WATER RESOURCES RESEARCH, VOL. 41, W02002, doi:10.1029/2004WR003682, 2005
Upscaling discrete fracture network simulations: An
alternative to continuum transport models
S. Painter
Center for Nuclear Waste Regulatory Analyses, Southwest Research Institute,
San Antonio, Texas, USA
V. Cvetkovic
Royal Institute of Technology,
Stockholm, Sweden
Abstract
[1] Particle tracking through stochastically generated networks of discrete fractures provides an
alternative to the conventional advection-dispersion description of transport in fractured rock.
However, discrete fracture network simulations are computationally intensive and usually
limited to small scales. An approach for direct upscaling of discrete fracture simulations is
described. Trajectories for nonreacting tracer particles are first extracted from relatively small
discrete fracture network simulations. Tracer-rock interaction is represented by also calculating a
cumulative reactivity parameter along each trajectory. The residence time/reactivity information
is then used in a Monte Carlo simulation to construct artificial particle trajectories of any length,
thereby achieving the upscaling objective. In its simplest form the procedure has the form of a
random walk evolving in a two-dimensional space. Tests using site-specific and generic
networks show that it is necessary to modify the random walk to produce sequential correlation
along the trajectories. We achieve this by using a discrete state Markov process to direct the
random walk. The procedure is computationally efficient, easily implemented, and compares
well with the network simulations.
Received 25 September 2004; accepted 10 November 2004; published 2 February 2005.
Keywords: fracture networks, retention, transport.
Index Terms: 1832 Hydrology: Groundwater transport; 1869 Hydrology: Stochastic hydrology.
Citation: Painter, S., and V. Cvetkovic (2005), Upscaling discrete fracture network simulations: An alternative to
continuum transport models, Water Resour. Res., 41, W02002, doi:10.1029/2004WR003682.
Copyright 2005 by the American Geophysical Union.
Article Map
Abstract
1. Introduction
2. Theory
2.1. Transport Model
2.2. Random Walk Representation for τ and B
2.3. Relationship to CTRW
3. Monte Carlo Simulation
3.1. Random Walk (RW)
3.2. Random Walk Directed by a Markov Process (MDRW)
4. Numerical Tests
4.1. Nonreacting Travel Time and Cumulative Reactivity
4.2. Transport
5. Example Simulations at the Geosphere Scale
6. Discussion
7. Conclusions
8. Acknowledgment
References
Figures
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Tables
Table 1. Half-Life, Diffusion Coefficient, and Distribution Coefficients for Three Radionuclides Used as
Examples in This Study
Table 2. Peak of the Impulse Response γ* and Total Mass Released μ for Three Radionuclides, as
Generated With the RW and MDRW Upscaling Methods Compared With Those From DFN Simulations
Introduction
[2] In hydrological applications involving low-permeability formations, flow and associated
advective transport in the host rock are often not significant, and interconnected networks of rock
fractures form the primary pathways for release of toxic chemical or radioactive wastes to the
accessible environment. In particular, fractures provide the most plausible pathways to the
biosphere for nuclear waste buried at depth, and therefore it is crucial to have accurate and
practical tools for modeling radionuclide movement through fracture networks.
[3] When applied to sparsely fractured rock, the conventional advective-dispersion equation
(ADE) description of transport falls short of predicting many behaviors observed in field or
numerical studies. For example, tracers are often transported over considerable distances in
highly localized channels in contrast to the more uniform dispersion predicted by the ADE
[National Research Council, 1996; Dverstorp et al., 1992; Neretnieks, 1993; Tsang and
Neretnieks, 1998]. Dispersivities appear to vary with the scale of the observation, which is in
direct conflict with the Fickian transport assumption underlying the ADE [e.g., National
Research Council, 1996]. Numerical simulations also show that fracture networks display
different effective porosities for transport in different directions [Endo et al., 1984] and highly
non-Gaussian breakthrough curves [Andersson and Dverstorp, 1987; Schwartz and Smith,
1988; Berkowitz and Scher, 1997, 1998; Painter et al., 2002]. These and other related
behaviors are not captured in the ADE model and may be due to the poorly connected nature of
many natural fracture networks. In general, the quality of the representation provided by the
ADE is better for highly and uniformly fractured rock as compared with sparsely fractured rock,
and may be questionable for the sparsely fractured rock favored for geological disposal.
[4] Discrete fracture network (DFN) models provide an alternative for situations where the
ADE is inadequate, and have been used in a variety of studies in two dimensions [Long et al.,
1982; Dershowitz, 1984; Endo et al., 1984; Robinson, 1984; Smith and Schwartz, 1984]
and three dimensions [e.g., Shapiro and Andersson, 1985; Elsworth, 1986; Andersson and
Dverstorp, 1987; Cacas et al., 1990; Long et al., 1992]. The DFN simulations avoid the
volume averaging required for traditional equivalent continuum models and can generally
represent a wider range of transport phenomena. DFN models play an important role in
fundamental conceptual model evaluations, but site-specific applications are generally limited to
near-field scales (50–100 m) due to the large computational demands. Many applications,
particularly those involving geological disposal of high-level nuclear waste, involve spatial
scales of a few hundred meters to a few kilometers in three dimensions. Thus there is a need for
methods to bridge the gap between the spatial scale at which DFN simulations are tractable and
the repository geosphere scale.
