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Statistical continuum modeling of mass transport through fractured media, in two and three dimensions Parney, Robert Wyn 1999

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STATISTICAL CONTINUUM M O D E L I N G OF M A S S TRANSPORT THROUGH F R A C T U R E D MEDIA, IN TWO A N D THREE DIMENSIONS by ROBERT W Y N P A R N E Y B.Sc , University of Calgary, 1986 M . Sc., University of Calgary, 1989 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE D E G R E E OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES Department of Earth and Ocean Sciences We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A April 1999 © Robert Wyn Parney, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of (JMAH A~>& Ocr/fr* GcXstA^*-* The University of British Columbia Vancouver, Canada DE-6 (2/88) A B S T R A C T The Statistical Continuum Method (SCM) provides a technique to model aqueous phase transport through a fractured rock mass at the field scale, while explicitly including the effects on transport of fracturing that are observed on the scale of a borehole or outcrop. The S C M approach models mass transport in two stages: (1) particles are first "educated" in a subdomain consisting of multiple discrete networks in order to capture the range of motion possible within a fracture system; and (2) particles are then moved in a random-walk through a larger continuum, obeying the range of motion "learned" within the subdomain. The use of discrete networks allows particle movements in the S C M continuum to honor the particle motion that occurs in the discrete subdomain, without the fundamental changes in the nature of the transport process necessary in most continuum approximations. The use of the continuum permits these movements to be extended into domains significantly larger or more complex than those that can be modeled by conventional discrete network simulations. The key element in the S C M method is the determination of the most appropriate methods for translating the motion of particles in the discrete subdomain into a set of statistical distributions that are then sampled in the continuum. To evaluate the effectiveness of the S C M approach the evolution of spatial moments through time for S C M models are compared with the evolution of moments for equivalent discrete network models. The S C M method is capable of reproducing mass transport in two and three-dimensional discrete fracture networks, as evidenced by the match between the trends in spatial moments through time for the discrete network and the spatial moments for the S C M model. A number of S C M modeling approaches produce moment values within one standard deviation of the average of the network realizations, although no one model works best for all fracture systems. 11 T A B L E OF CONTENTS Abstract " Table of Contents Hi List of Tables vii List of Figures ix Acknowledgements xix 1. INTRODUCTION A N D OVERVIEW 1 1.1. INTRODUCTION 1 1.2. O V E R V I E W OF T H E THESIS 3 2. REVIEW OF CONCEPTS, OBSERVATIONS A N D APPROACHES TO M O D E L I N G TRANSPORT IN F R A C T U R E D R O C K 6 2.1. INTRODUCTION 6 2.2. OBSERVATIONS AT EXPERIMENTAL SITES 6 2.2.1. Flow and transport experiments 6 2.2.2. Occurrence and mapping of natural fractures 8 2.3. M A T H E M A T I C A L BASICS IN MODELING FLOW A N D TRANSPORT 12 2.3.1. The Cubic Law 12 2.3.2. Percolation 14 2.3.3. Advection-Dispersion Equation 15 2.3.4. Particle tracking 15 2.3.5. Mixing 16 2.4. DISCRETE NETWORK MODELS IN TWO A N D THREE DIMENSIONS 17 2.5. C O N T I N U U M MODELS 21 2.6. H Y B R I D MODELS 22 2.7. C O N C L U S I O N 24 3. S C M METHOD IN TWO DIMENSIONS 27 3.1. INTRODUCTION 27 3.2. O U T L I N E O F T H E STATISTICAL C O N T I N U U M A P P R O A C H 27 3.3. G E N E R A T I O N O F TWO-DIMENSIONAL F R A C T U R E N E T W O R K S 28 3.3.1. Fractures from a single length scale 28 3.3.2. Fractures from a range of length scales 29 3.4. DISCRETE N E T W O R K F L O W SOLUTION A N D P A R T I C L E T R A C K I N G 31 3.5. M O T I O N STATISTICS 33 3.5.1. Introduction 33 3.5.2. Particle path geometry 35 3.5.3. Statistical testing of path-length models 36 3.5.4. Statistical testing of velocity models 38 3.5.5. Conditioning of velocity and length parameters 38 in 3.5.6. Assessment Of Parameter Variability and Convergence 42 3.6. C O N T I N U U M D O M A I N F L O W SOLUTION A N D P A R T I C L E T R A C K I N G 53 3.6.1. Method 53 3.6.2. Nomenclature for S C M models in two dimensions 54 3.7. C O N C L U S I O N 55 4. S C M M O D E L RESULTS IN TWO DIMENSIONS 78 4.1. INTRODUCTION 78 4.2. POISSON N E T W O R K S 79 4.2.1. Base-Case 79 4.2.2. Rotated gradient 91 4.2.3. Orthogonal network 93 4.2.4. High and reduced density networks 94 4.2.5. Poisson networks near the percolation threshold 96 4.3. L E V Y - L E E N E T W O R K S 100 4.3.1. Motion Statistics 101 4.3.2. Plumes 102 4.3.3. Moments 103 4.3.4. Levy-Lee B, fractal dimension 1.8 105 4.4. DISCUSSION OF S C M RESULTS IN T W O DIMENSIONS 106 4.5. S U M M A R Y 108 5. S C M METHOD IN THREE DIMENSIONS 142 5.1. INTRODUCTION 142 5.2. G E N E R A T I O N OF THREE-DIMENSIONAL FRACTURE NETWORKS 143 5.2.1. Outline of F R A C M A N 143 5.2.2. Modifications to F R A C M A N 147 5.3. M O T I O N STATISTICS 149 5.3.1. Introduction 149 5.3.2. Particle movement in three dimensions: concepts and definitions 149 5.3.3. Calculating motion statistics in three dimensions 152 5.3.4. Motion statistics under the long-path model 155 5.3.5. Summary 158 5.4. INTERDEPENDENCE OF MOTION STATISTICS 159 5.4.1. Introduction 159 5.4.2 Velocity as a function of the vertical trajectory (<j)) 159 5.4.3. Path-length as a function of the vertical trajectory ((()) 161 5.4.4. Velocity as a function of path-length 162 5.4.5. Summary 163 5.5. P A T H - L E N G T H MODELS INCLUDING A CUTOFF IN VERTICAL TRAJECTORY (§) 164 5.5.1. Introduction 164 5.5.2. Distributions of path-length vectors under a <j> cutoff 165 5.5.3. Motion statistics under a (j) cutoff 166 5.5.4. Summary 167 5.6. A S S E S S M E N T OF P A R A M E T E R VARIABILITY 167 iv 5.6.1. Introduction 167 5.6.2. Variation between networks 168 5.6.3. Variation of parameters with scale 169 5.7. C O N T I N U U M MODELS IN THREE DIMENSIONS 171 5.7.1. Introduction 171 5.7.2. Injection 172 5.7.3. Directional choice algorithms 172 5.7.4. Orientation 173 5.7.5. Velocity and path-length 174 5.7.6. Nomenclature for S C M models in three dimensions 174 5.8. S U M M A R Y 175 6. RESULTS IN THREE DIMENSIONS 213 6.1. INTRODUCTION 213 6.2. B A S E CASE: HORIZONTAL G R A D I E N T 214 6.2.1. Introduction 214 6.2.2. Moments for network realizations 214 6.2.3. S C M models: Varying models of velocity and path-length 215 6.2.4. S C M models: Direction choice algorithms 218 6.2.5. S C M models: Correlating velocity and path length to the vertical trajectory . 221 6.2.6. S C M models: Correlating vertical trajectory to horizontal trajectory 222 6.2.7. S C M models: Correlating velocity to path-length 223 6.2.8. S C M models: Applying a vertical trajectory (())) cutoff 224 6.2.9. Summary for the three-dimensional base-case model 227 6.3. B A S E CASE: R O T A T E D G R A D I E N T 228 6.3.1. Introduction 228 6.3.2. Moments for network realizations 228 6.3.3. Moments for S C M models 229 6.3.4. Summary 233 6.4. V A R I A B L E STRIKE A N D DIP FRACTURE S Y S T E M 233 6.4.1. Results with 4m cutoff applied 235 6.5. L O W E R DENSITY FRACTURE S Y S T E M 236 6.5.1. Introduction 236 6.6. L E V Y - L E E FRACTURE S Y S T E M 239 6.6.1. Networks 239 6.6.2. Statistics 240 6.6.3. Moments for network realizations 242 6.6.4. Moments for S C M models 243 6.7. S U M M A R Y OF THE S C M RESULTS IN THREE DIMENSIONS 245 7. DISCUSSION 300 7.1. INTRODUCTION 300 7.2. IS THE TEST AGAINST THE A V E R A G E V A L U E S OF T H E FIRST A N D SECOND SPATIAL MOMENTS FOR THE DISCRETE NETWORKS SUFFICIENT? 300 7.3. ERGODICITY A N D T H E S C M METHOD 302 7.4. PROBABILISTIC INTERPRETATION OF S C M PLUMES '. 307 7.5. EXTENSIONS TO T H E S C M 308 v 7.5.1. Variable motion statistics: Mixed gradient models and others 308 7.5.2. Predicting first arrivals 309 7.6. S U M M A R Y 310 7.6.1. General 310 7.6.2. Two-dimensional modeling 311 7.6.3. Three-dimensional modeling 312 8. REFERENCES 314 9. APPENDIX: MODIFICATIONS TO M A F I C 324 vi LIST OF TABLES Table 3-1: Input Parameters for the Poisson Fracture System 57 Table 3-2: Input Parameters for the Levy-Lee Fracture System 57 Table 3-3: Motion Statistics from 0 degree gradient, Poisson Network System, 10,000 realizations 58 Table 3-4: The five modeling methods of the S C M . 58 Table 3-5: Selected motion statistics in direction 2 versus domain size. Base-case Poisson fracture system. 99 % of all fractures are less than 4.32 m 59 Table 4-1: Errors in the slopes of the moments for the base-case Poisson network with horizontally applied gradient I l l Table 4-2: Motion Statistics for 45° case I l l Table 4-3: Errors in the slopes of the moments for the Poisson network with gradient applied 45 degrees from horizontal. Networks are the same as the base-case Poisson network, but the gradient orientation is rotated I l l Table 4-4: Errors in the slope of the moments for the Poisson network with perpendicular fracture sets. Networks are the same as the base-case Poisson network except that the average fracture orientation of fracture set two is 90 degrees instead of 70 112 Table 4-5: Errors in the slope of the moments for the low intensity Poisson network. Networks are the same as the base-case Poisson network except that fracture densities are 3.1 fractures / meter instead of 7.2 for both fracture sets 112 Table 4-6: Errors in the slopes of the moments for the high intensity Poisson network. Networks are the same as the base-case Poisson network except that the fracture intensity in both sets is 10.3 fractures / meter instead of 7.2 112 Table 4-7: Errors in slope of moments for Poisson network at percolation threshold. Network is the same as the standard Poisson network except that fracture densities are 1.5 fractures / meter 113 Table 4-8: Motion Statistics Calculated from Particle Movements within 10,000 Realizations of the Levy-Lee Network System 113 Table 4-9: Errors in slope of continuum moments for Levy-lee A. Levy-Lee network with fractal dimension 1.5 113 Table 4-10: Errors in slope of continuum moments for Levy-lee B. Levy-Lee network with fractal dimension 1.8 114 Table 4-11: Sum of the error in continuum slopes, compared to discrete slopes. Value is the summation of error in first and second x and y slopes. Cases in which the first y slope is almost zero do not include error in this slope 114 Table 5-1 Generation parameters for three-dimensional base-case fracture system 177 Table 5-2 Sample of a single particles movements as output from M A F I C . Residence time is in seconds. Particle locations are in meters 177 Table 5-3 Motion statistics for three-dimensional base-case network at § = 0° 178 Table 5-4 Motion statistics for three-dimensional base-case model under varying (j) cutoffs 179 Table 5-5 Motion statistics calculated under the long path-length definition for varying sizes of model domains. Subdomains used for all three-dimensional models are 10m by 10m by 10m 180 vii Table 6-1 Input parameters for the variable orientation fracture system 249 Table 6-2 Input parameters for the lower density fracture system 249 Table 6-3 Input parameters for three-dimensional Levy-Lee fracture system. 250 Table 6-4 Motion statistics for the Levy-Lee fracture system based on 1000 realizations of the discrete subdomain 250 Table 6-5 Errors in the slope of the moments for varying fracture systems. The average value represents the average of the absolute values of the errror. Each model presented gives the lowest average, absolute error 251 viii LIST OF FIGURES Figure 2-1 Comparison of particle movement probabilities at fracture intersections under two redistribution models (a) complete mixing (b) stream-tube routing. Fracture geometry and flow solutions are identical in both cases. Particles enter the fracture intersection from the left 26 Figure 3-1: Flow of Statistical Continuum Method 60 Figure 3-2: A large domain Poisson network. A single network realization based on the base case fracture system given in Table 3-1 61 Figure 3-3: Levy-Lee network. One realization of a discrete subdomain based on a fractal dimension D=1.5 62 Figure 3-4: Flow distribution for one realization of the base-case Poisson fracture system. Applied hydraulic gradient is horizontal. Line thickness is determined by the order of magnitude of flow. Flows are IO"4, 10~5and 10"6m3/day respectively. Fractures in which flow is less than 10"6 m3/day are not plotted 63 Figure 3-5: Fractures containing flows greater than 10"5 m3/day from Figure 3-4. Although these fractures carry over 80% of the flow, without the inclusion of the fractures containing flows less than 10"5 m3/day the network falls below the percolation threshold 64 Figure 3-6: Revised definition of path-length. Previously path-length was defined as the distance between fracture intersections. Under the new definition path-length is defined as the distance a particle travels along a fracture between the fracture intersection in which it enters and the intersection throughout which it exits 65 Figure 3-7 Path-length statistics for the base-case Poisson fracture system. Histogram, 3-parameter Gamma distribution, Normal distribution and Exponential distribution based on 10,000 realizations of a 10m x 10m discrete subdomain. Statistics are from the direction parallel to the applied hydraulic gradient 66 Figure 3-8 Velocities of particles for the base-case Poisson fracture system. Histogram, 3-Parameter Gamma distribution and Normal distribution based on 10,000 realizations of a lOmx 10m discrete subdomain. Statistics are from the direction parallel to the applied hydraulic gradient 67 Figure 3-9: Scatter plot of velocity versus path-length pairs. Each data point represents a single particle movement within a lOmx 10m discrete subdomain for the base-case Poisson network system. Movements are sampled from 20 realizations 68 Figure 3-10: Mean and standard deviation of path-length versus velocity. Data points are calculated over 10,000 realizations of a lOmx 10m discrete subdomain, based on the base-case Poisson network system 69 Figure 3-11 Mean and coefficient of variation for directional choice for the base-case Poisson network system 70 Figure 3-12: Mean and coefficient of variation for the mean, standard deviation and skew of path-length for the base-case Poisson network system 71 Figure 3-13: Mean and coefficient of variation for the mean, standard deviation and skew of the negative Logio velocity in the direction parallel to gradient for the base-case Poisson network system 72 ix Figure 3-14 Mean and coefficient of variation for the first and second order correlation coefficients for the base-case Poisson network system 73 Figure 3-15 Mean and coefficient of variation for directional choice after removal of networks with effective hydraulic conductivities in the highest and lowest quartiles 74 Figure 3-16: Mean and coefficient of variation for the mean, standard deviation and skew of path-length, after removal of networks with conductivities in the highest and lowest quartiles 75 Figure 3-17: Mean and coefficient of variation for the mean, standard deviation and skew of the negative Logio velocity in the direction parallel to gradient after removal of networks with effective hydraulic conductivities in the highest and lowest quartiles for the base-case Poisson network system 76 Figure 3-18 Mean and coefficient of variation for the first and second order correlation coefficients after removal of networks with effective hydraulic conductivities in the highest and lowest quartiles 77 Figure 4-1: Plumes from 3 individual discrete networks, for the base-case Poisson network system. Each plume contains 5000 particles. Each network is generated using the base-case Poisson fracture system (Table 3-1) 115 Figure 4-2: Plumes for the base-case model, using five different S C M models. Time = 200 days. Each plume contains 10,000 particles. See Table 3-4 for the model definitions 116 Figure 4-3: First moments for discrete and S C M models for the base-case Poisson fracture system. Values for the discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean of the moment for the discrete network realization 117 Figure 4-4: Second moments for discrete and S C M models for the base-case Poisson fracture system. Values for the discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean of the moment for the discrete network realization 118 Figure 4-5: First moments for discrete and S C M for the base-case Poisson fracture system. Moments are plotted against time in days. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is 45° from horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 119 Figure 4-6: Second moments for discrete and S C M for the base-case Poisson fracture system. Moments are plotted against time in days. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is 45° from horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 120 Figure 4-7: First moments for discrete and S C M models for orthogonal Poisson network system. Networks are similar to the base-case fracture system except that the vertical fracture set is aligned at 90° rather than 70°. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic x gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 121 Figure 4-8: Second moments for discrete and S C M models for orthogonal Poisson network system. Networks are similar to the base-case fracture system except that the vertical fracture set is aligned at 90° rather than 70°. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 122 Figure 4-9: One realization of the low-intensity Poisson fracture system. One realization of the large discrete subdomain. Fracture system is the same the base case except that fracture intensity in both fracture sets is 3.6 m"1 123 Figure 4-10 One realization of the high-intensity Poisson fracture system. One realization of the large discrete subdomain. Fracture system is the same the base case except that fracture intensity in both fracture sets is 10.8 m*1 124 Figure 4-11: First moments for discrete and S C M models for the low intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the intensity of both fracture sets is 3.5 m"1 instead of 7.2 m"1. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 125 Figure 4-12 Second moments for discrete and S C M models for the low intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the intensity of both fracture sets is 3.5 m"1 instead of 7.2 m"1. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 126 Figure 4-13: First moments for discrete and S C M models for the high intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical fracture set intensity is 10.2 m"1 instead of 7.2 m"1. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 127 Figure 4-14: Second moments for discrete and S C M models for the high intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical fracture set intensity is 10.2 m"1 instead of 7.2 m"1. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 128 Figure 4-15: One realization of the Poisson network at approximately the percolation threshold. Fracture intensity is 1.5 fractures m"1. A l l non-conducting fractures have been removed, leaving only the hydraulic backbone of the network 129 Figure 4-16: First moments for discrete and S C M models for the percolation threshold Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical and horizontal fracture intensity is 1.5 fractures per meter. Discrete values are based on 5,000 network realizations. xi Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 130 Figure 4-17 Second moments for discrete and S C M models for the percolation threshold Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical and horizontal fracture intensity is 1.5 fractures per meter. Discrete values are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 131 Figure 4-18 First moments for discrete and S C M models, for the percolation threshold Poisson fracture system, using continuum models based on 10,000 realizations of 20 x 15 m discrete subdomains 132 Figure 4-19 Second moments for discrete and S C M models, for the percolation threshold Poisson fracture system, using continuum models based on 10,000 realizations of 20 x 15 m discrete subdomains 133 Figure 4-20: Plumes from 4 individual Levy Lee networks, D=1.5. Time = 50 days. Each plume contains 5000 particles. Each network is from the network statistics in Table 3-2. Domain size is 30m x 12m 134 Figure 4-21: Plumes based on five different S C M models, for the Levy-Lee fracture system with D = 1.5. Domain size is 30m x 12m. Time = 50 days. Each plume contains 5,000 particles 135 Figure 4-22: Plume evolution through a single Levy-Lee network. Between 10 days and 30 days the plume expands in both the x sn&y directions. Between 30 and 50 days the plume continues to expand in the x direction, but mass does not move significantly in the y direction. This lack of spreading is partially because of the loss of mass through the upper boundary, but primarily it occurs because mass is forced down the limited number of channels between clusters 136 Figure 4-23: First moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.5. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 137 Figure 4-24: Second moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.5. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 138 Figure 4-25: Single realization of the Levy-Lee fracture system with fractal dimension 1.8 139 Figure 4-26: First moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.8. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 140 Figure 4-27: Second moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.8. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model 141 Figure 5-1 Axis definition for three dimensional domains 181 xii Figure 5-2 Description of fracture sets by poles and planes. One fracture plane from each set of the base-case fracture system. Fracture set orientation is described by the orientation of a pole, perpendicular to the fracture plane. The pole for set two points downward and towards the back of the cube (- x direction) 182 Figure 5-3 One realization of the three-dimensional base-case network 183 Figure 5-4 Particle movement across a fracture plane. Particles follow flow lines, so that the most likely path is directly between the two intersecting fractures. A small proportion of particles will follow the less direct flow lines. Arrows indicate the orientation of the particle trajectory 184 Figure 5-5 Simple definition of 9 and <|>. Note axis is rotated for display purposes 185 Figure 5-6 Schematic of particle movement through three intersecting fracture planes. Particle locations at elements of the mesh are converted to element to element movements. Then element-to-element movements are grouped into path-length vectors 186 Figure 5-7 Definition of set-directions on simple two fracture model 187 Figure 5-8 Schematic showing conversion of particle locations to path-length vectors, including definition of locations, elements, vectors, and set-directions 188 Figure 5-9 Raw movement vectors for a single realizations of a base-case discrete subdomain under a horizontally applied gradient. Vectors are not sorted into sets, (a) (j) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. (j), (e) Velocity vs. (j) 189 Figure 5-10 Wrap around effect. As § crosses 90 degrees 9 is changed by 180 degrees 190 Figure 5-11 Scatter plot of 9 and <j) with plots of the theoretical relationship overlaid. Vectors are from one realization of the base-case fracture system with a horizontally applied gradient 191 Figure 5-12 Movement vectors in set-direction 1, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) (j) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. (j), (e) Velocity vs. (j) 192 Figure 5-13 Movement vectors in set-direction 2, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) (j) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. (|>, (e) Velocity vs. § 193 Figure 5-14 Movement vectors in set-direction 3, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) (j) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. (j), (e) Velocity vs. <|> 194 Figure 5-15 Movement vectors in set-direction 4, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) (j) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. (j), (e) Velocity vs. (j) 195 Figure 5-16 Histograms of 9. Base-case model, gradient orientation 9 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 196 xin Figure 5-17 Histograms of (j). Base-case model, gradient orientation 9 = 0°, = 0°. Based on 1000 discrete subdomain realizations 197 Figure 5-18 Histograms of path-length. Base-case model, gradient orientation 6 = 0°, <|) = 0°. Based on 1000 discrete subdomain realizations 198 Figure 5-19 Histograms of velocity. Base-case model, gradient orientation 0 = 0°, <j> = 0°. Based on 1000 discrete subdomain realizations 199 Figure 5-20 Mean velocity versus <j). Base-case model, gradient orientation 9 = 0°, <() = 0°. Based on 1000 discrete subdomain realizations 200 Figure 5-21 Standard deviation of velocity vs. <j). Base-case model, gradient orientation 6 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 201 Figure 5-22 Mean path-length vs. (j). Base-case model, gradient orientation 9 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 202 Figure 5-23 Standard deviation of path-length versus <j). Base-case model, gradient orientation 9 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 203 Figure 5-24 Mean velocity as a function of path-length. Base-case model, gradient orientation 0 = 0°, <|> = 0°. Based on 1000 discrete subdomain realizations 204 Figure 5-25 Standard deviation of velocity as a function of path-length. Base-case model, gradient orientation 0 = 0°, <j) = 0°. Based on 1000 discrete subdomain realizations 205 Figure 5-26 Histograms of 0 under the 10 degree cutoff. Base-case model, gradient orientation 0 = 0°, = 0°. Based on 1000 discrete subdomain realizations 206 Figure 5-27 Histograms of under the ten degree cutoff. Base-case model, gradient orientation 0 = 0°, § = 0°. Based on 1000 discrete subdomain realizations 207 Figure 5-28 Length Histograms under a 10 degree cutoff. Base-case model, gradient orientation 0 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 208 Figure 5-29 Velocity histograms under a 10 degree cutoff. Base-case model, gradient orientation 0 - 0°, § = 0°. Based on 1000 discrete subdomain realizations 209 Figure 5-30 Mean and standard deviation of path-length vs. number of fracture networks run in the subdomain. Base-case model, gradient orientation 0 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 210 Figure 5-31 Mean and standard deviation of velocity as a function of the number of networks run in the discrete subdomain. Base-case model, gradient orientation 0 = 0°, <|> = 0°. Based on 1000 discrete subdomain realizations 211 Figure 5-32 Directional choice parameters as a function of the number networks run in the discrete subdomain. Base-case model, gradient orientation 0 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations 212 Figure 6-1 First moments for the base case model with applied gradient at 0 = 0° and <j) = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. Continuum models are H(v)H(l),H(v)G(l), N(v)G(l) and G(v)G(l) using long-path motion statistics 252 Figure 6-2 Second moments for the base case model with applied gradient at 0 = 0° and (j) = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. Continuum models are H(v)H(l),H(v)G(l), N(v)G(l) and G(v)G(l) using long-path motion statistics 253 xiv Figure 6-3 S C M models are G(v)G(l) with varying directional choice algorithms. First moments for the base case model with applied gradient at 9 = 0° and = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 254 Figure 6-4 S C M models are G(v)G(l) with varying directional choice algorithms. Second moments for the base case model with applied gradient at 9 = 0° and <j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 255 Figure 6-5 S C M models are H(v)H(l) with varying directional choice algorithms. First moments for the base case model with applied gradient at 9 = 0° and ()) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines '. 256 Figure 6-6 S C M models are H(v)H(l) with varying directional choice algorithms. Second moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 257 Figure 6-7 S C M models including velocity and path-length as a function of §. Second moments for the base case model with applied gradient at 9 = 0° and <j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 258 Figure 6-8 S C M models including velocity and path-length as a function of (j). Second moments for the base case model with applied gradient at 9 = 0° and <|> = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 259 Figure 6-9 S C M models in which (j) is calculated as a function of 9. First moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 260 Figure 6-10 S C M models in which (j) is calculated as a function of 9. Second moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 261 Figure 6-11 S C M models in which velocity as a function of path-length First moments for the base case model with applied gradient at 9 = 0° and <j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 262 Figure 6-12 S C M models in which velocity as a function of path-length Second moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 263 Figure 6-13 S C M models using motion statistics calculated using a § = 10° cutoff. First moments for the base case model with applied gradient at 9 = 0° and (j> = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 264 xv Figure 6-14 S C M models using motion statistics calculated using a <|> = 10° cutoff. Second moments for the base case model with applied gradient at 8 = 0° and <j> = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 265 Figure 6-15 S C M models using motion statistics calculated using a (|> = 5° cutoff. First moments for the base case model with applied gradient at 0 = 0° and § = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 266 Figure 6-16 S C M models using motion statistics calculated using a (j) = 5° cutoff. Second moments for the base case model with applied gradient at 9 = 0° and § = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 267 Figure 6-17 G(v)G(l) S C M models with varying directional choice algorithms, using ()) = 10° motion statistics. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 268 Figure 6-18 G(v)G(l) S C M models with varying directional choice algorithms, using <j) = 10° motion statistics. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 269 Figure 6-19 Base case model with applied gradient at 9 = 45°, (|) = 45°. S C M models are applied without correlations. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 270 Figure 6-20 Base case model with applied gradient at 9 = 45°, <j) = 45°. S C M models are applied without correlations. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 271 Figure 6-21 G(v)G(l) with varying probability algorithms. Base case model with applied gradient at 9 = 45°, <j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 272 Figure 6-22 G(v)G(l) with varying probability algorithms. Base case model with applied gradient at 9 = 45°, § = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 273 Figure 6-23 G(v)G(l) model with no-reversal/no-repeat algorithm using motion statistics with varying (j) cutoffs. Base case model with applied gradient at 0 = 45°, <j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 274 Figure 6-24 G(v)G(l) model with no-reversal/no-repeat algorithm using motion statistics with varying § cutoffs. Base case model with applied gradient at 0 = 45°, (j) = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 275 Figure 6-25 S C M models with velocity, path-length and (j) correlations. Base case model with applied gradient at 0 = 45°, <j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 276 Figure 6-26 S C M models with velocity, path-length and (j) correlations. Base case model with applied gradient at 9 = 45°, § = 45°. Second moments. Average for xvi the network realizations is the solid line with ± one standard deviation shown as dashed lines 277 Figure 6-27 S C M models with (j) calculated as a function of 6. Base case model with applied gradient at 9 = 45°, <j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 278 Figure 6-28 S C M models with <j) calculated as a function of 9. Base case model with applied gradient at 9 = 45°, (j) = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 279 Figure 6-29 Variable theta and phi network 280 Figure 6-30 Scatter plot of 9 vs. § for path-length vectors from one realization of the fracture system with variable strike and dip orientation 281 Figure 6-31 Variable strike and dip fracture system. First moments. Applied hydraulic gradient at 9 = 0° and § = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 282 Figure 6-32 Variable strike and dip fracture system. Second moments. Applied hydraulic gradient at 9 = 0° and § = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 283 Figure 6-33 S C M models are G(v)G(l) with varying directional choice algorithms. First moments for the variable strike and dip fracture system with applied hydraulic gradient at 9 = 0° and <) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 284 Figure 6-34 S C M models are G(v)G(l) with varying directional choice algorithms. Second moments for the variable strike and dip fracture system with applied hydraulic gradient at 9 = 0° and dp = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines 285 Figure 6-35 Variable theta and phi network, fractures truncated at an equivalent radius of 4 m 286 Figure 6-36 First moments for the V F V T Network with all fractures truncated at an equivalent radius of Am. S C M models are G(v)G(l) with varying directional choice algorithms 287 Figure 6-37 Second moments for the V F V T Network with all fractures truncated at and equivalent radius of Am. S C M models are G(v)G(l) with varying directional choice algorithms 288 Figure 6-38 Fracture networks for lower-density system. Networks contain 40% of the number of fractures in the variable orientation fracture system 289 Figure 6-39 First moments for the lower-density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are G(v)G(l) models with various directional choice algorithms 290 Figure 6-40 Second moments for the lower-density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are G(v)G(l) models with directional choice algorithms 291 xvii Figure 6-41 First moments for the lower-density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are all G(v)G(l) models with the no-reversal/no-repeat algorithm, with various injection band sizes 292 Figure 6-42 Second moments for the lower density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are all G(v)G(l) models with the no-reversal/no-repeat algorithm, with various injection band sizes 293 Figure 6-43 Sample of Levy-Lee fracture network:(a) Traces (b) Fractures 294 Figure 6-44 Scatter plot of element to element particle movements with the average relationship between theta and phi overlaid. Based on one network realization of the Levy-Lee fracture system 295 Figure 6-45 Histograms of 9 for the Levy-Lee fracture system, x axis is 9 in degrees, y axis is frequency 296 Figure 6-46 Histograms of <j) for the Levy-Lee fracture system, x axis is (j) in degrees, y axis is frequency 297 Figure 6-47 First moments for the Levy-Lee model. Various S C M models 298 Figure 6-48 Second moments for the Levy-Lee model. Various S C M models 299 xviii A C K N O W L E D G M E N T S As I was preparing for my final defense I reflected upon my time at U B C ; what occurred and what I accomplished. I came to the conclusion that the highlight of the last seven years is not the completion of my Ph.D. program, but rather the people I have had the opportunity to interact with during that time. First and foremost I would like to acknowledge the support and guidance provided by my supervisor Dr. Leslie Smith. Leslie has that rare ability to hone in on exactly what is important, and be uncompromising in the search to finding it. I would like to thank Tom Clemo for his assistance and permission to use and modify his DISCRETE, two-dimensional fracture network modeling software. I would also like to thank Tom Doe, B i l l Dershowitz, Paul La Pointe and Glori Lee of Golder Associates for their assistance, and for the use and modification of the F R A C M A N code for the three-dimensional network modeling. I would like to thank Roger Beckie and Brian Berkowitz for setting the standard earlier for teaching and research excellence, as well as all of the people who took supervisory roles during my time at U B C , including Oldrich Hungr, Michael Novak and Tad Ulrych. I would especially like to thank my classmates Peter Alt-Epping, Petros Gaganis, Craig Nichol, Jason Smolenski and Christoph Wels for their help, insight and friendship. None of what I have done in the last seven years could have been accomplished without the support of my friends, and at the risk of offending anyone by omission I would like to thank everyone I know in Vancouver's triathlon and music communities for just being themselves. Finally, I would like to dedicate this thesis to the memory of Rich Weiss, who died tragically soon after finishing his Doctoral work at U B C . The depth of Rich's personal integrity moved everyone with whom he came in contact. xix 1. Introduction And Overview 1.1. Introduction Contaminant transport through fractured-media is an issue of ongoing interest in many areas of earth science and engineering including the permanent geologic storage of high-level nuclear waste (e.g. Bear et al, 1993; National Research Council, 1996; Tsang and Neretnieks, 1998; Faybishenko, 1999). It has been observed at various experimental sites that the transport of solutes through fracture networks is a complex problem, with a number of chemical and physical processes affecting the movement of solute mass. Furthermore the location and geometry of fractures embedded within a volume of rock can only be inferred by indirect and incomplete information. As a result, it is difficult to predict the rates at which mass will be transported through a network of fractures. A combination of field experiments and computer models have been used to both understand and predict mass transport through fractured media. Several countries are exploring the possibility of permanent repositories for nuclear waste in geologic settings, often in crystalline bedrock. In order to test the feasibility of this concept experimental field programs have been developed at Stripa in Sweden (Rasmussen and Neretnieks, 1986; Dverstorp and Anderson, 1989; Abelin et al. 1991a; 1991b; Dverstorp et al, 1992), Spain (Alasandro et al., 1997), France at Fanay-Augeres {Cacas et al. 1990a,b), Canada (AECL, 1994; Vandergraaf et al. 1994, Chan et al, 1999), Finland (Niemi 1994), United States at Yucca Mountain (Bodvarsson, 1999), as well Switzerland and Japan. It has been recognized 1 that a critical point at many of these sites is the understanding of the flow and transport through fractures within crystalline rock.. Transport through fractured media has also been considered in: (1) petroleum reservoirs (e.g. Finley and Lorenz, 1987; Acuna and Yorstos, 1991); (2) fractured clay aquitards in near-surface waste disposal sites (e.g. Sudicky et al., 1983; Harrison et al, 1992; Schmidt 1993); and (3) as part of local and regional groundwater flow systems (e.g. Shapiro and Hsieh; 1991). Transport through fractured media can also be an issue in the remediation of contaminated sites (e.g. Parker and Sterling, 1999) and geothermal power (e.g. Garg and Nakanishi, 1999). Since Snow (1965) introduced the parallel-plate approximation, significant advances have been made in the understanding and modeling of flow and transport through fractured media. However, it is generally recognized that there has not been a major breakthrough that has "solved" the problem of fractured rock hydrology, and much is still unknown about the processes operating, and how to model them (Panel Discussion, Witherspoon Conference, February 1999: see Faybishenko, 1999). Numerical models are often applied in efforts to make predictions about flow and transport behavior through fracture networks. A key distinction within the modeling approaches is whether or not to directly include the geometry of the fractures, when representing the structure of the hydraulic conductivity fields found within natural fracture systems. The generation of field scale models which include the effects of fracture networks on contaminant transport remains an outstanding problem. Modeling solute transport through fracture networks is the subject of this thesis. The approach presented relies heavily on a discrete modeling approach, but by introducing a statistical continuum, allows models of transport at the field scale to include the effects of 2 fracture geometry, without explicitly including the fractures in the field-scale component of the model. The thesis is outlined in the following section. 1.2. Overview of the thesis The purpose of this thesis is to present the Statistical Continuum Method (SCM) in two and three-dimensions. The S C M provides a technique to model aqueous phase transport through a fractured rock mass at the field scale, while explicitly including the effects on transport of fracturing that are observed on the scale of a borehole or outcrop. The S C M approach models mass transport in two stages: (1) particles are first "educated" in a subdomain consisting of multiple discrete networks in order to capture the range of motion possible within a fracture system; and (2) particles are then moved in a random-walk through a larger continuum, obeying the range of motion "learned" within the subdomain. The use of discrete networks allows particle movements in the S C M continuum to honor the particle motion that occurs in the discrete subdomain, without the fundamental changes in the nature of the transport process necessary in most continuum approximations. The use of the continuum permits these movements to be extended into domains significantly larger or more complex than those that can be modeled by conventional discrete network simulations. The thesis is composed of six chapters beyond the introduction: The second chapter reviews the modeling approaches which have been applied for transport through fractured media. In the third and fourth chapters the S C M method and results in two-dimensions are presented. In the fifth and sixth chapters the S C M method and results in three-dimensions are presented. The seventh chapter concludes the thesis with a summary of the results and a discussion of the implications of this work. 3 In Chapter 2 the basics of modeling transport through fractured media are outlined. Some modeling approaches explicitly describe the location and geometry of each fracture within the network. These types of models reproduce the physics of flow and transport in fracture networks, but are limited by the computational burden required to include the detailed geometry of the network. Other models represent fracture networks as continuums, with flow and transport properties representing the cumulative properties of the fracture network averaged over a given length-scale. Continuum models tend to be less computationally intensive but do not attempt to reproduce the detailed structure of the hydraulic conductivity within the fracture network. Hybrid models, including the S C M approach, are an attempt to include the physics of flow through discrete networks while reducing the computational burden required in the discrete network models. In Chapter 3 the S C M method in two dimensions is outlined. The generation of discrete subdomains is outlined, and the capture of particle movements by the use of motion statistics is presented. The methods by which these motion statistics are applied in continuum models is outlined. In Chapter 4, the S C M method in two-dimensions is evaluated by comparing the results of S C M models to large-domain network realizations. Comparison is primarily achieved by evaluating the spatial moments of the mass distribution for the two modeling approaches. Several example fracture systems are used including Poisson networks of varying density, and a Levy-Lee fracture system in which the networks are composed of fractures from a range of length-scales. In Chapter 5, the S C M method in three dimensions is outlined. Differences between the two and three-dimensional approaches in the generation of discrete subdomains, the 4 calculation of motion statistics, and the application of motion statistics within a continuum domain are given. Applications of the S C M in three dimensions are presented in Chapter 6. The method is again evaluated by comparing mass transport in discrete network simulations to mass transport in S C M models. Comparisons are made by examining the spatial moments of the mass distribution for both of the modeling approaches. Baecher and Levy-Lee fracture systems are examined in order to include both single and multi-scale fracture networks. Networks generated using the Baecher model in three-dimensions have a similar type of fracture spacing as networks generated using the two-dimensional Poisson model. In Chapter 7, the two and three-dimensional results are compared and evaluated. The significance of comparing the spatial moments is discussed. The results of the method are then discussed in terms of Ergodic theory, which provides a framework for assessing the effectiveness of the S C M method. The purpose of the thesis is to develop the S C M method to enable the modeling of realistic fracture systems at the field scale, and to demonstrate the feasibility of doing so. Examples of fracture network systems are presented for the purpose of evaluating the S C M approach. These examples are meant to present a complexity similar to what might be encountered in natural fracture networks, but the parameters of these fracture systems are not rigorously controlled by site-specific data. This limitation of the discrete models may limit the validity of specific examples as comparisons to natural systems, but does not artificially enhance the tests of the effectiveness of the S C M models. These fracture systems should therefore be sufficient for the purpose of evaluating the S C M approach. 5 2. Review Of Concepts, Observations and Approaches to Modeling Transport in Fractured Rock. 2.1. Introduction Modeling of flow and transport through fractured rock networks can be subdivided into three broad categories based on the way in which the hydraulic conductivity within the fracture networks is represented: (1) Discrete models, in which fractures are represented explicitly within a model, (2) Continuum models, in which the average properties of fractures are represented using a continuum domain, and (3) Hybrids of discrete and continuum models (National Research Council, 1996). This chapter provides a brief review of the concepts, observations and approaches to transport in fractured rock. In section 2.2 selected observations of transport processes at a number of sites are discussed. In section 2.3 some mathematical background to the modeling, measurement and prediction of mass-transport through fractured media is discussed. In sections 2.4 through 2.6 discrete, continuum and hybrid modeling approaches are outlined. 2.2. Observa Hons a t experimen tal sites. 2.2.1. Flow and transport experiments An important objective in the development of high-level waste repositories has been to gather data and to build generic knowledge of transport processes in fractured rock (Dverstorp et al, 1992). At a number of experimental sites fractures have been mapped and hydrologic testing and tracer experiments have been conducted. A common observation has been that of all the fractures observed in a borehole, fewer than 10% are hydraulically active (Abelin et al, 1985, Rasmuson and Neretnieks, 1986; Cacas et al, 1990a.). One of the most 6 extensively studied sites is the Stripa mine in Sweden (e.g. Abelin et al. 1991a,b; Dverstorp et al, 1992). The Stripa experiment showed that there is an uneven spatial distribution of flow and tracers, suggesting channeling, and a complex and often multi-modal set of tracer breakthrough curves. At Stripa it has been demonstrated that the crystalline rock is highly heterogeneous with respect to the distribution of fracturing, and heterogeneous in both its hydraulic and transport properties (Dverstorp et al, 1992). One of the key conclusions of the work at Stripa (e.g. Abelin et al, 1991a,b) is that the flow and transport problems are three-dimensional in nature, thus a key to understanding the likelihood of success of a geologic repository is to understand the three-dimensional nature of the flow paths and channeling. There is evidence that transport can occur without mixing for distances from a few meters to as much as 150 m (e.g. Abelin et al; 1987). Tracer migration paths characterized by negligible mixing and low dispersion suggest to many researchers that flow is highly channelized. If a solute travels along a large number of channels then the behavior is expected to be highly dispersive; i f a transport path consists of a single channel then dispersion is minimized (Dverstorp et al, 1992). Channelized flow has been seen at a number of scales in fractured rock at a number of sites (e.g. Neretnieks., 1987; Cacas et al, 1990a,b). The preferential flow paths are thought to arise due to the spatially varying permeability within a medium. It was observed that the flux of water is unevenly distributed within each fracture (Abelin e. al, 1985), so that permeability variations within a single fracture lead to channeling within the fracture plane. Channeling also occurs within larger networks as one fracture will act as a more effective conduit than another. Flow channeling is significant to the performance assessment of underground repositories as i f channeling 7 results in a fast transport path connecting a repository to a major hydraulic conductor, such as a fracture zone, this will result in the fast movement of contaminants to the surface. Although many models of mass transport through fractured media consider only the movement of mass at the advective velocity of the groundwater, several other processes may affect this movement: (1) chemical sorption on surfaces; (2) diffusion into the rock matrix surrounding the open fracture aperture; and (3) dispersion. The importance of these three processes varies from medium to medium. An important first step is to understand transport due to advection through fractures, as advective transport at the network scale includes macro-dispersion due to the fracture geometry. A l l of the models in this thesis consider only mass transport by advection. It is clear that mass transport through fracture networks is a complex process. Channeling results in complex flow paths, but performance assessment of geologic disposal sites depends on understanding the nature of these flow paths before reliable predictions can be made. In the following section observations of naturally occurring fractures provide a key step towards understanding these flow paths. 