Abstract
Authors
N. Suciu
Institute of Applied Mathematics, Friedrich-Alexander University of Erlangen-Nuremberg, Erlangen, Germany
C. Vamos
T. Popoviciu Institute of Numerical Analysis, Romanian Academy,Cluj-Napoca, Romania
Keywords
Cite this paper as:
N. Suciu, C. Vamoş, Comment on ‘Nonstationary flow and nonergodic transport in random porous media’ by G. Darvini and P. Salandin, Water Resour. Res., 43 (2007), W12601,
doi: 10.1029/2007wr005946
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[1] Bellin, A., P. Salandin, and A. Rinaldo (1992), Simulation of dispersion in heterogeneous porous formations: Statistics, first‐order theories, convergence of computations, Water Resour. Res., 28(9), 2211–2227.
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[3] Darvini, G., and P. Salandin (2006), Nonstationary flow and nonergodic transport in random porous media, Water Resour. Res., 42, W12409.
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[4] Eberhard, J., N. Suciu, and C. Vamoş (2007), On the self‐averaging of dispersion for transport in quasi‐periodic random media, J. Phys. A Math. Theor., 40, 597–610.
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[8] Suciu, N., C. Vamoş, J. Vanderborght, H. Hardelauf, and H. Vereecken (2006a), Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res., 42, W04409,
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[9] Suciu, N., C. Vamoş, and J. Eberhard (2006b), Evaluation of the first‐order approximations for transport in heterogeneous media, Water Resour. Res., 42, W11504,
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Comment on ‘‘Nonstationary flow and nonergodic transport in
random porous media’’ by G. Darvini and P. Salandin
N. Suciu
1
and C. Vamos¸
2
Received 3 February 2007; revised 9 May 2007; accepted 18 September 2007; published 5 December 2007.
Citation: Suciu, N., and C. Vamos¸ (2007), Comment on ‘‘Nonstationary flow and nonergodic transport in random porous media’’ by
G. Darvini and P. Salandin, Water Resour. Res., 43, W12601, doi:10.1029/2007WR005946.
[1] Darvini and Salandin [2006] addressed the extremely
important problem of the implications of the statistical
inhomogeneity of the velocity field for the solute transport
in saturated aquifers. The authors considered the velocity
inhomogeneity due to the limited size of the computational
domain which occurs in numerical simulations for assumed
statistically homogeneous hydraulic conductivity. In this
comment we question the correctness of the estimation for
the expected second moment of the solute plume in the
paper of Darvini and Salandin [2006]. Even though the
explicit formula used to compute this quantity was not
provided, there are indications that relevant terms account-
ing for the inhomogeneity of the random velocity field were
ignored. Clarifying this issue is essential for an evaluation
of the results. We also show that the validity of the basic
relation for the expected value of the second spatial moment
of the plume [Darvini and Salandin, 2006, equation (19)]
extends beyond the frame of their first-order approach for
purely advective transport. Finally we argue that the
approach of Darvini and Salandin [2006] is well suited
and can be used to produce a detailed explanation for the
nonergodic behavior of the second moments caused by the
statistical inhomogeneity of the velocity field.
[2] With a finite element method using Taylor series
expansions, a realization of the velocity field was computed
as sum of a deterministic component (ensemble mean) and a
fluctuating one, v(x)= v
0
(x)+ v
0
(x)[Darvini and Salandin,
2006, equation [10]]. Because of the influence of the
boundaries for small computational domains, the ensemble
mean velocity v
0
= hvi is space dependent, thus the random
field is not statistically homogeneous. The local dispersion
is neglected and trajectories of the solute particles, starting
from x = a at time t = t
0
, are approximated by a first iteration
of the equation of motion
X
i
t ; a; t
0
ð Þ¼ X
0i
t ; a; t
0
ð Þþ
Z
t
t0
v
0
i
X
0
t
0
; a; t
0
ð Þ ð Þdt
0
; ð1Þ
where i = 1, 2, 3 and
X
0i
t ; a; t
0
ð Þ¼ a
i
þ
Z
t
t0
v
0i
X
0
t
0
; a; t
0
ð Þ ð Þdt
0
ð2Þ
is the deterministic trajectory of the mean velocity. This
consistent first-order approximation in velocity fluctuations
yields robust estimations [Suciu et al., 2006b] for dispersion
coefficients defined as time integrals of Lagrangian velocity
covariance [see, e.g., Salandin and Fiorotto, 1998]. The
equations (1) and (2), written for our convenience in a more
explicit form, correspond exactly to the Lagrangian
approach of Darvini and Salandin [2006, section 3].
