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Advances in Water Resources 106 (2017) 80–94
Contents lists available at ScienceDirect
Advances in Water Resources
journal homepage: www.elsevier.com/locate/advwatres
Anomalous transport in disordered fracture networks: Spatial Markov
model for dispersion with variable injection modes
Peter K. Kang
a , b , ∗, Marco Dentz
c , Tanguy Le Borgne
d , Seunghak Lee
a , Ruben Juanes b
a Korea Institute of Science and Technology, Seoul 02792, Republic of Korea b Massachusetts Institute of Technology, 77 Massachusetts Ave, Building 1, Cambridge, Massachusetts 02139, USA c Institute of Environmental Assessment and Water Research (IDÆA), Spanish National Research Council (CSIC), 08034 Barcelona, Spain d Université de Rennes 1, CNRS, Geosciences Rennes, UMR 6118, Rennes, France
a r t i c l e i n f o
Article history:
Received 25 May 2016
Revised 24 March 2017
Accepted 31 March 2017
Available online 3 April 2017
Keywords:
Discrete fracture networks
Injection modes
Anomalous transport
Stochastic modeling
Lagrangian velocity
Time domain random walks
Continuous time random walks
Spatial Markov model
a b s t r a c t
We investigate tracer transport on random discrete fracture networks that are characterized by the statis-
tics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport
through fractured media can be anomalous and particle injection modes can have major impact on dis-
persion, the incorporation of injection modes into effective transport modeling has remained an open
issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—
the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial dis-
tribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this
evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This
stochastic approach allows for the systematic incorporation of the initial velocity distribution and quanti-
fies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial
Markov model is characterized by the initial velocity distribution, which is determined by the particle in-
jection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity
distribution, and the spatial velocity correlation length, which is related to the characteristic fracture
length. This effective model leads to a time-domain random walk for the evolution of particle positions
and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the
proposed model can successfully predict anomalous transport through discrete fracture networks with
different levels of heterogeneity and arbitrary tracer injection modes.
© 2017 Elsevier Ltd. All rights reserved.
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1. Introduction
Flow and transport in fractured geologic media control many
important natural and engineered processes, including nuclear
waste disposal, geologic carbon sequestration, groundwater con-
tamination, managed aquifer recharge, and geothermal production
in fractured geologic media (e.g., Bodvarsson et al., 1999; Lewicki
et al., 2007; Tang et al., 1981; Chrysikopoulos et al., 2009; Pruess,
2006 ). Two dominant approaches exist for simulating flow and
transport through fractured media: the equivalent porous medium
approach ( Neuman et al., 1987; Tsang et al., 1996 ) and the discrete
fracture network approach (DFN) ( Bernabe et al., 2016; Cacas et al.,
1990; de Dreuzy et al., 2012; Frampton and Cvetkovic, 2011; Hy-
man et al., 2015a; Juanes et al., 2002; Karimi-Fard et al., 2004; Ki-
raly, 1979; Makedonska et al., 2015; Martinez-Landa and Carrera,
2005; Moreno and Neretnieks, 1993; Nordqvist et al., 1992; Park
∗ Corresponding author.
E-mail address: [email protected] (P.K. Kang).
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http://dx.doi.org/10.1016/j.advwatres.2017.03.024
0309-1708/© 2017 Elsevier Ltd. All rights reserved.
t al., 2003; Schmid et al., 2013 ). The DFN approach explicitly re-
olves individual fractures whereas the equivalent porous medium
pproach represents the fractured medium as a single continuum
y deriving effective parameters to include the effect of the frac-
ures on the flow and transport. The latter, however, is hampered
y the fact that a representative elementary volume may not exist
or fractured media ( Bear, 1972; de Marsily, 1986 ). Dual-porosity
odels are in between these two approaches, and conceptualize
he fractured-porous medium as two overlapping continua, which
nteract via an exchange term ( Arbogast et al., 1990; Barenblatt
t al., 1960; Bibby, 1981; Feenstra et al., 1985; Gerke and van
enuchten, 1993; Kazemi et al., 1976; Maloszewski and Zuber,
985; Pruess, 1985; Warren and Root, 1963 ).
DFN modeling has advanced significantly in recent years with
he increase in computational power. Current DFN simulators can
ake into account multiple physical mechanisms occurring in com-
lex 3D fracture systems. Recent studies also have developed
ethods to explicitly model advection and diffusion through both
he discrete fractures and the permeable rock matrix ( Geiger et al.,
P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94 81
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010; Houseworth et al., 2013; Sebben and Werner, 2016; Will-
ann et al., 2013 ). In practice, however, their application must ac-
ount for the uncertainty in the subsurface characterization of frac-
ured media, which is still an considerable challenge ( Chen et al.,
006; Dorn et al., 2012; Kang et al., 2016b ). Thus, there is a con-
inued interest in the development of upscaled transport models
hat can be parameterized with a small number of model param-
ters. Ideally, these model parameters should have a clear physical
nterpretation and should be determined by means of field exper-
ments, with the expectation that the model can then be used for
redictive purposes ( Becker and Shapiro, 2003; Kang et al., 2015b ).
Developing an upscaled model for transport in fractured me-
ia is especially challenging due to the emergence of anomalous
non-Fickian) transport. While particle spreading is often described
sing a Fickian framework, anomalous transport—characterized by
cale-dependent spreading, early arrivals, long tails, and nonlinear
caling with time of the centered mean square displacement—has
een widely observed in porous and fractured media across multi-
le scales, from pore ( Bijeljic et al., 2011; Gjetvaj et al., 2015; Kang
t al., 2014; Scheven et al., 2005; Seymour et al., 2004 ) to sin-
le fracture ( Detwiler et al., 20 0 0; Drazer et al., 20 04; Kang et al.,
016a; Wang and Cardenas, 2014 ) to column ( Cortis and Berkowitz,
004; Hatano and Hatano, 1998 ) to field scale ( Becker and Shapiro,
0 0 0; Garabedian et al., 1991; Haggerty et al., 2001; Hyman et al.,
015b; Kang et al., 2015b; Le Borgne and Gouze, 2008; McKenna
t al., 2001 ). The ability to predict anomalous transport is essen-
ial because it leads to fundamentally different behavior compared
ith Fickian transport ( Bouchaud and Georges, 1990; Metzler and
lafter, 20 0 0; Shlesinger, 1974 ).