2.2.2. Occurrence and mapping of natural fractures In order to model flow and transport through fracture networks, it is necessary to examine what is known about fractures themselves. One of the most striking features of fractures is that they occur everywhere and at almost all scales, from micro-cracks in quartz grains to mega-faults at the continental scale (e.g. Allegre et al, 1982). Fractures occur primarily in sets, which can be described by the distributions of their density, geometry and orientation. Fractures large enough to be mapped by satellite photos are usually treated as individual features - for example the San Andreas fault (Aviles et al, 1987), although even 8 faulting at the global scale can be described in terms of systems and sets (e.g. Kvet, 1982). In the geologic situations that arise in repository design it is often necessary to treat larger fractures individually, i f they can be mapped, and to treat small fractures as sets. The key problem with mapping fractures is that currently there does not exist a technology through which each fracture can be mapped explicitly, i.e. there is no remote-sensing geophysical tool that can be used to map the spatial extent of a fracture, or even the intensity of fracturing in a volume of rock. As a result all information on fractures is indirect and incomplete. Fractures can be imaged within boreholes, or mapped as traces on surficial outcrops within subsurface adits. In the development of a repository it is obviously not in the best interest of the final design to penetrate the medium, so that a trade-off exists between the physical penetration of the rock which weakens the barriers to flow, and the increase of ' information that arises as more of the medium is mapped. Therefore, descriptions of natural fractures are limited by the fact that fractures are essentially three-dimensional geometric objects, which can usually only be mapped extensively in two-dimensions, and are usually sampled by individual boreholes. So what can be mapped of a fracture population? Advantage can be taken of the fact that fractures occur within sets. Properties measurable on individual fracture traces can be assigned in a statistical sense to sets of fractures. For instance, the mean and standard deviation of the length of a series of fracture traces along an outcrop may be calculated, and the distribution of trace-lengths can then be described by a lognormal model. Much work has been done to characterize the statistical distribution of fractures in both two and three dimensions (e.g. Dershowitz and Einstein, 1988; Kulatilake et al, 1993; Gillespie et al, 1993; Yielding et al, 1996; and Marrett, 1996). 9 In this work the parameters used to describe the distribution of fractures in two-dimensions are: (1) the length of the trace; (2) the orientation of the trace; (3) the size of the aperture; (4) the density of fracturing; and (5) the distribution of fracturing. For most two-dimensional mapping and modeling, the three-dimensional shape of a fracture is assumed to be such that its intersection with a two-dimensional plane results in a linear feature or trace. Many authors have described the distribution of trace-lengths with either an exponential or lognormal distribution, although Kulatilake et al. (1993) found that a lognormal distribution was unsuitable for describing trace-length for a wide-range of previously published data sets. In two-dimensions the orientation of a fracture within a set can be described by the mean and variance of a single parameter 9, the orientation with respect to the x-axis within a two-dimensional plane. The distribution of these orientations in two-dimensions have been found to follow Bingham distributions (Kulatilake et al, 1993), and Fischer distributions (Dershowitz and Einstein, 1988). Aperture size in two-dimensional models is often described by a single value, the effective aperture, using either a normal, lognormal or exponential distribution. Fracture density can be described by the scan-line intensity, i.e. the average number of fractures that intersect a line perpendicular to the mean orientation of the fracture set (intersections per unit length). The description of fractures in three dimensions is more complicated. The key unknown parameter is the shape of the fractures themselves, i.e. curved or flat and square or spherical. This issue is an area of continued discussion (e.g. Crider and Pollard, 1998). Fractures have been described in three-dimensions as flat circular discs (e.g. Cacas et al, 1990a,b; Abelin et al, 1991; and Dverstorp et a., 1992). The orientation of the fracture planes in three dimensions must be described by two angles, and this is often done using the 10 distribution of the poles to the fracture plane identified by the dip and dip direction (e.g. Dershowitz and Einstein 1988). The aperture in a three-dimensional fracture can be described by a single value or by a distribution of aperture sizes throughout the fracture plane. In laboratory situations the aperture can be measured directly for small sections of a fracture, however the measurement process tends to destroy the sample (e.g. Pyrak-Nolte et al, 1987). It is more common to measure a flow parameter such as conductivity or transmissivity and use this to estimate an average or effective hydraulic aperture using the cubic law (see section 2.3). Much discussion has taken place about whether fractures in a single fracture set are from a single range of length scales, or from multiple ranges of length-scales (Barton and Hsieh, 1989; Velde et al, 1990; Gillespie et al. 1993; Belfield, 1994; Odling, 1994). This discussion is crucial, as measurements at field sites often occur at scales much smaller than those over which solute transport is to be predicted. It is therefore important to know whether the size of the largest fracture encountered will continue to increase with the size of the domain. Natural fracture patterns have been catalogued which appear to be self-similar over a range of length scales (Barton and Hsieh, 1989; Velde et al, 1990; Gillespie et al, 1993; Belfield, 1994; Odling, 1994). This self-similarity suggests that fracture patterns can be modeled over several scales using fractal geometry. With fractal geometry the space-filling properties of patterns are described by the fractal dimension (D), which is a non-integer or fractional version of Euclidean geometrical dimensions (Mandelbrot, 1977). For example, a pattern with a fractal dimension D = 2 entirely fills a two-dimensional plane, and similarly a pattern with D = 3 fills a volume. Barton and Hsieh (1989) found that fracture spacing at Yucca Mountain has a fractal dimension (D) between 11 1.6 and 1.8. For trace lengths, values were found to range between 0.8 to 1.3. Odling (1994) found that D ranged from 1.1 to 1.5 in the fracture profiles he surveyed. Fracture apertures also appear to be self-similar when measured over several length scales (e.g. Belfield, 1994). The ability to measure fractal dimension does not constitute proof that natural fracture patterns are fractal or contain fractures from a range of length scales. However, in this thesis, the fractal dimension is used to generate Levy-Lee models (section 3.3.2) so that the S C M method can be tested on fracture networks that include fractures from a range of length-scales. In this section parameters which can be used to describe the geometry of fractures have been presented. In chapters 3 and 5 these parameters are used to define fracture systems from which discrete fracture networks in two and three dimensions are generated. In the following section (2.3) some of the mathematical concepts used in the modeling of transport through fractured media are briefly outlined. 2.3. Mathematical basics in modeling flow and transport In this section four mathematical concepts used in the modeling of flow and transport in discrete fracture networks are briefly defined: (1) the cubic law; (2) the concept of percolation; (3) the advection-dispersion equation; and (4) the occurrence of complete mixing or stream-tube routing at fracture intersections. 2.3.1. The Cubic Law In most discrete network models, fluid flow through a single fracture is assumed to be analogous to laminar flow between two perfectly smooth parallel plates. Solving the Navier-Stokes equations for two parallel plates results in the cubic law: (Snow, 1965): 12 Q = -pgb3 dh 12// dl [2-1] where Q is the volumetric flow per unit thickness of the model, g is the gravitational constant, dh/dl is the hydraulic gradient b is the fracture aperture (j, is the viscosity of the fluid. The term pgb 112/u is the transmissivity of the fracture. It is generally not possible to measure the aperture directly. The cubic law can be used to convert a transmissivity measured using a packer test to an effective or average hydraulic aperture. There are, however, problems inherent with this approach that arise because most fractures do not contain a single aperture value (Smith et al. 1987, Tsang 1992). For more realistic geometries where the aperture size varies across the fracture plane, the Navier-Stokes equations cannot be solved directly and must be reduced to simpler equations such as the lubrication or Hele-Shaw equations (e.g. Zimmerman and Bodvarsson, 1996). Also, areas where the rock walls are in contact introduce tortuosity in the flow field. A review of analytical and numerical studies on the limitations of the cubic law found that in general, reasonably accurate predictions of the hydraulic aperture could be made by using either [2-2] 13 or the geometric mean of the hydraulic aperture, where <b> is the mean aperture and o2b is the variance of the aperture. Tortuosity in the flow path introduced by contact between the opposing faces of a fracture plane can be accounted for by using bl=bl(\-2c) P-3] where bo is the zero'th order approximation of bn, i.e. either the geometric mean or the estimate of b given by equation [2-2]. For modeling purposes the flow properties of individual fractures within a set can be represented directly by a distribution of transmissivities, or indirectly by a distribution of apertures, which are then converted to transmissivities via the cubic law. 2.3.2. Percolation A network is said to percolate i f there is at least one continuous connection from one side of the network domain to another. Networks are said to exist at the percolation threshold i f the network is conducting, but the conductive pathway could be broken by the removal of a single fracture from the system. Alternatively networks can be said to be exceed the percolation threshold i f on average they conduct (Bour and Davy, 1997) . The expression "on average" generally represents a spatial sampling of a network, so that i f approximately 50% of repeated samples of a constant area or volume percolate the network exceeds the percolation threshold. Percolation theory has been discussed extensively elsewhere (e.g. Stauffer, 1985; Sahimi, 1987; Berkowitz and Braester, 1991; Berkowitz and Balberg, 1993; and Bour and Davy, 1997). Renshaw (1996) has suggested that many fracture networks exist near the percolation threshold because the fracturing process may 14 terminate once fluid can flow across and out of the network. This view is not held universally, and many authors have modeled or interpreted natural fracture networks as existing significantly above the percolation threshold (e.g. Dershowitz, 1984; Endo et al, 1984; Smith and Schwartz, 1984; Long and Billaux, 1987; Dverstorp et al, 1992; Kulatilake et al, 1993; La Pointe et al, 1995b.; National Research Council, 1996; Clemo and Smith, 1997; Tsang and Neretnieks, 1998) 2.3.3. Advection-Dispersion Equation In porous media, the transport of non-reactive, non-decaying solutes through groundwater can be described by the advection dispersion equation (e.g. Bear 1972): ndC r / dC d „ 8C n R — + V,- D.. = 0 [2-41 dt ' dx: dx: lJ dx: L J The advective term (second) describes the movement of solutes at the velocity of the ground water. The dispersive term Ay (third) accounts for spreading of mass due to heterogeneity in the hydraulic conductivity of the porous medium. In fractured media the advection-dispersion equation can be solved at the fracture scale. In this work, however, instead of solving the A D E , mass transport is modeled using particle tracking. 2.3.4. Particle tracking In discrete network models the geometry of the fracture openings is known and hence so too is the detailed structure of the conductivity field. Solute mass can be treated as an ensemble of minute particles so that transport can be modeled by moving each particle at the velocity of the fluid within the fractures (e.g. Endo et al, 1984; Smith and Schwartz, 1984; 15 Long and Billaux, 1987; Dverstorp et al, 1992; Kulatilake et al, 1993; La Pointe et al, 1995b.; National Research Council, 1996; Clemo and Smith, 1997). Dispersion under this approach is introduced by the geometry of the fractures causing particles to travel along a range of pathways, with varying velocities and travel distances. 2.3.5. Mixing As mass is transported through fracture networks the issue of mass re-distribution at fracture intersections arises. Two models have been suggested: (1) complete mixing, and (2) stream-tube routing. Complete mixing (Travis 1984) suggests that mass entering a fracture intersection is not trapped within the stream-tubes in which it enters the intersection, and has probability of leaving the intersection or choosing to travel down a fracture based on the total flow out of the intersection within each fracture (Figure 2-la). This type of model is appropriate for situations in which mass resides within the intersection a sufficiently long time that it may diffuse across streamlines. At higher velocities it is thought that stream-tube routing prevails, mass travels through a fracture intersection without crossing streamlines, and in a two-dimensional model this concept is easy to understand (Figure 2-lb). Debate has occurred as to the flow-velocities at which either complete-mixing or stream-tube routing occur (Hull et al, 1987; Berkowitz et al, 1994; Stockman et al, 1997). Although it is possible to set up a simple fracture network model in which mass transport differs significantly depending upon the mass redistribution algorithm chosen, it has been the experience in the course of this work that mass transport through dense two-dimensional fracture systems, as expressed by the evolution of the spatial moments of the mass distribution, is nearly identical under both algorithms. In the three-dimensional work, 16 complete mixing is applied as it is the only option available within the software package that was adopted (Chapter 5). 2.4. Discrete network models in two and three dimensions. In this section a limited sampling of discrete network models is presented. Methods can be subdivided into models in two-dimensions (Sudicky and Frind, 1982; Schwartz et al, 1983; Dershowitz, 1984; Smith and Schwartz, 1984; Shapiro and Anderson, 1985; Shimo and Long, 1987; Long and Billaux, 1987; Smith et al, 1990; Wels and Smith, 1994; Renshaw, 1996; Clemo and Smith, 1997; Bour and Davy, 1997; Berkowitz and Scher, 1997) and models in three dimensions (Smith et al, 1987; Abelin et al, 1991b; Dverstorp et al, 1992; Cacas et al, 1990a,b; Moreno and Neretnieks, 1993; Dershowitz et al, 1995; Bour and Davy, 1998). Networks have also been represented by radial flow towards boreholes using AIDE'S (e.g. Rasmussen and Neretnieks, 1986; Allisandor, 1997). Discrete network models are primarily stochastic in nature, with fractures generated by sampling from statistical distributions of fracture geometry. In most two-dimensional models fractures are represented as linear features on a two-dimensional plane. Transmissivity values for each fracture are sometimes obtained by first assigning aperture values and then calculating the transmissivity using the cubic law (e.g. La Pointe et al. 1995), or by assigning transmissivity values directly (e.g. Cacas 1990a,b; Dverstorp et al. 1992). Mass-transport is modeled by first generating the fractures to form the network, finding a flow solution by applying hydraulic boundary conditions, and transporting mass by the use of particle-tracking or solving the A D E equation. Two-dimensional methods differ primarily in the algorithms used to generate the fractures. Fractures may be generated using random distributions of orientation and location, and power-law distributions of length (Bour and Davy 1997; Berkowitz and Scher 1997), or 17 from random samples of statistical distributions of geometric properties, based on field data (Schwartz et al 1983; Smith and Schwartz 1984; Long and Billaux 1987; Shimo and Long 1987; Dverstorp and Anderson 1989; Smith et al. 1990; and Wels and Smith 1994). Generation may not be entirely random as one fracture may be influenced by the location of a previously generated fracture (e.g. Clemo, 1994). Networks may also be generated using algorithms which are based on the physics of fracture formation and growth (e.g. Renshaw and Pollard, 1994; Renshaw, 1996, 1998), or are influenced by the response of a fracture network to applied stress (Gavrilenko and Gueguen, 1989; Degraff and Ay din, 1993; Brown and Bruhn, 1998). The algorithms by which two-dimensional fractures are generated in this work are discussed in detail in Chapter 3. Fracture generation in three dimensions is usually implemented using processes similar to fracture generation in two-dimensions, except that instead of representing fractures as linear features on a plane, fractures are usually represented as planar features within a three dimensional volume. In some three-dimensional fracture models the problem of calculating a flow solution is reduced into what might be called a 2.5 dimensional model. Fractures are first generated as two-dimensional discs in a three-dimensional volume. Once the intersections between the fracture planes are established, the conductive network is reduced to a set of one-dimensional tubes connecting the centres of the spheres through the intersections. The flow solution and subsequent transport is based on this reduced geometry (e.g. Cacas et al 1990a,b; Dverstorp et al. 1992). Moreno and Neretnieks (1993) have represented the entire flow domain as a network of channels in three dimensions. In the current work, based on the work of Dershowitz et al. (1995), flow is fully three-dimensional, 18 with finite element meshes covering each fracture plane, allowing fully two-dimensional flow within each plane. It has been noted by many authors that natural fracture planes contain a wide range of apertures, which leads to channeling of the flow within the fracture plane (Tsang and Tsang, 1987; Brown, 1987; Moreno et al, 1988; Tsang etal, 1988; Moreno et al,. 1990; Tsang, 1992; Vandergraaf et al, 1994). Many of these authors have then modeled a single fracture as containing a statistical distribution of apertures. Often this is done by generating a fracture plane and then assigning aperture values stochastically, so that aperture size varies as a function of position within the fracture plane (e.g. Smith et al, 1986; 1987; Moreno et al, 1988). Alternatively the opposing surfaces of the fracture void space are generated using a fractal algorithm, and then placed together to define the aperture geometry (e.g. Brown, 1987). Although approaches to the flow solution vary somewhat from one method to the next, most methods consist of applying the cubic law locally or assigning transmissivities to discrete elements within a fracture plane, and then solving the resulting flow field with a finite element or finite difference algorithm. Model results clearly show channeling within a single fracture plane (Moreno et al, 1990), although assumptions about the mixing of solute mass between grids can introduce numerical dispersion that is not consistent with the physical processes occurring (Goode and Shapiro, 1991). Although Smith et al. (1985a,b; 1987) included aperture variability within the fracture plane in a fully three-dimensional network model, most three-dimensional network models do not include aperture variability because the large computational burden limits the number of fractures that can be included in the model domain. 19 Two fundamental weaknesses of discrete network models are the difficulty obtaining field data on the geometry of the fractures, and the weak correlation that has been found at field sites between the density of fracturing and permeability determined by hydraulic testing. As a result network models are often calibrated to hydraulic data, either by rejecting networks with hydraulic conductivities outside of the range of measured conductivities (e.g. La Pointe et al, 1995b), or by adjusting the distribution of fracture apertures so that each network realizations falls within an acceptable range (e.g. Cacas et al 1990a,b). A third weakness of all discrete fracture network models is the high computational burden required by the detailed description the fracture network geometry. There are a finite number of fractures that can be represented within a two or three-dimensional model, so that modeling efforts depend on the size of the computer memory and time available to calculate flow solutions. This leads to two choices when using discrete models: (1) the maximum number of fractures can be utilized to represent a small fraction of the total domain of interest, but include all of the fractures expected within the domain, or (2) over a larger volume, only a fraction of the total number fractures expected can be included in the model. Hybrid modeling methods that try to combine these two approaches are discussed in section 2.6. As the complexity of the modeling process in each fracture is increased, i.e. as a more realistic model of flow and transport within each fracture is achieved, the total number of fractures that can be modeled is reduced. The computational burden of including this complexity has been a prohibitive barrier to the generation of field scale discrete network models of mass transport. 20 2.5. Con tinuum models A n alternative to modeling flow and transport within fractured rock has been to represent the rock mass as a continuum. The hydraulic properties of the network may be averaged over a model domain and the network treated as a porous medium with equivalent hydraulic properties. This is the simplest approach, and can be accomplished with standard modeling software such as MODFLOW. Alternatively the conductivity of the rock mass can be represented stochastically (Neuman 1987, 1988; Neuman and Depner, 1988; Niemi, 1994). Neuman (1988) suggests that because only tenuous connections have been made between fracture geometries and measurements of hydraulic conductivities, it is appropriate to discount geometric information, and to concentrate instead on the statistical structure of the hydraulic conductivity field. Once the statistical structure of the conductivity field is known, mass transport can be implemented by particle tracking or solving the advection-dispersion equation for a continuum domain in which the conductivity field is represented stochastically. One advantage of using continuum models is that in some cases continuum models are less computationally intensive than discrete models, allowing larger domains to be modeled. However, recent three-dimensional stochastic modeling of heterogeneous porous media by Naff et al. (1998a,b) indicates that when modeling heterogeneity which includes very large contrasts in the hydraulic conductivity, a "rather fine" computational grid is required. It seems likely that in order for a Stochastic Continuum model to realistically represent the heterogeneity occurring within a fracture network the continuum model will be nearly as computationally intensive as a discrete network model. 21 Stochastic continuum models also offer the advantage that the flow information upon which they are constructed is based on data that are directly measurable in the field. The results of borehole tests can be used to determine a transmissivity distribution, and the continuum models can then be generated based on the range of values measured. In discrete models transmissivity or aperture values are assigned to individual fractures, and then new transmissivity distributions are generated in each network by combining transmissivity information with fracture information. There are disadvantages to the use of this type of continuum models. Continuum models do not take into account the discrete nature of the hydraulic connections. The hydraulic properties of the network are averaged over the grid-spacing of a model, so that there is only a weak coupling between the fracture geometry and the hydraulic properties of the model. The link between the geometry of the fracture network and mass transport is also reduced in a continuum model, so that it seems unlikely that the effects of channeling on mass transport could be properly represented. Hybrid models, in which the effects of the fracture geometry is preserved without including all of the detailed geometry of the fracture networks, are discussed in the next section. 2.6. Hybrid models There is an obvious trade-off between discrete models which accurately represent the physics of flow and transport through fracture networks, but are limited in scope, and continuum models which have less limitations on domain size but which do not as accurately represent the physics of the transport process. Hybrid models attempt to merge the two approaches. One model, the Hierarchical model (Clemo, 1994; Clemo and Smith, 1997) 22 reduces the number of fractures within a network by solving for the lower-flow fractures separately in order to define a hydraulic backbone of the network. The higher-flow fractures, which make up this hydraulic backbone are represented as discrete features within the Hierarchical model. The flow and transport properties of the smaller fractures are solved separately to form blocks in which the residence time distributions are known. During transport particles can move through the fractures which compose the hydraulic backbone and also through the surrounding blocks. In this manner the effects of channeling in the hydraulic backbone are preserved, while including the dispersive effects of the slower fractures. A similar effort to reduce the number of fractures required is taken by La Pointe et al. (1995). Graph theory is used to find the primary connective pathways in fracture networks, reducing the number of fractures that need to be included in the flow solution. These methods have the advantage that by reducing the number of discrete fractures within the model, larger domains can be modeled. A problem with these approaches lies in the assumption that the lower flow fractures with smaller apertures have less influence on channeling. As will be seen in the plots of flow in the two-dimensional networks (section 3.4), these smaller fractures provide crucial links between the high-flow pathways. Other models represent flow and transport as a network of channels (Moreno and Neretnieks, 1993; Moreno et al, 1997; Tsang and Neretnieks, 1998). The use of channels allows for long transport paths with little mixing and transport paths with multiple channels in which extensive mixing occurs. However, these flow channels are not based on an understanding of the fracture network itself, but are rather a conceptual model of potential flow paths calibrated to the outputs of a field experiment. As in the continuum models, this approach does not make use of geometric information about the distribution of fractures. 23 An alternative approach, and the subject of this thesis, is the Statistical Continuum Method (Schwartz and Smith, 1988; Smith et al. 1990; Robertson 1990; Parney and Smith, 1995). Particle tracking through discrete fracture networks is used to gain an understanding of the influence of the network geometry on mass transport, and this understanding is applied in a larger continuum. Using discrete networks allows S C M models to include statistical information about the fracture geometry, measured values of the hydraulic conductivity, and the physics of flow and transport through the network. As a result the S C M approaches includes more information about the processes affecting transport in fractured media than either continuum models or channel models. By applying the statistical information gathered in discrete networks within a statistical continuum domain the S C M allows much larger models than can be implemented using discrete fracture networks. The S C M method is similar to channel models in the sense that the effects of channeling are included. However, the S C M approach offers the advantage that statistical representations of channeling are calculated by first examining how mass is channeled within a discrete subdomain. The S C M approach is fully described in Chapter 3 and Chapter 5. 2.7. Conclusion In this chapter key observations of flow and transport in fractured networks have been outlined. The organization of the hydraulic conductivity field of a fractured medium lends itself to the use of discrete fracture network models. Discrete models capture the physics of the flow process but are difficult to expand to the field scale. Continuum models do not include the geometry of individual fractures and rely on the correlation structure of the hydraulic conductivity field to reproduce the effects of the fracture geometry. The Statistical 24 Continuum Model is one of several hybrid approaches, and allows the up-scaling of discrete network models to the field scale, while including the effects of channeling on mass transport through fractured media. In the following chapter the S C M approach in two-dimensions is described. 2 5 (a) Complete Mixing 80 % Total Inflow 20 % Total Inflow Probability of exit = 0.2 Particle entrance 20% Total outflow 80 % Total outflow Probability of exit = 0.8 (b) Stream-tube routing Probability of exit = 0.0 Probability of exit =1.0 Figure 2-1 Comparison of particle movement probabilities at fracture intersections under two redistribution models (a) complete mixing (b) stream-tube routing. Fracture geometry and flow solutions are identical in both cases. Particles enter the fracture intersection from the left. 26 3. SCM Method In Two Dimensions •3.1. Introduction It is clear from Chapter 2 that a need exists to develop field-scale models of mass transport through fractured rock. However, because of limits on the total number of fractures that may be included in discrete network models, the larger the model domain, the lower the intensity of fracturing that can be modeled throughout that domain. As a result it may not be possible to include enough fractures to reproduce the intensity of fracturing visible at the scale of a bore-hole, for example, when using a discrete network to model mass-transport at field scale. Therefore, the goal of the Statistical Continuum Method (SCM) is to model the transport of dissolved solutes through fractured rock masses, at the field scale, while including the effects that fracturing at the scale of the bore-hole or outcrop may have on mass transport. In essence the goal of the S C M is to include the effects on mass transport of both the 10% of the fractures that determine the bulk of the flow, and the 90% of the less conductive fractures that control the tail of the mass distribution. In this chapter, the S C M is introduced. First, the concept of the S C M approach is outlined. Then each of the subcomponents - particle tracking through discrete subdomains, motion statistics and particle tracking in the continuum model - are explained. In the following chapter, the S C M method in two dimensions is evaluated by comparing the results of S C M models to the results for discrete network simulations. 3.2. Outline Of The Statistical Continuum Approach The approach of the S C M is to determine the rules governing mass transport in a fractured medium by using discrete fracture network models representative of the medium, 27 and then to apply these rules to mass transport in a continuum model. By using particle tracking (section 2.3), the S C M approach allows large-scale continuum models to include the details of the small-scale interactions occurring within discrete networks (Figure 3-1). In order to capture these interactions, particles are "educated" by first travelling through a discrete subdomain consisting of multiple network realizations, all of which are significantly smaller than field-scale (sec 3.3). As each particle travels through the subdomain; the velocities, distances and direction of motion are recorded (sec 3.4). Statistical models are fit to the distribution of each of these parameters (sec 3.5). Particle movement is reproduced in a larger continuum domain (sec 3.6) by first sampling statistical models of each motion parameter, and then converting these samples into particle motions. Mass transport in the continuum is then essentially a series of random walks using the motion statistics obtained in the discrete subdomain. 3.3. Generation Of Two-dimensional Fracture Networks 3.3.1. Fractures from a single length scale Discrete network simulations using particle tracking techniques involve three steps (National Research Council, 1996): (1) generating a fracture network (a network realization) by sampling from the distribution of the fracture population (a fracture system); (2) calculating the flow-field for the imposed boundary conditions; and (3) moving particles by advection through the network. For two-dimensional Poisson networks composed of a single length scale, the fracture system is defined by; (1) the number of fracture sets, (2) the mean and standard deviation of fracture length, (3) the mean and standard deviation of orientation, (4) the mean and standard deviation of aperture size, and (5) the scan-line intensity (Table 3-1). Networks are created 28 by randomly generating fracture midpoint locations within a domain which is extended beyond the intended model domain to avoid boundary effects, until the scan-line intensity of the input network system is reached. This approach results in a Poisson distribution of fracture spacing (Figure 3-2). Fracture-length and aperture are sampled from negative exponential and log normal distributions respectively, without correlation, or truncation. The parameters in Table 3-1 are referred to here as the two-dimensional base-case Poisson model. For this base-case model, equal numbers of fractures are generated for the vertical and horizontal fracture sets (Figure 3-2). If the average value for the fracture lengths are similar for the 2 fracture sets then application of the Poisson process in the manner described above leads to a fracture network with a single characteristic length scale. 3.3.2. Fractures from a range of length scales Networks composed of a range of length scales are created using a Levy-Lee flight process (Geier et al, 1989; Clemo, 1994; Dershowitz et al, 1995). A Levy-Lee flight is a random walk where each step of the walk determines the midpoint location of a fracture. The length and orientation of the fracture are then a function of the distance from the previous fracture midpoint. Each fracture set in the Levy-Lee network is generated as an independent random walk. Fracture centres are generated sequentially by first selecting a step length L ' from the fractal distribution (Bouchad and Georges, 1990): P L ( L ' > Z ) = L D 29 where D is the fractal dimension of the distribution of fracture centers. Step length L ' is selected using a uniform deviate U(0,1), an average length L^, and the fractal dimension D: U =[LM-U(0,l)f [3"2] The orientation of each step is chosen from a uniform circular distribution 2-TT-U(0,1), [3-3] and the new fracture mid-point placed at the end of this step. The length of the new fracture is proportional to the step size L ' , and the fracture orientation is sampled from N : N ( ^ f ) [3-4] where /u and cr are the mean and standard deviation of orientation, and L ' is the distance from the previous step of the random walk. This random walk results in a large number of small steps, and a small number of much larger steps. The large steps create long, widely-spaced fractures with, small orientation variations, and the small steps result in clusters of small, closely-spaced fractures, whose orientations vary widely (Figure 3-3). Levy-Lee network generation is controlled by 4 parameters (Table 3-2): (1) D, the fractal dimension of the point-field of fracture centers; (2) L^, the average length of the fractures; (3) ju, the orientation of the fracture set, and (4) cr, the standard deviation of orientation. Cluster intensity is proportional to D, and the orientation variation within the clusters is proportional to cr. Application of the Levy-Lee flight process in this manner results in a fracture network in which mid-point spacing and fracture lengths follow a fractal distribution (Bouchard and Georges, 1990). The algorithm used here is modified slightly so 30 that whenever the random walk exits the domain it is relocated within the domain, until the correct number of fractures have been generated. Because of this relocation the fractal dimension of the resulting distribution of fracture centers will differ somewhat from the fractal dimension used on input, but the network will still contain fractures from a range of length-scales. 3.4. Discrete Network Flow Solution and Particle Tracking In this thesis, it is assumed that fluid flow through the rock mass is confined to the fractures, and that the surrounding matrix is impermeable. In Chapters 3 and 4 fractures are represented by linear features within a two-dimensional plane, while in chapters 5 and 6 fractures are represented as planar features in three-dimensional volumes. The two-dimensional discrete modeling is based on modification of the code DISCRETE (Clemo 1994). Boundary conditions are defined by specified hydraulic head values along all domain boundaries. These hydraulic head values on the boundary are selected by specifying the average magnitude and orientation of a uniform mean gradient. The values of the hydraulic head in the interior of the network are found by solving the system of linear flow equations that arise when mass-balance is applied at each fracture intersection. With a solution for the hydraulic head at each fracture intersection, velocities within each fracture segment between intersections are calculated using the parallel plate approximation (section 2.3.1). Particles are injected along the upstream boundary through all the fractures that intercept a two-meter source band. Particles are injected in numbers proportional to the flow in the intersecting fractures, and are transported from intersection to intersection at the local fluid velocity. Only solute movement by advection is considered here, although inclusion of dispersion or 31 matrix permeability is not prohibited by the SCM. Stream-tube routing (section 2.3.5) is applied at each fracture intersection, with particles changing direction or continuing along a fracture depending upon both the flow volumes in the intersecting fractures, and the geometry of the intersections. Recent work by Stockman et al. (1997) suggests that complete mixing at fracture intersections may be a more appropriate model of solute redistribution at fracture intersections than stream-tube routing. However, particle-tracking through the base-case Poisson network system does not appear to be sensitive to the redistribution model chosen. In Figure 3-4, the flow distribution for one realization of the base-case Poisson network (Figure 3-2) is presented. The applied hydraulic gradient is horizontal, and the line thickness is determined by the order of magnitude of flow. These flows are IO'4, 10"5and 10"6 m3/day respectively. Fractures in which flow is less than 10~6 m3/day are not plotted. Less than 10% of all the fractures contain flows of 10"5 m3/day or larger (the thickest two sets of fractures plotted), yet these 10% of the total fractures carry over 80% of the total flow. However, these high-flow fractures do not form a conductive path from the left to right 5 3 boundaries of the domain. If all fractures containing flows less than 10* m /day are removed (Figure 3-5), the network composed of the remaining high-flow fractures would fall below the percolation threshold (section 2.3). It is apparent that fractures containing flows between 10"6 to 10"5 m3/day (the thinnest fractures plotted in Figure 3-4) provide essential links between the fractures containing higher flows. Because fluid mass must be conserved at 5 3 fracture intersections, each fracture carrying a flow larger than 10" m /day must feed into several fractures carrying lower flows. Therefore, flow-channeling in Poisson networks occurs as fluid travels through a multi-branched system of high, intermediate and low-flow 32 pathways. This channeling should not therefore be modeled as a single path of varying aperture. In order to model mass transport within a fractured rock mass, any modeling method must capture the effects of fluid channeling. If the S C M works correctly, then the influence of the entire range of flows will be recorded in the discrete subdomain, and then be reproduced in the continuum. Motion statistics provide the mechanism by which these effects are first recorded, and then reproduced. 3.5. Motion Statistics 3.5.1. Introduction Once the flow solution is calculated for an individual network, then all the velocities with which a particle can travel are pre-determined. However, the processes that determine the direction of particle movement at fracture intersections contain random elements, even in the case of stream-tube routing. Furthermore particle start locations are randomly selected along a source band. As a result, particles travelling across a network start at several different locations, and then follow numerous unique travel paths. Each pathway is composed of a series of straight-line steps between fracture intersections, with directional changes between each step. Particle motion in the discrete subdomain could be recorded by storing each pathway as a series of movement instructions defining every step and directional choice. Particle movement could then be recreated without the discrete network by executing these instructions in the order in which they occur. This approach would be redundant as it could only reproduce the exact particle motions that occurred in the discrete model. In the S C M , therefore, movement instructions are stored statistically. Particles are moved through a S C M continuum by randomly generating each step from a known range of instructions, 33 allowing particle-tracking to continue in domains much larger than the initial discrete subdomain. Motion statistics are at the core of the S C M , where thousands of movement instructions are reduced to the following parameters: (a) mean, standard deviation and skew of velocity; (b) directional choice at fracture intersections; (c) mean and standard deviation of direction orientation; and (d) mean, standard deviation and skew of path-length (Table 3-3). Correlations between adjacent velocities, and between velocity and path-length are also recorded. Path-length is defined below in section (3.5.2). Statistics are stored independently for each of the four directions in which a particle can travel from the intersection of two fractures (Table 3-3). In two dimensions, these set-directions are defined as: direction one -left; direction two - right; direction three - down; and direction four - up. In the two-dimensional model, the dimensions are referred to as x and y, so that each of the four directions represent trajectories of -180°, 0°, 90° and -90° respectively. The dip direction of the two-dimensional plane is fixed at 0°. The fifth column in Table 3-3 is discussed in section 4.2.1 The S C M method is based on the assumption that although the path traveled by an individual particle in the continuum does not correspond to the exact path traveled by an individual particle through a discrete network, when mass transport is averaged over multiple networks, the behavior of the plumes in the discrete and continuum models are statistically equivalent. In order for mass transport through a S C M model to be statistical equivalent to mass transport averaged over multiple network realizations, the S C M must reproduce the range of particle motions and velocities that occur in these networks. Particle motion through a small network realization can only sample small part of the range of particle 34 motions within a fracture system. In order to sample the entire range of particle motions, multiple network realizations are used in the discrete subdomain. In the following section, the range of particle motions or travel path geometries, is expressed as distributions of path-length, and directional choice at fracture intersections. Following this, the statistical models for both the travel path geometry and particle velocity are tested, and finally the variability of these motion parameters between network realizations is examined. 3.5.2. Particle path geometry Particle path geometry is composed of two components: (1) path length and (2) directional choice at the end of each path length. Every time a particle travelling down a fracture encounters a second, hydraulically active fracture that particle may leave the first fracture or continue along it. In the original S C M method {Schwartz and Smith 1988, Robertson 1990), this process was reproduced by first recording particle movements (directional choice) at each discrete fracture intersection in the subdomain (Figure 3-6a), and then recreating intersection to intersection particle movements in the continuum by sampling from the distribution of the spacing between intersections (path lengths). At the end of each continuum step, particles could continue to travel in the same direction, or change direction, based on the statistics gathered in the subdomain. Note that the spacing between fracture intersections reflects the network geometry, but is independent of the flow solution. In order to incorporate flow information into the path-length statistics, path-lengths are redefined here as the distance a particle travels along a fracture without changing direction (Figure 3-6b). Particle movements are now recorded only at those fracture intersections in which particles change direction, resulting in a longer measure of path-length. Revising the definition of 35 path-length has the following benefits: (1) path-length is affected by the flow field, and is therefore a function of the fluid velocity within a fracture (Parney and Smith, 1995); and (2) of secondary importance, computation time in the continuum is reduced because particles take a single long step in place of several shorter steps. Directional choice is also affected by the definition of path-length. The directional choice probabilities are determined by recording the number of steps in each direction that particles take within a discrete subdomain, without regard to the order in which each direction is chosen. Under the old definition of path-length a particle travelling down a long fracture is recorded as several directional choices in the same direction. When these directional choice statistics are applied in the S C M the particle is not required to take all of these steps sequentially, and the resulting particle motion may not be the same as in the discrete model. Under the new definition of path-length a particle travelling down a long fracture is recorded as a single directional choice. Particle motion in the S C M under the new definition of path-length should therefore be closer to particle motion in the discrete network, than under the old definition of path-length. 3.5.3. Statistical testing of path-length models In order to reproduce the geometry of the travel paths in the continuum, a statistical model capable of reproducing the distribution of path-lengths must be found. The simplest statistical model consists of a discrete histogram, determined by sorting path-lengths into a large number of bins (100 in this study). Histograms have a high degree of shape flexibility, but the computational burden is high because 100 variables are calculated instead of the two or three necessary for most continuous statistical models. Normal and exponential models are continuous, and depend upon the first and second moments (mean and variance), while 36 the three-parameter Gamma model includes information from the third moment (skew). Continuous distributions have a lower computational burden and are more effective at capturing the tails of the distributions. A l l of the statistical models used here are calculated separately for each of the four travel directions. Two approaches are used to assess the ability of a statistical model to reproduce the path-lengths in the subdomain: (1) continuous models are compared directly to histograms of the path-length data from which they are derived, and (2) mass movement within S C M models using the various models for path-length is compared to mass movement within large-scale discrete network models. The fit between the continuous statistical models and the histograms is examined here, while comparisons between the S C M models and the discrete network models are presented in Chapter 4. Three continuous models for path-length are examined: (1) Normal, (2) Gamma, and (3) Exponential. Testing is accomplished by comparing the cumulative distributions (CDF) to the histogram of path-lengths traveled within the discrete subdomain. In Figure 3-7, a sample is generated by using particles travelling through 10,000 realizations of the base-case Poisson system, i n a l O m b y l O w domain. The histogram illustrates the single-tail distribution of path-lengths, with frequency decreasing as a function of length. The double-tail nature of the Normal distribution makes it unsuitable as a model of path-length, which is indicated by the poor match between this model and the histogram (Figure 3-7). Both the exponential distribution and the Gamma distribution, which has the exponential distribution as a special case, produce excellent fits to the histogram of path-length. 