Ensemble Transport of Conservative Solutes in Simulated Fractured Media and the Correspondence to Operator-Stable Limit Distributions

Transcription

1 University of Nevada, Reno Ensemble Transport of Conservative Solutes in Simulated Fractured Media and the Correspondence to Operator-Stable Limit Distributions A dissertation submitted in partial fulfillment of the Requirements for the degree of Doctor of Philosophy in Hydrogeology by Donald M. Reeves Dr. David A. Benson, Dissertation Advisor May, 2006

2 Copyright by Donald M. Reeves 2006 All Rights Reserved

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4 i Abstract Transport of conservative solute particles in fracture networks with fractal length distributions is explored in the anticipation that complex transport behavior can be described by simple analytical equations. Synthetic particle plumes are produced using numerical simulations of fluid flow and solute transport through large-scale, random fracture networks. These two-dimensional networks are generated according to fracture statistics obtained from field studies which control fracture length, transmissivity, density and orientation. Characterization of solute plumes reveal that the ensemble particle motion converges to operator-stable densities that can be modeled using either classical integer-order or fractional-order advection-dispersion equations (ADE). Selection of the order of the ADE depends on distinguishing between Fickian and super-fickian transport regimes, which is primarily controlled by the distribution of fracture trace lengths. Lowto-moderately fractured networks with infinite variance fracture lengths produce solute plumes that exhibit power-law leading edge concentration profiles with super-fickian growth rates. For these network types, a multiscaling fractional-order ADE provides a multidimensional model of solute transport where different rates of plume growth are defined in multiple directions. This equation uses a matrix to describe a super-fickian growth process, in which the eigenvectors, or dominant plume growth directions, correspond to primary fracture group orientations. The eigenvalues of the matrix, which describe rates of plume growth, depend primarily on the distributional properties of fracture length and transmissivity. An arbitrary measure on the unit sphere accounts for the propensity for ensemble travel in any direction. The convergence of particle motion to a multi-gaussian (a subset of operator-stable) density occurs for densely fractured

5 ii networks with power-law fracture length exponent values greater than or equal to 1.9. Elliptical multi-gaussian plumes can be described using the classical integer-order ADE where Fickian growth rates occur along orthogonal plume directions. Deviations between ensemble plumes and individual plume realizations are significant for all network types, indicating that ADEs cannot adequately account for variability in a fractured medium from a single realization perspective. However, in the ensemble, ADE represent the full range of variability and remain useful for fracture media studies; particularly for designing a monitoring well network for a geologic repository.

6 iii Acknowledgements First and foremost, I need to thank David Benson, Mark Meerschaert, Peter Scheffler, and Steve Wheatcraft of the fractional research interest group for both introducing me to the field of theoretical solute transport and providing me with the necessary skill set to conduct research in this challenging discipline. This dissertation is a direct testament to their tutelage as they collectively mentored a student with very little background in either mathematics or contaminant transport. I am especially grateful for the countless hours of instruction that they provided. When I originally decided to pursue a Ph.D. in hydrogeology, I remember looking through the list of faculty in the Hydrologic Science Program at UNR. A DRI faculty member named David Benson described the use of fractional-order derivatives to account for the divergence of a non-gaussian solute flux vector. I had no idea of what he was referring to and made a mental note that I would never work with this guy. Besides, how could this stuff apply to real-world transport problems? Well, you know what happens when you say never. Five years later I am putting the finishing touches on this dissertation and am amazed at my progress during this time period. I can directly attribute much of my professional growth to Dave s mentoring skills and his open-door policy. Thank you for originally accepting me as a student and then friend, the free lunches, and the turns, or should I say research meetings at Mt. Rose. It is unfortunate that UNR students no longer have the opportunity to work with Dave as he is now at the Colorado School of Mines. Mark Meerschaert has been extremely helpful in teaching me the mathematics associated with the fractional advection-dispersion equation and heavy-tailed stochastic

7 iv processes. I am grateful that a mathematician of his caliber took the time to explain these advanced concepts to me. Our sessions were always intense as he could enhance my understanding of mathematics more in a couple of hours than a semester-long math class. I always felt like I went through a growth spurt after one of our meetings. Mark moved to University of Otago in New Zealand in January of In his absence, Peter Scheffler has helped answer my questions and further enhanced my understanding of operator-stable densities. Peter was always available to provide help and treated me as a colleague rather than a student. I also need to thank Steve Wheatcraft. When I first moved to Reno after obtaining a Master s in Hydrogeology from the University of Montana, I was the teaching assistant for his Ground Water Hydrology intro course. I was shocked at how much I learned as a TA in his class. His explanation of subsurface flow and transport theory is much different from my previous training and I appreciate the knowledge that I have acquired through his classes. It was Steve who first introduced me to the rigors of the advectiondispersion equation. He also taught me how to program in Fortran which was essential to the numerical aspects of this work. The remainder of my dissertation committee, Richard Schultz and Greg Pohll, deserve my gratitude. Richard Schultz taught me the mechanics behind fracture formation in rock. The addition of his expertise helped me formulate a well-rounded dissertation project. Greg Pohll was always available to answer my questions with MODFLOW or Fortran. He also helped me obtain a post-doc at the Desert Research Institute.

8 v The faculty and staff at Desert Research Institute provided me with a fantastic research environment. This research would have not been possible without the DRI ACES Supercomputer. A big thanks goes to Shulan Liu for teaching me how to use a supercomputer and for the Linux tips. While at DRI, I was fortunate to receive the DRI G.B. Maxey and NSF Nevada EPSCor ACES Fellowships and the DRI Collin Warden Award. Several others that I need to thank include: Eric LaBolle for making helpful revisions to his particle tracking code, Tom Fenstermacher for providing me with a Surfer script for plotting particle displacement plumes, and Yong Zhang for helping me with numerical problems that I encountered along the way. Rina Schumer also helped me by providing a copy of her dissertation and answering some of my questions regarding the multiscaling FADE. Justin Bartlett gave me a big hand in formatting this document. Last, but not least, I need to thank both my family and my wife, Molly, for their support and love through this long and difficult process. I cannot thank Molly enough for her endless patience as I was finishing this work. I am lucky to call her my wife and am grateful for our daughter Zoe.

9 vi TABLE OF CONTENTS Abstract... i Acknowledgements...iii Introduction... 1 Research Objectives... 3 References... 6 Multi-Scale Transport of Conservative Solutes in Simulated Fractured Media... 8 Abstract Introduction Characteristics of Operator-Stable Plumes Simulation of Fractures, Flow, and Transport Fracture Group Orientation Fracture Trace-Length Spatial Distribution of Fractures Fracture Spatial Density Fracture Transmissivity Fracture Implementation in MODFLOW Particle Tracking Results and Discussion Coordinate Transform Tail Estimation Methods Plume Scaling Rates Ensemble Plume Characteristics Set 1: Sparse Networks with Long Fractures Set 2: Medium Density and Intermediate Fracture Lengths Set 3: Dense Networks with Shorter Fractures Arrival of Particles at Model Boundaries Characteristics of Individual Solute Plume Realizations Conclusion References Ensemble Transport of Solutes Through Fracture Networks and the Correspondence to Operator-Stable Limit Distributions Abstract Introduction Particle Jumps and Operator-Stable Densities Synthetic Data Generation Results and Discussion Analysis of Ensemble Particle Displacement Plumes Estimation of Scaling Matrix, H Mixing Measure Weights Influences of Fracture Length Group 1: 1.0 a

10 vii Group 1 and Group 2 Mix Group 2: 1.9 α Group 2 and 3 Mix Group 3: 2.5 a Influence of Spatial Fracture Density Influence of Fracture Transmissivity Complex Networks Conclusions References On the Predictability of Solute Transport in Fractured Media Abstract Introduction Numerical Simulations of Flow and Transport Properties of the Hydraulic Backbone Probability of Entering the Hydraulic Backbone Transport of Conservative Particles Ensemble Particle Transport Fast Particle Transport Particle Retention Ergodicity and Concentration Field Variability Rock Mass Statistics and Geologic Repositories References Conclusions Recommendations References

11 viii LIST OF TABLES Multi-Scale Transport of Conservative Solutes in Simulated Fractured Media... 8 Table 1. Network Values of all Parameter Sets Table 2. Plume Scaling Rates Table 3. Values of α Based on Truncated Pareto MLE Ensemble Transport of Solutes Through Fracture Networks and the Correspondence to Operator-Stable Limit Distributions Table 1. Network Values for Parameter Sets Table 2. Parameters Required for the Computation of H Table 3. Plume Scaling Rates Table 4. Tail estimates of On the Predictability of Solute Transport in Fractured Media Table 1. Properties of the Hydraulic Backbone Table 2. Kolmogorov Distance Statistics

12 ix LIST OF FIGURES Multi-Scale Transport of Conservative Solutes in Simulated Fractured Media... 8 Figure 1. Fracture network samples representing: (a) sparsely fractured domains dominated by very long fractures (parameter set 1), (b) moderately fractured domains comprised of short and long fractures (parameter set 2) and (c) densely fractured domains dominated by short fractures (parameter set 3). For illustration purposes, the higher density values used in (b) and (c) require smaller subdomains. All values are in meters Figure 2. Finite-difference representation of a hypothetical rock fracture network. Rock fractures (line segments) are overlain onto a finite-difference grid. Fractureoccupied cells (gray) are assigned hydraulic properties based on fracture properties, while hydraulic properties of the matrix are assigned to non fracture-occupied cells (white). Grid discretization is for illustration purposes only Figure 3. Flow through a fracture equivalent on a finite-difference grid consists of both horizontal and vertical flow components (denoted by arrows) resulting in a ``stair step" pattern and a longer flow path than that of the original fracture,. The relationship between the equivalent flow path,, for a fracture of length,, oriented degrees from horizontal is: = [sin( )+cos( )] Figure 4. Porosity of fracture-occupied cell in relation to cell size and transport aperture,. For our simulations, cell porosity = / =, since = = = Figure 5. Idealized particle trajectories (dotted lines) in relation to straight-line distance (solid line) for a given orientation,. Mean curved segment length is assumed to be / Figure 6. Fracture network domain with shaded region representing area of domain subject to introduction of solute particles. The down gradient position of the box was selected to avoid potential boundary effects. All values are given in units of meters Figure 7. Ensemble results for parameter set 1 at an elapsed time of 10 years. (a) Ensemble concentration plume based on particle displacements with principal scaling directions, 1 and 2. Mandlebrot plots of largest ranked particle displacements (circles) along (b) 1 and (c) 2 with best-fit truncated power-law (TPL) model. Approximately every 1/1000 point is plotted. All spatial values are given in units of meters Figure 8. Ensemble results for parameter set 2 at an elapsed time of 46 years. (a) Ensemble concentration plume based on particle displacements with principal scaling directions, 1 and 2. Mandlebrot plots of largest ranked particle displacements (circles) along (b) 1 and (c) 2 with best-fit exponential (light) and truncated power-law (dark) models. Power-law motion occurs along both scaling directions with greater transport rates along the fracture group with the lower exponent value, 2. Note the divergence of the data from exponential model. Approximately every 1/1000 point is plotted. All spatial values are given in units of meters Figure 9. Ensemble results for parameter set 3 at an elapsed time of 2,154 years. (a) Ensemble concentration plume based on particle displacements with orthogonal principal scaling directions, 1 and 2. Mandlebrot plots of largest ranked particle displacements (circles) along (b) 1 and (c) 2. Exponential decay similar to Gaussian tailing,

13 x 2 [ > ], occurs along both scaling directions. Approximately every 1/1000 point is plotted. All spatial values are given in units of meters Figure 10. Values of 2 based on scaling of plume growth along 2 for Parameter Set 2 using all four normalized metrics. Subscripts 1 and 2 denote scaling along either 1 or 2. Slope of the regression lines is 1/. Particles leaving domain boundaries result in undefined quantile estimates for later time steps Figure 11. Particle displacement plumes for three individual realizations. (a) High fracture densities and length exponents result in Gaussian-like, elliptical plumes with little variability between individual realizations and the ensemble. (b) Particle motion along fractures with sparse to intermediate densities and lower fracture exponents can be dominated by one fracture group orientation (1) and (3) or a combination of the two (2) resulting in significant departures from the ensemble by individual realizations. For clarity, only 10% of particles are shown Figure 12. Preliminary correlation between fracture length exponent, fracture density and resultant particle displacement tail value (diamond) based on parameter sets 1-3. Values for both fracture exponent and particle displacement tail represent average values (i.e., 1+ 2]/2). Density values were converted to [m/m 2 ] to provide a more reliable estimate for fracture density as (4) is heavily influenced by fracture length exponent values. The dotted line denotes a crude division between Fickian and super-fickian plume growth. Future research efforts will further evaluate the transition between Fickian and super-fickian plumes in relation to fracture network statistics Ensemble Transport of Solutes Through Fracture Networks and the Correspondence to Operator-Stable Limit Distributions Figure 1. Fracture network domain with shaded region representing area of particle release. Note that a hydraulic backbone is not present in the particle release area for this realization. Spatial values are in meters Figure 2. Ensemble particle displacement plumes for Set 1 at transport times of (a) 4.6 (b) 10 and (c) 464 years. Fast particle transport is dependent on both fracture length and transmissivity values. Note the influence of large continuous fractures on solute transport. All values are given in units of meters Figure 3. Ensemble particle displacement plume for Sets 16 (a) and 17 (b) at transport times of 44,640 and 1000 years, respectively. The only difference between the fractured domains is that Set 16 (a) has a lower density than Set 17 (b). For lower values of spatial density, the tendency of particles to follow fracture orientations ±45º enhances transverse plume spreading and decreases plume migration rates. All values are given in units of meters Figure 4. Mandlebrot plots of largest ranked particle displacements (circles) for Set 1 along (a) 1 and (b) 2 with best-fit truncated power-law (TPL) model at an elapsed time of 0.1 years. Approximately every 1/1000 point is plotted. Values of are given in units of meters Figure 5. Values of based on scaling of plume growth along 1 for Parameter Set 2 using all four normalized metrics. Slope of the regression lines is 1/. Particles leaving domain boundaries result in undefined quantile estimates for later time steps. Estimates

14 xi of based on (stdev) are very sensitive to the loss of extreme values. The change of slope in (stdev) is caused by a significant loss of particles after the fourth time step more rapid rates of plume growth. Estimates of obtained from tail and plume spreading rate methods are both used for the computation of eigenvalues (Table 2) Figure 6. Jurek coordinate system where 1 = 1.0 and 2 = 1.3 lead to unequal directional scaling rates. The value of is defined as the position where the curved radius,, intersects the unit circle Figure 7. Histogram (a) and cumulative distribution plot (b) of mixing measure weights for Set 1 at a transport time of 0.46 years. Midpoint of bin interval is plotted. Note the concentration of weight in the direction of fracture group orientations, 30 and The uneven distribution of mixing measure weights along for eigenvectors is attributed to a greater number of large particle displacements transported in the fracture group oriented at Figure 8. Investigated regions (X) into the parameter space for fracture length exponents, 1 and Figure 9. Ensemble particle displacement plume from Set 11 at a transport time of 21 years. Fickian plume growth occurs along non-orthogonal axes Figure 10. Mandlebrot plots of largest ranked particle displacements (circles) for Set 12 along (a) 1 and (b) 2 with best-fit truncated power-law (TPL) model at an elapsed time of 1.0 years. Approximately every 1/1000 point is plotted. Values of are given in units of meters Figure 11. Histogram (a) and Gaussian probability plot (b) along 1 of Set 14 at a transport time of 100 years. The deviation between the theoretical Gaussian trend and marginal particle displacements is attributed to anomalous subdiffusion (slow particle movement). Approximately every 1/1000 point is plotted. Spatial values are given in units of meters Figure 12. Histogram (a) and Gaussian probability plot (b) along 2 of Set 14 at a transport time of 1000 years. Approximately every 1/1000 point is plotted. Spatial values are given in units of meters Figure 13. Mandlebrot plots of largest ranked particle displacements (circles) for Set 2 along (a) 1 and (b) 2 with best-fit truncated power-law (TPL) model at an elapsed time of 0.1 years. Approximately every 1/1000 point is plotted. Values of are given in units of meters Figure 14. Probability histogram of 10 5 randomly generated Fisher deviates according to dispersion parameters of (a) = 10 and (b) = 50. Note the effect of on deviations about the mean, Figure 15. Ensemble particle displacement plume from Set 20 at a transport time of 10 years. Note the deviation around the mean fracture group orientations, 1 and Figure 16. Histogram (a) and cumulative distribution plot (b) of mixing measure weights for Set 20 at a transport time of 100 years. Midpoint of bin interval is plotted. Fisher distribution curves are presented in (a) for comparison of mixing measure weights and distribution of fracture orientation. Note the clustering of mixing measure weights does not follow mean fracture group orientations of 30 and -60. Instead, weights are shifted towards the direction of the hydraulic gradient

