Probabilistic risk analysis and fault trees

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Probabilistic risk analysis and fault trees: Initial discussion of application to identiﬁcation of risk at a wellhead

C. Rodak

^⇑

, S. Silliman

Department of Civil Engineering and Geological Sciences, 156 Fitzpatrick Hall, University of Notre Dame, Notre Dame, IN 46556, USA

a r t i c l e i n f o

Article history:

Available online 3 March 2011 Keywords:

Probabilistic risk analysis Groundwater

Wellhead protection Health risk Fault tree Land use

a b s t r a c t

Wellhead protection is of critical importance for managing groundwater resources. While a number of previous authors have addressed questions related to uncertainties in advective capture zones, methods for addressing wellhead protection in the presence of uncertainty in the chemistry of groundwater contaminants, the relationship between land-use and contaminant sources, and the impact on health of the receiving population are limited. It is herein suggested that probabilistic risk analysis (PRA) combined with fault trees (FT) provides a structure whereby chemical transport can be combined with uncertainties in source, chemistry, and health impact to assess the probability of negative health outcomes in the population. As such, PRA-FT provides a new strategy for the identification of areas of probabilistically high human health risk. Application of this approach is demonstrated through a simplified case study involving flow to a well in an unconfined aquifer with heterogeneity in aquifer properties and contaminant sources.

1. Introduction

As the global population grows, our reliance on groundwater resources increases. The need to develop, manage, and protect these resources has become of increasing interest among both research- ers and government agencies. In the United States, for example, amendments to the 1986 Safe Drinking Water Act established the need to deﬁne wellhead protection programs (WHPPs) as a management strategy for protecting and managing the quality of groundwater supplies serving the public (e.g.,[39,37]).

Classically, the WHPP is based on a series of steps including both communication of risk to water managers and the public and technical delineation of a wellhead protection area (WHPA).

Over the past two decades, identification of the WHPA in the US has tended to focus on advective transport as assessed through one of several options, including: fixed radius, calculated fixed radius, analytical solutions to generalized hydrologic conditions, and numerical simulations including various levels of complexity in the groundwater system (e.g.,[28,13,2]). In the present manuscript, an alternative to advection-based identification of the WHPA is introduced in the form of probabilistic assessment of risk related to distribution of land use.

Four aspects of wellhead protection are of interest to the present discussion: (i) uncertainty in land-use practices (leading to uncertainty in chemical release), (ii) uncertainty in groundwater

ﬂow, (iii) variability in transport properties of contaminants, and (iv) estimated health risk from contaminant arrival at a well. As noted by other authors, uncertainties in land use and groundwater hydraulics can signiﬁcantly impact the estimation of the extent of capture zones to groundwater wells (e.g.,[8,40,15]). These uncertainties include, but may not be limited to, random variation in groundwater parameters (e.g., hydraulic conductivity, temporal/

spatial variation in recharge), uncertainties in the location and probability of occurrence of spills at potential contaminant sources (e.g., spatial variation in land-use, accidental releases), and varia- tions in stresses on the aquifer (e.g., rate of production at regional wells). Further, subsurface transport/chemical-reaction processes are of interest as they may inﬂuence the timing and concentration of different contaminants arriving at the production well: contaminants initially at higher concentration at a source may result in lower impact at the well due to combined retardation/decay/reaction processes. Finally, uncertainties exist relative to health risk associated with contaminants arriving at the production well and entering the water supply. Speciﬁcally, relatively high concentrations of low-risk contaminants may have substantially lower overall impact on health, and therefore on the delineation of necessary protection areas, than do high-risk contaminants present at low concentrations. Understanding the interplay of risk associated with the combination of different contaminants and their proximity to the well will provide the ability to identify land-use practices that reduce health risk at the well.

It is suggested herein that probabilistic risk analysis (PRA, e.g., [4,32]) combined with fault tree analysis (FT, e.g.,[4]) may provide 0309-1708/$ - see front matterÓ2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.advwatres.2011.02.005

⇑ Corresponding author. Tel.: +1 574 631 4306.

E-mail address:[email protected](C. Rodak).

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a viable option for incorporating these issues into wellhead protection and quantifying the probability of health risk associated with particular protection strategies. By using PRA-FT, the probability of sequences of events leading to a health risk at the groundwater well can be estimated and, as importantly, the contribution of individual events (location/mass/duration of contaminant events, flow field between contaminant source and well, etc.) to the overall probability of a health risk can be quantified. The objective of this application of PRA-FT is reduction in the overall probability of negative health impact based on land-use planning, as well as applying knowledge of the highest-probability threats to optimization of a WHPP. As such, it is suggested in this paper that application of PRA-FT to wellhead protection may provide a useful management tool for groundwater resources.

2. Motivation

While a number of tools have been introduced to the literature to delineate deterministic capture zones of groundwater wells for WHPA delineation, deterministic methods generally do not account for spatial variability in groundwater properties such as hydraulic conductivity, porosity and recharge. Quantiﬁcation of the uncertainty within groundwater systems using stochastic models for the purpose of wellhead protection has therefore been the focus of substantial research since the 1990s [8,40,3,35,15].

One common approach is through the use of Monte Carlo simulations to delineate wellhead capture zones under assumed statisti- cal structure of the underlying hydraulic conductivity field. While these stochastic, advection-based models provide the opportunity to account for spatial variability in the flow field, they do not account for differences in transport of different contaminants. Fur- ther, a common outcome of this approach, as shown in Cole and Silliman[8], is the estimation of relatively large WHPAs (as com- pared to deterministic methods) to account for uncertainty in the groundwater flow field: this approach may therefore lead to over- estimation of the size of the regions for which land-use practices must be restricted. It is suggested that Probabilistic Risk Analysis combined with fault trees (PRA-FT) may provide opportunity for more efficient regulation of land-use practices in the vicinity of a wellhead through estimation of the probability of health risk associated with specific land-use distributions and specific chemical contaminants.

PRA has developed into an important engineering tool used in the characterization of risk in natural and/or constructed systems (e.g.,[4,32]). PRA has been applied relatively extensively over the past two decades to questions of deﬁnition and management of health risks from contaminated water in surface and groundwater systems (e.g.,[31]): Lester et al.[16]provide a review of literature published since 2000 on the general question of PRA applied to health-risk related to water contamination. A common theme of the references cited by Lester et al. related to groundwater induced health risk, is the consideration of uncertainty in subsurface vari- ables and/or variability in individual physiological parameters to understand the impact of contamination on human health. These combined considerations have been addressed by a number of authors (e.g.,[5,17,19–21,27]) and provide promise that PRA will provide a valuable tool in deﬁning risk, and therefore the value of management strategies, associated with land-use planning in regions surrounding wellheads.

