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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 MEDIA  by  KAREN G. JARDINE  B. Sc., The University of Alberta, 1990  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geological Sciences  We accept this thesis as conforming to the required standard  THE UNIVERSITY  F BRITISH COLUMBIA  NOVEMBER 1993  © Karen G. Jardine, 1993  In presenting this thesis  in  partial fulfilment of the requirements  for degree at the University of British Columbia, I agree that the Library freely available for reference and study. I further agree that permission copying of this thesis for scholarly purposes may be granted by the department  or  his  by  or  her  representatives.  It  is  understood  that  an  advanced  shall make it for extensive head of my copying  or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department  of  oLoçcL  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  IiovEt-seR  5ciE1.)C_€  11  ABSTRACT  In this dissertation, a decision analysis framework is developed to assist in the design of monitoring networks at hazardous waste sites located above a fractured geologic 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 costs considered are those that are directly associated with the construction and operation of the monitoring network (actual costs). The risks considered are those that are associated with the detection of migrating contaminants and consequent costs of remediation, and the failure of the facility and the costs resulting from failure (expected costs). The benefits are considered to be the same regardless of the monitoring strategy adopted, and are neglected.  The fractured rock formation underlying the hypothetical landfill site is modelled in vertical section using a two-dimensional discrete fracture model. This model uses a particle tracking method to simulate the transport of a non-reactive solute through the fractured rock unit. Three fracture geometries are investigated, each with different hydrogeological behaviour. For each of these geometries, four monitoring schemes are considered: 1) monitoring the fractures that carry the highest volumetric flows, 2) monitoring the fractures that have the largest apparent apertures, 3) monitoring the areas of highest fracture density, and 4) placing the monitoring locations at predetermined depths. The effects of the distance of the monitoring network from the contaminant source, and the number of monitoring locations installed at each monitoring well site, are investigated for each of the four monitoring strategies in each of the three fracture geometries. The base case analysis is performed using a pseudo-three-dimensional approach that is adopted in an attempt to achieve consistency between the expected costs of remediation and failure, which assume a three-dimensional domain, and the costs of monitoring, which are calculated on the basis of each individual monitoring well site.  The best monitoring alternative in two of the three geometries investigated, and the highest probabilities of detection in all three fracture geometries occur when the fractures carrying the highest flows are monitored. However, the monitoring strategy that provides the highest probability of detection is not necessarily the best alternative.  111  In the geometries modelled, the probability of detection is influenced by the amount of vertical spreading the contaminant plume undergoes near the contaminant source as a result of the toruousity of the preferred flow paths through the fracture network. 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, the decision analysis developed in this study does not evaluate other functions that are potentially filled by a “backup” monitoring system.  The combination of monitoring options that provide the best monitoring alternative is insensitive to changes in the detection threshold and changes in the discount rate over the ranges investigated. The length of time between samples, and variations in the characteristics of the pseudo-three-dimensional analysis have only a small influence over the the combination of monitoring options that provide the best monitoring alternative.  iv TABLE OF CONTENTS  Abstract  ii  List of Tables  ix  List of Figures  x  Acknowledgement  xiv  1. Introduction  1  2. Decision Analysis Framework  5  2.1. Introduction  5  2.2. Objective Function  5  2.2.1.RiskTerm  8  2.3. Decision Scenario  13  2.3.1. Objective of Monitoring Network  13  2.3.2. Detection  14  2.3.2.1. Definition of Detection in a Monitoring Well  14  2.3.2.2. Consequences of Detection: Remedial Design  14  V  2.3.3. Failure  .15  2.3.3.1. Definition of Failure at the Compliance Surface  15  2.3.3.2. Consequences of Failure: Containment System  15  3. Review of the Flow and Transport Model  3.1. The Choice of Model  3.1.1. Generation of Fractures  3.1.1.1. Statistical Description of Fracture Geometry  16  17  17  18  3.1.2. Flow Solution  19  3.1.3. Transport  20  3.1.3.1. Introduction of Particles  20  3.1.3.2. Residence Time Distribution  21  3.1.3.3. Routing in Fracture Intersections  21  4. Choice of Model Domain  23  4.1. Horizontal Section  23  4.2. Vertical Section  24  vi  4.3. Description of Domain  .25  4.4. Pseudo-three-dimensional Analysis  26  5. Decision Model  5.1. Network Design  28  28  5.1.1. Well Siting  28  5.1.2. Monitoring Locations  28  5.1.3. Monitoring Interval  30  5.2. Defining Detection  30  5.2.1. Detection Threshold  30  5.2.2. Monitoring Period  31  5.2.3. Implementation  33  5.2.4. Limitations  34  5.3. Defining Failure  35  5.3.1. Assumptions Concerning the Compliance Surface and Failure  35  vii  6. Results  .37  6.1. Three Fracture Geometries  37  6.2. Base Case  44  6.2.1. Monitoring Parameters  44  6.2.2. Decision Analysis Parameters  44  6.3. Base Geometry  46  6.3.1. Monitoring Scheme Comparison  46  6.3.2. Sensitivity Studies  52  6.3.2.1. Detection Threshold  53  6.3.2.2. Monitoring Interval  56  6.3.2.3. Discount Rate  62  6.3.2.4. Cost of Failure  64  6.3.2.5. Pseudo-three-dimensional Analysis  65  6.3.2.6. Multiple Well Configurations  68  6.4.Geometry2  71  VII’  6.4.1. Monitoring Scheme Comparison  6.5.Geometry3  .72  80  6.5.1. Monitoring Scheme Comparison  80  6.5.2. Increased Cost of Failure  87  7. Conclusions  89  bibliography  94  ix LIST OF TABLES  Table  Page Statistical Input Parameters for the Three Fracture Geometries Investigated  37  6.2 Flow and Transport Characteristics from Preliminary Simulations of the Three Fracture Geometries Investigated  40  6.1  6.3  Decision Analysis Parameters for Base Case  44  x LIST OF FIGURES Figure  Page  1.1  Vertical cross-section of hypothetical landfill site  2.1  Decision tree based on a time independent version of the objective function  3.1  3  12  Fracture network from base geometry. a) generated fracture network, b) cleaned fracture network  18  3.2  Stream tubes in continuous intersection  22  4.1  Model domain representing hypothetical landfill site  25  4.2 Example of pseudo-three-dimensional domain with 10 slices 6.1  Fracture networks for the first realization of each geometry. a) base geometry, b) geometry two, c) geometry three  6.2  38  Cumulative probabilities of failure throughout compliance period for all three fracture geometries investigated  6.3  27  40  Volumetric flows through fractures in the fracture networks for the first realization of each geometry. a) base geometry, b) geometry two, c) geometry three  42  6.4 Total probabilities of detection over the compliance period vs. distance from the source for the base geometry 6.5  46  Cumulative probability of detection vs. time at 25 m and 75 m from contaminant source for base geometry with one monitoring location per monitoring well site. a) highest flow monitoring scheme, b) densest fracturing monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme  48  xi Figure  Page  6.6 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 monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme 6.7 Values of objective function for base geometry with base case analysis 6.8  6.9  49 50  Probability of detection vs. distance from the source for three different detection thresholds with base geometry  53  Values of objective function for three different detection thresholds with base geometry  54  6.10 Probability of detection vs. distance from the source for three different monitoring intervals with the base geometry and three different detection thresholds. a) 3 .83E6 particles/rn , b) 1.92 particles/rn 3 , 3 c) 1 particle per monitoring period 6.11  57  Arrival rate of particles at the compliance boundary in the fracture carrying the largest proportion of particles in the first realization of the base geometry with a total of 2000 particles injected  58  6.12 Values of the objective function for three different lengths of monitoring interval for the base geometry. a)threshold , b) threshold concentration 3 concentration of 1 .92E6 particles/rn 3 of 3.83E6 particles/rn 6.13  Values of objective function for the base geometry with three different discount rates  6.14 Values of objective function for two different costs of failure for the base geometry 6.15  Values of objective function for two-dimensional analysis for base geometry  60  62  64  65  xli  Figure  Page  6.16 Values of objective function for two-dimensional and two different pseudo-three-dimensional analyses for base geometry 6.17  67  Cumulative probability of detection vs. time at single well sites at 25 m and 75 m and a multiple well configuration with well at both  sites for base geometry 6.18 Values of objective function for single well sites and two multiple well configurations for base geometry. a) $5 million cost of failure, b) $10 million cost of failure  68  69  6.19 Total probabilities of detection over the compliance period vs. distance from the source for geometry two  72  6.20 Total probability of detection vs. distance for geometry two with one, two, and three monitoring locations per well site. a) highest flow monitoring scheme, b) densest fracturing monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme 6.21  73  Probability of detection vs. distance for both base geometry and geometry two. a)one particle per monitoring period and 3.83E6 , b) a)one particle per monitoring period and 1 .92E6 3 particles/m 3 particles/m  75  6.22 Values of the objective function for geometry two with base case analysis 6.23  77  Total probability of detection over the compliance period vs. distance for geometry three  80  6.24 Total probability of detection vs. distance for geometry three with one, two, and three monitoring locations per well site. a) highest flow monitoring scheme, b) densest fracturing monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme  81  xlii  Figure 6.25  Page Cumulative probability of detection vs. time for base geometry and geometry three. a) 25 m from source, b) 75 m from source  83  6.26 Values of the objective function for geometry three with base case analysis  85  6.27 Values of the objective function for geometry three with $10 million cost of failure  87  xiv ACKNOWLEDGEMENT  I would like to thank my supervisor, Leslie Smith, for his advice, guidance, and support throughout the process of preparing this thesis. I would also like to thank the other members of my supervisory committee, Brian Berkowitz, Tom Brown, and Roger Beckie, as well as Rosemary Knight and Dick Campanella who served on my examining committee. I am very grateful to have had the opportunity to learn from Al Freeze during his last year and my first year at UBC.  Special thanks go to Tom Clemo, who wrote the transport model I used and whose assistance and patience were invaluable, especially when I was debugging my computer code. The members of the groundwater group, both past and present, provided an enjoyable work environment, as well as feedback on technical problems and my interminable practise talks.  I would also like to thank my life partner, Markus Eymann, for looking after me and putting up with me throughout all of this ordeal.  Financial support was provided by both a postgraduate scholarship and an operating grant from the Natural Sciences and Engineering Research Council of Canada.  1  HYDROGEOLOGICAL DECISION ANALYSIS:  MONITORING NETWORKS FOR FRACTURED GEOLOGIC MEDIA  1. INTRODUCTION  As the population of the world grows, so does the amount of waste that we produce. The disposal of this waste has become a problem of major proportions. In the past few decades, the detrimental effects to the environment of indiscriminate, poor and/or ill-advised waste disposal practices have become clear. Harmful wastes that were buried in the ground are finding their way back to the biosphere and poisoning the water we drink, the air we breathe, and the ground that we walk on. The most common mode of transport of these wastes is in groundwater.  As the landfills we use are filled up, new sites must be found. One of the major concerns in siting a new landfill is to prevent leachate from the landfill from entering the groundwater system. One way in which this is done is to install synthetic or clay liners in landfill cells to prevent the migration of leachate. Another method is to locate the landfill in or above a geological unit that is relatively impermeable, such as clay or crystalline bedrock. Neither of these approaches is foolproof; liners may be breached and clay and bedrock contain fractures which may act as conduits for the transport of contaminants away from the landfill cells. Therefore, it is often required by law that the area surrounding a landfill be monitored in order to detect escaping contaminants. In British Columbia, all new landfills and expansions to existing landfills are required to include monitoring programs that address ground and surface water, landfill gas, and ambient air quality as a minimum, with the need for other environmental monitoring, such as vegetation 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 to assist in the design of monitoring networks in fractured media. Although similar  2 frameworks 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 the unique properties of fractured media. The prediction of contaminant transport in fractured rock is much more difficult than it is in a porous medium. In many fractured rock environments, groundwater flow occurs primarily in the fractures; there is virtually no flow in the rock matrix itself. Frequently, the bulk of the groundwater flow 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, resulting in a contaminant plume of a very different shape than is usually found in porous media. Consequently, monitoring strategies that are developed for use in porous media may be inappropriate for use in fractured media. Rock is being considered as a receiving environment for many types of waste, from municipal waste to high level radioactive waste. More and more waste disposal facilities are being constructed on or within fractured rock. Therefore, there is a real 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 an evaluation of prediction uncertainty within the decision making process. In this study, a risk-cost-benefit analysis is used to evaluate several alternative monitoring network designs. I carry out the decision analysis from the perspective of the owner-operator of a new landfill facility. However, the methodology developed in this study can easily be applied to other types of facilities, existing facilities, or sites that have already been contaminated. With some modification, it can also be applied from the perspective of a regulatory agency to facilitate the design of monitoring networks at compliance surfaces.  A vertical cross-section of the hypothetical landfill site under consideration is shown in Figure 1.1. A landfill cell is excavated through approximately 10 meters of low permeability surficial deposits and into the top 10 meters of a 50 meter thick fractured rock unit. This fractured rock unit is situated over a low permeability unit. The compliance 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 the area within which a degradation of the groundwater quality has been deemed acceptable; if a specified concentration of contaminant is detected at or beyond the compliance surface, the person or company responsible is liable to be penalized. The location of a compliance surface is usually set by a regulatory agency or negotiated by the regulatory  3  Figure 1.1: Vertical cross-section of hypothetical landfill site  agency and the owner/operator of a facility. The compliance surface for a landfill site can be any one of a number of possibilities including: 1) the zone between liners in a multiple liner 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 in this study. I have chosen to locate the compliance boundary well away from the facility to allow for the comparison of monitoring networks placed over a wide range of distances from the contaminant source.  The evaluation of prediction uncertainty is introduced into the decision analysis framework 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 the fractured rock unit. There are three possible consequences of this event that are of concern: 1) the contaminant plume will migrate past the compliance boundary and off of the landfill site, in which case failure is considered to have occurred, 2) the contaminant plume will be detected and remediated before it has crossed the compliance boundary, in which case failure will not occur, or 3) the contaminant will not be detected by the monitoring network but will remain within the landfill property boundaries. In the risk term, the costs associated with each of these consequences are multiplied by the probability of that consequence occurring. For the purposes of this study, the costs associated with failure are those of isolating the contaminant plume by the construction of  4 barrier walls. If the plume is detected before it has reached the compliance surface, a remedial program involving the installation and operation of horizontal interceptor wells is put into place. There are no costs associated with the third consequence, the contaminant plume remaining undetected and within the property boundaries. Because there is no indication that leachate has escaped from the landfill cell in this instance, no action is required. The probabilities of failure and of detection by the monitoring network are evaluated by running Monte Carlo simulations of fracture networks containing monitoring networks.  The object of locating a waste disposal facility in or on crystalline bedrock is to prevent the migration of contaminants away from the facility. A good site for a waste disposal facility is one in which a contaminant plume would migrate so slowly that it would 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 waste disposal facility is not faced with the high costs of remediating or containing a contaminant plume.  The objectives of this thesis are to investigate monitoring strategies in fractured media, and to do this within a decision analysis framework. Details of the decision analysis framework are discussed in Chapter Two. Chapter Three reviews the flow/transport model used for this study. A detailed discussion of the choice of model domain 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 Chapter Six. The conclusions are summarized in Chapter Seven.  5 2. DECISION ANALYSIS FRAMEWORK  2.1. INTRODUCTION  Decision 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 has been defined as “a formalization of common sense for decision problems which are too complex for informal common sense” (Massmann and Freeze, 1987a). By defining all variables in terms of either dollar figures or probabilities, decision analysis links the economic environment in which decisions are made and the technical information upon which these decisions are based (Massmann et. al., 1991). Decision analysis permits a consideration of prediction uncertainty within the decision making process. This is accomplished by assigning an economic value to the risk of not meeting the design objectives because of uncertainty in model predictions. Unlike optimization processes, such as linear and nonlinear programming, decision analysis does not provide an optimal solution across all decision variables. It allows only for the identification of the best solution from a finite number of options presented for investigation.  2.2. OBJECTWE FUNCTION  A risk-cost-benefit analysis is used to evaluate several design alternatives, and the alternative that provides the maximum value for the objective function is chosen. In this study, I am using one of the more general forms of objective function for the risk-cost benefit 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. The objective function is:  =  ±  l [B(t)_c(t)_R(t)] t=o(1+i)  (2.1)  6  where: 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 of construction 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 a regulatory agency, or the owner/operator is required to provide some guarantee against failure after the facility is decommissioned, the time horizon may be extended. The period of time encompassed by the time horizon is often referred to as the compliance period.  