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Hydrogeological decision analysis : monitoring networks for fractured geologic media Jardine, Karen G. 1993

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HYDROGEOLOGICAL DECISION ANALYSIS:MONITORING NETWORKS FOR FRACTURED GEOLOGIC MEDIAbyKAREN G. JARDINEB. Sc., The University of Alberta, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Geological SciencesWe accept this thesis as conformingto the required standardTHE UNIVERSITY F BRITISH COLUMBIANOVEMBER 1993© Karen G. Jardine, 1993In 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.Department of oLoçcL 5ciE1.)C_€The University of British ColumbiaVancouver, CanadaDate IiovEt-seRDE-6 (2/88)11ABSTRACTIn this dissertation, a decision analysis framework is developed to assist in thedesign of monitoring networks at hazardous waste sites located above a fracturedgeologic unit. The decision analysis framework is based upon risk-cost-benefit analysis,performed from the perspective of the owner/operator of the landfill facility. The costsconsidered are those that are directly associated with the construction and operation of themonitoring network (actual costs). The risks considered are those that are associated withthe detection of migrating contaminants and consequent costs of remediation, and thefailure of the facility and the costs resulting from failure (expected costs). The benefitsare considered to be the same regardless of the monitoring strategy adopted, and areneglected.The fractured rock formation underlying the hypothetical landfill site is modelledin vertical section using a two-dimensional discrete fracture model. This model uses aparticle tracking method to simulate the transport of a non-reactive solute through thefractured rock unit. Three fracture geometries are investigated, each with differenthydrogeological behaviour. For each of these geometries, four monitoring schemes areconsidered: 1) monitoring the fractures that carry the highest volumetric flows, 2)monitoring the fractures that have the largest apparent apertures, 3) monitoring the areasof highest fracture density, and 4) placing the monitoring locations at predetermineddepths. The effects of the distance of the monitoring network from the contaminantsource, and the number of monitoring locations installed at each monitoring well site, areinvestigated for each of the four monitoring strategies in each of the three fracturegeometries. The base case analysis is performed using a pseudo-three-dimensionalapproach that is adopted in an attempt to achieve consistency between the expected costsof remediation and failure, which assume a three-dimensional domain, and the costs ofmonitoring, which are calculated on the basis of each individual monitoring well site.The best monitoring alternative in two of the three geometries investigated, andthe highest probabilities of detection in all three fracture geometries occur when thefractures carrying the highest flows are monitored. However, the monitoring strategy thatprovides the highest probability of detection is not necessarily the best alternative.111In the geometries modelled, the probability of detection is influenced by theamount of vertical spreading the contaminant plume undergoes near the contaminantsource as a result of the toruousity of the preferred flow paths through the fracturenetwork. The increase in the probabilities of detection brought about by the installationof a “backup” monitoring network is insufficient to justify such an installation. However,the decision analysis developed in this study does not evaluate other functions that arepotentially filled by a “backup” monitoring system.The combination of monitoring options that provide the best monitoringalternative is insensitive to changes in the detection threshold and changes in the discountrate over the ranges investigated. The length of time between samples, and variations inthe characteristics of the pseudo-three-dimensional analysis have only a small influenceover the the combination of monitoring options that provide the best monitoringalternative.ivTABLE OF CONTENTSAbstract iiList of Tables ixList of Figures xAcknowledgement xiv1. Introduction 12. Decision Analysis Framework 52.1. Introduction 52.2. Objective Function 52.2.1.RiskTerm 82.3. Decision Scenario 132.3.1. Objective of Monitoring Network 132.3.2. Detection 142.3.2.1. Definition of Detection in a Monitoring Well 142.3.2.2. Consequences of Detection: Remedial Design 14V2.3.3. Failure .152.3.3.1. Definition of Failure at the Compliance Surface 152.3.3.2. Consequences of Failure: Containment System 153. Review of the Flow and Transport Model 163.1. The Choice of Model 173.1.1. Generation of Fractures 173.1.1.1. Statistical Description of Fracture Geometry 183.1.2. Flow Solution 193.1.3. Transport 203.1.3.1. Introduction of Particles 203.1.3.2. Residence Time Distribution 213.1.3.3. Routing in Fracture Intersections 214. Choice of Model Domain 234.1. Horizontal Section 234.2. Vertical Section 24vi4.3. Description of Domain .254.4. Pseudo-three-dimensional Analysis 265. Decision Model 285.1. Network Design 285.1.1. Well Siting 285.1.2. Monitoring Locations 285.1.3. Monitoring Interval 305.2. Defining Detection 305.2.1. Detection Threshold 305.2.2. Monitoring Period 315.2.3. Implementation 335.2.4. Limitations 345.3. Defining Failure 355.3.1. Assumptions Concerning the Compliance Surface andFailure 35vii6. Results .376.1. Three Fracture Geometries 376.2. Base Case 446.2.1. Monitoring Parameters 446.2.2. Decision Analysis Parameters 446.3. Base Geometry 466.3.1. Monitoring Scheme Comparison 466.3.2. Sensitivity Studies 526.3.2.1. Detection Threshold 536.3.2.2. Monitoring Interval 566.3.2.3. Discount Rate 626.3.2.4. Cost of Failure 646.3.2.5. Pseudo-three-dimensional Analysis 656.3.2.6. Multiple Well Configurations 686.4.Geometry2 71VII’6.4.1. Monitoring Scheme Comparison .726.5.Geometry3 806.5.1. Monitoring Scheme Comparison 806.5.2. Increased Cost of Failure 877. Conclusions 89bibliography 94ixLIST OF TABLESTable Page6.1 Statistical Input Parameters for the Three Fracture GeometriesInvestigated 376.2 Flow and Transport Characteristics from Preliminary Simulations ofthe Three Fracture Geometries Investigated 406.3 Decision Analysis Parameters for Base Case 44xLIST OF FIGURESFigure Page1.1 Vertical cross-section of hypothetical landfill site 32.1 Decision tree based on a time independent version of the objectivefunction 123.1 Fracture network from base geometry. a) generated fracture network,b) cleaned fracture network 183.2 Stream tubes in continuous intersection 224.1 Model domain representing hypothetical landfill site 254.2 Example of pseudo-three-dimensional domain with 10 slices 276.1 Fracture networks for the first realization of each geometry. a) basegeometry, b) geometry two, c) geometry three 386.2 Cumulative probabilities of failure throughout compliance period forall three fracture geometries investigated 406.3 Volumetric flows through fractures in the fracture networks for thefirst realization of each geometry. a) base geometry, b) geometrytwo, c) geometry three 426.4 Total probabilities of detection over the compliance period vs.distance from the source for the base geometry 466.5 Cumulative probability of detection vs. time at 25 m and 75 m fromcontaminant source for base geometry with one monitoringlocation per monitoring well site. a) highest flow monitoringscheme, b) densest fracturing monitoring scheme, c) largestaperture monitoring scheme, d) predetermined depth monitoringscheme 48xiFigure Page6.6 Total probability of detection vs. distance for base geometry with one,two, and three monitoring locations per monitoring well site. a)highest flow monitoring scheme, b) densest fracturing monitoringscheme, c) largest aperture monitoring scheme, d) predetermineddepth monitoring scheme 496.7 Values of objective function for base geometry with base case analysis 506.8 Probability of detection vs. distance from the source for three differentdetection thresholds with base geometry 536.9 Values of objective function for three different detection thresholdswith base geometry 546.10 Probability of detection vs. distance from the source for three differentmonitoring intervals with the base geometry and three differentdetection thresholds. a) 3 .83E6 particles/rn3,b) 1.92 particles/rn3,c) 1 particle per monitoring period 576.11 Arrival rate of particles at the compliance boundary in the fracturecarrying the largest proportion of particles in the first realizationof the base geometry with a total of 2000 particles injected 586.12 Values of the objective function for three different lengths ofmonitoring interval for the base geometry. a)thresholdconcentration of 1 .92E6 particles/rn3,b) threshold concentrationof 3.83E6 particles/rn3 606.13 Values of objective function for the base geometry with three differentdiscount rates 626.14 Values of objective function for two different costs of failure for thebase geometry 646.15 Values of objective function for two-dimensional analysis for basegeometry 65xliFigure Page6.16 Values of objective function for two-dimensional and two differentpseudo-three-dimensional analyses for base geometry 676.17 Cumulative probability of detection vs. time at single well sites at25 m and 75 m and a multiple well configuration with well at bothsites for base geometry 686.18 Values of objective function for single well sites and two multiplewell configurations for base geometry. a) $5 million cost offailure, b) $10 million cost of failure 696.19 Total probabilities of detection over the compliance period vs.distance from the source for geometry two 726.20 Total probability of detection vs. distance for geometry two with one,two, and three monitoring locations per well site. a) highest flowmonitoring scheme, b) densest fracturing monitoring scheme, c)largest aperture monitoring scheme, d) predetermined depthmonitoring scheme 736.21 Probability of detection vs. distance for both base geometry andgeometry two. a)one particle per monitoring period and 3.83E6particles/m3,b) a)one particle per monitoring period and 1 .92E6particles/m 756.22 Values of the objective function for geometry two with base caseanalysis 776.23 Total probability of detection over the compliance period vs. distancefor geometry three 806.24 Total probability of detection vs. distance for geometry three with one,two, and three monitoring locations per well site. a) highest flowmonitoring scheme, b) densest fracturing monitoring scheme, c)largest aperture monitoring scheme, d) predetermined depthmonitoring scheme 81xliiFigure Page6.25 Cumulative probability of detection vs. time for base geometry andgeometry three. a) 25 m from source, b) 75 m from source 836.26 Values of the objective function for geometry three with base caseanalysis 856.27 Values of the objective function for geometry three with $10 millioncost of failure 87xivACKNOWLEDGEMENTI would like to thank my supervisor, Leslie Smith, for his advice, guidance, andsupport throughout the process of preparing this thesis. I would also like to thank theother members of my supervisory committee, Brian Berkowitz, Tom Brown, and RogerBeckie, as well as Rosemary Knight and Dick Campanella who served on my examiningcommittee. I am very grateful to have had the opportunity to learn from Al Freeze duringhis last year and my first year at UBC.Special thanks go to Tom Clemo, who wrote the transport model I used and whoseassistance and patience were invaluable, especially when I was debugging my computercode. The members of the groundwater group, both past and present, provided anenjoyable work environment, as well as feedback on technical problems and myinterminable practise talks.I would also like to thank my life partner, Markus Eymann, for looking after meand putting up with me throughout all of this ordeal.Financial support was provided by both a postgraduate scholarship and anoperating grant from the Natural Sciences and Engineering Research Council of Canada.1HYDROGEOLOGICAL DECISION ANALYSIS:MONITORING NETWORKS FOR FRACTURED GEOLOGIC MEDIA1. INTRODUCTIONAs the population of the world grows, so does the amount of waste that weproduce. The disposal of this waste has become a problem of major proportions. In thepast few decades, the detrimental effects to the environment of indiscriminate, poorand/or ill-advised waste disposal practices have become clear. Harmful wastes that wereburied in the ground are finding their way back to the biosphere and poisoning the waterwe drink, the air we breathe, and the ground that we walk on. The most common mode oftransport of these wastes is in groundwater.As the landfills we use are filled up, new sites must be found. One of the majorconcerns in siting a new landfill is to prevent leachate from the landfill from entering thegroundwater system. One way in which this is done is to install synthetic or clay liners inlandfill cells to prevent the migration of leachate. Another method is to locate the landfillin or above a geological unit that is relatively impermeable, such as clay or crystallinebedrock. Neither of these approaches is foolproof; liners may be breached and clay andbedrock contain fractures which may act as conduits for the transport of contaminantsaway from the landfill cells. Therefore, it is often required by law that the areasurrounding a landfill be monitored in order to detect escaping contaminants. In BritishColumbia, all new landfills and expansions to existing landfills are required to includemonitoring programs that address ground and surface water, landfill gas, and ambient airquality as a minimum, with the need for other environmental monitoring, such asvegetation and soils to be assessed on a site specific basis (Ministry of Environment,Lands and Parks British Columbia, 1992).In this dissertation, I develop a decision analysis framework that is intended toassist in the design of monitoring networks in fractured media. Although similar2frameworks have been proposed for monitoring in porous media, (Massmann and Freeze,1987a and b) there has been no attempt to date to develop one that accounts for theunique properties of fractured media. The prediction of contaminant transport infractured rock is much more difficult than it is in a porous medium. In many fracturedrock environments, groundwater flow occurs primarily in the fractures; there is virtuallyno flow in the rock matrix itself. Frequently, the bulk of the groundwater flow occurs ina small fraction of the fractures, resulting in a channeling of the flow. Thus contaminantsmay become localized in a few major conduits, resulting in a contaminant plume of a verydifferent shape than is usually found in porous media. Consequently, monitoringstrategies that are developed for use in porous media may be inappropriate for use infractured media. Rock is being considered as a receiving environment for many types ofwaste, from municipal waste to high level radioactive waste. More and more wastedisposal facilities are being constructed on or within fractured rock. Therefore, there is areal need for some type of guide or framework such as the one I have developed.Decision analysis is a form of systems analysis. It enables one to choose the“best” alternative from a number of alternative courses of action and permits anevaluation of prediction uncertainty within the decision making process. In this study, arisk-cost-benefit analysis is used to evaluate several alternative monitoring networkdesigns. I carry out the decision analysis from the perspective of the owner-operator of anew landfill facility. However, the methodology developed in this study can easily beapplied to other types of facilities, existing facilities, or sites that have already beencontaminated. With some modification, it can also be applied from the perspective of aregulatory agency to facilitate the design of monitoring networks at compliance surfaces.A vertical cross-section of the hypothetical landfill site under consideration isshown in Figure 1.1. A landfill cell is excavated through approximately 10 meters of lowpermeability surficial deposits and into the top 10 meters of a 50 meter thick fracturedrock unit. This fractured rock unit is situated over a low permeability unit. Thecompliance surface is taken to be the downstream property boundary of the landfill site,200 meters from the landfill cell. A compliance surface represents the outer limit of thearea within which a degradation of the groundwater quality has been deemed acceptable;if a specified concentration of contaminant is detected at or beyond the compliancesurface, the person or company responsible is liable to be penalized. The location of acompliance surface is usually set by a regulatory agency or negotiated by the regulatory3agency and the owner/operator of a facility. The compliance surface for a landfill site canbe any one of a number of possibilities including: 1) the zone between liners in a multipleliner system, 2) the outside of the landfill cell itself, 3) the landfill property boundary, 4)a river or lake, or 5) a water supply well field (Domenico, and Palciauskas, 1982).Compliance boundaries are usually located closer to the facilities than the 200 m used inthis study. I have chosen to locate the compliance boundary well away from the facilityto allow for the comparison of monitoring networks placed over a wide range of distancesfrom the contaminant source.The evaluation of prediction uncertainty is introduced into the decision analysisframework through a risk term in the objective function of the risk-cost-benefit analysis.This term deals with the consequences of leachate from the landfill cell entering thefractured rock unit. There are three possible consequences of this event that are ofconcern: 1) the contaminant plume will migrate past the compliance boundary and off ofthe landfill site, in which case failure is considered to have occurred, 2) the contaminantplume will be detected and remediated before it has crossed the compliance boundary, inwhich case failure will not occur, or 3) the contaminant will not be detected by themonitoring network but will remain within the landfill property boundaries. In the riskterm, the costs associated with each of these consequences are multiplied by theprobability of that consequence occurring. For the purposes of this study, the costsassociated with failure are those of isolating the contaminant plume by the construction ofFigure 1.1: Vertical cross-section of hypothetical landfill site4barrier walls. If the plume is detected before it has reached the compliance surface, aremedial program involving the installation and operation of horizontal interceptor wellsis put into place. There are no costs associated with the third consequence, thecontaminant plume remaining undetected and within the property boundaries. Becausethere is no indication that leachate has escaped from the landfill cell in this instance, noaction is required. The probabilities of failure and of detection by the monitoringnetwork are evaluated by running Monte Carlo simulations of fracture networkscontaining monitoring networks.The object of locating a waste disposal facility in or on crystalline bedrock is toprevent the migration of contaminants away from the facility. A good site for a wastedisposal facility is one in which a contaminant plume would migrate so slowly that itwould not advance as far as the monitoring network for a very long time. In this instance,the facility has a minimal impact on the environment and the owner/operator of the wastedisposal facility is not faced with the high costs of remediating or containing acontaminant plume.The objectives of this thesis are to investigate monitoring strategies in fracturedmedia, and to do this within a decision analysis framework. Details of the decisionanalysis framework are discussed in Chapter Two. Chapter Three reviews theflow/transport model used for this study. A detailed discussion of the choice of modeldomain is presented in Chapter Four. In Chapter Five, I describe the decision model.The results of the simulations, analyses, and sensitivity studies are presented in ChapterSix. The conclusions are summarized in Chapter Seven.52. DECISION ANALYSIS FRAMEWORK2.1. INTRODUCTIONDecision analysis is a branch of systems analysis. It enables one to choose the“best” alternative from a number of alternative courses of action. Decision analysis hasbeen defined as “a formalization of common sense for decision problems which are toocomplex for informal common sense” (Massmann and Freeze, 1987a). By defining allvariables in terms of either dollar figures or probabilities, decision analysis links theeconomic environment in which decisions are made and the technical information uponwhich these decisions are based (Massmann et. al., 1991). Decision analysis permits aconsideration of prediction uncertainty within the decision making process. This isaccomplished by assigning an economic value to the risk of not meeting the designobjectives because of uncertainty in model predictions. Unlike optimization processes,such as linear and nonlinear programming, decision analysis does not provide an optimalsolution across all decision variables. It allows only for the identification of the bestsolution from a finite number of options presented for investigation.2.2. OBJECTWE FUNCTIONA risk-cost-benefit analysis is used to evaluate several design alternatives, and thealternative that provides the maximum value for the objective function is chosen. In thisstudy, I am using one of the more general forms of objective function for the risk-costbenefit analysis (Massmann and Freeze, 1987a, 1987b, Freeze et. al., 1990, 1992,Massmann et. al., 1991, Sperling et. al., 1992). This function discounts the risks, costs,and benefits over time, in other words, it converts everything to net present value. Theobjective function is:= ± l [B(t)_c(t)_R(t)] (2.1)t=o(1+i)6where:= objective function (dollars)t = time (year)T = time horizon (years)i = discount rate (decimal fraction)B(t) = benefits in year t (dollars)C(t) = costs in year t (dollars)R(t) = risks in year t (dollars)The time horizon is the length of time over which the analysis is carried out.When the analysis is carried out from the perspective of the owner/operator of a facility,the time horizon is usually the expected lifetime of the facility, from the beginning ofconstruction until the facility is decommissioned, usually between 10 and 50 years(Massmann and Freeze, 1987a). If the analysis is carried out from the perspective of aregulatory agency, or the owner/operator is required to provide some guarantee againstfailure after the facility is