[5] One approach for bridging this gap in scales is to perform DFN simulations for a subdomain
of the field-scale domain to deduce parameters for a simpler and less computationally demanding
model (such as a continuum model) for use at the field scale. The implicit assumption is that the
fracture network model used in the DFN simulation is representative of the fracture network of
the larger domain. This general approach has been referred to as the hybrid approach [National
Research Council, 1996] and can also be thought of as an upscaling problem. For example,
Long et al. [1982] used the results from a large number of DFN simulations to show that flow
in some (but not all) DFNs could be described by equivalent hydraulic conductivity tensors.
These permeability tensors are then available for use in a full field-scale simulation of flow using
conventional continuum flow models.
[6] Hybrid models have also been used for solute transport in fractured rock. Schwartz and
Smith [1988] collected statistics on particle velocities in DFN simulations, fitted lognormal and
gamma distributions to the velocity distributions, and then sampled these fitted distributions in a
random walk Monte Carlo simulation. The approach eliminates the need to assign numerical
values to a dispersion tensor, and accurately reproduced spreading patterns in two-dimensional
DFNs. Berkowitz and Scher [1997, 1998] also collected statistics on particle velocities from
DFNs and fitted a model distribution. Instead of using the fitted distribution in a Monte Carlo
simulation, they used the continuous time-random walk (CTRW) formalism and obtained
semianalytical results for plume evolution and important insights into anomalous (non-Fickian)
transport. Although not explicitly addressing fractured rock, recent work on CTRW [e.g.,
Berkowitz et al., 2002; Cortis et al., 2004] add to its potential utility as a hybrid model for
transport in fractured rock. Extending an approach originally proposed by Williams [1992,
1993], Benke and Painter [2003] used the results from DFN simulations to deduce parameters
in a linear Boltzmann transport equation and then used the Boltzmann equation as a continuum
model for transport at the field scale. This hybrid Boltzmann approach also successfully
reproduced spreading patterns and breakthrough curves calculated from DFN simulations.
[7] The hybrid models just discussed considered advective transport only and neglected
retention processes that occur primarily in the porous matrix. While purely advective transport is
a logical starting point for development of alternative models, mass exchanges between fractures
and matrix combined with retention processes (diffusion, sorption) in the matrix are important
for field applications and tend to dominate over advective transport at the time and spatial scales
relevant for nuclear waste disposal. However, mass exchange between matrix and fractures is
partly controlled by the local advective velocity [Cvetkovic et al., 1999]. Because of this strong
coupling between hydrodynamics and mass retention, models for field-scale transport in
fractured rock need to consider both the nonclassical phenomenology of advective transport in
random fracture networks and mechanistic models for solute retention in the host rock. Dentz
and Berkowitz [2003] recently extended the CTRW approach to account for retention process
that can be described as a multitude of first-order trapping rates that do not vary spatially.
Kosakowski et al. [2001] used breakthrough curves measured from a tracer test in a fractured
till to fit CTRW waiting time distributions, an approach that implicitly and phenomenologically
incorporates retention mechanisms. Using this approach, they successfully extrapolated
(upscaled) from a travel distance of 2.1 m to a travel distance of 3.6 m.
[8] In this paper, we describe a new approach for field-scale simulation of transport in fractured
rock based on upscaling the results of particle tracking on discrete fracture network simulations.
Specifically, we show how relevant information from nonreacting particle tracking in relatively
small DFN simulations can be extracted and used in a Monte Carlo random walk at the
geosphere scale. New aspects of this work are as follows. First, retention in the rock matrix is
described within a stochastic Lagrangian formalism, an approach that honors the hydrodynamic
controls on retention while accommodating a wide variety of retention models with random
spatial variations in retention parameters. Second, two new Monte Carlo procedures for
generating the controlling Lagrangian parameters at the field scale are described. Both make
direct use of information extracted from DFN simulations without fitting a distributional model.
Third, we show that sequential correlation (persistence) along the trajectory is necessary to
accurately reproduce the breakthrough curves. When this persistence is taken into account, the
upscaling method accurately reproduces the results of DFN simulations and provides a practical
and easily implemented alternative to continuum transport models.
Theory
2.1. Transport Model
[9] Consider a hypothetical solute source located in a fractured rock volume. Steady state
groundwater flow driven by a regional hydraulic gradient carries solutes toward a monitoring
boundary located downstream. Diffusive mass exchange with the host rock and sorption in the
host rock delay the downstream movement. We are interested in the time-dependent mass flux at
this monitoring boundary. We adopt a Lagrangian perspective and consider multiple meandering
transport pathways (trajectories) that connect each source location to the monitoring boundary.
In general, the flow velocities, fracture apertures, and possibly retention properties fluctuate
along each trajectory.
[10] Recent theory [Cvetkovic et al., 1998, 1999; Cvetkovic and Haggerty, 2002; Cvetkovic
et al., 2004] has shown that transport with a wide range of retention models can be represented
in a generic compact form in the Laplace domain. Specifically, it has been shown that the
fundamental solution (the impulse response function) for a Lagrangian representation of
transport with retention in fractured rock can be represented as
where λ is the radionuclide decay constant, τ is the water residence time along the trajectory, is
the Laplace transform of the memory function, and B is a cumulative reactivity parameter that
integrates retention properties along the trajectory. Equation (1) is rather general and includes
the widely used retention models with spatial variability in retention parameters as special cases
by appropriate selection of the memory function and cumulative reactivity. For example, limited
diffusion, unlimited diffusion, and multirate linear transfer models are included as special cases.