When the decision analysis is carried out from the perspective of the owner/operator of a facility, the current bank lending rate, also known as the market interest rate, is usually used for the discount rate (Massmann and Freeze, 1 987a, Freeze et. al., 1990). This discount rate generally ranges between 5% and 10%. If the decision analysis is carried out from the perspective of a regulatory agency, a social discount rate that is much lower than the market rate is used (Massmann and Freeze, 1987a). The benefits 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 the landfill (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 for a monitoring system constructed within the landfill property boundaries. There is no  7  analysis of the trade-off between design and monitoring, or of alternative methods of monitoring at the compliance boundary. To simplify the objective function, I am assuming that the revenues generated by this landfill would be the same regardless of the monitoring strategy adopted. Thus it is possible to neglect the benefits term. Since the trade-off between design and monitoring is not considered, the capital costs of constructing and operating the landfill itself are also the same regardless of the monitoring strategy chosen. The same holds true of the costs associated with monitoring at the compliance boundary. Consequently, I will neglect these costs and consider only those costs directly associated with the construction and operation of the monitoring system located within the landfill boundaries. These costs are the cost of installing the monitoring system, which includes the costs of materials and labour, and the annual cost of monitoring, which includes the costs of collecting, shipping and analyzing the groundwater samples.  The benefits term, which has been removed for the purposes of this study, is the only positive term in the objective function, all the remaining terms are negative. Rather than choose the maximum of a number of negative values, the signs are reversed and the alternative that provides the minimum value for the objective function, representing the minimum 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 year zero, the year before the landfill cell and monitoring system begin operation. Year zero is also the only year in which this one-time cost occurs. Consequently, the cost of installing the monitoring network can be moved outside the summation and the summation can now begin in year one rather than year zero.  The objective function becomes:  (I) =  + =i  (1 + 1)  [Cmon (t) + R(t)]  where: Cjt = the cost of installing the monitoring network (dollars) Cmon= the annual cost of monitoring (dollars)  (2.2)  8  Because the costs of installing the monitoring network occur in year zero, they are not affected by the discount rate chosen. The other costs of monitoring, those associated with sampling, are ongoing and don’t begin until year one. The total net present worth of the sampling costs is dependent on the discount rate. The relative impact of the two components of cost of monitoring on the total value of the objective function will vary with the discount rate chosen. The higher the discount rate, the smaller the impact of costs that are incurred in later years, and the larger the impact of the initial cost of constructing the monitoring system. It should also be noted that the time increments that are used in this analysis are years rather than months or quarters. Consequently, all of the costs that are accrued throughout any given year are not applied until the end of that year.  2.2.1. RISK TERM  The risk term represents the expected costs associated with the detection of contaminants by the monitoring network and those associated with failure. The costs associated with detection are those incurred by remediating the contamination. The costs associated with failure often include fines, penalties, and the costs of litigation as well as remediation. For simplicity, the only costs associated with failure that I consider in this study are those associated with the isolation of the contaminant plume. When applying this procedure, however, all of the costs associated with failure that an owner/operator is likely to incur should be taken into account.  For a site with no monitoring network, the risk term represents the expected costs associated 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 to allow for the possibility of the plume being detected and remediated before failure occurs:  9 R(t)  =  [Pfm (t)Crf (t)] + [Pd (t)Crd (t)]  (2.4)  where: probability of failure in year t with monitoring (decimal fraction)  fm = Crf(t) Pd(t)  =  =  Crd(t)  cost of remediation, given failure, in year t (dollars) probability of detection in year t (decimal fraction) cost of remediation, given detection, in year t (dollars)  The probability of failure in year t, with monitoring, is defined as 1 (t) = 1 Pf fnm (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 (decimal fraction)  Failure when a monitoring network is in place occurs when a contaminant plume from the landfill reaches the compliance surface without being detected by the monitoring network. The fact that both of these conditions must be met for failure to occur is reflected in equation 2.5, where the probability of failure without monitoring is multiplied by one minus the total probability of a plume being detected, to produce the probability of failure with monitoring. The probability of detection that is used in equation 2.5 represents 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 matter in what year the plume would arrive at the compliance boundary, or how long it would take to get there. This approach assumes that failure will not occur if the plume is detected; in other words, it assumes that the remediation strategy adopted will be completely successful. A monitoring network may not detect a contaminant plume if the contaminant concentration is below detection levels, if the plume travels through the fracture network along fractures that are not being monitored, or, in the case of a pulse release of contaminant, if it passes through a monitored location between the times at which that location is sampled.  10  There are two components of the cost of remediation, given detection, for the remediation strategy that I have chosen to use in this study: 1) the cost of installing the remediation network, and 2) the annual cost of operating this network. The details of the chosen remediation strategy are discussed later in this chapter. The cost of installing the remediation network is a one time cost that occurs only if a contaminant plume is detected. Therefore, the probability that the remediation network will be installed in any given year is Pd(t), the probability of a plume being detected by the monitoring network in that year. The operating costs for the remediation network begin once the remediation network is installed and are ongoing until the end of the compliance period. The probability that the annual operating cost will be paid in any given year is the probability that a plume has been detected by the monitoring network at any time up to and including the previous year.  Incorporating the definition of the probability of failure with monitoring presented in equation 2.5 and the two components of the cost of remediation given detection, along with their respective probabilities, into equation 2.4, the risk term becomes:  t—1  R(t) = ‘fnm (t)[1  —  ‘d ]Crf (t) + d (t)Crdinst (t) +  (2.6)  ‘d (i)Crdop (t)  For the purposes of this study, I am assuming that the costs associated with failure and both components of the cost of remediation given detection, remain constant with time. In other words, the cost of installing a remediation network in year ten will be the same as the cost of installing the same remediation network in year five, although the net present value of a system installed five years from now is more than one installed in ten years time, if a discount rate greater than zero is assumed. This approach assumes that there will be no technological advances that significantly reduce the cost of remediation. Given this assumption and the definitions of terms stated above, the full objective function used in this study is:  =  jst +  ±  1 (1+i)  t{mon + fnm (t)[1  —  d  lCrf + d (t)Crdinst +  d (rdop i=1  }  (2.7)  11  It should be noted that the annual cost of monitoring is unaffected by the probability of detection; it is independent of the risk term. Monitoring continues throughout the entire compliance period, even if a contaminant plume is detected and remediation 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 various components of the risk term are derived. For this example, I have assumed the following costs (in millions of dollars) and probabilities: cost of installing the monitoring network,  =  $0.1 M  total cost of monitoring, Cmon = $0.9M costs associated with failure (confinement), Cf = $5.OM costs associated with detection (remediation), Cr  $1.OM  probability of detection, d = 0.4 probability of failure (without monitoring), Pf= 0.8  These probabilities represent the total probability of the respective events occurring within the compliance period. Because there is no consideration of the times at which events occur, no distinction has been made between the two components of the cost 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 recorded above each path leading from an event node. Each path is also labeled with the consequence that it represents. The sum of the probabilities on all of the terminal paths of each decision leg must equal one. The initial node is the only decision node in this decision tree. This node represents the owner/operato?s decision whether or not to install a monitoring network. The first event node on each of the decision legs represents detection, and the second event node represents failure. There are no failure nodes on the paths on which detection has occurred, because I am assuming that failure will not occur  12 $O.40M  $2.40M  so so  so Figure 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, 1 d 0.4, Pf =0.8.  if the plume is detected. The dollar figure at the end of each terminal path is the value of that 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, his or her expected costs are the probability of the plume being detected (0.4) multiplied by the 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  13 the consequence of neither detection nor failure occurring, so this term is not included in the objective function. However, it must appear in the decision tree because it is a possible outcome, and there is a probability associated with it. If the owner/operator decides 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 for the decision to monitor is $2.8 million, and for the decision not to monitor the expected costs are $4.0 million. The owner/operator’s expected costs are higher if he or she decides 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 total value of the objective function for the decision not to monitor is $4.0 million. For the decision 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 monitoring network.  2.3. DEcisioN ScENARIo  2.3.1. OB.wcTIvE OF MONIToRING NETwoRK  Generally, the concerns of an owner/operator of a landfill facility are primarily monetary in nature. To him or her, the objective of a monitoring system is to detect a contaminant plume before it has reached the compliance boundary in order to enable early and less costly remediation of the plume, and thus avoid the potentially more costly consequences of failure. Only when the potential savings from detection, which include a less costly remediation scheme and a reduction in the probability of failure, are greater than the costs involved in constructing and operating the monitoring network plus the expected cost of failure, will the owner/operator consider it worthwhile to install a monitoring network. The decision analysis is intended to assist with the assessment of the relationship between the monetary factors mentioned above.  14  2.3.2. DETEcTIoN  2.3.2.1. Definition of Detection in a Monitoring Well  Detection is considered to occur at the earliest time at which a detectable amount of contaminant passes through a monitoring location during a monitoring period. Monitoring periods occur at regular intervals, for example every 60 days, throughout the compliance period. A detailed discussion of the method used to determine detection appears in Chapter Five of this thesis.  2.3.2.2. Consequences of Detection: Remedial Design  The remediation technique that I have chosen to use in this study involves the installation of horizontal interceptor wells at the time when detection occurs, and subsequent pumping and treating of contaminated water. Because of the long periods of time required to remediate groundwater contamination, I am assuming that, once the remediation network is installed, it remains in operation throughout the remainder of the compliance period. A horizontal interceptor well has been used with some success at Williams Air Force Base in Chandler, Arizona (Oakley et. al., 1992). It is believed that when one of these wells is pumped, its “capture zone” will be much larger than that of a conventional vertical well, resulting in more efficient plume recovery. The remediation strategy employed in this study involves the installation of these wells at approximately 50 meter intervals downstream from the contaminant source, up to and including the interval containing the monitoring well at which the plume was detected. For example, if the contaminant plume is detected at a monitoring well located less than 50 meters from the source, one interceptor well will be required. If, however, the monitoring well at which the plume is detected is located 125 meters from the contaminant source, three interceptor wells will be required to remediate the plume. This results in the cost of remediation increasing with the distance from the contaminant source at which the plume is detected. In other words, the larger the plume, the higher the cost of remediating that plume. Massmann (1987a, l987b) indexed the cost of remediation to the size of the plume by means of a linear function that related the cost of remediation to the distance between the contaminant source and the monitoring network.  15  2.3.3. FAILuRE  2.3.3.1. Definition of Failure at the Compliance Surface  As stated previously, failure is defined as a contaminant plume from the landfill cell 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 is introduced to the system as a pulse injection has reached or crossed the compliance surface. The major assumption inherent in this definition is that monitoring at the compliance surface is perfect. In other words, all of the groundwater that flows through the monitoring surface is analyzed and all of the contaminant in that water is detected.  2.3.3.2. Consequences of Failure: Containment System  The consequences of failure can be many and varied. They include regulatory penalties, the cost of litigation, remedial action, benefits foregone in the form of reduced revenues, 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 other consequences listed above. The remedial strategy that I have elected to use in the event of 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 the regulatory agencies involved that, should contaminants be detected at the compliance boundary, a grout barrier must be constructed to contain the contaminant plume. A grout barrier such as I envision has been built in fractured rock at an inactive hazardous waste landfill in Niagara Falls, New York (Gazaway et. al., 1991).  16 3. REVIEW OF THE FLOW AND TRANSPORT MODEL  Predicting groundwater flow and the transport of contaminants can be difficult. For the most part, the difficulties arise from problems associated with the characterization of heterogeneities that exist in an underground environment. Predicting the transport of contaminants is much more difficult in fractured media than in porous media. Because of the discrete nature of fractures, the characterization of a fractured medium is a complex problem. 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 groundwater flow 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 difficult and 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 elements of 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 an equivalent porous medium. Long et. al. (1982) found that in some cases, flow in a densely fractured medium could be adequately represented as an equivalent porous medium. Endo et. al. (1984) demonstrated that a fracture system that behaves as a continuum 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 and transport in discrete fractures on a variety of scales. These include models of a single fracture (e. g. Raven et. al., 1988, and Shapiro and Nicholas, 1989), and network models in which each fracture is modelled discretely, in either two dimensions (e. g. Andersson et. 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 three dimensions (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 computationally intensive. Consequently, at the field scale, it is usually impractical to attempt to model each fracture discretely, particularly in three dimensions. In an attempt to overcome this  17 difficulty, several people have begun developing continuum models of transport in fractured media. These include a stochastic continuum model that predicts the ensemble mean 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 and blocks containing the secondary fractures are modelled as continua (Smith et. al., 1990).  3.1. THE CHOICE OF MODEL  In order to address many of the issues I wished to investigate in this study, it was necessary to employ a discrete fracture model. The computer resources and the time available were insufficient to allow the use of a three-dimensional discrete model of a domain large enough for this study. Therefore, I chose to modify and use a twodimensional discrete fracture model, “Discrete”, written by Tom Clemo, that uses particle tracking to model solute transport (Clemo, in prep.).  This model simulates a rectangular domain containing a two-dimensional fracture network composed of planar fractures, each assigned a single aperture value. Groundwater flow is driven by a uniform, steady-state regional head gradient and occurs only in the fractures; the matrix is assumed to be composed of impermeable material with no open porosity. The model can be used to simulate a single network, or multiple realizations can be generated for a Monte Carlo study.  3.1.1. GENERATIoN OF FRAcTuREs  The fracture network includes both randomly generated multiple-fracture sets and individual, explicitly described fractures. The multiple-fracture sets are generated in a larger region than the simulation domain; in this way, fractures whose centres lie outside the 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 form the “generated fracture network”. This network then undergoes a “cleaning” process where fractures that do not form part of a connected path between two points on the domain boundary are removed. Fracture segments that are connected at only one end are also removed during this process. Fractures are not allowed to connect to impermeable boundaries. Figure 3.1 shows examples of a generated and a cleaned network. The  18 a) 50  S ‘---  C  40  0  30 20 S  o a) >  10 0 0  50  100 Horizontal Dimension  (  m  100 Horizontal Dimension  (  m  150  200  150  200  b) —  50  S C  40  0  30 20  J  0  50  Figure 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 in this study. When the model is operated in Monte Carlo mode, a new network is generated for each realization.  3.1.1.1. Statistical Description of Fracture Geometry  Each fracture set is generated from a statistical description consisting of six parameters, assumed to be constant throughout the domain:  19 1) The fracture density of the set. Fracture density is defined as the number of fractures per meter that are intersected by a line perpendicular to the mean orientation of the fracture set. 2) The mean fracture length of the set. The trace length of fractures within a set has been represented by either a lognormal distribution (Bridges, 1975, Barton, 1977, Baecher et. al., 1977, and Einstein et. al., 1980), or a negative exponential distribution (Robertson, 1970, Call et. al., 1976, Cruden, 1977, and Priest and Hudson, 1981). In this model, each fracture set is assumed to have a negative exponential distribution of fracture lengths. 3) The mean aperture of the fracture set. In their characterization of the Pikes Peak Granite near Manitou Springs, Colorado, Bianchi and Snow (1968) found that the apertures they measured were distributed very close to a lognormal distribution. This result was supported by the findings of pressure tests in a large number of fractured rocks throughout the world (Snow, 1970). The fracture sets in this model are assumed to have a lognormal distribution of apertures. 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 to have a Gaussian distribution of orientations. Studies have shown that the orientations of fractures in a set follow a Fisher-von-Mises distribution (Cacas et. al., 1990a), which is analogous to a Gaussian distribution for hemispherical projections. Many two-dimensional fracture models assume a Gaussian distribution of fracture orientations (Long et. al., 1982, Long and Witherspoon, 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 also be stipulated.  3.1.2. FLow SOLUTION  The finite element method is used to obtain the hydraulic head values for the flow solution. Each fracture segment is represented by a linear element with nodes at the  20 fracture intersections. The flow in the fracture segments is described by the cubic law for uniform, laminar flow between parallel plates:  Q= 1 b— Al l2ii  (3.1)  where:  Q  =  volumetric flow per unit depth of the domain  p  =  fluid density  g  =  acceleration due to gravity  =  dynamic viscosity of the fluid  =  fracture aperture  b  Ah= difference in head between nodes Al =distance between nodes  The model will accept various combinations of impermeable and prescribed head boundary conditions at the edges of the domain.  