DUAL PERMEABILITY MODELINGOFFRACTURED MEDIAbyTHOMAS MELDON CLEMOB.S., University of California, Davis, 1976M.S., University of California, Berkeley, 1978A THESIS SUBMITFED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR TF{E DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Geological SciencesWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJUNE 1994© Thomas Meldon Clemo, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_______Department of GcLOC (LThe University of British ColumbiaVancouver, CanadaDate 9 192DE-6 (2/88)ABSTRACTDual permeability is introduced as an approach to modeling flow and transport through fracturedmedia. The approach allows a large reduction in the number of fractures that are representedexplicitly in a discrete fracture network. The most important fractures in terms of fluid flow areidentified using their physical characteristics. These fractures form a sub-network that divides theentire fracture network into smaller domains. The fractures of the smaller domains are approximated.The approximations do not rely on continuum assumptions. They are determined individually andindependently for each small domain, resulting in a parallel structure to the calculations.An exploratory model is developed for steady state fluid flow and solute transport in twodimensional fracture networks and compared to a discrete fracture model that represents all of thefractures explicitly. Individual fractures are represented by finite lines with constant hydraulictransmissivity. Solute transport calculations use particle tracking. The dual permeability model isshown to provide acceptably accurate solutions while reducing the maximum number of simultaneousequations by well over an order of magnitude.The dual permeability approach takes advantage of the hydraulic behavior of highlyheterogeneous media. Channeling of flow that develops in such media is used to reduce errorsintroduced through the sub-continuum approximations. The dual permeability approach works betterwhen the fracture system has a broad range in the scale of fracturing as demonstrated by transportcalculations for fracture networks with a high degree of clustering or with a fractal nature.A study of sub-REV fracture networks reveals that averaged flow through fractured media canbe represented using tensor notation in much smaller domains than an REV. At these scales,heterogeneity within a fracture domain causes the tensor representation to be asymmetric, unlike thecontinuum representation of granular porous media. Hydraulic heads within a heterogeneous fracture11network do not vary in a smooth fashion. The hydraulic head reflects the distribution of heterogeneitywithin the domain. The approximations developed for the dual permeability model required that theboundary conditions imposed on the small domains reflect heterogeneity both within the domains andexternal to them.111Table of Contents ivList of Tables ixList of Figures ixAcknowledgements xvChapter One: SIMUCATION OF FRACTURE NETWORKS AT THE FIELD SCALE1.1 Introduction1.2 Theoretical Motivation1.3 Thesis overviewChapter Two: FRACTURE SYSTEM CHARACTERIZATION: A REVIEW 1313141517171919202223242828292931TABLE OF CONTENTSPageAbstract114112.1 Introduction2.2 Sources of Fracture System Information2.2.1 Sources of Specific Information . .2.2.2 Sources of Representative Information2.3 Statistical Characterization of Fractures2.3.1 Organization of Fractures into Sets2.3.2 Orientation2.3.3 Spacing2.3.4 Size2.3.5 Shape2.3.6 Aperture and Transmissivity2.3.7 Fracture Density2.3.8 Persistence2.4 Discrete Fracture Structural Models2.4.1 Structural Models2.4.2 Uniform Spacingiv2.4.3 Poisson Model2.4.4 Truncated Poisson Model2.4.5 War Zone Model2.4.6 Parent-Daughter Model2.4.7 Hierarchical Model2.4.8 Fractal Model2.4.9 Colonnade Model2.5 Discussion of Fracture System CharacterizationChapter Three: FRACTURE NETWORK GENERATION AND FLOWREPRESENTATION IN A DISCRETE FRACTURE MODEL 443.1 Introduction 443.2 Definition of a Fracture Network 443.3 Fracture Network Generation 453.4 Implementing the Structural Models 463.4.1 Poisson Model 463.3.2 Truncated Poisson Model 483.3.3 War Zone Model 523.3.4 Levy-Lee Model 563.4 Fracture Network Reduction 583.6 Modeling of Fluid Flow and Solute Transport 603.6.1 Fluid Flow 603.6.2 Solute Transport 663.7 Boundary Conditions 733.8 Program Discrete . 7576768080848888943232343737393941AND TRANSPORTChapter Four: DUAL PERMEABILITY: IMPLEMENTATION AND EVALUATION4.1 Introduction4.2 Selection of Primary Fractures4.2.1 Primary Fracture Selection Criteria4.2.2 Primary Fracture Selection Algorithm4.3 Dual Permeability Flow Model4.3.1 Effective Conductance of Network Blocks4.3.2 Hydraulic Head DistributionVTABLE OF CONTENTS4.4 Dual Permeability Transport Model 964.4.1 Transport Through a Network Block.. 964.4.2 Particle Movement 1044.5 Dual Permeability Model Evaluation 1064.5.1 A Single Realization 1084.5.2 Monte Carlo Evaluation 1214.5.2.1 Introduction 1214.5.2.2 Fracture System 1 1244.5.2.3 Fracture System 2 - Common Aperture Statistics 1274.5.2.4 Fracture System 3 1334.5.2.5 Fracture System 4 - Uniform Apertures 1374.5.2.6 A War Zone Model; Fracture System 5 1404.5.2.7 A War Zone Model; Fracture System 6 1444.5.2.8 The Levi-Lee model; Fracture System 7 1484.5.2.9 Potential Bias Introduced by Failure to Find Primary FractureNetwork 1524.5.2.10 Summary of Monte Carlo Evaluation 1544.5.3 Accuracy as a Function of Network Block Size 1544.5.4 Time and Memory 1554.6 Comparison to Reduced Fracture Simulation 1614.7 Alternate Approaches to Network Block Flow Models . 1734.7.1 Introduction 1734.7.2 Finite Element Network Block Model 1744.7.3 Effective Conductance with Linear Boundary Conditions. 1764.7.4 Iterated Flow Solution (Domain Decomposition) 1764.7.5 Improvements to the Current Network Block How Model 1794.7.6 Parallel Computing and the Dual Permeability Model . . 1814.8 Summary of Dual Permeability Model 181Chapter Five: A TENSOR DESCRIPTION OF FLOW 755.1 Introduction 1855.2 Conductivity Ellipse and Mohr’s Circle 187viTABLE OF CONTENTS5.3 Flow in a Fracture Network: An Example 192196204207211214218218222224226227227229229230233235238241243245Chapter Six: CONCLUDING COMMENTS6.1 Modeling Fracture Networks Below the REV Scale6.2 Dual Permeability ModelingNotation 251References 252Appendix A: FLOW CHARTS 2635.4 Hydraulic Conductance Tensor5.5 Short Circuit Fractures5.6 Asymmetry in Conductance Measures5.6.1 Asymmetry in Simple Fracture Networks5.6.2 Alternative Flow Measures5.7 Parameter Studies5.7.1 Introduction5.7.2 Transmissivity Variance5.7.3 Fracture Density5.7.4 Fracture Length5.7.5 Orientation5.7.6 Isotropy5.7.7 Measurement Scale5.8 Sensitivity to Fracture Network Structure5.8.1 War Zone Fracture Systems5.8.1.1 Sensitivity to Transmissivity Variation5.8.1.2 Non-orthogonal Fracture Systems .5.8.2 Levy-Lee Fracture Systems5.8.2.1 Sensitivity to Transmissivity Variation5.9 Discussion5.10 Summary247247249viiTABLE OF CONTENTSAppendix B: ASYMMETRY MEASUREMENT STATISTICS 275viiiLIST OF TABLESTable Page2.1: Fracture characteristics 182.2: Key features of structural models 313.1: Fracture system definition for the fracture network of Figure 3.1 504.1: Primary fracture selection criteria 814.2: Primary fracture selection criteria test 834.3: Primary fracture selection criteria test with unmodified network blocks 944.4: Network block residence time distribution evaluation 1034.5: Fracture system statistics for the dual permeability evaluation 1124.6: Net flows across the flow domain boundaries of the single realization comparison 1164.7: Computer memory storage and execution times 1585.1: Fracture system definitions for conductance study 1965.2: Fracture systems used to investigate parameter sensitivity 219ixLIST OF FIGURESPage1.1: Flow distribution within a fracture network 31.2: Structure of the dual permeability model concept 51.3: The Monte Carlo process 92.1: Two-dimensional Poisson fracture system representation 332.2: Truncated Poisson fracture network 352.3: Fracture network of a war zone structural model 362.4: Parent-Daughter fracture network generated from Fanay-Augeres fracture system 382.5: Fracture network of a Levy-Lee structural model 402.6: Colonnade structural model from Khaleel [1989) 423.1: Fracture network generated from a Poisson structural model 493.2: Fracture network generated from a truncated Poisson structural model 533.3: War zone definitions 543.4: Region boundaries used in the fracture reduction procedure 593.5: Reduced fracture network of the network shown in Figure 3.1 613.6: Reynolds and Péclet number distributions of flow and transport processes in singlefractures 633.7: Detail of fracture segments and intersections 723.8: Application of a constant hydraulic head gradient along the boundaries of the flowdomain 743.9: Flow chart of reference program, Discrete 764.1: Dual permeability model structure 774.2: Flow chart of the dual permeability model 794.3: Flow chart of the selection of a primary fracture network 854.4: Effective conductance approximation for network blocks 894.5: Calculation of the hydraulic connection between primary fracture intersections andprimary fracture segment midpoints 924.6: Front tracking procedure of Shimo and Long [1987] showing the combining ofupstream solute concentrations to form the concentration as a function of time at afracture intersection 974.7: Calculation of network block residence times 994.8: Comparisons of particle arrival rates of the dual permeability model using a singlexLIST OF FIGURESPagenetwork block and Discrete 1014.9: Flow chart of the particle tracking transport algorithm 1054.10: Fracture traces of the one realization of fracture system 1 of Table 4.6 1094.11: Fracture flow rates within in the fracture network of Figure 4.10 1114.12: Primary fracture network defined for the fracture network of Figure 4.10 1144.13: Flow rates within the primary fractures of the dual permeability model 1154.14: Particle arrival distribution at the downstream boundary of the fracture network ofFigure 4.10 1174.15: Particle arrival curves for entire downstream boundary and at the 14m location 1194.16: Particle location statistics 1204.17: Average cumulative particle arrival at downstream boundary for each of the primaryfracture network selection criteria used in dual permeability models of fracture system1 of Table 4.6 1254.18: The distributions of the initial, 50th, 75th and 90th arrival times using selection criteria6 in the dual permeability model and for the corresponding realizations of the fracturesystem using Discrete 1284.19: Average cumulative particle arrival at boundary 4, when the transmissivity of theboundary fractures of the network blocks are not increased to compensate for flowlosses through crossing fractures 1294.20: Flow plot of one realization of fracture system 2 1314.21: Average cumulative particle arrival at the downstream boundary for 100 realizations offracture system 2 1324.22: Fracture trace map and flow plot for one realization of fracture system 3 1344.23: Average cumulative particle arrival at the downstream boundary for 52 realizations offracture system 3 1364.24: Flow through one realization of fracture system 4, a fracture system without variationin fracture aperture 1384.25: Average cumulative particle arrival at the downstream boundary for 100 realizations offracture system 4 1394.26: Fracture trace map and flow plot for one realization of fracture system 5, a war zoneversion of fracture system 1 141xiLIST OF FIGURESPage4.27: Average cumulative particle arrival at the downstream boundary for 83 realizations offracture system 5 1434.28: Fracture trace map and flow plot for one realization of fracture system 6, a war zonestructural model version of fracture system 3 1454.29: Average cumulative particle arrival at the downstream boundary for 45 realizations offracture system 6 1474.30: Fracture trace map and flow plot for one realization of fracture system 7, generatedfrom the Levi-Lee structural model 1494.31: Average cumulative particle arrival at the downstream boundary for 70 realizations offracture system 7 1514.32: Average cumulative particle arrival for different network block size limits 1564.33: Fracture system 1: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures of fracture sets 3 and 4 1634.34: Fracture system 2: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures of fracture sets 3 and 4 1654.35: Fracture system 2: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures of all fracture sets 1664.36: Fracture system 3: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures of both fracture sets 1684.37: Fracture system 5: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures of war zone dependent sets 3 and 4 1694.38: Fracture system 6: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures of both sets 1704.39: Fracture system 7: mean particle arrival times for fracture networks that have beenreduced by removing small aperture fractures 1724.40: Flow chart of the conjugate gradient algorithm used to iterate the hydraulic headsolution along primary fractures 1785.1: Hydraulic conductivity drawn as a Mohr’s circle 1925.2: Trace map and flow plot of one fracture network used to investigate hydraulicconductance 1945.3: Hydraulic conductance ellipses and Mohr’s circle for the fracture network of FigurexiiLIST OF FIGURESPage5.2 2015.4: Hydraulic Conductance for the 2m by 2m region in the upper right corner of Figure5.2 2055.5: Hydraulic conductance tensor and Mohr’s circle for the modified versions of the samefracture network as Figure 5.4 2065.6: Diagram of the flow direction defined by fluid crossing two domain boundaries 2105.7: Fracture networks used to investigate alternative flow measures 2165.8: Conductance ellipses and Mohr’s circle based on using alternative flow measurement 3to measure the flows though network 4 of Figure 5.7 2175.9: Hydraulic conductance sensitivity to transmissivity variation 2235.10: Hydraulic conductance sensitivity to fracture density, e, length, f, and isotropy, k andm, variation 2255.11: Hydraulic conductance sensitivity to mean fracture orientation, and orientationvariation 2285.12: War zone fracture network 2315.13: Maximum directional hydraulic conductance sensitivity to transmissivity variation ofwar zone generated fracture networks 2345.14: Hydraulic conductance sensitivity to mean fracture orientation, and orientationvariation of war zone fracture networks 2365.15: Levy-Lee fracture network 2395.16: Maximum directional hydraulic conductance sensitivity to transmissivity variation ofLevy-Lee generated fracture networks 242A. 1: The structure of Discrete 264A.2: Fracture realization process 265A.3: Fracture network reduction 266A.4: Particle tracking in Discrete 267A.5: The structure of the dual permeability program 268A.6: Primary fracture network selection 269A.7: Formation of the dual permeability coarse scale flow model. 270A.8: Determination of the network block residence distributions 271A.9: Formation of the transport model 272xliiLIST OF FIGURESPageA. 10: Calculation of the primary fracture transport characteristics 273A. 11: Particle tracking algorithm 274B.1: Average deviation of the orthogonality of the semi major and semi minor axis of theconductance ellipses 277B.2: Average ratio of the difference in the off-diagonal elements to the average of thediagonal elements of the conductance tensor 278B .3: Average ratio of the asymmetry offset in the Mohr’ s circle to the radius of the Mohr’scircle 279xivACKNOWLEDGEMENTSBoth I and this thesis benefited greatly from the insight, patience, and editorial perseverance ofmy advisor Leslie Smith, whom I most heartily thank. I also thank Al Freeze, Rosemary Knight,Wayne Savigny, and Roger Beckie, for their help as members of my thesis committee. I amparticularly indebted to Roger Beckie for invaluable discussions and the introduction to domaindecomposition techniques.This thesis was improved through interaction and discussions with many of my coworkers. Inparticular, I benefited from discussions with Bruce James, Dan Walker, Christoph Wels, Karen Jardine,and Brian Berkowitz. Dr. Paul Hsieh, through a solicited review of an early version of Chapter 5,also had a substantial impact on this thesis. He introduced me to the Mohr’s circle development ofDr. Stephen Kraemer and suggested investigating asymmetry in the tensor representation of hydraulicconductance.The research was supported by EG&G Idaho Inc., the U.S. Department of Energy’sGeothermal Injection Program, The Natural Sciences and Engineering Research Council of Canada, theUniversity of British Columbia’s University Graduate Fellowship Program, and my mom.xvCHAPTER 1SIMULATION OF FRACTURE NETWORKSAT THE FIELD SCALE1.1 INTRoDUCTIONUnderstanding heterogeneous systems at a scale where the heterogeneity can not be representedexplicitly nor is amenable to homogenization through simple volume averaging is an active research topicin the hydrogeologic community. Many problems involving field scale simulation of flow in fracturedrock may lie in this region between the applicability of effective continuum models and the practicalityof explicit models. Solute transport exhibits even greater sensitivity to heterogeneity than does fluid flowand is less likely to be amenable to analysis using effective continuum models. It is not clear that theconsiderable advances that have been made in the past few years in understanding flow and transport inheterogeneous porous media[ e.g. Geihar and Axness, 1983; Dagan, 1988; Neuman, 1993] can be appliedto fracture dominated media. Given this uncertainty, there is a need to pursue techniques that may allowthe application of discrete fracture modeling at scales where the representation of each fracture is madeimpossible because of an overwhelming number of fractures.The dual permeability concept that is the focus of this thesis may provide a mechanism to developa simulation capability within the transition region between conventional discrete fracture modeling andthe treatment of fractured rock as an equivalent porous medium. The primary tenants of the dualpermeability approach are: 1) a relatively small number of fractures within a fracture system dominate thelarge scale hydraulic head distribution; 2) these same fractures carry the majority of the flow within thesystem; 3) the vast majority of fractures within the model domain have a subsidiary influence; and 4) theinfluence of these subsidiary fractures is, however, still important to the overall hydraulic behavior of the1fracture system because of their shear numbers and the effects of these fractures on the transport behaviorof the system.Figure 1.1 illustrates how a few fractures can dominate the flow system. In Figure 1.1 a, fracturetraces are shown for a twodimensional representation of a fracture network. In this particular fracturenetwork, the lengths of the fractures vary widely. The longer fractures also tend to have larger apertures.In Figure 1. lb, where the widths of lines are proportional to the flow rates within fractures, the long, widefractures dominate the flow system. Fractures with flow rates less than 3% of the largest flow rate arenot drawn in Figure 1. lb, revealing that most of the fractures carry relatively little flow. Figures 1.1 cshows the fracture traces of a sparse fracture network that has a smaller range of fracture lengths than thenetwork in Figure 1.1 a. The corresponding flow rates are shown in Figure 1.1 d. In this network,connections between fractures control the dominant flow paths suggesting that fracture connectivity canalso strongly influence the character of flow in fractured media.In both of these networks, a few interlinked fractures form dominant flow channels. However,these channels do not provide routes by which the majority of the fracture network is bypassed. Insteada clear interaction between the channels and the rest of the fracture network can be seen in both Figurel.lb and Figure l.ld by noting the variation of the flow rates along these channels. The flow in thesechannels tends to swell and then diminish indicating flow to and from other fractures in the network. Thishappens because groups of highly transmissive fractures form local clusters that do not interlink. Theseclusters are connected by individual or groups of less transmissive fractures.The importance and degree of channeling within fracture networks is dependent on the variabilityof fracture transmissivity and fracture connectivity in a complex manner. Densely fractured rock with alarge variation in fracture transmissivity will tend to have flow concentrated in fewer fractures thandensely fractured rock with more uniform fracture transmissivity. In contrast, within sparsely fracturedrock, flow dominance may be more dependent on the way fractures intersect than on the transmissivity2MetersCC00MetersC.)000000 I— 0C CCD CDCCI)C) C CCDo00000 CD PC 0• C-) PC ‘1 CD Ci,0 0 C CD Ci) j-h 0 0 1 11) CD CD F F0 PC CD CI) 0 P (ICCl)Cl)‘•CC-)CDCl)-P)—.CDCDCl)Cl).cF)CDoCDCD‘—,‘cFil--.IJOh—-Cl)- CDPCCD— )Cl)-zU)ZCDCDZr.’0I-F)0PCCD-CCCl) CDPCCDCDCl)-‘1Ci-1CDCD) CDCDCDCDCDCDCDPC CDCl)Crh0C Ci, x F C00 I-. x F 0toC’) x F— C 0 C’) x F-’ 0 0 x F-’ C 0 0) x F-’ 0 0 CD-x F 00Meters00 CMetersCC)hXj0 I-. CD PC 0 C-) PC ‘1 ‘1 CD F-I CI)Ct) 0F-’ 0CD CI,CA)000of those fractures.The name “dual permeability” arises from the structure that follows from flow dominance.Permeability is incorporated in a dual permeability model in two different ways. The dominating fracturesare modeled explicitly while the majority of fractures are represented, in some manner, as a group. Thephrase “in some manner” leaves a large degree of latitude to the actual implementation of the dualpermeability concept. The dual permeability structure is illustrated in Figure 1.2. This figure shows theselection of a few fractures as the dominant ones. The fractures were selected on the basis of both theirtransmissive character and their connections to other fractures; not by whether they actually dominate theflow regime which is unknown at the time of selection. These fractures are used to define a geometricstructure within the fracture network. The properties of the intervening regions of the fracture networkthat are defined by this structure are represented in the model without explicit representation of theindividual fractures within the region.Dual permeability modeling is an attempt to accommodate the heterogeneity of fractured rock infield-scale simulation models. The long term goal of this area of research is to provide a predictivecapability that can be applied at the field scale. The goals of this thesis are more limited. These goalsare: to translate the dual permeability concept into a flow and transport simulation model; to identifystrengths and weaknesses of the concept; to identify features of the concept that may limit its applicationto fractured rock investigations; and to use both the model itself and the issues which arise from itsimplementation to further our understanding of flow and transport in fractured media.1.2 THEORETICAL MOTIVATIONFlow and transport in fractured rock is strongly influenced by the three-dimensional aspects of thefracture network in most rock bodies. True three-dimensional fracture network models are presentlylimited to scales far smaller that most field scale problems and to scales below those required to4Fracture NetworkFigure 1.2: Structure of the dual penneability model concept. The fracture network is subdivided intosmall blocks by finding a network of dominant fractures. The hydraulic properties of thefractures within the small blocks, such as the one isolated in this schematic representation,are approximated in the representation of the blocks.Fractures Submerged intoNetwork Block5adequately address the issues of heterogeneity. At this time only two-dimensional representations can bothcapture the fundamental aspects of fractured rock heterogeneity and allow the representation of enoughdiscrete features so that the transition region, up to the scale of continuum behavior, can be explicitlyrepresented. Therefore, the emphasis of this research must be on concepts that arise in accommodatingheterogeneity in flow and transport processes rather than in providing a practical simulation model forflow and transport in fractured rock.The critical aspects of the hydrology of fractured mck that are captured in two-dimensionalrepresentations aref 1) the extreme contrast between the penneability of a fracture and that of the hostrock; 2) the strong local anisotropy introduced by the narrowness of fractures in comparison to theirextent in the other dimensions; 3) the important role that connections of fractures play in the networkscale hydraulic behavior of a fracture system; and 4) the size of some fractures may be so large that onan individual basis they noticeably effect the fluid flow in the rock mass.Another important aspect of heterogeneity in geologic media, and in fractured rock in particular,is the usual high degree of ignorance that exists about the hydraulic properties at specific locations withinthe flow domain. Discrete fracture models require a physical description of a fracture network in tennsof the location of the fractures and the space between the fracture walls. The identification of all thefeatures of a fracture network needed to create an explicit representation of a discrete fracture model isimpossible. Even in the most simple situations, only a crude knowledge of most of the fractures wifi beknown. Information from bore-holes or exposed surfaces may give some infomiation about the locationsof a small subset of the fractures, but the information will not be of sufficient detail to describe the entirefracture network. Geophysical techniques may be able to identify the locations of very large fractures orzones of intense fracturing, but they can not provide the detail that is needed for discrete fracturemodeling. Most of the information about the fracture system must come from characterization of the6fracture system in an indirect way; from investigations of bore-holes and surface exposures that areexpected to be representative of the interior of the rock.There are many possible approaches to accommodate the basic lack of knowledge about thestructure of a fracture system. Of these, two stand out as being generally accepted concepts. One isthrough repeated simulations of possible configurations of the fracture system, commonly called MonteCarlo analysis. The other approach is to develop large scale (macroscopic) representations of the hydraulicand transport properties of the fracture system. These representations account for the average effects ofheterogeneity but are insensitive to specific small scale heterogeneity. Macroscopic representations maythemselves be based on stochastic analysis and may be used in a Monte Carlo fashion to representheterogeneity on a scale that is larger than the scale at which the average properties are defmed.Macroscopic representations are appropriate only if the details of small scale heterogeneity are notimportant to the larger scale hydraulic behavior of the fracture network. The scale at which macroscopicrepresentations become appropriate has become known as the scale of the representative elementaryvolume (REV). The REV concept is fundamental to the issues addressed in this thesis. The concept hasbeen so well established in hydrogeology that a review here would be overly pedantic. The reader isreferred to Bear [1972] for a general discussion and Long et al. [1982] for specifics related to fluid flowin fractured media. The subject of the REV is discussed further in Chapter 5.The discrete fracture network models used in a Monte Carlo procedure require a completedescription of the network of fractures within the rock mass. A set of equally possible realizations aregenerated according to the probability distributions that describe the actual fracture system. Eachrealization will contain features that are consistent with the statistical representation of the fracture system,but do not exist in the actual fracture system. The essence of the Monte Carlo procedure is to createenough realizations to adequately estimate the frequency distribution of possible hydraulic behavior of thestatistical representation, thereby reducing the influence of unlikely occurrences in individual realizations.7This distribution can be used to anticipate the likely hydraulic behavior of the fracture system and toidentify possible behavior that may have important consequences. An important aspect of this process isthat it allows an analyst to associate a likelihood with possible but improbable hydraulic characteristics.The Monte Carlo process can also be described using the more abstract concept of infonnation.Figure 1.3 is included to aid in this description. The fracture network of a discrete fracture model containsa great deal of detailed geometrical information. A well understood fracture system wifi be characterizedmostly though a statistical representation in terms of probability distributions of fracture physicalcharacteristics. The Monte Carlo realization process creates the required detailed information of thefracture network from the statistical representations, step one in Figure 1.3. The discrete fracture modelthen transforms the detailed geometrical information about the fracture network into detailed informationabout the hydraulic behavior of the fracture network, such as the solute concentration as a function of timeat locations within the fracture network. Unfortunately, most of this detailed information is highlyunlikely as a representation of the hydraulic behavior at a specific point within the simulation domain andit must be weighted properly in the overall analysis. Forming a distribution over many realizationsreduces the importance of information from any individual realization so that the essential behavior thatis consistent with the statistical representation of the fracture system is enhanced over the multitude ofunlikely events that are imbedded in the detailed hydraulic behavior.Monte Carlo analysis is a tool that can be used to accommodate uncertainty in the knowledge ofa fracture network. A byproduct of using Monte Carlo analysis is a tolerance toward some of the errorsthat are introduced into the predictions of hydraulic behavior through the flow and transport models, inaddition to errors introduced through ignorance of the physical system. If the variations in predictedbehavior introduced through modeling errors are small compared to variations inherent in the generationof fracture network realizations, then the errors are not important to the overall results. In other words,if the errors introduced by the discrete fracture model are small compared to the variability introduced8CONCEPTUAL FRACTURE STUDYNon Specific FractureRepresentative StatisticsInformationStep 1REALIZATION PROCESS +Specific FractureDetailed NetworkInfromationStep2FLOW AND TRANSPORT +MODELINGSpecific TransientDetailed SoluteInfromation ConcentrationStep 3ACCUMULATION +OF REALIZATIONSNon Specific DistributionRepresentative ofInformation PredictionsFigure 1.3: The Monte Carlo process. On the right are the stages of a Monte Carlo analysis of afracture system. The boxes are processes. The area between boxes are stages resultingfrom preceeding process and/or input to the next process. On the left, the Monte Carloprocess has been abstracted to an information processing proceedure.9through random sampling of the statistical representation of the fracture system geometry they will notinfluence the post-averaging results of the Monte Carlo process.The same argument can not be used to dismiss biases in the hydraulic behavior introduced by adiscrete model. Biased errors introduced through a discrete fracture model will produce biased averagebehaviors and biased predictions. If, for example, the discrete fracture model consistently underestimatesthe transport time through individual fractures then transport predictions will be biased toward fastermovement of solute. The use of Monte Carlo analysis would not correct for this discrepancy.The conspicuous lack of detailed information about fracture networks at the field scale that ischaracteristic of hydrologic studies of fractured media makes the Monte Carlo process a standard approachto understand the hydraulic behavior of these systems. Conventional discrete fracture models provide anexact transformation of the physical description of a fracture network into a hydrologic one. Unbiasederrors in the results are tolerable because of the averaging process required in the Monte Carlo procedure.This tolerance provides an opportunity to intentionally allow errors in the fracture network model ifadditional capabilities are derived through the process. The major capability of interest at this time infracture network modeling is to be able to increase the scale of the problem that can be solved.This view of the Monte Carlo process provides justification for approximating the majority offractures in a dual permeability model, not in a manner that best retains the hydraulic behavior of thefracture network, but in a manner that allows the greatest number of fractures to be modeled while stillavoiding the introduction of biases. There should be a clear distinction here between the approximation.in terms of equivalent porous media modeling and approximation in terms of dual permeability modeling.In equivalent porous media modeling, the detailed hydraulic behavior is considered unimportant to thelarge scale behavior of the system. In dual permeability modeling, the detailed behavior is consideredimportant to the large scale behavior of a single realization, but not important to end use of the modelingresults.101.3 THEsIs OVERVIEWWe have already seen how accommodating the lack of knowledge of a fracture network at thefield scale influences the conceptual framework of this thesis. An important consideration, therefore, isunderstanding what constitutes a reasonable degree of knowledge about a fractured rock mass. Chapter2 addresses this issue with a review of how fracture system information is acquired. The information isneeded to identify physical characteristics of fractures that are usually included in discrete fracture models;the subject of Section 2.3. The chapter also includes a summary of some of the models of fracturenetwork geometry that could be used to incorporate the interrelationships of fractures into discrete fracturemodels.The fundamental modeling assumptions that are the basis of discrete fracture modeling used inthis thesis are presented in Chapter 3. The validity of many of these assumptions is investigated anddiscussed. Chapter 3 includes a description of the fracture network generation procedure that defines thefracture geometry for each fracture network realization. The chapter ends with a model description forthe discrete fracture model that is used to benchmark the dual permeability model.Chapter 4 presents the dual permeability model. This presentation uses the basic modeldescription of Chapter 3 as a starting point. An overview of the model structure is provided first and isfollowed by more detailed descriptions. An evaluation of the dual permeability model in terms of a MonteCarlo investigation shows that the dual permeability concept is a viable method of investigating thehydraulic behavior of fracture networks. The dual permeability model is compared to an alternate strategyof selectively ignoring fractures in a network in Section 4.6. The chapter concludes with some of thealternative implementations that were either tried or show promise. The last section focuses on the lessonsthat were learned from these trials that are important considerations in sub-REV models of fracturenetworks.11Chapter 5 turns to a characterisation of flow in fracture networks. This chapter presents a studyof the influence of fracture network heterogeneity on flow at sub-REV scales. The study is concernedwith representation of the aggregate flow behavior of a fracture network below the REV scale. The studydemonstrates that flow does not depend on the direction of the hydraulic gradient in an erratic manner,as has been implied by earlier studies. Instead, the flow dependence behaves in a predictable manner thatcan be represented using a tensor relationship.The concluding chapter provides an assessment of the dual penneability model and interprets thelessons learned from this one model in terms of the dual permeability concept. The chapter also pullstogether some of the disparate results from Chapters 4 and 5 into a discussion of some important aspectsof representing highly heterogeneous media below the scale of continuum behavior.12CHAPTER 2FRACTURE SYSTEM CHARACTERIZATION:A REVIEW2.1 INTRODUCTIONThis chapter describes the characterization of the physical attributes of fractures in a manner thatcan be useful to numerical simulation of fluid flow and solute transport within a fractured rock mass.Physical characterization is divided here into two subjects. 1) The statistical description of the physicalproperties of individual fractures. 2) The description of the relationships of individual fractures to eachother; referred to in this thesis as structural modeling. This division is not always straight forward andis not common in the relevant literature, but I believe it is useful.The description of the physical characteristics of fractures is a necessary part of any attempt tounderstand the hydraulic properties of fractured media through simulation of the hydraulic behavior ofdiscrete fracture networks. It is not, however, central to the research described in this thesis. The materialpresented in this chapter is a synthesis of some of the relevant literature on the topic, from the perspectiveof flow and transport simulation using discrete fracture networks.The description of the physical characteristics of fracture networks has been referred to as jointsystem modeling. Published descriptions of joint system models often include both the properties ofindividual fractures and the interrelationships between them. The two aspects are intentionally separatedhere without indication of the original linkage. Most joint system models have been presented as threedimensional models. The focus in this discussion is on representation of a fractured rock in twodimensions. In some instances however, three-dimensional aspects are included.13An overview of methods used to acquire data about fractures is given in Section 2.2. One of thetopics discussed in Section 2.2 is the difficulty in acquiring detailed information on the locations andcharacteristics of individual fractures within a rock mass. The scarcity of information about individualfractures in a fracture network is a critical feature of fracture system modeling and, as has been mentionedin Chapter 1, has influenced the course of this research. In Section 2.3 the characteristics of individualfractures are reviewed first, then the section focuses on relationships between fractures. Section 2.4discusses a limited number of structural models which represent the relationships between fractures in aquantitative manner.2.2 SOURCES OF FRACTURE SYSTEM INFORMATIONOne of the most basic aspects of characterizing the hydraulic behaviour of fractured rock is thescarcity of data that can be readily obtained about the geometry of individual fractures. This is truedespite the tremendous variety of techniques that have been developed to measure and characterizefractures. Measurement of fracture properties can be divided into two categories; direct and inferred.Direct measurements are the physical measurements of the fracture geometry, such as fracture orientationor trace length. Inferred measurements require interpretation of the influences of fracture geometry onthe measured parameter. An estimate of fracture transmissivity from analysis of pressure response to aninjection test is an example of inferred measurements. Such an estimate requires assumptions about theshape of the fractures and about the interconnection of the fracture intersected by the borehole with otherfractures within the rock body.Both measurement categories can be further sub-divided based on the type of acquired information.One type of information is specific, such as the location of individual fractures within a borehole. Theother type of information in representative. An example of representative information is data from themeasurement of fracture orientations on a surface exposure. These measurements give no specificinformation about the fractures within the rock mass, but may be indicative of those fractures. The14distinction between the terms specific information and representative information is not the informationitself but in how it is used. Specific information is used to identify the features of individual fracturesin a model. Representative information is used to identify the features of populations of fractures.2.2.1 SOURCES OF SPECIFIC INFORMATIONSpecific information is primarily available from borehole investigations. A large number of toolshave been developed to investigate the walls of boreholes. Some of these tools, such as callipers oracoustic televiewers, provide direct information on fracture spacing, orientation and aperture. Cores fromthe drilling of boreholes can also be used to provide this information. Correlating cores with otherborehole surveys may be useful in screening out fractures that have been created by the drilling processand that should not be used in characterizing the fracture system in the surrounding rock. Geophysicaltechniques, such as natural gamma logs, epithermal neutron logs, electrical resistivity logs, and radarsurveys, provide inferred information on the location and possibly the extent of individual fractures orzones of intense fracturing.Hydraulic tests using one or more boreholes can be conducted to gain understanding of the hydraulicproperties of localized regions of the fracture system and can give an indication of the connectivity of thefractures near the wellbore (e.g. Hsieh and Shapiro [1994]).Geophysical techniques can be separated into borehole techniques, cross-borehole tomographictechniques and regional techniques. Cross-borehole and regional techniques do not have the resolutionto provide the kind of detailed information about the fracture system that is available from boreholes andsurface exposures. Cross-borehole and regional techniques can, however, provide data, such as regionsof intense fracturing or the general direction of solute transport, that can be used to constrain discretefracture models.15Borehole geophysical techniques can give indications of the extent of fractures in the immediatevicinity of the boreholes. Geophysical methods can also identify the existence of water or injected tracerin nearby fractures. This information can be used to establish which fractures are actually participatingin the flow system and to give an indication of the channelling of flow within individual fractures [Olssonet al., 1991]. Tomographic imaging of multiple seismic or radar measurements made between boreholesor from boreholes to tunnels can give an indication of the existence of fractures or, more easily, zones ofintense fracturing [Majer et at., 1990; Black et at., 1991].On a scale larger than can be investigated through borehole investigations, measurements ofhydraulic head, flow rates into open weilbores or tunnels, or the movement of natural tracers may providespecific information that could be useful in establishing constraints on larger scale behaviour. These datamay provide checks on the large scale properties of the fracture network such as effective hydraulicconductivity. In making such checks, recognition of the influence of fracture system heterogeneity on themeasured data is important. Transport experiments provide data that may also prove useful. However,the uncertainty that accompanies the geometric description of a fracture network composed of more thana few fractures makes interpretation of the results of these tests difficult.The amount of specific information about the fractures, from even the most thorough siteinvestigation, would not be enough to fully define a discrete fracture network that could be used in a flowand transport simulation. We are forced to use structural models and individual fracture statistics basedon representative information to synthetically create the specific information required by discrete fractureflow and transport models. Representative information is, therefore, critical to modeling of the hydraulicbehavior of fracture systems.I2.2.2 SOURCES OF REPRESENTATIVE INFORMATIONRepresentative information often is gathered by both borehole investigations and surface surveys.The surfaces can be either outcrops, exposures from excavations, or the walls of tunnels. In all cases,changes in internal stresses within the rock mass resulting from the exposure of the surfaces may induceadditional fracturing that make the surfaces unrepresentative of the fracture system that exists within therock. Recent fractures and fractures that were not hydraulically active may be identifiable through thestudy of surface coatings or rock alteration near the fracture. Cores provide samples for measuringfracture roughness and fracture surface coatings that may be useful in delineating fracture sets [e.g.Herbert 19911. Surface coatings may also be useful in identifying which fractures were hydraulicallyactive prior to drilling a borehole.The expression of fractures on a surface is called a trace. Procedural techniques for carrying outtrace surveys are well established [e.g. Priest and Hudson 1976, Baecher and Lanney 1978, La Pointe andHudson 1985, Blin-Lacroix and Thomas 1990]. Surface surveys can give only partial information onfracture shape. An understanding of the shape of fractures can be acquired through careful excavation ofa fracture [Dershowitz and Einstein 1988].2.3 STATISTICAL CHARACTERIZATION OF FRACTURESTable 2.1, modified from Dershowitz and Einstein [1988], presents a list of identifiable fracturecharacteristics that are appropriate for statistical parameterization and are candidates for incorporation intojoint system models. The characteristics are divided into three groups: 1) characteristics of individualfractures, 2) fracture density which is a composite characteristic of all the fractures, and 3) characteristicsof the relationships of individual fractures. Most of these characteristics are discussed in the followingsub-sections.17Characteristic DescriptionOrientation Orientation of the fracture plane.