[11] The groundwater residence time can be written in integral form as
where x is the coordinate in the direction of mean flow, Vx is the velocity component in that
direction, and y = η(x), z = ζ(x) defines the particle trajectory. The cumulative reactivity B is
dependent on the choice for retention model, is highly correlated to τ, and generally integrates
retention properties along the trajectories.
[12] The function γl(t; τ, B) is the conditional impulse response, the time-dependent discharge at
a monitoring boundary located at distance l from a dirac-δ input for given values of τ and B. In
the situation of no decay, it can be thought of as an arrival time distribution at the monitoring
boundary. In general, it represents the discharge along a single trajectory characterized by τ and
B, and is not observable. In applications, unconditional quantities are needed and can be obtained
by averaging with respect to f(τ, B), the joint probability density for τ and B. For example, the
unconditional impulse response is
Under fully ergodic conditions γl(t) represents the breakthrough curve at the monitoring
boundary. Under nonergodic conditions, the breakthrough curve will vary from realization to
realization, and γl(t) represents the expected value. Other convenient measures of transport can
be defined [Cvetkovic et al., 2004] for the nonergodic or ergodic situations. In either case, the
density f(τ, B) is needed to calculate meaningful quantities.
2.2. Random Walk Representation for τ and B
[13] The two Lagrangian quantities τ and B characterize transport and retention along a given
trajectory. Specifically, given a value for τ and B, we can calculate the conditional timedependent discharge from equation (1). Given a distribution for τ and B, we can calculate the
unconditional discharge and measures of uncertainty in the predicted discharge. Thus the
challenge is to obtain a model or algorithm for the joint distribution of τ and B.
[14] Consider a discretization of the trajectory into a number of jumps, with each jump
corresponding to transit through an individual fracture. Let Δ1, Δ2, denote identically distributed
random variables that model the change in the x coordinate for the jumps. Similarly, let Δτ1, Δτ2,
and ΔB1, ΔB2, denote random variables modeling the change in τ and B, respectively. In
general, Δi, Δτi and ΔBi are correlated.
[15] After n jumps, the x position is L(n) =
Δi, and the τ and B values are τ(n) =
Δτi
and B(n) =
ΔBi. The random walk is to be executed until the particle hits a monitoring
boundary. The number of jumps required to hit the monitoring boundary is a random variable Nl
= min {n: L(n) ≥ l} and the corresponding τ and B are
The stochastic process {τ(Nl), B(Nl)}l>0 governs the desired distribution of τ and B for a given l. In
words, the process is a running sum of correlated random variables that is stopped when one of
these (L) reaches a predefined cutoff value l.
[16] The stochastic process defined by (3) with an additional assumption of independent jumps
has been studied by Meerschaert et al. [2002], who develop limit theories for the process in the
situation of power law tails in the distribution of jumps [see also Becker-Kern et al., 2004]. If
the number of terms in the sums in equation (3) was fixed, then (3) would be a simple random
walk. However, Nl is a random variable, and the stochastic process is thus a random walk
subordinated by the random process governing Nl [Meerschaert et al., 2002]. Note that x
position in (3) is assuming the role played by time in the process studied by Meerschaert et al.
[2002]. In particular, the particle executes the random walk until it reaches a specified
monitoring boundary, not a specified time level. In other words, our walk is developing in the
two-dimensional τ, B space with the x position triggering the termination of the walk. The
position in the direction of mean flow, x in (3) is not required to be strictly increasing as a time
would be.
[17] At this point, we have made no statement about the distribution of the jumps. An obvious
choice would be to assume independent jumps. However, careful review of DFN simulation
results clearly show correlation between successive segments along the trajectory [Painter et al.,
2002; Benke and Painter, 2003]. A particle that is in a high-velocity segment is more likely to
find itself in a high-velocity segment in the subsequent segment due to conservation of flux at the
fracture intersections. We use a Markov-type approximation to reproduce the sequential
correlation, wherein each jump is correlated to its previous jump. There are many ways that this
could be implemented in practice. A particularly convenient method to generate this sequential
correlation is to give each particle an internal state that changes from segment to segment
according to a discrete state Markov process. The selection of the internal state will be discussed
in Section 3; for now we denote the set of possible states as S. The internal state of the particle
is then used to alter the distribution of jumps. Specifically, let S1, S2, S3, denote the sequence of
internal states, which we assume to be a discrete state Markov process governed by the transition
matrix A. Usual rules for discrete state Markov processes apply: Amn is probability for
transitioning from state m to state n, Amn = Pn where Pn is probability for state n, Amn = 1. The
distribution for each jump is assumed to be dependent on the internal state only and has
conditional density denoted by f(Δ1, Δτ1, Δβ1|S1). The unconditional distribution for the first jump
is then
The joint distribution for the first and second jump is given by
and the joint density for all jumps along the trajectory can be written as
Such a model is an example of a subordination process with the Markov process directing the
random walk and can also be thought of as a random walk with an internal state. We refer to the
process as a Markov-directed random walk (MDRW).
[18] If we assume only one state, then the jumps become independent and identically
distributed:
This is referred to as the random walk (RW) in the following to underscore the independent
nature of the jumps. The MDRW model equation (3) with (6) is the primary focus in this
paper; the RW model (3) with (7) is also considered for comparison purposes.