decommissioned, the time horizon may be extended. Theperiod of time encompassed by the time horizon is often referred to as the complianceperiod.When the decision analysis is carried out from the perspective of theowner/operator of a facility, the current bank lending rate, also known as the marketinterest rate, is usually used for the discount rate (Massmann and Freeze, 1 987a, Freezeet. al., 1990). This discount rate generally ranges between 5% and 10%. If the decisionanalysis is carried out from the perspective of a regulatory agency, a social discount ratethat is much lower than the market rate is used (Massmann and Freeze, 1987a). Thebenefits to the owner/operator are mainly in the form of revenues for services rendered,and the costs are the capital costs and operating costs of construction and operation of thelandfill (Massmann and Freeze, 1 987a). The risks will be discussed later in this chapter.This study is concerned exclusively with the evaluation of design alternatives fora monitoring system constructed within the landfill property boundaries. There is no7analysis of the trade-off between design and monitoring, or of alternative methods ofmonitoring at the compliance boundary. To simplify the objective function, I amassuming that the revenues generated by this landfill would be the same regardless of themonitoring strategy adopted. Thus it is possible to neglect the benefits term. Since thetrade-off between design and monitoring is not considered, the capital costs ofconstructing and operating the landfill itself are also the same regardless of themonitoring strategy chosen. The same holds true of the costs associated with monitoringat the compliance boundary. Consequently, I will neglect these costs and consider onlythose costs directly associated with the construction and operation of the monitoringsystem located within the landfill boundaries. These costs are the cost of installing themonitoring system, which includes the costs of materials and labour, and the annual costof monitoring, which includes the costs of collecting, shipping and analyzing thegroundwater samples.The benefits term, which has been removed for the purposes of this study, is theonly positive term in the objective function, all the remaining terms are negative. Ratherthan choose the maximum of a number of negative values, the signs are reversed and thealternative that provides the minimum value for the objective function, representing theminimum of the actual costs plus the expected costs, will be chosen.The cost of installing the monitoring network is the only cost that occurs in yearzero, the year before the landfill cell and monitoring system begin operation. Year zero isalso the only year in which this one-time cost occurs. Consequently, the cost of installingthe monitoring network can be moved outside the summation and the summation can nowbegin in year one rather than year zero.The objective function becomes:(I) = + [Cmon (t) + R(t)] (2.2)=i (1 + 1)where:Cjt = the cost of installing the monitoring network (dollars)Cmon= the annual cost of monitoring (dollars)8Because the costs of installing the monitoring network occur in year zero, they arenot affected by the discount rate chosen. The other costs of monitoring, those associatedwith sampling, are ongoing and don’t begin until year one. The total net present worth ofthe sampling costs is dependent on the discount rate. The relative impact of the twocomponents of cost of monitoring on the total value of the objective function will varywith the discount rate chosen. The higher the discount rate, the smaller the impact ofcosts that are incurred in later years, and the larger the impact of the initial cost ofconstructing the monitoring system. It should also be noted that the time increments thatare used in this analysis are years rather than months or quarters. Consequently, all of thecosts that are accrued throughout any given year are not applied until the end of that year.2.2.1. RISK TERMThe risk term represents the expected costs associated with the detection ofcontaminants by the monitoring network and those associated with failure. The costsassociated with detection are those incurred by remediating the contamination. The costsassociated with failure often include fines, penalties, and the costs of litigation as well asremediation. For simplicity, the only costs associated with failure that I consider in thisstudy are those associated with the isolation of the contaminant plume. When applyingthis procedure, however, all of the costs associated with failure that an owner/operator islikely to incur should be taken into account.For a site with no monitoring network, the risk term represents the expected costsassociated with failure:R(t) = Pf (t)Cf (t) (2.3)where:Pf(t) = the probability of failure in year t (decimal fraction)Cf(t) = the costs that would arise as a consequence of failure in year t (dollars)For sites with monitoring networks, the above expression can be expanded toallow for the possibility of the plume being detected and remediated before failure occurs:9R(t)= [Pfm (t)Crf (t)] + [Pd (t)Crd (t)] (2.4)where:fm = probability of failure in year t with monitoring (decimal fraction)Crf(t) = cost of remediation, given failure, in year t (dollars)Pd(t) = probability of detection in year t (decimal fraction)Crd(t) cost of remediation, given detection, in year t (dollars)The probability of failure in year t, with monitoring, is defined asPf1 (t)= 1fnm (t)[l— ‘d 1 (2.5)where:Pf-(t) = probability of failure in year t without monitoring (decimal fraction)= total probability of detection over the compliance period (decimalfraction)Failure when a monitoring network is in place occurs when a contaminant plumefrom the landfill reaches the compliance surface without being detected by the monitoringnetwork. The fact that both of these conditions must be met for failure to occur isreflected in equation 2.5, where the probability of failure without monitoring is multipliedby one minus the total probability of a plume being detected, to produce the probability offailure with monitoring. The probability of detection that is used in equation 2.5represents the cumulative probability of a contaminant plume being detected at any time,during the compliance period, before it reaches the compliance surface. It does not matterin what year the plume would arrive at the compliance boundary, or how long it wouldtake to get there. This approach assumes that failure will not occur if the plume isdetected; in other words, it assumes that the remediation strategy adopted will becompletely successful. A monitoring network may not detect a contaminant plume if thecontaminant concentration is below detection levels, if the plume travels through thefracture network along fractures that are not being monitored, or, in the case of a pulserelease of contaminant, if it passes through a monitored location between the times atwhich that location is sampled.10There are two components of the cost of remediation, given detection, for theremediation strategy that I have chosen to use in this study: 1) the cost of installing theremediation network, and 2) the annual cost of operating this network. The details of thechosen remediation strategy are discussed later in this chapter. The cost of installing theremediation network is a one time cost that occurs only if a contaminant plume isdetected. Therefore, the probability that the remediation network will be installed in anygiven year is Pd(t), the probability of a plume being detected by the monitoring networkin that year. The operating costs for the remediation network begin once the remediationnetwork is installed and are ongoing until the end of the compliance period. Theprobability that the annual operating cost will be paid in any given year is the probabilitythat a plume has been detected by the monitoring network at any time up to and includingthe previous year.Incorporating the definition of the probability of failure with monitoring presentedin equation 2.5 and the two components of the cost of remediation given detection, alongwith their respective probabilities, into equation 2.4, the risk term becomes:t—1R(t)= ‘fnm (t)[1 — ‘d ]Crf (t) + d (t)Crdinst (t) + ‘d (i)Crdop (t) (2.6)For the purposes of this study, I am assuming that the costs associated with failureand both components of the cost of remediation given detection, remain constant withtime. In other words, the cost of installing a remediation network in year ten will be thesame as the cost of installing the same remediation network in year five, although the netpresent value of a system installed five years from now is more than one installed in tenyears time, if a discount rate greater than zero is assumed. This approach assumes thatthere will be no technological advances that significantly reduce the cost of remediation.Given this assumption and the definitions of terms stated above, the full objectivefunction used in this study is:= jst + ± 1 t{mon + fnm (t)[1 — d lCrf + d (t)Crdinst + d (rdop } (2.7)(1+i) i=111It should be noted that the annual cost of monitoring is unaffected by theprobability of detection; it is independent of the risk term. Monitoring continuesthroughout the entire compliance period, even if a contaminant plume is detected andremediation is implemented.Figure 2.1 shows an example of a decision tree based on a time independent(discount rate = 0) version of the objective function, which illustrates how the variouscomponents of the risk term are derived. For this example, I have assumed the followingcosts (in millions of dollars) and probabilities:cost of installing the monitoring network, = $0.1 Mtotal cost of monitoring, Cmon = $0.9Mcosts associated with failure (confinement), Cf = $5.OMcosts associated with detection (remediation), Cr $1.OMprobability of detection, d = 0.4probability of failure (without monitoring), Pf= 0.8These probabilities represent the total probability of the respective eventsoccurring within the compliance period. Because there is no consideration of the times atwhich events occur, no distinction has been made between the two components of thecost of remediation.There are two types of nodes in a decision tree: those that represent decisions(squares), and those that represent events that may occur as a result of those decisions(circles). The probabilities of the potential consequences of each event are recordedabove each path leading from an event node. Each path is also labeled with theconsequence that it represents. The sum of the probabilities on all of the terminal pathsof each decision leg must equal one. The initial node is the only decision node in thisdecision tree. This node represents the owner/operato?s decision whether or not to installa monitoring network. The first event node on each of the decision legs representsdetection, and the second event node represents failure. There are no failure nodes on thepaths on which detection has occurred, because I am assuming that failure will not occur12Figure 2.1: Decision tree based on a time independent version of the objective function.$0.1M, Cmon = $0.9M, Cf $5.OM, Cr = $1.OM, 1d 0.4, Pf =0.8.if the plume is detected. The dollar figure at the end of each terminal path is the value ofthat portion of the risk term, the expected cost, that is represented by that path.In this example, if the owner/operator decides to install a monitoring network, hisor her expected costs are the probability of the plume being detected (0.4) multiplied bythe cost of remediation ($1.0 million) plus the probability of failure with monitoring(0.48) multiplied by the cost of failure ($5.0 million). There are no costs associated with$O.40M$2.40Msososo13the consequence of neither detection nor failure occurring, so this term is not included inthe objective function. However, it must appear in the decision tree because it is apossible outcome, and there is a probability associated with it. If the owner/operatordecides not to install a monitoring network, his or her costs are the probability of failure(0.8) multiplied by the cost of failure ($5.0 million). The total of the expected costs forthe decision to monitor is $2.8 million, and for the decision not to monitor the expectedcosts are $4.0 million. The owner/operator’s expected costs are higher if he or shedecides not to monitor. The actual costs, however are higher for the decision to monitor.These are the costs of installing and operating the monitoring system ($0.1 million).There are no actual costs associated with the decision not to monitor. Therefore the totalvalue of the objective function for the decision not to monitor is $4.0 million. For thedecision to monitor, the total value of the objective function is ($0.1M + $0.9M + $2.8M)$3.8 million. In this case, it is to the owner/operator’s advantage to install the monitoringnetwork.2.3. DEcisioN ScENARIo2.3.1. OB.wcTIvE OF MONIToRING NETwoRKGenerally, the concerns of an owner/operator of a landfill facility are primarilymonetary in nature. To him or her, the objective of a monitoring system is to detect acontaminant plume before it has reached the compliance boundary in order to enableearly and less costly remediation of the plume, and thus avoid the potentially more costlyconsequences of failure. Only when the potential savings from detection, which include aless costly remediation scheme and a reduction in the probability of failure, are greaterthan the costs involved in constructing and operating the monitoring network plus theexpected cost of failure, will the owner/operator consider it worthwhile to install amonitoring network. The decision analysis is intended to assist with the assessment ofthe relationship between the monetary factors mentioned above.142.3.2. DETEcTIoN2.3.2.1. Definition of Detection in a Monitoring WellDetection is considered to occur at the earliest time at which a detectable amountof contaminant passes through a monitoring location during a monitoring period.Monitoring periods occur at regular intervals, for example every 60 days, throughout thecompliance period. A detailed discussion of the method used to determine detectionappears in Chapter Five of this thesis.2.3.2.2. Consequences of Detection: Remedial DesignThe remediation technique that I have chosen to use in this study involves theinstallation of horizontal interceptor wells at the time when detection occurs, andsubsequent pumping and treating of contaminated water. Because of the long periods oftime required to remediate groundwater contamination, I am assuming that, once theremediation network is installed, it remains in operation throughout the remainder of thecompliance period. A horizontal interceptor well has been used with some success atWilliams Air Force Base in Chandler, Arizona (Oakley et. al., 1992). It is believed thatwhen one of these wells is pumped, its “capture zone” will be much larger than that of aconventional vertical well, resulting in more efficient plume recovery. The remediationstrategy employed in this study involves the installation of these wells at approximately50 meter intervals downstream from the contaminant source, up to and including theinterval containing the monitoring well at which the plume was detected. For example, ifthe contaminant plume is detected at a monitoring well located less than 50 meters fromthe source, one interceptor well will be required. If, however, the monitoring well atwhich the plume is detected is located 125 meters from the contaminant source, threeinterceptor wells will be required to remediate the plume. This results in the cost ofremediation increasing with the distance from the contaminant source at which the plumeis detected. In other words, the larger the plume, the higher the cost of remediating thatplume. Massmann (1987a, l987b) indexed the cost of remediation to the size of theplume by means of a linear function that related the cost of remediation to the distancebetween the contaminant source and the monitoring network.152.3.3. FAILuRE2.3.3.1. Definition of Failure at the Compliance SurfaceAs stated previously, failure is defined as a contaminant plume from the landfillcell reaching the compliance surface without being detected by the monitoring network.The time of failure is determined to be the time at which 0.1% of the mass that isintroduced to the system as a pulse injection has reached or crossed the compliancesurface. The major assumption inherent in this definition is that monitoring at thecompliance surface is perfect. In other words, all of the groundwater that flows throughthe monitoring surface is analyzed and all of the contaminant in that water is detected.2.3.3.2. Consequences of Failure: Containment SystemThe consequences of failure can be many and varied. They include regulatorypenalties, the cost of litigation, remedial action, benefits foregone in the form of reducedrevenues, and loss of goodwill (Massmann and Freeze, 1987a, 1987b, Freeze et. al.,1990). I have concentrated on remedial action in this study and neglected the otherconsequences listed above. The remedial strategy that I have elected to use in the eventof failure involves containment of the plume by the construction of a grout barrier wall.It is assumed that an agreement exists between the landfill owner/operator and theregulatory agencies involved that, should contaminants be detected at the complianceboundary, a grout barrier must be constructed to contain the contaminant plume. A groutbarrier such as I envision has been built in fractured rock at an inactive hazardous wastelandfill in Niagara Falls, New York (Gazaway et. al., 1991).163. REVIEW OF THE FLOW AND TRANSPORT MODELPredicting groundwater flow and the transport of contaminants can be difficult.For the most part, the difficulties arise from problems associated with the characterizationof heterogeneities that exist in an underground environment. Predicting the transport ofcontaminants is much more difficult in fractured media than in porous media. Because ofthe discrete nature of fractures, the characterization of a fractured medium is a complexproblem. Groundwater flow in a fractured medium is largely confined to the fractures;there is virtually no flow in the rock matrix itself. Often, the bulk of the groundwaterflow occurs in a small fraction of the fractures, resulting in a channeling of the flow.Thus contaminants may become localized in a few major conduits. It can be very difficultand costly to identify these pathways in the subsurface.There are three approaches to modelling solute transport in fractured media:continuum models, discrete fracture models, and hybrid models which combine elementsof both of the above mentioned approaches. Long et. al. (1982) and Endo et. al. (1984)perfonned investigations to determine when a fractured medium can be modelled as anequivalent porous medium. Long et. al. (1982) found that in some cases, flow in adensely fractured medium could be adequately represented as an equivalent porousmedium. Endo et. al. (1984) demonstrated that a fracture system that behaves as acontinuum for fluid flow may not behave as a continuum for the transport of solutes.To date, several numerical models have been developed that simulate flow andtransport in discrete fractures on a variety of scales. These include models of a singlefracture (e. g. Raven et. al., 1988, and Shapiro and Nicholas, 1989), and network modelsin which each fracture is modelled discretely, in either two dimensions (e. g. Anderssonet. al. 1984, Endo et. al., 1984, Andersson and Thunvik, 1986, Dershowitz and Einstein,1987, Hull et. al., 1987, Long and Billaux, 1987, and Robinson and Gale, 1990) or threedimensions (e. g. Long et. al., 1985, Dershowitz and Einstein, 1987, and Cacas et. al.1990a, b).Modelling fluid flow in fractures on an individual basis can be computationallyintensive. Consequently, at the field scale, it is usually impractical to attempt to modeleach fracture discretely, particularly in three dimensions. In an attempt to overcome this17difficulty, several people have begun developing continuum models of transport infractured media. These include a stochastic continuum model that predicts the ensemblemean distribution of solute (Schwartz and Smith, 1988, Robertson, 1990, and Smith et.al., 1990), and a hybrid model where the major fractures are modelled discretely andblocks containing the secondary fractures are modelled as continua (Smith et. al., 1990).3.1. THE CHOICE OF MODELIn order to address many of the issues I wished to investigate in this study, it wasnecessary to employ a discrete fracture model. The computer resources and the timeavailable were insufficient to allow the use of a three-dimensional discrete model of adomain large enough for this study. Therefore, I chose to modify and use a two-dimensional discrete fracture model, “Discrete”, written by Tom Clemo, that uses particletracking to model solute transport (Clemo, in prep.).This model simulates a rectangular domain containing a two-dimensional fracturenetwork composed of planar fractures, each assigned a single aperture value.Groundwater flow is driven by a uniform, steady-state regional head gradient and occursonly in the fractures; the matrix is assumed to be composed of impermeable material withno open porosity. The model can be used to simulate a single network, or multiplerealizations can be generated for a Monte Carlo study.3.1.1. GENERATIoN OF FRAcTuREsThe fracture network includes both randomly generated multiple-fracture sets andindividual, explicitly described fractures. The multiple-fracture sets are generated in alarger region than the simulation domain; in this way, fractures whose centres lie outsidethe domain but extend into the domain are included, thus eliminating boundary effects.Once the fracture sets are generated, they are combined with the explicit fractures to formthe “generated fracture network”. This network then undergoes a “cleaning” processwhere fractures that do not form part of a connected path between two points on thedomain boundary are removed. Fracture segments that are connected at only one end arealso removed during this process. Fractures are not allowed to connect to impermeableboundaries. Figure 3.1 shows examples of a generated and a cleaned network. The18a)50S‘--- 40C03020So 10a)> 00 200b)— 50S40C03020J200Figure 3.1: Fracture network from base geometry. a) generated fracture network,b) cleaned fracture network.network shown is one realization of the fracture geometry that is used for the base case inthis study. When the model is operated in Monte Carlo mode, a new network is generatedfor each realization.3.1.1.1. Statistical Description of Fracture GeometryEach fracture set is