( Attitude)Spacing Distance between intersection points produced by a lineperpendicular to the mean orientation of the fracture set where aset consists of fractures that are parallel or sub-parallel to eachother.Fracture Size Extent of fracturing; generally expressed as trace length on two-dimensional surfaces such as outcrops or as the surface area ofindividual fractures.Shape Shape of boundaries can be polygonal, circular, elliptical orirregular.Planarity Character of the surface with respect to an ideal plane. Deviationsfrom planer surfaces can be on several scales.Roughness Deviation from planarity < 1 mmWaviness Deviation from planarity> 1 mmAperture The distance between fracture surfaces.Density Number of fractures per unit area or volume. A measurableindication of density such as total trace length per unit area ortotal surface area per unit volume may be used as a surrogate.Persistence The classifications persistent, sub-persistent, and non-persistent areused to indicate the degree of connectedness of the network.Termination The tendency of a fracture to end along the surface of anotherfracture. Commonly measured in outcrops in terms of the fractionof ends that terminate against another fracture.Co-planarity Co-planarity indicates that two or more fractures exist in the sameplane.Table 2.1: Fracture characteristics. The first seven characteristics describe individual fractures. Thelast three entries are properties describe the interrelationships of fractures. Densityseparating the two categories does not belong in either category. Modified fromDershowitz and Einstein [1988].182.3.1 ORGAMZATION OF FRACTURES INTO SETSPollard and Aydin [1988] state “Curiously, joints never occur alone, but as a series of subparallelfractures defining a set.” They also state that “Joint patterns comprising more than one joint set arecommon in nature.” Multiple sets may arise from a single deformation episode or from different events.The most common method of classifying fractures into sets is based on the orientation of the fractures.Unfortunately orientation alone can be a poor guide to classification. Often, the sets are not well defined[Barton and Hsieh 1989, see also Long and Billattr 1987]. Fractures with similar orientations may nothave been created in the same deformation episode and may belong to sets with much differentcharacteristics. Termination and cross-cutting relationships give indications that fractures of similarorientation belong to different sets. Surface coatings have also been used to relate individual fractureswith specific deformation episodes, allowing different sets with similar orientations to be distinguished.These different sets may have different aperture characteristics.2.3.2 ORIENTATIONFracture orientation indicates the plane through space that tends to correspond with a fracturesurface. The orientation measure that is most convenient to work with is the direction of the fracture pole,the line normal to the plane best aligned with the fracture surface. If the fracture surface is wavy, thenthe orientation is dependent on where the measurement is made. Generally, orientation measurements areaccurate to ±2° [Barton and Hsieh, 1987; see also Chernyshev and Dearman, 1991]. Poles are usuallyplotted on a Schmidt hemispherical projection [e.g. Davis, 1984]. Fractures of the same set tend to clusteron these diagrams. Identification of sets can be aided through the use of clustering algorithms that willseparate fracture pole data into sets [e.g. Shanley and Mahtab, 1975].Once sets have been identified, the orientations of fractures of each set must be characterized interms of statistical distributions. Dershowitz [1984] lists six candidate distributions that can be used to19represent the orientation of a set statistically: Unifon-n, Fisher, Elliptical Fisher, Bingham, Fisher-Bingham,and Bivariate NonTiaJ. These distributions are all hi-variate because orientation is represented by twoangles in the three-dimensional representations of interest to Dershowitz. Based on data from 25 sites withvarying geology, he concluded that none of the distributions could consistently fit the field data at the 95%confidence level indicating that an appropriate description for fracture orientations at a particular rock bodyshould not be a generic description but should be instead developed from the investigations of the rock.The Fisher distribution is most commonly used because it is analogous to a normal distribution onhemispherical projections [Dershowitz, 1984]. In two dimensions, orientation is defined by a single anglewhich can be represented by a normal distribution, albeit not necessarily accurately.2.3.3 SPACINGThe spacing of intersections with fractures along a line on an exposed surface has been studiedextensively. The line is commonly referred to as a scan-line and the density of intersections along the lineis called scan-line density. Spacing is an important issue in fracture system modeling because differentstructural models may have characteristic spacings. Spacing can therefore be used to assess theappropriateness of the different structural models. Nine investigations of fracture spacing are reported inKulatilake’s [1988] review of fracture system statistical models. Of the nine, four papers [Steffen et at.1975, Bridges 1975, Barton 1977, Sen and Kazi 1984] reported that spacing follows a lognonnaldistribution. Four papers reported that spacing follows a negative exponential distribution [Call et al.1976, Priest and Hudson 1976, Baecher et at. 1977, Wallis and King 1980]. Einstein et at. [19791reported that 80% of the distributions they investigated were best fit with a negative exponentialdistribution and that 20% were best fit with a lognormal distribution. Of the surveys reported byKulatilake, only Baecher et al. and Einstein et at. used goodness of fit tests to select the best distributions.Kulatilake et al. [1993] present a survey of fracture spacing in the Stripa mine. These authors divided thefracture network into sets based on fracture orientation prior to determining the spacing for each set. They20determined that a gamma distribution fit the data better than negative exponential, log-normal or normaldistributions based on goodness of fit tests.The lognormal and negative exponential distributions are very similar except for the frequency ofsmall values. Chernyshev and Dearman [19911 suggest that the studies that have reported lognormaldistributions may be neglecting the smallest fractures because they are not readily visible to the humaneye. As a caution, Priest and Hudson [19791 demonstrate that a mixture of sets composed of evenlyspaced fractures will combine to fonn intersections with a scan-line that have a negative exponentialdistribution. Gillespie et al. [19931 provide a clear illustration of this phenomena. These authorsdemonstrate that a scan-line survey that superficially indicates a negative exponential distribution offracture spacing also clearly supports the conclusion that fractures of a particular set are roughly equallyspaced and that this spacing is best represented using a normal distribution (a conclusion that seemsobvious from a visual inspection of the fracture traces).The conclusion that is supported by many studies of fracture spacing is that, within crystaliinerock, fractures occur randomly within a rock mass. The work of Gillespie et al. [1993] casts a shadowon this conclusion. The results of surveys that do not first distinguish fractures by set must be questioned.Following the logic of Chernyshev and Dearman [1991], original deformations result in a uniformdistribution of fractures with a relatively narrow range of orientations. Creation of fractures duringsubsequent deformation episodes are affected by the preexisting fractures. These subsequent fractures areoften regularly spaced with densities and orientations that are effected by the spacing between the preexisting fractures; causing clustering. The influence of pre-existing fractures causes a much greatervariation in the orientations of the subsequent fractures than the original fractures.The recent interest in the fractal nature of geologic media has expanded the debate on fracturespacing. It is possible that the results of spacing surveys have been constrained by the distribution modelsthat are considered. Barton and Larson [1985] performed a trace survey of three pavements at Yucca21Mountain in Nevada. They concluded that the fracture trace distributions can be represented fractally, withvery similar coefficients for each pavement. Barton [1993] also found a power law length distributionin the fracture traces observed on road cuts at the Mirror Lake site in New Hampshire. Geier et al. [1989]investigating a trace survey conducted along the floor of the TSE drift in Stripa, Sweden, found that thedistribution of trace lengths fit a power law distribution. The success of Barton and Larson, Barton, andGeier et al. in identifying fractal related characteristics in fractured rOck gives support to the fractalstructural models. However, an examination of the techniques used to test for fractal behaviour byGillespie et al. [1993] found that the techniques used in the above surveys do not discriminate fractal fromnon-fractal distributions well. These authors concluded that they found no evidence to support fractaldistributions of fracture joints but that some data of spacing of faults support fractal distributions. Theissue of how fractures are spaced in a rock mass is far from settled.2.3.4 SIZEThe length of traces on outcrops and other surface exposures has been extensively studied. Tracelength measurements are biased in three ways: 1) Traces are truncated along the edges of the exposuresurface. 2) The probability of intersection of a trace with a scan-line is inversely proportional to the tracelength. 3) Data is generally not collected below some length cut-off. Both Baecher and Lanney [1978]and Priest and Hudson [1981] provide techniques to correct length estimates of the traces for these biases.Chung [1986] provides a computer algorithm for this purpose.Kulatilake [1988] reviews nine studies of trace length. In five of these studies, the investigatorsfound that a lognormal distribution adequately represented the distribution of trace lengths [McMahon1974, Bridges 1975, Barton 1977, Baecher et at. 1977, Einstein et at. 1980]. In the other four studies,a negative exponential distribution was used to fit the data [Robertson 1970, Call et al. 1976, Cruden1977, Priest and Hudson 1981]. Only Baecher et at. and Einstein et at. compared the goodness of fitbetween the two distributions. The degree of bias correction prior to fitting the trace lengths to a22probability distribution differs in these studies. Priest and Hudson [19811 have shown that a negativeexponential distribution of true trace lengths will result in a lognormal distribution of trace lengths thatintersect a scan-line. This may have influenced the results of some of the surveys cited by Kulatilake.The trace length distribution gives an indication of the size of fractures within a rock mass. Largefractures are better represented in the exposures unless small fractures preferentially exist close to thesurface exposures as a result of stress relief. Fractures will also be better represented in the surfaceexposures if they intersect the exposure at large angles. These biases are correctable. However, thecorrections depend on assumptions about the shape of fractures. Research on the relationship of tracelengths to fracture dimensions has taken two paths: 1) Trace length distribution predictions have beenmade starting with assumptions of fracture shape and the distribution of fracture size [Baecher and Lanney1977, Baecher et at. 1977, Dienes 1979, Warburton 1980a, 1980b and Kulatilake and Wu 1986], and 2)The distribution of fracture sizes has been estimated from assumptions of fracture shape and distributionsof trace lengths [Dienes 1979 and Kulatilake and Wu 1986]. The first approach can be used to testhypotheses of fracture shape and size. The second approach can be used to estimate fracture sizedistributions from trace length data as demonstrated by Kulatilake et at. [19931.2.3.5 SHAPEFracture propagation and the resulting fracture shape is a complex topic that is touched onlybriefly here. Because of the major simplification that we must make to model fracture systems in twodimensions, the details of fracture shape are relatively unimportant to this thesis. Pollard and Aydin[1988] state “Although little is known about joint shape in massive rocks, rib marks suggest that anelliptical geometry may be common.” Rib marks are ridges or furrows on fracture surfaces that seem toradiate away from what is believed to be the location of fracture initiation. All current structural modelsassume that fractures are planar. This assumption seems quite good except along the edges of manyfractures where hackle marks may form at a small angle to the original fracture plane.23There are two dominant forms of fracture shape models, disc and polygonal. Polygonal modelsare well supported by surface exposures where the rock is broken into a multitude of separate blocks.Disc models seem more appropriate for rock masses that are not completely subdivided by fracturing[Dershowitz, 1984]. These discs are usually assumed to be either circular or elliptical.2.3.6 APERTURE AND TRANSMISSIVITYThe transmissivity of a fracture is strongly dependent on the spatially varying separation of afracture’s surfaces, i.e. the volume of the fracture, and on the possible existence of material filling thisvolume. The existence of infilling material has a strong influence of the transmissivity of a fracture. Forthe moment, consider fractures to be free of material between the fracture surfaces. Flow between twofaces with a constant aperture separating them is related to the cube of the aperture if laminar flow existswithin the fracture. Under natural conditions, flow in fractured rock is nearly always laminar. Theassumption that fractures have constant aperture is often made to take advantage of this cubic relationshipeven though it is well recognized that the surfaces of fractures are not constant and rarely smooth. Theyare rough and have spatially varying apertures. Indeed, to transfer stress across the surface of a fracture,the fracture must be in contact at some points. A constant aperture fracture would be in contacteverywhere. It is estimated that for crystalline rocks only about 3% of the surface area needs to be incontact to transfer stress [Chernyshev and Dearman 1991]. Pyrak-Nolte et al. [1987] found 10-15% ofthe surface area of granitic cores to be in contact for self-propped conditions and 30-40% for normalstresses of up to 85 MPa.Studies of the effects of aperture variation on flow in individual fractures have demonstrated thatflow is dominated by constrictions [Tsang and Tsang, 1987]. Aperture roughness and the resultantchanneling of flow induces dispersion into the transport of solute through a single fracture. Numericalstudies of the affects of aperture roughness have also demonstrated that most of the volumetric flow infractures tends to occur in a few channels [Moreno et at., 1988; Tsang et at., 19881. Observations of and24experiments on seepage into tunnels support the concept of channeling of flow in fractures [Neretniekset at., 19871.The importance of fracture aperture on the hydraulic properties of a fracture has spawned aplethora of techniques to determine apertures. Direct measurement of aperture from fracture traces onsurface exposures, cores, bore-holes, and cores can be performed using a wide variety of tools. Suchmeasurements have not been proven reliable in providing an estimate of the magnitude of apertures of insitu fractures. Bianchi and Snow [1968) used surface aperture measurements to suggest that apertures arelog-normally distributed. Chernyshev and Dearman [19911 reviewed a number of Russian studies offracture aperture distribution. These studies involving thousands of individual measurements resulted inthe conclusion that in some instances the apertures were better represented with log-normal distributionsand in other cases were closer to normal distributions.Profilometers and optical methods are used to measure the roughness distribution on exposedfracture surfaces [Brown, 1987; Huang et al., 1988; Miller et at., 1990; Voss and Shotwell, 1990].Injection of plastics into fractures prior to excavation can yield accurate information on the distributionof aperture [Gale, 1987; Pyrak-Nolte et al., 1987; Gentier et at., 1989; Cox et aL, 1990]. Differencesbetween in situ rock stress and the stress during injection can have a large influence on the aperturesmeasured by the injection technique and apertures within the rock body. One technique that does notrequire the separation of fracture surfaces to inspect the aperture distribution is tomographic imaging ofradioactive tracers [Wang et at., 1990].Investigations of the effect of stress on flow through fractures have been carried out by subjectinga fractured rock to a controlled confining pressure [Raven and Gale, 1985; Pyrak-Nolte et at., 1987].These studies are important because in situ stress conditions vary and may not be duplicated in laboratorytests or may be disturbed by wellbores. If corrections can be developed for stress variation then laboratoryor surface measurements of aperture may provide useful data for discrete fracture models. As it is, the25present state of knowledge about fracture apertures and the present technology developed to characterizethem makes the accuracy of direct incorporation of measured aperture data into discrete fracture networkmodels suspect.So far this discussion of apertures and fracture transmissivity has been based on the assumptionthat fracture volumes are filled only with fluid. The presence of mineral precipitation along fracturesurfaces is easily accommodated in this discussion by defining the surface of a fracture as the boundaryof the liquid volume rather than the boundary of the intact rock. In some instances, a fracture may befilled with fault gouge or clays that make the fracture less permeable than the host rock. Such fractureswill not contribute to the hydraulic behavior of a fracture network. Fractures may also be filled withmaterial that is more permeable than the host rock. The local transmissivity of these fractures isdependent on the permeability of the filling material and directly proportional to the fracture aperturerather than aperture cubed.Effective apertures or the transmissivity of in situ fractures must be inferred from hydraulic datasuch as borehole packer tests, flow into tunnels and regional flow and head gradient estimates. Hydraulictests of packed off intervals within a wellbore provides the most direct source of transmissivityinformation [e.g. Novakowski, 1989]. A large body of literature has been published on the subject ofhydraulic testing of fractured rock using wellbores and the subsequent evaluation of results [Barker, 1988;Karasaki et at., 1988; Black et at. 1991]. Attempts to find an equivalent single aperture measure todescribe the velocity of solute progressing though fractures have found that they are usually an order ofmagnitude, or more, larger than the equivalent aperture needed to match flow rates using a single aperture[Abelin et at., 1985; Novakowski et al., 1985; Tsang and Tsang, 1987] However, Raven et al. [1988]found the opposite relationship for monzonitic gneiss at Chalk River, Ontario. Sillimo.n [1989] hasexplained this apparent contradiction in terms of different definitions of equivalent aperture. The heartof the matter is that no single equivalent aperture is appropriate for both flow and transport In addition,26Smith et al. [1987] have demonstrated, using numerical modeling, that estimation of effective hydraulicapertures from pump tests is largely a measure of the aperture in the proximity of the welibore and is notrepresentative of the entire fracture surface.Flow into tunnels or regional flows provide data at a larger scale than is obtainable through well-bore testing. These data can be used to compare and calibrate aggregate behaviour of discrete networkmodels. Further calibration can be achieved through comparisons to tracer tests. These calibrationprocedures are constrained by uncertainties in the boundary conditions of physical experiments and in theexpense of carrying out in situ experiments at large scales.In summary, because of the sensitivity of both flow and transport to fracture apertures and becauseof the difficultly in estimating aperture/transmissivity, the uncertainty in model results introduced byaperture/transmissivity uncertainty dominates other sources of uncertainty in discrete fracture modeling[e.g. Herbert et at., 19911. The problem is so serious that some believe that discrete fracture modelingis not a viable approach [Neuman, 1987].To this rather negative assessment of the problems of incorporating aperture or transmissivity datainto discrete fracture models, I must add the success of Herbert et at. [1991] in simulation of flow atStripa, Sweden. These authors used data from single borehole hydraulic tests that are described in detailby Black et at. [19911. Fracture sets were identified and the geometric characteristics of each set wereestimated from bore-hole investigations. Despite this information, Herbert et at. decided thattransmissivity estimates of the entire population of fractures would be used for all of the fracture sets,instead of using the specific transmissivity estimates for each set. A lognonnal distribution was assumedover most of the transmissivity range but a linearly decreasing probability of highly transmissive fractureswas imposed on the tail of the distribution. The mean transmissivity used in the distribution was estimatedto be within 1.5 orders of magnitude of the true mean transmissivity of the fractures within the rock mass,based on comparisons of various fitted distributions of transmissivity. This uncertainty overstates the27uncertainty with respect to flow through a network of fractures because the estimated variance of the fitteddistributions was inversely proportional to the fitted mean. Both larger transmissivity variation and largermean transmissivity lead to larger flows through a network of fractures.A discrete fracture model based on these estimates was constructed by Herbert et at. [1991] andused to demonstrate the scale of continuum behaviour and to provide permeability estimates for REV-sizedfinite element blocks. Continuum representations of these blocks were later used in a larger scale flowmodel. The uncertainties in the mean flow though these blocks for a known hydraulic head gradient wereestimated to be a factor of two for dense networks and a factor of ten for sparse networks. Uncertaintyin transmissivity was cited as the dominant cause of the uncertainty in flow. However, to have estimatedthe flow rate to within a factor of two within the highly fractured regions of the Stripa site should beconsidered a significant achievement.2.3.7 FRACTURE DENSITYFor modeling purposes, fracture density is most usefully represented as fracture centers per unitvolume or unit area. Density is usually determined from trace surveys and is usually reported in termsof intersections per meter of scan-line, scan-line density. Scan-line density contains both fracture lengthinformation and information about the density of individual fractures. The density of intersections of afracture set with a scan-line has a correctable bias as a function of the angle between the scan-line andthe mean pole of the fracture set [Terzaghi, 1965].2.3.8 PERSISTENCEPersistence of a fracture system is a measure of the degree of interconnection of the individualfractures. ft is an important characteristic of a fracture system that should be reproduced by the fracturesystem model. The fracture system of a rock mass that is so intensely fractured that the rock is separatedinto blocks that are bounded on all sides by fracture surfaces is classified as persistent. If the fracturing28is so sparse that the fracture system does not connect sufficiently to provide a continuous pathway forfluid migration through the rock mass, then the fracture system is classed as non-persistent. In betweenthese two extremes is the condition where fractures are well connected in the sense of forming acontinuous network but not so dense as to actually form distinct blocks of rock. The sub-persistentclassification is not relevant in two dimensions because any continuous pathway will divide the rock intoseparate blocks. A fracture system that is sub-persistent in three dimensions will appear either persistentor non-persistent in any specific two-dimensional cross-section. The transformation of this qualitativemeasure of persistence into a measurable quantity has not been standardized. Dershowitz [1984] discussesa number of methods to measure persistence-like properties.One term closely allied with persistence is connectivity. Connectivity has been introduced as onemeasure of the degree of persistence of a fracture network. Connectivity has been quantified as theaverage number of intersections of a fracture with other fractures [Robinson, 1984; Charlaix et at., 1986,Hestir and Long, 1990]. Hestir and Long [1990] and Pike and Seager [1974] have shown that aconnectivity of 3.6 intersections per fracture is often related to the percolation threshold of two-dimensional Poisson fracture network models (See section 2.4.3 for the definition of a Poisson fracturenetwork model). The term percolation threshold is associated with the minimum density of fracturing ofa persistent fracture network (sub-persistent in 3D). Hestir and Long [1990] have also demonstrated afunctional dependence of fracture network hydraulic conductivity on connectivity.2.4 DISCRETE FRACTURE STRUCTURAL MODELS2.4.1 STRUCTURAL MODELSStructural models are a vehicle to incorporate knowledge of the fracture system that goes beyonda statistical representation of individual fracture properties. The way individual fractures are connected29is a very important aspect of a fracture system. Not surprisingly, the development history of a fracturesystem has a strong influence on the resulting connectivity of the system.It is through structural models that knowledge of fracture genesis and the possible episodic historyof the creation of the fracture system can be incorporated into the mathematical description of fracturesystems. Knowledge of the episodic history of deformation may provide dependencies of youngerfractures on preexisting fractures. This knowledge is reflected in different models of how the size andlocations of younger fractures are controlled by preexisting fractures.The following section briefly describes a few of the structural models that have been proposed.These are listed in Table 2.2. The emphasis in this section is on rationale behind each model and on theinterrelationships between fractures that are represented by the models. Some of the structural modelsStructure Model Key Features Used inthis StudyRegular Spacing Continuous fractures NoEqual spacingParallel fracturesPoisson Location and orientation independent YesParent-Daughter Spatially correlated density variation NoTruncated Poisson Poisson with truncated fractures YesHierarchical Location and truncation dependencies NoWar Zone Poisson with truncated fractures YesZones of high fracture densityFractal Power law fracture spacing YesPower law length distributionColonnade Hexagonal or distorted hexagonal fracture pattern NoTable 2.2: Key features of structural models.30mentioned in this section are used to generate fracture networks for the studies presented in later chapters.The implementation of these models is described in Chapter 3. Most structural models have beenproposed for incorporation into three-dimensional discrete network models to be used for hydrogeologicmodels and/or rock mechanics models. While the proposed models have generally included assumptionsabout the statistical characterization of the individual fracture properties, the organization used hereseparates the structural models from assumptions of individual fracture characteristics.2.4.2 UNIFORM SPACINGUniform spacing models assume that fractures are continuous and regularly spaced. The periodicnature of these models provides geometric symmetries that allow analytical analysis of the behavior of thefracture networks in fewer dimensions than those of the network. Models based on uniformly spacedfractures have been used in many studies and are the basis of a large number of numerical models. Ofhistorical note are the early orthogonal networks of Irmay [1955], Childs [1957j, and Snow [1965]. Inparticular, the dual porosity modeling approach of Warren and Root [19631 using a uniform spacing modelhas been extremely influential in the petroleum industry [Pruess and Narashimhan, 1982; Nelson, 1987;Zimmerman et al., 1993]. Dual porosity models, in general but not exclusively, focus on the pressureresponse within well-bores from pumping transients rather than transport processes. Pressure response isless sensitive to the way fractures interconnect than is solute transport.Even for transport models, regular spacing is in some instances well justified. For example,fractures in sedimentary rock fractures are often regularly spaced and the spacing often scales with thethickness of the fractured layer [Pollard and Aydin 19881. The use of continuous fractures is harder tojustify. For most fractured rock uniform spacing models do not adequately represent the heterogeneouscharacter of the fracture networks.312.4.3 PoissoN MODELPoisson structural models are based on the assumption that the location and orientation ofindividual fractures are independent of all other fractures. In a Poisson model, fracture centers areassumed be distributed randomly throughout the rock mass with a constant expected density. There issubstantial evidence to support the assumption of the independence of fracture location in many crystallinerocks [Snow, 1968; Priest and Hudson, 1976; Call et at., 1976, Wallis and King 1980, Chernyshev andDearman, 1991]. This evidence, supporting the independence of fracture location, does not mean that aconsensus has been reached on the applicability of Poisson models. Chernyshev and Dearman [19911 statethat the first episode of fracturing in crystalline rock tends to show independence of location, but fracturescreated between existing fractures are often regularly spaced.A three-dimensional Poisson model of circular or elliptically shaped fractures was proposed byBaecher et al. [1977]. Long et at. [1985] used the Beacher Poisson model with circular fractures to createa three-dimensional flow model. Two-dimensional Poisson models have been used in hydrogeologicalmodels by Long et at. [1982], Schwartz et at. [1983], Long and Witherspoon [1985], Herbert et al.[1991], and Hestir and Long [1990] among others.Figure 2.1 shows a two-dimensional fracture network generated by a Poisson structural model.The network is composed of two fracture sets. The mean orientations of the sets are orthogonal but eachset has a standard deviation in orientation of 100. The scan-line density of each set is 1.5 fractures permeter.2.4.4 TRUNCATED POISSON MODELA truncated Poisson model recognizes the tendency for fractures to stop growing when theyencounter another fracture. In a truncated Poisson model, fractures of some sets are preferentiallyterminated when they intersect fractures of assumed older sets. The truncated fracture network shown in32H 0 ct c,7 CI1 ctMeters001001Co 00 (71CD•CD0cJ- 01 Co 0Figure 2.2 has the same statistical properties as in Figure 2.1, except for the specification of a preferentialtermination of the sub-horizontal set at 50% of the intersections of the set on the vertical set. Theappearance of the two fracture networks is quite different.Geier et at. [1989] report that 1/3 to 2/3 of fracture terminations are truncations at preexistingfractures in 9 out of ten sites where truncation data was collected. At the other location, 15% of thefractures were truncated. Dershowitz et at. [1991] report 75% of terminations in densely fractured zonesand 44% of terminations in less densely fractured rock were truncations. The probability that a fracturetermination is a truncation at another fractures is related to but not the same as the truncation probabilityat each fracture intersection which is used to describe fracture truncation in the truncated Poisson model.Barton [1994] presents data from Yucca Mt., Nevada and the Mirror Lake site in New Hampshire showingtruncation probabilities of 50-80% at Yucca Mt. and 45-55% at Mirror Lake.2.4.5 WAR ZONE MODELThe war zone model, as proposed by Geier et at. [1989], attempts to incorporate the internalgeometry of shear zones as described by Segal and Pollard [1983] into a Poisson structural model. Theseshear zones have a high density of conjugate shear fractures between subparallel faults. The war zonemodel was originally proposed as a three-dimensional model. Figure 2.3 shows a network generated froma two-dimensional interpretation of Geier et al.’s proposal. The war zones are evident as the areas witha higher concentration of fracture traces.Fracture generation in the war zone model starts with the creation of independent fracture sets.From this initial fracture system, “war zones” are identified between closely spaced subparallel fractures.The zones are identified on the basis of three characteristics: 1) The angle between neighboring fractureplanes, 2) the percentage of the fractures areas that overlap between the fractures, and 3) the averagedistance between the fractures. Secondary fractures are created with a higher density within the war zones34r1 H CD CD C C,, C CD CMeters—0010(ii00 01CD (DO‘-1 (Il01 ro 0- 010.Meters0(710-I CD (-i)-1j C-) E 1 CD z CD C C 1 C C CD C,, C)C)C ci C)010 01CD•-•CD0‘•l cfro 0than outside the zones.2.4.6 PARENT-DAUGHTER MODELThe Parent-Daughter model was introduced by Long et al. [19871 to represent the fracture systemat the Fanay-Augeres uranium mine in France. Trace surveys of an 80m drift wall indicated that thefractures often occurred in clusters with similar orientation. The clustering could not be represented witha structural model that assumes that the statistics of the fracture network are constant throughout the rockmass, as is the case with the Poisson model. The Parent-Daughter model incorporates spatial variabilityin the distribution of fracture locations by using a Poisson model to generate fractures in small subregions. The values of the fracture system statistical parameters vary between sub-regions, but arespatially correlated with the values of the neighboring regions.Figure 2.4 was presented in Long et al. [19871. It shows a lOOm by lOOm domain containing65,740 fractures. The sub-regions for this domain were lOm on a side. Both the fracture density andmean length varied as a function of the sub-region. At first glance the density variation may not seemto be apparent. Close inspection reveals a number of sub-regions that are clearly different than theirneighbors.2.4.7 HIERARCHICAL MODELLee et al. [1990] describe a structural model that accommodates both spatial clustering of fracturesand the dependence of younger fracturing on preexisting fractures. The model generates locations ofthe first fracture sets through a random process referred to as a doubly stochastic Cox process [Diggle,1983]. Subsequent sets may incorporate a dependence of the fracture location on traces of the first sets,including truncation. The hierarchical model is an expansion of an earlier model of Conrad and Jacquin[1975].37Figure 2.4: Parent-Daughter fracture network generated from Fanay-Augeres fracture system FromLong etal. [1987]. The network is lOOm by lOOm, 25 times the area of figures Zi, 2.2and 2.3.382.4.8 FRACTAL MODELGeier et al. [1989] present a fractal-based model they refer to as the Levy-Lee model. In thismodel fracture centers are created sequentially by a Levy Flight process. The size of the fractures areproportional to the distance to the previous fracture. The distance, L’, between fractures is selected fromthe cumulative distribution:PL(L”> L) L -D (2.1)where D is a specified fractal dimension. A fracture network generated from a Levy-Lee structural modelis shown in Figure 2.5. The fractal dimension used to create thenetwork was 1.3. It is evident fromFigure 2.5 that the Levy-Lee structural model produces a fracture network in which there is a tendencyfor fractures to form clusters. A larger fractal dimension would have resulted in tighter clusters offractures. Geier et al. determined that a fractal dimension of 2.5 (1.5 in two dimensions) was appropriatefor trace survey performed on the floor of the Stripa TSE drift. Barton and Larson [1985] have calculateda fractal dimension between 1.12 and 1.16 from various trace survey maps. Barton [1994] calculated apower law distribution of fracture trace lengths to be 2.41 at the Mirror Lake site in New Hampshire.Barton and Hsieh [1985] report power law distributions in the range of 0.8 to 1.3 for trace length surveysat Yucca Mt. Nevada and fractal dimensions for spacing between 1.6 and 1.8. The Levy-Lee model usesthe same exponent for fracture trace length and spacing.2.4.9 COLONNADE MODELThe colonnade structure of large basalt flows that form long columns of roughly hexagonal crosssection may be the most appropriate fracture system for two-dimensional representation because of thesymmetry along the axis of the column. Khaleel [19891 used a mesh generation scheme to define a twodimensional pattern of fracturing that represented fractures of the colonnade structure in the flow interiors39C)tCEaC-)4-CCC)C)aC-)10(‘CiiS)1U)0Q)ci)Is)0C0(‘CiC)C)L()0LI)r0sJaaJAIn@4C)Iof the Columbia River Basalt Group. The patterns Khaleel achieved are shown in Figure 2.6.Dershowitz [1984] describes Voronoi tessellation [Voronoi, 19081 as a method to achieve similarpatterns. The Voronoi mosaic tessellation uses a Poisson point process. It is based on the center of therock blocks rather than on the center of the fractures as in a Poisson structural model. From the centers,the rock blocks are “grown” outward at variable rates. Growth stops at locations where two blocks meet.The resulting boundaries between the blocks define fracture surfaces.Although colonnade structural models are appropriate for two-dimensional modeling, they are notappropriate for this research. These fractures systems lack the flow dominating fractures that areconsidered essential to the dual permeability concept. As Khaleel [1989] demonstrates, these fracturessystems are amenable to representation using continuum approaches at large scales.2.5 DISCUSSION OF FRACTURE SYSTEM CHARACTERIZATIONIf viewed as a representation of fractured rock as a whole, much of the data and interpretationsof fracture systems are contradictory. There is undoubtedly no one set of either statistical parameters orstructural models that best represent fracture systems. Fracture networks are most likely a combinationof randomness and regularity that can not be completely captured by either a statistical description nor afixed set of structural model rules. There seems to be a trend toward more sophisticated structural modelsthat are attempting to combine these two features [e.g. the hierarchical model of Lee et at., 1990]. It maybe that structural model development has entered a stage of rapid proposal, investigation, rejection andreplacement with a more advanced proposal. This model development requires increasingly moresophisticated fracture characterization surveys.While considerable progress has been made in understanding the hydraulic behavior of individualfractures in the past ten years, the basic models are still in a state of flux. It is still not clear how to relateaperture and fracture density measurements from boreholes and surface surveys to undisturbed rock. To410Co000CC)-C)CC(-)(1C)-C)I0C00(I)0C00Co0000000’cJCOL()000C)LI)COC’J(w)UO!!SOd1make this connection, a greater understanding of the affects of stress variation and of fracture shapes isneeded. Of particular importance is the need to understand the nature of rough walled fractures and theaffects of aperture variations on both flow and transport.Four critical problems need to be solved in order to make field-scale, discrete fracture models aneffective tool for assessing contaminant transport in fractured media. These problems are: 1) the lack offield-scale three-dimensional fracture network flow and transport models, 2) the uncertainty inextrapolating borehole and surface measurements to undisturbed rock, 3) the primitive state of singlefracture flow and transport models, and 4) the lack of understanding of influence of connectivity andheterogeneity at field-scales. The relative importance of these issues to the hydraulic behavior of fracturedrock at the field-scale is not known. It may be that one or more of these problems will prevent thediscrete network approach to modelling fractured media from ever becoming an effective tool at the field-scale. The current limitations of discrete fracture models prevent us from resolving these issues.The emphasis of this thesis focuses on the fourth topic. The other three areas of uncertainty areaddressed in the following manner: 1) fracture systems are approximated as two-dimensional, 2) a fracturesystem characterization of a specific site is not used - instead statistical characterizations and structuralmodels are used in a generic sense to emulate the way fracture system information is incorporated intodiscrete fracture models, and 3) a constant aperture model is used to model individual fractures, isolatingthe affects of network scale heterogeneity from the effects of single fracture heterogeneity. Hopefully,the methods developed to incorporate network scale heterogeneity in two dimensions will provide insightto incorporating network scale heterogeneity in three-dimensional models. These methods are not likelyto be dependent on the assumption of constant aperture fractures.43CHAPTER 3FRACTURE NETWORK GENERATIONAND FLOW AND TRANSPORT REPRESENTATIONIN A DISCRETE FRACTURE MODEL3.1 INTRODUCTIONDetails of the discrete fracture model developed for this thesis are presented in this chapter. Thediscrete fracture model provides flow and transport simulations that are used to assess the performanceof the dual permeability model. Both the discrete fracture and dual permeability models are based on thesame assumptions describing the hydraulic properties of fracture networks. Both models share a commonalgorithm to generate fracture realizations. This chapter begins with a description of this procedure. Themathematical formulations of the simulated hydraulic processes are presented next. With the basicgroundwork common to both of these models described, the discrete fracture model is briefly described.The dual permeability model is described in Chapter 4.3.2 DEFINITION OF A FRACTURE NETWORKFracture networks are approximated using a two-dimensional representation. The followingassumptions and approximations describe how fractures are represented:• Fractures are represented as straight lines of finite length.• The aperture of each fracture is assumed constant and with no infiuing, resulting in aconstant transmissivity along its length.44• Fractures that overlap are assumed to be interconnected with no change in the aperture ofeither fracture.• Fractures are assumed to extend equally and symmetrically out of the plane of simulation.• The host matrix is treated as non-porous and impermeable.The fractures for a single realization are generated using a selected structural model from specifiedstatistical distributions of the characteristics of individual fractures. The fracture system description isassumed to be constant within the domain, with no dependence on depth or other varying stress conditions.The width, length, orientation and center of each fracture are generated stochastically using theseprobability distributions. These properties may be modified based on the rules of the selected structuralmodel. Additional fractures, or possibly the entire network, may be specified explicitly.3.3 FRACTURE NETWORK GENERATIONThe procedure for creating a fracture network realization is quite similar to others described in theliterature [e.g. Long et at., 1982]. There are some differences in the way fractures near the edges of themodeled region are treated, but these differences are fairly minor. A flow chart of the fracture generationprocedure is presented as Figure A.2 in the appendices. Four of the structural models that were describedin Chapter 2 have been implemented. These are the Poisson, truncated Poisson, war zone, and Levi-Leemodels. The Poisson, truncated Poisson, and war zone models form a sequence of models, in that eachcontains the previous model in the sequence as a sub-model.The generation of a fracture network realization has two components: fracture creation and fracturenetwork reduction. Fracture creation is the positioning and establishment of the physical properties ofindividual fractures within the fracture generation region. Fracture reduction is the elimination of45fractures, or segments of fractures, that cannot contribute to flow through the domain because they lackconnections to other fractures. The reduced fracture network is subsequently referred to as thehydraulically active network because the fractures of this network are all connected to the regional flowregime.An attempt has been made to implement the discrete fracture and dual permeability models in sucha way that the data inputs are statistical parameters that could be estimated from a rock body. The datainputs are:the number of fracture sets,scan-line density of each set,fracture length distribution of each set,fracture orientation distribution of each set,fracture aperture distribution of each set, andfor structural models that incorporate fracture truncation, dependence of sets on previous sets, andthe fracture intersection truncation probability for fractures of dependent sets on the fracture facesof earlier sets3.4 IMPLEMENTING THE STRUCTURAL MODELS3.4.1 POISSON MODELThe Poisson structural model is based on the assumption that the density of fractures is constantthrough the domain and that the position of any one fracture is independent of all other fractures. Eachfracture in the Poisson model is generated independently. The number of fractures that are generated foreach set is controlled by the scan-line density, the size of the model domain, the orientation distribution,and the length of the fractures for that set. Length enters the determination of the number of fractures that46are generated for each realization in two ways; first in the conversion of scan-line density to the numberof fractures per unit area, and second in determining the distance beyond the flow domain within whichfractures are generated.Scan line density is used as a measure of fracture density for each fracture set. If the fracturesof a set are all parallel, then the scan line density for the fracture set is=(3.1)where p is the density of fracture centers and 1 is the average length of the fractures. If the fractures varyabout the mean orientation, then Equation 3.1 must be adjusted. The correction, from Terzaghi [1965],isA, pE (3.2)whereE= f cos(O)g(O)dO (3.3)o is the deviation of the fracture orientation from the mean orientation, and g(O) is the probability densityfunction of fracture orientations. E has been included as a look-up table in the fracture generation routineas a function of the standard deviation of orientation, c, assuming a normal distribution of orientationabout the mean. The number of fractures generated for each set is given by47Ii! - - WW (3.4)2E(a)where W and W, are the dimensions of the fracture generation region. The fracture generation regionextends in each direction beyond the intended model domain by (2+cL)*max(1),where i indicates thefracture set. This extension incorporates most of the fractures that may be centered outside the flowdomain yet protrude into it. It is impossible to include all fractures that may protrude into the modeldomain because of the fmite probability of fractures larger than the largest specified generation region.The parameter cx is used to avoid an artificial increase in fracture density along the boundaries of thedomain. The use of cx will be explained in a later paragraph.The generation of each fracture proceeds by defining the fracture midpoint from a uniformdistribution of midpoints over the fracture generation region. Fracture length, aperture and orientation arethen independently selected from specified distributions. The fracture length may have a negativeexponential distribution, be log-normal, or be fixed as a constant. The aperture may be selected fromnormal, log-normal, or negative exponential distributions. Orientation may have a uniform distributionor be sampled from a normal distribution. Unless stated otherwise, the distributions used for this studyare; a negative exponential length distribution, log-normal aperture distribution and normal orientationdistribution.The fracture network of Figure 3.1 was generated using a Poisson structural model. The twofracture sets of Figure 3.1 were generated using the parameter definitions of Table 3.1.3.3.2 TRUNCATED POISSON MODEL48rIMeters0-)_C) 1-÷CDoCD CC)-0-ICl)C)C) Cl)CDCD C)CD0010ro010The truncated Poisson structural model adds fracture truncation to the Poisson structural model.Younger fractures may stop growing when they intersect the face of pre-existing fractures. The tendencyfor more recent fractures to stop growing when a preexisting fracture is encountered can be measured interms of a probability of truncation. Older fractures never truncate on the faces of younger fractures; anobservation that is often used to establish the relative ages of fractures. The age relationship isincorporated in the truncated Poisson model by specifying truncation dependencies.2 Number of fracture setsFracture set 11.5 fr/rn Fracture density2. rn Mean length6x105 m Mean aperture0. Standard deviation of log10 aperture900 Mean orientation10° Standard deviation of orientationFracture set 21.5 fr/rn Fracture density2. rn Mean length6x105 m Mean aperture0. Standard deviation of log10 aperture00 Mean orientation10° Standard deviation of orientationTable 3.1: Fracture system definition for the fracture network of Figure 3.1.50In the structural model, fractures that belong to sets that are dependent on pre-existing fracturesare truncated at intersections with pre-existing fractures with probability Tr1, the truncation probability ofthe fractures of set i on the faces of the fractures of set j. The procedure proceeds as follows. First, thefractures of the entire fracture network are generated independently. The lengths of the fractures to betruncated are increased to account for the anticipated truncation of these fractures. Then, for each fractureof the truncated sets, the truncation procedure starts at the midpoint of the fracture and works toward thetwo ends, testing for truncation at each intersection. A fracture that is truncated at an intersection endsat that intersection with all intersections with other fractures beyond the truncation being eliminated. Itcan be shown that the fraction of the length lost to fractures of set j because of the intersection of thesefractures with the fractures of set i can be written as:A Tr (3.5)where f(O) is the probability density function of fracture orientation. F1 is approximated in the truncationroutine byATr1 — —F ‘sinlo.