2.3. Relationship to CTRW
[19] In the special case of advection only (no retention), ΔBi = 0, the time domain
representation of equation (1) is γl(t; τ) = exp[−λτ]δ(t − τ), and the unconditional response
becomes γl(t) = fτ(t)exp[−λt] where fτ( ) is the travel time distribution for a nonreacting,
nondecaying tracer. In addition, (6) becomes
Making the further assumption of independent jumps,
Equation (3) with (9) is equivalent to the CTRW that Berkowitz and Scher [1997, 1998]
applied to advective transport in fractured rock, with f(Δ, Δτ) corresponding to the CTRW
waiting time density (denoted by ψ(t, l) in the notation of Berkowitz and Scher [1997, 1998]).
[20] It should be noted, however, that the CTRW framework contains additional flexibility that
Berkowitz and Scher [1997, 1998] did not use in their study of advective transport in fracture
networks. Retention in the CTRW framework is included in a very different way than in our
model. Specifically, retention is incorporated into the definition of the waiting density in CTRW,
as opposed to our Lagrangian approach, which effectively decomposes the problem into one of
determining nonreacting trajectories and then using equation (1) and (2) to incorporate the
effects of retention. Kosakowski et al. [2001] determined the waiting time distribution by fitting
breakthrough curves, an approach that phenomenologically incorporates retention into the
waiting time distribution. Although not specifically addressing fractured rock, Dentz and
Berkowitz [2003] incorporate mechanistic retention models in the situation of no spatial
variability in the mobile-immobile exchange rates. In fractured rock, the rates of exchange
between fractures and matrix are linked to the local velocity [Cvetkovic et al., 1999]; hence a
spatially variable velocity necessarily implies spatially variable exchange rates. V. Cvetkovic
and S. Painter (Tracer transport with transition and exchange disorder in random media,
submitted to Physical Review E, 2004) recently showed how general mechanistic models with
spatially variable exchange rates can be incorporated into the CTRW waiting time density, thus
achieving the same as equation (7). Extensions of CTRW that incorporate correlation between
one jump and the next jump in the sequence similar to equation (8) also exist [see, e.g.,
Hughes 1995, and references therein] but have not been applied to transport in the subsurface.
Berkowitz and Scher [1998] proposed to artificially extend the particle displacement to mimic
correlation along the trajectory, but they did not test this approximation nor did they specify how
to estimate the required extension. In principle, these various refinements and extensions of the
CTRW formalism could be combined with retention models appropriate for fractured rock to
represent the same processes as our model, but this has not yet been demonstrated. We choose
instead a simple Monte Carlo approach based on equation (3) with (6) or (7).
Monte Carlo Simulation
[21] For calculations, two steps remain: (1) determining the joint distribution of individual
fracture values Δ, Δτ, ΔB appropriate for a given site, and (2) determining the global distribution
of {τ, B} given the individual fracture distribution.
[22] At present, DFN simulation is the only viable method for determining the joint distribution
of individual fracture values. In this approach, realizations of the DFN at the near-field scale are
generated using standard fracture network modeling tools, taking into account as much sitespecific information as possible to constrain the network properties. Site-appropriate hydraulic
boundary conditions are applied and the resulting flow solved. Particles are then tracked through
the DFN velocity field. For each fracture segment traversed by each particle, the {Δ, Δτ, ΔB}
triplet is recorded. The set of these triplets represents Monte Carlo samples from the joint
distribution.
[23] The DFN simulations need to be large enough to minimize boundary effects that may
cause the jump statistics near the particle source to be different from those of the bulk DFN. In
the site-specific and generic simulations described later in this paper, such boundary effects were
observed but disappear quickly with distance from the source due to randomizing effects of
fracture intersections. In practice, it is easy to verify that the domain in large enough by dividing
the domain into upstream and downstream halves and verifying that the jump statistics are not
too different in the two halves.
[24] Note that an assumption of ergodicity has not been required. We simply require access to
the ensemble distribution of jumps. If the DFN simulations are large enough to ensure ergodicity
in the particle trajectories, then a single DFN simulation with multiple trajectories could be used
to generate the ensemble statistics. In the opposite extreme of fully nonergodic conditions, a
single trajectory from each of multiple realizations would be used. In practice, the situation is
likely to be intermediate between these two extremes, but experience suggests the situation to be
closer to the ergodic situation due to strong randomizing effects of fracture intersections. In the
numerical examples described later in this paper, multiple trajectories in a relatively small
number of realizations are used.
[25] The results of the DFN simulation can then be used to upscale the individual fracture
values to obtain global {τ, B}distributions. The procedure varies depending on what stochastic
process is assumed for the {τ, B}; that is, it depends on whether equation (6) or equation (7)
is used.
3.1. Random Walk (RW)
[26] If the individual jumps forming the running sums in equation (3) are assumed to
independent (i.e., governed by equation (7)) then the stochastic process {τ(Nl), B(Nl)}l>0 is
similar to the process studied by Meerschaert et al. [2002]. Our Monte Carlo procedure to
simulate this process is to form a particle trajectory by repeatedly selecting values at random
from the library of {Δ, Δτ, ΔB} triplets and forming the running sums in (3). A trajectory is
terminated once the x value exceeds l, indicating that the monitoring surface of interest has been
reached. The {τ, β} at that point becomes one Monte Carlo sample from the global distribution.