15 xii Figure 17. Ensemble particle displacement plume from Set 22 at a transport time of 100 years. Note the influence of the vertically orientated fracture group Figure 18. Preliminary correlation between fracture length exponent, fracture density and resultant plume growth rates (in the diamonds) for individual fracture groups within Sets A clear threshold between super-fickian and Fickian transport regimes is not present. However, super-fickian transport regimes may be defined in fracture rock masses where individual fracture groups have fracture length and density values less than 1.9 and 0.25 m/m 2, respectively On the Predictability of Solute Transport in Fractured Media Figure 1. Fracture network domain with shaded region representing area of particle release. Note that a hydraulic backbone is not present in the particle release area for this realization. Spatial values are in meters Figure 2. Influence of spatial density ( 2D) on the probability of a solute particle entering the hydraulic backbone, ( ). Values ( ) increase with density in a highly non-linear trend. A best-fit exponential trend line is presented for comparison Figure 3. The influence of mean fracture length ( ) exponent and spatial density ( 2D) on the probability of a solute particle traveling at least 1 km in 10,000 yeasrs ( ( )). The size of the dot is proportional to the value of ( ) Figure 4. The time of the fastest particle slug to reach 1km ( 1km) and mean value of based on rates of plume growth show a poor relationship Figure 5. Slow particle transport is characterized by (a) power-law decay of particles leaving the source area (b) and power-law residence time distributions of 1/. Values of β describe the slope of the power-law trends Figure 6. Computation of a bivariate CDF is similar to its univariate analog. Number of particles per cell (a) is adjusted to to reflect a cumulative particle total (b) where the cumulative number of particles per cell increases in response its row and column position on the grid. A bivariate CDF is obtained by dividing the cumulative number of particles in (b) by the total number of particles (in this case 149) Figure 7. Ensemble plumes with selected individual plume realizations for Set 5 at a transport time of 100 years. Sparse domains dominated by long fractures lead to a high degree of variability between ensemble and individual plumes. All particles have left the domain for the plume realization representing the maximum deviation from the ensemble. All spatial values are in units of meters Figure 8. Ensemble plumes with selected individual plume realizations for Set 5 at a transport time of 1000 years. Dense networks with short fractures lead to a lower degree of variability between ensemble and individual plumes. All spatial values are in units of meters Figure 9. Mean Kolmogorov distance (proportional to size of dots) in relation to mean fracture length exponent and spatial density 2D. Note that mean Kolmogorov distance decreases as mean fracture length exponent and density increase

16 1 Introduction Predicting the transport behavior of a dissolved solute through geologic media is an active area of research in ground water hydrology. Recent interest in long-term disposal of high-level radioactive waste in low-permeability rock masses has focused attention towards the ability to predict the migration of contaminants in fractured media [e.g., ;, 2004]. Naturally occurring fractures within a rock mass provide primary pathways for waste migration in an otherwise impermeable rock matrix. The advection-dispersion equation (ADE) is the classical governing equation for solute transport in subsurface flow systems. The ADE is derived for transport in relatively homogenous granular aquifers where a geologic medium can be approximated as an equivalent continuum and equation parameters describing advective and dispersive fluxes can be volume (or ensemble) averaged [, 1972]. In the case of a fractured medium, the equivalent continuum assumption may only be valid at high fracture densities [., 1982]. Instead of behaving like an equivalent continuum, fractured rock masses, especially those preferred for waste disposal, are sparsely fractured [, 1994;, 2004]. Low fracture densities restrict transport to an interconnected subset of fractures conducive to flow [, 1997;, 1999]. In contrast to the ADE, which describes solute transport using symmetric multi- Gaussian densities, both field and numerical studies indicate that localized transport through a subset of discrete fractures with varying length, orientation and permeability can result in asymmetric plumes and anomalous (non-gaussian) breakthrough tailing [, 1984;, 1997;, 2000;., 2002;, 2004].

17 2 Due to the inadequacy of the ADE for describing transport in fractured media, transport predictions are based on numerical simulations of ground water flow and solute transport [e.g.,, 1996;, 2005, and references therein]. The discrete fracture network (DFN) approach, in particular, is a based on a conceptual model where the majority of fluid flow and transport of solutes occur through a network of interconnected fractures [e.g.,, 1984;., 1991]. The primary drawback to numerical models is that they are site-specific and rely on extensive field characterization efforts to collect physical and hydraulic data on deterministic structures (i.e., rock fractures). Despite the limitations of the upscaled classical ADE for ensemble transport predictions in fractured media, analytical equations may have advantages over numerical models as quick, screening-level transport approximations constructed from limited field data. Current analytical transport models for fractured media include a fractional-order ADE [e.g.,., 2000, 2001;., 2003a,b], continuous time random walk models [e.g.,, 1997;., 2002] and the linear Boltzman transport equation [, 2003]. The applicability and parameterization of a multiscaling fractional-order ADE [., 1999;., 2003a] to fractured media is the primary research focus of this dissertation. The motivation behind the use of a fractional-order ADE for describing transport in fractured media is based on field studies that suggest trace-lengths of natural fracture networks are fractal and may lack a characteristic length scale [e.g.,, 1997;, 1999;., 2001]. Transport of conservative solute particles through networks lacking a characteristic length scale may lead to highly

18 3 asymmetric plumes with power-law probability decay of the largest particle displacements. A fractional-order ADE is based on fractals and may be able to describe this type of transport behavior. Research Objectives This research focuses on the following questions:????? The three papers included in this dissertation address each of these questions. The first paper, Multi-scale transport of conservative solutes in simulated fractured media establishes the methodology used in the stochastic generation of fracture networks, translation of these networks onto a highly-discretized finite-difference grid for solution of flow, and tracking of conservative solute particles through resultant velocity fields. Analyses of ensemble particle plumes are explained in detail. Preliminary data from 3 network types suggest that ensemble transport behavior may be represented by operator-

19 4 stable densities (multi-gaussian is a special case) which are solutions to either the classical integer-order or fractional-order ADE. The second paper, Ensemble transport of solutes thorough fracture networks and the correspondence to operator-stable limit distributions explores the transport behavior of a wide range of fracture networks types. A relationship between particle jumps and operator-stable random vectors is established. Ensemble particle plumes converge to either operator-stable densities with super-fickian growth rates and power-law leading plume edges or multi-gaussian densities with Fickian growth rates and exponential leading plume edges. Fracture network statistics are used to define a threshold between super-fickian and Fickian transport regimes. For networks with infinite variance fracture lengths and low-to-moderate densities, a multidimensional fractional-order ADE provides a good model of transport where eigenvectors of plume growth correspond to primary fracture group orientation and eigenvalues (plume growth rates) are related to distributional properties of fracture length and transmissivity. Networks with infinite variance fracture lengths close to the finite variance threshold with densities greater than specified criteria produce multi-gaussian densities where shorter fracture lengths and higher densities result in symmetric plumes. For these network types, spatial density controls the degree to which a plume spreads; lower densities result in more spreading. More complicated networks containing more than 2 fracture groups with variable orientation are also considered. The third paper, On the predictability of solute transport in fractured media evaluates the predictability of transport in the network types presented in the second paper. In an assessment of the utility of ADEs for fractured media, deviations between

20 5 ensemble plumes and individual realizations are analyzed. The largest variability between the ensemble and individual realizations occurs for ensemble plumes that resemble operator-stable densities and are lowest for ensemble plumes resembling multi- Gaussian densities. The lack of ergodicity implies upscaled solutions of ADEs cannot adequately account for all of the variability in a fractured medium from a single realization perspective. However, in the ensemble, ADEs represent the full range of variability in a fractured medium and may be useful for monitoring well network design. A probabilistic framework is adopted to recommend fracture statistics of rock masses that are most suitable for the disposal of high-level radioactive waste. This evaluation considers properties of the hydraulic backbone including: the probability that a solute particle will enter the hydraulic backbone and the probability, once a solute particle enters the backbone, it travels long distances. Fracture statistics that describe networks with relatively short fracture lengths at densities near the percolation threshold are most suitable for geologic repositories as this set of statistics promotes slow overall plume growth rates and arrival times and moderate variability between individual and ensemble plumes.

21 6 References Bear, J.,, Dover Publications, Inc., New York, Becker, M.W. and A.M. Shapiro, Tracer transport in fractured crystalline rock: Evidence on non-diffusive breakthrough tailing, (7), , Benke, R. and S. Painter, Modeling conservative tracer transport in fracture networks with a hybrid approach based on the Boltzmann transport equation, (11), 1324, doi: /2003wr001966, Benson, D.A., R. Schumer, and M.M. Meerschaert, Application of a fractional advectiondispersion equation, (6), , Benson, D.A., R. Schumer, M.M. Meerschaert, and S.W. Wheatcraft, Fractional dispersion, Lévy motion, and the MADE tracer tests, (1/2), , Berkowitz, B. and H. Scher, Anomalous transport in random fracture networks, (20), , Bonnet, E., O. Bour, N.E. Odling, P. Davy, I. Main, P. Cowie, and B. Berkowitz, Scaling of fracture systems in geologic media, (3), , Bour, O. and P. Davy, Connectivity of random fault networks following a power law fault length distribution, , Dershowitz, W., P. Wallmann, and S. Kindred, Discrete fracture network modeling of tracer migration experiments at the SCV site,, Swedish Nuclear Fuel and Waste Management Co. (SKB), Stockholm, Sweden, Ewing, R.C., M.S. Tierney, L.F. Konikow, and R.P. Rechard, Performance assessments of nuclear waste repositories: A dialogue on their value and limitations (5), , Kosakowski, G., Anomalous transport of colloids and solutes in a shear zone,, 23-46, Long, J.C.S., J.S. Remer, C.R. Wilson, and P.A. Witherspoon, Porous media equivalents for networks of discontinuous fractures, (3), , Meerschaert, M.M., D.A. Benson, and B. Baeumer, Multidimensional advection and fractional dispersion,, , 1999.

22 7 Munier, R., Statistical analysis of fracture data adapted for modeling discrete fracture networks Version 2,, Swedish Nuclear Fuel and Waste Management Co. (SKB), Stockholm, Sweden, National Research Council,, National Academy Press, Washington, D.C., Neuman, S.P., Trends, prospects and challenges in quantifying flow and transport through fractured rocks,, , doi: /s , Painter, S., V. Cvetkovic, and J.O. Selroos, Power-law velocity distributions in fracture networks: Numerical evidence and implications of tracer transport, (14), doi: /2002gl014960, Renshaw, C.E. and D.D. Pollard, Numerical simulations of fracture set formation: A fracture mechanics model consistent with experimental observation,, , Renshaw, C.E., Connectivity of joint networks with power-law length distributions, (9), , Scher, H., G. Margolin, and B. Berkowitz, Towards a unified framework for anomalous transport in heterogeneous media,, , Schumer, R.A., D.A. Benson, M.M. Meerschaert, and B. Baeumer, Multiscaling fractional advection-dispersion equations and their solutions, (1), , 2003a. Schumer, R.A., D.A. Benson, M.M. Meerschaert, and B. Baeumer, Fractal mobile/immobile solute transport, (10), 1-12, doi: /2003wr002141, 2003b. SKB, Research, Design and Development Programme 2004: Programme for research development and demonstration of methods for the management and disposal of nuclear waste, including social science research,, Swedish Nuclear Fuel and Waste Management, Co. (SKB), Stockholm, Sweden, Smith, L. and F.W. Schwartz, An analysis on the influence of fracture geometry on mass transport in fractured media (9), , 1984.

23 8 Multi-Scale Transport of Conservative Solutes in Simulated Fractured Media Donald M. Reeves Desert Research Institute, 2215 Raggio Parkway, Reno, Nevada David A. Benson Colorado School of Mines, Department of Geology and Geological Engineering, 1516 Illinois St., Golden, Colorado Mark M. Meerschaert Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand

24 9 Abstract An extended central limit theorem in multiple directions dictates that a large number of independent and identically distributed particle motions converges to an operatorstable density. If this is a good model of particle motion, then an ensemble plume in a fractured-rock domain may be described using the simple evolution equation of the operator-stable plume. Except for the classical multi-gaussian, which is a subset of the operator-stable, these plumes are characterized by power-law leading-edge concentration profiles and super-fickian growth rates. To check the correspondence of ensemble plumes to operator-stable densities, we use numerical simulations of fluid flow and solute transport through large-scale (2.5 km by 2.5 km), randomly generated fracture networks. These two-dimensional networks are generated according to the fracture statistics obtained from field studies that describe fracture length, transmissivity, density, and orientation. The first two of these fracture parameters are generated using power-law probability densities for the larger values. Fluid flow is solved simultaneously within the fracture network and rock matrix using MODFLOW, while a random walk tracking code, RWHet, is used to simulate the motion of conservative solute through the model domain. Both MODFLOW and RWHet require correction factors to account for differences in fluid flow and particle trajectories on a finite-difference grid. Low to moderate fracture density and long fracture lengths produce non-gaussian operator-stable plumes, while densely fractured networks with short fracture lengths lead to multi-gaussian plumes. Significant variability between ensemble operator-stable plumes and individual plume realizations is attributed to the presence of long fractures and sparsely fractured domains.

25 10 Much less variability was observed between individual plume realizations and ensemble multi-gaussian plumes.

26 Introduction The ability to predict large-scale transport of solutes in fractured rock is essential for evaluating the suitability of rock masses for the long-term disposal of waste including high-level radioactive waste. Fractures can serve as primary pathways for fluid flow and waste migration in a low-permeability rock mass. Transport in these ground water flow systems is dependent on rock fracture properties such as density, spatial location, permeability, length and orientation. The advection-dispersion equation (ADE) relies on volume (or ensemble) averaged parameters in an attempt to describe fractured media as an equivalent continuum [, 1972]. The equivalent continuum assumption may only be valid for fractured media at high fracture densities [., 1982]. Instead of behaving like an equivalent continuum, fractured rock masses, especially those preferred for waste disposal, are sparsely fractured [, 1994;, 2004]. Low fracture densities restrict transport to a small subset of interconnected fractures conducive to flow [, 1997;, 1999]. In contrast to the ADE, which describes solute transport using symmetric multi-gaussian densities, both field and numerical studies indicate that localized transport through discrete fractures can result in asymmetric plumes and non-gaussian breakthrough tailing [, 1997, 1998;, 2000;., 2002;, 2004;, 2004]. As an alternative to equivalent continuum models, the discrete fracture network (DFN) approach was developed to address network-scale fluid flow and solute transport behavior. This approach assumes that fluid flow through a low-permeability rock mass is controlled by interconnected fractures of a network with negligible contribution from the

27 12 rock matrix [e.g.,, 1984;., 1991]. Consequently, DFN simulations have been used to investigate flow and transport behavior in a wide range of fracture network types [e.g.,, 1984;, 1999;., 2001a,b;., 2001]. Since fluid flow can only occur within fractures for a DFN simulation, detailed site characterization is required for the identification and inclusion of deterministic structures into the model domain. To enhance connectivity of deterministic structures, stochastic introduction of background fractures is commonly required for DFN simulations [, 1996;, 2004]. Computational constraints associated with solving flow though a series of fractures with variable lengths and transmissivity values limit DFN simulations to applications where transport scales are 100 meters or less. Data from natural fracture networks suggest that above a certain lower cutoff size, fracture trace-lengths follow a power-law distribution: ( > ) = (1) where the probability of a fracture of length,, is dependent on, a constant that controls minimum fracture length, and power-law exponent,, that ranges between 1 and 3 for natural fracture networks [, 1993,, 1996;, 1996;, 1997;, 1999;., 2001]. Power-law distributions of fracture length lead us to examine other transport theories based on fractals. If < 2, the variance and standard deviation diverge, and these fractures do not have a distinct characteristic length scale. The transport of particles along infinite-variance pathways may result in power-law probability decline similar to (1) for the largest particle jumps. Particle jumps are defined as the distance solute particles travel over a given time interval. In this case, the

28 13 ensemble particle plumes would never converge to a multi-gaussian density, which is the Green function solution of the classical, second-order, multi-gaussian ADE. They may, however, converge to operator-stable plumes, which are the Green function solutions of a fractional-order ADE [., 2003a]. In addition to trace lengths, there are many other complicating factors, such as random fracture transmissivity and fracture density, that may change the statistics of the particle jumps. In this paper, we concentrate on influences of fracture trace-lengths and spatial density on solute transport to investigate whether ensemble particle plumes produced from multiple network realizations can be represented by operator-stable densities. 1.1 Characteristics of Operator-Stable Plumes Operator-stable densities provide a model of multidimensional solute transport where solute particles undergo power-law displacements with the possibility of different scaling exponents along multiple directions [., 2003a]. Lévy s general central limit theorem [, 2001]: lim ( r r r E ) r L (2) 1 2 describes convergence of an appropriately scaled sum of a large number of independent and identically distributed ( ) centered, heavy-tailed random particle displacement r r vectors,, to an operator-stable random vector,. The multi-gaussian follows this law and will be described shortly. Eigenvalues, 1/, of the scaling matrix E are used to describe rescaling, or growth rate in the case of a plume of particles, along the eigenvectors of the scaling matrix.