The continuous time random walk (CTRW) formalism ( Klafter
nd Silbey, 1980; Scher and Montroll, 1975 ) is a framework to
escribe anomalous transport through which models particle mo-
ion through a random walk in space and time characterized by
andom space and time increments, which accounts for variable
ass transfer rates due to spatial heterogeneity. It has been used
o model transport in heterogeneous porous and fractured me-
ia ( Berkowitz et al., 2006; Berkowitz and Scher, 1997; Dentz
t al., 2004; 2015; Geiger et al., 2010; Kang et al., 2011a; Wang
nd Cardenas, 2014 ) and allows incorporating information on flow
eterogeneity and medium geometry for large scale transport
odeling. Similarly, the time-domain random walk (TDRW) ap-
roach Benke and Painter (2003) ; Painter and Cvetkovic (2005) ;
ainter et al. (2008) models particle motion due to distributed
pace and time increments, which are derived from particle veloc-
ties and their correlations. The analysis of particle motion in het-
rogeneous flow fields demonstrate that Lagrangian particle veloc-
ties exhibit sustained correlation along their trajectory ( de Anna
t al., 2013; Benke and Painter, 2003; Cvetkovic et al., 1996;
atta et al., 2013; Gotovac et al., 2009; Kang et al., 2014; 2011b;
e Borgne et al., 2008; Meyer and Tchelepi, 2010 ). Volume con-
ervation induces correlation in the Eulerian velocity field because
uxes must satisfy the divergence-free constraint. This, in turn, in-
uces correlation in the Lagrangian velocity along a particle trajec-
ory. To take into account velocity correlation, Lagrangian models
ased on Markovian processes have been proposed ( de Anna et al.,
013; Benke and Painter, 2003; Kang et al., 2014; 2011b; 2015a;
015b; Le Borgne et al., 2008; Meyer and Tchelepi, 2010; Painter
nd Cvetkovic, 2005 ). Spatial Markov models are based on the
bservation that successive velocity transitions measured equidis-
antly along the mean flow direction exhibit Markovianity: a par-
icle’s velocity at the next step is fully determined by its current
elocity. The spatial Markov model, which accounts for velocity
orrelation by incorporating this one-step velocity correlation in-
ormation, has not yet been extended to disordered (unstructured)
FNs.
t
The mode of particle injection can have a major im-
act on transport through porous and fractured media ( Dagan,
016; Frampton and Cvetkovic, 2009; Hyman et al., 2015b;
ankovi c and Fiori, 2010; Kreft and Zuber, 1978; Le Borgne et al.,
010; 2007; Sposito and Dagan, 1994 ). Two generic injection
odes are uniform (resident) injection and flux-weighted injection
ith distinctive physical meanings as discussed in Frampton and
vetkovic (2009) . The work by Sposito and Dagan (1994) is one
f the earliest studies of the impact of different particle injec-
ion modes on the time evolution of a solute plume spatial mo-
ents. The significance of injection modes on particle transport
hrough discrete fracture networks has been studied for frac-
ured media ( Frampton and Cvetkovic, 2009; Hyman et al., 2015b ).
agan (2016) recently clarified the theoretical relation between in-
ection modes and plume mean velocity. Despite recent advances
egarding the significance of particle injection modes, the incorpo-
ation of injection methods into effective transport modeling is still
n open issue ( Frampton and Cvetkovic, 2009 ). The fundamental
hallenge is that the Lagrangian velocity distribution experienced
y tracer particles evolves with time from its initial distribution
hich is dictated by the injection mode to a stationary velocity
istribution ( Cvetkovic et al., 1996; Dentz et al., 2016; Frampton
nd Cvetkovic, 2009; Gotovac et al., 2009 ). In this paper, we ad-
ress these fundamental questions, in the context of anomalous
ransport through disordered DFNs.
The paper proceeds as follows. In the next section, we present
he studied random discrete fracture networks, the flow and trans-
ort equations and details of the different particle injection rules.
n Section 3 , we investigate the emergence of anomalous transport
y direct Monte Carlo simulations of flow and particle transport. In
ection 4 , we analyze Eulerian and Lagrangian velocity statistics to
ain insight into the effective particle dynamics and elucidate the
ey mechanisms that lead to the observed anomalous behavior. In
ection 5 , we develop a spatial Markov model that is character-
zed by the initial velocity distribution, probability density func-
ion (PDF) of Lagrangian velocities and their transition PDF, which
re derived from the Monte Carlo simulations. The proposed model
s in excellent agreement with direct Monte Carlo simulations. We
hen present a parsimonious spatial Markov model that quantifies
elocity correlation with a single parameter. The predictive capa-
ilities of this simplified model are demonstrated by comparison to
he direct Monte Carlo simulations with arbitrary injection modes.
n Section 6 , we summarize the main findings and conclusions.
. Flow and transport in discrete random fracture networks
.1. Random fracture networks
We numerically generate random DFNs in two-dimensional
ectangular regions, and solve for flow and tracer transport within
hese networks. The fracture networks are composed of linear frac-
ures embedded in an impermeable rock matrix. The idealized
D DFN realizations are generated by superimposing two differ-
nt sets of fractures, which leads to realistic discrete fracture net-
orks ( Geiger et al., 2010; Long et al., 1982 ). Fracture locations, ori-
ntations, lengths and hydraulic conductivities are generated from
redefined distributions, which are assumed to be statistically in-
ependent: (1) Fracture midpoints are selected randomly over the
omain size of L x × L y where L x = 2 and L y = 1 ; (2) Fracture orien-
ations for two fracture sets are selected randomly from Gaussian
istributions, with means and standard deviation of 0 ° ± 5 ° for the
rst set, and 90 ° ± 5 ° for the second set; (3) Fracture lengths are
hosen randomly from exponential distributions with mean L x /10
or the horizontal fracture set and mean L y /10 for the vertical frac-
ure set; (4) Fracture conductivities are assigned randomly from a
82 P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94
Fig. 1. (a) Example of a two-dimensional DFN studied here, with 20 0 0 fractures
(10 0 0 fractures for each fracture set). (b) Subsection of a spatially uncorrelated con-
ductivity field between 0 ≤ x ≤ 1 and 0.5 ≤ y ≤ 1. Conductivity values are assigned
from a lognormal distribution with σln K = 1 . Link width is proportional to the con-
ductivity value; only connected links are shown.
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predefined log-normal distribution. An example of a random dis-
crete fracture network with 20 0 0 fractures is shown in Fig. 1 .
The position vector of node i in the fracture network is denoted
by x i . The link length between nodes i and j is denoted by l ij . The
network is characterized by the distribution of link lengths p l ( l )
and hydraulic conductivity K . The PDF of link lengths here is ex-
ponential
p l (l) =
exp (−l/ l )
l . (1)
Note that the link length and orientation are independent. The
characteristic fracture link length is obtained by taking the average
of a link length over all the realizations, which gives l ≈ L x / 200 .
A realization of the random discrete fracture network is generated
by assigning independent and identically distributed random hy-
draulic conductivities K ij > 0 to each link between nodes i and j .
Therefore, the K ij values in different links are uncorrelated. The set
of all realizations of the spatially random network generated in this
way forms a statistical ensemble that is stationary and ergodic. We
assign a lognormal distribution of K values, and study the impact
of conductivity heterogeneity on transport by varying the variance
of ln ( K ). We study log-normal conductivity distributions with four
different variances: σln K = 1 , 2 , 3 , 5 . The use of this particular dis-
tribution is motivated by the fact that conductivity values in many
natural media can be described by a lognormal law ( Bianchi and
Snow, 1969; Sanchez-Vila et al., 2006 ).
2.2. Flow field
Steady state flow through the network is modeled by Darcy’s
law ( Bear, 1972 ) for the fluid flux u ij between nodes i and j , u i j =
K i j (� j − �i ) /l i j , where �i and �j are the hydraulic heads at
odes i and j . Imposing flux conservation at each node i , ∑
j u i j = 0
the summation is over nearest-neighbor nodes), leads to a linear
ystem of equations, which is solved for the hydraulic heads at the
odes. The fluid flux through a link from node i to j is termed in-
oming for node i if u ij < 0, and outgoing if u ij > 0. We denote by
ij the unit vector in the direction of the link connecting nodes i
nd j .