[3] To simplify the writing, in the following we note by
hf a ð Þi
a
¼
1
V
0
Z
V0
f a ðÞda;
the average of a function f over the initial plume of constant
concentration in the volume V
0
. So, the coordinate of the
center of mass is R
i
= hX
i
i
a
and the diagonal components of
the second moment can be written as S
ii
= h[X
i
R
i
]
2
i
a
. As
pointed out by Darvini and Salandin [2006, p. 4], in
statistically inhomogeneous velocity fields, both R
i
and S
ii
depend on the size, the shape, and the location of the initial
plume. The expectation of the second moment has the
following equivalent forms:
hS
ii
i ¼ hh X
i
R
i
½
2
i
a
i ¼ hhX
2
i
i
a
ihR
2
i
i
¼ hhX
2
i
i
a
ihR
i
i
2
R
ii
;
where R
ii
= hR
i
2
ihR
i
i
2
is the variance of the center of
mass. Assuming that the trajectory is continuous as a
function of the initial position, by virtue of Fubini’s theorem
the ensemble average permutes with the integral with
respect to a and one obtains
hS
ii
i¼hX
ii
i
a
R
ii
þ hhX
i
i
2
i
a
hR
i
i
2
; ð3Þ
where X
ii
= hX
i
2
ihX
i
i
2
is the moment computed by
ensemble averaging for a fixed initial position a; that is, it is
the one-particle displacements variance [Dagan, 1990]. The
relation (3), written here for diagonal components of the
expected second moment, is just the equation (19) presented
by Darvini and Salandin [2006] in the frame of their first-
order approximation. However, this relation is not based on
the approximate form of the equations (1)–(2) and is valid
under the only assumption that the averages over initial
position and velocity realizations permute. As already
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.
Copyright 2007 by the American Geophysical Union.
0043-1397/07/2007WR005946
W12601
WATER RESOURCES RESEARCH, VOL. 43, W12601, doi:10.1029/2007WR005946, 2007
1 of 4
shown by Suciu et al. [2006a, p. 9] a relation of the same
form also holds in the general case which considers local
dispersion.
[4] By using the displacement
e
X
i
= X
i
a
i
relative to the
initial position, the last two terms of (3) give S
ii
(0) + M
ii
+
Q
ii
, where S
ii
(0) = ha
i
2
i
a
ha
i
i
a
2
is the moment of the initial
plume and
M
ii
¼ 2h a
i
ha
i
i
a
h
e
X
i
ii
a
ð4Þ
Q
ii
¼ hh
e
X
i
i
2
i
a
hh
e
X
i
ii
2
a
: ð5Þ
With (4) and (5), (3) takes the equivalent form
hS
ii
i¼ S
ii
0 ð ÞþhX
ii
i
a
R
ii
þ M
ii
þ Q
ii
: ð6Þ
The term M
ii
is the mean correlation between initial
positions and particle displacements. Sposito and Dagan
[1994, p. 587] show that such correlations must be
incorporated into the prediction of both actual and expected
second moment of the plume. Because M
ii
carries the
information about the initial position of the particles, we call
it ‘‘memory term’’. The term Q
ii
occurring in (6) is a spatial
correlation of the expected relative displacement for
different initial positions and, as shown in the following,
it is a key tool which relates the time behavior of the
expected second moment to the velocity inhomogeneity. We
also note that hX
ii
i
a
+ Q
ii
= hh
e
X
i
2
i
a
i hh
e
X
i
i
a
i
2
=
e
X
ii
, which
gives another expression, equivalent with (3) and (6), for the
expectation of the second moment:
hS
ii
i¼ S
ii
0 ð Þþ
e
X
ii
R
ii
þ M
ii
: ð7Þ
The assumption of ‘‘Lagrangian stationarity’’ renders the
quantities h
e
X
i
i and X
ii
independent of the initial position a.