-OIii 4 junless the two mean angles are equal. The total fraction of length lost to truncation is(3.7)This truncation loss is a fraction of the fracture length that is initially generated. The fracture length priorto truncation, ? must beJ51_f (3.8)iFJfor the post truncated mean fracture length, , and scan-line density to be consistent with thespecifications for the setFigure 3.2 presents a truncated Poisson fracture network both prior to truncation, panel a, and aftertruncation, panel b. The second set is truncated on the first set with a truncation probability of 0.5. Theparameters for the fracture network in Figure 3 .2b are the same as for Figure 3.1. The pre-truncationaverage length of the second set has been increased by a factor of four to obtain the desired posttruncation length and scan line density resulting in a different appearance of the pre-truncated network inFigure 3.2a from that of the network in Figure 3.1.3.3.3 WAR ZONE MODELWar zones are areas of more intense fracturing than in the rest of the fracture network. The warzone model was proposed by Geier et al. [1989] for three dimensional fracture networks. A twodimensional version of the war zone model has been created for this thesis for two reasons; 1) the warzone model has two scales of fracturing, 2) the well defined war zones have very different hydrauliccharacteristics than the region surrounding them. Both of these features fit well into the structure of thedual permeability model described in Chapter 4. It has been recognized that most fractured rock hasfractures on many scales and the interaction of fractures at different scales may be an important aspectof fractured rock hydrology; an aspect that is missing in the Poisson model. An example of a war zonefracture network has been presented in Figure 2.3.War zones exist between pairs of closely spaced sub-parallel fractures as depicted in Figure 3.3a,modified from Geier et al. [1992]. Once the war zones are identified, additional fractures are then created52‘-I CDMetersMeters--001—C0CD-CD CCD‘—CJ101CDC-CD (DO•1-C),C-)CCD01CCI)NCDCd) CDCD-r—.CD—C.I)0OCDCD 1:1)CI)CDCDCD<0CD-i--rcO0100010(311’)00.abprojection NneIFigure 3.3: War zone definitions, a) Identification of war zones between close, subparallel,overlapping fractures (modified from Geier et al. [19891). b) Definition of war zonecharacteristic parameters A1 and A2 are the fracture lengths, A is the overlap betweenfractures, A is the average distance between fractures.A1AA254!within the zones. The fracture pairs defining war zones are selected based on the basis of three criteria:“parallelness’, ‘largeness’, and “closeness”. Parallelness W,, is measured as(3.9)where cc, is the angle between the two fractures. The largeness WL of a war zone is a measure of thefractional overlap between the fractures:WL = (3.10)A1÷A2where:A1,, is the projected overlap between the two fracturesA1 is the projection of the first fractureA2 is the projection of the second fracture.The projection is onto a line that bisects the difference in angle between the two fractures as shown inFigure 3.3b. The closeness W is:(3.11)where A is the average distance between the overlapping portions of the two fractures. A region isclassified as a war zone ifW,Z,, + WLZL + WZ > 3 (3.12)where Z,, ZL, and Z are empirically defined parameters based on investigation of a fracture system. Geieret al. suggested that 1.5 was adequate for each of these parameters for the Stripa SCV Project site. Thesevalues have been used for all of the war zone structural models used in this thesis.55War zones are areas of increased fracturing. The dependent fracture sets are first generatedthrough the entire fracture generation region using the input scan line density as in the Poisson model.Then, more fractures are added to each of the war zones. The resulting scan line density within the warzone becomes(3.13)where Jç is the war zone intensity factor. An intensity factor of 4.0 has been used for all war zones.Fractures that intersect the bounding fractures of a war zone are truncated at the intersection. These newfractures may retain the orientation disthbution specified in the input or they can be normally distributedabout the orientation perpendicular to the projection line.3.3.4 LEVY-LEE MODELA fracture generation algorithm has been developed based on the Levy-Lee model proposed byGeier et al. [1989]. As with the war zone model, the Levy-Lee model is a useful tool to investigate issuesof multiple scales of fracturing. The Levy-Lee flight process is used to generate a fracture network thathas a continuum of scales, both in terms of fracture length and in the spacing between fractures.The report of Geier et al. is not a formally published document and describes the model inconceptual rather than detailed form. Therefore, the implementation described in this section may deviatefrom the intentions of the original creators. As described by Geier et al., the spatial distribution offractures generated using the Levy-Lee model is fractal. Fracture lengths are proportional to the spacingdistance, resulting in a power law distribution of fracture length. A power law distribution less than -1,which is recommended, must be truncated at some minimum fracture spacing. The resulting minimumlength determines the scan-line density of the generated fractures.56The Levy-lee flight process results in a random walk in the locations of the generated fractures.Each new fracture center is located a random distance and random orientation away from the previousfracture. The distance in determined by dividing a minimum spacing distance by a uniform randomnumber that has been raised to the power law exponent. The displacement orientation is uniformlydisthbuted between 0° and 360°. When the locations move outside of the fracture generation region, thefracture is relocated randomly within the region; resulting in a series of over-printed random walks.Fractures are generated until the specified scan-line density is reached. In the realizations that are usedin this study, the fractures are relocated 30 to 40 times. The Levy-Lee structural model generated highlyclustered fracture systems with power law length distributions but the spacing distributions are probablyconsistent with a power law distribution over a limited range of scale. The actual spacing distributionshave not been determined.Geier et al. proposed a coffelation between fracture spacing and orientation. A linear relationbetween the standard deviation of fracture orientation and the spacing distance is used for this study. Thisrelationship results in sub-parallel long fractures with little orientation variability. Smaller fractures havemuch greater variability in orientation and are highly clustered. Clustering results in concentrations ofhighly connected fractures embedded in a larger scale framework.A fracture network generated using the Levi-Lee model has been presented earlier in Figure 2.5.The fracture sets have the same orientation and fracture scan-line densities as those used for the Poissonmodel of Figures 3.1. The mean length of the fractures in Figure 2.5 is half of that in Figure 3.1. Theminimum fracture length of the Levy-Lee model is proportional to the specified mean fracture length. Afractal dimension of 1.3 and a minimum spacing distance of 0.25m were used to generate the network.573.4 FRACTURE NETWORK REDUCTIONOnce generated, fractures go through a series of screening processes that reduce the number offracture segments that define the fracture networks of the flow and transport models. Fracture segmentsthat are eliminated cannot have fluid flowing through them and thus are not important hydrau]ically.Eliminating these fracture segments makes the solution of the hydraulic head distribution easier todetermine. A flow chart of the fracture reduction process is included as Figure A.3 in the appendices.Figure 3.4 is used to aid the explanation of the fracture reduction procedure. Figure 3.4 showsthree rectangular regions and a number of fractures. The outer rectangle defines the fracture generationregion that is used to place fracture centers. Fracture centers are shown as dots in the figure. The nextlargest rectangle defines the region within which fractures are needed to provide a hydraulic connectionbetween the fractures of the model domain and the surrounding fracture network. This rectangle isreferred to below as the influence boundary. The model domain, or flow domain, is drawn as the innerrectangle.The first screening process is to discard fractures that are not likely to be needed to provide ahydraulic connection between the fractures of the model domain and the surrounding fracture network.The portions of fractures that are outside the influence boundary, a distance a*max(1), from the modeldomain are discarded. These fractures are shown as dashed lines in Figure 3.4.The next stage in the reduction process is to find all the intersections of each fracture with otherfractures. Fractures that do not intersect other fractures or which have only one intersection cannot be apathway for flow. These fractures are discarded from further consideration. In addition, fractures thatcannot be connected through two distinct pathways to the outer edge of the zone of influence cannot havefluid flowing in them and are discarded. The portions of each fracture that are not between fractureintersections and/or intersections with the domain boundary are eliminated by shortening the fractures.58Fractures• Hydraulically ActiveOutside BoundaryInfluence Region• InactiveErroneously Activewithout BoundaryInfluence RegionFigure 3.4: Region boundaries used in the fracture reduction procedure. The fracture generationregion defines the outer limit of fracture center placement. The boundary influence regiondefines the limit of the region where fractures connect to form a hydraulically activefracture network. The model domain identifies the extent of the flow domain.59Fracture segments removed at this stage are shown as both shaded thick lines and thin lines in Figure 3.4.The procedure described in the two preceding paragraphs is repeated using the flow domainboundary rather than the outer edge of the zone of influence. The fractures that remain after the removaland truncation process make up the hydraulically active network, the solid thick lines in the figure. Thehydraulically active network contains all of the fracture segments that are flow pathways. If the screeningprocess were perfect then the remaining fractures would be limited to the active network. It is not. Somefracture segments are retained that are not flow pathways, but are hydraulically connected to the activenetwork as is the group of fractures labelled A in Figure 3.4. These fracture segments slow the flowsolution but do not influence the results.The use and rationale for the first pass through the screening process based on truncation at theinfluence boundary requires some explanation. In Figure 3.4 the shaded thick fractures would be retainedif the parameter a were zero. These fractures form pathways that cross the domain boundary but do notintersect the influence boundary. For sparse fractures systems, setting a to zero results in an increase infracture density and hence hydraulic conductivity near the edges of the domain. In the study describedin Chapter 5, the average hydraulic conductivity of the domain was sensitive to this increase inpermeability; especially when fractures such as the otherwise isolated fracture in the lower left corner ofFigure 3.4, labelled B, were included. For dense fracture systems, changing the influence boundary haslittle effect. The reduced fracture network of the Poisson fracture network presented in Figure 3.1 isshown in Figure 3.5.3.6 MODELING OF FLUID FLOW AND SOLUTE TRANSPORT3.6.1 FLUID FLOWThe fluid flow model is based on the following set of assumptions.1) parallel sided, unfilled fractures60T1Meters0010- 01ro 0r%.) 02) non-porous, impermeable host rock3) fully saturated conditions4) steady state, laminar flow5) Fluid density,p, and viscosity,p, are assumed to be insensitive to any pressure, temperature orsolute concentration variations within the flow domain. The values used for all simulations area fluid density of 998.0 kg/ni3 and a viscosity of 0.001 kglms.The assumption of laminar flow is supported by investigations of the Reynolds numberrepresentation of flow within the fractures. The Reynolds number is a dimensionless measure of the ratioof inertial forces to viscous forces. The Reynolds number for a fracture is given by(3.14)where V is the average fluid velocity, b is the fracture aperture, p is the fluid density, and i is theviscosity. In circular pipes, the flow is laminar if Re is less than 2x103 [Saberslcy et al., 1971]. A similarthreshold should apply to fractures. The fracture network realization of the fracture system defined byTable 3.1 and shown in Figure 3.1 was used to estimate typical Reynolds numbers for the simulations thatare performed during the course of this research. Figure 3 .6a shows the distribution of Reynolds numbersfor the flow solution obtained by applying a constant hydraulic head gradient of 0.01 m/m along theboundaries of the fracture network. The maximum Reynolds number is l0 indicating the assumption oflaminar flow is well supported.62Figure 3.6: Reynolds and Pdclet number distributions of flow and transport processes in singlefractures. Distributions are from solution to the fracture network shown in 1. Thehydraulic head gradient was 0.01. a) Reynolds number. b) Péclet numbers calculated forlongitudinal diffusion. c) Weighted Péclet numbers calculated for transverse diffusion asdefined by Equation 3.23. d) Longitudinal dispersion coefficients. Pdclet numberscalculated for fracture intersections.1 501 00500b>C-)Ca)C0)ciC-)Ca)C0)-4 -2 0 2Péclet Number [ log1 ]4a Flow within Fractures200150kC)C100-C)0jdIIiIReynolds Number [log10 ]CC.) 100Ca)C500-10-8 -6 -4 -2 0Weighted Péclet Number [ log10 Ie Diffusion in Fracture Intersections3020100______________-5100500-22 -20-18 -16 -14 -12 -10Long. Dispersion Coefficient [m2/s]>.‘C).) I I I 11111F-4 -3 -2 -1Péclet Number [ log10063Under laminar flow conditions the average flow within a fracture is described by the cubic lawfor parallel-sided fractures [Snow, 1965]:-pgb38h (3.15)12 8LwhereQ = volumetric flow per unit thickness of the model,g = gravitational constant,= hydraulic head gradient along the fracture.For a fracture with a constant aperture the hydraulic head gradient is constant between fractureintersections. Equation 3.15 can be rewrittenQ _pgb3Ahaj (3.16)‘ 12i ALUwhere is the flow from intersection i to intersection j, Ah is the hydraulic head difference betweenintersections i and j and zXL is the distance between them. An equivalent expression ish.Q.. -T L (3.17)ZJkiwhere T is the transmissivity of the fracture between the two intersections.Equation 3.17 is conceptually more satisfying than Equation 3.16. The use of Equation 3.17allows the assumption of constant aperture to be relaxed. That isT.. (3.18)j 12.L64The constant fracture aperture, b, in Equation 3.16 is replaced by an effective hydraulic aperture, bh, thatyields the appropriate T over the length of the fracture segment between intersections. The effectivehydraulic aperture is defined as [Silliman, 1989]AL4 -113b = _L —L (3.19)hwhere Q is a position along the fracture segment. To be fully consistent with the notion of the fractureaperture representing an effective hydraulic aperture, need not be the same for each segment fracture,unlike the constant aperture assumption for each fracture in this model.By symmetry, no flow occurs into or out of the plane of the simulation. There are also no sourcesor sinks for fluid such as wells. At each fracture intersection the fluid mass must be conserved. The flowinto each intersection must be equal to the flow out of the intersection. The mass balance equation foreach intersection is-T.0 (3.20)i iiIn equation 3.20, the index j is limited to interior intersections but index i can also include fractureintersections with the fracture network boundary. The set of equations 3.20 for all intersections withinthe flow domain can be written in matrix notation as= 0 (3.21)65where A is a symmetric matrix with diagonal coefficients and off diagonal coefficientsI ifI is a vector of hydraulic heads for each intersection, and 6 is the vector containing boundary conditionsfor each intersection.Intersections that are directly connected to the domain boundary have boundary conditions definedbyb1=(3.22)where Tm is the transmissivity of the fracture segment connecting intersection i to the boundary locationn and h is the hydraulic head specified at n. The other components of l are all zero.The that satisfies Equation 3.21 defines the hydraulic head at all intersections of the fracturenetwork and through Equation 3.17 defines the flow in each fracture segment. In the simplest terms, acentral mission of the research has been to investigate methods of finding an approximate solution toEquation 3.21 that avoids the direct use of matrix A, in hopes of increasing the size of fracture networksthat can be studied.3.6.2 SOLUTE TRANSPORTAt the level of individual fractures, two processes interact to effect the movement of solute. Oneprocess, advection, is driven by flow within the fractures. The other process, diffusion, is driven by soluteconcentration gradients. It is assumed that the solute concentration is always small so that the perturbationof fluid density by the solute has no influence on the advective process.66The character of solute transport in a fracture network is strongly dependent on the relativeimportance of advection compared to diffusion at the scale of individual fractures. For example, ifadvection dominates the flow within intersections of fractures then the solute from one inlet fracturesegment will not mix into the fluid of another inlet fracture. If diffusion dominates then the fluids do mixacross streamlines in an intersection and solute may enter fractures that disperse the solute to a muchgreater extent.Diffusion may be an important process at the scale of individual fractures because of thesensitivity of network scale behavior to these local conditions. The relative importance of advectioncompared to diffusion is typically measured in terms of a dimensionless Pclet number. The Pcletnumber can be defined byVL (3.23)Pe =Dwhere V is the fluid velocity, L is a representative distance, and D is the solute diffusion coefficient. Arepresentative diffusion coefficient of io m2/s is used in the Péclet number calculations. The Pécletnumber can be thought of as the ratio of the time needed for solute to diffuse a distance L, over the timeneeded to be moved by advection across that distance. If the Pdclet number is much greater than one, thendiffusion can be neglected in modeling transport. I am not aware of an accepted value of the Pécletnumber that would indicate negligible diffusion. The number is probably between 100 and 1000. If thePéclet number is much less than one, diffusion dominates and advection can be ignored.The advection distance may be different than the diffusive distance in some situations. In thissituation, diffusion will dominate if the Péclet number is much greater than the ratio of the diffusivedistance to the advective distance. To retain one as the indication of transition region between advectionand diffusion, the Péclet number can be weighted by the inverse of the distance ratio.67For transport along a fracture segment, V is taken as the average velocity within the fracture(Q / b1), and L is the length of the fracture segment, Figure 3.6b presents the Pdclet numbers fortransport along the fractures for the network of Figure 3.1. Most of the Pëclet numbers are greater than1, but roughly half of the Péclet numbers are less than 100 indicating that longitudinal diffusion along afracture influences solute transport in many fractures.The fluid velocity within a fracture is not unifonn. The fluid near the center of the fracture movesfaster than fluid near the fracture walls. The distribution of the velocity introduces dispersion into themovement of solute through fractures. The dispersion is influenced by the interaction of diffusion acrossthe streamlines within the fracture as the solute moves along the fracture segment. The relativeimportance of diffusion across the width of the fracture relative to advection along the fracture ischaracterized using a weighted Pdclet number defined byFe = (3.24)b DLIf the weighted Péclet number is less than 8 then the transverse distribution of tracer is nearly constantacross the fracture [Hull et al., 1987]. The weighted Péclet number calculated for the network of Figure3.1 (shown in Figure 3.6c) provides assurance that transverse diffusion homogenizes the tracer across thewidth of the fracture. Under these conditions dispersion is Fickian and can be described by a longitudinaldispersion coefficient defined as [Hull et aL, 1987]D = (yb)2 (3.25)2lODFigure 3.6d shows the longitudinal dispersion coefficient calculated for the same netwo± as the otherfigures. The figure reveals that this longitudinal dispersion coefficient is much less than moleculardiffusion under these conditions and can be neglected.68In the discrete fracture and dual permeability models, both diffusion and longitudinal dispersionare assumed to be negligible in all fractures. It is not clear to what extent that neglecting diffusioninfluences temporal dispersion at the scale of the fracture network. By neglecting solute longitudinaldispersion in individual fractures, temporal dispersion caused by the variation of fracture characteristicsand the way fractures are connected can be investigated more clearly.In a parallel sided fracture, the time to move between intersections is given bybiL (3.26)1)As mentioned earlier, the assumption of parallel sided fractures is not supported by fracture investigations.This assumption requires some discussion because it is an important limitation of the models used for thisthesis. In a rough fracture, the time to move between intersections is given by= 1 (3.27)Q10After substituting Equation 3.16 for Q, an effective aperture for transport could written as [Silliman, 1989]-1/2bT = fb()d (3.28)bAL, oThis effective aperture is not equal to the effective hydraulic aperture, bh, calculated in Equation 3.19.To represent the effective aperture of a rough walled fracture, the representation should be different forthe flow process and the transport process. The direct linkage of the flow and transport processes through69the assumption of constant aperture fractures may have an influence on the predictions of network scaledispersion. For a more thorough discussion of effective apertures, see Silliman [1989] and Tsang [1992].Fracture aperture roughness also introduces dispersion into the transport process [Moreno et al.,1985; Tsang and Tsang, 1987]. Dispersion arises from the tortuous paths solute takes through a fractureplane. No attempt to model dispersion within a single fracture is made here. Dispersion could potentiallybe introduced through the use of statistical transport times through a single fracture in a manner similarto the three dimensional flow and transport model described by Nordqvist et a!. [19921. These authorsfirst created a library of possible transport times by stochastically generating a large number of twodimensional single fractures. Transport times through individual fractures in the fracture network modelwere detemiined by sampling from the library. The relative importance of dispersion in a rough-walledfracture compared to dispersion from fracture network heterogeneity is presently unknown.Another important source of dispersion in fracture networks arises from mixing at fractureintersections. Diffusion across stream lines in fracture intersections causes solute to enter fractures thatit otherwise would not enter if it did not diffuse. For fracture intersections a Péclet number can be definedasFe (3.29)Dwhere r is defined asr______(3.30)and b1 and b2 are the apertures of the intersecting fractures. The distribution of fracture intersection Pécletnumbers is shown in Figure 3.6e. These Péclet numbers indicate that transport in nearly all intersectionsof this simulation are diffusion dominated. A complete mixing model of transport in diffusion dominatedfractures assumes that the solute is homogenized within the fracture intersection but leaves the intersection70into the downstream fractures in proportion to fluid advection. Berkowitz et al. [19931 have shown thata condition of complete mixing will never be reached because diffusion causes much of the solute to enterthe downstream fractures before the solute reaches a state of homogenization. The resulting mixing is lessthan complete. To model this transport behavior would require a diffusive transport model thatincorporated the exact geometry of the fracture intersections. No such model exists at present and thedevelopment of such a model for this work is not justified by the anticipated gain in accuracy of modelingnetwork dispersion.The transport model is based on a particle tracking algorithm. It is similar to the particle trackingmodel described by Schwartz et al. [1983]. It is assumed that the aggregate behaviorof a large numberof particles mimics the advection of a dilute, non-reactive solute. Particles are assumed to move throughfractures at the average fluid velocity within the fractures (plug flow). There are no explicit diffusion ordispersion tems in the particle tracking algorithm. Diffusion of solute into the host matrix and into non-flowing fractures is assumed to be negligible. The complete mixing model is used at fracture intersections.Figure 3.7 is used to explain the salient features of the transport algorithm. In Figure 3.7, fourfracture intersections are shown along with the connecting fracture segments. Particles that pass throughintersection 1 may be transported though the fracture segments that carry flow away from intersection 1.The fracture segment that the particle enters is chosen randomly( the complete mixing assumption) witha probability equal to the ratio of the flow through the fracture segment to the total flow through theintersection. In Figure 3.7, a particle that enters intersection 1 at time t from fracture segment 1 has a 70percent chance of traveling to intersection 4 and a 30 percent chance of moving towards intersection 2.A particle traveling from intersection 1 to intersection 2, will arrive at intersection 2 at time t + At12.Where At12 is71Q=3=8Figure 3.7: Detail of fracture segments and intersections. The parallel lines represent fracturesegments. The dots represent fracture intersections, where the solute is completely mixed.The Q’s are relative flow rates within the segments.720=Q=1101=3 °1,4 = 74.4= 4=400 =4hALAt1,2 1,1,2 (3.31)Here Q12 is the flow rate determined from Equation 3.16, b12 is the aperture, and AL12 is the length ofthe fracture segment respectively.The source of solute is based on a pulse injection. All particles are assumed to arrive at theupstream edge of the fracture domain at the same time, t=O. Transport for more complex situations canbe determined from the breakthrough histories of the pulse injection by convolving the breakthroughhistories with the desired concentrations at the upstream boundaries as a function of time.Particles are introduced into the domain either from a point source or a line source along anupstream boundary. A uniform concentration is assumed along line sources. Particles start at eachfracture intersecting a line source with probabilities that are equal to the fraction of total flow across theline source that is carried by the intersecting fracture segment.A single particle is moved at a time in the algorithm. This one particle is moved exclusively untilit reaches another edge of the domain boundary. The positions of the particles at specific times may berecorded to provide the data for particle position plots and dispersion calculations. The location and timeof arrival at a downstream boundary may be recorded, providing particle arrival histories at specificlocations and spacial distributions of arrival along the boundaries.3.7 BOUNDARY CONDITIONSThe models developed during this research are all limited to either specified head boundaryconditions or a combination of specified head and impermeable boundary conditions. The head specifiedalong permeable boundaries is consistent with a constant hydraulic head gradient along the boundaries ofthe flow domain. Figure 3.8 depicts the hydraulic head specified along each boundary for a gradient73Figure 3.8: Application of a constant hydraulic head gradient along the boundaries of the flowdomain.-000.500.250.00I0.000.250.500.751.00I 1 0.00BOUNDARY 3 —0.25ziBOUNDARY 10.0030Head Gradient74oriented 30° to boundary 1. The designations of the boundaries shown in the figure are used throughoutthis thesis. The boundary conditions used for these models becomes important in Chapter 5.3.8 PROGRAM DISCRETEThe program Discrete was developed to provide a reference calculation that the dual permeabilitymodel could be tested against. The structure of Discrete is outlined in the flow chart labeled Figure 3.9.The basic structure, as with any of the models described in this thesis, is fracture network generation andreduction, solution for the hydraulic head distribution, and particle transport. The hydraulic headdistribution is determined by solving Equation 3.21 using a conjugate gradient iterative solver for sparsematrices that uses incomplete Cholesky decomposition to precondition the A matrix.75Program Discrete*A2Fracture Network RealizationFracture Network Reduction A.3Determine Flow Equations*1Solve Flow ProblemDetermine Fracture IntersectionFlow Splits andFracture Residence TimesParticle Tracking * A.4Transport SimulationFigure 3.9: Flow chart of reference program, Discrete. Procedures with an in the upper right cornerare expanded in the indicated figure of Appendix A.CHAPTER 4DUAL PERMEABILITY:IMPLEMENTATION AND EVALUATION4.1 INmoDuc’rIoNAs discussed in Chapter 1, the dual permeability flow and transport model is based on the conceptthat in many cases a relatively small number of the fractures dominate the flow system. The criticalassumption underpinning the dual permeability model is that these fractures should treated in detail,whereas the majority of fractures in the network should be approximated, not neglected. The name dualpermeability is intended to be evocative of this two level treatment of the fracture system.Two groupings of fractures are defined in the dual permeability model. Group 1 contains fracturesthat are anticipated to dominate flow through the fracture system. This group forms the primary fracturenetwork. Group 2 contains all the other fractures in the network. Most group 1 fractures are identifiedin the model through a set of rules that link fracture characteristics with large flow rates. Additionalfractures are added to the group to ensure that group 1 fractures form a connected network. Finally, a fewmore fractures may be added to group 1 to force the regions defined by these fracture to be within amaximum size constraint. Figure 4.1 demonstrates the subdivision of the fracture network. The entirenetwork is shown in Figure 4. la. Figure 4. lb is a schematic of the primary fracture network that has beenidentified for this network. Identifying a primary fracture network requires inferences about the featuresof a fracture network that influence flow dominance, including scale effects that are critical tounderstanding flow and transport in fractured media.The regions enclosed by the primary fracture network are called network blocks. Figure 4.lcshows a network block that has been defined by the primary fracture network of Figure 4. lb. This block76i)-4-)Figure 4.1: Dual permeability model structure. a) The hydraulically connected fracture network. b)The primary fracture network.. c) A network block consisting of internal fractures and thebounding primary fractures is labeled B in Panel b of 1 near the location (15,32).c A Network Blocka Hydraulically Connected Fractures4030201000 10 20Meters30 40C’]00Meters.077is located near the point (15m, 30m) and is labeled B in Figure 4. lb. Network blocks define a sub-network of fracture segments that include both those primary fracture segments on the boundaries of theblock and all other fracture segments located within the block boundaries.Before discussing individual model components in detail, I will provide an overview of the methodused to create a dual permeability model. Figure 4.2 is a flow chart that shows how division of thefracture network into primary fractures and network blocks is used to create a flow and transport model.Creating a realization of the fracture network and subsequent fracture network reduction is the first stageof forming a dual permeability model. The next stage is to find a primary fracture network and to definethe internal sub-networks of all the network blocks. The effective hydraulic conductance of each of thenetwork blocks is calculated and then combined with the transmissivity of the fractures of the primaryfracture network to form an approximate set of flow equations for the entire fracture network. This setof equations is solved to provide an estimate of a coarse-scale hydraulic head distribution that is definedat each intersection of the primary fracture network. One advantage of solving for the hydraulic headdistribution at this coarse-scale is that the number of equations is reduced from the number needed todescribe the head distribution in the entire network. Using the hydraulic head estimates at the primaryfracture intersections as boundary conditions, a detailed flow solution that provides hydraulic headestimates throughout the fracture network is calculated by treating each network block as an independentsub-domain. Thus, the flow through the entire network is determined but the calculations have beenbroken up so that the flow equations for the entire network (Equation 3.21) are not solved directly.At this point in the modeling process, the flow solution could be used in conjunction with thetransport model described in Chapter 3. Instead, the dual permeability structure is also used to create asolute transport model. Particle tracking is used to simulate transport in the dual permeability model.Unlike the particle tracking routine described in Chapter 3, particles are moved across the regions definedby the network blocks in single steps. These large steps are made possible by precalculating residence78Dual Permeability ProgramFracture Network:RealizationPrimary Fracture* A.6Network SelectionDetermine Effective Conductance A.7of Network Blocks*1*A7Incorporate Network Blocksinto Primary Network toForm Flow Model*1Solve Flow Model forHydraulic Head Distribution*A8Find Network BlockResidence Time Distributions*1A.9Form Transport Model fromNetwork Blocks and Primary Fractures*1Particle TrackingTransport SimulationFigure 4.2: Flow chart of the dual permeability model.79time distributions for all possible transport paths through the blocks. Residence time distributions linklocations of inflow to locations of outflow of each network block. For each of these links, theprobabilities of particle arrival at discrete times are determined from the flows calculated individually foreach block.Sections 4.2 through 4.4 provide a detailed description of the components of the dual permeabilitymodel. In some instances, the results of dual permeability simulations are presented to help explain certainelements of the model. In Section 4.5, the flow and transport behavior calculated using the dualpermeability model is compared with that of Discrete. In addition, Section 4.5.4 presents an efficiencyevaluation of the model in comparison to Discrete. In Section 4.6, dual permeability simulations arecompared to Discrete simulations in which some of the less important fractures have been completelyneglected. This comparison facilitates a discussion of differing modeling philosophies; one in which theless important fractures are approximated, and the other in which these fractures are neglected. Adiscussion of alternatives to the methods described in Sections 4.2 through 4.4 are presented in Section4.7.4.2 SELECTION OF PRIMARY FRACTURES4.2.1 PRIMARY FRACTURE SELECTION CRITERIAPrimary fractures are either those fractures that are anticipated to dominate local flow conditions;those that have been explicitly defined as important fractures through the program input; or those that arerequired to link the primary fracture network together creating network blocks. Flow dominance is asubjective term. A fracture is considered to dominate the local flow if the flow within the fracture is largein proportion to the average flow rates within fractures in the immediate vicinity of the fracture. Bothfracture transmissivity and fracture connectivity influence the tendency for a fracture to dominate the flowregime in its vicinity.80The initial stage of the selection process for identifying primary fractures is based strictly on thegeometric characteristics of individual fractures. The individual characteristics of fractures are: position,length, orientation, and aperture (transmissivity). Of these characteristics, the orientation of a fracture doesnot seem to be strongly related to flow dominance. The position of a fracture relative to the boundariesof the domain can be important, however the interaction of the boundary conditions with the behavior ofthe fracture network is an artifact of limited domain sizes and should not be used as the basis of selectingprimary network fractures.A set of seven criteria have been examined to determine an appropriate method to choose primaryfractures. Each of these criteria weight the relative importance of transmissivity and length differently.The seven selection criteria are listed in Table 4.1. The criteria are used to rank the fractures in anordered list. Using the ranking, a specified percentage of the total number of fractures are identified asprimary fractures. This fraction is specified in the model definition but may be increased during theselection process if an adequate primary fracture network cannot be found using the specified fraction (seeSection 4.2.2).Criteria Criteria DescriptionNumber1 Aperture2 Length3 Aperture with minimum length4 Aperture * length5 Aperture2 * length6 Aperture3 * length7 Aperture6 * lengthTable 4.1: Primary fracture selection criteria.The symbol * indicates multiplication.81The primary fractures must form a completely connected network. Each end of every primaryfracture must have a flow pathway through other primary fractures that leads eventually to a domainboundary. Additional fractures are added to those selected strictly by individual geometric properties inorder to provide the required interconnection. These fractures are selected strictly due to connectivity,without regard to transmissivity.An additional factor is introduced into the selection process for primary fractures by a need tolimit the size of the tables used to calculate and store the particle residence time distributions of thenetwork blocks. Network blocks that have too many internal fracture intersections are subdivided usingprocedures described in detail in Section 4.2.2.In Section 4.5.2, a suite of seven fracture systems are used to investigate aspects of the dualpermeability model, including the merit of the various selection criteria. Results from the investigationof one of these fracture systems (fracture system 1 of Table 4.5) are summarized in Table 4.2. Theseresults are included here to provide some perspective on issues that need to be considered in determiningthe primary fracture network. An example of a fracture network realization for this fracture system isshown later in Figure 4.10. A hydraulic head gradient parallel to the bottom boundary is imposed on theboundaries of the fracture network. Particles are introduced over a ten meter region centered at the middleof the left boundary and are transported to the downstream boundary.The comparison is based on three measures; the total flow across the domain boundaries, the timeof the first particle arrival at the downstream boundary of the fracture network, and the time of the 50thpercentile particle arrival. Table 4.2 presents a comparison of the average deviation of the results of adual permeability calculation from the Discrete. The last column of Table 4.2 lists the percentage ofrealizations where an acceptable primary fracture network could not be found using the selection criteria.One hundred realizations are used in this analysis.82Criteria Flow Error Time Delay Time Delay P.F.N.( %) Initial Median Failure(%) (%) (%)1 -8.2 2.5 9.1 222 -9.9 1.8 8.9 33 -5.7 0.7 5.2 114 -4.3 -2.2 3.6 55 -3.6 -3.1 2.7 86 -3.6 -4.4 1.6 87 -4.5 1.0 4.0 11Table 4.2: Primary fracture selection criteria test. The results are based on one hundred realizations.The comparisons are calculated for individual fracture networks, that is the results do notinclude fracture network variability. Flow error is based on the total flow across thedomain boundaries. The time delays of the dual penneability model are for initial solutearrival and median particle arrival. P.F.N Failures indicate the number of realizationswhere the primary fracture selection routine gave up trying find an acceptable network.The table shows that there is reasonably close agreement between the two models and that theagreement is sensitive to the selection criteria. If the median arrival time is given emphasis in comparingthe selection criteria, then criterion six ,product of length and transmissivity, is the best selection criteria.The last column of the table reveals that an acceptable primary fracture network could not be identifiedfor some of the fracture network realizations.It is emphasised that acceptability of a primary fracture network is not based on the results of theflow and transport calculations. Unacceptable primary fracture networks either require too many primaryfractures or have network blocks that are too large and cannot be reduced in size. An upper limit isimposed on the number of primary fractures in order to limit the number of coarse-scale flow equations.The need to limit the network block size is discussed later in this chapter. Failure to find an acceptableprimary fracture network was more common using criteria based primarily on transmissivity, i.e. criteria1 and 3. Criteria that exclusively weight either transmissivity or length do not work as well as criteriathat weight both connectivity and transmissivity.A second category of failures in primary fracture selection has not been listed in Table 4.2 becausethese failures are not sensitive the selection criteria. The selection algorithm makes a mistake ininterpreting the fracture network geometry in roughly 5% of the realizations. These mistakes are usuallyassociated with the accuracy of the trigonometric calculations. Despite extensive efforts to make thecalculations of fracture geometry both precise and robust, some geometric calculations, such asdetennining what network blocks contains which portions of a particular fracture, are not totally reliable.Creating an algorithm to select the primary fracture network has proven to be one of the mostdifficult problems in developing the dual permeability model. A complex algorithm evolved from thesedifficulties, yet the number of failures indicated in Table 4.2 demonstrates that the algorithm is still notflexible enough to deal with all of the arbitrary fracture geometries that arise from stochastically generatedrealizations.4.2.2 PRIMARY FRACTURE SELECTION ALGORITHMThis section describes, in detail, the major steps in the fracture selection procedure. A simplifiedflow chart of the primary fracture selection routine is given in Figure 4.3.The first fractures to be included as primary fractures are those that may have been preselected.These fractures may be known a priori and are being used to condition the fracture network, or they maybe fractures that have been selected by the modeler from those fractures generated through the networkrealization process. The latter case may arise if an initial attempt of the primary fracture selectionalgorithm failed to find an adequate primary fracture network. In this case, the pre-selection of specificfractures may enable the selection process to create a network that incorporates these key fractures. Preselection of primary fractures is an option that allows greater control of the primary fracture network, but84Primary Fracture Network SelectionIncrease Percentageof Primary FracturesSelect Primary FracturesFrom lndMdual CharacteristicsFind Primary Fracture Intersections.Connect Ends UsingAdded Primary Fracture Segments.4,.Remove UnconnecedPrimary FracturesFind Internal Network BlockFractures and IntersectionsCant DivideSelect New1Primary FractureToo LargeITest Block SizeI. Divide BlockJ OK OKSelect New___Primary FractureFigure 4.3: Flow chart of the selection of a primary fracture network.