generated from a statistical description consisting of sixparameters, assumed to be constant throughout the domain:50 100 150Horizontal Dimension ( m0 50 100 150Horizontal Dimension ( m191) The fracture density of the set. Fracture density is defined as the number offractures per meter that are intersected by a line perpendicular to the meanorientation of the fracture set.2) The mean fracture length of the set. The trace length of fractures within a sethas been represented by either a lognormal distribution (Bridges, 1975, Barton,1977, Baecher et. al., 1977, and Einstein et. al., 1980), or a negativeexponential distribution (Robertson, 1970, Call et. al., 1976, Cruden, 1977, andPriest and Hudson, 1981). In this model, each fracture set is assumed to have anegative exponential distribution of fracture lengths.3) The mean aperture of the fracture set. In their characterization of the PikesPeak Granite near Manitou Springs, Colorado, Bianchi and Snow (1968) foundthat the apertures they measured were distributed very close to a lognormaldistribution. This result was supported by the findings of pressure tests in alarge number of fractured rocks throughout the world (Snow, 1970). Thefracture sets in this model are assumed to have a lognormal distribution ofapertures.4) The standard deviation of the log of the fracture apertures.5) The mean orientation of the set. In this model, the fracture sets are assumed tohave a Gaussian distribution of orientations. Studies have shown that theorientations of fractures in a set follow a Fisher-von-Mises distribution (Cacaset. al., 1990a), which is analogous to a Gaussian distribution for hemisphericalprojections. Many two-dimensional fracture models assume a Gaussiandistribution of fracture orientations (Long et. al., 1982, Long andWitherspoon, 1985, Long and Billaux, 1987).6) The standard deviation of the orientation.A minimum fracture length and a minimum fracture aperture for each fracture set can alsobe stipulated.3.1.2. FLow SOLUTIONThe finite element method is used to obtain the hydraulic head values for the flowsolution. Each fracture segment is represented by a linear element with nodes at the20fracture intersections. The flow in the fracture segments is described by the cubic law foruniform, laminar flow between parallel plates:Q=1-b— (3.1)l2ii Alwhere:Q = volumetric flow per unit depth of the domainp = fluid densityg = acceleration due to gravity= dynamic viscosity of the fluidb = fracture apertureAh= difference in head between nodesAl =distance between nodesThe model will accept various combinations of impermeable and prescribed headboundary conditions at the edges of the domain.3.1.3. TRANsPoRTThe transport of solute is simulated using a particle tracking method. This methodemploys a large number of particles to characterize the movement of solute through aflow domain. Each particle moves in discrete steps from node to node. The particles areintroduced into the domain individually and each particle completes its transit of thedomain before the next particle is introduced. Since each particle is introduced at timezero, the model simulates a pulse injection of solute rather than a continuous source.3.1.3.1. Introduction of ParticlesThe particles may be introduced along any side of the domain, either at aparticular fracture specified as the th fracture along the chosen side, or over a rangespecified by a minimum and a maximum coordinate. When the solute is introduced over21a range, the choice of entrance fracture for each particle is determined probabilistically.The probabilities of a particle entering the domain through any of the fractures containedwithin the injection range are proportional to the flow entering those fractures;effectively, each fracture receives the same concentration of particles.3.1.3.2. Residence Time DistributionOnce a particle enters the domain, it is moved from node to node until it exits thedomain. The particle’s travel time is an accumulation of the residence times within eachfracture segment through which the particle has travelled. The residence time for a givenfracture segment is calculated from the bulk fluid velocity divided by the distancebetween the nodes.3.1.3.3. Routing in Fracture IntersectionsThe model contains two options for particle routing at fracture intersections:complete mixing and stream tube routing. With complete mixing, the probability of aparticle leaving an intersection through an outflow segment equals the proportion of thetotal outflow that is carried by that segment, and it is independent of the segment throughwhich the particle entered the node. Laboratory experiments show, however, that whenflow is laminar, there is little or no mixing in fracture intersections that have two inflowsand two outflows (Wilson and Witherspoon, 1976, Hull and Koslow, 1986, andRobinson and Gale, 1990). In a study using both a physical model and a numericalmodel, Hull et. al. (1987) found that the transfer of solute across streamlines contributedsignificantly to the dispersion of solute in fracture systems. In most fracture systems,neither the complete mixing model nor the streamtube routing model of mass transfer atfracture intersections is completely valid; a combination of both processes is occurring.The Peclet number, the ratio of the fluid velocity multiplied by a characteristic radius ofthe fracture intersection to the diffusion coefficient, can be used to characterize therelative importance of advection and diffusion within a fracture intersection. For Pecletnumbers greater than approximately 0.1, advection dominates solute transport through anintersection, and stream tube routing is the appropriate model for mass transfer (Smithand Schwartz, 1993).22Figure 3.2: Streamtubes in continuous intersection.In this study, mass transfer at fracture intersections is represented usingstreamtube routing. When a particle has entered a continuous intersection with twoinflows and two outflows (Figure 3.2) the probability of that particle exiting through agiven outflow segment is a function of that outflow segment’s position and the volume offlow it carries relative to the inflow segment through which the particle arrived, hereafterreferred to as the entry segment. A particle will tend to exit an intersection through theoutflow segment that is adjacent to the entry segment. If the adjacent outflow segmentcarries more flow than the entry segment, the probability of the particle exiting throughthat segment is one. If the adjacent outflow segment carries less flow than the entrysegment, the probability of the particle exiting through that segment is the ratio of theflow in the outflow segment to the flow in the entry segment. The probability of theparticle exiting through the opposite outflow segment is one minus the probability of theparticle exiting through the adjacent outflow segment. This approach assumes a uniformdistribution of particles across the entire width of a fracture.234. CHOICE OF MODEL DOMAINOnce the decision was made to use a two-dimensional model for the study, thequestion of whether to model the domain in vertical or horizontal section remained.4.1. H01Uz0NTAL SECTIONSince the fractures in a two-dimensional model are assumed to be of great extentin the direction orthogonal to the plane of the section, the fractures represented by ahorizontal section would be subvertical. Fracture networks composed chiefly ofsubvertical fracture sets occur in situations where the least principal stress was in thehorizontal plane when the fractures were formed. The most common geologicenvironment composed of subvertical fractures is columnar fractures in basalt. Thesefractures are formed as a result of tension in the horizontal plane caused by contraction ofthe rock mass as it cools (Davis, 1984).The effects of varying a number of decision and model variables can beinvestigated regardless of whether the section is oriented in the horizontal or verticalplane including: 1) the distance from the source to the monitoring network, 2) thefrequency of monitoring, and 3) the fracture geometry within the domain. However, onlya horizontal section can be used to investigate the relationship between the lateraldispersion of the solute and the spacing of the monitoring wells perpendicular to the meandirection of flow. The geometrical configuration of the monitoring wells, whether, forinstance, the wells are arranged in a line or an arc, can also be investigated only inhorizontal section.There are two major disadvantages in using a horizontal section: 1) the probabilityof intersecting a subvertical fracture with a vertical well is extremely low, and 2) one isunable to model subhorizontal fractures, which are often important conduits for themigration of contaminants. It would probably be necessary to resort to installing angledboreholes in order to raise the probability of intersecting a fracture to a point where theprobability of detection would become measurable.244.2. VERTICAL SECTIONUsing a vertical section allows one to model fracture geometries that are morecommonly found than the subvertical systems portrayed by a horizontal section. Thesegeometries consist of two or more sets of fractures with orientations varying from thesubhorizontal to the subvertical. In such systems, many of the subhorizontal fractures areclosed (nonconducting) as a result of the weight of the overburden, but the fewsubhorizontal fractures that remain open often form major conduits which can dominatethe flow pattern and control offsite migration of solute. This behavior has been observedat many sites, including the University of Arizona test site near Oracle, Arizona (Jones etal., 1985), and Environment Canada’s National Hydrology Research Institute field sitenear Chalk River, Ontario (Raven, 1986).There are some decision and model variables the effects of which can beinvestigated only with a vertical section. These include: 1) the criteria for choosingmonitoring locations within the wells, such as choosing to monitor the fractures with thelargest apertures or those that carry the highest flows, and 2) the number of monitoringlocations per well.One advantage to using a vertical section is that the probability of a monitoringwell intersecting at least one conducting fracture is high if the fracture network ishydraulically connected from the source to the compliance surface. The majordisadvantage is the inability to investigate the relationship between the lateral dispersionof the solute and the spacing of the monitoring wells perpendicular to the mean directionof flow. With a vertical section, one must assume that the solute moves in the plane ofthe section being monitored.I chose to use a vertical section to model the hypothetical landfill site. Theadvantage of being able to investigate the relationship between the lateral dispersion ofthe solute and the spacing of the monitoring wells with a horizontal section is outweighedby the disadvantages of being confined to investigating only fracture systems composedof subvertical fractures and the low probability of intersecting a subvertical fracture witha vertical well. The ability to investigate different criteria for choosing monitoring25locations within a monitoring well, as well as the ability to vary the number ofmonitoring locations in each well, are as important as the ability to vary the number andspacing of monitoring wells perpendicular to the mean direction of flow.4.3. DEscRIPTIoN OF DoMAINThe model domain representing the hypothetical landfill site is shown in Figure4.1. The model domain consists of a rectangular section of fractured rock, 200 m in the xdirection and 50 m in the z direction, with impermeable boundaries along the top andbottom of the domain. The left and right edges of the domain are constant headboundaries, with a higher head along the left boundary than the right. This scenario isrepresentative of a fractured hydrogeologic unit confined above by a layer of clays andsilts and underlain by relatively unfractured crystalline bedrock. The contaminant sourceis along the upper ten meters of the left boundary, representing the downstream edge of alandfill cell that has been excavated into the fractured rock unit to a depth often meters. Ihave assumed a thickness often meters for the upper confining layer, bringing the totaldepth of the landfill cell to 20 m. The right boundary of the domain represents thedownstream property boundary of the landfill site. This property boundary has beendesignated as the compliance boundary. If solute from the landfill cell crosses thisboundary, failure is considered to have occurred.///////////////////////////////////////////////////—lOm source 6h/6z=O‘1’h=h1 h=h, 50m6h/6z=O///////////////////////////////////////////////////200mFigure 4.1: Model domain representing hypothetical landfill site.264.4. PsEuDo-THREE-DIMENsIoNAL ANALYSISIn order to achieve consistency between the costs of monitoring and the costsassociated with detection and with failure, I found it necessary to devise a pseudo-three-dimensional approach to the decision analysis. The costs of installing and operating ahorizontal interceptor well network, as well as the cost of installing a grout barrier wall,both assume a three-dimensional domain; therefore, the cost of monitoring should alsoreflect a three-dimensional domain. A pseudo-three-dimensional analysis can beestablished by having each monitoring well that is depicted in the plane of the modelrepresent more than one well in the plane orthogonal to the model plane. I implementthis concept by dividing the hypothetical landfill site into a number of “strips”, or slices,all parallel to the plane of the modelled section as shown in Figure 4.2. I have arbitrarilychosen to use 10 slices as the base in the analyses. Any number of these two-dimensionalslices may be monitored, and the cost of monitoring in the slice that is simulated ismultiplied by the number of monitored slices. The main assumption made with thispseudo-three-dimensional approach is that the contaminant enters and travels in one sliceonly, consequently detection occurs in only one of the slices. The contaminant has anequal chance of entering any slice, but once it has entered a slice, it does not leave thatslice; the contaminant continues to travel within that slice until it reaches the compliancesurface. The probability that the slice in which the contaminant travels is monitored isthe ratio of the number of monitored slices to the total number of slices. The probabilityof the contaminant being detected becomes the probability of detection determined by thesimulation multiplied by the ratio of monitored slices. The probability of failure withmonitoring is also affected by this change in the probability of detection. I will bepresenting results from both two-dimensional and pseudo-three-dimensional analyses.Figure 4.2: Example of pseudo-three-dimensional domain with 10 slices.27285. DECISION MODEL5.1. NETwoRK DESIGNIn the design of a contaminant monitoring network, there are a number ofdecisions that must be made. Three of the most important considerations are:1) the number of wells to be installed and their locations (well siting)2) where, in each well, to position discrete monitoring zones (monitoringlocations)3) how often to take samples from the wells (monitoring interval).5.1.1. WELL SITINGThe use of a two-dimensional, vertical section restricts the investigation of theeffects of monitoring well location to a comparison of the effectiveness of monitoringwells placed at different distances from the contaminant source. With a vertical section,it is not possible to investigate the relationship between the spacing of monitoring wellsperpendicular to the mean direction of flow and transverse spreading of the contaminantplume. The investigation of the effect of the number of monitoring wells in the networkis restricted to whether there are one or two wells in the plane of the section, and to thenumber of slices that are monitored in the pseudo-three-dimensional analysis. Twomonitoring wells in the plane of the section represent two rows of monitoring wellsperpendicular to the plane of the section. The rationale behind installing two rows ofwells is that the row farther from the source can act as a backup system to detect amigrating contaminant plume if it is missed by the closer row of wells.5.1.2. MoNIToRiNG LOCATIONSOne or more discrete monitoring locations are usually isolated within amonitoring well. Unless individual fractures or zones of fractures are isolated,contaminated water entering the well from one fracture may be diluted byuncontaminated water entering the well from other fractures. This dilution can result in29deceptively low concentration readings at monitoring wells within contaminant plumes.Another reason to isolate monitoring locations within a well is the potential for thecontamination of previously uncontaminated sections of the aquifer. For this reason, theisolation of monitoring locations is often required by law.Monitoring locations are isolated within monitoring wells by means of packers,usually bentonite seals, installed in the well borehole both above and below themonitoring location. In this study, I am assuming a packer spacing, or length ofmonitoring location, of three meters. Because of the potential for error in placingpackers accurately when installing multiple packers in one borehole, it is commonpractice in the industry to drill a separate borehole for each monitoring location installedat a given well site. This is the approach that I have taken when determining the cost ofinstalling the monitoring network.Once a monitoring well site has been chosen, the question of where to position themonitoring locations remains. In porous media, monitoring locations are usuallypositioned in areas of relatively high hydraulic conductivity. These areas carry a largervolume of flow than areas of lower hydraulic conductivity, thus one would be more likelyto encounter migrating contaminants in the high hydraulic conductivity areas. Theaverage linear velocity of the groundwater is higher in high hydraulic conductivity areas,therefore contaminants travelling in these areas are likely to arrive at the monitoring wellsooner than those being transported through the lower hydraulic conductivity strata. Ihave investigated four criteria for selecting monitoring locations within a well bore infractured media: 1) monitoring the fractures along the borehole wall that carry the highestvolumetric flows, 2) monitoring the fractures along the borehole wall with the largestapparent apertures, 3) monitoring the areas of densest fracturing, and 4) placing themonitoring locations at predetermined depths. The first three of these criteria can all beconsidered to be analogous to some degree with areas of higher hydraulic conductivity inporous media.In the hypothetical field situation which I am portraying, once a well site had beenchosen, a borehole that fully penetrated the hydrogeologic unit, or area of concern, wouldbe drilled. If the fourth placement criterion, predetermined depths, was chosen, the initialborehole would be drilled to half the packer spacing below the depth of the deepest30monitoring location rather than fuiiy penetrating the hydrogeologic unit. A geophysicallog of the borehole would then be performed. A borehole TV and/or an acousticteleviewer can be used to identify the fractures with the largest apertures and the areas ofdensest fracturing. An examination of the drill core can also assist in locating areas ofdense fracturing. A borehole flow meter can be used to determine the locations of thefractures that carry the highest volumetric flows. It should be noted that borehole flowmeters that have a fine enough resolution to discriminate different volumes of flow at thelow levels that are often present in fractured rock are an emerging technology at thepresent time. In porous media, lithologic information and injection tests are used tolocate the areas of highest hydraulic conductivity.5.1.3. MoNIToRING INTERVALThe monitoring interval is the length of time between the collection of watersamples from the monitoring network. The annual cost of monitoring is directly tied tothe length of the monitoring interval; the more frequently water samples are collected andanalysed, the higher the annual cost of monitoring. However, the longer the monitoringinterval, the higher the risk of not detecting a contaminant plume. If the contaminantsenter the flow domain intermittently or in a pulse, the possibility of a concentration peakpassing through a monitoring location between monitoring periods increases with thelength of the monitoring interval. In this study, I investigate a number of monitoringintervals ranging from 60 to 360 days.5.2. DEFINING DETECTION5.2.1. DETEcTIoN THREsHoLDIn the transport simulation model used for this study, particle tracking methodssimulate the migration of a non-reactive solute through the model domain. Rather thansimply assuming detection to occur if a single particle passes through a monitoringlocation during a sampling time (monitoring period), I have incorporated the concept of adetection threshold in my determination of detection. The detection threshold is anattempt to mimic what happens in the field. The water samples that are collected fromeach monitoring location are used to determine whether or not contaminants have31migrated from the source. A key issue in interpreting the data acquired from the samplesis the concept of a threshold concentration at which it can be determined that acontaminant species has been positively identified at a monitoring location. If a solutethat is being used as an indicator species exists in the groundwater prior to the installationof the landfill facility, the detection threshold must be a large enough concentration to beeasily distinguishable from natural fluctuations in the background level of that species.5.2.2. MoNIToRING PERIODWhen a water sample is taken in the field, the water removed comes from avolume of the medium surrounding the monitoring location. The region of the domainthus affected is known as the sampling volume. Under natural gradient conditions, ittakes a certain period of time for the volume of water contained in the sampling volumeto pass through a monitoring location. When the detection threshold approach is used incombination with particle tracking, it is necessary to relate a monitoring period overwhich the particles passing through the monitoring location are counted to either thesampling volume around a monitoring location or the volume of water contained in thesample. In this study, I have chosen to relate the monitoring period to the samplingvolume.In the course of