WATER RESOURCES RESEARCH, VOL. 39, NO. 9, 1270, doi:10.1029/2002WR001721, 2003
Flow and transport in highly heterogeneous formations:
3. Numerical simulations and comparison with theoretical results
I. Janković
Department of Civil, Structural and Environmental Engineering, State University of New York at Buffalo, Buffalo,
New York, USA
A. Fiori
Dipartimento di Scienza dell’Ingegneria Civile, Università di Roma Tre, Rome, Italy
G. Dagan
Department of Fluid Mechanics and Heat Transfer, Tel Aviv University, Tel Aviv, Israel
Received 16 September 2002; revised 3 July 2003; accepted 16 July 2003; published 27 September 2003.
[1] In parts 1 [Dagan et al., 2003] and 2 [Fiori et al., 2003] a multi-indicator model of
heterogeneous formations is devised in order to solve flow and transport in highly
heterogeneous formations. The isotropic medium is made up from circular (2-D) or
spherical (3-D) inclusions of different conductivities K, submerged in a matrix of effective
conductivity. This structure is different from the multi-Gaussian one, even for equal log
conductivity distribution and integral scale. A snapshot of a two-dimensional plume in a
highly heterogeneous medium of lognormal conductivity distribution shows that the model
leads to a complex transport picture. The present study was limited, however, to
investigating the statistical moments of ergodic plumes. Two approximate semianalytical
solutions, based on a self-consistent model (SC) and on a first-order perturbation in
the log conductivity variance (FO), are used in parts 1 and 2 in order to compute the
statistical moments of flow and transport variables for a lognormal conductivity pdf. In this
paper an efficient and accurate numerical procedure, based on the analytic-element method
[Strack, 1989], is used in order to validate the approximate results. The solution
satisfies exactly the continuity equation and at high-accuracy the continuity of heads
at inclusion boundaries. The dimensionless dependent variables depend on two parameters:
the volume fraction n of inclusions in the medium and the log conductivity variance sY2. For
inclusions of uniform radius, the largest n was 0.9 (2-D) and 0.7 (3-D), whereas the largest
sY2 was equal to 10. The SC approximation underestimates the longitudinal Eulerian
velocity variance for increasing n and increasing sY2 in 2-D and, to a lesser extent, in 3-D, as
compared to numerical results. The FO approximation overestimates these variances, and
these effects are larger in the transverse direction. The longitudinal velocity pdf is
highly skewed and negative velocities are present at high sY2, especially in 2-D. The main
results are in the comparison of the macrodispersivities, computed with the aid of the
Lagrangian velocity covariances, as functions of travel time. For the longitudinal
macrodispersivity, the SC approximation yields results close to the numerical ones in 2-D
for n = 0.4 but underestimates them for n = 0.9. The asymptotic, large travel time values
of macrodispersivities in the SC and FO approximations are close for low to moderate
sY2, as shown and explained in part 1. However, while the slow tendency to Fickian
behavior is well reproduced by SC, it is much quicker in the FO approximation. In 3-D the
SC approximation is closer to numerical one for the highest n = 0.7 and the different sY2 = 2,
4, 8, and the comparison improves if molecular diffusion is taken into account. Transverse
macrodispersivity for small travel times is underestimated by SC in 2-D and is closer to
numerical results in 3-D, whereas FO overestimates them. Transverse macrodispersivity
asymptotically tends to zero in 2-D for large travel times. In 3-D the numerical simulations
lead to a small but persistent transverse macrodispersivity for large travel times, whereas it
tends to zero in the approximate solutions. The results suggest that the self-consistent
semianalytical approximation provides a valuable tool to model transport in highly
heterogeneous isotropic formations of a 3-D structure in terms of trajectories statistical
Copyright 2003 by the American Geophysical Union.
0043-1397/03/2002WR001721$09.00
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JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
moments. It captures effects like slow transition to Fickian behavior and to Gaussian
trajectory distribution, which are neglected by the first-order approximation.
INDEX
TERMS: 1829 Hydrology: Groundwater hydrology; 1831 Hydrology: Groundwater quality; 1832 Hydrology:
Groundwater transport; 1869 Hydrology: Stochastic processes; KEYWORDS: transport, self-consistent,
inclusion, dispersion, analytic element, supercomputer
Citation: Janković, I., A. Fiori, and G. Dagan, Flow and transport in highly heterogeneous formations: 3. Numerical simulations and
comparison with theoretical results, Water Resour. Res., 39(9), 1270, doi:10.1029/2002WR001721, 2003.
1. Introduction and Background
[2] This paper concludes the sequence of articles (Dagan
et al. [2003], part 1, and Fiori et al. [2003], part 2) dealing
with the solution of flow and transport through highly
heterogeneous porous formations of lognormal conductivity
distribution. Its main objectives are to report numerical
experiments, to compare the results with the approximate
semianalytical ones of the previous parts and to conclude
the series. For ease of reading and for the sake of completeness we recall here briefly the essence of the method,
as exposed in the preceding parts.
[3] The model of the heterogeneous structure we adopt,
coined as a multi-indicator one, is a collection of M blocks
of constant conductivity K( j) ( j = 1, . . ., M) embedded in a
matrix of conductivity K0, within the flow domain (part 1,
Figures 1 and 2). The blocks of a prescribed shape do not
overlap and their centroids are set at random, with no
correlation between their conductivities. To achieve simple
semianalytical solutions and accurate numerical ones, we
selected circular (2-D) and spherical (3-D) inclusions of
constant radius R, to represent isotropic formations. The
inclusions are confined in , a circle of diameter L (2-D) or
a spheroid of axis L (3-D) with L R. The medium of
conductivity K0 extends to infinity, where uniform flow of
velocity U prevails.
[4] The conductivities obey a lognormal distribution, i.e.,
Y = ln K is normal and defined by the mean hYi = ln KG (the
geometric mean) and the variance sY2. The conductivity of
the matrix K0 is taken equal to the effective conductivity
Kef. The latter is equal to KG in 2-D and to the selfconsistent approximation (given by equation (21), part 1)
in 3-D. Under this choice the mean velocity within is
equal to U and the flow outside is not disturbed by the
presence of the heterogeneous medium. The spatial distribution of the conductivity K/KG is therefore completely
determined by the parameters R, n and sY2, where n is the
volume fraction of inclusions in . We are interested in
values of n up to unity; however, the circular and spherical
shapes of inclusions of uniform radii limit the value to about
0.9 and 0.7, respectively, as described in the sequel.
[5] The two point autocorrelation function of K or Y, can
be determined analytically and are given by equation (13)
of part 1. In particular, the integral scales IY are given by
(8/3p)R and (3/4)R in 2-D and 3-D, respectively.
