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WATER RESOURCES RESEARCH, VOL. 46, W05502, doi:10.1029/2009WR008220, 2010
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Significance of higher moments for complete characterization
of the travel time probability density function in heterogeneous
porous media using the maximum entropy principle
Hrvoje Gotovac,1,2 Vladimir Cvetkovic,1 and Roko Andricevic3
Received 18 May 2009; revised 25 October 2009; accepted 17 December 2009; published 1 May 2010.
[1] The travel time formulation of advective transport in heterogeneous porous media is
of interest both conceptually, e.g., for incorporating retention processes, and in
applications where typically the travel time peak, early, and late arrivals of contaminants
are of major concern in a regulatory or remediation context. Furthermore, the travel time
moments are of interest for quantifying uncertainty in advective transport of tracers
released from point sources in heterogeneous aquifers. In view of this interest, the travel
time distribution has been studied in the literature; however, the link to the hydraulic
conductivity statistics has been typically restricted to the first two moments. Here we
investigate the influence of higher travel time moments on the travel time probability
density function (pdf) in heterogeneous porous media combining Monte Carlo simulations
with the maximum entropy principle. The Monte Carlo experimental pdf is obtained by the
adaptive Fup Monte Carlo method (AFMCM) for advective transport characterized by a
multi‐Gaussian structure with exponential covariance considering two injection modes
(in‐flux and resident) and lnK variance up to 8. A maximum entropy (MaxEnt)
algorithm based on Fup basis functions is used for the complete characterization of the
travel time pdf. All travel time moments become linear with distance. Initial nonlinearity is
found mainly for the resident injection mode, which exhibits a strong nonlinearity within
first 5IY for high heterogeneity. For the resident injection mode, the form of variance
and all higher moments changes from the familiar concave form predicted by the first‐
order theory to a convex form; for the in‐flux mode, linearity is preserved even for high
heterogeneity. The number of moments sufficient for a complete characterization of the
travel time pdf mainly depends on the heterogeneity level. Mean and variance completely
describe travel time pdf for low and mild heterogeneity, skewness is dominant for lnK
variance around 4, while kurtosis and fifth moment are required for lnK variance higher
than 4. Including skewness seems sufficient for describing the peak and late arrivals.
Linearity of travel time moments enables the prediction of asymptotic behavior of the
travel time pdf which in the limit converges to a symmetric distribution and Fickian
transport. However, higher‐order travel time moments may be important for most practical
purposes and in particular for advective transport in highly heterogeneous porous media
for a long distance from the source.
Citation: Gotovac, H., V. Cvetkovic, and R. Andricevic (2010), Significance of higher moments for complete characterization
of the travel time probability density function in heterogeneous porous media using the maximum entropy principle, Water
Resour. Res., 46, W05502, doi:10.1029/2009WR008220.
1. Introduction
[2] Under given boundary conditions, groundwater flow
and advective transport are controlled by the heterogeneity of
the hydraulic properties. A key challenge then is in providing
1
Department of Land and Water Resources Engineering, KTH
Royal Institute of Technology, Stockholm, Sweden.
2
Also at Department of Civil and Architectural Engineering,
University of Split, Split, Croatia.
3
Department of Civil and Architectural Engineering, University of
Split, Split, Croatia.
Copyright 2010 by the American Geophysical Union.
0043‐1397/10/2009WR008220
a stochastic quantification of heterogeneity effects on flow
and transport variables, such as hydraulic head, velocity,
solute flux, concentration, displacement, or travel time [e.g.,
Dagan, 1989; Rubin, 2003]. In this paper we adopt the travel
time concept, first introduced by Shapiro and Cvetkovic
[1988] and Dagan et al. [1992], which considers Lagrangian
analysis of particle movement and advective travel time
needed for tracers or contaminants to reach some particular
location in the aquifer, from a known source. In an application context, we require a complete travel time probability
density function (pdf) in order to analyze early arrivals
particularly important for reliability of waste disposal sites
[e.g., Zimmerman et al., 1998], travel time peak related to
maximum concentration [Bellin and Rubin, 2004], and
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exposure for risk assessment [Andricevic and Cvetkovic,
1996; Maxwell et al., 1999], but also late arrivals that are
of interest, e.g., in a remediation context [Berglund and
Cvetkovic, 1995].
[3] Statistically, a complete characterization of the travel
time is given by a pdf clearly defining early and late arrivals
as well as travel time peak. Alternatively, travel time can be
described by the first few travel time moments. The standard
problem in subsurface hydrology is quantification of uncertainty using only the second moment or variance. In that
way, uncertainty is defined with a corresponding interval of
confidence representing it as fluctuations around the mean
travel time value hti ± kst , where k 2 N presents an interval
width. However, using only the variance for uncertainty
estimation clearly neglects the influence of higher‐order
travel time moments, implying a normal or lognormal distribution. Loss of information can be vital for accurately
capturing the travel time peak, early, and late arrivals.
[4] A number of studies have considered travel time
moments in the context of the first‐order theory, focusing on
the first two moments and assuming a lognormal travel time
pdf [Dagan et al., 1992; Cvetkovic et al., 1992]. Shapiro
and Cvetkovic [1988] and later Cvetkovic and Shapiro
[1990] presented the first‐order mean and variance of the
travel time by neglecting backward flow. More recently,
Guadagnini et al. [2003] and Sanchez‐Villa and Guadagnini
[2005] generalized their results for mean and variance including the second‐order correction terms. Ezzedine and
Rubin [1996] showed that alternative and more accurate
means for calculating the mean and variance can be obtained
with the cumulative distribution function (CDF) of the travel
time. Cvetkovic and Dagan [1994] and later Woodbury and
Rubin [2000] calculated the CDF from the pdf of the
longitudinal Lagrangian position X1(t). They assumed a
normal pdf of X1(t) which in turn yields a nonsymmetric but
Fickian travel time CDF and pdf. Indeed, all aforementioned
analytic studies considered the resident injection mode.
Demmy et al. [1999] presented the influence of the injection
mode (resident particles are injected uniformly and in‐flux
particles are injected proportionally to the flux) on the first
two travel time moments.
[5] Monte Carlo simulations can provide both higher
travel time moments and a complete pdf. However, all
previous studies considered only first two moments and/or a
pdf without analysis of the influence of the higher moments
including the recent study of Gotovac et al. [2009b]. Bellin
et al. [1992] and later Bellin et al. [1994] showed that even
in low heterogeneity cases with s2Y = 0.25, the travel time pdf
close to the injection source is non‐Gaussian. Cvetkovic et al.
[1996] presented travel time statistics for highly heterogeneous porous media and resident injection mode (s2Y 4).
They showed that travel time variance changes the form
from the familiar concave predicted by the first‐order theory
to a convex form in case of high heterogeneity. Recently,
Gotovac et al. [2009b] presented travel time statistics for
highly heterogeneous porous media and in‐flux injection
mode (s2Y 8). They showed that travel time mean and
variance are linear after a relatively short distance from the
source, even in the case of high heterogeneity.
[6] The principle of maximum entropy (MaxEnt) was
formulated by Jaynes [1957] based on Shannon information
entropy [Shannon, 1948] such that the actual pdf is presented only by the first few moments. More precisely, the
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MaxEnt pdf describes the actual pdf with highest uncertainty or maximum entropy among all possible pdf’s satisfying given moment constraints. In that way, the actual
travel time pdf can be presented by MaxEnt pdf, clearly
exposing how many moments are needed for its accurate
representation and, more important, providing a correct
physical interpretation of travel time moments with respect
to peak, early, and late arrivals.
[7] In subsurface hydrology, three important applications
of the entropy formalism have been proposed. Kitanidis
[1994] defined a new parameter: a dilution factor for representation of the plume dilution which enables its distinction from the plume spreading in contaminant transport.
Woodbury and Ulrych [1993, 2000] and Woodbury et al.
[1995] introduced minimum relative entropy (MRE) for
the inverse modeling of the groundwater flow and transport
processes by combining the Bayesian theorem and the
MaxEnt principle with the aid of the prior pdf. If the prior
pdf is uniform, then the posterior MRE and MaxEnt pdf
become equivalent. Another promising tool is the Bayesian
maximum entropy (BME) methodology introduced by
Christakos [2000] which consists of three main steps:
(1) prior stage, which generates prior pdf based on epistemic or general knowledge using the MaxEnt principle
(variograms, differential equations, empirical relationships,
and so on); (2) metaprior stage, expressing site‐specific
knowledge into the appropriate stochastic form; and (3) posterior stage, which incorporates together general knowledge
(step 1) with site‐specific knowledge (step 2) in the form of
the posterior final pdf at each space/time point using the
Bayesian theorem. Among others, this approach was applied
for flow [Serre et al., 2003] as well as for transport analysis
[Kolovos et al., 2002].