this study, I investigate three different fracture geometries, eachwith its own statistical description. Details of these fracture geometries are presented inChapter Six. Each fracture geometry exhibits different hydrogeological behaviour.Therefore, if the size of the sampling volume is held constant, the length of themonitoring period must vary between fracture geometries. I have adopted the followingprocedure to determine the length of monitoring period to be used.For each fracture geometry, a preliminary Monte Carlo simulation of 25realizations is performed. From the results of this simulation, the average dailyvolumetric flow through the domain, Q, the average daily flow rate through a monitoringlocation, Vf, and the average linear groundwater velocity, v, are computed. In eachrealization, 21 monitoring locations are sampled. These monitoring locations are locatedon seven monitoring wells in which the three intersected fractures carrying the highestflows are monitored. The average linear groundwater velocity is computed using the32average arrival time of the fiftieth percentile breakthrough fraction, t50, at the complianceboundary:200 mv= (5.1)t50Using this velocity and the average daily flow rate through the domain, theeffective porosity (ne) of the fracture system can be estimated:(5.2)Avwhere:A = the cross-sectional area of the downstream boundary of the domainI assume the shape of the sampling volume to be a cylinder surrounding and of thesame height, h, as the monitoring location. For a monitoring period of one day duration,the sampling volume for an “average” monitoring location would have a radius, r1, of:IVf/r1=1I /e (5.3)‘d ithI have chosen to use a monitoring period, mp, that corresponds to the travel timefor solute to pass through a sampling volume of 0.5 m radius:(5.4)r1It should be noted that the selection of the monitoring period for a given fracturegeometry is based upon certain average hydraulic properties of 25 realizations of thatfracture geometry. The selection is not based upon the properties of any individualrealization of that fracture geometry, nor is it based upon the behavior of the mediumsurrounding any specific monitoring location.335.2.3. IMPLEMENTATIONThe detection concentration threshold is set arbitrarily at a specific number ofparticles, 20 for example, in a fluid volume corresponding to the average daily flowthrough a monitoring location as calculated from the preliminary 25 realization MonteCarlo simulation of the base case fracture geometry. No attempt is made to defineabsolute concentration values by assigning a specific quantity of mass to each particle. Inthis study, I investigate several values for the detection threshold, including a singleparticle per monitoring period. I look at the sensitivity to the detection threshold of boththe probability of detection and the decision analysis objective function. The results ofthis investigation are discussed in detail in Chapter Six.The probability of detection at any given monitoring well site for a particularmonitoring strategy in any of the fracture geometries is determined by performing aMonte Carlo simulation of 200 realizations of the fracture geometry with the chosenmonitoring strategy implemented. The probability of detection at that monitoring wellsite is the proportion of realizations in which detection occurred at the well site withinthe compliance period. For example, if detection occurred at a monitoring well sitelocated 25 m from the contaminant source in 75 of the 200 realizations, the probability ofdetection at that well site would be 37.5%.To determine whether detection has occurred at a given monitoring location inany one realization, the particle concentration for each monitoring period is compared tothe detection threshold concentration. To calculate the particle concentration, the numberof particles that pass through a monitoring location during a monitoring period, is dividedby the volume of fluid that has flowed through that monitoring location during themonitoring period. The time of detection at a given monitoring location is set at themidpoint of the earliest monitoring period during which the detection thresholdconcentration has been met or exceeded in that monitoring location. Since the detectionthreshold concentration of particles is based on the “average” flow through a monitoringzone, the number of particles required to constitute detection varies with each monitoringlocation. For monitoring locations with a daily volumetric flow that is less than the“average”, fewer particles are required for detection, and, conversely, more particles are34required to constitute a detection in monitoring locations with a greater than “average”flow. To account for the “noise” that may be present as a result of using a particletracking method for the transport simulation, I have imposed a lower cutoff on theabsolute number of particles required for detection in monitoring locations with low flowrates. In most cases, this lower cutoff is one particle per day.Because the volume of water passing through a monitoring location during amonitoring period varies from one monitoring location to another, the sampling methodthat has been adopted for this study results in a different number of particles beingrequired for detection in different monitoring locations. Under ideal conditions, in a fieldsituation, the same volume of fluid would be collected from each monitoring location,thus detection would be based on the same mass of contaminant for each monitoringlocation.Values for the detection threshold concentration are based on simulations of thebase case fracture geometry with 20 000 particles injected at the source. This number ofparticles produces stable transport characteristics. Were a greater or lesser number ofparticles to be injected at the source in the base case geometry, the detection thresholdconcentration would be adjusted accordingly. However, different fracture geometrieshave different effective porosities. To keep the particle concentration per unit volume ofthe domain constant between different fracture geometries, I have adjusted the number ofparticles injected at the source, but have retained the same detection thresholdconcentrations. This approach assumes that the effective porosity calculated for the 25preliminary realizations of the entire domain accurately reflects conditions in the regionsurrounding the source.5.2.4. LIMITATIoNsIn the course of this study, I have found that the probabilities of detection aresensitive to the procedures I have used to convert particle counts to detection thresholdconcentrations and to determine the length of monitoring period, despite attempts toretain the same particle concentration at the source. For this reason, it is best to confinecomparisons of the probability of detection to those made between different monitoring35strategies within the same fracture geometry rather than attempt to carry out comparisonsof the probabilities of detection between different fracture geometries.I have attempted to mimic the sampling process used in a field setting by using adetection threshold concentration rather than an absolute particle count to indicatedetection. However, because of limitations inherent in the particle tracking method andthe use of a two-dimensional model to represent a three-dimensional situation, thedetection threshold concentrations used in this study bear only a qualitative relationshipto those that would be used in a field setting. Therefore, the probabilities of detectionobtained in this study must be viewed in relative terms, and not be construed to representvalues that could be expected at a specific field site.5.3. DEFINING FAiLu1uAs stated in Chapter Two, failure is defined as a contaminant plume from thelandfill cell reaching the compliance surface without being detected by the monitoringnetwork. The time of failure is determined to be the time at which 0.1% of the mass thatis introduced to the system has reached or crossed the compliance surface. As eachparticle crosses the compliance surface, the particle count for the time interval duringwhich the particle crosses is incremented. Once transport is complete, the time of failureis set at the midpoint of the time interval during which the cumulative total of particlesthat have passed the compliance surface reaches or exceeds 0.1% of the total number ofparticles injected at the source. The probability of failure for a given fracture geometry isthe proportion of the total number of realizations in which failure occurs during thecompliance period.5.3.1. AssuMPTioNs CoNcERNING THE COMPLIANCE SuRFAcE AND FAILuREThe major assumption inherent in this definition of failure is that the monitoringat the compliance surface is perfect. In other words, all of the groundwater that flowsthrough the monitoring surface is analyzed and all of the contaminant in that water isdetected. Although monitoring at the compliance surface is unlikely to be this completein a field setting, the probability of detection at the compliance surface is a separate issuethat I have chosen not to investigate in this study.36Because every fracture that crosses the compliance surface is assumed to bemonitored constantly, it is not convenient to set a threshold concentration for failure. Thecriterion for failure is arbitrarily set at 0.1% of the injected mass of contaminant. Thisfigure is assumed to be large enough that failure will not be triggered by the arrival of afew anomalous particles that might arrive at the compliance surface well in advance ofthe rest of the plume. At the same time, it is assumed to be small enough to signal thebeginning of the arrival of the main body of the plume.376. RESULTS6.1. THREE F1cTu1u GE0METIUEsIn the course of this study, I investigate three different fracture geometries. Thestatistical input parameters for all three geometries are listed in Table 6.1. All threegeometries contain three fracture sets: one subhorizontal, one subvertical, and one with amean orientation 45 degrees counterclockwise from horizontal. In both the basegeometry and geometry two, the subhorizontal fractures are considerably longer than thefractures forming the other two sets. Geometry two is identical to the base geometryexcept that the mean aperture of the subhorizontal fracture set is twice that in the basegeometry. In geometry three, all three fracture sets have the same density, length andaperture parameters. The fracture networks generated for the first realization of eachfracture geometry are shown in Figure 6.1. This figure shows that the fracture networksfor geometry three are considerably denser than those for the other two geometries, andthat they do not contain the long horizontal fractures seen in the base geometry andgeometry two.It is advantageous in the numerical procedure to introduce the monitoring wellsinto the domain as long vertical fractures. These fractures, located at 50 meter intervalsthroughout the domain, have an aperture of one micron, at least an order of magnitudesmaller than the mean aperture of any of the fracture sets in any of the geometriesinvestigated. The vertical fractures used to simulate a monitoring well are assigned smallapertures so that they will have as little effect as possible on the natural flow conditionswithin the domain. When the vertical fractures used to simulate a monitoring well weregiven apertures of less than one micron, the model often had problems arriving at a flowsolution. The vertical fractures used to simulate a monitoring well are entered into themodel explicitly. They are neglected in the first part of the cleaning process, when theunconnected fractures are removed, to avoid the situation in which a monitoring wellfracture provides a connection in a network that would otherwise be unconnected.38Table 6.1: Statistical Input Parameters for the Three Fracture Geometries InvestigatedFracture Parameter Base Geometry 2 Geometry 3Set Geometryfracture density (m4) 0.35 0.35 0.50mean fracture length (m) 16.0 16.0 5.0mean fracture aperture (m) 20.OE-6 40.OE-6 20.OE-6a log aperture 0.40 0.40 0.60mean fracture orientation (deg.) 0.00 0.00 0.00a orientation (deg.) 10.00 10.00 10.002 fracture density (rn-i) 0.40 0.40 0.50mean fracture length (rn) 4.0 4.0 5.0mean fracture aperture (rn) 40.OE-6 40.OE-6 20.OE-6a log aperture 0.60 0.60 0.60mean fracture orientation (deg.) 90.00 90.00 90.00a orientation (deg.) 15.00 15.00 15.003 fracture density (rn-i) 0.50 0.50 0.50mean fracture length (m) 7.0 7.0 5.0mean fracture aperture (m) 10.OE-6 10.OE-6 20.OE-6a log aperture 0.60 0.60 0.60mean fracture orientation (deg.) 135.00 135.00 135.00a orientation (deg.) 40.00 40.00 40.003a)—S 50S•— 40030200b)—S 50S— 400030ci)20ccio 10Ii)> 0200c)50S40030II)S 20o 10ci)> 00 200Figure 6.1: Fracture networks for the first realization of each geometry. a) basegeometry, b) geometry two, c) geometry three.50 100 150Horizontal Dimension ( m0 50 100 150Horizontal Dimension ( m50 100 150Horizontal Dimension ( m40The fracture networks generated from each fracture geometry exhibit differenthydrogeological behavior. Table 6.2 lists values of different flow and transportcharacteristics of the different fracture geometries obtained from the preliminary MonteCarlo simulations of 25 realizations that were performed for the monitoring periodcalculations. (The data required to calculate most of these characteristics were generatedin the preliminary simulations only; they were not generated in the larger productionsimulations of 200 realizations.) On average, the networks generated from geometry twohave approximately four times the flow through the network and an effective porositymore than one and a half times greater than do those generated from the base geometry.These networks have a mean linear groundwater velocity almost three times as large as inthe base geometry. The relatively higher velocity in geometry two is reflected inFigure 6.2, a plot of the cumulative probabilities of failure for all three fracturegeometries, based on Monte Carlo simulations using 200 realizations. During the first tenyears of the compliance period, geometry two has a much larger probability of failurethan does the base geometry, although the probabilities of failure for both geometries arerelatively close by the end of the compliance period. This behaviour is the result of thehigher groundwater velocities and consequent shorter travel times through the geometrytwo networks. Because the contaminants are travelling through the domain more quicklyin the geometry two networks, failure occurs at earlier times in a larger proportion of therealizations than in the networks forming the base geometry. Fracture networksgenerated from geometry three have an average flow almost five times smaller than thatin the base geometry, and the groundwater in these networks has an average linearvelocity that is smaller by a factor greater than five. The average effective porosity of thenetworks, however, is almost the same for both geometries. The low velocity in thegeometry three networks is evident from the late failure times and relatively lowcumulative probability of failure within the compliance period as illustrated in Figure 6.2.Figure 6.3 is a representation of the volumetric flows through the fractures in thefracture networks of the first realization of each of the three geometries. Although thesefigures are based on a single realization of each fracture geometry, they are representativein a general way of the flux distributions in the networks generated from the respectivegeometries. The line thickness with which each fracture is drawn is proportional to theflow through that fracture. Those fractures carrying the highest flows are drawn with the41Table 6.2: Flow and Transport Characteristics from Preliminary Simulations of the ThreeFracture Geometries InvestigatedParameter Base Geometry Geometry 2 Geometry 3LL000.EC.)mean volumetric flow through 5.22E-6 2.29E-5 l.12E-6monitoring location1(m3/day)mean volumetric flow through 1 .74E-5 7.43E-5 3.49E-6network (m3lday)mean travel time of 50tli 8960percentile of plume (days)mean linear velocity of flow .022(mlday)mean effective porosity of 1 .56E-5 2.52E-5 1 .68E-5network1The fractures carrying the highest flows are monitored.3390 48020.059 .004— ——10.80.60.40.20////— —— geometry twobase geometrygeometry three0 5 10 15 20 25Year30Figure 6.2: Cumulative probabilities of failure throughout compliance period for all threefracture geometries investigated.42thickest line width, and those fractures drawn with the narrowest line width carry lessthan 1/30 of the maximum flow in the fracture network. The maximum flow carried inany fracture in the base geometry network is 2.7E-5 m3/day. The fractures in thegeometry two network carry a maximum flow of 3.3E-5 m3/day, slightly higher than thebase geometry, and the maximum flow in the geometry three network is 1 .5E-6m3/day,smaller than the base geometry by a factor of almost 20. Consequently, for relativecomparisons between the three geometries, the line widths can be compared onlyqualitatively. Each fracture network has one or more preferred flow paths which wouldbe expected to act as major conduits for transport. The preferred flow paths through thebase geometry network and the geometry two network are very similar, although there area smaller number of fractures carrying high flows in the geometry two network, creating aslightly less tortuous, more direct, preferred flow path. Both of these fracture networkshave preferred flow paths that are considerably less tortuous than those in the geometrythree network.The tortuosity of the preferred flow paths in fracture networks generated fromgeometry three is a contributing factor to the lower average linear velocity and longercontaminant plume travel times that have been observed in these networks. Thehydraulic head difference between the left and right boundaries of the domain is the samefor all of the networks generated from all three geometries. The more tortuous thepreferred flow paths, the lower the effective permeability within a network.Consequently, although the networks generated from the base geometry and geometrythree have similar average effective porosities, the average effective permeability isconsiderably lower in the geometry three networks. This lower average effectivepermeability results in a lower volumetric flow through the boundaries. Although the lesstortuous flow paths in the networks generated from geometry two contribute to a higheraverage effective permeability than in the base geometry networks, the increased meanaperture of the subhorizontal fractures plays a role as well. The permeability of a fractureis proportional to the square of its aperture. The higher average effective permeability ofthe networks in geometry two brought about by the less tortuous preferred flow paths,combined with the increase in the permeability of the preferred flow paths brought aboutby a higher mean aperture, results in a larger volumetric flow through the domainboundaries in geometry two than in the base geometry.S0D)a)SC.)a)—S 50S0a)SC)a)— 50S0a)SC)a)43504030201000b)50 100 150= 2.7e—05 m3/day Horizontal Dimension ( m200403020100c)o 50 100 150—= 3.3e—05 m3/day Horizontal Dimension ( m2004030201000 50 100 150= 1.5e—06 m3/day Horizontal Dimension ( m200Figure 6.3: Volumetric flows through fractures in the fracture networks for the firstrealization of each geometry. a) base geometry, b) geometry two, c)geometry three.446.2. BASE CASE6.2.1. MoNIToRING PARAMETERsAll of the monitoring locations in all of the Monte Carlo simulations for all threefracture geometries have a packer spacing of three meters. For the base geometry andgeometry two, the monitoring interval used for the base case is 60 days. Because the lowvolume of flow through the networks in geometry three necessitates the use of a longermonitoring period, 180 days is used as the monitoring interval for the base case forgeometry three. The detection threshold for the base case is 3. 