Consequently, the terms (4) and (5) vanish and (6) becomes
the well known relation of Dagan [1990, equation (11)],
hS
ii
i¼ S
ii
0 ð Þþ X
ii
R
ii
: ð8Þ
Following Dagan [1990], Darvini and Salandin [2006, p. 5]
denote by ‘‘ergodicity’’ the relevance of the one particle
variance X
ii
for the expected second moment of the plume (8),
which, in this case, is ensured by a negligible small variance
R
ii
of the center of mass.
[5] Before proceeding with the inhomogeneous case, let
us shortly discuss the relation (7) from the perspective of the
two-dimensional simulations of Suciu et al. [2006a]. In
that paper, accurate simulations of advection-dispersion in
statistically homogeneous random velocity fields, for log-
hydraulic conductivity of variance 0.1 and correlation
length l = 1 m, were presented in detail. With a constant
mean velocity U = 1 m/d and an isotropic local dispersion
coefficient D = 0.01 m
2
/d, the Pe´clet number was fixed at
Ul/D = 100, a reasonably high value which is representa-
tive for real aquifer systems. The accuracy of the transport
simulations was ensured by tracking 10
10
particles in a
given velocity realization with the ‘‘global random walk’’
algorithm [Vamos¸ et al., 2003]. By using 256 velocity
realizations and 6400 periodic modes in the Kraichnan
routine the resulting statistical ensemble allows reliable sim-
ulations of the self-averaging transport process over thousands
of dimensionless times Ut/l [Eberhard et al., 2007].
[6] In the case of nonvanishing local dispersion the
relations presented above hold true with the only difference
that the averages over initial positions and velocity realiza-
tions are preceded by the average over the realizations of the
local dispersion process. It was found that for point and slab
sources oriented across the i axis, for which the memory
term (4) vanishes or is negligible, the moment
e
X
ii
is
independent of initial conditions and (7) reduces to (8).
On the contrary, for sources oriented along the i axis the
relation (8) is no longer verified. Assuming that
e
X
ii
is
practically the same as that for point sources in all cases,
we estimate the transverse memory terms for transverse
sources from relation (7). The result presented in Figure 1
shows significant memory terms, which increase with the
source dimension. The same increase with the source
dimensions of the single realization memory terms (i.e.,
defined by (4) without ensemble averaging) was found by
Fiori and Jancovic´ [2005, Figure 5] by simulations of the
purely advective transport.
[7] Since the mean displacement h
e
X
i
i in (4) is a time
integral of the ensemble mean of the velocity field sampled
on trajectories, hv
i
(X(t; a, t
0
))i, nonvanishing ensemble
average memory terms occur only if hv
i
i is not constant
as a function of a. Figure 2 shows that, for the simulations
presented in Figure 1, the mean hv
2
i varies with the initial
position a of the simulated advection-dispersion processes.