[8] In this paper we will present a complete characterization of the travel time pdf for advective transport in a
mean uniform flow, subject to both injection modes and
instantaneous injection, and for a multi‐Gaussian structure
of lnK ranging from low to high heterogeneity (s2Y 8). We
implemented the adaptive Fup Monte Carlo method
(AFMCM) [Gotovac et al., 2009a] and travel time statistics
[Gotovac et al., 2009b] on the one hand and the Fup MaxEnt
algorithm [Gotovac and Gotovac, 2009] on the other hand.
To the best of our knowledge, this is the first analysis which
links higher travel time moments with the pdf through the
MaxEnt principle. This work shows how many moments
accurately describe first and last arrivals as well as the travel
time peak with respect to the injection mode, heterogeneity
level, and distance from the source. Also, extension of the
travel time statistics over the computational domain as well
as the asymptotic behavior is discussed.
2. Theory
2.1. Advective Travel Time
[9] Let a dynamically inert and indivisible tracer parcel
(or particle) be injected into the transport (inner computational) domain at the source line (say, at origin x = 0) for a
given velocity field (Figure 1). The tracer advection trajectory can be described using the Lagrangian position vector
as a function of time X(t) = [X1(t), X2(t)] [e.g., Dagan,
1984] or, alternatively, using the travel (residence) time
from x = 0 to some control plane at x, t (x), and transverse
displacement at x, h(x) [Dagan et al., 1992]. The t and h are
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tical stationarity, i.e., integral functions in (3) are assumed
independent of a and x. Moreover, the third moment is
related to the three‐point statistics of the slowness, the
fourth moment is related to the four‐point statistics of the
slowness, and so on. Analytical solutions for mean and
variance are given by Shapiro and Cvetkovic [1988],
Guadagnini et al. [2003], and Sanchez‐Vila and Guadagnini
[2005].
[12] The CDF of the travel time is to be computed as
F ðt; xÞ ¼ EðH ðt ð xÞÞ ¼
Figure 1. Simulation domain needed for global flow analysis and inner computational domain needed for flow and
transport ensemble statistics.
Lagrangian (random) quantities describing advective transport along a streamline. The advective tracer flux [M/TL] is
proportional to the joint probability density function
R (pdf)
[Dagan
et
al.,
1992].
Marginal
pdf’s
f
=
fth dh
fth (t, h; x)
t
R
and fh = fth dt separately quantify advective transport in
the longitudinal and transverse directions, respectively.
[10] Let l denote the intrinsic coordinate (length) along a
streamline/trajectory originating at y = a and x = 0; we shall
omit a in the following expressions for simplicity. The
trajectory function can be parameterized using l as [Xx(l),
Xy(l)], and we can write travel time as
Z
ð xÞ ¼
l ð xÞ
0
1
d :
v Xx ðÞ; Xy ðÞ
ð1Þ
Our focus in the computations will be on the first‐passage
time, which is the travel time required for a particle to reach
the control plane defined at x for the first time. This means
that a possible multiple crossing at x due to backward flow
will not be recorded. We can introduce a simple scaling
x = (l(x)/x) & ≡ l(x) &, whereby
Z
ð xÞ ¼
0
x
ð xÞ
d& v Xx ð& Þ; Xy ð& Þ
Z
0
x
ð&; xÞ d&:
ð2Þ
In (2), a is referred to as the “slowness” or the inverse
Lagrangian velocity [T/L]. It may be noted that in this
approach, all Lagrangian quantities depend on space rather
than time as in the traditional Lagrangian approach [e.g.,
Taylor, 1921; Dagan, 1984]. Scaled velocity defined by
w(&) ≡ v/l will be referred to as “Langrangian velocity.”
Note that l(x) is unique as the trajectory length for the first
crossing at x. This means that backward flow is included
along the particle streamline/trajectory, but only first passages are considered at x.
[11] The first two moments of t can be computed as
Z
A ð xÞ Eð Þ ¼
0
nMC
NP X
1 X
ð H ðt ð xÞÞÞ; ð4Þ
NP nMC i¼0 j¼1
where H is a Heaviside function, NP is the number of particles, nMC is the number of Monte Carlo realizations, while
travel time in (4) has the form (2) for each particular particle
and realization; therefore the expectation that appears in (4) is
obtained over all realizations and particles from the source.
The pdf is simply obtained as ft (t;x) = ∂(Ft(t;x))/∂t. Moreover, travel time moments can be computationally more
efficiently obtained with the travel time pdf rather than
slowness statistics. The travel time mean is computed as
Z1
A ð xÞ ¼
t f ðt; xÞ d t:
0
Higher central travel time moments (as variance) are obtained
directly from the pdf as
Mi ð xÞ ¼
Z1
ðt A ð xÞÞ i f ðt; xÞ d t ; i ¼ 2; 3; . . . ; 1:
0
Noncentral moments mti can be directly obtained from the
central moments and vice versa. In this paper, the term
“moment” refers to the central moment if it is not particularly
specified as in section 2.2.
2.2. Maximum Entropy Algorithm
[13] The MaxEnt principle is widely recognized as a
powerful inference tool, especially in information theory.
Furthermore, MaxEnt is particularly useful in pdf characterization with respect to showing how many conventional
statistical moments are required to accurately describe important pdf properties such as its shape, tailing, peakedness,
number of peaks, skewness, or kurtosis. Shannon entropy
[Shannon, 1948] is defined in a broader sense as
Zxmax
Sð f Þ ¼ ln ð f ð xÞÞ f ð xÞ d x;
ð5Þ
xmin
x
A ð Þd;
xZ
h
i Z
2 ð xÞ E ð A Þ2 ¼
0
0
x
C ð 0 ; 00 Þd 0 d 00 :
ð3Þ
Travel time moments are completely defined by the statistics
of the slowness; implicit in (3) is the assumption of statis-
where I = ln( f(x)) is a quantity of information measuring the
amount of uncertainty associated with a corresponding random variable, while f is its pdf. Shannon information entropy
is then the expected information quantity. The logarithm is
chosen arbitrarily according to the desired entropy properties:
(1) decreasing of probability causes increase of information,
(2) higher uncertainty or information causes higher entropy,
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and (3) total entropy of two independent events is equal to the
sum of particular entropies [Tung et al., 2006]. This means
that more information is contained in extreme events or realizations with low pdf values that define pdf tailings, namely,
early and late arrivals. Mean and variance do not describe
these extreme events, so higher moments are required for the
description of more uncertain realizations and complete
characterization of pdf. Note that another bound case is a
deterministic event where there is no uncertainty for which
the entropy is zero.
[14] The MaxEnt principle is defined by Jaynes [1957]
such that the pdf with highest entropy is selected among
all other possible pdf’s that satisfy known constraints. In
other words, the MaxEnt principle states that among the
probability distributions that satisfy our incomplete information about the system, the probability distribution that
maximizes entropy is the least biased estimate that can be
made. It agrees with everything that is known but carefully
avoids anything that is unknown. If these constraints are
known noncentral or standard statistical moments, MaxEnt
can be defined as the following optimization problem:
Zxmax
max S ð f Þ subject to
x j f ð xÞ d x ¼ mj ; j ¼ 0; . . . ; k:
ð6Þ
xmin
This optimization problem can be solved by introducing
a Lagrangian function and corresponding Lagrangian
multipliers:
Lð f ; Þ ¼ S ð f Þ k
X
0
j @
j¼0
Zxmax
1
x j f ð xÞ d x mjA:
ð7Þ
xmin
Therefore problem (6) is reduced to finding a global minimum of the Lagrangian function for all pdf’s that satisfy
moment constraints (see Berger et al. [1996] for a more
complete discussion):
@ L ð f ; Þ
¼
@f
Zxmax
ð1 ln ð f ð xÞÞ Þd x
k
X
1 ln ð f ð xÞÞ k
X
j
j x
j
d x ¼ 0:
ð8Þ
!
j x
j
:
ð9Þ
Practically, the optimization problem (8) requires solving
the (m+ 1) Lagrangian multipliers from the nonlinear system, which are obtained if the pdf solution (9) is substituted
into the moment constraints (6):
x exp 1 0 xmin
k
X
!
