PROBABILITY OF FAILURE ANALYSIS AND CONDITION ASSESSMENT OF CAST IRON PIPES DUE TO INTERNAL AND EXTERNAL CORROSION IN WATER DISTRIBUTION SYSTEMS

Transcription

1 PROBABILITY OF FAILURE ANALYSIS AND CONDITION ASSESSMENT OF CAST IRON PIPES DUE TO INTERNAL AND EXTERNAL CORROSION IN WATER DISTRIBUTION SYSTEMS by Hamidreza Yaminighaeshi B.Sc, Iran University of Science and Technology, 1993 M.Eng, University of British Columbia, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2009 Hamidreza Yaminighaeshi, 2009

2 ABSTRACT Corrosion of cast iron ies in distribution systems can lead to the develoment of corrosion its that may reduce the resistance caacity of the ie segment, resulting in mechanical failure. These ies have a tendency to corrode externally and internally under aggressive environmental conditions. The mechanical failure of ies is mostly the result of this structural weakening couled with externally, environmental, and internally, oerational, imosed stresses. While external corrosion has been shown to significantly affect the likelihood of mechanical failure, the risk of failure may be further heightened if internal corrosion is occurring. This thesis develos a methodology for estimating the robability of mechanical failure of cast iron ies due to internal corrosion that incororates the relationshi between chlorine consumtion and the rate of internal corrosion in a cast iron ie. A robability analysis is develoed that incororates the internal corrosion model as well as the Two-hase nonlinear external corrosion model to calculate the overall robability of mechanical failure. Monte Carlo Simulation (MCS), First Order Reliability Method (FORM)-, and Second Order Reliability Method (SORM)-based aroaches are used to estimate the robability of mechanical failure. Next, a methodology is develoed for analyzing ie condition based on the data resulting from the robability of mechanical failure analysis incororating internal and external corrosion. A modeling strategy insired by survival analysis is used to obtain the redicted number of ie breaks for a given exosure time. The likelihood of failure at a given residual ie wall thickness is estimated and couled with the redicted number of ie breaks as surrogates for ie condition. These condition indices may suort decisions regarding relacement lanning and can be couled with economic assessment models in the develoment of future asset management strategies. ii

3 TABLE OF CONTENTS Abstract. ii Table of contents... iii List of tables....vii List of figures viii List of symbols.x Acknowledgements...xiv Co-authorshi statement...xv 1 Introduction Motivation, thesis questions, and objectives Background Electrochemical corrosion Corrosion roblems Corrosion rates Pie failure mechanisms Parameters affecting internal and external corrosion Modeling of corrosion External corrosion models Internal corrosion models Internal and external stresses Circumferential stresses Longitudinal stresses...11 iii

4 1.2.8 Pie strength caacity Probability of failure estimation Probability of failure definitions Monte Carlo Simulations (MCS) First and Second Order Reliability Methods (FORM and SORM) Probabilistic models of ie failure Asset management of ie systems Asset management definitions Long-term lanning of ie relacement and condition assessment Contributions of thesis Structure of thesis 22 References Probability of failure analysis due to internal corrosion in cast iron ies...28 Preface Introduction Corrosion decay- corrosion rate model Corrosion models Probabilistic analysis Results Sensitivity analysis Summary and conclusions References.. 45 iv

5 3 Probability of failure analysis due to internal and external corrosion in cast iron Pies.. 47 Preface Introduction External and internal stresses and corrosion models Pie strength caacity Probability of failure due to internal corrosion Probability of failure due to internal and external corrosion Results and discussion Sensitivity analysis Summary and conclusions References Condition assessment of cast iron ies in water distribution systems. 67 Preface Introduction Estimating the number of ie failures due to corrosion Estimates of the cumulative robability of failure due to corrosion Estimates of the number of ie failures Estimates the likelihood of failure for a given residual ie wall thickness Results and discussion Toward assessing ie condition and relacement strategies Model validation Conclusions. 81 v

6 References Conclusions Summary Limitations Alicability to ractice Time-variant variables with a stochastic rocess The effects of water quality arameters on the robability of failure Asset management in ractice Imrovement to the model System reliability Selection of arameters to reflect real world engineering Conclusions Future work.. 94 References.. 96 Aendix A: List of ublications. 97 vi

7 LIST OF TABLES Table 2.1. Characteristics of inut variables for the robabilistic model of ie failure.. 37 Table 3.1. Probability distributions of inut variables for calculating robability of failure. 58 vii

8 LIST OF FIGURES Figure 1.1. Schematic of a internal corrosion cell... 5 Figure 1.2. Comarison of redictive models for corrosion it deths... 9 Figure 1.3. Circumferential failure mode for cast iron ie Figure 1.4. Longitudinal failure mode for cast iron ie 12 Figure 1.5. Schematic descrition of FORM in a bivariate u-sace.. 15 Figure 2.1. Section of a ie segment with outer and inner diameters Figure 2.2. Probability of failure throughout the exosure time, for a normal distribution using MCS and FORM- and SORM-based aroaches (203 mm ie diameter and failure criteria * of d / 2 ). 38 t P Figure 2.3. Probability of failure throughout the exosure time, for all statistical distributions * using a SORM-based aroach (203 mm ie diameter and failure criteria of d / 2) 39 Figure 2.4. Probability of failure throughout the exosure time, for a normal distribution and * various ie diameters using a SORM-based aroach (failure criteria of d / 2 ) 40 Figure 2.5. Probability of failure throughout the exosure time, for a normal distribution and two failure criteria using a SORM-based aroach (203 mm ie diameter).. 40 Figure 2.6. Probability of failure throughout the exosure time, for a normal distribution with different correlation assumtions using a SORM-based aroach (203 mm ie diameter * and failure criteria of d / 2 ) t P Figure 2.7. Sensitivity factors for inut arameters throughout the exosure time, for a normal distribution using a SORM-based aroach (203 mm ie diameter and failure criteria of * d / 2).. 42 t P Figure 3.1. Predictive external and internal corrosion models for corrosion deths...53 t P t P viii

9 Figure 3.2. Probability of failure due to the thinness of the ie wall caused by internal corrosion throughout the exosure time using MCS, FORM- and SORM- based aroaches (failure criteria of F Y ) 59 Figure 3.3. Probability of failure due to the thinness of the ie wall caused by external and internal corrosion throughout the exosure time using MCS, and FORM- and SORM-based aroaches (failure criteria of Y ). 60 Figure 3.4. Sensitivity factors for inut variables throughout the exosure time using a SORMbased aroach (internal corrosion, failure criteria of F Y ) 61 Figure 3.5. Sensitivity factors for inut variables throughout the exosure time using a SORMbased aroach (external and internal corrosion, failure criteria of Y ) Figure 4.1. Probability of failure due to external and internal corrosion throughout the exosure time, based on statistical distributions using a SORM-based aroach for a 203 mm ie diameter. 71 Figure 4.2. Discrete robability density function fit to the Two-arameter Weibull distribution function (=1.552, k=0.017).. 73 Figure 4.3. Discrete hazard function fit to the Two-arameter Weibull distribution function (=1.552, k=0.017) 74 Figure 4.4. Survival function based on the hazard function for first order breaks. 75 Figure 4.5. Cumulative number of ie failures based on the results of SORM analysis fitted to the Weibull/Exonential distributions Figure 4.6. Cumulative robability of failure vs. residual ie wall thickness, for a SORM-based aroach, for different exosure times (ie ages) Figure 4.7. Comaring the survival functions for three municialities with survival function associated with the Weibull Distribution using the SORM-baroach. 80 ix