Significant differences occur at less than 100 dimensionless
times. The multiplication by [a
2
ha
2
i
a
] in (4) explains the
mean memory terms shown in Figure 1 and their increase
with the source dimension L. The average of hv
2
(X(t; a, t
0
))i
over a (solid line in Figure 2) is also nonvanishing in the
preasymptotic regime and differs from the numerically
estimated Eulerian mean velocity hv
2
(x)i, which belongs
to the range [0.00072, 0.00031] [Suciu et al., 2006a,
Table B1]. A comparison of the curves presented in Figure 2
indicates that the term Q
22
, which depends on the spatial
variance of the ensemble mean velocity on trajectories, is
about two orders of magnitude smaller than the local
dispersion. Therefore the term
e
X
22
in (7) is practically
independent of the initial conditions. This example shows
that, even for accurate simulations of transport in statisti-
cally homogeneous fields, the Lagrangian stationarity can-
not be assumed and the mean memory terms M
ii
are
nonvanishing for asymmetric initial plumes. Since the
ensemble average statistics of the velocity simulated by
the finite element method varies from point to point, even in
the core region nonaffected by boundaries [Bellin et al.,
1992, p. 2217], one expects that in the paper commented
here the simulated second moments hS
ii
i also contain
significant mean memory terms M
ii
.
[8] The ergodicity issue was found to be more subtle than
one can expect from analyzes on the basis of relation (8)
only. Though R
ii
can be neglected for large plumes [Suciu et
al., 2006a, Figure 12] the one-particle moment X
ii
’
e
X
ii
becomes relevant for the ensemble average of the actual
moment hS
ii
i S
ii
(0) only when the memory term M
ii
also
becomes negligible (see Figure 1). Moreover, the standard
deviation of S
ii
increases with the extension L of the source
on the i axis [Suciu et al., 2006a, Figure 8] and it can be
shown that for large sources (longitudinal and transverse
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slabs and squares) it is well estimated by the standard
deviation of M
ii
(Suciu et al., Memory effects induced by
dependence on initial conditions and ergodicity of transport
in heterogeneous media, submitted to Water Resources
Research, 2007). Summarizing the facts, we see that the
ergodicity of the actual moment S
ii
S
ii
(0) with respect to
the one-particle moment X
ii
is quantified by both the bias of
its ensemble average, which according to (7) is M
ii
R
ii
,
and by its standard deviation [see Suciu et al., 2006a,
definition (5)]. It is noteworthy to point out that such
ergodic properties, often formulated in the hydrological
literature, differ from the ergodicity of a given random
function. The latter denotes the convergence toward the
ensemble mean of an unbiased estimator defined by time or
space averages of a single realization of the random
function.
[9] The expectation of the second moment in the inho-
mogeneous case is given explicitly by (6). The terms hX
ii
i
a
and R
ii
can be expressed in the above first-order approxi-
mation with the aid of the one- and two-particle Eulerian
velocity covariances evaluated at points on the trajectory
X
0
(t; a, t
0
) of the mean velocity [Darvini and Salandin,
2006, equations (22) and (23)]. Using (1) and (2), the last
two terms of (6) are obtained from (4) and (5) as
M
ii
t ; t
0
ð Þ¼ 2
Z
t
t0
h a
i
ha
i
i
a
v
0i
X
0
t
0
; a; t
0
ð Þ ð Þi
a
dt
0
ð9Þ
Q
ii
t ; t
0
ð Þ¼
Z
t
t0
Z
t
t0
hv
0i
X
0
t
0
; a; t
0
ð Þ ð Þv
0i
X
0
t
00
; a; t
0
ð Þ ð Þi
a
hv
0i
X
0
t
0
; a; t
0
ð Þ ð Þi
a
hv
0i
X
0
t
00
; a; t
0
ð Þ ð Þi
a
dt
0
dt
00
: ð10Þ
Thus the first-order approximation (1) – (2) describes con-
tributions to the expected second moment (6) due to the
spatial inhomogeneity of the mean Eulerian velocity field,
sampled on the zeroth-order trajectories, by the spatial
correlation of the mean velocity (10) and by its correlation
with the initial positions of the solute particles (9).
[10] As the authors stated in the abstract, the objective of
the paper was ‘‘to describe nonergodic transport of inert
solutes by spatial moments in a domain of finite size.’’ In
other words, according to the sense of ergodcity in their
paper, the aim was to characterize the deviation, due to
velocity inhomogeneity in a domain of finite size, of the
expected second moment from the one particle variance X
ii
.