37 3.5.4. Statistical testing of velocity models The ability of the various statistical distributions to model the velocities encountered by particles moving through the discrete subdomain is evaluated in a manner similar to the evaluation of the statistical models of path-length. In this section, continuous statistical models are compared directly to histograms of the velocities encountered within the discrete subdomain. An evaluation of the velocity models based on comparing mass transport within the varying S C M models to mass transport within a large scale network model is left to Chapter 4. Figure 3-8 is a plot of the frequency of occurrence of velocities encountered by particles travelling within 10,000 realizations of the base-case Poisson network system, plotted versus the logio of velocity. Velocities are shown for particles travelling in direction 2. Similar results are obtained in directions 3 and 4. Because the velocities range over four orders of magnitude all statistical models are based on the distribution of the logio of velocity (Table 3-4). In Figure 3-8, two continuous models, the Log-Gamma and Log-Normal are compared to the histogram of the velocities. When the Log-Gamma and Log-Normal models are applied to velocities within the SCM, these models are referred to simply as the Gamma and Normal models. The Gamma model gives a better fit to the histogram of the data than does the Normal model, as the Gamma model matches the asymmetry around the median value that is apparent in the histogram. 3.5.5. Conditioning of velocity and length parameters Sections 3.5.3 and 3.5.4 describe statistical models in which path-length and velocity are treated as independent random variables. Methods are now examined in which particle velocities in the continuum are conditioned on either the path-length chosen, or the velocities 38 during previous steps along the flow path. In the first part of this section, two approaches to conditioning are outlined: (1) velocity correlation between sequential steps using a second order auto-regressive model, and (2) using the path-length of each step within the continuum to determine the distribution from which the velocity is generated. In the later part of this section, two additional S C M models using a velocity conditioning that were abandoned are briefly outlined for completeness. It is likely that the velocity at which a particle travels out of an intersection between two fractures is often similar or correlated to the velocity at which a particle enters that intersection. Robertson (1990) quantified this similarity by calculating the lag(l) to lag(4) correlation coefficients, of particle velocities along flow paths within the discrete subdomain. The lag(&) correlation coefficient is given by: where /uv and crv are the mean and standard deviation of the velocity. These correlation coefficients are then applied in the continuum domain by using a standard auto-regressive (AR) correlation structure for a Normal random variable (Box and Jenkins, 1976), and converting by a probability transformation, to a gamma deviate (Rubenstein, 1981). For two-lag correlation (Rk2), i f Z is the deviation in velocity from the mean velocity (juv) i.e. RK(k) = E[(v, -Mv)(vt+k - / Q ] [3-5] Z = V-ju [3-6] 39 then the velocity deviate at time t (Z,) is calculated using a normal random deviate at: N(0, crfl), and conditioning this random deviate on the deviates of the two previous steps, and the lag one and lag two correlation coefficients: Z, = ^1)Z,_, + *(2)Z_2 + (1 - RK<\)4>(1) - RK(2)fi(2))at P"7] (RK(W-RK(2)) [ 3 . 8 ] (l-RK(V)f {RK(2)-RK(lY) \-RK(\f [3-9] a] =r>(\-u(\)RKQ-)7mRK(2)) [3"10] The gamma-distributed deviate x : F(/j,a,A), is found from the Normal distributed deviate by applying the transformation Z [3-H] where Fl is the two-parameter gamma (ju, o~) inverse Cumulative Distribution Function (CDF), and O is the standard normal CDF. Robertson (1990) found that a strong correlation existed between velocities of particles travelling in the same direction, but that this correlation was weaker when a particle changed direction at an intersection. Velocity-to-velocity correlation using a second-order A R correlation structure between path-lengths in the same direction is referred to as the Rk2 model in the present work. Further details of the Rk2 algorithm for correlating velocities between adjacent steps are outlined by Robertson (1990). 40 The distance a particle travels along a fracture may depend in part upon the fluid velocity in that fracture. At fracture intersections, particles are most likely to travel in the fracture containing the highest fluid flow, as under either complete mixing or stream-tube routing, directional choice is proportional to flow volume. This process causes particles to travel further in fractures in which the fluid velocity is higher than in fractures containing lower fluid velocities, so that a relationship exists between path-length and velocity (Parney and Smith, 1995). A scatter plot of particle velocity versus path length for particles travelling through 20 subdomains from the base case Poisson network is shown in Figure 3-9. Mean particle velocity is calculated for each path length by weighting the velocity of each segment by the length of that segment. Short-path lengths occur at all velocities, while long paths occur within only a limited range of velocities. In this example, the longest path length is on the order of 5 meters, and occurs in a fracture where the fluid velocity is approximately 4 x 10"6m/s. The maximum particle velocities are eight times higher than the average, but the path lengths associated with these velocities are less than half a meter. At low velocities, path lengths are short. As velocity increases toward 4 x 10"6 m/s the maximum path length increases, but shorter paths also occur. Similar behavior is seen in scatter plots for fracture systems of lower and higher fracture intensities (Parney and Smith, 1995). In order to quantify this velocity/path-length relationship, the mean and standard deviation of particle velocities are plotted as a function of path-length (Figure 3-10). The average velocity rises quickly with path-length between 0 and 1.0 m, and continues to increase, but at a lower rate, towards the longest paths traveled (Figure 3-10a). More than 99% of all the paths traveled are less than 4 m in length (Figure 3-7); beyond this length the mean and standard deviation of velocity are based on a limited number of data points and 41 scatter appears in the trends (Figure 3-10). As expected from the scatter plot, the standard deviation in the particle velocity decreases with longer path lengths (Figure 3-10b). In order to apply the velocity-length correlation in the continuum domain, the mean, standard deviation and skew of velocity are calculated for each bin of the length histogram. When a length bin is selected within the continuum, a corresponding distribution of velocities is defined by the statistical model chosen, and the velocity parameters calculated for that length bin. The velocity is then generated from this conditioned distribution. A third method of conditioning velocities and path-lengths is possible by reversing the process discussed in the previous paragraph; particle velocity is selected first, and then path length is determined as a function of this velocity. This algorithm was initially thought to correspond to the physical processes in discrete networks which determines the relationship between velocity and path-length (Parney and Smith, 1995). Defining path-length based on velocity however, did not work as well as the previous method in initial testing, and is mentioned here only for completeness. A fourth conditioning algorithm was also attempted. In this approach the mean and standard deviation of velocity were determined based on the previous travel velocity, and a new particle velocity was chosen at random from the CDF defined by these parameters. This method differs from the first correlation method where all velocities are chosen from the identical distributions, and then modified by the A R process. As with the third algorithm this approach was not effective and is mentioned only here for completeness. 3.5.6. Assessment Of Parameter Variability and Convergence The statistical parameters estimated using the discrete subdomain must accurately represent the full range of velocities and path-lengths that solutes encounter within a field-42 scale, fractured-rock network. Each sub-domain is composed of a limited number of fractures, which are only a sample of the entire fracture population described by a fracture system. Consequently, motion statistics generated by particles travelling within a discrete subdomain represent only a subset of the population of velocities and path-lengths that particles would travel at within a larger network system or field site. Two methods are available for calculating motion statistics that are representative of a fracture system: (1) the size of the discrete sub-domain can be increased until the fracture network is large enough to include the full range of velocities and path-lengths occurring within a fracture system, or (2) particle movement can be simulated in multiple, smaller network realizations until a sufficient range of velocities and path-lengths are encountered. These two approaches produce equivalent statistics under the ergodic theory (section 7.3), as long as the subdomains are large enough to capture the longest path-length that can be traveled, i.e. that particle travel is not censored by the limited domain size. As mentioned previously, the size of a discrete network model is limited by the absolute number of fractures that can be included. Multiple subdomains must therefore be used because a single subdomain cannot be made large enough that the motion of particles travelling through it will be fully representative of the range of motions possible within a fracture system. It is also computationally more efficient to solve multiple small networks rather than a single large network. If multiple network realizations are to be used to capture the full range of particle motions occurring within a fracture system, each network must contain a unique, but statistically equivalent fracture geometry. Because of the unique fracture geometry each network realization has a different effective hydraulic conductivity, and particles travelling 43 through one network will move at a different mean and standard deviation of velocities, and path-lengths, than in another network. In order to obtain the statistics for the continuum, motion statistics from each subdomain are averaged over a large number of network realizations. In the remainder of sections 3.5, the limits on the size and number of networks required to calculate representative motion statistics are examined. In order to determine the number of subdomains that are necessary to find motion statistics representative of the fracture system: (1) average motion statistics are calculated as a function of the number of networks generated; (2) the variation of parameters as a function of the size of the subdomain is then examined and (3) the variability of motion statistics as a function of the effective hydraulic conductivity of the network is explored. Variation Between Networks. In order to determine the number of the subdomains needed to represent the fracture system, the average and the coefficient of variation of the following motion statistics are plotted as a function of the cumulative number of realizations from which they are calculated: (1) the probability of moving in direction two; (2) the mean, standard deviation and skew of path-length in direction two; and (3) the mean, standard deviation and skew of velocity in direction two. Results for directions 3 and 4 are nearly identical to those from direction 2 and are therefore not included here. Results for direction 1 are also similar, but are somewhat more irregular due to the low probability of travelling directly against the applied gradient. Motion statistics are calculated for a total of 10,000 realizations, in 10m x 10m subdomains of the base-case Poisson network system (Table 3-1). For all of the motion statistics, as the number of realizations is increased, the average value should converge towards the expected value for a given fracture system and applied 44 hydraulic gradient. The rate at which the average of any motion parameter converges towards the expected value depends upon how widely that parameter varies from realization to realization. This rate is a function of the fracture system and the size of the subdomain used. Convergence is measured here by plotting the coefficient of variation for each parameter as a function of the number of realizations from which it is calculated. The coefficient of variation (COV), i.e. the standard deviation of a given motion statistic normalized by its average value, is used to allow direct comparison of the variability of the motion statistics. Figure 3-1 la shows the average value for the probability of choosing to move in direction 2 at an intersection, while Figure 3-1 lb shows the corresponding COV, plotted as a function of the number of realizations. Below 2000 realizations the average probability oscillates, and then slowly converges towards a constant value. After 5000 realizations the average probability remains constant at a value of 0.507. The COV of this average is initially close to 0.01 and then decreases, dropping below 0.001 at 10,000 realizations. The low COV indicates that the probability of travel in direction 2 does not vary significantly between realizations, so that the average of the probability converges rapidly. After 2000 realizations little fluctuation is seen, so that average probability can be used at this point. In Figure 3-12, the average mean value and the COV of the mean path-length in direction 2 (Figure 3-12a,b), its standard deviation (Figure 3-12c,d) and the skew (Figure 3-12e,f) are presented. The average of the mean path-length (Figure 3-12a) converges towards a constant value at a slower rate than the average of the probability of travelling in direction 2 (Figure 3-1 la), reaching a stable value at around 7000 realizations. The average values of the standard deviation and skew of path-length converge more rapidly than the 45 mean path-length, with only minor fluctuations occurring in the standard deviation and skew after 3000 realizations. The coefficients of variation do not converge until 8000 realizations. One striking feature of Figure 3-12 is the high COV of the mean path-length relative to the COV for the standard deviation in path-length. This behavior suggests that the magnitude of the range of path-lengths traveled by particles within each subdomain is similar from realization to realization, but that the mean path-length changes from one discrete subdomain to the next. Because the motion of particles is a function of path-length, directional choice and velocity, it is possible that changes in the mean path-length are compensated for or enhanced by changes in the distribution of velocities traveled. However, it may be that each realization has a characteristic mean path-length, a feature that could be important when modeling behavior at a field site. In Figure 3-13, the average value and the COV of the mean of the particle velocity (Figure 3-13a,b), its standard deviation (Figure 3-13c,d), and skew (Figure 3-13e,f) are plotted as a function of the number of realizations. Unlike the estimate of mean path-length, the estimate of mean velocity converges rapidly. The average of the standard deviation of velocity does not appear to converge until nearly 10,000 realizations, and the COV of the velocity standard deviation is on the order of 0.01. After 10,000 realizations, the average of the skew of velocity is not constant (Figure 3-13e), although the magnitude of this parameter does not vary by a large amount after 5000 realizations. The C O V of the skew of velocity is initially larger than the COV for the standard deviation, but does not change significantly after 2500 realizations. The key difference between the velocity parameters and those describing the geometry of the travel-path is that the largest COV for the path-length occurs in the mean, 46 with the standard deviation and skew of path-length being relatively constant between realizations, while the mean of the velocity has a very low COV, relative to the standard deviation and skew of the velocity. From one network to the next the distribution of path-lengths has a similar shape, but the distribution is shifted up or down according to the change in the mean. Particle velocities are distributed around a similar mean value, but the width and skew of the velocity distribution vary between realizations. The average and COV of the first (Rkl) and second order (Rk2) correlation coefficients are plotted in Figure 3-14. For both correlation coefficients, the average value appears to converge between 2500 and 5000 networks (Figure 3-14a). The COV for both R k l and Rk2 decreases towards approximately 0.01 at 10,000 networks, with the magnitude of the COV slightly higher for the second order correlation coefficient (Figure 3-14b), but then continues to decrease slightly after this point. Both the correlation coefficients and the standard deviation are based upon second moment statistics, so that the variability in the correlation coefficients is similar to the variability in the standard deviation of velocity (Figure 3-13c,d). Of all the parameters examined here, the mean path-length requires the largest number of realizations (approximately 8000) before the estimate converges. Even though the estimate of the standard deviation of velocity and the estimate of the correlation coefficients continue to vary slightly after 8000 realizations, both parameters appear to have converged by 10,000 realizations. The convergence of the motion statistics when averaged over multiple realizations can only be used as a secondary indicator when determining i f a sufficient number of networks have been run in the discrete subdomain. Because mass transport is a complicated 47 function of all the motion statistics, a better test of sufficiency is the comparison of the spatial moments for mass transport through an S C M model, to the spatial moments for mass transport through discrete networks. It may turn out that motion statistics averaged over fewer networks than are needed for parameter stability are sufficient for use in the S C M , i f the match between the moment values for the S C M models and the discrete networks is not improved by averaging over larger numbers of networks. Variation of Parameters with Scale The variation of motion statistics as a function of domain size is examined here because it is important that each subdomain be large enough that motion statistics are not censored to any significant degree by boundary effects. It must also be determined whether or not motion statistics change with scale, as one of the key assumptions within the S C M is that the movements of individual particles at fracture intersections (i.e. particle velocities and path-lengths) do not change significantly with changes in the size of the model domain. Scale invariance of the motion statistics is necessary as it allows application of these statistics to particle movements in continuum domains larger than the network realizations from which these statistics are originally calculated. For example, in the base-case Poisson system, fracture length is generated using an exponential distribution with a mean of 1.6 meters (Table 3-1). Although application of this exponential distribution can result in extremely long fractures, the longest path traveled by particles in 10m by 10m subdomains was only on the order of 5 m (Figure 3-9). Subdomains must therefore be large enough to allow particles to enter and exit long fractures without encountering boundaries, so that the path-length traveled by particles is a function of the network geometry and the distribution of flow, but not the size of the domain. In order to 48 examine the issues of maximum and minimum domain size, motion statistics are generated for 6 different subdomain sizes, using 10,000 network realizations. In Table 3-5, motion statistics in direction 2, from network realizations with length dimensions in the direction of flow varying from 3 m to 30 m are listed. These dimensions represent domains significantly smaller and significantly larger than the 10 m by 10 m domain used in the previous section. Each subdomain is created from the base-case fracture system (Table 3-1), in which the mean fracture length is 1.6 m, so that 99% of all fractures created are less than or equal to 4.3 m. Changes in motion statistics due to changes in subdomain size can be subdivided into two broad classes: (1) flow-field threshold changes, where a minimum size of subdomain is necessary to avoid having flow dominated by fractures which intersect constant head boundaries at each end, and (2) scale changes, where parameters change systematically with changes in domain size, independent of boundary effects. Velocity parameters fall into the first category. In the 3 m by 3 m subdomain, the correlation coefficients are significantly higher than in any of the other domains. Also, the negative logio velocity is slightly lower (i.e. the velocity is slightly higher) in this small domain. Boundary effects dominate a domain of this size, so that direct hydraulic connections between the upstream and downstream boundaries are frequent enough to cause particle motion to deviate significantly from the particle motion that occurs in a larger domain. Above a minimum domain size, the velocity parameters are similar, suggesting that once a threshold size is reached, velocity statistics are not affected by changes in scale. However, under the second broad class of size effects, path-lengths increase with scale, and the probability of choosing direction two decreases with scale (Table 3-5). The mean and standard deviation of path-length both increase approximately 2 % from the 10 m 49 by 10 m domain to the 30 m by 15 m domain, with similar increases occurring in directions three and four (not given). The probability of choosing direction two decreases by a marginal 0.7 %. Although motion statistics appear to change as a function of subdomain size, the S C M method can still be used to model the base-case Poisson system using 10 mx 10 m subdomains as: (1) a large change in subdomain size results in only a minor change in-statistical parameters, for example the mean path-length increases approximately 2 % for a 3 fold increase in domain length from 10 to 30 m (Table 3-5); and (2) changes in one parameter are compensated for by changes in other parameters. For larger subdomains, the path-length in direction two increases, while the probability of choosing to travel in direction two decreases. Although particles choose direction 2 less often in larger networks, each step in that direction is longer, so that one change counter-acts the effect of the other, and the distance each particle travels in direction 2 is approximately preserved. Therefore, above a minimum domain size, particle motion is not significantly altered by changes in subdomain size. Understanding the relationship between changes in the directional choice probability, and changes in the mean path-length, is complicated by the increase in the standard deviation of path-lengths as the size of the domain increases. The first possibility is that a larger realization represents a larger sample of the fracture system, so that wider ranges of path-lengths are available for particle travel in a larger realization. Second, as discussed in section 3.4, long fractures containing high flows are linked by shorter fractures containing lower flows. Therefore, a larger realization increases the likelihood that transport will be dominated by more than one channel; which in turn increases the likelihood that particles will travel through shorter fractures, in order to make the connection between the high-flow 50 channels; all of which contributes to a wider range of travel-paths. Whatever the physical process behind the increase in the standard deviation with the size of the realization, the small magnitude of the change in parameters with change of domain size does not suggest that the method is compromised by this change. In light of the above discussion it must again be noted that the viability of the S C M approach can only be determined by the ability of the method to reproduce mass transport at a field location, regardless of how motions statistics vary as a function of network size, or the processes behind this variation. However, the results of Table 3-5suggest that for base-case Poisson system: (1) motion statistics calculated in subdomains 7 m by 7 m and above are a good approximation to motion statistics in larger (e.g. 30 m by 15 m) domains, and (2) 10 m by 10 m subdomains are large enough to avoid boundary effects when modeling the base-case Poisson system. How closely the S C M models reproduce mass transport through discrete network models is examined in Chapter 4. Variation of Parameters as A Function of Network Conductivity In the previous sections, the variation of motion statistics between random, although statistically equivalent networks was examined. In field applications, statistically equivalent networks are sometimes accepted or rejected based on flow or transport information at the site (e.g. La Pointe et al, 1995a). At a field location, a range of conductivity values can be measured using a series of pump tests. This range of conductivities can then be used to condition a series of network realizations. One approach to conditioning the network realizations with the field measurements is to reassign the distribution of fracture apertures within the network so that the effective hydraulic conductivity of the network falls within the range of conductivity values measured within the field (e.g. Cacas et al, 1990). A second 51 method of conditioning discrete models is to reject all realizations with effective conductivity's outside of the range of conductivity values measured in the field (La Pointe et al, 1995a). To determine i f conditioning networks using this second approach will change either the average or COV of the motion statistics, the previous results are redisplayed after application of an effective conductivity filter. Fifty percent of the networks are removed from the previous plots by calculating the effective hydraulic conductivity for the 10,000 subdomains generated previously, and rejecting those networks with an effective hydraulic conductivity in the lowest and highest quartiles. This approach only approximates the conditioning of realizations with actual conductivity measurements, but the 50 percent value is consistent with the rejection rate found by La Pointe et al (1995a) when conductivity measurements were available. The results of filtering networks based on hydraulic conductivity are given in Figure 3-15 to Figure 3-18. The behavior can be summarized as follows: (1) average values of all parameters converge toward values similar to the average values calculated without filtering, but at a faster rate, and (2) the COV of all parameters decrease at approximately twice the rate as the C O V before filtering. The average values of the motion statistics do not change because the median value of the effective conductivities is the same for both the filtered and unfiltered networks. Conditioning the networks based on an expected value for the effective hydraulic conductivity other than the average value for the unfiltered networks would likely change the estimated values of the parameters. Because the average values of the motion statistics are unaffected by the current filtering process, a reduction in the C O V is due entirely to a reduction in the standard deviation of each motion parameter. 52 Variability in the motion statistics is linked to variability in the network conductivity between realizations. As a result, i f flow information can be used to constrain the networks in the S C M , motion statistics averaged over subdomains will converge more rapidly toward the mean values for network systems, and therefore fewer subdomains wil l be required. 3.6. Continuum Domain Flow Solution And Particle Tracking 3.6.1. Method The flow solution for the continuum domain is calculated by assuming a uniform hydraulic gradient applies in all areas of the domain. The S C M method is also, however, designed to allow more complicated flow system, with spatial variations in the magnitude and orientation of the hydraulic gradient. With a uniform hydraulic gradient, a single set of motion statistics applies to particle movements at all locations within the domain. In previous work on the S C M (Schwartz and Smith 1988) the gradient applied to the discrete subdomain is rotated through seven orientations, so that seven complete sets of motion statistics are generated for use in the continuum. In order to apply these separate motion statistics, a flow solution must first be calculated for the continuum domain, and then as mass moves through the domain, the appropriate motions statistics must be chosen based on the local magnitude and orientation of the hydraulic gradient. This flow solution for the nonuniform gradient problem must be determined outside of the S C M program. Mass transport in the continuum domain is initiated by injecting particles at random along a predetermined zone, on the upstream border, which represents the source area for the plume. Particle movement is then carried out through a random-walk procedure, using the motion statistics described above as input. The random-walk consists of first selecting one of the four set-directions, then a fracture length, and then a velocity. The particle is moved to 53 the end of this new pseudo-fracture segment and this process is repeated until a pre-determined period of time has elapsed. Particle locations are used to generate spatial plots of mass distribution (plume plots), or to calculate spatial moments as a function of time. By modeling mass transport in the continuum in this manner, the S C M approach is capable of capturing the anisotropic character of the dispersion. In modeling studies Schwartz et al. (1983) and Smith and Schwartz (1984) noted the relationship between the pattern of spreading and the fracture geometry: (1) the centre of mass of a moving tracer often diverges from the flow direction defined by the mean hydraulic gradient; (2) the major and minor principal axes of the plume may be rotated from the direction of the hydraulic gradient; and (3) the direction of maximum spreading need not be greatest in the direction most closely aligned with the mean hydraulic gradient. The orientation of each particle movement in the continuum reflects the orientation of fractures within the discrete network, and the discrete nature of the hydraulic connections, so that the anisotropic character of the dispersion is captured. In Chapter 4, plume plots and spatial moments are used to compare mass transport in the S C M model with mass transport in discrete networks. 3.6.2. Nomenclature for SCM models in two dimensions Velocity and path-length can be generated using a number of algorithms within S C M models. A system of nomenclature is introduced in order to keep track of the large number of permutations that result when these algorithms are combined. S C M modeling approaches are identified by the velocity and path-length models used: If a histogram is used, the name combines the capital " H " with the parameter chosen, e.g. H(v) and/or H(l). A normal distribution is designated by an " N " , and a gamma distribution by a " G " . For example an 54 H(v)G(l) model is one in which the velocity is sampled from a histogram and the path-length is sampled from a Gamma distribution. Models in which velocities are correlated to path-length are indicated by including an "1" within the velocity distribution brackets. A N(v,l)H(l) is then a model in which the velocity is based on a normal distribution whose parameters are correlated to path-length, and the path-length is generated by sampling a histogram. In section 5.7.6 this nomenclature is expanded to include additional modeling approaches in the three-dimensional models. 3.7. Conclusion In this chapter the methods used in the generation of S C M models in two-dimensions were outlined. The processes by which motion statistics are calculated in the discrete subdomain were given, and the stability of these motion statistics with respect to the size and number of networks used in the subdomain were analyzed. For the base-case Poisson fracture system with a mean fracture length of 1.6 m subdomains of 10 m by 10 m were found to be sufficient to generate meaningful motion statistics. The parameter most sensitive to domain size was the mean path-length traveled by particles, and this parameter continued to increase slightly as a function of domain size for domains larger than 10 m by 10 m. Velocity parameters were less sensitive to domain size, beyond a minimum size of 7 m by 7 m. Motion statistics obtained by averaging over 10 m by 10 m subdomains tended to converge by about 5000 realizations for the base-case Poisson fracture system. Statistical models to represent the velocity and path-length were examined, and it appears that both the path-length and the logio of the velocity can be effectively modeled using three-parameter gamma distributions. Correlations were observed between the 55 distance along a fracture a particle traveled and velocity of the fluid within that fracture. In the next chapter mass transport in S C M models is compared to mass transport through discrete network realizations. 56 Table 3-1: Input Parameters for the Poisson Fracture System. Parameter Horizontal Fracture Set Vertical Fracture Set Number of Particles 5000 Number of Realizations 10000 Applied Gradient -0.001 Orientation of Gradient 0° Injection Width 2 m. Scan-line density 7.2 lm 7.2 lm Mean Fracture Length 1.6 1.6 Standard Deviation of Length 1.0 1.0 Mean Fracture Aperture -4.5 -4.5 Standard Deviation of Aperture 0.3 0.3 Mean Orientation 0.0° 70.0° Standard Deviation of Orientation 3.0° 3.0° Table 3-2: Input Parameters for the Levy-Lee Fracture System. Parameter Mean Horizontal Mean Vertical Fracture Set Fracture set Number of particles 5000 Monte Carlo Realizations 5000 (high failure) Applied Gradient -0.01 Injection width 1.5 m Scan-line density 10 lm 10 lm Mean fracture length 4.0 m 4.0 m Fractal Dimension 1.5 1.5 Log Mean Aperture -4.5 (m) -4.5 (m) Aperture Standard 0.0 0.0 Deviation Orientation 0.0° 90.0° Orientation Standard 10.0° 10.0° Deviation 57 Table 3-3: Motion Statistics from 0 degree gradient, Poisson Network System, 10,000 realizations. Direction: i 2 3 4 Injection Orientation: 3.14 0.00 4.37 1.21 0.26 a 0.048 0.052 0.052 0.052 0.48 (Degrees) M- 179.7° -0.043° 250.4° 70.0° 15.3° a 2.75° 2.99° 2.99° 2.99° 2.75° Velocity: (logio m/s) -6.4 -5.7 -6.1 -5.9 -5.7 a 0.45 0.34 0.43 0.38 0.26 K 0.89 1.07 0.83 0.99 2.21 Correlation: R k l 0.16 0.70 0.26 0.45 Rk2 0.008 0.49 0.008 0.25 Path-length (m): 0.09 0.58 0.20 0.35 0.66 a 0.09 0.60 0.20 0.38 0.46 K 1.72 1.86 1.79 2.02 0.79 Directional Choice: 0.01 0.51 0.12 0.36 Aperture: (logio m) -4.44 -4.15 -4.25 -4.16 -4.11 O" 0.20 0.23 0.23 0.23 0.14 Table 3-4: The five modeling methods of the S C M Model Name Velocity Distribution Path-Length Distribution Notes: 1 H(v)H(l) Histogram Histogram 2 N(v)G(l) Log-Normal Gamma 3 G(v)G(l) 3-Parameter Log-Gamma Gamma 4 G(v,Rk2)G(l) 3-Parameter Log-Gamma Gamma Second order auto-regressive correlation of velocities 5 N(v,l)H(l) Log-Normal Histogram Velocity selected from a distribution whose mean and standard deviation are correlated to the path-length. 58 Table 3-5: Selected motion statistics in direction 2 versus domain size. Base-case Poisson fracture system. 99 % of all fractures are less than 4.32 m. Domain Length (m). /j Length: cr. Probability Velocity (logio m/s): ju Rkl Rk2 3x3 0.556 0.484 0.523 5.62 0.84 0.73 7x7 0.578 0.579 0.512 5.66 0.69 0.48 10x10 0.585 0.601 0.507 5.66 0.68 0.47 12x12 0.588 0.617 0.506 5.66 0.68 0.48 20x10 0.594 0.635 0.500 5.64 0.67 0.47 30x15 0.597 0.647 0.500 5.65 0.68 0.48 59 Process Result Fracture Data Acquisition Discrete Subdomain Discrete Network Generation Flow Solution Particle Tracking Continuum Domain Particle Tracking In Continuum Fracture System Definition Plumes Spatial Moments Figure 3-1: Flow of Statistical Continuum Method 60 Figure 3-2: A large domain Poisson network. A single network realization based on the base case fracture system given in Table 3-1. 61 Figure 3-3: Levy-Lee network. One realization of a discrete subdomain based on a fractal dimension D=1.5. 62 Figure 3-4: Flow distribution for one realization of the base-case Poisson fracture system. Applied hydraulic gradient is horizontal. Line thickness is determined by the order of magnitude of flow. Flows are 10"4, 10"5 and 10"6 m3/day respectively. Fractures in which flow is less than 10"6 mVday are not plotted. 63 Figure 3-5: Fractures containing flows greater than 10"5 m3/day from Figure 3-4. Although these fractures carry over 80% of the flow, without the inclusion of the fractures containing flows less than 10"5 m3/day the network falls below the percolation threshold. 64 New definition of path-length Figure 3-6: Revised definition of path-length. Previously path-length was defined as the distance between fracture intersections. Under the new definition path-length is defined as the distance a particle travels along a fracture between the fracture intersection in which it enters and the intersection throughout which it exits. 65 0.15 T T T _ 3 a* a u -#> P i 0.1 0.05 h 0.0 Exponential Three parameter Gamma Hv. ^ Histogram Normal 0.0 1.0 2.0 3.0 4.0 5.0 Path Length (m) Figure 3-7 Path-length statistics for the base-case Poisson fracture system. Histogram, 3-parameter Gamma distribution, Normal distribution and Exponential distribution based on 10,000 realizations of a lOmx 10m discrete subdomain. Statistics are from the direction parallel to the applied hydraulic gradient. 66 0.1 0.05 Gamma Normal 0.0 Histogram -8.0 -7.5 -7.0 -6.5 -6.0 Log 1 0 of Velocity -5.5 -5.0 -4.5 Figure 3-8 Velocities of particles for the base-case Poisson fracture system. Histogram, Parameter Gamma distribution and Normal distribution based on 10,000 realizations of 10m x 10m discrete subdomain. Statistics are from the direction parallel to the applied hydraulic gradient. 67 Figure 3-9: Scatter plot of velocity versus path-length pairs. Each data point represents a single particle movement within a 10m x 10m discrete subdomain for the base-case Poisson network system. Movements are sampled from 20 realizations. 68 ^ (b) Standard deviation a .IT °-7 Path-length (m) Figure 3-10: Mean and standard deviation of path-length versus velocity. Data points are calculated over 10,000 realizations of a 10m x 10m discrete subdomain, based on the base-case Poisson network system. 6 9 (a) 0.51 O 03 QJ _© .2 > o = o U .505 0.50 (b) 0.1 0.01 0.001 — 0.0001 2500 5000 7500 10,000 Number of Networks 2500 5000 7500 10,000 Number of Networks Figure 3-11 Mean and coefficient of variation for directional choice for the base-Poisson network system. 70 Mean Coefficient of Variation (b) 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 (d) 0.001 \ -0.0001 _L CZ5 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 CD 0.57 0.1 0.01 0.001 0.0001 _L 0 2500 5000 7500 10,000 Number of Networks 0 2500 5000 7500 10,000 Number of Networks Figure 3-12: Mean and coefficient of variation for the mean, standard deviation and skew of path-length for the base-case Poisson network system. 71 (a) ca o 2 '> a Q -a c SI C/3 Mean > O a 5.65 \ -(b) Coefficient of Variation o . i o . o i h o . o o i o . o o o i 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 (c) 0.36 0.35 0.34 0.33 (d) o . i 0.01 0.001 0.0001 J L 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 (e) l . i (f) r jg 1-05 I-C/3 1.0 i r _i_ _i_ 0.01 0.001 h 0.0001 _L _L 0 2500 5000 7500 10,000 Number of Networks 0 2500 5000 7500 10,000 Number of Networks Figure 3-13: Mean and coefficient of variation for the mean, standard deviation and skew the negative Logio velocity in the direction parallel to gradient for the base-case Poisson network system. 72 Mean Coefficient of Variation (a) 0.7 0.69 0.68 h 0.67 0.66 | -0.65 ( c ) 0.50 0.49 h 0.48 0.47 h 0.46 0.45 H b 0 2500 5000 7500 10,000 _L _J_ > d 0 2500 5000 7500 10,000 (b) 0.1 0.01 0.001 h 0.0001 (d) 0.1 0.01 0.001 h-0.0001 JL 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 Number of Networks Number of Networks Figure 3-14 Mean and coefficient of variation for the first and second order correlation coefficients for the base-case Poisson network system. 73 (a) 0.510 e 0> 0.505 fl #o .2 > o fi . w ' 3 S o U 0.500 (b) 0.1 0.01 1 1 1 0 2500 5000 7500 10,000 Number of Networks 0.001 0.0001 2500 5000 7500 10,000 Number of Networks Figure 3-15 Mean and coefficient of variation for directional choice after removal o networks with effective hydraulic conductivities in the highest and lowest quartiles. 74 Mean Coefficient of Variation S (a) (b) 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 2500 5000 7500 10,000 (d) o. i 0.01 0.001 0.0001 0 2500 5000 7500 10,000 (0 0 2500 5000 7500 10,000 Number of Networks 0 2500 5000 7500 10,000 Number of Networks Figure 3-16: Mean and coefficient of variation for the mean, standard deviation and skew of path-length, after removal of networks with conductivities in the highest and lowest quartiles. 75 (a) a, ^ 5.7 o 5.65 I-W> fl « 5.6 Mean 1 T J L X (b) Coefficient of Variation o.i o.oi V-o.ooi -0.0001 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 0.001 0.0001 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 (0 0.1 o.oi h o.ooi \-o.oooi 0 2500 5000 7500 10,000 Number of Networks 0 2500 5000 7500 10,000 Number of Networks Figure 3-17: Mean and coefficient of variation for the mean, standard deviation and skew of the negative Logio velocityin the direction parallel to gradient after removal of networks with effective hydraulic conductivities in the highest and lowest quartiles for the base-case Poisson network system. 76 Mean Coefficient of Variation (a) 0.7 0.69 0.68 0.67 0.66 \ -0.65 JL 0 2500 5000 7500 10,000 (b) 0 2500 5000 7500 10,000 (d) 0 2500 5000 7500 10,000 0 2500 5000 7500 10,000 Number of Networks Number of Networks Figure 3-18 Mean and coefficient of variation for the first and second order correlation coefficients after removal of networks with effective hydraulic conductivities in the highest and lowest quartiles. 77 4. SCM Model Results in Two Dimensions 4.1. Introduction In this chapter, the S C M method is evaluated in two dimensions using examples based on two fracture systems: (1) Poisson systems, in which network realizations are composed of fractures characterized by a single length scale, and (2) Levy-Lee systems, in which network realizations are fractal in nature, and are composed of fractures from a range of length scales. In each case motion statistics are calculated within a series of small (10 m by 10 m) discrete subdomains, and then the continuum simulations are generated using H(v)H(l), N(v)G(l), G(v)G(l), G(v,Rk2)G(l) and N(v,l)H(l) models (see section 3.6.2 for nomenclature). The S C M method is evaluated by comparing how closely mass transport within the S C M models matches mass transport in a large-domain network model, consisting of multiple network realizations. The network realizations and S C M models used for evaluation are significantly larger than the subdomains in which the motion statistics are generated. Evaluations are based on visual inspection of plumes, and comparisons of the evolution of the spatial moments through time. Mass transport within the S C M approximates the average behavior of mass travelling through a fracture system, and not the behavior that can be expected in any one realization. In order to evaluate the effectiveness of this approximation, spatial moments for S C M models are compared to spatial moments averaged over multiple realizations of the large-domain network models. The spatial moments for each of the network realizations are calculated individually, and then averaged, so that both the average rate of mass-transport through a fracture system and the variation of transport rates between realizations are determined. The variation between networks is presented as the standard deviation of the moments, and acts as a reference by which 78 to gauge how closely moments for the S C M models match those for the discrete model. The slopes of the moment versus time curves are also calculated, which allows differences between the discrete and continuum simulations to be quantified. 4.2. Poisson Networks 4.2.1. Base-Case In the previous work by Schwartz and Smith (1988) and Robertson (1990), the fracture systems examined were composed of orthogonal fracture sets with constant orientation. Here, a more general model is considered by investigating a fracture system with non-orthogonal fracture sets, in which the fracture orientations vary around a mean value (Table 3-1). For the first test, dense well-connected networks (Figure 3-2) are generated using the base-case Poisson fracture system (Table 3-1). In this section mass transport under a hydraulic gradient applied parallel to the x axis is examined. In the following section (4.2.2) this modeling is repeated after rotating the gradient by 45°. In section 4.2.3, mass-transport is modeled through a fracture system in which the fracture sets are orthogonal, but which is otherwise identical to the base case Poisson fracture system. Motion Statistics Motion statistics for the base-case fracture system were given in Table 3-3. These statistics are based on particle movements within 10,000 discrete subdomains. Each subdomain consists o f a l O m x l O m network realization, through which 5000 particles are tracked. Particles are injected in a zone between 2 and 4 meters from the lower corner of the up-stream boundary. Motion statistics are recorded separately for each travel direction. 79 Directions 1 and 2 are defined as horizontal left and right respectively, and direction 3 and 4 are down and up in directions sub-parallel to the v axis. The fifth column is based on the initial particle movement after injection. Injection statistics are calculated independently to determine i f early particle motions are influenced by the injection process. The probability of choosing direction 2 is 51%, so that approximately half of the directional changes result in particles travelling in direction 2. Direction 4 is chosen three times as often as direction 3 with the remaining 1 percent of particle movements at fracture intersections in direction 1. The applied hydraulic gradient directly opposes direction 1, but because of the geometry of the networks, instances occur locally where mass travels opposite to the predominant flow direction. The fracture system is composed of two fracture sets which are identical other than in the mean orientation, so that the probability of choosing a given direction is dictated by how closely that direction is aligned with the hydraulic gradient. Path-length is similarly affected by the orientation of the hydraulic gradient, so that the longest average path-lengths occur when particles travel parallel to the hydraulic gradient (direction 2). The length of the path a particle travels immediately after injection is on average longer than usually occurs in direction two (column 5). Longer paths occur because particles are injected into fractures in numbers proportional to flow volumes, and higher flow volumes are associated with longer flow paths (Parney and Smith, 1995). Comparisons of S C M results with and without the use of separate injection statistics are discussed further below. Orientation statistics describing the particle motion (Table 3-3) deviate little from orientation statistics describing the fracture sets (Table 3-1). It might be expected that particles choose those fractures within a set which are more closely aligned with the gradient 80 direction. However, most intersections contain only one fracture from each conjugate set, so that particles do not usually have the potential to choose between fractures within the same set. As a result, the average orientation at which a particle travels within a fracture set is not influenced by the flow solution. On average, particles travel at the highest velocities in direction 2, and at the lowest velocities in direction one. The range of velocities encountered is larger in the directions in which the average velocity is lower, that is the standard deviation in particle velocity is inversely related to the average velocity. The standard deviation of particle velocities immediately following injection is lower than the standard deviation of velocities for all particles travelling in direction 2. By injecting more particles into fractures containing higher flows, the injection process reduces the standard deviation in velocities by not sampling the full range of flow velocities equally. Velocity correlation coefficients are higher in directions with a higher average velocity; so that coefficients in directions 2 and 4 are the largest. Fracture apertures are recorded along with particle velocity, although this statistic is not used in the continuum component of the SCM. The mean aperture of all fractures contained within the fracture system is 10"4 5 m or 32 microns. However, the average aperture of those fractures through which particles travel (while moving in direction 2) is significantly larger, 79 microns. Furthermore the standard deviation in aperture size through which particles travel is smaller than the standard deviation for the fracture system. This selective sampling of the larger-aperture fractures offers insight into the processes by which mass channeling occurs, and is discussed further in Parney and Smith (1995). 81 Plumes Once the motion statistics have been calculated using the discrete subdomains, large-domain discrete and continuum models are compared. Large-domain network realizations from the base-case fracture system are generated in a 40 m x 20 m domain. In Figure 4-1, the distribution of mass at 200 days is plotted for 3 network realizations. Particles are injected between 2 and 4 meters on the left boundary, and travel under a hydraulic gradient of 0.001, applied horizontally left to right. Values for the first and second, x and y spatial moments are given for each of the three plumes (Figure 4-1). Mass transport varies significantly from network to network, resulting in visible differences in the plume distribution, and in the values of the spatial moments. The discrete plumes represent the identical modeling process applied to 3 unique, but statistically equivalent, network realizations. Each discrete plume is irregular in shape, although the area covered by the particles can be bounded approximately by an ellipse. Between realizations, these ellipses would vary in size and orientation, specifically the orientation of the major axes and the dispersion of mass about these major axes would change. Clearly, however, the plumes in Figure 4-1 are not elliptical. It is therefore difficult to define the location of an ellipse for the discrete plumes, and it is especially difficult to determine a valid and robust algorithm for calculating the orientation of the principal axes. Although a graphical approach could be applied to the three networks in Figure 4-1, application of this approach to particle clouds in each of the 10,000 network realizations is not feasible. This observation is significant to the moment calculation, and is revisited in the next section. 