29 14 According to (2), plume growth in ú is described by -eigenvectors. The largest particle displacements dominate the growth of the process. For fractured media in this study, we focus on two-dimensional (2-D) particle motion where only two eigenvectors r are used to appropriately scale particle displacements,. Orientation of these eigenvectors is unrestricted, as the scaling matrix E need not be symmetric. Thus, the shape of operator-stable plumes can range from elliptical to highly asymmetric. If eigenvectors lie along orthogonal coordinate axes, 1/ 1 E 0 = 1/ 2 0. If the particles are conservative and do not partition to an immobile phase, then time is linear with the number of particle jumps in (2) so that particle motion along the eigenvector scales according to 1/ [., 2003a]. Particle motion, or plume growth, in noneigenvector directions scales according to a mixture of the two scaling rates. Values of need not be equal along both eigenvectors, as plume growth may be more rapid in one direction than another. The methodology used in this study could allow particles to move into the low-permeability matrix; however, for simplicity, the matrix will be impervious to chemical transport. If every component of r has finite variance, then E= (1/2,1/2), and (2) is equivalent to the well-known central limit theorem, where particle displacements are r r described by a multi-gaussian random vector, (, Σ), where r is a mean shift vector and Σ is a covariance matrix. Multi-Gaussian plumes are characterized by elliptical plume geometry with orthogonal, Fickian scaling rates proportional to 1/2 along major and minor axes of an ellipse.

30 15 The application of operator-stable densities to 2-D transport in fractured media is based on a simple model: r r r = (3) where particle displacement, r, is based on marginal distributions of particle jumps,, along principal plume growth directions as defined by coordinate vectors (eigenvectors), r. If the magnitudes of the particle jumps are heavy-tailed enough to converge to - stable random variables, then by definition, r is an operator-stable random vector. Conversely, if r is multi-gaussian, each will converge to a Gaussian [, 2001]. For operator-stable plumes, eigenvectors may correspond to principal fracture group orientations. Multi-Gaussian plumes require orthogonal plume growth directions that will not necessarily correspond to non-orthogonal principal fracture group orientations [, 2004]. In summary, the distinguishing properties of a non-gaussian, 2-D, operator-stable plume are 1) the marginal distributions along the eigenvectors of the particle jumps have power-law tails, with exponents 1 and 2, and 2) the growth rates along the eigenvectors 1/ 1 are super-fickian at rates proportional to 1/ 2 and. The purpose of this study is to investigate the operator-stable (including multi-gaussian) properties of ensemble particle motions in large-scale fractured media simulations. If the plumes have operator-stable properties, then there is a possibility that ensemble transport in fractured media may be described by simple analytical equations of fractional-order [., 2003a]. We focus on three individual parameter sets to represent 1) sparsely fracture networks dominated by very long fractures, 2) densely fractured networks comprised of short

31 16 fractures and 3) moderately fractured networks comprised of both short and long fractures. These parameter sets were selected as a preliminary attempt to investigate the full range of network variability. 2.0 Simulation of Fractures, Flow, and Transport The numerical experiments consist of the generation of random fracture networks, solving for the velocity distributions with the fracture networks and rock matrix, and tracking particle trajectories through the model domain. Monte Carlo methodology is implemented to produce ensemble particle plumes from multiple realizations of fluid flow and solute transport through the 2-D fracture networks. The fracture network geometry for each realization is based on probability distributions for fracture placement and trace-length, and an equation used to control fracture spatial densities in the model domain. Each fracture network consists of two independent fracture groups with different orientations, spatial densities, and/or powerlaw exponent for trace-lengths (Table 1 and Figure 1). We refer to the sparse network dominated by longer fractures as Set 1; the moderately dense network with long and short fractures as Set 2; and the densely fractured network dominated by shorter fractures as Set Fracture Group Orientation These 2-D fracture network simulations are meant to approximate horizontal (mapview) flow, so that each fracture is assumed to be vertical. The fracture orientations refer to the angle between the fracture and the hydraulic gradient applied at the boundaries of the model domain. Natural networks typically consist of two [e.g.,

32 17, 1985;, 1995;, 2000] or several fracture groups [e.g.,, 1989;., 1993;, 1997], with most fractures in a group oriented in nearly the same direction. To test a simple case, our fracture networks consist of two fracture groups with fixed orientations. Future simulations will involve complex networks with more than two fracture groups where the orientation of individual fractures can deviate from the dominant fracture group direction. Table 1. Set a Network Values of all Parameter Sets a θ Refer to (4) for the computation of spatial density. 2.2 Fracture Trace-Length Several analyses of field data from natural rock fracture networks indicate trace-lengths of natural rock joints (opening/mode I) and faults (shear/mode II) are distributed θ 2 according to power-law models [, 1993;, 1996;, 1996;, 1997;, 1999;., 2001]. A Pareto probability distribution (1) is used in this study to assign fracture trace-length. The use of a power-law distribution for fracture trace-length results in higher frequencies of smaller fractures with decreasing frequencies of longer fractures. To be consistent with the field studies of [1993], [1999] and. [2001], values for the power-law exponent range between 1 and 3. A minimum fracture length of five times the cell length of 1 m was This value is used in (1) for the computation of.

33 Figure 1. Fracture network samples representing: (a) sparsely fractured domains dominated by very long fractures (parameter set 1), (b) moderately fractured domains comprised of short and long fractures (parameter set 2) and (c) densely fractured domains dominated by short fractures (parameter set 3). For illustration purposes, the higher density values used in (b) and (c) require smaller subdomains. All values are in meters. 18

34 Spatial Distribution of Fractures Numerical and field studies suggest that mechanical crack interaction plays a central role during fracture propagation and may control both fracture lengths and spacing in natural fracture networks [, 1983;, 1993;, 1997;., 2003]. As a stochastic model, we chose to ignore the complexities of mechanical crack interaction as fracture lengths in natural fracture networks follow power-law distributions. Based on field studies that suggest spacing is an exponentially distributed random variable [., 1992;., 1996;, 2002], a joint uniform (1,2500) distribution is used to randomly assign the location of fracture centers within the model domain as a Poisson process [, 1985]. This is because the spacing between uniform variables are in fact exponentially distributed [, 1985]. 2.4 Fracture Spatial Density Fractures are placed into the model until specified spatial density criteria are fulfilled. Spatial fracture density is computed by [, 1965]: 1 2 = 1 2 = 2 (4) where the density of fractures in a two-dimensional domain is computed from the sum of fracture trace-lengths,, and normalized by area,. Spatial density values for the model domain are divided into three groups: minimum, intermediate, and maximum, based on the value of the power-law exponent. Minimum values correspond to networks that are at or just above the percolation threshold and were determined visually from MODFLOW solutions. Based on field mapping studies in fractured granite, a spatial density value of

35 [, 1983] and a fractal dimension of 1.8 [, 2000] were used to assign maximum spatial density. Fractal dimensions of the network were computed using a standard box counting method [, 1995]. The use of both spatial density and fractal dimension was required due to the large range in power-law exponents used to control fracture lengths. In general, the spatial density value was used as the maximum spatial density criteria for networks with lower power-law exponent values, while fractal dimension was used for networks with higher power-law exponents. Intermediate values lie directly between maximum and minimum spatial densities. 2.5 Fracture Transmissivity We are unaware of any study that measures both the length and transmissivity of fractures. Therefore, we assume that these quantities are uncorrelated. Based on results from recent hydraulic testing on boreholes at the Äspo Hard Rock Laboratory [, 2005], we use a transmissivity distribution similar to (1) with a power-law exponent of = 0.4 along with minimum and maximum values of m 2 /s and 10-2 m 2 /s to randomly assign transmissivity values to individual fractures. The upper limit on transmissivity is maintained by discarding values greater than 10-2 m 2 /s. 2.6 Fracture Implementation in MODFLOW MODFLOW [, 1988] is used to solve for two-dimensional fluid flow in a large-scale (2.5 km by 2.5 km) model domain. The simulations were made similar to the DFN methodology since a majority of fluid flow occurs through discrete fractures, and the details of the generated networks are preserved across all scales. However, the use of a continuum allows us to include the effects of matrix

36 21 interaction in future studies (Figure 2). Each cell measures 1 m 1 m 1 m. Since any amount of matrix flow is allowed, each simulation has 6.25 million cells. To assess the contributions of the fractures to plume growth, cells occupied by fractures are assigned corresponding fracture transmissivity values, while a matrix transmissivity of m 2 /s is assigned to cells not occupied by fractures to minimize matrix interaction. A continuum model enhances network connectivity as the discretization of individual fractures into cells connects fractures that otherwise would not communicate in a DFN simulation. A small cell size is used to minimize this influence. At cells containing more than one fracture, two rules are used: 1) when fracture orientations are equal, individual fractures are parallel and fracture transmissivity values are added together to form a large equivalent fracture; and 2) when fractures of different orientations intersect, the resultant transmissivity is based on the largest individual fracture transmissivity value. The addition of an active low-permeability matrix dramatically decreases computational demands for solving flow in the interconnected fracture network while adding only minor contributions of flow to the model domain. The use of a finite-difference grid to simulate discharge in a fracture that is not aligned with the grid requires an adjustment to account for longer flow paths (Figure 3). Although head values in the model domain are unaffected by the configuration of the fracture equivalents, longer flow paths due to both horizontal and vertical flow components reduce the hydraulic gradient from cell to cell along the stair step pattern. Transmissivity values must be increased to correct for the gradient so that proper discharge value can be obtained in each fracture. In 2-D, the transmissivity input into

37 22 Figure 2. Finite-difference representation of a hypothetical rock fracture network. Rock fractures (line segments) are overlain onto a finite-difference grid. Fracture-occupied cells (gray) are assigned hydraulic properties based on fracture properties, while hydraulic properties of the matrix are assigned to non fracture-occupied cells (white). Grid discretization is for illustration purposes only. Figure 3. Flow through a fracture equivalent on a finite-difference grid consists of both horizontal and vertical flow components (denoted by arrows) resulting in a ``stair step" pattern and a longer flow path than that of the original fracture,. The relationship between the equivalent flow path,, for a fracture of length,, oriented degrees from horizontal is: = [sin( )+cos( )].

38 23 MODFLOW is: [ sin cos ] = + (5) where is fracture orientation from the horizontal gradient. Once fracture transmissivity is adjusted, the measured error between flow for a straight-line fracture and its MODFLOW equivalent is less than 0.2%. All model domain boundaries are constant head, inducing fluid flow from left to right according to a linear hydraulic gradient of The boundary configuration represents an unbounded fractured rock mass where both fluid and solutes can exit any down gradient boundary. Due to the large-scale nature of these simulations (over 6 million cells), the Advanced Computing in the Environmental Sciences (ACES) supercomputer located at Desert Research Institute, Reno, Nevada, is used to solve the steady-state ground water flow equation for multiple fracture network realizations. 2.7 Particle Tracking The Random Walk Particle Method for Simulating Transport in Heterogeneous Permeable Media (RWHet) solves an advection-dispersion equation on a finite-difference grid using a random walk particle method [., 1996;, 2000]. The advective motion implemented by RWHet consists of the calculation of a series of independent, conservative solute particle trajectories based on bilinear interpolation of the velocity field with respect to particle location [., 1996]. We use RWHet to track particles through the flow fields using advective transport only. The bilinear interpolation scheme introduces micro-dispersion as differential velocities within single fractures, while macro-dispersion increases with transport

39 24 distance and varies according to orientation. At a transport scale of 100 meters, withinfracture longitudinal dispersivity ranges from 10-8 m for fractures aligned with the horizontal gradient to a maximum of 10-1 m for fractures oriented at 45º from the gradient. These values are low enough that flow in a simple fracture is piston-like. Particle location is recorded for 16 time steps ranging from 10 to 10 6 years based on equal log-cycle time increments. Emphasis is placed on the analysis of ensemble particle plumes at early time steps when particles with rapid trajectories are still within the model domain. A unique transmissivity value is assigned to each finite-difference cell containing at least one fracture. Since a rock fracture physically occupies only a small volume of a 1 m by 1 m by 1 m cell, a porosity relationship for each fracture-occupied cell is computed based on cell size and transport aperture (Figure 4). For our simulations, porosity is equivalent to transport aperture. We use law: [1999] empirical quadratic 1 2 = 0.25 (6) where [m] is the transport aperture, and [m 2 /s] is the transmissivity of a fracture occupied cell, which the authors argue gives better estimates of transport aperture than the cubic law. Similar to the calculation of discharge in a finite-difference setting, an adjustment is needed to correctly calculate the particle velocity. Since the particle must travel a longer path in the model than in the real fracture, it must be sped up by the length ratio of the hypotenuse to the sides of a right triangle. This speed is adjusted through each cell s

40 25 porosity. The correction factor used in (5) over estimates a typical particle s path length since it assumes that a particle goes around corners at right angles. We correct the distance by assuming that mean particle path is semi-circular around corners (Figure 3) with a length of /4 times the cell length. The total correction for a fracture oriented degrees from horizontal (Figure 5) is: / 2 + (tan = = ) 1 tan + 1 ( ) 2 (7) where = -1 when -45º 45º and = 1 when 45º < < 90º or -90º < < -45º. Using (7), effective porosity values for RWHet are then computed: = (8) where effective porosity,, used by RWHet is the product of and transport aperture,, based on (6). Numerical tests on individual fractures show that errors for velocity using (8) vary according to the position of a particle relative to the cell boundaries but are less than 10%. The placement of particles in the model domain is intended to be representative of a repository scenario where the possibility of the release of contaminants over a large spatial area is likely. For each realization, 25,000 conservative particles are input into the model domain in the form of a 100 m by 100 m box that extends from 100 m to 200 m in the -direction and 1200 m to 1300 m in the -direction (Figure 6). Particles are only placed into active cells located within the previously defined zone. The hydraulic backbone, which consists of the interconnected fracture network where fluid flow occurs, was not rigorously defined in this study. Criteria for active cell designation consist of

41 26 Figure 4. Porosity of fracture-occupied cell in relation to cell size and transport aperture,. For our simulations, cell porosity = / =, since = = = 1. Figure 5. Idealized particle trajectories (dotted lines) in relation to straight-line distance (solid line) for a given orientation,. Mean curved segment length is assumed to be /4.

42 27 both occupation of a cell by a fracture and cell Darcy velocities that are at least one order of magnitude greater than the average matrix value. Figure 6. Fracture network domain with shaded region representing area of domain subject to introduction of solute particles. The down gradient position of the box was selected to avoid potential boundary effects. All values are given in units of meters. 3.0 Results and Discussion The total number of realizations per parameter set is limited to 500 due to the largescale nature of the simulations. All 500 individual realizations are used to form a single ensemble data set for the study of ensemble particle behavior and construction of ensemble concentration plumes (Figures 7a-9a). The distributional properties of the plumes are analyzed by transforming particle displacements from joint (2-D) densities to marginal densities along the eigenvectors of plume growth. The marginal densities of the ensemble plumes are analyzed for growth rate and weight in the leading edges, or tails.

43 Coordinate Transform The process of subtracting initial particle location from final particle location deconvolves the shape of the initial particle input from a relatively large spatial area to a point source. These deconvolved particle displacements represent the Green function of the motion process. Due to the deviation of fracture group orientation from the coordinate axes of a Cartesian coordinate system, ensemble particle displacements represent joint probability densities where particle motion along coordinate axes are dependent and cannot be analyzed separately. To study particle motion along primary plume growth directions, a projection matrix is required for the transformation of particle displacements, r, from Cartesian coordinates ( 1, 2) onto a new coordinate system ( 1, 2). This new coordinate system is defined by eigenvectors that correspond to primary plume growth directions. Transformation to a new coordinate system is described by: = (9) where eigenvectors are denoted as individual components,. Based on visual analysis of ensemble particle plumes, eigenvectors correspond to fracture group orientations for parameter sets 1 (Figure 7a) and 2 (Figure 8a), while eigenvectors for parameter set 3 (Figure 9a) were found to be orthogonal and different from fracture group orientations. After coordinate transform, individual components of the random vectors displacements, r, along the new coordinate axes, ( 1, 2), are marginal distributions and can be analyzed individually. Scalar values along 1 and 2 correspond to 1 and 2 in (3).