[6] Two approximate solutions of flow and transport are
developed in parts 1 and 2. The first one, of a semianalytical
nature, is coined as the self-consistent approximation herein.
It consists of deriving the velocity field by superimposing
the disturbances to the uniform flow associated with each
inclusion. The disturbance velocity potentials are derived
analytically by regarding each inclusion as an isolated one
surrounded by the matrix of conductivity K0. This embedding matrix model [Dagan, 1979] assumes that the impact
of the surrounding inclusions is taken into account by the
choice K0 = Kef. Transport is solved subsequently in a
Lagrangian framework by particle tracking, leading to
trajectories moments and values of macrodispersivities in
terms of a few quadratures. Various results are presented in
part 2 and in particular the dependence of the longitudinal
and transverse dispersivities on travel time. Asymptotically,
the trajectories pdf become normal and transport is completely characterized in the mean by the constant, asymptotic, dispersivities. The latter are proportional to IY and n,
but depend in a nonlinear fashion upon sY2.
[7] The second type of solution is the first-order approximation in sY2 (linearized solution). It depends only on the
two-point covariance CY and both flow and transport problem can be solved analytically [see, e.g., Dagan, 1984].
Thus one of the simplest results is for the asymptotic
longitudinal dispersivity aL ! sY2IY. It was shown in part 2
that the self-consistent model, valid for any sY2, leads to the
first-order approximation under a power expansion and
retaining terms O(sY2).
[8] In the present article precise and efficient numerical
solutions are obtained for the multi-indicator conductivity
structure by using a novel numerical technique. They are
devised as numerical experiments to validate the approximate solutions.
[9] The first part of the paper contains the description of
the numerical flow solution. It is based on the analyticelement method introduced by Strack [1989], and it can be
used to precisely simulate the flow for any value of the
variance of log conductivity (values up to sY2 = 10 are
considered here). At the same time, flow domains more than
1800 (2-D) and 250 (3-D) times the integral scale of log
conductivity long were simulated.
[10] The essence of the flow solution is the principle of
superposition: disturbance velocity potential due to each
inclusion is expressed separately and the solution is
obtained by adding these velocity potentials. The expressions do not require any discretization of the flow domain:
flow solution (head and velocity) are available at any
location without interpolation. The flow solution of the
self-consistent model is a special case, in which nonlinear
interactions between inclusions are not accounted in a direct
manner.
[11] The present method is different from previous
numerical solutions of flow in heterogeneous formations
[e.g., Tompson and Gelhar, 1990; Bellin et al., 1992; Burr et
al., 1994; Chin, 1997; Salandin and Fiorotto, 1998] that
used finite differences or finite elements methods that
limited the range of sY2 values and the domain size,
especially in 3-D simulations.
[12] The remaining part of the paper contains the details
of the experimental setup, of the implementation of the
numerical method and presentation and discussion of
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
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analytic element method of Strack [1989], expressions for
the disturbance velocity potential due to individual inclusions are superimposed on the velocity potential due to
uniform flow to yield the total potential. Expressions for the
velocity potential for each inclusion, expressed in spherical
coordinates (r, q, y), are constructed to meet Laplace
equation exactly. Here r is referred to the center of the
particular inclusion. The disturbance velocity potential in
the interior of a generic inclusion is given (up to an additive
constant) by
fðinÞ ¼
1 X
k
X
rk Tm:k ðcos qÞ½am:k cosðmyÞ þ bm:k sinðmyÞ ð1Þ
k¼1 m¼0
and, in the exterior by
fðexÞ ¼
1 X
k
X
ck rðkþ1Þ Tm:k ðcos qÞ½am:k cosðmyÞ þ bm:k sinðmyÞ
k¼1 m¼0
ð2Þ
Figure 1. Geometry and dimensions of the flow domain
(square and rectangle) in terms of domain length (L) for 2-D
numerical simulations (top) and 3-D numerical simulations
(bottom) with streamlines shown for 2-D flow.
results. Unlike the self-consistent model, the numerical
simulations take a considerable computer time. It is found
important to carry out a few numerical experiments, of an
exploratory nature, to examine the performance of the more
efficient self-consistent model, for prediction purposes.
where Tm:k is Ferrer’s function [e.g., Abramowitz and
Stegun, 1965], and am:k, bm:k and ck are unknown
coefficients, that have to be determined by velocity and
head matching conditions at inclusions boundaries. The
earliest reference to the functional form of (1) and (2) is
reportedly included by Laplace in Memoires des Savants
Etranges (1785). A more recent reference is given by Byerly
[1893] and Hobson [1931].
[16] The coefficients ck are computed exactly by requiring continuity of the normal (r) component of the velocity
[Fitts, 1991]. This gives:
k
R2kþ1
ck ¼ ðkþ1Þ ¼ kþ1
dr
dr
R
drk
dr
ð3Þ
where R is the radius of the inclusion. Expression (3)
guarantees continuity of the normal component of the
2. Numerical Flow Solution
[13] The following is an overview of analytic-based
solution for three-dimensional potential flow with inclusions in hydraulic conductivity that are shaped as spheres.
Equivalent solution for circular inclusions is presented by
Barnes and Janković [1999] and is included in part 1.
Solution for inclusions shaped as rotational ellipsoids are
given by Janković and Barnes [1999]. Detailed derivation
of all solutions is presented by Janković [1997].
[14] Each inclusion can have a unique size, conductivity,
ratio of the long and the short axis (rotational ellipsoids
only) and may be arbitrarily placed as long as it does not
intersect or touch any other inclusion. Solutions are valid
for general flow conditions including far-field uniform flow
(present study), point and line sinks.
[15] The governing equation for three-dimensional flow
is the Laplace equation, r2f = 0 where the velocity
potential is given by f = KH/q, H is head, K is hydraulic
conductivity, and q is a constant porosity. Following the
Figure 2. Snapshot of a plume made of 40,000 particles in
input zone (rectangle) of 30R 8R of 2-D flow with sY2 = 4
and n = 0.9 at tU/IY ’ 15.
16 - 4
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JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
velocity regardless of the values of coefficients am:k and
bm:k. It is emphasized the conservation of mass and
continuity of normal velocity component on inclusion
boundaries are satisfied exactly if the series in (1) and (2)
are truncated at a finite k, provided the same number of
terms are kept in both interior and exterior potentials. This is
a distinctive feature of the solution as compared with the
ones based on conventional numerical methods.
