w MODELING CONTAMINANT TRANSPORT IN FRACTURED ROCK WITH A HYBRID APPROACH BASED ON THE BOLTZMANN TRANSPORT EQUATION Roland Benke and Scott Painter ABSTRACT Fractures often represent the primary transport pathways within low-permeability rock. Continuum models based on the advective-dispersionequation have difficulty representing the complex transport phenomenology observed in field studies. Discrete fracture network models provide greater flexibility at the price of being computationally intensive, which severely limits their application to small rock volumes. To overcome these shortcomings for simulating transport in fractured rock, the linear Boltzmann transport equation from the kinetic theory of gases includes an additional dependencies on the particle speed and direction of travel and was evaluated for modeling more complex transport behaviors. Parameters appearing in the Boltzmann equation were calibrated using small-scale discrete fracture networks. The calibrated Boltzmann model can simulate at spatialhemporal scales that would be prohibitively expensive with a discrete fracture network. This hybrid approach was successfully tested in two-dimensions for isotropic and anisotropic fracture orientations through comparisons to discrete fracture network simulations. Keywords: Contaminant transport, Discrete fracture network, Monte Carlo, Boltzmann transport equation, Fractured rock
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MODELING CONTAMINANT TRANSPORT IN FRACTURED ROCK WITH
A HYBRID APPROACH BASED ON THE BOLTZMANN TRANSPORT EQUATION
Roland Benke and Scott Painter
ABSTRACT
Fractures often represent the primary transport pathways within low-permeability rock.
Continuum models based on the advective-dispersion equation have difficulty representing the
complex transport phenomenology observed in field studies. Discrete fracture network models
provide greater flexibility at the price of being computationally intensive, which severely limits
their application to small rock volumes. To overcome these shortcomings for simulating
transport in fractured rock, the linear Boltzmann transport equation from the kinetic theory of
gases includes an additional dependencies on the particle speed and direction of travel and was
evaluated for modeling more complex transport behaviors. Parameters appearing in the
Boltzmann equation were calibrated using small-scale discrete fracture networks. The calibrated
Boltzmann model can simulate at spatialhemporal scales that would be prohibitively expensive
with a discrete fracture network. This hybrid approach was successfully tested in two-dimensions
for isotropic and anisotropic fracture orientations through comparisons to discrete fracture
network simulations.
Keywords: Contaminant transport, Discrete fracture network, Monte Carlo, Boltzmann transport
equation, Fractured rock
INTRODUCTION
Transport in fractures is a key issue in many hydrogeological applications involving low-
permeability rock. Flow and associated advective transport in the host rock are often not
significant in these applications, and interconnected networks of rock fractures form the primary
pathways for release of toxic chemical or radioactive wastes to the accessible environment.
Applications include safety assessments of land disposal sites and deep disposal wells for
chemical or low-level radioactive wastes, protection of fractured aquifer water wells, and safety
assessments of proposed geological repositories for high-level nuclear waste. Fractures provide
the most plausible pathways to the biosphere for nuclear waste buried at depth, and therefore, it
is crucial to have quantitative tools for modeling radionuclides movement through these fracture
networks.
The advective-dispersion equation (ADE), the conventional approach to modeling
transport in granular aquifers, is a questionable approximation at the scales of interest in many
The travel time to the next interaction is calculated as
di ti+l = ti t -
vi (A-2)
where vi represents the outgoing speed of the ifh fracture intersection and incoming speed of the
(i+l)'h fracture intersection. This travel time, ti+l, is compared to the time remaining in the
simulation. If the travel time is less than the time remaining in the simulation, the particle
reaches the (i+l)'h fracture interaction, whose position is calculated as
xi+l = xi t di . COS(^) yi+, = yi t di * sin(B,)
where 6 represents the outgoing angle of the ith fracture intersection and incoming angle of the
(i+l)'h fracture intersection. The particle speed and angle emitted from the fracture intersection
are extracted by sampling from the results of the discrete fracture network. Specifically, a
random real number is sampled between zero and one, and the element corresponding to location
of the value of in the discrete cumulative distribution function of the transition probabilities is
selected. The selected element is converted into the corresponding speed and direction in the
discrete probability distribution function for the transition probabilities. Using Eq. A-1, the
distance to the next fracture intersection is calculated and the process is repeated until the
particle travel time to the next fracture intersection exceeds the time remaining in the simulation,
T. When the particle does not reach the next fracture interaction before the end of the simulation,
the final location of the particle is calculated based on the time remaining and particle's current
speed and direction for a straight-line path from the location of the last fracture intersection. The
final position of the particle is calculated as
Xend = x i t (T - t i ) * v i . cos(B,)
Y e n d
(A-4)
where i represents the number of the last interaction before the simulation time is exceeded.