However, the authors do not explain how they obtained the
results on the expected second moment presented in their
Figures 10–12. The only indication is that ‘‘plume statistics
are computed by numerical quadrature on the same grid
adopted in the FE solution by relationships described in
section 3’’ [Darvini and Salandin, 2006, p. 5]. These are (3)
and (8) discussed here and explicit forms for X
ii
and R
ii
.
[11] The moments hS
ii
i S
ii
(0) were found to be in
excellent agreement with Monte Carlo simulations [Darvini
and Salandin, 2006, Figure 11] developed following the
approach previously used by Salandin and Fiorotto [1998].
The method described in the latter paper consisted in
simulating the advective displacement of 40 particles,
equally spaced by a heterogeneity scale, in 500 Monte
Carlo runs. The statistics was evaluated ‘‘by averaging the
single realization results on all the Monte Carlo runs’’
[Salandin and Fiorotto, 1998, pp. 953–954]. Finally, dis-
persion coefficients ‘‘were computed by the integration of the
Lagrangian velocity covariance’’ [Salandin and Fiorotto,
1998, p. 958]. That means, first evaluating the velocity
covariance on the particle trajectory X
i
(t; a, t
0
) in a given
velocity realization (Monte Carlo run) by an average hv
i
v
i
i
a
hv
i
i
a
hv
i
i
a
over all particles identified by their initial position
(and possibly by a time average along particles trajectories,
which for the sake of simplicity we do not consider here),
then averaging over velocity realizations and integrating in
time to obtain the moments
Y
ii
t ; t
0
ð Þ¼
Z
t
t0
Z
t
t0
hhv
i
X t
0
; a; t
0
ð Þ ð Þv
i
X t
00
; a; t
0
ð Þ ð Þi
a
i
hhv
i
X t
0
; a; t
0
ð Þ ð Þi
a
hv
i
X t
00
; a; t
0
ð Þ ð Þi
a
i
dt
0
dt
00
: ð11Þ
Figure 1. Transverse memory terms for slab sources (l,
Ll) oriented across the mean flow, computed from data
published by Suciu et al. [2006a, Figure 13].
Figure 2. Ensemble average of the transverse velocity on
the trajectories of the advection-dispersion processes
starting from different initial positions a; the solid line
represents the space average of hv
2
i with respect to a for a
transverse slab source with dimensions (l, 10l).
W12601 SUCIU AND VAMOS¸: COMMENTARY
3 of 4
W12601
19447973, 2007, 12, Downloaded from https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2007WR005946 by Cochrane Romania, Wiley Online Library on [28/08/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Since the particle trajectory is given by
X
i
t ; a; t
0
ð Þ¼ a
i
þ
Z
t
t0
v
i
X t
0
; a; t
0
ð Þ ð Þdt
0
;
from (11) and (4) it follows that
Y
ii
¼ hhX
2
i
i
a
ihR
2
i
i ha
2
i
i
a
ha
i
i
2
a
h i
M
ii
¼hS
ii
i S
ii
0 ð Þ M
ii
: ð12Þ
[12] The agreement of the first-order estimations with the
Monte Carlo simulations suggests that the quantity pre-
sented in Figures 10–12 of Darvini and Salandin [2006]
might be an estimation of Y
ii
. With the first-order approx-
imation (1)–(2), this estimation is obtained from (11) by
replacing the argument X of v
i
by X
0
. However, as follows
from the definition of the uncertainty of the second moment
with respect to the one-particle displacements covariance
[Darvini and Salandin, 2006, p. 12], one can also suppose
that the authors estimated hX
ii
i
a
R
ii
. This can be achieved
by replacing in (11) v
i
by v
i
0
and X by X
0
, as in relations
(22) and (23) of Darvini and Salandin [2006]. From
(12) and (6) we have Y
ii
= hX
ii
i
a
R
ii
+ Q
ii
. Thus the
estimation of the second moment by hX
ii
i
a
R
ii
disregards
not only the memory term M
ii
(as the numerical approach
based on (12) does) but also the correlation of the mean
velocity Q
ii
(accounted for by the numerical simulations).