[17] The coefficients am:k, bm:k are determined using a
two-step process. The first step is to expand the velocity
potential due to uniform flow and all inclusions other than
the examined one, f0, on the boundary of the inclusion
itself. Following Byerly [1893] and MacRobert [1967], the
orthogonal uniformly convergent expansion is (up to an
additive constant):
f0 ¼
1 X
k
X
Tm:k ðcos qÞ½am:k cosðmyÞ þ bm:k sinðmyÞ
ð4Þ
k¼1 m¼0
Coefficients am:k and bm:k may be computed by integrating
the products of f0 and basis functions, or, more efficiently,
using over-specification [Janković and Barnes, 1999]. If a
single inclusion is placed in uniform flow, only a0:1, a1:1
and b1:1 have nonzero values; all other coefficients are zero.
In the general case the coefficients am:k and bm:k depend
linearly on the unknown am:k, bm:k pertaining to all other
inclusions.
[18] Coefficients am:k and bm:k are obtained, following
continuity of head H at r = R, by requiring f/K to be
continuous across the boundary of the inclusion. Using (1),
(2) and (4), this continuity condition may be written as:
Kint
K0
ð5Þ
¼ dk bm:k ; dk ¼ K
k
int
k
þ1
R
K0 k þ 1
1
am:k ¼ dk am:k ; bm:k
where K0 is the background hydraulic conductivity and Kint
is the conductivity of the inclusion. Because of the
aforementioned dependence of am:k and bm:k upon am:k,
bm:k of the other inclusions, (5) constitutes a linear system of
equations. Once it is solved, substituting (5) and (3) into (1)
and (2) gives exactly
f
ðinÞ
Kint
r k
K0
¼
Tm:k ðcos qÞ
Kint k
R
k¼1 m¼0
þ1
K0 k þ 1
½am:k cosðmyÞ þ bm:k sinðmyÞ
ð6Þ
Kint
r ðkþ1Þ
K0
¼
Tm:k ðcos qÞ
Kint k þ 1 R
k¼1 m¼0
þ
K0
k
½am:k cosðmyÞ þ bm:k sinðmyÞ
ð7Þ
1 X
k
X
1
and:
f
ðexÞ
1 X
k
X
1
3. Experimental Setup
[19] Two-dimensional simulations were carried out in a
circular domain of diameter L (referred to as the simulation
domain) as shown on Figure 1. Far-field velocity U was
uniform and oriented along x1 axis. The conglomerate of
inclusions behaves in the mean as an equivalent inclusion of
radius L/2 of an equivalent conductivity. Since the analytic
solution for flow past a circular inclusion of constant
conductivity yields uniform flow inside the inclusion, the
circular geometry has thus ensured that the mean velocity
inside the simulation domain was uniform. Three-dimensional simulations were carried out using a domain shaped
as a prolate ellipsoid (Figure 1). The mean flow inside the
simulation domain was again uniform, following the solution for a prolate inclusion in uniform flow at infinity.
Furthermore, by selecting the matrix and exterior conductivity to be equal to the effective one, the mean flow inside
domains was equal to U, as discussed in the sequel.
[20] The aim of the simulations was to derive the statistical moments of flow and transport variables pertaining to
stationary fields, as assumed in the approximate solutions of
parts 1 and 2. However, the detailed results (see Figure 2)
permit one to investigate additional features, e.g., the mean
concentration. These and other topics are not examined
here.
[21] To eliminate boundary effects along the border of the
simulation domain, flow and transport were examined in a
domain (referred to as the flow domain) which is smaller
than the simulation domain. The boundary effects, due to
the transition from the exterior matrix to the heterogeneous
medium, were assessed by computing the velocity variance
along various horizontal and vertical lines along the procedure of Bellin et al. [1992]. The experiments show that the
boundary effects were confined to the zone which is about
5 – 10 inclusions wide. The flow domain was shaped as a
square in 2-D and as a parallelepiped in 3-D, as shown on
Figure 1, for ease of computation of various flow and
transport statistics. The constancy of the velocity variance
determined by sampling along lines, is regarded as a
diagnostic of the given realization to represent a stationary
Eulerian velocity field.
[22] Hydraulic conductivity of inclusions, K, was generated using lognormal distribution in a dimensionless form
by normalizing by KG, which was set equal to unity. Twodimensional simulations were carried out with variance of
log conductivity (sY2), of 0.1, 0.5, 1, 2, 4, 6, 8 and 10. Threedimensional simulations were limited to sY2 = 2, 4 and 8 due
to significantly larger computational effort. Background
conductivity (K0 from equation (5)) was set to the effective
conductivity of the conglomerate: KG for 2-D flow and
effective conductivity computed using self-consistent
model for 3-D flow (numerical solution of equation (23),
part 1): 1.291KG, 1.558KG and 2.095KG for sY2 = 2, sY2 =
4 and sY2 = 8 respectively.
[23] In the discussion of part 1 (as shown in Figure 2
there) the representation of the medium is one of inclusions
of radius R, embedded in a matrix of inclusions of much
smaller radii, which was replaced by a matrix of conductivity Kef. The degree of freedom introduced by the
parameter n, volume fraction of inclusions, was of no
consequence for the semianalytical and first-order solutions
of parts 1 and 2: all the results of interest were scaled by n.
This is not the case for the numerical solutions and to check
the effect of varying n we adopted the values 0.4, 0.6, 0.8
and 0.9 in 2-D whereas 3-D simulations were limited to n
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
equal to 0.4, 0.6 and 0.7. However, our main interest
resides with the largest n, for which the approximate
solution is subjected to the most severe test.
[24] In previous numerical works Monte Carlo simulations were used primarily in 2-D flows. Furthermore, due to
computer time limitations, the dimension of the flow
domain was quite limited. In the present simulations we
wished to use a large flow domain, to be able to analyze
transport for sufficiently large travel times. This requirement
and the large computer time (detailed below) restricted the
number of simulations, especially for large sY2 and in 3-D.
Hence we have used mostly single realizations and tried to
identify statistical moments, by assuming stationarity and
ergodicity, from space distributions. This was considered as
sufficient for the main aim of this exploratory study, namely,
validation of the approximate solutions. Thus each 2-D and
3-D simulation was carried out with 50,000 equally spaced
inclusions. The radius of inclusions, R, was selected to yield
desired volume fraction: it was set to L/707, L/577, L/500
and L/471 in 2-D simulations of n = 0.4, n = 0.6, n = 0.8 and
n = 0.9 respectively, and to L/159, L/139 and L/132 in 3-D
simulations of n = 0.4, n = 0.6 and n = 0.7.