83E6 particles/m3,whichis the equivalent of an average of 20 particles/day passing through an average monitoringlocation in the base geometry when the fractures carrying the highest flows aremonitored. The depths of monitoring locations chosen for the predetermined depthmonitoring scheme are 16.7 m, 25.0 m, and 33.3 m from the bottom of the fractured rockunit. When one monitoring location is installed at a monitoring well site when thepredetermined depth monitoring scheme is implemented, it is installed at mid depth of thedomain, 25 m from the bottom. When two monitoring locations are required, they areinstalled at 16.7 and 33.3 m from the bottom of the domain. When any of the othermonitoring schemes are implemented, the monitoring locations are installedhierarchically. For instance when the highest flow monitoring scheme is implementedand two monitoring locations per well site are required, the monitoring locations arecentred about the two fractures carrying the highest flows. Sensitivity studies of thedetection threshold and length of the monitoring interval are carried out using the basegeometry. The results of these studies are discussed later in this chapter.6.2.2. DEcisioN ANALYSIS PARAMETERsThe decision analysis parameters that are used for the base case are listed inTable 6.3. The values used for the monitoring costs were obtained from various localconsulting engineers, geophysical firms, and well drilling contractors, and from a localwater testing laboratory. An order of magnitude estimate for the cost of constructing ahorizontal interceptor well was arrived at through conversations with various consultingengineers and equipment suppliers at a 1992 EPA conference. The cost of constructing a45Table 6.3: Decision Analysis Parameters for Base CaseBasic Objective Function Parameters:discount rate (%) 5.0compliance period (years) 30.0total number of slices in pseudo-three-dimensional domain 10number of monitored slices 3Monitoring Cost Parameters:advance rate for drilling (mlhr) 5.0drill rig chargeout rate ($/br) 145technician chargeout rate ($fhr) 60cost of 3m length of well casing ($) 30cost of 3m length of well screen ($) 50cost of 25 1 bag of bentonite ($) 35cost of 30 1 bag of sand ($) 10cost per well site of borehole logging ($) 1000cost of collecting, shipping and analysing each water sample 525($)Costs Associated with Detection:cost of constructing each interceptor well ($) 500 000annual cost of operating each interceptor well ($) 20 000Costs Associated with Failure:cost of constructing grout barrier wall ($) 5 000 00046grout barrier wall is based on the cost of a similar wall constructed in fractured rock atNiagara Falls, New York. Sensitivity studies of the discount rate, the costs associatedwith failure, and the number of monitored layers are carried out using the base geometry.The results of these studies are discussed later in this chapter.6.3. BAsE GEOMETRYThe monitoring period calculated for the base geometry is seven days. Sevendays is the amount of time required for the water contained in a three meter longcylindrical sampling volume with a diameter of one meter to pass through an averagemonitoring location when the fractures carrying the highest flows are monitored. Theminimum number of particles required for detection in monitoring locations with lowvolumetric flow rates is seven, which corresponds to one particle per day throughout themonitoring period. The total number of particles injected in each realization is 20 000. Adetailed discussion of detection and the method used to determine the length of themonitoring period was presented in Chapter Five.6.3.1. MoNIToRING SCHEME CoMPARIsoNThe total probabilities of detection over the compliance period vs. the distancefrom the source are plotted in Figure 6.4 for all four monitoring schemes. Theseprobabilities are those obtained from 200 Monte Carlo realizations in a two-dimensionalanalysis performed with the basic monitoring parameters and one monitoring location perwell site. Confidence intervals are not relevant because each probability determined inthis study is a single data point representing the proportion of realizations in which theevent occurs. The highest flow monitoring scheme consistently provides the highestprobability of detection, followed by the densest fracturing, largest aperture, andpredetermined depth schemes respectively. The curves for both the highest flow and thepredetermined depth monitoring schemes are smooth convex curves that peak in thevicinity of the well site located 75 m from the contaminant source. The anomalies in thedensest fracturing and largest aperture monitoring scheme curves are likely the result ofthe limited number of Monte Carlo simulations performed. When 200 differentrealizations were performed with the densest fracturing monitoring scheme, the plot ofthe total probabilities of detection vs. the distance from the source took on the same47100_____• high flow monitoring—°—-— density monitoring80aperture monitoring60—X— predetermined depth40e 20xx00 25 50 75 100 125 150 175 200Distance from Source (m)Figure 6.4: Total probabilities of detection over the compliance period vs. distance fromthe source for the base geometry.convex shape as the curves for the highest flow and the predetermined depth monitoringschemes in Figure 6.4. The difference in the probabilities of detection between the twosets of realizations was between 3% and 4%. It is likely that if more than 200 MonteCarlo realizations were performed, all of the curves would take on the smooth convexshape of the curves for the highest flow and the predetermined depth monitoring schemesin Figure 6.4.The probabilities of detection closer to the source than the peak are lower becausethe contaminant plume has not yet had much opportunity to disperse vertically. Thecontaminant, which is introduced in the top ten meters of the domain, is more likely to betravelling in fractures located in the upper portion of the domain in the region close to thecontaminant source. With the exception of the predetermined depth monitoring scheme,a single monitoring location in any of the well sites is just as likely to be located in thelower portion of the domain as it is in the upper portion. When the predetermined depthmonitoring scheme with one monitoring location per well site is employed, themonitoring location is located at mid depth, 25 m from the bottom of the domain. Whilethe plume is small and travelling in the upper portion of the domain, it is more probablethat the bulk of the contaminant may be localized in fractures that are not monitored, and48consequently not be detected. As the plume continues to travel through the fracturenetwork it becomes more disperse as it extends over a larger portion of the domain. Asthe plume spreads, it is more likely to be travelling in fractures that are monitored. Oncea certain amount of spreading has occurred, however, the plume may become diluted tothe point where the contaminant concentrations are lower than the detection thresholdconcentration. This dilution is reflected in the tailing off of the probabilities of detectionat the monitoring well sites located farther than 100 m from the contaminant source.The time at which detection occurs affects the value of the objective function.Early detection times will result in a high value for the expected cost of detection in twoways: the earlier a plume is detected, the longer the remedial network must be inoperation, and the earlier detection occurs the less the costs associated with detection arediscounted with time. The cumulative probabilities of detection vs. time at both 25 m and75 m from the source for all four monitoring schemes are depicted in the plots inFigure 6.5. These plots are for the case where there is one monitoring location per wellsite. In the high flow, densest fracturing, and the large aperture monitoring schemes, theprobability of detection at 25 m is higher than that at 75 m in the early times. By the endof the compliance period, the probability of detection is higher at 75 m in the high flowand largest aperture schemes, and the probability of detection at both monitoring wellsites are similar in the densest fracturing monitoring scheme. This behaviour is a result ofthe time lag in the time at which the contaminant plume first reaches the monitoring wellsite at 25 m and when it arrives at the monitoring well site located 75 m from the source.The probability of detection in the predetermined depth monitoring scheme ishigher at 75 m from the source than at 25 m throughout the entire compliance period.The monitoring location in this monitoring scheme is located at mid depth of the domain.The probability of a plume being detected is higher at 75 m from the source because theplume has had more opportunity to disperse by the time it has travelled that far, and ismore likely to have spread into the middle of the domain. There is no apparent time lagwhen this monitoring scheme is implemented. The most likely explanation of thisbehaviour is that in those realizations where the plume is detected at the 25 m well site,the paths through which the plume has travelled are tortuous enough to cause sufficientdispersion that the plume has spread into the middle of the domain. The travel timesthrough these tortuous paths are high in comparison to the travel times in most of therealizations. When the other three monitoring schemes are implemented, some of thea) b)• 10.8- .2 0.60.4CoD 0.2E80.0Co.0 o0DEC.)Figure 6.5: Cumulative probability of detection vs. time at 25 m and 75 m fromcontaminant source for base geometry with one monitoring location permonitoring well site. a) highest flow monitoring scheme, b) densest fracturingmonitoring scheme, c) largest aperture monitoring scheme, d) predetermineddepth monitoring scheme.plumes that are detected at the 25 m well site are still relatively small and have travelledrelatively quickly. They are detected because the monitoring location is in the upperportion of the domain and in the path of the plume. With all four monitoring schemes,detection occurs during the first 10 or 15 years at both well sites in the majority of therealizations. The near horizontal slope of both curves beyond 15 years in all four plotsindicates that few detections occur beyond this time.The plots in Figure 6.6 provide a comparison of the total probabilities of detectionobserved with one, two and three monitoring locations per well site for each monitoringscheme. With all four monitoring schemes, the more monitoring locations at each wellsite, the higher the probability of detection. The difference between the probabilities of75 m fromsource—— 25 m from49— 75 m fromsource—— 25 m fromsource0 5 10 15 20 25 30c)0.80.60.40.20d)0.80.60.40.200.80.60.40.2075 m fromsource—— 25 m fromsource0 5 10 15 20 25 30— 75 m fromsource—— 25 m fromsourceYear0 5 10 15 20 25 30 0 5 10 15 20 25 30Yeara) b)100-80604020o 000 25 50 75 100 125 150 175 200Figure 6.6: Total probability of detection vs. distance for base geometry with one, two,and three monitoring locations per monitoring well site. a) highest flowmonitoring scheme, b) densest fracturing monitoring scheme, c) largestaperture monitoring scheme, d) predetermined depth monitoring scheme.detection from one to two monitoring locations is greater than the difference from two tothree monitoring locations at every well site for each of the four monitoring schemes. Allof the curves in all of the plots exhibit lower probabilities close to the source, the highestprobability of detection at 50 or 75 meters from the source, and a trailing off of theprobability of detection at larger distances. This behaviour is particularly evident whenthe highest flow monitoring scheme is employed. When this monitoring scheme is used,the difference in the probability of detection between one monitoring location and twomonitoring locations at the 25 m well site is considerably larger than that at any of thewell sites that are farther from the source. This large spread in the probabilities ofdetection implies that, while the plume is small and located in the upper portion of thedomain, the chances of detecting a contaminant plume increase much more by theaddition of a second monitoring location at each well site than they do once the plume ismore dispersed. When two monitoring locations are installed at a well site, there is ahigher probability that there will be at least one monitoring location in the upper half ofC0a)0000100806040200500 25 50 75 100 125 150 175 200c)100806040200d)1008060402003 locationS/wellSite2 locations/wellSitetocatlon!weltile0 25 50 75 100 125 150 175 200Distance from Source(m)0 25 50 75 100 125 150 175 200Distance from Source(m)51addition of a second monitoring location at each well site than they do once the plume ismore dispersed. When two monitoring locations are installed at a well site, there is ahigher probability that there will be at least one monitoring location in the upper half ofthe domain than there is if oniy one monitoring location is installed. Whether the increasein the probability of detection resulting from the installation of additional monitoringlocations at any given well site is worth the additional expense of installing and samplingfrom those monitoring locations is a question that must be evaluated within the decisionanalysis framework.A histogram of the values obtained for the objective function with the base caseanalysis is shown in Figure 6.7. There are three values for the objective function given ateach well site. These represent the values obtained for that well site with one, two, andthree monitoring locations installed. The column closest to the viewer represents thevalue of the objective function that is obtained with one monitoring location installed atthe well site; the middle column is for two monitoring locations, and the farthest columnis for three monitoring locations. The highest flow monitoring scheme consistentlyprovides the lowest values for the objective function, while the predetermined depthscheme provides the highest. The values provided by the largest aperture and the densestfracturing monitoring schemes are close but the values for the largest aperture scheme areusually slightly higher. The value of the objective function that is obtained for thisgeometry when no monitoring is undertaken is $2.64 million. The predetermined depthmonitoring scheme is the only one that does not provide at least one option with ano •4,.— C0I.I-depthaperturedensityflow22.50SU-,c’JFigure 6.7: Values of objective function for base geometry with base case analysis.52In almost every instance, one monitoring location per well site provides a lowervalue for the objective function than do two or three monitoring locations per well site.The higher values obtained for the objective function when more than one monitoringlocation is installed indicate that the reduction in the expected costs associated withfailure, incurred as a result of the increase in the probability of detection, are not as greatas the additional expenses incurred. These expenses include both the cost of installingand taking samples from the additional monitoring locations and an increase in theexpected costs associated with detection that results from an increase in the probability ofdetection. The two instances where one monitoring location per well site does notprovide the lowest value at the well site occur at the 25 m well site when either thehighest flow or the predetermined depth monitoring scheme is implemented. In theseinstances, the difference in the probability of detection between one and two monitoringlocations is considerably larger than in any of the other situations. Thus in these cases thereduction in the expected costs of failure outweigh the increase in both the costsassociated with monitoring and the expected costs associated with detection.The lowest value for the objective function overall, $2.41 million, occurs at thewell site 50 m from the source, with one monitoring location centred about the fracturecarrying the highest flow. This is not the situation that provides the highest probability ofdetection. The highest probability of detection, 88.5%, occurs with the flow monitoringscheme at the well site located 50 m from the source with three monitoring locationsinstalled. When one monitoring location is installed per well site, the highest probabilityof detection occurs at the well site located 75 m from the source with the fracturescarrying the highest flows monitored. This does not provide the lowest objectivefunction, however, because two horizontal interceptor wells are required for theremediation program if the contaminant plume is detected 75 m from the source. If thecontaminant plume is detected 50 m from the source, only one interceptor well isrequired. Any advantage gained by the higher probability of detection at 75 m is morethan offset by the additional expected expense of installing and operating two interceptorwells rather than one.6.3.2. SENsITIvITY STUDIES53A number of sensitivity studies involving both monitoring parameters anddecision parameters are carried out using the base geometry. Unless otherwise indicated,the results presented are those obtained when the highest flow monitoring scheme isimplemented. This monitoring scheme was chosen for the sensitivity analyses, because,in the base case analysis of the base geometry, it provided the highest probabilities ofdetection and it is the preferred monitoring scheme with the lowest values for theobjective function. This is the monitoring scheme that the owner/operator of the landfillfacility should choose to install on a site located above a fractured rock unit with thisfracture geometry. When the sensitivity study analyses were carried out using othermonitoring schemes, the same basic behaviour was exhibited as when the highest flowmonitoring scheme was implemented. Except where explicitly stated, only one parameteris varied in each study; the remaining parameters are equal to those in the base caseanalysis. As in the monitoring scheme comparison, the probabilities of detection reportedare those obtained from a two-dimensional analysis.6.3.2.1. Detection ThresholdOne of the sensitivity studies performed with the base geometry is the effect ofdifferent detection concentration thresholds on both the probability of detection and thevalue of the objective function. A plot of the probability of detection vs. distance fromthe source for three different detection thresholds is shown in Figure 6.8. This figurereports the total probabilities of detection over the compliance period for two thresholdconcentrations, 1.92E6 and 3.83E6 particles/rn3,and a detection threshold of one particleper monitoring period. The two threshold concentrations represent, respectively, 10 and20 particles per day passing through an average monitoring location in the base geometry,when the highest flow monitoring scheme is implemented. These probabilities are thoseobtained when one monitoring location is installed at each well site.54100a- 800.4-’600‘I040• I particle per monitoringperiod20—°-—— I .92E6 particlesIm3A 3.83 particleslm**30 I0 25 50 75 100 125 150 175 200Distance from Source (m)Figure 6.8: Probability of detection vs. distance from the source for three differentdetection thresholds with base geometry.At all monitoring well sites, the higher the detection threshold, the lower theprobability of detection. The probabilities of detection obtained with each of the threedetection thresholds are relatively close to each other at the well sites located at 25 m and50 m from the source, but diverge at greater distances. When only one particle permonitoring period is required for detection, the probability of detection increases withdistance to a distance of 125 m from the source, where it levels off. This steady rise withdistance reflects the increasing likelihood, as the plume disperses, that contaminants willbe transported in fractures that are monitored. With a detection threshold of only oneparticle per monitoring period, no dilution effects are evident. The effects of dilutionwith dispersion become evident in the curves representing the two thresholdconcentrations. With the lower of the two threshold concentrations, the probability ofdetection increases until 100 m from the source, and then decreases slowly with distance.When an average of 20 particles per day are required for detection, the probability ofdetection decreases rapidly with distance beyond a distance of 100 m. The more particlesrequired to constitute detection, the more rapid the decrease in the probability of detectionas the contaminant plume becomes more disperse and the contaminants more dilute. The55distance from the source at which dilution becomes an issue is smaller with a higherthreshold concentration.Figure 6.9 is a histogram of the values obtained for the objective function with thethree detection thresholds. At all well sites located more than 25 m from the source withall three detection thresholds, the lowest value for the objective function is achieved whenone monitoring location is installed. The lower the detection threshold, the lower thevalue obtained for the objective function. When the same number of monitoringlocations per well site are installed, a lower detection threshold will produce a higherprobability of detection. In this situation, the actual costs are the same, regardless of thedetection threshold, but the expected costs vary; the costs associated with detectionincrease, and the costs associated with failure decrease. The value of the objectivefunction decreases with the detection threshold because the decrease in the expected costof failure brought about by the increased probability of detection outweighs the increasein the expected cost of detection. In general, the differences in the value of the objectivefunction are less between detection thresholds than they are between different numbers ofmonitoring locations. Even though the increase in the probability of detection at a givenwell site brought about by the addition of another monitoring location may be greater. 0.0Cu-383 parUcles/m**3I .92E6 particles/m**31 parbcle/mon. penodmFigure 6.9: Values of objective function for three different detection thresholds with basegeometry.2.7100m75m50m2556than the difference between any two detection thresholds when the same number ofmonitoring locations are installed, the additional cost of monitoring is much greater thanthe net reduction of the expected costs brought about by the increased probability ofdetection.The lowest value of the objective function, $2.34 million, occurs at 50 m from thesource when one monitoring location is installed and one particle per monitoring period isrequired for detection. The rise in the values of the objective function at 75 m from thesource reflects the increased expected cost of remediation brought about by therequirement of installing and operating a second horizontal interceptor well. The valuesobtained with the two lower detection thresholds decrease again at 100 m from thesource. This decrease is probably brought about by two factors: an increase in the totalprobability of detection and a later time of detection. When the plume is detected at alater time, the expected costs of remediation are reduced because of both discounting withtime and the shorter period of time over which the remediation network must be inoperation.It should be remembered that the detection thresholds used in this study arearbitrary. No attempt is made to define absolute concentration values by assigning aspecific quantity of mass to each particle. This sensitivity study shows that the objectivefunction is relatively insensitive to the detection threshold chosen, as is the probability ofdetection at well sites close to the contaminant source. Dilution may conceivably play arole when a metal or salt that occurs naturally in the groundwater is used as an indicatorspecies. In this case, a relatively high concentration may be required as a detectionthreshold in order to distinguish between the existence of contaminant from the landfillfacility and natural fluctuations in the background level of that species. Organiccompounds are usually detectable at very low concentrations. Consequently,concentration reductions brought about by dispersion are usually not an issue whenorganic compounds are used as indicator species.6.3.2.2. Monitoring IntervalThe sensitivities of both the probability of detection and the value of the objectivefunction to the length of the monitoring interval are investigated using the base geometry.57Three different monitoring intervals are investigated: 60 days, 180 days, and 360 days.The effects of the length of the monitoring interval on the total probabilities of detectionare examined for each of the three detection thresholds mentioned in the above section.The resulting probabilities of detection, when one monitoring location is installed at eachwell site, are shown in the plots in Figure 6.10.For all three detection thresholds, the total probability of detection decreases asthe monitoring interval increases. It is thought that because of the many differentpathways that can be taken from the contaminant source to each of the monitoringlocations, there is a wide variation in the amount of contaminant that passes through amonitoring location from one monitoring period to another, and the less frequently asample is taken, the lower the probability that a sample will be taken during a monitoringperiod when the contaminant concentration is large enough to constitute detection. Thisbehaviour may be exaggerated to some extent by the discrete nature of particles. Anindication of the variation in the number of particles passing through a specified locationwithin a discrete time period can be gained from the breakthrough curve shown inFigure 6.11. This plot shows the arrival rate of particles at the compliance boundarythrough the fracture carrying the largest proportion of particles in a single realization ofthe base geometry. 