—‘1BlockDefine NetworkDivide BlockI NoC-)0C-)w0I-i-4’ Cant Divide85is not required to fonn the network. Pre-selection of primary fractures has not been used in any of thesimulations reported in this thesis.The next group of fractures are selected from the generated fracture network using one of thecriteria listed in Table 4.1. The first step in this process is to order the individual fractures based on theirphysical characteristics using the specified selection criteria. Fractures are added to the primary fracturenetwork from this ordered list until these fractures reach a fixed percentage of fractures in the entirenetwork. The appropriate percentage is dependent on the characteristics of the fracture system and theselection criteria used. An initial guess for this percentage is supplied by the modeller. As shown in theflow chart, this percentage may be increased during the selection process if the initial attempts to forma primary fracture network are unsuccessful. The initial primary fracture percentage was set to 1.4% inthe cases presented in Table 4.2. Selection criteria that weight transmissivity more heavily tend to requiremore primary fractures.After the primary fractures have been selected based on individual fracture characteristics,additional fractures are added to completely connect the primary fracture network. Each primary fracturemust be connected at each end to another primary fracture. Starting at the end of a primary fracture, theintersections with other fractures are investigated. If the intersecting fracture is a primary fracture thenthe other end or next primary fracture is checked. If the intersecting fracture is not a primary fracture thenthe intersecting fracture is checked to see if it is connected to another primary fracture. If it is connectedto another primary fracture then the segment of fracture between the two primary fractures is itself madea primary fracture segment. If the intersection does not lead to a connection to another primary fracturethen the next closest intersection to the end is checked. This procedure continues until either a link ismade or the end of the fracture is reached. The checking procedure is repeated for both sides at each endof all of the primary fractures. The connection procedure may result in some primary fractures that are86not connected to the rest of the primary fracture network or to fractures that are connected at only onelocation. These fractures are removed from the primary fracture network.Once a primary fracture network is formed from the primary fractures, it is no longer meaningfulto describe the network in terms of individual fractures. Instead, the number of fracture segments thatconnect the intersections becomes a much more meaningful description. For example, a primary fracturenetwork selected for one of the realizations used to obtain the results presented in Table 4.2 contained 58primary fractures and 734 primary fracture segments. Of these 734 segments, 151 segments are portionsof fractures that were included to provide the necessary connectivity. In this realization, 4,035 fracturessurvived the fracture reduction procedure (Section 3.3).Once the primary fracture network has been determined then the boundaries of each network blockare identified. The last stage in the primary fracture selection routine is to subdivide network blocks thatare too large. The number of intersections within the block is estimated by assuming that the density ofintersections is constant across the model domain. The total number of intersection is calculated in thefracture network generation portion of the program. Any block-that is estimated to contain too manyintersections is divided by finding a fracture that connects two boundaries of the block. The fracture thatcomes closest to the centroid of the block and has the largest transmissivity is added to the primaryfracture network. If no one fracture connects two sides of the block then the block is searched for twofractures that provide the connection. If this procedure also fails, then the fracture that is ranked highestusing selection criteria 5 and is at least partly within the network block is added to the primary fracturenetwork. Additions of new primary fractures necessitates the redetermination of the interconnectedprimary fracture network and subsequent identification of network blocks.If, after a number of fractures have been added to the primary fracture list without successfulidentification of a primary fracture network, then the percentage of initial primary fractures can beincreased. This outer loop in the procedure can continue until the requested percentage exceeds an upper87limit on the primary fractures. An increase in the limit on primary fractures tends to both reduce thenumber of failures to find an adequate primary fracture network and increase the accuracy of the dualpermeability model, but increase the computer memory requirements of the model.4.3 DUAL PERMEABILITY FLOW MODELDetermining the flow within the fracture network is accomplished in three distinct stages. Thefirst stage is to find a hydraulic conductance approximation that can be used to represent the effectivehydraulic conductance of the network of fracture segments that are contained within each of the networkblocks. These approximations are then combined with the conductance of the primary fractures to formthe coarse-scale flow problem. The hydraulic head distribution that results from solution of the coarse-scale flow problem provides the boundary conditions that are used to estimate the flow within the fracturesrepresented by the network blocks.4.3.1 EFFECTIVE CONDUCTANCE OF NETWORK BLOCKSThe hydraulic conductances of the fractures within the network blocks are approximated usingeffective conductances for the blocks. The effective conductance of each network block is calculatedindependently. Figure 4.4 is used to illustrate the concept of effective conductances. This figure showsa schematic drawing of a single network block. In the figure, the segments of primary fractures thatdefine the boundaries of the block are represented but the internal fractures are not drawn. At the endsof each segment of a primary fracture, filled circles represent the intersections of the bounding primaryfracture segments. Between each pair of intersections is an open circle. These are referred to as fracturemidpoint nodes. Fracture midpoints are located half way between the two ends of the fracture segment.The drawing also shows two kinds of arrows that are labeled ‘two way connections” and “flow tomidpoint”. These arrows will be explained shortly. First, however, is an explanation of the fracturemidpoint nodes and the role they play in the effective conductance representation of the network block.88 .IIIIIII ,iit!oIPrimary FractureIntersectionFigure 4.4: Effective conductance approximation for network blocks.Two way ConnectionsI — —I4;IFlow!IIMi\dpoint‘IIIIIIIacture MidpointII ‘89The fracture midpoint nodes provide an estimate of the hydraulic head within the primary fracturesegments. This head estimate is needed to calculate the flow from each primary fracture intersection into(or out 00 each primary fracture segment. This flow is needed in the transport algorithm to determinethe probability that a solute particle will enter the respective fracture segments from the primary fractureintersection. A flow balance exists at the midpoint nodes that corresponds to the net flow into the primaryfractures. This flow consists of two terms; 1) the net flow exchange between the primary fracture andthe interior of the network block, and 2) flow entering the primary fracture segment directly through itsends, the primary fracture intersections. The outer edges of the primary fracture segments are treated asimpermeable boundaries for the individual network block flow calculations, as depicted in the Figure 4.4.Two types of arrows are used in Figure 4.4 to emphasize the difference in the hydraulicconnections of the midpoint nodes from those of the primary fracture intersections. These arrows havebeen drawn for one of the midpoint nodes. Double headed arrows depict the flow exchange between theintersections of the primary fractures and the primary fracture segments.Single headed arrows, drawn as dashed lines, represent the hydraulic connection between theprimary fracture intersections and a primary fracture segment through the fracture segments that are withinnetwork block. The single headed arrows depict connections that are “unidirectional”. That is, thehydraulic connection between the primary fracture intersections and the primary fracture segment isincluded in the mass balance of the primary fracture midpoint nodes, but is not included in the massbalance of the primary fracture intersections. There is no direct flow between the primary fractureintersections and the network block unless a fracture within the network block directly connects to theintersection. The flow balance equations for the primary fracture intersections only include the direct flowexchange with the primary fracture segments through the ends of the segments (i.e. the double headedconnections). The use of unidirectional connections ensures that the correct apportionment of thevolumetric flow into (out of) the primary fractures from the network blocks is maintained.90In order to gain a full description of the effective conductance between all primary fractureintersections and midpoint nodes, multiple simulations of flow through the fracture network in the networkblock are required. A separate flow problem, which focuses on the hydraulic connections of an individualprimary fracture intersection, is solved for each primary fracture intersection of the network block. Figure4.5 shows how these flow calculations are used to determine the hydraulic connection through the networkblocks of the primary fracture intersection and the midpoint nodes. For each calculation, Dirichletboundary conditions are imposed at all of the primary fracture intersections. All of the boundary headsare assumed to be 0.0 m except at one intersection. Here, a unit head of 1.0 m is imposed. Row throughfractures within the network blocks is calculated using these boundary conditions.The flow calculations are performed in isolation from other blocks. The solution for flow withineach network block neglects any influence of fractures within neighboring network blocks on the hydraulichead distribution within the primary fractures. The boundaries just outside the primary fractures segmentsare treated as impermeable except at the primary fracture intersections.The flow through each intersection of a primary fracture with the internal fractures is summed toform a response of the primary fracture midpoint to a unit change in the hydraulic head at the primaryfracture intersection. This unit response is the effective conductance between the primary fractureintersection and the primary fracture midpoint. A single flow calculation provides the effectiveconductances of all of the primary segment midpoint nodes to one primary fracture intersection.Combining the effective conductances into a matrix for the series of flow solutions creates a local stiffnessmatrix for the block that is analogous to the local stiffness matrices of a finite element model.If the primary fractures truly dominate the flow system then, almost by definition, flow exchangebetween the primary fractures and the network blocks is small relative to the flow along the primaryfractures and hence has little influence on the hydraulic head distribution along the fractures. For thesefractures, neglecting flow into neighboring network blocks is a reasonable approximation. Unfortunately,91Internal fracture flow int&Primary fracture segmentFlow to Midpoint--Unit HeadFigure 4.5: Calculation of the hydraulic connection between primaryprimary fracture segment midpoints.= Zero Headfracture intersections and(vthe fractures added to the primary fracture network to ensure that the network is completely connectedalmost always include fractures that do not truly dominate the local flow system. A significant flow errorcan result from forcing the flow, that should cross a primary fracture into an adjacent block, to remain inthe fracture until a primary fracture intersection is reached.A heuristic modification has been added to the calculation of effective conductances to reducethe influence of primary fractures with low transmissivity on the calculation of flow within the networkblocks. In the flow calculations of a network block, the transmissivities of the primary fracture segmentsare adjusted by adding the transmissivity of all fractures that connect each segment to adjacent networkblocks. If the primary fracture transmissivity is large relative to the combined transmissivities of thesefractures, then the addition is minor. If the primary fracture transmissivity is not large, then the additionof additional hydraulic transmissivity to the primary fractures can have a significant influence. One canview this addition as a relaxation of the assumption that the outer boundaries of the network block areimpermeable. The conductance of neighboring network blocks is approximated by adding to thetransmissivity of the primary fracture segments. The addition of extra transmissivity is a recognition thatmuch of the flow into the primary fracture crosses the fracture into the adjacent block.Table 4.3 presents the results from flow and transport simulations of the same fracture networksused to study the primary fracture selection criteria. In these cases, however, the transmissivities of theprimary fracture segment were not increased for the network block conductance calculations. Acomparison of the results in Table 4.3 to those of Table 4.2 demonstrates that increasing the transmissivityof primary fractures in the network block conductance calculations significantly improves the performanceof the dual permeability model.93Criteria Flow Error Time Delay Time Delay( %) Initial Median(%) — (%)1 -17.8 16.1 26.92 -24.2 15.5 30.13 -13.1 9.3 16.74 -12.6 2.5 14.45 -10.2 1.0 10.96 -9.6 1.7 10.47 -10.9 8.7 14.7Table 4.3: Primary fracture selection criteria test with unmodified network blocks. The networkswere identical to those used for Table 4.2. The errors are calculated for individualfracture networks, that is the results do not include fracture network variability. Flowerror is based on the total flow across the domain boundaries. The time errors are forinitial solute arrival and median particle arrival.4.3.2 HYDRAULIC HEAD DISTRIBUTIONThe coarse-scale flow problem is formed by combining the local network block stiffness matriceswith the two way connections between the midpoint nodes of the primary fracture segments and primaryfracture intersections at each end of the segments (See Figure 4.4). The hydraulic head values at theprimary fracture intersections provided by the coarse-scale flow calculations are used as boundaryconditions for the network blocks. One more set of calculations of flow within each of the network blocksprovides a fine scale description of flow within the entire fracture network.While the fine scale hydraulic head distribution calculated within the network blocks is unique,it is not single valued along the primary fracture segments. Adjacent network blocks share common94primary fracture segments. The hydraulic head distribution along these common primary fracture segmentswill be different in each block. This disagreement in the value of the hydraulic head is acceptable becauseit is the flow rates within the network blocks that are used for solute transport, not the hydraulic headvalues. After the flow calculations, the flow distribution is known throughout the network blocks and ateach primary fracture intersection. The final step in calculating flow within the fracture network involvesthe determination of the flow rates within the primary fracture segments.The flow rates within the primary fractures are determined by forcing mass conservation at eachintersection of the primary fracture with fractures of the network blocks. The flow entering each primaryfracture segment from the intersections of primary fractures is calculated using Equation 3.17 applied tothe half of the primary fracture segment between intersection and the fracture midpoint. The flow fromthe primary fracture intersection is determined strictly from the results of the coarse-scale flow model,ensuring a mass balance at each primary fracture intersection.The calculation proceeds along the primary fracture segment from the primary fractureintersection with the highest hydraulic head to that with the lowest. At each intersection of the primaryfracture segment with the internal network block fractures, the flow into the intersection that wascalculated in the network block flow calculations is added to the flow in the fracture segment.Calculation of flow downstream of the last intersection determines the flow into the primaryfracture intersection with the lowest value of hydraulic head. A comparison of flow rate at the end of thefracture segment calculated using the above procedure and the flow rate determined by applying Equation3.17 provides a check of mass conservation between the network blocks and the primary fractures.The coarse-scale / fine-scale formulation of the flow problem within the fracture network has theadvantage that the largest set of flow equations that must be solved simultaneously is either the size ofthe largest network block or the size of the coarse-scale flow model for the primary fracture network.95This advantage is reduced by the disadvantage of the need to solve for the network block flow severaltimes to obtain the effective conductance of the network blocks and the overhead burden of defining theprimary fracture network geometry. For small domains all calculations can be performed using a directsolver. Large domains require an iterative solver. This issue is pursued in more detail in Section 4.5.4.4.4 DUAL PERMEABILITY TRANSPORT MODEL4.4.1 TRANSPORT THROUGH A NETWORK BLOCKThe primary fracture network / network block structure is also used in the particle tracking portionof the dual permeability model. Network blocks are represented in the particle transport routine byresidence time distributions. The residence time distributions are tables of probability that link locationsof inflow to the network blocks to locations of outflow. For each inlet-outlet pair there may be manypaths a particle may take through the network block. Different paths lead to different residence timeswithin the block. Each inlet I outlet pair has a set of residence times, each of which have differentprobabilities of occurrence. This leads to a three dimensional table of probabilities. One dimension isneeded for the inlet locations. One dimension is needed for the outlet locations. And, one dimension isneeded to represent the different times of arrival. The values in the table are pairs of probabilities andarrival times. A front tracking method introduced by Shimo and Long [1987] has been modified todetermine the residence times and their associated probabilities of occurrence.The Shimo and Long front tracking method follows the rise in concentration along a fracturenetwork from a step function injection of solute at an upstream boundary. The rise in concentrationdownstream of the injection location is followed from intersection to intersection in descending order ofhydraulic head at the intersections. The concentration at an intersection is the sum of the concentrationsat intersections that are directly upstream, with appropriate time delays in the upstream concentrations toaccount for the travel times through the fracture segments. Figure 4.6, from Shimo and Long [1987],96CFigure 4.6: Front tracking procedure of Shimo and Long [1987] showing the combining of upstreamsolute concentrations to form the concentration as a function of time at a fractureintersection. From Shimo and Long [1987].97Path iitC Path jk-ttdepicts this combination of concentrations.The implementation used here replaces the step function rise in concentration with the pulseinjection of a single particle. The increase of solute concentration at a downstream intersection at a giventime becomes the probability that the particle will pass through the intersection at that time.Figure 4.7a, which shows four fracture intersections that are part of a larger fracture networkwithin a network block, is used to illustrate the fomiation of residence time distributions for a networkblock. The intersections are drawn as filled circles. Fractures are presented as parallel lines. Arrows givethe direction of flow within the fracture segments. The variables Q1 in the figure are the flow rates withinthe fracture segments connecting the intersections. The rest of Figure 4.7 contains plots of the possibleparticle arrival time probabilities at these four intersections, for a particle that is injected into the fracturenetwork upstream of the first intersection. To illustrate the calculation, panel b presents an assumedprobability of arrival of solute at intersection 1 from an unidentified upstream location. The two values,at t’1 and 2’ indicate that there are two paths from the inlet location to intersection 1 that are not shownin Panel a. In Panel c, the residence time distribution of intersection 2 is displayed. The probabilities atintersection 2 are 30% of those at intersection 1 because 30% of the flow through intersection 1 goes tointersection 2. The times t and i2 have increased from t’1 and t’2 by the particle travel time fromintersection 1 to intersection 2. Panel e, for intersection 3, has a similar explanation. Panel d,representing intersection 4, shows four probability values resulting from the combination of possible pathsthrough intersection 2 and intersection 3. The offset in time of t41 from t43 is caused by the differencein travel time along the two paths.Figure 4.7 demonstrates an increase in the number of distinct probability-time pairs as the numberof potential paths from the block inlet to outlet increases. The number of computations required toestablish the residence time distribution increases in proportion to the increase in the number ofprobabilities. This behavior places a practical limit on the number of intersections that can exist in a980.4)0.35ccs0o 0.140 •IO17abQ =30 =8=4Intersection 10.50.2t12tilII III I Iii 111111d0 0.2 0.4 0.6 0.8Intersection 41C Intersection 220.15 20.0611111111 1111110 0.2 0.4 0.6 0.81e Intersection 330.15 3t20.06 tjI I I i i I I0 0.2 0.4 06 0.8Figure 4.7:10 0.2 0.4 0.6 0.8time timeCalculation of network block residence times. a) Schematic of four fracture intersectionsand their contiguous fracture segments. The intersections are drawn as dots. Theintersecting fractures are presented as parallel lines. Arrows give the direction of flowwithin the fracture segments. The variables Q are relative flow rates within the fracturesegments connecting the intersections. b) Probability of a particle arriving at theintersection 1 at specific times. c) Probabilities for intersection 2. d) Probabilities forintersection 3. e) Probabilities for intersection 4.99network block. The limit is highly dependent on the actual flow distribution within the network block,however 400 intersections seems to be reasonable upper limit to the size of the blocks without some formof reduction in the number of distinct arrival times.Two techniques, again following Shimo and Long, are used to reduce the number of distinct arrivaltimes at the outlets from the network block. One technique is to combine probabilities of arrival for timesthat are close together. The second technique is to remove low probability arrivals by combining themwith higher probability arrival events at an intersection. If the probability of a particle arriving at aintersection from an inlet is below a specified threshold then the probability-time pair is combined withanother pair that is the closest in time. For both of these techniques, the mean time is used to representboth paths with a probability that is the sum of the two probabilities. The mean time is determined byweighting the two times by their respective probabilities.The two techniques can have a significant influence on the efficiency of the particle trackingalgorithm. This point is demonstrated using a fracture network composed of two orthogonal sets ofcontinuous fractures. While the trace map of this fracture network in Figure 4.8a looks like evenly ruledgraphpaper, transmissivity variations make the network heterogeneous. Both Discrete and the dualpermeability model were used to determine transport through the domain. The input to the dualpermeability model was specified so that there were no primary fractures selected for the fracture network.All of the fractures were included in a single network block.Table 4.4 contains statistics from various dual permeability simulations of transport through thedomain in which the techniques for enhancing efficiency were applied. The first column of the tableidentifies the simulation by number. The second column contains values for the minimum probabilitythreshold (plim) of distinct probability-time pairs. The third column lists the value of the maximum timedifference (tdfJ) between probability-time pairs that are combined into a single pair at an intersection.The fourth column lists the number of distinct probability-time pairs needed to represent the inlet-to-outlet100a CC,)Q)4)a)(12a)C)0000 0.1C,)q)C)—400(12C)4J(c3(.100Discrete — — Dual PermeabilityFigure 4.8: Comparisons of particle arrival rates of the dual permeability model using a singlenetwork block and Discrete. a) The fracture network model as a single network block.b) Arrival at peak location with no storage reduction. c) Arrival at peak location withp_urn = 0.01. d) Arrival at peak location with p_urn = 0.0001 and t dtff= 0.001 days.0.40.30.20.1: : :: ::0.2 0.3Metersb0.4 0.06 0.08Daysd0.1201510500.04201510500.04201510500.04 0.06 0.08Days0.1 0.06 0.08Days0.1101residence time distribution for the block. The final column is the time needed to form and solve the flowproblem, form the transport model, and then move 50,000 particles using an SGI R4000 computer.The first simulation had both plim and t duff set to zero. There are almost 710,000 distinctarrival time, inlet-outlet combinations for this extremely well connected network. The time needed forthe transport calculation was 730s for the dual permeability model and 12.5s for the program Discrete.Figure 4.8b shows the cumulative arrival of particles at the location where the greatest number ofparticles crossed the downstream boundary. These arrival patterns are not significantly different.Setting p_urn to 106 had only a minor influence on these results. The probability limit wasincreased by an order of magnitude for each successive run up to a value of 102. At the highest valuethe number of distinct points in the residence time distribution was reduced to 2,780 and the time of thedual permeability model simulation was reduced to 2.8s. Figure 4.8c shows a noticeable discrepancy inthe cumulative arrival curves, but the results are still quite close. For the rest of the test, the minimumtime difference, tdiff was increased while plim remained l0 in order to evaluate the influence ofcombining the probabilities of similar times. For run 7, the minimum time difference was 0.0001 days.At this level, the dual permeability model took 7.3s and needed 42,000 points for the residence timedistribution of the block. The arrival curve was nearly identical to the curve from Discrete.The final run used a minimum time difference of 0.001 days, the same size time interval used tocollect the particle arrival rates. This run required only 2.9s of computer time. The results of the dualpermeability model shows a raggedness in Figure 4.8d, signifying that the large minimum time differencehas shifted the arrival of a significant number of particles. The value of t duff is larger than the timeinterval between points used to represent the cumulative arrival causing a stair step pattern to the dualpermeability results.102Run plim tdiff RTD Size Run Timedays s1 0.0 0.0 708270 731.12 1.e-6 0.0 642892 652.43 1.e-5 0.0 431029 366.54 1.e-4 0.0 107905 44.55 1.e-3 0.0 16274 7.46 1.e-2 0.0 2780 2.87 1.e-4 0.001 42135 7.38 1.e-4 0.0025 22310 3.99 1.e-4 0.005 13687 3.310 1.e-4 0.01 7291 2.9Table 4.4: Network block residence time distribution evaluation. The parameters p_urn and t_dffare explained in the text. RTD size is the number of probability/time data pairs neededto define the residence time distribution. The run times are for a Silicon Graphics R4000computer. These times include the time needed to set up and solve the flow problem aswell as particle transport. The reference program, Discrete, required 12.5s to perform thesame transport simulation.The use of minimum probabilities and time differences had a substantial influence on both thememory requirements and computation time of these simulations. Both the time and memory requirementsvaried by a factor of more than 200. The particle arrival curves did not show a strong sensitivity to thevalue of p tim. The use of large values of t diff only had a noticeable influence if t duff was larger thanthe time interval used to determine the particle arrival rate. In no case were the results stronglyinfluenceed by the use ofplim and/or t_diff A substantial performance increase can be obtained throughthe use of these approximations without the need for tremendous caution in their application.In summary, this limited evaluation of the use of residence time distributions to simulate transportthrough network blocks indicates that a substantial time savings can be obtained with negligible103degradation in the transport predictions. Later, in Section 4.5.4, it is shown that in practice the timeneeded to set up the residence time distribution and then move 50,000 particles is about 50% of the timeneeded by the particle tracking algorithm of Discrete.4.4.2 PARTICLE MOVEMENTFigure 4.9 is a flow chart for the particle movement routine. I will explain the particle trackingroutine by following a particle. The top of the flow chart represents the introduction of a particle at thedomain boundary. For this discussion, the boundary location is at one end of a primary fracture, but itcould also be a network block inlet location. The particle enters the fracture. The time that the particletook to reach the first intersection of the primary fracture with a fracture of a network block is recordedas the cumulative transport time. At each intersection of the primary fracture with fractures of the networkblocks, a random number is used to test whether the particle enters the network block. The probabilityof entering the block is based on the fraction of flow through the node that enters the block( i.e. completemixing).If the particle continues to move down the fracture instead of entering the block, the cumulativetransport time of the particle increases as each intersection along the primary fracture segment is reached.After the last intersection of the segment is passed, the particle enters a primary fracture intersection.From here the particle transfers to a new primary fracture segment, again based on relative flow rates fromthe intersection.When a particle transfers from an intersection along a primary fracture segment to a networkblock, the intersection corresponds to an inlet location of the network block. A random number isgenerated from the uniform distribution between 0 and 1. Starting with the exit location that is the mostprobable, the probabilities of a particle leaving an exit location are summed until the next location wouldcause the sum to exceed the random number. This location is the exit location from the block for this104Particle Tracking SimulationTIntroduce Particle atInjection NodeExit at intersection to a Network Block?IsYesthe Node a Boundary Node?Save ParticleExit Time and LocationIs the Node a YesBoundary Node?Locate Exit onPrimary Fracture SegmentFigure 4.9: Flow chart of the particle tracking transport algorithm.For each ParticleI Randomly Select an Exit FractureAdd time to Reach Next IntersectionNoNoYesNo- End of Fracture?YesSelect Exit Locationfrom Network Blockand add Residence Time105particle. The sum is subtracted from the random number to create a new random number that is unifonmlydisthbuted between 0 and the probability of exiting at the selected exit location. The procedure ofsumming probabilities, in the order of decreasing probabilities, is repeated for the probability-time pairsuntil the random number would again be exceeded. The time of the last pair is selected as the residencetime for the particle in the network block. The exit location corresponds to a network block intersectionon some downstream bounding primary fracture segment of the network block.The particle movement through the newly entered primary fracture segment proceeds from theselected intersection. The movement may be into the adjoining network block or along the primaryfracture. The particle movement continues until a domain boundary is reached. When the boundary isreached, the location and total transport time of the particle is recorded, providing particle arrival historiesat the domain boundaries.The distributions of particles at a common time can be obtained from the particle transport routine.Each time a particle moves, the time is tested against a list of recording times. If the addition to the totaltransport time causes that time to exceed one of the times on the list then the position of the particle isestimated from a linear interpolation of its position at the beginning and end of the particular movement.Along a primary fracture the interpolated position is precise. Within network blocks, the precision of theinterpolated position is limited by the size of the block. The interpolated position is along a lineconnecting the inlet location to the outlet location. The actual travel path would rarely coincide with thisestimate.4.5 DUAL PERMEABILITY MODEL EVALUATIONThe purpose of the dual permeability model evaluation is to assess the degradation in the accuracyof using a discrete fracture representation when the model is transformed from a complete discretefracture model to a dual permeability model. Discrete represents that complete discrete fracture model.106Any differences between the hydraulic characteristics of a hypothetical fractured rock mass representedby the fracture system statistical characterization and those predicted by Discrete are beyond the scopeof this evaluation. The calculations using Discrete are treated as the truth.The evaluation is composed of four components. The first component is a comparison betweenthe dual permeability model and Discrete of flow and transport through a single realization of a fracturenetwork. Detailed results from one realization are used for three purposes: 1) to reiterate the majorfeatures of the dual permeability model, 2) to give an indication of types and magnitude of errors that areintroduced through the decomposition of the network into primary fractures and network blocks, and 3)to demonstrate information that can be obtained from the model.The second and most important component in this evaluation is in the context of a Monte Carlostudy. A premise of the dual permeability concept was that the variability and inherent uncertainty in thephysical characteristics of a fractured rock body strongly compels a stochastic investigation of the fracturesystem through a Monte Carlo analysis. Hence, it is from the results of a Monte Carlo investigation thatone should judge the merits of the dual permeability model. While the single realization comparison islimited to one network, the Monte Carlo evaluation uses a wide variety of fracture systems because theperformance of the dual permeability model depends, to a degree, on the nature of the fracture system.A study of the influences of network block size on the accuracy of the dual permeability modelis presented as the third component of the evaluation. The question addressed is: “Does the accuracy ofthe dual permeability model degrade quickly as the scale of the network blocks increases?’. This is aminor consideration compared to the broader issue of the efficacy of the dual permeability concept yet itmust be addressed in order to put the results presented in the second and fourth components of thisevaluation in perspective.107The fourth component of this evaluation looks at the advantages of the dual permeability modelrather than focusing on the inaccuracies that it introduces. It addresses the basic questions of: 1) “Howlarge of a fracture network can be simulated?”, and 2) “How long does it take to solve the problem?”.These investigations compare two variations of the dual permeability model, that have different memorystorage requirements and speed attributes, to Discrete. The two variations of the dual permeability modelare distinguished by the solver used to find the coarse-scale head distribution.4.5.1 A SINGLE REAUZATIONThe fracture system chosen for this demonstration is one that was known to provide excellentresults. Therefore, the results presented here, in some ways, represent a best case scenario. However, thespecific fracture network was not selected from a suite of realizations. The results presented in this sectionshould not be considered abnormally accurate. They are in fact, slightly less accurate than an “average”realization for this fracture system. The realization was created from fracture system 1 of Table 4.5 usinga Poisson structural model. The dimensions of the flow domain were 40m by 40m. The fracture tracemap of the 8,400 fractures within the flow domain is shown in Panel a of Figure 4.10. Tn Panel b, thereduced fracture network of just over 4,000 fractures is shown. Flow rates calculated for a hydraulic headgradient of 0.01 in the horizontal direction of the figure are presented as line widths in Figure 4.11. Onlyfractures with flow rates above 5.1x10” m3/s are drawn, giving the appearance that some fractures areisolated from the rest of the network. The cumulative flow across the boundary on the right was 3.2x109108CC-)C)ONaC)C-)V-CHa)‘CtC)-cCiHCaC)CiC/Co232c-2C)Oj—C0—C—)C.)ON0C)0—I00H603C0-Jr2p2C)CCHCuC0)0C000C(0(NJ100C0)L1SJ9NMeters- CD000C c_)0cICDcCD_CoCDc1’)C(DO•CD CCo-0C-)1 CD C.,)CD -0C H C C CDBoundary 3C\2-dz0 0Boundary 10 2x10’° 4x101° 6x10’° 8x101°I I I I I I IFlow Rate (m3/s)Minimum flow drawn = 3.0 e— 11 m3/sFigure 4.11: Fracture flow rates within in the fracture network of Figure 4.10. Fractures are drawnwith a line width proportional the flow rates calculated using Discrete. The scaling offlow rate to line width is given by the scale below the main figure. Only flow rate above3.0x10’1 are drawn. The hydraulic head gradient has a magnitude of 0.01 and is orientedat 00 from the lower boundary.illFracture Systems______Parameters Sets 1 2 3 4 5 6 [ 7Density 1 0.8 2.8 2.8 2.8 2.8(Fractures/m ) 2 0.8 2.8 2.8 0.3* 0.8* 2.83 2.0 0 0 0.3* 0 04 2.0 0 0 0 0Length 1 10 3.6 3.6 3.6 l.8(m) 2 10 3.6 3.6 3.6* 18t3 14 1Aperture 1 60 40 40 40Mean 2 60 40 40 40(urn) 3 40 404 20 40Aperture 1 0.2 0.3 0.3 0 0.3Standard 2 0.2 0.3 0.3 0 0.3Deviation 3 0.2 0.3(Log10) 4 0.2 0.3Orientation 1 0Mean 2 90( Degrees) 3 04 90Orientation 1 2 10 10 10 10Standard 2 2 10 10 10 10Deviation 3 10(Degrees) 4 10* Scan-line density outside of war zones. The actual scan-line density is larger.Minimum fracture length generated from a power law distribution with an exponent of -1.3.Table 4.5: Fracture system statistics for the dual permeability evaluation. Entries in the table areonly made when the parameter differs from system 1.112m3/day. The flow was largely controlled by the first two of four fracture sets, which both have a lOmmean fracture length and have higher transmissivity than the rest of the fractures in the network.Fifty-seven fractures were initially selected to be primary fractures using the selection criteriabased on the product of length and aperture, selection criteria 4. The primary fracture network shown inFigure 4.12 resulted from forming an interconnected network composed of these 57 fractures and otherfracture segments needed to define closed network blocks. The primary fracture percentage was initiallyspecified as 1.4% and was not increased during the primary fracture selection process. The interconnectednetwork has 518 primary fracture segments and divides the fracture network into 175 network blocks.The coarse-scale hydraulic head distribution is defined by 302 locations in the dual permeability modelcompared to 13,787 locations required to define the head distribution within the entire network. Thecoarse-scale head distribution required the solution of 731 simultaneous equations. Recall that the numberof equations is greater than the number of hydraulic head locations in the coarse-scale hydraulic headdistribution because of the inclusion of the primary fracture midpoints in the flow equations.Figure 4.13 is a plot of the flow rates within the primary fracture network. The plot is made usingthe same scale of line widths as in Figure 4.11. Only the flow in the primary fractures is shown. Most,but not all, of the fractures that have larger flow rates in the Discrete simulation are represented in theprimary fracture network. While the general flow pattern is similar in Figures 4.11 and 4.13, there aresome noticeable differences. Primary fractures tend to have greater flow rates in this dual permeabilitymodel than in Discrete. This tendency for large flows results in a five percent higher net flow throughthe dual permeability flow domain and earlier particle arrival at the downstream boundaries. In most ofthe realizations encountered in the Monte Carlo portion of the evaluation, the calculated flow using thedual permeability model was less than the flow calculated using Discrete. So, the estimation of higherflow in the dual permeability model is atypical, but the primary fractures do tend to have high flow ratesin most realizations.11312Figure 4.12: Primary fracture network defined for the fracture network of Figure 4.10.40302002a)-I-)U)1000 10 20 30Meters4011440Cl)c.L)I I I I I I I I I I I IFlow Rate (ms/s)Minimum flow drawn = 3.Oe—11 m3/sFigure 4.13: Flow rates within the primary fractures of the dual permeability model.30201000 10 20 30 40Meters0 2x101° 4x101° 6x10’° 8x10’°115The net fluid flow across the domain boundaries for both models is presented in Table 4.6. Thefirst row lists the net flow through the entire domain. The other four rows of the table list the net flowacross each boundary. The flow difference for each boundary is of the same order of magnitude and doesnot correlate strongly with the net flow across the boundary.Boundary Discrete Dual Permeability Difference( m3/s ) ( m3/s ) ( m3/s )Net Flow 3.38x109 3.56x1(X9 l.8x10’°1 3.21x109 3.05x109 l.5x10’°2 4.2x10’° l.8x10’° 2.5x10’°3 2.95x10 3.38xl09 4.3x10’°4 l.7x10’° 5.1x10’° 3.3x10’°Table 4.6: Net flows across the flow domain boundaries of the single realization comparison.Fifty thousand particles were used to simulate transport in both the dual permeability model andDiscrete. The particles were introduced into the fracture network over a range from 15m to 25m alongboundary 2. Figure 4. l4a depicts the particle arrival locations along the downstream boundary in aperspective plot. The height of the columns represent the cumulative number of particles that crossed theboundary at the location of the column. The columns representing data from Discrete are shaded. Panelb is a plot of the cumulative arrival along boundary 4. The solid line is a plot of the particle arrivalspredicted by the program Discrete. The dashed line represents the particle arrivals predicted by the dualpermeability model. The abscissa in Panel b corresponds to position along boundary 4 as defined inFigure 4.11. The arrow above the columns in Panel a points in the direction of increasing distance alongboundary 4. The ordinate in Panel b is the cumulative number of particles that crossed the boundary withincreasing distance along the boundary. The particles crossed the downstream boundary 4 over a rangeof lOm to 30m as shown in Figure 4.14. The results presented in Figure 4.14 show that dispersion116ab 100i2a)C)cE0E-•1.0Injection Range8060402000Figure 4.14:10 20 30 40MetersParticle arrival distribution at the downstream boundary of the fracture network of Figure4.10. a) A perspective view. Particles were released over the interval from 15 m to 25m along upstream boundary. The percentage of particles that crossed the boundary at aspecific location is presented as columns along the downstream boundary. b) The particlearrivals are integrated from the lower right corner of the flow domain to the upper rightcorner. The arrow above the columns in Panel a points in the direction of integration.117perpendicular to the hydraulic head gradient was closely matched but that there were variations in thenumber of particles that crossed the boundaries at any particular location. The results are typical in thatthe particle transport predictions of the models differ in the details but retain the same general behavior.Figure 4.15 contains graphs of particle arrival as a function of time. In Panels a and b, the arrivalof particles at boundary 4 through the fracture located 20.2 m from bottom is plotted. Panel a shows therate of arrival and Panel b shows the cumulative arrival over time. Both plots are in terms of thepercentage of the 50,000 particles injected. The general timing of the particle arrival is quite similar.Discrete predicted a larger number of particles crossing the boundary at this location.In Panels c and d, the particle arrival rate andcumulative arrival at boundary 4 are presented forall of the particles that crossed the boundary. Panels c and d are a better representation of the generaltransport behavior of the network than the results from a particular point. Panels c and d both reveal aslightly larger longitudinal dispersion in the particle transport of the dual permeability model. The fastestparticles reach boundary 4 slightly earlier than in Discrete. Slower particles arrive a bit later. In general,the shapes of the arrival curves are very similar.Particle transport is characterized using the mean and standard deviation of particle positions inFigure 4.16. Using statistical moments of solute plume behavior is a popular method of characterizingtransport behavior in heterogeneous porous media (e.g. Graham and McLaughlin [19891). The dualpermeability model is not well suited to gathering these statistics. Position statistics are influenced by theresidence time distribution method of modeling transport through the network blocks. Particle locationswithin the network blocks are not known. The positions used to collect the statistics presented in Figure4.16 are estimated by linearly interpolating the particle positions within the network blocks from theparticle entrance location and time and the downstream exit location and time. The linear interpolationintroduces inaccuracy into the position estimates. It should be noted, however, that fracture networkheterogeneity introduces variability in the particle statistics at the scale of the network blocks. The118isDiscrete— — — Dual PermeabilityFigure 4.15: Particle arrival curves for entire downstream boundary and at the 14m location. Thecurves represent the arrival along the entire boundary. a) The rate of particle arrival atthe downstream boundary. b) Cumulative arrival over time at the downstream boundary.c) Particle arrival rate at 14m. d) cumulative arrival at 14m.0.1ana) 0.15a,a)a)0.050Ca)a)a)a)C)a)1000 2000b10080a)C)a)a)4.)40.4-02003000d2015a)0.ia)— 10a)4.)0E‘-50e 501000 2000 30000.060.040.0201000 2000Days3000 1000 2000Days3000119Figure 4.16: Particle location statistics. a) Mean particle position parallel to the gradient direction. b)Standard deviation of particle location parallel to the gradient direction. c) Mean particleposition along the axis perpendicular to the gradient direction. d) Standard deviation inparticle position perpendicular to the gradient direction.anE0C)CSa)C)C)10502423222120bd3.55 32.5b1.5I I I I’—’——’ IH11111 I liii III0 20 40 60 80 100TimeI Till F I I I1FF Ill liii liii Iii0 20 40 60 80 100Timec Discrete0 20 40 60 80 100Time0 20 40 60Time80 100t- -a—A Dual Permeability120uncertainty in particle statistics introduced by the network block structure occurs at the same scale orsmaller than the large variability in plume statistics caused by fracture network heterogeneity.Panel a in Figure 4.16 shows the average particle position along the axis of boundary 1. The positionsare tracked only to 100 days. The first particle reached boundary 4 at 115 days, limiting the validity ofthe position statistics to the period prior to this time. Even after 100 days, most of the particles are nearthe particle source region. Panel a shows the mean particle position. Because at these times mostparticles are still concentrated in a few network blocks, the mean positions determined from the dualpermeability model and Discrete are not in as close agreement as one would expect from the resultspresented in Figure 4.15. Panel b confirms the increased longitudinal dispersion in the dual permeabilitymodel. Panelc, a plot of the mean particle position transverse to the regional hydraulic gradient, showsa disagreement in position that is not evident in the cumulative arrival along boundary 4( Figure 4.14).The affect is also evident in the standard deviation of particle positions that are transverse to the gradientdirection shown in Panel d. As the plume of particles spreads away from the source