[25] Placement of inclusions was uniformly random for
the low volume fraction n = 0.4 in 2-D, i.e., the center of
each additional inclusion was set at random in the space
between existing inclusions. Such a setting is in complete
agreement with the pdf of centroids forwarded in part 1. For
higher values of n and in 3-D simulations the inclusions
centers were placed in a periodic fashion to achieve the high
volume fractions adopted in the study. Thus hexagonal
packing of inclusions was used for 2-D simulations and
face-centered cubic lattice (also referred to as cubic closest
packing) for 3-D. The maximum volume
pffiffiffifraction that may
be obtained
using
these
schemes
is
p/2/
3 = 0.9069 in 2-D
pffiffiffi
and p/3/ 2 = 0.7405 in 3-D. As explained in part 1, due to
the random generation of conductivities, the pdf of a
centroid in this setup has a discrete representation in terms
of a sum of Dirac functions, rather than the uniform one
assumed in the various theoretical derivations. For the high
n values considered here, this setting leads to same conductivity statistics as the uniform one, as shown in the
sequel.
[26] As mentioned above, most of the cases were run as a
single realization due to large computational expense of
each simulation. Nevertheless, a limited number of cases
was run as two realizations to check the assumed ergodic
behavior. Results of two realizations were very similar. We
therefore assumed that single realization of each case was
sufficient to obtain reliable estimates of flow and transport
statistics.
[27] The inclusions were subject to uniform flow in x1
direction. Subsequent to the flow solution, the conductivity
and velocity statistics of 2-D simulations were computed
using a 2000 2000 grid placed over the flow domain. This
grid was not used during the solution process and does not
influence the solution precision; it was used only for
computing flow statistics. Grid size for 3-D simulations
was 100 100 100.
[28] Advective transport was simulated using particle
tracking. Thus 2000 equally spaced particles were released
at x1 = L/3 between x2 = L/3 and x2 = L/3 in 2-D
simulations. Similarly, 1296 equally spaced particles
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(36 particles in each direction) were released at x1 =
0.275L between x2 = 0.1L and x2 = 0.1L, and between
x3 = 0.1L and x3 = 0.1L in 3-D simulations. Particle
locations and velocities were reported at approximately
5000 (2-D) and 500 (3-D) equally spaced time instances
and used to compute various transport statistics. Particle
tracking was terminated once the fastest particle reached the
end of the flow domain.
[29] Additional 2-D simulations with 250,000 inclusions
were carried out for several n = 0.4 and n = 0.9 simulations
(sY2 = 2, sY2 = 4 and sY2 = 8). Particles were distributed in
input zone 6R (n = 0.4) and 4R (n = 0.9) long, and 1050R
(n = 0.4) and 700R (n = 0.9) wide. The larger size did not
affect the velocity variance; but transport statistics for sY2 =
4 and sY2 = 8 n = 0.9 simulations were more than 1% larger
than those obtained for domain with 50,000 inclusions.
Results from other simulations were virtually identical.
All 2-D transport statistics presented in this paper are based
on simulations with 250,000 inclusions.
[30] Additional 3-D simulations with 100,000 inclusions
and 4,900 particles (70 rows and columns) distributed in
input zone 40R (n = 0.4), 35R (n = 0.6) and 33R (n = 0.7)
wide (in both transverse directions) were carried for sY2 = 4
3-D simulations examined in this paper (n = 0.4, n = 0.6 and
n = 0.7). All transport statistics computed using these new
simulations were within 1% of transport statistics for
original simulations (with 50,000 inclusions and 1296
particles).
4. Implementation of Numerical Method
[31] Implementation of infinite series (6) and (7) (and
corresponding series for circular inclusions) requires truncation that may results in head discontinuities across the
boundary of inclusions, while continuity of velocity is
satisfied exactly. High truncations levels were selected to
eliminate head discontinuities. The truncation level, Nmax,
for 2-D simulations with 50,000 inclusions was 40 for n =
0.4 and 0.6, 70 for n = 0.8 and 100 for n = 0.9. All 3-D
simulations with 50,000 inclusions were carried out with
Nmax = 14 (truncation level for the outer sum of (6) and (7)).
Number of degrees of freedom for each circular inclusion is
2 Nmax + 1 (e.g., 201 for n = 0.9), and (Nmax + 1)2 for each
spherical inclusion (225 for all n). All 2-D simulations with
250,000 inclusions were carried out with Nmax = 35; 3-D
simulations with 100,000 inclusions were carried out with
Nmax = 16 yielding 2.89 107 degrees of freedom.
[32] Flow problems were solved using an iterative superblock algorithm of Strack et al. [1999]. Iterations were
terminated once the largest relative change in coefficients
(between two subsequent iterations) over all coefficients of
all inclusions was under 108. The relative change in a
coefficient of an inclusion was computed as the absolute
change in the coefficient divided by the sum of absolute
values of all coefficients of the inclusion.
[33] Particle tracking was carried out using a fourth-order
Runge-Kutta method with a combination of a variable time
step (used everywhere except across the boundary of
inclusions) and a constant space-step (used when a particle
crosses inclusion boundary). The velocity, required by the
particle tracking procedure, was obtained analytically following Fitts [1990], by differentiating (6) and (7). The time
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JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
step size was adjusted for each particle during the simulation to limit the velocity change over each time step to 2%
of the velocity magnitude. The space-step was set to R/100.
Few simulations were also run with an order of magnitude
tighter tolerances; the results were identical to those presented later. Similarly, a simulation in which flow was
reversed resulted in the plume returning practically to its
initial position, indicating lack of numerical dispersion.
[34] Simulations were carried out on three Linux-based
distributed-memory clusters of 1 GHZ Pentium III and
2.4 GHZ Pentium IV dual-processor nodes at the Center
for Computational Research, University at Buffalo. Most
2-D simulations with 50,000 inclusions were conducted on
a single processor, without parallel processing. The required
CPU effort has depended on the volume fraction and
variance of log conductivity. Simulation have lasted up to
42 CPU days on Pentium III nodes.
[35] Each 3-D simulation with 50,000 inclusions was run
in parallel on 40 Pentium III processors (at 20 nodes). Each
processor was responsible for a number of super-blocks
(during solve phase) and a number of particles (during
tracking phase). Coefficients were exchanged between
processors at the end of each iteration (approximately every
6 hours) during the solve phase. The speed-up was linear
with the number of processors for both the solve and the
tracking phase. The simulations ran up to 12 CPU days; this
corresponds to 480 CPU days on a single processor.
[36] Simulations with 250,000 inclusions (2-D) and
100,000 inclusions (3-D) were run on 128 Pentium IV
nodes (256 processors). Two-dimensional simulations have
lasted up to 1.5 days (corresponding to 384 days on a single
processor); 3-D simulations have lasted up to 7 days
(corresponding to 1792 days on a single processor).
[37] In Figure 2 we have represented a snapshot of a
plume in 2-D flow for sY2 = 4, an input zone of 30R 8R
and a travel time tU/R = 13 i.e., tU/IY ’ 15.
[38] Flow nets from a 2-D simulation with 50,000 inclusions, n = 0.9 and sY2 = 8 are presented for illustration in
Figure 3. The largest scale on Figure 3 corresponds to the
flow domain. It is seen that the gap between inclusions was
extremely small and they practically touch. Flow nets were
visually perfect for all 2-D simulations on all investigated
scales: from 103R to 103R.