82 S C M models are also generated within 40 m by 20 m domains for comparison to the large-domain network realizations. Each plume in Figure 4-2 shows the mass distribution at 200 days as predicted by the S C M method. Plumes are calculated using the base-case motion statistics (Table 3-3), but 5 different velocity models are adopted within the S C M to represent particle motion (Table 3-4). The plumes resulting from application of the S C M models (Figure 4-2) are more evenly distributed than the plumes based on the discrete network simulation (Figure 4-1). Each S C M model generates an approximately elliptical plume, with mass concentrated in the centre, and particle intensity thinning towards the edges. Differences in mass transport in the 5 S C M models result in different average plume travel distances, and different magnitudes of the dispersion. These differences are evident in both the plume plots and spatial moments (Figure 4-3 and Figure 4-4). The slowest, least-dispersed plume is generated by the H(v)H(l) model, and the plume with the greatest dispersion is generated by the G(v,Rk2)G(l) model (see Table 3-4 for model definitions). For the N(v)G(l) model the plumes are less dispersed than when the G(v,Rk2)G(l) model is applied. Mass remains behind the main plume in a long tail stretching back to the injection band when the G(v)G(l), G(v,Rk2)G(l) and H(v)H(l) models are applied, but this tail is not present when the N(v,l)H(l) and the N(v)G(l) models are used. Because the motion statistics are based on 10,000 realizations (section 3.5), S C M plumes reflect the influence of the entire fracture system on mass transport, while each discrete plume reflects only the influence of only the portion of a single realization that is traversed by the mass. It is therefore necessary to use the results of multiple network realizations to evaluate the S C M models, but this approach does not work for a visual 83 comparison of the plume geometry. Plumes can be generated by including particles from several network realizations (e.g. Robertson 1990), but this approach is not taken because the resulting plumes contain artificial dispersion. This dispersion arises because the major axes of the plume changes between network models. When plumes with differing major axes are summed together, the width of the resulting plume is greater than the width of any of the component plumes. As a result, dispersion within plumes generated by this summation is not representative of average behavior within the fracture system. Spatial moments are used instead of plumes in order to compare mass transport within S C M models to mass transport within multiple network realizations. Moments Mass transport through a large-domain (40 m by 20 m) discrete network model is quantified by calculating the first and second spatial moments in the x and y directions. The first moments are the average spatial location of the plumes in each network realization and the second moments are the spreading around these mean values. These moments are calculated individually for particles travelling through each discrete network, and then averaged over 5,000 network realizations for the base-case Poisson fracture system (Table 3-1). Both the mean values and the standard deviation of the moments are plotted as a function of time (Figure 4-3 and Figure 4-4). It must be noted that in several of the examples presented, the forward motion of the plume is not necessarily aligned parallel to the x axis. As a result values for the lateral and transverse dispersivity (OCL and OCT) cannot be directly estimated from the slope of the second moment vs. time curves. In porous media modeling it is common practice to align the measurement axis parallel to the primary transport direction, allowing direct calculation of 84 a L a n d df. However, in the discrete models examined here the orientation of the plumes varies significantly from one network realization to the next, so that the coordinate system would have to be rotated for each of the 5000 large-domain realizations individually. The coordinate system could be rotated by defining an elliptical envelope around each plume and rotating the axis until they were parallel to the primary axis of this ellipse. The problem with rotating the axis in this manner is that the spatial distribution of mass travelling through discrete networks is often only vaguely elliptical. Substantial efforts were made to determine a robust method of numerically defining an elliptical envelope around each particle cloud before it was concluded that not only are the plumes not elliptical, but the primary axis of the each plume can rotate through time. The motivation for calculating the spatial moments is to allow a quantifiable comparison between the S C M models and the discrete modeling approach. As long as the method of calculating the moments is consistent between all of the models, this end will be achieved. For this reason the axis orientation is kept constant for all moment calculations. It is also important for consistency to avoid introducing dispersion when averaging the moments for the network realizations. For this reason the second moments for the network realizations are calculated about the individual mean values for each plume, and then averaged. Under this approach the second moment values are consistent with the dispersion in each individual plume. Moment values are also calculated for the five S C M models. Unlike the network realizations, however, multiple realizations are not used for the S C M models as there is no functional difference between one continuum simulation and the next. In this section, moments for the S C M models are compared to the average of the moments for the network 85 realizations. An exact match between the moments for the network realizations and the S C M models is not expected because of the approximations involved in the S C M model. The standard deviation of the moments for the network realizations is included to quantify the range of moment values that occur between networks. Because the fracture network at a field site is equivalent to a single realization from a fracture system, the variability of the moments between realizations can also be used to gauge how closely a moment obtained by averaging over multiple network realizations might relate to mass transport at a field site. For example, in a sparse fracture system, the standard deviation of the moments between network realizations can be large. In this case, moments for an S C M model may easily fall within one standard deviation of the average moments for a discrete model, and yet not be sufficiently accurate to achieve the goals of a given modeling study. Although any increased variability in the moments from the network realizations results in increased latitude for the acceptance of an S C M model, these changes also indicate the accuracy to which predictions can be made using the averages for the network realizations. Therefore, by using the one standard deviation criteria, the magnitude of the differences between the moments for the S C M model, and the average moments for the network realizations, are given perspective. In the course of developing these methods, it was observed that an additional step is required to allow a direct comparison between the moments for the network realizations and S C M models in two dimensions. In the network realizations, particles are injected between 4 m and 6 m on the upstream boundary. It was found empirically that by reducing the source zone in the S C M models by 30%, the second y moment curve for the S C M model can be shifted downwards to more closely match the early values for the network realizations. This shift is necessary in part because on injection, particles within the continuum can move 86 immediately in any of the 4 travel directions, while particles injected into a network realization tend to travel only in the x direction (direction 2). As a result of this difference in particle behavior on injection, spreading in the y direction at t = 0 is larger in the S C M models than the network models when equal injection bands are used. Changing the size of the source zone in the S C M model does not, however, change the rate at which mass disperses. Because the bulk-shift can be used to improve the match between moment values at early times, it is important to focus on the differences in the slope of the moment curves when comparing results from the two modeling methods. The first and second, x and y moments for both the network realizations and S C M models are plotted versus time (Figure 4-3 and Figure 4-4). The solid lines are the averaged values of the moments for the network realizations, and the dashed lines represent one standard deviation above and below the average values. The moment curves from the network realizations can be visualized in terms of early, asymptotic and late periods defined relative to the slope of the moment versus time plots. These periods correspond to the near-field, far-field and late behavior of the mass plume: (1) In the early period, the slopes of the second moment curves increase with time until reaching a constant value; (2) The asymptotic period is defined as the period in which the moment versus time curves are approximately linear. The boundary between the early and asymptotic periods is gradational, and occurs at different times in each moment curve; (3) The late period begins when the slopes of all the moment curves decrease as particles begin to exit the domain. The early period reflects the transient development of the dispersion process, until the plume samples a representative set of the heterogeneity. The late periods mark the appearance of boundary effects in the models. Comparisons between the network realizations and S C M models are 87 restricted to the asymptotic periods, as during this period plumes are fully developed and mass transport is free from boundary effects. Most of the first x moments for the S C M models fall within one standard deviation of the average first x moments for the network realizations. However, after 100 days, the H(v)H(l) model underestimates the average value of the first x moment for the network realizations by more than one standard deviation. By 300 days, the average value of the first x moment for the network realizations is on the order of 20 m. The one standard deviation error bounds are on the order of 4 m, and the error in the first x moment for the H(v)H(l) model is on the order of 7 m. Of the three S C M models for which moments are larger than the average for the network realizations, the G(v)G(l) model most closely matches the average first x moments, followed by the G(v,Rk2)G(l) model, and the N(v)G(l) model. The N(v,l)H(l) model underestimates the first x moment for the network realizations. However, the values for the continuum model fall within one standard deviation of the network averages, and the error is of a similar magnitude as the error for the N(v)G(l) model. By calculating the slopes of moment curves during the asymptotic period, the differences between the S C M models can be quantified. The comparison between the slopes of the moment curves for the various models is given in Table 4-1, presented as the ratio of the slopes of the moment curve for the S C M model to the slope of the moment curve for the network realizations, minus 1.0. This ratio gives the error between the network realizations and continuum models, with a value of 0 indicating that the slope of the moment for the S C M model is equal to the average slope of the moment for the network realizations. Because the moment versus time curves only approach a straight line during the asymptotic period, the 88 time range from which to calculate the slope of the curves must be chosen carefully. In Table 4-1 the slopes are calculated between 100 and 200 days, using a linear regression. For the first moments, the error ratio is smallest for the G(v)G(l) model as would be expected from the close match of the moment values (Figure 4-3). However, the error ratio of the N(v,l)H(l) model is lower than the error ratio given by the N(v)G(l) model. The reason that the match of the N(v,l)H(l) model to the network average appears to improve when the error ratios are compared instead of the moment curves, is that differences in the near-field period shift the moment curves such that each starts the asymptotic period at a different time and moment value. As mentioned previously, evaluating the ability of the S C M to reproduce mass transport in a discrete network is best done during the asymptotic period. Therefore, based on the error ratios, the G(v)G(l) model gives the best match to the network averages for the first x moments. Results for the first y moment are similar to those for the first x moment. Relative to each other, the magnitudes of the moments for the S C M models occur in the same order for both the first x and first y moments. Although the relative ranking of the moments from the S C M models is the same as for the first JC moment, for the first y moments, the N(v,l)H(l) model gives results closest to those for the network realizations (Figure 4-3). A l l of the S C M models other than the H(v)H(l) model generate first y moment values that fall within one standard deviation of the average first y moment for the network realizations. Based on the error ratio of the first y moment curve (Table 4-1) the N(v,l)H(l) model is closest to the network realizations, underestimating the slope of the moment versus time curve by only 5 percent. The error ratio for the H(v)H(l) model is the largest, being on the order of 30 percent. 89 Only two of the S C M models generate second x moments that fall within one standard deviation of the second x moments for the network realizations (Figure 4-3). Of these models, the G(v)G(l) model underestimates spreading in the x direction while the G(v,Rk2)G(l) model overestimates spreading in the x direction. After approximately 125 days, three of the S C M models underestimate the second x moment by more than one standard deviation. When the error ratios are compared (Table 4-1), only the G(v)G(l) model yields values within 10% of the average for the network realizations, and all the other models have errors in the slope ranging from 6% to 49 %. Results for the second y moment are somewhat better than those for the second x moment, as all the S C M models other than the H(v)H(l) model generate moments within one standard deviation of the moments for the network realizations (Figure 4-3). This marks a significant improvement when compared to results for the second x moments as the one-standard deviation error bounds are smaller for the second y moment. Moments for the G(v)G(l) model come closest to the average second y moment for the network realizations, while moments for the H(v)H(l) model show the largest deviation from this average. Errors in the slopes (Table 4-1) range from 10% for the G(v)G(l) model, to 40% for the H(v)H(l) model. For both of the first moments, all the S C M methods except the H(v)H(l) model predict moment values within one standard deviation of the average for the network realizations. However, only the G(v)G(l) and G(v,Rk2)G(l) models are within one standard deviation of the average second x moment for the network realizations. The G(v)G(l) model without correlation appears to produce the best overall results for this fracture system, based on three criteria: (1) visual examination of the moment versus time curves (Figure 4-3 and 90 Figure 4-4); (2) having the smallest error in the slopes in four out of the five moments examined; and (3) having the smallest average error (Table 4-1). The averages that appear in Table 4-1 are determined by calculating the mean of the absolute values of the error for each of the five moments. A more appropriate average might weigh the second moment errors more heavily than the first moments, as prediction of the second moments or spreading of the plume is more difficult than predicting the first moments or average location of the mass. There is no obvious weighting function that can be easily justified, and it has been found that averages determined using the current method are consistent with the best-fit as determined by the moment plots. 4.2.2. Rotated gradient A second experiment is conducted by rotating the mean direction of the applied hydraulic gradient by 45°, and repeating both the continuum and discrete modeling. With the rotation of the hydraulic gradient, the orientation of the principal axes of the plume is also rotated. This rotation decreases the rate of mass transport in the x direction, and increases the rate of mass transport in the y direction (Figure 4-5 and Figure 4-6). Because the mean travel direction is also rotated, mass now exits primarily through the top boundary of the model domain, and because the domain is longer in the x direction than the y direction, moments are affected by the boundary at earlier times. For the discrete subdomain, 5000 network realizations are used, as results in Chapter 3 indicate that estimates of the motion statistics calculated using 5000 realizations (Table 4-2) are not significantly different from statistics calculated using 10,000 realizations. First moment results in the 45° gradient example are similar to those for the example using a horizontal gradient except that both the H(v)H(l) and the N(v,l)H(l) models 91 underestimate the moments for the network realizations by more than one standard deviation (Figure 4-5). The magnitude of the first x moments predicted by the S C M models occur in the same order relative to each other as was found for the 0° gradient experiment, with the largest x moment for the N(v)G(l) model and the lowest x moment for H(v)H(l) model. However, in the 45° case, moments for all five S C M models underestimate the average first x moment observed in the network realizations. At later times, the N(v)G(l) model produces the closest match to the average first x moment for the network realizations (Figure 4-5), although the error ratio is only slightly less than that for the G(v,Rk2)G(l) model (Table 4-3). The G(v)G(l) model, which has the lowest error for the 0° case (Table 4-1), produces errors on the order of 13% for the 45° case (Table 4-3). S C M models based on the Gamma distribution, with and without Rk2 correlation of velocities, produce the only second x moment values within one standard deviation of the second x moment values for the network realizations (Figure 4-6). However, moments for the G(v)G(l) model barely fall within one standard deviation of the average second x moment for the network realizations. Unlike the results from the 0° case, the N(v)G(l), H(v)H(l), and N(v,l)H(l) models underestimate the average moments for the network realizations by more than one standard deviation, for both the second x and second y moments. Based on the error ratios, the G(v,Rk2)G(l) model has the lowest error for both the x and y second moments, with errors in the x andy components at 1 and 6 % respectively (Table 4-3). Although the N(v)G(l) model produces the lowest error ratio for 3 out of the 4 moments (Table 4-3), errors in the magnitude of the second moments exceed one standard deviation (Figure 4-6). Of all four of the moments examined here, only moments for the G(v)G(l) and G(v,Rk2)G(l) models are within one standard deviation of the average 92 moments for the network realizations. Note that in both the 0° gradient example (Figure 4-3 and Figure 4-4) and the 45° gradient example (Figure 4-5 and Figure 4-6), the addition of Rk2 velocity correlation to the Gamma distribution increases both the velocity of the plume (first moments) and the spreading (second moments) estimated by the S C M . However in the 45° example this increase improved the match to the network averages, whereas in the 0° example the addition of Rk2-correlation degraded the results in comparison to the G(v)G(l) model. 4.2.3. Orthogonal network Previous work on the S C M has examined only networks composed of orthogonal fracture sets, with no variability in orientation within each set (Schwartz and Smith, 1988; Robertson, 1990). As a comparison to the previous work, a fracture system is created using parameters identical to the base case, except that the conjugate fracture set is oriented at 90° from the horizontal instead of 70°. This fracture system is similar to that examined by Robertson (1990) except that fracture orientation is allowed to vary around the mean values. Because the second fracture set is now oriented perpendicular to a horizontally-aligned gradient, no net travel occurs in the y direction, and all spreading information is contained within the second x and y moments. Moment results (Figure 4-7 and Figure 4-8) are similar to those in the previous two examples. Only the H(v)H(l) model generates first x moments more than one standard deviation below the value of the moments for the network realizations. Because there is no net travel in the y direction, the first y moment is unchanged from the mean y value at the injection location, and since the slopes of the first y moments are all approximately zero, the error ratios of the first y moments are not included in Table 4-4. The G(v)G(l) and 93 G(v,Rk2)G(l) models produce the smallest error in the slope of the first x moment (Table 4-4). Moments for the G(v)G(l) model are the only second x moments that fall within one standard deviation of the averages for the network realizations (Figure 4-8). A l l S C M models other than the H(v)H(l) model generate second y moments within one standard deviation of the average for the network realizations. The G(v)G(l) model has the smallest error in the slopes for the second x moment (Table 4-4), while the N(v)G(l) model has the smallest error in the slope for the second y moment. However, the error in the slopes of the second y moment is greater than 20% for all of the S C M models (Table 4-4). The lowest average absolute error is given by the G(v,Rk2)G(l) model (Table 4-4). This result is consistent with Robertson (1990) who concluded that for orthogonal networks, moments for S C M models most closely matched average moments for network realizations when velocities were modeled using a Gamma distribution, and a Rk2 correlation. However, less than a 1% difference exists in the average error between the G(v)G(l) model, and the G(v,Rk2)G(l) model. Furthermore, the addition of Rk2 correlation causes the estimates of the second x moments to exceed the one standard deviation bound (Figure 4-8). Based on the results here, it is not clear that the use of Rk2 correlation for velocity is a significant improvement over other velocity models when using the S C M to simulate mass-transport in an orthogonal network. 4.2.4. High and reduced density networks Two additional Poisson models are examined to analyze the ability of the S C M methodology to reproduce mass transport in fracture systems in which fracture densities are higher and lower than those in the base-case. The fracture system statistics are the same as in 94 the base-case except: (1) in one case, fracture density is reduced in both fracture sets from 7.2 Im to 3.6 Im (Figure 4-9); and (2) in a second case, fracture density is increased in both fracture sets to 10.8 Im (Figure 4-10). A reduced (30 m by 12 m) area is used for the large domain, high-density fracture network, as computer memory limits the number of fracture intersections allowed. Moments for the reduced-density fracture system are given in Figure 4-11 and Figure 4-12, and moments for the high-density fracture system are given in Figure 4-13 and Figure 4-14. The effective hydraulic conductivity of a fracture network increases with fracture intensity. This increase in effective hydraulic conductivity results in an increase in the slope of the first x and y moments between the reduced density (Figure 4-1 land Figure 4-12), base-case (Figure 4-3 and Figure 4-4), and high density (Figure 4-13 and Figure 4-14) fracture systems. The magnitude of the standard deviation in the first moments decreases with increasing fracture density. In the base-case and reduced-density network systems, the G(v)G(l) and G(v,Rk2)G(l) models have low error ratios for the first moment values (Table 4-1 and Table 4-5). However, the N(v,l)H(l) model has the lowest error in the slopes for the high-intensity network system (Table 4-6). In modeling these three Poisson systems, there is no clear relationship between fracture intensity and the ability of the S C M approach to predict first moments. The rate at which mass disperses, as measured by the second moments at 100 days, increases with fracture intensity from the reduced-density system (Figure 4-12), to the base-case network systems (Figure 4-4). However the degree of dispersion remains reasonably constant between the base case system and the higher-density fracture system (Figure 4-14). For the second x moments, all five S C M models for the reduced-density fracture system 95 predict moments within one standard deviation of the average for the network realizations. For the base case and high-density fracture system, only the G(v)G(l) and G(v,Rk2)G(l) models predict second x moments within one standard deviation of the network averages. For all three fracture densities only the H(v)H(l) model fails to estimate second y moment values within one standard deviation of the network averages. For all three fracture systems composed of non-orthogonal fracture sets, the addition of Rk2 correlation increases the error ratios relative to the G(v)G(l) model (Table 4-6). 4.2.5. Poisson networks near the percolation threshold Based on the results presented so far in this chapter, S C M models are capable of reproducing mass transport in two-dimensional fracture systems of varying fracture density. However, even the reduced density fracture system produces well connected networks, i.e. networks that are significantly above the percolation threshold. Renshaw (1996) has suggested that many natural fracture networks exist near the percolation threshold, while others hold a contrary opinion (see section 2.3 for the definition of percolation threshold). Although the S C M was originally thought to be applicable only in well connected fracture networks, the effectiveness of the S C M method in fracture systems near the percolation threshold is examined here using an extremely low intensity fracture system. In Figure 4-15, a network realization from a fracture system in which the fracture intensity in both the sub-horizontal and sub-vertical fracture sets is 1.5 fracture per meter is plotted. A l l of the fractures that are not connected to the flow system have been removed, leaving only the hydraulically-active portion of the network. A l l parameters other than the fracture intensity are identical to those for the base-case Poisson fracture system. Connection 96 between the left and right boundaries can be broken by removing a single fracture, and hence the network is at the percolation threshold. With the network realizations from the fracture system all near the percolation threshold, only 4939 out of 10,000 networks are hydraulically conductive over a 40 m by 20 m model domain. Moments are based on mass transport through the conducting networks only, and as a result these values reflect only a subset of all possible realizations of the fracture system. The average and standard deviation of the moments for the large-domain network realizations and the S C M models for the fracture system at the percolation threshold are given in Figure 4-16 and Figure 4-17. The low density of fracturing reduces the effective conductivity of the networks so that mass transport is substantially slower than in the previous examples. As a result, the length of time over which moments are plotted is increased. The low density also increases the variability of mass transport between individual realizations so that the standard deviation of each of the moments is larger than in the previous examples. The continuum models in Figure 4-16 and Figure 4-17 are based on motion statistics generated using 10,000 realizations of a 10m by 10m discrete subdomain. There are large discrepancies between the first x moments for the continuum models and the average moments for the discrete networks, with the exception of the moments for the N(v,l)H(l) model at early time. The first x moment for the network realizations and the N(v,l)H(l) model are almost identical until approximately 2000 days, after which they diverge. The moments for the G(v,Rk2)G(l) model are within one standard deviation of the average first x moment for the network realizations. Moments for the G(v,Rk2)G(l) model are more than one-standard deviation larger than the average first x moment at early times. Although they 97 don't match the average for the network realizations, the G(v)G(l), G(v,Rk2)G(l) and N(v,l)H(l) models predict moment values within one standard deviation this average at later times. The H(v)H(l) model and the N(v)G(l) model overestimate the value of the first x moment, although the H(v)H(l) model stays within one standard deviation of the network average for the first 2000 days. For the first y moment, moments for the G(v)G(l), G(v,Rk2)G(l), N(v,l)H(l), and H(v)H(l) models are within one standard deviation of the average for the network realizations until approximately 3000 days. The N(v)G(l) model significantly overestimates movement in the y direction. For the second x moment, only the N(v,l)H(l) model stays within one standard deviation of the average for the discrete networks, as all of the other S C M models significantly overestimate this average. Mass transport modeled with the N(v)G(l) model reaches a boundary at approximately 2500 days, and the second x moment decreases beyond this time. The N(v,l)H(l) model and the G(v)G(l) model both predict moment values within one standard deviation of the average for the discrete networks for the second y moment. The G(v,Rk2)G(l), H(v)H(l) and N(v)G(l) models all predict moment values outside of the one-standard deviation limit. Overall there is a much larger discrepancy between the moments for the S C M models and the average moments for the network realizations than in previously examined Poisson fracture systems. The method is less effective for the fracture system at the percolation threshold because with a very sparse network an insufficient number of fracture intersections, i.e. statistical events, are encountered when generating the motion statistics. A larger discrete 98 subdomain is used here in an attempt to improve the match. S C M models were re-rerun using statistics generated in a 20 m by 15 m discrete subdomain (Figure 4-18 and Figure 4-19). The match between the moments for the discrete models and the moments for the continuum models is significantly improved by the use of a larger discrete subdomain. However, the N(v)G(l) model still exceeds the one-standard deviation limit for the first x moment and first y moment. The N(v,l)H(l) model tends to underestimate the average moment values for the network realizations, although it appears that at later times the slope of the moment vs. time curves are similar for the network averages and this model. The poorest match between the continuum models and the network realizations occurs in the second x moment, with most continuum models exceeding the network averages by more than one standard deviation by 4000 days. The errors in the slope of the moments are given in Table 4-7. Overall, errors in the slope of the moments for all of the models are much higher than errors in the previously examined Poisson networks. The G(v)G(l) and G(v,R22)G(l) models give the lowest average error in the slopes at 0.36 and 0.41 respectively, but both models exceed the average for the network realizations for the second x moment by more than one standard deviation. Although the magnitude of the error in the N(v,l)H(l) model is the highest, it would appear to give one of the best fits to the average moments for the network realizations. The errors for the H(v)H(l) and N(v)G(l) models are lower than the average error for the N(v,l)H(l) model, but both the H(v)H(l) and N(v)G(l) models exceed the network averages by more than one standard deviation for the second x moment by around 5000 days. It appears that moment values can be estimated using the S C M method for networks near the percolation threshold. These estimates are not as close to the average moments for 99 network realizations as estimates for previous Poisson networks are, but because of the large variation between these discrete networks, the large standard deviations gives a larger margin for error. The drastic changes in the behavior of mass transport in differing networks near the percolation threshold suggests that using average values of discrete networks may not be appropriate. In these cases the S C M method may still provide an estimate which is as significant as estimates obtained by averaging over Monte Carlo network realizations. Many different fracture system models exist, and the purpose of this section is not to examine the validity of the S C M method for all models, but rather to demonstrate that S C M models will reproduce mass-transport for a range of Poisson models. In the following section, the ability of the S C M to model mass transport in a fracture system geometrically different from the Poisson model is tested. 4.3. Levy-Lee Networks Levy-Lee fracture systems are examined in this section in order to test the ability of the S C M to model mass transport in networks that are structurally different than the Poisson fracture systems modeled in section 4.2. Networks in this section are generated using a Levy-Lee flight process (section 3.3.2), so that each realization is composed of fractures from a range of length-scales (section 2.2.2). Model parameters are given in Table 3-2. Levy-Lee networks are primarily composed of dense clusters of closely-spaced fractures, with widely-variable orientations (Figure 3-3). At a larger length scale, long sub-horizontal and sub-vertical fractures act as conduits between the clusters of smaller length-scale fractures. Two well-connected fracture systems are examined here, with the first generated by using a point field with a fractal dimension of 7.5, and the second a point field with a fractal dimension of 100 1.8. Due to the stochastic nature of the generation process, the actual value of D for each realization varies. Although methods are available to calculate D for a fracture network, the calculation process required is numerically intensive and hence time-consuming. Because the exact value of D is not significant for the purposes of this study, these calculations were not done. Fractures within these Levy-Lee networks are generated using the same mean aperture as in the Poisson networks. A l l fractures in the Levy-Lee network system have equal aperture so that differences in flow rate are smaller than occur in the Poisson fracture system. 4.3.1. Motion Statistics Motion statistics are calculated over 10,000 network realizations i n a l O / w b y l O m discrete subdomain. This size and number of the subdomains are sufficient that average values of the parameters have converged, and this was determined by testing similar to that outlined in section 3.5.6 for the base-case Poisson fracture system. The average velocity of particle movement is higher in the Levy-Lee networks than in the Poisson networks (Table 4-8). Higher particle velocities occur because flow and transport in the Levy-Lee networks is dominated by the small number of long fractures that connect the clusters. Because all fractures have the same aperture there is a much lower standard deviation in particle velocities within Levy-Lee network realizations. The average and standard deviation in the path-lengths in the Levy-Lee network realizations are larger than that in the Poisson network realizations. An increase in standard deviation in the path-length is consistent with the change from a network containing fractures from a single length scale to a network containing fractures from a range of length scales. The increase in the mean path-length is 101 consistent with the presence of the long fractures through which all particles travel between the clusters of short fractures. 4.3.2. Plumes Plumes generated from 4 realizations of the discrete Levy-Lee networks (Figure 4-20) and plumes from the corresponding S C M model (Figure 4-21) are strikingly different. Continuum plumes for the Levy-Lee model fill space in a manner similar to continuum plumes for the Poisson networks. In order to illustrate the nature of transport channeling in a Levy-Lee network, plumes at three successive times are presented in Figure 4-22. The plume through the Levy-Lee network realization appears to first spread in both the x and y directions from 10 days to 30 days, and then by 50 days the plume contracts in the y direction as it spreads in the x direction between x — lm and x = 10 m (Figure 4-22). Mass tends to disperse upon entering clusters of short fractures. Fluid can exit a fracture cluster only through a limited number of longer fractures, so that a plume which is dispersed within a cluster will reduce its cross-section as it enters one of these longer fractures. Effectively the dispersion process can and does reverse for this kind of multi-scale fracture network. The values of the second moments would show this reversal for a single network realization, but when the second moments are averaged over multiple networks the resulting values increase continuously. However, this reversal of dispersion illustrates one aspect of mass movement that is not reproduced by the S C M , as in S C M modeling, independent random walks of each particle cause plumes to disperse continuously. The importance of including the reversal of the dispersion process when modeling mass transport through fractured rock is not yet clear, as: (1) although mass transport through an individual network realization may be characterized by expansions and contraction in a 102 direction transverse to the hydraulic gradient, when averaged over multiple realizations mass transport may be characterized by continuous dispersion at a slower rate, and (2) this type of dispersion and contraction of the mass plume may be restricted to multi-scale networks. In a fracture network near the percolation threshold there will occur instances in which two regions of the network are connected by a single fracture. Under these circumstance plumes will be forced to contract in a manner similar to that occurring in the Levy-Lee network. For this reason the possibility exists that contraction occurs in all types of fracture systems. The effect of this process is quantified in the next section on moments. 4.3.3. Moments The average first x moment curve for the Levy-Lee network realizations with D = 1.5 does not show the breaks into early, asymptotic and late periods evident for the Poisson networks (Figure 4-23). Instead, the slope of the average first x moment versus time curve decreases continuously. The standard deviation for the first x moment curves for the Levy-Lee network realizations is significantly larger than the standard deviation for the moments for the Poisson network realizations. Levy-Lee networks vary significantly from realization to realization, resulting in large variations in the spatial distribution of mass (Figure 4-20), and therefore a large standard deviation in the moments. Because the standard deviation for the moments for the network realizations is large, initially most of the S C M models produce moment estimates within one standard deviation of the average moments for the network realizations. However, only moments for the N(v)G(l) model appear to closely match the average first x moments for the network realizations. After approximately 40 days, moments for all of the other S C M models 103 underestimate the average first x moment for the network realizations by more than one standard deviation. Comparing the slopes of the moment versus time curves is difficult for the Levy-Lee fracture system because the moment versus time curves are not straight lines. For the Levy-Lee network system with D = 1.5, the interval in which slopes of the moment curves are calculated is between 40 and 80 days. This period is selected to avoid times during which the moment curves are affected by particles exiting the model domain. However, results might vary somewhat i f the slopes of curves within different periods were compared, and this variability reduces the reliability of the error measurements for this Levy-Lee example. The smallest error in the slope of the first x moment is 9 %, and occurs for the N(v)G(l) model. The H(v)H(l) and N(v,l)H(l) models underestimate the average first x moment of the network realizations by 32 % and 37 % respectively, while errors for the G(v)G(l) and G(v,Rk2)G(l) models are on the order of 70 % (Table 4-9). The error in predicting the slope of the first moments for the D = 1.5 Levy-Lee network realizations (Table 4-9) is significantly larger than the error in predicting the slope of the average first x moment of the Poisson network realizations (compare Table 4-3 to Table 4-5). For the Levy-Lee network realizations the standard deviation of the second x moment is larger than the average value, so that the lower bound for the second x moment is truncated at zero (Figure 4-24). As a result of the large variation, all the S C M models fall within the plus or minus one standard deviation of the average second x moment for the network realizations, even though several of the S C M models are visibly poor fits. The lowest error in the slopes occurs with the G(v,Rk2)G(l) model at around 4 % (Table 4-9). For the second y moments only the N(v)G(l) model predicts values that are more than one standard deviation 104 larger than the average moments for the discrete networks. Errors in the slope of the second y moment range from 16 % for the G(v,Rk2)G(l) model, to 76 % for the N(v)G(l) model. Only the N(v)G(l) model produces acceptable results for the first x moments, but both the second moments for this model are inadequate. Therefore, it does not appear that any S C M method would be valid for this particular Levy-Lee network system. 4.3.4. Levy-Lee B, fractal dimension 1.8 By increasing the fractal dimension from 1.5 to 1.8, and increasing the average fracture length, a more intensely fractured and therefore better connected fracture system is obtained (Figure 4-25). The purpose of examining a Levy-Lee fracture system with a higher D is to determine whether the problems in the previous example are due to the multi-scale nature of the network or to the limited number of connections between the fractures. Moment estimates for the S C M models are generally closer to the average moments for the D = 1.8 Levy-Lee network realizations than the D = 1.5 Levy-Lee network realizations, with the exception of the second y moments (Figure 4-26 and Figure 4-27). Improvements are seen in the first and second x moments, and the errors in the slope of these moments are on the order of one third the error for the previous Levy-Lee example. Second y moments are not predicted as effectively as the first moments by the S C M models. Only moments for the H(v)H(l) model fall within one standard deviation of the network averages, with an error of approximately 29 % (Table 4-10). However, both the value and slope of the second y moments are so small that the differences between any of the models are probably insignificant. In the second Levy-Lee system, the H(v)H(l) model produces the lowest average error, which is in striking contrast to the Poisson fracture systems examined 105 previously in which the histogram model consistently underestimated the average moments for the network realizations. 4.4. Discussion of SCM Results in Two Dimensions The purpose of comparing S C M models to network realizations is to evaluate the ability of the S C M method to predict the transport of mass through fracture networks, with the eventual goal of modeling mass transport through fractured rock at the field scale. In the examples examined in this chapter it was demonstrated that the choice of statistical models to represent velocity and path-length control how well the S C M models work. In none of the examples examined were there exact matches between the average moments for the network realizations and the moments for any S C M model. Moment values for each of the S C M models are often, but not always, within one standard deviation of the average moments for the network realizations. The values of the moments generated by applying the five velocity distributions within the S C M models frequently occur in the same order with the H(v)H(l) model producing the smallest moments, and the G(v,Rk2)G(l) model producing the largest moments. The relative ranking of the S C M moments is observed in all of the fracture systems examined above, with the exception of the fracture system at the percolation threshold, and the Levy-Lee fracture systems. Despite the consistency of the ranking of the moments generated by the S C M approaches, no single S C M approach produced moment values that were consistently the closest to the average moments for the network realizations. Over all the fracture systems examined in this chapter, the G(v)G(l) model most frequently produces the smallest error (Table 4-11). However, for example, the H(v)H(l) model produced the smallest error for the second Levy-Lee fracture system. 106 Results based on orthogonal network simulations lead Robertson (1990) to suggest that Rk2 correlation of velocities in the S C M consistently improved the match between moments from the S C M models and average moments for the network realizations. For all the fracture systems examined to date, the addition of velocity correlation consistently increases the magnitude of the moments over the G(v)G(l) models, but this increase only occasionally improves the match to the average moments for the network realizations. For the base-case Poisson system, the moments for the S C M model underestimate the moments for the discrete model, so that a slight increase in the moment values due to the inclusion of the velocity correlation would improve the match between the moments. However, the increase in the value of the moments due to the velocity correlation.is so large that the match between the moments is worse with Rk2 velocity correlation than without this correlation. In other cases, both examined previously in this chapter, and not included here, the use of an S C M model using Normal or Gamma velocity distribution overestimates the average moments for the network realizations, so that the inclusion of velocity correlation only worsens the match between the S C M models and network averages. Caution is required when interpreting a connection between the use of correlation as a numerical modeling device, and the physical processes occurring during mass transport through discrete networks. Exact duplication of the average moment values by the S C M models was not expected, and did not occur. Therefore continuum models are judged to be successful i f the moment estimates fall within one standard deviation of the average for network realizations. The one standard deviation limit may or may not be sufficiently discerning, depending upon the objectives of the modeling study. However, the large standard deviation in the moments 107 for the network realizations suggests that exact reproduction of the average moment values may not be necessary, or even significant. This leads to the question of what either the S C M method or the network realizations are capable of predicting. This issue of prediction is addressed in Chapter 7. Also, the fact that no S C M model is consistently closest to the average for the network realizations suggests that the remaining differences between the moments for the S C M models and the average moments for the network realizations are not a function of the statistical models chosen, but are in some manner due to the approximating assumptions implicit within the S C M . As will be seen in section 7.3, however, the effectiveness of each model can be predicted by Ergodic theory. 4.5. Summary It was demonstrated in this chapter that the Statistical Continuum Method is capable of reproducing mass transport within an equivalent discrete network model, within certain bounds. Five S C M approaches were assessed, each using a unique set of statistical models and correlations to generate particle movement. Comparisons of the evolution of the spatial moments through time indicated that moment values for most S C M models were within one standard deviation of the mean moments for the discrete network realizations. By providing a measure of the variation in mass transport from one discrete network to the next, the one standard deviation range provides a useful gauge of the difference between the moments for the S C M and DISCRETE models. Best results were most often obtained by applying a three-parameter gamma distribution to both the path-length and velocity, without correlation of parameters. However, the G(v)G(l) model was not the best approach for all fracture systems examined. The key to the S C M is the selection of the method by which statistics describing 108 the motion of particles in the discrete subdomain are translated to particle movements in the continuum. For the base-case fracture system, densely-fractured Poisson networks were created from two non-orthogonal fracture sets. In the discrete subdomain motion statistics gathered within 10 m by 10 m fracture networks converged to average values within 10,000 realizations. Moments for the S C M approach using Gamma distributions for velocity and path-length were closest to the average moments for 40 m by 20 m network realizations. Most models produced moment values within one standard deviation of the average moments for the network realizations for the base-case, although only the G(v)G(l) and G(v,Rk2)G(l) approaches were within this range for the second x moment. Errors in the slopes of the moment vs. time curves were calculated in order to compare the rate of mass transport in the S C M approaches, to the average rate of mass transport through the network realizations. For the base-case example the error in the slope of the moment curves is on the order of 6 percent. When the applied hydraulic gradient was rotated 45°, moments for the G(v,Rk2)G(l) model were closest to the average for the discrete networks, with an error of 7 percent. For the networks composed of fractures sampled from orthogonal fracture sets both the G(v)G(l) and G(v,Rk2)G(l) approaches produced the best results with errors around 12 %. Correlating velocity and path-length works best for the low density network (8 % error) and the G(v)G(l) model works best for the high density network (5 % error). For all five of the Poisson fracture systems modeled in this chapter the S C M method predicted moments within 12 percent of the average moments for the discrete network realizations, as measured by the error in slope of the moment vs. time curves. 