An important outcome of the work outlined above is that pre- dicting the impact of uncertainty in groundwater transport and the associated health risk involves a complex interplay among multiple processes. Commonly associated with PRA, fault tree (FT) analysis provides a visual tool for organizing the interplay among processes potentially leading to contamination and associ-

ated health risk of a water source (e.g.,[7]). Several authors have applied FT analysis to describe failure of complex water distribution systems. For example, Risebro et al.[30]applied FTs to known disease outbreaks across Europe focusing the work on causation of the outbreaks. Similarly, Lindhe et al.[18]applied FT analysis to investigation of quantity (the consumer did not receive water) or quality (the water delivered to the consumer was contaminated) failures of a public water system in Sweden. In interpreting the resulting fault tree in terms of probabilistic risk, these authors combined hard data and expert judgment to show that the major cause of failure was related to failure in the distribution system and not the raw water source.

With respect to groundwater systems, Tartakovsky [36] provided an early application of PRA-FT to the question of groundwater contamination: system failure was deﬁned as contamination of a speciﬁc region of an aquifer due to a surface source of contamination. Tartakovsky suggested that the probability of aquifer contamination can be determined by combining the probability of the occurrence of a spill and the probability of failure of natural attenuation or remediation efforts to reduce the contaminant concentration in the subsurface. Based on this logic, Tartakovsky constructed a basic fault tree in which the events in the fault tree were used to construct Boolean logic statements capable of quantifying the probability of failure.

Bolster et al.[6]expanded on the work of Tartakovsky by applying PRA-FT to the assessment of the probability of failure of a per- meable reactive barrier for protection of a groundwater resource.

The authors define 3 possible event paths representing interaction of the contaminant with the reactive barrier and the protection zone and calculated the probability of occurrence of each of these flow paths. As such, the authors use PRA-FT as a means of de-con- voluting a relatively complicated flow and transport scenario so as to provide an estimate of the probability of failure of a specific remediation technology.

3. Scope of present discussion

Based on this brief review of the literature, particularly the recent work of Tartakovsky[36]and Bolster et al.[6], it is suggested that the use of PRA-FT as a tool in deﬁning required wellhead protection for groundwater wells is both reasonable and appropriate.

Further, this approach, combined with Monte Carlo simulation, will allow incorporation of variable contaminant sources, uncertainty in transport properties, and consideration of health risk associated with alternative land-use strategies. That is, it is argued that by combining Monte Carlo analysis with PRA-FT, it will be possible to address the four key WHPP issues raised in the introduction: (i) uncertainty in land-use practices, (ii) uncertainty in groundwater ﬂow, (iii) variability in transport properties of contaminants, and (iv) uncertainty in health risk from contaminant arrival at a well.

In order to explore this application of PRA-FT, a general discussion is presented on PRA-FT applied to wellhead protection. Fol- lowing this discussion, a very simplified, hypothetical case study is presented to demonstrate how PRA-FT might be of value in an aquifer system. Specifically, the case study is based on management of a single production well in an unconfined aquifer under two scenarios: (i) regional planning of land-use constraints that might help to protect water quality at the well, and (ii) site specific planning in which a land owner is requesting a change of zoning to a land-use with potential to result in release of a new chemical contaminant to the groundwater. For both scenarios, the analysis is limited to the case in which detection and remediation are not feasible and in which spatial heterogeneity in the hydraulic conductivity follows a very simple model. Further, the case study is

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limited to consideration of two hypothetical chemicals with known transport and decay properties. Finally, consideration of health risk will be limited to consideration of oral consumption of water (versus inhalation or skin contact) for average body mass and an average volume of consumption (that is, uncertainties in health risk within the exposed population are not considered in the present example). While it is recognized that the case study is in itself not representative of any real ﬁeld site, nor is it complete in its consideration of all uncertainties that will impact real systems, it nevertheless presents an illustration of the concepts discussed below.

4. PRA-FT applied to wellhead protection

4.1. Overview of PRA-FT applied to wellhead protection

Underlying the identification of threats to a well is the concept of groundwater flow and chemical transport between a contaminant source and a groundwater well.Fig. 1is presented as a basic visualization of the problem involved. Here, three possible contamination sources (A, B, and C) are identified, each with a defined probability of causing groundwater contamination with a mixture of chemical constituents unique to each source. Questions of interest relative to protection of the wellhead include: (i) whether the mix of land-use in the region results in a probability of a contamination event occurring, (ii) whether flow pathways will connect any of the potential sources to the well (e.g., possible flow paths FLA1FLB1and FLC1versus alternative flow paths such as FLA2FLB2

and FL_C2which bypass the well) and (iii) whether natural attenuation processes may attenuate the impact of a contamination release with respect to water quality at the well. Questions of interest relative to protection of public health include those above plus questions of the potential health risk associated with chemical contaminants arriving at the well.

PRA-FT, as applied to this system, involves identification of processes (e.g., a chemical spill, groundwater flow due to regional recharge, etc.) that may lead to increased health risk at the well related to the arrival of one or more contaminants. While a general discussion of PRA-FT as applied to the most general field conditions (multiple contaminants, multiple land-use scenarios, monitoring/remediation scenarios, etc.) is possible, such a general discussion leads to unnecessarily complicated diagrams for an

initial discussion of the underlying concepts. As a result, the presentation below is based on a number of simpliﬁcations. The ﬁrst of these, as noted above, is that we ignore for the present discussion issues related to detection/treatment of the water at the well or remediation efforts in the aquifer; if the contaminant is released at the water table, it is assumed for the present discussion that only advection/dispersion, natural attenuation, and mixing at the well can impact the concentration in water withdrawn from the well.

A second simplification, we limit our consideration of land-use scenarios, for this introduction to PRA-FT, to two general management scenarios. The first scenario is herein termed the ‘‘planning scenario’’ in which a planning process is being developed whereby land-use constraints are being imposedin three pre-specified re- gionssurrounding anewgroundwater well. Failure in this scenario is defined as contamination at the well derived from one or more sources resulting in a health risk above a predefined limit for the average person consuming the water (average mass and average volume consumed). This scenario will lead to defining required land-use management based on health risk to the population including all potential sources contributing to the well. The second scenario is herein termed the ‘‘zoning scenario’’ in which a change in zoning is proposed for agiven parcelof land to allow a use that introduces a new possibility of groundwater contamination. Fail- ure in this scenario is defined as an incremental increase in health risk resulting from the proposed change in zoning for the average person consuming water from the well. This scenario will lead to individual zoning decisions based on incremental health risk to the population resulting from introduction of the individual source.