5. Derivation of Flow and Transport Statistics
From Numerical Simulations
[39] This section presents expressions that were used to
compute various flow and transport statistics for 3-D
simulations. Expressions for 2-D simulations can be
obtained in a similar manner.
5.1. Conductivity and Eulerian Velocity Statistics
[40] Velocity variance, sii2, was computed in the longitudinal (i = 1) and transverse (i = 2 and i = 3) direction using
the grid placed over the flow domain as follows
with Vi(l, m, n) the ith component of the velocity at grid
node (l, m, n) and N the number of grid nodes in each
direction (100 for all simulations).
[41] Autocorrelation, rii, of the ith component of the
velocity in j direction was computed by
NX
kdj1 NX
kdj2 NX
kdj3
1
1
rii k xj ¼ 2 2
½Vi ðl; m; nÞ Vi sii N ð N k Þ l¼1 m¼1 n¼1
ð9Þ
Vi l þ kdj1 ; m þ kdj2 ; n þ kdj3 Vi
where k = 0, . . ., N 1, d is Kronecker delta, and xj is the
flow domain size in j direction divided by the number of
divisions (N 1). Autocorrelation of hydraulic conductivity
or log conductivity was obtained by replacing Vi in (9) by K
or Y, respectively.
5.2. Transport Variable Statistics
[42] The numerical experiments described here were used
primarily to determine the time dependent longitudinal and
transverse macrodispersivities aL and aT in advective transport and their comparison with those based on the selfconsistent model (part 2).
[43] As a first step, we determined the trajectory of the
centroid of the swarm of particles in each simulation
N
xpi ðt Þ ¼
s2ii ¼
1
N3
1
Vi ¼ 3
N
N
X
N
X
l¼1 m¼1 n¼1
N X
N X
N
X
l¼1 m¼1 n¼1
2
½Vi ðl; m; nÞ Vi ð8Þ
Vi ðl; m; nÞ
ð10Þ
where xpi( j, t) is the xi coordinate of particle j at time
instance t and Np is the number of particles. It was found
that the mean trajectory is linear at high accuracy and the
slope was very close to the mean velocity U.
[44] A few options were considered in order to determine
the macrodispersivities in each realization.
[45] The first one is to determine the spatial moments Sii
(i = 1, 2, 3) of the swarm of particles that represent the
plume,
N
Sii ðt Þ ¼
p
2
1 X
xpi ð j; t Þ xpi ðt Þ ;
Np j¼1
ð11Þ
and subsequently to derive aL = (1/2) dS11/dt and aT = (1/2)
dS22/dt or aT = (1/2) dS33/dt. This procedure was found to
lead to oscillatory behavior for aT that suggests that the
plume was not large enough to ensure a smooth, ergodic,
behavior.
[46] A second procedure was to determine the trajectories
second moment Xii(t) by computing the fluctuation of the
trajectory of each particle with respect to its mean value and
averaging for all particles,
N
Xii ðt Þ ¼
p
2
1 X
xpi ð j; t Þ xpi ð j; 0Þ Vpi t ;
Np j¼1
ð12Þ
where
N
Vpi ¼
N
X
p
1 X
xpi ð j; t Þ
Np j¼1
Nt
p X
X
1
Vpi ð j; m t Þ
Np ðNt þ 1Þ j¼1 m¼0
ð13Þ
In (13)Vpi( j, t) is the ith velocity component of particle j at
time instance t, t is the time increment used in reporting
particle positions and velocities and Nt is the number of time
increments. This achieved considerable smoother results,
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
SBH
16 - 7
Figure 3. Flow nets and three stream tubes at four spatial scales with particle positions at a time
instance (largest scale only) for 2-D flow with sY2 = 8 and n = 0.9 and 50,000 inclusions; solid lines are
streamlines, and dashed lines are lines of constant head.
but the differentiation of Xii in order to determine transverse
macrodispersivity still resulted in some small oscillations.
[47] The third procedure, that led to the most stable
results, that were considered representative for the ensemble, was to determine first the Lagrangian velocity covariances wii(t) (i = 1, 2, 3). They are given by
was employed in order to determine macrodispersivities.
The numerical experiments show that two last approaches
yield close results, typical difference being under 5%. Since
the dispersivity computed by (15) was smoother for both aL
and aT, it was selected in the following presentations.
N
wii ðk t Þ ¼
p N
t k
X
X
1
Vpi ð j; m t Þ Vpi
Np ðNt þ 1 k Þ j¼1 m¼0
Vpi ð j; m t þ k tÞ Vpi
6. Presentation and Discussion of Results
ð14Þ
where k = 0, . . ., Nt 1. Subsequently, the well known
Taylor formula
1
ai ðt Þ ¼ Vp1
Zt
wii ðtÞdt ðaL ¼ a1 ; aT ¼ a2 or aT ¼ a3 Þ ð15Þ
0
6.1. Conductivity Statistics
[48] The autocorrelation of hydraulic conductivity, rK,
was quite independent of volume fraction, variance of log
conductivity, setting and direction, in all simulations. Hence
hydraulic conductivity was isotropic and the results are
consistent with theoretical ones obtained in part 1, that are
based on independence of coordinates of inclusion centers
and on continuous variation of the distance between two
SBH
16 - 8
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
Figure 4. Autocorrelation of hydraulic conductivity (rK)
in terms of separation (r/R) for a 2-D (n = 0.9) and a 3-D
(n = 0.7) numerical simulation with sY2 = 2.
inclusions. We have represented in Figure 4 the theoretical
conductivity autocorrelations (equation (13) of part 1) with
results from two simulations with highest volume fractions.
It is seen that the agreement is very good and the same is
true for the autocorrelation of log conductivity rY, which
was the same as rK.
[49] This check strengthens the confidence in both theoretical model of the structure and in the setup adopted in the
numerical simulations. The simple analytical expressions of
rY (equation (13) part 1) can be used to derive directly the
first-order approximation of macrodispersivities. However,
we shall use in the sequel the results of part 2.
6.2. Eulerian Velocity Field Statistics
6.2.1. Average Velocity
[50] The average transverse velocity (V 2 for 2-D, and V 2
and V 3 for 3-D simulations) was zero for all simulations.
The average longitudinal velocity (V 1), was within 2.2% of
far-field uniform velocity U in all 2-D and 3-D simulations.
The velocity field outside the domain of heterogeneity was
not disturbed and the choice of the self-consistent Kef for the
embedding matrix proved to be correct. The issue of
effective conductivity is discussed in detail in a separate
paper [Janković et al., 2003].