j x
j
i ¼ 0; . . . ; k;
ð11Þ
where "i(x) is a residual between the polynomial and its
Fup2 approximation, and dij is the connection matrix,
which depends mostly on the approximation method and
the number of moments and/or basis functions. The main
idea behind the algorithm is to use a low order of Fup2 basis
functions to accurately describe higher‐order polynomials
to significantly reduce numerical problems in the nonlinear
system (10). This implies the presence of more balanced
nonlinearities and a weaker interdependence between different Lagrangian multipliers. As a consequence, the
MaxEnt algorithm exactly describes only moments up to
the second order, while all higher moments are obtained
approximately, causing an iterative scheme of the algorithm
in the following way:
ðl1Þ
mi mi
¼
k
X
Fup2 ðlÞ
dij mj
;
i ¼ 0; . . . ; k ; l ¼ 1; . . . ;
ð12Þ
j¼1
Zxmax
dij Fup2 j ð xÞ þ "i ð xÞ ;
xmin
The final solution is the pdf, which can be written in the
analytical form
f ð xÞ ¼ exp 1 0 k
X
j¼0
x j f ð xÞ d x
!
k
X
xi ð xÞ ¼
j¼0
j¼0
xmin
moments and different sensitivities of monomials x j in the
exponent of the solution (9). Therefore the existence of
many local minima decreases the efficiency of the classic
damped line search Newton method, especially due to the
choice of the initial guess, and the applicability of the
MaxEnt principle is reduced [e.g., Abramov, 2007, 2009].
The common way to overcome these difficulties is usage
of the orthogonal polynomials Qj(x) instead of x j such as
shifted Chebyshev [Bandyopadhyay et al., 2005], Lagrangian
[Turek, 1988], or generalized orthogonal multidimensional
polynomials [Abramov, 2007, 2009]. In this paper we use
the recently developed Fup MaxEnt algorithm [Gotovac and
Gotovac, 2009] based on Fup basis functions with compact
support [Gotovac et al., 2009a], as well as a relationship
between polynomials and these basis functions as given
by
Zxmax
j¼0
xmin
Zxmax
¼
i
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GOTOVAC ET AL.: CHARACTERIZATION OF THE TRAVEL TIME PDF
d x ¼ mi ; i ¼ 0; . . . ; k: ð10Þ
j¼1
Note that the above nonlinear system is not trivial due to a
high and unbalanced nonlinearity introduced by higher
Fup2(l)
where Dm(l)
are residual and Fup moments,
i and mi
respectively, and l is an iteration step. An algorithm starts
with an initial pdf guess (l = 0). In each iteration step,
residual moments are first calculated from the previous
iteration or initial conditions, and then Fup moments are
obtained from the system (12). Finally, the MaxEnt nonlinear system (10) is solved with respect to Fup2 moments
using an improved iterative scaling, an unconditionally
stable iterative procedure irrespective to the initial guess
that solves only one moment equation in each nonlinear
step and finds a correction of the corresponding Lagrangian
multiplier [Bandyopadhyay et al., 2005]. Nonlinear solver
is now more efficient because each Fup2 basis function in
system (10) changes only few moment equations which
belong to neighboring basis functions due to the existence
of compact support of Fup basis functions. In that way, all
basis functions and consequently multipliers are similar;
there is no difference as in a case of monomials. Abramov
[2007] discussed other optimization algorithms for solving the system (10), such as Newton or gradient methods.
The procedure is repeated until convergence is achieved.
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The optimal MaxEnt Fup approximation of the pdf has the
form
f * ð xÞ ¼ exp 1 k
X
j Fup2 j ð xÞ ;
ð13Þ
where moments are satisfied exactly according to the
relations (11) and (12):
mi ¼
x f * ð xÞ d x ¼
i
m
X
Z1
dij
j¼0
0
Z1
þ
"i ð xÞ f * ð xÞ d x ;
tistical properties of the Fup basis functions for step 6
[Gotovac et al., 2009a].
!
j¼0
Z1
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Fup2j ð xÞ f * ð xÞ d x
0
i ¼ 0; . . . ; m:
ð14Þ
0
A more complete description of the Fup MaxEnt algorithm
is given by Gotovac and Gotovac [2009].
2.3. Monte Carlo Methodology
[15] Our recently presented simulation methodology, referred to as the adaptive Fup Monte Carlo method (AFMCM)
[Gotovac et al., 2009a], supports the Eulerian‐Lagrangian
formulation, which separates the flow from the transport
problem and consists of the following common steps [Rubin,
2003]: (1) generation of as high a number as possible of log
conductivity realizations with predefined correlation structure, (2) numerical approximation of the log conductivity
field, (3) numerical solution of the flow equation with prescribed boundary conditions to produce head and velocity
approximations, (4) evaluation of the displacement position
and travel time for a large number of the particles, (5) repetition of steps 2–4 for all realizations, and (6) statistical
evaluation of flow and transport variables such as head,
velocity, travel time, transverse displacement, solute flux, or
concentration (including their cross moments and pdf’s).
[16] The AFMCM methodology is based on Fup basis
functions with compact support (related to the other localized basis functions such as splines or wavelets) and the Fup
collocation transform (FCT), which is closely related to the
discrete Fourier transform. It can simply represent, in a
multiresolution way, any signal, function, or set of data
using only a few Fup basis functions and resolution levels
on nearly optimal adaptive collocation grids that are capable
of resolving all spatial and/or temporal scales and frequencies. Fup basis functions and the FCT are presented in detail
by Gotovac et al. [2007]. Other improved MC methodology
aspects are (1) the Fup regularized transform (FRT) for data
or function (e.g., log conductivity) approximations in the
same multiresolution way as FCT, but computationally more
efficient, (2) adaptive Fup collocation method (AFCM) for
approximation of the flow differential equation, (3) particle
tracking algorithm based on the Runge‐Kutta‐Verner explicit
time integration scheme and FRT, and (4) Monte Carlo
statistics represented by Fup basis functions. All aforementioned Monte Carlo methodology components are presented by Gotovac et al. [2009a].
[17] Finally, AFMCM uses a random field generator
HYDRO_GEN [Bellin and Rubin, 1996] for step 1, FCT or
FRT for log conductivity approximation (step 2), AFCM for
the differential flow equation (step 3), a particle tracking
algorithm for transport approximations for step 4, and sta-
3. Simulation Setup
[18] For illustrating the application of the MaxEnt principle for the travel time pdf characterization, we consider a
two‐dimensional steady state and “uniform‐in‐the‐average”
flow field with a basic configuration as illustrated in
Figure 1, imposing the following flow boundary conditions:
left and right boundaries are prescribed a constant head,
while the top and bottom are no‐flow boundaries. Moreover,
we use “classic” multiGaussian lnK heterogeneity fields,
which are completely defined by the first two statistical
moments and three basic parameters: mean, variance, and
integral scale. This flow configuration as related to a multiGussian lnK field has been extensively studied in the literature [e.g., Bellin et al., 1992; Cvetkovic et al., 1996;
Salandin and Fiorotto, 1998; Hassan et al., 1998; Janković
et al., 2003, 2006; Fiori et al., 2006; De Dreuzy et al., 2007;
Gotovac et al., 2009b].
[19] Transport simulations are performed in the inner
computational domain to avoid nonstationary influence of
the flow boundary conditions (Figure 1). The injection tracer
mass is divided into a given number of particles that carry an
equal fraction of total mass. Particles are injected along the
source line and followed downstream such that travel time
and transverse displacement are monitored at arbitrary
control planes denoted by x. Two different injection modes
are considered: uniform resident and uniform in flux [Kreft
and Zuber, 1978; Demmy et al., 1999]. For brevity, we use
the terms “resident” and “in‐flux” injection mode. For both
modes, “uniform” refers to the homogeneous mass density in
the source. “Resident” refers to the volume of resident fluid
into which the solute is introduced, while “in‐flux” refers to
the influent water that carries the solute into the flow domain.
Particles are separated by equal distance within the source
line for resident and by a distance inversely proportional to
the specified flow rate between them for the in flux mode
[Demmy et al., 1999]. In this study, inert tracer particles are
injected instantaneously according to both injection modes,
allowing for the observations of their similarities and differences in the sense of travel time statistics [Cvetkovic et al.,
1996; Gotovac et al., 2009b].