We study a uniform flow setting characterized by constant
ean flow in the positive x -direction parallel to the principal set
f factures. No-flow conditions are imposed at the top and bot-
om boundaries of the domain, and fixed hydraulic head at the
eft ( � = 1 ) and right ( � = 0 ) boundaries. The overbar in the fol-
owing denotes the ensemble average over all network realizations.
he one-point statistics of the flow field are characterized by the
ulerian velocity PDF, which is obtained by spatial and ensemble
ampling of the velocity magnitudes in the network
p e (u ) =
∑
i> j l i j δ(u − u i j )
N � l . (2)
here N � is the number of links in the network. Link length and
ow velocities here are independent. Thus, the Eulerian velocity
DF is given by
p e (u ) =
1
N �
∑
i> j
δ(u − u i j ) . (3)
ven though the underlying conductivity field is uncorrelated, the
ass conservation constraint together with heterogeneity leads to
he formation of preferential flow paths with increasing network
eterogeneity ( Bernabe and Bruderer, 1998; Kang et al., 2015a ).
his is illustrated in Fig. 2 a and b, which show maps of the relative
elocity magnitude for high velocities in networks with log- K vari-
nces of 1 and 5. As shown in Fig. 2 c and d, for low heterogeneity
ost small flux values occur along links perpendicular to the mean
ow direction, whereas low flux values do not show directionality
or the high heterogeneity case. This indicates that fracture geome-
ry dominates small flux values for low heterogeneity and fracture
onductivity dominates small flux values for high heterogeneity. An
ncrease in conductivity heterogeneity leads to a broader Eulerian
elocity PDF, with significantly larger probability of having small
ux values as illustrated in Fig. 3 , which shows p e ( u ) for networks
f different heterogeneity strength.
.3. Transport
Once the fluxes at the links have been determined, we simulate
ransport of a passive tracer by particle tracking. Particles are in-
ected along a line at the inlet, x = 0 , with two different injection
ethods: (1) uniform injection, and (2) flux-weighted injection.
niform (resident) injection introduces particles uniformly through-
ut the left boundary; this means that an equal number of parti-
les is injected into each inlet node i 0 ,
i 0 =
N p ∑
i 0
, (4)
here N i 0 is the number of particles injected at node i 0 , N p is
he total number of injected particles. Flux-weighted injection in-
roduces particles proportional to the total incoming flux Q i 0 at the
njection location i 0
i 0 = N p
Q i 0 ∑
i 0 Q i 0
. (5)
niform injection simulates an initial distribution of tracer parti-
les extended uniformly over a region much larger than the char-
cteristic heterogeneity scale, and flux-weighted injection simu-
ates a constant concentration pulse where the injected mass is
P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94 83
Fig. 2. Normalized flow field ( | u i j | / u ) showing high and low flux zones for a log-normal conductivity distribution with two different heterogeneities. Link width is propor-
tional to the magnitude of the normalized flow. (a) σln K = 1 . Links with the flux value smaller than u / 5 are removed. (b) σln K = 5 . Links with the flux value smaller than
u / 5 is removed. Preferential flow paths emerge as conductivity heterogeneity increases. (c) σln K = 1 . Links with the flux value larger than u / 5 are removed. Most of low flux
values occur at the links perpendicular to the mean flow direction. (d) σln K = 5 . Links with the flux value larger than u / 5 are removed. Low flux values show less spatial
correlation than high flux values.
Fig. 3. Eulerian flux probability density functions for four different levels of con-
ductivity heterogeneity. Increase in conductivity heterogeneity significantly in-
creases the probability of small flux values.
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roportional to the local injection flux at an inlet boundary that
s much larger than the heterogeneity scale. For the uniform in-
ection, the initial velocity distribution is then equal to the distri-
ution of the Eulerian velocities. For the flux-weighted injection,
he initial velocity distribution is equal to the flux-weighted Eule-
ian distribution. In general the initial velocity distribution may be
rbitrary and depends on the conditions at the injection location.
ore detailed discussions can be found in Sections 4 and 5 .
Injected particles are advected with the flow velocity u ij be-
ween nodes. To focus on the impact of conductivity variability
n particle transport, we assume porosity to be constant. This is
reasonable assumption because the variability in porosity is sig-
ificantly smaller than the variability in conductivity ( Bear, 1972;
reeze and Cherry, 1977 ).
At the nodes, we apply a complete mixing rule ( Berkowitz et al.,
994; Park et al., 2001a; Stockman et al., 1997 ). Complete mix-
ng assumes that Péclet numbers at the nodes are small enough
hat particles are well mixed within the node. Thus, the link
hrough which the particle exits a node is chosen randomly with
ux-weighted probability. A different node-mixing rule, streamline
outing, assumes that Péclet numbers at nodes are large enough
hat particles essentially follow the streamlines and do not tran-
ition between streamlines. The complete mixing and streamline
outing rules are two end members. The local Péclet number and
he intersection geometry determine the strength of mixing at
odes, which is in general between these two end-members. The
mpact of the mixing rule on transverse spreading can be signifi-
ant for regular DFNs with low heterogeneity ( Kang et al., 2015a;
ark et al., 2001b ). However, its impact is much more limited for
andom DFNs ( Park et al., 2001a ). Since our interest in this study is
he longitudinal spreading in random DFNs, we focus on the case
f complete mixing. Thus, the particle transition probabilities p ij rom node i to node j are given by
p i j =
| u i j | ∑
k | u ik | , (6)
here the summation is over outgoing links only, and p i j = 0 for
ncoming links. Particle transitions are determined only by the out-
oing flux distribution.
84 P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94
Fig. 4. Particle distribution at t = 20 t l for a given realization after the instantaneous release of particles at the inlet, x = 0 . t l is the median transition time to travel L x /100.
(a) The low heterogeneity case ( σln K = 1 ) with flux-weighted injection. (b) The low heterogeneity case ( σln K = 1 ) with uniform injection. (c) The projected particle distribution
in the longitudinal direction for the low heterogeneity case ( σln K = 1 ). (d) The high heterogeneity case ( σln K = 5 ) with the flux-weighted injection. (e) The high heterogeneity
( σln K = 5 ) with the uniform injection. (f) The projected particle distribution in the longitudinal direction for the high heterogeneity case ( σln K = 5 ). For the high heterogeneity
case, the injection method has significant impact on particle spreading. The uniform injection method leads to more anomalous spreading.
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The particle pathways and times are obtained by the following
recursion relations
x n +1 = x n + � n e n , (7a)
n +1 = t n +
� n
u n , (7b)
where x n ≡ x i n is the particle position after n random walk steps,
� n ≡ l i n i n +1 the particle displacement and e n ≡ e i n i n +1
its orienta-
tion; the particle velocity at the n th step is denoted by u n ≡| u i n i n +1
| . The particle displacement, orientation and velocity deter-
mined by the transition probability p i n j from node i n to the neigh-
boring nodes j given by Eq. (6) . Eq. (7) describe coarse-grained
particle transport for a single realization of the spatial random
network. Particle velocities and thus transition times depend on
the particle position. The particle position at time t is x (t) = x i n t ,
where
n t = sup (n | t n ≤ t) (8)
denotes the number of steps needed to reach time t . We solve
transport in a single disorder realization by particle tracking based
on Eq. (7) with the two different injection rules (4) and (5) at the
inlet at x = 0 . The particle density in a single realization is
p(x , t) = 〈 δ(x − x n t ) 〉 , (9)
where the angular brackets denote the average over all injected
particles. As shown in Fig. 4 , both network heterogeneity and injec-
tion rule have significant impact on particle spreading. An increase
in network heterogeneity leads to an increase in longitudinal parti-
cle spreading, and the uniform injection rule significantly enhances
ongitudinal spreading compared to flux-weighted injection. The
mpact of network heterogeneity and injection method can be
learly seen from projected concentration profiles, f τ ( ω). Arbitrary
njection modes are also studied and discussed in Section 5.2 .