6.2.2. Velocity Variance
2
2
) and transverse (s22
)
[51] Variance of longitudinal (s11
velocity component, made dimensionless with respect to
nU 2, are presented on Figures 5 and 6. Results for the selfconsistent approximation (SC) and first-order approximation (FO), derived in part 2, are included for comparison. In
both cases, due to linear dependence on n, these are unique
curves. Variance of transverse velocity component for 3-D
simulations was obtained as an average of variance in x2 and
x3 direction which were only a few percent different. The
same averaging was used for all other flow and transport
transverse statistics of 3-D simulations.
[52] The flow solution based on the self-consistent
approximation (part 2) has the same functional form as
the first-order (k = 0 and k = 1) terms of (6) and (7). The
differences between numerical simulations and self-consistent model observed on Figures 5 and 6 are due to highorder terms of (6) and (7), that are necessary to ensure
continuity of head, and a different method of computing
Figure 5. Variance of the longitudinal component of the
2
) as a function of log conductivity variance (sY2)
velocity (s11
and volume fraction (n) for numerical simulations (NS),
self-consistent model (SC), and first-order approximation
(FO).
first-order coefficients (a0:1, a1:1, b1:1): the first-order
coefficients in the self-consistent model are function of
inclusion conductivity only, while in the numerical solution
they also depend on local flow conditions.
[53] Comparison of the numerical solution (NS) and the
SC approximation in 2-D for the variance of the longitudinal velocity (Figure 5) shows a consistent trend: they are
close for sY2 < 1; they are close for any sY2 for n 0.6 and
the NS is larger than the SC for n > 0.6. The latter difference
is discussed in the next paragraph.
[54] The 3-D results show a similar trend, though the SC
approximation is in better agreement with NS for a larger
range of sY2 and n. This is understandable in view of the
more localized effect of neighboring inclusions. In contrast,
the FO approximation is overestimating largely the NS for
say sY2 > 0.25.
[55] The results for the variance of the transverse velocity
(Figure 6) display the same trends, but the difference
between the NS and the SC values is larger than for the
longitudinal component. This difference is attributed to the
Figure 6. Variance of the transverse component of the
2
) as a function of sY2 and n; for notation, see
velocity (s22
Figure 5.
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
nonlinear effects resulting from interaction between neighboring inclusions that are present only in the NS. For
instance, in the SC approximation the interior velocities
are parallel to the mean flow, whereas in the NS they can be
slanted.
6.2.3. Probability Density Function of Velocity
[56] For purpose of illustration, the probability density
function of longitudinal velocity component (V1/U) for a
2-D (n = 0.9) and a 3-D (n = 0.7) numerical simulation with
sY2 = 4 is shown on Figure 7. These pdf are highly skewed,
in contrast with the Gaussian distribution of the FO approximation. This large difference is manifesting in the higherorder statistical moments of V1, that are not presented here.
It is interesting to note the presence of a small fraction of
negative velocities. Negative velocities were present in 3-D
only at highest volume fraction (n = 0.7) while in 2-D they
were also present for lower volume fractions due to more
constrained flow patterns observed in 2-D flow.
[57] Another interesting feature is the presence of velocities larger than those predicted by the SC approximation
SC
(V1,max
= 2U in 2-D and 3U in 3-D, respectively). This
effect is attributed to the nonlinear interaction between
neighboring inclusions that is neglected in the SC approxSC
imation. The existence of the tails of the pdf for V1 > V1,max
explains the differences between velocity variances shown
in Figure 5. The tail for 3-D flow is much smaller than the
one for 2-D flow, resulting in better agreement with the SC
approximation. This is understandable in view of the
stronger interactions existing in the 2-D configurations.
[58] It is interesting to compare these results with those
obtained by Bellin et al. [1992] and extended by Salandin
and Fiorotto [1998] for 2-D flow in media of multiGaussian log conductivity structure. A qualitative comparison between the pdf in Figure 7 with Figure 9a of Salandin
and Fiorotto [1998] for their largest sY2 = 4 reveals a high
degree of similarity. Nevertheless, the velocity variance
(their Figure 3a) is in excess of the FO approximation
whereas the NS results (Figure 5) are below. Our main
interest resides in the 3-D flow, for which the effect of the
tail of the pdf is largely reduced.
6.2.4. Velocity Autocorrelation
[59] Autocorrelations (rii(k xj)) of the longitudinal (i = 1)
velocity component in longitudinal ( j = 1) and transverse ( j
Figure 7. Probability density function (PDF) of Eulerian
longitudinal velocity component (V1) for a 2-D (n = 0.9) and
a 3-D (n = 0.7) numerical simulation with sY2 = 4.
SBH
16 - 9
Figure 8. Autocorrelation (rii(rj)) of the longitudinal (i = 1)
and transverse (i = 2) velocity component in longitudinal
( j = 1) and transverse (j = 2) direction as a function of
separation rj/R for a 3-D sY2 = 4 n = 0.7 simulation; for
notation, see Figure 5.
= 2) direction, and of transverse (i = 2) velocity component
in longitudinal ( j = 1) direction for a 3-D simulation with
n = 0.7, sY2 = 4 are shown for illustration in Figure 8.
The agreement with the SC values is very good for the
longitudinal velocity in both longitudinal and transverse
direction. Similar trends, but larger differences are present
for the transverse velocity in the longitudinal direction.
Two-dimensional simulations display the same features.
It seems therefore that the SC approximate model captures
correctly the range of lags for which velocities are correlated
(approximately 6R) and the screening effect of the medium
surrounding each inclusion.
6.3. Transport Statistics
6.3.1. Qualitative Discussion
[60] Before embarking on a detailed analysis of the
statistical moments of trajectories, it is worthwhile to
examine the illustrative example of the plume displayed in
Figure 2. It is seen that the multi-indicator model leads to
the complex behavior found in previous works [e.g., Wen
and Gomez-Hernandez, 1998]. Although the topic of mean
concentration distribution is the object of a future study, it is
worthwhile to point out to a few important features. First, it
is observed that channels of high velocity are created by
strings of high conductivity inclusions, leading to ‘‘fingers’’
of solute that move forward (the presence of transverse
pore-scale dispersion, neglected here, may cause some
mixing). Secondly, the retention of a large portion of the
plume in the zones of low conductivity, in agreement with
the discussion of part 1 on the asymmetry of the transport, is
also evident. Last but not least, the extreme delay in zones
of low conductivity is illustrated by the blobs of the plume
retained in the injection zone. This effect reminds, at least in
a qualitative manner, the findings at the MADE field test
[Boggs et al., 1992]. We limit the present study to the first
statistical moments of ergodic plumes, much larger than the
one of Figure 2, that characterize their global behavior.