3.1.3. TRANsPoRT  The transport of solute is simulated using a particle tracking method. This method employs a large number of particles to characterize the movement of solute through a flow domain. Each particle moves in discrete steps from node to node. The particles are introduced into the domain individually and each particle completes its transit of the domain before the next particle is introduced. Since each particle is introduced at time zero, the model simulates a pulse injection of solute rather than a continuous source.  3.1.3.1. Introduction of Particles  The particles may be introduced along any side of the domain, either at a particular fracture specified as the th fracture along the chosen side, or over a range specified by a minimum and a maximum coordinate. When the solute is introduced over  21 a 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 contained within 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 Distribution  Once a particle enters the domain, it is moved from node to node until it exits the domain. The particle’s travel time is an accumulation of the residence times within each fracture segment through which the particle has travelled. The residence time for a given fracture segment is calculated from the bulk fluid velocity divided by the distance between the nodes.  3.1.3.3. Routing in Fracture Intersections  The model contains two options for particle routing at fracture intersections: complete mixing and stream tube routing. With complete mixing, the probability of a particle leaving an intersection through an outflow segment equals the proportion of the total outflow that is carried by that segment, and it is independent of the segment through which the particle entered the node. Laboratory experiments show, however, that when flow is laminar, there is little or no mixing in fracture intersections that have two inflows and two outflows (Wilson and Witherspoon, 1976, Hull and Koslow, 1986, and Robinson and Gale, 1990). In a study using both a physical model and a numerical model, Hull et. al. (1987) found that the transfer of solute across streamlines contributed significantly 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 at fracture 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 of the fracture intersection to the diffusion coefficient, can be used to characterize the relative importance of advection and diffusion within a fracture intersection. For Peclet numbers greater than approximately 0.1, advection dominates solute transport through an intersection, and stream tube routing is the appropriate model for mass transfer (Smith and Schwartz, 1993).  22  Figure 3.2: Streamtubes in continuous intersection.  In this study, mass transfer at fracture intersections is represented using streamtube routing. When a particle has entered a continuous intersection with two inflows and two outflows (Figure 3.2) the probability of that particle exiting through a given outflow segment is a function of that outflow segment’s position and the volume of flow it carries relative to the inflow segment through which the particle arrived, hereafter referred to as the entry segment. A particle will tend to exit an intersection through the outflow segment that is adjacent to the entry segment. If the adjacent outflow segment carries more flow than the entry segment, the probability of the particle exiting through that segment is one. If the adjacent outflow segment carries less flow than the entry segment, the probability of the particle exiting through that segment is the ratio of the flow in the outflow segment to the flow in the entry segment. The probability of the particle exiting through the opposite outflow segment is one minus the probability of the particle exiting through the adjacent outflow segment. This approach assumes a uniform distribution of particles across the entire width of a fracture.  23 4. CHOICE OF MODEL DOMAIN  Once the decision was made to use a two-dimensional model for the study, the question of whether to model the domain in vertical or horizontal section remained.  4.1. H01Uz0NTAL SECTION  Since the fractures in a two-dimensional model are assumed to be of great extent in the direction orthogonal to the plane of the section, the fractures represented by a horizontal section would be subvertical. Fracture networks composed chiefly of subvertical fracture sets occur in situations where the least principal stress was in the horizontal plane when the fractures were formed. The most common geologic environment composed of subvertical fractures is columnar fractures in basalt. These fractures are formed as a result of tension in the horizontal plane caused by contraction of the rock mass as it cools (Davis, 1984).  The effects of varying a number of decision and model variables can be investigated regardless of whether the section is oriented in the horizontal or vertical plane including: 1) the distance from the source to the monitoring network, 2) the frequency of monitoring, and 3) the fracture geometry within the domain. However, only a horizontal section can be used to investigate the relationship between the lateral dispersion of the solute and the spacing of the monitoring wells perpendicular to the mean direction of flow. The geometrical configuration of the monitoring wells, whether, for instance, the wells are arranged in a line or an arc, can also be investigated only in horizontal section.  There are two major disadvantages in using a horizontal section: 1) the probability of intersecting a subvertical fracture with a vertical well is extremely low, and 2) one is unable to model subhorizontal fractures, which are often important conduits for the migration of contaminants. It would probably be necessary to resort to installing angled boreholes in order to raise the probability of intersecting a fracture to a point where the probability of detection would become measurable.  24  4.2. VERTICAL SECTION  Using a vertical section allows one to model fracture geometries that are more commonly found than the subvertical systems portrayed by a horizontal section. These geometries consist of two or more sets of fractures with orientations varying from the subhorizontal to the subvertical. In such systems, many of the subhorizontal fractures are closed (nonconducting) as a result of the weight of the overburden, but the few subhorizontal fractures that remain open often form major conduits which can dominate the flow pattern and control offsite migration of solute. This behavior has been observed at many sites, including the University of Arizona test site near Oracle, Arizona (Jones et al., 1985), and Environment Canada’s National Hydrology Research Institute field site near Chalk River, Ontario (Raven, 1986).  There are some decision and model variables the effects of which can be investigated only with a vertical section. These include: 1) the criteria for choosing monitoring locations within the wells, such as choosing to monitor the fractures with the largest apertures or those that carry the highest flows, and 2) the number of monitoring locations per well.  One advantage to using a vertical section is that the probability of a monitoring well intersecting at least one conducting fracture is high if the fracture network is hydraulically connected from the source to the compliance surface. The major disadvantage is the inability to investigate the relationship between the lateral dispersion of the solute and the spacing of the monitoring wells perpendicular to the mean direction of flow. With a vertical section, one must assume that the solute moves in the plane of the section being monitored.  I chose to use a vertical section to model the hypothetical landfill site. The advantage of being able to investigate the relationship between the lateral dispersion of the solute and the spacing of the monitoring wells with a horizontal section is outweighed by the disadvantages of being confined to investigating only fracture systems composed of subvertical fractures and the low probability of intersecting a subvertical fracture with a vertical well. The ability to investigate different criteria for choosing monitoring  25 locations within a monitoring well, as well as the ability to vary the number of monitoring locations in each well, are as important as the ability to vary the number and spacing of monitoring wells perpendicular to the mean direction of flow.  4.3. DEscRIPTIoN OF DoMAIN  The model domain representing the hypothetical landfill site is shown in Figure 4.1. The model domain consists of a rectangular section of fractured rock, 200 m in the x direction and 50 m in the z direction, with impermeable boundaries along the top and bottom of the domain. The left and right edges of the domain are constant head boundaries, with a higher head along the left boundary than the right. This scenario is representative of a fractured hydrogeologic unit confined above by a layer of clays and silts and underlain by relatively unfractured crystalline bedrock. The contaminant source is along the upper ten meters of the left boundary, representing the downstream edge of a landfill cell that has been excavated into the fractured rock unit to a depth often meters. I have assumed a thickness often meters for the upper confining layer, bringing the total depth of the landfill cell to 20 m. The right boundary of the domain represents the downstream property boundary of the landfill site. This property boundary has been designated as the compliance boundary. If solute from the landfill cell crosses this boundary, failure is considered to have occurred.  ///////////////////////////////////////////////////  lOm  ‘1’  source  —  6h/6z=O  1 h=h  h=h, 6h/6z=O  / //////////////////////////////////////////////////  200m Figure 4.1: Model domain representing hypothetical landfill site.  50m  26  4.4.  PsEuDo-THREE-DIMENsIoNAL ANALYSIS  In order to achieve consistency between the costs of monitoring and the costs associated with detection and with failure, I found it necessary to devise a pseudo-threedimensional approach to the decision analysis. The costs of installing and operating a horizontal 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 also reflect a three-dimensional domain. A pseudo-three-dimensional analysis can be established by having each monitoring well that is depicted in the plane of the model represent more than one well in the plane orthogonal to the model plane. I implement this 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 arbitrarily chosen to use 10 slices as the base in the analyses. Any number of these two-dimensional slices may be monitored, and the cost of monitoring in the slice that is simulated is multiplied by the number of monitored slices. The main assumption made with this pseudo-three-dimensional approach is that the contaminant enters and travels in one slice only, consequently detection occurs in only one of the slices. The contaminant has an equal chance of entering any slice, but once it has entered a slice, it does not leave that slice; the contaminant continues to travel within that slice until it reaches the compliance surface. The probability that the slice in which the contaminant travels is monitored is the ratio of the number of monitored slices to the total number of slices. The probability of the contaminant being detected becomes the probability of detection determined by the simulation multiplied by the ratio of monitored slices. The probability of failure with monitoring is also affected by this change in the probability of detection. I will be presenting results from both two-dimensional and pseudo-three-dimensional analyses.  27  Figure 4.2: Example of pseudo-three-dimensional domain with 10 slices.  28 5. DECISION MODEL  5.1. NETwoRK DESIGN  In the design of a contaminant monitoring network, there are a number of decisions 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 (monitoring locations) 3) how often to take samples from the wells (monitoring interval).  5.1.1. WELL SITING  The use of a two-dimensional, vertical section restricts the investigation of the effects of monitoring well location to a comparison of the effectiveness of monitoring wells 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 wells perpendicular to the mean direction of flow and transverse spreading of the contaminant plume. The investigation of the effect of the number of monitoring wells in the network is restricted to whether there are one or two wells in the plane of the section, and to the number of slices that are monitored in the pseudo-three-dimensional analysis. Two monitoring wells in the plane of the section represent two rows of monitoring wells perpendicular to the plane of the section. The rationale behind installing two rows of wells is that the row farther from the source can act as a backup system to detect a migrating contaminant plume if it is missed by the closer row of wells.  5.1.2. MoNIToRiNG LOCATIONS  One or more discrete monitoring locations are usually isolated within a monitoring well. Unless individual fractures or zones of fractures are isolated, contaminated water entering the well from one fracture may be diluted by uncontaminated water entering the well from other fractures. This dilution can result in  29 deceptively low concentration readings at monitoring wells within contaminant plumes. Another reason to isolate monitoring locations within a well is the potential for the contamination of previously uncontaminated sections of the aquifer. For this reason, the isolation 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 the monitoring location. In this study, I am assuming a packer spacing, or length of monitoring location, of three meters. Because of the potential for error in placing packers accurately when installing multiple packers in one borehole, it is common practice in the industry to drill a separate borehole for each monitoring location installed at a given well site. This is the approach that I have taken when determining the cost of installing the monitoring network.  Once a monitoring well site has been chosen, the question of where to position the monitoring locations remains. In porous media, monitoring locations are usually positioned in areas of relatively high hydraulic conductivity. These areas carry a larger volume of flow than areas of lower hydraulic conductivity, thus one would be more likely to encounter migrating contaminants in the high hydraulic conductivity areas. The average 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 well sooner than those being transported through the lower hydraulic conductivity strata. I have investigated four criteria for selecting monitoring locations within a well bore in fractured media: 1) monitoring the fractures along the borehole wall that carry the highest volumetric flows, 2) monitoring the fractures along the borehole wall with the largest apparent apertures, 3) monitoring the areas of densest fracturing, and 4) placing the monitoring locations at predetermined depths. The first three of these criteria can all be considered to be analogous to some degree with areas of higher hydraulic conductivity in porous media.  In the hypothetical field situation which I am portraying, once a well site had been chosen, a borehole that fully penetrated the hydrogeologic unit, or area of concern, would be drilled. If the fourth placement criterion, predetermined depths, was chosen, the initial borehole would be drilled to half the packer spacing below the depth of the deepest  30 monitoring location rather than fuiiy penetrating the hydrogeologic unit. A geophysical log of the borehole would then be performed. A borehole TV and/or an acoustic televiewer can be used to identify the fractures with the largest apertures and the areas of densest fracturing. An examination of the drill core can also assist in locating areas of dense fracturing. A borehole flow meter can be used to determine the locations of the fractures that carry the highest volumetric flows. It should be noted that borehole flow meters that have a fine enough resolution to discriminate different volumes of flow at the low levels that are often present in fractured rock are an emerging technology at the present time. In porous media, lithologic information and injection tests are used to locate the areas of highest hydraulic conductivity.  5.1.3. MoNIToRING INTERVAL  The monitoring interval is the length of time between the collection of water samples from the monitoring network. The annual cost of monitoring is directly tied to the length of the monitoring interval; the more frequently water samples are collected and analysed, the higher the annual cost of monitoring. However, the longer the monitoring interval, the higher the risk of not detecting a contaminant plume. If the contaminants enter the flow domain intermittently or in a pulse, the possibility of a concentration peak passing through a monitoring location between monitoring periods increases with the length of the monitoring interval. In this study, I investigate a number of monitoring intervals ranging from 60 to 360 days.  5.2. DEFINING DETECTION  5.2.1. DETEcTIoN THREsHoLD  In the transport simulation model used for this study, particle tracking methods simulate the migration of a non-reactive solute through the model domain. Rather than simply assuming detection to occur if a single particle passes through a monitoring  location during a sampling time (monitoring period), I have incorporated the concept of a detection threshold in my determination of detection. The detection threshold is an attempt to mimic what happens in the field. The water samples that are collected from each monitoring location are used to determine whether or not contaminants have  31 migrated from the source. A key issue in interpreting the data acquired from the samples is the concept of a threshold concentration at which it can be determined that a contaminant species has been positively identified at a monitoring location. If a solute that is being used as an indicator species exists in the groundwater prior to the installation of the landfill facility, the detection threshold must be a large enough concentration to be easily distinguishable from natural fluctuations in the background level of that species.  5.2.2. MoNIToRING PERIOD  When a water sample is taken in the field, the water removed comes from a volume of the medium surrounding the monitoring location. The region of the domain thus affected is known as the sampling volume. Under natural gradient conditions, it takes a certain period of time for the volume of water contained in the sampling volume to pass through a monitoring location. When the detection threshold approach is used in combination with particle tracking, it is necessary to relate a monitoring period over which the particles passing through the monitoring location are counted to either the sampling volume around a monitoring location or the volume of water contained in the sample. In this study, I have chosen to relate the monitoring period to the sampling volume.  In the course of this study, I investigate three different fracture geometries, each with its own statistical description. Details of these fracture geometries are presented in Chapter Six. Each fracture geometry exhibits different hydrogeological behaviour. Therefore, if the size of the sampling volume is held constant, the length of the monitoring period must vary between fracture geometries. I have adopted the following procedure to determine the length of monitoring period to be used.  For each fracture geometry, a preliminary Monte Carlo simulation of 25 realizations is performed. From the results of this simulation, the average daily volumetric flow through the domain, Q, the average daily flow rate through a monitoring location, Vf, and the average linear groundwater velocity, v, are computed. In each realization, 21 monitoring locations are sampled. These monitoring locations are located on seven monitoring wells in which the three intersected fractures carrying the highest flows are monitored. The average linear groundwater velocity is computed using the  32 average arrival time of the fiftieth percentile breakthrough fraction, t , 5 0 at the compliance boundary:  v=  200 m 50 t  (5.1)  Using this velocity and the average daily flow rate through the domain, the effective porosity (ne) of the fracture system can be estimated:  Av  (5.2)  where: A = the cross-sectional area of the downstream boundary of the domain  I assume the shape of the sampling volume to be a cylinder surrounding and of the same 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, r , of: 1  IVf/  =1I‘d 1 r  /e  ith  (5.3)  I have chosen to use a monitoring period, mp, that corresponds to the travel time for solute to pass through a sampling volume of 0.5 m radius:  r1  (5.4)  It should be noted that the selection of the monitoring period for a given fracture geometry is based upon certain average hydraulic properties of 25 realizations of that fracture geometry. The selection is not based upon the properties of any individual realization of that fracture geometry, nor is it based upon the behavior of the medium surrounding any specific monitoring location.  