. Average solute spreading behavior
We first study the average solute spreading behavior for the
our different levels of conductivity heterogeneity and the two dif-
erent injection methods described above. We first illustrate the
ersistent effect of the particle injection method on particle trans-
ort, with the two different injection modes. To investigate the av-
rage spreading behavior, we average over all particles and net-
ork realizations. The average particle density is given by
(x , t) = 〈 δ(x − x n t ) 〉 , (10)
here the overbar denotes the ensemble average over all realiza-
ions. We run Monte Carlo particle tracking simulations for 100 re-
lizations for each combination of conductivity heterogeneity and
article injection rule. In each realization, we release 10 4 particles
t the inlet ( x = 0 ) with the two different injection methods.
.1. Breakthrough curves
The average particle spreading behavior is first studied with
he first passage time distribution (FPTD) or breakthrough curve
BTC) of particles at a control plane located at x = x c . The FPTD
s obtained by averaging over the individual particle arrival times
a (x c ) = inf (t n | | x n − x 0 | > x c ) as
f (τ, x c ) = 〈 δ[ τ − τa (x c )] 〉 . (11)
P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94 85
Fig. 5. (a) FPTDs for σln K = 1 , 2 , 3 , 5 with flux-weighted injection at x c = 200 l . Increase in conductivity heterogeneity leads to larger dispersion and stronger late-time tailing.
(b) FPTDs for σln K = 1 , 2 , 3 , 5 with uniform injection (black solid lines). Uniform injection leads to significantly larger dispersion and late-time tailing compared to the flux-
weighted injection (red dashed lines). FPTDs are normalized with the peak arrival time. (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
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Fig. 5 shows FPTDs at the outlet, f (τ, x c = 200 l ) , for different
onductivity heterogeneities and injection rules. Conductivity het-
rogeneity has a clear impact on the FPTD by enhancing longitudi-
al spreading. This is so because stronger conductivity heterogene-
ty leads to broader particle transition time distribution, which in
urn leads to enhanced longitudinal spreading. The injection rule
lso has a significant impact on FPTDs especially for high con-
uctivity heterogeneity. FPTDs between the two different injection
ules are similar for σln K = 1 , but uniform injection shows signif-
cantly stronger tailing for σln K = 2 , 3 , 5 ( Fig. 5 (b)). As conductivity
eterogeneity increases, the flux values at the inlet also becomes
roader. For flux-weighted injection, most of particles are injected
t the nodes with high flux values. However, for uniform injection,
articles are uniformly injected across the injection nodes and rel-
tively large number of particles are released at the nodes with
ow flux values. This leads to notable difference between the two
njection rules and the difference grows as the conductivity het-
rogeneity increases.
.2. Centered mean square displacement
We also study longitudinal spreading in terms of the cen-
ered mean square displacement (cMSD) of average particle den-
ity, P (x , t) . For the longitudinal direction ( x ), the cMSD is given
y σ 2 x (t) = 〈 [ x (t) − 〈 x (t) 〉 ] 2 〉 where 〈 · 〉 denotes the average over
ll particles for a given realization. In Fig. 6 , we show the time evo-
ution of the longitudinal cMSDs. The time axis is normalized with
he mean travel time along the characteristic fracture link length,
. For both injection methods, spreading shows a ballistic regime
∼ t 2 ) at early times, which then transitions to a preasymptotic
caling in an intermediate regime and finally to a final asymptotic
egime. The time evolutions of cMSDs for the two injection cases
re notably different as conductivity heterogeneity increases, while
he asymptotic late-time scalings are very similar.
The asymptotic power-law scaling can be understood in the
ramework of a continuous time random walk (CTRW) description
f dispersion. At large times the Lagrangian velocity distributions
re in their steady states and subsequent particle velocities are in-
ependent. Thus, at large times horizontal particle dispersion can
e described by the CTRW
n +1 = x n + � 0 , t n +1 = t n + τn , (12)
ith the transition time τn = � 0 / v n . The velocities v n are dis-
ributed according to p s ( v ) which is space Lagrangian velocity
DF, and � 0 is a distance along the streamline that is suffi-
iently large so that subsequent particle velocities may be con-
idered independent. Thus, the distribution of transit times τ n is
iven in terms of the space Lagrangian and Eulerian velocity PDFs
s Dentz et al. (2016)
(τ ) =
� 0
τ 2 p s (� 0 /τ ) =
� 0
τ 3 v p e (� 0 /τ ) , (13)
here v is the average Eulerian velocity, see also Section 4 . Specif-
cally, for the scaling p e ( v ) ∝ v α at small velocities, the transit time
DF scales as
(τ ) ∝ τ−1 −β, β = 2 + α. (14)
From Fig. 3 , we estimate for σln K = 5 that α ≈ −0 . 55 , which
orresponds to β = 1 . 45 . CTRW theory ( Berkowitz et al., 2006;
entz et al., 2004 ) predicts that the cMSD scales as t 3 −β, which
ere implies t 1.55 . This is consistent with the late-time scaling of
he cMSD shown in 6 for σln K = 5 .
The Monte Carlo simulations show that, in the intermediate
egime ( t/ t l approximately between 1 and 100), the longitudi-
al cMSD increases linearly with time for flux-weighted injection
Fig. 6 (a)). For uniform injection, cMSD increases faster than lin-
arly (i.e., superdiffusively) for intermediate to strong heterogene-
ty in the intermediate regime ( Fig. 6 (b)). The stronger heterogene-
ty led to the increase in the late-time temporal scaling for both
ux-weighted and uniform injection cases. The Monte Carlo sim-
lations also show that there is no noticeable difference between
he uniform injection and the flux-weighted injection for the low
eterogeneity case whereas the difference increases as heterogene-
ty increases ( Fig. 6 (b), inset).
In summary, both the increase in conductivity heterogeneity
nd the uniform injection method enhance longitudinal spread-
ng. For low heterogeneity, the two different injection rules do not
ffect particle spreading significantly. The difference, however, be-
omes significant as the conductivity heterogeneity increases. Both
he magnitude of the cMSD and the super-diffusive scaling behav-
or are notably different for the two different injection rules at
igh heterogeneity. We now analyze the Lagrangian particle statis-
ics to understand the underlying physical mechanisms that lead
o the observed anomalous particle spreading.
. Lagrangian velocity statistics
The classical CTRW approach Berkowitz et al. (2006) ;
erkowitz and Scher (1997) —see Eq. (12) —relies on the inde-
endence of particle velocities at subsequent steps and thus
86 P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94
Fig. 6. Time evolution of longitudinal MSDs for σln K = 1 , 2 , 3 , 5 obtained from Monte Carlo simulations (solid lines), and the model predictions from the Markov-Chain CTRW
(32) and (35) with the full transition PDF (dashed lines). Increase in conductivity heterogeneity leads to higher dispersion, and the Markov-chain CTRW model is able to
accurately capture the time evolution of the MSDs for all levels of heterogeneity and injection rules. (a) Flux-weighted injection. (b) Uniform injection. Inset: Comparison
between flux-weighted and uniform injection for σln K = 1 , 5 . Impact of injection rule is significant for high conductivity heterogeneity.