This choice is justified if a highly accurate numerical
solution for the zeroth-order potential head [Darvini and
Salandin, 2006, equation (6)] was obtained or if no numer-
ical solution was computed at all and a constant head
gradient, as resulting from boundary conditions, was im-
posed. This leads to a constant zeroth-order velocity and to
the cancellation of both M
ii
and Q
ii
in (9) – (10). In this case,
the comparison with the simulation results [Darvini and
Salandin, 2006, Figure 11] indicates that the variance of
the mean Lagrangian velocity near the left boundary,
corresponding to the finite element solution of the exact flow
equations in the Monte Carlo simulations, does not produce
significant terms Q
ii
.
[13] If, instead of following exactly the same approach as
Salandin and Fiorotto [1998], the expected second moment
was computed from the mean square displacement of the
simulated particles trajectories, then the Monte Carlo results
correspond to the exact relation (6). Hence, even if Q
22
can
be negligible, significant memory terms M
22
, as in our
example presented in Figure 1, can occur for the transverse
line source placed near the left boundary. Figure 11 of
Darvini and Salandin [2006] shows that this was not the
case. To help the interested reader to understand their
results, the authors should explain how the expected second
moment was computed in both the first-order stochastic
finite element approach and Monte Carlo simulations. In
addition, we suggest that in a reply to this comment the
authors of the paper commented here should present some
results on the path line analysis of the mean Lagrangian
velocity field. This can be achieved by evaluations of the
correlation terms (9) – (10), which are readily obtainable by
their approach, or, as in our example from Figure 2, by
presenting the dependence of the numerically derived mean
Lagrangian velocity on initial positions of the particles. If
one proves that such dependencies can be neglected, the
effect of spatial inhomogeneity of the random velocity field
caused by the finite size of the domain, considered by
Darvini and Salandin [2006], can be completely described
in terms of one- and two-particle velocity covariances.
[14] For physically relevant statistical inhomogeneity of
the velocity field, as that caused by a trend in the mean
hydraulic conductivity, the bias of the expected second
moment hS
ii
i S
ii
(0) with respect to the space average of
the one-particle variance hX
ii
i
a
is given, according to (6), by
M
ii
+ Q
ii
R
ii
. Since in the first-order approximation M
ii
(9)
and Q
ii
(10) quantify the effect of the space variation of the
mean Eulerian velocity, these terms are particularly relevant
for the inhomogeneous case. To complete the assessment of
ergodicity with respect to hX
ii
i
a
, in addition to the bias of
the expectation, the standard deviation of S
ii
must be
computed as well [Suciu et al., 2006a]. Both tasks can be
achieved with the first-order stochastic finite element method
of Darvini and Salandin [2006]. By solving a deterministic
problem for given mean and correlation functions of the
hydraulic conductivity, such an approach avoids cumber-
some repeated Monte Carlo computations.
[15] Acknowledgments. This work was supported by Deutsche
Forschungsgemeinschaft (grant SU 415/1-2) and Romanian Ministry of
Education and Research (grant 2-CEx06-11-96). We thank F. Radu for
helpful discussions.
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N. Suciu, Institute of Applied Mathematics, Friedrich-Alexander
University of Erlangen-Nuremberg, Martensstrasse 3, D-91058 Erlangen,
Germany. (suciu@am.uni-erlangen.de)
C. Vamos¸, ‘‘T. Popoviciu’’ Institute of Numerical Analysis, Romanian
Academy, P. O. Box 68-1, 400320 Cluj-Napoca, Romania. (cvamos@
ictp.acad.ro)
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