33  5.2.3. IMPLEMENTATION  The detection concentration threshold is set arbitrarily at a specific number of particles, 20 for example, in a fluid volume corresponding to the average daily flow through a monitoring location as calculated from the preliminary 25 realization Monte Carlo simulation of the base case fracture geometry. No attempt is made to define absolute concentration values by assigning a specific quantity of mass to each particle. In this study, I investigate several values for the detection threshold, including a single particle per monitoring period. I look at the sensitivity to the detection threshold of both the probability of detection and the decision analysis objective function. The results of this investigation are discussed in detail in Chapter Six.  The probability of detection at any given monitoring well site for a particular monitoring strategy in any of the fracture geometries is determined by performing a Monte Carlo simulation of 200 realizations of the fracture geometry with the chosen monitoring strategy implemented. The probability of detection at that monitoring well site is the proportion of realizations in which detection occurred at the well site within the compliance period. For example, if detection occurred at a monitoring well site located 25 m from the contaminant source in 75 of the 200 realizations, the probability of detection at that well site would be 37.5%.  To determine whether detection has occurred at a given monitoring location in any one realization, the particle concentration for each monitoring period is compared to the detection threshold concentration. To calculate the particle concentration, the number of particles that pass through a monitoring location during a monitoring period, is divided by the volume of fluid that has flowed through that monitoring location during the monitoring period. The time of detection at a given monitoring location is set at the midpoint of the earliest monitoring period during which the detection threshold concentration has been met or exceeded in that monitoring location. Since the detection threshold concentration of particles is based on the “average” flow through a monitoring zone, the number of particles required to constitute detection varies with each monitoring location. For monitoring locations with a daily volumetric flow that is less than the “average”, fewer particles are required for detection, and, conversely, more particles are  34 required 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 particle tracking method for the transport simulation, I have imposed a lower cutoff on the absolute number of particles required for detection in monitoring locations with low flow rates. In most cases, this lower cutoff is one particle per day.  Because the volume of water passing through a monitoring location during a monitoring period varies from one monitoring location to another, the sampling method that has been adopted for this study results in a different number of particles being required for detection in different monitoring locations. Under ideal conditions, in a field situation, 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 monitoring location.  Values for the detection threshold concentration are based on simulations of the base case fracture geometry with 20 000 particles injected at the source. This number of particles produces stable transport characteristics. Were a greater or lesser number of particles to be injected at the source in the base case geometry, the detection threshold concentration would be adjusted accordingly. However, different fracture geometries have different effective porosities. To keep the particle concentration per unit volume of the domain constant between different fracture geometries, I have adjusted the number of particles injected at the source, but have retained the same detection threshold concentrations. This approach assumes that the effective porosity calculated for the 25 preliminary realizations of the entire domain accurately reflects conditions in the region surrounding the source.  5.2.4. LIMITATIoNs  In the course of this study, I have found that the probabilities of detection are sensitive to the procedures I have used to convert particle counts to detection threshold concentrations and to determine the length of monitoring period, despite attempts to retain the same particle concentration at the source. For this reason, it is best to confine comparisons of the probability of detection to those made between different monitoring  35 strategies within the same fracture geometry rather than attempt to carry out comparisons of the probabilities of detection between different fracture geometries.  I have attempted to mimic the sampling process used in a field setting by using a detection threshold concentration rather than an absolute particle count to indicate detection. However, because of limitations inherent in the particle tracking method and the use of a two-dimensional model to represent a three-dimensional situation, the detection threshold concentrations used in this study bear only a qualitative relationship to those that would be used in a field setting. Therefore, the probabilities of detection obtained in this study must be viewed in relative terms, and not be construed to represent values that could be expected at a specific field site.  5.3. DEFINING FAiLu1u  As stated in Chapter Two, failure is defined as a contaminant plume from the landfill cell 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 is introduced to the system has reached or crossed the compliance surface. As each particle crosses the compliance surface, the particle count for the time interval during which the particle crosses is incremented. Once transport is complete, the time of failure is set at the midpoint of the time interval during which the cumulative total of particles that have passed the compliance surface reaches or exceeds 0.1% of the total number of particles injected at the source. The probability of failure for a given fracture geometry is the proportion of the total number of realizations in which failure occurs during the compliance period.  5.3.1. AssuMPTioNs CoNcERNING THE COMPLIANCE SuRFAcE AND FAILuRE  The major assumption inherent in this definition of failure is that the monitoring at the compliance surface is perfect. In other words, all of the groundwater that flows through the monitoring surface is analyzed and all of the contaminant in that water is detected. Although monitoring at the compliance surface is unlikely to be this complete in a field setting, the probability of detection at the compliance surface is a separate issue that I have chosen not to investigate in this study.  36  Because every fracture that crosses the compliance surface is assumed to be monitored constantly, it is not convenient to set a threshold concentration for failure. The criterion for failure is arbitrarily set at 0.1% of the injected mass of contaminant. This figure is assumed to be large enough that failure will not be triggered by the arrival of a few anomalous particles that might arrive at the compliance surface well in advance of the rest of the plume. At the same time, it is assumed to be small enough to signal the beginning of the arrival of the main body of the plume.  37 6. RESULTS  6.1. THREE F1cTu1u GE0METIUEs  In the course of this study, I investigate three different fracture geometries. The statistical input parameters for all three geometries are listed in Table 6.1. All three geometries contain three fracture sets: one subhorizontal, one subvertical, and one with a mean orientation 45 degrees counterclockwise from horizontal. In both the base geometry and geometry two, the subhorizontal fractures are considerably longer than the fractures forming the other two sets. Geometry two is identical to the base geometry except that the mean aperture of the subhorizontal fracture set is twice that in the base geometry. In geometry three, all three fracture sets have the same density, length and aperture parameters. The fracture networks generated for the first realization of each fracture geometry are shown in Figure 6.1. This figure shows that the fracture networks for geometry three are considerably denser than those for the other two geometries, and that they do not contain the long horizontal fractures seen in the base geometry and geometry two.  It is advantageous in the numerical procedure to introduce the monitoring wells into the domain as long vertical fractures. These fractures, located at 50 meter intervals throughout the domain, have an aperture of one micron, at least an order of magnitude smaller than the mean aperture of any of the fracture sets in any of the geometries investigated. The vertical fractures used to simulate a monitoring well are assigned small apertures so that they will have as little effect as possible on the natural flow conditions within the domain. When the vertical fractures used to simulate a monitoring well were given apertures of less than one micron, the model often had problems arriving at a flow solution. The vertical fractures used to simulate a monitoring well are entered into the model explicitly. They are neglected in the first part of the cleaning process, when the unconnected fractures are removed, to avoid the situation in which a monitoring well fracture provides a connection in a network that would otherwise be unconnected.  38  Table 6.1: Statistical Input Parameters for the Three Fracture Geometries Investigated  Fracture  Parameter  Set  Geometry 3  fracture density (m ) 4  0.35  0.35  0.50  mean fracture length (m)  16.0  16.0  5.0  20.OE-6  40.OE-6  20.OE-6  a log aperture  0.40  0.40  0.60  mean fracture orientation (deg.)  0.00  0.00  0.00  a orientation (deg.)  10.00  10.00  10.00  fracture density (rn-i)  0.40  0.40  0.50  mean fracture length (rn)  4.0  4.0  5.0  40.OE-6  40.OE-6  20.OE-6  a log aperture  0.60  0.60  0.60  mean fracture orientation (deg.)  90.00  90.00  90.00  a orientation (deg.)  15.00  15.00  15.00  fracture density (rn-i)  0.50  0.50  0.50  mean fracture length (m)  7.0  7.0  5.0  10.OE-6  10.OE-6  20.OE-6  0.60  0.60  0.60  mean fracture orientation (deg.)  135.00  135.00  135.00  a orientation (deg.)  40.00  40.00  40.00  mean fracture aperture (rn)  3  Geometry 2  Geometry  mean fracture aperture (m)  2  Base  mean fracture aperture (m) a log aperture  3  a) —S  50  S •—  40  0  30  50  100 Horizontal Dimension  (  m  100 Horizontal Dimension  (  m  100 Horizontal Dimension  (  m  150  200  150  200  150  200  b) —S  50  S —  0  40  0  30 ci)  20 cci  o  10  Ii)  >  0 0  50  c) 50  S 40 0  30 II)  S o  20 10  ci)  >  0 0  50  Figure 6.1: Fracture networks for the first realization of each geometry. a) base geometry, b) geometry two, c) geometry three.  40  The fracture networks generated from each fracture geometry exhibit different hydrogeological behavior. Table 6.2 lists values of different flow and transport characteristics of the different fracture geometries obtained from the preliminary Monte Carlo simulations of 25 realizations that were performed for the monitoring period calculations. (The data required to calculate most of these characteristics were generated in the preliminary simulations only; they were not generated in the larger production simulations of 200 realizations.) On average, the networks generated from geometry two have approximately four times the flow through the network and an effective porosity more 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 in the base geometry. The relatively higher velocity in geometry two is reflected in Figure 6.2, a plot of the cumulative probabilities of failure for all three fracture geometries, based on Monte Carlo simulations using 200 realizations. During the first ten years of the compliance period, geometry two has a much larger probability of failure than does the base geometry, although the probabilities of failure for both geometries are relatively close by the end of the compliance period. This behaviour is the result of the higher groundwater velocities and consequent shorter travel times through the geometry two networks. Because the contaminants are travelling through the domain more quickly in the geometry two networks, failure occurs at earlier times in a larger proportion of the realizations than in the networks forming the base geometry. Fracture networks generated from geometry three have an average flow almost five times smaller than that in the base geometry, and the groundwater in these networks has an average linear velocity that is smaller by a factor greater than five. The average effective porosity of the networks, however, is almost the same for both geometries. The low velocity in the geometry three networks is evident from the late failure times and relatively low cumulative 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 the fracture networks of the first realization of each of the three geometries. Although these figures are based on a single realization of each fracture geometry, they are representative in a general way of the flux distributions in the networks generated from the respective geometries. The line thickness with which each fracture is drawn is proportional to the flow through that fracture. Those fractures carrying the highest flows are drawn with the  41 Table 6.2: Flow and Transport Characteristics from Preliminary Simulations of the Three Fracture Geometries Investigated  Parameter  Base Geometry  Geometry 2  Geometry 3  5.22E-6  2.29E-5  l.12E-6  1 .74E-5  7.43E-5  3.49E-6  8960  3390  48020  .022  .059  .004  1 .56E-5  2.52E-5  1 .68E-5  mean volumetric flow through monitoring location 1 (m /day) 3 mean volumetric flow through network (m lday) 3 tli mean travel time of 50 percentile of plume (days) mean linear velocity of flow (mlday) mean effective porosity of network  The fractures carrying the highest flows are monitored. 1  1 LL 0.8  — — —  0 —  — —  /  0.6  base geometry  /  0  0. 0.4  geometry two  geometry three  / /  0.2  E C.)  0 0  5  10  15  20  25  30  Year  Figure 6.2: Cumulative probabilities of failure throughout compliance period for all three fracture geometries investigated.  42 thickest line width, and those fractures drawn with the narrowest line width carry less than 1/30 of the maximum flow in the fracture network. The maximum flow carried in any fracture in the base geometry network is 2.7E-5 m /day. The fractures in the 3 geometry two network carry a maximum flow of 3.3E-5 m /day, slightly higher than the 3 base geometry, and the maximum flow in the geometry three network is 1 .5E-6 m /day, 3 smaller than the base geometry by a factor of almost 20. Consequently, for relative comparisons between the three geometries, the line widths can be compared only qualitatively. Each fracture network has one or more preferred flow paths which would be expected to act as major conduits for transport. The preferred flow paths through the base geometry network and the geometry two network are very similar, although there are a smaller number of fractures carrying high flows in the geometry two network, creating a slightly less tortuous, more direct, preferred flow path. Both of these fracture networks have preferred flow paths that are considerably less tortuous than those in the geometry three network.  The tortuosity of the preferred flow paths in fracture networks generated from geometry three is a contributing factor to the lower average linear velocity and longer contaminant plume travel times that have been observed in these networks. The hydraulic head difference between the left and right boundaries of the domain is the same for all of the networks generated from all three geometries. The more tortuous the preferred flow paths, the lower the effective permeability within a network. Consequently, although the networks generated from the base geometry and geometry three have similar average effective porosities, the average effective permeability is considerably lower in the geometry three networks. This lower average effective permeability results in a lower volumetric flow through the boundaries. Although the less tortuous flow paths in the networks generated from geometry two contribute to a higher average effective permeability than in the base geometry networks, the increased mean aperture of the subhorizontal fractures plays a role as well. The permeability of a fracture is proportional to the square of its aperture. The higher average effective permeability of the 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 about by a higher mean aperture, results in a larger volumetric flow through the domain boundaries in geometry two than in the base geometry.  43 a) 50  S 40 0 D)  30  a)  S C.)  a)  20 10 0 0  50 =  2.7e—05 m /day 3  100 Horizontal Dimension  (  m  100 Horizontal Dimension  (  m  (  m  150  200  150  200  150  200  b) —S  50  S 40 0  30 a)  S C)  a)  20 10 0  o  50  —=  3.3e—05 m /day 3  c) —  50  S 40 0  30 a)  S C)  20 10 0 0  50 =  1.5e—06 m /day 3  100 Horizontal Dimension  Figure 6.3: Volumetric flows through fractures in the fracture networks for the first realization of each geometry. a) base geometry, b) geometry two, c) geometry three.  44  6.2. BASE CASE  6.2.1. MoNIToRING PARAMETERs  All of the monitoring locations in all of the Monte Carlo simulations for all three fracture geometries have a packer spacing of three meters. For the base geometry and geometry two, the monitoring interval used for the base case is 60 days. Because the low volume of flow through the networks in geometry three necessitates the use of a longer monitoring period, 180 days is used as the monitoring interval for the base case for geometry three. The detection threshold for the base case is 3. 83E6 particles/m , which 3 is the equivalent of an average of 20 particles/day passing through an average monitoring location in the base geometry when the fractures carrying the highest flows are monitored. The depths of monitoring locations chosen for the predetermined depth monitoring scheme are 16.7 m, 25.0 m, and 33.3 m from the bottom of the fractured rock unit. When one monitoring location is installed at a monitoring well site when the predetermined depth monitoring scheme is implemented, it is installed at mid depth of the domain, 25 m from the bottom. When two monitoring locations are required, they are installed at 16.7 and 33.3 m from the bottom of the domain. When any of the other monitoring schemes are implemented, the monitoring locations are installed hierarchically. For instance when the highest flow monitoring scheme is implemented and two monitoring locations per well site are required, the monitoring locations are centred about the two fractures carrying the highest flows. Sensitivity studies of the detection threshold and length of the monitoring interval are carried out using the base geometry. The results of these studies are discussed later in this chapter.  6.2.2. DEcisioN ANALYSIS PARAMETERs  The decision analysis parameters that are used for the base case are listed in Table 6.3. The values used for the monitoring costs were obtained from various local consulting engineers, geophysical firms, and well drilling contractors, and from a local water testing laboratory. An order of magnitude estimate for the cost of constructing a horizontal interceptor well was arrived at through conversations with various consulting engineers and equipment suppliers at a 1992 EPA conference. The cost of constructing a  45 Table 6.3: Decision Analysis Parameters for Base Case  Basic Objective Function Parameters: discount rate  (%)  5.0  compliance period (years)  30.0  total number of slices in pseudo-three-dimensional domain  10  number of monitored slices  3  Monitoring Cost Parameters: advance rate for drilling (mlhr)  5.0  drill rig chargeout rate ($/br)  145  technician chargeout rate ($fhr)  60  ($)  30  cost of 3m length of well screen ($)  50  cost of 3m length of well casing  cost of 25 1 bag of bentonite cost of 30 1 bag of sand  ($)  35  ($)  cost per well site of borehole logging  10  ($)  1000  cost of collecting, shipping and analysing each water sample  525  ($) Costs Associated with Detection: cost of constructing each interceptor well  ($)  annual cost of operating each interceptor well  500 000  ($)  20 000  Costs Associated with Failure: cost of constructing grout barrier wall  ($)  5 000 000  46 grout barrier wall is based on the cost of a similar wall constructed in fractured rock at Niagara Falls, New York. Sensitivity studies of the discount rate, the costs associated with 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 GEOMETRY  The monitoring period calculated for the base geometry is seven days. Seven days is the amount of time required for the water contained in a three meter long cylindrical sampling volume with a diameter of one meter to pass through an average monitoring location when the fractures carrying the highest flows are monitored. The minimum number of particles required for detection in monitoring locations with low volumetric flow rates is seven, which corresponds to one particle per day throughout the monitoring period. The total number of particles injected in each realization is 20 000. A detailed discussion of detection and the method used to determine the length of the monitoring period was presented in Chapter Five.  6.3.1. MoNIToRING SCHEME CoMPARIsoN  The total probabilities of detection over the compliance period vs. the distance from the source are plotted in Figure 6.4 for all four monitoring schemes. These probabilities are those obtained from 200 Monte Carlo realizations in a two-dimensional analysis performed with the basic monitoring parameters and one monitoring location per well site. Confidence intervals are not relevant because each probability determined in this study is a single data point representing the proportion of realizations in which the event occurs. The highest flow monitoring scheme consistently provides the highest probability of detection, followed by the densest fracturing, largest aperture, and predetermined depth schemes respectively. The curves for both the highest flow and the predetermined depth monitoring schemes are smooth convex curves that peak in the vicinity of the well site located 75 m from the contaminant source. The anomalies in the densest fracturing and largest aperture monitoring scheme curves are likely the result of the limited number of Monte Carlo simulations performed. When 200 different realizations were performed with the densest fracturing monitoring scheme, the plot of the total probabilities of detection vs. the distance from the source took on the same  _____  47 100  • —°—-—  80  high flow monitoring density monitoring aperture monitoring  —X—  60  predetermined depth  40  e  20 xx  0 0  25  50  75  100  125  150  175  200  Distance from Source (m)  Figure 6.4: Total probabilities of detection over the compliance period vs. distance from the source for the base geometry.  