N
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spatial positions. Recent studies, however, have shown that the
underlying mechanisms of anomalous transport can be quantified
through an analysis of the statistics of Lagrangian particle veloc-
ities such as velocity distribution and correlation ( de Anna et al.,
2013; Dentz and Bolster, 2010; Kang et al., 2014; 2011b; 2015a;
Le Borgne et al., 2008; Meyer and Tchelepi, 2010 ). In the following,
we briefly introduce two viewpoints for analyzing Lagrangian
velocities—equidistantly and isochronally along streamlines—and
the relation between them Dentz et al. (2016) . We then proceed to
a detailed analysis of the Lagrangian velocity statistics measured
equidistantly along streamlines.
4.1. Lagrangian velocities
Particle motion is described here by the recursion relations (7) .
In this framework, we consider two types of Lagrangian velocities.
The t(ime)–Lagrangian velocities are measured at a given time t ,
v t (t) = u n t , (15)
where n t is defined by (8) . The s(pace)–Lagrangian velocities are
measured at a given distance s along the trajectory. The distance
s n traveled by a particle along a trajectory after n steps is given
by
s n +1 = s n + � n . (16)
The number of steps needed to cover the distance s is described by
n s = sup (n | s n ≤ s ) . Thus, the particle velocity at a distance s along
a trajectory is given by
v s (s ) = u n s . (17)
The PDF of t–Lagrangian velocities sampled along a particle
path is given by
p t (v ) = lim
n →∞
∑ n i =1 τi δ(v − u i ) ∑ n
i =1 τi
, (18)
where we defined the transit time,
τi =
� i
u i
(19)
The PDF of s–Lagrangian velocities sampled along a particle path
are defined analogously as
p s (v ) = lim
n →∞
∑ n i =1 � i δ(v − u i ) ∑ n
� i . (20)
i =1
ote the difference with respect to Eq. (2) , which samples veloci-
ies in the network uniformly, while in Eq. (20) velocities are sam-
led along trajectories. Using the definition of the transit time τ i
n (19) , the PDFs of the s– and t–Lagrangian velocities are related
hrough flux weighting as Dentz et al. (2016)
p s (v ) =
v p t (v ) ∫ dv v p t (v )
. (21)
urthermore, for flux-preserving flows and under ergodic condi-
ions, the Eulerian and t-Lagrangian velocity PDFs are equal,
p e (v ) = p t (v ) . (22)
hus, under these conditions, the s–Lagrangian and Eulerian veloc-
ty PDFs are related as Dentz et al. (2016)
p s (v ) =
v p e (v ) ∫ dv v p e (v )
. (23)
his means that the stationary s-Lagrangian velocity PDF can be
etermined from the Eulerian velocity PDF. Fig. 7 illustrates this
elation by comparing the s-Lagrangian velocity PDFs measured
rom the numerical simulation to the flux-weighted Eulerian ve-
ocity PDFs shown in Fig. 3 .
.2. Evolution of Lagrangian velocity distributions
It is important to emphasize that the above definitions of the
agrangian velocity PDFs refer to stationary conditions. We now
efine the PDFs of t– and s–Lagrangian velocities through sam-
ling between particles and network realizations at a given time
t-Lagrangian) or space (s-Lagrangian) velocities
ˆ p t (v , t) = 〈 δ[ v − v (t)] 〉 , ˆ p s (v , s ) = 〈 δ[ v − v (s )] 〉 . (24)
n general, these quantities evolve in time and with distance along
he streamline and are sensitive to the injection conditions because
vidently for t = 0 and s = 0 both are equal to the PDF of initial
article velocities ˆ p t (v , t = 0) = ˆ p s (v , s = 0) = p 0 (v ) , but their re-
pective stationary PDFs are different, namely
p t (v ) = lim
t→∞
ˆ p t (v , t) , p s (v ) = lim
s →∞
ˆ p s (v , s ) . (25)
Let us consider some further consequences of these properties.
irst, we notice that under (Eulerian) ergodicity the uniform injec-
ion condition (4) corresponds to an initial velocity PDF of
p 0 (v ) = p e (v ) = p t (v ) , (26)
P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94 87
Fig. 7. s-Lagrangian velocity PDFs for σln K = 1 and σln K = 5 . The measured s-
Lagrangian velocity PDF agrees very well with the PDF obtained by transforming
the Eulerian velocity PDF using Eq. (23) .
t
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Fig. 8. Lagrangian flux distributions at the inlet and outlet for uniform and flux-
weighted injection rules, for σln K = 5 . Note that Lagrangian flux distributions at
the outlet are identical regardless of the injection method. Inset: the same plot for
σln K = 1 .
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hat is, the initial velocity PDF is equal to the Eulerian and thus t–
agrangian velocity PDFs. This means that for the uniform injection
ethod, the t–Lagrangian velocity PDF is steady, ˆ p t (v , t) = p t (v ) ,hile the s–Lagrangian velocity PDF is not. It evolves from its ini-
ial distribution p s (v , s = 0) = p e (v ) to the steady state distribu-
ion (23) .
The flux-weighted injection condition, on the other hand, corre-
ponds to the initial velocity PDF
p 0 (v ) = p s (v ) , (27)
ue to relation (21) . The initial velocity PDF is equal to the s–
agrangian velocity PDF. This means that under flux-weighting,
he s–velocity PDF is steady, ˆ p s (v , s ) = p s (v ) . Under these condi-
ions, the t–Lagrangian velocity PDF ˆ p t (v , t) evolves from the initial
istribution ˆ p t (v , t = 0) = p s (v ) towards the asymptotic p t (v ) =p e (v ) , which is equal to the Eulerian velocity PDF. These are key
nsights for the qualitative and quantitative understanding of the
verage transport behavior.
.3. Space-Lagrangian velocity statistics
We analyze particle velocities along their projected trajecto-
ies in the longitudinal direction. Spatial particle transitions may
e characterized by the characteristic fracture link length, l . The
agrangian velocity v s ( s n ) at a distance x n = n l along the pro-
ected trajectory is approximated by the average velocity v n ≡ /τn where τ n is the transition time for the distance l at step
. In the following, we investigate the statistical characteristics
f the s-Lagrangian velocity series { v n }. For the uniform flow
onditions under consideration here, the projected distance x n s a measure for the streamwise distance s n , and v n for the s-
agrangian velocity v s ( s n ). Spatial Lagrangian velocities have been
tudied by Cvetkovic et al. (1996) and Gotovac et al. (2009) for
ighly heterogeneous porous media and by Frampton and
vetkovic (2011) for 3D DFNs in view of quantifying particle travel
ime statistics and thus modeling effective particle motion.
We first study the convergence of the s–Lagrangian veloc-
ty PDFs towards a stationary distribution and the invariance of
ˆ p s (v , s ) for a stationary (flux-weighted) initial velocity PDF. We
onsider the two injection conditions (4) and (5) and record the
istribution of particle velocities at a line located at the control
oint x c . Under ergodic conditions, we expect ˆ p s (v , s ) to converge
owards its steady state distribution (23) for uniform injection and
o remain invariant for the flux-weighted injection. Fig. 8 shows
ˆ p s (v , s = 0) and ˆ p s (v , s = x c ) for uniform and flux-weighted in-
ection conditions and two different heterogeneity strengths. We
learly observe that ˆ p s (v , x c ) = p s (v ) is invariant for flux-weighted
njection. For uniform injection, ˆ p s (v , x c ) has already evolved to-
ards its steady limit after x c = 200 l . This is an indication that the
ow and transport system is in fact ergodic. Note that in terms of
omputational efficiency, this observation gives a statistically con-
istent way of continuing particle trajectories through reinjection
t the inlet. If the outlet is located at a position x c large enough so
hat ˆ p s (v , s = x c ) = p s (v ) , particles are reinjected at the inlet with
ux-weighted probability, this means that the velocity statistics are
reserved. Furthermore, this method ensures that the domain is
arge enough to provide ergodic conditions. In the following, we
nalyze the statistical properties of streamwise velocity transitions
ith the aim of casting these dynamics in the frame of a Markov
odel for subsequent particle velocities.