2000 particles are injected into the domain in this realization and thearrival array is divided into 500 time periods of 50 day duration. Although the overallshape of the breakthrough curve is relatively smooth, there is a wide variation in thenumber of particles passing through the compliance boundary from one time period toanother. A plot of a breakthrough curve at the compliance boundary is used to illustratethis point because the program used for the model does not have the capability ofgenerating a similar curve at a monitoring location. It should be noted that thiscomparison is also limited by the fact that there are only 2000 particles used to modeltransport in this realization, and 200 000 particles are used in each realization of theproduction runs.For the two threshold concentrations, 1 .92E6 and 3.83E6 particles/rn3,thedifference between the total probabilities of detection for the three lengths of monitoringinterval remains more or less constant as the distance from the source increases.However, when only one particle per monitoring period is required for detection, theprobabilities of detection approach each other as the distance from the source increases.a) 58—a--— 60 day monitoring interval100180 day monitoring friterval80—a--— 360 day monitorhg interval605).340200 25 50 75 100 125 150 175 200Distance from Source (m)b)10080Ca. 60ymonitoringtel0.340—0--— 180 day monitoritig interval20 360 day monitori-ig interval&00 25 50 75 100 125 150 175 200Distance from Source (m)c)10080Caaymonitoring interval. 605)0.340 —0—— 180 day monitoring i,terval—*—— 360 day monitoring iiterval4520&00 25 50 75 100 125 150 175 200Distance from Source (m)Figure 6.10: Probability of detection vs. distance from the source for three differentmonitoring intervals with the base geometry and three different detectionthresholds. a) 3. 83E6 particles/rn3,b) 1.92 particles/rn3,c) 1 particle permonitoring period.5940- 300-oC)4-CDCD20C.)4-CD0100Time (days)Figure 6.11: Arrival rate of particles at the compliance boundary in the fracturecarrying the largest proportion of particles in the first realization of the basegeometry with a total of 2000 particles injected.5000 io4 1.5X104 2X10460However, when only one particle per monitoring period is required for detection, there islittle difference in the probabilities of detection between the different lengths ofmonitoring interval. The probability of sampling during a time period when the numberof particles passing through the monitoring location is large enough to be detected ishigher when fewer particles are required to constitute a detection.Histograms of the value of the objective function for the two thresholdconcentrations are included in Figure 6.12. Both histograms show that with a fewexceptions, the value of the objective function decreases as the length of the monitoringinterval increases. This decrease reflects the lower cost of monitoring associated withless frequent sampling. When one monitoring location is installed 25 m from thecontaminant source, however, the value of the objective function for both thresholdconcentrations increases when the monitoring interval is lengthened from 180 to 360days. This increase indicates that the reduction in monitoring costs afforded by the dropin sampling frequency is smaller than the increase in the expected costs brought about bythe decrease in the probability of detection. When the monitoring interval is 60 days, forboth detection criteria the lowest value for the objective function at each well site, exceptfor the one 25 m from the contaminant source, occurs when one monitoring location isinstalled. As the length of the monitoring interval increases, the lowest value for theobjective function shifts to situations where more than one monitoring location isinstalled at each well site. This trend is particularly apparent when the thresholdconcentration is 3 .83E6 particles/m3 Here, when the length of the monitoring interval isincreased to 180 days, the lowest value for the objective function at each well site occurswhen there are two monitoring locations installed. When the monitoring interval isfurther increased to 360 days, the lowest value for the objective function at three of thefour well sites shown occurs when there are three monitoring locations. With lessfrequent sampling, the ongoing costs of sampling are reduced to the point that theincreased monitoring costs incurred by the addition of monitoring locations areoutweighed by the reduction in the expected costs that result from the increasedprobability of detection that accompanies the addition of monitoring locations. Thesehistograms are a good illustration of the way in which a decision analysis framework canresolve complex design issues into a straightforward evaluation.a)b)0G .2‘s 2.4-LIF.isc.2U.6Odaysl8odays36odays6Odays180 days36OdaysFigure 6.12: Values of the objective function for three different lengths of monitoringinterval for the base geometry. a)threshold concentration of 1 .92E6particles/rn3,b) threshold concentration of 3.83E6 particles/rn3.6125m62This analysis shows a decrease in the value of the objective function when thelength of the monitoring interval is increased. If the decrease in the probability ofdetection is to some extent an artifact of the particle tracking method, the decrease in thevalue of the objective function with a longer monitoring interval would be even greater ina field situation. The lower probabilities of detection obtained in the model, increase thevalue of the objective function because of the effect of the probability of detection on theexpected cost of failure. However, when deciding upon a sampling frequency, careshould be taken to consider the velocity with which a contaminant plume could betransported in the fracture network. The analysis used in this study assumes that when acontaminant plume is detected, it has not spread much beyond the monitoring network. Ifthe monitoring interval is long, and the velocity of the front of the plume is high, acontaminant plume could extend some distance beyond the monitoring network before itis detected, resulting in a much higher cost of remediation than anticipated. In the basegeometry, failure occurs during year two in 1% of the realizations. This means that in thefracture networks generated in those realizations, the front of the contaminant plumecould travel more than 100 m between samples if a monitoring interval of 360 days isadopted. In a situation such as this, it is possible for failure to occur before or at the sametime as the plume is detected, even though the contaminant plume has travelled throughone or more monitoring locations. In 50 % of the realizations, failure has occurred by theend of year ten. In half of the realizations generated, the front of the contaminant plumecan travel 20 m or more in 360 days. Because of longitudinal dispersion, the velocity ofthe front of the plume is greater than that of the centre of the plume. In networksgenerated from the base geometry, the mean velocity of the centre of the plume isapproximately 8 rn/year, less than half that of the plume front.6.3.2.3. Discount RateA comparison of the values obtained for the objective function when discountrates of 0%, 5%, and 10% are used is shown in the histogram in Figure 6.13. Changingthe discount rate changes the relative weight of costs incurred early in the complianceperiod and those that are incurred in later years. The value of the objective function isreduced when the discount rate is increased, because the present day value of ongoingcosts, such as the cost of sampling from the monitoring network, and costs arising later in63C) 4• Cow0% discount rate5% discount rate10% discountrateFigure 6.13: Values of objective function for the base geometry with three differentdiscount rates.the compliance period, such as the expected costs of remediation, decrease as the discountrate increases. The values obtained with a discount rate of 10% are less than half thoseobtained when the costs are not discounted over time. For all of the well sites except theone located 25 m from the source, one monitoring location per well site provides thelowest value for the objective function, regardless of the discount rate used. At this wellsite, the value for the objective function provided by the installation of two monitoringwells is slightly lower than that provided by one monitoring location when a discount rateof 5% or 10% is used. In these cases, discounting the ongoing costs of sampling overtime reduces the overall cost of monitoring so that the reduction in the expected cost offailure brought about by the increase in the probability of detection outweighs the cost ofinstalling and sampling from a second monitoring location. The lowest value for theobjective function occurs at 50 m from the source for each one of the three discount ratesused. In the design scenario used for this study, the same option provides the lowestvalue for the objection function regardless of the discount rate chosen. Other thanchanging the absolute value of the objective function, changing the discount rate does nothave much impact on this analysis.m646.3.2.4. Cost of FailureThe effects of a higher cost associated with failure are shown in the histogram inFigure 6.14. Increasing the cost associated with failure throws more weight on theexpected cost of failure relative to the actual costs in the objective function. A highercost associated with failure should promote a more conservative design for a monitoringnetwork by increasing the effects of the probability of detection. An increase in theprobability of detection will cause a larger reduction in the expected cost of failure, whenthe costs associated with failure are high. In this way, any additional costs incurred toprovide an increase in the probability of detection may be offset by the large reduction inthe expected cost of failure. The decision analysis framework provides a basis for theowner/operator to determine just how conservative a design he or she should choose.When the cost associated with failure is $5 million, the lowest value of the objectivefunction is obtained with one monitoring location installed per well site at every well sitebut the one located 25 m from the source. As discussed previously, this behaviour occursbecause the reduction in the expected costs of failure brought about by the increase in theprobability of detection, when more than one monitoring well is installed, is not as largeas the additional cost of installing and sampling from the additional monitoring wells plusthe increased expected cost of remediation that is also a result of the higher probability ofdetection. When the cost associated with failure is increased to $10 million, the lowestvalue for the objective function at each well site occurs when there are two monitoringlocations installed. With the larger cost associated with failure, the amount of reductionin the expected cost of failure is now large enough to offset the increased costs incurredwhen two monitoring locations are installed but not those incurred when three monitoringlocations are installed. The lowest value for the objective function does not occur whenmore than one monitoring location is installed at the well site located 100 m from thesource, where the difference in the probability of detection between one and twomonitoring locations is not as great as it is at the closer well sites (see Figure 6.6). Forboth cases, the lowest value of the objective function occurs at the well site 50 m from thesource.65LL. $10 Million Cf$5 Million CfFigure 6.14: Values of objective function for two different costs of failure for the basegeometry.6.3.2.5. Pseudo-three-dimensional AnalysisThe pseudo-three-dimensional approach to the decision analysis that is adoptedfor this study is an attempt to achieve consistency between the costs of monitoring andthe expected costs associated with detection and failure. In this approach, thehypothetical landfill site is divided into several slices parallel to the plane of the model,and each monitoring well site in the model domain represents several well sites in thehypothetical landfill site arranged along a plane perpendicular to the plane of the model.The main assumption made with this approach is that the solute enters and travels in oneslice only. The likelihood that well sites have been installed in the slice through whichthe contaminant is travelling is the ratio of monitored slices to the total number of slicesin the hypothetical landfill site. The investigation of the effects of the pseudo-three-dimensional analysis on the objective function involves varying both the total number ofslices considered and the proportion of those slices that are monitored.The histogram depicted in Figure 6.15 shows the resulting values of the objectivefunction when the base case analysis is performed from a two-dimensional perspective foreach of the four monitoring schemes. In this perspective, there is only one slice in the100 75 50 25m m m m66hypothetical landfill site, and each monitoring well site in the simulation represents onlyone well site. At each well site, the lowest value of the objective function occurs whenthree monitoring locations are installed, regardless of the monitoring schemeimplemented. This behaviour is the result of two factors: the costs of installing andsampling from additional monitoring wells are 1/3 those involved in the pseudo-three-dimensional analysis adopted for the base case, and the probabilities of detection are 3.3times as great. As in the base case, the highest flow monitoring scheme provides thelowest values for the objective function, and the predetermined depths scheme thehighest. The lowest overall value for the objective function occurs at the well site that is50 m from the source when the highest flow monitoring scheme is implemented, the samelocation and monitoring scheme as in the base case.The histogram of objective function values in Figure 6.16 is a comparison of thetwo-dimensional analysis discussed above, the base case in which three of ten layers aremonitored, and a pseudo-three-dimensional analysis in which the hypothetical landfill siteis represented by 20 slices, three of which are monitored. The values obtained for theobjective function with the two dimensional analysis are consistently lower than thoseobtained with either of the pseudo-three-dimensional analyses. The highest values for theo 54.QOLIdepthaperturedensity2.5mmFigure 6.15: Values of objective function for two-dimensional analysis for base geometry.67objective function are obtained when the domain is divided into 20 slices, because theprobabilities of detection in this case are lower than in either of the other two cases.These lowest probabilities of detection result in the highest expected costs of failure. Inthe two-dimensional analysis, the lowest value for the objective function at each well siteoccurs when three monitoring locations are installed, and in both pseudo-three-dimensional analyses, the lowest value for the objective function at each well site usuallyoccurs when one monitoring location is installed. The reasons for this behaviour havebeen discussed previously. When the domain is divided into 20 slices, and three of themare monitored, the cost of installing and sampling from each monitoring location is thesame as in the base case, but the probabilities of detection that are used in the objectivefunction are half those of the base case. These two factors have the effect of increasingthe expected costs associated with failure, when the landfill site is divided into 20 slices,while the costs of installing and sampling from additional monitoring locations remainthe same as in the base case. This increase in the expected cost of failure is evident at the25 m well site where the addition of a second monitoring location results in a slightlowering of the value of the objective function in the base case but an increase in the 20slice case. There is a considerably smaller increase in the values of the objectivefunction between the two pseudo-three-dimensional analyses than there is between thetwo-dimensional analysis and the base case. This increase is smaller because it is theresult of only the higher expected cost of failure brought about by the lower probabilitiesof detection that are used when the domain is divided into 20 slices. There is no increasein the cost of monitoring between the two pseudo-three-dimensional analyses as there isbetween the two-dimensional analysis and the base case.It should be noted that this pseudo-three-dimensional analysis is only an attemptto introduce a three-dimensional element into this decision analysis. The thresholdconcentrations used, and the decisions concerning the total number of slices into whichthe domain is divided, and the proportion of those slices which are monitored, arearbitrary. The probabilities of detection observed are not neccessarily those that would beobtained in a field situation or if a three-dimensional transport model were used tosimulate the transport of contaminants through the hypothetical landfill site.68o 4• C.CU.3120 layers monitored3/10 layers monitored1/1 layers monitoredFigure 6.16: Values of objective function for two-dimensional and two different pseudo-three-dimensional analyses for base geometry.6.3.2.6. Multiple Well ConfigurationsThe multiple well configurations considered in this study consist of two rows ofmonitoring well sites, represented by two well sites in the plane of the model situated atdifferent distances from the contaminant source. The reason behind installing two rowsof monitoring wells is that the row situated farther from the source can act as a backupsystem to detect plumes that may be missed by the closer row of wells. Two multiplewell configurations are considered in this sensitivity study: one with well sites located25 m and 50 m from the source, and one with well sites at 25 m and 75 m. The plot of thecumulative probabilities of detection for single well sites at 25 m and 75 m and a multiplewell configuration with wells at both sites (Figure 6.17) shows that, throughout thecompliance period, the probability of detection for the multiple well configuration ishigher than for either of the individual well sites, but not as great as the sum of the two.The probability of detection for the multiple well configuration is not as great as the sumof the probabilities of detection at the two individual well sites, because the plumes insome of the realizations are detected at both well sites. The probabilities shown in this2.5-0.5-75m25m69II0 0.8YearFigure 6.17: Cumulative probability of detection vs. time at single well sites at 25 m and75 m and a multiple well configuration with well at both sites for basegeometry.figure are those obtained when one monitoring location is installed at each well site. Itshould also be noted that the curve for the multiple well configuration roughly parallelsthat for a single well at 75 m from the source.The values of the objective function for the individual well sites as well as the twomultiple well configurations are presented in the histograms in Figure 6.18. Figure 6.18arepresents the values obtained with the base case analysis and the highest flowmonitoring scheme. Because this is a three-dimensional histogram and the columnsrepresenting the two multiple well configurations are not located in the same row, care isrequired when interpreting this histogram. The values for the objective function obtainedwith both of the multiple well configurations are larger than the values obtained at any ofthe individual well sites. As with the installation of additional monitoring locations in thebase case, the reduction in the expected cost of failure provided by a higher probability ofdetection is not sufficient to outweigh the costs of installing and sampling from7000.