109 For the network at the percolation threshold, as long as the size of the discrete subdomain was large enough, the moments for the N(v,l)H(l) model were close to the average for the network realizations. The S C M approach was also shown to be capable of reproducing mass transport in selected multi-scale networks. In the lower D Levy-Lee fracture system the slope of the moments for the N(v)G(l) approach deviated from the slope of the average moments by at least 51%. For the higher density Levy-Lee fracture system the slope of the moments for the H(v)H(l) approach was within 17% of the slope of the moments for the network realizations. The nature of the transport channeling varied significantly between the Poisson and Levy-Lee fracture systems. It was shown that channeling could cause dispersion to reverse in some Levy-Lee networks. By predicting moment values close to the average moments for the network realizations, in most of the models examined the S C M method reproduced the influence of channeling on mass transport through both Levy-Lee and Poisson fracture systems. This concludes the two-dimensional modeling. In the next chapter the S C M method in three-dimensions is outlined, and in Chapter 6 the results in three-dimensions are presented. The overall results are discussed in Chapter 7 110 Table 4-1: Errors in the slopes of the moments for the base-case Poisson network with horizontally applied gradient. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x -0.316 0.135 0.043 -0.071 0.092 First y -0.300 0.167 0.073 -0.048 0.142 a 2 x -0.486 -0.402 -0.063 -0.448 0.331 2 CT y -0.403 -0.186 -0.096 -0.204 0.156 Average 0.400 0.206 0.058 0.193 0.167 Table 4-2: Motion Statistics for 45° case Direction: 1 2 3 4 Injection Orientation: 3.14 0.00 4.37 1.22 0.78 CT 0.050 0.052 0.050 0.052 0.37 Velocity: V- -6.1 -5.7 -6.1 -5.7 -5.8 CT 0.43 0.37 0.44 0.36 0.19 K 0.83 0.95 0.85 0.96 3.11 Correlation: R k l 0.21 0.55 0.19 0.59 Rk2 0.06 0.34 0.07 0.38 Path Length: V- 0.15 0.43 0.12 0.49 0.51 CT 0.15 0.47 0.12 0.53 0.31 K 1.75 2.08 1.76 2.11 0.50 Directional Choice: 0.05 0.44 0.03 0.48 Aperture: -4.31 -4.15 -4.35 -4.16 -4.20 CT 0.22 0.23 0.22 0.23 0.11 Table 4-3: Errors in the slopes of the moments for the Poisson network with gradient applied 45 degrees from horizontal. Networks are the same as the base-case Poisson network, but the gradient orientation is rotated. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x -0.435 -0.059 -0.128 -0.269 -0.072 First y -0.448 -0.062 -0.139 -0.276 -0.084 a 2 x -0.615 -0.593 -0.309 -0.592 -0.011 CT2Y -0.528 -0.390 -0.209 -0.430 0.058 Average 0.507 0.276 0.196 0.392 0.056 111 Table 4-4: Errors in the slope of the moments for the Poisson network with perpendicular fracture sets. Networks are the same as the base-case Poisson network except that the average fracture orientation of fracture set two is 90 degrees instead of 70. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x -0.351 0.062 -0.019 -0.128 0.026 a 2* -0.497 -0.365 -0.163 -0.441 0.187 o\ -0.533 -0.207 -0.272 -0.241 -0.232 Average 0.46 0.211 0.151 0.27 0.148 Table 4-5: Errors in the slope of the moments for the low intensity Poisson network. Networks are the same as the base-case Poisson network except that fracture densities are 3.1 fractures / meter instead of 7.2 for both fracture sets. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x -0.406 0.126 0.011 -0.115 0.024 First y -0.382 0.163 0.040 -0.080 0.039 a 2 x -0.560 -0.499 -0.159 -0.519 0.318 o\ -0.374 -0.170 0.047 -0.168 0.262 Average 0.437 0.23 0.057 0.229 0.157 Table 4-6: Errors in the slopes of the moments for the high intensity Poisson network. Networks are the same as the base-case Poisson network except that the fracture intensity in both sets is 10.3 fractures / meter instead of 7.2. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x -0.215 0.117 0.141 -0.078 0.212 First y -0.181 0.219 0.245 0.030 0.323 2 O- x -0.141 0.024 0.155 -0.119 0.482 o\ -0.273 0.067 0.136 -0.023 0.267 Average 0.211 0.081 0.143 0.075 0.293 112 Table 4-7: Errors in slope of moments for Poisson network at percolation threshold. Network is the same as the standard Poisson network except that fracture densities are 1.5 fractures / meter. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x 0.15 0.668 0.048 -0.594 0.063 First y -0.125 0.318 -0.136 -0.649 -0.12 a 2* 1.198 0.708 1.079 -0.438 1.169 a 2 y 0.302 0.363 0.268 -0.533 0.345 Average 0.45 0.602 0.361 0.54 0.41 Table 4-8: Motion Statistics Calculated from Particle Movements within 10,000 Realizations of the Levy-Lee Network System. Direction: 1 2 3 4 Injection Orientation: 3.13 0.00 4.96 1.32 0.70 a 0.652 0.338 0.314 0.319 0.739 Velocity: l i -6.0 -5.2 -5.7 -5.7 -5.4 a 0.32 0.23 0.30 0.30 0.14 K 1.21 1.26 1.09 1.04 2.27 R k l 0.03 0.70 0.24 0.22 Rk2 0.02 0.37 0.17 0.15 Path Length: ^ i 0.16 1.23 0.47 0.48 1.67 a 0.13 1.19 0.41 0.41 0.80 K 1.26 2.60 2.31 2.29 1.16 Directional Choice: 0.01 0.53 0.23 0.23 Aperture: -4.5 -4.5 -4.5 -4.5 -4.5 a 0.0 0.0 0.0 0.0 0.0 Table 4-9: Errors in slope of continuum moments for Levy-lee A. Levy-Lee network with fractal dimension 1.5. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x -0.324 -0.087 -0.708 -0.368 -0.715 a 2 x -0.435 -0.711 0.064 -0.429 0.043 a 2 y -0.407 -0.763 0.177 -0.379 0.158 Average 0.389 0.52 0.316 0.392 0.305 113 Table 4-10: Errors in slope of continuum moments for Levy-lee B. Levy-Lee network with fractal dimension 1.8. H(v)H(l) N(v)G(l) G(v)G(l) N(v,l)H(l) G(v,Rk2)G(l) First x 0.119 0.278 0.248 0.334 0.253 a 2 x 0.15 -0.329 -0.226 0.179 -0.129 o\ 0.259 0.338 0.279 0.812 0.247 Average 0.176 0.315 0.251 0.442 0.21 Table 4-11: Sum of the error in continuum slopes, compared to discrete slopes. Value is the summation of error in first and second x and y slopes. Cases in which the first y slope is almost zero do not include error in this slope. Model/Error First x First y cfx Average Base-Case Poisson G(v)G(l) Gradient: 9 = 0° 0.043 0.073 -0.063 -0.096 0.058 Base-Case Poisson G(v,Rk2)G(l) Gradient: 0 = 45° -0.072 -0.084 -0.011 0.058 0.056 Perpendicular Fracture Sets G(v)G(l) Gradient: 9 = 0° 0.026 N / A 0.187 -0.232 0.148 Low-Intensity Fracture System G(v)G(l) 0.011 0.040 -0.159 0.047 0.057 High-Intensity Fracture System N(v,l)H(l) -0.078 0.030 -0.119 -0.023 0.075 Percolation Threshold Fracture System G(v)G(l) 0.048 -0.136 1.079 0.268 0.361 Levy-Lee A Fracture System G(v,Rk2)G(l) -0.715 N / A 0.043 0.158 0.305 Levy-Lee B Fracture System H(v)H(l) 0.119 N / A 0.15 0.259 0.176 114 (1) x = 15.16 m y = 7.63 m CT2x= 5.07 m cr 2 y= 2.24 m 20 10 1 I i _ .;, •- / ' i " i' I 10 20 30 40 Figure 4-1: Plumes from 3 individual discrete networks, for the base-case Poisson network system. Each plume contains 5000 particles. Each network is generated using the base-case Poisson fracture system (Table 3-1). 115 (1) H(v)H(l) x =9.99 m y = 7.93 m a 2 x = 3.86 m .2 = 1.75 m (2) N(v)G(l) x =16.29 m y = 9.82 m a 2 x = 4.03 m a 2 = 2.03 m (3) G(v)G(l) x = 15.16 m y = 9.49 m a 2 x = 4.96 m a 2 y = 2.15 m 1 '* •»•'' V 1 1 _ _ 1 (4) G(v,Rk2)G(l) x = 15.86 m y = 9.70 m a 2 x = 6.00 m a 2 y = 2.36 m (5) N(v,l)H(l) 13.40 m x y 8.93 m .2 = x a 2 y = 2.01 m Figure 4-2: Plumes for the base-case model, using five different S C M models. Time = 200 days. Each plume contains 10,000 particles. See Table 3-4 for the model definitions. 116 0 L I I I I 0 200 400 Time (days) Figure 4-3: First moments for discrete and S C M models for the base-case Poisson fracture system. Values for the discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean of the moment for the discrete network realization. 117 Time (days) Figure 4-4: Second moments for discrete and S C M models for the base-case Poisson fracture system. Values for the discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean of the moment for the discrete network realization. 118 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) 20 I 1 1 r 0 L I I I 1 0 100 200 Time (days) Figure 4-5: First moments for discrete and S C M for the base-case Poisson fracture system. Moments are plotted against time in days. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is 45° from horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 119 CN 0 40 30 20 0 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)G(l) X H(v)H(l) l ^ J 1 C N ^ 20 0 0 200 400 Time (days) Figure 4-6: Second moments for discrete and S C M for the base-case Poisson fracture system. Moments are plotted against time in days. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is 45° from horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 120 10 0 0 200 400 Time (days) Figure 4-7: First moments for discrete and S C M models for orthogonal Poisson network system. Networks are similar to the base-case fracture system except that the vertical fracture set is aligned at 90° rather than 70°. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 121 0 200 400 Time (days) Figure 4-8: Second moments for discrete and S C M models for orthogonal Poisson network system. Networks are similar to the base-case fracture system except that the vertical fracture set is aligned at 90° rather than 70°. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 122 40 m Figure 4-9: One realization of the low-intensity Poisson fracture system. One realization of the large discrete subdomain. Fracture system is the same the base case except that fracture intensity in both fracture sets is 3.6 m"1 123 30 m Figure 4-10 One realization of the high-intensity Poisson fracture system. One realization of the large discrete subdomain. Fracture system is the same the base case except that fracture intensity in both fracture sets is 10.8 m"1 124 30 20 10 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) o N(v)°(1) X H(v)H(l) 0 10 h 0 0 100 200 300 400 500 Time (days) Figure 4-11: First moments for discrete and S C M models for the low intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the intensity of both fracture sets is 3.5 m"1 instead of 7.2 m"1. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 125 40 30 20 0 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)GG) X H(v)H(l) (N ^ 20 0 0 100 200 300 400 500 Time (days) Figure 4-12 Second moments for discrete and S C M models for the low intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the intensity of both fracture sets is 3.5 m"1 instead of 7.2 m"1. Values for the discrete model a based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 126 30 20 h 10 h O + • o x N(v,l)H(l) G(v)G(l) G(v,Rk2)G(l) N(v)G(l) H(v)H(l) 0 Figure 4-13: First moments for discrete and S C M models for the high intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical fracture set intensity is 10.2 m"1 instead of 7.2 m"1. Values for the discrete model based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 127 40 h 30 20 0 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)G(l) X H(v)H(l) ° H •+- ^ o ° x J I L ^ 20 0 0 100 200 Time (days) Figure 4-14: Second moments for discrete and S C M models for the high intensity Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical fracture set intensity is 10.2 m"1 instead of 7.2 m"1. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 128 Figure 4-15: One realization of the Poisson network at approximately the percolation threshold. Fracture intensity is 1.5 fractures m"1. A l l non-conducting fractures have been removed, leaving only the hydraulic backbone of the network. 129 30 20 i r O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(V)G0) X H(v)H(l) 0 o o o o X X x 10 h 0 j — A r 20 0 t> i i i i I 0 1000 2000 3000 4000 5000 Time (days) Figure 4-16: First moments for discrete and S C M models for the percolation threshold Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical and horizontal fracture intensity is 1.5 fractures per meter. Discrete values are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 130 CN 200 150 100 50 0 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)G(l) X H(v)H(l) X + x -t-X CN ^ 20 0 0 1000 2000 3000 4000 5000 Time (days) Figure 4-17 Second moments for discrete and S C M models for the percolation threshold Poisson fracture system. Networks are similar to the base-case fracture system except that the vertical and horizontal fracture intensity is 1.5 fractures per meter. Discrete values are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 131 30 O N(v,l)H(l) - + G(v)G(l) • G(v,Rk2)G(l) O N(v )GG) X H(v)H(l) 0 L I I I I I 0 1000 2000 3000 4000 5000 Time (days) Figure 4-18 First moments for discrete and S C M models, for the percolation threshold Poisson fracture system, using continuum models based on 10,000 realizations of 20 x 15 discrete subdomains. 132 200 CN X 150 100 h O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)G(l) X H(v)H(l) 50 0 Time (days) Figure 4-19 Second moments for discrete and S C M models, for the percolation threshold Poisson fracture system, using continuum models based on 10,000 realizations of 20 x 15 discrete subdomains. 133 (1) "x~ = 8.50 m 10 7 = 0.98 m G 2 x = 6.13 m 5 a 2 = 0.837 m (2) T x = 9.43 m y = 2.40 m a 2 = 5.89 m 0.38 m (3) x = 5.46 m y = 2.15 m .2 = x r2 1.67 m i o h 20 (4) ¥ = 7.23 m y = 1.96 m 10 -1 , ••j..Jp « i . . . . . 1 1 I . .. ...1 i.. . -a 2 x = 5.92 m 5 — — a 2 y = 1.50 m 1 1 1 5 10 15 20 Figure 4-20: Plumes from 4 individual Levy Lee networks, D=1.5. Time = 50 days. Each plume contains 5000 particles. Each network is from the network statistics in Table 3-2. Domain size is 30m x 12m. 134 H(v)H(l) N(v)G(l) G(v)G(l) G(v,Rk2)G(l) N(v,l)H(l) X = 5.59 m y = 2.85 m *\ = = 4.99 m °\ = = 1.64 m X = 10.24 m y = 3.80 m °\ = = 5.09 m -v-= 2.14 m X = 5.87 m y = 4.13 m = 4.96 m = 1.68 m X = 5.89 m y = 4.17 m = 4.98 m °\= = 1.66 m X = 5.29 m y = 3.45 m °\ = : 5.35 m °v = 1.68 m -1 1 1 St! • ' 0 5 10 15 20 1 . ja- ' i i j . i ' . i • 1 • lifp Sill *X .ilvv"^ '',!•'•, ,;: r't.'',^.-'-". •'• "l. ' Figure 4-21: Plumes based on five different S C M models, for the Levy-Lee fracture system with D = 1.5. Domain size is 30m x 12m. Time = 50 days. Each plume contains 5,000 particles 135 1 0 D a y s x =2.12 m y = 0.44 m G 2 x = 5.17 m a 2 = 0.49 m 10 15 3 0 D a y s x =4.98 m y = 1.04 m a 2 x = 7.04 m a 2 = 1.11 m 5 0 D a y s x = 6.65 m y = 1.60 m o-2y: 7.37 m 1.28 m 10 10 15 Figure 4-22: Plume evolution through a single Levy-Lee network. Between 10 days and 30 days the plume expands in both the x andy directions. Between 30 and 50 days the plume continues to expand in the x direction, but mass does not move significantly in the y direction. This lack of spreading is partially because of the loss of mass through the upper boundary, but primarily it occurs because mass is forced down the limited number of channels between clusters. 136 30 20 10 0 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) S o o O N(V)G(!) X H(v)H(l) / / / / / / / / / 20 10 o 0 50 Time (days) 100 Figure 4-23: First moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.5. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 137 CN 0 40 h 30 20 h O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)G(l) X H(v)H(l) . 0 CN ^ 20 L 0 0 50 100 Time (days) Figure 4-24: Second moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.5. Values for the discrete model are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 138 Figure 4-25: Single realization of the Levy-Lee fracture system with fractal dimension 1.8 139 30 20 O N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(V)G(1) X H(v)H(l) 10 0 20 10 h 0 0 50 Time (days) 100 Figure 4-26: First moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.8. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 140 C N 40 30 0 N(v,l)H(l) + G(v)G(l) • G(v,Rk2)G(l) O N(v)G(l) X H(v)H(l) 20 0 CN ^ 20 L 0 0 50 100 Time (days) Figure 4-27: Second moments for discrete and S C M models for the Levy-Lee fracture system with D = 1.8. Values for discrete models are based on 5,000 network realizations. Applied hydraulic gradient is horizontal. Cross-hatched curve represents +/- one standard deviation in the Mean for the discrete model. 141 5. SCM Method In Three Dimensions 5.1. Introduction In the preceding two chapters the Statistical Continuum Method was presented, in two dimensions. In the following two chapters the Statistical Continuum Method is developed into a fully three-dimensional modeling tool. Many sites at which mass may be transported through fractured rocks are three-dimensional by nature of their geometry. For example radionuclides escaping from a repository below a mass of rock may do so in any direction, allowing mass to reach highly conductive pathways by a number of routes. This is especially true when the predominant gradient direction is unknown or may change substantially over the time frame of the model. Furthermore there is little correlation between the transport properties of a two-dimensional network representation of a three dimensional network, and the transport properties of that three-dimensional network. A two-dimensional slice of a three-dimensional fracture network has a much lower connectivity, and hence conductivity than the original three-dimensional network. Even i f the two-dimensional slice is aligned along the maximum average forward movement direction and maximum average spreading direction within a three-dimensional network it seems unlikely that mass movement would be contained within the plane. It is difficult to imagine how mass moving through a large channel out of the plane could be effectively modeled by movement restricted to the plane. Two-dimensional fracture models have value in theoretical studies, but it seems unlikely that two-dimensional discrete networks could be sufficient to predict mass transport at a field location. In this chapter the mechanics of extending the method from two to three dimensions is outlined. First the three-dimensional modeling of fracture networks is discussed. Then 142 extensions to the procedures used to calculate motion statistics are outlined. Three sections discussing the motion statistics follow: first the interdependence of the motion statistics; second the sensitivity of the motion statistics to the way in which the path-length is defined; and third the variability of the motion statistics as a function of the size and number of networks in the subdomains. The final two sections of this chapter outline extensions to the particle tracking process in the continuum. 5.2. Generation of three-dimensional fracture networks 5.2.1. Outline of FRACMAN Three-dimensional fracture network modeling is implemented using the F R A C M A N suite of programs developed by Golder Associates (Dershowitz et al., 1995). These programs are designed to model the geometry of discrete features, and can be used to analyze raw data, and to stochastically simulate fracture patterns for exploration simulation, and flow and transport modeling. Four of the F R A C M A N programs are used in the three-dimensional modeling: (1) F R A C W O R K S is used to generate the fracture networks; (2) MESHMONSTER is used to generate a finite element mesh; (3) E D M E S H is used to edit the finite element mesh; and (4) MAFIC is used to calculate the hydraulic head distribution and implement particle-tracking. The discrete network model is only briefly reviewed here, emphasis in this chapter is placed on the determination of the motion statistics needed for S C M analysis. Only minor modifications to the network modeling software are implemented here. FRACWORKS In order to generate fracture networks it is necessary to choose a fracture-generation algorithm and set limits on the size of the domain within which fractures are created. The 143 fracture system parameters are chosen so that networks: (1) have transport properties similar to naturally occurring fracture networks; (2) fit within the limits imposed by both the computer hardware and software; and (3) have sufficient complexity to test the S C M approach. A three-dimensional, base-case model consisting of two non-orthogonal sets in which fractures are sub-vertical is created (Table 5-1). One thousand fractures are generated in each of two sets in a cubic domain measuring 21 m on each side (Figure 5-1). Each network used in the discrete subdomain is a 10 m on each side sample from the centre of the generation domain, while the large-domain networks used to evaluate the S C M are 20 m on each side. Networks are sampled from volumes larger than the flow region in order to reduce boundary effects. It is important to note that although there are some similarities, this base-case is not the three-dimensional equivalent of the two-dimensional fracture system considered in the previous two chapters. For example, the scan-line density for each fracture set in the three-dimensional base-case model is on the order of 1.1 m'J, based on repeated measurements, while the scan-line density for each fracture set in the two-dimensional base-case is 7.2 m"1. Networks for the base-case fracture system are composed of fractures generated using an Enhanced Baecher model (Baecher, et al, 1978). Square, two-dimensional fracture planes are generated by first sampling center locations from a uniform spatial distribution. Networks generated using a uniform spatial distribution have a fracture spacing similar to networks generated using the two-dimensional Poisson model. Square fractures are used in part because this shape promotes stability in the calculation of the flow solution. The area of each fracture is sampled from a lognormal distribution of the equivalent radius, which has a 144 mean of 1.4 m and a standard deviation of 0.8 m. The equivalent radius is the radius of a circle with an area equivalent to the area assigned to the fracture. The orientation of the fracture is sampled from a distribution of the trace and plunge of the poles to the fracture plane (Figure 5-2). For the base-case model the first fracture set has a trace and plunge of 90° and -20°, respectively, and the second set has a trace and plunge of -10° and 20°, respectively. For the base-case fracture system the trace and plunge of each fracture set is constant, but for later models variability in the orientations is introduced. The resulting networks are dense and well connected (Figure 5-3). For the base-case fracture system the transmissivity of each fracture is sampled from a lognormal 6 2 7 2 distribution with a mean value of 1.0 x 10" (m /s) and a standard deviation of 5x10" (m /s). Transmissivity varies from fracture to fracture but is constant across an individual fracture plane. This approach differs from the two-dimensional modeling where fractures are assigned aperture values from a lognormal distribution and transmissivities are calculated by applying the cubic law. However, the input parameters are chosen so that similar conductivity distributions result. Once all the fractures have been generated, F R A C W O R K S finds the intersections of the fractures and prepares the network for the forming of the finite-element mesh. MESHMONSTER AND EDMESH After the networks are generated finite element meshes using triangular elements are created using the MESHMONSTER routine. The output of the meshing programs is a list of nodes containing their x, y and z coordinates, and a list of elements which define the three nodes comprising each element. The meshing process is complicated and imperfect, and the resulting mesh must be edited in order to remove nodes that are likely to interfere with the 145 calculation of a flow solution. For example, elements with high aspect ratios must be subdivided or merged into other elements in the E D M E S H program. The details of the mesh generating and editing processes are proprietary to Golder Associates. MAFIC After the application of the MESHMONSTER and E D M E S H programs, the resulting finite-element mesh and boundary conditions are processed within M A F I C . The first step in MAFIC is the calculation of a steady-state flow solution. Constant head boundary conditions are applied by specifying the head values, based on the application of a linear hydraulic gradient. As in the two-dimensional model, flow and transport are restricted to the fracture planes in all of the three-dimensional models presented here. A n Incomplete Cholesky Conjugate Gradient (ICCG) algorithm is used to solve the equation of flow. Once the flow solution is calculated, and the head values at each node are determined, the fluid velocity within each element is calculated. Particles are introduced to the model domain at specified source nodes within an injection band on the upstream boundary. The introduction of mass to the system is based on the flux at each injection node. This is done according to (Dershowitz et al, 1995, equation 2-48) Ms=Qs-tpm-Cs-(Qe/Qg) [5-1] where Qs = inflow rate at source node (L7T) t = time step size (T) pffi = density of injected fluid (M/L 3 ) 146 Cs = specified source concentration Ms = total mass of solute released (M) Qe = net inflow to the nodal group Qg= sum of positive inflows to nodes in the group The number of particles is given by the total mass released divided by the mass of a single particle. Particles are injected throughout a time step, but for this work, that time step is set to such a small value that all particles are effectively injected at t = 0. Mass transport is implemented by moving particles by advection from element to element. There is an option within MAFIC to include a dispersive component of travel, but in the current model, transport is restricted to advection by setting the coefficients of dispersion to 0. There are additional options within M A F I C to account for reactive and radioactive solutes, but these options are not applied here. Once a particle enters an element the particle wil l select an element boundary through which to leave, based on the relative magnitude of fluxes in the adjoining elements. This procedure is analogous to the complete mixing approach at fracture intersections in two-dimensions. Particles travel from element to element in this manner until they exit at a downstream boundary. The S C M model was solved using a 300 MHz, Pentium II computer, with 128 Meg of R A M . It takes approximately two weeks to solve 1000 realizations of the discrete subdomain along with 1000 realizations of the large-domain discrete network. 5.2.2. Modifications to FRACMAN In the two-dimensional model (Chapters 3 and 4), particles move either forward or backward along fractures in one of the two fracture sets, i.e. particles move in one of the four set-directions. The set-direction is stored at every fracture intersection during the generation 147 of a network, and then as particles move through the network the set-direction is used to determine the beginning, end, and set of each path-length. The addition of set-direction information required substantial modification to the DISCRETE modeling software and dramatically increased the memory requirements for each simulation. A different approach is taken in three-dimensions. In order to avoid increasing the memory required for the three-dimensional network modeling, and to reduce the likelihood of compromising the integrity of the Golder Associates program M A F I C , modifications are limited to the insertion of two WRITE statements. As particles move from element to element, the position of the particle and the residence time in the element are recorded, along with the particle number and the number of the element within which the particle is travelling. Table 5-2 illustrates a limited sample of this output for a single particle, including: (1) the first five particle locations starting at injection along the upstream boundary; then (2) five consecutive steps near the middle of the particle's path; and finally (3) the last five steps the particle takes before the reaching the downstream boundary. The particle starts at the x - 5 boundary at v = 3.8 and z = 2.41. The particle moves through the network and exits at they =5 boundary, at x = -4.3 and z = 3.18. In the first five particle steps there are components of movement in all three directions, while in the middle five steps particle movement is only in the x direction. As the particle exits along the downstream boundary there are again components of movement in all three directions. Because of the limited modifications to the three-dimensional network modeling software, there is no a priori knowledge of the set-direction during particle tracking. It is therefore not known in which fracture set a particle is travelling, and as a result it is not known explicitly which steps correspond to particle movement within a fracture plane and 148 which steps correspond to particle movements from one fracture plane to another. Therefore an additional processing step is required before the sequence of particle locations output from M A F I C can be converted into motion statistics. This conversion process is not trivial, and is discussed further in the next section. 5.3. Motion statistics 5.3.1. Introduction The framework for calculating motion statistics in three-dimensional models is similar to the framework for calculating motion statistics in two dimensions. In both two and three-dimensional modeling, particle movement in discrete subdomains is analyzed statistically to find the average, standard deviation and skew of the velocity, path-length, and orientation of particle travel. In three dimensions, the variables describing the orientation are expanded to include particle motion in the third dimension. Correlations between the orientation variables, velocities and path-lengths are also introduced in the three-dimensional model. In section 5.3.2 the concepts and definitions surrounding particle movement in three-dimensions, and the conversion of these movements into motion statistics, are outlined. The algorithms used for the conversion process are discussed in section 5.3.3. In section 5.3.4 the motion statistics calculated for the three-dimensional base-case model are examined. 5.3.2. Particle movement in three dimensions: concepts and definitions Although for the most part the motion statistics calculated in the three-dimensional modeling are similar to the motion statistics calculated in the two-dimensional model, there are four significant differences: (1) additional motion statistics are necessary to capture the 149 greater range of particle motions within each fracture; (2) the definition of path-length must be refined to accommodate this greater range of motion; (3) the method of calculating the path-length must be refined to reflect both the new definition of path-length; and (4) the concept of the set-direction must be modified. In two-dimensional networks a particle can move either directly towards one end or the other of a linear fracture. Upon reaching an intersecting fracture the particle may either continue along the initial fracture or exit into the intersecting fracture. As a result, the motion of a particle traveling along a fracture in a two-dimensional network can be completely described by: (1) the orientation of the fracture (9); (2) the fracture-set to which the fracture belongs; and (3) the direction in which a particle is moving. In three-dimensional networks particles may travel in a range of directions within a single fracture plane, and then leave that fracture at the intersection with another fracture plane(Figure 5-4). Attributes assigned to particle movement vectors must now include a second term to describe the geometry of movement in three dimensions. Also, movements within a single fracture plane can no longer be described by a simple linear vector between the points at which a particle enters and exits the fracture. Therefore, particle movements are described by the horizontal trajectory angle 9, which is the angle of the particle movement vector with respect to the x-axis in the x-y plane, and the vertical trajectory angle <|>, which is the angle of the particle movement vector with respect to the z-axis (Figure 5-5). Path-lengths in two-dimensional discrete subdomain are defined by the distance a particle travels between the entry and exit points along a fracture (Chapter 3). For the three-dimensional model, particles follow the flow lines within each fracture plane and as a result rarely travel a significant distance in a straight line. Path-lengths in the three-dimensional 150 discrete subdomain are initially defined as the linear distance a particle travels within a single fracture plane before a deflection in 9 occurs. This is referred to as the long-path algorithm. In this way particle trajectories are subdivided into a series of linear steps (Figure 5-6). For particles traveling through sub-vertical fractures, significant changes in 9 correspond to particle movement from one fracture plane to another. Simplifying the path taken by a particle in this manner reduces the amount of movement information captured. A forthcoming section (5.5) discusses models in which the path-lengths are more finely defined based on changes in <j). During particle tracking through the two-dimensional discrete subdomain the set-direction defines both the set of the fracture through which a particle is moving, and the direction of that movement along that fracture. In three dimensions, particles are not limited to travel directly towards one end of a fracture or another but may follow a 360° range of trajectories within the fracture plane. As the fractures in the base-case model are sub-vertical the set-direction is defined by both the set of the fracture within which the particle is travelling, and by the orientation of trajectory in the x-y plane (0). Particles travelling in fractures from fracture-set one are assigned to set-direction one i f they are travelling in the positive x direction (-90° < 0 < 90°), or set-direction two i f they are travelling in the negative x direction (90° < 0 < 270°). Set-directions three and four are similarly defined by particles travelling in the positive y (0° < 9 < 180°), and negative y (180° < 9 < 360°) directions, through fractures from fracture-set two (Figure 5-7). Determination of the fracture set and then the set-direction is discussed in the following section in which the methods by which particle locations at fracture elements are converted into motion statistics are outlined. 151 5.3.3. Calculating motion statistics in three dimensions Particle motions are converted from a series of particle locations and residence times recorded at element boundaries into motion statistics in three steps: (1) particle locations are converted into element to element particle movements; (2) element to element particle movements are summed into path-length vectors; and (3) path-length vectors are assigned to fracture-set directions (Figure 5-8). Element-to-element particle movements are the vectors formed between each of the particle locations recorded during particle tracking in the discrete subdomain. The difference between the spatial locations is used to calculate the length (1) and orientation (0 and <j)  of the particle movement. The residence time and length are then used to calculate the particle velocity during this movement. Once all of the element-to-element movements have been calculated, these movements can be grouped into path-length vectors. Path-length vectors are determined by summing along the element-to-element movements of each particle until there is a significant change in the 9 component of the trajectory of the particle. These break-points define the beginning and end of the path-length vectors. Values of 9, (j), velocity and path-length are recalculated along the full length of each path-length vector, and the vector is then assigned to a set-direction. A sample distribution of the path-length vectors for a single realization of the base-case fracture system is given in Figure 5-9. In order to determine the set of the fracture in which a particle is travelling, and then determine the set-direction for each path-length vector, the relationship between (|) and 9 for each fracture set must be determined. In a model composed of sub-vertical fractures the relationship between 9 and <j> is complex, and there are pairs of 9 and (j) which can occur in 152 more than one fracture set (Figure 5-9a) . This is further complicated by the wrap-around effect where changes in § result in changes in 9 (Figure 5-10). For example, i f the trajectory ((()) of a particle travelling in the positive x direction increases beyond 90° the path-length vector is assigned to the opposite fracture set-direction, and the value of 9 is now 180° from its previous value. It is possible, however, to define the relationship between 9 and (j) for any linear movement along a fracture plane, i f the orientation of the plane is known (e.g. Ragan, 1973). Because the scatter plot in Figure 5-9 is from the base-case fracture system (Table 5-1), the strike and dip of each fracture set is known, and can be thought of as the true strike and dip of the fracture. Path-length vectors have orientations described by 9 and d) which act as a measurements of strike and dip of the fracture, but which are not the true strike and dip unless they are oriented directly along the true strike and dip directions. The relationship between the measured or apparent strike and dip and the true strike and dip is defined by (Ragan, 1973): tan a = tan S • sin /3 ^5-2] where a is the apparent dip, 8 is the true dip, and (3 is the angle between the strike and the apparent dip direction. Noting that p = the strike of the fracture set - 9, and 5 = the dip of the fracture set, then [5-1] can be rearranged so that [5-31 (f) = arctan(tan(setdip) • sm(setstrike — 0)) L J For the base-case model the first fracture set has a strike of 0° and a dip of 80°, and the second fracture set has a strike of 290° and a dip of 70° (Table 5-1). In Figure 5-11 there is 153 an excellent match between the theoretical curves for 9 and § and the vectors, but there are a few data points which fall off of the expected relationship. These data points are idiosyncrasies of the way in which particle locations are output from the M A F I C program. Occasionally particles appear to move in directions that do not seem possible given the constraints of the network. These vectors have minute length and occur to due to round-off error when subtracting two particle positions that should be identical. Since the distances moved are minute these data points can be considered noise. Once the relationship between 9 and <|> is used to determine within which set a particle is moving, it is necessary to further separate vectors into movements which are either forward or backward along a fracture. In Figure 5-9c there is a grouping of the vectors by velocity into three clusters, defined by 9, and in Figure 5-9b a corresponding clustering of the path-lengths. Each vector is assigned to a fracture set based on the relationship between 9 and §, and because 9 can be used to define these clusters, it is used to subdivide movement by set-direction. In Figure 5-12, Figure 5-13, Figure 5-14 and Figure 5-15 all of the vectors seen in Figure 5-9 are sorted into four set-directions. Set-direction one is predominantly against the average hydraulic gradient (Figure 5-12). As a result there are few vectors in direction one, and the shape of the distributions of path-length and velocity are unclear. In the lower part of Figure 5-12 there is some correspondence between velocity and with the highest velocities shown for <|) values around -90° (Figure 5-12e). Vectors from set-direction 2, the set most closely oriented with the applied hydraulic gradient, are presented in Figure 5-13. It is clear that a peak in the number of vectors occurs around 9 = 180°, (j) = 0°, with the frequency of occurrence tapering off in either direction. Peak values occur in the distributions of both path-lengths and velocities (Figure 5-13b,c,d and e). The break between 154 sets one and two is clear, even from Figure 5-9. The break between sets three and four is not as clearly defined in Figure 5-9, although there is some clustering in the velocity as a function of 0. Therefore, although there is some overlap between sets 3 and 4, the break between the two is set at 0 = 180°. In directions 2, 3 and 4 there is some correlation between both path-length and velocity to 0 and (j). The maximum and average velocities appear to be generally higher (i.e. the -logio of velocity is lower) towards (j) = 0°, with peak values at 0 = 90°, 180° and 270° for all set-directions. For the path-lengths, the longest values appear to occur coincident with the most frequently occurring 0 and <j) values, which vary from set to set. In section 5.4 models which include these apparent correlations are examined. Despite the complexity introduced in the three-dimensional models by the need to convert element-to-element particle movements to vectors, it appears that the algorithms used are effective. Although particle movements sometimes fall outside of the clear definition of 0 and (j), these movements represent a very small amount of computational noise in the larger signal. In the following section the distributions of 0, <j), path-length and velocity are examined. 5.3.4. Motion statistics under the long-path model In three-dimensional modeling motion statistics are parameterized in a similar manner as was done in two-dimensional modeling. The parameters calculated for each of the four set-directions are: (1) the directional choice, i.e. the number of times particles move in each of the four fracture set-directions; (2) the directional choice based on the previous direction of travel: (3) the mean, standard deviation and skew of the velocity; (4) the mean, standard 155 deviation and skew of the path-length; (5) the mean value of 9; and (6) the mean value of <|> (Table 5-3). Histograms are also calculated for 0, <j), velocity and path-length. The statistics in Table 5-3, Figure 5-16, Figure 5-17, Figure 5-18, and Figure 5-19 are based on 1000 realizations of the discrete subdomain for the base-case fracture system. It is clear from Table 5-3 that the most common travel set-direction is direction 2, as 48% of all particle motions are in this direction. Direction one is the least frequently chosen set-direction as only 3% of particle motion is in this direction. As expected, the most frequent travel directions are the set-directions closest to the orientation of the applied hydraulic gradient. A substantial number of movements are in direction 3 (37%), with the remaining 11% of the movements in direction 4. Examining the directional choice probabilities conditioned on the previous travel direction it is clear that particles are not likely to travel in the same direction on consecutive steps, but will instead choose between the remaining directions based again on how closely the set-directions are aligned to the average gradient direction. As in the two-dimensional modeling, grouping the motion statistics by set-direction is used to capture the anisotropic character of dispersion caused by the geometry of the fracture networks. Although average values are calculated for 9 and <|), these parameters are not sufficient to describe the distribution of motions within each of the four travel directions. In Figure 5-16 histograms of 9 are plotted for each of the four travel directions. Each histogram peaks at or near the average value of 9. Although histograms in all four set-directions are unimodal, the shapes of the distributions vary substantially from direction to direction, and are difficult to reproduce with a single type of continuous model. Therefore, in the three-dimensional continuum models, histograms are used for 9. 156 As shown in Figure 5-17, <j) values are less evenly distributed than 9 values. In directions 2 and 3 histograms of (j) are unimodal, in directions 1 and 4 histograms are bimodal with sharp truncations at the vertical angle (+/- 90°). The truncation occurring at +/- 90° marks the transition from the forward to backward travel direction along a fracture. In order to capture the complexity of the (j) distribution, histograms are used to model (j) for all of the three-dimensional S C M models in the present work. It is possible to rearrange the definitions of the set-directions to reduce or eliminate the abrupt truncations of in the distributions of 9 and (j), but the present definition provides a simple and robust method of separating vectors into sets. Histograms of path-length from the three-dimensional model follow a pattern similar to histograms of path-length in the two-dimensional model (Figure 5-18). However, the shapes are not perfectly defined by smooth curves as there is a slight roughness in the shape of each fracture set. This roughness suggests that although the way in which path-length vectors are assigned to fracture set-directions is effective, it is not perfect. The shape of the histograms suggests that the gamma distribution, used extensively for path-length in two-dimension, might be an effective model in three-dimensions. The range of path-lengths in each direction is correlated to how closely a given set is oriented to the direction of the applied hydraulic gradient. Direction 2, which is aligned closest to the hydraulic gradient, has the greatest range of path-lengths, including the maximum path-lengths overall. Direction 3, which is the next closest to the gradient orientation has the next longest maximum path lengths, and the downward trend in the maximum path-lengths and closeness to the hydraulic gradient continues through directions 4 and 1. 157 Velocity histograms are unimodal in all directions, although the distributions are slightly uneven in directions one and four (Figure 5-19). Unevenness in the velocity histogram for the first set-direction is not as significant as unevenness in the fourth set-direction as there is a significantly lower probability of particles moving in direction one. As it is the logio of velocity that is plotted, it would appear that log-Normal or log-Gamma models of velocity might be as effective in three-dimensional modeling as they were in two-dimensional modeling. The regular shape of the velocity histograms suggests that the algorithm for assigning the set-directions provides a reasonable grouping of the path-length vectors. In the two-dimensional model it was clear that the selection of statistical models used to represent velocity and path-length was critical to the successful implementation of the SCM. Specifically, how well an S C M method performs under the application of a given statistical model of velocity or path-length is critical, and not the fit of a given model to the histogram of the data from the discrete subdomain. Evaluation of the relative effectiveness of each of the statistical models is therefore left to Chapter 6 in which moments for S C M models are compared to moments for discrete models. In three dimensions, both Gamma models and histograms are used to model the distribution of path-lengths. Histograms, Normal distributions and three-parameter Gamma distributions are used to represent the distribution of the velocities. 5.3.5. Summary For the three-dimensional base-case model the S C M method can effectively convert particle movements recorded as locations and residence times, to element-to-element movements, and subsequently group movements into path-length vectors. These path-length 158 vectors are assigned to a fracture set-direction, and then statistics similar to those used in two-dimensional modeling are calculated. The shape of the velocity and path-length histograms indicates that they can be modeled using similar statistical models as were applied in the two-dimensional modeling. Distributions of the horizontal and vertical trajectories (Band <j>) are not as easily reproduced with continuous models so that only histograms are used for the orientation parameters in the S C M . The distributions of the motion parameters indicate that the algorithm for grouping path-length vectors by set-direction is effective. In the following section the relationships between the parameters is examined. 5.4. Interdependence of motion statistics 5.4.1. Introduction In section 5.3, the collection of motion statistics describing the path-length vectors is discussed. Scatter plots of 9, <j), velocity, and path-length for the base-case fracture system suggest that the values of these parameters follow similar trends, and are potentially correlated. In this section the relationships between the values of d) and velocity (section 5.4.2), and d) and path-length (section 5.4.3) are examined. In section 5.4.4 the relationship between velocity and path-length is explored. These relationships are then applied in S C M models in Chapter 6. 