The Monte Carlo simulation allows computations of the particle locations for multiple
user-specified times. For each particle interaction progressing through the simulation, if the time
to the next interaction exceeds any of the additional times and if it is the first occurrence for that
additional time, the particle position at the additional time is calculated from the time difference
between the last interaction and the additional time, the outgoing direction and speed from the
last interaction, and the location of the last interaction.
As an output of the Monte Carlo simulation, the arrival time of particles at an imaginary
barrier with a predetermined x-coordinate were calculated. Particles were only allowed to arrive
once. The arrival time for each particle is initialized to the simulation time and is overwritten
when the particle arrives. Therefore, those particles that never arrive during the simulation time
retain an arrival time equal to the simulation time and can be identified by the vertical line at the
tail of the arrival time distribution.
APPENDIX B: Testing of the Two-Dimensional Boltzmann Monte Carlo Simulator
Arbitrary units were used in testing of the Boltzmann Monte Carlo simulator and should
be assumed throughout this section only. A one-speed, two-dimensional simulator was
developed for isotropic scattering with a mean free path length of 200 (arbitrary units) and
compared to analytical solutions of the diffusive problem. After a long simulation time of 20,000
(arbitrary units), the simulator computed the number of particles per unit area in six radial rings
for comparison against the flux shape calculated from the solution of diffusion differential
equation with a constant source at the origin releasing one particle per unit of time with a
vacuum boundary located at a radial distance of 1000 (arbitrary units). Because speed was set
equal to 1 .O (arbitrary units), the number of particles per arbitrary unit of area is equivalent to the
particle flux for the 1 -speed simulation. A well-known solution to the steady-state neutron
diffusion equation in cylindrical coordinates is a Bessel function (Lamarsh, 1983) of the form:
where JO represents an ordinary Bessel function of the first kind and A and B represent constants.
A boundary condition on the diffusion equation was established based on the fact that the
particle flux must be zero at a radius just larger than the product of the largest particle speed,
v,,,,, and the total simulation time, T. Therefore, the constant B can be replaced by vo/(v,,,, T),
where vo represents the first zero of Jo (approximately 2.4048). The flux shape, q5, from the
neutron diffusion equation becomes
2.404 8r ( v,,T ~ r ) = A . J,,
where the constant A was determined by minimizing the chi-squared fit to the simulated data.
The statistical error in the simulated data was approximated by the square root of the number of
particles divided by the area of the radial ring. Figure B-1 displays the results of the comparison.
Because the diffusion equation is not an exact solution and breaks down close to the source and
medium boundary (Lamarsh, 1983), the Boltzmann Monte Carlo simulator was compared to the
diffusion equation at radii of 300,400, 500, 600, 700, and 800.
ACKNOWLEDGMENTS
This work was completed under an internal research and development project at the
Southwest Research Institute. The authors wish to thank 0. Pensado for his technical review and
B. Sagar for his programmatic review.
REFERENCES
Andersson, J., and B. Dverstrop, Conditional simulation of fluid flow in three-dimensional
networks of discrete fractures, Water Resour. Res., 23, 1876-1 886, 1987.
Berkowitz, B., and H. Scher, Theory of anomalous chemical transport in random fracture
networks, Phys. Rev. E, 57(5), 5858-5869, 1997.
V as/
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Buckley, R.L., S.K. Loyalka, and M.M.R. Williams, Numerical studies of radionuclide migration
in heterogeneous porous media, NucL Sci. and Eng., 1 18, 1 13-144, 1994.
Cacas, M.C., E. Ledoux, G. de Marisly, A. Barbreau, P. Calmels, B. Gaillard, and R. Margritta,
Modelling fracture flow with a stochastic discrete fracture network: Calibration and
validation, 2, The transport model, Water Resour. Res. , 26,491-500. 1990.
Dershowitz, W.S. Rock joint systems. Ph.D. Thesis, Massachusetts Institute of Technology,
Cambridge. 1984.
Duderstadt, J.J., and L.J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, Inc., New
York, 1976.
Dverstrop, B., and J. Andersson, Application of the discrete fracture network concept with field
data: possibilities of model calibration and validation, Water Resour. Res., 25(3), 540-550,
1989.
Dverstrop, B., J. Andersson, and W. Nordqvist, Discrete fracture network interpretation of field
tracer migration in sparsely fractured rock, Water Resour. Res., 28,2327-2343, 1992.
Endo, H.K., J.C.S. Long, C.K. Wilson, and P.A. Witherspoon, A model for investigating
mechanical transport in fractured media, Water Resour. Res. , 20( lo), 1390-1400, 1984.
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Lamarsh, J. R., Introduction to Nuclear Engineering, 2nd edition, Addison-Wesley Publishing
Company, Inc., Reading, Massachusetts, 1983.