region, and growsin size, the influence of uncertainty in the particle positions within network blocks would become lessevident.The comparison made for this single realization demonstrates that the dual permeability model isin close agreement with Discrete, at least for this particular fracture network. A more appropriatecomparison is how the results of a series of simulations might affect the estimates of expected behaviorof a fracture system.4.5.2 MONTE CARLO EVALUATION4.5.2.1 INTRODUCTIONFor the Monte Carlo evaluation, the performance of the dual permeability model was evaluatedusing the seven fracture systems defined in Table 4.5. The evaluation is based on one hundred transport121calculations for each fracture System. The seven systems represent a broad spectrum of differingheterogeneity. The fractures systems were chosen in part to test the importance of a clearly definabledominant fracture type on the performance of the model and to test whether the dual permeability modelis influenced by changing the fracture system structural model. The fracture systems were also chosento explore the robustness of both the dual permeability model and, by inference, the concept of flowdominance. Of particular interest is the relationship of the range in the scale of fracturing to flowdominance and transport characteristics.The fracture systems used in the evaluation are briefly reviewed in this paragraph. They aredescribed in more detail later in separate sub-sections. Fracture system 1, composed of four fracture sets,was designed to provide easily identifiable primary fracture networks of long and highly transmissivefractures. Fracture network realizations are generated using a Poisson structural model. The fracturesystem has two dominate length scales because two of the fracture sets have mean lengths of ten metersand the other two have mean fracture lengths of one meter. In fracture system 2, the sets of long fracturesare retained, but the aperture statistics have been adjusted so that the long fractures are not necessarily themost transmissive; all of the fracture sets have the same transmissivity statistics. The long fracture setsare removed completely in fracture system 3, leaving a fracture system composed of two sub-orthogonalsets. The scan-line density and mean transmissivity of the fractures is the same for fracture system 3 asfor fracture system 2, but the distribution of lengths is different. Fracture system 4 is the antithesis ofwhat the dual permeability concept was designed to simulate. In this fracture system, the transmissivityvariability has been completely removed, leaving all the fractures with the same transmissivity. Fracturesystem 5 returns to the general characteristics of the first fracture system, but the realizations are generatedusing a war zone structural model rather than a Poisson model. The two long fracture sets define the warzones. The long length and near parallelism of these fractures define large narrow zones of densefracturing. The sixth fracture system is based on the fracture sets of fracture system 3 using a war zonestructural model to generate the fracture network realizations. In this network, the war zones are clusters122of high-density fracturing rather than the linear features of the other war zone fracture system. Theseventh system is implemented with a Levy-Lee structural model. This fracture system has the samefracture density and transmissivity variation as fracture system 3 but, because of the Levy-Lee model, thefractures have a board spectrum of fracture length, with shorter fractures being tightly clustered.Before discussing the results for each fracture system in detail, I will provide a brief overview ofconclusions that can be drawn from the entire exercise. Transport characteristics are the focus of thecomparisons because particle transport is a sensitive indicator of both errors in the flow solution and inthe particle tracking routine. I believe that the differences between transport characteristics of the dualpermeability and Discrete are dominated by differences in the flow solutions. As demonstrated in Section4.4.1, the transport portion of the dual permeability model introduces little error into the calculations. Themajor conclusions drawn from this evaluation are:1) The dual permeability model requires a fracture network where the flow rates within somefractures are much higher than most of the other fractures that they intersect. Given this situation,the existence of relatively long fractures is crucial in the creation of the primary fracture network.2) Transport predictions of the dual permeability model are not strongly sensitive to the selectioncriteria listed in Table 4.1. However, selection criteria for identifying the primary fracturenetwork that use both fracture transmissivity and length are usually better than selection criteriathat focus primarily on one of these characteristics.3) Selection criteria 5 and 6 always resulted in average net flows through the domain that werewithin five percent of the average net flow calculated using Discrete, except for fracture systems3 and 4 which fail to meet condition 1 above. These criteria also led to an estimate of the meanarrival time at the downstream boundary that was also within five percent of the Discrete results.1234) The fastest particles tend to move faster in the dual permeability model than in Discrete but mostparticles tended to be retarded in the dual permeability model introducing a slight increase intemporal dispersion. The simulations for fracture system 2 are an exception to this, as discussedlater.5) Primary fracture networks could not be determined for all realizations. Failure to find a primaryfracture network is highly dependent on both the selection criteria and the nature of the network.Using selection criteria that weight transmissivity more strongly is less likely to result in asuccessful search for a primary fracture network.4.5.2.2 FRACTURE SYSTEM 1Fracture system 1 fits the assumptions that underlie the dual permeability concept better than anyof the other fracture systems listed in Table 4.5. That is, the hydraulic behavior of this fracture systemis dominated by a small proportion of fractures with high transmissivity that form a well connectednetwork by themselves. The fracture system is based on a Poisson structural model with four fracture sets.There are two scales of fracturing in fracture system 1. Two of the fracture sets are composed of fracturesthat are much longer and wider than the fractures of the other two sets, but these sets make up only somefive percent of the total number of fractures. The mean orientations of the two sets are orthogonal. Thisfracture system was used earlier to investigate the primary fracture network selection criteria and for thesingle realization comparison. A fracture trace map and flow distribution for a realization of this fracturesystem were presented in Figures 4.10 and 4.11.A comparison flow and transport predictions of the dual permeability model and Discrete forfracture system 1 has already been presented in Table 4.2 for each of the selection criteria. Table 4.2shows that the differences in mean behavior in terms of both flow and transport is small and that thesedifferences are sensitive to the selection criteria. Figure 4.17 presents this comparison in terms of the124Fracture System 110.80.60.40.200Figure 4.17:Arrival Time ( Days )Average cumulative particle arrival at downstream boundary for each of the primaryfracture network selection criteria used in dual permeability models of fracture system 1of Table 4.6. The curves are limited to the initial arrival time and the 25th, 50th, 75thand 90th percentile arrival times. The solid line represents the cumulative arrival for 70realizations of the fracture system using Discrete. The dashed lines represent thecorresponding average cumulative arrivals using the dual permeability model. Thecorrelation between line type and selection criteria is given in the upper left of the figure.200 400 600 800125cumulative arrival of particles along the downstream boundary and provides a visual indication of howdispersion is influenced by the dual permeability model.In Figure 4.17 the average time when the leading edge of the particle plume crosses the boundaryis the first point plotted for the cumulative arrival. Other points on the curve represent the average timeswhen 25%, 50%, 75% and 90% of the particles have reached the boundary. The results from Discreteare plotted as a solid line. The results from the dual permeability model using the various primary fracturenetwork selection criteria are plotted as dashed lines that are defined by the legend in the upper leftportion of the figure. Sc. 1 refers to selection criteria 1.The data presented in Figure 4.17 is slightly different than the data presented in Table 4.2. Table4.2 lists the average differences for individual fracture network realizations where a primary fracturenetwork could be found. For each selection criteria the realizations may have been slightly different. InFigure 4.17, the cumulative arrivals are averages using only the 70 realizations for which all of theselection criteria resulted in an acceptable primary fracture network.The initial particle arrival at the downstream boundary is close to the Discrete prediction, withsome of the selection criteria resulting in earlier arrivals and some resulting in later arrivals. After 15%of the particles have crossed the downstream boundary all of the dual permeability simulations tend to lagbehind the Discrete results. Use of any of the selection criteria provided good transport predictions. Theuse of selection criteria 1, 2, and 3 which primarily use just one geometric property tend to lead to longerdelays in the particle arrival than the use of selection criteria 5 or 6 which use both length andtransmissivity. The increasing time lag in the dual permeability results is an indication of an increase intemporal dispersion that was described for the single realization. Calculated in terms of the half widthof the arrival curve, the difference between the 75th and 25th percentile arrival, the dual permeability modelincreases the temporal dispersion of the particles by 3 to 7 percent.126The distribution of the initial, 50th, 75th and 90th percentile arrival times, using selection criteria6, along with the corresponding distribution from Discrete are presented as bar charts in Figure 4.18. Thedata from Discrete are lightly shaded. The dual permeability results are shaded with thicker and morewidely-spaced lines. An example of Discrete is shown between Panels c and d and an example of thedual permeability plot is shown between Panels a and b. Panel a represents the initial arrival times. Panelb shows the median arrival time. Panel c contains the 75th percentile times and Panel d the 90th percentilearrival. This figure clearly shows that the inaccuracies introduced by the dual permeability model aresmall compared to the variations in arrival times that are inherent to the fracture system. The variabilityin transport through different realizations is large. This variability is a feature that is lost in the averagingthat is used for the rest of the figures presented in this section. Recall from the discussion in Chapter 1that if inaccuracies introduced by approximating the fracture system would not change the conclusions thatan analyst might draw from a Monte Carlo investigation of the fracture system, then the decrease inaccuracy is not important. The conclusions about transport in fracture system 1 drawn from thedisthbutions of arrival times from the dual permeability model as presented in Figure 4.18 would, in alllikelihood, not be significantly different than those drawn from the Discrete results.The final figure presented for fracture system 1, Figure 4.19, is the arrival data from the dualpermeability model for the case where the transmissivities of the network block boundary fractures werenot adjusted in the effective hydraulic conductance calculations of the network blocks. Is clear fromFigures 4.17 and 4.19 that the adjustment of the boundary fractures, in the effective conductancecalculations, is required for an accurate dual penneability model.4.5.2.3 FRACTURE SYSTEM 2 - COMMON APERTURE STATISTICSThe major difference between fracture system 2 of Table 4.5 and fracture system 1 is that meantransmissivity of the four fracture sets is the same in fracture system 2, whereas the fractures of the longerfracture sets of system 1 were also considerably wider. The transmissivity of all of the fractures are not127Initial arrival time0a)a)1-475% arrival time11111. ilil200 400 600 80010001200Daysa20151050C10550 100 150 200 250Dual Permeability200 400 600 8000a)a).4’ .4’ 4’,“-I :.‘•,4 .,•.,.1..t..ESii4 ‘i ‘,i ‘4 ‘,f/i 4’,” Ii’ l, 41 ‘i04:’ 4 ‘/‘A 44 ‘A A,4%44l S444:‘I4.4Uki ‘..44.øY,b.0500 1000 1500 2000DaysFigure 4.18:DiscreteThe distributions of the initial, 50th, 75th and 90th arrival times using selection criteria6 in the dual permeability model and for the corresponding realizations of the fracturesystem using Discrete.128Fracture System 1 — Without Boundary Fracture ModificationFigure 4.19: Average cumulative particle arrival at downstream boundary, when the transmissivity ofthe boundary fractures of the network blocks are not increased to compensate for flowlosses through crossing fractures.c1.4)0.40 200 400 600 800 1000Arrival Time ( Days )129all the same, however. Each fracture set has a standard deviation of the log10 aperture of 0.3. In fracturesystem 2, the long fractures do not necessarily dominate the local hydraulic behavior of the fracturesystem. Figure 4.20 shows the flow distribution in one fracture network of fracture system 2. Thenetwork differs from the network shown in Figure 4.10 only in the transmissivity of the fractures. Thearrangement of the fractures is identical. A comparison of Figure 4.20 with Figure 4.11 shows that alarger number of fractures are drawn in Figure 4.20, indicating that the flow in fracture system 2 is lessconcentrated within channels than the flow in fracture system 1.The cumulative downstream particle arrival curves for Discrete and for the dual permeabilitymodel using various selection criteria are presented in Figure 4.21. With the decrease in the transmissivityof the long fractures, particles remain in the fracture network longer in fracture system 2 than they didin fracture system 1. The average cumulative arrival times predicted by the dual permeability results tendto be shifted forward roughly 5% to 10% relative to Discrete.Again, all of the selection criteria resulted in good transport predictions. Selection criteria 1,which used only fracture transmissivity, was not included in figure 4.21 because of unsuccessful primaryfracture network searches for a large number of realizations. The use of a minimum fracture length of1.5m overcame this problem using selection criteria 3. In contradiction to the conclusions drawn fromfracture system 1, it is not clear whether any of the selection criteria are superior. The use of selectioncriteria 2 and 3 resulted transport predictions that were just as accurate as those using selection criteria6. The performance of the dual permeability model using selection criteria 4 relative to the model usingselection criteria 6 presented in figures 4.19 and 4.21 may indicate that the modification to thetransmissivity of the primary fractures in the effective conductance calculations is too large for fracturesystem 2 but too small for fracture system 1.The good performance of the dual permeability model for fracture system 2 indicates that afracture system is not required to have a sub-system of long, high transmissivity fractures in order for130403Flow Rate (m /s)Minimum flow drawn = 2.Oe— 11 m3/sFigure 4.20: Flow plot of one realization of fracture system 2. The fracture trace map for thisrealization is identical to the map shown for fracture system 1 in Figure 4.10.U)ci)30201000 10 20 30Meters0140I I I2x 10T I I4x 10_jo13110.80.60.40.20Fracture System 2Arrival Time ( Days )Figure 4.21: Average cumulative particle anival at the downstream boundary for 100 realizations offracture system 2.DiscreteSC.2— SC.3SC.4SC.5SC.60 500 1000132some fractures to dominate the flow in the fracture system and that the selection criteria were able toidentify the most important fractures.4.5.2.4 FRACTURE SYSTEM 3 - COMBINED FRACTURE SETSFracture system 3 of Table 4.5 focuses on the importance of connectivity in the primary fracturenetwork and, by inference, in the. dominance of local flow. Fracture system 3 is composed of only twofracture sets and has a only one scale of fracturing. Each fracture set retains the vertical and horizontalscan-line density, 2.8 fractures per meter, and mean length, 3.6 m, as the composite of the twocorresponding fracture sets in fracture systems 1 and 2. A fracture network from this system is shownin Figure 4.22a. Fracture system 3 lacks the long, nearly orthogonal fractures of systems 1 and 2. Basedon a visual comparison, the plot of flow rates in Figure 4.22b has a degree of channeling that isintermediate to that of fracture systems 1 and 2. In the upper right of the figure a few fractures withextremely large flows are evident. The flow rates in these fractures are influenced by the proximity ofthe fractures to the boundaries of the fracture network. These large flows are not important to theevaluation of the dual permeability model but are a concern in the development of models sub-REVfracture networks. The causes and effects of this interaction with the boundaries are pursued further inChapter 5.The dual permeability model is much less effective in simulating transport in fracture system 3than for the earlier fracture systems, as shown in Figure 4.23. The curves in Figure 4.23 were plottedfrom 52 common realizations. Because of the difficulty in establishing the primary fracture network foreach realization using selection criteria that tend to ignore fracture length, only selection criteria 2, 4, 5,and 6 are presented for fracture system 3. All of these criteria resulted in transport simulations that laggedbehind the results of Discrete. Using criteria 4, 5, or 6, the median arrival time is delayed 20% from themean arrival predicted by Discrete. The net flows through the fracture network are roughly 10% lowerin the dual permeability simulations. Using selection criteria 2, the corresponding errors are closer to 50%33CDCDCD CD CD-I-i CD CD CDCD CD 0 -J 0 CD CD CD 1 CD Cl) CD CD H CD-CD -3 CD CD -3 CD0Meters(D [ID CO S000r\) 0CA)000 0(O c-roCD0LI]CA)0U)0)0)3Flow Rate (m /s)Minimum flow drawn 3.3e— 11 m3/sFigure 4.22:Cont. Fracture trace map and flow plot for one realization of fracture system 3. b) Flowrates within fractures.4030201000 10 20 30 400 2x101°r ii’I IMeters4x10’° 6x10’° 8x10’°. I I I I I I I1og. I135Fracture System 310.8El 0.60.40.20Arrival Time ( Days )1500Figure 4.23: Average cumulative particle arrival at the downstream boundary for 52 realizations offracture system 3.0 500 1000136delay in the cumulative arrival times and a 20% flow error.The large difference in the performance of the dual permeability model revealed by theinvestigations of fracture systems 2 and 3 may indicate that a larger scale structure must exist in thefracture network for the creation of the dual permeability model to be effective. It may be that the flowchannels that are evident in Figure 4.22 include too few fractures and thus are not well enough connectedto be represented properly by the primary fracture network structure. Flow dominance may be so localin this fracture system that it cannot be adequately represented using the conductance approximations ofthe network blocks.4.5.2.5 FRACTURE SYSTEM 4 - UNIFORM APERTURESFracture system 4 is the antithesis of fracture system 1. This system has neither a widedistribution in fracture length nor variation in fracture transmissivity. The negative exponential distributionin fracture length provides a range in length scales but the larger lengths of sets 1 and 2 of fracture system1 is lacking. Other than the variation in fracture transmissivity, fracture system 4 is identical to fracturesystem 3. The fracture network shown in Figure 4.22 also serves as an example of fracture system 4.Dominant flow channels do not form in fracture system 4 as shown in flow plot of Figure 4.24. Thereare variations in the flow rates of different fractures but these are small.Since there is no transmissivity variation in fracture system 4, only selection criteria 2 was usedto find the primary fracture networks in the dual permeability model. The dual permeability model is lesseffective simulating fracture system 4 than simulating fracture system 3. The average net flow ratecalculated for the fracture networks is 20% less in the dual permeability simulations compared to Discrete.As shown in Figure 4.25, the downstream arrival times average a delay of roughly 30% in the medianarrival time. The test of fracture system 4 provides further evidence that the dual permeability conceptis limited to fracture systems that have dominant flow pathways.137—--:5Flow Rate (m /s)Minimum flow drawn 4.5e— 12 m/sFigure 4.24: Flow through one realization of fracture system 4, a fracture system withoui variatio:: infracture aperture. The fracture trace map is idemical to Figure 4.22a.C.)C.)403020.1000 10 20 30Meters0r405xj11T T101o-E1382S-l—I$-i—4UFigure 4.25: Average cumulative particle arrival at the downstream boundary for 100 realizations offracture system 4.Fracture System 410.80.60.40.200 500 1000Arrival Time ( Days )15001394.5.2.6 A WAR ZONE MODEL; FRAcTURE SYSTEM 5The bimodal range of fracture lengths that exists in fracture systems 1 and 2 is an inherent featureof the war zone structural model( Sections 2.4.5 and 3.4.3). In the war zone model there is a length scaledefined by the fracture set parameters and a length scale defined by the size of the war zones.Fracture system 5 is nearly the same as fracture system 1, except that a war zone structural modelis used to generate the fracture network realizations. The war zones are defined by the two sets of longerfractures. The other two sets define the fractures created within the war zones. Some of these fracturesin these latter two sets are generated without respect to the war zones and others are generated exclusivelywithin each war zone. The density of these two fracture sets was reduced so that the sets have the samescan-line densities in fracture system 5 as in fracture systems 1 and 2. Figure 4.26a is a trace map of arealization from fracture system 5. The particular arrangement of war zones makes the fracture densityappear quite blocky in this figure. There are large regions of both high and low densities of fractures.The flow plot presented in Figure 4.26b shows the strong influence that the region of low fracture densityin the lower right of the domain has on the flow pattern. Much of the flow entering the lower part of thedomain is drained off by a highly transmissive fracture that defines the boundary of the low densityregion.The averaged transport characteristics of the war zone fracture networks are nearly identical tothose of fracture system 1; an indication that the larger fractures truly dominate the flow within thesefracture systems. For the selection criteria that were tested, the choice of selection criteria have littleinfluence on the dual permeability results unless fracture transmissivity is completely ignored, as shownin Figure 4.27.1400Meters00c) 00-nCC) C) 1 C) C-) C) 1 C) 1 C-) C) -J C) 0-D 0 C) N C) 0 0 -j 1 C) C) C) —I C-) 1, C) C) C) 1 N 0 C) C)0 0CD ÷roCD0 0 0C) 0 0 -j - C) C) d C) -J C) H C) C C) C) C)-p 0 C-) J) C)- C) C) C) C) 1 C) (I)Flow Rate (m3/s)Minimum flow drawn = 9.8e— 11 m3/sFigure 4.26:Cont. Fracture trace map and flow plot for one realization of fracture system 5, a warzone version of fracture system 1. b) Flow rates within fractures.U)a)a)4030201000 10 20 30Meters0r40I I I Iic-9T I I2x109I I142270.80.60.40.20Figure 4.27: Average cumulative particle arrival at the downstream boundary for 76 realizations offracture system 5.Fracture System 5 — War Zone Structural Model1DiscreteSC.2SC.4SC.5SC.6—////////0 200 400 600 800Arrival Time ( Days )1434.5.2.7 A WAR ZONE MODEL; FRACTURE SYSTEM 6Fracture system 6 closely mimics fracture system 3 except for the use of a war zone structuralmodel to generate the realizations. Fractures of the first fracture set define the war zones in fracturesystem 6. The second fracture set represents the later fractures that are created preferentially within thewar zones. Figure 4.28a presents the fracture trace map for one realization of this fracture set. ComparingFigures 4.28a and 4.24a reveals that the war zone fracture system has a much greater variation in theclustering of the fractures. There is a dumpiness to the distribution of fractures in Figure 4.28a that isnot evident in Figure 4.24. The size of the war zones is related to the size of the fractures that definethem, leading to war zones of a smaller scale in fracture system 6 compared to those of fracture system5. The difference in the relative size of the fractures within the war zones compared to the war zonedefining fractures is much smaller in fracture system 6 compared to fracture system 5. As was done forfracture system 5, the scan-line density of set 2 was reduced so that the scan-line density of all the set 2fractures is equivalent to the scan line density of set 2 in fracture system 3. The war zone model,however, creates fractures that have smaller lengths than the fractures defined by the set parameters.There is a similarity between the flows rates depicted in Figure 4.28b for fracture system 6 andthe flow rates of Fracture system 3, in part because the fractures of fracture set 1 are identical in the tworealizations. The flows in fracture system 6 appear more channeled than in fracture system 3, but thedifferences are slight. More importantly, these channels are more continuous in fracture system 6 thanin fracture system 3.Despite the similarity in the flow diagrams for fracture systems 3 and 6, the performance of thedual permeability model for these two fracture systems is radically different. As shown in Figure 4.29,the arrival times predicted using selection criteria 4, 5 or 6 are within five percent of the arrival timespredicted using Discrete. The difference may be a result of fundamental differences in the nature of theflow systems in fractures systems 3 and 6; namely, the larger scale of dominate flow channels with respect144Meters(I)11300 0 000000ii oNCD0 CDr\)CA)a000Flow in the Major Carriersci)Flow Rate (m3/s)Minimnm flow drawn = 3.3 e— 11 m3/sFigure 4.28:Cont. Fracture trace map and flow plot for one realization of fracture system 6, a warzone structrual model version of fracture system 3. b) Row rates within fractures.40301000 10 20 30 40Meters0 2x10’° 4x10’° 6x101° BxlQ’° 10rI I I I I I I1462,0.80.60.40.20Figure 4.29: Average cumulative particle arrival at the downstream boundary for 45 realizations offracture system 6.Fracture System 6 — War Zone Structural Model10 200 400 600 800 1000Arrival Time ( Days )147to the fracture scale in fracture system 6.Reviewing the first six fracture system indicates the bimodal length scales of fracture systems 1,2, 5, and 6 leads to flow dominance that has a larger scale than the majority of fractures and that thissituation is well suited to simulation using the dual permeability concept. In fracture systems 1, 2, and5, two fracture scales were defined explicitly through the fracture set definitions. In fracture system 6,and to a lessor degree in fracture system 5, the two scales were an inherent feature of the war zonestructural model.4.5.2.8 THE LEVI-LEE MODEL; FRACTURE SYSTEM 7Fracture system 7 is based on the Levy-Lee structural model (Sections 2.4.8 and 3.4.4). Thefracture system is composed of two orthogonal fracture sets. The scan-line density of each set is the sameas in fracture system 3. The fracture spacing follows a negative power law distribution with an exponentof -1.3. Fracture lengths and the variation of orientation are linearly related to the fracture spacing. Thedisthbution of fractures is highly clustered as shown in Figure 4.30a. Shorter fractures are more highlyconcentrated than the longer fractures making the larger scale structure defined by the longer fracturesmore evident in Figure 4.30a than any of the other fracture systems. Channeling with the Levy-Leefracture system (Figure 4.30b) is much stronger than in fracture system 1 (Figure 4.11).The transport predictions of the dual permeability model are in close agreement with Discrete forthe Levy-Lee fracture system (Figure 4.31). Using selection criteria 4 the dual permeability model iswithin 2 percent of Discrete for all of the mean arrival times other than the initial arrival. All of theresults show small increases in the temporal dispersion of the particles.There is no correlation between fracture transmissivity and length in the Levy-Lee fracture system.The excellent agreement of the transport predictions of the dual permeability model with those of Discretemay indicate that the large scale connectivity of the long fractures is more important to flow dominance148Meters0I’)aC)0403020100MetersFlow Rate (ma/s)Minimum flow drawn = 3.9e—11 m3/s40Figure 4.30:Cont. Fracture trace map and flow plot for one realization of fracture system 7,generated from the Levi-Lee structural model. b) Flow rates within fractures.0 10 20 300 2x101°4x10 ° 6x10’° 8x10’°I I I I I I I I I—910I I1500.60.40.20Figure 4.31: Average cumulative particle arrival at the downstream boundary for 70 realizations offracture system 7.Fracture System 7— Levy—Lee Structural Model10 500 1000 1500Arrival Time ( Days )151than the linkage of highly transmissive fractures. The structure of a fracture network generated using theLevy-Lee model meets the requirements of the dual permeability model well because the flow dominanceis strongly associated with fracture connectivity and hence length. As with two scales of fracturing, thedual permeability concept is well suited to the broad range of fracture scales in the Levy-Lee structuralmodel.4.5.2.9 POTENTIAL BIAs INTRODUCED BY FAILURE TO FIND PRIMARY FRACTURE NETWORKThe dual permeability solute transport calculations for fracture system 1 are only presented for the70 realizations for which all seven selection criteria resulted in an acceptable primary fracture network.For this fracture system, there were 130 other realizations where the dual permeability model could notdetermine a primary fracture network for one or more of the selection criteria. The failures to find anacceptable primary fracture network occurred in different realizations for different selection criteria,sugjesting a weak correlation between the failures of most of the selection criteria. However, failure toidentify an acceptable primary fracture network must be dependent on the nature of the fracture network.A correlation may exist between the failure of the primary fracture selection algorithm and the transportcharacteristics of the fracture network. If so, neglecting these networks introduces a bias into samplingof the fracture system which effects the flow and transport characteristics of the both the Discretecalculations and those of the dual permeability model. This bias is unrelated to the accuracy of the dualpermeability model for individual networks, the purpose of the present evaluation. The bias may beimportant if the results of a Monte Carlo study are to be considered representative of the fracture system.The possibility of bias introduced by failure to find an acceptable primary fracture network wasinvestigated by comparing the arrival statistics for realizations that were used in the Monte Carloevaluation to those that were rejected because one or more of the selection criteria caused the primaryfracture selection algorithm to fail. The difference in mean arrival times of the 70 realizations of theprogram Discrete were compared to the other 130 realizations using a Student’s t test. The comparison152indicated that the mean arrival times of the 70 realizations were earlier than the 130 other realizations, butthese differences could be due to random fluctuation. The difference in the averages is less than the delayintroduced by the dual permeability model. Therefore, arrival time calculations of the dual permeabilitymodel for the 70 realization are delayed in comparison to the Discrete results for all 200 realizations, butless so than presented in Figure 4.17.The investigation of bias was also performed for the other fracture systems. The comparisons formost of the fracture systems (3, 4, 5 and 7) do not indicate a statistically significant bias in the realizationsused for the Monte Carlo analysis. However, the results from the 100 realizations of fracture system 2are biased towards earlier times. More detailed comparisons were made by comparing successful tounsuccessful realizations for each selection criteria. These individual comparisons indicate that the biasin the fracture system 2 results is predominantly due to section criteria 3 and 6 which weight fractureaperture most heavily. The bias introduced using these selection criteria decreases the average arrivaltimes by roughly five percent. The data for fracture system 6, Figure 4.29, is also biased toward earlierarrival times. The differences in the mean arrival times between the realizations used in Figure 4.29 andall of the realizations are larger than the delays introduced by the dual permeability model. Thus, the dualpermeability arrival times shown in Figure 4.29 are earlier than the mean arrival times of Discrete usingall of the realizations. If the realizations that were rejected because of selection criteria 6 were added tothose used in the Monte Carlo evaluation then the bias would be reduced to below a statistically significantlevel.This analysis of bias is based on too few realizations to be considered definitive. It also relies onan assumption of a normal distribution in the arrival times, even though the data presented in Figure 4.18indicates that the arrival times are not normally distributed about the mean. The analysis does demonstratea possible bias introduced by primary selection failures is a potential source of error introduced by thedual permeability model; an error that is not reduced through the use of Monte Carlo analysis. This bias153is independent of the inaccuracies introduced by the dual permeability calculations for individualrealizations.4.5.2.10 SUMMARY OF MONTE CARLO EVALUATIONThis evaluation of the dual permeability model demonstrates that the technique can provideaccurate predictions of flow and transport through fracture networks composed of thousands of individualfractures. The failure of the dual permeability model to adequately simulate the hydraulic behavior offracture systems 3 and 4 indicates that the existence of a flow structure that is somewhat larger than mostof the fractures is required for the modeling approach to work. The success of the dual penneabilitymodel in simulating fracture systems 2 and 6 indicates that the fractures that dominate flow do not haveto be extremely different than the majority of fractures, nor are they necessarily those that are the mosttransmissive. Indeed, the distinction between the flow regimes of fracture system 6 and fracture system3 is subtle.The dual permeability approach requires that the fractures within the fracture netwofic vary intransmissivity. This is not a constraint that conflicts with our general view of fracture systems. Theapproach also requires a larger separation of fracture lengths than is defined by a negative exponentialdistribution. Certainly some fractured rock have broader length ranges than this but some may havesmaller ranges. It is uncertain at this time whether the length constraint seriously limits the applicabilityof the dual permeability approach.4.5.3 ACCURACY AS A FUNCTION OF NETWORK BLOCK SIZEOne of the controlling factors of the primary fracture network is the limit on the maximum sizeof the network blocks. The number of primary fracture intersections and hence the memory storagerequirement of the model decreases with increasing network block size. In this section, the influence ofmaximum block size on model accuracy is investigated. The maximum network block size used in the154Monte Carlo investigations of section 4.5.2 was never more than 600 fracture intersections and never lessthan 400. One hundred realizations of fracture system 1 using selection criteria 6 were repeated usingblock size limits of 200, 400, 600, 800, 1000, and 1500 fracture intersections. The results presented inFigure 4.32 show that the accuracy of the dual permeability model is influenced by the maximum blocksize. The figure presents the average percentile arrival times that are familiar from the Monte Carloinvestigations. Allowing for some variations due to limited sample size, the results show a monotonicdecrease in accuracy as the network block size increases.The legend in the upper left of the figure relates the block size limit to the different curves. Afterthe arrow in the legend, the number fractures that were selected to be primary fractures based on theirgeometric characteristics is listed as a percent of the total number of fractures in the flow domain. Thetotal number of fractures that are at least partly included in the primary fracture network is larger becauseof the addition of connecting fractures to the network (see Section 4.2). The entire primary fracturenetwork includes about one and a half to three times as many fractures than the percentages listed. Thenumber of primary fractures is directly related to the block size limit. Recall that the primary fractureselection routine attempts to limit the number of primary fractures to the specified percentage. For the800 and 1500 block sizes the specified number of primary fractures was large enough to bias the finalnumber of primary fractures slightly. Only a small percentage of the fractures in a network are includedin the primary fracture network. As this percentage increases, the results become more accurate. In thelimit, as the number of primary fractures approaches all of the fractures in the fracture network, the flowand transport predictions of the dual permeability model would approach those of Discrete.4.5.4 TIME AND MEMORYThe fundamental purpose of the dual permeability approach is to provide a modelling capabilitythat allows the investigation of fracture systems that cannot be studied using the conventional discretefracture approach because of computer resource limitations. These may be limitations of either execution155fl0.80.60.40.20Fracture System 1 — Network Block Limit1Figure 4.32: Average cumulative particle arrival for different network block size limits. The curvelegends indicate the maximum number of fracture intersections included in a networkblock. Following the arrows in the legend, the percentage of fracture that were selectedto be primary fractures is listed.Discrete—-—-200-’ 2.8%400-’ 1.3%600-’ 1.1%800-’ 1.0%1000 —‘ 0.5%. 1500 0.4%0 200 400 600 800 1000Arrival Time ( Days )156time or memory storage requirements. An implicit assumption is made here that some form ofapproximation must be introduced in order to simulate flow and transport at the field scale. The emphasisof the model developed for this thesis, however, has been on investigating the dual permeability concept,not in maximizing computational efficiency.The execution time and memory storage requirements of the dual permeability model and Discretewere determined for a series of fracture networks of increasing scale. The scale is measured in terms ofthe number of fracture intersections that remain after the fracture network is reduced to only the fracturesegments that carry flow. The number of fracture intersections is the number of equations that arerequired to solve for flow within Discrete. Calculations were perfonned, for fracture system 1 usingselection criteria 6 and a maximum block size of 800, on a Hewlett-Packard 735 workstation.Table 4.7 presents four sets of data. Two sets of data are presented for the dual permeabilitymodel. One of these sets is for a model that uses a sparse direct solver to find the coarse-scale headdistribution. The other dual permeability data set is from calculations using a sparse, conjugate gradientiterative solver based on LU decomposition, DILUCG. The results of Discrete are presented next. Thefinal set of data is for a modified version of fracture system 1 where the variability of the log10 of theaperture has been increased 0.5. For this latter case two results are presented, one for the dualpermeability model using the direct solver and one for Discrete. The dual permeability model usingDILUCG did not converge.157The first column of the table lists the dimensions of the domain. The second column is the sizeof the domain measured in terms of the number of fracture intersections in the reduced fracture network.The total computer memory requirement is next, followed by the memory needed for the flow solution.The memory required by the dual permeability model for the residence time distributions of the networkblocks is listed under the heading Trans. The memory required for the flow solution listed for the dualpermeability models is the maximum of either the coarse flow solution or the largest network block flowcalculation.The dual permeability model that used the direct solver requires more memory than either of theother two models. The program requires about 40% more memory for each calculation than doesDiscrete. The dual permeability model using the iterative solver requires a bit less, about 20% less thanDiscrete. The memory requirements of the network block residence time distributions are about 10-15%of the total requirements of the dual permeability model. This requirement is significant but does notdominate the total memory needed.The total time required to calculate both flow and transport is listed in the sixth column. Thefollowing columns list estimates of the times taken to perform specific functions within the simulations.The first of these is the time needed to create and then reduce the fracture network to just the fracturesthat have flow within them. This procedure dominates the execution time of the larger networks. Thenext two columns list the time needed to determine the primary fracture network, PFN, and to find theeffective conductances of the network blocks, NB. Both dual permeability models use identical proceduresfor this stage of the calculations. The time estimates are approximate and should be considered accurateto roughly 10% of the times listed. The table shows that determining the primary fracture networkrequires less than 10% of the total simulation time. Finding the effective conductances of the networkblocks is a time consuming process, however. The next column lists the time used to calculate the coarsescale head distribution in the dual permeability models and the time used to calculate the full head159distribution in Discrete. For large domains the dual permeability model is significantly faster thanDiscrete if the direct solver is used and significantly slower if the iterative solver is used. The timerequired for the iterative solution of the coarse-scale flow solution increases with domain size at a muchfaster rate than the ICCG routine used within Discrete, a feature that may be caused by the lack ofsymmetry in the coarse-scale flow equations.The next two columns of the table list the times used in the transport calculations. These timeare based on using 50,000 particles. The column titled RTD is the time needed to calculate both the flowdistribution within the network blocks of the dual permeability model and the time required to create theresidence time distribution tables. The time to calculate the residence time distribution table is about 10%of the total. The transport column lists the time used to calculate particle movement. The dualpermeability model uses approximately one half to one third of the time to move particles, but thetransport time is small compared to the total simulation time. The time needed to set up the residencedistribution tables in the dual permeability model is roughly 10% of the time spent moving the particles.The final two rows of Table 4.7 are data from calculations of a modified fracture network. Forthis network the standard deviation of the log10 aperture was increased from 0.2 to 0.5 for all sets offracture system 1. This standard deviation in aperture is large but not unreasonable for fractured rock.The dual permeability model using the direct solver is not sensitive to this change in the fracture system.Iterative solvers are sensitive to this change. The dual permeability model using DILUCG did notconverge in 10,000 iterations. Discrete did converge but the time required by the ICCG solver increasedfrom l,286s to 5,660s. The dual permeability model with the direct solver took about one third of thetime need by Discrete to solve the problem.For fracture system 1, the dual permeability model using the direct solver does not have asignificant computational advantage over Discrete even though the calculation of the coarse-scale headdistribution and subsequent determination of flow in the network blocks requires much less time than the160solution of the hydraulic head distribution for the entire network. The time spent determining the primaryfracture network and calculating the effective conductances of the network blocks more than compensatesfor this time advantage. The time requirements of the iterated solver of dual permeability model increasesmore rapidly than those of Discrete making this version much slower for the simulation of flow in largedomains. The last set of calculations indicates that the dual permeability model can have a significantadvantage for some problem defmitions.The reader is reminded that the development of the current dual permeability has not emphasisedcomputational efficiency. Some improvements to the current model that could have a significant influenceof model efficiency are suggested in Sections 4.7.5 and 4.7.6. However, the true purpose of modeldeveloped for this thesis is to provide a tool to explore the dual permeability concept and the importanceof flow dominance to forming sub-continuum approximations. The current model serves as a platformfrom which further approximation of fracture network flow and transport behavior can be isolated fromthe more basic influences of separating a fracture network into blocks.4.6 COMPARISON TO REDUCED FRACTURE SIMULATIONHow do the inaccuracies introduced through the dual permeability model compare to othermethods of approximating the fracture network? One such approach is to reduce the fracture network byignoring small aperture fractures. This method was used by Herbert et al. [1991] in their threedimensional simulations of the Stripa test site (See Section 2.3.6.). Herbert et al. investigated theinfluences on both flow and transport of simulating only the largest fractures. They found that using 70%of the fractures reduced the net flow though a cube 12.5 on a side by 10%. Using only 30% of thefractures reduced the net flow by 30%. The selection of fractures to be retained in the model was basedboth on the length of fractures and on the aperture. The fracture system at the Stripa site was modeledusing a standard deviation in natural logarithm of aperture in the range of 0.8 (0.35 log10). This variationin aperture is slightly larger than the aperture variations of the fracture systems listed in Table 4.5. More161importantly, the fracture system simulated by Herbert et al. is denser and more highly connected than thenetworks simulated in this study which makes the fracture system of Herbert et al. less sensitive tofracture reduction than the fracture systems used in this investigation.The transport simulations at Stripa were made using 30% of the fractures in the fracture tietwork.The applicability of the transport model was tested using 5m cubes. The results of these tests indicateda general agreement in the transport times between the complete and reduced network calculations, withless