44 Figure 7. Ensemble results for parameter set 1 at an elapsed time of 10 years. (a) Ensemble concentration plume based on particle displacements with principal scaling directions, 1 and 2. Mandlebrot plots of largest ranked particle displacements (circles) along (b) 1 and (c) 2 with best-fit truncated power-law (TPL) model. Approximately every 1/1000 point is plotted. All spatial values are given in units of meters. 29

45 30 Figure 8. Ensemble results for parameter set 2 at an elapsed time of 46 years. (a) Ensemble concentration plume based on particle displacements with principal scaling directions, 1 and 2. Mandlebrot plots of largest ranked particle displacements (circles) along (b) 1 and (c) 2 with best-fit exponential (light) and truncated power-law (dark) models. Power-law motion occurs along both scaling directions with greater transport rates along the fracture group with the lower exponent value, 2. Note the divergence of the data from exponential model. Approximately every 1/1000 point is plotted. All spatial values are given in units of meters.

46 31 Figure 9. Ensemble results for parameter set 3 at an elapsed time of 2,154 years. (a) Ensemble concentration plume based on particle displacements with orthogonal principal scaling directions, 1 and 2. Mandlebrot plots of largest ranked particle displacements (circles) along (b) 1 and (c) 2. Exponential decay similar to Gaussian tailing, 2 [ > ], occurs along both scaling directions. Approximately every 1/1000 point is plotted. All spatial values are given in units of meters.

47 Tail Estimation Methods Tail estimation methods involve the use of order statistics to rank the largest particle displacements in descending order. The ranked particle displacements (1), (2),, ( ) are then used for visual and numerical determination of the possible presence and exponent of power-law probability decay for the largest particle displacements. If the probability falls off like a power-law, then a Mandlebrot plot of particle displacement values versus their corresponding empirical probabilities in log-log coordinates provides a visual means to estimate the exponent (Figures 7b,c-9b,c). Linear trends on these plots indicate powerlaw particle excursion probabilities with slope. It should be noted that power-law exponents and were used to assign fracture lengths and transmissivity values according to a Pareto distribution, while the use of to denote the slope of the power-law tails of particle displacement assumes that particle displacements are in an -stable domain of attraction where 0 < 2 [, 2001]. A truncated Pareto distribution (or truncated power law (TPL) [., 2005] describes the power-law content with an upper bound: ( > ) = ( ) 1 (10) where and are the minimum and maximum values of, and is the exponent of the power-law portion of the distribution. A maximum likelihood estimation (MLE) estimator [., 2005] was coded and used to solve for the truncated Pareto estimated parameters ˆ and ˆ for each of the marginal distributions:

48 33 ˆ ln + ln ln = 0 (11) 1 (1) ( + 1) ( + 1) (1) (1) ˆ ( ) ( 1) ˆ + = 1 ( + 1) where ˆ = (1) is the largest observed data value and 1 ˆ 1 ˆ ˆ ( + 1) ˆ ( ). ( ( + 1) ) = (1) (12) Based on (11) and (12), estimated values for ˆ and ˆ were found to be sensitive to the selection of the number of largest tail data. To address this problem, ˆ and ˆ were computed using different values of between 5% and 25% of the largest particle displacements. Within this range, the standard deviation of ˆ was found to vary between 0.l4 and 0.25 for our simulations. The chi-square test was used to determine best-fit estimated parameters for the upper tail of particle displacements [, 2002]: 2 2 χ = (13) = 1 where the value of 2 is a function of the cumulative difference between the observed frequency of the experimental data,, and the expected frequency,, based on (10), over a total of bins. Bin intervals for each displacement subset were selected every 0.05 th quantile of the tail data ( largest displacements) for a total of 20 bins. The best-fit estimated parameters were determined by the lowest value of 2.

49 Plume Scaling Rates The scaling of marginal particle displacements along primary plume growth directions can be described by [., 2000;., 2003a]: 1/ = (2 ) (14) where is some measure of the plume size and is a constant dispersion coefficient. If < 2, particle displacements scale according to a super-fickian model of dispersion. When = 2, particles scale according to a Fickian model of dispersion and 2 is equal to particle displacement variance. Four metrics were used to study rates of plume evolution over time. The first is the standard deviation of particle jump magnitude, (stdev), while the other three define sections along marginal distributions according to quantile pairs (0.16,0.84), (0.05,0.95) and (0.01,0.99) of ranked particle displacements. To compute, the linear distance between each quantile pair is plotted against time in log-log coordinates (Figure 10). The slope of these plots is equal to 1/ (Table 2). The computation of quantiles is normalized to account for particles that have left the model domain, and is valid until the time that the particle corresponding to the quantile itself has left the domain.

50 35 Figure 10. Values of 2 based on scaling of plume growth along 2 for Parameter Set 2 using all four normalized metrics. Subscripts 1 and 2 denote scaling along either 1 or 2. Slope of the regression lines is 1/. Particles leaving domain boundaries result in undefined quantile estimates for later time steps. Table 2. Plume Scaling Rates Set 1(stdev) 1(0.16,0.84) 1(0.05,0.95) 1(0.01,0.99) 2(stdev) 2(0.16,0.84) 2(0.05,0.95) 2(0.01,0.99) N/A N/A Unreliable due to lack of extreme values 2 Non linear trend Table 3. Set Values of α Based on Truncated Pareto MLE

51 Ensemble Plume Characteristics Plume concentration profiles for all ensemble plumes are skewed (or elongated) in the main directions of flow (Figures 7a, 8a, and 9a). In part, the asymmetric concentration profile can be explained by the selection criteria of active cells subject to particle introduction. A large mass of particles remains near the origin for all time steps. Transport rates of particles that remained near the source indicate that these particles were introduced into cells that are not connected to the hydraulic backbone of active cells suggesting that the velocity criteria used to determine active cells are insufficient. Instead of traveling through MODFLOW fracture equivalents, these particles are essentially contained in the rock matrix and are considered immobile. Subsequent simulations will use a higher cell velocity criteria to avoid the tendency of some particles to remain nears the source as an immobile phase. After this adjustment, slow particle motion should be solely attributed to the wide range of fracture transmissivity values. However, slow particles do not enter significantly into subsequent calculations of advective statistics. Displacement plumes for parameter sets 1 and 2 exhibit non-gaussian plume geometries (Figures 7a and 8a). The resultant shape of ensemble plumes for both parameter sets is attributed to the correspondence between primary plume growth directions and the two fracture group orientations, where ( 1, 2) are (15º,-30º) and (30º,- 30º), respectively. The strong correlation between fracture group orientation and plume growth directions is caused by sparse hydraulic backbones and low to moderate length exponent values which tend to focus flow along longer fractures.

52 Set 1: Sparse Networks with Long Fractures At the time that the first particle exits the domain, the particle displacement distributions in both eigen-directions are clearly heavy tailed, with more weight than an exponential (Figures 7b and c). The particle displacement distributions are best fit by power-law probability decay with a sharp truncation. The truncation of the power law trend naturally arises from a finite sampling of heavy-tailed distributions of fracture lengths and velocities within a finite model domain by solute particles. Additional factors include: 1) maximum particle jump distance may be restricted by the truncation of the transmissivity distribution and 2) lack of dispersion within a few dominant fractures results in piston-flow at the leading plume edge. Estimated tail parameter values, ˆ, along both 1 and 2 for parameter set 1 (sparse network with low fracture length exponents) range between 0.7 and 1.1 for all time steps (Table 3). These values are quite low compared to those for porous media where has been reported in the range of 1.1 to 2.0 [., 2000; 2001]. The measured ˆ values ( ) are also somewhat lower (i.e., heavier-tailed) than the trace length exponents 1 = 1.0, 2 = 1.3. We attribute the lower values to the further influence of the low exponent value for the transmissivity distribution ( = 0.4). Lower values for particle displacements suggest flow fields in fractured media may be more conducive to super-fickian dispersion than in porous media due to a higher potential for rapid transport along a few, highly conductive fractures with long path lengths. The hierarchial (fractal) nature of both fracture length and permeability may strongly influence transport characteristics within connected fractures along a hydraulic backbone.

53 38 Based on quantiles (0.05,0.95) and (0.01,0.99), values of along 1 and 2 are 1.5 and , respectively (Table 2). These values support the super-fickian particle rates based on tail estimation methods, although the plume spreads more quickly (higher values), especially in the 2 direction. Scaling estimates based on the standard deviation of particle jump magnitude are very sensitive to the loss of extreme values from a data set and were not useful for this parameter set. A reason for the lack of a power-law trend for (0.16,0.84) is unknown Set 2: Medium Density and Intermediate Fracture Lengths Tails of the distributions of particle displacements for parameter set 3 networks (intermediate density and fracture length exponents) demonstrate trends of power-law probability decay in the range previously reported for porous media with ˆ α values ranging between along 1 and along 2 (Figures 8b and c; Table 3). These values are only slightly less than the exponents specified for fracture trace lengths ( 1 = 2.2, 2 = 1.6). Power-law trends for the leading edge of the particle displacement plume disappear in later time steps after a sufficient number of particles that describe the tail leave the model domain. The more rapid growth rate along 2 corresponds to the fracture group with the lower power-law exponent value ( 2 = 1.6) and greater probability of longer fracture lengths. Scaling rates based on quantiles and standard deviation of particle jump magnitude are in general agreement with the presence of super-fickian dispersion and ˆ α values determined from tail analysis (Table 2; Figure 10). Higher estimates of according to (0.01,0.99) may be caused by the truncation of particle displacements at the tails (Figure 8). There is considerable variation in estimating the

54 39 plume growth rates using quantile and variance estimation methods (Figure 10) much more than form the tail distributional estimates. Parameter set 2 networks are comprised of two fracture groups, one with finitevariance fracture lengths ( 1 = 2.2) and the other with infinite variance fracture lengths ( 2 = 1.6). The particle displacements in both directions show power-law content and growth rates consistent with an infinite variance model. This relationship is most likely due to the influence of the heavy-tailed transmissivity distribution, which increases the likelihood of rapid transport along some of the fractures included on the hydraulic backbone. A set of realizations with a thin-tailed transmissivity distribution (such as lognormal) would confirm this inference Set 3: Dense Networks with Shorter Fractures The displacement plume for parameter set 3 (Figure 9a) most closely resembles a Gaussian plume. First, the displacement plume has a classical elliptical shape, with the major plume axis, 1, oriented directly between fracture group orientations, 1 = 30º and 2 = 10º, with a secondary plume growth direction, 2, oriented along the minor axis of the ellipse, orthogonal to 1. Second, tail analyses along the leading plume edge for 1 and 2 (Figure 9b and c) indicate a strong deviation from a power-law model and the particle displacement distributions decay at a rate similar to a Gaussian: 2 [ > ]. However, slightly super-fickian plume spreading rates ( = 1.4 to 1.8) in the longitudinal ( 1) direction represent a departure from the Fickian model (Table 2). Normally, the presence of super-fickian rates of plume growth and exponential decay of largest particle jumps in the longitudinal direction would indicate anomalous superdiffusion where dispersivity increases with distance [e.g.,., 1992;, 2001]. For

55 40 our simulations, the presence of super-fickian scaling rates is caused by anomalous subdiffusion [e.g.,, 1997;., 2002;., 2003b;, 2003], where the tendency of particles to remain near the source increases interquantile distances, and consequently, rates of scaling. According to quantile estimates, particle transport in the transverse direction, 2, is Fickian or slightly sub-fickian (Table 2). The super-fickian scaling rate computed by the standard deviation of plume jump magnitude in the transverse direction, 2, is inconsistent with quantile estimates, and we do not consider this to be a reliable indicator of super-fickian scaling. The Gaussian-like plumes produced by parameter set 3 can be attributed to both high fracture densities and high power-law exponent values,. First, power-law distributions where 2 are finite variance. Second, high fracture densities result in the truncation of pathways for particle transport along longer fractures due to the influence of fracture intersections. The fracture intersections allow a particle to move in other directions; hence, the probability of large excursions in one direction is further reduced. Despite the use of a heavy-tailed transmissivity distribution ( = 0.4), the presence of Gaussian-like leading tails demonstrates that fracture trace-lengths have greater influence on solute transport rates than the transmissivity distribution itself Arrival of Particles at Model Boundaries The absence or presence of power-law motion during transport of contaminants has significant ramifications for geologic repositories. For comparison, 21% of solute particles left the model domain (2.5 km) in less than 10 years for parameter set 1 networks, while it took approximately 50 years before any particles left parameter set 2

56 41 model domains. Furthermore, it took over 2100 years for solute particles to leave the densely fractured domains dominated by short fractures (parameter set 3). The dominance of fracture trace-lengths on solute particle transport rates becomes clear since all of the networks, regardless of parameters, were assigned hydraulic properties based on the same distribution of transmissivity. 3.5 Characteristics of Individual Solute Plume Realizations Classical stochastic theory for solute transport in porous media dictates that particle motion in the ensemble can be replaced by spatial averaging of particle motions for a single plume after a sufficient number of particle motions occur, enabling particles to experience the full range of aquifer heterogeneity [e.g.,, 1993]. Individual solute plume realizations for parameter set 3 (high density and length exponents) have a strong connection to traditional stochastic theory where individual plumes strongly resemble the ensemble plumes (Figure 11a). This is attributed to high fracture density and finite variance fracture lengths that have some characteristic length scale. Conversely, individual solute plume realizations for parameter sets 1 and 2 typically differ markedly from ensemble plumes. A single plume may have little or no transport in a certain direction (Figure 11b). Instead, long fracture lengths and sparse domains lead to irregular hydraulic backbone geometries for individual fracture networks realizations. Resultant particle motion is then dominated by one fracture group orientation or a combination of both fracture groups. For these cases, determination of probabilities associated with the position of individual plumes may be as important as the prediction of overall ensemble motion. Furthermore, the role of conditioning will certainly be

57 42 Figure 11. Particle displacement plumes for three individual realizations. (a) High fracture densities and length exponents result in Gaussian-like, elliptical plumes with little variability between individual realizations and the ensemble. (b) Particle motion along fractures with sparse to intermediate densities and lower fracture exponents can be dominated by one fracture group orientation (1) and (3) or a combination of the two (2) resulting in significant departures from the ensemble by individual realizations. For clarity, only 10% of particles are shown.

58 43 enhanced in the sparser networks. These examinations are underway. 4.0 Conclusion Standard numerical simulators designed for porous media were used to investigate the influence of power-law fracture trace-length distributions and fracture density on largescale behavior in fractured media. Diffusion into the rock matrix was not allowed in these simulations to isolate the effects of the networks on advective transport. Three sets of 500 fracture networks were generated to represent low to high values of density and proportion of very long fractures. Hydraulic properties for all networks were randomly assigned according the same heavy-tailed transmissivity distribution. Particle tracking results from the densest and sparsest end members suggest that transport in fractured media may be Fickian-like or strongly super-fickian, respectively (Figure 12). Furthermore, [1999] proposed that the most likely range for power-law exponents of the fracture length distribution is The presence of super- Fickian motion for parameter set 2 ( 1 = 2.2, 2 = 1.6) shows that super-fickian dispersion may be more common in fractured media than less rapid Fickian dispersion. We anticipate that the addition of results from more simulations to Figure 12 will further aid in distinguishing between Fickian and super-fickian transport regimes. The importance of identifying the fracture trace-length distribution and overall density is demonstrated by the approximately three order of magnitude difference in the earliest arrival of particles at the boundary of the hypothetical domain. Ensemble particle plumes produced by networks dominated by short fractures and high densities resemble multi-gaussian densities. The principal directions of plume

59 44 Figure 12. Preliminary correlation between fracture length exponent, fracture density and resultant particle displacement tail value (diamond) based on parameter sets 1-3. Values for both fracture exponent and particle displacement tail represent average values (i.e., 1+ 2]/2). Density values were converted to [m/m 2 ] to provide a more reliable estimate for fracture density as (4) is heavily influenced by fracture length exponent values. The dotted line denotes a crude division between Fickian and super-fickian plume growth. Future research efforts will further evaluate the transition between Fickian and super-fickian plumes in relation to fracture network statistics.