10 LIST OF SYMBOLS A = ie effective length on which load is estimated (mm); a = final itting rate constant (0.009 mm/yr); a 1 = constant used to determine ie strength caacity; a n = lateral dimension of it deth (mm); b = itting deth scaling constant (6.27 mm); b 1 = constant used to determine ie strength caacity; B d = width of the ditch (mm); C 0 = the initial chlorine concentration at time t = 0 (mg/l); C d = calculation coefficient for soil ressure which needs to be selected for a given situation on the basis of imerfect information (tyically between 1.12 and 1.52); Cl 2 = chlorine concentration (mg/l); C r = internal corrosion rate (μm/year); C r(int) = internal corrosion rate (μm/year); C r(ext) = external corrosion rate (μm/year); C t = chlorine concentration at any time t (mg/l); C s = surface load coefficient that describes the distribution of traffic loads and needs to be selected for a given situation (tyically between 0.96 and 0.144); C t = chlorine concentration ratio; C 0 c = corrosion rate inhibition factor (0.14/yr); D = ie diameter (m); d* = corrosion it deth (mm); d* ext = external corrosion it deth (mm); d* int = internal corrosion deth (mm); d = maximum accetable decrease in ie wall thickness; x

11 E = modulus elasticity of the ie (Ma); F = wheel load of traffic (N); F(T) = cumulative robability function of ie failure; F(T 1,2 ) = cumulative robability function of ie failure at times T 1 and T 2 ; f(t) = robability distribution function of ie failure; f 1 (T) = robability distribution function of ie failure associated with Weibull distribution; f frost = frost load multilier; I c = imact factor coefficient that determines the imact of surface loads on a ie segment for a given situation (tyically between and 1.875); K = chlorine decay constant (mg/l); K a = constant used in emirical ower model (tyically=2); K d = deflection coefficient that deends on the distribution of the vertical loads on the to of the ie and associated reaction on the bottom of the ie for a given situation (tyically between and ); K m = bending moment coefficient that deends on the distribution of vertical load on the to of the ie and associated reaction on the bottom of the ie for a given situation (tyically between and 0.285); K n = constant; K q = measurable fracture toughness (constant value for cast iron ie=10) (MPa m 1/2 ); k = arameter of Weibull distribution; k 2 = arameter of Exonential distribution; L = ie length (m); n a = constant used in emirical ower model (tyically=0.3); n = soil redox otential (related to soil oxidation); xi

12 = arameters of Weibull distribution; i = internal fluid ressure (MPa); H = soil acidic and alkaline nature; S(T) = survival function associated with the Weibull distribution at time T; S(T 1,2 ) = survival functions associated with the Weibull distribution at times T 1 and T 2 ; T = exosure time (year); t = time of travel within the ie over its entire length under a given design flow regime (s); t = original ie wall thickness (mm); t res = residual ie wall thickness (mm); t res = residual thickness (mm); V = water velocity (m/s); σ θ = total circumferential stress (MPa); σ F = stress due to internal fluid ressure (MPa); σ S = stress due to soil ressure (MPa); σ L = stress due to frost ressure (MPa); σ V = stress due to traffic loads (MPa); σ X = total longitudinal stress (MPa); σ Y = ie strength caacity (MPa); σ F = stress due to internal ressure in longitudinal direction (MPa); σ T = stress related to temerature differences along the ie (MPa); γ = unit weight of the soil (N/mm 3 ); λ(t) = hazard function (failure rate); ρ soil = soil resistivity; υ = ie material Poisson ratio; 6 α = constant thermal exansion coefficient of ie (tyically11 10 ); xii

13 β = geometric factor; ΔT = temerature differential ( 0 C); α, S = constants used in fracture toughness equations. xiii

14 ACKNOWLEDGMENTS Here I want to exress my areciation to all of the eole who have been suorting and heling me through my PhD study. I would like to thank my suervisor, Dr. Barbara Lence. Her constant suort, kind guidance, and informative instructions are a key basis for this thesis, always bringing me insiration. My writing skills have been substantially imroved under her training. Strong suort from my family, esecially my father and mother, are my siritual ro. I dedicate this thesis to my lovely wife, Saghar. I highly areciate her great understanding during my graduate studies. I could not have comleted this research without her suort. Thanks are also given to my advisor and final examination committees, including Dr. Sue Baldwin, Dr. Ricardo Foschi, Mr. James Atwater, Dr. Rehan Sadiq, and Dr. Walter Grayman. This thesis benefits from their constructive suggestions and recommendations. Also, all of my friends here rovide areciable hel to me. I cherish those unforgettable hours I sent in working with them and sharing joy with them. Particularly, I am grateful to my UBC research mates, Ali Naghibi, Yi Li, and Andrew Wood, for constructive discussions and Mohammad Seasi for his great hel. My ersonal effort is another key to the success of this research. The careful and creative work gives me confidence. I areciate all financial suort for my PhD study, including NSERC Discovery Research Grant ( ) awarded to my suervisor, UBC PhD Tuition Fee Award, and UBC Bursary. This suort not only financially heled me, but also, more imortantly, siritually encouraged me to comlete my study. xiv

15 CO-AUTHORSHIP STATEMENT Hamidreza Yaminighaeshi was the lead and rincial researcher of the work contained in the thesis titled Probability of failure analysis and condition assessment of cast iron ies due to internal and external corrosion in water distribution systems. Dr. Barbara Lence, the Research Suervisor of the thesis, rovided insiration and suorted the writing of the aers included in thesis. xv

16 CHAPTER 1 INTRODUCTION 1

17 1.1 MOTIVATION, THESIS QUESTIONS, AND OBJECTIVES Our understanding of the failure behavior of a ie segment can be imroved by redicting the likelihood of failure of a water main for a given exosure time. Analyzing the robability of failure in ies can serve the following imortant uroses. It can be used to: 1) determine the contribution of internal corrosion to the overall robability of mechanical failure, 2) determine the arameters that have the most significant effect on the overall robability of failure, and 3) identify ie condition indices to assist in the long-term lanning of the renewal of water distribution systems. This information may, in turn, be used to erform economic evaluation to determine an otimum relacement time for water mains that are failing due to corrosion and the budgetary needs for future reairs under various relacement and rehabilitation strategies as art of a comrehensive asset management analysis. The research questions that will be addressed in this thesis are: 1. How may the robability of mechanical failure due to internal corrosion during the exosure time be determined? 2. What corrosion arameters significantly affect the failure robability due to internal corrosion? 3. How may uncertainties in corrosion arameters be accounted for when analyzing mechanical ie failures due to internal corrosion? 4. How may the robability of mechanical failure due to internal and external corrosion during the exosure time be determined? 5. How may one estimate the redicted annual number of failures of cast iron ies due to internal and external corrosion? 6. How may one use the robabilistic analysis for assisting in condition assessment and longterm lanning for relacement of water mains in a distribution system? These questions are designed to increase our understanding of failure behavior of cast iron ies and the condition of assets, e.g., water mains, to assist in develoing future lanning strategies. The high degree of uncertainty associated with arameters that affect ie failures warrants a robabilistic-based analytical aroach. A detailed uncertainty analysis is required to quantify the robability of ie failures at a given time and to lan maintenance, relacement, and reair strategies. 2