We first consider the distribution ψ τ ( t ) of transition times
long particle trajectories through sampling the transition times
long all particle trajectories and among network realizations. To
his end, we consider a flux-weighted injection because it guar-
ntees that the s-Lagranagian velocities are stationary. Fig. 9 illus-
rates the PDF of transition times for different variances of ln ( K ).
s σ ln K increases, the transition time PDFs become broader. The
ransition time closely follows a truncated power-law distribution.
Next we consider two-point velocity statistics to gain insight
nto the velocity correlations along a streamline. To this end, we
onsider the velocity auto covariance for a given lag �s = s − s ′ .s pointed out above, for flux-weighted injection, the streamwise
elocities here are stationary and therefore
s (s − s ′ ) = 〈 [ v s (s ) − 〈 v s (s ) 〉 ][ v s (s ′ ) − 〈 v s (s ′ ) 〉 ] 〉 . (28)
n order to increase the statistics, we furthermore sample along
treamlines over a distance of 10 2 l . The velocity variance is σ 2 v =
s (0) . The velocity autocorrelation function χs (s ) = C s (s ) /σ 2 v . The
88 P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94
Fig. 9. (a) Lagrangian transition time distributions for σln K = 1 , 2 , 3 , 5 with flux-
weighted injection. As the network conductivity becomes more heterogeneous, the
transition time distribution becomes broader. Inset: the effective correlation length
increases with increasing network heterogeneity: 3 . 8 l , 5 . 6 l , 8 . 2 l , 11 . 5 l . The correla-
tion step ( n c ) is computed by integrating velocity autocorrelation function in space.
s
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correlation length scale � c is defined by
� c =
∫ ∞
0
ds χs (s ) . (29)
The inset in Fig. 9 shows the increase in the velocity corre-
lation length scale with increasing ln ( K ) variances for a flux-
weighted injection case. This can be attributed to the emer-
gence of preferential flow paths, as shown in Fig. 2 . Painter and
Cvetkovic (2005) and Frampton and Cvetkovic (2011) also re-
ported the existence of clear velocity correlation between succes-
sive jumps in DFNs and showed that this correlation structure
should be captured for effective transport modeling.
The existence of a finite correlation length along the particle
trajectories indicates that subsequent velocities, when sampled at
a distance much larger than the correlation length � c , may be con-
sidered independent. In order to study this feature, we characterize
the series of s-Lagrangian velocities { v n } in terms of the transition
probabilities to go from velocity v m
to velocity v m + n . We deter-
mine the transition probabilities under flux-weighted particle in-
jection because, as detailed above, under these conditions, the s-
Lagrangian velocity is stationary. Thus, the transition probability is
only a function of the number n of steps,
r n (v | v ′ ) = 〈 δ(v − v m + n ) 〉 | v m = v ′ (30)
Numerically, the transition probability is determined by discretiz-
ing the s-Lagrangian velocity PDF into N velocity classes C i =(v s,i , v s,i + �v s,i ) and recording the probability for each class given
the previous velocity class. This procedure gives the transition ma-
trix T n ( i | j ) from class j to i after n steps such that r n ( v | v ′ ) is ap-
proximated numerically as
r n (v | v ′ ) =
N ∑
i, j=1
I C i (v ) T n (i | j) I C j (v ′ ) �v i
, (31)
where the indicator function I C i (v ) is 1 if v ∈ C i and 0 otherwise.
Fig. 10 shows the one-step transition matrix T 1 ( i | j ) for equidis-
tant and logarithmically equidistant velocity classes for differ-
ent network heterogeneity. Higher probabilities along the diago-
nal than in the off-diagonal positions indicate correlation between
subsequent steps, which, however, decreases as the number of
teps along the particle trajectory increases, as indicated by the ex-
stence of a finite correlation scale � c .
. Stochastic particle motion and effective transport model
In the following, we describe the evolution of the s-Lagrangian
elocities by a Markov-chain, which is motivated by the existence
f a finite spatial correlation scale (see inset of Fig. 9 ). This leads to
spatial Markov-chain random walk (which we also termed spa-
ial Markov Model) formulation of particle dispersion that is valid
or any initial velocity distribution, and thus for any injection pro-
ocol. This modeling approach is in line with the time-domain ran-
om walk (TDRW) and continuous time random walk (CTRW) ap-
roaches discussed in the Introduction and below.
.1. Markovian velocity process
Along the lines of Le Borgne et al. (2008) and Kang et al.
2011b ), we model the velocity series { v n } as a Markov-chain,
hich is a suitable model to statistically quantify the evolu-
ion of the s-Lagrangian velocities based on the existence of a
nite correlation length. In this framework, the n –step transi-
ion probability r n ( v | v ′ ) satisfies the Chapman–Kolmogorov equa-
ion Risken (1996)
n (v n | v 0 ) =
∫ dv k r n −k (v n | v k ) r k (v k | v 0 ) . (32a)
The velocity process is fully characterized in terms of the one-
tep transition PDF r 1 ( v | v ′ ) and the steady state PDF p s ( v ) of
he s-Lagrangian velocity. Consequently, the evolution of the s-
agrangian velocity PDF ˆ p s (v , s n ) is given by
p s (v , s n ) =
∫ dv ′ r 1 (v | v ′ ) p s (v ′ , s n ) , (32b)
ith the arbitrary initial PDF p s (v , s 0 = 0) = p 0 (v ) . The number of
teps to decorrelate this Markov-chain is given by n c = � c / l . Fig. 11
hows the evolution of the PDF of s-Lagrangian velocities for the
niform injection (4) . Recall that the uniform injection mode corre-
ponds to the initial velocity PDF p e ( v ). Thus, the numerical Monte
arlo simulations are compared to the predictions of (32b) for the
nitial condition ˆ p s (v , s 0 = 0) = p e (v ) . The transition PDF r 1 ( v | v ′ ) is
iven by (31) with the velocity transition matrix shown in Fig. 10 .
s shown in Fig. 11 , the prediction of the Markovian velocity
odel and the Monte Carlo simulation are in excellent agreement,
hich confirms the validity of the Markov model (32) for the evo-
ution of s-Lagrangian velocities. Velocity transition dynamics are
ndependent of the particular initial conditions and thus allow pre-
icting the evolution of the Lagrangian velocity statistics for any
nitial velocity PDF and thus for any injection protocol.