[20] Table 1 presents all input data needed for Monte
Carlo simulations. The experimental setup presented here is
based on the convergence and accuracy analysis of Gotovac
et al. [2009a]. Figure 1 shows a 2‐D computational domain
for steady state and unidirectional flow simulations of
64IY × 32IY (IY is the integral scale). The random field
generator HYDRO_GEN [Bellin and Rubin, 1996] generates lnK fields for four discrete values of lnK variance: 1,
4, 6, and 8; for simplicity, the porosity is assumed uniform.
Gotovac et al. [2009a] defined discretization or resolution
for lnK and head field for a domain 64IY × 32IY to get
accurate velocity solutions for the particle tracking algorithm
(Table 1).
[21] Gotovac et al. [2009b] showed particular analysis
that finds an inner computational domain implying flow
criterion that each point of the inner domain must have a
constant Eulerian velocity variance and transport criterion
that injected particles do not fluctuate outside the inner
domain. All simulations use up to NP = 4000 particles, nMC =
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Table 1. All Input Data for Monte Carlo Simulations
Flow Domain, 64IY × 32IY
s2Y
nY/IY
nh/IY
Simulation 1
Simulation 2
Simulation 3
Simulation 4
1
4
16
4
8
32
6
8
32
8
8
32
40IY × 26IY
40IY × 20IY
40IY × 18IY
40IY × 16IY
1000
500
12
in‐flux,
resident
4000
500
12
in‐flux,
resident
4000
500
12
in‐flux,
resident
4000
500
12
in‐flux,
resident
Inner Domain
Np
nMC
y0/IY
Injection
mode
500 Monte Carlo realizations, source area (or line; y0 = 12IY)
and relative accuracy of 0.1% for calculating t in each realization in order to minimize statistical fluctuations [Gotovac
et al., 2009b, Table 1]. A detailed description of Monte
Carlo statistics using AFMCM is presented by Gotovac et al.
[2009a].
4. Travel Time Moments
[22] The dimensionless mean travel time is presented in
Figure 2. It is closely reproduced with t A = x′ = x/IY for in‐
flux injection mode and all considered s2Y values, following
results of Demmy et al. [1999] and Gotovac et al. [2009b].
The second‐order prediction t A = x′ − s2Y((3/x′3)(exp(−x′) −
1) + (3/x′2) exp(−x′) + 3/2x′ − 1) by Guadagnini et al. [2003]
is quite accurate for low and mild heterogeneity (s2Y < 3) and
for the resident injection mode. Initial nonlinearity is caused
by an injection of tracer particles into the mainly low‐velocity
zones, which therefore produces a larger mean travel time for
the resident mode. However, after 5–15IY, all curves become
linear with the nearly same slopes [Cvetkovic et al., 1996].
[23] The dimensionless travel time variance is shown as a
function of distance in Figure 3, where a comparison is
made with analytical solutions [Shapiro and Cvetkovic,
1988; Sanchez‐Villa and Guadagnini, 2005]. The simulated
variance is a nonlinear function of the distance from the
source only for the first 5–15IY, after which it attains a near‐
linear dependence for both modes. This behavior is explained
in detail by Gotovac et al. [2009b] for the in‐flux injection
mode with respect to the slowness correlation (covariance)
function (equation (3)). The slowness correlation length and
integral scale are relatively small, being approximately equal
to the integral scale of log conductivity. Because of a rapid
decrease of the correlation function close to the origin, the
integration of equation (3) yields a near‐linear travel time
variance only after a few IY. After 30IY, the slowness correlation reaches zero for all considered values of s2Y. According
to equation (3), the travel time variance asymptotically
reaches a linear form after about 60IY. Because of a
decrease in the slowness correlation with increasing s2Y, the
nonlinear features of s2t significantly diminish with distance as s2Y increases.
[24] The resident injection mode changes the form of
variance from the familiar concave form predicted by the
first‐order theory to a convex form in the case of high
heterogeneity (s2Y > 3), which is consistent with results of
Cvetkovic et al. [1996]. In this case, because of tracer injec-
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tions mostly in slow streamlines, slowness, as well as
Lagrangian velocity, are nonstationary for the first 5–15IY,
until the particles reach the nearly asymptotic Lagrangian
velocity. This velocity is the same as the flux‐averaged
Eulerian velocity imposed by the in‐flux injection mode
(see discussion by Le Borgne et al. [2007] and Gotovac et al.
[2009b]). Comparisons with analytical solutions indicate,
consistent with earlier studies, that up to s2Y = 1, the first‐order
theory reproduces simulated values reasonably well, although
some deviations are visible even for s2Y = 0.25 [Gotovac et al.,
2009b]. With increasing s2Y, the deviations are significantly
larger, especially for high heterogeneity and resident injection mode.
[25] Figures 4a–4d show the third and fourth travel time
moments for both modes and for s2Y = 1 and 8. For low and
mild heterogeneity (e.g., s2Y = 1, Figures 4a–4b), higher
travel time moments for both modes maintain nonlinear
behaviors along the first 10–20 integral scales, but after that
they exhibit a near‐linear dependence as in the case of the
travel time variance. At high values of heterogeneity (e.g.,
s2Y = 8, Figures 4c–4d), higher travel time moments are
practically linear for all control planes and for both modes
after approximately 5–10 integral scales. Small deviations
are present only due to statistical fluctuations, i.e., a finite
number of Monte Carlo realizations. Differences between
modes are similar as observed in the variance case, but
initial nonlinearity increases for the resident injection mode,
higher values of s2Y, and higher travel time moments. Third
and fourth moments are completely defined by three‐ and
four‐point slowness statistics, respectively. Furthermore,
linearity of the travel time moments enables the possibility
of extending present simulations to the infinite domain, or at
least to control planes at large distances where advective
transport may converge to the Fickian regime; this possibility will be explored in sections 5 and 6.
Figure 2. The dimensionless travel time mean for in‐flux
and resident injection mode and for s2Y = 1, 4, 6, and 8.
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Figure 3. Dimensionless travel time variance of adaptive Fup Monte Carlo method (AFMCM) and
analytic solutions for both injection modes and (a) s2Y = 1 and 4 and (b) s2Y = 6 and 8.
[26] Influences of the higher travel time moments on the
travel time pdf are better illustrated by skewness and kurtosis coefficients for different control planes and for s2Y
(shown in Tables 2 and 3 for in‐flux and resident injection
modes, respectively). Skewness and kurtosis in the log‐
travel time domain are significantly smaller, which implies
that a travel time pdf should be considered in the log
domain. Table 2 indicates that for the in‐flux mode and
low and mild heterogeneity, log skewness decreases with
distance. After 40IY, log skewness is close to zero while
log kurtosis is close to three, indicating a consistency with
the lognormal distribution. For high heterogeneity, skewness and kurtosis take on large positive values, which imply
extremely long tailings (late arrivals) and sharp peaks. On
the other side, log skewness has small positive values indicating slightly skewed log pdf values with log kurtosis values
being relatively close to three. Within the first 40IY, log
skewness decreases very slowly and it is difficult to estimate
when and if the simulated pdf becomes a lognormal pdf.
[27] Table 3 shows similar behaviors of skewness and
kurtosis for the resident injection mode. Values are comparable for low and mild heterogeneity, implying that there
are no important differences between modes in these cases.
For high heterogeneity cases, the trend is similar, with lower
values for x/IY < 10 and higher values for x/IY > 10 as a
consequence of the initial nonlinearity and very long tailings.
Note that skewness and kurtosis are still of significant
magnitude within x/IY = 40 for s2Y = 6–8 and show very
clearly the impact of higher travel time moments.
5. Travel Time Distribution
5.1. Flux Injection Mode
[28] The travel time pdf’s are illustrated on a log‐log plot
(Figure 5) for the in‐flux injection mode, different control
planes, and for s2Y values of 1 and 8. Figure 5 presents the
Monte Carlo experimental AFMCM pdf [Gotovac et al.,
2009b] as well as its MaxEnt approximation pdf (according to the Fup MaxEnt algorithm in section 3) which uses
the travel time moments up to the sixth order.