18 This research develos a robability analysis of mechanical failure due to internal and external corrosion and identifies ie condition indices to assist in the long-term lanning of the renewal of water distribution systems. The objectives of the research are: To develo a methodology for estimating the robability of mechanical failure of cast iron ies due to internal corrosion. A robabilistic analysis is develoed that identifies and incororates the relationshis between water quality arameters, e.g., the chlorine concentration ratio, hydraulic erformance, e.g., the velocity, and the rate of internal corrosion. Monte Carlo Simulation (MCS), First Order Reliability Method (FORM)-, and Second Order Reliability Method (SORM)-based aroaches are used for estimating the robability of mechanical failure due to internal corrosion. The corrosion deth is estimated based on the assumtion that the rate of corrosion is constant throughout the exosure time, at least after the ie is 18 months old (Snoeyink and Wagner 1996). Mechanical failure is defined as the oint at which the corrosion deth is more than the maximum accetable decrease in ie wall thickness set by the utility manager. The robability of failure for a given exosure time is determined and may be thought of as a surrogate for the service life of a ie. A sensitivity analysis is conducted to determine the sensitivity of the estimates of the failure robability to each of the uncertain inuts of internal corrosion. To estimate the robability of mechanical failure of cast iron ies due to both internal and external corrosion. A robabilistic analysis is used that incororates the Two-hase external corrosion model as well as the internal corrosion model develoed in the first objective to calculate the robability of ie failure. The failure definitions based on the difference between the stresses acting on a ie segment and the ie strength caacity may be comared with the failure definition for the robability of failure due to the external corrosion estimated by Rajani et al. (2000). A sensitivity analysis is conducted to determine the sensitivity of the estimates of the failure robability to each of the uncertain inuts of both internal and external corrosion. 3

19 To develo an aroach for assisting in redicting condition of water mains over time using the robability of failure due to corrosion and a modeling strategy insired by survival analysis. This aroach uses the robability of failure due to internal and external corrosion and incororates a modeling strategy to obtain the condition of water mains. A modeling strategy, insired by survival analysis, is emloyed to redict the structural behavior of a ie over time. Survival analysis, which describes the likelihood of a ie surviving at a given exosure time, is a statistical technique that draws uon the oututs of robability of failure analyses to redict the annual number of ie breaks over time. Probability density, hazard, and survival functions are estimated based on the results of robability analyses due to internal and external corrosion. The estimated survival function develoed in this work comares well with other survival functions based on case studies develoed by Pelletier et al. (2003). The estimated robability density, hazard, and survival functions are then used to estimate the number of ie breaks that may be exected throughout the exosure time. The annual number of ie failures for a given ie segment may be considered as a first indicator of ie condition in water distribution systems. The likelihood of failure for a given residual ie wall thickness, e.g., as a second indicator of ie condition, and the number of redicted breaks may be used to suort decisions regarding ie relacement and can be couled with economic assessment models to develo future asset management strategies. This Chater reviews the corrosion henomena in cast iron ies, including roblems associated with corrosion, corrosion rates, ie failure mechanisms, and factors affecting internal and external corrosion. It describes the current models for redicting corrosion it deths and for estimating loads acting on cast iron ies, the existing robabilistic models for estimating the robability of failure at a given exosure time, and condition assessment and aroaches for long-term relacement lanning of cast iron ies. The structure of the thesis is also described herein. 4

20 1.2 BACKGROUND Electrochemical Corrosion Pie material deterioration is due to the internal and external chemical, biochemical, and electrochemical environment. The most effective deterioration mechanism of the exterior of cast iron ies is electrochemical corrosion (Rajani and Kleiner 2001). Also, Benjamin et al. (1996) reort that the deterioration of the interior surface of such ies due to corrosion is governed by electrochemical reactions. This tye of corrosion damages the ie surface and creates corrosion its. Electrochemical corrosion is the destruction of a metal by electron transfer reactions. For this tye of corrosion to occur, all comonents of an electrochemical cell must be resent. These comonents include an anode and a cathode, which are sites on the metal that have different electrical otential; a connection between the anode and cathode for transorting electrons, i.e., an internal circuit; and an electrolyte solution that will conduct ions between the anode and cathode, i.e., an external circuit. The oxidation of metal occurs at the anode. In the case of internal corrosion, the electrons generated by the anodic reaction transort through the internal circuit, i.e., the ie surface, to the cathode, where they are discharged to a suitable electron accetor such as oxygen or chlorine, by a chemical reduction reaction. The ositive ions generated at the anode will tend to be transorted through solution, i.e., the bulk water, to the cathode, and the negative ions generated at the cathode will tend to be transorted to the anode. The distribution of anodic and cathodic areas over the corroding material is defined as electrochemical corrosion. Figure 1.1 shows a schematic of an internal corrosion cell in a ie segment. 2e - 1 e O2 4 HOCl 2e 1 2 H 2 O Cl OH OH Anode Me 2+ OH - Cathode Figure 1.1. Schematic of a corrosion cell (Adated from Snoeyink and Wagner, 1996) 5

21 1.2.2 Corrosion roblems Two tyes of electrochemical corrosion considered in this research are external and internal corrosion. External corrosion of ies in distribution systems can lead to the develoment of corrosion its that may reduce the resistance caacity of the ie segment, resulting in mechanical failure. Internal corrosion may lead to two major roblems. The first roblem is water leakage and loss of hydraulic caacity caused by the loss of ie wall thickness and build u of corrosion roducts, i.e., scale layers. The second roblem is the change in water quality as water travels through corroding ies in the distribution system. This imact is caused by corrosion roducts entering the water, causing, for examle, red water, and reducing the chlorine residual. In this research, the mechanical failure of water mains due to external and internal corrosion is considered and the second tye of failure, i.e., the water quality roblem, due to internal corrosion will be reserved for future research Corrosion rates The loss of ie wall thickness due to the corrosion rocess can be either uniform around the ie or localized. The corrosion henomenon is comlex, and the rate of wall loss due to both internal and external corrosion, as well as whether the rate is constant or variable, has been the subject of debate (Melchers 1996 and Snoeyink and Wagner 1996). Corrosion may be considered as a self-inhibiting rocess whereby the rotective roerties of corrosion roducts reduce the corrosion rate over time and then stabilize the rate at a secific value thereafter. The most common external and internal corrosion models will be reviewed in Section of this chater. These models relate the corrosion deth with the ie age and may be used to redict the remaining service life of water mains Pie failure mechanisms The hysical mechanisms that lead to ie breakage involve three rincial asects: 1) internal loads due to oerational fluid ressure and the external loads due to soil ressure, frost ressure, traffic loads, stress due to temerature differences along the ie and third arties, 2) material deterioration due to internal and external corrosion, and 3) failure due to ie structural roerties, tye of material, ie-soil interaction, and quality of ie installation. 6