In each scenario, failure will result from a combination of events including the presence of a land-use activity with the potential for contaminant release, the actual release of a contaminant to the environment (from one or more release points), movement of groundwater from the contaminant source(s) to the vicinity of the well, transport of the contaminant to the well via the groundwater and during a time period of signiﬁcance, and a resulting increase in health risk from consumption of the water containing the contaminant. As shown below, PRA provides the probabilistic framework from which to analyze this problem, while FT aids in visualizing and organizing the possible processes that could lead to system failure.

Fig. 1.Conceptual image of an aquifer with 3 possible contaminant sources and two hypothetical ﬂow paths (one entering the well and one bypassing the well) for each contaminant source.

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Given these definitions of failure, it is possible to present events leading to failure in each of the scenarios. For theplanning scenario, events leading to failure are shown inFig. 2. In this situation, the distribution of land-use within the region surrounding the well must provide the opportunity for release of one or more contaminants with the potential to pose a health risk at the well. Following this distribution of land use, one or more contaminant releases to the groundwater must occur within this region. A release does not, however, guarantee arrival of contaminants at the well; rather there must be a transport connection (including advection and dispersion) between a source and the well for there to be potential for contamination of the well from that source. Due to the possibility of multiple releases in this scenario, the question of transport of contaminants to the well must involve consideration of potential flow pathways from each of the sources. In addition to transport, contaminants can be impacted by natural attenuation in the subsurface (including retardation, decay, biological degradation, and chemical reactions); failure requires that natural attenuation not reduce the mass of chemical per time arriving at the well to the degree that the potential health risk is reduced below the level of failure. Following transport, a mixture of contaminants must enter the well in sufficient mass per time to result in increased health risk to the consuming population. At this point, we assume (for both scenarios) that no detection of the contaminant occurs such that concentration at the well can be used directly in health risk calculations. The possibility of detection/treatment/remediation could be added to the analysis.

However, this would add little to the utility of the case study in demonstrating the use of PRA-FT, and this possibility is therefore omitted from the present discussion.

The sequence is somewhat simpler for thezoning scenario. In order for the new land-use to result in increased health risk, the first event that must occur is a contaminant release at the site of interest (i.e., there is no question regarding the distribution of multiple land uses as was the first stage in the planning scenario). Once in the groundwater, the contaminant must be transported with the groundwater. Finally, this transport (advection and dispersion) combined with any natural attenuation must result in sufficient mass per time arriving at the well to result in increased health risk to the population. A flow chart of these events is shown inFig. 3.

ExaminingFigs. 2 and 3, it is noticed that there is uncertainty associated with each of the events included in the case study: ﬁve uncertain events are identiﬁed (with only the latter 4 appropriate for thezoning scenario):

1. whether the land-use distribution in the region bounding the well results in a probability of release(s) of a particular contaminant of interest

2. whether a release (or multiple releases) occurs with sufﬁcient mass to potentially cause negative health impact

3. whether, through consideration of only advection and dispersion, transport path(s) will result in contaminants potentially arriving at the well (in the absence of natural attenuation) at sufﬁcient combined mass per time to cause negative health impact

4. whether natural attenuation reduces chemical mass sufﬁciently to avoid negative health impact at the well

5. whether the average individual in the population consumes the water as a primary source

Here we differentiate transport and attenuation as two distinct events based on the argument that two chemicals might be transported by advection and dispersion to a well, but that their impact on water quality at the well may be quite different due to differences in the adsorption, decay, biological degradation and reaction processes governing the two chemicals. Hence, in identifying the events that might contribute to increased health risk at the well, these two sets of processes are considered separate and having different potential impacts on system behavior.

These flow charts can be used as a conceptual foundation from which to construct FTs for both of the scenarios within a generic groundwater system. Recognizing that the fault tree for the zoning scenario will be a subset of the fault tree for the planning scenario (specifically, there is no uncertainty as to the proposed land-use), only the planning scenario fault tree is discussed and illustrated as provided inFig. 4. From this fault tree we can define base events, represented by the circles inFig. 4, which are necessary for failure of the system. Among the base events: (i) water is a primary source to the population (i.e., primary source implies that the individual regularly consumes water from this source such that there can be expectation of regular exposure to any contaminants), (ii) the distribution of land-use practices must introduce a non-zero probability of one or more contaminant releases leading to a total mass per time release sufficient to represent a potential health risk at the well, (iii) one or more contaminant releases must occur (at a minimum of one source location) with a total mass per time over all releases sufficient to cause a potential health risk, (iv) the contaminant(s) must be transported via groundwater from the source Fig. 2.Flow chart illustrating the required events to occur to result in an unacceptably high health risk from contaminants released in a region surrounding a well.

Fig. 3.Flow chart illustrating the required events for an increased health risk to occur based on contamination related to a new land-use practice on an isolated piece of land.

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location(s) to the well with sufficient mass per time to potentially lead to health risk, and (iv) natural attenuation (adsorption, decay, biological degradation, chemical reactions) does not prevent sufficient mass per time of the chemical from arriving at the well so as to eliminate the probability of health risk. These base events are similar, in several respects, to those defined by other authors (e.g.,[36,6]). A summary of the base events and their abbreviations can be found inTable 1.

It is customary at this point in the assessment of a fault tree to deﬁne ‘‘minimal cut sets’’, or sets of events that will lead to failure [4]. Similar to the work of Tartakovsky[36]where the fault tree remained relatively simple, in the absence of detection/remediation efforts, there is only one minimal cut set in our case studies. Spe- ciﬁcally, each of the base events discussed above must occur in order for negative health impacts to be realized at the well.

Next, the fault tree can be represented using Boolean logic.

Fig. 4is represented by:

Failure¼WCLUROTPAT ð1Þ

The convention is to treat the base events as independent. It is recognized that, at least with respect to transport by advection–disper-

sion versus natural attenuation, independence may not be appropriate in real systems. Sadiq et al.[33]have discussed this is- sue and suggested that interdependencies of events may be incorporated into the FT analysis. Hence, we will assume independence among these events for simplicity of this presentation, but recog- nize that interdependence of base events will likely need to be addressed in actual application. Under this assumption, the Boolean string can be translated into the probability statement

PðFailureÞ ¼PðWC\LU\RO\TP\ATÞ

¼PðWCÞ PðLUÞ PðROÞ PðTPÞ PðATÞ ð2Þ

Note that the deﬁnition of events leading to failure combined with visualization via the fault tree has here allowed clear delineation of the mechanisms leading to uncertainty in health-risk failure at the well. To the degree that the individual probabilities can be estimated (and any interdependencies can be identiﬁed and accounted for in the Boolean string), this expression also provides a strategy for estimating the total probability of failure.