As mentioned above, the Markov-chain { v n } is fully character-
zed by the stationary PDF of the s-Lagrangian velocities and the
ransition PDF r 1 ( v | v ′ ). The behavior of the latter may be charac-
erized by the number of steps n c needed to decorrelate, i.e., the
umber of steps n c such that r n ( v | v ′ ) for n > n c converges to the
tationary PDF r n ( v | v ′ ) → p s ( s ). The number of steps for velocities
o decorrelate can be quantified by
c =
� c
l , (33)
he simplest transition PDF that shares these characteristics
s Dentz et al. (2016) ; Kang et al. (2016a) ; 2015a) ; 2015b)
1 (v | v ′ ) = aδ(v − v ′ ) + (1 − a ) p s (v ) , (34)
ith a = exp (−l /� c ) . This transition PDF is thus fully determined
y one single parameter n c . Note that the latter increases with the
evel of heterogeneity, as illustrated in the inset of Fig. 9 . This pa-
ameter is estimated here from the simulated Lagrangian velocities.
P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94 89
Fig. 10. (a) One-step velocity transition matrix T 1 ( i | j ) with linear equiprobable binning for N = 50 velocity classes for σln K = 1 . (b) Velocity transition matrix with linear
equiprobable binning for σln K = 5 . The color-bar shows the logarithmic scale. (c) Velocity transition matrix with logarithmic binning for σln K = 1 . (d) Velocity transition
matrix with logarithmic binning for σln K = 5 . Increase in conductivity heterogeneity leads to higher probability close to diagonal entries. (For interpretation of the references
to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 11. Evolution of the PDF of s-Lagrangian velocities for uniform injection , i.e.,
for an initial velocity PDF ˆ p s (v , s 0 = 0) = p e (v ) . The symbols denote the data ob-
tained from the direct numerical simulation, the dashed lines show the predictions
of (32b) with the transition matrix shown in Fig. 10 . Inset: Evolution of the PDF of
s-Lagrangian velocities for a flux-weighted injection . In this case, the initial velocity
PDF is identical to the stationary s-Lagrangian velocity PDF.
I
(
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t may also be measured in the field from multiscale tracer tests
Kang et al., 2015b ). In the following, we study particle dispersion
n the Markovian velocity model for the full transition PDF shown
n Fig. 10 and the reduced-order Markov model (34) .
.2. Particle dispersion and model predictions
We consider particle motion along the mean pressure gradient
n x –direction, which is described by the stochastic regression
n +1 = x n + l , t n +1 = t n +
l
v n . (35)
he velocity transitions are determined from the Markovian veloc-
ty process (32) . Note that the { v n } process describes equidistant
elocity transitions along particle trajectories, while (35) describes
article motion projected on the x -axis. In this sense, (35) approxi-
ates the longitudinal travel distance x n with the distance s n along
he streamline, which is valid if the tortuosity of the particle tra-
ectories is low. As indicated in Section 3.2 , for travel distances � 0 arger than � c , or equivalently, step numbers n � n c ≡ � c / l , subse-
uent velocities may be considered independent and particle dis-
ersion is fully characterized by the recursion relation (12) and the
ransition time PDF (13) . Thus, as shown in Section 3.2 , the CTRW
f Eq. (12) correctly predicts the asymptotic scaling behavior of the
entered mean square displacement. This is not necessarily so for
he particle breakthrough and the preasymptotic behavior of the
MSD. As seen in Section 3.1 , the late time tailing of the BTC de-
ends on the injection mode and thus on the initial velocity PDF.
n fact, the slope observed in Fig. 5 for uniform particle injection
an be understood through the persistence of the initial velocity
DF. The first random walk steps until decorrelation at n = n c are
haracterized by the transit time PDF
0 (t ) =
l
t 2 p 0 ( l /t ) . (36)
hus, for an initial velocity PDF p 0 (v ) = p e (v ) , the initial tran-
it time PDF is given in terms of the Eulerian velocity PDF,
hich is characterized by a stronger probability weight towards
ow velocities than the PDF of the s-Lagrangian velocities, which
90 P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94
Fig. 12. Particle BTCs from Monte Carlo simulations and the predictions from the Markov-chain CTRW model with the full velocity transition matrix for (a) flux-weighted
injection, and (b) uniform injection at x c = 200 l .
Fig. 13. Particle BTCs from Monte Carlo simulations and predictions from the Markov-chain CTRW model with the reduced-order velocity transition matrix for (a) flux-
weighted injection, and (b) uniform injection at x c = 200 l .
s
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is given by (23) . The space-time random walk (35) together
with the Markov model (32b) is very similar to the TDRW ap-
proach Benke and Painter (2003) ; Painter and Cvetkovic (2005) and
can also be seen as a multi-state, or correlated CTRW approach be-
cause subsequent particle velocities and thus transition times are
represented by a Markov process Chechkin et al. (2009) ; Haus and
W. (1987) ; Le Borgne et al. (2008) ; Montero and Masoliver (2007) .
The joint distribution p ( x, v, t ) of particle position and velocity at a
given time t is given by Le Borgne et al. (2008)
p(x, v , t) =
∫ t
0
dt ′ H( l / v − t ′ ) R (x − v t ′ , v , t − t ′ ) , (37)
where H ( t ) is the Heaviside step function; R ( x, v, t ) is the frequency
by which a particle arrives at the phase space position ( x, v, t ). It
satisfies
R (x, v , t) = R 0 (x, v , t) +
∫ dv ′ r 1 (v | v ′ ) R (x − l , v ′ , t − l / v ′ ) , (38)
where R 0 (x, v , t) = p 0 (x, v ) δ(t) with p 0 (x, v ) = p(x, v , t = 0) . Thus,
the right side of (37) denotes the probability that a particle arrives
at a position x − v t ′ where it assumes the velocity v by which it
advances toward the sampling position x . Eqs. (37) and (38) can be
combined into the Boltzmann equation
∂ p(x, v , t) ∂t
= −v ∂ p(x, v , t)
∂x − v
l p(x, v , t)
+
∫ dv ′ v
′ ¯
r 1 (v | v ′ ) p(x, v ′ , t) , (39)
lee Appendix A . This result provides a bridge between the
DRW approach Benke and Painter (2003) ; Painter and
vetkovic (2005) and the correlated CTRW approach.
As illustrated in Fig. 3 , the Eulerian velocity PDF can be char-
cterized by the power-law p e ( v ) ∝ v α . Thus, the first CTRW steps
ntil the decorrelation at n = n c are characterized by the tran-
it time PDF ψ 0 (t) ∝ t −2 −α . The corresponding tail of the BTC is
f (t, x c ) ∝ t −2 −α . The observed value of α = −0 . 55 explains the tail-
ng of the BTC in Fig. 5 as t −1 . 45 , which shows the importance of
he initial velocity distribution. We also observe decrease in BTC
ailing (larger absolute slope) with travel distance as initial veloc-
ty distribution converges to stationary Lagrangian velocity distri-
ution and as tracers sample more velocity values. This implies
hat one needs to be careful when inferring a β from single BTC
easurement because the slope can evolve depending on the in-
ection method, velocity PDF and velocity correlation.
First, we compare the results obtained from Monte Carlo sim-
lation in the random DFN to the predictions of the Markov-
hain CTRW (32) and (35) with the full transition PDF of Fig. 10 .
ig. 6 shows the evolution of the cMSD for different levels of het-
rogeneity and different injection modes. As expected from the
bility of the Markov model to reproduce the evolution of the s-
agrangian velocity PDF for both uniform and flux-weighted injec-
ion conditions, the predictions of particle spreading are in excel-
ent agreement with the direct numerical simulations. In Fig. 12
e compare breakthrough curves obtained from numerical simu-
ations with the predictions by the Markov model for the uniform
P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94 91
Fig. 14. Lagrangian flux distributions at the inlet and outlet with two arbitrary ini-
tial velocity distributions for σln K = 5 . The two initial velocity distributions come
from uniform injections into regions of the 20-percentile highest, and 20-percentile
lowest velocities. Flux values are normalized with the mean flux value. Note that
flux distributions at outlet are identical regardless of the initial velocity distribu-
tion. Inset: same plot for σln K = 1 .