22 Pie failure is likely to occur when the total ressure, including internal water ressure, soil ressure, frost ressure, traffic loads, and ressure gradients due to temerature differences along the ie act on ies in which structural integrity has been reduced by corrosion or inadequate installation. These arameters contribute to reducing the life of the ie and result in mechanical failure. In this research, the effect of external and internal loads and external and internal corrosion will be modeled to determine the robability of mechanical ie failure. Bending stress due to soil movement and ie installation quality has not been modeled due to the varied, uncertain, and comlex arameters involved in installation and in modeling the installation rocess Parameters affecting internal and external corrosion Several arameters affect external corrosion including the soil acidity, alkalinity, and temerature. Similarly, a number of arameters affect ie internal corrosion including the water quality, i.e., H, alkalinity, concentration of disinfectants, and flow conditions, i.e., water velocity. There is a high degree of uncertainty associated with all of the arameters contributing to ie mechanical failure, and esecially with corrosion rates because of the large satial and temoral variability of these arameters. The uncertainty related to many of these arameters is addressed in this study through the use of robability and sensitivity analyses Modeling of corrosion This section reviews existing external and internal corrosion models which are used in this study to determine the robability of mechanical failure of cast iron ies in water distribution systems External corrosion models Four emirically-based external corrosion models are used in ractice. They are the Twohase, linear, and two tyes of ower models. Rajani et al. (2000) develo the Two-hase emirical corrosion model which is based on the assumtion that the corrosion rate is generally high during early ie ages and then stabilizes at a certain value thereafter. In the first hase of the Two-hase model, a raid exonential it 7

23 growth occurs and in the second hase a slow linear it growth occurs due to the corrosion selfinhibiting rocess. This model is defined as: ct d * at b(1 e ) (1.1) The emirical linear model is develoed by Sheikh et al. (1990) for the determination of corrosion it deth as a function of ie age. Here, the value of the corrosion rate is considered constant over time. The model is defined as: d* C T (1.2) r ( ext) The emirical ower model develoed by Kucera and Mattson (1987) uses a ower function to relate the deth of the corrosion it with ie age. A moderate nonlinear relationshi between it deth and ie age is exhibited for the first 50 years of ie life and this relationshi is stabilized as a linear function thereafter. This model can be defined as: n d* K a a T (1.3) Rossum (1968) rooses a general emirical ower function that relates corrosion it deth with ie age to redict the remaining wall thickness of cast iron ie. This equation is a function of corrosion it deth, ie age, soil H, soil resistivity, and ie surface area exosed to corrosion. There have not been many documented studies that validate Rossum s (1968) Power function. This model is defined as: Where n d K n C r ( ext) * (1.4) (10 H) T C [ ] (1.5) r( ext) The first three of these models are develoed and validated based on a data set collected from utilities across North America including the Cities of Edmonton, Winnieg, Quebec, Boston, and Philadelhia. Figure 1.2 shows a lot of the corrosion it deth versus ie age for these models based on the data collected from these utilities (adated from Sadiq et al. 2004). soil 8

24 Corrosion it deth (mm) 8.0 Two-hase model (Rajani et al. 2000) 6.0 Power model (Kucera and Mattson 1987) Linear model (Sheikh et al. 1990) Exosure time (year) Figure 1.2. Comarison of redictive models for corrosion it deths (Adated from Sadiq et al. 2004) The choice of a articular model largely deends on the available data for a secific case and whether these data can be accurately described with one of the external corrosion models Internal corrosion models Internal corrosion may be considered as a self-inhibiting rocess whereby the rotective roerties of corrosion roducts, i.e., generally iron-oxides, reduce the corrosion rate throughout the exosure time, e.g., within 18 months in cast iron (Snoeyink and Wagner 1996), and then stabilize the rate at a constant value thereafter. Chlorine decay modeling in distribution systems is based on the consumtion of chlorine due to the oxidation reactions within the bulk water, a first-order reaction, and to the oxidation reactions occurring at the ie wall, a first or zeroth-order reaction. A theoretical zeroth-order model which defines the relationshi between the chlorine decay rate and the corrosion rate is develoed by Kiene and Wable (1996). This model is defined as: d[ Cl dt 2 ] C K D r(int) (1.6) 9

25 A study regarding the relative imortance of arameters influencing chlorine consumtion in distribution systems by Kiene et al. (1998) shows that the corrosion of cast iron ies is one of the major arameters. They undertake a ilot-scale study to determine the relationshi between the free chlorine decay rate and corrosion rate under ractical exerimental conditions. They verify Equation 1.6 with values of K (chlorine decay constant) which are different from the theoretical value Internal and external stresses Internal loads and external loads may cause stress in both the circumferential and longitudinal directions. Ahammed and Melchers (1994) show that if a ie is uniformly loaded and suorted along its length, then circumferential stresses can dominate. A large number of arameters influence the loads acting on ie walls including the width of the trench, the unit weight of the soil, and the surface load. Rajani et al. (2000) develo a model of external and internal stresses including all circumferential and longitudinal stresses. They define the hoo or circumferential stress as a combination of stresses due to internal fluid ressure, soil ressure, frost ressure, and traffic loads. Similarly the axial or longitudinal stress is defined as a combination of stresses related to internal fluid ressure, soil ressure, frost ressure, traffic loads, and ressure gradients due to temerature differences along the ie. In this research, the overall likelihood of ie failure over time is determined under the assumtion that the ie is uniformly loaded in the circumferential direction and suorted along its length Circumferential stresses Rajani et al. (2000) develo the following formulation for total circumferential stress as: Where F S L V (1.7) F = i 2 D t (1.8) S = 3 K E m t 3 B 2 d 3 C K d d E i t D 3 D (1.9) 10

26 L = f frost S = 2 3 K m Bd Cd E t D f frost (1.10) 3 E t 3 K D 3 d i V = 3 A K m [ E I c 3 t C s 3 F K E d i t D D 3 ] (1.11) Figure 1.3 shows the circumferential failure mode due to environmental and oerational loads acting on a ie segment. The effect of internal fluid ressure is significant comared with environmental ressures for determining circumferential stress. Frost Pressure Traffic Loads Soil Pressure Internal Fluid Pressure Circumferential Break Poor bedding Figure1.3. Circumferential failure mode for cast iron ie Longitudinal stresses Rajani et al. (2000) develo a formulation for total longitudinal stresses as: ' X ( F S L V ) P T (1.12) Where ' F i ( 2 t D i ) 2 (1.13) T = EP P T (1.14) Figure 1.4 shows the axial failure mode resulting from longitudinal stresses. The effect of internal fluid ressure is minor comared with environmental ressures. The ressure gradient due to temerature differences along the ie is significant for determining longitudinal stress. 11

27 Frost Pressure Traffic Loads Soil Pressure Internal Fluid Pressure Axial Break `` Poor bedding Thermal Exansion Figure 1.4. Longitudinal failure mode for cast iron ie Pie strength caacity Rajani et al. (2000) establish that the structural strength caacity, Y, of cast iron is related to the corrosion it deth. They show that increasing corrosion it deth caused decreased ie residual strength over exosure time. This model is described by the following relationshi: Y ( d */ t K res q a n ) S (1.15) Residual ie thickness is defined as: t t d * res (1.16) 1.3 PROBABILITY OF FAILURE ESTIMATION Probability of failure definitions Probability of failure is defined based on a load-resistance interference reresented by a erformance function G, which incororates a set of random variables (x) and deterministic arameters. The erformance function of a ie segment is defined as the difference between the resistance, R, and load, L. The resistance is the strength or caacity of the system, e.g., a maximum accetable decrease in the thickness of the ie; whereas the load is the result of external conditions or influences uon a system, e.g., the corrosion deth. States of the erformance function may have more, but tyically consist of two categories, safety (G>0) or failure (G 0). The boundary between both states corresonds to the surface in 12