[29] Generally, deviations from a symmetrical distribution
(e.g., lognormal) or from MaxEnt pdf with the first two
moments decrease with distance from the source, and
increase significantly with increasing s2Y. For low and mild
heterogeneity (e.g., s2Y = 1, Figures 5a and 5b), small
deviations from a symmetric distribution occur only within
the first 10–20 integral scales, while almost complete symmetry is attained after 40 integral scales. This implies that the
higher travel time moments only slightly influenced the pdf
close to the source area. This agrees with the findings of
Gotovac et al. [2009b] (see their Figure 2 and pdf results for
s2Y = 0.25) and provides further evidence that the first‐order
theory yields robust and efficient travel time statistics in
media with low heterogeneity (s2Y < 1) where mean and
variance completely describe advective transport. For mild
heterogeneity (1 < s2Y < 3), asymmetry of the travel time pdf
becomes more apparent, while the first‐order theory is only
partially adequate due to deviations in the variance. The mild
heterogeneity range presents a transition zone where the
higher travel time moments start to play a more important
role in defining the travel time pdf.
[30] For high heterogeneity (e.g., s2Y = 8, Figures 5c and
5d), the computed AFMCM pdf is increasingly asymmetric;
both early and late arrivals are shifted to later times with
respect to the lognormal pdf (MaxEnt with two moments).
Although the asymmetry in the pdf diminishes with increasing distance, it is still maintained over the entire considered
domain of 40IY for high heterogeneity. Generally, the main
influence is by the third moment, which quantifies the pdf
skewness. Moreover, Figure 6 demonstrates that the third
moment also plays a crucial role for accurately describing
the travel time peak. Fourth and other higher travel time
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GOTOVAC ET AL.: CHARACTERIZATION OF THE TRAVEL TIME PDF
Figure 4. Dimensionless third travel time moment for both injection modes and (a) s2Y = 1 and (c) s2Y = 8.
Dimensionless fourth travel time moment for both injection modes and (b) s2Y = 1 and (d) s2Y = 8.
Table 2. Skewness and Kurtosis of the Travel Time Probability
Density Function for In‐Flux Injection Mode, Different lnK
Variances, and Control Planes
s2Y
1
4
6
8
St
Kt
Slnt
Klnt
St
Kt
Slnt
Klnt
St
Kt
Slnt
Klnt
St
Kt
Slnt
Klnt
x/IY = 5
x/IY = 10
x/IY = 20
x/IY = 30
x/IY = 40
1.852
9.934
0.223
3.035
7.394
171.321
0.518
3.226
17.948
1094.63
0.379
3.266
34.063
4331.58
0.416
3.218
1.461
7.128
0.219
3.000
4.739
69.047
0.298
3.186
12.964
639.044
0.343
3.321
17.511
1149.30
0.415
3.226
1.112
5.344
0.157
3.032
3.288
33.544
0.224
3.140
9.378
343.880
0.348
3.432
10.873
387.492
0.417
3.375
0.891
4.572
0.079
3.029
2.497
19.407
0.216
3.107
6.872
186.733
0.356
3.464
8.991
247.548
0.412
3.404
0.739
4.022
0.047
3.046
2.185
14.451
0.208
3.154
6.555
184.696
0.331
3.501
7.795
188.47
0.412
3.458
Table 3. Skewness and Kurtosis of the Travel Time Probability
Density Function for Resident Injection Mode, Different lnK
Variances, and Control Planes
s2Y
1
4
6
8
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St
Kt
Slnt
Klnt
St
Kt
Slnt
Klnt
St
Kt
Slnt
Klnt
St
Kt
Slnt
Klnt
x/IY = 5
x/IY = 10
x/IY = 20
x/IY = 30
x/IY = 40
1.871
10.260
0.213
3.025
6.737
109.461
0.371
3.115
17.354
649.594
0.399
3.264
21.583
1093.08
0.463
3.356
1.309
6.033
0.201
3.020
5.161
71.021
0.343
3.160
15.183
527.608
0.463
3.451
18.957
885.820
0.472
3.485
1.029
4.889
0.159
3.012
3.768
42.073
0.270
3.204
12.333
376.557
0.499
3.577
16.462
714.602
0.491
3.712
0.881
4.447
0.089
3.009
3.035
28.310
0.263
3.198
10.351
285.087
0.489
3.631
14.646
598.740
0.492
3.782
0.748
4.084
0.055
3.026
2.522
20.955
0.234
3.188
9.096
229.902
0.478
3.684
13.399
522.786
0.479
3.858
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GOTOVAC ET AL.: CHARACTERIZATION OF THE TRAVEL TIME PDF
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Figure 5. Maximum entropy (MaxEnt) travel time probability density function (pdf) for s2Y = 1 using
the first four travel time moments, in‐flux mode, and two control planes: (a) x/IY = 10 and (b) x/IY =
40. MaxEnt travel time pdf for s2Y = 8 using the first six travel time moments, in‐flux mode, and two
control planes: (c) x/IY = 5 and (d) x/IY = 40.
moments only improve the MaxEnt pdf with respect to the
early arrivals, but the peak (Figure 6) and late arrivals remain
almost the same (Figures 5c and 5d). Because the MaxEnt pdf
defines a pdf with the highest degree of uncertainty among all
possible pdf’s which satisfy certain moment constraints,
fourth and higher moments describe only early arrivals which
are subject to the greatest uncertainty. Figures 5c and 5d
suggests that for a very high heterogeneity (s2Y = 8), the
complete convergence of the MaxEnt pdf to the actual
AFMCM pdf is quite slow and requires more than six
moments depending on the distance from the source.
5.2. Resident Injection Mode
[31] Figure 7 shows experimental and corresponding
MaxEnt pdf’s which use up to six moments in the case of
the resident injection mode. Figures 7a and 7b indicates that
the travel time pdf is similar for mild heterogeneity (s2Y = 1)
and for both modes. According to the moment similarities,
pdf for both modes can be completely described by the first
two moments. Simulations indicate that the travel time pdf
for a high heterogeneity case with s2Y = 4 is almost completely determined by the first three moments for both
injection modes. The travel time pdf for very high heterogeneity cases (s2Y = 8) and resident injection mode require
between three and four moments for both the peak and late
arrivals (Figures 7c and 7d). Late arrivals for the resident
injection mode depend on particles which have been injected in low‐conductivity zones and hence exhibit significantly longer tailing than in the flux mode. Thus late arrivals
require an additional fourth moment in order to be accurately
described by the pdf for close control planes (x/IY < 20) as
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Figure 6. Central part (around peak) of the MaxEnt travel
time pdf for s2Y = 8 using the first six travel time moments at
x/IY = 5 and in‐flux mode.
obtained in Figure 7c. However, for more distant control
planes (x/IY = 40, Figure 7d), three moments accurately
describe late arrivals. Both modes deviate markedly from
the lognormal distribution for high heterogeneity. Finally,
early arrivals are subjected to the greatest uncertainty as in the
flux mode, which demonstrates similar behavior.
6. Discussion
6.1. Methodological Issues
[32] Advective transport in heterogeneous porous media
is completely characterized by transverse displacement and
travel time statistics [Dagan et al., 1992]. For a multi‐
Gaussian heterogeneity field, the transverse displacement
becomes nearly normal after only x/IY = 20, even for high
heterogeneity; however, the travel time shows more complex
behavior depending on the injection mode and heterogeneity
level [Gotovac et al., 2009b]. MaxEnt approximation of the
experimental AFMCM pdf relates its properties with statistical moments. This study enables the travel time pdf to be
represented by only a few travel time moments. In particular,
higher travel time moments completely characterize the peak,
early, and late arrivals.
[33] Typically, the stochastic methods such as the first‐
order perturbation theory [e.g., Dagan, 1989], spectral
methods [e.g., Gelhar, 1993], or moment methods [e.g.,
Winter et al., 2003] consider only the first two moments,
implying normality. Other more powerful, but computationally demanding methods such as Monte Carlo [Michalak
and Kitanidis, 2000; Janković et al., 2003; De Dreuzy et
al., 2007] or the probabilistic collocation method [Li and
Zhang., 2007] yield a complete pdf and all higher moments,
but do not relate pdf characteristics to their moments. The
approach presented here bridges the gap between methodologies which are either focused on only the travel time
moments and those focused entirely on the pdf.
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6.2. Travel Time Moments
[34] One of the main findings in the present study is that
the linearity of all travel time moments is maintained after a
given distance from the source or injection control plane.