5.4.2. Velocity as a function of the vertical trajectory (<))) In many of the scatter plots it appears that there is a correlation between the velocity at which a particle travels, and the vertical trajectory, (j). For example in Figure 5-13e it appears that a subtle peak or most frequent value of velocity occurs at around (j) = 0°, and that 159 the values of velocity decrease away from this peak value. In Figure 5-20, the average logio velocity (p.v) for each of the four travel directions is plotted as a function of (j). In direction 2 the highest velocities are seen at (j) = 0° and as d> increases towards 90°, p,v decreases. For (j) values below d> = 0° (i.e. 270° to 360°) this trend is not as consistent. Similar trends are seen in direction four, in which the highest values of u.v occur at around d> = 0°, and then decrease away from this orientation. In direction one the opposite is true as the highest velocities occur around d> = ± 90°, and velocities decrease towards § = 0°. This pattern is consistent with the fact that the applied hydraulic gradient is directly opposite to the d) = 0° orientation in set-direction one. Overall it is clear from Figure 5-20 that there is a relationship between mean velocity and vector orientation. In Figure 5-21 the standard deviation of velocity (o~v) in each of the four travel directions is plotted as a function of (j). The relationship between a v and d> is more complicated than the relationship between u.v and d>. In direction 2 the largest standard deviation of velocities occurs around ± 45°, with o~v decreasing for d) values away from 45°. The lowest range of velocities occurs at around d) = 0°, in direction 2. In directions 3 and 4 the relationship between a v and d) is less clear, although it can be seen that there is a peak value of d> at which either the highest or lowest values of o~v occur. There is some correlation between CTv and (j) but there does not appear to be a simple linear correlation between \xv and G v to (j). However, models are tested in the chapter 6 that use histograms to correlate particle velocity in the continuum to dh 160 5.4.3. Path-length as a function of the vertical trajectory (<()) Relationships exist between the distance a particle will travel within a fracture and the <j) orientation at which it travels. In direction 2 the relationship is clear; the average travel path-length (u,/) is longest around § = 0° and decreases as § deviates from 0° (Figure 5-22b). The standard deviation of path-length (a/) in direction two reaches a maximum around § — 0°, and then decreases toward 90° (Figure 5-23b). For set-directions 3 and 4, u./ peaks at around §=±45°, and decreases with orientation away from 45°. The longest average paths overall are those in direction 2 at (j) = 90°, the orientation closest to the applied hydraulic gradient. At (j) angles in which the mean length is greatest, the standard deviation of length is also greatest for all set-directions. However, it is only for set-direction 2 that the p,/ and CT/ both peak at (j) = 0°. There is a general trend towards having peak values of p v, av, u./ and o/ at either § = 0° or at § = 45°. These peaks occur because of the process by which particle transport is optimized through the fracture system. Because of the orientation of the applied hydraulic gradient, solute velocities and flow-volumes are generally highest in set-direction 2 at around (j) = 0°. Particles preferentially travel in the highest flow-paths available so that the longest path-lengths and highest velocities occur at this orientation. Particle velocities in directions three and four are generally slower than those in direction 2 because fractures in these sets deviate from the gradient orientation. When particles are forced to travel in these less optimal directions the longest travel paths occur around (j) = 45°. If we examine the standard deviations of the path-lengths as a function of <j) (Figure 5-23), similar peak values are seen at either 0 or 45°. Particle travel in set-direction one is the least frequent, especially at (j) = 161 0°, as this orientation is directly opposite to the applied hydraulic gradient. As a result the shortest path-lengths and slowest velocities overall occur at this orientation. The relationship between u./, a/, and d) is more consistent than the relationship between u.v, a v and d> as the peak values of u.v and a v . do not necessarily occur at coincident angles of d). It is clear that there is some relationship between d> and velocity, path-length and velocity, and 9 and the velocity, but there are no two parameters which are consistently correlated for all of the set-directions. There is a strong correlation between p,/ and a/, and although there is a correspondence between values of a/ and av for individual set-directions, these relationships are not consistent from one set-direction to the next. For example when c„ increases as a function of <|> in direction 3, o~/ also increases. However in direction two as d) deviates from 0° cr/ decreases, while av increase, peaks at <j) = ± 45 °, and then decreases again. Although the relationships between velocity, path-length and (j) are not simple, they can be captured using histograms. S C M models using correlations between velocity and <j), and path-length and d> are examined in chapter 6. 5.4.4. Velocity as a function of path-length With the two-dimensional S C M model, it can be seen that there is a relationship between the fluid velocity within a fracture and the distance a particle travels along that fracture (Parney and Smith 1995). In sections 5.4.2 and 5.4.3 a relationship was shown to exist between the velocity, path-length and (|). In this section the direct relationship between velocity and path-length is examined. In Figure 5-24 the relationship between the mean velocity for each of the four set-directions is plotted. Note that the mean velocity plotted is 162 actually the -logio of velocity so that the highest velocities are the lowest values along the y axis in the figure. In directions two and three, the relationship between velocity and path-length is consistent with the two-dimensional models, where the mean velocity increases as a function of the path-length. In direction four, however, there is little change in the average velocity as a function of the path-length. In direction one there is considerable scatter in the distribution of velocities and as a result no discernable correlation between velocity and path-length is seen. In Figure 5-25 the standard deviation of velocities is plotted as a function of the path-length. The relationship between the range of velocities and the path-length is clearly defined: the highest range of velocities occurs at the shortest path-lengths, and the range of velocity decreases for the longer path-lengths. This is the same pattern seen in the two-dimensional modeling. Models are applied in Chapter 6 in which the mean and standard deviation of velocity are correlated to the path-length. 5.4.5. Summary In sections 5.4.2, 5.4.3 and 5.4.4 the relationships between velocity, path-length 6 and (j) are examined. It is seen that there are relationships between velocity and (j), relationships between path-length and <j), and also relationships between velocity and path-length. The highest mean velocities and longest paths are seen at trajectories closest to the orientation of the mean hydraulic gradient. The standard deviations of both velocity and path-length also tend to be highest at these gradient orientations. Although strong trends are apparent for individual parameters and set-directions, these trends are not consistent across all set-directions. S C M models incorporating the relationships between velocity, path-length and § are tested in Chapter 6. 163 A l l of the statistics in the preceding sections are based on the path-length vectors defined as the continuous distance a particle travels across a fracture between significant changes in 9. In the next section statistics are calculated using shorter definitions of the path-length which take into account changes in <j) orientation of particle travel. 5.5. Path-length models including a cutoff in vertical trajectory (<j)) 5.5.1. Introduction In section 5.3 an approach was described in which particle motions were converted into linear vectors based on significant changes in the 9 orientation of the particle trajectory. Changes in the <j) orientation of the particle trajectory along the fracture plane are discounted in this approach, resulting in a loss of information about particle movement. As a result, motion statistics calculated under this approach may not be a sufficient to represent particle movement through discrete networks. In this section, an approach is described where path-length vectors are further subdivided based on changes in the <j) orientation of the particle trajectory. This subdivision increases the number of steps and decreases the average path-length, resulting in changes to both the distribution of path-lengths and the frequency with which each set-direction is chosen. The minimum required change in (j) acts as a break-point between path-length vectors, and is referred to here as the § cutoff. A (j) cutoff of 5° causes the algorithm that determines path-length to be more sensitive to changes in the direction of particle movement than a <j) cutoff of 10°, so that decreases in the value of the § cutoff are referred to as increases in the <j) sensitivity. 164 5.5.2. Distributions of path-length vectors under a § cutoff In the first example, element-to-element particle movements are grouped into path-length vectors defined by breaks in the <j) orientation of a particle's trajectory greater than 10°. Any breaks in the 0 orientation continue to define new path-length vectors. Histograms of 0 and (j) in each of the four set-directions under a 10° d> cutoff are plotted in Figure 5-26 and Figure 5-27. These figures are comparable to Figure 5-16 and Figure 5-17 in which the histograms of 0 and d) are plotted when no d) cutoff is applied (long-path statistics). There does not appear to be any substantial difference in the shape or magnitude of 0 and (j) histograms between the 10° cutoff and long-path statistics. The largest changes occur in direction one due to the low frequency of particle movement in that direction. None of these changes appear to significantly alter the mean or standard deviation of 0 or § In Figure 5-28 histograms of the path-lengths calculated using a d> = 10° cutoff are plotted. The histograms of path-length are smoother under a 10° cutoff than without the application of a cutoff (Figure 5-18). The maximum path-length is reduced under the § = 10° cutoff, especially in directions 2, 3, and 4. In Figure 5-29 histograms of velocities under a 10° cutoff are plotted. The introduction of the d> cutoff does not significantly change the distribution of the velocities of the path-length vectors (Figure 5-29), from the distribution of velocities without the cutoff (Figure 5-19), although there is a slight smoothing of the distributions for directions 1 and 4. Histograms of velocity and path-length as a function of (j), and velocity as a function of path-length are almost identical under the 10° cutoff and long-path model, and are not included here. Only the distribution of the path-lengths is affected, and not the relationships between velocity, path-length and <j). 165 5.5.3. Motion statistics under a § cutoff In Table 5-4 comparisons are made between using long-path statistics, statistics calculated using a (j) cutoff of 10°, and statistics calculated using a § cutoff of 5°. Increasing the (j) cutoff sensitivity increases the proportion of steps in direction 2 from 48% to 56 % to 60%, while the proportion of travel in the other three set-directions decreases. Because these directional choice parameters are ratios of the total number of steps taken, the number of steps taken in all directions must increase and only the relative frequency of travel in directions 1, 3, and 4 decreases. Therefore, most of the subdivision of paths occurs in direction 2, which, having the longest paths initially, has the greatest potential for subdivision. Under the application of a (j) cutoff, there is some variation in the velocity parameters, although the mean and standard deviation of velocities are essentially unchanged. With increasing cutoff sensitivity there is a decrease in the mean path-length in set-direction 2 from 1.8, to 0.9, to 0.6. The largest changes in path-length occur in set-direction 2, consistent with the largest changes in directional choice occurring in set-direction 2. As the (j) cutoff sensitivity is increased, there is a reduction in the standard deviation of the path-length in each of the set-directions. Changes in the (j) cutoff do not significantly change the mean value of 9 in set-directions 2 and 3. Changes in the values of 9 in set-direction 1 occur due to the limited frequency of travel in this direction. There is a consistent shift in the average 9 orientation in set-direction 4 with changes in the § sensitivity. Shifts in the mean value of 9 are due to changes in the shape of the distribution rather than shifts in the peak value (compare Figure 166 5-16d to Figure 5-26d). It is not clear what causes this to occur but it is coincident with a reduction in the number of vectors in set-direction 4 with a <j) below 0°. Changes in the 0 distribution are a consequence of a more detailed sampling of particle movements in set-direction 4. Similar changes occur in the average value of d), with the largest changes in § occurring in directions one and four. 5.5.4. Summary The introduction of the d> = 5° and 10° cutoffs has primarily changed the directional choice and path-length parameters. These changes are predictable: With the increased sensitivity to changes in the orientation of the vectors, particle movements are captured using an increased number of steps, resulting in a decrease in the average path-length and an off-setting increase in the probability of travelling in set-direction 2. The standard deviation of the path-lengths in all directions is reduced. There are some slight changes to the velocity, 0 and (j) parameters with changes in the § cutoff. The effect of changing all of the statistical parameters becomes significant when the resulting statistics are applied under the varying S C M models. These effects are described in Chapter 6. 5.6. Assessment of parameter variability 5.6.1. Introduction In section 5.3 the movements of particles through a three-dimensional discrete subdomain was captured statistically by parameters describing the velocity, path-length and orientation of particle motion. In this section the variability of these parameters is examined (1) as a function of the number of networks generated within the discrete subdomain, and (2) with the scale of the discrete subdomain. Understanding the variation of the parameters 167 between networks is critical because it is necessary to run enough network realizations of a sufficient size that the motion statistics reflect the range of particles motions within the fracture system. Also it is necessary that the movement of particles at the scale of the distance between fracture intersections does not change significantly when the size of the fracture domain is changed. In section 5.6.2 the convergence of motion statistics as a function of the number of networks in the discrete subdomain is discussed. In section 5.6.3 the variation of parameters as a function of domain size is presented. 5.6.2. Variation between networks In the modeling of solute transport through fracture networks, complexity increases substantially from two-dimensional modeling to three-dimensional modeling. As a result the generation of three-dimensional networks and finding solutions of the flow equations is a far more computationally intensive and time-consuming task. However, this increase in complexity also increases the number of fracture intersections per realization. As a result a much larger number of path-length vectors are generated in a three dimensional network realization than in a two-dimensional network realization. Therefore, statistical parameters converge at a much higher rate when averaged over three-dimensional networks. In Figure 5-30a and Figure 5-30b, the convergence of the mean and standard deviation of the path-length as a function of the number of networks is plotted. Although the length parameters vary for low numbers of realizations, by approximately 150 networks the average values of the mean and standard deviation of velocity have converged to stable values. There is some slight variability in the average path-length in set-direction 1 up to 400 realizations, but the magnitude of the variation does not appear to be significant. 168 In Figure 5-31 the variation of the mean and standard deviation of velocity as a function of the number of networks is plotted. For set-directions 2, 3, and 4 the velocity parameters converge within 150 networks. Direction 1 has the highest variability and there continues to be some slight variation in the mean velocity up to 700 networks. There is a larger variation in the standard deviation of the velocity in direction 1, and this parameter does not converge until approximately 600 networks have been run. Averaging the velocity parameters over 200 networks should provide a sufficient estimate of the parameters for the base-case model. Further variations occur in both the mean and standard deviation of velocity in direction one after 200 networks. However, these variations are slight, direction one is chosen infrequently and the path-lengths in direction one are short so that the influence of any variation should be minimal. Directional choice parameters in all four set-directions converge quickly, stabilizing within 25 networks (Figure 5-32). 5.6.3. Variation of parameters with scale In order for the statistical continuum method to work it is important that the statistics describing particle motion at the scale of the distance between fracture intersections do not change significantly as a function of the size of the model domain. Discrete networks are run varying the size of the domain from an 8 m by 8 m by 8 m cube, t o a l 2 w b y l 2 m b y l 2 m cube t o a l 6 m b y l 6 m b y l 6 m domain. Motion statistics averaged over 1000 realizations of the domain are compared to statistics for the l O m b y l O m b y l O m domain used previously. The l O m b y l O m b y l O / n domain should be sufficiently large, as the mean radius of the fractures for both sets is 1.4 m, and the largest mean path-length under the long-path motion statistics is 1.8m. 169 The frequency with which particles move in set-direction 2 decreases from 49.7%, to 48%, to 47%, to 46% with increasing domain size (Table 5-5). In set-direction 3, the frequency of particle movement decreases from 38%, to 37.7%, to 37.3% to 36.7%, and in direction 4, the frequency varies from 9%, to 13%, to 12% to 13.6% as the size of the domain increases. The magnitude of these changes does not appear to be large enough to be significant. Although the probability of travelling in direction two decreases, the average length of the path in direction 2 increases from 1.76 to 1.80 to 1.81 to 1.84 m with an increase in the size of the model domain. Again in direction three, as the size of the domain increases the probability of travelling in this direction decreases, but the length of the paths increases from 0.95 to 1.04 to 1.06 to 1.08 m. However, in direction one and four both the mean path-length and frequency of travel increase with the size of the domain. The standard deviation of the path-length in direction two increases from 1.39 to 1.47 to 1.49 to 1.53 m. In direction four the standard deviation of the path-length expands from 0.504 to 0.548 to 0.580 to 0.625 m, an increase of approximately 20%. With an increase in the model domain the average velocity in direction two increases from 2.16 to 2.18 to 2.20 to 2.21 (-logio m/s). As these values are the -logio of velocity, this increase in values represents a decrease in the average velocity from the smallest model to the largest. The velocity in direction three, however, is consistently around 2.4 (-logio m/s) for all size of domains. The velocity in direction one decreases somewhat between the 8 m domain to the 12 m domain, but does not change between the 12 m and 16 m domains. The velocity in the fourth direction varies slightly, but there is no systematic change. There is an 170 increase in the standard deviation of velocity in direction two, but in direction three the standard deviation of the velocity is relatively unaffected by the size of the model domain. Although there are systematic changes in the velocity and path-length, the magnitude of these changes is relatively small, and the changes in path-length tend to be offset by the changes in directional choice. The magnitude of these changes does not, however, suggest that the S C M method cannot effectively model mass transport in domains larger than the discrete subdomain. In Chapter 6 moments for large-domain discrete network models are compared to moments for large-domain S C M models. 5.7. Continuum models in three dimensions 5.7.1. Introduction In the previous sections of Chapter 5 the method by which motion statistics are calculated was outlined. In this section the algorithms by which these motion statistics are used to generate particle motion in the continuum domain are described. These algorithms are then tested in chapter 6. The application of these statistics in a three-dimensional continuum is very similar to the application of motion statistics in two-dimensions. First a flow solution within the model domain is found. Particles are then introduced into the model domain by an injection process, and each step of a particle is generated from distributions of 0, (j), velocity and path-length. Each parameter may be generated independently, or may be correlated to a previously generated parameter. The simplest flow solution is to assume that the magnitude and orientation of the mean hydraulic gradient are the same in both the discrete subdomain and continuum domain. A constant gradient allows a single suite of motion statistics to apply at every point within the continuum domain. It is possible to expand the method so that more complex flow 171 solutions can be modeled, using the local hydraulic gradient to determine the appropriate suite of motion statistics to be applied at each location within the model. 5.7.2. Injection Particle movement in the continuum model is initiated by the selection of an injection location. Particles are introduced across a square area on the upstream boundary by setting the x location equal to the x location of the boundary, and then generating the y and z locations from independent, uniform distributions. As in the two-dimensional models the injection band for the continuum domain is slightly smaller than the injection band for the large-domain discrete model. For most of the examples in Chapter 6, a 6 m by 6 m source band is used for the S C M models, and a 7 m by 7 m source band used for the larger domain discrete models. The implications of differences in the size of the injection band are discussed further in Chapter 6. 5.7.3. Directional choice algorithms Choosing a set-direction initiates each random-walk step. Four directional choice algorithms have been applied in the three-dimensional S C M approach: (1) the unconditional algorithm, where the probability of travel in any direction is based on the cumulative probabilities (row 2 in Table 5-3). For example, in Table 5-3, the probability of a particle travelling in set-direction two is 0.48, so that in the first algorithm a particle will travel in set-direction 2, 48% of the time, independent of the previous travel direction; (2) the conditional probability algorithm, where the probability of travelling in any direction is conditioned on the previous direction traveled by using the directional choice statistics grouped by the previous travel direction (rows 9 to 12 in Table 5-3). Under the statistics for the base-case model, a particle travelling in set-direction 2 has a 6% probability of taking a subsequent step 172 in set-direction 2. However a particle travelling in set-direction 3 has an 82% probability of taking its next step in set-direction 2. The third and fourth directional choice algorithms are based on the unconditional directional choice used in the first algorithm with some movements disallowed; (3) the no reversal/no repeat algorithm, is based on the cumulative probabilities, with movement restricted by not allowing particles to either reverse direction or repeat a direction in sequential steps. A particle travelling in set-direction 2 cannot travel in set-direction 2 a second time, nor can the particle reverse direction and travel in set-direction 1. Under the first algorithm a particle would have a 48% probability of travelling in set-direction 2, but under the third algorithm a particle which has just traveled in set-direction 2 has zero probability of travelling in either set-direction 1 or 2. Particles select either set-direction 3 or 4 based on the relative probability of travelling in each of those directions. (4) the no reversal algorithm, based on the cumulative probabilities but with particles not allowed to reverse directions. The choice of a set-direction strongly influences particle motion, as all of the other statistical parameters from which each step in the continuum is generated are based on the set-direction chosen. 5.7.4. Orientation Orientation is determined by two angles, 9 and d>. There are two algorithms by which orientation is chosen: In the first algorithm 9 and (j) are selected by randomly sampling from histograms of the distribution of these two parameters, and the values of 9 and <j) are generated independent of one another. However, the relationship between 9 and d) is known for particles moving through a fracture plane of known orientation (section 5.3.3). The second orientation model makes use of this relationship by first generating a value for 9, and then calculating d> from 9 using Equation 5-3. 173 5.7.5. Velocity and path-length Several of the continuum approaches allow velocity and path-length to be generated independently. Velocity and path-length are both modeled using histograms, normal distributions and three-parameter gamma distributions. As noted previously, it is the -loglO of velocity that is represented by these distributions so that normal and gamma distributions of velocity are effectively lognormal and log-gamma distributions. Sampling from these distributions is the same in three-dimensional modeling as it was in two-dimensional modeling (section 3.5). Velocity can also be correlated to § or path-length, and path-length can be correlated to § (section 5.4). The method by which velocity and path-length are correlated to § is similar to the method by which velocity was correlated to path-length in the two-dimensional model. For each bin of the <j) histogram in the discrete subdomain, a mean and standard deviation of velocity, and a mean and standard deviation of path-length are calculated. Then at each step in the continuum a § value is sampled from a histogram, and the values of u.v, a v , u./ and 07 corresponding to that § value are used to generate the velocity and path-length. Several correlation methods are tested in chapter 6. Velocity is correlated to path-length using the same algorithm adopted in the two-dimensional model (section 3.5): A path-length is chosen from a histogram of path length, and a mean and standard deviation of velocity corresponding to that path-length bin are generated. A velocity is then generated using a Normal model based on the path-length correlated parameters. 5.7.6. Nomenclature for SCM models in three dimensions Each of the parameters used to create particle motion in the continuum domain, i.e. 6, <j>, velocity and path-length, can be generated using a number of algorithms. The system of 174 nomenclature introduced in section 3.6.2 for two-dimensional models is expanded here to include the parameters introduced in three dimensions. As before, S C M modeling approaches are identified by the velocity and path-length models used: If a histogram is used, the name combines the capital " H " with the parameter chosen, e.g. H(v) and/or H(l). A normal distribution is designated by an " N " , and a gamma distribution by a " G " . If the length or velocity is correlated to § this is included within the brackets. For example in N(v,d))G(l,(|)), both velocity and path-length are influenced by the selection of <)), with velocity generated from a normal distribution and path-length generated from a Gamma distribution. With the four different directional choice models, the two orientation models, the five velocity models and the four path-length models, there are potentially 160 combinations of algorithms for a given suite of motion statistics. Statistics can be calculated by varying the (j) cutoff, further multiplying the number of S C M models available for a single network system. 5.8. Summary The calculation and modeling of motion statistics in the three-dimensional models are for the most part an expansion of the approach used in two dimensions. Motion statistics are gathered based on particle movements through a discrete subdomain consisting of multiple realizations of three-dimensional networks. One thousand cube-shaped subdomains measuring 10 m along each side were sufficient to generate motion statistics for the base-case fracture system, in which the mean equivalent radius was 1.4 m. A n additional step is introduced in three-dimensional modeling in order to group element-to-element particle movements into path-length vectors, and assign these vectors to set-directions. Motion statistics are expanded from the two-dimensional statistics to include both horizontal (9) and vertical (())) angles of trajectory. 175 It was demonstrated that there are relationships between velocity and (j), relationships between path-length and §, and also relationships between velocity and path-length. The highest mean velocities and longest paths are seen at trajectories closest to the orientation of the mean hydraulic gradient. The standard deviations of both velocity and path-length also tend to be highest at these gradient orientations. Although strong trends are apparent for individual parameters and set-directions, these trends are not consistent across all set-directions. There are systematic changes in the velocity and path-length parameters, as a function of the size of the discrete subdomain. The magnitude of these changes is relatively small, and the changes in path-length tend to be offset by the changes in directional choice. As a result the occurrence of these changes does not suggest that the S C M method cannot effectively model mass transport in domains larger than the discrete subdomain. The introduction of the § = 5° and 10° cutoffs in the path-length algorithm primarily changed the directional choice and path-length parameters. There are also some slight changes to the velocity, 6 and § parameters with changes in the § cutoff. Particle movement in the three-dimensional continuum domain is similar to particle movement in the two-dimensional continuum domain with the addition of sampling from the <j) distribution for movement in the third dimension. The algorithms used to generate particle movements are identified by a simple system of nomenclature. In the following chapter the S C M method is evaluated in three-dimensions by comparing spatial moments for the varying S C M approaches to the average moments for large-domain discrete models. In the following chapter S C M model results in three-dimensions are presented. 176 Table 5-1 Generation parameters for three-dimensional base-case fracture system. Parameter Fracture set one Fracture set two Generation region 21m x 21m x 21m 21m x 21m x 21 m Number of fractures 1000 1000 Fracture Model Enhanced Baecher Enhance Baecher Generation Mode Centers Centers Truncation Mode Off Off Number of Sides 4 4 Pole: (trace, plunge) 90,-10 -20,20 Pole distribution Constant Constant Size distribution Lognormal Lognormal Mean Radius 1.4 m 1.4m Standard deviation 0.8 0.8 Aspect Ratio 1 1 Transmissivity Distribution Lognormal Lognormal Mean Transmissivity 1.0e-06m2/s 1.0e-06 m2/s Standard deviation 5.0e-07 5.0e-07 Table 5-2 Sample of a single particles movements as output from M A F I C . Residence time is in seconds. Particle locations are in meters. Particle # Element it-Residence Time Vx x Location y Location z Location 1 391 2 00E-01 0.00E+00 0.00E+00 5.00E+00 3 28E+00 2 41E+00 1 391 2 65E-01 -2.88E-03 -2.24E-03 5.00E+00 3 28E+00 2 41E+00 1 441 1 32E+02 -2.96E-03 -2.11E-03 4.77E+00 3 58E+00 2 71E+00 1 447 3 31E+01 -2.29E-03 -2.75E-03 4.72E+00 3 65E+00 2 78E+00 1 402 1 05E+02 2.75E-03 -2.29E-03 4.54E+00 3 89E+00 3 02E+00 "l" 9747 1 61E+01 2.57E-03 9.12E-03 -2.80E-01 3 97E+00 2 79E+00 1 9758 1 56E+01 6.87E-03 6.34E-03 -4.25E-01 3 97E+00 2 78E+00 1 9776 3 79E+00 -7.80E-03 -6.12E-03 -4.62E-01 3 97E+00 2 78E+00 1 9768 1 85E+01 2.04E-03 -9.57E-03 -6.42E-01 3 97E+00 2 76E+00 1 17956 4 26E+00 2.14E-03 -9.52E-03 -6.84E-01 3 97E+00 2 76E+00 "i" 18213 1 23E+00 1.70E-04 -1.03E-02 -4.26E+00 4 93E+00 3 14E+00 I 18423 2 52E+00 1.70E-04 6.96E-03 -4.26E+00 4 94E+00 3 15E+00 I 18418 2 14E+00 3.44E-03 8.67E-03 -4.27E+00 4 96E+00 3 16E+00 I 18216 5 21E-03 -2.44E-03 6.18E-03 -4.27E+00 4 96E+00 3 16E+00 I 18417 2 86E-02 -4.63E-03 -6.18E-03 -4.27E+00 4 96E+00 3 16E+00 i 18214 7 47E+00 4.33E-03 -6.07E-03 -4.30E+00 5 00E+00 3 18E+00 177 Table 5-3 Motion statistics for three-dimensional base-case network at d) = 0°. Set-direction 1 2 3 4 Directional Choice 0.029 0.481 0.377 0.113 Velocity: Mean (-logio m/s) 2.65 2.18 2.42 2.64 V : Standard deviation 0.65 0.24 0.26 0.37 V:Skew -2.37e-02 -1.24e-03 3.7e-04 -1.68e-03 Length: Mean (m) 0.24 1.80 1.04 0.49 L: Standard deviation 0.33 1.47 0.96 0.55 L:Skew 1.02 0.32 0.43 0.71 Direction after 1. 0.019 0.368 0.468 0.146 Direction after 2. 0.049 0.057 0.717 0.177 Direction after 3. 0.02 0.818 0.050 0.112 Direction after 4. 0.02 0.684 0.261 0.032 Mean 0: -16.9 -179.5 124.6 -100.0 Mean d): 36.0 1.6 30.1 39.8 178 Table 5-4 Motion statistics for three-dimensional base-case model under varying d> cutoffs. Parameter Path-Length Algorithm Set-Direction 1 2 3 4 Directional Choice Long Path 0.029 0.481 0.377 0.113 4> = 10° 0.018 0.558 0.355 0.075 <j> = 5° 0.008 0.602 0.331 0.059 Velocity: Mean Long Path 2.65 2.18 2.42 2.64 (-logio m/s) <|> = 10° 2.71 2.18 2.43 2.65 <|) = 5 0 2.75 2.21 2.46 2.67 V: Standard Long Path 0.65 0.24 0.26 0.37 deviation <j) = 10° 0.48 0.23 0.26 0.35 § = 5° 0.42 0.24 0.27 0.34 V:Skew Long Path -2.37e-02 -1.24e-03 3.70e-04 -1.68e-03 <) = 10o -1.07e-02 -2.13e-05 -3.77e-04 -4.17e-03 j) = 5° -4.82e-03 9.65e-04 4.09e-04 -2.5e-03 Length: Mean (m) Long Path 0.24 1.80 1.04 0.49 4> = 10° 0.21 0.91 0.60 0.34 <j> = 5° 0.20 0.60 0.43 0.30 L: Standard Long Path 0.33 1.47 0.96 0.55 deviation <) = 10o 0.25 1.00 0.64 0.41 d> = 5° 0.22 0.68 0.46 0.35 L: Skew Long Path 1.02 0.32 0.43 0.71 <|> = 10° 0.82 0.78 0.71 0.91 (j) = 5° 0.71 0.92 0.77 0.89 Mean 6 Long path -16.9 -179.5 124.6 -100.0 (j) = 10° -24.2 -178.9 125.4 -105.3 (j) = 5° -27.7 -179.1 125.8 -109.1 Mean d>: Long Path 36.0 1.6 30.1 39.8 d> = 10° 47.7 3.0 31.3 43.7 (j) = 5 0 51.5 2.1 32.2 46.4 179 Table 5-5 Motion statistics calculated under the long path-length definition for varying sizes of model domains. Subdomains used for all three-dimensional models are 10m by 10m by 10m. Parameter Domain Size Set-direction 1 2 3 4 Directional Choice 16m x 16m x 16m 0.035 0.462 0.367 0.136 12m x 12m x 12m 0.018 0.473 0.373 0.123 10m x 10m x 10m 0.029 0.481 0.377 0.113 8m x 8m x 8m 0.024 0.498 0.383 0.095 Velocity: Mean: 16m x 16m x 16m 2.70 2.21 2.43 2.63 (-log io m/s) 12m x 12m x 12m 2.70 2.20 2.42 2.64 10m x 10m x 10m 2.65 2.18 2.42 2.64 8m x 8m x 8m 2.64 2.16 2.42 2.64 V : Standard deviation 16m x 16m x 16m 0.62 0.24 0.27 0.36 12m x 12m x 12m 0.61 0.24 0.27 0.38 10m x 10m x 10m 0.65 0.24 0.26 0.37 8m x 8m x 8m 0.64 0.24 0.25 0.37 V:Skew 16m x 16m x 16m -2.76e-02 -0.81e-03 4.10e-04 -0.28e-03 12m x 12m x 12m -2.29e-02 -1.10e-03 1.58e-04 -1.75e-03 10m x 10m x 10m -2.37e-02 -1.24e-03 3.70e-04 -1.68e-03 8m x 8m x 8m -2.35e-02 -1.35e-03 1.39e-04 -3.08e-03 Length: Mean (m) 16m x 16m x 16m 0.27 1.84 1.08 0.56 12m x 12m x 12m 0.25 1.81 1.06 0.52 10m x 10m x 10m 0.24 1.80 1.04 0.49 8m x 8m x 8m 0.22 1.76 0.95 0.46 L: Standard deviation 16m x 16m x 16m 0.35 1.53 1.06 0.62 12m x 12m x 12m 0.34 1.49 1.01 0.58 10m x 10m x 10m 0.33 1.47 0.96 0.55 8m x 8m x 8m 0.31 1.39 0.87 0.50 L: Skew 16m x 16m x 16m 0.93 0.35 0.56 0.74 12m x 12m x 12m 0.98 0.33 0.49 0.69 10m x 10m x 10m 1.02 0.32 0.43 0.71 8m x 8m x 8m 1.08 0.27 0.42 0.66 Mean 9 16m x 16m x 16m -19.2 -179.4 124.1 -98.8 12m x 12m x 12m -17.8 -179.4 124.5 -99.6 10m x 10m x 10m -16.9 -179.5 124.6 -100.0 8m x 8m x 8m -15.5 -179.7 124.9 -100.9 Mean cj) 16m x 16m x 16m 41.7 1.7 28.7 38.8 12m x 12m x 12m 38.9 1.9 29.6 39.4 10m x 10m x 10m 36.0 1.6 30.1 39.8 8m x 8m x 8m 32.7 1.1 31.0 40.8 180 Figure 5-1 Axis definition for three dimensional domains. 181 y -Fracture Set 2 Pole to Fracture Set 1 Fracture Set 1 Fracture planes Set 1: 90° y + z + Set 1: -10° x-y plane Set 2: -20° Set 2: 20° x + z -Poles: Trace Poles: Plunge Figure 5-2 Description of fracture sets by poles and planes. One fracture plane from each set of the base-case fracture system. Fracture set orientation is described by the orientation of a pole, perpendicular to the fracture plane. The pole for set two points downward and towards the back of the cube (- x direction). 182 Fracture Traces Figure 5-3 One realization of the three-dimensional base-case network 183 Particle travel paths in fracture plane Figure 5-4 Particle movement across a fracture plane. Particles follow flow lines, so that the most likely path is directly between the two intersecting fractures. A small proportion of particles will follow the less direct flow lines. Arrows indicate the orientation of the particle trajectory. 184 +z A Figure 5-5 Simple definition of 9 and d). Note axis is rotated for display purposes. 185 Figure 5-6 Schematic of particle movement through three intersecting fracture planes. Particle locations at elements of the mesh are converted to element to element movements. Then element-to-element movements are grouped into path-length vectors. 186 Figure 5-7 Definition of set-directions on simple two fracture model. 187 Fracture Plane (1) Particle Locations At Elements (2) Element to Element Movements Set-Direction 1: 0c,d)c,vc,lc Set-Direction 3: 6b,<j>b,vb,lb Set-Direction 1: 6a,<|>a,va,la (3) Path-Length Vectors (4)Vector Statistics Assigned to Set Direction Figure 5-8 Schematic showing conversion of particle locations to path-length vectors, including definition of locations, elements, vectors, and set-directions. 188 Figure 5-9 Raw movement vectors for a single realizations of a base-case discrete subdomain under a horizontally applied gradient. Vectors are not sorted into sets, (a) (j) as a function of 6, (b) Path-length vs. 0, (c) Velocity vs. 9, (d) Path-length vs. (j), (e) Velocity vs. 4>. 189 Figure 5-10 Wrap around effect. As (j) crosses 90 degrees 0 is changed by 180 degrees. 190 Figure 5-11 Scatter plot of 9 and <j) with plots of the theoretical relationship overlaid. Vectors are from one realization of the base-case fracture system with a horizontally applied gradient. 191 Length (m) (c) 3.5 e -logjo velocity (m/s) (a) 90 45 -45 -90 0 9 0 (d) 4 <_ (e) i i i r - x « — -©-X ; X XX X 1 1 1 180 270 3 6 0 6 0 7.5 15 0 3.5 7 Length (m) -log 1 0 velocity (m/s) Figure 5-12 Movement vectors in set-direction 1, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) (j) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. <j), (e) Velocity vs. §. 192 Figure 5-13 Movement vectors in set-direction 2, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) d> as a function of 6, (b) Path-length vs. 0, (c) Velocity vs. 0, (d) Path-length vs. <(), (e) Velocity vs. 193 (b) 15 7.5 U e (c) 3.5 360 Length (m) -log10 velocity (m/s) (a) 90 45 (j) 0 -45 -90 (d) 1—i—r J I L (e) 0 90 180 270 360 9 0 7.5 15 Length (m) J I L 0 3.5 7 -log10 velocity (m/s) Figure 5-14 Movement vectors in set-direction 3, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) d) as a function of 9, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. d>, (e) Velocity vs. d). 194 (b) 15 7.5 (C) 3.5 360 9 *>< * x x X e Length (m) -log10 velocity (m/s) (a) 90 45 (() 0 -45 -90 0 90 180 270 360 6 (d) n — i — r 0 7.5 15 Length (e) 0 3.5 7 -log10 velocity Figure 5-15 Movement vectors in set-direction 4, for a single realizations of a base-case discrete subdomain under a horizontally applied gradient, (a) (j) as a function of 8, (b) Path-length vs. 9, (c) Velocity vs. 9, (d) Path-length vs. <j), (e) Velocity vs. <j). 195 Set-direction 1 Set-direction 2 180 360 0 180 360 Set-direction 3 = in 0.2 0.1 180 360 Set-direction 4 0.11 1 1 r 0.05 I 0 180 360 6 9 Figure 5-16 Histograms of 9. Base-case model, gradient orientation 9 = 0°, <j> = 0°. Based on 1000 discrete subdomain realizations. 196 Set-direction 1 180 360 Set-direction 2 0.2 i 1 r 0.1 0 180 360 Set-direction 3 = =3 QJ U 0.1 0.05 180 360 Set-direction 4 0.2, 1 1 r 0.1 I I M 180 360 Figure 5-17 Histograms of (j). Base-case model, gradient orientation 9 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations. 197 Set-direction 1 o S cr U 0.1 0.05 Set-direction 2 0.1 i r 0.05 6 8 10 0 2 4 6 8 10 Set-direction 3 Set-direction 4 C cr 0.2 0.1 0 2 4 6 8 10 Length (m) 0 2 4 6 8 10 Length (m) Figure 5-18 Histograms of path-length. Base-case model, gradient orientation 0 = 0°, § Based on 1000 discrete subdomain realizations. 198 Set-direction 1 S3 s cr Li-ft. 1 0.05 1 1 1 . J -5 0 10 Set-direction 2 0.2 0.1 h 1 1 1 . L. i 10 Set-direction 3 63 cr u to 0.2 0.1 Set-direction 4 0.1 0.05 10 -log10 velocity (m/s) -log1 0 velocity(m/s) Figure 5-19 Histograms of velocity. Base-case model, gradient orientation 9 = 0°, § = 0°. Based on 1000 discrete subdomain realizations. 199 Set-direction 1 Vi o > © 1—1 o 180 360 Set-direction 2 4 i 1 1—:—r 0 180 360 Set-direction 3 Vi WD o 180 360 Set-direction 4 4 180 360 Figure 5-20 Mean velocity versus <j>. Base-case model, gradient orientation 9 = 0°, <j> Based on 1000 discrete subdomain realizations. 200 o Set-direction 1 0.4 0.2 0 Set-direction 2 0.4 0.2 180 360 180 360 Set-direction 3 0.4 i 1 r-h x 0.2 1* _ J L 180 360 Set-direction 4 0.4 0.2 h 180 4> 360 Figure 5-21 Standard deviation of velocity vs. d>. Base-case model, gradient orientation 6 0°, d> = 0°. Based on 1000 discrete subdomain realizations. 201 ox S3 OH Set-direction 1 2.0 Set-direction 2 2.0 1.0 h 360 180 360 Set-direction 3 Set-direction 4 S3 a 2.0 1.0 180 360 180 360 Figure 5-22 Mean path-length vs. (j). Base-case model, gradient orientation 8 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations. 202 = I Set-direction 1 2.0 I 1 1 r 1.0 h J L 0 180 360 Set-direction 2 2.0 360 WD C a b Set-direction 3 2.0 | 1 1 r 1.0 X X JLl I U 180 Set-direction 4 360 360 Figure 5-23 Standard deviation of path-length versus d). Base-case model, gradient orientation 9 = 0°, d> = 0°. Based on 1000 discrete subdomain realizations. 203 W3 Set-direction 1 Set-direction 2 2.5 h o 13 > 2 2 I 0 2 4 6 8 10 2.5 b 0 2 4 6 8 10 Set-direction 3 Set-direction 4 WD o i-H OX) o 2.5 0 2 4 6 8 10 2.5 h 0 2 4 6 8 10 Length (m) Length (m) Figure 5-24 Mean velocity as a function of path-length. Base-case model, gradient orientation 0 = 0°, § = 0°. Based on 1000 discrete subdomain realizations. 204 Set-direction 1 a , & 0.5 « J cu > 0 Set-direction 2 1 i 1 1 1 r 0.5 \-0 I —i L 0 2 4 6 8 10 0 2 4 6 8 10 Set-direction 3 Set-direction 4 X 0.5 CJ "3 > o o 0.5 0 2 4 6 8 10 0 2 4 6 8 10 Length (m) Length (m) Figure 5-25 Standard deviation of velocity as a function of path-length. Base-case model, gradient orientation 0 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations. 205 Set-direction 1 180 360 Set-direction 2 1 r 180 360 o S3 s cr u to Set-direction 3 0-2 , , , r 0.1 l- I  I L 180 360 Set-direction 4 o.o5 r 180 360 e 0 Figure 5-26 Histograms of 0 under the 10 degree cutoff. Base-case model, gradient orientation 0 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations. 206 Set-direction 1 C QJ a cr 0.2 0.1 180 360 Set-direction 2 0.2 i 1 r 180 360 w S3 = cr QJ U Li. Set-direction 3 0.05 180 360 Set-direction 4 180 360 4> 4> Figure 5-27 Histograms of (j) under the ten degree cutoff. Base-case model, gradient orientation 8 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations. 207 S3 cu ss cr cu u to Set-direction 1 0 2 4 6 8 10 Set-direction 2 0.2 0.1 h 0 2 4 6 8 10 Set-direction 3 Set-direction 4 eu S3 CU = cr cu to 0.4 0.2 6 8 10 0 2 4 6 8 10 Length (m) Length (m) Figure 5-28 Length Histograms under a 10 degree cutoff. Base-case model, gradient orientation 9 = 0°, <j> = 0°. Based on 1000 discrete subdomain realizations. 208 Set-direction 1 Set-direction 2 cu fi cu 3 cu 0.1 0.05 0.2 0.1 1 1 1 i 10 -5 0 10 Set-direction 3 cu fi cu cr cu 5-0.15 0.1 1 1 1 1 ., Set-direction 4 0.1 0.05 1 1 1 • -5 0 10 -5 0 5 10 -log1 0 velocity (m/s) -log1 0 velocity (m/s) Figure 5-29 Velocity histograms under a 10 degree cutoff. Base-case model, gradient orientation 9 = 0°, § = 0°. Based on 1000 discrete subdomain realizations. 209 Set direction 2 Set direction 3 Set direction 4 Set direction 1 J L 200 400 600 800 C/3 1 1 1 1 1 1 1 -1 1 1 1 1 1 1 200 400 600 800 Number of networks Figure 5-30 Mean and standard deviation of path-length vs. number of fracture networks run in the subdomain. Base-case model, gradient orientation 6 = 0°, (j) = 0°. Based on 1000 discrete subdomain realizations. 210 "i i i i i i r Set direction 1 2.8 Set direction 4 Set direction 3 2.4 Set direction 2 2.0 J L J I I L 200 400 600 800 Figure 5-31 Mean and standard deviation of velocity as a function of the number of networks run in the discrete subdomain. Base-case model, gradient orientation 9 = 0°, (j> = 0°. Based on 1000 discrete subdomain realizations. 211 CU *© U c o cu 5-0.4 0.2 n i 1 1 1 r Set direction 2 Set direction 3 Set direction 4 I Set direction 1 J I I L 200 400 600 800 Number of networks Figure 5-32 Directional choice parameters as a function of the number networks run in the discrete subdomain. Base-case model, gradient orientation 6 = 0°, <j) = 0°. Based on 1000 discrete subdomain realizations. 212 6. Results In Three Dimensions. 6.1. Introduction Chapter 6 presents the results of the three-dimensional modeling. Mass transport in a continuum domain is compared to mass transport through large-domain discrete network models, that are four times the volume of the discrete subdomains. The two modeling approaches are compared by plotting the evolution of the spatial moments as a function of time. For the discrete network realizations, both the average and standard deviation of these moments are plotted. The variation between networks is presented as the standard deviation of the moments, and acts as a reference by which to gauge how closely moments for the S C M models match those for the discrete model. Six examples are analyzed in this chapter. Networks used in the first two examples are generated from the three-dimensional base-case fracture system (Table 5-1). In this dense fracture system networks are composed of fractures sampled from two sub-vertical, non-orthogonal sets, in which the orientation of the fractures within each set is constant (Figure 5-3). In the first example the applied hydraulic gradient is oriented at a horizontal angle (9) of 0° and a vertical angle (())) of 0°. In the second example a gradient oriented at 9 = 45° and d> = 45° is applied to the same set of networks used in the first example. In the third example, variation in fracture orientation is introduced to the networks, with all of the other fracture parameters equal to the base case model. The fourth example is similar to the third, but with a 4 m cutoff applied to the fracture length. The fifth model is a lower-density version of the variable orientation fracture system (example 3), containing 40% of the 213 fractures within the base-case model. Finally, a network based on a Levy-Lee algorithm is presented. 6.2. Base case: Horizontal Gradient 6.2.1. Introduction This section presents the results for the base-case fracture system, in which networks are composed of sub-vertical fractures from non-orthogonal sets, with no orientation variation within each set. A plot of a single network realization from this fracture system is given in Figure 5-3. Motion statistics for the S C M models are calculated i n a l O m b y l O m by 10 m discrete subdomain. In order to evaluate the S C M models, mass transport in a 20 m by 20 m by 20 m continuum domain is compared to mass transport in 1000 network realizations of the same volume. The first and second, x, y and z spatial moments for the S C M models are compared to the average and standard deviation of the moments for the network realizations. A l l the moments are plotted as a function of time in days. In the first example, the applied hydraulic gradient is at a horizontal angle (9) of 0° and a vertical angle (40 = o °. 6.2.2. Moments for network realizations The average and standard deviation of the first and second moments for the network realizations are plotted in Figure 6-1 and Figure 6-2. The average moment values are plotted as solid lines, with one standard deviation above and below the average moments plotted as dashed lines. The discrete moments are discussed first. Although the applied hydraulic gradient is horizontal, the orientation of the fracture sets is such that the centre of mass has components of motion in all three directions. Mass is injected in a zone centrally located on 214 the upstream boundary, at x = 0 m and travels towards the downstream boundary at x = 20 m. However, particles begin to exit the model domain before the centre of mass reaches the downstream boundary. The evolution of the first moments is linear until just before 4000 days, at which point the centre of mass appears to slow slightly as the particles in the front of the plume leave the model domain through the downstream boundaries. Boundary effects are more distinctly visible in the second moment plots (Figure 6-2). Starting at the mid-point of the injection zone aty = 0 m, the first y moment increases in a linear fashion until at 4000 days the loss of mass influences the moment curve. The first z moment also starts at 0 m, but increases at a slightly faster rate than the first y moment. Although the standard deviation of the first x moments increases with time, the standard deviation of the first y and z moments is relatively constant. The standard deviations of the first y and z moments are initially non-zero due to the width of the injection band in the y and z directions. The second x moment starts at 0 m as the injection zone is confined in the x direction to the x = 0 m boundary (Figure 6-2). The second x moment increases linearly, reaching a magnitude of approximately 17 m2 at 4000 days. Both the second y and second z moments, start at approximately 3 m , as mass is injected across a source band withy and z values ranging from +3 m to -3 m. Spreading increases at similar rates in both the second y and z moments. The standard deviation in the second y moment remains relatively constant, while the standard deviation in the second z moment increases slightly over time. 6.2.3. SCM models: Varying models of velocity and path-length In the following sections moments for a number of continuum models are compared to the moments for the network realizations. The difference between the average moments for the network realizations and the moments for an S C M model are subsequently referred to 215 as the deviation, so that the smaller the deviation the better the match between the two modeling approaches. In this section the various statistical models are used in the S C M to represent velocity and path-length, and in section 6.2.4 the influence of the directional-choice algorithm is examined. In section 6.2.5, models in which velocity and/or path-length are correlated to the vertical trajectory (d>) are examined, followed in section 6.2.6 by models in which the vertical trajectory ((j)) is correlated to the horizontal trajectory (9). In section 6.2.7 models in which velocity is correlated to path-length are presented. In the first five sections all of the models are based on motion statistics calculated using the long-path algorithm (section 5.3.3), and in section 6.2.8 a number of the previously examined modeling approaches are repeated using statistics calculated using varying <j) cutoffs. In Figure 6-1 and Figure 6-2 the four statistical continuum models examined are: model (1) the H(v)H(l) model; model (2) the H(v)G(l)model; model (3) the G(v)G(l) model; and model (4) the N(v)G(l) model (see section 5.7 for a summary of the naming convention). The behavior of each of these models can be summarized: (a) The H(v)H(l) model works well for the first x and y moments, but for the first z moment, mass movement is too slow. The second x moment for the H(v)H(l) model spreads too rapidly, exceeding the average for the network realizations by one standard deviation after 1000 days. For the second y and z moments the H(v)H(l) model works well, although the starting value of the second z moment is slightly lower than the average second z moment for the network realizations. (b) Similar results are obtained for the H(v)G(l) model. There is a slightly larger deviation in the first x and z moments for the H(v)G(l) model than for the H(v)H(l) model. The first z moment is more than one standard deviation below the average for the network realizations. The H(v)G(l) model works well for the first y moment. There is a slight 216 improvement in the second x moment, when using the H(v)G(l) model, although these moments are still not within one standard deviation of the average second x moments for the network realizations. There is an improvement over the first model for the second y moments, and the H(v)G(l) model works well for the second z moments, although the starting value of the second z moments is lower than the average for the network realizations. Moment values for models 1 and 2 are similar, with both failing to reproduce the second x moment. (c) In the third model, both velocity and path-length are represented using a three parameter gamma model. The first x moment for the G(v)G(l) model is within one standard deviation of the average first x moment for the network realizations. The magnitude of the error in the first x moments is similar to that in models 1 and 2, although in the opposite direction. However, this model does not work as well for the first y moment, producing movement in they direction that exceeds the average for the network realizations by 3500 days. For the first z moment the G(v)G(l) model improves the results to the point where moments are just within one standard deviation of the network average. The G(v)G(l) model works well for the second x moment, and unlike the first two models produces values within one standard deviation of the average for the network realizations. The G(v)G(l) model provides an excellent match to the second y moment for the network realizations, and improves upon the results of the previous two models for the second z moment. (d) Results for fourth S C M approach, the N(v)G(l) model, are similar to results for the G(v)G(l) model, other than a slight improvement in the match of the first x moments, and a slight increase in the deviation for the second z moment. 