convex shape as the curves for the highest flow and the predetermined depth monitoring schemes in Figure 6.4. The difference in the probabilities of detection between the two sets of realizations was between 3% and 4%. It is likely that if more than 200 Monte Carlo realizations were performed, all of the curves would take on the smooth convex shape of the curves for the highest flow and the predetermined depth monitoring schemes in Figure 6.4.  The probabilities of detection closer to the source than the peak are lower because the contaminant plume has not yet had much opportunity to disperse vertically. The contaminant, which is introduced in the top ten meters of the domain, is more likely to be travelling in fractures located in the upper portion of the domain in the region close to the contaminant 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 the lower portion of the domain as it is in the upper portion. When the predetermined depth monitoring scheme with one monitoring location per well site is employed, the monitoring location is located at mid depth, 25 m from the bottom of the domain. While the plume is small and travelling in the upper portion of the domain, it is more probable that the bulk of the contaminant may be localized in fractures that are not monitored, and  48 consequently not be detected. As the plume continues to travel through the fracture network it becomes more disperse as it extends over a larger portion of the domain. As the plume spreads, it is more likely to be travelling in fractures that are monitored. Once a certain amount of spreading has occurred, however, the plume may become diluted to the point where the contaminant concentrations are lower than the detection threshold concentration. This dilution is reflected in the tailing off of the probabilities of detection at 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 two ways: the earlier a plume is detected, the longer the remedial network must be in operation, and the earlier detection occurs the less the costs associated with detection are discounted with time. The cumulative probabilities of detection vs. time at both 25 m and 75 m from the source for all four monitoring schemes are depicted in the plots in Figure 6.5. These plots are for the case where there is one monitoring location per well site. In the high flow, densest fracturing, and the large aperture monitoring schemes, the probability of detection at 25 m is higher than that at 75 m in the early times. By the end of the compliance period, the probability of detection is higher at 75 m in the high flow and largest aperture schemes, and the probability of detection at both monitoring well sites are similar in the densest fracturing monitoring scheme. This behaviour is a result of the time lag in the time at which the contaminant plume first reaches the monitoring well site 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 is higher 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 the plume has had more opportunity to disperse by the time it has travelled that far, and is more likely to have spread into the middle of the domain. There is no apparent time lag when this monitoring scheme is implemented. The most likely explanation of this behaviour 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 sufficient dispersion that the plume has spread into the middle of the domain. The travel times through these tortuous paths are high in comparison to the travel times in most of the realizations. When the other three monitoring schemes are implemented, some of the  49  a)  b) •  75 m from  1 0.8  — —  -  Co  D  E  —  source  75 m from  source 0.8  25 m from  — —  .2 0.6  0.6  0.4  0.4  0.2  0.2  8  25 m from  source  0 0  5  10  15  20  25  30  c)  0  5  10  15  20  25  30  d) 75 m from  0 .0 Co .0 0  D  E C.)  source 0.8  — —  —  25 m from  source  0.8 —  source  o 0.6  —  0.6  0.4  0.4  0.2  0.2  0  75 m from 25 m from source  0 0  5  10  15  20  25  Year  30  0  5  10  15  20  25  30  Year  Figure 6.5: Cumulative probability of detection vs. time at 25 m and 75 m from contaminant source for base geometry with one monitoring location per monitoring well site. a) highest flow monitoring scheme, b) densest fracturing monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme.  plumes that are detected at the 25 m well site are still relatively small and have travelled relatively quickly. They are detected because the monitoring location is in the upper portion 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 the realizations. The near horizontal slope of both curves beyond 15 years in all four plots indicates that few detections occur beyond this time.  The plots in Figure 6.6 provide a comparison of the total probabilities of detection observed with one, two and three monitoring locations per well site for each monitoring scheme. With all four monitoring schemes, the more monitoring locations at each well site, the higher the probability of detection. The difference between the probabilities of  a)  b) C 0  50  100  100  80  80  60  60  40  40  20  20  a) 0 0  0 0  0  0 0  25  50  75  100  125  150  175  200  c)  0  25  50  75  100  125  150  175  200  d) 100  100  -80  80  60  60  40  40  20  20  o 0  3 locationS/well Site 2 locations/well Site tocatlon!welt ile  0  0 0  25  50  75  100  125  150  Distance from Source(m)  175  200  0  25  50  75  100  125  150  175  200  Distance from Source(m)  Figure 6.6: 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 monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme.  detection from one to two monitoring locations is greater than the difference from two to three monitoring locations at every well site for each of the four monitoring schemes. All of the curves in all of the plots exhibit lower probabilities close to the source, the highest probability of detection at 50 or 75 meters from the source, and a trailing off of the probability of detection at larger distances. This behaviour is particularly evident when the highest flow monitoring scheme is employed. When this monitoring scheme is used, the difference in the probability of detection between one monitoring location and two monitoring locations at the 25 m well site is considerably larger than that at any of the well sites that are farther from the source. This large spread in the probabilities of detection implies that, while the plume is small and located in the upper portion of the domain, the chances of detecting a contaminant plume increase much more by the addition of a second monitoring location at each well site than they do once the plume is more dispersed. When two monitoring locations are installed at a well site, there is a higher probability that there will be at least one monitoring location in the upper half of  51 addition of a second monitoring location at each well site than they do once the plume is more dispersed. When two monitoring locations are installed at a well site, there is a higher probability that there will be at least one monitoring location in the upper half of the domain than there is if oniy one monitoring location is installed. Whether the increase in the probability of detection resulting from the installation of additional monitoring locations at any given well site is worth the additional expense of installing and sampling from those monitoring locations is a question that must be evaluated within the decision analysis framework.  A histogram of the values obtained for the objective function with the base case analysis is shown in Figure 6.7. There are three values for the objective function given at each well site. These represent the values obtained for that well site with one, two, and three monitoring locations installed. The column closest to the viewer represents the value of the objective function that is obtained with one monitoring location installed at the well site; the middle column is for two monitoring locations, and the farthest column is for three monitoring locations. The highest flow monitoring scheme consistently provides the lowest values for the objective function, while the predetermined depth scheme provides the highest. The values provided by the largest aperture and the densest fracturing monitoring schemes are close but the values for the largest aperture scheme are usually slightly higher. The value of the objective function that is obtained for this geometry when no monitoring is undertaken is $2.64 million. The predetermined depth monitoring scheme is the only one that does not provide at least one option with an  2  •4,  o .—  C  2.50  0I.I-  S  depth aperture density flow  U-,  c’J  Figure 6.7: Values of objective function for base geometry with base case analysis.  52  In almost every instance, one monitoring location per well site provides a lower value 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 monitoring location is installed indicate that the reduction in the expected costs associated with failure, incurred as a result of the increase in the probability of detection, are not as great as the additional expenses incurred. These expenses include both the cost of installing and taking samples from the additional monitoring locations and an increase in the expected costs associated with detection that results from an increase in the probability of detection. The two instances where one monitoring location per well site does not provide the lowest value at the well site occur at the 25 m well site when either the highest flow or the predetermined depth monitoring scheme is implemented. In these instances, the difference in the probability of detection between one and two monitoring locations is considerably larger than in any of the other situations. Thus in these cases the reduction in the expected costs of failure outweigh the increase in both the costs associated with monitoring and the expected costs associated with detection.  The lowest value for the objective function overall, $2.41 million, occurs at the well site 50 m from the source, with one monitoring location centred about the fracture carrying the highest flow. This is not the situation that provides the highest probability of detection. The highest probability of detection, 88.5%, occurs with the flow monitoring scheme at the well site located 50 m from the source with three monitoring locations installed. When one monitoring location is installed per well site, the highest probability of detection occurs at the well site located 75 m from the source with the fractures carrying the highest flows monitored. This does not provide the lowest objective function, however, because two horizontal interceptor wells are required for the remediation program if the contaminant plume is detected 75 m from the source. If the contaminant plume is detected 50 m from the source, only one interceptor well is required. Any advantage gained by the higher probability of detection at 75 m is more than offset by the additional expected expense of installing and operating two interceptor wells rather than one.  6.3.2. SENsITIvITY STUDIES  53  A number of sensitivity studies involving both monitoring parameters and decision parameters are carried out using the base geometry. Unless otherwise indicated, the results presented are those obtained when the highest flow monitoring scheme is implemented. This monitoring scheme was chosen for the sensitivity analyses, because, in the base case analysis of the base geometry, it provided the highest probabilities of detection and it is the preferred monitoring scheme with the lowest values for the objective function. This is the monitoring scheme that the owner/operator of the landfill facility should choose to install on a site located above a fractured rock unit with this fracture geometry. When the sensitivity study analyses were carried out using other monitoring schemes, the same basic behaviour was exhibited as when the highest flow monitoring scheme was implemented. Except where explicitly stated, only one parameter is varied in each study; the remaining parameters are equal to those in the base case analysis. As in the monitoring scheme comparison, the probabilities of detection reported are those obtained from a two-dimensional analysis.  6.3.2.1. Detection Threshold  One of the sensitivity studies performed with the base geometry is the effect of different detection concentration thresholds on both the probability of detection and the value of the objective function. A plot of the probability of detection vs. distance from the source for three different detection thresholds is shown in Figure 6.8. This figure reports the total probabilities of detection over the compliance period for two threshold , and a detection threshold of one particle 3 concentrations, 1.92E6 and 3.83E6 particles/rn per monitoring period. The two threshold concentrations represent, respectively, 10 and 20 particles per day passing through an average monitoring location in the base geometry, when the highest flow monitoring scheme is implemented. These probabilities are those obtained when one monitoring location is installed at each well site.  54  100  a -  80  0 .4-’  60 0 ‘I  0 40 •  I particle per monitoring period  20 —°-——  I .92E6 particlesIm3  A  3.83 particleslm**3  0  I  0  25  50  75  100  125  150  175  200  Distance from Source (m)  Figure 6.8: Probability of detection vs. distance from the source for three different detection thresholds with base geometry.  At all monitoring well sites, the higher the detection threshold, the lower the probability of detection. The probabilities of detection obtained with each of the three detection thresholds are relatively close to each other at the well sites located at 25 m and 50 m from the source, but diverge at greater distances. When only one particle per monitoring period is required for detection, the probability of detection increases with distance to a distance of 125 m from the source, where it levels off. This steady rise with distance reflects the increasing likelihood, as the plume disperses, that contaminants will be transported in fractures that are monitored. With a detection threshold of only one particle per monitoring period, no dilution effects are evident. The effects of dilution with dispersion become evident in the curves representing the two threshold concentrations. With the lower of the two threshold concentrations, the probability of detection 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 of detection decreases rapidly with distance beyond a distance of 100 m. The more particles required to constitute detection, the more rapid the decrease in the probability of detection as the contaminant plume becomes more disperse and the contaminants more dilute. The  55 distance from the source at which dilution becomes an issue is smaller with a higher threshold concentration.  Figure 6.9 is a histogram of the values obtained for the objective function with the three detection thresholds. At all well sites located more than 25 m from the source with all three detection thresholds, the lowest value for the objective function is achieved when one monitoring location is installed. The lower the detection threshold, the lower the value obtained for the objective function. When the same number of monitoring locations per well site are installed, a lower detection threshold will produce a higher probability of detection. In this situation, the actual costs are the same, regardless of the detection threshold, but the expected costs vary; the costs associated with detection increase, and the costs associated with failure decrease. The value of the objective function decreases with the detection threshold because the decrease in the expected cost of failure brought about by the increased probability of detection outweighs the increase in the expected cost of detection. In general, the differences in the value of the objective function are less between detection thresholds than they are between different numbers of monitoring locations. Even though the increase in the probability of detection at a given well site brought about by the addition of another monitoring location may be greater  2.7 .  0  .0  Cu100 m 75 m  383 parUcles/m**3 I .92E6 particles/m**3 1 parbcle/mon. penod  50 m 25 m  Figure 6.9: Values of objective function for three different detection thresholds with base geometry.  56 than the difference between any two detection thresholds when the same number of monitoring locations are installed, the additional cost of monitoring is much greater than the net reduction of the expected costs brought about by the increased probability of detection.  The lowest value of the objective function, $2.34 million, occurs at 50 m from the source when one monitoring location is installed and one particle per monitoring period is required for detection. The rise in the values of the objective function at 75 m from the source reflects the increased expected cost of remediation brought about by the requirement of installing and operating a second horizontal interceptor well. The values obtained with the two lower detection thresholds decrease again at 100 m from the source. This decrease is probably brought about by two factors: an increase in the total probability of detection and a later time of detection. When the plume is detected at a later time, the expected costs of remediation are reduced because of both discounting with time and the shorter period of time over which the remediation network must be in operation.  It should be remembered that the detection thresholds used in this study are arbitrary. No attempt is made to define absolute concentration values by assigning a specific quantity of mass to each particle. This sensitivity study shows that the objective function is relatively insensitive to the detection threshold chosen, as is the probability of detection at well sites close to the contaminant source. Dilution may conceivably play a role when a metal or salt that occurs naturally in the groundwater is used as an indicator species. In this case, a relatively high concentration may be required as a detection threshold in order to distinguish between the existence of contaminant from the landfill facility and natural fluctuations in the background level of that species. Organic compounds are usually detectable at very low concentrations. Consequently, concentration reductions brought about by dispersion are usually not an issue when organic compounds are used as indicator species.  6.3.2.2. Monitoring Interval  The sensitivities of both the probability of detection and the value of the objective function to the length of the monitoring interval are investigated using the base geometry.  57 Three 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 detection are 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 each well site, are shown in the plots in Figure 6.10.  For all three detection thresholds, the total probability of detection decreases as the monitoring interval increases. It is thought that because of the many different pathways that can be taken from the contaminant source to each of the monitoring locations, there is a wide variation in the amount of contaminant that passes through a monitoring location from one monitoring period to another, and the less frequently a sample is taken, the lower the probability that a sample will be taken during a monitoring period when the contaminant concentration is large enough to constitute detection. This behaviour may be exaggerated to some extent by the discrete nature of particles. An indication of the variation in the number of particles passing through a specified location within a discrete time period can be gained from the breakthrough curve shown in Figure 6.11. This plot shows the arrival rate of particles at the compliance boundary through the fracture carrying the largest proportion of particles in a single realization of the base geometry. 2000 particles are injected into the domain in this realization and the arrival array is divided into 500 time periods of 50 day duration. Although the overall shape of the breakthrough curve is relatively smooth, there is a wide variation in the number of particles passing through the compliance boundary from one time period to another. A plot of a breakthrough curve at the compliance boundary is used to illustrate this point because the program used for the model does not have the capability of generating a similar curve at a monitoring location. It should be noted that this comparison is also limited by the fact that there are only 2000 particles used to model transport in this realization, and 200 000 particles are used in each realization of the production runs.  