28 which random variables cause G=0 called limit-state surface. The robabilistic x-sace is then categorized into two domains, i.e., the failure domain F where G 0, and the safety domain S where G>0. The likelihood of being an event is in a safe domain S is referred to as reliability. The comlement of reliability is known as the robability of failure. The exact robability of failure may be obtained by an analytical method using integration of joint robability density function, f(x), over the failure domain or the event in which G<0 as follows: P f ( x) dx (1.17) f F The Equation 1.17 is the direct integration method. The exact robabilistic distribution of G may be derived based on f(x). Analytical methods to estimate the robability of failure are generally imractical because either f(x) or the derived distribution is not easy to estimate. Different methods for estimating the robability of failure include 1) samling methods such as those based on Monte Carlo Simulation (MCS); 2) aroximate methods based on calculation of a oint estimate of a erformance function or a reliability index, β, such as First Order Reliability Method (FORM) and the Second Order Reliability Method (SORM); 3) combinations of FORM and SORM; and 4) FORM or SORM with Imortance Samling (Schueller et al in which only on the region of most imortance (e.g., near the surface of G=0) are samled, or investigated, in order to estimate the robability of failure. The reliability index, β, is the ratio of the mean of G and the standard deviation of G. (Ang and Tang 1984; Madsen et al. 1986; Tung 1996; Melchers 1999; Lian and Yen 2003). In this research, MCS, and FORM- and SORM-based aroaches are used to estimate the robability of mechanical failure throughout the exosure time (T) due to the ie corrosion. T is defined as the time eriod in which the ie has been installed u to the time of analysis which corresonds to the ie age Monte Carlo Simulation (MCS) MCS is a samling method in which a large number of simulations are erformed using random variables to calculate the robability of events in the failure domain G 0 and in the safety domain G>0. In MCS, multile samled values of x are randomly generated according to their resective robabilistic distributions. Then G is calculated for each set of generated samled values. The statistics and a robabilistic distribution of G may be obtained based on the 13

29 calculated values of G. If G 0, then the combination of samled random variables leads to the failure event. Similarly, the combination of samled random variables leads to the robability of non-failure (reliability) where G>0. The robability of failure, P f, may then be aroximated by the ratio of the number of failure events, N f, where G<0, and the total number of failure lus non-failure events, N, where G 0 and G>0. It is defined as: N f Pf (1.18) N The larger the value of N, the more accurate the estimation of the robability of failure is. Desite their accuracy, samling methods require high comutational loads. Another drawback of this method is that it cannot be used to estimate the sensitivity of the reliability estimates to the various random variables First and Second Order Reliability Methods (FORM and SORM) Aroximations are efficient methods of estimating the robability of failure. The most ractical aroximation aroaches are the reliability index-based methods (e.g., FORM and SORM) that aroximate G using normally distributed and uncorrelated random variables. Under a FORM analysis, the original x-sace is transformed into a standardized uncorrelated normal sace, u. Thus u-sace has a zero mean and unit standard deviation. To simlify the solution, FORM transforms the original G in the u-sace using the first-order Taylor Series exansion of G at the design oint, u*, as indicated in Figure 1.5. As a local otimum on the limit-state surface of G, u* is the closest oint to the origin of the u-sace. If G>0, β is equal to the distance from the u origin to the limit-state surface at the design oint u*. For G<0, β is the negative of that distance between the u origin and the limit-state surface. Because u* is a local otimum, the following relationshi holds: β = u*. α (1.19) Where α = the negative of the normalized gradient vector of G at u*. FORM aroximates P f as follows: Pf Φ(-β) (1.20) Where Φ( ) = the cumulative distribution function of the standard normal distribution. Equation 1.20 imlies that the reliability index is required to estimate the robability of failure. FORM has been shown to be adatable for comlex roblems in comarison with the analytical 14

30 method, and it is comutationally efficient in comarison with MCS. In some comlex roblems where G is not linear, the first-order Taylor Series exansion of G may be inaccurate and may make estimating u* difficult. Therefore, the robustness of FORM is decreased for these roblems. FORM may be modified by assuming that the limit-state surface where G=0 is a quadratic function with curvatures equal to those of the real limit-state surface at the design oint u*. The resulting method is called SORM. G G G Figure 1.5 Schematic descrition of FORM in a bivariate u-sace (Adated from Yi and Lence 2007) In this research, an internal corrosion rate model is develoed and combined with a Twohase external corrosion model to estimate the effects of internal corrosion and robability of failure due to this throughout T. MCS, FORM-, and SORM-based aroaches are used for this robability analysis. The efficiency of FORM-, and SORM- based aroaches are evaluated herein. One of the main advantages of FORM-, and SORM-based aroaches is that they may be used to investigate the sensitivity of the overall robability of failure to each random variable. Sensitivity analysis is conducted to identify the variables that have the most significant effect on the reliability index, β, and subsequently on the robability of failure. The sensitivity of the 15

31 16 reliability index to each of the random variables is evaluated by the rate of change of β with resect to each of the variables, and this is referred to as sensitivity factor, n(x). These rates are defined as the artial derivatives of β with resect to each of the variables as follows: x x n ) ( (1.21) Probabilistic models of ie failure Sadiq et al. (2004) estimate the reliability of a ie in the face of external corrosion- associated failure in cast iron water mains. They define the failure time as the time at which the external stresses exceed the ie strength caacity. They also define the erformance function so as to relate all the arameters of the stresses acting on ie, the ie residual strength, and the external corrosion. The erformance functions associated with circumferential stresses, G (X), and longitudinal stresses, G (X), are exressed as Equations 1.17 and 1.18, resectively: Y X G ) ( (1.22) X Y X G ) ( ' (1.23) Substituting Equations in Equation 1.22 for considering circumferential stresses, Equations , 1.13, 1.14 in Equation 1.12 for considering longitudinal stresses, and then Equations 1.7, 1.12, 1.15 in Equations 1.22 and 1.23 for describing the erformance function due to total circumferential, and to longitudinal stresses resectively, yields Equations 1.24 and 1.25, resectively: ] [1 ] 3 [ 3 2 ) * / ( ) ( D K t E D t E C B K f D K t E A D t FE C I K t D t d K X G i d d d m frost i d s c m i S n q (1.24) } ] 3 3 ] [1 ] 3.[ 3 1) ( 2 {[ ) */ ( ) ( ' T E D K t E D t E C B K f D K t E A D t FE C I K t D t d K X G P P i d d d m frost i d s c m i S n q (1.25)

32 For a given age of ie, or time of exosure, if G (X) or G (X) is less than zero, then the ie is considered to have failed. The robability of failure, as in all reliability analyses, is defined as: ' P (G) 0 Or P ( G ) 0 (1.26) The effect of external corrosion on ie mechanical failure is discussed extensively by Sadiq et al. (2004) but the effect of internal corrosion due to the variability in water quality arameters on the ie mechanical failure is not addressed. 1.4 ASSET MANAGEMENT OF PIPE SYSTEMS Asset management definitions Vanier (2001) identifies six comonents of an asset management rogram. These are: asset inventory, asset value, deferred maintenance, asset condition, rediction of service life, and rioritization of reair and relacement of assets. Inventory, as the first comonent of an asset management rogram, is the act of determining the tye, quantity, and extent of the assets (e.g., water mains). In water resources management, the rimary caital assets that are owned by utilities include water suly reservoirs, all hydraulic structures related to dams, water treatment lants, and water distribution systems with their comonents such as ies, valves (e.g., ressure reducing valves), fire hydrants, tanks, um stations, and meters. Comuter tools such as sreadsheets, Geograhic Information Systems (GISs), Comuter Aided Drafting Systems (CAD), and Comuterized Maintenance Management Systems (CMMSs) are used to track and maintain inventories of utilities. Wood and Lence (2006) reort that archival records are still the redominant sources of data for utilities that serve the oulation of less than 50,000. Use of CMMs is growing in recent years but these systems are still exensive to imlement and require significant resources to maintain (Wood 2007). The total value of utility assets must be known to lan for relacement. Utility managers must have an overall understanding of the system worth and the budgetary needs for ie relacement. In many utilities, in the case of estimating value for water mains, the value is simly estimated by multilying an average cost of construction er meter of ie by the total length of the ies in the network (Wood 2007). 17