217 Of the four models in Figure 6-3 and Figure 6-4, the G(v)G(l) and N(v)G(l) models produce the best results. Although the H(v)G(l) and H(v)H(l) models produce better results for the first y moment, these models fail because values for both the second x moment, and the first z moments differ by more than one standard deviation from the average moments for the network realizations. The largest difference between these S C M approaches occurs between the models in which a histogram is used for velocity and those in which a continuous distribution is used for the velocity. Errors in the slopes of the moments for all of the three-dimensional models are discussed in section (6.7) 6.2.4. SCM models: Direction choice algorithms S C M models can be improved by modifying the algorithm that determines the direction of travel at the end of each step. Particles choose a set-direction based on one of four approaches (section 5.7.3): (1) the unconditional algorithm, where the probability of travel in any direction is based on the cumulative probabilities (row 2 in Table 5-3), and is independent of the previous travel direction; (2) the conditional probability algorithm, where the probability of travelling in any direction is conditioned on the previous direction traveled by using the directional choice statistics grouped by the previous travel direction (rows 9 to 12 in Table 5-3); (3) the no reversal/no repeat algorithm, based on the cumulative probabilities, with movement restricted by not allowing particles to either reverse direction or repeat a direction in sequential steps; or (4) the no reversal algorithm, based on the cumulative probabilities, but with particles not allowed to reverse directions. The unconditional algorithm was used in the previous four models. Moments for S C M models using the G(v)G(l) approach with varying directional choice algorithms are plotted in Figure 6-3 and Figure 6-4. The unconditional model in 218 Figure 6-3 is a repeat of the G(v)G(l) model in Figure 6-1 and Figure 6-2. For the first x moment, the unconditional model produces moment values nearly one standard deviation below the average for the network realization. For the conditional probability model the first x moments are just above the one standard deviation line. The no-reversal/no-repeat model is a slight improvement on the no-restrictions model, but the best results are obtained for the G(v)G(l) model in which no reversal is allowed. For the first y moment, all of the conditioned directional choice models significantly improve the results over the no-restrictions model. The first z moments are unaffected by modifying the directional choice algorithm, with all moments just within one standard deviation of the average moment for the network realizations. There is a change but no net improvement from the no-restrictions model to the no-reversal model, in the second x moments. Moments for the no-reversal/no-repeat model and the conditional probability model are reduced enough that they are less than one standard deviation below the average second x moment for the network realizations. For the second y moment the results for the conditional model are similar to the no-restrictions model, although there is a slight worsening of the results for the no-reversal/no-repeat model and the no-reversal model. For the second z moment, modification of the directional choice algorithm decreases the spreading in the z direction, slightly degrading the results. The best overall results are obtained using the G(v)G(l) model in which particles are not allowed to reverse direction at the end of each random walk step. This algorithm is consistent with the expected behavior of particles in the three-dimensional discrete subdomain. Applying the no-reversal algorithm brings all of the moments for the G(v)G(l) model within one standard deviation of the average moments for the network realizations. 219 In Figure 6-5 and Figure 6-6 the directional choice algorithms applied within Figure 6-3 and Figure 6-4 are applied to the H(v)H(l) model. The first model in Figure 6-5 is the no-restrictions model and is the same as the H(v)H(l) model in Figure 6-1 and Figure 6-2. Models two through four are the H(v)H(l) models with the conditional probability, no-reversal/ no-repeat, and no-reversal algorithms applied. For the first x moment the no-reversal model is the only clear improvement over the no-restrictions approach. Conditional probability and the no-reverse/no-repeat model do not change the results much for the first x moment. For the first y moment the introduction of the directional choice rules only slightly affect the moment values. A l l three of the directional choice models improve the results for the first z moment, although all models are more than one standard deviation below the average first z moment for the network realizations. For the second x moment the no-reversal model hardly changes the results at all, and the conditional probability and the no-reversal/no-repeat models slightly improve the results for the second x moment. Although the conditional probability models slightly change the second y moment values there is no net improvement or worsening of the continuum results. Results for the second z moment are improved by the introduction all of the directional choice algorithms. Introduction of the directional choice algorithms does not reduce the difference between the average moments for the network realizations and moments for the H(v)H(l) model enough to provide satisfactory results. As with the G(v)G(l) model the best results were obtained when using H(v)H(l) with the no-reversal model. However moments for the G(v)G(l) model with no-reversal are much closer to the network averages than the H(v)H(l) model with no-reversal, especially for the second x moment. 220 6.2.5. SCM models: Correlating velocity and path length to the vertical trajectory With the introduction of the third dimension it is now the possible to correlate velocity and/or path-length to the vertical trajectory (<j>) (section 5.4). Four models in which various combinations of velocity and path-length are correlated to <j) are examined in Figure 6-7 and Figure 6-8: (1) N(v,<t>)N(l,<j)) ; (2) N(v,d>)G(l); (3) N(v)N(l,d>) ; and (4) H(v)N(l,<t>) . (See section 5.7 for nomenclature). There were no correlations to d) in any of the previously examined models. A l l four (j) correlated models produce moments barely within one standard deviation of the average first x moments for the network realizations. Although moments for model 4 in which path-length is correlated to <j), while velocity is not correlated to (j), are different than moments for the other four models, these moments deviate from the averages by similar amounts. However the H(v)N(l,d>) model produces an excellent match to the first y moments for the network realizations. First y moments for the other three models move mass at a faster rate than occurred in the network realizations, exceeding the network average by over one standard deviation by 4000 days. The first z moments for the H(v)N(l,(j)) model are again different than the moments for the other three models, but in this case the other three models are closer to the network averages. The H(v)N(l,d>) model produces moments that are more than one standard deviation larger than the average second x moments for the network realizations, while moments for the other three models are all close to the average for the network realizations. For the second y moment all models produce excellent matches to the average moments for the network realizations. Moments for all four models are within one standard deviation of the network average for the second z moments. Overall, when long-221 path motion statistics are used, correlating velocity or path-length to the vertical trajectory ((j)) does not appear to improve the S C M results. 6.2.6. SCM models: Correlating vertical trajectory to horizontal trajectory Because fractures are oriented obliquely to the x-y-z coordinate axis, there is a correlation between the values of the vertical and horizontal trajectories that are allowable within a given fracture set (5.7.4). Moments for S C M models in which the vertical trajectory (<j)) is calculated as a function of the horizontal trajectory (9) are displayed in Figure 6-9 and Figure 6-10. The models used in Figure 6-9 and Figure 6-10 are the same models used in Figure 6-1 and Figure 6-2 ,with the addition of the correlation between § and 9. The first and second x and y moment are only slightly modified by the introduction of the 9-(j) correlation. However, there is a marked slowdown in mass movement in the z direction, and the spreading in the z direction increases with introduction of correlation between 9 and §. Although there is only a slight deviation in second z moments, models without the 9-<j) correlation are just as effective. Overall the introduction of the 9-<j) correlation was not a significant improvement Two points explain the lack of improvement: (1) Correlation of 9 and (j) was not as essential to the S C M as first thought. Certain vector orientations are more common within the discrete subdomain, and other vector orientations are less common. However the most frequently-occurring pairs of 9 and (j) are comprised of the most frequent values of 9 and the most frequent values of <j). Therefore, even when 9 and § are sampled independently, the most common combinations of 9 and <j), and hence the most common vectors, result. (2) The 222 method by which (j) is correlated to 6 introduces an approximation into the distribution of (j) that overwhelm any improvement made due to the correlation. In order correlate 9 and (j), 9 is generated at random from a histogram of 9 values, and then (j) is calculated directly using equation (5-3), the average relationship between the two. The simplification in the § distribution occurs because there are a wider range of § values than 9 values. Intuitively it seems that the 9-<j) correlation might be necessary for some fracture systems. Improvements to the correlation algorithm may be necessary in these cases. 6.2.7. SCM models: Correlating velocity to path-length There is some correlation between velocity and path-length because particles tend to travel farther in fractures in which the fluid velocity is higher (section 5.4.4). Two models in which velocity is correlated to path-length are plotted in Figure 6-11 and Figure 6-12. Two further models in which directional choice algorithms are applied in addition to the velocity and path-length correlation are also presented. The first and second models are the N(v,l)H(l) and the G(v,l)H(l) with no conditions on directional choice. The third and fourth models are the G(v,l)H(l) models, the same approach as the second model, but the third model uses conditional probability, and the fourth model uses the no-reversal/no-repeat algorithm. For all of the models the values of the first x and y moments are higher than the average moments, although this result is not significantly different from models which do not correlate velocity and path-length. Good results are obtained for the first z moments, in fact models incorporating v-1 correlation and those in which the velocity and path length are correlated to § are the only models which produce good estimates of the first z moments. 223 The best models for the second x moment are the first and second models, i.e. those with no constraints on directional choice, although results are improved for all models. Results for both the second y and z moments show a good match to the average moments for the network realizations. There is a similarity between the x and z moments for the v-1 correlated models and the x and z moments for models in which v and 1 are correlated to (j). The similarity in the moments for these models is not surprising as when v and 1 are both correlated to (j), they are indirectly correlated to each other. Although correlating velocity to path-length significantly improves the z moments, overall inclusion of this correlation is not an improvement of the method. 6.2.8. SCM models: Applying a vertical trajectory (())) cutoff In the previous sections of Chapter 6 S C M models were examined in which the particle motions in the continuum are generated using differing algorithms, all applied using the same set of motion statistics. In this section, these same models are reexamined but there are changes in the motion statistics used. As discussed in section 5.5 there are a number of ways to describe the path-length as particles travel through a discrete subdomain. In the previous sections all statistics were based on the long path-length algorithm. In this section particles paths are subdivided at breaks in (j), so that a particle's trajectory across a fracture plane that was previously recorded as a single step is now subdivided into several steps. The S C M models used in Figure 6-13 and Figure 6-14 are the same models used in Figure 6-1 and Figure 6-2, only using statistics calculated with a 10° threshold in (j) instead of the long-path statistics. Applying the H(v)H(l) and H(v)G(l) models to the 10° statistics slows the evolution of the first x moment, degrading results compared to the models using long-path statistics. However, the first x moments for the G(v)H(l) and N(v)H(L) models are 224 changed only slightly by the change in the motion statistics. For the first y moments, models one and two are slower and are therefore degraded by the use of the 10° statistics, while models three and four are also slower which acts as a slight improvement in the match to the network averages. For the first z moment the first and second models are degraded by use of the 10° cutoff statistics, but the third and fourth models are improved with these statistics. For the second x moments there is slower spreading observed in all of the models, generally improving the results. However, models three and four disperse slowly enough that values for the second x moments are more than one standard deviation below the average second x moments for the network realizations by 4000 days. Spreading is also slower in the second y moments with the 10° statistics, but slowing degrades the results in this case. Dispersion is significantly less for the second z moments, dropping below one standard deviation of the average for the network realizations. With the introduction of the 10° cutoff moments for the G(v)G(l) models are slightly further from the average moments for the network realizations. In Figure 6-15 and Figure 6-16, the same series of models that were run in Figure 6-1, Figure 6-2 and in the previous section are applied, but now using a set of statistics based on a d) = 5 ° cutoff. A 5° cutoff further subdivides the path-lengths relative to a 10° cutoff, forcing particles to take a larger number of shorter steps to cover the same distance. Results for the first x andy moments are similar to the results using the 10 ° statistics. First z moments for models one and two are similar to the moments using the 10° statistics. Second moments show an even greater reduction in spreading in all directions when a 5° cutoff is used instead of a 10° cutoff. Second x moments are marginal to poor for all four S C M approaches, and results for the second y and z moments decrease enough that these moments fall more than one standard deviation below the average moments for the network realizations. Applying 225 statistics calculated using a 5° cutoff does not improve the results over using the long-path statistics case. Although neither the 10° nor the 5° statistics improved the results for the models discussed so far, the use of cutoffs will be shown to improve the S C M results in further cases discussed below. For this reason one further example using a § = 10° cutoff is given, followed by a brief summation of the effects of using a § cutoff observed in other results not presented. Moments for the G(v)G(l) model in which the directional choice algorithms are varied, and in which 10° cutoff statistics are used, are presented in Figure 6-17 and Figure 6-18 . The first x and y moments are almost unchanged from the long-path statistics (Figure 6-3 and Figure 6-4) to the 10° statistics, but there is a slight increase in the closeness of the match for the first z moments when the 10° statistics are used. Generally, the rate of mass movement and spreading is reduced from the long-path statistics to the 10° statistics. This trend was seen in all S C M models used including most of the models in which correlations were introduced between 9, (j), v and 1. The one exception is that models in which velocity was correlated to path-length are almost unaffected by using a § cutoff. Overall in the base-case fracture system, with the horizontally applied hydraulic gradient, the introduction of (j) cutoffs in the calculation of the motion statistics did not improve the S C M results. However, moments for the best model, the G(v)G(l) model with no-reversal, change only slightly with the introduction of a 10° cutoff. 226 6.2.9. Summary for the three-dimensional base-case model The most effective S C M approach for the base-case model is the G(v)G(l) model using the long-path statistics without correlations, and the no-reversal directional choice algorithm. This model accurately predicts the first x and y moments but produces first z moments slower than the average first z moments for the network realizations. This model is also effective at predicting the second x and y moments, with results slightly degraded from the non-directionally correlated models for the second z moments. Although the G(v)G(l) model with no-reversal is effective, other models which are not as effective overall produce better results for the z moments. It appears from Figure 6-7, Figure 6-8, Figure 6-11 ,and Figure 6-12 that models in which velocity and path-length are correlated to (j), or models in which velocity is correlated to path-length, produce the best results for both the first and second z moments. Models in which velocities are correlated to path-length using the long-path statistics produce the best overall results for the z moments (Figure 6-11). These models are not as effective at reproducing the first x and y moments as the G(v)G(l) model, but they produce first x and y moments within one standard deviation of the average moments for the network realizations. Results for the G(v)G(l) model with no reversal are acceptable for all of the moments when the 10° cutoff statistics are used, although the second x moments are slightly slow. However, results using the 10° cutoff statistics are not generally as good as when the G(v)G(l) model is used with the long-path statistics. Models in which velocity is correlated to path-length also produce acceptable moment values when using the 10° statistics, although the spreading for the second x moment is again slow. In all cases using motion statistics calculated with increased sensitivity of the d) cutoff, i.e. going from long-path statistics to 10° 227 to 5° etc., reduces the rate of spreading. In models in which velocity is described using a histogram, first moments in all directions were slower using 10° statistics than when using long-path statistics, but little change occurs between using 10° statistics and using 5° statistics. The S C M method appears to work well for the base-case fracture system. The best results are obtained using the long-path statistics with the G(v)G(l) model with no reversal. Using the 10° cutoff statistics with this model also produces acceptable results. The use of 5° statistics does not appear to improve the model. In the next section results for the base-case model are expanded to a second orientation of the applied hydraulic gradient. 6.3. Base case: Rotated Gradient 6.3.1. Introduction One of the potential strengths of the S C M model is the analysis of situations in which the orientation of the hydraulic gradient varies through the model domain. In this section the hydraulic gradient applied to networks from the base-case fracture system is rotated by 45° in the horizontal direction (9), and 45° in the vertical direction (§). As before motion statistics are calculated in a 10m x 10m x 10m discrete subdomain, and continuum moments are compared to the average and standard deviation of moments for a 20m x 20m x 20m large-domain fracture-network model. 6.3.2. Moments for network realizations Discrete moments are plotted as a function of time for the 9 = 45°, <j) = 45 ° gradient model in Figure 6-19 and Figure 6-20. Changing the orientation of the hydraulic gradient significantly affects both the magnitude and direction of mass transport within the networks. 228 Although the centre of mass still moves forward in the x direction, the rate at which the mass moves in this direction is significantly slower than under the influence of the horizontal gradient. Under the (45°,45°) gradient there is very little movement in the first y moment. The z component of the mass movement changes direction so that the plume has a downward movement under the (45°,45°) gradient instead of the upward movement seen under the horizontal gradient. Mass encounters a boundary at a much earlier time under the (45°,45°)gradient. The standard deviation of the first moments is larger under the (45°,45°) gradient, indicating that there is a greater variability in the mean plume direction than when mass was moved by a horizontal gradient. The magnitude of the second moments is related to the magnitude of the first moments. With less movement in the x andy directions and more in the z direction, spreading is slower in the x and y directions and faster in the z direction. The standard deviation of the x, y and z second moments is significantly larger than the standard deviation of these moments under the horizontal gradient. 6.3.3. Moments for SCM models In this section moments from a range of the S C M modeling approaches are compared to the moments for the network realizations. In the first example various statistical models of path-length and velocity are applied without correlation, using long-path motion statistics. Then, the effect of applying the four directional choice algorithms is presented. This section is concluded with a limited sample of S C M models using motion statistics calculated with various (j) cutoffs, both with and without parameter correlations. It is apparent that the continuum models using the four uncorrected models for velocity and path-length are not as effective at matching the average moments for the 229 network realizations under the (45,°45°) gradient (Figure 6-19 and Figure 6-20). Using the four uncorrelated models, the match for the first and second z moments under the (45,°45°) gradient is better than the match under the (0°,0°) gradient (Figure 6-1 and Figure 6-2). However the S C M results for the first and second x moments are more than one standard deviation away from the average moments for the network realizations under the (45,°45°) gradient. Of all the three-dimensional fracture systems and gradient orientations examined to date, the base case fracture system under the (45,°45°) hydraulic gradient has proven to be one of the most difficult to match. Results can be improved by applying the four directional choice algorithms to the G(v)G(l) model (Figure 6-2land Figure 6-22). When the G(v)G(l) model includes either the no-reversal/no-repeat algorithm the first x moments are within one standard deviation of the average for the network realizations. Application of the unconditional rule gives the largest differences between the moments. Conditional probability and no-reversal/no-repeat algorithms improve the first y moment to within one standard deviation of the network average, and first z moments for all of the S C M approaches except the unconditional model are close to the average first z moment for the network realizations. Conditional probability and no-reversal/no-repeat algorithms improve the second x moments of the G(v)G(l) model to within one standard deviations of the average second x moments for the network realizations. Second y and second z moments are almost equal to the average for the network realizations, when the conditional probability and no-reversal/no-repeat algorithms are used. The G(v)G(l) model with the no-reversal/no-repeat algorithm produces moments that are within one standard deviation of the averages for each of the x,y 230 and z first and second moments. This model produces barely acceptable results for the first and second x moments, but excellent results for all of the y and z moments. The results for the G(v)G(l) model with the no-reversal/no-repeat algorithm can be improved by changing the method by which the statistics are calculated. In Figure 6-23 and Figure 6-24 the G(v)G(l) model is rerun using statistics based on the long-path definition and § cutoffs of 30°, 10°, and 1°. Each decrease in the cutoff from the long-path to the 30° to the 10° and finally the 1° increases the number of steps used to describe as particle movement through the discrete subdomain. Each increase in the § sensitivity results in a slowing of mass movement for the first x and z moments, but a gradual increase in the rate at which the centre of mass moves in the first y moment. Increasing the § sensitivity also reduces the spreading seen in the second x moment. The best results are obtained using the 10° cutoff, with excellent results for most of the moments. Changing the § sensitivity results in a tradeoff between the representation of particle movement from one fracture to the next, and the representation of particle motions within a single fracture plane. With low (j) sensitivity, the path-lengths are long and capture the net trajectory of a particle traveling across a fracture plane. Each particle step in the continuum then represents the entire path of a particle between the point of entry and exit within a fracture plane. By increasing the (j) sensitivity, the smaller path-lengths can better describe the curved trajectories taken by particles across a fracture plane (see Figure 5-4), but particle motion in the continuum will not exactly reproduce particle motion in the discrete subdomain unless particles take repeated steps in the same set-direction. 231 Although no other S C M approaches work well using either the long-path statistics or the 10° statistics, a collection of the models which came the closest or which work well for individual moments are included (Figure 6-25, Figure 6-26, Figure 6-27 and Figure 6-28). In each there are two models using long-path statistics, which are then repeated using 10° statistics. The first two models are based on correlations of velocity and/or path-length to (j), and the second two models are based on correlation of velocity to path-length. In Figure 6-25 and Figure 6-26 the first model is the H(v)N(l,(j)) based on the long-path statistics and model two is the same model using the 10° statistics. Model one is included because although it does not work well for the first and second x moments or the first y moment, it provides a good match for the second y moment and both the z moments. Several of these matches are improved with 10° statistics. The match for the first x moment is good, and the match for the second x moment is improved but is still approximately one standard deviation larger than the average for the network realizations. The match is improved for the first y moment, but made slightly worse for the second y moment when the 10° statistics are used. Movement in the z direction is slow, as seen in the first z moment, but the match for the second z moment improves when the 10° statistics are used. The third and fourth models are the N(v,l)H(l) with the no-reversal-no repeat algorithm applied to directional choice. Overall the results of this approach are not that good but the N(v,l)H(l) model does provide a good estimate of the second x and second y moments. In Figure 6-27 and Figure 6-28 models in which d) is calculated as a function of 6 are given. Models one and two are calculated using the long-path statistics, while models three and four are calculated using 1° cutoff statistics. Model one and three are the H(v)H(l) approach with d> a function of 0, while models two and four are the H(v)G(l) approach with d> 232 a function of 6. The long-path models provide good estimates of the first x, y and z moments. Improvements are seen using the 10° statistics in the calculation of the second x, y and z moments, but this is at the expense of the first y and z moments. The best model is the H(v)G(l) approach with the long-path statistics. 6.3.4. Summary Models can be found for which moments are within one standard deviation of the average moments for the network realizations. The best models for the 9 = 45°, (j) = 45° gradient orientation are not necessarily also the best models for the base-case fracture system with the 9 = 0°, <j) = 0° gradient orientation. The best model is the G(v)G(l) model with the no-reversal/no-repeat algorithm applied, as long as the 10° cutoff statistics are used. Notably this model produced reasonable but not the best results under the application of the horizontal gradient. It is therefore possible that the G(v)G(l) model could be used in a model in which the gradient orientation varies. It is important to keep in mind that applying the (45°,45°) gradient to the base-case fracture system was the most difficult case seen in all of the three-dimensional models examined to date. 6.4. Variable strike and dip fracture system In the previous sections S C M and discrete models of dense, well-connected fracture systems with non-orthogonal sets of sub-vertical fractures were examined. In this section a further level of complexity is introduced into the fracture system by allowing both the strike and dip of each fracture to vary with a standard deviation of 3° (Table 6-1). The resulting networks are again dense and well connected (Figure 6-29). A horizontal hydraulic gradient 233 is applied in both the 10 m x 10m x 10 m discrete subdomains, and the 20 m x 20 m x 20 m large domains. Although the variation in the fracture orientations is not large, it causes the particle movement vectors to stray from the average 9 and § relationship line (Figure 6-30). Strike and dip variation of the fractures introduces a level of uncertainty when assigning vectors to fracture sets. It will be shown that the algorithm by which movements are assigned to fracture sets is, however, robust enough to calculate motion statistics in this case. Moments for the discrete network realizations and moments for the four uncorrelated S C M models are plotted in Figure 6-31 and Figure 6-32. The average and standard deviation of the moments for the network realizations are almost identical to the moments for the base-case network realizations, in which the strike and dip do not vary. Variation in the fracture orientation does not significantly affect the way in which mass is transported through the fracture networks. Because the high density of fracturing results in such well-connected fracture networks, the averaging process eliminates the variations with individual fracture networks. Comparing the moments for the continuum methods to earlier results, it is clear that the S C M method is as effective for the fracture system with variable strike and dip as it was for the base-case fracture system, with perhaps a slight improvement of the match for the first x moments. For the G(v)G(l) model with the four directional choice algorithms (Figure 6-33 and Figure 6-34) the moments for the S C M models are as close to the average moments for the discrete networks as they were for the same gradient applied to the base-case fracture system (Figure 6-3 and Figure 6-4). In fact the match in the second x moment appears to be 234 slightly improved over the base-case model. However, the match for the first z moment is still borderline. A l l of the other S C M models, including the various correlations of velocity and path-length to both (j) and each other, produced nearly identical results to those for the base-case network system. 6.4.1. Results with 4m cutoff applied There is some possibility that the longest fractures within a network realization might dominate the flow field by providing direct connections from one boundary to another. If this occurs then neither the subdomain nor the large-domain networks may contain a sufficient volume of the network to be considered a REV. To test for this possibility the model with the variable 9 and (j) is rerun with all parameters identical except that all fractures are truncated at an effective radius of 4 m. A realization of a truncated network is given in Figure 6-35, and is comparable to the non-truncated network in Figure 6-29. Application the 4m cutoff appears to reduce the longest fractures visible on the surface trace plot, but direct comparison is difficult because the networks are not identical. It is difficult, however, to detect any substantial difference between the interior connectedness of the two networks. Application of the four meter cutoff does not appear to significantly alter the match between the moments for both the network realizations and continuum models (Figure 6-36 and Figure 6-37) from the base-case model (Figure 6-33 and Figure 6-34). Reducing the length of the longest fractures reduces the connectedness of the fracture system, slightly reducing the rate of change of the first and second moments. However, the change in the relative magnitudes of the moments is not large enough to suggest that larger domains are necessary for the Baecher models. 235 6.5. Lower density fracture system 6.5.1. Introduction In order to test the limits of the S C M approach, an additional fracture system is tested in which the fracture density of the previously examined fracture systems is reduced. In this model the fracture density is reduced to 40% (Table 6-2) of the fracture density in the variable orientation fracture system. A l l modeling parameters other than the density of fracturing remain the same, including the size of the models and subdomains, the magnitude and orientation of the applied hydraulic gradient, the geometry of the fracture sets, and the distribution of the transmissivities. The resulting networks, although sparser than for the realizations with the previous fracture systems, are still dense and well connected (Figure 6-38). Reducing the fracture density results in a slower rate of mass transport in both the average first and average second moments for the network realizations (Figure 6-39 and Figure 6-40). Simulations are extended to longer times than in the previous fracture systems, before the loss of mass through the downstream boundary begins to affect the moments. By 4000 days the first x moment moves only 6 m in the lower density networks and 10 m in the base-case networks. The first y moments moves approximately 2 mm both the base-case and lower-density fracture systems by 4000 days, and the first z moment moves 3 m and 2 m for the base-case and lower-density systems, respectively. Movement is also reduced for the second moments in the lower-density fracture systems, and the second y and z moments are smaller at t = 0. The standard deviation of the moments is significantly larger for all of the moments as the lower density networks exhibit a higher degree of variability from one network to the next. 236 A l l of the continuum models examined previously in this chapter were applied to the lower-density fracture system, and motion statistics were calculated using various § cutoffs. Using a G(v)G(l) model, with the no-reversal/no-repeat algorithm, and using statistics calculated with a § cutoff of 10° provides the best match between the moments for the S C M approaches and the average moments for the network realizations. The first x andy moments increase faster for the continuum models than for the network realizations but are within one standard deviation of the network averages. The first z moments for this S C M model are almost identical to the average first z moments for the network realizations. Matches for the second x and second z moments are excellent, and although there are similar expansion rates in the second y moments, the initial second y moment for the S C M model is noticeably larger than the initial second y moment for the network averages. Overall the matches between moments for the G(v)G(l), no-reverse/no-repeat model are good, but not as good as the matches for the higher density fracture networks. A better match to the first moments can be obtained by using a less-sensitive (j) cutoff, but this increases the spreading rates of the second moments to an inappropriate level. It is possible to change the starting values of the second moments by adjusting the size of the injection band used in the continuum model (Figure 6-41 and Figure 6-42). Changing the width of the injection band has no effect on the first x, y and z moments. There is also no significant change in the second x moments. However, the initial values of both the secondy and second z moments are reduced by reducing the injection band. In the network realizations a 7 m by 7 m injection band is used. Reducing the injection band in the continuum domain to 4 m by 4 m improves the match in the second y moments. Although 237 reducing the injection zone reduces the initial values of the second z moments, the match to the average moments for the network realizations is not significantly improved. Differences in the injection band used in the network realizations and S C M models are necessary because: (1) a limited number of fractures fall within the injection zone in any individual network realization, so that the net areal extent of the injection nodes in each discrete network is smaller than the total areal extent of the injection zone; (2) in the discrete networks the particles are injected into fractures based on the magnitude of the flow within the fractures, which biases the early particle movements towards the higher-flow fractures; and (3) the first step taken by particles in the continuum is governed by the average motion statistics for the fracture system, so that particles may initially move in the y or z directions before moving in the x direction. In the discrete networks, the first step taken by particles is most likely in the mean flow direction. Because the differences between the second moments for the S C M models and network realizations increases for the lower-density network, it seems likely that the first effect dominants. Overall, the S C M method is as'effective at modeling mass-transport through the lower density fracture system as it was in the fracture systems examined previously. Although the lower-density fracture system generates networks containing a substantially lower number of fractures, these networks are still dense and well connected. As in all of the previous examples, the fractures have been from a single length-scale. In the following section, the S C M method is compared to discrete models of a Levy-Lee fracture system, in which the fractures are from a range of length-scales. 238 6.6. Levy-Lee fracture system 6.6.1. Networks In this section three-dimensional networks containing fractures from a range of length scales are created using a Levy-Lee fractal model (e.g. Dershowitz et al. 1996). The algorithm for two-dimensional network generation is discussed fully in section 3.3.2 with a brief outline of the approach in three dimensions given here. Levy-Lee fracture networks are generated using a Levy flight process in which the distribution of random-walk step lengths is described by a power-law probability function. Each step determines the location of the centre of a square planar fracture, the equivalent radius of the fracture, and the range of orientations from which the actual fracture orientation is selected. The Levy-Lee flight process results in a small number of large steps, and a large number of small steps creating networks consisting of clusters of smaller fractures connected by larger individual fractures (Figure 6-43). When applying the Levy-Lee algorithm in three dimensions the fractal dimension (D) of the random walk, and hence the D of the point-field of fracture centres, can range from 0.5 to 5. Networks generated using a D value of 0.5 have a weak tendency to cluster and networks generated using a D value of 5 have a strong tendency to cluster. The input parameters for the Levy-Lee fracture system are given in Table 6-3. As with the Baecher models examined previously, the discrete subdomains are 10 m by 10 m by 10 m cubes, and the large domain continuum and discrete models are 20m x 20m x 20m cubes. Two hundred and fifty fractures are generated in each of two fracture sets, with average values of 9 and (j) of 90 and -10° for the first and -20° and 20° for the second. Although the average orientation of each of these fracture sets is the same as the base case model, there is a much larger variation in the orientation within each set in order to generate 239 networks similar to the Levy-Lee networks generated in two-dimensions (Figure 3-3 and Figure 4-25). The mean radius of each fracture set is 1.8 m but because of the power-law distribution of the fracture radii, many much larger fractures are created. Transmissivities are generated using a log-normal distribution with the same mean and standard deviation used in the base-case fracture system. 6.6.2. Statistics In Figure 6-44 the 9 angle of particle movements within fracture elements is plotted as a function of the § angle. Superimposed upon this plot are the curves defining the relationship between the average value of 9 and the average value of § for the two fracture sets. Assigning a set-direction to the element-to-element particle movements is not as straightforward for the Levy-Lee fracture system as it is for the base-case fracture system (Figure 5-11). However, path-length vectors are still assigned a set-direction using this curve, despite the potential for problems. In order to examine whether the set assignment is likely to be effective, histograms for the 9 and (j) components of the particle vectors are plotted (Figure 6-45 and Figure 6-46), and these can be compared to the histograms for the base-case system (Figure 5-16 and Figure 5-17). There is no question that the distributions of 9 and (j) are different for the Levy-Lee fracture system. The distribution of 9 for direction one is almost uniform, but as there is a low probability of travel in direction one there are probably not sufficient samples to represent the actual distribution. The distribution of 9 in directions two and three are unimodal for the Levy-Lee networks, but cover a wider range of values than for the base-case fracture system. However the peak value for 9 in direction three is only slightly lower than 240 the peak value for the trajectory in direction two, whereas in the base-case model there is a clear difference between the average orientation of the two set-directions. The direction four 0 histogram is strongly skewed and uneven, and although the histograms for direction four are also strongly skewed for the base-case model, this skew is in the opposite direction. For direction 4 it appears the histograms peak near 180°, and that this is just a truncated sample of the direction 2 histogram. From the 9 histograms it could be argued that the particle movements assigned to direction four are composed of the upper tails of the distributions from directions two and three, although discussion of the <j) distribution (below) will demonstrate that this is not the case. Histograms of the <j) values for the Levy-Lee system are similar to those for the base-case fracture system. In directions 2 and 3 particle travel most frequently at a § value of 0°. In directions one and four the peak values of § are close to 90°. Although there is some unevenness in the histograms of 0 and it appears that the algorithm used for sorting element-to-element particle movements into path-length vectors is robust enough to calculate motions statistics in the Levy-Lee fracture system. Motion statistics for the Levy-Lee fracture system using a 10° d) cutoff are given in Table 6-4. The directional choice statistics are of similar magnitudes as those seen using a (j) = 10° cutoff for the base-case model (Table 5-4). The mean velocities are generally lower for the Levy-Lee fracture system, and the standard deviation of velocity is higher in direction two and three, approximately equal in direction four, and lower in direction one. The skew in the particle velocity is extremely small for both fracture systems. The mean path-length is considerably larger in the Levy-Lee fracture system than in the base-case fracture system and the standard deviation in path-length is approximately double that for the base-case. The 241 skew in the path-lengths is approximately 50% larger for the Levy-Lee fracture system. The large variability in path-lengths appears consistent with particles moving through fractures from a range of length-scales. The mean direction of 9 is similar between the Levy-Lee and base case systems in direction 2, but the mean 9 in directions three and four is closer to 180° in the Levy-Lee system. The mean values of (j) are similar between the fracture systems, except that movement in directions three and four is closer to horizontal in the Levy-Lee. 6.6.3. Moments for network realizations In Figure 6-46 and Figure 6-47 moments for the discrete and continuum models of the Levy-Lee fracture system are plotted as a function of time. These moments are similar in magnitude to the average moments for the lower-density Baecher model, probably because the number of fractures and hence the connectivity in each network realization is similar. The first x moments for the Levy-Lee system are slightly large than the first x moment for the lower-density model, but not as large as the first x moment for the base-case model. The first y and z moments for the Levy-Lee are somewhat smaller than for the lower-density model. There is little net movement in the y and z directions, which is surprising as the fracture sets are not on average perpendicular so that some net movement iny and z would be expected. The variation in the first moments is similar between the lower-density model and the Levy-Lee model although there is a slightly higher standard deviation in the first x moment in the Levy-Lee model. Spreading seems to be of a similar magnitude in all three of the second moments, and the variation in the spreading is comparable between the lower-density model and the Levy-Lee. Mass moves through the Levy-Lee networks at a similar rate as Baecher networks of a similar fracture density, although in the Levy-Lee model there is slightly less net movement in the first y and first z moments. 242 6.6.4. Moments for SCM models Also shown on Figure 6-46 and Figure 6-47 are the moments for 4 S C M models. Two of these models are based on the long definition of path-length statistics, and the other models use statistics calculated using a 10° (j) cutoff. The best model, model three, is a G(v)G(l) model using conditional probabilities, and the motion statistics are calculated under a 10° (j) cutoff. The injection band for the discrete domain is 7m x 7m, and although most S C M models use a 6m x 6m band the best match to the average moments for the discrete networks occurs when the injection band is reduced to 4m x 4m. With this reduction, matches are excellent for all of the moment values except the second z moment. A l l three of the first moments are closely matched although the values of the first x moments appear to evolve at a slightly different rate than the average for the network realizations. Model three also provides an excellent match to the average second x and second y moments. However, although model three appears to match the rate of change of the second z moment, the initial value of spreading in the z direction is lower than the average for the network realizations. Model two is identical to model three except that a 6m x 6m injection band is used. Increasing the injection band does not substantially change the first x ,y and z moments. The larger injection band results in slightly larger spreading for the second x moment, although the match is still fairly good. For the second y moment the rate of change is very similar to the average for the networks, but the initial value is too high. The rate of change for the second z moment is very similar to the network average at later times but the initial value is slightly high. Although some fine tuning of the injection band size and shape might improve the match to the second moments at early times, there appears to be some trade off between matching the second y and matching the second z moments. 