4.2. Estimating the necessary probabilities and the cumulative health impact

The goal of the above analysis is the estimation of the impact on probability of health risk from a planned regional zoning distribution (planning scenario) or a proposed change in zoning at a specific location (zoning scenario). This goal is based, for this simplified scenario, on defining health risk from consumption of water containing contaminants. It is noted that such health risk could result from short-term, high concentration exposure to a contaminant or chronic exposure to lower concentration of one or more contaminants. For the present discussion, we focus on chronic exposure.

In order to once again ensure simplicity of the following discussion, an additional assumption is made for the following analysis:

it is assumed that the health risk of interest is the cancer risk as de- ﬁned by the USEPA [38]. As discussed above, other measures of health risk could be employed in place of, or in addition to, this particular measure without impact on the process as described.

The USEPA[38]recommends a total cancer risk (referred here by Fig. 4.Fault tree forPlanning Scenario. ForZoning Scenario, the basic event of land-use distribution is omitted as the land-use requested will be known with certainty.

Table 1

Summary of base events for the fault tree inFig. 4.

Base event Abbreviation

Water is a primary source to the population WC Land-use practices introduce potential for chemical release to

occur

LU Release(s) occurs (single release for zoning scenario/multiple

are possible for planning scenario) at mass per time above that required for potential negative health impact

RO

One or more transport pathway(s) will result in the contaminant(s) being produced at the well above the combined mass per time required for a negative health impact (one source for zoning, multiple possible sources for planning)

TP

Natural attenuation is insufﬁcient to reduce the contaminant mass per time below the combined level necessary for a possible negative health impact

AT

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the symbol, R) of 10⁶as the threshold measure and we therefore adopt this as the measure of failure for the planning scenario (where total contamination of the well is considered). For the zoning scenario, an incremental increase in the cancer risk is adopted as the measure of failure.

Using the EPA guidelines (1989), cancer risk is calculated as,

R¼1expðCDIxSFÞ ð3Þ

where CDI is the chronic daily intake averaged over 70 years (mg/

kg d) and SF is the slope factor expressed in (mg/kg d)¹. The SF is determined via laboratory tests and extrapolated to human applica- tions[1]. The estimates for contaminant speciﬁc slope factors are listed in the Integrated Risk Information System (IRIS) database accessible electronically though the EPA website[12]. It is argued that the total cancer risk from multiple chemicals, if each is at relatively low risk, is approximately additive such that the total risk can be estimated as the sum of the risk from each individual contaminant.

Recent literature [19–25]suggests the CDI can be estimated using the equation,

CDI¼CCREFED

BWTAV

c

_ki ð4Þ

where, C is the average concentration over exposure period (mass/

volume), CR represents the contact rate (volume/day), EF is the exposure frequency (days/year), ED is the exposure duration (years), BW is the average body mass during the exposure period (kg), TAV is the period over which exposure is averaged (days), and

c

kirepresents the ‘‘intermedia transfer function’’ and is dimen- sionless. The intermedia transfer function relates groundwater contaminant concentrations to tap water contaminant concentrations [19,23–25]. Of these, C is related to the contaminant release, transport path, and natural attenuation whereas the other parameters are related to relative health risk as characterized by the chemical properties, water distribution, water consumption, and physical characteristics of the population. It is noted, however, that concentration of chemical in the water consumed (C) and the rate of consumption of water (CR) are closely related in terms of impact on health risk as deﬁned in Eqs.(3) and (4). Speciﬁcally, the question arises (as implied inFig. 4) as to whether the population relies on the well water as their primary source of water for direct consumption: if this is the primary source, then relatively low concentrations will result in relative high exposure whereas if the well water represents only a small portion of consumed water (due, for example, to the use of bottle water in the household), then an equivalent exposure may only result if the concentration in the well becomes elevated. While this interplay represents an interesting opportunity for further study, we assume for the present discussion a simple 0/1 model in which the well water either is the primary source of water for consumption or the well water is only a negligible source (e.g., consumption can be assumed equal to 0). Hence, the probability associated with WC inFig. 4, for the present discussion, is a simple Bernoulli random variable (Yes/No) with an associated probability that the well is the primary water source. A number of authors have discussed other important aspects of the health risk equations (see, for example,[11,19,38]). This portion of the probability estimation process is therefore not discussed further in the present paper: focus is on the estimation of the probability distribution of C. This probability distribution will depend on the probabilities of appropriate land-use practices (providing opportunity for contaminant release), an environmental release within the given land-use distribution, an appropriate transport route, and the lack of attenuation processes eliminating contaminants prior to water arriving at the well.

In terms of probabilities of land-use practice at individual locations in the region bounding the well (as required for theplanning scenario), it is anticipated that three data/information sources may

be of interest. First, historical land-use records (e.g., land title history) as well as historical satellite imagery may provide evidence of the historical distribution of land-use practices (e.g.,[42]). If such records are complete for the entire region surrounding the well, there may be little uncertainty in terms of historical land use.

However, it is anticipated that, in many cases, such information will be incomplete or uncertain such that a combination of public records and expert information (e.g., the recollections of historians for a region) may be required in order to provide a probabilistic description of historical land use.

Second, current land-use practices are expected to be estimated from a combination of current and historical records of land use, regional planning documents, GIS data bases for the region of interest, satellite imagery, and expert opinion (e.g.,[42]). Depen- dent on the availability of this information, there may be a range of levels of uncertainty in the distribution of land-use under current conditions (with highest uncertainties commonly being either in rural regions with minimal land-use planning/control or in high-density urban regions containing a mix of land-use practices within minimal zoning control).

Third, a planning group may pursue prediction of future land- use practices. In this case, a number of modeling approaches have been suggested in the literature (e.g.,[41,26]). Such models will provide relatively objective approaches (when required parameters are available) to delineating probabilities of future land-use distributions under various zoning and socio-economic conditions.

Although beyond the scope of this manuscript, such models hold the promise to add a unique new planning attribute to wellhead protection efforts.