-o 25&75m25&50 msingle wella)0o 25&75m25&50msingle wellb)Figure 6.18: Values of objective function for single well sites and two multiple wellconfigurations for base geometry. a) $5 million cost of failure, b) $10 millioncost of failure.additional monitoring locations plus the increased expected cost of remediation. Evenwhen the cost of failure is doubled to $10 million (Figure 6.18b), doubling the size of thereduction in the cost of failure, the single well configurations provide lower values for theobjective function than do either of the multiple well configurations. In both histograms,the lowest values for the objective function for the multiple well configurations occurwhen one monitoring location is installed at each well site.75m 50m 25m71Although this sensitivity study indicates that there is no advantage to installing a“backup” monitoring network, located farther from the source than the “primary”monitoring network this may not be the case in all situations. As with the pseudo-three-dimensional analysis sensitivity study, the results obtained in this sensitivity study are notnecessarily those that would be obtained from a similar study conducted in a fieldsituation or with a three-dimensional transport model. There are other functions that canbe filled by a “backup” monitoring network that can not be evaluated in the decisionanalysis framework developed for this study. A “backup” monitoring network can assistin the detennination of the extent of a contaminant plume once it has been detected. Ifthe rate at which contaminants are transported through a fracture network is higher thananticipated, and, consequently, the monitoring interval that is used is too long, thecontaminant plume may have spread well beyond the monitoring network by the time it isdetected. If the owner/operator of the landfill assumes that the plume does not extendmuch beyond the monitoring location at which it was detected, an inadequate remediationsystem may be implemented. If the contaminant plume is also detected by the “backup”monitoring network, the owner/operator of the landfill facility is alerted to the fact thatthe contaminant plume has spread farther than anticipated. A “backup” monitoringnetwork can also function as an indicator of the effectiveness of a remediation strategy.If the remediation network is not successful in capturing all of the contaminant plume,escaping contaminant may be detected by the “backup” monitoring network.6.4. GEOMETRY 2The second fracture geometry which I investigate is identical to the basegeometry, except that the mean aperture of the horizontal fracture set is 40 microns, twicethat of the base geometry. The larger apertures of the horizontal fractures increase theaverage effective porosity of the fracture networks as well as the average velocity withwhich contaminants are transported through the domain. The probability of failureduring the compliance period is higher for this fracture geometry than for the basegeometry and failure tends to occur earlier. Because of the earlier failure times andhigher probability of failure, this geometry represents a less desirable site for a landfillfacility.72The monitoring period calculated for geometry two is three days. The networks inthis geometry have a higher average volumetric flow and a higher average groundwatervelocity than do the networks in the base geometry, consequently less time is required forthe volume of water contained in an average sampling volume to pass through an averagemonitoring location. The minimum number of particles required for detection inmonitoring locations with low volumetric flow rates is three, which corresponds to oneparticle per day throughout the monitoring period. The total number of particles used tomodel transport in each realization is 32 256. By using this number of particles, Imaintain the same average particle concentration per unit volume in the source area of thedomain as in the base geometry.6.4.1. MoNIToRING ScHEME C0MPAIUs0NThe total probabilities of detection over the compliance period vs. the distancefrom the source for geometry two, for all four monitoring schemes, are plotted inFigure 6.19. These probabilities are those obtained in a two-dimensional analysis withthe monitoring parameters used for the base case and one monitoring location per wellsite. As with the base geometry, the highest flow monitoring scheme yeilds the highestprobability of detection and the predetermined depth monitoring scheme the lowest.With the exception of the well site located 75 m from the source, the densest fracturingmonitoring scheme provides a higher probability of detection than does the largestaperture monitoring scheme. The highest probability of detection for three of the fourmonitoring schemes occurs at the well site 75 m from the source. The exception is thedensity monitoring scheme, which has a slight dip in its curve at this well site. Thegeneral behaviour of the probability of detection in this geometry is similar to that in thebase geometry. Close to the source, the probability of detection increases with distanceas vertical dispersion spreads the plume over more of the domain. Farther from thesource, the plume becomes more dilute and the probability of detection decreases withdistance.A comparison of the total probabilities of detection observed with one, two andthree monitoring locations per well site for each monitoring scheme is provided by theplots in Figure 6.20. The behaviour of the probability of detection in geometry two with73100.00 • high flow monitoring—°----- density monitoring80.00 £ aperture monitoring0—X— predetermined depth60.00 monitoring40.00.0Cs.02 20.0000.00-,0 25 50 75 100 125 150 175 200Distance from Source (m)Figure6.19: Total probabilities of detection over the compliance period vs. distance fromthe source for geometry two.different numbers of monitoring locations installed at each well site is also similar to thatin the base geometry. With all four monitoring schemes, the more monitoring locationsat each well site, the higher the probability of detection. The difference between theprobabilities of detection from one to two monitoring locations is greater than thedifference from two to three monitoring locations at most well sites for each of the fourmonitoring schemes. The difference in the probabilities of detection between onemonitoring location and three monitoring locations is greater close to the source than it isat the well site at 150 m. This reduction in the difference in the probabilities of detectionbetween one monitoring location and three monitoring locations with distance illustratesthat, as is the case in the base geometry, the addition of monitoring locations has moreimpact on the probability of detecting a contaminant plume while the plume is small, thanonce the plume has had more opportunity to disperse.The major differences in the behaviour of the probability of detection withdistance between this geometry and the base geometry are ones of degree. The effects ofthe spreading of the plume due to vertical dispersion are apparent over a greater distancein geometry two than in the base geometry. When the highest flow or largest aperturemonitoring schemes are employed, the increase in the probability of detection between74a) b)3 locatioesiwell. 100 100 siteC. 80 80 2 lOcatioss/wellsite60 60 flocatlonlwell0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200c) d)3 Iscalions/well 3locations/weII, 100 sIte 100 site2 locatIons/well 2 locatIons/wellsite site60 1 location/well 60 1 location/well40sIte40site- 20 2000 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200Distance from Source (m) Distance from Source (m)Figure 6.20: Total probability of detection vs. distance for geometry two with one, two,and three monitoring locations per well site. a) highest flow monitoringscheme, b) densest fracturing monitoring scheme, c) largest aperturemonitoring scheme, d) predetermined depth monitoring scheme.50 m and 75 m from the source is larger in geometry two than in the base geometry. Themost likely explanation for this behaviour is that because the preferred flow paths ingeometry two are more direct, and the flow more channellized, the rate of verticaldispersion with distance is lower. In the networks in the base geometry, the plume hasalready spread over a substantial portion of the height of the domain by the time it hastravelled 50 m from the source; the additional spreading between 50 and 75 m from thesource is smaller, as reflected by the relatively small increase in the probability ofdetection between these two monitoring well sites. In the networks in geometry two,however, the plume has not spread over as large a portion of the height of the domain bythe time it has travelled 50 m from the source, and it continues to spread at close to thesame rate when it travels between 50 and 75 m from the source as it was spreading whenit travelled between the well sites at 25 m and 50 m from the source. Another indicationthat the plume continues to spread at greater distances in geometry two than in the basegeometry is given in the plots in Figure 6.21. These plots are a comparison of the3 locatisns/weIlsite2 locatIons/wellsite1 locatios/wellsite75geometry is given in the plots in Figure 6.21. These plots are a comparison of theprobabilities of detection for the different detection thresholds in both geometry two andthe base geometry. The probabilities of detection reported in this figure are thoseobtained in a two-dimensional analysis when one monitoring location is installed at eachwell site, and the fractures carrying the highest flows are monitored. Figure 6.21a showsthe probabilities of detection for both one particle per monitoring period and the higher ofthe two threshold concentrations, 3.83E6 particles/rn3,for both geometries, and Figure6.2 lb shows the same comparison for one particle per monitoring period and the lowerthreshold concentration, 1 .92E6 particles/rn3.In these plots, when one particle permonitoring period is required for detection, the probability of detection for geometry twocontinues to rise after the probability of detection for the base geometry has levelled off.This continued rise in the probability of detection indicates that the plume is stillspreading vertically at distances greater than 100 m from the source in the networks ingeometry two, but not in the networks in the base geometry.The distance over which the highest probabilities of detection occur is shorter ingeometry two than it is in the base geometry. This behaviour may also be a result of thelower rate of vertical dispersion with distance travelled of the contaminant plume ingeometry two. The plots in Figure 6.21 show that in the base geometry, most of theincrease in the probability of detection that results from vertical dispersion of the plumeoccurs between 25 and 50 m from the source for both of the threshold concentrations.There is relatively little change in the probability of detection at the well sites located 50to 100 m from the contaminant source. When the lower of the threshold concentrations isused in the base geometry, the high probabilities of detection extend all the way to thewell site 150 m from the source; there is only a small decrease in the probability ofdetection between 100 and 150 m. In geometry two, however, the rate of increase in theprobability of detection remains almost constant from the well site at 25 m from thecontaminant source through to the well site at 75 m for both concentration thresholds. Arapid decline in the probability of detection due to dilution of the contaminant beginsafter the plume has travelled 100 m from the source when the lower thresholdconcentration is used in geometry two, and after the plume has travelled only 75 m fromthe source in the case of the higher concentration threshold. In the base geometry, forboth threshold concentrations, there is a distance of at least 50 m where the contaminantplume has spread over a large enough portion of the height of the domain that theprobability of detection is relatively unaffected by further spreading of the plume anda) 76030.1.0C00.020.00 25 50 75 100 125 150 175 200Distance from Source (m)Figure 6.21: Probability of detection vs. distance for both base geometry and geometrytwo. a)one particle per monitoring period and 3.83E6 particles/rn3,b) a)oneparticle per monitoring period and 1.92E6 particles/rn.before the dilution of the contaminant due to dispersion begins to affect the probability ofdetection to a large extent. In geometry two, the sharp inflection point in the curverepresenting the probabilities of detection for the larger of the two concentrationthresholds indicates that the area of high probabilities of detection is most likely less thanperiodgeometry two• 1 particle per monitoring periodbase geometryper m3 basegeometry83EO6 particles per m3geometry two10080604020010080604020b)0 25 50 75 100 125 150 175 2001 particle per monitoring periodgeometry two1.Epacles perm base1 particle per monitoring periodbase geometrygeometry—&--— 1.92EpartrcIes perm3geometry two7725 m in extent in this case. When the lower threshold concentration is used, the area ofhigh probabilities of detection is also shorter in geometry two than in the base geometry.There are two differences in the behaviour of the probability of detection withdistance between this geometry and the base geometry, whose causes remain unresolved.The fall in the probability of detecting a contaminant plume at greater distances is morepronounced in geometry two than it is in the base geometry, and the probabilities ofdetection for geometry two are lower for all four of the monitoring schemes throughoutthe entire length of the domain. The more pronounced fall in the probability of detectinga contaminant plume at greater distances in this geometry is the opposite to what I wouldhave expected. With the channellization of the flow and consequent lower rate of verticaldispersion that occurs in geometry two, I would have expected there to be less dilution inthis geometry resulting in higher probabilities of detection at the larger distances than inthe base geometry. Although the lower rate of dispersion with distance may be acontributing factor to the lower probabilities of detection observed in geometry two, Iwould have expected higher probabilities of detection in this geometry when the highestflow monitoring scheme is implemented, as a result of the contaminant becominglocalized in the fractures carrying the largest flows. Preliminary investigations of thebehaviour of this geometry that were carried out in an attempt to determine the cause ofthe lower probabilities of detection indicate that neither increasing the length of themonitoring period, varying the length of the monitoring interval, nor changing thenumber of particles introduced into the domain has a significant effect on the rate ofdecline in the probabilities of detection at large distances from the contaminant source.When the length of the monitoring period for geometry two was increased to the samelength as for the base geometry, the probabilities of detection increased but were stilllower than those observed in the base geometry.The values obtained for the objective function with the base case analysis ofgeometry two are shown in the histogram in Figure 6.22. As in the base geometry, thehighest flow monitoring scheme consistently provides the lowest values for the objectivefunction, while the predetermined depths scheme provides the highest values. The valueof the objective function that is obtained for this geometry when no monitoring isundertaken is $3.48 million, almost one million dollars higher than in the base geometry.The value obtained for the objective function when no monitoring is undertaken is higherin geometry two than in the base geometry for two reasons: geometry two has a higher78o 3.4-Ccli-densityU,cjFigure 6.22: Values of the objective function for geometry two with base case analysis.probability of failure than does the base geometry, and failure tends to occur earlier ingeometry two than in the base geometry. The net present value of the expected cost offailure is higher the earlier that failure occurs because of the effects of discounting withtime, The predetermined depth monitoring scheme is the only monitoring scheme thatdoes not provide at least one option with an objective function value lower than the valueobtained when no monitoring is undertaken in both the base geometry and in geometrytwo. In both geometries, when the predetermined depth monitoring scheme is employed,the probability of detection at all well sites is so low that the costs of monitoring are notoffset by the reduction in the expected cost of failure afforded by detection.In all but five instances, the lowest value for the objective function occurs whenthere is one monitoring location installed at the well site. In the other five instances, thereis a substantial increase in the probability of detection when a second monitoring well isinstalled. The highest probability of detection, 80.5 %, occurs at the 75 m well site whenthree monitoring locations are installed and the highest flow monitoring schemeimplemented. As in the base geometry, the combination of options that provides thehighest probability of detection does not provide the lowest value for the objectivefunction. The lowest value obtained for the objective function with geometry two occursat the well site 50 m from the source when the highest flow monitoring scheme isimplemented and two monitoring locations are installed at this well site. The highestdepthaperture2 flow79probability of detection at this well site occurs when three monitoring locations areinstalled. The increase in the probability of detection afforded by the installation of athird monitoring location at this well site does not provide a sufficient reduction in theexpected cost of failure to offset the additional monitoring costs as well as the increase inthe expected cost of detection. The highest probability of detection when two monitoringlocations are installed at each well site occurs at the monitoring well site located 75 mfrom the contaminant source. The lowest value for the objective function does not occurat this well site because the cost associated with detection at this well site is twice that atthe 50 m well site. If the contaminant plume is detected 50 m from the contaminantsource, one horizontal interceptor well is required for the remediation network, but if theplume is detected 75 m from the source, two horizontal interceptor wells are required.The lowest value obtained for the objective function with geometry two is$3.20 million. This value is approximately $0.8 million higher than the lowest valueobtained for the objective function in the base geometry. All of the values for theobjective function that are obtained in geometry two are larger than those obtained in thebase geometry. Four factors contribute to the higher objective function values ingeometry two: 1) a higher probability of failure, 2) earlier times of failure, 3) lowerprobabilities of detection, and 4) earlier times of detection. The higher probability offailure provides a higher expected cost of failure, especially in conjunction with lowerprobabilities of detection. When failure occurs at an earlier time, the discounting withtime is smaller, resulting in a higher net present value for the expected cost of failure.While lower probabilities of detection result in lower expected costs of detection, theyalso result in higher expected costs of failure. Earlier times of detection result inconsiderably higher expected costs of detection in two ways: the discounting with time isless, and the length of time over which the remediation network must be in operation islonger, resulting in a higher cost of operating the remediation network. The hydraulicbehaviour of geometry two makes this geometry representative of a less desirable site onwhich to locate a landfill facility than sites represented by the base geometry. The lowersuitability for landfill facilities of sites represented by geometry two is reflected by thehigher values obtained for the objective function.806.5. GEoMETRY 3In the third geometry which I investigate, all three fracture sets have the samedensity, length, and aperture parameters. The networks for this geometry are moredensely fractured than the previous two geometries and they do not contain the longhorizontal fractures seen in the other two geometries. The average effective porosity ofthe geometry three networks is similar to that in the base geometry, but the averagegroundwater velocity and volumetric flow are both smaller than in the base geometry by afactor of approximately five. The probability of failure during the compliance period ismuch lower in this fracture geometry than in either the base geometry or geometry two,and failure occurs later. Because of the later failure times and the low probability offailure, this geometry represents a more desirable site for a landfill facility than do eitherof the other two geometries.The monitoring period calculated for geometry three is 35 days. The networksfor this geometry have a considerably lower average groundwater velocity and volumetricflow and a slightly higher average effective porosity than do the networks for the basegeometry, therefore a greater period of time is required for the volume of water containedin a sampling volume to pass through a monitoring location. Because this monitoringperiod extends over more than half of the monitoring interval of 60 days used in the basecase analysis of the other two geometries, the base case monitoring interval for geometrythree is set at 180 days. One particle is the minimum number required for detection inmonitoring locations with low volumetric flow rates. The total number of particles usedto model transport in each realization is 21 510. This number of particles reflects anaverage effective porosity that is slightly higher than that of the base geometry, and lowerthan that of geometry two.6.5.1. MoNIToRING SCHEME COMPARISONFor this geometry, the total probability of detection over the compliance period vs.distance from the source is plotted in Figure 6.23. This figure shows the probabilitiesobtained in a two-dimensional analysis with the basic monitoring parameters and onemonitoring location per well site for each of the four monitoring schemes. The highest81flow monitoring scheme consistently provides the highest probability of detection,followed by the densest fracturing monitoring scheme. The probabilities of detectionprovided by the largest aperture and the predetermined depth monitoring schemes aresimilar; these monitoring schemes provide the lowest probabilities of detection. Thehighest probability of detection for each of the monitoring schemes occurs at the well site50 m from the contaminant source. The networks generated from this geometry are moredensely fractured and the horizontal fractures are shorter than in either of the other twogeometries, hence the contaminant plume experiences a higher rate of dispersion, bothvertically and horizontally, with distance travelled. The representation of the flowsthrough a sample network for geometry three (Figure 6.3c) shows that the preferred flowpaths through the networks for geometry three are more tortuous than those in the othergeometries. Contaminants being transported along tortuous paths encounter morefracture intersections than do those travelling along more direct paths and, consequently,experience more opportunity for dispersion. Because of this enhanced dispersion, thecontaminant plumes in the networks for geometry three reach the point that furtherspreading has little impact on the probability detection while closer to the source than docontaminant plumes in the networks from the other two geometries. The effects of the100.00 • high flow monitoring—°—— density monitoring80.00 aperture monitoring.2—X— predetermined depth60.00 monitoring40.00.0.02 20.000.000 25 50 75 100 125 150 175 200Distance from Source (m)Figure 6.23: Total probability of detection over the compliance period vs. distance forgeometry three.82higher rate of dispersion with distance travelled are evident in the fact that the highestprobabilities of detection for each monitoring scheme occur closer to the contaminantsource in this geometry than in either the base geometry or geometry two. The increasesin the probabilities of detection between the monitoring well site 25 m from thecontaminant source and the well site at 50 m are smaller in geometry three than in eitherof the other two geometries, indicating that the contaminant plume in geometry three islikely to have undergone substantial vertical spreading by the time it has travelled only25 m from the contaminant source.The effects of the greater dispersion that occurs close to the source