243 Model one and model four are the same as models two and three except that statistics calculated under the long path-length definition are used instead of the <j) = 10° cutoff used for models two and three. Model one is G(v)G(l) with conditional probability using a 6m x 6m injection band. This model works well for all of the first moments, as do most models. The rate of spreading is slightly high for the second x moment, and although the rate of change is similar for the second y and second z moments, the early values of these moments are higher than the averages for the networks. Model four is the same as model one with a smaller injection band. Reducing the injection band causes the secondy moment to start at a level similar to the average for the networks but the spreading then increases at a higher rate than the average until about 2000 days . After 2000 days there is a break in the slope, so that the spreading evolves at a rate similar to the average for the networks. A similar break in the curve occurs in the second z moment, but in this case model 4 starts with a lower second z moment and then levels off close to the average around 2000 days. A l l other moments are unaffected by changes in the size of the injection band The statistical continuum method produced moments close to the average moments for the Levy-Lee network realizations. Particle movement through the LeVy-Lee fracture system is fundamentally different from movement through the Baecher fracture systems, and this is most clearly demonstrated by the lack of trends in the element-to-element scatter plots. However, the multi-scale nature of the networks may not significantly influence the transport of mass as the networks are still dense and well connected. It is interesting that the S C M method works despite the potential failings of the algorithm used to separate particle movements into fracture sets . 244 6.7. Summary of the SCM results in three dimensions In this chapter moments for three-dimensional S C M models were compared to average moments for three-dimensional network realizations from six fracture systems. For all of the fracture systems examined, an S C M model can be found which produces moments within one standard deviation of the average moments for the network realizations. It does not appear that one common set of parameters is best for all moments in each of the six fracture systems, but the G(v)G(l) model in which particles are not allowed to reverse direction on adjacent steps produces the best results more frequently than any other. Matches between the moments are acceptable for the single length-scale fracture systems, over a range of fracture densities. The match between the moments for the S C M model and moments for the Levy-Lee fracture system were also acceptable. The match between the moments for the S C M model and the average moments for the network realizations is not perfect for any of the fracture systems. A perfect match is not expected because of the loss of information between the discrete and continuum domains. In order to evaluate the closeness of the match, the standard deviation of the moments for the network realizations is calculated. The difference between the moments for the continuum model and the average moments for the network realizations is less than one standard deviation for the best model in all fracture systems. Not only are the values of the moments for the S C M models close to the average values for the networks but the moments change at similar rates, even when values are not equal. The best S C M model for each fracture system is given in Table 6-5, along with the errors in the slope of the moments. As was done in the two-dimensional analysis the error in the slope of the moments is calculated by taking the difference between the slopes of the 245 moments for the discrete and S C M models, and dividing by the average moments for the network realizations. A l l of the slopes are taken between 1000 and 3000 days. For all of the three-dimensional fracture-systems, the G(v)G(l) model provides the best results, although the best algorithm for directional choice and the best value of the (j) cutoff varies from model to model. Magnitudes of the average errors are around 12% for all of the models with the largest average error in the lower-density model (13.9%) and the smallest average error in the Levy-Lee model (11.3%). In the two-dimensional modeling (Table 4-11) the average errors ranged from approximately 6% to 36%. For the two and three-dimensional base-case models the errors in the first x and y moments are of similar magnitudes, while the error in the first z moment is substantially larger in the three-dimensional model. The two-dimensional S C M models appear to produce better results than the three-dimensional models. This is possibly due to the uncertainty introduced into the three-dimensional modeling during the calculation of motion statistics, when particle movements within fracture planes cannot be separated from particle movements from one fracture to the next (section 5.3.3). Although the errors in the slopes of the moments were large for the three-dimensional models, the values of the moments were quit close to the average values of the moments for the three-dimensional network realizations. It appears that over the 20m x 20m x 20m model domain tested the S C M can represent average mass transport through discrete fracture systems, at least to the degree that can be measured by the first and second spatial moments. It may be that at higher moments (e.g. the third moment measuring the skew of the mass distribution) there are significant differences between the continuum and discrete models. However, it would seem unlikely 246 that good agreement between the first and second moments would occur i f there are large discrepancies in the third moments. Moments for S C M models are influenced by the methods under which the motion statistics are calculated, and by the choice of algorithms that translate motion statistics are into particle movements. Moments vary when using histograms, Normal models or Gamma models for either the path-length or the velocity. In an effort to compensate for the information lost by reducing particle movements to motion statistics, correlations between the velocity, path-length and directional choice parameters are examined. Although these correlations affect the moment values, there is not a consistent enough improvement in the match to the moments for network models to suggest that these correlations cause particle movement in the continuum to be close to particle movement in the discrete networks. Changing the size of the injection band also changes the values of the second moments at early times, but at later times the effect of the injection band are diminished. Changes to the cutoff value of § at which a change in particle trajectory is treated as a new particle step also affects the moments for the S C M models. A lower § threshold causes particle paths to be broken into a larger number of smaller steps than a higher threshold, or the long-definition of path-length. In some case results are improved by applying a <j) = 10° cutoff, but this is not consistently true for all models and all moments. The issue of the (j) cutoff arises in part because it is not possible with the current modifications to the F R A C M A N software to determine when particles move from one fracture to another in the discrete subdomain. If the method is modified to allow this determination then the § cutoff may no longer be as significant an issue. 247 This ends the presentation of the S C M modeling results. The final chapter discusses the implications of the S C M modeling results, and suggests where and how the method might be applied. 248 Table 6-1 Input parameters for the variable orientation fracture system. Parameter Fracture set one Fracture set two Generation region 21 mby 21 mby 21 m 21 mby 21 mby 21 m Number of fractures 1000 1000 Fracture Model Enhanced Baecher Enhance Baecher Generation Mode Centers Centers Truncation Mode Off Off Number of Sides 4 4 Pole: (trace, plunge) 90,-10 -20,20 Pole distribution Bivariate Normal Bivariate Normal K1,K2,K12 3,3,0 3,3,0 Size distribution Lognormal Lognormal Mean Radius 1.4 (m) 1.4 (m) Standard deviation 0.8 0.8 Aspect Ratio 1 1 Transmissivity Distribution Lognormal Lognormal 1.0e-06(m2/s) Mean Transmissivity 1.0e-06(m2/s) Standard deviation 5.0e-07 5.0e-07 Table 6-2 Input parameters for the lower density fracture system. Parameter Fracture set one Fracture set two Generation region 21 m by 21 m by 21 m 21 m by 21 m by 21 m Number of fractures 400 400 Fracture Model Enhanced Baecher Enhance Baecher Generation Mode Centers Centers Truncation Mode Off Off Number of Sides 4 4 Pole: (trace, plunge) 90,-10 -20,20 Pole distribution Bivariate Normal Bivariate Normal K1,K2,K12 3,3,0 3,3,0 Size distribution Lognormal Lognormal Mean Radius 1.4 (m) 1.4 (m) Standard deviation 0.8 0.8 Aspect Ratio 1 1 Transmissivity Distribution Lognormal Lognormal Mean Transmissivity 1.0e-06 (m2/s) 1.0e-06(m2/s) Standard deviation 5.0e-07 5.0e-07 249 Table 6-3 Input parameters for three-dimensional Levy-Lee fracture system. Parameter Fracture set one Fracture set two Generation region 21 m by 21 m by 21 m 21 m by 21 m by 21 m Number of fractures 250 250 Fracture Model Levy-Lee Levy-Lee Generation Mode Centers Centers Truncation Mode Off Off Number of Sides 4 4 Pole: (trace, plunge) 90,-10 -20,20 Pole distribution Bivariate Normal Bivariate Normal K1,K2,K12 20,20,0 20,20,0 Size distribution Truncated Exponential Truncated Exponential Mean Radius 1.8 (m) 1.8 (m) Maximum Radius 30 30 Minimum Radius 0.1 0.1 Fractal Dimension 1.0 1.0 Minimum Step 0.1 0.1 Aspect Ratio 1 1 Transmissivity Distribution Lognormal Lognormal Mean Transmissivity 1.0e-06(m2/s) 1.0e-06(m2/s) Standard deviation 5.0e-07 5.0e-07 Table 6-4 Motion statistics for the Levy-Lee fracture system based on 1000 realizations of the discrete subdomain. Set-direction 1 2 3 4 Directional Choice 0.008 0.511 0.336 0.145 Velocity: Mean (-logio ni/s) 2.69 2.22 2.37 2.41 V : Standard deviation 0.38 0.31 0.27 0.34 V:Skew -6.47e-03 -0.98e-03 0.46e-03 -0.64e-03 Length: Mean (m) 0.48 1.21 0.95 0.73 L: Standard deviation 0.59 1.53 1.21 0.97 L: Skew 1.00 1.11 1.16 1.22 Direction after 1. 0.088 0.349 0.300 0.267 Direction after 2. 0.007 0.574 0.264 0.155 Direction after 3. 0.025 0.352 0.518 0.104 Direction after 4. 0.015 0.434 0.214 0.036 Mean 9: -36.8 179.7 144.6 -129.0 Mean §: 51.9 -3.6 22.7 21.5 250 Table 6-5 Errors in the slope of the moments for varying fracture systems. The average value represents the average of the absolute values of the errror. Each model presented gives the lowest average, absolute error. Model/Error jUX /JZ Average Base-Case G(v)G(l) -0.062 0.044 -0.163 -0.341 0.001 -0.135 0.125 no reversal/ no repeat Gradient: 0 = 0°, (j) = 0° Long-path Base-Case G(v)G(l) -0.288 0.026 0.042 -0.027 -0.228 -0.181 0.132 no reversal/ no repeat Gradient: 0 = 45°, d> = 45° Long-path Variable strike and dip 0.00 0.098 -0.258 -0.163 0.085 -0.084 0.115 G(v)G(l) no reversal Long-path Four m cutoff 0.00 0.04 -0.275 -0.187 -0.086 -0.263 0.142 G(v)G(l) no reversal Long-path Lower Density 0.053 0.542 0.034 -0.120 -0.054 0.020 0.139 G(v)G(l) no reversal Long-path Levy-Lee G(v)G(l) -0.016 -0.021 -0.169 -0.032 -0.092 -0.348 0.113 Conditional Prob. <) = 10° cutoff 251 _ O H(v)H(l) 2000 4000 Time (days) Figure 6-1 First moments for the base case model with applied gradient at 9 = 0° and § = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. Continuum models are H(v)H(l),H(v)G(l), N(v)G(l) and G(v)G(l) using long-path motion statistics. 252 30 0 I i i i I 0 2000 4000 Time (days) Figure 6-2 Second moments for the base case model with applied gradient at 9 = 0° and § 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. Continuum models are H(v)H(l),H(v)G(l), N(v)G(l) and G(v)G(l) using long-path motion statistics. 253 <y Unconditional 0 2000 4000 Time (days) Figure 6-3 S C M models are G(v)G(l) with varying directional choice algorithms. First moments for the base case model with applied gradient at 0 = 0° and d> = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 254 . 1 I I <0> Unconditional - f Conditioned on previous • No repeat - No reverse Q No reverse 0 I i i i I 0 2000 4000 Time (days) Figure 6-4 S C M models are G(v)G(l) with varying directional choice algorithms. Second moments for the base case model with applied gradient at 9 = 0° and § = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 255 Figure 6-5 S C M models are H(v)H(l) with varying directional choice algorithms. First moments for the base case model with applied gradient at 0 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 256 0 I i 1 1 1 0 2000 4000 Time (days) Figure 6-6 S C M models are H(v)H(l) with varying directional choice algorithms. Second moments for the base case model with applied gradient at 0 = 0 ° and (j) = 0 ° . Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 257 16 12 8 4U 0 O N(v,(|))N(l,(|>) + N(v,4>)G(l) • N(v)N(l,4>) O H(v)N(M>) 0 2000 Time (days) 4000 Figure 6-7 S C M models including velocity and path-length as a function of §. Second moments for the base case model with applied gradient at 6 = 0° and § = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 258 o O N(v,<|>)N(M>) + N(v )^G(l) 0 12 0 I i i 1 1 0 2000 4000 Time (days) Figure 6-8 S C M models including velocity and path-length as a function of (j). Second moments for the base case model with applied gradient at 0 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 2 5 9 0 2000 4000 Time (days) Figure 6-9 S C M models in which § is calculated as a function of 9. First moments for the base case model with applied gradient at 9 = 0° and <J> = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown dashed lines. 260 CN X CN CN 30 24 18 12 -0 O H(v)H(l) + H(v)G(l) • G(v)G(l) O N ( V ) G ( ! ) 0 12 6 -0 12 6 0 2000 Time (days) 4000 Figure 6-10 S C M models in which <\> is calculated as a function of 9. Second moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 261 Figure 6-11 S C M models in which velocity as a function of path-length First moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 262 b CN CN 30 24 18 12 O N(v,l)H(l) Unconditional + G(v,l)H(l) Unconditional • G(v,l)H(l) No rev. - no repeat O G(v,l)H(l) No reverse 0 12 rjo00 0 12 6 h 0 0 2000 Time (days) 4000 Figure 6-12 S C M models in which velocity as a function of path-length Second moments for the base case model with applied gradient at 0 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 263 0 2000 4000 Time (days) Figure 6-13 S C M models using motion statistics calculated using a (j) = 10° cutoff. First moments for the base case model with applied gradient at 0 = 0° and § = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 264 CN CN CN 30 24 18 12 O H(v)H( l ) + H(v)G( l ) • G(v)G( l ) 0 2000 Time (days) 4000 Figure 6-14 S C M models using motion statistics calculated using a (j) = 10° cutoff. Second moments for the base case model with applied gradient at 6 = 0° and <j) = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines.. 265 Time (days) Figure 6-15 S C M models using motion statistics calculated using a d> = 5° cutoff. First moments for the base case model with applied gradient at 9 = 0° and (j> = 0°. Average for network realizations is the solid line with ± one standard deviation shown as dashed lines. 266 30 O H(v)H(l) + H(v)G(l) • G(v)G(l) 12 i 1 1 r 0 I I | L 12 l 1 1 r 0 I i 1 1 1 0 2000 4000 Time (days) Figure 6-16 S C M models using motion statistics calculated using a (j) = 5° cutoff. Second moments for the base case model with applied gradient at 9 = 0° and (j) = 0°. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 267 16 12 8 o-Unconditional 4- Conditioned on previous 0 • No repeat - No reverse o ^ Q -Q No reverse 0 Is? 0 2000 Time (days) 4000 Figure 6-17 G(v)G(l) S C M models with varying directional choice algorithms, using d> = 10° motion statistics. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 268 CN CN CN 30 24 18 12 r -6 L 0 12 0 Unconditional -|- Conditioned on previous • No repeat - No reverse Q No reverse 0 12 0 0 2000 Time (days) 4000 Figure 6-18 G(v)G(l) S C M models with varying directional choice algorithms, using <j) = 10 motion statistics. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 269 0 j 1 1 r 0 2000 4000 Time (days) Figure 6-19 Base case model with applied gradient at 9 = 45°, (j) = 45°. S C M models are applied without correlations. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 270 (N CN 0 CN 30 24 18 -12 h O H(v)H(l) + H(v)G(l) • G(v)G(l) O N(V)G0) 0 12 er0-Bo"[J 0 12 + 0 +o+o u O o 4 0 0 2000 Time (days) 4000 Figure 6-20 Base case model with applied gradient at 6 = 45°, d) = 45°. S C M models are applied without correlations. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 271 Is? 12 8 |_ 0 Unconditional -|- Conditioned on previous • No repeat - No reverse 0 <^  <b_ Q No reverse <> o 0 2000 Time (days) 4000 Figure 6-21 G(v)G(l) with varying probability algorithms. Base case model with applied gradient at 9 = 45°, § = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 272 CN CN CN 30 24 18 L 12 0 12 0 <0> Unconditional - f Conditioned on previous • No repeat - No reverse Q No reverse 0 12 •<r^  o o 0 2000 Time (days) 4000 Figure 6-22 G(v)G(l) with varying probability algorithms. Base case model with applied gradient at 9 = 45°, (j) = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 273 IN? 12 8 h 0 <0> Long path + (j) = 30° cutoff • (j) = 10° cutoff o o o o O H 4 O 4> = 1° cutoff o o 0 o ^ " ^ r + n D I I I 0 0 2000 Time (days) 4000 Figure 6-23 G(v)G(l) model with no-reversal/no-repeat algorithm using motion statistics with varying (j) cutoffs. Base case model with applied gradient at 0 = 45°, (j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 274 CN b CN b CN b 30 24 18 |- O Long path + (j) = 30° cutoff • (j) = 10° cutoff O i = l° cutoff 12 -6 U 0 12 o o o o o - r ^ ^ ^ n < > i . ^ r T ^ n t P - - I 3 - — " ~ ~ ^ n O O O 0 12 T * d o o o o o o o 0 0 2000 Time (days) 4000 Figure 6-24 G(v)G(l) model with no-reversal/no-repeat algorithm using motion statistics with varying (j) cutoffs. Base case model with applied gradient at 0 = 45°, (j) = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 275 Figure 6-25 S C M models with velocity, path-length and (j) correlations. Base case model with applied gradient at 0 = 45°, <j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 276 CN CN 30 24 18 U 12 6 k 0 12 .0 H(v)N(l,(|)) long-path + H(v)N(l,(|)) (j) = 10° cutoff N(v,l)H(l) long-path O N(v,l)h(l)(|) = 10° cutoff o • o • 0 12 JT^^^^^ •__o- 1 a- — — T + + +- +• 0 o o ° + - r ^ — . + + + 0 2000 Time (days) 4000 Figure 6-26 S C M models with velocity, path-length and d> correlations. Base case model with applied gradient at 9 = 45°, <j) = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 277 12 8 O H(v)H(l) (j>;f('e) long-path + H(v)G(l) (b;f(G) long-path • H(v)H(l)<|> = 10° cutoff O H(v)G(l) (j) - 10° cutoff 0 4 T + + + -^ 4- + + + + _L _L IN? 0 -4 -8 1 1 1 , Do no no oo no oo no oo no no DO no no cog 0 2000 Time (days) 4000 Figure 6-27 S C M models with <j) calculated as a function of 9. Base case model with applied gradient at 0 = 45°, (j) = 45°. First moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 278 <N 0 CN 0 CN 30 24 18 -12 -6 -0 12 1 1 1 — O H(v)H(l) (j);f(0) long-path + H(v)G(l) d);f(0) long-path • H(v)H(l)(j)-10° cutoff O H(v)G(l)(j) = 10° cutoff 0 12 ^ ^ ^ ^ o_ _<> O-6 h 0 T T * o o o+. 0 2000 Time (days) 4000 Figure 6-28 S C M models with d) calculated as a function of 6. Base case model with applied gradient at 9 = 45°, d) = 45°. Second moments. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 279 Fracture Traces 20 m Fracture Planes 20 m 20 m Figure 6-29 Variable theta and phi network. 280 Figure 6-30 Scatter plot of 0 vs. d) for path-length vectors from one realization of the fracture system with variable strike and dip orientation. 281 0 2000 4000 Time (days) Figure 6-31 Variable strike and dip fracture system. First moments. Applied hydraulic gradient at 9 = 0° and § = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 282 CN X CN CN 30 24 O H(v)H(l) + H(v)G(l) • G(v)G(l) o N( v)G( ]) o_| + 0 2000 Time (days) 4000 Figure 6-32 Variable strike and dip fracture system. Second moments. Applied hydraulic gradient at 0 = 0° and d) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 283 16 - 0 Unconditional 0 2000 4000 Time (days) Figure 6-33 S C M models are G(v)G(l) with varying directional choice algorithms. First moments for the variable strike and dip fracture system with applied hydraulic gradient at 9 = 0° and (j) = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 284 CN b CN CN b 30 24 18 L 12 0 12 0 12 6 t ~ o 0 2000 Time (days) 4000 Figure 6-34 S C M models are G(v)G(l) with varying directional choice algorithms. Second moments for the variable strike and dip fracture system with applied hydraulic gradient at 8 = 0° and § = 0°. Long-path motion statistics. Average for the network realizations is the solid line with ± one standard deviation shown as dashed lines. 285 Fracture Traces Figure 6-35 Variable theta and phi network, fractures truncated at an equivalent radius of 4 m. 286 <y Unconditional -(- Conditioned on previous 0 2000 4000 Time (days) Figure 6-36 First moments for the V F V T Network with all fractures truncated at an equivalent radius of 4m. S C M models are G(v)G(l) with varying directional choice algorithms. 287 30 CN b b CN 24 18 12 <3> Unconditional -|- Conditioned on previous • No repeat - No reverse O No reverse 0 12 0 0 12 6 - ^ 0 2000 Time (days) 4000 Figure 6-37 Second moments for the V F V T Network with all fractures truncated at and equivalent radius of Am. S C M models are G(v)G(l) with varying directional choice algorithms. 288 Fracture Traces 20 m Fracture Planes 20 m 20 m Figure 6-38 Fracture networks for lower-density system. Networks contain 40% of the number of fractures in the variable orientation fracture system. 289 _ <^> Unconditional 0 2000 4000 Time (days) Figure 6-39 First moments for the lower-density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are G(v)G(l) models with various directional choice algorithms 290 30 I i i r ~ <3> Unconditional 24 - + Conditioned on previous • No repeat - No reverse - Q No reverse H 18 -CN b 12 I 1 1 r 0 I 1 1 1— 1 0 2000 4000 Time (days) Figure 6-40 Second moments for the lower-density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are G(v)G(l) models with directional choice algorithms. 291 IN? 0 2000 Time (days) 4000 Figure 6-41 First moments for the lower-density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are all G(v)G(l) models with the no-reversal/no-repeat algorithm, with various injection band sizes. 292 30 CN X b CN b CN b 24 18 12 0 12 6 -o + • o 6 ra x 6 ra 4 m x 4 m 2 m x 2 m 1 ra x 1 m 0 12 0 0 2000 Time (days) 4000 Figure 6-42 Second moments for the lower density fracture system. Continuum models are based on statistics calculated using a 10 degree phi cutoff. S C M models are all G(v)G(l) models with the no-reversal/no-repeat algorithm, with various injection band sizes. 293 Figure 6-44 Scatter plot of element to element particle movements with the average relationship between theta and phi overlaid. Based on one network realization of the Levy-Lee fracture system 295 Set-direction 1 Set-direction 2 C cu S3 cr CU u to 0.04 0.0 180 360 0.1 0.0 180 360 Set-direction 3 360 0.15 Set-direction 4 0.0 180 360 e e Figure 6-45 Histograms of 9 for the Levy-Lee fracture system, x axis is 0 in degrees, y is frequency. 296 o S3 33 cr 360 cs 33 cr Li. Set-direction 3 360 Set-direction 4 0.1 0.0 i r j i I-I Mil 180 360 Figure 6-46 Histograms of d> for the Levy-Lee fracture system, x axis is d> in degrees, y axis is frequency. 297 12 10 8 _<> G(v)G(l) Conditional prob. + G(v)G(l)CP, <j) = 10° 0 0 • G(v)G(l)_C?^jn injection I ' D"G(v)G(l) CP, 4 min], § = 10° ^ 4 h 0 0 2000 4000 Time (days) Figure 6-47 First moments for the Levy-Lee model. Various S C M models. 298 CN b CN PS CN 30 24 18 12 0 G(v)G(l) Conditional prob. + G(Y)G(1) CP, <|> = 10° • G(v)G(l) CP, 4 m2 injection O G(v)G(l) CP, 4 m2inj, <|> = 10c X N(v)N(l,(t)) 6 m 2 inj, d) = 10° 0 12 r 0 12 T 6 h O <> • ° 0 0 2000 Time (days) 4000 Figure 6-48 Second moments for the Levy-Lee model. Various S C M models. 299 7. Discussion 7.1. Introduction In this thesis the Statistical Continuum Method has been developed in two and three dimensions. By using particle tracking, the S C M approach allows large-scale continuum models to include the details of the small-scale interactions occurring within discrete networks. In order to capture these interactions, particles are "educated" by first travelling through a discrete subdomain consisting of multiple network realizations, all of which are significantly smaller than field-scale. The approach is based on the premise that solute movement by advection from one fracture intersection to the next is independent of the scale of the network. The method is capable of including the effects of fracture networks in the continuum because each step the particle takes in the continuum is consistent with the restrictions the geometry of a fracture network places on aqueous phase transport. This consistency has been demonstrated by using S C M models to match the spatial moments of mass distribution in both two and three-dimensional discrete network models. 7.2. Is the test against the average values of the first and second spatial moments for the discrete networks sufficient? Although the S C M can match the average spatial moments for larger-domain discrete network simulations, the sufficiency of this test must be assessed in terms of the application for which the S C M model is intended. There are two issues regarding this sufficiency: (1) Is the match between the moments sufficiently close that the method can be applied to predict mass transport in field situations? (2) Is the size of the discrete network domain large enough 300 to demonstrate that matching the moments is sufficient to validate application of the method in a substantially larger domain? The moment results presented in Chapters 4 and 6 suggest that a S C M model will reasonably approximate the average rate of mass transport and spreading predicted by a discrete network model. Reproducing the average moments for a discrete network model may be sufficient for theoretical studies, but could be problematic in field applications. At a field site a natural fracture network is statistically equivalent to a single realization of a fracture system, so that even a successful prediction of the first and second moments may not provide sufficient information to characterize the rate of mass transport at that site. The large standard deviations around the average value of the moments in the discrete network realizations are indicative of the wide range of advective velocities and spreading rates from one network realization to the next. The moments provided by the S C M approach are no more or less applicable to site-assessment than the moments determined by averaging over multiple discrete network realizations. Each approach may be sufficient for predicting the average response at a field site, but neither can predict the exact response. Multiple discrete networks can also be used to predict the uncertainty in the average moments. Currently the S C M model does not generate a measure of the uncertainty in the average moments because only the average values for each of the motion statistics are calculated. It may be possible to quantify the uncertainty in the moments for S C M models i f the variability of the motion statistics from one subdomain network to another were calculated, and this may be a subject of future research. A significant issue is that the distances over which the prediction of mass transport is likely to be made are significantly larger than the large-domain network realizations used to 301 evaluate the method. It is reasonable to expect that moments for the S C M method will continue to be close to the average for the network realizations over larger distances but this cannot be assumed to be true. The network realizations act as a surrogate for a field site, but reproducing the results of discrete network models is not a guarantee that S C M models will reproduce mass transport at a field site of similar size, or larger. Therefore, the key issue is not whether moments for S C M models will continue to match average moments for larger discrete network realizations, but i f moments for S C M models will match mass transport measured at a larger field site. This question can only be answered by applying S C M modeling as part of a field investigation. A perfect test of the sufficiency of the S C M method is not available, and as a result the comparison of mass transport in S C M models to mass transport in discrete networks cannot substantiate the validity of the S C M model in all situations. The success or failure of an S C M model is discussed in terms of ergodic theory in the following section. 7.3. Ergodicity and the SCM method Previous chapters have compared the results of S C M modeling with the results of discrete network modeling. In densely-fractured networks in two and three dimensions, the S C M method predicts the first and second spatial moments within one standard deviation of the average moments for the network realizations. However, for the two-dimensional Levy-Lee model with a fractal dimension of 1.5, the S C M model failed to produce moments within one standard deviation of the average moments for the network realizations. For the three-dimensional Levy-Lee model the S C M method is effective. Ergodic theory provides a frame of reference in which to interpret the S C M results, and helps to predict the situations in which the S C M may be effective. 302 The ergodic hypothesis can be said to hold i f in a given realization of a stochastic process, all states of a system are represented (Beran, 1968; Dagan, 1984, 1986, 1989, 1990, 1991, 1994). In stochastic modeling of porous media Naff et al. (1998) suggest that: Ergodic behavior implies that as a tracer cloud from a single realization grows to encompass multiple length-scales of the medium, the moments of this single realization cloud will come to mimic the ensemble moments. In discrete fracture models the states of the system are the range of velocities, path-lengths and directional choices along particle travel paths. Therefore, when comparing mass transport in S C M models to mass transport in large-domain discrete fracture models or field sites, two requirements must be met in order for the ergodic hypothesis to hold: (1) particles must sample a sufficient range of velocities, path-lengths and orientations (states) in the discrete subdomain so that the motion statistics calculated within the subdomain are representative of the ensemble motion statistics for the fracture system, (2) when comparing moments for the S C M models to moments for the large-domain network realizations or at a field-site, mass must sample a sufficient range of states in the large-domain network or field site that plume movement can be considered representative of the ensemble average for the fracture system. With some restrictions, under the ergodic hypothesis, it is possible to interchange spatial and ensemble averaging (Freeze, 1975; Dagan, 1990). It is therefore possible to satisfy the first ergodic condition (above) by using either a sufficiently large network, or a large number of statistically equivalent network subdomains. Meeting the second condition depends on the field site or fracture system being modeled. For example, mass moving through a dense fracture network encounters a wide range of velocity and path-length combinations after a relatively short travel distance, and the 303 values of the spatial moments rapidly approach representative values for the fracture system. In a sparse network, fracture intersections are less spatially frequent, so that moment estimates in the discrete network are less likely to approach representative moments for the fracture system over the same transport distance, or will approach representative moments only after mass has traveled further within the network. In the two-dimensional Levy-Lee fracture systems satisfying either of the ergodic conditions may be difficult. In multi-scale fracture networks, the larger the domain the greater the maximum fracture length it contains. Similarly, the further through a multi-scale network that a particle travels, the greater the number of length-scales of fracturing that are encountered. If the length-scales encountered increase with domain size then it may not be possible to find an average or ensemble behavior for mass travelling through a multi-scale fracture system. For example, i f longer path-lengths are encountered in larger domains then the average path-length may become a function of the size of the network. If the average mass motion varies as a function of network size, then motion statistics from a subdomain are not equivalent to motion statistics from a larger network. However, the S C M method samples networks at the scale of the path-lengths, the length-scale of which are on the order of the distance between fracture intersections. If a network is sufficiently dense, the addition of a fracture from a higher length-scale will not significantly alter the length-scale of the path-lengths. As a result the S C M method effectively models mass transport in the large domain for the D = 1.8 Levy-Lee fracture system and the three-dimensional Levy-Lee fracture system, because in these situations the path-length is not significantly affected by the domain size. 304 The likelihood of satisfying the second ergodic condition is a measure of the likelihood that moments estimated using the S C M method will be similar to moments from a discrete model or field location. The S C M method is based on the assumption that the movement of mass through the continuum will be equivalent to the movement of mass through a large-network realization, only if the statistical distribution of velocities and path-lengths traveled by individual particles in the large network is reproduced by particles travelling in the continuum. In order to insure that the range of particle movements in the continuum remains statistically equivalent to the range of particle movements in the large-domain network, the distributions of the motion statistics calculated within the discrete subdomain are sampled at random in the continuum domain. This independent, random sampling of motion statistics implicitly assumes that any combination of velocity and path-length is allowable in the continuum domain, i.e. that all states within the fracture system are available to particles travelling in the continuum. In other words, particles move through a S C M model under the assumption that the ergodic hypothesis holds. Therefore, since satisfying the second condition determines how closely the ergodic hypothesis holds in the discrete model, and since the continuum produces the movement of mass that occurs when the ergodic hypothesis holds, the degree to which the second ergodic condition is satisfied in the discrete network also determines how closely moment values for S C M models will match moment values for the large-domain discrete model or field site. Ergodic theory also suggests an explanation for the observation that S C M models using uncorrelated directional choice statistics generally produce better moment estimates than S C M models in which directional choice is correlated to the previous travel direction. In the two-dimensional examples all models use uncorrelated probabilities, and in three dimensions all 305 models except the conditional probability model use uncorrected probabilities. Path-length is defined as the distance a particle travels between directional changes in the discrete network. Correlation of the directional choice probabilities to the previous direction traveled is consistent with the path-length definition, and should improve the S C M results because the correlation algorithm includes additional information about particle movement. However, S C M models in which directional choice is not correlated frequently produce moment estimates closer to the average moment for the network realizations. Results improve because uncorrected directional choice allows a wider range of combinations of velocity and path-length to occur in the continuum, so that a particle encounters a wider range of states, and the ergodic hypothesis is more likely to hold. Furthermore, uncorrected particles are allowed to take multiple steps in the same direction so that path-lengths in excess of those encountered within the discrete subdomain may occur in the continuum. Moving particles in the S C M without directional choice correlation is therefore consistent with particle motion in the network realizations because these longer paths also occur in the large-domain discrete networks. It may not be necessary to completely satisfy the first ergodic condition for the S C M method to work effectively. Using continuous statistical distributions (e.g. Gamma or Exponential) it is possible to generate extreme but infrequent combinations of path-length and velocity. These combinations are encountered in larger networks with a calculable probability, but are less likely to be encountered by particles travelling within the discrete subdomain. Some of the information necessary for particle travel therefore comes from the choice of the distribution model. This suggests that not all path-lengths and velocities need to be encountered within the discrete subdomain in order to be successfully predicted by an 306 S C M model, as long as appropriate distributions are used with sufficient estimates of parameters. 7.4. Probabilistic interpretation of SCM plumes For the base-case Poisson fracture system, spatial moments for the S C M models are statistically equivalent to the average spatial moments for the network realizations (e.g. Figure 4-3), so that it appears that the first and second ergodic conditions have been satisfied (section 7.3). However, i f mass movement in S C M models (e.g. plumes in Figure 4-2) is only statistically equivalent to mass movement through discrete networks (e.g. plumes in Figure 4-1), what do plumes generated by the continuum method represent? Two possible interpretations of the meaning of plumes are examined: (1) as probability fields, and (2) as maps of expected values of concentration. One interpretation is to consider a plume in the continuum model as a spatial probability field, at a fixed point in time t=T. If N particles are released from the source, each particle location (/= 1...N) within the continuum plume represents a l/Nprobability that mass travelling through a statistically equivalent fracture network wil l be located at point xt at time T. It is not, however, individual particle locations x, that are important, but instead it is their intensity that is significant. In order to interpret plumes as probability fields, two options appear viable: (1) define a compliance boundary so that the probability of non-compliance is given by the percentage of particles passing this boundary; or (2) produce a map of probability density by calculating the number of particles per unit area or volume. Probability plumes have been generated for solute movement through fractured clay aquifers (Schmidt, 1993) by directly solving the advection-dispersion equation and calculating concentration variation between realizations. Similar approaches have been applied in porous 307 media (e.g. Smith and Schwartz, 1981; Rubin, 1991). Schmidt (1993) calculates concentrations at specific locations within each realization, and then calculates both an average and standard deviation of concentration for each location. The distribution of concentrations is then used to calculate the probability that the average concentration will occur in any given realization. So then, does particle density in a continuum plume represent probability or concentration? Following Schmidt (1993), plumes from continuum models represent average mass movement, with the density of particles at any location being equal to the average concentration that could be expected at that location. 7.5. Extensions to the SCM 7.5.1. Variable motion statistics: Mixed gradient models and others Uniform hydraulic gradients were applied in all of the two and three-dimensional S C M models examined in Chapters 3 through 6. As a result only one set of motion statistics is necessary for each example. In this section conceptual models in which complexities in either the network geometry or the flow field necessitate the calculation of several sets of motions statistics are outlined. The common theme of these complexities is that particle motion changes as a function of location within the model domain. Models in which the local gradient changes in magnitude and orientation as a function of location are potentially one of the most powerful applications of the S C M . Statistics can be calculated over a suite of applied hydraulic gradient orientations. Then as particles move through the continuum the motion statistics used at each random walk step can be keyed to the local hydraulic gradient. This concept has been implemented previously for a simple orthogonal network in two-dimensions (Schwartz and Smith, 1988). 308 Determining the local gradient requires an estimate of the flow-field in the continuum. The flow field could be estimated using a porous media model with an estimate of the bulk transmissivity of the network, for the specified hydraulic boundary conditions. The problem with this approach is that the highly heterogeneous structure of the conductivity field within the fracture network is not represented in the flow solution. The effect of the conductivity structure is however included in the motion statistics, so that transport will still reflect the influence of the fracture geometry at the scale of the fracture intersections, even though flow does not. A second example in which more than one set of motion statistics would be applied is the case in which fracture properties change as a function of location. For example the density of fracturing in a network may decrease as a function of depth. A different fracture density could be used in each of a series of discrete subdomains, and motion statistics representative of each fracture density could be created. Then in the continuum the location of a particle would be used to determine the appropriate set of motion statistics. Using the S C M in this manner would allow models of transport in fracture networks to include levels of complexity significantly beyond what has previously been possible. 7.5.2. Predicting first arrivals One of the critical predictions in the assessment of waste disposal sites is the first arrival of contaminants at a compliance boundary. It is reasonable then to consider if and/or how the S C M model could be used to predict these first arrivals. Previous discussion has focused on using particle tracking in the S C M model to predict average moments values. However, as discussed in section 7.4 the particle locations within the S C M represent a probability field, with the outer boundaries of the plume defining to a quantifiable probability 309 the maximum travel distance within a given fracture system. A correlation can be made between the furthest extent of a plume and the.probability that mass will travel a given distance in a given time. With the appropriate statistical models reasonable estimates of the maximum travel distance and the maximum affected volume could be made. The S C M approach could therefore be used to predict the arrival of mass at a boundary in a probabilistic framework. 7.6. Summary 7.6.1. General • The statistical continuum method has been developed as a technique to model aqueous phase transport through a fractured rock mass at the field scale, while explicitly including the effects on transport of fractures that are observed on the scale of a borehole or outcrop. • The approach of the S C M is to determine the rules governing mass transport in a fractured medium by using discrete fracture network models representative of the medium, and then to apply these rules to mass transport in a large continuum model. • The key to the S C M approach is the selection of the statistical models and algorithms by which motion statistics in the discrete subdomains are converted to particle motions in the continuum. • The S C M method proved effective at modeling mass transport in a number of two and three-dimensional fracture systems, capturing both the mean velocity and the anisotropic dispersion of mass travelling through discrete fracture network models, as described by the first and second spatial moments. 310 Reproducing the spatial moments for the network realizations suggests that the influence of the fracture geometry on aqueous phase transport, including channeling and the anisotropic nature of dispersion, have been included in the S C M model. The success of the method depends upon particles sampling a sufficient range of velocities, path-lengths and orientations in the discrete subdomain that the motion statistics calculated within the subdomain are representative of the ensemble motion statistics for the fracture system. There are systematic changes in the velocity and path-length parameters, as a function of the size of the discrete subdomain, for both the two and three-dimensional models. The magnitude of these changes is relatively small, and the changes in path-length tend to be offset by the changes in directional choice. There is a correlation between the distance a particle wil l travel in a fracture and the fluid velocity within that fracture. Strong trends were also observed between the velocity, path-length and orientation parameters, in both two and three dimensions. However there were no correlations between velocity, path-length and orientation that consistently improved the S C M results. 7.6.2. Two-dimensional modeling Fracture systems examined included well-connected Poisson networks with varying fracture density; a Poisson fracture system near the percolation threshold; and two Levy-Lee fracture systems. S C M models were found to produce first and second moment values within one standard deviation of the average moments for network realizations, for all of the models except the Levy-Lee fracture system with D= 1.5. 311 No single S C M algorithm produced the best results for every fracture system. However, the best results were most frequently obtained using a S C M model in which velocity and path-length were modeled using three-parameter Gamma distributions (G(v)G(l)), and directional choice was not correlated to the previous direction traveled. Average errors in the slope in the asymptotic region of the moments vs. time curves ranged from 6% to 17% in models in which the S C M method was effective. When the fractal dimension of the Levy-Lee fracture system was raised to 1.8, S C M models were found that produced moment values within one standard deviation of the average moments for the network realizations. This suggested that the density of fracturing was a more significant factor in the success or failure of the S C M approach, than the range of length-scales of fractures within the network. 7.6.3. Three-dimensional modeling The three-dimensional models examined included Baecher models of varying density; models in which the orientation of fractures varied within the sets; and a Levy-Lee fracture system. S C M models were found to produce first and second moment values within one standard deviation of the average moments for network realizations, for all of the models presented. Average errors in the slope in the asymptotic region of the moments vs. time curves ranged from 11% to 14% in models in which the S C M method was effective. 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Neeham, T. and Jones, H. , Sampling of fault populations using sub-surface data: a review, Journ. of Struct. Geol, 18(2/3), 135-146, 1996. Zimmerman, R.W. and Bodvarsson, G.S., Hydraulic conductivity of rock fractures, Transport in Porous Media, 23, 1-30, 1996. 323 9. Appendix: Modifications To MAFIC In this appendix comments are made about the modifications to the F O R T R A N code used for the three-dimensional network modeling. The modification to M A F I C consists of the addition of write statements in two subroutines (1) M O V E . F for particle data at node locations and (2) TRAP3D.F for the injection location. Each write statement consists of the particle number, the element number, the time spent by the particle within the element, the x andy elemental velocities (which are never used), and the x, y and z coordinates of the particle. An additional modification is necessary to calculate the coordinates of the injection location of the particle. The subroutine global.f, which is routinely called within the move.f subroutine, is called within trap3d.f to convert the local coordinates, that is the coordinates of the particle measured relative to the triangular element, to the global coordinates, which are the model or true coordinates (italics indicate Golder definition). There is an intermediate step before the output from M A F I C is processed to calculate the vectors. When M A F I C is used for advective transport in non-porous media there is a numerical artifact introduced to the movement of some particles. Particles occasionally get trapped within the mesh, so that some particles return to an element within the mesh through which they have previously traveled. Before vectors can be calculated it is necessary to eliminate particles which have become stuck or bounce. Files named bounce.dat are files in which the output from M A F I C has all the bounced particles removed. The algorithm for removing particles is a simple check for repeated elements. Particles are allowed to repeat elements in adjacent steps, as this occasionally occurs when particles move from one side of an element to the next. However, i f a particle returns to an element with intermediate steps at 324 other elements then that entire particle is removed from the calculations of both statistics and moments. 325 

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