Long J.C.S., J.S. Remer, C.R. Wilson, and P.A. Witherspoon., Porous media equivalents for
networks of discontinuous fractures, Water Resour. Res., 18(3), 645-658, 1982.
Long J.C.S., P. Gilmour, and P.A. Witherspoon, A model for steady state flow in random, three-
dimensional networks of disk-shaped fractures, Water Resour. Res., 2 1 @), 1 1 15-1 150, 1992.
National Research Council. Rock Fractures and Fluid Flow, Contemporary Understanding and
Applications, National Academy Press, Washington, DC, 1996.
Neretnieks, I., Diffusion in the rock matrix: An important factor in radionuclide retention, J.
Geophys. Res., 85,4379-4397, 1980.
Neretnieks, I., Solute transport in fractured rock: Applications to radionuclide waste repositories,
in Flow and Contaminant Transport in Fractured Rock, pp. 39-128, Academic, San Diego,
California, 1993.
Painter, S., Cvetkovic, V., and Selroos, J-O., Power-law velocity distributions in fracture
networks: Numerical evidence and implications for tracer transport, Geophys. Res. Lett.,
29(14), 21-1-21-4,2002.
Robinson, P. Connectivity, flow and transport in network models of fractured media, Ph.D.
Thesis, Oxford University, Oxford, 1984.
Schwartz, F.W., and L. Smith, A continuum approach for modeling mass transport in fractured
media, Water Resour. Res., 24(8), 1360-1372, 1988.
Shapiro, A.M. and J. Anderson, Simulation of steady state flow in three-dimensional fracture
networks using the boundary element method, Advances in Water Research, 8(3), 106-1 10,
1985.
Smith, L. and F.W. Schwartz, An analysis of the influence of fracture geometry on mass
transport in fractured media, Water Resour. Res., 20(9), 1241-1252, 1984.
Williams, M.M.R., Radionuclide transport in fractured rock. A new model: Application and
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Williams, M.M.R., Stochastic problems in the transport of radioactive nuclides in fractured rock,
Nucl. Sci. andEng., 112,215-230, 1992.
FIGURE CAPTIONS
V
Figure 1. Example of a two-dimensional discrete fracture network simulation. Shown are: (a) the
set of discrete fractures modeled as line elements, (b) the network backbone after trimming dead-
end elements, and (c) six particle trajectories. In (b) the segments are color coded according to
the amount of flux each carries; warm colors indicate large flux. In (c) each trajectory is assigned
a different color. Some trajectories share common segments and the last trajectory to be plotted
obscures the previous ones.
Figure 2. Comparison plots for a simulation time of 11 days with an isotropic fracture network:
(a) Monte Carlo Particle Tracker with 11 linear speed bins, (b) Monte Carlo Particle Tracker
with 11 logarithmic speed bins, and (c) discrete fracture network.
Figure 3. Comparison plots for a simulation time of 22 days with an isotropic fracture network:
(a) Monte Carlo Particle Tracker with 11 linear speed bins, (b) Monte Carlo Particle Tracker
with 11 logarithmic speed bins, and (c) discrete fracture network.
Figure 4. Comparison plots for a simulation time of 55 days with an isotropic fracture network:
(a) Monte Carlo Particle Tracker with 11 linear speed bins, (b) Monte Carlo Particle Tracker
with 11 logarithmic speed bins, and (c) discrete fracture network.
Figure 5. Residence (arrival) time distributions calculated by the DFN simulations (individual
points) and the Boltzmann method with logarithmic speed bins (solid curves). The cumulative
and complementary cumulative distributions are both shown to emphasize both tails of the
distribution. The Boltzmann method produces a residence time distribution that is a little
U
narrower than the DFN simulation, but the discrepancy is relatively small. The vertical line at the
tail of the CCDF for the Boltzmann method is an artifact of the simulation ending before the last
few particles arrived.
Figure 6. Example extrapolation of small-scale DFN simulations using the Boltzmann method
with logarithmic speed bins. Parameters appearing in the Boltzmann model were extracted from
the small-scale DFN simulations (40 m x 40 m), and the Boltzmann model was then used to
extrapolate to much larger spatial scales. The plume is shown at 6 years. Note that a DFN of this
scale could not have been simulated because of the enormous computational requirements.
Figure 7. Comparison plots for a simulation time of 14 days with an anisotropic fracture
network: (a) Monte Carlo Particle Tracker with 12 logarithmic speed bins and (b) discrete
fracture network.
Figure 8. Comparison plots for a simulation time of 28 days with an anisotropic fracture
network: (a) Monte Carlo Particle Tracker with 12 logarithmic speed bins and (b) discrete
fracture network.