60 45 growth are not aligned with the fracture directions. Little variability was observed between individual plume realizations and ensemble multi-gaussian plumes. The ensemble particle plumes produced by networks with low to moderate fracture density and length exponent values have the characteristics of operator-stable densities, including power-law leading edges and super-fickian growth rates. The exponents of these two indicators did not match exactly, although considerable variability exists in the metrics. For these networks, fracture orientation corresponds to eigenvectors of the plume growth rates, while the rates themselves are described by eigenvalues that are influenced by both length exponent values and transmissivity. Significant contrasts between ensemble plumes and individual plumes for the operator-stable plumes demonstrate the challenge of predictability in fractured media. Subsequent research will further explore the trend between fracture length and density and resultant transport rates. It is anticipated that these data will be used for the development and parameterization of analytical equations for predictions of solute transport based on fracture network statistics [., 2003a]. Solutions from these equations will then be compared to the synthetic particle tracking data to assess predictability of both ensemble and individual plumes.

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66 51 Ensemble Transport of Solutes Through Fracture Networks and the Correspondence to Operator-Stable Limit Distributions Donald M. Reeves Desert Research Institute, 2215 Raggio Parkway, Reno, Nevada David A. Benson Department of Geology and Geological Engineering, Colorado School of Mines 1516 Illinois St., Golden, Colorado Mark M. Meerschaert Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand Hans-Peter Scheffler Department of Mathematics, University of Nevada, Reno, Nevada 89512

67 52 Abstract In networks where individual fracture lengths follow a fractal distribution, ensemble transport of conservative solute particles converge to operator-stable densities. These densities have, as their governing equations of transport, either fractional-order or integer-order advection-dispersion equations. Selection of the order of an advectiondispersion equation depends on the identification of either Fickian or super-fickian transport regimes, which in turn depends on the threshold between infinite variance and finite variance distributions of fracture length. Low-to-moderately fractured networks with power-law fracture length exponents less than or equal to 1.9 (i.e., infinite variance), produce solute plumes that exhibit power-law leading edge concentration profiles and super-fickian plume growth rates. For these network types, a multiscaling fractional advection-dispersion equation (MFADE) provides a model of multidimensional solute transport where different rates of power-law particle motion are defined along multiple directions. The MFADE relies on a scaling matrix to describe the super-fickian growth process, in which the eigenvectors of the dominant plume growth directions correspond to primary fracture group orientations and the growth rate eigenvalues depend primarily on fracture length and transmissivity. An arbitrary measure of the unit sphere accounts for the propensity for ensemble travel in any direction. The convergence of particle motion to a multi-gaussian (a subset of the operator-stable) for densely fractured networks with power-law fracture length exponent values greater than or equal to 1.9 (finite variance fracture length occurs at 2.0) can be described using the classical ADE where Fickian scaling rates occur along orthogonal plume growth directions. Particle arrival times at model boundaries for multi-gaussian plumes are highly variable and

68 53 depend on values of spatial density. However, all things held equal, arrival times are much shorter for sparsely-fractured domains with lower fracture length power-law exponents.

69 Introduction The use of an analytical solution an equation such as the advection-dispersion equation (ADE) would be very convenient for screening-level predictions of solute transport in fractured media. Unfortunately, the ADE has been shown to be inadequate for describing solute transport behavior in sparsely to moderately fractured rock masses [, 1983;, 2000;, 2005;, 2002;, 1984, 2004;, 1997;, 2004;, 2006]. At the network scale, non-gaussian transport behavior may be partially explained by the fractal nature of fracture networks where the distribution of permeability and/or trace-lengths of individual fractures or fracture zones can vary over several orders of magnitude [e.g.,, 2001; 2001;, 1997;, 2005], resulting in highly heterogeneous flow fields where equivalent properties cannot be defined over a larger continuum [ 1982]. Instead of converging to elliptical multi-gaussian densities, transport of particles through a restricted subset of interconnected fractures can lead to highly asymmetric solute plumes with anomalous scaling rates [e.g.,, 2001;, 2004;, 2006]. [2006] introduced the use of operator-stable densities to describe ensemble transport behavior of conservative solute particles in simulated fractured media. Based on Monte Carlo realizations for three different network types, they found that certain fracture network characteristics, such as low to moderate densities coupled with heavy-tailed distributions of fracture length and transmissivity, lead to super-fickian rates of dispersion and heavy leading edges of plumes (i.e., the probability of largest

70 55 particle jumps follow a power-law). Eigenvectors and eigenvalues of the scaling matrix were determined from ensemble particle plumes. A direct correlation between eigenvectors of ensemble plume growth and fracture group orientations was observed, while eigenvalues (measures of the degree of super-fickian growth) were dependent on both the distributions of fracture length and transmissivity. In the case of a highly fractured domain dominated by short fractures, transport was adequately described by a multi-gaussian density, which is a subset of the operator-stable densities. Based on these findings, [2006] proposed a crude relationship to delineate between Fickian and super-fickian transport regimes based on fracture density and distribution of fracture trace lengths. This research expands the work of [2006] by examining the convergence of ensemble particle plumes to operator-stable densities for the application of a fractional-order advection-dispersion equation for solute transport predictions in fractured media. Estimation of equation parameters describing ensemble plume growth rates and primary directions is based on relationships between fracture network statistics and ensemble particle plumes. An extensive array of fracture network types are analyzed to identify influences of fracture length, transmissivity, orientation, density and presence of multiple fracture groups on transport behavior of a conservative solute. 2.0 Particle Jumps and Operator-Stable Densities Solute particle motion in multiple dimensions can be described by a random walk model in which each jump takes the form: r H = Θ r (1)

71 56 where a random particle jump vector r is the product of two independent random variables: a particle jump size matrix H and a direction unit vector Θ r [, 2003]. The distribution of the scalar, where ( > ) 1, is heavy-tailed [, 2003]. The matrix H rescales the distribution of jump lengths to account for larger jumps in certain directions: 1 H = SH S (2) where S and H are, respectively, the eigenvector and eigenvalue matrices for H [Schumer et al., 2003]. We address plume growth in 2-D, which requires only two eigenvectors to represent dominant plume growth directions and two eigenvalues to scale transport (i.e., describe plume growth rates) in all directions. Eigenvector and eigenvalue matrices have the form S = and 1/ 1 0 H = 0 1/ where column 2 r r eigenvectors 1 and 2 are defined in terms of their vector coordinates and eigenvalues (1/ ) assign rates of scaling along these directions. The probability decay of particle jump magnitude along the th eigenvector,, follows a power-law: ( > ). Particle motion in non-eigenvector directions scale according to a mixture of scaling coefficients [ 2003]. The directional vector Θ r in (1) is described by its distribution on a unit circle for two-dimensional particle jumps. If particle motion were in 3-D, Θ r would be distributed r along a unit sphere. The distribution of direction unit vectors, ( Θ), is called the mixing measure [, 2001; r, 2003]. ( Θ) contains the

72 57 weights or relative intensities of directional particle movement such that 2 0 r ( Θ) = 1 [, 1999]. In 1-D, mixing measure weights describe the skewness of an α-stable distribution [, 1998]. Although mixing measure weights can be either isotropic or anisotropic, it is assumed that the distribution of directional particle movement is anisotropic for solute transport in ground water flow systems due to the influence of a regional hydraulic gradient [, 2003]. [ r Particle jumps,, as described by (1) are in an operator-stable domain of attraction, 2001]. The normalized sum of these particle jumps converge in the limit to operator-stable random vectors [, 2001]. Operator-stable densities are the most general case of joint α-stable distributions where the rates of scaling (1/ ) are different along each primary scaling direction. If rates of scaling are equal in all directions, as in the case of a multivariate -stable distribution, H reduces to a scalar equal to 1/ [, 1999]. Multi-Gaussian densities, a subset of operator-stable densities, result when H = plume growth. (1/2,1/2) for two-dimensional Closed-form analytical solutions do not exist for operator-stable densities. The logcharacteristic function of a centered non-gaussian operator-stable density, ( r ), can be represented as [, 2003]: r r r r ( ) 1 r ( Θ) (3) 1+ Θ 2 r r H H Θ Θ = 2 2 H 0 0

73 & & 58 r r where H 1/ 1 1/ 2 > 0 provides scale to the density and Θ = cos( ) + sin( ) 1 2 if H is diagonal. The inside integral describes the direction and scaling of probability mass for an operator-stable density according to H, while the outside integral distributes r probability mass according to weights of the mixing measure, ( Θ). The density, r ( ), of this operator-stable random vector can be computed by taking the inverse Fourier transform [, 2003]: r r r r r 2 ( ) ( ) = 2. r 2 R (4) The fast Fourier transform [, 1988] can also be used to calculate (4). A multiscaling fractional advection-dispersion equation (MFADE) models solute transport according to operator-stable densities [! 2003]: ' ( r, ) r ' ( r 1 H ' ( r =, ) + *, ) ".# 2001;! $ % "., (5) ' ( r where solute concentration,, ) changes due to a combination of advective, and super-fickian dispersive, flux. The multiscaling fractional derivative, & H 1, uses scaling r matrix H and mixing measure (same as ( Θ) ) to describe directional particle movement. H is equal in (1), (3) and (5). Values of + in H are spatial fractional-order derivatives where Note that [! r r r [ ]( ) = ( ) ˆ( ), 1 H ' - ' "., 2001]: i.e., the Fourier transform of a multiscaling fractional derivative is equal to the log characteristic function of an operator-stable density (3). Linear advective velocity, ), in (6)

74 59 (5) provides a vector shift to the operator-stable density, while a constant dispersion coefficient, *, provides scale to the mixing measure. Note that in (3) and * in (5) are r proportional. A goal of this study is to identify the parameters H and ( Θ) to discern the parameters from fracture statistics and hydraulic data. in an effort 3.0 Synthetic Data Generation Realizations of fractured media are numerically generated according to probability distribution functions (pdfs) for fracture length, transmissivity, placement of fracture centers, and orientation, while an algorithm is used to control fracture densities in the model domain. Parameter sets, defined as a group of values used to define fracture length, density, and orientation, are used to generate 500 statistically equivalent fracture network realizations according to Monte Carlo methodology (Table 1). Distributions of fracture placement and transmissivity are held constant. However, two distributions are used to assign fracture transmissivity. The primary distribution is heavy-tailed and assigns transmissivity values to individual fractures according to a truncated Pareto distribution with a power-law exponent of 0.4. A lognormal distribution is used in 4 parameter sets (Sets 4, 8, 15 and 19) to investigate the effects of a thin-tailed distribution on transport rates. A continuum approach using MODFLOW [ * " $!, 1988] solves for large-scale (2.5km by 2.5km) two-dimensional fluid flow in both finite-difference fracture equivalents and the rock matrix. Flow and particle transport in finite-difference fracture equivalents requires the use of correction factors to account for deviations in path ) length [ "., 2006]. To minimize the role of the rock matrix on flow and

75 60 Table 1. a Network Values for Parameter Sets Set T a 1 b * 1 c 1 d 2 * TP 1.0 max/ max/ TP 1.0 max/ max/ TP 1.3 int/ int/ LN 1.3 int/ int/ TP 1.0 min/ max/ TP 1.3 max/ max/ TP 1.6 max/ int/ LN 1.6 max/ int/ TP 1.9 int/ int/ TP 1.9 min/ max/ TP 1.9 max/ max/ TP 1.9 int/ max/ TP 2.5 min/ min/ TP 2.5 max/ max/ LN 2.5 max/ max/ TP 2.8 min/ min/ TP 2.8 int/ int/ TP 3.0 max/ max/ LN 3.0 max/ max/ Transmissivity distribution where TP and LN denote truncated Paerto and lognormal, respectively. b Fracture length power-law exponent value. c Spatial fracture trace-length density [m/m 2 ]. d Fracture group orientation from horizontal in degrees.

76 61 transport, cells representing the rock matrix are assigned transmissivity values (10-15 m 2 /s) that are four orders-of-magnitude less than the lowest fracture transmissivity (10-11 m 2 /s). Each realization consists of 6.25 million cells, with each cell measuring 1 m 1 m 1 m. RWHet [ " ", 2000] is used to track conservative particle trajectories through the resultant velocity field by advection only. To mimic a geologic repository where the release of contaminants over a large spatial area is likely, 25,000 particles are input into each realization in a 100 m 100 m box that extends from 100 m to 200 m in the ( -direction and 1200 m to 1300 m in the -direction (Figure 1). Within this release area, particles are input only into cells that both contain a fracture and satisfy a minimum Darcy cell velocity criteria to restrict transport to the hydraulic backbone. The majority of simulations used a Darcy cell criteria of three orders-of-magnitude greater than the average matrix values, although a two orders-of-magnitude criteria was used for simulations (Sets 6, 10-12, 15-16) computed earlier in the study. Though Sets 3, 7, and ) 18 contain the same parameters previously presented by ". [2006], particle motion is not the same due to the establishment of this new active cell criteria. An equal number of particles is placed into each active cell. Particle location is recorded for 16 time steps based on equal log-cycle time increments for 6 log cycles. Actual log-cycle times depend on distributions of fracture trace-length. Particle positions are first recorded at 0.1 years for sets dominated by longer fractures and 10 years for all other ) sets. A comprehensive explanation of the numerical simulations is presented by ". [2006].

77 62 Figure 1. Fracture network domain with shaded region representing area of particle release. Note that a hydraulic backbone is not present in the particle release area for this realization. Spatial values are in meters. 4.0 Results and Discussion A total of 23 parameter sets, consisting of 500 possible individual realizations each, are analyzed for convergence of particle jumps to either operator-stable or multi- Gaussian random vectors. The density of the particle displacements (or jumps), defined as the difference between final and initial particle locations at a given time increment, represents the Green function of the motion process. The first 19 parameter sets (Table 1) consist of simple networks comprised of only two fracture groups with constant orientation. The remaining 4 parameter sets are used to investigate more complex networks where fracture orientation is allowed to deviate around a mean orientation and/or more than two fracture groups are present. MFADE parameters H (Table 2) and r ( Θ) are estimated from ensemble particle displacement plumes that fit a super- Fickian/operator-stable growth process. Plume spreading rates are computed for all

78 " 63 ensemble particle plumes (Table 3). Relationships between fracture network statistics such as fracture length, density, transmissivity, variable fracture group orientation and the influence of multiple fracture groups and resultant particle transport are identified. Table 2. Parameters Required for the Computation of H r r Set Analysis of Ensemble Particle Displacement Plumes Ensemble particle plumes (e.g., Figure 2) represent a 2-D joint density and consist of all particle jumps for a given time step of a single parameter set (Table 1). Examining the convergence of ensemble particle displacement vectors to operator-stable random vectors requires analysis of the marginal distributions. By definition, if joint ensemble particle displacement vectors converge to an operator-stable random vector, then marginal distributions of ensemble particle jumps converge to α-stable random variables [!!, 2001]. If joint ensemble particle displacement vectors converge to a multi-gaussian random vector, then marginal distributions of ensemble particle jumps converge to Gaussian random variables.

79 64 Figure 2. Ensemble particle displacement plumes for Set 1 at transport times of (a) 4.6 (b) 10 and (c) 464 years. Fast particle transport is dependent on both fracture length and transmissivity values. Note the influence of large continuous fractures on solute transport. All values are given in units of meters.

80 " " " 65 To evaluate the marginal distributions of joint ensemble particle plumes, the coordinate directions (eigenvectors) that dominate plume growth must be identified; otherwise the variation in tail behavior will be masked by the plume growth direction with the heaviest tail [!!, 2003].!! [2003] propose the use of a sample covariance matrix to compute the eigenvectors of an elliptical operator-stable density. However, the computation of eigenvectors from a sample covariance matrix leads to eigenvectors with orthogonal orientations. Orientations of fracture groups are not always orthogonal [e.g., $, 2004], and our simulations demonstrate that non-orthogonal fracture group orientations can lead to nonorthogonal directions of plume growth. In the absence of a rigorous method to identify dominant plume growth directions, eigenvector directions are identified by visual analysis of ensemble particle displacement plumes. A transformation along eigenvectors of plume growth (denoted as column vector components, ) maps particle displacement r vectors,, from Cartesian coordinates ( 1, 2) onto a new coordinate system ( 1, 2) [!!, 2003]: = (7) Values of 1 and 2 are marginal distributions and are analyzed independently for powerlaw content and plume scaling rates. The labeling convention for 1 and 2 depends on plume type. For operator-stable plumes, 1 and 2 represent random variables along eigenvectors that are the most and least positive in orientation relative to the hydraulic gradient (e.g., Figure 2a), while 1 and 2 represent marginal random variables along major and minor plume axes, respectively for multi-gaussian plumes (e.g., Figure 3).

