Comment on ‘‘Comparison of Fickian and temporally nonlocal
transport theories over many scales in an exhaustively sampled
sandstone slab’’ by Elizabeth Major et al.
A. Fiori,
1
G. Dagan,
2
and I. Jankovic
3
Received 1 December 2011 ; revised 12 April 2012 ; accepted 24 May 2012 ; published 21 July 2012.
Citation: Fiori, A., G. Dagan, and I. Jankovic (2012), Comment on ‘‘Comparison of Fickian and temporally nonlocal transport theories
over many scales in an exhaustively sampled sandstone slab’’ by Elizabeth Major et al., Water Resour. Res., 48, W07801, doi:10.1029/
2011WR011706.
[1] The paper by Major et al. [2011] presents a compre-
hensive analysis of a laboratory flow and transport experi-
ment carried out by Klise et al. [2008] on a slab of
sandstone of spatially variable conductivity K. The study
comprises data analysis, geostatistical characterization, nu-
merical simulation of flow and transport, and theoretical
developments. The present comment addresses only one
issue of the extensive study of Major et al. [2011], namely,
the adequacy of the advection-diffusion equation (ADE) to
model transport. For ease of reading we recapitulate briefly
the developments related to this issue solely.
[2] A slab of sandstone of 30.5 30.5 2.1 cm
3
was
exhaustively sampled by carrying out 17,238 measurements
of K, with samples having a support volume of around
0.45 cm
3
[Klise et al., 2008]. The slab was relatively homo-
geneous with variance
2
Y
of Y ¼ ln K equal to 0.69. The
slab was sealed on four faces, and a mean uniform flow
was created by connecting it to two constant head reser-
voirs at the remaining opposite sides. It was saturated with
KI-bearing water and subsequently flushed with fresh water,
with a mean pore velocity of the order of 0.007 cm s
1
for
24 min, increased to 0.02 cm s
1
subsequently. The concen-
trations were exhaustively sampled and transport was quanti-
fied by the breakthrough curve (BTC) at four control planes
(CP) at distances x ¼ 7:5; 15; 22:5; 30 cm from the inlet
[Klise et al., 2008]. The experimental points are displayed in
Figure 1, which reproduces Figure 5 of Major et al. [2011].
[3] The authors have carried out simulations of flow by
generating 100 realizations of the K field and numerically
solving the flow equations by the MODFLOW and PAR-
FLOW codes to determine the specific discharge and the
associated velocity field VðxÞ at a dense set of points. Sub-
sequently, they have solved numerically the ADE
@C
@t
¼ r ðV C D rCÞ (1)
by a particle tracking code. The local diffusion coefficient
D was taken equal to the molecular one (set at 1.8 10
5
cm
2
s
1
). Hence, equation (1) models essentially an advec-
tive transport, and solute spreading resulted almost entirely
from the spatial variability of VðxÞ. The BTC averaged
over the 100 realizations of the velocity field pertaining to
the initial K ðxÞ distribution (of variance
2
Y
¼ 0:69Þ is rep-
resented in Figure 1 by a thin line, as depicted in Figure 5
of Major et al. [2011]. Since the simulated BTC did not
reproduce the tailing of the measured BTC (Figure 1), the
authors assumed that a possible explanation might have
been the low value of
2
Y
; which was due to the K upscaling
involved in the measurements over the aforementioned sup-
port volume. Subsequently, they carried out a downscaling
of the K field by imposing variances
2
Y
five and ten times
larger than the original ones and generated realizations that
honor measurements [Major et al., 2011, Figure 4]. Numer-
ical solutions of the flow problem and of equation (1) for
these new K fields lead to the BTC depicted by a dashed
line and a thick line in Figure 1, respectively. While the
simulated BTC for the tenfold increased variance matches
the measurement pertinent to the large time tail, it misses
the bulk of the BTC at shorter times. The authors con-
cluded that the convex shape of the ADE solutions cannot
reproduce the concave tail of the measured BTC and there-
fore the ADE is an inadequate model of solute spreading
for permeability fields of a log conductivity variogram
Y
of the type displayed by Major et al. [2011, Figure 1]. It
is worthwhile to mention that the logarithmic scale of
Figure 1 amplifies the impact of the point representing the
late time solute arrival at around 7600 s (Figures 1c and
1d), whose mass represents the small fraction of about
0.005 of the total mass.
[4] The authors’ conclusion is in variance with our
works [e.g., Fiori et al., 2006, 2007] on flow and transport
in highly heterogeneous formations in which we found that
the advective transport model can capture the long tailing
of the BTC, like the one appearing in Figure 1. We wish to
apply our model to the conditions of the experiment of
Figure 1 and for the sake of completeness and easiness of
reproducing the results, we briefly review the approach.
[5] The medium is modeled as an ensemble of spherical
(in 3-D) or circular (in 2-D) inclusions, of radius R and of
different and independent conductivity K , whose values are
drawn from a lognormal pdf f(K). With the integral log
conductivity scale given by I
Y
¼ð3=4Þ R; the medium is
completely characterized by three parameters : K
G
(the
1
Dipartimento di Scienze dell’Ingegneria Civile, Universita ` di Rome
Tre, Rome, Italy.
2
Department of Fluid Mechanics and Heat Transfer, Tel Aviv Univer-
sity, Ramat Aviv, Israel.
3
Department of Civil, Structural and Environmental Engineering, State
University of New York at Buffalo, Buffalo, New York, USA.
Corresponding author: A. Fiori, Dipartimento di Scienze dell’Ingegneria
Civile, Universita ` di Rome Tre, via Vito Volterra 62, I-00146, Rome, Italy.
(aldo@uniroma3.it)
©2012. American Geophysical Union. All Rights Reserved.
0043-1397/12/2011WR011706
W07801 1 of 4
WATER RESOURCES RESEARCH, VOL. 48, W07801, doi :10.1029/2011WR011706, 2012