33 Asset management requires knowledge related to the condition of assets. Asset managers are facing roblems associated with determining how and what to evaluate, defining what constitutes condition and indices, and determining how to use data related to condition of assets. There are no standard condition indices for assets (Grigg, 2004). A general diagnosis of the structural behavior of ies is required, as are condition indices, to redict the ie failure behavior over time and to imrove the level of accuracy of ie condition assessment that does not require a long history of data (De Silva et al. 2002). Also, field-based condition assessment is costly and only rovides information for a ie segment at a articular oint in time. Therefore, the redicted number of ie failures may be considered herein as one of the indicators for assessing the ie condition and for undertaking the economic analysis to develo an asset management rogram. The maintenance in water utilities that has not been erformed or has been deferred is known as deferred maintenance. For deferred utility maintenance, the comounding effect of maintenance that has not been erformed (e.g., not cleaning or reairing ies) must be identified and managed. It is very difficult to quantify the deferred maintenance and there is very little information available for municialities regarding deferred maintenance of water distribution systems (Wood 2007). Estimating the service life of the ies and undertaking long term lanning for their relacement deends on rediction of the future conditions of the assets (e.g., water mains) in the system. To manage water utility assets and lan for relacement, the estimates of technical and economical service life of the assets is required. Rajani and Makar (2000) define the time of death of a ie segment as the time in which its mechanical factor of safety is less than an accetable value set by the utility manager. Kleiner and Rajani (1999) roose that the service life of a ie segment is a function of the costs of deterioration rate and relacement and suggest that ie failure coincides with the otimal time of relacement. Prioritization of asset relacement is the last comonent of an asset management rogram. In this stage, managers should be able to determine what to relace and when to relace it. These are related to the financial and technical issues that lead to the decisions of whether or not to reair or relace an asset. Utilities have rioritized ie relacements based on a combination of current management ractices and historical breakage data. 18

34 In this research, the likelihood of failure for a given residual ie wall thickness at a given ie age is estimated using robabilistic analysis and the number of redicted breaks are estimated using survival analysis. Theses may serve as surrogates for condition in develoing a comrehensive relacement strategy and may be couled with economic assessment models to analyze future asset management decisions Long-term lanning for ie relacement and condition assessment The most exensive comonent of a water suly system is the water mains in the distribution network (Shamir and Howard 1979). Aging water mains, couled with the environmental and oerational loads due to continuous stresses acting on ies, leads to the system deterioration. Resulting ie breaks may, in turn, lead to otential contamination of the water suly. Deterioration of a ie network also increases oeration and maintenance costs and water losses, and reduces the quality of service over time. Water utilities are facing the roblem of maintaining deteriorating networks in the most efficient way ossible to meet existing and future levels of service. Give the system deterioration and otential shortages of future caital resources, it is essential to develo a comrehensive methodology to assist lanners and decision- makers in selecting the most cost-effective rehabilitation olicy that addresses the issues of reliability, safety, quality, and efficiency. That is, to assist them in undertaking long-term lanning to maintain or imrove water distribution system reliability and oeration regularity within a secified budget. Water distribution system relacement and rehabilitation has been the subject of research over the ast two decades. Shamir and Howard (1979) aly regression analysis to obtain an exonential relationshi for ie breakage as a function of year of installation. This relationshi is used to find the otimal time of ie relacement at which the total cost of reair and relacement is minimized. Walski and Pelliccia (1982) roose a similar exonential relationshi but add two other factors related to the historical ie breaks and observed differences in breakage rates in larger diameter ies. Walski (1987) adds the cost of water losses through ie leakage and the cost of broken valve relacement to this decision model. In subsequent studies, the failure rate and the redicted number of ie failures are used as essential arameters for undertaking an economic analysis as art of a long-term lanning strategy 19

35 (Andreou et al., 1987a, b; Su and Mays, 1988; Kim and Mays, 1994; Schneiter et al., 1996). Failure rate corresonds to the instantaneous robability of having a break between time t and (t+ t) conditional to survival u to time t. Condition assessment and the rediction of system reliability are required to determine the long-term lanning of the renewal of water distribution systems. It is defined as the readiness of the asset, e.g., the water main, to erform its function (Grigg, 2004). Predictive modeling based on the statistical analysis of the recorded ie breaks has been used in order to roject future breaks (Andreou et al., 1987a, b). Survival analysis is the most common aroach used for redicting ie breakage behavior over the ast two decades (Su and Mays, 1988, Andreou et al. 1987a; Kim and Mays, 1994). The statistically-based survival function, S (T), resented by Kalbfleisch and Prentice (1980) is used to redict the behavior of breaks in a ie segment throughout the exosure time. For examle, the survival function associated with the Weibull distribution defined by two arameters, k and, for describing the behavior of first break at time T, is defined as: S ( T ) ex[ ( kt) ] (1.27) The hazard function (failure rate) is also used to estimate ie breakage rate. It is the robability that a ie exeriences a failure between time T 1 and T 2 given that the ie segment is oerating at time zero and has survived to time T 1 (Kaur and Lamberson 1977). The hazard function is determined by Equation 1.28 that shows the relationshi between the hazard function, the robability distribution function (PDF), f (T), and cumulative robability function, F (T), of ie failure. f ( T) ( T ) (1.28) 1 F( T) Pelletier et al. (2003) estimate the hazard and survival functions for cast iron ies using ie break data for the municialities of Chicoutimi, Gatineau and Saint-Georges, Quebec. They also determine the survival function for first breaks, based on their data. Pelletier (2000) and Mailhot et al. (2000) resent a modeling strategy that uses the Weibull and Exonential distributions to model the different break orders. The Weibull distribution is associated with the behavior of first break and the Exonential distribution is used to describe the behavior of subsequent breaks. 20

36 The model develoed by Pelletier (2000) requires an estimation of the PDF, hazard, and the survival functions. They estimate the number of ie breaks for a given ie segment between time T 1, and T 2, n (T 1, T 2 ), as: n( T1, T2 ) [ S( T1 ) S( T2 )] k2{ T2[1 S( T2 )] T1[1 S( T1 )] dt t f1( T)} (1.29) The likelihood of failure for a given residual ie wall thickness is determined based on the results of the robability analysis due to the internal and external corrosion. The average number of ie failures and the likelihood of failure for a given residual ie wall thickness may then be considered as condition indices of water mains in water distribution systems. T T CONTRIBUTIONS OF THESIS The main contribution of this research is to link a variety of model alications including a chlorine consumtion model, an internal corrosion rate model, internal and external corrosion rate models, a robability-based redictive ie breakage model, and a model for estimating the total number of ie failures, to determine a ie condition assessment (e.g., condition indices) of water mains in water distribution systems. Data related to condition indices may be used to erform the economic evaluation for estimating an otimum relacement time for water mains that are failing as a result of both internal and external corrosion, and the budgetary needs for future reairs under various relacement and rehabilitation strategies as a art of asset management rogram. The first contribution of this research is to use a chlorine consumtion model (Kiene and Wable 1996) for develoing an internal corrosion rate model that determines the relationshi between water quality arameters (e.g., the chlorine concentration ratio) and hydraulic conditions (e.g., water velocity) with an internal corrosion rate, to estimate the robability of failure due to internal corrosion throughout the exosure time. The second contribution is to use internal (Yamini and Lenc 2009) and external (Rajani et al. 2000) corrosion rate models to determine the overall robability of failure in water mains and to identify the variables that have the most significant effect on the robability of failure. All of the variables in internal and external corrosion models (e.g., the chlorine concentration ratio, and external corrosion scaling factor), environmental and oerational loads (e.g., traffic loads, and 21