dispersion in the reduced fracture network. A more specific influence of the fracture reduction ontransport could not be detennined from the results published in Herbert et al. [19911 becauseapproximations to particle transport within individual fractures were also included in the reduced networksimulations. These single fracture particle transport approximations were not included in the completenetwork simulations.In this section, the dual penneabiity model is compared to the simulation strategy of reducing thenumber of fractures in the networks by eliminating the least transmissive fractures. For each of the sixfracture systems with variable transmissivity, transport sensitivity to fracture reductions of 10%, 25%,50%, and 75% were investigated, using fracture domains of 40m by 40m. The dual permeability modelrepresents about 2% of the fractures using the primary fracture network and approximates the rest usingthe network blocks.In Figure 4.33, the results reducing the fracture networks of fracture system 1 are presented, usingdashed lines as labeled in the upper left corner of the figure. In addition, the results from using the fullfracture system are plotted using a solid line and the dual permeability model results, for selection criteria6, are shown using a dotted line. Only the smaller fracture sets were reduced for this fracture systemleaving the dominant fractures uneffected. This reduction strategy contains an a priori recognition of theimportance of the long fractures. All of the reduced networks have initial arrival times that are close tothe initial arrival of the full networks. Reducing the number of fractures causes later particles to arrive1623310.8El 0.60.40.20Fracture System 1Arrival Time ( Days )Figure 4.33: Fracture system 1 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures of fracture sets 3 and 4.0 200 400 600 800163earlier than they do in the full networks; decreasing the temporal dispersion of the particles. This resultis in agreement with the findings of Herbert et al., but is in contrast to the dual permeability model whichhas a slight increase in temporal dispersion. Except for the initial arrival time, the cumulative arrival timesof the dual permeability model are closer to the full network than the arrival times of any of the reducedfracture simulations.The results from reducing fracture system 2, which has the same transmissivity statistics for eachof the four fracture sets, are presented in Figures 4.34 and 4.35. In Figure 4.34, only fracture sets 3 and4, the short fractures, are subject to fracture reduction. For the smaller reductions in the number offractures, 10% and 25%, the influences on transport are similar to those of fracture systems 1. With a50% reduction, and more clearly with a 75% reduction, a different behavior is evident. The arrival timesusing a 75% reduction in the number of fractures can be later than using a 50% reduction. This reversalin trends is an indication that fractures are being removed that provide important connectivity to thefracture network, a feature that is more clearly displayed in the investigation of fracture system 3 (Figure4.36).Applying the fracture reduction to all of the fracture sets results in the cumulative arrival curvesshown in Figure 4.35. Removing the longer low transmissivity fractures results in a significantly largerreduction in dispersion. A comparison of Figure 3.34 and 4.35 indicates that although the long fractureshave a more significant impact on dispersion, the highly transmissive fractures that were removed fromsets 3 and 4 instead of these long fractures have greater influence on flow in the dominant channels ofthe full network. It should be noted that for these and all other simulations in this section, realizationsthat resulted in greater than 10% of the particles crossing the top or bottom boundaries were not used.The cumulative arrival times are based solely on particles that crossed boundary 4, the right side.Rejecting realizations in which these constraints were violated led to a slightly different set of realizationsfor Figures 3.34 and 3.35. Note that the cumulative arrival times of the dual permeability model match1640.8E 0.60.40.20Arrival Time ( Days )1500Figure 4.34: Fracture system 2 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures of fracture sets 3 and 4.165Fracture System 2 — Common Aperture Statistics10 500 1000Fracture System 2 — All sets reduced10.80.60.40.20Figure 4.35: Fracture system 2 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures of all fracture sets.Network////Complete- SC.610%25%-—-—--50%75%//V/ VVV/////IV///II ////I/I///IIIIIIIIII////0 500 1000Arrival Time ( Days )166those of Discrete for the full fracture system closer than the cumulative arrival times of a 10% reductionin the fracture network at later arrival times but not for the early times in both Figure 4.34 and 4.35.Figure 4.36 presents the arrival curves for fracture system 3. The difference in character betweenthe influence of a 10% and 25% reduction in the number of fractures and the 50% and 75% reductionsis a vivid demonstration that at some point removing fractures starts to influence the dominant flowchannels. When this occurs the major influence on transport of removing fractures changes from primarilydecreasing dispersion to substantially delaying the arrival of particles. The dual permeability model whichcontains less than five percent of the fractures in the primary fracture network provides a betterrepresentation of transport compared to a simulation where 25% of fractures remain after fracturereduction, but for less drastic reductions predictions of the dual permeability model are worse than thosefrom the reduced networks.The fracture reduction of fracture system 5 which used a war zone structural model was limitedto the fracture sets than were preferentially created in the war zones. This method of reducing the fracturenetwork recognizes the important role that fractures that define the war zones play in connecting the zonestogether. As shown in Figure 4.37, this war zone fracture system was less sensitive to the reduction infractures than fracture system 1.For fracture system 6, fracture reduction was applied to all the fracture sets which resulted in theloss of some war zones. Fracture removal was applied prior to the identification of war zones. Becauseof the way the fracture networks are generated, the 100 realizations used to determine the cumulativearrival curves shown in Figure 4.38 differ for each curve. This introduces a variation between curves thatdoes not exist in the other figures of this section. I do not believe that the influences on the plots aresignificant, but it may explain why even the 10% and 25% fracture reductions result in a slight delay inthe initial arrival times. This delay may instead indicate that this fracture system was sensitive to a minorloss of connectivity between war zones. The initial arrival times for the 50% reduction are delayed by167Fracture System 3 — Combined Fracture Sets10.80.60.40.20Figure 4.36: Fracture system 3 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures of both fracture sets.0 500 1000ArrivaI Time ( Days )16810.80.60.40.20Fracture System 5 — War Zone Structural ModelArrival Time ( Days )Figure 4.37: Fracture system 5 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures of war zone dependent sets 3 and 4.0 200 400 600 800169Fracture System 6 — War Zone Structural Model10.80.60.40.20Figure 4.38: Fracture system 6 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures of both sets.0 200 400 600 800 1000Arrival Time ( Days )170about 30% from the full network, much more than can be attributed to variations between realizations.With a 75% reduction in the number of fractures, the 25th, 50th, 75th, and 90th percentile arrival times arein close agreement with the full network. Based on an interpretation of Figure 4.36 from fracture system3, the arrival times are increasing rapidly with increasing fracture reduction at the 75% level and, bycoincidence, agree with the full network at this reduction level. For this fracture system, the dualpermeability model is superior to a fracture reduction strategy.Fracture reduction on fracture system 7 which uses a Levi-Lee structural model has a greaterinfluence of the estimated dispersion than on any of the other fracture systems. The cumulative arrivaltimes displayed in Figure 4.39 show that the average initial arrival times are not influenced by fracturereduction, even with 75% of the fractures removed. Recall from Figure 4.30 that flow through thisfracture system is strongly dominated by long wide fractures that span the domain. One interpretation ofthe arrival times presented in Figure 4.39 is that although fracture reduction leaves the dominant flowpaths intact the interaction between these dominant fractures and the less insignificant fractures has beenlost. The dual penneability model contains far fewer fractures within the primary fracture network thanremain in even the most highly reduced fracture networks but, since the other fractures are approximatedwithin the network blocks, the essential interaction between the dominant fractures and the rest of thefracture network is still retained.In summary: 1) Reducing the fracture network reduces particle dispersion leading to earlier arrivaltimes for later particles, 2) The initial arrival times are not significantly influenced by fracture reductionbased on aperture until the 50% reduction is reached, 3) The dual permeability model tends to result incloser agreement in the cumulative arrival times compared to the arrival times of a 10% reduction in thefracture network; fracture system 3 excepted. 4) The interaction between the dominant fractures and theleast important fractures is lost through fracture reduction but not in the dual permeability model.171Fracture System 7 — Levy—Lee Structural. Model10.80.40.20Arrival. Time ( Days )1500Figure 4.39: Fracture system 7 mean particle arrival times for fracture networks that have been reducedby removing small aperture fractures.1720 500 10004.7 ALTERNATE APPROACHES TO NETWORK BLOCK FLOW MODELS4.7.1 INTRODUCTIONIn the introductory chapter, it was emphasised that the dual permeability model presented in thischapter is one implementation of the dual permeability concept; there are a variety of possible approachesto stmcturing the hierarchy of primary fractures and network blocks and to approximating the flow andtransport behavior in the fracture networks within the network blocks. The present formulation of thenetworks blocks (Section 4.3) requires repeated solution of the network block flow equations to fullydefine the hydraulic characteristics of the blocks. The calculated flows have been demonstrated to provideenough accuracy to be used for-solute transport. Early attempts to formulate a flow model for the networkblocks were based on computationally more efficient algorithms, but these formulations were either notcompatible with the particle tracking transport algorithm or suffered from unacceptable inaccuracy at thelevel of individual realizations.The first of these formulations is presented in Section 4.7.2. Network blocks were representedwithin the dual permeability model using finite elements with a conductivity defined by the tensorrepresentation of conductance for each block. The tensor representation presented in Chapter 5 emergedfrom the development of this network block model. This network block flow model is presented toexplain why a more specialized formulation was needed and to reinforce the notion that the modelpresented in this chapter does not define the dual permeability concept.A third network block approach is presented in Section 4.7.3. This approach is similar to thepresent model except that the head distribution along the bounding primary fractures was treated as aboundary condition to the network blocks. The head gradient along the boundaries of the blocks wasassumed to vary linearly between the primary fracture intersections. The failure of this model to provideadequate transport predictions points to some serious limitations in sub-REV scale modeling that rely on1 13boundary conditions that are not sensitive to heterogeneity within the domain. This issue is pursuedfurther in the concluding chapter of this thesis.A fourth alternative is presented in Section 4.7.4. The major innovation in this alternative, overthe present model of Section 4.3, is the use of iteration to improve the estimate of the hydraulic headprovided by the flow model. The flow solution provided by this alternative is identical, within numericalprecision, to that provided by Discrete. However, the number of iterations needed to reach an adequateflow solution is larger than the number of iterations that Discrete requires to reach a flow solution. Thisalternative has not been explored in great detail. An improvement in primary fracture selection, in thealgorithm used in the iteration, or the use of parallel processors may make iteration of,the head distributionan attractive alternative.4.7.2 FINITE ELEMENT NETWORK BLOCK MODELThe REV needed to represent flow may be smaller than the REV needed to represent transport.A dual permeability model that represents the fracture network as a continuum for the purposes ofestablishing regional flow rates but discretely for the purposes of estimating local flow rates and solutetransport has the potential to be extremely efficient. The network block formulation described in thissection uses triangular finite elements with homogeneous hydraulic conductivity to represent theconductance of the network blocks and line elements to represent primary fractures. See Baca et al.[1983] for a description of line element representation of fractures. The line elements were located alongthe edges of the network blocks.The triangular elements were formed using the centroid of a network block as one vertex of allthe triangles within the block. The other vertices of a triangle were located at the intersections of theprimary fracture segments that define the boundaries of the blocks. An equivalent conductance matrix(stifthess matrix) was then formed by combining the finite element stiffness matrices. The centroid of174each of the blocks, which is not a primary fracture intersection, was removed in the procedure that formedthe coarse-scale flow model.To obtain the conductivity for the triangular finite elements, each network block was tested usingconstant hydraulic head boundary conditions along the boundaries of the network block that wereconsistent with a regional gradient oriented first at 0° and then at 90°. The directional flow through theblocks was measured by summing the directional components of flow at each boundary fracture asdescribed in procedure 1 of Section 5.4.2.The use of triangular finite elements required that the network blocks be convex. By convex, Imean that the shape of a network block must be such that any line passing through the block wouldintersect the block boundaries at only two locations. To ensure convexity some boundaries of the blockswere defined by straight lines connecting primary fractures rather than always being actual fractures.These boundary lines were treated as primary fractures of zero aperture.The finite element representation of the network blocks was used only to estimate the hydraulicheads at the primary fracture intersections, just as the coarse-scale flow model is presently used. Finescale flow within individual network blocks was determined by using the hydraulic head values of theprimary fracture intersections as boundary conditions for the network blocks. Linear head gradients alongthe primary fractures were assumed. The use of linear gradients along the network block boundaries isconsistent with the assumptions made in approximating the network blocks using triangular finite elementswith linear basis functions, but is incompatible with the particle transport algorithm used to determine theresidence time distributions of the network blocks.Combining the flows from the network blocks with flows calculated for the primary fractures ledto local mass balance errors that were unacceptable for use with the particle tracking transport algorithm.Efforts to eliminate these local mass balance errors by modifying the common hydraulic heads along the175zero aperture boundary lines of adjacent network blocks led to the creation of fictitious circular flowpatterns. The finite-element based formulation of network block conductance does not provide anacceptable flow estimate for the particle tracking routine described in section 4.5. It may provide anadequate flow estimate if a transport algorithm that is less sensitive to local mass balance errors were used.One possibility might be a continuum based transport routine such as the transport model proposed bySchwartz and Smith [1988].4.7.3 EFFECTIVE CONDUCTANCE WiTH LINEAR BOUNDARY CONDITIONSIn a second network block model, the effective conductance of the network blocks was calculateddirectly, in order to increase the accuracy of the network block flow approximation compared the use offinite elements. The technique was the same as the one described in Section 4.3 except that the networkblock flow model did not include the primary fractures. Instead, a linear hydraulic head gradient wasassumed to exist along the primary fractures.This formulation successfully eliminated local mass balance errors. Despite this fact, this networkblock model led to poor transport characteristics when viewed over the entire fracture network domain.The problem with this model formulation is a fundamental difficulty in forming sub-REV flow models.There is a strong correlation between local hydraulic head gradients and local heterogeneity. In fracturedmedia, if the correlation is ignored on the boundaries of modeling blocks then excessive flow rates maybe estimated within large aperture fractures that intersect the boundaries. Further implications of ignoringinternal heterogeneity in specifying the boundary conditions are pursued in Chapter 5 and addressed ina discussion in Section 6.1.4.7.4 ITERATED FLOW SOLUTION (DOMAIN DECOMPOSITION)The formulation of the dual permeability model rests on the subdivision of the fracture networkinto distinct parts. This concept relates the dual permeability approach to the concepts of domainI lbdecomposition. The term domain decomposition encompasses numerous techniques that are based on theseparation of domains. See for example Chan et al. [1988] and Keyes et al. [1992]. To my knowledge,none of these techniques use approximate solutions based on a non-iterated coarse-scale calculation.Therefore, I am using the term domain decomposition to describe only the technique of iterating theprimary fracture hydraulic head distribution to reach a hydraulic head solution that agrees, withinnumerical precision, with that of Discrete.The composition of the network blocks differs from the model described in Section 4.3. In thestandard dual permeability formulation, the network blocks include the primary fractures along the blockboundaries. Thus, two contiguous network blocks may have different hydraulic heads at locations thatrepresent the same intersection of fractures. The dual permeability formulation forces a mass balance atthese points but not continuity of hydraulic head. In the iterated procedure continuity of hydraulic headis maintained by excluding the primary fractures from the network blocks and using the entire hydraulichead distribution along the primary fractures to define the boundary conditions of the blocks.A mass balance error is calculated at each intersection of the boundary primary fracture segmentswith fractures within the network block. The error is the sum of the flows entering the intersections fromthe network blocks and flows entering the intersection from the primary fracture segment. The massbalance error is the error residual used in a conjugate gradient iteration procedure. Local conservation ofmass at each intersection along the primary fracture ensures that the hydraulic head solution is consistentwith Discrete. The solution for hydraulic head within the model domain must be found through iterationbecause the primary fracture hydraulic head distributions depend on the flows calculated within thenetwork blocks.Figure 4.40 is a flow chart of the iteration procedure. The first step is to provide an initialestimate of hydraulic head in the primary fractures by creating the coarse-scale model described earlier.The hydraulic head calculated along the boundaries of two adjacent network blocks is averaged to obtain177Note‘r and p are scalersr,q,x, and are vectorsUpdate Residual= r —Convergence? NoYesNew Descent Vector‘ç, 2 ,c’ç 2— r= —r1 + ‘yO)Figure 4.40: Flow chart of the conjugate gradient algorithm used to iterate the hydraulic head solutionalong primary fractures.Calculate Primary FractureIntersection Hydraulic HeadsAverage Hydraulic HeadsAlong Primary Fractures FromNetwork Block Flow Calculationsxo*1Recalculate Network Block FlowsUsing x0 as Boundary ConditionsCalculate MassBalance ResidualsInitial Descent Vectorw0=—rtn_OE0c,)4-,Ca)0a)0)•C00RecalculateUsing co,, asNetwork Block FlowsBoundary Conditionstn — n+1Calculate MassBalance Sensitivity to oqCalculate Descent Scaler2, 2pUpdate Primary Fracture Hydraulic Headsxn+1 = Xr- Pn0n178a head distribution along each primary fracture segment. The network block flows are recalculated usingthese averaged hydraulic heads as new boundary conditions.The conjugate gradient iteration approach applied to domain decomposition was developed in avariational framework by Glowinski and Wheeler [1988] and applied in a fmite element domaindecomposition model of groundwater flow by Beckie et al. [19931. The conjugate gradient method usedin the dual permeability model follows the application of Bec/de et al. As pointed out by Beckie et al.,solution by conjugate gradient based domain decomposition (DDCG) is much slower than by a conjugategradient technique with convergence acceleration enhanced by preconditioning the entire A matrix usingincomplete Cholesky decomposition (ICCG). These authors found the DDCG technique to take on theorder of 45 times as long as ICCG, for the problems they investigated. For the one fracture network usedto test this model, the iterated hydraulic head solution of the dual permeability model took approximately27 times as long to obtain a flow solution as Discrete, which uses an ICCG solver. The iterated solutionuses a direct solver to obtain flow solutions within the network blocks which probably explains its relativespeed in comparison to the results of Beckie et al. They were able to reduce the solution times to 5 to10 times the ICCG solution time by applying a multigrid-based convergence acceleration technique.Multigrid acceleration was not attempted in the dual permeability model because of difficulty applyingthe multigrid technique to the irregular geometry of the network blocks.For the investigations addressed in this thesis, computer execution time rather than memory storagerequirements have been the constraining resource. Because the initial investigations of the iterated versionindicated that this version of the dual permeability was much slower than Discrete, it has not been pursuedto the point where its utility in comparison to the non-iterated version or Discrete can be conclusivelyestablished. For other problems, where only a few realizations of a large model domain are required, thelower memory requirements of the iterated version may be an advantage over Discrete. The memorystorage requirements for the coarse-scale hydraulic head distribution are smaller for the iterated version179than for the non-iterated version of the dual permeability model using an iterative solver to find thecoarse-scale head distribution. It is unclear at this point what the relative speed of these two versionswould be for extremely large domain sizes. Therefore, while the iterated or domain decompositionapproach to the dual permeability model has been abandoned for this research, there is a possibility thatit has the potential to be a useful tool in investigating extremely large fracture domains.4.7.5 IMPROVEMENTS TO THE CURRENT NETWORK BLOCK FLOW MODELThe memory usage and problem execution time study of Section 4.5.4 have revealed two aspectsof the effective conductance approximation that show obvious room for improvement. The first aspectis the time required to determine the effective conductance of each block. The second aspect is the slowconvergence of the iterative solution of the coarse-scale head distribution.In Table 4.7, the time required to determine the effective hydraulic conductances of the networkblocks is much larger than the time required to calculate the coarse-scale head distribution. Additionalinvestigations, that were not reported in Section 4.5.4, have shown that longer times are required todetermine the effective conductances of fewer network blocks with many primary fracture intersectionsalong their boundaries than to find the conductances of more blocks but with fewer primary fractureintersections along each block boundary. A substantial reduction in these times may be brought about bydetermining the block conductances in a single calculation rather than by performing an separate flowcalculation for each primary fracture intersection as is described in Section 4.3.1. To perform thiscalculation an effective means of inverting a portion of the complete stiffness matrix describing thenetwork block flow equations must be determined. This inversion need not be exact; the effectiveconductance of a network block is already an approximation because of the way the block boundaryconditions are treated.180The equations that describe the effective conductance of a network block are non-symmetric. Thenon-symmetry arises from the difference in the equations describing the hydraulic head response of theprimary fracture intersections from the equations of the midpoints. The non-symmetry of the coarse-scaleflow equations can have a large influence on the time required to find the coarse-scale head distributionusing an iterative solver and roughly doubles the memory storage requirements using the direct solver.In hindsight, it may be more efficient to create the effective hydraulic conductance approximation usingonly the primary fracture intersections. The flow into each end of the primary fracture segments wouldthen need to be determined from the final flow calculations the network blocks.4.7.6 PARALLEL COMPUTING AND THE DUAL PERMEABILITY MODELI would be remise in not pointing out that the structure of the dual permeability approach isinherently parallel and in not speculating on the ramifications of the application of parallel computingtechniques to the dual permeability model. The independence of the effective conductance calculationsfor the network blocks allows these calculations to be done simultaneously. This stage of the dualpermeability model could be reduced from the most time consuming part of the coarse-scale flowcalculation to an insignificant portion through the use of parallel processors. The calculation of theresidence time distributions of the network blocks are also independent and, thus, inherently parallel. Theparticle transport calculation is another aspect of the model that is inherently parallel. Finally, the fracturegeneration and reduction procedures could be divided into regional sub-problems that could be solvedindependently, or nearly so. The use of parallel processing in the dual permeability approach wouldtremendously speed the calculations with very little change to the way the models are structured.4.8 SUMMARY OF DUAL PERMEABILITY MODELIn this Chapter, one implementation of the dual permeability modeling concept has been presentedin detail. The model was shown to be effective at simulating both flow and transport within fracture181networks where a primary fracture network of hydraulically important fractures could be identified. Acrucial feature of this implementation is the approximation of the effective conductance of a small regionof the fracture network.The effective conductance of the network blocks is used in a calculation of a coarse-scalehydraulic head distribution within the fracture network. The hydraulic conductance matrix that describesthe equivalent conductance through the network block from a primary fracture intersection to a fracturemidpoint is not an approximation. The network block conductance retains all of the complex hydraulicconductance structure of the fractures within the network block in a condensed form. However, it wasfound that this description of the network block was not enough to fully capture the mutual influences ofthe small portion of the fracture network that the block represents and the rest of the entire network onhydraulic behavior. The way boundary conditions are applied to the network blocks is the essentialapproximation in the network block model.Neither impermeable boundaries nor completely open boundaries with linear fixed head gradientswere adequate descriptions of the network block boundary conditions. The current network block modeluses “responsive” boundary conditions. There are two important features to these boundary conditions;1) the hydraulic head on the boundaries is allowed to respond to the heterogeneity of the internal fracturenetwork, and 2) the equivalent of a drain is modeled along the boundaries of the network blocks. Thedrain accounts for the hydraulic conductance of the bounding primary fractures and includes a roughapproximation of the hydraulic conductance of fractures that are near, but still exterior, to the networkblock. Thus, the boundary conditions of the network block reflect, in a crude sense, the heterogeneity ofthe fracture network that surrounds the network blocks. Figures 4.17 and 4.19 demonstrate that therecognition of the exterior fracture network is an important aspect of the network block model.The network block model allows the fracture network to be divided and modeled in independentsteps rather than being treated as a whole; potentially providing a computational advantage at the cost of182decreased accuracy in the flow and solute transport calculations. The computational advantage is limitedto special situations in the model presented in this chapter. In Section 4.7.5, some changes are proposedthat may increase the computational efficiency of the model.A key feature in the dual permeability concept is the identification of a primary fracture network.This network defines the geometry of the network blocks. The primary fracture network provides animportant hydraulic buffer between adjacent network blocks, allowing flow in the network blocks to becalculated independently of other network blocks. An algorithm has been developed that, in conjunctionwith the selection criteria, determines which fractures should be included in the primary fracture network.It was found that both the transmissivity and connectivity of fractures are important charadteristics for thefractures of the primary fracture network.The algorithm for determining the primary fractures evolved into a complex element of the dualpermeability model. The complexity is potentially a serious liability that may limit the extension of thedual permeability model to three dimensions. However, much of the complexity involved in defining theprimary fracture network could be avoided if the primary fracture network were generated first and thenthe network blocks were filled with the less important fractures. While the primary fracture selectionprocess is complex, it is also relatively fast compared to defining the fracture geometry and solving forflow within the fracture network.In summary, the model described in this chapter has been used both to explore the dualpermeability concept and to increase understanding of the influence of flow channeling on solute transportin fracture networks. The primary fracture selection algorithm is able to identify a primary fracturenetwork in a wide variety of fracture networks - shedding light on the importance of fracture connectivity.The dual permeability approach has been shown to be effective in simulating flow and transport in fracturenetworks where a small subset of fractures tend to dominate the flow field. An important characteristicof the model is that the losses in the accuracy of flow and transport simulations can be reduced to183insignificance compared to the inherent uncertainty in the fracture networks. While the model presentedin this chapter neither captures all of the important physical processes that control flow and transport infracture media nor exhibits a decided advantage over conventional discrete fracture network models, theresults of this first investigation into sub-continuum approximations of fractured media are sufficientlypromising to warrant additional research to identify methods for improving computational efficiency andefficacy of the modeling approach.184CHAPTER 5A TENSOR DESCRIPTION OF FLOW5.1 INTRODUCTIONAre fractured media special cases of porous media? Do fractured media behave differently frommedia dominated by primary porosity? Conversely, can the framework used to describe flow and transportin porous media adequately describe the behavior of fractured media? These issues are central to aconceptual understanding the hydraulic properties of fractured media and to the development of modelingtools that can be used to represent fluid flow and solute transport in fractured media. Measurements ofthe hydraulic properties of fractured media typically exhibit a high degree of spatial variability. Fracturedmedia are heterogenous over a broad range of scales. To some, this heterogeneity forms the basis for thesupposition that fractured media are fundamentally different from porous media. To others the differenceis one of degree. This latter group argues that the tools being developed to describe flow in heterogeneousgranular porous media can adequately capture the behavior of flow in fractured media.A number of studies of fractured media have focused on determining the conditions needed toenable fractured media to be treated as classical porous media. To meet these conditions, a fracturedmedium must behave as a uniform continuum at the scale of interest. The conventional concept used todescribe this behavior is that of the representative elemental volume (REV), the volume over which thedetailed structure of the medium must be averaged to represent the hydraulic properties in terms of a muchsimpler, uniform continuum. The scale of the REV is dependent on the property of interest Forpredictions of volumetric flow through a fracture network, at the scale of an REV, the flow is insensitiveto the precise structure of the local fluid pathways within the averaging volume.In field investigations, hydraulic conductivity measurements from borehole pressure tests tend toshow a high degree of variability, suggesting that the portion of the fracture system investigated from185single boreholes is below the scale of an REV [e.g. Raven 1986, Neuman 19871. Hsieh and Neuman[1985] developed a technique using cross-borehole pressure measurements to sample a larger volume ofa rock mass than is possible using single boreholes. The cross-borehole measurements yield directionalhydraulic conductivity estimates. The technique combines the measurements from a series of closelyspaced cross-borehole tests. Even though the measurements of single borehole or individual cross-borehole tests inayindicate sub-REV behavior, the entire volume tested in the series may be larger thanthe REV for flow. If the volume that is tested is larger than an REV, the directional hydraulicconductivity estimates provide a basis for constructing a porous medium model of the fracture system.Hsieh et at. [1985] and Hsieh [1987] present case studies of using cross-borehole pressure teststo define the directional hydraulic conductivity of fractured rock. Two criteria were used to examinewhether the scale of an REV for flow had been reached. At the Oracle site in Arizona, individual pressureresponses of the cross-borehole tests could be fit to the analytical solution for a point source in anisotropic porous media of infinite extent, indicating that the flow in the vicinity of the weilbores was notdominated by individual fractures. These tests gave indications that the fracture system is well connectedand not extremely heterogeneous at the scale of the packer tests. Given this behavior Hsieh et at. thendemonstrated that, although the measurements show considerable scatter, directional conductivitymeasurements from cross-borehole tests were consistent with a tensor description of the hydraulicconductivity. Thus, at the test scale of lOm, they concluded that it is reasonable to describe the hydraulicproperties of the fractured Oracle granite in terms of a uniform, anisotropic porous medium. Neuman[1987] later noted that the tensor description of the directional hydraulic conductivity may change if cross-borehole data were obtained at a larger scale because the cross-borehole data from the Oracle site covereda depth interval that is smaller than the estimated vertical correlation length of hydraulic conductivityderived from the evaluation of single-borehole tests.186Hsieh [1987] describes a similar set of pressure tests carried out at Mirror Lake in NewHampshire. Although the site is densely fractured, single-borehole packer tests revealed that most of thefractures are not transmissive. The permeable fractures had a large variation in hydraulic conductivity.In the Mirror Lake tests, pressure responses of individual cross-borehole tests could not be fit to theanalytical solution for a uniform porous media, indicating flow was predominantly limited to a smallnumber of fractures. A hydraulic conductivity tensor was not defmed. These results indicate that theMirror Lake site is extremely heterogeneous at the scale of both the single and cross-borehole tests andthat the volume tested by the cross-borehole tests was below the scale of an REV for flow.Numerical studies have been used to address the issue of whether, or at what scale, a fracturedmedium can be treated as a uniform continuum. The investigations of Long et at. [1982], Long andWitherspoon, [1985], and Hestir and Long [1990] have established an understanding of the importantcriteria that control hydraulic conductivity in two-dimensional fracture system representations. Of themany contributions of these papers, two are of particular relevance: 1) Long et at. [1982] establishedtechniques to determine whether a fracture network was above or below the scale of an REV. 2) Longet at. [1982] surmised that sparse fracture systems may not be amenable to modeling as a continuum atany scale.One of the criteria used to establish the scale of an REV in numerical studies has been thesmoothness of the hydraulic conductivity ellipse. In the study of Long et at. [1982], and more recentinvestigations by Dershowitz and Einstein [1987], Khaleel [19891, and Odling and Webman [19911, plotsof the inverse of the square root of the directional hydraulic conductivity as a function of the angle of thehydraulic head gradient (1/fk) are erratic below the scale of the flow REV for the fracture network.In each of these studies, the model domain was rotated within the fracture system as a means of changingthe hydraulic gradient, which was aligned with the domain boundaries. A consequence of the rotation ofthe model domain is that the fracture network within the model domain changes. As the size of the187domain increases, directional hydraulic conductivity estimates will converge toward a smooth curve thatplots as an ellipse. Sensitivity of the directional hydraulic conductivity to rotation of the model domainprovides a simple and direct indication that the domain is below the scale of an REV for flow.While rotation of the domain boundaries provides a technique to establish the REV size for afractured medium, it does not provide a clear indication of the properties of the medium below the scaleof the REV. The observations that for some fracture networks the REV size can be larger than thecapacity of current fracture network models and that other networks may not have an REV at any scaleindicates the need for further understanding of sub-REV scale behavior.An alternative procedure is examined in this chapter for evaluating the size of an REV that isbased on calculations using a fixed model domain. This approach highlights aspects of flow in fracturedmedia that are observed when only the direction of the hydraulic head gradient changes rather than boththe gradient and the fracture domain. Such an approach permits an evaluation of the scale-dependentproperties of a hydraulic tensor describing fluid flow through the fracture network and allows anestimation of variability in the directional aspects of the hydraulic conductivity. Perhaps more importantly,the properties of fracture networks that are revealed through this simple change in the treatment of modelboundaries may affect our view of fractured media in relation to granular porous media.Many researchers have come to accept the notion that directional hydraulic conductivity measuredbelow its REV scale is inherently erratic in fractured media. This chapter will demonstrate that: 1) at thesub-REV scale, flow through a two-dimensional fracture system can be represented by a tensor equation(i.e. the directional hydraulic conductivity tensor does not behave in an erratic manner); 2) this tensor canbe asymmetric - a result of averaging the flow through a heterogeneous medium; 3) while a radial plotof the inverse square root of the directional hydraulic conductivity with respect to the hydraulic gradientmay be a smooth ellipse, one cannot conclude from this that the fracture system necessarily behaves asa porous medium.1885.2 CoNDUCTIViTY ELLIPSE AND MOHR’s CIRCLEAt the continuum scale, the hydraulic conductivity of an anisotmpic porous medium is a smoothlyvarying function of the direction of the hydraulic head gradient. There are many possible methods ofrepresenting anisotropic hydraulic conductivity. Four methods are used here. First, the tensor can bedescribed in matrix notation. Second, a graphical representation of a hydraulic conductivity ellipse canbe defined in terms of the direction of the specific discharge vector, O. Third, the inverse hydraulicconductivity ellipse can be defined in terms of the direction of the hydraulic head gradient,Og. Fourth,the hydraulic conductivity tensor can be represented graphically as a Mohr’s circle [Kraemer, 1990]. Eachof the three graphical representations is briefly reviewed in this section.To define the hydraulic conductivity ellipse, the hydraulic conductivity in a direction parallel tothe specific discharge vector can be written [Hsieh et al., 19851K= 1 (5.1)n1zJK)where n and ii,, are directional cosines of flow along the i and j axes, respectively. The term (K’)represents the elements of the inverse of the hydraulic conductivity tensor. Summation over both i andj is assumed in Equation 5.1 and in the following equations with two or more symbols with commonindices that are multiplied together. Defining(5.2)as coordinates in conductivity space results inI 391 = p, pJK’) (5.3)Equation 5.3 defines an ellipse in two dimensions, an ellipsoid in three dimensions, where the semiaxesare aligned with the principal directions of hydraulic conductivity.To define the inverse conductivity ellipse, the hydraulic conductivity in a direction parallel to thehydraulic head gradient is defined asKg = K,ñJ4, (5.4)where ñ1 are direction cosines of the hydraulic gradient along the j and i axes, respectively. Defininga new set of coordinatesn11 (5.5)results in1 = KQIJ?51 (5.6)Using Equation 5.5, 1//Kg(Og) plots as an ellipse with respect to the direction of the hydraulic headgradient, rather than the direction of flow which is used to plot /K3(o).The hydraulic conductivity tensor of a homogeneous porous medium is symmetric, that is =Kfi. As shown later, the tensors describing flow through fracture domains below the scale on an REV maynot be symmetric. The hydraulic conductivity ellipses (Equations 5.3 and 5.6) are not sensitive indicatorsof asymmetry in the hydraulic conductivity tensor. Kraemer [19901 has presented a graphical method,that is sensitive to asymmetry, based on a Mohr’s circle and used it to characterize the relationshipbetween the direction of flow and the direction of the hydraulic head gradient. Kraemer incorporates190asymmetries in the hydraulic response of the fracture network, a feature that distinguishes his work fromthe more familiar Mohr’s circle development of Bear [1972].The coordinates of the Mohr’ s circle are the components of flow normalized per unit hydraulichead gradient and aligned parallel and orthogonal to the direction of the head gradient. This set ofcoordinates is:- -(5.7)Pg IVhI ‘ IVhIwhere q8 is the specific discharge in the direction of the hydraulic head gradient, q is the specificdischarge in the direction normal to the head gradient, and IVh is the magnitude of the head gradient.Following Kraemer, the equations describing the coordinates areK+K K—Kr K+RPg = 112 + cos(28) + 12 1 Sifl(26g)(5.8)K12 - K12+K21cos(2O) + K11 K22 sin(Og)These coordinates define a circle centered on the locationK11+K22 K12—K2 (5.9)2 ‘ 2The radius, , of the circle is+ (K12+ i)2 (5.10)191If the hydraulic conductivity tensor is symmetric, then the Mohr’s circle wifi be centered along the axisparallel to the gradient direction. Figure 5.1 depicts a Mohr’s circle plot for an asymmetric tensor. ThePgaxis is horizontal. The P axis is vertical. There are four points on the circle that are of special interest.The first two, labelled n, are the center of gradients. The two center of gradients points are located at(K,-K21) and (K11, 2). The center of gradients are useful because a line drawn through one of thesepoints and any other point on the Mohr’s circle will be at an angle with respect to the Pg axis that isequivalent to the angle formed by the direction of the hydraulic head gradient and one of the coordinateaxes of the hydraulic conductivity tensor. The reader is referred to Kraemer [19901 for a proof of theproperties of the center of gradients.The other two points of interest, labelled a and b in Figure 5.1, are the intersections of the Mohr’scircle with the Pg axis. These points correspond to the situation where the flow direction is parallel tothe hydraulic gradient. These directions are referred to as the principal directions, ea and °b in Figure5.1, and each of the directional hydraulic conductivities along these directions is called a principalhydraulic conductivity. If the hydraulic conductivity tensor is symmetric then the principal directions areorthogonal to each other and a hydraulic gradient aligned along these directions will give the maximumand minimum specific discharges. It follows that these directions also define the alignment of the majorand minor axes of the hydraulic conductivity ellipse. The principal directions are not orthogonal if thetensor is asymmetric, nor do they correspond with the major and minor axes of the directional hydraulicconductivity effipses, which are always orthogonal.5.3 FLow IN A FRACTURE NETwoRK: AN ExAMPLEFigure 5.2 depicts the flow in a fracture network that is typical of the fracture systems that wereinvestigated for this chapter. Figure 5.2a shows the fracture network. The fracture network realization192‘i)w0C)0-— Flow parallel to head gradientFigure 5.1: Hydraulic conductivity drawn as a Mohr’s circle. The K’s correspond to the elements ofthe hydraulic conductance tensor. The points label n are the center of gradients. °a and°b define the principal directions.Mohr’s circle representationK22193MetersCD-0010cD0C)HC C)-C) CDC) C)ci,C) C)D’QC) CDC)(0C-)—&CD0(00(I]0-C)01g 0C)C) CI)-CD 0C)roC0CD J)CD CI) (C) PD C) 0- - PD C)- 012015cl2ci)50Meters0 5x101 10_b 1.5x10’°I I I I I I I I I I IFlow Rate (m3/s)Minimum flow drawn = 5.8e—12 m3/sFigure 5.2:Cont. Trace map and flow plot of one fracture network used to investigate hydraulicconductance. The hydraulic head gradient is oriented along the bottomboundray. Fractures with flow rates less than 5.8x10’2m3/s are not shown.1950 5 10 15 20was generated using a Poisson structural model. The fracture system has two orthogonal fracture sets.The statistical parameters of the fractures sets are defined in Table 5.1 under the column for fracturesystem a. In Figure 5.2b the fracture network is redrawn with line widths proportional to the flow rateswithin each fracture. Only flow rates greater than 5.8x10’2m3/s are drawn, making some of the fracturesappear to be isolated from the rest of the network. One can see that the flow is not evenly distributedthrough the fracture network and that there are local channels of high flow.5.4 HYDRAULIC CONDUCTANCE TENSORTo avoid the use of the term hydraulic conductivity, which is a continuum property, the flowbehavior is described here in terms of a conductance across the domain boundaries when referring to theflow properties of a sub-REV flow domain. The conductance is a description of an averaged flowbehavior. For a single realization of a fracture system, an equation describing flow across each of thedomain boundaries can be written:q =-E _i (5.11)mmXa 8zwhere m corresponds to the boundaries as designated in Figure 5.2, and the terms 1i, are hydraulicconductances. Two calculations of flow using unit hydraulic gradients oriented 00 and 90° from the x axisprovide enough information to determine the eight components iç, of L With the gradient aligned withthe x axis, the volumetric flow across the four boundaries determines the values t,,. With the gradientaligned with the z axis, the volumetric flow across the four boundaries determines the values t,,. In196Fracture Systemsarameters Se a 1:Density 1 3 1.5(fr/rn) 2 3 1.5£ngth 1 1 2(m) 2 1 1[‘ransmissivity 1 52.2 6.5vlean( x10 m2/s) 2 6.5 6.5[‘ransrnissivity 1 0.6 0.0 1.0 1.5 0.0Standard Deviation( log10) 2 0.6 0.0 1.0 1.5 0.0Orientation 1 0 30 30 30Vlean( Degrees) 2 90 60 60 60Orientation 1 10 0 30 0 0 30StandardDeviation( Degrees) 2 10 0 30 0 0 30Table 5.1: Fracture system defmitions for conductance study. Entries in the table are only madewhen the parameter differs from system a.197expanded form, Equation 5.11 isq1 1x’q2 =-K2x (5.12)q3 E3INote that below the scale of an REV, flow across parallel boundaries on either side of a network domainwill not necessarily be equal. For example, the flow across boundary 2 of Figure 5.2b is larger than theflow across boundary 4. Also of note in Figure 5.2b is the tendency for high flow rates near theboundaries, as demonstrated by the short wide lines along boundaries 1 and 3. These high flow rates donot influence ,i because flow that enters and leaves along the same domain boundary does not contributeto the conductance calculation, only the net flow across a boundary is used in the analysis. An exampleof a very large flow that is excluded is represented by the large spot under the boundary 3 label of Figure5.2b.The eight component tensor provides a complete description of the average fluid flow across theboundaries imposed on the fracture domain, for any orientation of the hydraulic gradient. In terms ofdescribing average flows through a given volume of rock, it fully characterizes sub-REV behavior.Arguments can be made that one need not proceed any further in attempting to simplify the eightcomponent tensor. However, in this form it is not amenable to comparison with the more traditional four-component tensor used in porous media studies.It is possible to reduce the conductance terms in Equation 5.12 to a two by two conductancetensor, K’, by working with the average flows across parallel boundaries of the flow domain. To do this,a second set of specific discharges is defined198= -(q2÷4)q = !