For the two threshold concentrations, 1 .92E6 and 3.83E6 particles/rn , the 3 difference between the total probabilities of detection for the three lengths of monitoring interval remains more or less constant as the distance from the source increases. However, when only one particle per monitoring period is required for detection, the probabilities of detection approach each other as the distance from the source increases.  58  a) —a--— 60 day monitoring interval  100  180 day monitoring friterval 80  5)  —a--— 360 day monitorhg interval  60  .3  40  20  0  25  50  75  100  125  150  175  200  Distance from Source (m)  b) 100  80 C  a .  60  0  .3  ymonitoring tel  40  —0--—  20  180 day monitoritig interval 360 day monitori-ig interval  & 0 0  25  50  75  100  125  150  175  200  Distance from Source (m)  c) 100  80 C  a .  5)  60 ay monitoring interval  0  .3  40  —0——  180 day monitoring i,terval  —*——  360 day monitoring iiterval  45  20  & 0 0  25  50  75  100  125  150  175  200  Distance from Source (m)  Figure 6.10: Probability of detection vs. distance from the source for three different monitoring intervals with the base geometry and three different detection thresholds. a) 3. 83E6 particles/rn , b) 1.92 particles/rn 3 , c) 1 particle per 3 monitoring period.  59  40  -  0  30  -o  C)  4-  CD CD  C.)  20  4-  CD  0  10  0  5000  4 io  4 1.5X10  4 2X10  Time (days)  Figure 6.11: Arrival rate of particles at the compliance boundary in the fracture carrying the largest proportion of particles in the first realization of the base geometry with a total of 2000 particles injected.  60 However, when only one particle per monitoring period is required for detection, there is little difference in the probabilities of detection between the different lengths of monitoring interval. The probability of sampling during a time period when the number of particles passing through the monitoring location is large enough to be detected is higher when fewer particles are required to constitute a detection.  Histograms of the value of the objective function for the two threshold concentrations are included in Figure 6.12. Both histograms show that with a few exceptions, the value of the objective function decreases as the length of the monitoring interval increases. This decrease reflects the lower cost of monitoring associated with less frequent sampling. When one monitoring location is installed 25 m from the contaminant source, however, the value of the objective function for both threshold concentrations increases when the monitoring interval is lengthened from 180 to 360 days. This increase indicates that the reduction in monitoring costs afforded by the drop in sampling frequency is smaller than the increase in the expected costs brought about by the decrease in the probability of detection. When the monitoring interval is 60 days, for both detection criteria the lowest value for the objective function at each well site, except for the one 25 m from the contaminant source, occurs when one monitoring location is installed. As the length of the monitoring interval increases, the lowest value for the objective function shifts to situations where more than one monitoring location is installed at each well site. This trend is particularly apparent when the threshold concentration is 3 .83E6 particles/m . Here, when the length of the monitoring interval is 3 increased to 180 days, the lowest value for the objective function at each well site occurs when there are two monitoring locations installed. When the monitoring interval is further increased to 360 days, the lowest value for the objective function at three of the four well sites shown occurs when there are three monitoring locations. With less frequent sampling, the ongoing costs of sampling are reduced to the point that the increased monitoring costs incurred by the addition of monitoring locations are outweighed by the reduction in the expected costs that result from the increased probability of detection that accompanies the addition of monitoring locations. These histograms are a good illustration of the way in which a decision analysis framework can resolve complex design issues into a straightforward evaluation.  61  a)  0  G  .2  ‘s  2.4-  LI  25 m  6Odays l8odays 36odays  b)  F  .isc .2 U.  6Odays 180 days 36Odays  Figure 6.12: Values of the objective function for three different lengths of monitoring interval for the base geometry. a)threshold concentration of 1 .92E6 , b) threshold concentration of 3.83E6 particles/rn 3 particles/rn . 3  62  This analysis shows a decrease in the value of the objective function when the length of the monitoring interval is increased. If the decrease in the probability of detection is to some extent an artifact of the particle tracking method, the decrease in the value of the objective function with a longer monitoring interval would be even greater in a field situation. The lower probabilities of detection obtained in the model, increase the value of the objective function because of the effect of the probability of detection on the expected cost of failure. However, when deciding upon a sampling frequency, care should be taken to consider the velocity with which a contaminant plume could be transported in the fracture network. The analysis used in this study assumes that when a contaminant plume is detected, it has not spread much beyond the monitoring network. If the monitoring interval is long, and the velocity of the front of the plume is high, a contaminant plume could extend some distance beyond the monitoring network before it is detected, resulting in a much higher cost of remediation than anticipated. In the base geometry, failure occurs during year two in 1% of the realizations. This means that in the fracture networks generated in those realizations, the front of the contaminant plume could travel more than 100 m between samples if a monitoring interval of 360 days is adopted. In a situation such as this, it is possible for failure to occur before or at the same time as the plume is detected, even though the contaminant plume has travelled through one or more monitoring locations. In 50 % of the realizations, failure has occurred by the end of year ten. In half of the realizations generated, the front of the contaminant plume can travel 20 m or more in 360 days. Because of longitudinal dispersion, the velocity of the front of the plume is greater than that of the centre of the plume. In networks generated from the base geometry, the mean velocity of the centre of the plume is approximately 8 rn/year, less than half that of the plume front.  6.3.2.3. Discount Rate  A comparison of the values obtained for the objective function when discount rates of 0%, 5%, and 10% are used is shown in the histogram in Figure 6.13. Changing the discount rate changes the relative weight of costs incurred early in the compliance period and those that are incurred in later years. The value of the objective function is reduced when the discount rate is increased, because the present day value of ongoing costs, such as the cost of sampling from the monitoring network, and costs arising later in  63  4  C) •  C  ow  0% discount rate 5% discount rate 10% discountrate m  Figure 6.13: Values of objective function for the base geometry with three different discount rates.  the compliance period, such as the expected costs of remediation, decrease as the discount rate increases. The values obtained with a discount rate of 10% are less than half those obtained when the costs are not discounted over time. For all of the well sites except the one located 25 m from the source, one monitoring location per well site provides the lowest value for the objective function, regardless of the discount rate used. At this well site, the value for the objective function provided by the installation of two monitoring wells is slightly lower than that provided by one monitoring location when a discount rate of 5% or 10% is used. In these cases, discounting the ongoing costs of sampling over time reduces the overall cost of monitoring so that the reduction in the expected cost of failure brought about by the increase in the probability of detection outweighs the cost of installing and sampling from a second monitoring location. The lowest value for the objective function occurs at 50 m from the source for each one of the three discount rates used. In the design scenario used for this study, the same option provides the lowest value for the objection function regardless of the discount rate chosen. Other than changing the absolute value of the objective function, changing the discount rate does not have much impact on this analysis.  64  6.3.2.4. Cost of Failure  The effects of a higher cost associated with failure are shown in the histogram in Figure 6.14. Increasing the cost associated with failure throws more weight on the expected cost of failure relative to the actual costs in the objective function. A higher cost associated with failure should promote a more conservative design for a monitoring network by increasing the effects of the probability of detection. An increase in the probability of detection will cause a larger reduction in the expected cost of failure, when the costs associated with failure are high. In this way, any additional costs incurred to provide an increase in the probability of detection may be offset by the large reduction in the expected cost of failure. The decision analysis framework provides a basis for the owner/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 objective function is obtained with one monitoring location installed per well site at every well site but the one located 25 m from the source. As discussed previously, this behaviour occurs because the reduction in the expected costs of failure brought about by the increase in the probability of detection, when more than one monitoring well is installed, is not as large as the additional cost of installing and sampling from the additional monitoring wells plus the increased expected cost of remediation that is also a result of the higher probability of detection. When the cost associated with failure is increased to $10 million, the lowest value for the objective function at each well site occurs when there are two monitoring locations installed. With the larger cost associated with failure, the amount of reduction in the expected cost of failure is now large enough to offset the increased costs incurred when two monitoring locations are installed but not those incurred when three monitoring locations are installed. The lowest value for the objective function does not occur when more than one monitoring location is installed at the well site located 100 m from the source, where the difference in the probability of detection between one and two monitoring locations is not as great as it is at the closer well sites (see Figure 6.6). For both cases, the lowest value of the objective function occurs at the well site 50 m from the source.  65  $10 Million Cf $5 Million Cf  LL. 100 m  75 m  50 m  25 m  Figure 6.14: Values of objective function for two different costs of failure for the base geometry.  6.3.2.5. Pseudo-three-dimensional Analysis  The pseudo-three-dimensional approach to the decision analysis that is adopted for this study is an attempt to achieve consistency between the costs of monitoring and the expected costs associated with detection and failure. In this approach, the hypothetical 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 the hypothetical 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 one slice only. The likelihood that well sites have been installed in the slice through which the contaminant is travelling is the ratio of monitored slices to the total number of slices in the hypothetical landfill site. The investigation of the effects of the pseudo-threedimensional analysis on the objective function involves varying both the total number of slices considered and the proportion of those slices that are monitored.  The histogram depicted in Figure 6.15 shows the resulting values of the objective function when the base case analysis is performed from a two-dimensional perspective for each of the four monitoring schemes. In this perspective, there is only one slice in the  66  hypothetical landfill site, and each monitoring well site in the simulation represents only one well site. At each well site, the lowest value of the objective function occurs when three monitoring locations are installed, regardless of the monitoring scheme implemented. This behaviour is the result of two factors: the costs of installing and sampling from additional monitoring wells are 1/3 those involved in the pseudo-threedimensional analysis adopted for the base case, and the probabilities of detection are 3.3 times as great. As in the base case, the highest flow monitoring scheme provides the lowest values for the objective function, and the predetermined depths scheme the highest. The lowest overall value for the objective function occurs at the well site that is 50 m from the source when the highest flow monitoring scheme is implemented, the same location and monitoring scheme as in the base case.  The histogram of objective function values in Figure 6.16 is a comparison of the two-dimensional analysis discussed above, the base case in which three of ten layers are monitored, and a pseudo-three-dimensional analysis in which the hypothetical landfill site is represented by 20 slices, three of which are monitored. The values obtained for the objective function with the two dimensional analysis are consistently lower than those obtained with either of the pseudo-three-dimensional analyses. The highest values for the  2.5  o  54  .Q  OLI  m m  depth aperture density  Figure 6.15: Values of objective function for two-dimensional analysis for base geometry.  67 objective function are obtained when the domain is divided into 20 slices, because the probabilities 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. In the two-dimensional analysis, the lowest value for the objective function at each well site occurs when three monitoring locations are installed, and in both pseudo-threedimensional analyses, the lowest value for the objective function at each well site usually occurs when one monitoring location is installed. The reasons for this behaviour have been discussed previously. When the domain is divided into 20 slices, and three of them are monitored, the cost of installing and sampling from each monitoring location is the same as in the base case, but the probabilities of detection that are used in the objective function are half those of the base case. These two factors have the effect of increasing the 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 remain the same as in the base case. This increase in the expected cost of failure is evident at the 25 m well site where the addition of a second monitoring location results in a slight lowering of the value of the objective function in the base case but an increase in the 20 slice case. There is a considerably smaller increase in the values of the objective function between the two pseudo-three-dimensional analyses than there is between the two-dimensional analysis and the base case. This increase is smaller because it is the result of only the higher expected cost of failure brought about by the lower probabilities of detection that are used when the domain is divided into 20 slices. There is no increase in the cost of monitoring between the two pseudo-three-dimensional analyses as there is between the two-dimensional analysis and the base case.  It should be noted that this pseudo-three-dimensional analysis is only an attempt to introduce a three-dimensional element into this decision analysis. The threshold concentrations used, and the decisions concerning the total number of slices into which the domain is divided, and the proportion of those slices which are monitored, are arbitrary. The probabilities of detection observed are not neccessarily those that would be obtained in a field situation or if a three-dimensional transport model were used to simulate the transport of contaminants through the hypothetical landfill site.  68  2.5-  o •  4 C  .  CU.  0.5-  75 m  25 m  3120 layers monitored 3/10 layers monitored 1/1 layers monitored  Figure 6.16: Values of objective function for two-dimensional and two different pseudothree-dimensional analyses for base geometry.  6.3.2.6. Multiple Well Configurations  The multiple well configurations considered in this study consist of two rows of monitoring well sites, represented by two well sites in the plane of the model situated at different distances from the contaminant source. The reason behind installing two rows of monitoring wells is that the row situated farther from the source can act as a backup system to detect plumes that may be missed by the closer row of wells. Two multiple well configurations are considered in this sensitivity study: one with well sites located 25 m and 50 m from the source, and one with well sites at 25 m and 75 m. The plot of the cumulative probabilities of detection for single well sites at 25 m and 75 m and a multiple well configuration with wells at both sites (Figure 6.17) shows that, throughout the compliance period, the probability of detection for the multiple well configuration is higher 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 sum of the probabilities of detection at the two individual well sites, because the plumes in some of the realizations are detected at both well sites. The probabilities shown in this  69  II  0  0.8  Year  Figure 6.17: Cumulative probability of detection vs. time at single well sites at 25 m and 75 m and a multiple well configuration with well at both sites for base geometry.  figure are those obtained when one monitoring location is installed at each well site. It should also be noted that the curve for the multiple well configuration roughly parallels that 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 two multiple well configurations are presented in the histograms in Figure 6.18. Figure 6.18a represents the values obtained with the base case analysis and the highest flow monitoring scheme. Because this is a three-dimensional histogram and the columns representing the two multiple well configurations are not located in the same row, care is required when interpreting this histogram. The values for the objective function obtained with both of the multiple well configurations are larger than the values obtained at any of the individual well sites. As with the installation of additional monitoring locations in the base case, the reduction in the expected cost of failure provided by a higher probability of detection is not sufficient to outweigh the costs of installing and sampling from  70  0  0.-  o  25&75m 25&50 m single well 75m  50m  25m  a)  0  o  25&75m 25&50m single well  b) Figure 6.18: Values of objective function for single well sites and two multiple well configurations for base geometry. a) $5 million cost of failure, b) $10 million cost of failure.  additional monitoring locations plus the increased expected cost of remediation. Even when the cost of failure is doubled to $10 million (Figure 6.18b), doubling the size of the reduction in the cost of failure, the single well configurations provide lower values for the objective function than do either of the multiple well configurations. In both histograms, the lowest values for the objective function for the multiple well configurations occur when one monitoring location is installed at each well site.  71  Although 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-threedimensional analysis sensitivity study, the results obtained in this sensitivity study are not necessarily those that would be obtained from a similar study conducted in a field situation or with a three-dimensional transport model. There are other functions that can be filled by a “backup” monitoring network that can not be evaluated in the decision analysis framework developed for this study. A “backup” monitoring network can assist in the detennination of the extent of a contaminant plume once it has been detected. If the rate at which contaminants are transported through a fracture network is higher than anticipated, and, consequently, the monitoring interval that is used is too long, the contaminant plume may have spread well beyond the monitoring network by the time it is detected. If the owner/operator of the landfill assumes that the plume does not extend much beyond the monitoring location at which it was detected, an inadequate remediation system 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 that the contaminant plume has spread farther than anticipated. A “backup” monitoring network 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 2  The second fracture geometry which I investigate is identical to the base geometry, except that the mean aperture of the horizontal fracture set is 40 microns, twice that of the base geometry. The larger apertures of the horizontal fractures increase the average effective porosity of the fracture networks as well as the average velocity with which contaminants are transported through the domain. The probability of failure during the compliance period is higher for this fracture geometry than for the base geometry and failure tends to occur earlier. Because of the earlier failure times and higher probability of failure, this geometry represents a less desirable site for a landfill facility.  