The main reason for initial nonlinearity is the influence of
the injection mode where the resident injection mode involves strong initial nonlinearity for high heterogeneity
within the first 5IY due to injection of tracer particles in the
slow streamlines. The initial nonlinearity is closely related
to the slowness statistics as shown by the relationship between the travel time variance and slowness correlation in the
case of the in flux mode [Gotovac et al., 2009b]. Equation (3)
shows that after the double slowness integral scale, the travel
time variance reaches an asymptotic linear shape. Figure 4
confirms this behavior for higher moments and for both
injection modes. Le Borgne et al. [2007] analyzed the conditional Lagrangian velocity distribution related to the initial
velocity in the source. They showed that after approximately
10IY, the conditional velocity correlation is very small, while
after 100IY, particles completely lose their memory about
initial velocity and reach the asymptotic unconditional
Lagrangian distribution which is relevant for the travel time
statistics.
[35] For the in‐flux injection mode, slowness statistics are
stationary because the Lagrangian velocity distribution in
the source and in all other control planes is the same as the
asymptotic distribution, i.e., flux‐averaged Eulerian velocity
distribution imposed by in flux mode in the source [Gotovac
et al., 2009b]. Moreover, the slowness correlation decreases
with increasing s2Y, which implies that initial nonlinearity is
longer for lower heterogeneity values. For the resident
injection mode, the slowness statistics are nonstationary
within the first 10IY until the Lagrangian velocity reaches
the asymptotic distribution. After the first 10IY, the same
slope of all higher travel time moments in both modes
confirms the similarity of the slowness and Lagrangian
velocity statistics. Finally, we can conclude that initial nonlinearity is mainly defined by slowness nonstationarity for the
resident injection mode.
6.3. Asymptotic Behavior
[36] The presented approach enables the use of the computationally demanding AFMCM only in a relatively small
domain (x/IY 20) until all travel time moments become
linear. Moreover, because of the linearity, it is possible to
extrapolate travel time moments outside of the computational domain using the MaxEnt principle and thus obtain a
travel time pdf only from its moments. This leads to the
following MaxEnt formulation relating known noncentral
travel time moments (directly related to the presented central
moments in Figures 2–4) in the real t domain to the unknown
MaxEnt travel time pdf in the T = lnt domain:
Z1
exp ði T Þ exp 1 1
i ¼ 0; . . . ; k:
k
X
!
j
d T ¼ mi x=IY ; 2Y ;
j exp T
j¼0
ð15Þ
System (15) presents the maximum entropy problem over
exponential moments in the T domain that is extremely
difficult numerical task. However, it is worthwhile to note
that (15) also presents an exponential transform in which
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GOTOVAC ET AL.: CHARACTERIZATION OF THE TRAVEL TIME PDF
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Figure 7. MaxEnt travel time pdf for s2Y = 1 using the first four travel time moments, resident mode, and
two control planes: (a) x/IY = 10 and (b) x/IY = 40. MaxEnt travel time pdf for s2Y = 8 using the first six
travel time moments, resident mode, and two control planes: (c) x/IY = 5 and (d) x/IY = 40.
discrete real t moments can be related to the continuous
moment generating function, i.e., mts = E(s) [Tung et al.,
2006]. Hence the MaxEnt travel time pdf in the T domain
can be easily obtained by described Fup MaxEnt algorithm
[Gotovac and Gotovac, 2009] with T moments using the
relation mTi = [di E(s)/dsi]s=0. Therefore linearity of travel
time moments enables the exact computation of the asymptotic behavior. For linear travel time variance and third‐order
central moment, skewness has the following asymptotic form:
S ¼
M3
a3 ðx=IY Þ
a3
1
aS
¼
¼ 3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ;
3=2
3
ðx=IY Þ
ðx=IY Þ
ða2 ðx=IY ÞÞ
a2
ð16Þ
where a2 and a3 are linear parameters (slopes) of the variance and third moment, respectively, and aS is a skewness
parameter. All of these parameters can be directly calculated
from Figures 2–4 for both injection modes. Using the simple
relation (16), it is possible to check values from Tables 2 and
3 within the first 40IY, but skewness can also be calculated
for control planes at larger distances. Skewness (16) asymptotically converges to zero and consequently kurtosis converges to three for x/IY ! ∞, but it requires a very large
distance for transforming an extremely skewed initial travel
time pdf to a symmetric pdf. For example, when s2Y = 4
(Table 2; in‐flux mode), the skewness parameter is aS =
15.81 and the distance required for the skewness to reduce to
0.5 is approximately 1000IY, but the distance needed for the
skewness to reduce to 0.1 (practically zero) is 25,000IY.
More drastically, for s2Y = 8 (Table 2; in‐flux mode), the
skewness parameter is aS = 49.21 and the distance needed
for skewness to reduce to 0.1 is 242,000IY. Because of the
significant influence of the initial nonlinearity, these values
are much larger for the resident injection mode. Consequently, the coefficient of variation (CV) is also proportional
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Table 4. First Arrivals for 10−4 of Total Injected Mass Predicted
by AFMCM and MaxEnt Probability Density Function for Both
Injection Modes, Six Travel Time Moments, and s2Y = 8
tu/IY
Mode
In‐flux
Resident
s2Y x/IY AFMCMa
k=2
k=3
k=4
k=5
k=6
8
8
8
8
8
8
0.0660
0.7158
2.3811
0.0764
0.8092
2.5710
0.1215
1.2428
3.8501
0.1571
1.3467
3.8559
0.1533
1.3823
4.0655
0.1933
1.5808
4.1825
0.1657
1.4632
4.1716
0.2049
1.6284
4.2245
0.1738
1.6442
4.5919
0.2230
1.8755
4.8336
5
20
40
5
20
40
0.1981
1.7839
5.2237
0.2609
1.9495
5.5491
a
Adaptive Fup Monte Carlo method.
pffiffiffiffiffiffiffiffiffi
to 1/ x=IY as x/IY ! ∞, which is consistent with Fickian
transport.
6.4. Travel Time PDF, Early, and Late Arrivals
[37] Tables 2 and 3 show that the travel time pdf transforms more quickly to a nearly lognormal pdf at low and
mild heterogeneity values. The high heterogeneity case with
s2Y = 4 still shows relatively regular behavior in the lnt
domain and skewness decreases with distance. However,
very high heterogeneity cases with s2Y values of 6 and 8 show
that skewness, and especially kurtosis, initially increases in
the lnt domain within the first 40IY. Because of the very slow
decrease in the t domain, it appears that higher‐order
moments play an important role over a long distance, even
in the lnt domain. Note that skewness and kurtosis satisfy
the general relationship of K S2 + 1, which holds for all
pdf’s. In particular, Tables 2 and 3 demonstrate that the
actual relationship is relatively far from the lower bound of
K = S2 + 1 due to influence of the sharp peak and long tails
in the travel time pdf.
[38] MaxEnt enables an analysis of three basic parts of the
travel time pdf: peak and early arrivals important to risk
assessment and late arrivals relevant for remediation. We
can conclude that the higher travel time moments are very
important for the complete characterization of the travel
time pdf, especially in highly heterogeneous porous media.
The number of moments depends mainly on the heterogeneity level and injection mode. Roughly speaking, mean and
variance completely describe the travel time pdf for s2Y < 3,
skewness is dominant for s2Y = 4, while kurtosis and fifth
moments are needed for s2Y values of 6 and 8. Generally, the
resident injection mode requires more moments due to initial nonlinearity. This is particularly true for late arrivals
which require the fourth moment for s2Y values of 6 and 8 for
closer control planes. For the peak and late arrivals, the most
important moment is the third moment, implying that high
heterogeneity mostly changes the skewness of the travel
time pdf.
[39] On the other hand, the largest uncertainty is related to
the first arrivals. The same conclusion was pointed out by
Rubin [2003] when considering the total mass that crossed
the control plane at time t, M(t). The coefficient of variation
of M is ((1 − Ft )/Ft )1/2, where Ft is the cumulative distribution function (CDF) of the travel time. Because Ft
approaches zero for early travel times, estimation of first
arrivals is related to the largest uncertainty. Moreover, the
MaxEnt pdf presents the pdf with largest uncertainty of all
possible pdf’s for given moment constraints. Early arrivals
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require more moments than the other parts of the travel time
pdf; they are also related to the largest uncertainty in the
sense of maximum entropy principle. The present study can
quantify uncertainty of the early arrivals with respect to
related travel time moments as shown in Table 4 for s2Y = 8,
for different control planes, and for both injection modes.
Early arrivals for 10−4 of total injected mass predicted by
MaxEnt pdf are always shorter than the early arrivals calculated by AFMCM, which is conservative from the risk
assessment point of view. Indeed, the actual error is small
relative to the mean travel time. For example, the difference
between first arrivals predicted by AFMCM and MaxEnt
with four moments relative to the corresponding mean travel
time is less than 3% for all cases and both modes in Table 4.