Because this procedure is very fast, it can be executed for large l, and is not limited by the same
computational constraints as DFN simulations. Thus it can be used to upscale the results of DFN
simulations, provided the fracture network model used in the DFN simulations is representative
of conditions throughout the larger region of interest.
3.2. Random Walk Directed by a Markov Process (MDRW)
[27] To account for persistence along the trajectory, the Markov directed random walk defined
by equations (3) and (6) can be used. The Monte Carlo procedure for simulating the process is
similar to that described in section 3.1 except that we give each particle an internal state, and
then use this state to control the selection of the next {Δ, Δτ, ΔB} triplet in the sequence. The
selection of the internal state is discussed in the following paragraph. We record, in addition to
the {Δ, Δτ, ΔB} triplets, the internal state for each segment of each particle trajectory in the DFN
simulation. When executing the Monte Carlo simulation, the state of the current segment is used
to constrain the selection of the next {Δ, Δτ, ΔB} triplet. Specifically, if the current segment has
state designated K, then only those segments that have K as the preceding state are eligible for
selection as the next segment. Using this algorithm, the sequence of internal states represent a
discrete state Markov process governed by the transition matrix A (see section 2.2), which
directs the random walk simulation of the particle trajectory in the τ, B space. With this sampling
method, it is not necessary to explicitly construct the A matrix from the DFN simulations,
although it is easily accomplished [Benke and Painter, 2003].
[28] The definition of internal states requires some further discussion. Given that the goal is to
create some sequential correlation in the sequence of {Δτ, ΔB} selected, a logical choice would
be to discretize the{Δτ, ΔB} space, and use this discretization to define the internal state. This
would have the desired effect of correlating each {Δτ, ΔB} pair with the preceding pair in a
classic Markov-type approximation, provided such correlation is manifest in the DFN
trajectories. This general approach has been used successfully in a fully discretized sense by
Benke and Painter [2003]. A closely related but simpler approach is to use the particle speed to
define the internal state. Because {Δτ, ΔB} are both closely related to local speed, this choice has
a similar effect of building in correlation along the trajectory. We take this latter approach.
Specifically, we discretize the particle speed into a small number of classes and use these speed
classes as the internal particle state. Unless otherwise noted, we use 20 classes and partition the
range of observed speeds so that each class has the same number of observations.
Numerical Tests
[29] Site specific and generic three-dimensional DFN simulations were used to test the
upscaling methods. The site-specific DFN and particle tracking simulations [Outters, 2003]
were designed to mimic the fracture network near the Äspö hard Rock Laboratory, Sweden and
used the FracMan [Dershowitz et al., 1998] software with the MAFIC module [Miller et al.,
1994] for transport. The computational domain was a 100 m × 100 m × 100 m cube. Each of the
20 realizations contained about 20000 disk shaped fractures tessellated into triangular finite
elements. In each of the generated realizations, generic boundary conditions were prescribed so
as to obtain a globally unidirectional flow. Once the flow field was computed, a large number of
inert particles were released from a 50 m × 50 m square on the upstream boundary and traced
though the network, assuming perfect mixing at each intersection.
[30] The generic simulations [Painter et al., 2002] were also computed in three dimensions,
but used a simplification that approximates flow between any two fracture intersections as onedimensional pipe flow instead of using a finite element mesh on each fracture plane. Similar
approximations have been used previously [e.g., Outters and Shuttle, 2000]. For example, it is
available as one option in the MAFIC software [Miller et al., 1994]. Despite the simplified
representation of flow on each fracture plane, this approximate model does represent some threedimensional effects. Specifically, it represents the coordination number (average number of
neighbors for each node) correctly for three-dimensional systems, and is thus considered to be
adequate for the purposes of testing the upscaling algorithm. Nevertheless, in applications full
finite element discretization of the fracture plane is to be preferred.
[31] Two sets of generic simulations were performed, one using a 30 m × 30 m × 30 m domain
and one using a 90 m × 30 m × 30 m domain. For each set, 100 realizations were created. The
fracture network model was identical in the two sets and used disk-shaped fractures with an
isotropic model for fracture orientation. Fracture radii follow a lognormal distribution with log
variance of 0.25 and geometric mean of 1 m. Fracture apertures were also selected from a
lognormal distribution (log variance of 0.25 and geometric mean of 100 micron). The density of
fracture centers was 0.0185 m−3, corresponding to 500 fractures for the smaller size and 1500 for
the larger simulations.
[32] In order to proceed with the numerical tests, we need to specify a retention model. The
unlimited diffusion model [e.g., Neretnieks, 1980] is the most widely used model for transport
in sparsely fractured rock and is the logical first model to consider for the Äspö site. In this case,
the memory function is = s−1/2 and the impulse response function in the time domain is
where H is the Heaviside function. The cumulative reactivity for this model can be written as B =
( DR)1/2
where , D, R, b are the matrix porosity, effective diffusion coefficient,
retardation factor, and fracture half aperture, respectively. For simplicity, we assume , D, R are
spatially constant and known. In this case, variability in B arises from variability in aperture and
velocity along the pathway, B = ( DR)1/2 β = ( DR)1/2 ∑ βi, and the new variable β =
captures this variability. This assumption of constant retention properties is reasonable given that
little information on retention property variability is available. However, we emphasize that
spatial variability in the retention properties can easily be accommodated in the upscaling
procedure, if adequate site-specific information is available.