6.3.2. Average Lagrangian Velocity
[61] The average Lagrangian velocity determined directly
as the rate of change of xpi(t) was constant in time and
different by less than 1% from the Eulerian mean V 1.
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JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
Figure 9. Lagrangian longitudinal velocity covariance
(w11) as a function of time for a few 2-D and 3-D sY2 = 4
simulations with infinite and finite Peclet numbers; for
notation, see Figure 5.
Figure 10. Longitudinal dispersivity (aL) as a function of
time for 2-D n = 0.4 simulations with sY2 = 2, 4, 8; for
notation, see Figure 5.
6.3.3. Lagrangian Longitudinal Velocity Covariance
[62] As mentioned above, we have used the numerically
determined Lagrangian velocity covariances (14) to compute macrodispersivities, the variable of major interest of
this study. However, for the sake of conciseness and for
illustration we display only one case of such a covariance
in Figure 9. The case we chose is the 3-D flow with n = 0.7,
sY2 = 4.
[63] The first-order approximation of the Lagrangian
covariance is identical with the Eulerian one, after replacing
the spatial lag r1 in the mean flow direction by Ut [see, e.g.,
Dagan, 1989]. Therefore the difference between the FO and
NS graphs in Figure 9 reflect the differences between the
Eulerian and Lagrangian covariances. It is seen that the
variance of FO is much larger, which was already found and
illustrated in Figure 5. In contrast, the Lagrangian covariance is persistent over a much larger time, due to the large
residence time of fluid particles in zones of low velocity, as
discussed at length in parts 1 and 2.
[64] Since the longitudinal asymptotic dispersivity results
from the integration of the covariance, the compensatory
effect analyzed in parts 1 and 2 is evident in Figure 9 as
well. The FO approximation overestimates the effect of
large conductivity zones, which determined the shape at
small travel time, and underestimates that of low conductivity ones. The effect on time dependent macrodispersivity
will be illustrated in the following figures.
[65] The SC approximation, though underestimating the
NS at small and moderate travel times, captures altogether
the trend of NS. The tailing of both curves suggests that
Fickian behavior may be reached only after a considerable
travel time, as revealed in part 2.
6.3.4. Longitudinal Dispersivity
[66] Dimensionless longitudinal dispersivity aL/R for
n = 0.4 and n = 0.9 2-D simulations for three values of
sY2 each is shown on Figures 10 and 11, respectively. Results
from the self-consistent model and the first-order approximation are included. The dispersivity was computed by
integration of Lagrangian longitudinal velocity covariance
(equation (15) for i = 1). Because of the low volume
fraction, n = 0.4 simulations could be carried out for large
dimensionless travel time.
[67] The numerical (NS) and the self-consistent model
(SC) results are overall in excellent agreement, even for
sY2 = 8. This is understandable in view of the diminishing
nonlinear interaction between inclusions as n decreases.
[68] Results for the more interesting case of n = 0.9 are
displayed in Figure 11. Both the NS and SC models exhibit
growth of dispersivity beyond tU/R = 150 for sY2 = 4 and
sY2 = 8, while dispersivity reaches a constant value at about
tU/R = 10 in the first-order solution regardless of the
variance of log conductivity. At tU/R = 150, the selfconsistent model and the first-order approximation yield
similar dispersivity for sY2 = 2. This is because overestimation of velocity variance (by the first-order approximation)
is compensated by shorter integral scale of Lagrangian
longitudinal velocity as discussed already.
[69] Numerical simulations produce larger dispersivity
than both self-consistent model and first-order approximation. The overall agreement between numerical simulations
and self-consistent model is better for n = 0.4 than for n = 0.9
due to lesser influence of high-order terms. For example, for
sY2 = 4 at tU/R = 150, the self-consistent model predicts
Figure 11. Longitudinal dispersivity (aL) as a function of
time for 2-D n = 0.9 simulations with sY2 = 2, 4, 8; for
notation, see Figure 5.
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
Figure 12. Longitudinal dispersivity (aL) as a function of
time for 3-D n = 0.7 simulations with sY2 = 2, 4, 8; for
notation, see Figure 5.
dispersivity which is 5% larger than that predicted by
numerical simulation for n = 0.4 and 30% smaller for
n = 0.9. Salandin and Fiorotto [1998, Figure 11a] arrived
at similar results for the their NS relative to the FO
approximation. However, their Monte Carlo simulations
were carried out for 2-D, multi-Gaussian structure and
travel time limited to tU/IY = 20.
[70] Results for four 3-D simulations with n = 0.7 are
shown on Figure 12. The overall agreement of numerical
simulations and self-consistent model is better than in 2-D,
despite the proximity of inclusions (volume fraction of
n = 0.7 is close to the maximum volume fraction of about
n = 0.74). For example, for sY2 = 4 at tU/R = 40, the selfconsistent model predicts dispersivity which is 10% smaller
than that predicted by numerical simulation.
[71] As already found in part 2, the main discrepancy
between the first-order and nonlinear results manifest in the
transient behavior of the longitudinal macrodispersivity: the
FO predicts a much quicker tendency to Fickian behavior.
[72] In parts 1 and 2 we have discussed in a semiqualitative manner the impact of molecular diffusion, which
is important in reducing the residence time in zones of very
low conductivity. Although this is a topic of future investigations, we have tried to assess in an exploratory manner
the effect of finite Pe = UR/Dm number. Toward this aim a
few 3-D simulations were carried out with small amounts of
molecular diffusion. This has been achieved by supplementing advective displacements by random walk steps. The
effect of molecular diffusion, displayed in Figure 12, makes
the comparison between NS and SC even more favorable,
especially at highest log conductivity variance (sY2 = 8)
where the number of low-conductive inclusions that trap
particles is the largest. Molecular diffusion has smaller
impact for sY2 = 2 and sY2 = 4. Molecular diffusion of
magnitude examined here (Pe = 1000) does not affect selfconsistent model results for timescales shown on Figure 12,
as shown in part 2. While this was not a comprehensive
study of the impact of molecular diffusion, the results show
that it plays a significant role in formations of large log
conductivity variance.
6.3.5. Transverse Dispersivity
[73] Dimensionless transverse dispersivity, obtained by
integration of Lagrangian transverse velocity covariance, is
SBH
16 - 11
Figure 13. Transverse dispersivity (aT) as a function of
time for 2-D sY2 = 4 simulations; for notation, see Figure 5.
presented on Figures 13 and 14 for several 2-D and 3-D
simulations of varying n and constant sY2 = 4. The general
shapes of the NS are reproduced by the approximate methods, though at different degree of quantitative agreement.