This means that four moments quite accurately describe early
arrivals and significantly reduce the estimation uncertainty.
7. Concluding Remarks
[40] In this paper we used the simulation methodology
AFMCM with its high accuracy presented by Gotovac et al.
[2009a], to study the travel time pdf and its higher moments
for advective transport in a multi‐Gaussian structure under a
mean uniform flow [Gotovac et al., 2009b]. We show the
complete characterization of the travel time pdf using two
injection modes: in‐flux and resident. Fup MaxEnt algorithm [Gotovac and Gotovac, 2009] is used for an accurate
representation of the experimental Monte Carlo pdf with
respect to only a few first travel time moments. The main
conclusions can be summarized as follows:
[41] 1. All travel time moments become linear after a
given distance from the source. Initial nonlinearity is caused
mainly by the resident injection mode.
[42] 2. The resident injection mode changes the form of the
variance and all higher travel time moments from the familiar
concave predicted by the first‐order theory to a convex form
in case of a high heterogeneity.
[43] 3. The number of moments needed for an accurate
description of the travel time pdf mainly depends on the
heterogeneity level. Mean and variance completely describe
the travel time pdf for s2Y < 3, skewness is dominant for
s2Y = 4, while kurtosis and fifth moment are required for s2Y
values of 6 and 8.
[44] 4. Peak and late arrivals are mainly described by
skewness. The MaxEnt pdf requires a fourth moment only
for the resident injection mode, close control planes and s2Y
values of 6 and 8.
[45] 5. The highest uncertainty is found for the early
arrivals because it requires a larger number of moments
than other parts of the travel time pdf. In particular, we
show that four moments describe quite accurately early
arrivals with time error relative to the mean travel time less
than 3% needed for crossing 10−4 of total injected mass
through the control plane.
[46] 6. The travel time pdf is well approximated by the
lognormal distribution up to 40IY, for an lnK variance less
than around 3 (weak to moderate heterogeneity); for larger
lnK variance, travel time pdf presumably becomes lognormal
at a distance larger than 40IY. Since travel time moments are
linear with the distance,
pthe
ffiffiffiffiffiffiffiffifficoefficient of variation (CV) is
also proportional to 1/ x=IY as x/IY ! 1, the travel time
pdf converges to a normal distribution in the limit. Higher‐
order travel time moments however may be important in
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many practical cases, in particular for characterizing early
arrivals.
[47] The presented analysis couples the AFMCM with the
MaxEnt principle and can be efficiently applied to more
general (non‐Gaussian) heterogeneity structures. Of particular interest can be analysis of the full travel time pdf in a non‐
Gaussian structure with same point pdf as a multi‐Gaussian
field, but with more correlated and connected high‐ and/or
low‐permeability zones [Fiori et al., 2006; Le Borgne et al.,
2008]. Zinn and Harvey [2003] showed that first arrivals are
approximately 10 times faster for such non‐Gaussian
structures with connected high‐permeability zones than in
the classical multi‐Gaussian field. It is also of interest to
analyze how pore‐scale dispersion (especially velocity‐
dependent) affects higher travel time moments depending
on the Peclet number; such analysis will be left for future
investigations.
[48] Furthermore, analysis of different detection scales
and modes [Selroos and Cvetkovic, 1992] also requires
attention because it is important for concentration and solute
flux statistics. Since a finite detection scale implies more
variability and uncertainty, presented analysis and higher
moments may be more relevant than implied by this study.
Andricevic [2008] showed, for instance, that the concentration pdf requires at least four moments in order to accurately describe a bimodal pdf and the influence of pore‐scale
dispersion, even for low‐heterogeneity values.
References
Abramov, R. (2007), An improved algorithm for the multidimensional
moment‐constrained maximum entropy problem, J. Comput. Phys.,
226, 621–644, doi:10.1016/j.jcp.2007.04.026.
Abramov, R. (2009), The multidimensional moment‐constrained maximum
entropy problem: A BFGS algorithm with constraint scaling, J. Comput.
Phys., 228, 96–108, doi:10.1016/j.jcp.2008.08.020.
Andricevic, R. (2008), Exposure concentration statistics in the subsurface
transport, Adv. Water Res., 31(8), 714–725.
Andricevic, R., and V. Cvetkovic (1996), Evaluation of risk from contaminants migrating by groundwater, Water Resour. Res., 32(3), 611–621.
Bandyopadhyay, K., A. Bhattacharya, P. Biswas, and D. Drabold (2005),
Maximum entropy and the problem of moments: A stable algorithm,
Phys. Rev. E, 71(5), doi:10.1103/PhysRevE.71.057701.
Bellin, A., and Y. Rubin (1996), HYDRO_GEN: A spatially distributed
random field generator for correlated properties, Stochastic Hydrol. Hydraul.,
10(4), 253–278, doi:10.1007/BF01581869.
Bellin, A., and Y. Rubin (2004), On the use of peak concentration arrival
times for the inference of hydrogeological parameters, Water Resour.
Res., 40, W07401, doi:10.1029/2003WR002179.
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,
doi:10.1029/92WR00578.
Bellin, A., Y. Rubin, and A. Rinaldo (1994), Eulerian‐Lagrangian approach
for modeling of flow and transport in heterogeneous geological formations, Water Resour. Res., 30(11), 2913–2924, doi:10.1029/94WR01489.
Berger, A. L., S. A. Della Pietra, and V. J. Della Pietra (1996), A maximum
entropy approach to natural language processing, Comp. Linguistics, 22(1),
39–71.
Berglund, S., and V. Cvetkovic (1995), Pump and treat remediation of heterogeneous aquifers: Effects of rate‐limited mass transfer, Ground Water,
33(4), 675–685, doi:10.1111/j.1745-6584.1995.tb00324.x.
Christakos, G. (2000), Modern Spatiotemporal Statistics, Oxford Univ.
Press, New York.
Cvetkovic, V., and G. Dagan (1994), Transport of kinetically sorbing solute
by steady random velocity in heterogeneous porous formations, J. Fluid
Mech., 265, 189–215, doi:10.1017/S0022112094000807.
Cvetkovic, V., and A. M. Shapiro (1990), Mass arrival of sorptive solute in
heterogeneous porous media, Water Resour. Res., 26, 2057–2067.
W05502
Cvetkovic, V., A. M. Shapiro, and G. Dagan (1992), A solute‐flux approach to transport in heterogeneous formations: 2. Uncertainty analysis,
Water Resour. Res., 28(5), 1377–1388, doi:10.1029/91WR03085.
Cvetkovic, V., H. Cheng, and X.‐H. Wen (1996), Analysis of linear effects
on tracer migration in heterogeneous aquifers using Lagrangian travel
time statistics, Water Resour. Res., 32(6), 1671–1680, doi:10.1029/
96WR00278.
Dagan, G. (1984), Solute transport in heterogeneous porous formations,
J. Fluid Mech., 145, 151–177, doi:10.1017/S0022112084002858.
Dagan, G. (1989), Flow and Transport in Porous Formations, Springer, Berlin.
Dagan, G., V. Cvetkovic, and A. M. Shapiro (1992), A solute‐flux approach
to transport in heterogeneous formations: 1. The general framework,
Water Resour. Res., 28(5), 1369–1376, doi:10.1029/91WR03086.
De Dreuzy, J. R., A. Beaudoin, and J. Erhel (2007), Asymptotic dispersion in
2‐D heterogeneous porous media determined by parallel numerical simulations, Water Resour. Res., 43, W10439, doi:10.1029/2006WR005394.
Demmy, G., S. Berglund, and W. Graham (1999), Injection mode implications
for solute transport in porous media: Analysis in a stochastic Lagrangian
framework, Water Resour. Res., 35(7), 1965–1974, doi:10.1029/
1999WR900027.
Ezzedine, S., and Y. Rubin (1996), A geostatistical approach to the conditional estimation of spatially distributed solute concentration and notes
on the use of tracer data in the inverse problem, Water Resour. Res.,
32(4), 853–861, doi:10.1029/95WR02285.
Fiori, A., I. Janković, and G. Dagan (2006), Modeling flow and transport in
highly heterogeneous three‐dimensional aquifers: Ergodicity, Gaussianity, and anomalous behavior: 2. Approximate semianalytical solution,
Water Resour. Res., 42, W06D13, doi:10.1029/2005WR004752.