4.1. Nonreacting Travel Time and Cumulative Reactivity
Figure
Figure 1
1. Cumulative distribution (CD) and complementary cumulative
distribution (CCD) of global τ and β from discrete fracture network simulations
compared with upscaling results based on a random walk (RW) and Markovdirected random walk (MDRW) simulation.
[33] The first step is to compare the distributions of τ and β from the DFN simulations with
those obtained from Monte Carlo upscaling simulations. Marginal distributions of τ and β for the
Äspö reference case are compared in Figure 1. The data points represent the results of the DFN
simulation, the dashed lines are from the random walk (RW) upscaling procedure, and the solid
lines are the result of the Markov-directed random walk (MDRW). In both upscaling results, we
took single-fracture distributions generated from FracMan/MAFIC and attempted to reproduce
the global τ and β distributions. The RW upscaling method generally reproduces the distributions
of τ and β in the right tail and in the bulk of the distribution, but generally predicts slightly higher
values of τ and β in the left tail. The MDRW produces a better fit to the left tail because it better
captures the weak correlation in velocity along the particle trajectories, which reduces overall
travel time and cumulative reactivity in the leading edge of the distribution.
Figure 2. Cross plots of nonreacting travel time τ and cumulative reactivity
parameter β (left) from discrete fracture network simulations and (right) upscaled
from the single-fracture statistics by the Markov-directed random walk algorithm. The high
correlation between τ and β is reproduced by the upscaling method.
[34] Cross plots of τ and β obtained from the MDRW and the DFN simulations are compared in
Figure 2. The similarity between the two cross plots demonstrates that the upscaling method
reproduces the strong correlation between τ and β.
Figure 3. Detail of cumulative distribution for global β from discrete fracture
network simulations compared with upscaling results based on a random walk
(RW) and random walk directed by 2-state, 5-state, and 20-state Markov
processes. The mismatch between the Markov-directed random walk (MDRW)
and the DFN simulations improves as the number of states is increased, but most of the
improvement occurs when going from one state (the RW situation) to two states in the directing
process.
[35] The MDRW results used in the comparisons in Figures 1 and 2 were obtained by using
the local speed to define the particle's internal state as described in Section 3. The particle speed
was binned into 20 speed classes in this case. An interesting issue to address is how the results
vary as the number of speed classes is varied from 1 (the RW case) to 20. This sensitivity is
shown in Figure 3. In going from one bin to two bins, there is a relatively large decrease in the
mismatch between the DFN and the upscaling results. Comparatively smaller improvements are
noted when going from two to five bins and from 5 to 20 bins. Even with 20 bins, there are still
some differences between the DFN and the upscaling results. One possible cause of this
mismatch is longer-range persistence in the sequence that is not captured with the Markov-type
approximation. Another possible cause is weak nonstationarity, especially near the source region.
Whatever the cause, the upscaled results appear to provide a reasonable approximation. This
adequacy of the fit is explored in the next subsection.
[36] The upscaling algorithms were also tested on an Äspö DFN simulation that had fracture
density reduced by a factor of two compared with the reference case. Results (not shown) were
similar to the reference case. The only significant differences between the upscaling simulations
and the DFN simulations were in the left tail of the distribution, where the MDRW performs
better than the RW. Interestingly, both the RW and the MDRW perform better in the lower
density case as compared with the reference case. One possible explanation for this better
performance is that there is some residual longer-range correlation along the trajectory that is not
capture by the MDRW, and that this longer-range correlation is stronger in the denser networks.
This issue does, however, require further study.
[37] In the site-specific simulations just described, an assumption of perfect mixing at the
fracture intersections was used. Other assumptions for particle redistribution at intersections are
possible (i.e., streamline routing), and the transport consequences of the mixing assumption have
been studied [e.g., Küpper et al., 1995; Park et al., 2001]. For the Äspö site-specific
simulations, the perfect mixing and streamline routing assumptions produce nearly identical {τ,
β} distributions [Cvetkovic et al., 2004]. The choice of mixing assumption is not expected to
affect the performance of the upscaling method, but this needs to be verified using DFN particle
tracking simulations with streamline routing.
Figure 4. Cumulative distribution (CD) and complementary cumulative
distribution (CCD) of global τ and β from generic discrete fracture network
simulations compared with upscaling results based on a random walk (RW) and Markov-directed
random walk (MDRW) simulation. In this example, statistics extacted from 30 × 30 × 30 m DFN
simulations were used in the two upscaling algorithms to predict the distributions at the 90 × 30
× 30 m scale.
[38] In the final test of the ability to upscale the τ and β distributions, generic DFN simulations
are considered. In this situation, we start with DFN simulations at the 30 × 30 × 30 m scale and
attempt to use single-fracture distributions derived from these to reproduce larger simulations (90
× 30 × 30 m). Results are shown in Figure 4. As with the site-specific simulations, the RW
produces significant errors in the left tail of the distribution. In the generic simulations, the RW
also has significant errors in the right tail of the distribution. The MDRW reproduces the DFN
simulations well across the entire distribution, with the only significant errors occurring in the
extreme right tail of the distribution which is highly uncertain because of the very few samples
there.
4.2. Transport
[39] Figures 1, 2, 3, and 4 suggest that the MDRW upscaling procedure provides reasonably
accurate approximations to the τ and β distributions. To assess the adequacy of the
approximations in a more quantitative way, we need go beyond the travel time/cumulative
reactivity distributions and consider transport. To make this assessment, we focus on the Äspö
reference case and consider the unconditional response equation (2). Three radionuclides were
considered with properties as shown in Table 1.