[74] The results concerning the NS are similar to those of
Salandin and Fiorotto [1998, Figure 11b] for sY2 = 4 and
tU/IY < 20, as far as the relation between the NS and the FO
approximation is concerned. The underestimation of the NS
results by the SC approximation was explained above: due
to nonlinear interaction between neighboring inclusions the
interior flow is slanted with respect to the X1 axis. Although
these deviations are small, they make a relatively large
accumulated contribution. Still, the transverse dispersivity is
very small compared to the longitudinal one.
[75] The asymptotic (large time) transverse dispersivity
tends to zero for all 2-D simulations regardless of volume
fraction and log conductivity variance, as demonstrated on
Figure 13. The transverse dispersivity was proportional to 1/t
(corresponding to logarithmic grow of X22). The behavior
was hence in agreement with that of the self-consistent
model (part 2).
[76] The asymptotic transverse dispersivity was nonzero
in all 3-D simulations in the travel time interval achieved in
this study. The dispersivity has diminished with a decrease
in n, but in all cases a small nonzero limit is present.
Figure 14. Transverse dispersivity (aT) as a function of
time for 3-D sY2 = 4 simulations; for notation, see Figure 5.
SBH
16 - 12
JANKOVIĆ ET AL.: TRANSPORT IN HIGHLY HETEROGENEOUS FORMATIONS, 3
Because of the numerical nature of the flow and transport
solution and of the finite plume size it is premature to reach
definite conclusions based on present simulations. However,
consistency of results and a reverse particle tracking test that
has resulted in identical trajectories as the original forward
particle tracking suggest that the phenomenon may be
physical.
[77] In addition, the transverse dispersivity results for 3-D
sY2 = 4 simulations (n = 0.4, n = 0.6 and n = 0.7) with twice
as many inclusions (100,000), almost four times as many
particles (4,900), more degrees of freedom per inclusion
(289), and an order of magnitude more precise particle
tracking were generally within 1% of those presented on
Figure 14. This suggests that numerical results obtained
here are not affected by the number of inclusions, number of
particles and precision of flow and transport solution.
[78] Although this topic is of interest in principle, it is
worthwhile to emphasize that in any case the large time
transverse dispersivity was several orders of magnitude
smaller than longitudinal dispersivity.
[79] The transverse spreading may also be assessed,
though with some more noise, as half the slope of X22(t)
and X33(t), determined directly from simulated trajectories.
In contrast, the slopes of the plumes spatial moments S22(t)
and S33(t) have exhibited almost an order of magnitude
larger oscillations than X22(t) and X33(t) both in 2-D and in
3-D and can not be used to draw any conclusions. This has
an important practical significance since S22(t) and S33(t) are
usually the only available measures of transverse spread.
This topic is the object of a different study.
7. Summary and Conclusions
[80] The main aim of the present sequence of articles was
to investigate flow and advective transport in highly heterogeneous formations. This has been achieved by adopting
a multi-indicator structure model, i.e formations made up
from blocks of different conductivities. This selection was
motivated by a few considerations: it may reflect the
observed geological features of some aquifers, it is flexible,
it allows for efficient and accurate numerical simulations
and for semianalytical approximations. Even in its simplest
version adopted here, of isotropic structures made up from
inclusions of constant radii, it can reproduce the lognormal
univariate conductivity distribution and a given integral
scale. The multi-indicator structure is different from the
multi-Gaussian one that has been used in most numerical
simulations in the past. To our best knowledge field
observations did not favor this particular choice.
[81] In the simple representation adopted here, the structure is represented by three parameters: the volume fraction
n and variance sY2 of correlated conductivity and the length
scale R, proportional to the integral scale IY. Thus three
parameters characterizing field conductivity measurements
can be fitted: a nugget, sY2 and IY. In the absence of a nugget,
i.e., n = 1, we believe that the results obtained in the
numerical simulations for the maximal n, are representative
since the inclusions are quite dense.
[82] The study has revealed many facets of flow and
transport that were discussed and summarized in the three
parts of this study. Here are a few general conclusions.
[ 83 ] The illustrative example of a plume snapshot
(Figure 2) reveals the complex features observed in other
simulations in the literature, in a qualitative manner. In
particular, the trailing of the plume due to low conductivity
zones in the injection area, is in qualitative agreement with
the field findings in the MADE experiment.
[84] The value of the asymptotic, long travel time, longitudinal macrodispersivity, determined by the first-order
approximation in the log conductivity variance and by the
self-consistent approximation are in fair agreement with the
numerical simulations even for large sY2, especially in 3-D.
This agreement was observed also in the past, but it was
explained here by the cancelling effects associated with a
symmetric conductivity distribution.
[85] In contrast, the nonlinear solution, obtained by both
numerical simulation and self-consistent approximate model,
display a few important features that are not captured by the
linearized one: (1) the tendency to Fickian behavior, i.e.,
constancy of the longitudinal macrodispersivity, is much
slower and the ‘‘setting travel time’’ increases with sY2. The
important implication is that in applications in which the
travel time is not sufficiently large, transport may be perceived as anomalous; (2) the tendency to Gaussianity, i.e.,
normalcy of trajectories pdf and existence of an advectiondispersion equation for the mean concentration with constant
coefficient, is even slower. The mean concentration is skewed
during a long period, again depending on the magnitude of
sY2; (3) although the numerically determined transverse dispersivity reaches a maximal value near the source, it does not
decay to zero for large travel times in 3-D. This persistent
effect has yet to be investigated and understood. Still, the
value is smaller, by orders of magnitude, than the longitudinal
macrodispersivity; (4) the transverse spatial moment dependence on time in the numerical simulations is very oscillatory,
even for the large initial area of the plume. This finding
suggests that in applications it may be subjected to large
uncertainty (nonergodic behavior); (5) molecular diffusion
may play an important role in transport through highly
heterogeneous formations by removing solute captured by
zones of very low conductivity.
[86] The present study is a first and exploratory attempt to
investigate in a systematic manner flow and transport in
highly heterogeneous formations of three-dimensional
structure. Many topics, e.g., effect of anisotropy, nonergodic
behavior, characterization of uncertainty of spatial moments
and local concentrations, effect of diffusion and pore-scale
dispersion, call for further investigation.
[87] Acknowledgments. This material is based on work supported by
the National Science Foundation under grant EAR-0218914. The authors
wish to thank the Center of Computational Research, State University of
New York at Buffalo, in particular Matt Jones, for assistance in running
numerical simulations and for a very large amount of computer time.
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G. Dagan, Department of Fluid Mechanics and Heat Transfer, Tel Aviv
University, Ramat Aviv, 69978 Tel Aviv, Israel.
A. Fiori, Dipartimento di Scienza dell’Ingegneria Civile, Università di
Roma Tre, via Volterra 62, 00146 Roma, Italy.
I. Janković, Department of Civil, Structural and Environmental
Engineering, State University of New York at Buffalo, Buffalo, NY
14260-4400, USA. ([email protected])