It is anticipated that, in practice, multiple zoning strategies will be simulated (thus impacting probabilities of different land-use distributions) to determine the impact of competing zoning practices on probable health risk associated with the well. The zoning strategies may be based on simple constructs (e.g., a series of con- centric radii centered on the well), historical land-use patterns, geopolitical boundaries (e.g., physical features such as lakes or riv- ers, or jurisdictional boundaries), or hydrogeologic analysis (e.g., simulation of capture zone). Given multiple management strategies, Monte Carlo simulation can be used to estimate the resulting distribution of actual land-use and, therefore, the relative probabilities for each management strategy to result in contaminant release(s) and rise in health risk from the well.

With respect to estimating the probability of contaminant release given a speciﬁc distribution of land use practices, these will likely be estimated based on historical records and expert testi- mony, but with different emphasis. In theplanning scenario, one can imagine a wide variety of land-use practices including both private (e.g., residential, light industrial, etc.) and public (e.g., parks, roads, etc.) uses that will lead to potential releases of contaminants to the environment. The resulting description of land- use will therefore lead to description of one or more contaminant releases to the environment that will include location, composi- tion, mass ﬂux, initial time of release, and duration of that release.

These probabilities associated with the type, timing and mass of the release will likely be derived from a combination of historical records of releases on similar land uses and expert opinion regarding likelihood of release characteristics.

In contrast to this relatively elaborate description required for the planning scenario, thezoning scenariois based on the concept that an individual land owner has requested a change in zoning.

Thus there is no uncertainty or spatial variation relative to land use. In this situation, it is likely that estimation of probability of a release will be based both on historical records for the specific zoning classification requested by the land owner as well as the environmental history (if it is public record) of that particular land owner. Where public records are insufficient to develop appropriate

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probability estimates, expert opinion will need to be used to supplement the public record in deﬁning probability of a release, mass ﬂux associated with a release, and timing/duration of a release.

With respect to the groundwater flow and chemical transport, it is anticipated that classic methods will be employed. This will commonly be through numerical modeling combined with a Monte Carlo approach, genetic algorithm, or similar method for incorporating uncertainty into the solution of the differential equations of groundwater flow and transport (e.g.,[19,34]). Such uncertainty will likely include consideration of spatial/temporal variation in such parameters as: hydraulic conductivity, porosity, local well production rates, and recharge. When detailed field data are available, probabilistic description of these parameters may be discernable from analysis of the field data. In many cases, however, the probability models will be derived from a combination of literature review for similar sites/sediments and expert opinion.

5. Illustrating the process: a simpliﬁed case study 5.1. The simulated groundwater system

It is recognized that the process described above involves a relatively complex integration of hydrologic, land-use, and health risk considerations. Therefore, a case study is used to illustrate this process, but for a significantly simplified scenario. For this case study, an unconfined aquifer of dimension 21.7 km10.3 km130 m (depth) is simulated (as shown inFig. 5). Inflow to this aquifer is derived from uniformly distributed recharge and outflow is to a river simulated at the left end of the figure. The mean values for the hydraulic conductivity, recharge and porosity for this aquifer are assumed equal to 90 m/d, 0.001 m/d, and 0.3, respectively. A single well withdraws water from the aquifer at a rate of

2500 m³/d at a location approximately 5.2 km upgradient from the river. Transport of chemicals was based on deterministic longi- tudinal and transverse dispersivities of 10 m and 1 m, respectively, zero sorption, and ﬁrst-order decay with a coefﬁcient equal to either 0/d (no decay) or 0.015/d depending on whether the simulation represented a conservative (0/d) or non-conservative (0.015/

d) chemical contaminant. It is assumed that the well represents the primary source of water for the population (P[WC] = 1.0 in Fig. 4) and is consumed at an average rate of 2 l/day for a body mass of 70 kg over a 70 year exposure period[38].

The case study is developed through Monte Carlo simulations using MODFLOW and MODPATH for ﬂow/advection and MT3D for dispersion/decay [14,29,43]. As shown inFig. 5, the porous medium is divided into a grid containing 80805 elements.

River cells were added to the third layer on the left end of the grid and recharge was uniformly applied to the uppermost non-dry layer of the grid. Thus there is a ﬂow ﬁeld established that, on average, moves from right to left relative to the orientation inFig. 5.

The well is located in layer 3, approximately 5200 m from each of the vertical grid boundaries closest to the well. Discretization is reduced in the cells bounding the well with minimum cell dimensions of 1 m1 m10 m (10 m in the vertical direction):

the maximum horizontal cell dimensions are 200 m400 m at the edges of the grid.

As discussed below, random variation is introduced to select simulations in two forms: (i) random variation in the hydraulic conductivity, and (ii) random location of land use/chemical release.

The hydraulic conductivity is simulated as a correlated random field using a standard LU decomposition method[10]based on a mean hydraulic conductivity of 90 m/d, a variance in the conductivity of 1.0, and a range of the variogram of 1000 m. The conduc- tivities were allowed to vary only in the horizontal direction (i.e., the correlated field was calculated once for each simulation and applied to each of the five grid layers), thus implying a correlation

Fig. 5.Unconfined aquifer used in the case study. All boundaries with the exception of the river cells (layer 3) are no-flow boundaries. Recharge is uniformly distributed in space. The well is located in layer 3 at the intersection of the refinement in the numerical grid (minimum horizontal cell dimension is 1 m1 m). The three regulation zones used in the planning scenario are shown (zone 3 is everything outside of zones 1 and 2). The parcel of land used in the zoning scenario is shown. This aquifer was simulated as a 5-layer MODFLOW finite-difference grid with a single well simulated in the third layer. The overall dimension of the simulated aquifer was approximately 21.7 km10.3 km130 m.

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length scale in the vertical direction substantially greater than the depth of the aquifer resulting in K being constant over the depth of the aquifer. We acknowledge that this is not realistic to many ﬁeld sites. However, given that inclusion of vertical variation in K would not involve a substantially greater amount of numerical effort, but would add an additional level of complexity in illustrating this simpliﬁed case study, we opted to make this assumption to main- tain clarity of explanation for the case study.

Simulating random land use and chemical release was performed as uncorrelated random ﬁelds. In this case, a simple uniform random number generator was used to generate a value in the range [0, 1] for each element in the uppermost saturated layer in the numerical grid to determine both land use and chemical release realizations: details are provided below.

For this case study, neither spatial variation in recharge nor porosity were included. Once again, inclusion of random variability in these two parameters (including cross-correlations with land- use and hydraulic conductivity) could easily be incorporated into the simulation process. Further, variation in these parameters, particularly recharge, could have substantial impact on the result probability estimates for a real field site and would therefore be important parameters to consider in any field application. How- ever, as argued for other simplifications outlined above, inclusion of randomness in these parameters would add substantial complexity to the results of this case study without adding new illus- trative value to the case study. Hence, the decision was made to limit variation within each realization to hydraulic conductivity and land-use for the present study.