Figure 5. Unconditional discharge (breakthrough curves) for three radionuclides
as calculated from DFN simulations and the two upscaling methods. The travel
distance is 100 m in this example. Recall that γ(t) is the breakthrough
corresponding to a dirac-δ input of unit strength and is thus dimensionless.
[40] The normalized unconditional response γ(t) (breakthrough) as calculated from the DFN,
RW and MDRW representations of the τ and β distributions are shown in Figure 5 for the three
radionuclides. The RW representation underestimates the breakthrough curve by a significant
amount for all three radionuclides (as much as a factor of 10 for Cs). In all cases the MDRW
representation performs much better, underestimating γ(t) by about 25% at the worst point and
by much smaller amounts for most of the curves. This error is likely to be acceptable for
applications.
[41] Some more compact measures of transport can also be compared. Specifically, the mass
fraction released μ, which is obtained by integrating equation (1) over all times, and the peak
value for γ(t), denoted γ* are listed in Table 2. Both upscaling methods represent μ better than
γ*, but even for γ* the MDRW values are within 25% of the DFN values.
Example Simulations at the Geosphere Scale
Figure
after
6. Normalized unconditional discharge for two radionuclides
upscaling to the 500 m scale. DFN simulations at this scale are
not feasible because of computational limitations.
Figure 2
[42] The computational requirements for the MDRW algorithm are smaller than those of a full
DFN simulation by many factors of ten, and once a small-scale DFN simulation has been
completed the MDRW can be used to extrapolate to larger scales. An example is shown in
Figure 6. In constructing Figure 6 the MDRW algorithm was executed for l = 500 m. The
resulting τ, β joint distribution was then used to construct γ(t) for 126Sn and 129I. The γ(t) for 135Cs
was very small (10−18) and is not plotted. Recall that γ(t) is the breakthrough corresponding to a
dirac-δ input of unit strength, and is thus dimensionless.
[43] The results in Figure 6 demonstrate that it is feasible to use direct upscaling of DFN
simulations to estimate geosphere-scale transport. Note that DFN simulations at the geosphere
scale are not feasible due to computational limitations. The 100 m × 100 m × 100 m DFN
simulations required a few days to complete. Relevant scales for the geosphere are 500–1000 m
and larger. DFN simulations at the 1000 m scale would increase the memory requirements by a
factor of 103, and the computational requirements would increase by a much larger factor because
the simulation time in such simulations increases nonlinearly with the number of fractures.
Discussion
[44] Our approach for upscaling the τ.B distribution to field scales considers that particles hop
from fracture intersection to fracture intersection. This conceptualization is intuitive and has
been used in other studies of transport in fractured rock.
[45] One fundamental difference between this work and previous works that used a similar
conceptualization is in how retention is incorporated. Specifically, we use the random walk to
determine the distribution for the auxiliary variables τ.B. This determination is but one step in the
calculation of breakthrough curves; the joint distribution f(τ, B) must then be combined with a
mechanistic model for the retention mechanisms (see (2)). This approach allows mechanistic
understanding of retention processes to be incorporated in a direct way. Once the τ.B distribution
is calculated, multiple retention models can then be evaluated. This latter feature is particularly
convenient for nuclear waste repository studies, which often require that alternative models be
evaluated.
[46] Another new aspect of this work is that we use individual segment properties directly from
a DFN simulation in a Monte Carlo simulation, without fitting a model distribution. Schwartz
and Smith [1988] also use a Monte Carlo simulation, but they first fit a model distribution and
do not address retention. The advantage of our Monte Carlo option is that it is extremely simple
to implement. Only a few lines of code are required, and the simulation executes very quickly.
Of course, Monte Carlo approaches do not yield the same level of insight that can be obtained
through analytical methods, and we regard the approach as a practical tool that can be used to
estimate transport at a given site. For more generic insights into the transport process, methods
like CTRW and its extensions are available.
[47] The Monte Carlo method is also able to include correlation between successive segments
in the trajectory through the fractured rock mass. This correlation is clear from our generic and
site-specific DFN simulations. The results in Figure 5 demonstrate that large errors are
introduced if this sequential correlation is neglected. The particular variants of CTRW that have
been used to date to study transport in fractured rock are based on an assumption that the
segments along the trajectory are mutually independent; correlation between successive
segments in the trajectory is not included. Variants of CTRW that do include this sequential
correlation have been developed in the physics literature [e.g., Hughes, 1995], but application
of such methods to transport in fractured rock has yet to be demonstrated.
Conclusions
[48] In conclusion, direct upscaling of DFN simulations provides an alternative to site-scale
continuum transport models. The suggested procedure is to first perform small-scale DFN
simulations utilizing site-specific information on the fracture network, and then use the results
collected from these DFN simulations in a Monte Carlo calculation to obtain transport results at
the field scale. The approach avoids volume averaging and other assumptions inherent in the
continuum approach. It preserves the highly non-Gaussian velocity statistics and the spatial
correlation in velocity that are observed in DFN simulations. It also allows mechanistic models
for retention processes to be incorporated directly, including the effects of spatial variability in
retention properties. We demonstrated that these processes can be modeled in combination and at
the geosphere scale with relatively modest computational effort.