72  The monitoring period calculated for geometry two is three days. The networks in this geometry have a higher average volumetric flow and a higher average groundwater velocity than do the networks in the base geometry, consequently less time is required for the volume of water contained in an average sampling volume to pass through an average monitoring location. The minimum number of particles required for detection in monitoring locations with low volumetric flow rates is three, which corresponds to one particle per day throughout the monitoring period. The total number of particles used to model transport in each realization is 32 256. By using this number of particles, I maintain the same average particle concentration per unit volume in the source area of the domain as in the base geometry.  6.4.1. MoNIToRING ScHEME C0MPAIUs0N  The total probabilities of detection over the compliance period vs. the distance from the source for geometry two, for all four monitoring schemes, are plotted in Figure 6.19. These probabilities are those obtained in a two-dimensional analysis with the monitoring parameters used for the base case and one monitoring location per well site. As with the base geometry, the highest flow monitoring scheme yeilds the highest probability 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 fracturing monitoring scheme provides a higher probability of detection than does the largest aperture monitoring scheme. The highest probability of detection for three of the four monitoring schemes occurs at the well site 75 m from the source. The exception is the density monitoring scheme, which has a slight dip in its curve at this well site. The general behaviour of the probability of detection in this geometry is similar to that in the base geometry. Close to the source, the probability of detection increases with distance as vertical dispersion spreads the plume over more of the domain. Farther from the source, the plume becomes more dilute and the probability of detection decreases with distance.  A comparison of the total probabilities of detection observed with one, two and three monitoring locations per well site for each monitoring scheme is provided by the plots in Figure 6.20. The behaviour of the probability of detection in geometry two with  73 100.00  • —°-----  80.00  £  0  —X—  60.00  high flow monitoring density monitoring aperture monitoring predetermined depth monitoring  40.00 .0 Cs .0  2  20.00  0  0.00  -,  0  25  50  75  100  125  150  175  200  Distance from Source (m)  Figure6.19: Total probabilities of detection over the compliance period vs. distance from the source for geometry two.  different numbers of monitoring locations installed at each well site is also similar to that in the base geometry. With all four monitoring schemes, the more monitoring locations at each well site, the higher the probability of detection. The difference between the probabilities of detection from one to two monitoring locations is greater than the difference from two to three monitoring locations at most well sites for each of the four monitoring schemes. The difference in the probabilities of detection between one monitoring location and three monitoring locations is greater close to the source than it is at the well site at 150 m. This reduction in the difference in the probabilities of detection between one monitoring location and three monitoring locations with distance illustrates that, as is the case in the base geometry, the addition of monitoring locations has more impact on the probability of detecting a contaminant plume while the plume is small, than once the plume has had more opportunity to disperse.  The major differences in the behaviour of the probability of detection with distance between this geometry and the base geometry are ones of degree. The effects of the spreading of the plume due to vertical dispersion are apparent over a greater distance in geometry two than in the base geometry. When the highest flow or largest aperture monitoring schemes are employed, the increase in the probability of detection between  a)  74  b) .  3 locatisns/weIl site  100  C .  3 locatioesiwell site  100  80  2 locatIons/well site  80  2 lOcatioss/well site  60  1 locatios/well site  60  flocatlonlwell  0  25  50  75  100  125  150  175  200  c)  0  25  50  75  100  125  150  175  200  d) ,  3 Iscalions/well sIte  100  3locations/weII site  100  2 locatIons/well site  60  -  1 location/well sIte  2 locatIons/well site  60  40  40  20  20  1 location/well site  0 0  25  50  75  100  125  150  Distance from Source (m)  175  200  0  25  50  75  100  125  150  175  200  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 monitoring scheme, b) densest fracturing monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme.  50 m and 75 m from the source is larger in geometry two than in the base geometry. The most likely explanation for this behaviour is that because the preferred flow paths in geometry two are more direct, and the flow more channellized, the rate of vertical dispersion with distance is lower. In the networks in the base geometry, the plume has already spread over a substantial portion of the height of the domain by the time it has travelled 50 m from the source; the additional spreading between 50 and 75 m from the source is smaller, as reflected by the relatively small increase in the probability of detection 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 by the time it has travelled 50 m from the source, and it continues to spread at close to the same rate when it travels between 50 and 75 m from the source as it was spreading when it travelled between the well sites at 25 m and 50 m from the source. Another indication that the plume continues to spread at greater distances in geometry two than in the base geometry is given in the plots in Figure 6.21. These plots are a comparison of the  75 geometry is given in the plots in Figure 6.21. These plots are a comparison of the probabilities of detection for the different detection thresholds in both geometry two and the base geometry. The probabilities of detection reported in this figure are those obtained in a two-dimensional analysis when one monitoring location is installed at each well site, and the fractures carrying the highest flows are monitored. Figure 6.21a shows the probabilities of detection for both one particle per monitoring period and the higher of the two threshold concentrations, 3.83E6 particles/rn , for both geometries, and Figure 3 6.2 lb shows the same comparison for one particle per monitoring period and the lower threshold concentration, 1 .92E6 particles/rn . In these plots, when one particle per 3 monitoring period is required for detection, the probability of detection for geometry two continues 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 still spreading vertically at distances greater than 100 m from the source in the networks in geometry two, but not in the networks in the base geometry.  The distance over which the highest probabilities of detection occur is shorter in geometry two than it is in the base geometry. This behaviour may also be a result of the lower rate of vertical dispersion with distance travelled of the contaminant plume in geometry two. The plots in Figure 6.21 show that in the base geometry, most of the increase in the probability of detection that results from vertical dispersion of the plume occurs 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 50 to 100 m from the contaminant source. When the lower of the threshold concentrations is used in the base geometry, the high probabilities of detection extend all the way to the well site 150 m from the source; there is only a small decrease in the probability of detection between 100 and 150 m. In geometry two, however, the rate of increase in the probability of detection remains almost constant from the well site at 25 m from the contaminant source through to the well site at 75 m for both concentration thresholds. A rapid decline in the probability of detection due to dilution of the contaminant begins after the plume has travelled 100 m from the source when the lower threshold concentration is used in geometry two, and after the plume has travelled only 75 m from the source in the case of the higher concentration threshold. In the base geometry, for both threshold concentrations, there is a distance of at least 50 m where the contaminant plume has spread over a large enough portion of the height of the domain that the probability of detection is relatively unaffected by further spreading of the plume and  a)  76 100  period geometry two  80 •  1 particle per monitoring period base geometry  0  3 60  per m3 base geometry 83EO6 particles per m3 geometry two  0 .1  40  .  0  20  0 0  25  50  75  100  125  150  175  200  b) 100  80 C 0 1 particle per monitoring period geometry two  60  1 particle per monitoring period base geometry  0  40  .0  2  1.Epacles perm base geometry —&--—  20  0.  1.92EpartrcIes perm3 geometry two  0 0  25  50  75  100  125  150  175  200  Distance from Source (m)  Figure 6.21: Probability of detection vs. distance for both base geometry and geometry two. a)one particle per monitoring period and 3.83E6 particles/rn , b) a)one 3 particle per monitoring period and 1.92E6 particles/rn . 3  before the dilution of the contaminant due to dispersion begins to affect the probability of detection to a large extent. In geometry two, the sharp inflection point in the curve representing the probabilities of detection for the larger of the two concentration thresholds indicates that the area of high probabilities of detection is most likely less than  77 25 m in extent in this case. When the lower threshold concentration is used, the area of high 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 with distance 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 more pronounced in geometry two than it is in the base geometry, and the probabilities of detection for geometry two are lower for all four of the monitoring schemes throughout the entire length of the domain. The more pronounced fall in the probability of detecting a contaminant plume at greater distances in this geometry is the opposite to what I would have expected. With the channellization of the flow and consequent lower rate of vertical dispersion that occurs in geometry two, I would have expected there to be less dilution in this geometry resulting in higher probabilities of detection at the larger distances than in the base geometry. Although the lower rate of dispersion with distance may be a contributing factor to the lower probabilities of detection observed in geometry two, I would have expected higher probabilities of detection in this geometry when the highest flow monitoring scheme is implemented, as a result of the contaminant becoming localized in the fractures carrying the largest flows. Preliminary investigations of the behaviour of this geometry that were carried out in an attempt to determine the cause of the lower probabilities of detection indicate that neither increasing the length of the monitoring period, varying the length of the monitoring interval, nor changing the number of particles introduced into the domain has a significant effect on the rate of decline 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 same length as for the base geometry, the probabilities of detection increased but were still lower than those observed in the base geometry.  The values obtained for the objective function with the base case analysis of geometry two are shown in the histogram in Figure 6.22. As in the base geometry, the highest flow monitoring scheme consistently provides the lowest values for the objective function, while the predetermined depths scheme provides the highest values. The value of the objective function that is obtained for this geometry when no monitoring is undertaken 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 higher in geometry two than in the base geometry for two reasons: geometry two has a higher  78  o  3.4C  cli-  2  depth aperture density flow  U,  cj  Figure 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 in geometry two than in the base geometry. The net present value of the expected cost of failure is higher the earlier that failure occurs because of the effects of discounting with time, The predetermined depth monitoring scheme is the only monitoring scheme that does not provide at least one option with an objective function value lower than the value obtained when no monitoring is undertaken in both the base geometry and in geometry two. 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 not offset 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 when there is one monitoring location installed at the well site. In the other five instances, there is a substantial increase in the probability of detection when a second monitoring well is installed. The highest probability of detection, 80.5 %, occurs at the 75 m well site when three monitoring locations are installed and the highest flow monitoring scheme implemented. As in the base geometry, the combination of options that provides the highest probability of detection does not provide the lowest value for the objective function. The lowest value obtained for the objective function with geometry two occurs at the well site 50 m from the source when the highest flow monitoring scheme is implemented and two monitoring locations are installed at this well site. The highest  79 probability of detection at this well site occurs when three monitoring locations are installed. The increase in the probability of detection afforded by the installation of a third monitoring location at this well site does not provide a sufficient reduction in the expected cost of failure to offset the additional monitoring costs as well as the increase in the expected cost of detection. The highest probability of detection when two monitoring locations are installed at each well site occurs at the monitoring well site located 75 m from the contaminant source. The lowest value for the objective function does not occur at this well site because the cost associated with detection at this well site is twice that at the 50 m well site. If the contaminant plume is detected 50 m from the contaminant source, one horizontal interceptor well is required for the remediation network, but if the plume 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 value obtained for the objective function in the base geometry. All of the values for the objective function that are obtained in geometry two are larger than those obtained in the base geometry. Four factors contribute to the higher objective function values in geometry two: 1) a higher probability of failure, 2) earlier times of failure, 3) lower probabilities of detection, and 4) earlier times of detection. The higher probability of failure provides a higher expected cost of failure, especially in conjunction with lower probabilities of detection. When failure occurs at an earlier time, the discounting with time 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, they also result in higher expected costs of failure. Earlier times of detection result in considerably higher expected costs of detection in two ways: the discounting with time is less, and the length of time over which the remediation network must be in operation is longer, resulting in a higher cost of operating the remediation network. The hydraulic behaviour of geometry two makes this geometry representative of a less desirable site on which to locate a landfill facility than sites represented by the base geometry. The lower suitability for landfill facilities of sites represented by geometry two is reflected by the higher values obtained for the objective function.  80  6.5. GEoMETRY 3  In the third geometry which I investigate, all three fracture sets have the same density, length, and aperture parameters. The networks for this geometry are more densely fractured than the previous two geometries and they do not contain the long horizontal fractures seen in the other two geometries. The average effective porosity of the geometry three networks is similar to that in the base geometry, but the average groundwater velocity and volumetric flow are both smaller than in the base geometry by a factor of approximately five. The probability of failure during the compliance period is much 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 of failure, this geometry represents a more desirable site for a landfill facility than do either of the other two geometries.  The monitoring period calculated for geometry three is 35 days.  The networks  for this geometry have a considerably lower average groundwater velocity and volumetric flow and a slightly higher average effective porosity than do the networks for the base geometry, therefore a greater period of time is required for the volume of water contained in a sampling volume to pass through a monitoring location. Because this monitoring period extends over more than half of the monitoring interval of 60 days used in the base case analysis of the other two geometries, the base case monitoring interval for geometry three is set at 180 days. One particle is the minimum number required for detection in monitoring locations with low volumetric flow rates. The total number of particles used to model transport in each realization is 21 510. This number of particles reflects an average effective porosity that is slightly higher than that of the base geometry, and lower than that of geometry two.  6.5.1.  MoNIToRING SCHEME COMPARISON  For 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 probabilities obtained in a two-dimensional analysis with the basic monitoring parameters and one monitoring location per well site for each of the four monitoring schemes. The highest  81 flow monitoring scheme consistently provides the highest probability of detection, followed by the densest fracturing monitoring scheme. The probabilities of detection provided by the largest aperture and the predetermined depth monitoring schemes are similar; these monitoring schemes provide the lowest probabilities of detection. The highest probability of detection for each of the monitoring schemes occurs at the well site 50 m from the contaminant source. The networks generated from this geometry are more densely fractured and the horizontal fractures are shorter than in either of the other two geometries, hence the contaminant plume experiences a higher rate of dispersion, both vertically and horizontally, with distance travelled. The representation of the flows through a sample network for geometry three (Figure 6.3c) shows that the preferred flow paths through the networks for geometry three are more tortuous than those in the other geometries. Contaminants being transported along tortuous paths encounter more fracture intersections than do those travelling along more direct paths and, consequently, experience more opportunity for dispersion. Because of this enhanced dispersion, the contaminant plumes in the networks for geometry three reach the point that further spreading has little impact on the probability detection while closer to the source than do contaminant plumes in the networks from the other two geometries. The effects of the  100.00  • —°——  80.00  high flow monitoring density monitoring aperture monitoring  .2  —X—  60.00  predetermined depth monitoring  40.00 .0 .0  2  20.00  0.00 0  25  50  75  100  125  150  175  200  Distance from Source (m)  Figure 6.23: Total probability of detection over the compliance period vs. distance for geometry three.  82 higher rate of dispersion with distance travelled are evident in the fact that the highest probabilities of detection for each monitoring scheme occur closer to the contaminant source in this geometry than in either the base geometry or geometry two. The increases in the probabilities of detection between the monitoring well site 25 m from the contaminant source and the well site at 50 m are smaller in geometry three than in either of the other two geometries, indicating that the contaminant plume in geometry three is likely to have undergone substantial vertical spreading by the time it has travelled only 25 m from the contaminant source.  The effects of the greater dispersion that occurs close to the source with this geometry can be seen in the plots in Figure 6.24. These plots represent a comparison of the total probabilities of detection achieved with one, two, and three monitoring locations installed at each well site. When the highest flow monitoring scheme is employed, and  a)  b) ,  .2  3 locations/well site  100  2 locationslwell  80  3 locations/well site  100  2tocations/well  80  I:: 20  20  -‘3  0 0  25  50  75  100  125  150  175  200  c)  0  25  50  75  100  125  175  150  200  d) .  3 loCations/welt site  100  3tncations/welt site  100  .2  80  2 locationS/well Site  80  2tocationS/welt site  ti)  60  1 location/welt Site  60  1 tocatioo/wetl Site  0  25  50  75  100  125  150  Distance from Source (m)  175  200  : 0  25  50  75  100  125  150  175  200  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 monitoring scheme, b) densest fracturing monitoring scheme, c) largest aperture monitoring scheme, d) predetermined depth monitoring scheme.  