(q1f3) (5.13)II =For a unit hydraulic gradient first aligned in the positive x direction, and then in the positive z direction,the components of the conductance tensor K’ are calculated using the relationôh-- K K, (5.14)qj KKEquation 5.14 can also be written in a form that more clearly shows its relationship to the eight componenttensorL4+c E2z:+E4 8h—— 2 2 (5.15)— ic + K3 K1 + K3 h2 2Below the scale of an REV, the off-diagonal terms in Equation 5.14 may not be equal.The values t and K’ are unique to each realization of the fracture system. These are theconductance values that preserve the total flow across each of the domain boundaries; that is, they are theequivalent homogeneous values that describe the average flow across the fracture network. The tensorreplicates the different flows across parallel boundaries, while K’ replicates the average flow across parallelboundaries. For a given fracture density, K’ should approach the fracture system’s equivalent porousmedium conductivity tensor as the size of the fracture network approaches the scale of the REV.The components of , and K’, are computed using only two orientations of the hydraulic gradient.The ability of ii to replicate the flow across the domain boundaries for any orientation of the hydraulicgradient has been verified. The verification was performed by comparing the flow across each boundary,199calculated using Equation 5.11, with the observed flow rates in a corresponding discrete networksimulation. The q,,, for gradient orientations of 00 to 1800 in 15° increments were calculated, for tenfracture network realizations generated from each of five of the fracture systems listed in Table 5.1. Ineach case, the use of iç, predicted the flow rates across each of the boundaries to within the accuracyof the numerical results. The tensor 1 contains all the information that is needed to predict the flow acrossthe four boundaries and thus, the average flow through the fracture network from which it was derived,for any orientation of the hydraulic gradient.As noted earlier, directional conductance of a fracture network can be plotted either as a functionof the direction of the specific discharge vector (Ks), or the direction of the hydraulic gradient (Kg). Ator above the scale of the flow REV, the first method illustrates the hydraulic conductivity ellipse for anequivalent porous medium, the second method the inverse hydraulic conductivity ellipse. Below the scaleof a flow REV, these methods result in elliptical figures that characterize the directional conductance ofthe fracture network domain. Panel a in Figure 5.3 shows a plot of the inverse hydraulicconductance, 1//Og) determined for the network in Figure 5.2. The directional hydraulic conductanceas a function of the gradient direction has been definedK(O) IqIcos(05—Og) (5.16)Vhwhere q Icos(03-O8)is the flow in the direction of the gradient, and O iso cos__ (5.17)IIValues of the inverse hydraulic conductance as a function of can be estimated in two ways: (1) fromthe conductance tensor K’, and (2) from solutions of the discrete network model. Method (2) provides200a l/VKg(B)Cbo Simulation Resultsx Conductance Tensorv’Kje)(rn/see)Figure 5.3: Hydraulic conductance ellipses and Mohr’s circle for the fracture network of Figure 5.2.a) Hydraulic conductance ellipse. b) Inverse hydraulic conductance. Xs are plotted fromthe hydraulic conductance tensor K’ , listed below the hydraulic conductance ellipse. 1sare plotted from the results of the numerical simulations. c) Hydraulic conductance tensorwith the principal axes listed below. d) Mohr’s circle plot of the hydraulic conductance.2011 = 10000 [m/sec]2 1 = 0.0001 [m/sec]”1 d Mohr’s circle representationConductance Tensor K’(rn/sec )2.42e—08 —3.Ole—09—1.14e—08 1.15e—08Principal Directions= 37.2 Degrees02 = 101.3 DegreesMajor axes155.8 Degrees65.8 Degrees0—1xl08a check on values calculated directly from the conductance tensor. Both methods proceed by firstdetermining q in the direction of the hydraulic gradient, and then solving for Kg for various orientationsof the hydraulic gradient.The X’s on Figure 5.3 are the conductance values calculated using the tensor K’. These values havebeen plotted at 5° increments for a hydraulic gradient orientation from 00 to 360°. The D’s are values ofKg determined from the flow solution provided by the discrete network model. The components ofhydraulic conductance tensor K’ are listed in Panel c of Figure 5.3. The tensor is nonsymmetric. Theseresults indicate: (1) a smooth and regular pattern is seen in the directional dependence of the hydraulicconductance, as opposed to the erratic behavior that occurs when the model domain is rotated, (2) the non-symmetric conductance tensor correctly predicts the directional conductivity of the fracture network, and(3) the regular nature of K8 occurs even though the network is below the scale of an REV. Theasymmetry in the conductance tensor is one indication that the flow domain is not at the scale of an REV.Additional evidence for sub-REV behavior will be cited later in the examination of this fracture systemas the scale of the flow domain is changed.Panel b of Figure 5.3 shows a plot of the hydraulic conductance as a function of the direction ofthe specific discharge vector. The X’s are conductance values of K, using the tensor K’. The D’s are thevalues of K, determined from the flow solution provided by the discrete network model. The directionalconductance is plotted with respect to the direction of specific discharge, which is calculated asK(O) = (5.18)hi cos(03—O)The behavior of K,(O,) is consistent with that observed for Kg(eg).202The orientations of the major and minor axes of the conductance ellipse are listed in Panel c.These values can be determined graphically, or as done in a later sensitivity study, by calculating thedirection of the hydraulic gradient,Om, that yields maximum/minimum values of q. The direction Om isgiven by0 = tan1 IiI--ii1 (5.19)24 ))whereb =___________(5.20)IcK. ÷KKThe Mohr’s circle representation of the conductance tensor is shown on Panel d of Figure 5.3.The El symbols are the values of (p8, fl) calculated using the conductance tensor. The values lie on acircle, with no indication of an erratic behavior as a function of orientation. The offset of the center ofthe circle from the Pg axis is an indication of the asymmetry in the conductance tensor for this fracturenetwork. The principal directions of the conductance tensor can be calculated graphically from the Mohr’ scircle (Figure 5.1), or algebraically by finding eigenvectors associated the roots of the characteristicequation for the hydraulic conductance tensor1(iC-i) = o (5.21)The principal directions for this realization of the fracture network are listed in Panel c of Figure 5.3.Note that they are not orthogonal to each other, differing from orthogonality by 25.9°, and that theprincipal directions are not aligned with the major and minor axes of the directional hydraulic conductanceellipse. These features again point to sub-REV behavior.2035.5 SHORT CIRCUIT FRACTURESThe boundary flows used in calculating the directional conductances can be sensitive to thepresence of fractures that directly connect two boundaries, such as found near the left corner of Figure5.2b. With values of hydraulic head being specified at both ends of these fractures, flow through these“short circuits” is effectively independent of the connectivity of the fracture network. If these fractureshave a high transmissivity, then the flow through them will dominate the calculation of the average flowsacross the domain boundaries. As shown in a calculation that follows, these short-circuiting fractures haveno effect on the characteristic smooth variation observed in the directional conductance. They do howeverincrease the magnitude of the measured conductance, and the degree of asymmetry in the conductancetensor.As a demonstration of the affects of short circuiting fractures, a conductance tensor wasdetermined for a 2m by 2m flow domain composed of the fracture network in the upper right hand cornerof Figure 5.2. Figure 5.4 shows the hydraulic conductance tensor, ellipses and Mohr’s circle for thisblock. Of the four fractures in the small network that directly connect adjacent boundaries, two havetransmissivities greater than the expected average for their set. These fractures were modified in twoexperiments. The transmissivities are first set to the average value for the fracture network. In the secondexperiment, the transmissivity of these fractures is increased rather than reduced.In the first experiment, the change corresponds roughly to a reduction in transmissivity by a factorof two for both of the fractures. The conductance tensor, principal directions and semi-major axes of theconductance ellipses are presented in Panel a of Figure 5.5. The result is less than a ten percent decreasein the absolute magnitude (determinant) of the conductance tensor, but a change from 29° to 210 in thedeviation from orthogonality of the principal directions of the conductance tensor. The plot of the Mohr’scircle in Figure 5.5b reveals a small shift of the center toward the Pg axis.CFlow parallel to head gradient ( rn/sec)Figure 5.4: Hydraulic Conductance for the 2m by 2m region in the upper right corner of Figure 5.2.a) Hydraulic conductance ellipse. b) Inverse hydraulic conductance. Xs are plotted fromthe hydraulic conductance tensor K’ , listed below the hydraulic conductance ellipse. flsare plotted from the results of the numerical simulations. c) Hydraulic conductance tensorwith the principal axes listed below. d) Mohr’s circle plot of the hydraulic conductance.205a i/v) b1 1000 [rn/sec]’2 1 0.0001 [m/sec]”d Mohr’s circle representationConductance Tensor K’( rn/sec )4.92e—08—4.98e—09[5.91e_09 2.71e—08Principal Directions= 164.1 Degrees02 = 103.5 DegreesMajor axes1.2 Degrees91.2 Degrees0Figure 5.5:Li c1—4ci)0-0-ci)00Hydraulic conductance tensor and Mohr’s circle for the modified versions of the samefracture network as Figure 5.4. a) The hydraulic conductance tensor when two of the shortcircuiting fractures have been reduced to the mean transmissivity of the fracture sel b)Mohr’s circle plot. c) The hydraulic conductance tensor for a doubling of the aperturesof the corner cutting fractures. d) Mohr’s circle plot for the doubled aperture case.2062x108Conductance Tensor K’(rn/sec )4.73e—08 —3.51e—094.42e—09 2.53e—08Principal Directions01 168.3 Degrees02 = 99.3 DegreesMajor axes1.2 Degrees91.2 Degrees—2x 100CConductance Tensor K’( rn/sec )1.42e—07 —1.70e—08[9.43e_08 5.79e—08Principal Directions01 = 120.6 Degrees02 = 106.9 DegreesMajor axes21.2 Degrees111.2 Degrees1 HI I I I5x10d—1x107Flow parallel to head gradient ( rn/see)5x108 ii.5xi02—5x108Figure 5.5c is a listing of the conductance tensor, principal directions and semi-major axes of theconductance ellipses when the apertures of the two fractures are doubled rather than reduced. Thetransmissivity of the two fractures increased by a factor of eight. The magnitude of the conductancetensor more than doubles in this case. The deviation from orthogonality of the principal directionsincreases to 76° and the center of the Mohr’s circle moved far from the Pg axis. The modification alsoresults in a 10° shift in the orientation of the semi-major axes, an effect that was not evident when thetransmissivities are reduced. Although not presented here, the agreement of the flow calcu]ations fromthe tensor description and the numerical model remains exact.These two experiments conducted on a small domain demonstrate the sensitivity of theconductance tensor to fractures that directly connect two boundaries of the flow domain. If these fractureshave transmissivities near or below the mean transmissivity then their influence on the conductance tensormay be noticeable but is not significant. If these fractures are extremely wide then the affect may bedramatic. In the second experiment, the fracture transmissivities were roughly sixteen fold greater thanthe mean of the sets and accounted for more than half the flow though the small domain. For a fractureset with a standard deviation of log10 transmissivity of 0.6 as in fracture system a of Table 5.1, there isless than a 0.1 percent probability that any one short circuiting fracture would have such a largetransmissivity. However, for fracture system d, with a standard deviation in log10 transmissivity of 1.5,over five percent of such fractures would have transmissivities greater than sixteen times the mean.Within the flow domain the influence of these wide fractures is reduced by the narrower fractures thatconnect to them. The flow through short-circuiting fractures is not influenced by other fractures. Thelarge flow through these fractures is a result of imposing a constant hydraulic head gradient along thedomain boundaries. These short-circuiting fractures are not a representative measure of the directionalconductance at the scale of the model domain.207One method of reducing the influence of the short circuiting fractures would be to allow thehydraulic head values on the boundary to respond to the high flows. This approach has been appliedthrough different methods in Hestir and Long’s study [1990] and by Herbert et al. [1991]. These authorsimpose constant head boundary conditions on a larger extension of the fracture domain that surrounds theflow domain they investigate, allowing the head gradient along the boundaries of the flow domain to beinfluenced by heterogeneity both inside and outside the flow domain. In the case of a highly transmissivefracture crossing the corner of the flow domain, the local hydraulic head gradient within the fracture wouldmost likely be smaller than the regional gradient and the flow within the fracture would be smaller thanif the regional gradient were applied to the flow domain boundaries. The approaches of Hestir and Longand Herbert et al. result in hydraulic head variations along the boundaries that respond to heterogeneityboth interior and exterior to the model domain.While the imposition of a linear hydraulic head gradient is more restrictive than an approach thatallows the boundary heads to vary, in order to investigate the properties of a hydraulic conductance tensorwe must be able to uniquely defme the hydraulic heads along the boundaries of the domain whenspecifying the gradient direction. To reduce the influence of high transmissivity fractures that form shortcircuits, their transmissivity is reset to be equal to the mean value of the fracture set. Note that none ofthe fractures were modified in the network presented in Figure 5.2, nor were any fractures modified in thetesting of the eight component tensor. During the sensitivity study presented later, the number of fracturesthat were modified was monitored. It will be shown that a small percentage of the fractures were affected.Returning to the broader issue of flow behavior in fractured media, the efficacy of the conductancetensor to describe the variation of flow with respect to gradient direction demonstrates that flow in a two-dimensional fracture network can be examined in a way that does not lead to erratic variation indirectional conductance at a scale smaller than the REV for flow. The erratic variation documented inearlier studies occurs as a result of changes in fracture geometry in the corner regions of the flow domain208as the model domain is rotated within the larger fracture network. Given boundary conditions that areconsistent with the assumption of a uniform regional hydraulic gradient, it is possible to describe theaverage flow behavior in sub-REV domains using a tensor notation for hydraulic conductance. Unliketensors describing flow in a homogeneous medium, the four-component conductance tensor formed byaveraging the flow across the boundaries of the domain may not be symmetric for domains smaller thana flow REV.5.6 ASYMMETRY IN CONDUCTANCE MEASURESThe conductance tensor relates the average flow through a fracture network to the direction of thehydraulic gradient. There are many ways to average flow and the method chosen can affect the propertiesof the conductance tensor. In using Equation 5.12, the q,,, are determined by summing the volumetric flowfrom each fracture crossing the boundary and dividing by the length of the boundary. By doing so, anyfluid that enters and then leaves the same boundary is neglected. The direction that fluid is moving inindividual fractures is lost in this averaging process. The net flow direction through the model domaincan be estimated, but is limited by the averaging process.To guide a discussion on the affect of averaging on the estimated direction of flow, we can partition thetotal flow into 12 groups on the basis of the two boundaries the flow crosses (Figure 5.6). We know themagnitude of the flow and the identity of the two boundaries of each group. The two faces are used todefine a flow direction. The 12 groups define eight unique flow directions as shown in Figure 5.6. Asan example, flow entering boundary 2 and leaving across boundary 3 would be assigned a flow directionof 450 The net flow direction is the vector sum of these eight directions with each vector weighted bythe magnitude of flow between the appropriate faces. The eight directions are basis vectors for the flowmeasurement. If all of the flow entered boundary 2 and left across boundary 3 then the estimated flowdirection can only be 45°. It is shown below that if the net flow direction is not 450 then a tensorrepresentation of the measured flow direction is asymmetric. It is also demonstrated that a four component209Boundary 3Boundary 1Figure 5.6: Diagram of the flow direction defined by fluid crossing two domain boundaries. The flowdirection is given by the arrows pointing from the flow entrance boundary to the flow exitboundary.C1iJ>as-DCD0>asVC0210tensor representing flow in a “homogeneous” fracture network will be symmetric.5.6.1 AsYMMETRY IN SIMPLE FRACTURE NETWORKSConsider a square domain with one fracture connecting boundary 2 to boundary 3. The boundaryconditions on the domain are the same as in the numerical simulations, a constant hydraulic head gradientalong the boundaries of the domain. Applying Equation 3.17, the flow along the fracture isQ = _Tj,LCOS(og_O? (5.22)where O is the orientation of the fracture. The average boundary flows are = - for m=2,3 whereW is the length of a boundary. Substituting for q in Equation 5.11 results in-l’cl,z-dh -cM————cos(O) = —K —cos(O ) —K —sin(6) (5.23)W dOg ‘ dOg dOg gUsing the trigonometric relationcos(A —B) = cos(A)cos(B) — sin(A)sin(B) (5.24)leads to- TK = —Lcos(Onix j(5.25)- TK = —tsin(ftnixFrom Equations 5.14 and 5.15, the four component tensor is211T T._.L cos(O) _L sin(O)K’2W 2W- T tcos(O) _L_ sin(O)2W 2WFor a single fracture connecting boundaries 2 and 3, K’ will be symmetric if and only if O, is 450• Nowlet the fracture connect boundaries 1 and 3 instead of 2 and 3. The four component tensor becomesNow consider a network composed of n parallel fractures. Let the aperture of each fracture varyTsuch that the transmissivity of each fracture is _L• First consider the case of n=2 where one fracturenconnects boundaries 2 and 3 and the other connects boundaries 1 and 3. For this network the fourcomponent tensor isIf 450< O< 90° thenCOS(O)of the fractures connect boundary 2 to boundary 3,sin(O? +cos(O)(5.26)K’ = (5.27)K’ will be symmetric if and only if O is 90°. From these two cases, it is clear that K’ will be symmetricif and only if the fracture is oriented parallel to the direcfion of flow as given by the basis vector for thetwo faces connected by the fracture.K’ will be symmetric if Of is tan’(3). Next let the fractures be evenly spaced and let n-.oo.(5.28)212n (sin(O) -cos(O))of the fractures connect boundary 1 to boundary 3, andsin(O) + cos(O)n (cos(O?of the fractures connect boundary 1 to boundary 4. The factor sin(ef)-i-Cos(Of) accountssin(O? +cos(O?for the change in fracture spacing required by the constraint of a fixed number of fractures, n, within thedomain.For this network,T 2(OK _L________2Wsin(O)÷cos(O?T cos(e,)sin(eK =__L2W sin(O? + cos(O?(5.29)KT( 2(o? + (sin(O? —cos(O))cos(O)2 Wt (sin(O) + cos(O) sin(O? + cos(6)K = cs(0?(0?+ sin(O)(sin(Of) - cos(e))2Wsin(O)+ COS(O? cos(O)Combining terms results in_________T cos(O?sin(O?2W sinCe? +cos(9? 2W () ÷cos(O? (5.30)Tf cos(6)sin(O) Tf sin2(O?2W sin(O) +cos(O) 2W sin(Of) +cos(O)K’ is symmetric for all O.These networks indicate that a network of only a few fractures leads to an asymmetric conductancetensor unless the fracture orientation coincides with the sum of the basis vectors defined by the boundaryconnections. Two networks composed of the same fractures, but positioned differently within a domain,can result in different conductance descriptions. For example a domain that contains ten equally spaced213fractures with equal traflSmissivities and all oriented at 600 could have principal directions of conductanceoriented 56° and 150° or 62° and 150° depending on the positioning of the fractures within the netwo&.Note that for the ten fractures, the deviation of the principal directions from orthogonality is less than 5°.If the network is composed of many parallel fractures of equal transmissivity, then the conductance tensorwould be very close to being symmetric.These simple examples lead to a supposition that the asymmetry in K’ is a result of the flowaveraging process. The last example indicates that the K’ tensor of a homogeneous medium will besymmetric. Asymmetry in K’ is a product of an interaction between the way the direction of flow throughthe domain is calculated and inhomogeneities within the flow domain. The asymmetry is affected byheterogeneity within the fracture network and is, therefore, an indicator of sub-REV behavior.5.6.2 ALTERNATIVE FLOW MEASURESIf the asymmetry in the conductance tensor is a result of averaging flow across the flowboundaries, can a flow averaging procedure be found that does not introduce asymmetry into theconductance measurements? In this section three alternative measurements of flow through the fracturenetwork domain are investigated. These are:1) Summation of the vector components of the flow across each boundary such that= (Qi (5.31)where j represents locations of flow across the domain boundaries and i is an ordinal direction.2) Summation of the nonrial component of flow across each boundary.3) A magnitude of flow defined by the net volumetric inflow, and a flow direction defined as the directionfrom the centroid of the flow weighted inlet locations to the centroid of the flow weighted outlet locations.214A sequence of four networks are used to evaluate each of these four average flow measures. Thefirst network contains only one fracture oriented 600 to boundary 1 (Figure 5.7, Panel a). The secondnetwork contains two fractures that intersect the domain boundary at the same locations as the fracturein the first network (Panel b). The apertures of these fractures are larger than the aperture of the singlefracture such that the magnitude of flow within the fractures would be equivalent to the single fractureif the boundary conditions of the two networks were equivalent. The third network contains threefractures (Panel c). Again the intersections with the boundaries and fracture apertures are such that forequivalent boundary conditions the magnitude of flow through this network would be equivalent to theflow through the first two networks. The two fractures intersecting the domain boundaries are parallelin this network resulting in flow in the two fractures being equal but in opposite directions. The lastnetwork has the same set of three fractures as the third network plus an additional fracture that connectsthe network to boundary 4 (Panel d).As we have seen before, the first network results in a asymmetric tensor when flow is calculatedusing the original flow measure. The flow in this case is always calculated to be at 45° to boundary 1.The other three measures provide an estimated flow direction of 60° to boundary 1 and yield a symmetrichydraulic conductivity tensor. When applied to the second network, measures 1 and 2 yield identicalresults to the original measure, that is flow at 45°. Measure 3 yields an average flow direction of ()° anda symmetric tensor.When applied to the third network, measure 1 results in zero flow and no conductance, a resultthat is not acceptable. Measure 2 provides an estimated net flow direction of 30° rather than 60°. Appliedto the third network, measure 3 results in a 60° flow direction, consistent with the other networks.Measure 3, which works wonderfully for networks that have only one entrance and one exit, fails toproduce consistent results for more complicated situations, as shown below for the fourth network. Figure5.8 presents the conductance tensor and graphical plots from the results of measure 3 applied to the fourth215a bC/2ci)J]ci)ci)Figure 5.7: Fracture networks used to investigate alternative flow measures. a) One 600 fracture. b)Two orthogonal fractures. c) Three fractures, -30°, 60°, and 150°. d) Four fractures withthree intersections with the boundary.216I I I I I I I I I I I I I1111111! I 1111110 5 10 15 200 5201510502015105010 15 20C2015105020151050d0 5 10 15 20 0 5 10meters meters15 20a 1//J)oSirnulatjon Resultsx Conductance TensoroJFigure 5.8: Conductance ellipses and Mohr’s circle using alternative flow measurement 3 to measurethe flows though network 4 of Figure 5.7. The figure demonstrates the inconsistent natureof this flow measurement.1 = 10000 [rn/secjd1 = le—05 [m/sec]tMohrs circle representationConductance Tensor K’ ( rn/sec2.51e—09 6.13e—106.54e—10 1.06e—09Principal Directionse1= 158.8 Degreese2 = 70.1 DegreesMajor axes20.6 Degrees110.6 Degrees000a0.a -0.0F’low parallel to head gradient ( rn/sec217network of these examples. The graphical plots reveal that the conductance tensor developed fromgradient directions of 0° and 90° does not adequately represent the flow for other gradient directions.Figure 5.8 reveals that the flow as determined by measure 3 can no longer be described using a tensorrepresentation.Although each of the three alternate flow measures can be considered correct lbr the case of asingle fracture, they are all inferior to the measure used in this research. Measures 1 and 3 are inadequatefor our purposes because of the problems identified above. Measure 2 is sensitive to the orientation ofthe fractures that intersect the domain boundaries. The measurement changes for each of the threenetworks, indicating a sensitivity to the angle of the fractures intersecting the boundaries. A small changein any of these networks, such as a short wide fracture intersecting the network near the domain boundary,could change the calculated flow direction. This sensitivity to the local conditions on the boundary couldresult in a gross misrepresentation of the bulk of the domain, a situation that is not acceptable for thisresearch. The flow averaging procedure used in the conductance tensor calculations is superior to all ofthe alternates considered in this section even though it has been demonstrated to be a contributing factorin the asymmetry of the conductance tensors.5.7 PARAMETER STUDIES5.7.1 INTRODUCTIONThis section describes the results of simulations carried Out to investigate the sensitivity of theconductance tensor and the principal axes to changes in the fracture system statistics and to the scale ofmeasurement. Using these observations, can the tensors . or K’ be used as a guide in establishing theminimum scale of a REV? A number of criteria can be examined to test whether a fracture network ata given scale of observation can be approximated by a uniform continuum. Long et al. [1982] establishedthe use of the deviation of the directional hydraulic conductivity from a best-fitting ellipse, where218directional conductivity is determined by rotating the model domain within the fracture system. Asdemonstrated earlier, plots of Kg(eg) and K3(O) are smooth and regular below the scale of a REV whenthe hydraulic head gradient is rotated about a fixed fracture network domain. The behavior of the eight-component tensor . (Equation 5.12) could be examined to see if there are equal flows across parallelboundaries, as expected if the domain encompasses an REV for the fracture system. Instead, the flowbehavior is analyzed in terms of K” because of the large body of work that has focused on the fourcomponent tensor (Equation 5.14). The following three criteria are used to assess whether or not the scaleof observation is above the scale of the REV. The first focuses on the conductance tensor of individualnetworks, the latter two emphasize how the tensor varies with changes in the scale of measurement.1) The tensor K’ should be symmetric, and the principal axes should be orthogonal. The deviationfrom orthogonality of the principal axes is used as the measure of asymmetry. Two othermeasures of asymmetry have been investigated. In Appendix B, the results of these measures arepresented and compared to the results using the orthogonality of principal axes as a measure ofasymmetry.2) If the scale of the fracture network domain is above that of a flow REV, then the tensor K’ shouldnot change appreciably with an increase in the scale of measurement.3) In that K’ is a description of the average hydraulic conductivity of the fracture network, then atthe scale of an REV, K’ should show only small variations between different realizations of thefracture system. This independence refers to the magnitude of the conductance, the orientationof the principal directions of the conductance tensor, and the orientation of the major and minoraxes of the hydraulic conductance ellipse. The variation in the orientation of the major and minoraxes of the directional hydraulic conductance ellipse will be presented, as it is intuitively easierto visualize than the orientation of the principal directions. If K’ is not reasonably independentof specific realizations, this behavior suggests that the local scale inhomogeneities remain219important in determining the average hydraulic properties of the network at the scale of theanalysis.The fracture systems listed in Table 5.1 have been used to study the sensitivity of the hydraulicconductance tensor to fracture system parameters. The fracture systems were created to study theinfluence of six parameters on the nature of the fracture system conductance as listed in Table 5.2. Theinfluence of each of these parameters on the conductance is discussed below under separate sub-headings.Parameter racture SystemsEransmissivity a b c d and j kracture density a evlean fracture length a fVlean fracture orientation a IiOrientation variance a j n and g h iIsotropy k mTable 5.2: Fracture systems used to investigate parameter sensitivity.For each of the fracture systems listed in Table 5.1, conductance tensors are calculated for 150realizations of a fracture system for square domains 5m, lOm, 20m, and 40m on a side. The number ofrealizations that could be investigated was constrained by computer resource time limitations. The tensordetermination for a 40m x 40m domain requires from 40 minutes, for fracture system a, to eight hours,for fracture system d, on an SGI R4000 computer. The progression of domain sizes provides an indicationof the behavior of the conductance as the domain approaches the size of an REV. For the density of openfractures used in this study, the 5m by 5m domains are on the order of the size one might expect toinvestigate using a single borehole pressure test. These small domains show a very large variation in the220hydraulic conductance measures. An appreciation for the domain size, relative to fracture density, can begained by referring back to the network shown in Figure 5.2.As mentioned in the Section 5.5, short circuiting fractures that directly connect two boundariesof a flow domain may inappropriately dominate the average flow through the domain. The calculated flowthrough these fractures is not consistent with the notion that calculated flow is related to the size of thedomain because the fixed head boundary conditions for these fractures are imposed over a short distance.It is not reasonable to assume that the flow rates through these fractures could be maintained through thefractures immediately surrounding the domain. The effect has been reduced by adjusting the transmissivityof these fractures; effecting about 0.5% of the fractures generated in the smallest domains(5m by 5m) and0.01% of the fractures in the largest domains(40m by 40m). The adjustment of most of these fractureshas a small effect because most of these fractures have transmissivities that are less than one standarddeviation from the average transmissivity. The occasional short-circuiting fracture has a traiismissivitythree or four standard deviations above the average and can have a tremendous impact on the average flowthrough the domain.Despite the reduction of the transmissivities of the short circuiting fractures, anomalously largevalues of conductance still occur in a small number of the fracture network realizations. These large valuesmay be attributed to the interconnection of two or more large transmissivity fractures that, combined, alsoprovide a direct connection between adjacent domain boundaries near a corner. As an additional filter ofthese small scale features, fracture network realizations that have log values of the magnitude of theconductance tensor that are more than four standard deviations above the mean log conductance valueshave been removed from the data base. As with discarding the short circuiting fractures, these realizationswere not used because the calculated flow rates are highly unlikely to be consistent with the possible flowsin the surrounding fracture network. Given the large conductances of these infrequent events, to haveretained these values would have biased the averages of the fracture systems towards these small scale221phenomena and masked the essential character of the scale dependence of the fracture systems. At mostthree realizations have been discarded for any domain because of anomalously large conductances. Inaddition some realizations of fracture system e were discarded because, for this sparse system, no fracturesconnected the boundaries of the domain resulting in non-conducting networks. Because our conductancemeasure is in log conductance a conductance of zero could not be used in the averages.5.7.2 TR1sIIssIvITY VARIANCEFracture systems b, a, c and d have standard deviations in log10 transmissivity of individualfractures of 0., 0.6, 1.0, 1.5 respectively. Figure 5.9a presents a plot of the average maximum directionalhydraulic conductance for these simulations. The average conductance decreases as the size of the flowdomain increases, a clear indication that for these fracture systems even the largest domains are below theREV scale of flow.The decrease in conductance observed in Figure 5.9a may be related to channeling similar to thatevident in Figure 5.2b. If the flow is highly charinelized and dominated by a few pathways, then the flowthrough these pathways would be limited by the narrowest fracture in the pathway. Dominate pathwaysthat connect opposite boundaries require more fractures as the domain size increases. Pathways requiringmore fractures are less likely to contain only the largest transmissivity fractures of the transmissivitydistribution. On statistical grounds, the conductance should decrease with increasing domain size.Figure 5.9a also shows that the average measured conductance decreases with increasingtransmissivity variation. It is important to note that the average transmissivity of each set is constant inthese studies, not the average of the log transmissivity. As the transmissivity variance increases fewer ofthe fractures have transmissivities that are larger than the average. Following the logic of the previousparagraph, the decrease in log conductance with respect to transmissivity variation shown in Figure 5.9ais consistent with the notion of dominant pathways and channeling of flow.222Figure 5.9: Hydraulic conductance sensitivity to transmissivity variation, a) log10 mean of themaximum directional conductance. b) Log10 standard deviation of the maximumdirectional conductance. c) Standard deviation of the direction of the principal axes ofthe conductance tensor. d) Orthogonality deviation of the principal directions ofconductance.I I I I I I I I I Ia7C(1200)30. 20III 111111 111111---::-b-ad-°—I I I I I I I Io io 20 30 40I I I I I I I I I I I I IdCS. d-- -e- -.a-. - - - - -a-.b..II III liii T IIb-.. —‘:-- --e -I I I I i i I i !iiII,II2100.80 10 20 30 40I I I I I I I I I I I I I I I I I20--‘c-_S__S15 a.5---5------C.10 - “a...., -‘b-.5- --5------ -o - --. -0 -- I III I 11111111111 Iii00 10 20 30 40Domain size ( m )10 20 30 40Domain size ( m )223Figure 5.9b presents the standard deviation in the distribution of the log10 of the maximumdirectional hydraulic conductance (SDK). As one would expect, the SDK decreases with increasingdomain size and with smaller transmissivity variation of individual fractures.Fracture systems with a large transmissivity variation show a large variation in the direction ofthe major axes of the conductance ellipse. The variation decreases with increasing domain size (Figure5.9c). Even in the 40m by 40m realizations, systems c and d have large variations in the direction ofmaximum conductance. Clearly, the scale of a REV increases with increasing variability in the individualfracture transmissivities. It is also evident that these domains are smaller than a flow REV.Figure 5.9d presents an asymmetry measure in terms of the deviation from orthogonality of theprincipal directions of the conductance tensor. Figure 5.9d shows that the asymmetry decreases withdomain size and that the asymmetry is dependent on the variance of fracture transmissivity. If one setsan arbitrary threshold of five degrees to be significant, then only system b in the largest realization doesnot have significant asymmetry. To put a five degree threshold in perspective, consider the channelizedflow in the fracture network depicted in Figure 5.2b, a single realization of fracture system a. Theorthogonality deviation in Figure 5.3, from the network of Figure 5.2, is sixteen degrees.5.7.3 FRc’ruRE DENSITYThe scan-line density was halved in fracture system e, from 3 fractures per meter to 1.5 fracturesper meter, for each fracture set. Fracture system e is close to the percolation threshold and is quite sparse.The investigation of fracture system e demonstrates a strong sensitivity of the conductance to fracturedensity in a system near the threshold of percolation. Comparing system e with a, in Figure 5. lOa, revealsthat not only does e have a lower mean conductance, as one would expect, but that the conductancedecreases much more rapidly with domain size than it does for the other fracture systems. This isconsistent with the notion of channel-dominated flow and the consequent sensitivity to pathways composed224Figure 5.10: Hydraulic conductance sensitivity to fracture density, e, length, f, and isotropy, k and m,variation, a) log10 mean of the maximum directional conductance. b) Log10 standarddeviation of the maximum directional conductance. c) Standard deviation of the directionof the principal axes of the conductance tensor. d) Orthogonality deviation of theprincipal directions of conductance.225‘° a7‘7:8.50cin- -- -- - - - -0 10 20 30 400.800c 10‘4-,‘ I I I I I I I-fl’.- ‘w-- - - -0 10 20 30 40111111111 III liii-- -- ‘S1S -. - -L .‘: -0U2a)a)a)0.—.4-,C)a)I4I0C50403020100I I I I I I I I I I I I I I]fil—g5 ‘.gH0 10 20 30 40Domain size ( m )10 20 30Domain size ( m40)of larger than average transmissivity. As the domain size increases, fewer clusters of highly transmissivefractures are large enough to connect the domain boundaries. Individual realizations of system e containcases that do not have any fractures that connect the domain boundaries; these cases are not included inthe conductance calculations.The orientation of the fracture sets is the same in fracture system e and the reference system a.However, the standard deviation in the orientation of the major axes of the conductance ellipse indicatesthat for fracture system e the distribution of the major axes varies so widely as to be reasonablyapproximated by a uniform distribution. The difference in the variation of the orientation of major axesbetween systems e and a is an indication that the lack of connectivity due to the sparseness of system eis more important to the direction of the maximum conductance than the orientation variability of thefracture sets. System e has a relatively large degree of asymmetry at all the scales investigated (Figure5. lOd), which is consistent with the supposition that the sparseness of system e is an important feature andthat asymmetry is linked to a few dominant flow pathways.5.7.4 FRACTURE LENGTHThe mean fracture length of the first fracture set of fracture system f is doubled from that insystem a to 2m. The mean length of the set 2 fractures remains at lm. Since the scan-line density ofsystem f is the same as the scan-line density of system a, the density of fractures expressed in terms ofnumbers of fracture centers per unit area is smaller in fracture systemf than in the reference system. Theconductance statistics for fracture system fare presented in Figure 5.10. The longer fractures in systemf, in comparison to the reference system a, mean that fewer fractures are required to form a dominantchannel. The result is a larger mean maximum conductance that decreases less rapidly. The SDK isroughly equivalent. The major axes of the conductance ellipses are more closely aligned with the x axisthan those of system a. For the 40m by 40m domain, system fhas a standard deviation in the major axesdirection of 5 degrees. The longer fractures in set 1 also result in less asymmetry in the conductance226tensor. In summary of these characteristics, while system f has longer fractures and a lower density intenns of areal density, this system has a higher connectivity and a smaller REV than the reference system.5.7.5 ORIENTATIONThe effects of the orientation of the fracture sets on hydraulic conductance have been investigatedusing two groupings of the fracture systems. Both groups, j, a and n and g, h and i, have systems withstandard deviations about the mean orientation of 0°, 10°, and 30° respectively. The mean orientationsof the group g, h and i are 30° for set 1 and 60° for set 2. The group j, a and n are orthogonal fracturesystems. Figure 5.11 contains the statistical results from these systems. The non-orthogonal systems g,h, and i have a lower connectivity, resulting in smaller conductances and slightly larger SDKs. Fracturesystem i has such a large variation in fracture orientation that the lack of orthogonality in the meanorientation yields a conductance behavior that similar to the sub-orthogonal reference simulation. Whilethe mean conductance of the non-orthogonal systems are sensitive to the variability in orientation, theorthogonal systems show limited sensitivity to this parameter. As might be expected, the larger variationin fracture orientation tends to result in larger variations in the direction of major axes of the conductanceellipses. System j, with no orientation variability, is an exception. System j exhibits a slightly largerdirectional variability in the major axes of the conductance ellipse than the reference system. Figure 5.1 iddoes not reveal a clear link between asymmetry of the conductance tensor and the variance in fractureorientation. The asymmetry is larger for the sub-orthogonal fracture systems.5.7.6 IsoTRoPYFracture system m is composed of two orthogonal sets with identical statistical parameters, leadingto an isotropic hydraulic conductance. In this case, the conductance ellipse is nearly circular. Theorientation of the major axis of the conductance ellipse and the deviation in the principal directions of theconductance tensor from orthogonality are sensitive to small variations in the flow through the domain.227Figure 5.11: Hydraulic conductance sensitivity to mean fracture orientation, and orientation variation.a) log10 mean of the maximum directional conductance. b) Log10 standard deviation ofthe maximum directional conductance. c) Standard deviation of the direction of theprincipal axes of the conductance tensor. d) Orthogonality deviation of the principaldirections of conductance.- 1 1 I I I I I J Ili1]ri— .