81 Figure 3. Ensemble particle displacement plume for Sets 16 (a) and 17 (b) at transport times of 44,640 and 1000 years, respectively. The only difference between the fractured domains is that Set 16 (a) has a lower density than Set 17 (b). For lower values of spatial density, the tendency of particles to follow fracture orientations ±45º enhances transverse plume spreading and decreases plume migration rates. All values are given in units of meters. 66

82 Estimation of Scaling Matrix, H An upper truncated Pareto (power-law) model is used to estimate tail thickness of the largest marginal particle jumps for the super-fickian plumes [ "., 2005]: ( > ) = ( ) 1 (8) where (1), (2),, ( ) are marginal particle displacements ( 1 or 2) ranked in descending order, and are the minimum and maximum values of and + describes the power-law tail of the distribution. The use of a truncated Pareto, which has finite moments of all orders, appears to contradict the use of heavy-tailed (infinite variance) statistics required by (5). However, the truncation of the power-law trend observed in the distribution of marginal particle displacements naturally arises from a finite sampling of heavy-tailed distributions of fracture lengths within a finite model domain. Since particle transport occurs exclusively within a subset of fractures of a network called the hydraulic backbone, solute particles only experience a small subset of fracture lengths and velocities. This is particularly true for the sparsely to moderately fracture networks with long fracture lengths that promote the formation of operator-stable densities. Additional limitations on maximum particle distance are imposed by the distribution of fracture transmissivity and lack of within fracture dispersion which leads to piston flow along ) dominant transport fractures [ "., 2006]. Values of + based on (8) are sensitive to the selection of tail data, so is allowed to range between the greatest 5% to 25% of marginal particle jumps. Final estimates of + and represent the best fit between ) empirical distributions of marginal particle jumps and (8) (Table 4) [ "., 2006].

83 68 Mandlebrot plots are used to verify tail thickness estimates of all power-law trends (Figure 4). Analysis of spreading rates along primary plume growth directions can be estimated using [, 2000;! $ % "., 2003]: 1/ = (2* ) (9) where is some measure of plume size and * is a constant dispersion coefficient [Table 3]. For an operator-stable density following (5), estimates of + (0 < + <2) from tail (8) and plume spreading rates (9) should be equal. A Fickian growth process has + = 2 and 2 is equal to particle displacement variance. Several metrics are used to compute plume size, including the standard deviation of particle jump magnitude, (stdev), and quantile pairs (0.16,0.84), (0.05,0.95) and (0.01,0.99) where the subscript refers to fraction of particles behind and ahead (Table 3). The linear distance between each quantile pair is plotted against time in log-log coordinates (Figure 5). A linear trend on this plot will have a slope of 1/+ according to (9). For some parameter sets, quantiles are undefined at later time steps after a sufficient fraction of particles have left the model domain. The scaling matrix H in (5) uses eigenvectors and eigenvalues to describe anisotropic plume evolution over time as a super-fickian growth process. The eigenvectors, r, denote primary plume growth directions and are computed visually from ensemble particle displacement plumes. Primary plume growth directions are presented in degrees r from horizontal where = [cos( )sin( )] [Table 2]. Eigenvalues (1/+ ) describe plume growth rates and are computed from marginal distributions along r. Due to the inverse relationship between + and rates of plume spreading (1/+ ), lower values of + indicate

84 Figure 4. Mandlebrot plots of largest ranked particle displacements (circles) for Set 1 along (a) 1 and (b) 2 with best-fit truncated power-law (TPL) model at an elapsed time of 0.1 years. Approximately every 1/1000 point is plotted. Values of are given in units of meters. 69

85 70 Table 3. Plume Scaling Rates Set 1(stdev) 1(0.16,0.84) 1(0.05,0.95) 1(0.01,0.99) 2(stdev) 2(0.16,0.84) 2(0.05,0.95) 2(0.01,0.99) N/A N/A Non-linear trend in log-log space.

86 + 71 Table 4. Tail estimates of + Set Figure 5. Values of + based on scaling of plume growth along 1 for Parameter Set 2 using all four normalized metrics. Slope of the regression lines is 1/+. Particles leaving domain boundaries result in undefined quantile estimates for later time steps. Estimates of + based on (stdev) are very sensitive to the loss of extreme values. The change of slope in (stdev) is caused by a significant loss of particles after the fourth time step.

87 " " 72 more rapid rates of plume growth. Estimates of + obtained from tail and plume spreading rate methods are both used for the computation of eigenvalues (Table 2) Mixing Measure Weights When + < 2, the variance of particle jumps, hence covariance matrix, is undefined. r Instead, weights and directions of the mixing measure, ( Θ), describe the dependence r r structure between eigenvectors of plume growth ( 1, 2 ) and heavy-tailed particle jumps, r [!! r eigenvectors are correlated, ( Θ), 2001]. Since orientations of primary fracture groups and describes the influence of each fracture group on particle transport. If mixing measure weights in the ensemble tend to be uniform over some interval of the unit circle, transport of solute particles in individual realizations is significantly dependent on each fracture group. If mixing measure weights in the ensemble are concentrated in eigenvector directions, then particle transport in individual realizations predominately occurs along one fracture group or the other, not a mixture of both. The formation of a hydraulic backbone suggests that particle transport is dependent on each fracture group, although the degree of dependence may be associated with the distribution of trace-lengths for each fracture group (i.e., a fracture group with longer fracture lengths may exert more influence on transport than a fracture group with shorter fracture lengths). A heavy-tailed random vector in the generalized domain of attraction for some operator-stable law, r, has the relationship [!, 1999]: r = ( r ) Θ( r ) (10) H r

88 " 73 r r which describes the dependence structure for directional particle transport, Θ( ) r r r according to some radius, ( ) > 0. Values of Θ( ) are defined as the position where a, particle jump vector, r, with scaling properties, circle of a Jurek coordinate system (Figure 6) [ $ H 1/ 0 1 = 0 1/ 2, intersects the unit, 1984]. A Jurek coordinate system is an anisotropic polar coordinate system where coordinate axes are scaled according to power-law coefficients where and + =. Due to unequal scaling rates, r ( ) is curved. If + 1 = + 2, scaling is isotropic and a Jurek coordinate system reduces to a symmetric polar plot. Since primary plume growth directions are not necessarily aligned with axes of a Cartesian coordinate system for our simulations, it is advantageous to correct for this before applying (10). By applying the change of coordinates, represents an eigenvector matrix of plume growth, (10) can be expressed as: r Values of ( ) r 1 = S r where S r = ( r ) Θ( r ). (11) r H are then computed using a method presented in!! [2003] where Pareto distributions, ( > x) = (, are used to appropriately scale r along coordinate axes. Values of + for the Pareto distributions are estimated using (8) (Table 4), while values of are computed using the relationship r "., 2005]. Next, values of ( ) [ r r r are ranked in descending order, ( 1), ( 1), K, ( ), where is based on percentages used in (8) for the computation of + r r r values of Θ( ), computed from the largest ( ) and. Finally, values, are transformed back to the

89 r r r r r r original coordinate system using the relationship, Θ( ) = SΘ( ). Values of Θ( ) are r binned into 5º increments and normalized to provide relative weights for ( Θ) Figure 7). 74 (e.g., Figure 6. Jurek coordinate system where + 1 = 1.0 and + 2 = 1.3 lead to unequal directional scaling rates. The value of is defined as the position where the curved radius,, intersects the unit circle.

90 $ Influences of Fracture Length Trace-lengths of natural rock joints and faults are often observed to be distributed according to power-law models [e.g., $ ) *, 1997;!, 1999; "., 2001]. A Pareto probability distribution is used to assign fracture trace lengths above a certain cutoff in our simulations: ( ) = (12) where fracture length ( ) depends on a power-law exponent,, which ranges between 1 and 3 [e.g., $ ) *, 1997;!, 1999; "., 2001] and the scalar controls minimum fracture length (5m). In general, sample mean fracture length and are inversely related (i.e., mean fracture length increases as decreases). For < 2, variance and standard deviation of fracture length diverge and a characteristic fracture length is undefined. If an absolute correlation exits between and estimates of + based on both plume spreading rates and power-law probability decay of largest particle jumps, then is the only statistic necessary to distinguish between super-fickian ( < 2) and Fickian ( 2) transport regimes. We investigate the influence of fracture length exponent over the range of power-law exponent values (Figure 8). Based on network connectivity studies [e.g.,!, 1999; * * "., 2001], parameter sets 1-19 are divided into 3 groups based on fracture length exponent values. These groups define domains where network properties are dominated by either long fractures with (Sets 1-5), a combination of both short and long fractures (Sets 9-11), or short fractures (Sets 13-19). Parameter sets 6, 7 and 12 are used to explore overlaps between groups.

91 Figure 7. Histogram (a) and cumulative distribution plot (b) of mixing measure weights for Set 1 at a transport time of 0.46 years. Midpoint of bin interval is plotted. Note the concentration of weight in the direction of fracture group orientations, 30 and -60. The uneven distribution of mixing measure weights along for eigenvectors is attributed to a greater number of large particle displacements transported in the fracture group oriented at

92 77 1/ X 1.3 X 1.6 X 1.9 X X 2.2 X X 2.5 X 2.8 X 3.0 X Figure 8. Investigated regions (X) into the parameter space for fracture length exponents, 1 and Group 1: 1.0 a 1.6 For low values ( ) of trace-length exponents (Sets 1-5), connectivity and transport properties are dominated by a few, very long fractures that span a large area of the model domain. The dominance of these fractures is apparent from the ensemble particle plumes that preserve the features of individual transport realizations (Figure 2). Early arrival times of particles at model boundaries are dependent on both the presence of very long fractures and high fracture transmissivity values. The first particles leave the model domain for Set 1 ( 1 = 1.0, 2 = 1.0) at approximately 2 years in a realization in which all 25,000 particles leave the northern domain boundary. All of the particles in this realization are transported through a single, highly transmissive fracture that spans the entire model domain.

93 78 Tail estimates of + for Set 1 range between for 1 and for 2 (Table 4, Figure 4). Though fracture length exponents 1 and 2 are identical, differences in fracture orientation relative to the hydraulic gradient ( 1 = 30º, 2 = -60º) may explain higher values of + along 2. The presence of very heavy leading edge distribution tails along 1 for early time steps matches the power-law exponent value assigned to fracture transmissivity ( = 0.4), suggesting that tail estimates of + for networks dominated by very long fractures primarily reflect the distribution of fracture transmissivity. Early-time tail estimates for Set 1 are dependent only on the distribution of fracture transmissivity, since the upper tail of marginal particle displacements consists of realizations where particle transport occurs through only one or a few fractures that span the entire domain. Interaction between these long, dominant fractures and other less significant fractures with shorter lengths and lower transmissivity values are minimal. This is further suggested by estimates of + based on plume spreading rates and the distribution of mixing measure weights. Values of + based on plume spreading rates are near 1.0, indicating ballistic transport along long fractures where little or no mixing occurs between flow paths. This is analogous to differential advection in a stratified aquifer [, 1967]. Mixing measure weights are highly concentrated along eigenvector directions for all time steps (Figure 7), suggesting that one fracture group or the other dominates particle transport for individual realizations. Parameter Sets 3 ( 1 = 1.3, 2 = 1.0) and 5 ( 1 = 1.0, 2 = 1.6) within this group emphasize the influence of different fracture length exponents on solute transport. The separation distance between fracture length exponents is 0.3 for Set 3 and 0.6 for Set 5. Tail values of + for 1 and 2 are similar for both sets and range between for Set 3

94 79 and for Set 5. Again, low tail estimates reflect the influence of a heavy-tailed transmissivity distribution ( = 0.4). Relatively equal tail estimates of + along 1 and 2 indicate that differences in fracture length exponents for fracture networks, where fracture length exponents for both sets are in group 1, do not significantly influence rates of particle transport. Estimates of + based on plume spreading rates ( ) for Sets 3 and 5 are higher than tail estimates and indicate ballistic transport where minimal flow path mixing occurs Group 1 and Group 2 Mix Sets 6 ( 1 = 1.3, 2 = 1.9) and 7 ( 1 = 1.6, 2 = 2.2) explore the transition between fracture length exponent group 1 ( ) and group 2 ( ). Networks for Sets 6 and 7 both contain fracture groups where one fracture group primarily consists of long fractures while the other consists of a combination of short and long fractures. To study contributions of each fracture group on overall particle transport, fracture group orientations ( 1 = 60º, 2 = -10º) for Set 6 are intended to minimize the influence of the hydraulic gradient along the fracture group with the lower fracture length exponent value ( 1 = 1.3), while maximizing the influence of the hydraulic gradient along the fracture group with the higher fracture length exponent value ( 2 = 1.9). Tail estimates for Set 6 do not reflect contrasts in hydraulic gradient where values of + along 1 and 2 range between and , respectively. Lower tail estimates along 2 relative to the fracture length exponent ( 2 = 1.9) may be attributed to either influences from the transmissivity distribution or longer fractures from the other fracture group ( 1 = 1.3). With the exception of (0.01,0.99), plume spreading + estimates (+ 1 = , + 2 = )

95 80 are in general agreement with tail values along 1 and are generally lower than predicted by the tail for 2. Since the primary metrics used to quantify plume spreading rates are based on the distance between particles representing quantile pairs ( (0.16,0.84), (0.05,0.95), (0.01,0.99)), lower estimates of + for plume spreading rates than tail estimation methods are most likely caused by particle retention in low velocity fractures. Retention of particles within the source area increases interquantile distance, which results in lower estimates of + (i.e., the plume appears to grow more rapidly). Although particle retention near the source area is observed for all network types regardless of fracture network statistics such as fracture length and density, the influence of these particles on plume spreading rates does not contradict tail estimates of + until networks contain at least on fracture group that is near or above the finite variance threshold ( = 2.0) for fracture length. The correlation between fracture trace length and particle retention may be misleading as the spatial density at which a network percolates is dependent on the distribution of fracture lengths [!, 1999]. Instead, the greater number of fractures at higher spatial densities increase the likelihood of particles entering low velocity fractures. All ensemble particle plumes containing at one fractures group where 1.9 have spreading rates that are influenced by particle retention in low velocity fractures. Set 7 ( 1 = 1.6, 2 = 2.2) contains one fracture group with finite variance lengths and another with infinite variance. Estimates of α reflect this contrast as tails along 1 vary between a very weak power-law (1.9) and exponential decay, while tail estimates for 2 range between 1.4 and 1.6. Since a hydraulic backbone consists of both fractures groups, particle excursions along 2 (fracture group with infinite variance) may contribute to the

96 81 formation of a weak power-law tail for 1. However, the influence of the finite variance fracture group did not influence particle motion along 2. This may indicate that the heavy-tailed distribution of transmissivity can promote the formation of a weak powerlaw trend. Estimates of + based on plume spreading rates along 2 ( ) show large variability but are in general agreement with tail values. With the exception of (0.01,0.99), lower values of + for plume spreading rates along 1 (1.5) indicate particle retention in fractures with low transmissivity values Group 2: 1.9 α 2.2 Group 2 ( ) serves as a transition between fracture length exponent groups 1 and 3 as fracture groups can have either infinite variance or finite variance fracture length distributions, although the infinite variance fracture length exponent ( = 1.9) is near the finite variance threshold ( = 2.0). Parameter Sets 9 and 11 are used to further investigate the transition between super-fickian and Fickian transport. While results from Sets 9 and 11 indicate that, similar to group 1 networks, eigenvectors of plume growth correspond to fracture group orientations, the primary difference between group 1 and group 2 networks is that growth rates along eigenvectors can be either Fickian or super-fickian. Tail estimates ( ) for Set 9 ( 1 = 1.9, 2 = 1.9) confirm that particles traveling through networks containing fracture groups with fracture length exponents near the finite variance cutoff ( = 2.0) can experience super-fickian motion. Exponential decay of the probability distribution of the largest particle displacements for Set 11 ( 1 = 1.9, 2 = 2.2) suggest Fickian growth rates along non-orthogonal eigenvectors (Figure 9). In this case, a symmetric multi-gaussian density is not an adequate model of solute transport. The convergence of ensemble particle displacements

97 82 to multi-gaussian densities for group 2 networks most likely occurs only if fracture group orientations are orthogonal. Other restrictions, such as spatial density, will be discussed in the next section. Figure 9. Ensemble particle displacement plume from Set 11 at a transport time of 21 years. Fickian plume growth occurs along non-orthogonal axes Group 2 and 3 Mix Only one parameter set, Set 12 ( 1 = 1.9, 2=2.5) was chosen to investigate networks containing fracture groups with fracture length exponent values in the range of group 2 and group 3. The resultant ensemble particle plume for this parameter set is elliptical, with a strong power-law trend of the decay of largest particle displacements (+ = ) in the longitudinal direction ( 1) and exponential tailing in the transverse ( 2) direction (Figure 10). The different type of tail decay in each direction can be attributed to the difference that exists between fracture length exponents (0.6). The fracture group with

98 83 shorter fractures ( 2 = 2.5) essentially enhances connectivity between longer fractures of the other group ( 1 = 1.9), allowing transport to occur predominately along 1. Thus, the influence of longer fractures on solute transport is preserved. An ADE with fractionalorder derivatives in the longitudinal direction and integer-order derivatives in the transverse direction can describe this behavior [! $ % Group 3: 2.5 a 3.0 "., 2003]. Fickian transport occurs in networks where fracture length exponents are in group 3 ( ) (Sets 13-19). Differences between Fickian and super-fickian transport regimes are easily observable. First, correlations between fracture group orientations and eigenvectors of plume growth become weak to non-existent. This leads to orthogonal scaling rates according to the major and minor plume axes (Figure 3). Second, exponential or Gaussian tails are observed for all groups (not shown). Estimates of + based on plume spreading rates for all parameter sets in group 3 (13-19) suggest that fractures with low transmissivity values act as a retention mechanism, and especially affect rates of solute transport in the longitudinal direction, 1 (Table 3). Spreading rates in the transverse direction, 2, are not as heavily influenced. The fit between empirical distributions of marginal particle jumps and a theoretical Gaussian varies and is heavily influenced by the retention of particles in short fractures with low transmissivity values (Figures 11 and 12).