37 water ressure), and ie strength caacity (e.g., ie thickness) are considered in the estimates of the robability of failure. The third contribution of this research is to link the data resulting from the robabilistic analysis to a robability-based redictive ie breakage model (Pelletier 2000; and Mailhot et al. 2000), and a total number of ie failure model (Pelletier 2000) to estimate the two condition indices of water mains. This methodology integrates the robability of failure analysis with an aroriate robability-based redictive ie breakage model, i.e., based on a Weibull distribution of first breaks and an Exonential distribution of subsequent ie failures, to estimate the total number of ie failures throughout the exosure time, as the first condition index of a ie segment. The second condition index is defined as the likelihood of failure for a given residual ie wall thickness at a given ie age. To validate the estimates of the survival function, which is based on the continuous robability density and hazard functions from the SORM analysis, the estimated survival function for all exosure times are comared with those determined by Pelletier et al. (2003) based on ie break data for the municialities of Chicoutimi, Gatineau and Saint-Georges, Quebec. The rimary contributions of this study is to link all the model alications to determine the robability of failure due to internal and external corrosion and to determine the two condition indices for understanding the structural behavior of water mains. A secondary contribution is the evaluation of the efficiency of FORM and SORM aroximation aroaches for this alication. 1.6 STRUCTURE OF THESIS This thesis is organized as a series of three manuscrits that have been submitted to or are tentatively acceted in journals. As a aer manuscrit, each of the three chaters (Chaters 2 to 4) can be read and understood indeendently. Each of them contains a reface that links that chater to the others, a main body, and references. In each of the chaters, figures and tables are embedded in the main body. All chaters adot the reference style alied in American Society of Civil Engineers (ASCE) journals. Chater 2 introduces a method for estimating the robability of failure due to internal corrosion. This method emloys a erformance function based on the failure criteria assumed by 22

38 the authors. A journal article manuscrit resulting from this chater has been acceted for ublication in ASCE Journal of Infrastructure Systems, and is in rint for December issue. Chater 3 discusses the exansion of this method for estimating the robability of failure due to internal and external corrosion and resents a comrehensive method for defining ie failure at a given exosure time. A technical note resulting from this chater was submitted to the ASCE Journal of Infrastructure Systems, was tentatively acceted for ublication, and is in final review. Chater 4 rooses a methodology to evaluate the behavior of ie breaks over time that may be used to estimate the condition of assets, i.e., water mains. This methodology integrates the failure robability analysis with an aroriate robability-based redictive ie breakage model, i.e., based on a Weibull distribution of first breaks and an Exonential distribution of subsequent ie failures, to estimate the condition of assets in a redefined lanning eriod. This aroach may be used to assist in long-term lanning of ie relacement in water distribution systems. A journal article manuscrit resulting frok/m this chater will be submitted to the ASCE Journal of Infrastructure Systems. The final chater, Chater 5, resents the summary, conclusions, future work, and the limitation of the methods develoed herein. 23

39 References Ahammed, M., and Melchers, R.E. (1994). "Reliability of underground ielines subject to corrosion." ASCE Journal of Transortation Engineering, 120 (6). Andreou, S. A., D. H. Marks, and Clark R.M. (1987 a). "A new methodology for modelling break failure atterns in deteriorating water distribution systems: Theory."Advance Water Resources, 10, Andreou, S. A., D. H. Marks, and Clark R.M (1987b). "A new methodology for modeling break failure atterns in deteriorating water distribution systems: Alications." Advance Water Resources, 10, Ang, A. H. S., and Tang, W. H. (1984). Probability Concets in Engineering Planning and Design, Vol. II: Decision, Risk, and Reliability, John Wiley, New York. Benjamin, M.M., Sontheimer, H., and Leroy, P. (1996). "Internal Corrosion of Water Distribution Systems." Chater 2 (Cooerative Research Reort, Denver, Colo.) American Water Works Association (ISBN: ). DeSilva, D., Davis, P., Burn, L. S., Ferguson, P., Massie, D., Cull, J., Eiswirth, M. and Heske, C. (2002). "Condition Assessment of Cast Iron and Asbestos Cement Pies by In-ie Probes and Selective Samling for Estimation of Remaining Service Life." Proceedings of the 20 th International No-Dig Conference, Coenhagen, Denmark. Grigg, N.S. (2004). "Assessment and renewal of water distribution systems." American Water Works Association and Research Foundation (AWWARF), 6666 West Quincy Avenue, Denver, CO Kalbfleisch, J. D., and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data, Wiley, New York. Kaur, K. C., and Lamberson, L. R. (1977). Reliability in Engineering Design, OSTI ID: , John Wiley and Sons, Inc., New York, NY Karaa, F. A., and Marks, D. H. (1990). "Performance of water distribution networks: Integrated aroach." J. Perform. Constr. Facil., 4(1), Kiene, L., Lu, W. and Levi, Y. (1998). "Relative imortance of the henomena resonsible for chlorine decay in drinking water distribution systems." Water Science and Technology, 38 (6),

40 Kiene, L., and Wable, M. (1996). "Characterization and Modeling of Chlorine Decay in Distribution System." Chater 2, Denver, Colo. Water Works Association and Research Foundation and American Water Works Association (ISBN: ). Kim, J. H., and Mays, L.W. (1994). "Otimal rehabilitation model for water distribution systems" Journal of Water Resources and Planning, Manage. Div. Am. Soc. Civ. Eng., 120, Kirmeyer, G.J., Richards, W. and Smith, CD. (1994). "An Assessment of Water Distribution Systems and Associated Research Needs." American Water Works Association Research Foundation, (90658), Denver, Colorado. Kucera, V., and Mattson, E. (1987). Atmosheric corrosion, Corrosion mechanics, F. Mansfeld, Ed., Mercel and Dekker, Inc., New York, NY. Lian, Y., and Yen, B. C. (2003). "Comarison of Risk Calculation Methods for a Culvert." ASCE Journal of Hydraulic Engineering, 129(2), Madsen, H. O., Krenk, S., and Lind, N. C. (1986). Methods of Structural Safety, Prentice-Hall, Englewood Cliffs, NJ. Melchers, A.M. (1996). "Reliability of underground ielines subject to corrosion." ASCE Journal of Transortation Engineering, 120 (6), Melchers, R. E. (1999). Structural Reliability Analysis and Prediction, John Wiley & Sons Ltd., Chichester. Mailhot, A., Pelletier, G., Noel, J. F., and Villeneuve, J. P. (2000). "Modeling the Evolution of Structural State of Water Pie Networks with Brief Recorded Pie Break Histories: Methodology and Alication." Water Resources Research, 36(10), O Day, D. K., Weiss, R., Chiavari, S., and Blair, D. (1986). "Water main evaluation for rehabilitation/relacement."american Water Works Association Research Foundation, Denver, and U.S. Environmental Protection Agency, Cincinnati. Pelletier, G. (2000). "Imact du Remlacement des Conduites D aqueduc sur la Nombre Annuel de Bris." PhD thesis, Institut National de la Recherche Scientifique, Sainte-Foy, Que., Canada (in French). Pelletier, G., Mailhot, A., and Villeneuve, J.P. (2003) "Modeling Water Pie Breaks Three Case Studies." ASCE Journal of Water Resources Planning and Management, Vol. 129, No.2, ISSN /2003/