For clarity of presentation (as this case study is used only for demonstration), only 10 realizations were performed per group of simulations involving randomness in the conductivity or land- use/chemical release: this limitation provides the opportunity to list all simulation outcomes and thus provides a more direct explanation of how random variation enters the health risk calculation.

It is recognized that substantially larger numbers of simulations would be required in actual practice. For each realization, MOD- PATH [29]was used to estimate the inﬁnite-time capture zone.

MT3D [43] was used to estimate the concentration versus time at the well for a period of 70 years (assuming either conservative transport or transport with decay).

5.2. Estimating health risk

In estimating health risk, a number of parameters will commonly add uncertainty to the estimation. For example, it is recognized that both the concentration of a contaminant arriving at a well and an individual’s rate of consumption will vary over time such that the product of CCR should be integrated as a product.

The health risk will also be a function of the slope factor of the contaminant(s) involved. Finally, the period of exposure will vary by individual (e.g., people may move in/move out of a neighborhood served by the well) with the EPA recommended 70-year period being a conservative extreme (conservative from the viewpoint of maximizing the estimated health risk). While each of these factors could be incorporated into Monte Carlo studies (e.g., integrat- ing the product of volume of consumption and concentration over an individual’s period of exposure), such calculations represent only modifications to the final estimated risk, not adjustment in the method being suggested to produce that estimate. In order to further reduce the complexity of results in the case study, health risk is based, for the case study, on an assumed average rate of consumption of 2 liters/day times the time average concentration of chemical from all active sources as simulated at the well over the 70 year simulation period (for each given realization of land- use and groundwater flow). This product is assumed constant over a period of 70 years for an individual with an assumed body mass

of 70 kg[38]. The slope factors utilized are 1.5 (considered a chemical with high risk of causing cancer) and 7.510³(considered a chemical with low risk of causing cancer), both in units of kg day/

mg. These slope factors were found in the IRIS database and were based on inorganic arsenic as a high risk compound at 1.5 (mg/

kg d)¹ and Dichloromethane as a low risk compound at 7.510³(mg/kg d)¹[12].

5.3. Deﬁning failure

For this case study, failure is defined in terms of cancer risk according to Eq.(3). For total health risk (the planning scenario), failure for this case study is arbitrarily defined as a probability greater than 10% that R (the measure of cancer risk) is 10⁶or higher. For incremental increase in health risk (the zoning scenario), failure is arbitrarily defined as a probability greater than 10% that R due to a chemical release at the specified parcel of land is 10⁷or higher (i.e., that the parcel of land contributes at least 10% to the limit on total health risk).

5.4. Description of problems addressed

In running this case study to completion, it is necessary to identify planning questions/planning scenarios in which PRA-FT might be applied. While the range of possible questions/scenarios is quite broad, it is necessary for this case study to limit consideration to a reduced number of specific scenarios. As with the other assump- tions underlying the case study presented here, alternative planning questions/scenarios could be identified and examined with the same method. Significantly, this type of broad comparison among multiple planning questions/scenarios is facilitated by PRA-FT, a significant strength of the PRA-FT approach. For example, suggesting an additional planning scenario and determining the probability of failure under that scenario would take little more effort under the PRA-FT framework.

The case study presented here is focused on the following limited set of scenarios and questions related to protection of the wellhead in the unconﬁned aquifer:

-1- Problem Statement #1: A planning scenario is defined whereby three zones are identified for possible regulation of land use above a heterogeneous aquifer (as shown in Fig. 5). Within this scenario, it is assumed that the probability of an environmental release of the chemical of interest is equal to 0 in regions that are regulated, but that it is ubiquitous (probability of 1.0) in unregulated regions. The problem is considered for two chemicals (one without decay and one with decay), each with an initial concentration, where present, of 10 ppb. Based on the definition of three zones provided above, three representative results will be obtained under both the no-decay and decay conditions: the chemical is present over the entire grid (all three zones remain unregulated), the chemical is present in zones 2 and 3 (zone 1 is regulated), and the chemical is present only in zone 3 (zones 1 and 2 are regulated).

This scenario focuses on the impact of these three regulation options on the probability of an unacceptable health risk from water derived from the well.

-2- Problem Statement #2: The second scenario presented is based on the assumption that the aquifer is hydraulically homogeneous, but that the distribution of land-use is random: this leads to random release(s) of the chemical of con- cern with the probability of release varying in space (with land-use). Two land-use types are deﬁned. The ﬁrst land use, which has a probability of 0.90 of occurring in the absence of regulation, results in a zero probability (at that

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location) of a release of the chemical of interest. The second land use, with a probability of occurrence of 0.10 in the absence of regulation, results in a probability equal to 0.20 of a release of the chemical of interest occurring at that site.

It is assumed that the distribution of land-use is uncorrelated in space. Thus, any element on the numerical grid that is not regulated has a 0.100.20 = 0.02 probability of being the site of a release of the chemical of interest (we make the additional simplification here that this probability is independent of cell dimension in the model). For those locations where the release has occurred, it is once again assumed that the release is present in the recharge water with a concentration of 10 ppb. Under regulation, only the first land-use would be permitted such that the probability of a chemical release reduces to 0. Using the same three zones as for the first problem statement, the same three possibilities for zoning regulation as used for the first problem are then com- pared. This scenario focuses on the impact of zoning on the probability that cancer risk will be greater than 10^6.

-3- Problem Statement #3: In this problem, a request for a change in zoning of a particular parcel of land is being considered. Here, it is assumed that the zoning change will lead, with probability 1.0, to the release on the parcel of land of the chemical of interest. This chemical is assumed present at the water table at a concentration of 1.0 ppm. As with the ﬁrst problem, the results are considered both with and without chemical decay and under conditions of hydraulic heterogeneity. This scenario shows the utility of PRA-FT for the problem of regulation of a proposed change in zoning with associated probability of an unacceptable increasein the health risk at the well.

These three problems each represent potential for application of a simplified form of the PRA-FT structure represented inFig. 4. For all three, the probability of water consumption has been set equal to 1.0 with, as noted above, a rate of consumption of 2 l/day; we do not deal here with the question of whether the population has other water sources available for consumption. Additionally for the first problem, there is no uncertainty included either in the land use or the chemical release: both P(LU) and P(RO) are set equal to 1.0. For the second question, P(LU) and P(RO) are included (0.10 and 0.20, respectively), but the uncertainty in the flow field and natural attenuation is eliminated (i.e., neither TP nor AT are uncertain as it is assumed both that the system is homogeneous and that the chemical is conservative). For the third question, the problem is simplified by assuming that P(LU) and P(RO) are both equal to 1.0. The three scenarios are summarized inTable 2.