Figure 9. Comparison plots for a simulation time of 69 days with an anisotropic fracture
network: (a) Monte Carlo Particle Tracker with 12 logarithmic speed bins and (b) discrete
fracture network,
w V
Figure B-1. Comparison of the One-Speed Monte Carlo Particle Tracker to the Flux Shape from
the Solution of the Diffusion Equation for a Constant Source at the Origin with a Vacuum
Boundary at a Radius of 1000.
Ir,
FIGURES
a
U
b
0 ' - - - - - - - . -
0 IO m t
Macroscopic Flow
Figure 1. Example of a two-dimensional discrete fracture network simulation. Shown are: (a) the set of discrete fractures modeled as line elements, (b) the network backbone after trimming dead-end elements, and (c) six particle trajectories. In (b) the segments are color coded according to the amount of flux each carries; warm colors indicate large flux. In (c) each trajectory is assigned a different color. Some trajectories share common segments and the last trajectory to be plotted obscures the previous ones.
5
0
-5
-10
D
-15 I 1
5 10 15 20 25 30 35 4
b
20
-15 I I
5 10 15 20 25 30 35 C
5
0
-5
-15 I .. ..
.. . I
5 10 15 20 25 30 35 4
3
D
Figure 2. Comparison plots for a simulation time of 11 days with an isotropic fracture network: (a) Monte Carlo Particle Tracker with 11 linear speed bins, (b) Monte Carlo Particle Tracker with 11 logarithmic speed bins, and (c) discrete fracture network.
a 2 0 ,
. *: , .: *. . . . ...
5 10 15 20 25 30 35
b 3
20 1
I I
5 10 15 20 25 30 35 40 C
2 0 I
1 5 10 15 20 25 30 35 8
~’
0
Figure 3. Comparison plots for a simulation time of 22 days with an isotropic fracture network: (a) Monte Carlo Particle Tracker with 11 linear speed bins, (b) Monte Carlo Particle Tracker with 11 logarithmic speed bins, and (c) discrete fracture network.
a
-15.
20 1 1
. . . . . . . . . . . .. . . . . . *. .':. .
. . . I .. . . I
b
20
. . . . *. . . . . . . . I
5 1 0 1 5 20 25 30 35 40 C
20 I I
. . .- .. -15
.. 5 10 15 20 25 30 35 40
Figure 4. Comparison plots for a simulation time of 55 days with an isotropic fracture network: (a) Monte Carlo Particle Tracker with 11 linear speed bins, (b) Monte Carlo Particle Tracker with 1 1 logarithmic speed bins, and (c) discrete fracture network.
0. I 0.2 0.5 1 2 5 10 Arrival Time (years)
Figure 5. Residence (arrival) time distributions calculated by the DFN simulations (individual points) and the Boltzmann method with logarithmic speed bins (solid curves). The cumulative and complementary cumulative distributions are both shown to emphasize both tails of the distribution. The Boltzmann method produces a residence time distribution that is a little narrower than the DFN simulation, but the discrepancy is relatively small. The vertical line at the tail of the CCDF for the Boltzmann method is an artifact of the simulation ending before the last few particles arrived.
Boltzmann Monte Carlo
0 200 400 600 800 x position (meters)
Figure 6. Example extrapolation of small-scale DFN simulations using the Boltzmann method with logarithmic speed bins. Parameters appearing in the Boltzmann model were extracted from the small-scale DFN simulations (40 m x 40 m), and the Boltzmann model was then used to extrapolate to much larger spatial scales. The plume is shown at 6 years. Note that a DFN of this scale could not have been simulated because of the enormous computational requirements.
a
0
-20
10 20 30 40
b
20
10
-10
-20 ’
I I
10 20 30 40 I
Figure 7. Comparison plots for a simulation time of 14 days with an anisotropic fracture network: (a) Monte Carlo Particle Tracker with 12 logarithmic speed bins and (b) discrete fracture network.
W
a
I 10 20 30 40 I
b
0
-lD/ 10 20 30 40
Figure 8. Comparison plots for a simulation time of 28 days with an anisotropic fracture network: (a) Monte Carlo Particle Tracker with 12 logarithmic speed bins and @) discrete fracture network.
a
V
10 20 30 4 0
b
7
.. . z . :
-201 , , , - - ,
10 20 30 40 I
Figure 9. Comparison plots for a simulation time of 69 days with an anisotropic fracture network: (a) Monte Carlo Particle Tracker with 12 logarithmic speed bins and @) discrete fracture network.
0.0025
0.0020 Y .I E 1
t .= 0.0015 P
: 0.0010 .I Y
4-r 0 L 2 0.0005 E i
i -Fitted Diffusion Equatior
0.0000 0 200 400 600
Radius
800 1000
Figure B-1. Comparison of the One-Speed Monte Carlo Particle Tracker to the Flux Shape from the Solution of the Diffusion Equation for a Constant Source at the Origin with a Vacuum Boundary at a Radius of 1000.