83 three monitoring locations are installed at each well site, the highest probability of detection 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 point where the plume is spread across enough of the vertical extent of the domain that there is no further increase in the probability of detection by further spreading that may occur as the plume travels onward to the monitoring well site at 50 m. As with the other two geometries, the more monitoring locations at each well site, the higher the probability of detection for all four monitoring schemes. The difference between the probabilities of detection from one to two monitoring locations is greater than the difference from two to three monitoring locations at all well sites for each of the four monitoring schemes. The difference in the probabilities of detection between one monitoring location and three monitoring locations is greater close to the source than it is at the well site located 150 m from the contaminant source.  As in both the base geometry and geometry two, the probability of detection falls off 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 most likely not the major contributing factor to this behaviour. The plots in Figure 6.25 are comparisons of the cumulative probabilities of detection vs. time between the base geometry and geometry three at the well sites located 25 m and 75 m from the contaminant source. These probabilities of detection are obtained when one monitoring location is installed at each well site and the highest flow monitoring scheme is implemented. These plots show that in both geometries, the contaminant plume has passed the monitoring well site located 25 m from the contaminant source by the end of the compliance period in most of the realizations. The slope of the curves for both geometries approach horizontal before the end of the compliance period. At the monitoring well at 75 m, however, the curve representing the probability of detection in geometry three is still rising at the end of the compliance period. The fact that this curve has not levelled off indicates that the contaminant plumes in a number of the realizations for this geometry have not yet arrived at this monitoring well site by the end of the compliance period. Therefore, the fall in the probability of detection that occurs at distances greater than 50 m from the contaminant source in this geometry may be more the result of the lower average groundwater velocity in geometry three than the result of dilution of the contaminant plume due to dispersion.  84 a)  base geometry 0  geometry three  0.8  0.6 Ø4  0.4 a)  E 0 0 0  5  10  15  20  25  30  Year  b) base geometry 0  geometry three 0.8  .0 0.6  a)—. 0.4  0.2  0 0  5  10  15  20  25  Year  Figure 6.25: Cumulative probability of detection vs. time for base geometry and geometry three. a) 25 m from source, b) 75 m from source.  30  85  The probabilities of detection in this geometry are higher near the source and lower at the well sites located at greater distances from the contaminant source than those in the base geometry, except when the arbitrary depth monitoring scheme is implemented. When the arbitrary depth monitoring scheme is implemented, the probabilities of detection are higher throughout the domain in geometry three. The higher probabilities of detection in geometry three are most likely due in a large part to the higher rate of dispersion with distance travelled that occurs in this geometry.  A histogram of the values obtained for the objective function with the base case analysis for this geometry is shown in Figure 6.26. The objective function values for each monitoring scheme are similar, but the predetermined depth monitoring scheme consistently provides the lowest values, followed by the largest aperture, densest fracturing and highest flow monitoring schemes respectively. This is the reverse order of monitoring schemes than in the other two geometries. The probability of failure is much lower for networks generated from this geometry than it is for those generated from either of the other geometries. The low probability of failure in this geometry is a result of the low groundwater velocities. Failure occurs later in geometry three than in either of the other two geometries investigated, and does not occur during the compliance period in many of the realizations for geometry three. If a longer compliance period were used for this analysis, the probability of failure in geometry three would be higher. As a consequence of the low probability of failure observed in this geometry, the objective function is dominated by the cost of monitoring and the expected cost of detection when the base case analysis is performed. Unlike the results obtained with either of the other two fracture geometries, the monitoring scheme that provides the highest probability of detection also provides the highest values for the objective function. Because the probability of failure is so low, the reduction in the expected cost of failure provided by an increase in the probability of detection is overshadowed by the increase in the expected cost of detection. Thus, monitoring schemes that provide a lower probability of detection lead to smaller values for the objective function. When no monitoring is undertaken in geometry three, the value obtained for the objective function is $0.42 million. This value is much lower than the $2.64 million that is obtained in the base geometry when there is no monitoring undertaken. The lower value for the objective function that is obtained in geometry three reflects both the lower probability of failure in this geometry and the later failure times. The lower probability of failure results in a  86  w .2 L1  m  25  m  Figure 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 present value 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 one monitoring location is installed. The variation in the values for the objective function at all well sites is greater between the different number of monitoring locations within any given monitoring scheme than the variation provided between any two monitoring schemes with the same number of monitoring locations installed. The expenses incurred by the addition of monitoring locations are larger than any change in the total expected costs that result from a change in the probability of detection between either different monitoring schemes or different numbers of monitoring locations at a given monitoring well site. The lowest value obtained for the objective function, $0.47 million, occurs at the well site located 50 meters from the source, when either the predetermined depth, largest aperture, or densest monitoring scheme is implemented and one monitoring location is installed. The probabilities of detection for the predetermined depth and the largest aperture monitoring schemes are similar in this case, but the difference in the probabilities of detection between the predetermined depth and the densest fracturing monitoring schemes is 19.5%. In this instance, the reduction in the expected cost of failure brought about by the higher probability of detection with the densest fracturing monitoring scheme is approximately equal to the resulting increase in the expected cost of detection. Consequently, the total value of the objective function is the same for all three  87 monitoring schemes. This value is higher than the $0.42 million obtained when no monitoring is undertaken.  6.5.2. INCREASED COST OF FAILuRE  In this fracture geometry, the probability of failure is so low that the objective function is dominated by the cost of monitoring and the expected cost of detection when the base case analysis is performed. Increasing the cost associated with failure will give more weight to the expected cost of failure term in the objective function by increasing the reduction in the expected cost of failure that results from an increase in the probability of 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 the objective 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 monitoring schemes are similar with no one monitoring scheme consistently providing the lowest or the highest values. The variation in the value of the objective function provided by different numbers of monitoring locations per well site is still greater than the variation between the different monitoring schemes. However, the overall lowest value provided for the objective function, $0.81 million, is now lower than the value that is obtained when no monitoring is undertaken, $0.84 million. All of the values obtained for the objective function, for each monitoring scheme as well as the no monitoring option, are lower than the values obtained for the base geometry in the base case analysis, despite the doubling of the costs associated with failure. The lowest value for the objective function in this case is obtained at the 50 m well site, when the highest flow monitoring scheme is implemented and one monitoring location installed. This is the option that provides the highest probability of detection when there is one monitoring location installed at each well site. Doubling the costs incurred as a consequence of failure results in an expected cost of failure that is large enough that, in some cases, the reduction to this expected cost brought about by an increase in the probability of detection is now larger than the resulting increase in the expected cost of detection. When more than one monitoring location per well site is installed, however, the reduction to the expected cost of failure that is brought about by the increased probability of detection is overshadowed by the increase in the cost of monitoring.  88  >  .2 Udepth ape ritre  density flow  m  Figure 6.27: Values of the objective function for geometry three with $10 million cost of failure.  The investigation of fracture geometry three indicates that for landfill facilities constructed on sites with low groundwater velocities and consequent low probabilities of failure, it may be to the advantage of the owner/operator not to install any monitoring network. It should be remembered, however, that the detection thresholds used in this study and the decisions concerning the pseudo-three-dimensional analysis are arbitrary. For this reason, care should be taken in making comparisons between values obtained for the objective function when monitoring is undertaken and those obtained when there is no monitoring 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 the objective function. Therefore, it is to the advantage of the owner/operator of a landfill facility 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 the facility within the expected lifetime of the facility.  89 7. CONCLUSIONS  In this dissertation, I develop a decision analysis framework to assist in the design of monitoring networks at hazardous waste sites located above a fractured geologic unit. The decision analysis framework is based upon risk-cost-benefit analysis, performed from the perspective of the owner/operator of the landfill facility. In this analysis I consider those costs that are directly associated with the construction and operation of the monitoring network (actual costs). The risks considered are those that are associated with the detection of migrating contaminants and consequent costs of remediation, and the failure of the facility and the costs resulting from failure (expected costs). The benefits are considered to be the same regardless of the monitoring strategy adopted, and are neglected. Therefore, the objective of the analysis is to fmd the monitoring strategy that provides the lowest value for the objective function, minimizing the sum of the actual and expected costs. This monitoring strategy is hereafter referred to as the “best” monitoring strategy.  The fractured rock formation underlying the hypothetical landfill site is modelled in vertical section using a two-dimensional discrete fracture model. This model uses a particle tracking method to simulate the transport of a non-reactive solute through the fractured rock unit. I investigate three fracture geometries, each with different hydrogeological behaviour. For each of these geometries, I investigate four monitoring schemes: 1) monitoring the fractures that carry the highest volumetric flows, 2) monitoring the fractures that have the largest apparent apertures, 3) monitoring the areas of highest fracture density, and 4) placing the monitoring locations at predetermined depths. I also investigate the effects of the distance of the monitoring network from the contaminant source, and the number of monitoring locations installed at each monitoring well site, for each of the four monitoring schemes in each of the three fracture geometries. This base case analysis is performed using a pseudo-three-dimensional approach that is adopted in an attempt to achieve consistency between the expected costs of remediation and failure, which assume a three-dimensional domain, and the costs of monitoring, which are calculated on the basis of each individual monitoring well site.  Monitoring networks in fractured media should focus on those fractures that carry the highest flows. The”best” monitoring strategy in two of the three geometries  90 investigated, and the highest probabilities of detection in all three fracture geometries occur when the fractures carrying the highest flows are monitored. However, the monitoring strategy that provides the highest probability of detection is not necessarily the “best” monitoring strategy. For options that provide a higher probability of detection than the “best” monitoring strategy, the expected cost of remediation is higher because the probability of detection is higher and, perhaps, because a larger remediation network may be required if the plume is detected farther from the contaminant source. In these cases, the higher expected cost of remediation, when combined with any increased cost of monitoring that may be required to provide the higher probability of detection, outweighs the reduction in the expected cost of failure brought about by a higher probability of detection.  Fracture geometries that have a low probability of failure provide low values for the objective function. Therefore, it is to the advantage of the owner/operator of a new landfill to locate the facility on a site where, should contaminants escape from a landfill cell, there is a small probability of these contaminants migrating any great distance within the expected lifetime of the facility.  In the three geometries investigated, the highest probability of detection occurs closer to the contaminant source when the preferred flow paths are more tortuous. The preferred flow paths through a fracture network become more tortuous the lower the connectivity of the fracture network. The more tortuous the preferred flow paths, the higher the rate of dispersion with distance travelled. A higher rate of dispersion means that the contaminant plume will spread over the height of the domain closer to the contaminant source, thus increasing the probability that there will be contaminant travelling in those fractures that are monitored. The rate of vertical spreading of the contaminant plume with distance travelled controls the probability of detection in the region near the contaminant source. Farther from the source, the probability of detection is controlled by dilution of the contaminant plume, in the geometries modelled in this study. In the base geometry and geometry two, the well site that provides the highest probabilities of detection is located 75 m from the contaminant source, and in geometry three the well site located 50 m from the source provide the highest probabilities of detection. In all three geometries, however, the”best” monitoring strategy occurs at the well site located 50 m from the contaminant source. More research must be done on a wider range of fracture geometries before any conclusions can be drawn concerning the  91 relationship between statistical descriptions of fracture geometries and the optimal distance at which to locate a monitoring network.  In addition to the base case analysis, I also investigate the viability of two different multiple well configurations, and conduct a number of sensitivity studies using the base geometry. Two of the monitoring parameters, the detection threshold and the length of the monitoring interval, are varied and the resulting effects on both the probability of detecting a contaminant plume and the value of the objective function are investigated. The sensitivity of the value of the objective function to the discount rate and the cost of failure are investigated and the characteristics of the pseudo-threedimensional 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, the decision analysis developed in this study does not evaluate other functions that are potentially filled by a “backup” monitoring system. These functions include assisting in the determination of the extent of a detected contaminant plume and functioning as an indicator of the effectiveness of a remediation system.  Of the parameters varied in the sensitivity studies, the cost of failure is the one to which the decision analysis is most sensitive. In the two geometries where the”best” monitoring strategy is provided when the fractures carrying the highest flows are monitored, the decision analysis is dominated by the expected cost of failure. In these two geometries, the expected cost of failure dominates the decision analysis because the probability of failure is high and the cost associated with failure is the largest of the costs involved. Only when the probability of failure is very small, as in the third fracture geometry, does the expected cost of failure not dominate the objective function. In this case, changes in the probability of detection do not affect the expected cost of failure very much; changes in the expected cost of detection brought about by variations in the probability 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 of failure by producing a larger change in the expected cost of failure than in the expected cost of detection for each percent of change in the probability of detection. This  92 increased influence of the expected cost of failure results in the lowest value for the objective function being provided by a monitoring strategy that provides a high probability of detection. Therefore the cost of failure is a potential administrative tool available to regulators who wish to promote the use of more conservative monitoring networks to ensure a high rate of compliance with environmental standards.  The results of the decision analysis are less sensitive to variations in the other parameters investigated in the sensitivity studies. The combination of monitoring options that provides the “best” monitoring strategy is insensitive to changes in the detection threshold and changes in the discount rate over the ranges investigated. However, the probability of detection is sensitive to the detection threshold in the region farther from the contaminant source, where dilution effects predominate. Of the combination of monitoring options that provide the “best” monitoring strategy, only the number of monitoring locations per well site is affected by changes in the length of time between samples, or variations in the characteristics of the pseudo-three-dimensional analysis. As the 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 reduces the impact on the value of the objective function of the additional cost of monitoring incurred when more monitoring locations are installed at each well site. The reduced impact of the additional cost of monitoring when a long monitoring interval is used allows the lowest value for the objective function to occur when a greater number of monitoring locations are installed. The number of monitoring locations per well site that provides the lowest value for the objective function varies with the characteristics of the pseudo-three-dimensional analysis, because the cost of additional monitoring relative to the probability of detection and the expected cost of failure vary with the characteristics of the pseudo-three-dimensional analysis.  This study represents a first attempt to develop a decision analysis framework for the design of contaminant monitoring networks that accounts for the unique properties of fractured media. Consequently, the emphasis of the investigation is on the relationships between the different components of the decision analysis objective function with respect to changes in the probability of detecting a contaminant plume. Many questions remain unanswered. Potential areas for future research include: 1) the relationship between fracture geometries and the optimal distance from a contaminant source at which to install a monitoring network, 2) investigations using three-dimensional fracture models, and  93 3) methods of evaluating the worth of a “backup” monitoring system. I hope that this dissertation provides a solid foundation for further investigations into the design of monitoring networks in fractured media.  94 BIBLIOGRAPHY Andersson, J., A. M. Shapiro and J. Bear. 1984. A stochastic model of a fractured rock conditioned by measured information. Water Resources Research, v. 20, no. 1, pp. 79-88. Andersson, J., R. Thunvik. 1986. Predicting mass transport in discrete fracture networks with the aid of geometrical field data. Water Resources Research, v. 22, no. 13, pp. 1941-1950 Baecher, G. B., N. A. Lanney, and H. H. Einstein. 1977. Statistical description of rock th U. S. 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