5.5. Results of the case studies

Results for the three problem statements are provided in Table 3. In order to understand this table, it is necessary to deﬁne the use of four chemical species. As noted in the descriptions of problems 1 and 2 above, interest is expressed in whether a chemical distributed in zones 1, 2, and 3 will impact the cancer risk at the well. Here we take advantage of a feature in MT3D [43]:

MT3D allows definition of multiple chemical species and solves the transport equation for each of the species in the same simulation. In the current application, four species are defined. The first three represent the same chemical, but are associated with chemical releases in different zones. Specifically, ‘‘species 1’’ is present only within zone 1, ‘‘species 2’’ is present only within zone 2, and ‘‘species 3’’ similarly only in zone 3. In this manner, total concentration at the well was determined through addition of the concentrations of these three species. Further, the contribution of a particular zone to the total concentration at the well was determined by looking at the concentration of the species recharged in that zone. In this way, three planning strategies (no regulation, elimination of contamination in zone 1, and elimination of contamination in zones 1 and 2) were evaluated within a single simulation by: (i) adding contributions of all three species, (ii) adding the contributions solely of species 2 and 3, and (iii) the contribution solely of species 3. Species 4 is used to represent the chemical released at the particular parcel of land of interest in thezoning scenario.

As a significant simplification for this case study, it is assumed both that there is no contamination in the groundwater system prior to the start of the simulation period and that all sources of contamination become active at the start of the simulation. While these conditions will clearly not be present in real systems, they provide for easily interpretable concentration results for this case study. For example, Fig. 6shows typical breakthrough curves of the chemical at the well for each of the three problem scenarios defined above (normalized by the concentration at the end of the 70 year simulation period), illustrating both the increase in concentration over time as chemical is transported from source to the well and the fact that a chemical may not, even under these simple conditions, achieve steady-state concentration within the 70 years of simulation. The results, as provided inTable 3, include the average concentration at the well (estimated by integrated simulated concentration at the well, as derived from all contamination sources, over the 70 year simulation period), the calculated cancer risk associated with each realization, and the mean/variance of the health risk over the ten realizations associated with each problem. These results provide several insights into the potential benefits of using PRA-FT in connection with wellhead protection efforts.

-1- Problem Statement #1 – Ubiquitous contamination in a heterogeneous aquifer: the ﬁrst observation from these realizations is that the slope factor of the chemical (level of cancer risk associated with the chemical) will have a substantial impact on the effectiveness of regulating zones 1 and 2.

Viewing the results in Table 3a, it is observed that if the chemical has a high slope factor, the health risk at the well is unacceptably high even if both zones 1 and 2 are regulated (i.e., the probability that cancer risk is greater than 10⁶ approaches 100% for all three options for regulation). In contrast, regulating zone 1 for the chemical with the low slope factor appears to be adequate to drop the probability to 10%

(within the limits of the small number of realizations) of a cancer risk greater than 10⁶. Speciﬁcally, the probability of exceeding 10⁶ appears to be substantial if zone 1 is

Table 2

Summary of the three proposed scenarios and the corresponding parameters.

Problem Combined land-use/release probability Contaminant concentration Hydraulic conductivity Slope factor (kg d/mg) Decay rate (1/d) Low risk High risk

1 100% 10 ppb heterogeneous 7.510³ 1.5 0.015

2 2% 10 ppb homogeneous 7.510³ 1.5 –

3 Parcel 1 ppm heterogeneous 7.510³ 1.5 0.015

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unregulated (9 out of 10 realizations exceeded this threshold), however only one of the realizations exceeded this threshold when zone 1 was regulated and zones 2 and 3 remained unregulated.

A similar observation is made with respect to natural attenuation as observed through comparison ofTable 3a and b.

Whereas the probability of elevated cancer risk for the large slope factor was high in the absence of decay (even for regulation of both zones 1 and 2), this probability was dramat- ically reduced in the presence of chemical decay.

Speciﬁcally, it can be anticipated from these limited results that the probability of exceeding 10⁶for a chemical that Table 3

Simulation results for 3 scenarios. Species 1–3 are identical chemicals, but distributed in zones 1–3, respectively, inFig. 5. Species four is associated with individual land parcel involved in the zoning scenario. (a) Heterogeneous hydraulic conductivity with uniform concentration across the three regulation zones (Fig. 5), no decay, and initial concentration of 10 ppb, (b) Heterogeneous hydraulic conductivity with a uniform concentration across the three regulation zones with decay and initial concentration of 10 ppb, (c) Homogeneous hydraulic conductivity with random distribution of land-use/environmental release and initial concentration of 10 ppb, (d) Heterogeneous hydraulic conductivity with contamination at individual land parcel (Fig. 5), with/without decay and initial concentration of 1 ppm. ‘‘None’’ refers to no regulation such that concentrations from all three zones is included in risk calculation; ‘‘2 + 3’’ refers to regulating land use in zone 1 such that concentrations at the well are summed only over zones 2 and 3; ‘‘3’’

refers to regulating land use in zones 1 and 2 such that only concentrations from zone 3 are included in the risk calculations. Risk is reported to only one digit.

Realization Concentrations (ppb) High Risk (slope = 1.5 kg day/mg) Low Risk (slope = 7.510³kg-day/mg)

Cancer risk Cancer risk

Species 1 Species 2 Species 3 None 2 + 3 3 None 2 + 3 3

(a)

1 5.2 2.2 1.0 3.6e4 1.4e4 4.3e5 1.8e6 7.0e7 2.2e7

2 6.0 1.7 1.0 3.7e4 1.1e4 4.2e5 1.9e6 5.7e7 2.1e7

3 4.6 2.8 1.1 3.7e4 1.7e4 4.6e5 1.8e6 8.4e7 2.3e7

4 0.2 2.4 0.3 1.2e4 1.1e4 1.2e5 6.2e7 5.7e7 5.8e8

5 3.8 2.7 1.6 3.5e4 1.9e4 6.9e5 1.7e6 9.3e7 3.5e7

6 2.7 3.2 1.5 3.1e4 2.0e4 6.5e5 1.6e6 1.0e6 3.2e7

7 5.1 2.1 1.0 3.5e4 1.3e4 4.4e5 1.8e6 6.6e7 2.2e7

8 6.0 0.9 1.0 3.4e4 7.9e5 4.1e5 1.7e6 4.0e7 2.0e7

9 1.7 3.3 1.3 2.7e4 2.0e4 5.6e5 1.4e6 9.9e7 2.8e7

10 6.4 1.4 0.9 3.7e4 9.9e5 4.1e5 1.9e6 4.9e7 2.0e7

Mean 4.2 2.3 1.1 3.2e4 1.4e5 4.6e5 1.6e6 7.2e7 2.3e7

St. Dev. 2.0 0.8 0.4 7.6e5 4.3e5 1.6e5 3.8e7 2.2e7 8.0e8

(b)