[49] Although not specifically addressed in the numerical tests presented here, variations in
fracture aperture within each fracture can also be accommodated. Specifically, if internal
variability in fracture apertures is included in the small-scale DFN simulations, then the effect of
this variability will be embedded in the single-fracture statistics of travel time and cumulative
reactivity, and will thus be carried forward into the upscaled results. DFN simulation with
internal aperture variability has been demonstrated previously [e.g., Nordqvist et al., 1992,
1996], and is now available as an option in commercial software. For τ, internal variability is
anticipated to be less important than fracture-to-fracture variability. However, internal variability
may result in channeling and alter the cumulative retention parameter B. Additional study is
required to assess the magnitude of the effect.
[50] One key finding of this study is that correlation between successive segments along the
particle trajectory is an important control on the breakthrough curves. Random walk models for
transport in fractured rock need to incorporate this sequential correlation. The Monte Carlo
algorithm described here achieves this in a simple and direct way.
[51] Two straightforward modifications of the upscaling approach may be needed for
applications. The method is based on the assumption that the jump statistics from the DFN
simulations are representative of the jump statistics for the larger region of interest. This would
not be true if the statistical properties of the networks vary significantly over the larger region of
interest. Such nonstationarity is not uncommon and can be addressed in applications by simply
dividing the larger region of interest into subregions with approximately constant network
properties in each. For anisotropic networks, the jump statistics may also depend on the direction
of macroscopic gradient relative to the principal directions of the network. If the direction of the
macroscopic gradient varies significantly over the larger region of interest, similar subdivision
may also be needed. This would, of course, require a separate set of DFN simulations to obtain
the jump statistics in each modeled subregion.
Acknowledgment
[52] The authors thank Jan-Olof Selroos, the Swedish Nuclear Fuel and Waste Management
Company (SKB), and the Southwest Research Institute Advisory Committee for Research for
supporting this research.
WATER RESOURCES RESEARCH, VOL. 41, W02002, doi:10.1029/2004WR003682, 2005
Figures
Figure 1. Cumulative distribution (CD) and complementary cumulative distribution (CCD) of
global τ and β from discrete fracture network simulations compared with upscaling results based
on a random walk (RW) and Markov-directed random walk (MDRW) simulation.
Figure 2. Cross plots of nonreacting travel time τ and cumulative reactivity parameter β (left)
from discrete fracture network simulations and (right) upscaled from the single-fracture statistics
by the Markov-directed random walk algorithm. The high correlation between τ and β is
reproduced by the upscaling method.
Figure 3. Detail of cumulative distribution for global β from discrete fracture network
simulations compared with upscaling results based on a random walk (RW) and random walk
directed by 2-state, 5-state, and 20-state Markov processes. The mismatch between the Markovdirected random walk (MDRW) and the DFN simulations improves as the number of states is
increased, but most of the improvement occurs when going from one state (the RW situation) to
two states in the directing process.
Figure 4. Cumulative distribution (CD) and complementary cumulative distribution (CCD) of
global τ and β from generic discrete fracture network simulations compared with upscaling
results based on a random walk (RW) and Markov-directed random walk (MDRW) simulation.
In this example, statistics extacted from 30 × 30 × 30 m DFN simulations were used in the two
upscaling algorithms to predict the distributions at the 90 × 30 × 30 m scale.
Figure 5. Unconditional discharge (breakthrough curves) for three radionuclides as calculated
from DFN simulations and the two upscaling methods. The travel distance is 100 m in this
example. Recall that γ(t) is the breakthrough corresponding to a dirac-δ input of unit strength and
is thus dimensionless.
Figure 6. Normalized unconditional discharge for two radionuclides after upscaling to the 500 m
scale. DFN simulations at this scale are not feasible because of computational limitations.
Citation: Painter, S., and V. Cvetkovic (2005), Upscaling discrete fracture network simulations: An alternative to
continuum transport models, Water Resour. Res., 41, W02002, doi:10.1029/2004WR003682.
Copyright 2005 by the American Geophysical Union.
WATER RESOURCES RESEARCH, VOL. 41, W02002, doi:10.1029/2004WR003682, 2005
Tables
Table 1. Half-Life, Diffusion Coefficient, and Distribution Coefficients for Three Radionuclides
Used as Examples in This Studya
Tracer t1/2, years D, m2/yr Kd, m3/kg
126
Sn
1.0e5
1.3e-6
1.0e-3
129
I
1.6e7
3.9e-6
0
135
Cs
2.3e6
1.3e-6
5.0e-2
a
Read 1.0e5 as 1.0 × 105.
Table 2. Peak of the Impulse Response γ* and Total Mass Released μ for Three Radionuclides,
as Generated With the RW and MDRW Upscaling Methods Compared With Those From DFN
Simulationsa
γ*
Tracer RW
μ
MDRW DFN
RW
MDRW DFN
126
Sn
7.1e-7 1.4e-6
2.0e-6 0.086 0.12
0.14
129
I
4.5e-4 7.4e-4
9.2e-4 0.97
0.97
135
Cs
2.6e-9 7.4e-9
1.1e-8 0.013 0.029
a
0.97
0.037
Scale is 100 m.
Citation: Painter, S., and V. Cvetkovic (2005), Upscaling discrete fracture network simulations: An alternative to
continuum transport models, Water Resour. Res., 41, W02002, doi:10.1029/2004WR003682.
Copyright 2005 by the American Geophysical Union.
Figure 3