Gelhar, L. W. (1993), Stochastic Subsurface Hydrology, Prentice Hall,
Englewood Cliffs, N. J.
Gotovac, H., and B. Gotovac (2009), Maximum entropy algorithm with
inexact entropy bound based on Fup basis functions with compact support,
J. Comput. Phys., 228, 9079–9091, doi:10.1016/j.jcp.2009.09.011.
Gotovac, H., R. Andricevic, and B. Gotovac (2007), Multi‐resolution adaptive modeling of groundwater flow and transport problems, Adv. Water
Res., 30, 1105–1126, doi:10.1016/j.advwatres.2006.10.007.
Gotovac, H., V. Cvetkovic, and R. Andricevic (2009a), Adaptive Fup multi‐
resolution approach to flow and advective transport in highly heterogeneous porous media: Methodology, accuracy and convergence, Adv.
Water Res., 32, 885–905, doi:10.1016/j.advwatres.2009.02.013.
Gotovac, H., V. Cvetkovic, and R. Andricevic (2009b), Flow and travel time
statistics in highly heterogeneous porous media, Water Resour. Res., 45,
W07402, doi:10.1029/2008WR007168.
Guadagnini, A., X. Sanchez‐Villa, M. Riva, and M. De Simoni (2003), Mean
travel time of conservative solutes in randomly unbounded domains under
mean uniform flow, Water Resour. Res., 30(3), 1050, doi:10.1029/
2002WR001811.
Hassan, A. E., J. H. Cushman, and J. W. Delleur (1998), A Monte Carlo
assessment of Eulerian flow and transport perturbation models, Water
Resour. Res., 34(5), 1143–1163, doi:10.1029/98WR00011.
Janković, I., A. Fiori, and G. Dagan (2003), Flow and transport in highly
heterogeneous formations: 3. Numerical simulation and comparison with
theoretical results, Water Resour. Res., 9(9), 1270, doi:10.1029/
2002WR001721.
Janković, I., A. Fiori, and G. Dagan (2006), Modeling flow and transport in
highly heterogeneous three‐dimensional aquifers: Ergodicity, Gaussianity, and anomalous behavior: 1. Conceptual issues and numerical simulations, Water Resour. Res., 42, W06D12, doi:10.1029/2005WR004734.
Jaynes, E. T. (1957), Information theory and statistical mechanics, Phys.
Rev., 106, 620–630, doi:10.1103/PhysRev.106.620.
Kitanidis, P. K. (1994), The concept of the dilution index, Water Resour.
Res., 30(7), 2011–2026, doi:10.1029/94WR00762.
Kolovos, A., G. Christakos, M. L. Serre, and C. T. Miller (2002), Computational Bayesian maximum entropy solution of a stochastic advection‐
reaction equation in the light of site‐specific information, Water Resour.
Res., 38(12), 1318, doi:10.1029/2001WR000743.
Kreft, A., and A. Zuber (1978), On the physical meaning of the dispersion
equation and its solution for different initial and boundary conditions,
Chem. Eng. Sci., 33, 1471–1480, doi:10.1016/0009-2509(78)85196-3.
Le Borgne, T., J.‐R. de Dreuzy, P. Davy, and O. Bour (2007), Characterization of the velocity field organization in heterogeneous media by conditional correlation, Water Resour. Res., 43, W02419, doi:10.1029/
2006WR004875.
Le Borgne, T., M. Dentz, and J. Carrera (2008), Lagrangian statistical model
for transport in highly heterogeneous velocity fields, Phys. Rev. Lett., 101,
090601, doi:10.1103/PhysRevLett.101.090601.
13 of 14
W05502
GOTOVAC ET AL.: CHARACTERIZATION OF THE TRAVEL TIME PDF
Li, H., and D. Zhang (2007), Probabilistic collocation method for flow in
porous media: Comparisons with other stochastic methods, Water Resour.
Res., 43, W09409, doi:10.1029/2006WR005673.
Maxwell, R., W. E. Kastenberg, and Y. Rubin (1999), A methodology to
integrate site characterization information into groundwater‐driven
health risk assessment, Water Resour. Res., 35(9), 2841–2855.
Michalak, A. M., and P. K. Kitanidis (2000), Numerical investigations of
mixing in physically heterogeneous porous media using the one‐ and
two‐particle covariance, in Computational Methods in Water Resources
XIII, vol. 1, Computational Methods for Subsurface Flow and Transport,
edited by L. R. Bentley et al., pp. 423–429, A. A. Balkema, Rotterdam,
Netherlands.
Rubin, Y. (2003), Applied Stochastic Hydrogeology, Oxford Univ. Press,
New York.
Salandin, P., and V. Fiorotto (1998), Solute transport in highly heterogeneous aquifers, Water Resour. Res., 34(5), 949–961, doi:10.1029/
98WR00219.
Sanchez‐Villa, X., and A. Guadagnini (2005), Travel time and trajectory
moments of conservative solutes in three dimensional heterogeneous
porous media under mean uniform flow, Adv. Water Resour., 28,
429–439, doi:10.1016/j.advwatres.2004.12.009.
Selroos, J. O., and V. Cvetkovic (1992), Modeling solute advection coupled with sorption kinetics in heterogeneous formations, Water Resour.
Res., 28(5), 1271–1278, doi:10.1029/92WR00011.
Serre, M. L., G. Christakos, H. Li, and C. T. Miller (2003), A BME solution of the inverse problem for saturated groundwater flow, Stochastic
Environ. Res. Risk Assess., 17(6), 354–369.
Shannon, C. E. (1948), The mathematical theory of communication, Bell
Syst. Tech. J., 27, 379–423.
Shapiro, A., and V. Cvetkovic (1988), Stochastic analysis of solute arrival
time in heterogeneous porous media, Water Resour. Res., 24(10), 1711–
1718, doi:10.1029/WR024i010p01711.
Taylor, G. I. (1921), Diffusion by continuous movements, Proc. London
Math. Soc., 20, 196–212, doi:10.1112/plms/s2-20.1.196.
Tung, Y.‐K., B.‐K. Yen, and C. S. Melching (2006), Hydrosystems Engineering Reliability Assessment and Risk Analysis, 495 pp., McGraw‐Hill,
New York.
W05502
Turek, I. (1988), A maximum‐entropy approach to the density of states
with the recursion method, J. Phys. C, 21, 3251–3260, doi:10.1088/
0022-3719/21/17/014.
Winter, C. L., D. M. Tartakovsky, and A. Guadagnini (2003), Moment differential equations for flow in highly heterogeneous porous media, Surv.
Geophys., 24(1), 81–106, doi:10.1023/A:1022277418570.
Woodbury, A., and Y. Rubin (2000), A full‐Bayesian approach to parameter inference from tracer travel time moments and investigation of scale
effects at the Cape Cod Experimental Site, Water Resour. Res., 36(1),
159–171, doi:10.1029/1999WR900273.
Woodbury, A., and T. J. Ulrych (1993), Minimum relative entropy: Forward probabilistic modeling, Water Resour. Res., 29(8), 2847–2860,
doi:10.1029/93WR00923.
Woodbury, A., and T. J. Ulrych (2000), A full‐Bayesian approach to the
groundwater inverse problem for steady state flow, Water Resour.
Res., 36(8), 2081–2093, doi:10.1029/2000WR900086.
Woodbury, A., F. Render, and T. Ulrych (1995), Practical probabilistic
ground‐water modeling, Ground Water, 33(4), 532–538, doi:10.1111/
j.1745-6584.1995.tb00307.x.
Zimmerman, D. A., et al. (1998), A comparison of seven geostatisticalally
based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow, Water Resour. Res., 34(6),
1373–1414, doi:10.1029/98WR00003.
Zinn, B., and C. F. Harvey (2003), When good statistical models of aquifer
heterogeneity go bad: A comparison of flow, dispersion, and mass transfer in connected and multivariate Gaussian hydraulic conductivity fields,
Water Resour. Res., 39(3), 1051, doi:10.1029/2001WR001146.
R. Andricevic, Department of Civil and Architectural Engineering,
University of Split, Matice hrvatske 15, 21000 Split, Croatia.
V. Cvetkovic and H. Gotovac, Department of Land and Water Resources
Engineering, KTH Royal Institute of Technology, Brinellvagen 32, SE‐
10044 Stockholm, Sweden. ([email protected])
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