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nd flux-weighted injection modes. Again, the impact of the injec-
ion mode and thus initial velocity PDF is fully quantified by the
arkov model.
We now apply the Markov model (32) –(35) , i.e., employing a
arsimonious parameterization of the velocity transition PDF, with
single parameter n c ( Eq. (33) ), which is estimated here from ve-
ocity correlations along streamlines (see inset of Fig. 9 ). We first
ompare the reduced-order Markov model to the cases of uniform
nd flux-weighted injection, and conclude that the proposed par-
imonious stochastic model provides an excellent agreement with
he direct numerical simulations ( Fig. 13 ). This implies that the
imple correlation model (34) can successfully approximate the ve-
ocity correlation structure. Hence it appears that high order cor-
elation properties, quantified from the full transition probabili-
ig. 15. (a) Particle BTCs from Monte Carlo simulations for injection into high-velocity regi
rder velocity transition matrix (dashed line). (b) Corresponding results for injection into
ies ( Fig. 10 ), are not needed for accurate transport predictions in
he present case. This suggests promising perspective for deriv-
ng approximate analytical solutions for this Markov-chain CTRW
odel Dentz et al. (2016) . Furthermore, as discussed in Kang et al.
2015b ), the velocity correlation parameter n c can be estimated in
he field by combining cross-borehole and push-pull tracer experi-
ents.
Finally, we consider the evolution of the particle BTC and the
MSD for arbitrary injection modes . For real systems both flux-
eighted and uniform injections are idealizations. A flux-weighted
ondition simulates a constant concentration pulse where the in-
ected mass is proportional to the local injection flux at an inlet
oundary that is extended over a distance much larger than the
orrelation scale during a given period of time. A uniform injection
epresents an initial concentration distribution that is uniformly
xtended over a region far larger than the correlation length. In
eneral, the initial concentration distribution may not be uniform,
nd the injection boundary may not be sufficiently large, which
eads to an arbitrary initial velocity distribution, biased maybe
o low or high flux zones, as for example in the MADE exper-
ments, where the solute injection occurred into a low perme-
bility zone ( Harvey and Gorelick, 20 0 0 ). For demonstration, we
tudy two scenarios representing injection into low and high flux
ones: uniform injections into regions of the 20-percentile high-
st, and 20-percentile lowest velocities. The initial velocity PDF for
he low velocity mode shows the power-law behavior which is the
haracteristic for the Eulerian PDF, and the initial velocity PDF for
he high velocity mode shows narrow initial velocity distribution
Fig. 14 ). Eventually, the s-Lagrangian PDFs evolve towards the sta-
ionary flux-weighted Eulerian PDF as discussed in the previous
ection.
Fig. 15 shows the predictive ability of the effective stochastic
odel for these different injection conditions. The reduced-order
arkov velocity model compares well with the direct Monte Carlo
imulation in the random networks. As expected, the BTCs for in-
ection into low velocity regions have a much stronger tailing than
or injection into high velocity regions. In fact, as the initial ve-
ocity shows the same behavior at low velocities as the Eulerian
elocity PDF, the breakthrough tailing is the same as observed in
ig. 12 . We also observed that the reduced-order Markov veloc-
ty model can capture important features of the time evolution of
MSDs. This demonstrates that the proposed model can incorpo-
ate arbitrary injection modes into the effective modeling frame-
ork.
ons (solid line) and predictions from Markov-chain CTRW model with the reduced-
low-velocity regions.
92 P.K. Kang et al. / Advances in Water Resources 106 (2017) 80–94
U
>
R
A
B
B
B
B
B
B
B
B
B
B
B
B
C
C
C
C
C
D
6. Conclusions
This study shows how the interplay between fracture geomet-
rical properties (conductivity distribution and network geometry)
and tracer injection modes controls average particle transport via
Lagrangian velocity statistics. The interplay between fracture het-
erogeneity and tracer injection methods can lead to distinctive
anomalous transport behavior. Furthermore, the injection condi-
tions, for example, uniform or flux-weighted, imply different ini-
tial velocity distributions, which can have a persistent impact on
particle spreading through DFNs. For uniform injection, the s-
Lagrangian velocity distribution evolves from an Eulerian velocity
distribution initially to a stationary s-Lagrangian distribution. In
contrast, for flux-weighted injection, the s-Lagrangian velocity dis-
tribution remains stationary.
We have presented a spatial Markov model to quantify anoma-
lous transport through DFNs under arbitrary injection conditions.
We derive an analytical relation between the stationary Lagrangian
and the Eulerian velocity distribution, and formally incorporate the
initial velocity distribution into the spatial Markov model. The pro-
posed model accurately reproduces the evolution of the Lagrangian
velocity distribution for arbitrary injection modes. This is accom-
plished with a reduced-order stochastic relaxation model that cap-
tures the velocity transition with a single parameter: the effective
velocity correlation � c . The agreement between model predictions
and direct numerical simulations indicates that the simple veloc-
ity correlation model can capture the dominant velocity correlation
structure in DFNs.
In this study, we investigated the particle transport and the
impact of the injection condition for idealized 2D DFN using
a Markov velocity model. These findings can be extended to
3D DFNs, for which similar behaviors regarding the injection
mode have been found Hyman et al. (2015b ). Also, Frampton and
Cvetkovic (2011) reported similar velocity correlation structures for
3D DFNs as in 2D, which suggests that a velocity Markov model
such as the one presented in this work can be used for the mod-
eling of particle motion in 3D DFNs.
Acknowledgements
PKK and SL acknowledge a grant ( 16AWMP-B0 6 6761-04 ) from
the AWMP Program funded by the Ministry of Land, Infrastructure
and Transport of the Korean government and the support from Fu-
ture Research Program ( 2E27030 ) funded by the Korea Institute of
Science and Technology (KIST). PKK and RJ acknowledge a MISTI
Global Seed Funds award. MD acknowledges the support of the
European Research Council (ERC) through the project MHetScale
( 617511 ). TLB acknowledges the support of European Research
Council (ERC) through the project ReactiveFronts ( 648377 ). RJ ac-
knowledges the support of the US Department of Energy through
a DOE Early Career Award (grant DE-SC0 0 09286 ). The data to re-
produce the work can be obtained from the corresponding author.
Appendix A. Boltzmann equation
The time derivative of (37) gives
∂ p(x, v , t) ∂t
= −v ∂ p(x, v , t)
∂x + R (x, v , t) − R (x − l , v , t − l / v ) .
(A.1)
Note that R ( x, v, t ) denotes the probability per time that a particle
has the velocity v at the position x . It varies on a time scale of v .Thus we can approximate (37) as
p(x, v , t) ≈ l R (x − l , v , t − l / v ) . (A.2)
v
sing this approximation and combining (A.1) with (38) gives for t
0
∂ p(x, v , t) ∂t
= −v ∂ p(x, v , t)
∂x − v
l p(x, v , t)
+
∫ dv ′ r 1 (v | v ′ ) v
′
l p(x, v ′ , t) . (A.3)
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