N =
10
10
particles at the origin of the grid [Suciu et al., 2006]. In
order to simulate the advective-dispersive transport in un-
bounded domains, the grid, with constant step of 0.1 m, is
chosen to be larger than the maximum extension of the plume.
The restriction to the two-dimensional case enables us to
perform accurate GRW simulations over thousands of corre-
lation lengths, which are necessary to describe the conver-
gence toward the macrodispersion limit. We evaluate
numerically, for 512 realizations (8) of the velocity, the
explicit EPM and LIM expressions for D
1
given respectively
in Appendices A1 and B1 of Eberhard [2004], adapted for
two-dimensional problems. Comparisons are done between
the approximated coefficient D =
1
t
R
0
t
D
1
(t
0
)dt
0
and the
coefficient D given by GRW simulations.
[13] Before we address the evaluation of EPM and LIM,
we give an evaluation of the GRW procedure. This is done
by comparisons using a recently developed method [Suciu
et al., 2005] called ‘‘biased global random walk’’ (BGRW).
Unlike GRW and particle tracking procedure, BGRW sim-
ulates the advective displacements by a bias in the proba-
bility associated with the diffusive jumps of the particles.
Since in a single computation step the particles move only
to the nearest grid sites, BGRW has no overshooting errors.
However, as BGRW requires larger computing resources,
GRW is more suitable for large-scale computations.
[14] The overshooting errors in GRW procedure are
estimated by the percentage relative error e( b) =
100[b(GRW) b(BGRW)]/b(BGRW), where b stands,
respectively, for the average hDi
v
of the dispersion coeffi-
cient over the ensemble of velocity realizations and for the
relative standard deviation s
D
/h Di
v
. The comparison with
BGRW was limited to 100 days, N
p
= 64 modes in the
Kraichnan routine and 256 realizations. The results pre-
sented in Figure 1 show that after about 35 days the mean as
well as the fluctuations are reproduced in GRW procedure
with errors of the order of 5% (indicated by horizontal lines
in Figure 1).
[15] The ensemble mean dispersion coefficients hDi
v
given by EPM and LIM are compared to the GRW results
in Figure 2 and the corresponding sample-to-sample fluc-
tuations s
D
/hDi
v
are presented in Figure 3. Additionally, by
curves labeled EPM
1
and LIM
1
, we represented in Figures 2
and 3 the means and the fluctuations of the primary data D
1
given by the two approximations. The consistent first-order
approximation of the average coefficient (7) was computed
by numerical integration of the Eulerian velocity correla-
Figure 1. Error estimations for the GRW computations.
Figure 2. Comparisons for the mean dispersion coeffi-
cients given by different first-order approximations and by
the GRW simulations.
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tion, estimated as follows. First, the correlations are com-
puted for ensembles of 512 velocity realizations generated
by (8) on straight lines starting from all grid points in a slab
with dimensions l 100l, oriented across the mean flow.
Then, in order to obtain a better estimation for the theoret-
ical statistical homogeneous Gaussian field, the correlation
was estimated by an average over all initial grid points. The
coefficient (7) is also represented in Figure 2 and compared
to D
1
av
(LIM), which represents the average coefficient
1
2
dS/
dt derived by LIM method, using (3), (4) and (6).
4. Interpretation of the Results
[16] The numerical results for LIM and EPM, presented
in Figures 2 and 3, are very similar, indicating that the two
methods give equivalent asymptotic expansions in velocity
fluctuations. The output D
1
of LIM and EPM behaves
noisily and shows much larger sample-to-sample fluctua-
tions than its time average D. Excepting the noise, the
coefficient D
1
av
(LIM) is practically identical with the aver-
age coefficient (7) given by the consistent first-order ex-
pansion (Figure 2). This is another indication for the
robustness of the first-order approximation of Dagan
[1984] for the second moments (6) of the ensemble aver-
aged concentration, which completes the conclusion drawn
by Chin and Wang [1992] for the purely advective transport.
Since the average coefficients D
1
av
highly underestimate the
traveltime necessary to approach the macrodispersive be-
havior (see Figure 2), the moments (5) and the associated
coefficients D and D
1
have to be used to describe a typical
solute plume [Bouchaud and Georges, 1990, section 2.1].
As we have seen in the section 2 above, the consistent
expansion truncated at the first order in velocity fluctuations
does not capture the behavior of the typical dispersion
coefficient D and its fluctuations. By leaving the effect of
the local dispersion in the zeroth-order term, the LIM and
EPM methods avoid the complexity of higher-order approx-
imations and provide useful analytical expressions for the
effective coefficients in given realizations.
[17] The deviations of the first-order approximations
from the results given by simulations are significantly larger
than the error estimations for GRW presented in Figure 1.
This is clearly shown by Figure 4, where the relative errors
with respect to GRW simulations, for the mean and the
fluctuations computed by LIM, are shown in the same
percentage representation as in Figure 1. For instance, at
100 days which corresponds to one dispersion timescale l
2
/
D
0
, the mean coefficient is underestimated by approximately
20% and the fluctuations are overestimated by 80%. For large
times the errors decrease and approach the overshooting error
estimations for GRW (horizontal lines in Figure 4).