- -e- -- f. --CS, -ino io 20 30 40—b,ThI I I I I 1c liii0II ILong Fr. -r-f087.5 OrthogoaL0.6-8m otrPi 0.48.5--- e- Sparse :e—0 -‘—•• —9ITIIIIIII,IIIIIII0 10 20 30 405C I’ll lilIII 11111111-I°: : 20f40-e:.215h ;::-- 1 10ii,Iii- 00 HI,, I0I I I I I I I I I I I Ia.10 20 30 40Domain size ( m )I I I i i I i I i i I0 10 20 30 40Domain size ( m )228The large variations shown in Figure 5. lOc and d for fracture system m are not indicative of sub-REVbehavior. The small variation in the mean log conductance with domain size, the magnitude of thestandard deviation in log k, and its small variation with domain size, all indicate that the flow REV forsystem m is the smallest of all those examined.5.7.7 MEASuREMENT SCALEFigures 5.9, 5.10 and 5.11 show very consistent dependencies on scale. The mean conductancesand the SDKs all decrease with an increasing scale of measurement. The variation in the direction of themajor axes of the conductance ellipse and the degree of asymmetry in the conductance tensor also decreasewith domain size. Of all the fracture systems investigated, system m seems to have the smallest REVscale for flow. This scale is near or slightly larger than 40m. Hestir and Long [1990] have proposed anempirical formula to estimate the scale of a REV for two-dimensional Poisson networks with zerotransmissivity variance. System m is the only fracture system in this study that satisfies the assumptionsadopted in Hestir and Long’s analysis. Applying their formula to fracture system m predicts the scale ofthe REV to be close to 50m, in apparent agreement with the conclusions drawn from Figure 5.10.5.8 SENsmvrrY TO FRACTURE NETWORK STRUCTUREThe sensitivity investigation in Section 5.7 focused on the effects of changes in the characteristicsof the fracture sets on the hydraulic conductance tensor of a fracture network. The fracture networks weregenerated using a Poisson structural model. In this section, the effects of using different structural modelsto generate the fracture networks are investigated. The following two questions are addressed. 1) Do thesensitivities to parameter changes differ with the structural model? 2) How does the scale of a REVdepend on the structural model?Two structural models are examined; the war zone model and the Levy-Lee model. These twostructural models were described in Sections 3.3.3 and 3.3.4, respectively. For both structural models, the229fundamental changes in the nature of the fracture systems can be described as an increased clustering offractures and in a broadening of the length scale of the fractures. The length scales of fractures of thepoisson model are determined by the parameters of the fracture sets. In the war zone model, fracturelength scales are determined by the size of the war zones in addition to the fracture set parameters. TheLevy-Lee model has a broader range in the scale of the fracture lengths than either of the other twomodels.5.8.1 WAR ZONE FRACrURE SYSTEMSWar zones can be though of as regions where pre-existing fractures have changed the local stressconditions promoting a higher frequency of fracturing within the zones. A war zone network of fracturesystem a is shown in Figure 5.12a. The distribution of fractures is more heterogeneous than in thecorresponding Poisson model (Compare with Figure 5.2a). To apply the war zone structural model tofracture system a, it is assumed that fracture set 1 represents the pre-existing fractures and fracture set 2represents later fractures. Fractures of set 2 are more dense than a corresponding Poisson network withinthe war zones, but outside of the war zones they are less dense. The lower density of fractures outsidethe war zones follows from an assumption that the scan-line density for fracture set 2 includes both thefractures created within the war zones and those outside the war zones. The specified scan-line densityof set 2 was halved in order to keep the overall scan-line density of fracture set 2 equal to the scan-linedensity in the Poisson model.The assumption that the scan-line density includes the war zones is arbitrary. It is also possibleto assume that the measured scan-line density does not include war zone fractures, in which case thedecrease in fractures outside the war zones would not occur. The issue is mentioned because many of thedifferences in the results of the war zone sensitivity from those of the Poisson structural model can beattributed to the lower fracture density outside of the war zones.230C CD C-)CD 0 C) —U)—C) CD B-p C 0 CD C) C)T1 C CDMeters00100 0101N)0CD CD0 01 N) 012Cl)a)a)Figure 5.12: War zone fracture network b) Flow rates within fractures.201510500 5 10 15Meters0 2x101 4x1O’1 6x101 8x101 10_to20I I I I I I I I I I I I I I I I I I IFlow Rate (m3/s)Minimum flow drawn 3.8e—12 m3/s232A diagram of flow through the war zone network is shown in Figure 5.12b for a hydraulic headgradient oriented parallel to boundary 1. Flow appears to be concentrated in fewer channels in Figure5.12b, compared to the flow plot of the Poisson network shown in Figure 5.2b. War zones are often apart of these channels. In war zones, such as the one near the location (18,9), the flow tends to disperseand then reconcentrate as the flow leaves the zone.5.8.1.1 SENSITIVITY TO TRANSMISSIVITY VARIATIONThe changes in hydraulic conductance with the scale of measurement, that are seen from the warzone model (Figure 5.13), are similar to those observed from the Poisson model. These results, whichare plotted for fracture systems with different standard deviations in the log of the fracture transmissivity,should be compared with the corresponding calculations for the Poisson model shown in Figure 5.9. Thesensitivity of the hydraulic conductance to increases in fracture transmissivity variance observed in Figure5.13 are also similar to the sensitivity of the Poisson based fracture systems. However, the magnitudesof the hydraulic conductance are lower in the war zone based networks than in the corresponding Poisonnetworks and the variability between realizations is larger. These differences are much like the changethat occurred when the fracture density was decreased in the Poisson fracture system.The results of halving the fracture density of the Poisson fracture system a to form fracture systeme are presented in Figure 5.10. The differences are more pronounced in Figure 5.10 than the differencebetween the war zone generated fracture system a and the Poisson generated fracture system a, consistentwith a larger decrease in fracture density from fracture system a to fracture system e compared to thefracture density decrease outside the war zones. The changes between fracture system a and fracturesystem e can be used as a reference in understanding the effects of the increased fracture clustering in thewar zone model.233wC)-C)•ti8.0U00.—201o‘s:--.. O-I 111,1111111111110 10 20 30 40Figure 5.13: Hydraulic conductance sensitivity to transmissivity variation of war zone generatedfracture networks. a) log10 mean of the maximum directional conductance. b) Log10standard deviation of the maximum directional conductance. c) Standard deviation of thedirection of the principal axes of the conductance tensor. d) Orthogonality deviation ofthe principal directions of conductance.234a7I I I j I I I I I IIIIIIIIIIIIIIIIIIIIII.•IbII II! IllIlIlili I,C;o.s0.610j50 10 20 30 40:1 I I I I I I I I I I I I:cL-:‘ci,. - -—C.,. d..—0,•:e..C‘b. a--,.- ___-.bI I I I I I I I i i i I—.I I I I I I I I I I I Ib. a..— ,.._-----.----III II 1111111 iii0 10 20 30 40Domain size ( m )0 10 20 30 40Domain size ( m )The variation in orientation of the major axes of the hydraulic conductance tensor for the war zonefractures systems shows a slight increase over that of the corresponding Poisson fracture systems, but notthe large increase in the variation in orientation of fracture system e. Evidently, the change in fracturedensity outside the war zones is not enough to make the connectivity of the fractures a stronger influencein the orientation of the conductance than the higher transmissivity in the first fracture set, as is the casein fracture system e.In comparing the war zone and Poisson models, the deviation from orthogonality of the principaldirections of conductance is somewhat higher for fracture systems a and b but virtually the same for thefracture systems with greater aperture variability. This behavior may indicate a transition from a casewhere connectivity is the dominant form of heterogeneity in fracture systems a and b, to one wheretransmissivity variability dominantly influences the conductance.In summary, the basic response of the hydraulic conductance of a fracture network to changes intransmissivity variations is not effected by the introduction of the war zone model. The first fracture set,which is the same in the war zone networks and the Poisson networks, controls the orientation of theconductance tensor but the network scale connectivity provided by the second fracture set has beenreduced by concentrating the fractures in the war zones. This reduction in connectivity lowers the averageeffective conductance of the network and causes a greater variability in the hydraulic conductance betweendifferent fracture networks. The war zone fracture systems have a larger scale of an REV than thecorresponding Poisson fracture systems.5.8.1.2 NON-ORTHOGONAL FRACTURE SYSTEMSThe averages calculated from the hydraulic conductance tensors of the war zone generated fracturenetworks with non-orthogonal fracture sets are presented in Figure 5.14. Fracture systems g, h, and i havefracture sets with mean orientations that differ by 300 rather than 90° as in fracture system a. The235C-:50000C2010Figure 5.14: Hydraulic conductance sensitivity to mean fracture orientation, and orientation variationof war zone fracture networks. a) log10 mean of the maximum directional conductance.b) Log10 standard deviation of the maximum directional conductance. c) Standarddeviation of the direction of the principal axes of the conductance tensor. d)Orthogonality deviation of the principal directions of conductance.23614a7C)-C)0C.)to0—‘bIIIIIIIIIIIIIIIIIIlII IIIIIIlIIIIr,,IiiiI,,0 10 20 30 40 0 10 20 30 40‘‘.i-_ 10 --‘.\a...’-., - .‘ -. 1--____. ----,--- —------,“- - 5---::--- goIIIIIIIIlIIIIII1 Q IIIIIIIIIIIIIIIIIfII00 10 20 30 40 0 10 20 30 40Domain size ( m ) Domain size ( m )standard deviation of orientation distribution for fracture systems g, h, and i is 00, 100, and 30°respectively. More war zones and war zone fractures are created in models with less variation in thefracture orientation. Fracture system g has roughly 20% more war zones than system h and fracturesystem i has about 50% less. The density of fractures outside the war zones is the same for each fracturesystem. Thus, the scan-line density of fractures of set 2 is approximately 10% higher in fracture systemsg and about 25% less in fracture system i compared to the corresponding Poisson fracture systems.The sensitivity of the hydraulic conductance to changes in the variability in the orientation of thefractures of the non-orthogonal fracture systems is the same in the war zone fracture systems as in thePoisson model based fracture systems. Indeed, unlike the orthogonal fracture systems the non-orthogonalwar zone fracture systems have only a slight sensitivity to the change in structural models. The mostnoticeable difference is a lesser sensitivity of the average hydraulic conductance to domain size in the warzone fracture systems and an increase in the variability in the magnitude of the hydraulic conductance thatdecreases more rapidly with domain size. The average magnitude of the conductance tensor is slightlylarger at the 40m domain size in the war zone systems but somewhat smaller in the smallest domains.The similarity in the changes of these fracture systems to the war zone structural model indicatesthat the differences in the frequency of the war zones is relatively unimportant. The magnitude of thehydraulic conductance in fracture system h is much less sensitive to the clustering in the war zone modelthan is the orthogonal networks of fracture system a. The variability in the magnitude of the conductance,however, increases about the same degree as the reference system. Fracture system h has a slight increasein the variability in orientation of the major axes but little change in the deviation between principalconductance directions. The shift in the relative magnitude of the mean conductance can’t be explainedby the decrease in fracture density outside the war zones. It may be due to increased connectivity withinthe war zones themselves in fracture system h.237Differences in the overall density of fractures is not a dominate influence of the conductance offracture systems g, h and i. Each of these fracture systems have a slight decrease in the magnitude of thehydraulic conductance. The variability of the conductance increases with all three fracture systems, withfracture systems of less orientation variability having a larger variance. It may be that the variation isbeing effected by the relative high conductivity in the war zone regions but the effect may also be dueto the decrease in fracture density outside of the war zones.The standard deviation in the orientation of the principal directions of the hydraulic conductancetensor shows much less variation between these non-orthogonal fracture networks generated using the warzone structural model, when compared to networks generated with the Poisson model. The fracturesystems with little orientation variation are more sensitive to the decrease in fracture density outside thewar zones. The deviation from orthogonality of the principal directions of the conductance tensor is notappreciably effected by the structural model change for these fracture systems.In summary, the major influence of the greater clustering of the fractures in the war zone modelseems to be from the reduction in the density of fracturing outside of the war zones required to maintaina constant overall fracture density. The characteristics of fracture systems g, h, and i indicate that thevariability in local conductivity of the war zones may increase the variability in hydraulic response of anetwork at all scales. The scale at which the war zone fractures systems may be regarded as a continuumis larger than the scale of the REV when a Poisson structural model is used to generate the networks.5.8.2 LEVY-LEE FRACTURE SYSTEMSThe predominant feature of fracture networks generated using the Levy-Lee structural model isthat a few fractures completely span the fracture network domain no matter what is the scale of thedomain. One realization of the Levy-Lee fracture system is shown in Figure 5. iSa. This figure showsthat the fracture network has local clusters of fractures that are connected by a few long fractures, at this238C) C) CD C) C) 0 1N)C) 2 - C) -, C) CD B 0 Ii 0 C) cD C) C) -i C)CC) C) 1 C)Meters0010- 01F\)00 01CD_Lcoo(1201 1’J020ri)a)a)j I I I I I I I IFigure 5.15:Cont. Levy-Lee fracture network. b) Flow rates within fractures. Line widths areproportional to the flow rate.1510500 5 10 15020Meters1010 2x101°Flow Rate (ms/s)Minimum flow drawn 9.7e—12 m3/s240scan-line density. These long fractures tend to control flow through the network as shown in Figure 5.15bwhere three wide, long fractures carry most of the flow acmss the fracture network. The net flow throughthe fracture network shown in Figure 5. 15b, and in general for the Levy-Lee fracture system, is mostlydependent on the apertures of the few fractures that span the network. The remaining flow is dependenton a small number of additional connected fractures.The scan-line density of regions within the trace map of Figure 5.15a has a large variation. Thisis an important feature that motivated a change in the method used to generate fracture networks for thisportion of the study. All of the fracture networks for the Levy-Lee fracture systems were generated ina 45m x 45m domain. This ensures that the scan-line density of the smaller domains fluctuates in amaimer that is representative of the Levy-Lee structural model. The standard method of generating thefracture network ensures that the scan-line density within the flow domain is close to the specified scan-line density.5.8.2.1 SENSiTiVITY TO TRANSMISSIVITY VARIATIONThe most striking feature of the network conductance of the Levy-Lee fracture system is that themean conductance has little if any dependence of the variability of the transmissivity as shown in Panela of Figure 5.16. The mean conductance of the fracture networks is strictly a function of the meantransmissivity of the individual fractures because the flow through the fracture networks is controlled byindividual fractures. The fracture network conductance of the Levy-Lee structural model also has adifferent dependence on domain size than the other fracture networks. In Panel a, the mean conductanceof the networks shows little change with domain size. The slight decrease in conductance with increasingdomain size may be due to the limited fracture generation region. Long fractures are probably underrepresented in the larger domain sizes.24116CoC)C)C)5—0.—C)C)‘4-I0ci-iFigure 5.16: Maximum directional hydraulic conductance sensitivity to transmissivity variation ofLevy-Lee generated fracture networks. a) log10 mean of the maximum directionalconductance. b) Log10 standard deviation of the maximum directional conductance. c)Standard deviation of the direction of the principal axes of the conductance tensor. d)Orthogonality deviation of the principal directions of conductance.242a7‘7:8.500CI I I I I I I I I10.8 -0.60-.- 5* ‘S•a11)- III,,I 111111,10 20 30 40- I_lII III I 111111? liii Ii0 10 20 30 40I I I I I I I I I I I I 1’- d- - -d-‘S.SSc--___H-- a- - --•a:El I I I I IIII1P Io 10 20 30 40Domain size ( m )50403020100c10J5d.d‘S‘SIS0 10 20 30 40Domain size ( m )The mean conductance of the Levy-Lee model is not totally independent of scale because of thedependence on fracture orientation variation with fracture length. Recall that the standard deviation in theorientation of the fractures generated using the Levy-Lee model is inversely proportional to the fracturelength. It appears that this feature changes the character of the Levy-Lee generated fracture networksbetween the Sm and lOm domain scale. The 5m x 5m fracture networks may also be effected by the O.Smminimum fracture length in the Levy-Lee model.As one would expect from a stronger dependence on individual fractures, the standard deviationof conductance of the Levy-Lee fracture networks shown in Panel b is dependent on the variability of thefracture transmissivity and is larger than that of the other fracture systems. Neither the variability in thedirection of the conductance (Panel c) nor the orthogonality deviation of the principle directions ofconductance (Panel d) show significant differences in their dependence on transmissivity variation or scalefrom the Poison fracture systems (Figure 5.9). None of the data presented in Figure 5.16 indicates asensitivity to the variability in scan-line density. That the variability in the scan-line density of the smallerfracture networks is not reflected in Figure 5.16 is a further indication that the conductance of the Levy-Lee fracture networks is dominated by a few long fractures.5.9 DiscussioNIn this chapter, an approach to evaluating the scale of an REV for fluid flow in fractured mediahas been presented that provides a tensor characterization of sub-REV flow. The tensor representationfully describes the dependence of average fluid flow on the orientation of the hydraulic gradient. Themagnitude of the hydraulic conductance tensor has an inverse dependence on the scale of the flow domain.Asymmetries in the conductance tensor representations of flow behavior are sensitive to heterogeneitywithin the flow domain.243It should be emphasized that sensitivity to rotation of the model domain, as used in previousinvestigations, is a clear indication that the scale of the domain is below the scale the REV. The erraticbehavior noted by previous investigators provides a sensitive indication that the domain is below the scaleof an REV. The importance and use of the erratic behavior below the scale of an REV has led to theassumption on the part of many people that the erratic behavior is an inherent property of fractured mediaand that it distinguishes fractured media from granular porous media. The dependence of flow in fracturedmedia on the direction of the hydraulic head gradient is not inherently erratic but that it can appear erraticusing a specific method of analysis. Thus, while the hydraulic behavior of a fractured medium does havesome special attributes that arise from its geometric structure, the medium behaves in a manner analogousto a granular pomus medium.This issue brings us back to the field evaluations of the cross-borehole method described by Hsiehet al. [1985]. A direct comparison of cross-borehole directional hydraulic conductance measurements tothe numerical calculations of directional hydraulic conductance for two-dimensional fracture networks isnot possible. Not only is the difference in dimensionality important but the method used to characterizethe flow behavior influences the conclusions one draws, as demonstrated by both the smooth behavior ofthe conductance tensor and the asymmetry introduced by the flow averaging. Fitting the square root ofdirectional conductance measurements to an ellipse is not, in and of itself, an indication that the scale ofa REV has been reached in both two and three-dimensional fracture networks. Even though flow throughan individual realization of a fracture system can be described by a conductance tensor, different fracturenetworks of the same fracture system can have yery different tensor descriptions.In a field setting there is only one fracture network. However, a local detennination of directionalhydraulic conductance samples only a portion of the network. If the volume sampled in the investigationis a REV of the fracture network then, if the macroscopic character of the fracture system does not changespatially, the results of the investigation can be applied to the rest of the network. If the volume sampled244is below the scale of a REV then the results of the study described in this chapter indicate that a hydraulicconductivity representative of the fracture network may differ in both magnitude and in the orientationof anisotropy from the hydraulic conductance determined for the sampled volume. If determinations ofdirectional hydraulic conductivity from field experiments, such as the cross-borehole pressure measurementtechnique, are best represented by an asymmetric hydraulic conductivity tensor then the asymmetry maybe an indication that the averaging volume of the field experiments is smaller than an REV. To make thelink between tensor asymmetry and sub-REV behavior requires an understanding of the relationship ofasymmetry in the conductance tensor of a measured volume of rock to the method of analysis used todetermine the directional hydraulic conductivity.5.10 SUMMARY1. Flow through fractured media below the REV scale can be described using tensor notation. Thetensor representation fully describes the average flow behavior of the fracture network. Forfracture networks defined with fixed boundaries and constant hydraulic head gradient boundaryconditions, plots of the square root of directional hydraulic conductance as a function of thedirection of flow, and plots of the square root of the inverse of the directional hydraulicconductivity as a function of the direction of the hydraulic gradient, are smooth and predictablefrom the hydraulic conductance tensor. The erratic behavior of conductance estimates noted inprevious papers [Long et al., 1982; Long and Witherspoon, 1985; Dershowitz and Einstein, 1987;Khaleel, 1989; Hestir and Long, 1990; Odling and Webman, 19911 is not a fundamental propertyof flow through fracture systems. The erratic behavior is a result of the rotation of the modeldomains that are used in these studies to change the direction of the hydraulic gradient. As such,it provides a useful method for evaluating the scale of an REV for flow.2. The hydraulic conductance tensors for the two-dimensional representations of fracture systemsmodeled in this paper need not be symmetric. The asymmetry is, however, an artifact of the245averaging process used to measure flow across the domain boundaries. The asymmetry is anindication of sub-REV behavior.3. A consistent scale dependence is observed in the hydraulic behavior of the two-dimensionalfracture systems below the scale of an REV. Measurements of mean conductances, the standarddeviation in maximum directional conductance and the deviation from orthogonality of principaldirections of the conductance all decrease with an increase in domain size. These dependenciesmay be different in three dimensional fracture systems.4. The variation in the orientation of the major axes of the conductance ellipse, calculated frommultiple realizations of the same fracture system, is a sensitive indicator of the scale of an REV.246CHA1I’ER 6CONCLUDING COMMENTS6.1 MODELING FRACTURE NETWORKS BELOW THE REV SCALEA constant theme has mn through earlier chapters. The theme has been the hydraulic behaviorof fractured regions smaller than the scale appropriate for equivalent homogeneous continua representation.It is axiomatic to say that the heterogeneity in the hydraulic characteristics of fractures within a sub-REVdomain must be considered; it is not clear how closely the heterogeneity must be retained. Theimportance of the strong interaction of internal heterogeneity and the boundaries conditions of the domainis a less obvious but equally important issue.In Chapter 5, it was demonstrated that the erratic character attributed to fractured media below thescale of the REV is not an inherent property of fractured media but rather a hypersensitivity to smallchanges in the fracture network in the corners of a fracture domain. The sensitivity is amplified byboundary conditions that do not respond to fractures that cross domain boundaries twice in a shortdistance. The tensor relationship developed in Chapter 5 shows that the flow within fracture networks isnot erratic and behaves in a smooth and predictable manner. At the same time, the asymmetric nature ofthe hydraulic conductance tensors further demonstrates that ignoring the influence of heterogeneity, be itinternal or external, creates behavior in the model domain that is not representative of a natural situation.The elimination of corner cutting fractures in some of the networks in Chapter 5 was required tokeep the hydraulic behavior of a relatively large domain from being obscured by high flow rates throughshort fracture segments that connected with boundaries with fixed hydraulic heads. While the boundaryconditions could be considered reasonable for the domain as a whole, the lack of response of theseboundary conditions to those short-circuiting fractures could not be considered reasonable. Hestir andLong [1990] and Herbert et al. [1991] addressed the sensitivity of fractured media to the boundary247conditions by limiting their analysis of a fracture network to an interior region; buffering the flow domainboundaries from the specified boundary conditions with an intervening zone of the fracture network.Network block fractures could not be eliminated to rid the network blocks of short-circuitfractures, nor could the network blocks be completely buffered from their neighbors. The treatment ofthe boundary conditions in the dual permeability model evolved toward boundary conditions that not onlyrespond to the heterogeneity inside the network blocks but also incorporated some sense of theheterogeneity of the fractures surrounding the network blocks. The ability of the dual permeability modelto simulate flow and transport in a wide variety of fracture networks demonstrates that the method usedto characterise the network blocks retains the most important characteristics of these sub-REV regions.There are three essential features to the way the boundary conditions of the network blocks aretreated. The first is the partial buffering from the rest of the fracture network that the primary fracturesprovide. These fractures are selected, in part, because they are anticipated to have a strong influence ontheir local flow regime. It was initially thought that this feature alone would be sufficient to allow a linearhydraulic head gradient to be assumed within these fractures. The second major feature of the networkblock boundary conditions is inclusion of the primary fractures in the effective conductance representationof network blocks. This approach allows the hydraulic head distribution along these fractures to respondedto the heterogeneity within the network blocks. It is appropriate to consider the primary fractures as partof the boundary conditions. As such, they provide a responsive aspect to the boundary conditions. Thethird feature is the enhancement of the transmissivity of the primary fractures as a crude method ofincluding the heterogeneity of the surrounding fracture network in the boundary conditions. These threeaspects of the way boundary conditions have been treated in the network blocks are the key element toeffectively incorporating these sub-REV domains in the dual permeability model.2486.2 DUAL PERMEABILITY MODELINGThe main thrust of this thesis has been the proposal and investigation of a new concept ofmodeling fluid flow and solute transport in fractured media at the field scale. The main proposition ofthe concept is that it is acceptable to increase the uncertainty of the predicted hydraulic behavior of therock mass through approximations to the hydraulic behavior of sub-continuum regions of a fracturenetwork, because the knowledge of the fracture network of any rock body will, in practice, have a largedegree of uncertainty. These approximations are used to determine a coarse-scale hydraulic headdistribution within the fracture network. Solute transport properties of the sub-continuum regions aredetermined using the coarse-scale head distribution as boundary conditions to the regions. The transportproperties of the sub-continuum regions are used in a model of transport for the entire fracture network.An implementation of this concept has been presented in Chapter 4. The emphasis in thedevelopment of the dual permeability model has been on maximizing the accuracy of the flow andtransport predictions in a model that contains the essential features of a dual permeability model. Thesefeatures are: 1) the subdivision of the fracture network into smaller domains based on the heterogeneityof the fracture network, 2) sub-continuum models of flow and transport within these small domains, 3)coarse-scale models of flow and transport that utilize approximations of the small domains, 4) theindependence of the small domains from adjacent domains.The dual permeability model succeeds in reproducing the hydraulic behavior of a wide variety offracture systems as demonstrated by the agreement in the particle transport predictions of the dualpermeability model and Discrete for most of the fracture systems examined in Chapter 4. The modelseems to be particularly well suited to the Levy-Lee structural model which uses a power law model offracture scaling. The model is limited to fracture networks that have a broad range of fracture lengths asdemonstrated by the poor agreement for fracture system 3. The two aspects of the dual permeabilityapproach that proved to be the most difficult to address were how to subdivide the domain and how to249approximate the hydraulic conductance of these domains. The approximation of the network blocks wasdiscussed in the previous section. The development of an automated procedure to subdivide the fracturedomain on the basis of the geometric properties of the fractures has been a difficult task. The routine thathas been developed is very complex and is not universally successful in creating an acceptable primaryfracture network. It is this phase of the dual penneability model that may be the most difficult transitionto three dimensional fracture system modeling. 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Res., 17(3), 555-564, 1981.Terzaghi, R., Sources of error in joint surveys, Geotechnique, 15, 287-303, 1965.Tsang, Y.W. and P.A. Witherspoon, The dependence of fracture mechanical and fluid flow propertieson fracture roughness and sample size, J. of Geophys. Res., 88 (B3), 2359-2366, 1983.261REFERENCES - Cont.Tsang, Y.W., The effect of tortuosity on fluid flow through a single fracture, Water Resour. Res.,20(9), 1209-1215, 1984.Tsang, Y.W. and C.F. Tsang, Channel model of flow through fractured media, Water Resour. Res.,23(3), 467-479, 1987.Tsang, Y.W., C.F. Tsang, I. Neretnieks, and L. Moreno, Flow and tracer transport in fractured media:a variable aperture channel model and its properties, Water Resour. Res., 24(12), 2049-2060, 1988.Tsang, Y.W., Usage of “Equivalent Apertures” for rock fractures as derived from hydraulic and tracertests, Water Resour. Res., 28(5), 145 1-1455, 1992.Terzaghi, R., Sources of error in joint surveys, Geothechnique, 15(3), 287-304, 1965.Voos, C.F. and L.R. Shotwell, An investigation of the mechanical and hydraulic behavior of tufffractures under saturated conditions, Proc. High Level Radioactive Waste Management, Am.Nuclear Soc., 825-834, 1990.Wallis, P.F. and M.S. King, Discontinuity spacings in a crystalline rock, mt. J. Rock Mech. Mi Sci.& Geomech. Abstr., 17, 63-66, 1980.Wang, J.S.Y., Y.W. Tsang, F.V. Hale, M.M. Colina, S.E. Derenzo, and T.F. Budinger, Feasibility ofusing positron emission tomography for flow and transport studies of fractured, porous media,Geophysics Rev. Lett., ???, 1990.Warburton, P.M., A stereological interpretation of joint trace data, mt. J. Rock Mech. Mm. Sci. andGeomech. Abstr., 17, 18 1-190, 1980a.Warburton, P.M., Stereological interpretation of joint trace data: influence of joint shape andimplications for geological surveys, mt. J. Rock Mech. Mm. Sci. and Geomech. Abstr., 17, 305-316,1980b.Warren, J.E., and Root P.J., The behaviour of naturally fractured reservoirs, Society of PetroleumEngineers Journal, 3, 245-255, 1963.Wilson, C.R., An investigation of laminar flow in fractured porous rocks, Ph.D. thesis, Univ. of Calif.,Berkeley, 1970.Zienkiewicz, O.C., and K. Morgan, Finite Elements and Approximation, p. 181, John Wiley, NewYork, 1983.Zimmerman, R.W., G. Chen, T. Hadgu, and G.S. Bodvarsson, A numerical Dual-porosity model withsemianalytical treatment of fracture/matrix flow, Water Resour. Res., 29(7), 2127-2137, 1993.262APPENDIX AFLOW CHARTSAppendix A contains two series of flow charts that describe the program Discrete and the dualpermeability program. Four charts (Figures A. 1 - A.4) provide an overview of Discrete.seven additional charts( Figures A.5 -A. 11) are used to present the more complicated dual permeabilityprogram. The first chart in each series is an overview of the structure of the entire program. Tasksthat are described in more detail in later flow charts are indicated with the appropriate figure numberin the upper right corner of the task description box. The flow charts describing the fracture networkgeneration process are identical for both models.The flow chart figures are:Figure A. 1: The structure of Discrete;Figure A.2: Fracture realization process;Figure A.3: Fracture network reduction;Figure A.4: Particle tracking in Discrete;Figure A.5: The structure of the dual permeability program;Figure A.6: Primary fracture network selection;Figure A.7: Formation of the dual permeability coarse scale flow model;Figure A. 8: Determination of the network block residence distributions;Figure A.9: Formation of the transport model;Figure A. 10: Calculation of the primary fracture transport characteristics;Figure A. 11: Particle tracking algorithm.263Program Discrete4FractureNetwork RealizationDetermine Flow EquationsSolve Flow Problem*1Determine Fracture IntersectionFlow Splits andFracture Residence TimesParticle TrackingTransport SimulationFigure A. 1: The structure of Discrete.264Fracture Network RealizationAdd Explicit Fractures Read Fracture NetworkiRead Fracture StatisticsI Generate Fracture NetworkitIdentify Fracture IntersectionsSet Boundary toInfluence BoundaryReduce NetworkIrSet Boundary toFlow Boundary*1Reduce NetworkFigure A.2: Fracture realization process.265Reduce NetworkRemove IntersectionsOutside Boundaries4Identify FracturesConnected to Boundaries.0C)0C0C)0C004-C)00C0.0004-Is the Fracture Connectedto a Boundary by theIntersecting Fracture?0IC4-C)000C00I0•0NolYosMark Fracture as Connected__!!J Have new Connections[_Found?00C) —0,00u-00)0CMark Intesection to be Deletedfrom Intersecting Fracture00I0 0CCh.. —h..C)00u-CC0C)Figure A.3:Remove Deleted IntersectionsTruncate Retained Fracturesat Outer IntersectionsFracture network reduction.266Particle Tracking Simulation00I0.c000U-Add to Position StatisticsFigure A.4:Is the Intersectionon the Boundary ?L“YesSave ParticleExit Time and LocationtParticle tracking in screte.Introduce Particle atInjection Node4Randomly Select an Exit Fracture4Add time to Reach Next Intersection*1’Is Time Later than___Position Monitoring Time? YesNo267Dual Permeability ProgramFracture Network:Realization*1Primary Fracture A.6Network Selection*1Determine Effective Conductance A.7of Network Blocks*A7Incorporate Network Blocksinto Primary Network toForm Flow ModelSolve Flow Model forHydraulic Head Distribution*A8Find Network BlockResidence Time Distributions*1*A9Form Transport Model fromNetwork Blocks and Primary Fractures* A.11Particle TrackingTransport SimulationFigure A.5: The structure of the dual permeability program.268Primary Fracture Network SelectionJ Increase Percentageof Primary FracturesSelect Primary FracturesFrom Individual CharacteristicsFind Primary Fracture Intersections.Connect Ends UsingAdded Primary Fracture Segments.Remove UnconnecedPrimary FracturesBlockDefine Network Block Geometry I Failure10K-:t________Too LargeTest Block Sizej Divide BlockOK OK Can’t U&__ _Select New______ _____________Primary Fractur*Yesy New Primary Fractures)4NoFind Internal Network BlockFractures and IntersectionsLU— Too Large______________ITest Block SizeL Divide Block 110K OKSelect New IPrimary FractureFigure A.6: Primary fracture network selection.269Form Dual Permeability Flow ModelC.)0.00ci)0ci,0C0C)ci)ci)Ca)C.)CcicciEI0-cC)ccici)10+Allocate Space forStiffness MatrixFigure A.7: Fonnation of the dual permeability coarse scale flow model.Form Network BlockStiffness MatrixAssign Unit HeadSet others to Zero HeadSolve Flow EquationDetermine Flow into eachPrimary Fracture Segment*1Add Flows as conductanceterms to Dual PermeabilityStiffness Matrix+Add Conductance ofPrimary Fracture Segmentsto Stiffness Matrix*1270.0.c 00 .L00>u-.0c0EE0.0Co000oCu_ .,.-Form Transport ModelFigure A.8: Formation of the transport model.frCalculate Network Block* A9Residence Time DistributionCalculate Transport Characteristicsof Primary Fracture SegmentAlOCalculate Exit Probabilitiesof Primary Fracture Intersections*1271Recalculate Network BlockStiffness MatrixSolve Flow ProblemCalculat Flow at eachBoundary Node0________o ‘ Order Internal Nodes by Head0Dm— Determine the Inlet andOutlet Boundary Nodes00Determine Flow Splits-,—. C) at each Internal NodeF- VC) Find Arrival time at NodeConnected to Boundary Inlet— Calculate Probabilities and.Q co Times of Entering NodeC) from Upstream NodesCCalculate Probabilities andArrival Times at OutletsFigure A.9: Determination of the network block residence distributions.272Calculate Transport Characteristicsof Primary Fracture SegmentCalculate Flow at Upstream Endfrom Coarse Scale HeadsVI Calculate Time to Reach Intersection ICalculate Probability of .2leaving Primary FractureCalcu late Flow toLrseCalculate Flow at Downstream Endfrom Coarse Scale HeadsCalculations Ag‘:Figure A. 10: Calculation of the primary fracture transport characteristics.273Particle Tracking SimulationFor each ParticleIntroduce Particle atInjection NodeExit Time and LocationSave ParticleRandomly Select an Exit FractureIs the Node a LBoundaryf______Locate Exit on 1Primary Fracture SegmentFigure A. 11: Particle tracking algorithm.Add time to Reach Next IntersectionNoExit at intersection to a Network Block?YesNoSelect Exit Locationfrom Network Blockand add Residence TimeIs the Node a Boundary Node?NoH Yes274APPENDIX BASYMMETRY MEASUREMENT STATISTICSIn Appendix B, three alternative methods of statistically characterizing the sensitivity ofconductance tensor asymmetry to fracture network geometry and the scale of the flow domain areinvestigated. These statistics are:1) The deviation in degrees from orthogonality of the principal directions of the conductance tensor.2) The ratio of the positive difference in the off-diagonal elements of the conductance tensor toaverage of the diagonal elements of the tensor.3) The ratio of the positive difference in the off-diagonal elements to the radius of the Mohr’s circle.As noted earlier, the principal directions of a symmetric tensor are orthogonal. The first statistic ismotivated by considering asymmetry as a rotation of the principle directions of the conductance tensor.The second statistic can be thought of as the fraction of the magnitude of the conductance thatcontributes to asymmetry. The third measure treats asymmetry as a fraction of the anisotropy in theconductance.Figure B. 1 merges the average orthogonality deviation for all of the cases studied into a singlefigure. As a measure of sub-REV behavior, the deviation from orthogonality behaves in a similarmanner to the variation in the orientation of the major axes of the conductance tensor. It appears to beslightly more erratic than the orientation of the major axes. The orthogonality deviation tends todecrease as the domain size increases, and more fractures are included in the flow domain. Theexceptions are cases m and e which are insensitive to domain size, a characteristic that is shared withthe standard deviation of the orientation of the major axes of the conductance tensor.275Figure B.2 shows the average ratio of the difference in the off-diagonal elements to the average ofthe diagonal elements of the conductance tensor, relative to domain size. This measure is sensitive toincreasing domain size in all cases. Of particular interest in comparing measure 1 and measure 2 arethe results for cases m and n. Case m along with case e shown the largest asymmetry using measure1. In contrast, case m has the lowest ratio of off-diagonal element difference to average diagonalelements. Case n shows a similar shift from a high value for measure 1 to a median value formeasure 2. Case m is isotropic and case n is nearly so because of the variation in the meanorientation of the fractures of each set. The ratio of the difference in the off-diagonal elements to theaverage of the diagonal elements of the conductance tensor is a superior measure of sub-REV behaviorcompared to the orthogonality deviation of the principal directions of hydraulic conductanceFigure B .3 shows the results of averaging the ratio of difference the off-diagonal elements and theradius of the Mohr’s circle. These plots do not show a strong correlation between the measured valuesand domain size. The highest individual values result from networks that are nearly isotropic, that isnetworks with a small Mohr’s circle radius. The ratio of difference the off-diagonal elements and theradius of the Mohr’s circle is the poorest measure of sub-REV behavior of the three we haveinvestigated. The link between asymmetry and anisotropy is poor.276Sa,a,I,0a,C,a,00a,0.00Figure B.1: Average deviation of the orthogonality of the semi major and semi minor axis of theconductance ellipsi. a) Sensitivity to transmissivity variation. b) Sensitivity to fracturedensity,e, length,f, and isotropy, k and m, variation. c) Sensitivity to mean fractureorientation, and orientation variation.277b25201 15V11:0aC•1 I I I-—j I I I I I I I I I I I I I I--a-...- )_______I I I I Ia,a,a,a,000Va,VC00.0025:1I I I I I I I I I I20—15—a’10 a..-50 I I0 10 20 30 40 0 10 20 30 4025 I I I I I I I I200liii 11111111111 Iii0 10 20 30 40Domain Size ( m )a bD.250.2i150.1J.05C0Figure B.2: Average ratio of the difference in the off-diagonal elements to the average of the diagonalelements of the conductance tensor. a) Sensitivity to transmissivity variation. b)Sensitivity to fracture density,e, length,f, and isotropy, k and m, variation. c) Sensitivityto mean fracture orientation, and orientation variation.1II I III liii II III-S.’C..ciUciU0.30.2C0.1-Sd--S..e-.13. “ -a.UUCciUCgciI-.a,0.2i0.1>5cibtill?I I I I I I I I I I Ie.e-m--.m.30 40C0 10 20 30 40 0 10 20I I i I i i , I i i0 10 20 30 40Domain Size ( m )278a b0.64, 0.4O.2V• 0.60.4V—liii liii liii liii 1111 liii, ‘VE; :1 :- I I i i i i i I I I I I 0 - I i i i I i i i i i i iiiC 0.80 10 20 30 40 0 10 20 30 40III 1111111111111111NgaIII II 111111 I0 10 20 30 40Domain Size ( m )Figure B.3: Average ratio of the asymmetry offset in the Mohr’s circle to the radius of the Mohr’scircle, a) Sensitivity to transmissivity variation. b) Sensitivity to fracture density,e,length,f, and isotropy, k and m, variation. c) Sensitivity to mean fracture orientation, andorientation variation.279
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Dual permeability modeling of fractured media Clemo, Thomas Meldon 1994
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Title | Dual permeability modeling of fractured media |
Creator |
Clemo, Thomas Meldon |
Date Issued | 1994 |
Description | Dual permeability is introduced as an approach to modeling flow and transport through fractured media. The approach allows a large reduction in the number of fractures that are represented explicitly in a discrete fracture network. The most important fractures in terms of fluid flow are identified using their physical characteristics. These fractures form a sub-network that divides the entire fracture network into smaller domains. The fractures of the smaller domains are approximated. The approximations do not rely on continuum assumptions. They are determined individually and independently for each small domain, resulting in a parallel structure to the calculations. An exploratory model is developed for steady state fluid flow and solute transport in two dimensional fracture networks and compared to a discrete fracture model that represents all of the fractures explicitly. Individual fractures are represented by finite lines with constant hydraulic transmissivity. Solute transport calculations use particle tracking. The dual permeability model is shown to provide acceptably accurate solutions while reducing the maximum number of simultaneous equations by well over an order of magnitude. The dual permeability approach takes advantage of the hydraulic behavior of highly heterogeneous media. Channeling of flow that develops in such media is used to reduce errors introduced through the sub-continuum approximations. The dual permeability approach works better when the fracture system has a broad range in the scale of fracturing as demonstrated by transport calculations for fracture networks with a high degree of clustering or with a fractal nature. A study of sub-REV fracture networks reveals that averaged flow through fractured media can be represented using tensor notation in much smaller domains than an REV. At these scales, heterogeneity within a fracture domain causes the tensor representation to be asymmetric, unlike the continuum representation of granular porous media. Hydraulic heads within a heterogeneous fracture network do not vary in a smooth fashion. The hydraulic head reflects the distribution of heterogeneity within the domain. The approximations developed for the dual permeability model required that the boundary conditions imposed on the small domains reflect heterogeneity both within the domains and external to them. |
Extent | 5576501 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0052916 |
URI | http://hdl.handle.net/2429/6994 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geological Sciences |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1994-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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