99 Figure 10. Mandlebrot plots of largest ranked particle displacements (circles) for Set 12 along (a) 1 and (b) 2 with best-fit truncated power-law (TPL) model at an elapsed time of 1.0 years. Approximately every 1/1000 point is plotted. Values of are given in units of meters. 84

100 Influence of Spatial Fracture Density Fracture spatial density,, is highly dependent on the distribution of fracture lengths in a model domain [Renshaw, 1999] and is defined as the ratio between the sum of individual fracture lengths, " and domain area, : 2 1 =. (13) = 1 For each power-law exponent, spatial density values are assigned to represent sparsely (min), moderately (int) and densely (max) fractured domains (Table 1). Values for spatial density range from at or slightly above the percolation threshold for the sparely fracture domains to maximum reported density values [, 1983;, 2000] for the densely fractured domains. Density values assigned to moderately fractured domains lie directly between values representing sparsely and densely fractured domains. To investigate the influence of spatial density on particle transport, values of fracture length exponents and orientations are kept constant for each fracture group for the following groups of parameter sets: Sets 1-2, 9-10, 13-14, and Only values of spatial density are changed. Sets 1 and 2 are used to analyze influences of density values for networks dominated by very long fractures (group 1, ). To test the hypothesis that truncation of particle pathways may lead to Fickian transport at very high spatial densities for group 1 networks, the spatial density value assigned to Set 2 is well beyond the maximum density assigned to this distribution of fracture length exponents. When spatial density is dramatically increased, tail exponent estimates along 1 and 2 increase from and for Set 1 to and for Set 2 (Figures 4 and 13). Higher tail estimates

101 86 Figure 11. Histogram (a) and Gaussian probability plot (b) along 1 of Set 14 at a transport time of 100 years. The deviation between the theoretical Gaussian trend and marginal particle displacements is attributed to anomalous subdiffusion (slow particle movement). Approximately every 1/1000 point is plotted. Spatial values are given in units of meters.

102 87 Figure 12. Histogram (a) and Gaussian probability plot (b) along 2 of Set 14 at a transport time of 1000 years. Approximately every 1/1000 point is plotted. Spatial values are given in units of meters.

103 88 of for Set 2 are caused by the greater number of fractures that are connected to the hydraulic backbone within the release area. The release of particles into more fractures for Set 2 leads to a greater sampling of fracture flow paths so that values ( 1 = , 2 = ) more closely represent the distribution of fracture lengths ( 1 = 1.0, 2 = 1.0). If truncation of fracture lengths was a controlling factor on transport rates, much higher tail estimates (i.e., closer to the finite variance threshold) would be expected due to the extremely high spatial density assigned to the network. Values of based on plume spreading rates for Set 2 match both tail estimates and fracture length exponent values. The influence of density between Sets 1 and 2 is also reflected in the timing and location of particles exiting model domain boundaries. For a total simulation time of 10,000 years, more than twice the number of particles leave the model domain boundary for Set 2 due to the increased number of high velocity pathways available for transport. The distribution of mixing measure weights is unaffected by spatial density (not shown). The influence of density for fracture length exponent group 2 ( ) on super-fickian and Fickian transport regimes is observed in Sets 9 and 10 where both 1 and 2 equal 1.9. Set 9 contains intermediate density values for both fracture groups, while Set 10 contains a network where fracture groups contain minimum ( 1) and maximum ( 2) densities. Tails of marginal particle displacements for Set 9 are powerlaw with tail thickness estimates of By increasing spatial density for one of the fracture groups in parameter Set 10, power-law trends along both 1 and 2 are lost and the decay of the largest marginal particle displacements follows an exponential trend.

104 Figure 13. Mandlebrot plots of largest ranked particle displacements (circles) for Set 2 along (a) 1 and (b) 2 with best-fit truncated power-law (TPL) model at an elapsed time of 0.1 years. Approximately every 1/1000 point is plotted. Values of are given in units of meters. 89

105 90 Two groups of parameter sets, Sets and Sets 16-17, are used to investigate the influence of spatial density on solute transport for networks dominated by short fractures. These simulations show a general trend where spatial density controls both the degree to which a plume spreads transverse to the hydraulic gradient and transport times to model boundaries for group 3 networks. Networks with lower values of spatial density (Sets 13 and 16, e.g., Figure 3a) exhibit a higher degree of spreading than more densely fractured networks (Sets 14 and 17, e.g., Figure 3b). Increasing the network density from 0.53 m/m 2 in Set 13 to 0.95 m/m 2 in Set 14 decreases the time it takes particles to leave the down gradient model boundary from 10,000 years (Set 13) to 2154 years (Set 14). A greater contrast in times to model boundaries (approximately two orders of magnitude decrease) is observed when the network density is increased from 0.60 m/m 2 for Set 16 to 0.80 m/m 2 for Set 17 as transport times to model boundaries decrease from 100,000 to 2154 years. Lower spatial densities promote both a higher degree of spreading transverse to the hydraulic gradient and longer travel times to model boundaries, as lower spatial densities increase the tendency of particles to stay within individual fracture segments, allowing fracture orientation to exert more influence. At higher spatial densities, the intersection of individual fractures is enhanced, resulting in the truncation of pathways for solute migration and less transverse dispersion since more pathways aligned in the direction of the hydraulic gradient are available. This allows particles to move more rapidly towards the down gradient model boundary.

106 Influence of Fracture Transmissivity Two substantially different pdfs were selected to identify the role of the distribution of transmissivity values on ensemble solute transport rates. The primary transmissivity distribution is based on hydraulic testing on boreholes at the Äspo Hard Rock Laboratory, where transmissivity values recorded in 3 meter intervals match a Pareto distribution similar to (12) with a power-law exponent of = 0.4 along with minimum and maximum values of and 10-2 m 2 /s [, 2005]. For the same data, parameters for a lognormal distribution were estimated by forcing the data to a lognormal model resulting in a log 10 ( ) mean of -9.0 and log 10 ( ) standard deviation of 1.1. These values are very similar, especially in terms of standard deviation, to another hydraulic testing data set at Äspo Hard Rock Laboratory described by [2001] where the log 10 ( ) mean and standard deviation were estimated at -8.2 and 1.05, respectively. The truncated Pareto and lognormal (-9.0,1.1) distributions are used to. randomly assign fracture transmissivity to individual fractures. Fracture length and transmissivity are uncorrelated. Four groups of parameter sets, Sets 3-4, 7-8, and 18-19, are used to study the role of fracture transmissivity on particle transport rates. The first parameter set in each group assigns fracture transmissivity according to a truncated Pareto pdf (e.g., Set 3), while the second parameter set assigns transmissivity using a lognormal pdf (e.g., Set 4). With the exception of the transmissivity distribution, parameter sets and fracture realizations for each group are identical, even down to the random seed used to position the random number sequence.

107 92 Estimates of for Sets 3 and 4 are almost identical for both tail and plume spreading methods. In the fracture length discussion (Section 4.2.1), we attributed the very heavy tailing observed for Set 3 to both fracture length exponent values and a heavy-tailed distribution for fracture transmissivity. Set 4 indicates that this may only be partially correct. It is true that the very low fracture length exponent values ( 1 = 1.0, 2 = 1.0) result in very long fractures that span the entire domain. Particles moving in these fractures are not influenced by other fractures so that fracture transmissivity governs particle transport rates. However, both distributions allow transmissivity values to vary by orders of magnitude. Since particles traveling through only a few, very long fractures with high transmissivity values create the tail of the distribution, we hypothesize that for group 1 networks, heavy-tails of marginal particle displacements can result from any transmissivity distribution that encompasses several orders of magnitude. Even a loguniform distribution may be able to reproduce this behavior. The effect of different transmissivity distributions result in significant contrasts in tail estimates for Sets 7 and 8 ( 1 = 2.2, 2 = 1.6). While tail estimates for Set 7 (Pareto- ) range between 1.9-exponential and for 1 and 2, tail estimates for Set 8 (lognormal- ) are significantly lower, and For Set 8, tail estimates may be unreliable as lower values of are not reflected for early times of particles leaving model boundaries. Particles first leave the model domain boundary at 10 years for both sets. Plume growth rates for both sets are very similar indicating that differences in tail estimates have little effect on overall plume growth. The major differences between the two parameter sets is demonstrated by the loss of particles at model domain boundaries. Over the course of 10,000 years, 2.5 times more particles leave the model domain for Set

108 93 8 than for Set 7. This is caused by the lognormal distribution having a higher median transmissivity than the truncated Pareto distribution. This effect was also observed for Sets where 20% more particles leave the model domain boundary for Set 19. Upper tails of marginal particle displacements for Sets and follow a similar exponential probability decay. 4.5 Complex Networks Natural fracture networks typically consist of two [e.g.,., 1995;., 1993;, 2000] or more fracture groups [e.g.,, 1982;., 1989;, 1997] with different mean orientations. Fracture networks for Sets 1-19 are restricted to two fracture groups with constant fracture orientation. Four additional parameter sets (Sets 20-23) are used to evaluate more complex fracture networks. Fractures are allowed to deviate around a mean fracture group orientation for Sets 20, 21 and 23. Parameter sets 22 and 23 contain three fracture groups. Instead of investigating these influences over a wide range of fracture network statistics, we focus on network statistics that promote super-fickian plume growth rates. This way the applicability of (5) to describe more complex ensemble particle displacement plumes as operator-stable densities is evaluated. Besides, previous simulations demonstrate that the convergence of ensemble particle displacements to symmetric, multi-gaussian densities is limited to very dense networks with short fracture lengths. Deviations in fracture orientation about the mean are assigned according to a Fisher distribution: [, 1953;, 1992]: cos( ) ( ) = (14) 4 sinh( )

109 94 where the deviation of fracture orientation from the mean, -π/2 < < π/2, is related to a dispersion parameter,. Low values of in (14) describe a large variability of fracture orientation from the mean, while large values indicate a tight clustering around the mean orientation (Figure 14). Values of for natural rock fractures range between 10 and 300 [, 2003;., 2004]. A positive constant,, ensures that / 2 ( ) = 1. Since we use the rejection method [ / 2, 1985] to generate fisher random variables, is not needed. Sets 20 and 21 allow for deviations about two mean fracture orientations (based on fracture group orientations for Set 1, 1 = 30º, 2 = -60º) according to Fisher dispersion parameters of = 10 and = 50, respectively. With the exception of variable fracture group orientations, networks for Sets 20 and 21 are identical to Set 1 (Table 1). Parameters representing Set 1 were selected as these set of statistics describe very sparse networks ( = m/m 2 ) with very long fracture lengths ( 1 = 1.0, 2 = 1.0). Thus, variability in fracture orientation should have the most pronounced effect on these networks. As expected, plume growth directions for ensemble plumes for Sets 20 and 21 are variable about the mean orientations, with more pronounced variability for the group with the lowest (Figure 15). Since (14) symmetrically describes variability in fracture orientation about a mean orientation according to a negative exponential decay of probability (Figure 14), we assume that eigenvector coordinates and mean fracture group orientations are correlated. Tail estimates of for both Sets 20 and 21 range for 1 and for 2, respectively. The range of based on plume growth rates is narrow

110 Figure 14. Probability histogram of 10 5 randomly generated Fisher deviates according to dispersion parameters of (a) = 10 and (b) = 50. Note the effect of on deviations about the mean, 0. 95

111 96 Figure 15. Ensemble particle displacement plume from Set 20 at a transport time of 10 years. Note the deviation around the mean fracture group orientations, 1 and 2. with estimates of for each eigenvector. Estimates based on tail and plume growth methods are identical to results for Set 1, indicating that deviations in fracture group orientation do not influence rates of particle transport. However, the influence of variability in fracture orientation is reflected in the distribution of the mixing measure, where a greater variability of weights occurs along each eigenvector (Figure 16). Networks for Sets 22 and 23 contain three fracture group oriented at ±45º and 90º with fracture length exponents of 1.6 and equal values of spatial density ( = 0.05) for a total network density of 0.15 m/m 2. Fracture orientations for Set 22 are constant, while fracture group orientations for Set 23 are allowed to deviate according to a Fisher dispersion constant of 50. The orientation of the third fracture group (90º) relative to the hydraulic gradient is intended to investigate the role of a third fracture group on

112 Figure 16. Histogram (a) and cumulative distribution plot (b) of mixing measure weights for Set 20 at a transport time of 100 years. Midpoint of bin interval is plotted. Fisher distribution curves are presented in (a) for comparison of mixing measure weights and distribution of fracture orientation. Note the clustering of mixing measure weights does not follow mean fracture group orientations of 30 and -60. Instead, weights are shifted towards the direction of the hydraulic gradient. 97

113 98 connectivity between the two fracture groups that are more preferably aligned with the hydraulic gradient. Ensemble particle displacement plumes (Figure 17) confirm that the fracture group oriented at 90º does enhance connectivity (and solute mixing) between the other two fracture group oriented at ±45º that are responsible for the majority of particle transport. However, a lower percentage of solute particles are transported normal to the gradient due to the combination of pipe flow methodology which propagates a regional gradient through all interconnected fractures regardless of orientation, and constant head conditions at all lateral boundaries. Since particle movement is two-dimensional, the two directions with the heaviest tails (lowest values) are used describe plume growth. This occurs along the fracture groups oriented at ±45º. Estimates of based on both tail ( ) and plume growth rate ( ) methods are similar for both Sets 20 and 21. Again, this supports the conclusion based on Sets 20 and 21 that deviations of fracture orientations from the mean in any group do not affect plume scaling rates. However, the primary difference between ensemble particle displacement plumes and operator-stable densities is the loss of particles through vertical transport in the fracture group oriented at 90º. Due to the model dimensions, particles undergoing vertical transport have smaller transport distances than particles transported along the other fracture groups oriented at ±45º. Consequently, these particles are not represented in the mixing measure (not shown) which computes directional weights based on the tail of the greatest particle displacements. This signifies a limitation on operator-stable densities for providing a full description of ensemble particle behavior for these networks. However, our simulations represent the most complex scenario where solutes are allowed to leave any down

114 99 Figure 17. Ensemble particle displacement plume from Set 22 at a transport time of 100 years. Note the influence of the vertically orientated fracture group. Figure 18. Preliminary correlation between fracture length exponent, fracture density and resultant plume growth rates (in the diamonds) for individual fracture groups within Sets A clear threshold between super-fickian and Fickian transport regimes is not present. However, super-fickian transport regimes may be defined in fracture rock masses where individual fracture groups have fracture length and density values less than 1.9 and 0.25 m/m 2, respectively.

115 100 gradient boundary. Limitations imposed on a flow system by the presence of shear zones with fault gauge or permeability contrast between lithographic could result in no flow boundary conditions at the north and south model boundaries and particle transport would be concentrated towards the eastern (down gradient) boundary. Furthermore, it is unlikely that three fracture groups would have equal densities. Instead, it is more likely that dominant transport directions can be inferred from the two fracture groups containing the greatest fracture densities. 5.0 Conclusions Data from fluid flow and particle tracking simulations in networks with power-law length distributions demonstrate that ensemble particle displacement vectors have many characteristics of operator-stable plumes including: 1) power-law probability tails of the largest particle displacements, 2) super-fickian plume growth rates, 3) different growth rates in each coordinate, where coordinates correspond to the two main fracture orientations, 4) non-elliptical plumes with distinct (and discrete) directional probability measures describing plume shape. Particle motion in densely fractured domains with short fracture lengths resembles a multi-gaussian density (an operator-stable subset) where elliptical plumes have exponential tailing of largest particle displacements and Fickian (or near Fickian) plume growth rates along plume axes. The operator-stable densities can be modeled using either integer-order or fractional-order ADEs, which describe ensemble transport according to multi-gaussian (a special case of operatorstable) and operator-stable densities, respectively.