[18] The to some extent similar two-dimensional simu-
lations of Dentz et al. [2003] (performed by tracking 100
particles in 2500 realizations of Kraichnan fields with N
p
=
64 modes, for Pe = 100 and s
2
= 1) indicate that the EPM
analytical approximation for Gaussian velocity fields under-
estimates the mean longitudinal dispersion coefficient hD
1
i
v
at one dispersion timescale by about 200%. As expected,
the higher variance of the log hydraulic conductivity field
results in larger velocity fluctuations and deteriorates the
accuracy of the approximation. However, the large devia-
tions from simulations can also be explained by trapping
phenomena related to closed streamlines in the velocity field
generated by the Kraichnan method [Dentz et al., 2003].
5. Conclusions
[19] The evaluation presented in this technical note is
based on statistically stable GRW simulations (obtained
with N = 10
10
particles in one realization), for the param-
eters Pe = 100 and s
2
= 0.1 which minimize possible
artifacts in such two-dimensional numerical simulations,
and uses 512 Kraichnan velocity realizations with N
p
=
6400 periodic modes which ensure reliable simulations of
transport in Gaussian fields [Suciu et al., 2006, sections A2,
B1, and B2]. Moreover, the LIM and EPM approximations
are computed for the same ensemble of Kraichnan velocities
and the GRW procedure itself is also independently evalu-
ated. Under these conditions, the differences shown in
Figures 2–4 have to be caused by the first-order approx-
imations of the transport equations. The approximation
errors given in Figure 4 provide useful minimal estimations
Figure 3. Comparison between the fluctuations of the
dispersion coefficient given by the different first-order
approximations and GRW simulations.
Figure 4. Error estimations for the mean and the
fluctuations of the dispersion coefficient computed by the
first-order approximations.
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for larger s
2
and/or Pe, when the accuracy is expected to be
worse [Dentz et al., 2003; Morales-Casique et al., 2006b].
For moderate variability of the hydraulic conductivity these
approximations exhibit reasonable predictions for times
larger than the dispersion scale. The convergence of the
errors for the mean value and the fluctuations of the
longitudinal coefficient to the precision of the numerical
method further indicates that the LIM and EPM approx-
imations are mainly useful in investigations on large-scale
behavior and self-averaging properties of the transport.
[20] On the basis of this evaluation the LIM method has
been used to investigate the performance of the Kraichnan
routine in more detail. It has been found that the fluctuations
of the dispersion coefficients of transport in Kraichnan
fields grow logarithmically, after a time that increases with
the number of periodic modes. However, using a number of
modes of the order of the total simulation time and hundreds
of realizations, the simulations based on the Kraichnan
method reproduce quite well the self-averaging behavior
of transport in Gaussian fields with finite correlation range
(J. Eberhard et al., On the self-averaging of dispersion for
transport in random media, submitted to Journal of Physics
A: Mathematical and General, 2006).
[21] Insofar as the advection-dispersion local-scale pro-
cess, described by (1) or (2), is an acceptable description for
real world contamination scenarios, the approximations
evaluated here could be very useful in applications. EPM
and LIM provide dispersion quantities D and D
1
equivalent
to second moments of the plume, without solving the
transport equations and regardless the Fickian or non-Fickian
behavior of the process. The best fit of theoretical and field-
scale measured quantities allows, in principle, the estimation
of the parameters s
2
and l. Dagan [1984] determined s
2
and
l by a best fit of the time averaged coefficient (7) and the
coefficient
1
2
S/t measured in the Borden tracer test. Since, as
shown in Figure 2, the coefficient (7) overestimates the time
behavior of typical solute plumes at early times, the results
obtained in this way (s
2
= 0.19 and l = 1.4 m for horizontal
correlation scale) underestimate the range of parameters
obtained by different methods (s
2
= 0.25 0.39 and l =
2.7 11.6 m) [Rajaram and Gelhar, 1991]. By compar-
ing the measured slope of the longitudinal second mo-
ment with the EPM theoretical expression of hD
1
i
v
, Dentz
et al. [2000] found a good agreement, when parameters
relevant for Borden site were used. The evaluation of
first-order approximations presented in this note suggests
an improvement of the parameter identification by using
the mean slope D =
1
2
S/t instead of the local one, D
1
=
1
2
dS/dt. The former is affected by much smaller fluctua-
tions, as shown in Figure 3, and is therefore more
representative for single realizations such as a field
experiment.
[22] Acknowledgments. The research reported in this paper was
supported in part by the Deutsche Forschungsgemeinschaft grant SU 415/
1-1, awarded to the first author. Part of the work is a contribution to the
‘‘Interdisciplinary Programme for the Prevention of the Major Risk Phe-
nomena at National Level’’ of the Romanian Academy and to computing
project JICG41 at Institute of Agrosphere/Research Center Ju¨lich, Ger-
many. The authors thank A. Guadagnini for helpful suggestions.
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