1 3.5 2.4e1 7.1e6 1.6e04 1.0e05 3.1e10 8.0e07 5.1e08 1.5e12

2 3.8 1.5e2 2.4e6 1.6e04 6.6e07 1.0e10 8.1e07 3.3e09 5.2e13

3 3.2 6.4e2 2.6e7 1.4e04 2.7e06 1.1e11 6.9e07 1.4e08 5.7e14

4 0.0 5.0e6 1.1e7 3.5e10 2.2e10 4.6e12 1.8e12 1.1e12 2.3e14

5 2.3 1.1e2 4.7e6 9.9e05 4.6e07 2.0e10 4.9e07 2.3e09 1.0e12

6 0.2 7.9e4 1.8e7 7.3e06 3.4e08 7.8e12 3.7e08 1.7e10 3.9e14

7 3.1 8.0e3 1.2e6 1.3e04 3.4e07 5.3e11 6.7e07 1.7e09 2.7e13

8 3.2 4.4e5 1.4e9 1.4e04 1.9e09 6.1e14 6.8e07 9.5e12 0.0

9 0.0 2.8e4 2.7e10 4.7e07 1.2e08 1.2e14 2.3e09 5.9e11 0.0

10 3.4 5.4e3 2.1e7 1.4e04 2.3e07 8.8e12 7.2e07 1.1e09 4.4e14

Mean 2.3 3.4e2 1.6e6 9.8e05 1.5e06 7.0e11 4.9e07 7.3e09 3.5e13

St. Dev. 1.6 7.3e2 2.4e6 6.8e05 3.1e06 1.0e10 3.4e07 1.6e08 5.2e13

(c)

1 0.0 1.5e02 3.5e02 2.1e06 2.1e06 1.5e06 1.1e08 1.1e08 7.5e09

2 9.6e02 4.5e04 1.5e02 4.8e06 6.8e07 6.6e07 2.4e08 3.4e09 3.3e09

3 2.4e02 3.4e02 1.3e02 3.1e06 2.0e06 5.7e07 1.5e08 1.0e08 2.9e09

4 0.0 1.1e02 3.0e02 1.7e06 1.7e06 1.3e06 8.7e09 8.7e09 6.4e09

5 5.5e02 1.1e02 1.3e02 3.4e06 1.0e06 5.4e07 1.7e08 5.1e09 2.7e09

6 0.0 1.0e02 4.4e03 6.2e07 6.2e07 1.9e07 3.1e09 3.1e09 9.5e10

7 0.0 6.9e02 2.0e02 3.8e06 3.8e06 8.7e07 1.9e08 1.9e08 4.4e09

8 6.1e02 0.0 2.1e02 3.5e06 9.0e07 9.0e07 1.8e08 4.5e09 4.5e09

9 2.9e02 2.6e02 1.3e03 2.4e06 1.2e06 5.6e08 1.2e08 5.9e09 2.8e10

10 4.2e02 2.0e02 2.5e03 2.7e06 9.5e07 1.1e07 1.4e08 4.7e09 5.4e10

Mean 3.1e02 2.0e02 1.6e02 2.8e06 1.5e06 6.7e07 1.4e08 7.6e09 3.3e09

St. Dev. 3.3e02 2.0e02 1.1e02 1.2e06 9.8e07 4.8e07 5.9e09 4.9e09 2.4e09

Realization Concentrations (ppm) of species 4 High risk (slope = 1.5 kg-day/mg) Low risk (slope = 7.510³kg day/mg)

Cancer risk Cancer risk

(d)

1 2.0e04 8.5e08 4.3e10

2 7.9e02 3.4e05 1.7e07

3 1.6e02 7.0e06 3.5e08

4 9.5e04 4.1e07 2.0e09

5 2.2e03 9.6e07 4.8e09

6 3.0e03 1.3e06 6.4e09

7 9.3e03 4.0e06 2.0e08

8 1.8e03 7.8e07 3.9e09

9 4.1e04 1.7e07 8.7e10

10 7.7e02 3.3e05 1.6e07

Mean 1.9e02 8.2e06 4.1e08

St. Dev. 3.1e02 1.3e05 6.7e08

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undergoes the simulated level of decay is extremely small if zones 1 and 2 are regulated: this result is related to extended travel times, and therefore extended time for decay, for transport from any location within zone 3. For the low risk chemical in the presence of decay, the results suggest that no regulation is required.

As an indication of the impact of the hydraulic heterogeneity,Fig. 7shows an approximate outer boundary containing the range of capture zone boundaries observed over the ten realizations (as well as the capture zone for the homogeneous conductivity ﬁeld). It is apparent from this plot that there is considerable variability across the ten realizations in terms of the advective transport behavior within the simulated boundaries. This results in variability from one realization to the next in the mass of each ‘‘species’’ captured by the well.

-2- Problem Statement #2: Homogenous hydraulic conductivity and random contamination:Table 3c shows the results for this scenario and reinforces the observation from the heterogeneous simulations – the slope factor will have substantial impact on the probability of exceeding a cancer risk greater than 10⁶. These results also demonstrate that, due to the larger total surface area associated with zone 3, the probability of failure is dependent to a signiﬁcant degree on the distribution of contaminant release in this outer region and the proximity of a contaminant release to the homogenous capture zone fromFig. 7. Note, for example, that for a contaminant with high slope factor, the cancer risk is the same for no regulation and for regulation of zone 1 for four of the ten realizations, thus implying that all health risk is derived from contaminants originating outside of zone 1.

Also note that two realizations for the high risk slope factor Fig. 6.Typical breakthrough curves of the chemical at the well for each of the three problem scenarios normalized by the maximum (year 70) concentration of each scenario.

Scenario 1: ubiquitous contamination (solid line); scenario 2: random contamination (dashed line); scenario 3: parcel zoning change (alternating dashes).

Fig. 7.Capture zone to the well under homogeneous conditions (red zone) and approximate limit of capture zone variation over ten realizations of correlated, random hydraulic conductivity ﬁeld (black dashed line). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)