116 101 Selection of a representative ADE depends on the transport regime. Quantifiable properties of the fractured medium, such as distributional properties of fracture length and values of spatial density, can be used to distinguish between Fickian and super- Fickian transport regimes (Figure 18). The transport regime is most heavily influenced by the distribution of fracture trace-lengths, while spatial density plays a secondary role when fracture lengths are near, or just above, the finite variance threshold for fracture lengths. Based on comparisons between a heavy-tailed and thin-tailed transmissivity distribution, the distribution of fracture transmissivity does not significantly influence transport regime as long as fracture transmissivity is allowed to vary over several orders of magnitude. Particle retention in low velocity fractures within the particle release area was observed for all network types. The transport characteristics of these particles is investigated in a companion study. Super-Fickian transport exclusively occurs when power-law fracture length exponent is in the range , even when spatial density values exceed those typically found in the field. Although not specifically tested, this range most likely extends to For these network types, estimates of the operator-stable tail index are lower than fracture length exponents and reflect the added influence of the wide distribution(s) of fracture transmissivity. Estimates of based on plume spreading rates are higher than tail estimates and indicate ballistic transport where is at or slightly above 1.0. Eigenvectors of plume growth are correlated with the orientation of fracture groups. Fickian transport exclusively occurs when fracture lengths are in the range , where the combination of short fracture lengths and high fracture densities promote the formation of multi-gaussian densities with orthogonal plume growth directions. The

117 102 range of fracture length exponent values favorable for multi-gaussian densities is approximately in the range For shorter fracture lengths, values of spatial density exert a significant amount of control over particle arrivals at model boundaries where particle arrival times can vary up to two orders of magnitude with all other characteristics held constant. The threshold between Fickian and super-fickian transport regimes closely follows the boundary between infinite variance and finite variance distributions of fracture length. Moderately fractured networks with fracture lengths in the range can lead to transport that can be either Fickian, super-fickian or a combination of the two. For networks where values of fracture length exponents equal 1.9, lower spatial densities ( < 0.25 m/m 2 ) preserve the infinite variance nature of solute pathways resulting in super- Fickian plume growth, while higher spatial densities ( > 0.25 m/m 2 ) result in the truncation of solute pathways leading to Fickian growth rates and exponential probability decay at leading plume edges. However, the presence of Fickian growth rates does not assure multi-gaussian ensemble plumes. If some correlation exists between plume growth directions and fracture group orientations, ensemble plumes can have nonorthogonal scaling directions. In this case, a classical ADE is inadequate even if Fickian scaling is observed. Our findings demonstrate that the distribution of fracture length exerts a strong control over solute movement in rock fracture networks, where unequal fracture length exponents for individual fracture groups can lead to plumes with dramatically different plume spreading rates along eigenvectors of plume growth. This is a departure from previous studies which only assign a single length exponent value to the entire network

118 103 [, 1999;., 2003]. [1999] proposed that fracture length exponent values for natural fracture networks are in the range This suggests that a super-fickian model of transport such as the multiscaling fractional advection-dispersion equation may be applicable to more field sites than the conventional ADE, which has shown poor performance for sparsely fractured domains dominated by long fractures. The use of analytical equation for solute transport predictions provides advantages over numerical simulations as less intensive field characterization is needed to produce screening-level predictions. We do not address deviations between ensemble and individual plumes here. Fulfillment of the ergodic hypothesis for these simulations is the subject of a companion paper.

119 104 References Aban, I.B., M.M. Meerschaert and A.K. Panorska, Parameter estimation methods for the truncated Pareto distribution,,, Barton, C.C., Fractal analysis of scaling and spatial clustering of fractures,, C.C. Barton and P.R. LaPointe, eds., Plenum Press, New York, Becker, M.W. and A.M. Shapiro, Tracer transport in fractured crystalline rock: Evidence of non-diffusive breakthrough tailing, (7), , Benson, D.A., The fractional advection-dispersion equation: Development and Application, Ph.D. Thesis, Univ. of Nev., Reno, Benson, D.A., S.W. Wheatcraft, and M.M. Meerschaert, Application of a fractional advection-dispersion equation, (6), , Berkowitz, B. and H. Scher, Anomalous transport in random fracture networks, (20), , Billaux, D.M., J.P. Chiles, and C. Hestir, Three-dimensional statistical modelling of a fractured rock mass- An example from the Fanay-Augéres Mine, (3/4), , Bonnet, E., O. Bour, N.E. Odling, P. Davy, I. Main, P. Cowie, and B. Berkowitz, Scaling of fracture systems in geologic media, (3), , Bour, O. and P. Davy, Connectivity of random fault networks following a power law fault length distribution,, , Butler, R.F., Scientific Publications, Oxford, 1992., Blackwell DeDreuzy, J.-R., P. Davy, and O. Bour, Hydraulic properties of two-dimensional random fracture networks following a power law length distribution. 1. Effective connectivity, (8), , Ehlen, J., Fractal analysis of joint patterns in granite, , Fisher, R., Dispersion on a sphere, Sci., 37,, ,

120 105 Gillespie, P.A., C.B. Howard, J.J. Walsh, and J. Watterson, Measurement and characterization of spatial distributions of fractures,, , Gustafson, G. and A. Fransson, The use of the Pareto distribution for fracture transmissivity assessment,, doi: /s y, Jiménez-Hornero, J.V. Giráldez, A. Laguna, and Y. Pachepsky, Continuous time random walks for analyzing the transport of a passive tracer in a single fracture, (W04009), doi: /2004wrr003852, Jurek, Z.J., Polar coordinates in Banach Spaces,, 61-66, Kemeny, J. and R. Post, Estimating three-dimensional rock discontinuity orientation from digital images of fracture traces, and Geosciences, 29, 65-77, Kosakowski, G., Anomalous transport of colloids and solutes in a shear zone,, 23-46, LaBolle, E.M., RWHet: Random Walk Particle Model for Simulating Transport in Heterogeneous Permeable Media,, LaPointe, P.R., and J.A. Hudson, Characterization and interpretation of rock mass joint patterns,, Long, J.C.S., J.S. Remer, C.R. Wilson, and P.A. Witherspoon, Porous media equivalents for networks of discontinuous fractures, (3), , McDonald, M.G. and A.W. Harbaugh, A modular three-dimensional finite-difference ground-water flow model,, Meerschaert, M.M., D.A. Benson, and B. Baeumer, Multidimensional advection and fractional dispersion,, , Meerschaert, M.M., D.A. Benson, and B. Baeumer, Multidimensional advection and fractional dispersion,, , Meerschaert, M.M. and H.P. Scheffler,, John Wiley and Sons, Inc., New York, Meerschaert, M.M. and H.P. Scheffler, Nonparametric methods for heavy tailed vector data: A survey with applications from finance to hydrology,

121 106 Science, Amsterdam, 2003., M.G. Akritas and D.N. Politis, eds., Elsevier Mercado, A., The spreading pattern of injected water in a permeability-statified aquifer,, 23-36, Munier, R., Statistical analysis of fracture data adapted for modelling discrete fracture networks- Version 2,, Swedish Nuclear Fuel and Waste Management, Co. (SKB), Stockholm, Sweden, Odling, N., Scaling and connectivity of joint systems in sandstones from western Norway,, , Painter, S., Cvetkovic, V. and J.O. Selroos, Power-law velocity distributions in fracture networks: Numerical evidence and implications of tracer transport, (14), doi: /2002gl014960, Reeves, D.M., D.A. Benson, and M.M. Meerschaert, Multi-scale transport of conservative solutes in simulated fractured media,, Renshaw, C.E., Connectivity of joint networks with power law length distributions, (9), , Ross, S.M., Florida, 1985., Academic Press, Orlando, Scheffler, H.-P., On estimation of the spectral measure of certain nonnormal operator stable laws,, , Schumer, R., D.A. Benson, M.M. Meerschaert, and B. Baeumer, Multiscaling fractional advection-dispersion equations and their solutions, (1), , Schwartz, F.W., L. Smith, and A.S. Crowe, A stochastic analysis of macroscopic dispersion in fractured media, Water (5), , Segall, P. and D.D. Pollard, Joint formation in granitic rock in the Sierra Nevada,, , Smith, L., and F.W. Schwartz, An analysis of the influence of fracture geometry on mass transport in fractured media, (9), , Stigsson, M., N. Outters, and J. Hermanson, Äspö Hard Rock Laboratory, Prototype Repository Hydraulic DFN Model no:2,, Swed. Nucl. Fuel and Waste Management Co. (SKB), Stockholm, 2001.

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123 108 On the Predictability of Solute Transport in Fractured Media Donald M. Reeves Desert Research Institute, 2215 Raggio Parkway, Reno, Nevada David A. Benson Department of Geology and Geological Engineering, Colorado School of Mines 1516 Illinois St., Golden, Colorado Mark M. Meerschaert Department of Mathematics and Statistics, University of Otago, Dunedin, New Zealand Hans-Peter Scheffler Department of Mathematics, University of Nevada, Reno, Nevada 89512

124 109 Abstract Numerical simulations based on realistic fracture network statistics are used to explore the transport of conservative solute particles released from a hypothetical geologic repository situated in a low permeability rock mass where both fluids and solutes can exit any down-gradient boundary. Particle transport results are used to determine both the applicability of analytical equations for solute transport predictions and to identify network statistics that are most suitable for a geologic repository. Depending on fracture network statistics, characteristics of ensemble solute particles that travel away from the source release area resemble either operator-stable or multi- Gaussian probability density functions. Though operator-stable and multi-gaussian densities can be modeled using fractional and classical integer-order advection-dispersion equations (ADEs), respectively, these equations assume a dissolved solute undergoes ergodic transport. An analysis of variability between individual realizations and the ensemble indicates that pre-ergodic transport conditions occur for all networks. The largest variability between the ensemble and individual realizations occurs for ensemble plumes that resemble operator-stable densities and are lowest for ensemble plumes resembling multi-gaussian densities. The lack of ergodicity implies that upscaled solutions of ADEs cannot adequately account for variability in a fractured medium from a single realization perspective. However, in the ensemble, ADEs represent the full range of variability in a fractured medium and may be useful for monitoring well network design. A probabilistic framework is adopted to identify networks statistics that are most suitable for a geologic repository. Criteria for the analysis include: the probability that a solute particle enters the hydraulic backbone, and once in the backbone, the probability of

125 110 a solute particle undergoing large transport distances. Other considerations include the potential for fast particle arrival times, characteristics of particle retention within the backbone, and the degree of variability between ensemble and individual plumes. Rock masses with relatively short fractures, at densities near the percolation threshold, are most suitable for geologic repositories. These types of random fracture networks engender slow overall plume growth rates, later arrival times, and moderate variability between individual and ensemble plumes.

126 Introduction Proposals for the long-term disposal of high-level radioactive wastes emphasize the role of a geologic barrier in isolating these wastes from the biosphere [e.g.,, 2004;, 2004]. Upon release from a repository, the likelihood of a dissolved solute leaving a fractured rock mass is dependent on flow and transport properties of a fractured medium. The high degree of spatial variability in rock fracture properties, such as fracture length, density, permeability, and network connectivity result in highly heterogeneous subsurface flow systems. Complexities associated with fluid flow and solute transport in fractured media have led to a reliance on numerical models for transport predictions [e.g.,, 1996;, 2005 and references therein]. Site-specific numerical models depend on extensive field characterization efforts to collect physical and hydraulic data on deterministic structures. Analytical equations that model transport as an advective-dispersive process may have advantages over numerical models as quick, screening-level solute transport approximations constructed from limited field data. Analytical equations can also describe solute transport at scales that exceed computational constraints of most numerical models. The reliability of predictions based on advection-dispersion equations (ADEs) depends on the fulfillment of the ergodic hypothesis; ergodicity is implied through the use of mathematical limit theorems that describe convergence of a normalized sum of independent and identically distributed ( ) random vectors to a limit distribution [, 1990;, 2001]. A generalized form of the central limit theorem describes the convergence of random vectors (i.e., particle displacement vectors) to an operator-stable density, while a multi-

127 112 Gaussian density is a special case of an operator-stable distribution that occurs when the magnitude of random vectors has a finite variance restriction [, 2001]. Operator-stable and multi-gaussian densities are solutions to fractionalorder and classical integer-order ADEs, respectively. In an effort to evaluate the performance of both classical integer-order and fractionalorder ADEs in describing ensemble transport of conservative particles in fractured media,. [2006a,b] produced synthetic plumes from multiple realizations of fluid flow and solute transport through large-scale, randomly generated fracture networks with statistically realistic features. The ensemble average of relative concentration for those particles that have traveled away from their initial location closely resembles operatorstable [Figure 1] or multi-gaussian probability densities. Ensemble plumes produced from sparsely to moderately fractured networks with infinite variance distributions of fracture length were found to have many characteristics of operator-stable densities including power-law decay of largest particle displacements, super-fickian plume growth rates where different growth rates in each coordinate correspond to primary fracture orientations, and non-elliptical plumes with distinct probability measures describing plume shape. Moderately to densely fractured networks with finite variance fracture length distributions were found to produce multi-gaussian densities where elliptical plumes have exponential rates of probability decay for the largest particle displacements and Fickian (or near Fickian) growth along plume axes. For all networks, transport in low velocity fracture segments enhances particle retention; retention of solute particles is most prevalent at the particle release area.

128 113 The use of ADEs for solute transport predictions assumes that deviations between ensemble and individual plumes are minimal. Based on only 3 network types,. [2006a] suggested that deviations from the ensemble are largest for sparsely fractured domains dominated by very long fractures and are smallest for densely fractured domains dominated by short fracture. This work is a continuation of the study by. [2006a,b]. In addition to an analysis of deviations between ensemble plumes and individual realizations to assess the degree that plumes are ergodic, a probabilistic framework based on particle transport is adopted to recommend fracture statistics of rock masses that are most suitable for the disposal of high-level radioactive waste. For this evaluation, properties of the hydraulic backbone are evaluated, including: 1) the probability that a solute particle will enter the hydraulic backbone, 2) probability that solute particles undergo large transport distances ( 1 km), 3) the presence or absence of fast particle arrival times based on a radial transport distance of 1 km, and 4) characteristics of particle retention. 2.0 Numerical Simulations of Flow and Transport A comprehensive explanation of the numerical simulations was given by. [2006a]. The large-scale (2.5 km by 2.5 km) fluid flow and solute transport simulations presented by. [2006b] provide synthetic data used for all the presented analyses here. A total of 15 parameter sets, defined as a group of values used to assign fracture length, density and orientation, represent a wide variety of fracture network types [Table 1]. The ranges of values used are taken from reported values measured at largescale field sites. Monte Carlo methodology is used to generate 500 equi-probable fluid

129 114 flow and solute transport realizations for each parameter set. Ensemble plumes are constructed from all individual realizations of a parameter set. A Pareto distribution, ( > ) =, is used to control fracture trace-lengths (above a certain cutoff) by allowing the value of the power-law exponent to vary between 1 and 3 [, 1993;, 1999;., 2001]. In general, fracture length decreases as increases. The distributional properties of fracture transmissivity and placement (or spacing) are held constant for all parameter sets. Transmissivity values assigned to individual fractures are heavy-tailed, based on an upper truncated Pareto distribution where values are allowed to range from to 10-2 m 2 /s according to a power-law exponent of 0.4 [, 2005; Poisson process using joint uniform (0,2500) distributions [., 2006]. A, 1985] is used to place fracture centers in the model domain, resulting in exponential spacing between fractures [., 1992;., 1996;, 2002]. Fractures are input into the model domain until a specified spatial density ( ), defined as the sum of fracture lengths normalized by area of the domain, is reached. Networks with longer fractures (lower values of ) typically have lower values of spatial density [, 1997; 1999]. This may be related to the role of fluid pressure during fracture propagation. If fluid pressure is a driving force behind fracture propagation, the release of excess fluid pressure at the percolation threshold may cause fracturing to cease [ Using a novel algorithm [, 1996].., 2006a], the fracture networks are translated onto a finite-difference grid of equal cell size (1m by 1m by 1m) and MODFLOW [, 1988] is used to solve for fluid flow in both fractures and the less permeable matrix. To restrict the transport of solutes to fractures, a

130 Figure 1. Fracture network domain with shaded region representing area of particle release. Note that a hydraulic backbone is not present in the particle release area for this realization. Spatial values are in meters. 115