with thisgeometry can be seen in the plots in Figure 6.24. These plots represent a comparison ofthe total probabilities of detection achieved with one, two, and three monitoring locationsinstalled at each well site. When the highest flow monitoring scheme is employed, anda) b)3 locations/well 3 locations/well, 100 site 100 site.2 80 2 locationslwell 80 2tocations/wellI::20 20-‘300 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200c) d)3 loCations/welt 3tncations/welt. 100 site 100 site.2 80 2 locationS/well 80 2tocationS/weltSite siteti) 60 1 location/welt 60 1 tocatioo/wetlSite::Site0 25 50 75 100 125 150 175 200 0 25 50 75 100 125 150 175 200Distance from Source (m) Distance from Source (m)Figure 6.24: Total probability of detection vs. distance for geometry three with one, two,and three monitoring locations per well site. a) highest flow monitoringscheme, b) densest fracturing monitoring scheme, c) largest aperturemonitoring scheme, d) predetermined depth monitoring scheme.83three monitoring locations are installed at each well site, the highest probability ofdetection occurs at the monitoring well site located 25 m from the contaminant source.This behaviour indicates that the contaminant plume has already dispersed to the pointwhere the plume is spread across enough of the vertical extent of the domain that there isno further increase in the probability of detection by further spreading that may occur asthe plume travels onward to the monitoring well site at 50 m. As with the other twogeometries, the more monitoring locations at each well site, the higher the probability ofdetection for all four monitoring schemes. The difference between the probabilities ofdetection from one to two monitoring locations is greater than the difference from two tothree monitoring locations at all well sites for each of the four monitoring schemes. Thedifference in the probabilities of detection between one monitoring location and threemonitoring locations is greater close to the source than it is at the well site located 150 mfrom the contaminant source.As in both the base geometry and geometry two, the probability of detection fallsoff at the monitoring well sites located at greater distances from the contaminant source.In this geometry, however, dilution of the contaminant plume due to dispersion is mostlikely not the major contributing factor to this behaviour. The plots in Figure 6.25 arecomparisons of the cumulative probabilities of detection vs. time between the basegeometry and geometry three at the well sites located 25 m and 75 m from thecontaminant source. These probabilities of detection are obtained when one monitoringlocation is installed at each well site and the highest flow monitoring scheme isimplemented. These plots show that in both geometries, the contaminant plume haspassed the monitoring well site located 25 m from the contaminant source by the end ofthe compliance period in most of the realizations. The slope of the curves for bothgeometries approach horizontal before the end of the compliance period. At themonitoring well at 75 m, however, the curve representing the probability of detection ingeometry three is still rising at the end of the compliance period. The fact that this curvehas not levelled off indicates that the contaminant plumes in a number of the realizationsfor this geometry have not yet arrived at this monitoring well site by the end of thecompliance period. Therefore, the fall in the probability of detection that occurs atdistances greater than 50 m from the contaminant source in this geometry may be morethe result of the lower average groundwater velocity in geometry three than the result ofdilution of the contaminant plume due to dispersion.a)b)00.6Ø40.4a)E000.00.6a)—.0.40Figure 6.25: Cumulative probability of detection vs. time for base geometry andgeometry three. a) 25 m from source, b) 75 m from source.0.80.80.2303084base geometrygeometry three0 5 10 15 20 25Yearbase geometrygeometry three0 5 10 15 20 25Year85The probabilities of detection in this geometry are higher near the source andlower at the well sites located at greater distances from the contaminant source than thosein the base geometry, except when the arbitrary depth monitoring scheme is implemented.When the arbitrary depth monitoring scheme is implemented, the probabilities ofdetection are higher throughout the domain in geometry three. The higher probabilities ofdetection in geometry three are most likely due in a large part to the higher rate ofdispersion with distance travelled that occurs in this geometry.A histogram of the values obtained for the objective function with the base caseanalysis for this geometry is shown in Figure 6.26. The objective function values foreach monitoring scheme are similar, but the predetermined depth monitoring schemeconsistently provides the lowest values, followed by the largest aperture, densestfracturing and highest flow monitoring schemes respectively. This is the reverse order ofmonitoring schemes than in the other two geometries. The probability of failure is muchlower for networks generated from this geometry than it is for those generated from eitherof the other geometries. The low probability of failure in this geometry is a result of thelow groundwater velocities. Failure occurs later in geometry three than in either of theother two geometries investigated, and does not occur during the compliance period inmany of the realizations for geometry three. If a longer compliance period were used forthis analysis, the probability of failure in geometry three would be higher. As aconsequence of the low probability of failure observed in this geometry, the objectivefunction is dominated by the cost of monitoring and the expected cost of detection whenthe base case analysis is performed. Unlike the results obtained with either of the othertwo fracture geometries, the monitoring scheme that provides the highest probability ofdetection also provides the highest values for the objective function. Because theprobability of failure is so low, the reduction in the expected cost of failure provided byan increase in the probability of detection is overshadowed by the increase in theexpected cost of detection. Thus, monitoring schemes that provide a lower probability ofdetection lead to smaller values for the objective function. When no monitoring isundertaken in geometry three, the value obtained for the objective function is $0.42million. This value is much lower than the $2.64 million that is obtained in the basegeometry when there is no monitoring undertaken. The lower value for the objectivefunction that is obtained in geometry three reflects both the lower probability of failure inthis geometry and the later failure times. The lower probability of failure results in a86w .2L1Figure 6.26: Values of the objective function for geometry three with base case analysis.lower expected cost of failure, and the later failure times result in a lower net presentvalue for this expected cost because of the effects of the discounting with time.At each well site, the lowest value for the objective function is provided when onemonitoring location is installed. The variation in the values for the objective function atall well sites is greater between the different number of monitoring locations within anygiven monitoring scheme than the variation provided between any two monitoringschemes with the same number of monitoring locations installed. The expenses incurredby the addition of monitoring locations are larger than any change in the total expectedcosts that result from a change in the probability of detection between either differentmonitoring schemes or different numbers of monitoring locations at a given monitoringwell site. The lowest value obtained for the objective function, $0.47 million, occurs atthe well site located 50 meters from the source, when either the predetermined depth,largest aperture, or densest monitoring scheme is implemented and one monitoringlocation is installed. The probabilities of detection for the predetermined depth and thelargest aperture monitoring schemes are similar in this case, but the difference in theprobabilities of detection between the predetermined depth and the densest fracturingmonitoring schemes is 19.5%. In this instance, the reduction in the expected cost offailure brought about by the higher probability of detection with the densest fracturingmonitoring scheme is approximately equal to the resulting increase in the expected cost ofdetection. Consequently, the total value of the objective function is the same for all threem 25m87monitoring schemes. This value is higher than the $0.42 million obtained when nomonitoring is undertaken.6.5.2. INCREASED COST OF FAILuREIn this fracture geometry, the probability of failure is so low that the objectivefunction is dominated by the cost of monitoring and the expected cost of detection whenthe base case analysis is performed. Increasing the cost associated with failure will givemore weight to the expected cost of failure term in the objective function by increasingthe reduction in the expected cost of failure that results from an increase in the probabilityof detection. When the costs associated with failure are doubled, from $5.00 million to$10.00 million, the highest flow monitoring scheme provides the lowest value for theobjective function at the two well sites closest to the contaminant source (Figure 6.27).In general, though, the values provided for the objective function by the four monitoringschemes are similar with no one monitoring scheme consistently providing the lowest orthe highest values. The variation in the value of the objective function provided bydifferent numbers of monitoring locations per well site is still greater than the variationbetween the different monitoring schemes. However, the overall lowest value providedfor the objective function, $0.81 million, is now lower than the value that is obtainedwhen no monitoring is undertaken, $0.84 million. All of the values obtained for theobjective function, for each monitoring scheme as well as the no monitoring option, arelower than the values obtained for the base geometry in the base case analysis, despite thedoubling of the costs associated with failure. The lowest value for the objective functionin this case is obtained at the 50 m well site, when the highest flow monitoring scheme isimplemented and one monitoring location installed. This is the option that provides thehighest probability of detection when there is one monitoring location installed at eachwell site. Doubling the costs incurred as a consequence of failure results in an expectedcost of failure that is large enough that, in some cases, the reduction to this expected costbrought about by an increase in the probability of detection is now larger than theresulting increase in the expected cost of detection. When more than one monitoringlocation per well site is installed, however, the reduction to the expected cost of failurethat is brought about by the increased probability of detection is overshadowed by theincrease in the cost of monitoring.88>.2U-depthape ritredensityflowFigure 6.27: Values of the objective function for geometry three with $10 million cost offailure.The investigation of fracture geometry three indicates that for landfill facilitiesconstructed on sites with low groundwater velocities and consequent low probabilities offailure, it may be to the advantage of the owner/operator not to install any monitoringnetwork. It should be remembered, however, that the detection thresholds used in thisstudy and the decisions concerning the pseudo-three-dimensional analysis are arbitrary.For this reason, care should be taken in making comparisons between values obtained forthe objective function when monitoring is undertaken and those obtained when there is nomonitoring network in place. One conclusion that can be drawn from this investigation,however, is that the lower the probability of failure, the lower the values obtained for theobjective function. Therefore, it is to the advantage of the owner/operator of a landfillfacility to locate the facility on a site where, should contaminants escape from the facility,there is a small probability of these contaminants migrating any great distance from thefacility within the expected lifetime of the facility.m897. CONCLUSIONSIn this dissertation, I develop a decision analysis framework to assist in the designof monitoring networks at hazardous waste sites located above a fractured geologic unit.The decision analysis framework is based upon risk-cost-benefit analysis, performed fromthe perspective of the owner/operator of the landfill facility. In this analysis I considerthose costs that are directly associated with the construction and operation of themonitoring network (actual costs). The risks considered are those that are associated withthe detection of migrating contaminants and consequent costs of remediation, and thefailure of the facility and the costs resulting from failure (expected costs). The benefitsare considered to be the same regardless of the monitoring strategy adopted, and areneglected. Therefore, the objective of the analysis is to fmd the monitoring strategy thatprovides the lowest value for the objective function, minimizing the sum of the actual andexpected costs. This monitoring strategy is hereafter referred to as the “best” monitoringstrategy.The fractured rock formation underlying the hypothetical landfill site is modelledin vertical section using a two-dimensional discrete fracture model. This model uses aparticle tracking method to simulate the transport of a non-reactive solute through thefractured rock unit. I investigate three fracture geometries, each with differenthydrogeological behaviour. For each of these geometries, I investigate four monitoringschemes: 1) monitoring the fractures that carry the highest volumetric flows, 2)monitoring the fractures that have the largest apparent apertures, 3) monitoring the areasof highest fracture density, and 4) placing the monitoring locations at predetermineddepths. I also investigate the effects of the distance of the monitoring network from thecontaminant source, and the number of monitoring locations installed at each monitoringwell site, for each of the four monitoring schemes in each of the three fracturegeometries. This base case analysis is performed using a pseudo-three-dimensionalapproach that is adopted in an attempt to achieve consistency between the expected costsof remediation and failure, which assume a three-dimensional domain, and the costs ofmonitoring, which are calculated on the basis of each individual monitoring well site.Monitoring networks in fractured media should focus on those fractures that carrythe highest flows. The”best” monitoring strategy in two of the three geometries90investigated, and the highest probabilities of detection in all three fracture geometriesoccur when the fractures carrying the highest flows are monitored. However, themonitoring strategy that provides the highest probability of detection is not necessarilythe “best” monitoring strategy. For options that provide a higher probability of detectionthan the “best” monitoring strategy, the expected cost of remediation is higher becausethe probability of detection is higher and, perhaps, because a larger remediation networkmay be required if the plume is detected farther from the contaminant source. In thesecases, the higher expected cost of remediation, when combined with any increased cost ofmonitoring that may be required to provide the higher probability of detection, outweighsthe reduction in the expected cost of failure brought about by a higher probability ofdetection.Fracture geometries that have a low probability of failure provide low values forthe objective function. Therefore, it is to the advantage of the owner/operator of a newlandfill to locate the facility on a site where, should contaminants escape from a landfillcell, there is a small probability of these contaminants migrating any great distance withinthe expected lifetime of the facility.In the three geometries investigated, the highest probability of detection occurscloser to the contaminant source when the preferred flow paths are more tortuous. Thepreferred flow paths through a fracture network become more tortuous the lower theconnectivity of the fracture network. The more tortuous the preferred flow paths, thehigher the rate of dispersion with distance travelled. A higher rate of dispersion meansthat the contaminant plume will spread over the height of the domain closer to thecontaminant source, thus increasing the probability that there will be contaminanttravelling in those fractures that are monitored. The rate of vertical spreading of thecontaminant plume with distance travelled controls the probability of detection in theregion near the contaminant source. Farther from the source, the probability of detectionis controlled by dilution of the contaminant plume, in the geometries modelled in thisstudy. In the base geometry and geometry two, the well site that provides the highestprobabilities of detection is located 75 m from the contaminant source, and in geometrythree the well site located 50 m from the source provide the highest probabilities ofdetection. In all three geometries, however, the”best” monitoring strategy occurs at thewell site located 50 m from the contaminant source. More research must be done on awider range of fracture geometries before any conclusions can be drawn concerning the91relationship between statistical descriptions of fracture geometries and the optimaldistance at which to locate a monitoring network.In addition to the base case analysis, I also investigate the viability of twodifferent multiple well configurations, and conduct a number of sensitivity studies usingthe base geometry. Two of the monitoring parameters, the detection threshold and thelength of the monitoring interval, are varied and the resulting effects on both theprobability of detecting a contaminant plume and the value of the objective function areinvestigated. The sensitivity of the value of the objective function to the discount rateand the cost of failure are investigated and the characteristics of the pseudo-three-dimensional analysis are varied.The increase in the probabilities of detection brought about by the installation of a“backup” monitoring network is insufficient to justify such an installation. However, thedecision analysis developed in this study does not evaluate other functions that arepotentially filled by a “backup” monitoring system. These functions include assisting inthe determination of the extent of a detected contaminant plume and functioning as anindicator of the effectiveness of a remediation system.Of the parameters varied in the sensitivity studies, the cost of failure is the one towhich the decision analysis is most sensitive. In the two geometries where the”best”monitoring strategy is provided when the fractures carrying the highest flows aremonitored, the decision analysis is dominated by the expected cost of failure. In thesetwo geometries, the expected cost of failure dominates the decision analysis because theprobability of failure is high and the cost associated with failure is the largest of the costsinvolved. Only when the probability of failure is very small, as in the third fracturegeometry, does the expected cost of failure not dominate the objective function. In thiscase, changes in the probability of detection do not affect the expected cost of failure verymuch; changes in the expected cost of detection brought about by variations in theprobability of detection outweigh the changes in the expected cost of failure and the”best”monitoring strategy tends to be the one that provides the lowest probability of detection.Increasing the cost associated with failure puts more weight on the expected cost offailure by producing a larger change in the expected cost of failure than in the expectedcost of detection for each percent of change in the probability of detection. This92increased influence of the expected cost of failure results in the lowest value for theobjective function being provided by a monitoring strategy that provides a highprobability of detection. Therefore the cost of failure is a potential administrative toolavailable to regulators who wish to promote the use of more conservative monitoringnetworks to ensure a high rate of compliance with environmental standards.The results of the decision analysis are less sensitive to variations in the otherparameters investigated in the sensitivity studies. The combination of monitoring optionsthat provides the “best” monitoring strategy is insensitive to changes in the detectionthreshold and changes in the discount rate over the ranges investigated. However, theprobability of detection is sensitive to the detection threshold in the region farther fromthe contaminant source, where dilution effects predominate. Of the combination ofmonitoring options that provide the “best” monitoring strategy, only the number ofmonitoring locations per well site is affected by changes in the length of time betweensamples, or variations in the characteristics of the pseudo-three-dimensional analysis. Asthe length of the monitoring interval is increased, the probability of detection decreases,and the cost of monitoring is reduced. The reduction in the cost of monitoring reducesthe impact on the value of the objective function of the additional cost of monitoringincurred when more monitoring locations are installed at each well site. The reducedimpact of the additional cost of monitoring when a long monitoring interval is usedallows the lowest value for the objective function to occur when a greater number ofmonitoring locations are installed. The number of monitoring locations per well site thatprovides the lowest value for the objective function varies with the characteristics of thepseudo-three-dimensional analysis, because the cost of additional monitoring relative tothe probability of detection and the expected cost of failure vary with the characteristicsof the pseudo-three-dimensional analysis.This study represents a first attempt to develop a decision analysis framework forthe design of contaminant monitoring networks that accounts for the unique properties offractured media. Consequently, the emphasis of the investigation is on the relationshipsbetween the different components of the decision analysis objective function with respectto changes in the probability of detecting a contaminant plume. Many questions remainunanswered. 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