41 Rajani, B., and Kleiner, Y. (2001). "Comrehensive review of structural deterioration of water mains: hysically based models."institute for Research in Construction, National Research Council Canada (NRC), Ottawa, Ontario, Canada. Rajani, B., Makar, J., McDonald, S. Zhan, C., and Kuraoka, S., Jen C-K., and Veins, M. (2000). "Investigation of grey cast iron water mains to develo a methodology for estimating service life." American Water Works Association Research Foundation, Denver, CO. Rossum, J. R. (1968). "Prediction of itting rates in ferrous metals from soil arameters." American Water Works Association, 61(6), Schueller, G.I., Bucher, C.G and Bourgund, U. (1989) "On Efficient Comutational Schemes to Calculate Structural Failure Probabilities" Probabilistic Engineering Mechanics, Vol.4, No.1, Page Shamir, U., and Howard, (1979) "An analytical aroach to scheduling ie relacement." Journal of American Water Works Association, 71(5), Sadiq, R., Rajani, B. and Kleiner, K. (2004). "Probabilistic risk analysis of corrosion associated failures in cast iron water mains." Reliability Engineering and System Safety, 86, Schneiter, C. R., Haims, Y.Y, Li, D, and Lambert, J.H. (1996). "Caacity reliability of water distribution networks and otimum rehabilitation decision making." Water Resources Research, 32, Sheikh, A.K., Boah, J.K., and Hensen, D.A. (1990). Statistical modeling of itting corrosion and ieline reliability, Corrosion Handbook, 46(3): Snoeyink, V.L. and Wagner, I. (1996). "Internal Corrosion of Water Distribution Systems.", Chater 1, (Cooerative Research Reort, Denver, Colo.) American Water Works Associations (ISBN: ). Su, Y. C., and Mays, L.W. (1998). "New methodology for determining the otimal rehabilitation and relacement of water distribution system comonents." Hydraulic Engineering: Proceedings of the ASCE National Conference, edited by S. R. Abt and J. Gessler, , Am. Soc. of Civ. Eng., Reston, Va. Tung, Y. K. (1996). Uncertainty and Reliability Analysis, Water Resources Handbook, McGraw- Hill, Vanier, D.J., (2001). "Asset Management: A to Z." American Public Works Association Annual Congress and Exosition- Innovations in Urban Infrastructure Seminar, Philadelhia,

42 Walski, T. M., (1987). "Relacement rules for water mains." Journal of American Water Works Association, 79, 33 37, Walski, T. M., and Pelliccia, A., (1982). "Economic analysis of water main breaks." Journal of American Water Works Assoccition., 74, Wood, A. (2007). "Data Assessment and Utilization for Imroving Asset Management of Small and Medium Size Water Utilities.", Ph.D thesis, University of British Columbia. Wood, A. and Lence, B.J., (2006). "Assessment of Water Main Break Data for Asset Management." Journal of AWWA, 98:07. Yamini, H. and Lence, B.J. (2009a). "Probability of Failure Analysis due to Internal Corrosion in Cast Iron Pies." Acceted in ASCE Journal of Infrastructure Systems, IS/2007/ In rint for December. Yamini, H. and Lence, B.J. (2009b). "Probability of Failure Analysis due to Internal and External Corrosion in Cast Iron Pies." Acceted in ASCE Journal of Infrastructure Systems, IS/2007/ Yamini, H. and Lence, B.J. (2009c). "Condition Assessment of Cast Iron Pies in Water Distribution Systems." Submitting to the ASCE Journal of Pie Systems. Li, Y., and Lence, B. J. (2007). "Mean Points of Failure and Safety Domains: Estimation and Alication." Probabilistic Engineering Mechanics, 22(3),

43 CHAPTER 2 PROBABILITY OF FAILURE ANALYSIS DUE TO INTERNAL CORROSION IN CAST IRON PIPES A version of this chater has been acceted for ublication in ASCE Journal of Infrastructure Systems. H. Yamini and B. J. Lence. Probability of Failure Analysis due to Internal Corrosion in Cast Iron Pies. 28

44 PREFACE In cast iron ies, chlorine consumtion due to internal corrosion is high comared with consumtion caused by other water quality arameters in such systems, and may be considered as an aroximate indicator of the rate of internal corrosion. In this chater a methodology is develoed for estimating the robability of mechanical failure of cast iron ies due to internal corrosion. A relationshi is develoed that determines the rate of internal corrosion as a function of the chlorine concentration ratio, chlorine decay constant, and the velocity of the water in a ie segment with a given length and diameter. These arameters may be considered uncertain at any given time and the robability of mechanical failure due to internal corrosion of a ie segment throughout the ie life is thus estimated using robabilistic methods including Monte Carlo Simulation (MCS), the First Order Reliability Method (FORM)-, and the Second Order Reliability Method (SORM)-based aroaches to account for these uncertainties. In this chater a mechanical failure is defined as the oint at which the corrosion deth is more than the maximum accetable decrease in ie wall thickness set by the utility manager. The results indicate that the likelihood of failure is nearly 50% by the 80 th year for a cast iron ie of length 6.1 m and diameter of 203 mm, and that the affect of internal corrosion on mechanical failure is lower than the effect of external corrosion. Also, a sensitivity analysis is conducted to determine the sensitivity of the estimates of the robability of failure to each of the water quality and hydraulic conditions. In the next chater, an alternate failure mode is defined to estimate the robability of failure due to internal corrosion. Then the robability of failure due to both internal and external corrosion is estimated based on the environmental and oerational imosed stresses acting on a ie segment. 29

45 2.1 INTRODUCTION Water distribution systems must be efficiently maintained, rehabilitated, and relaced to guarantee customers a sufficient suly of high quality and affordable water. The long-term lanning of the renewal of ies in water distribution systems requires assessment of the system reliability and estimation of the time to failure (Andreou et al., 1987a, b). A large ortion of the total ie length in water distribution systems in Canada and the USA are aging cast iron ies. A survey of 21 cities reresenting 11% of the Canadian oulation, conducted by Rajani and McDonald (1995), reveals that in 1993 aroximately 50% of all water distribution ies are gray cast iron. Kirmeyer et al. (1994) also reort that 48% of water distribution ies in the USA are gray cast iron. These ies are also the oldest ies, and have a tendency to corrode externally and internally under aggressive environmental conditions. External corrosion can lead to the develoment of corrosion its that may reduce the resistance caacity of the ie segment, resulting in mechanical failure. The internal corrosion of ies in distribution systems may lead to two major roblems. Mechanical failure may result in water leakage and build u of corrosion roducts, i.e., scale layers, which may cause loss of hydraulic caacity. The second roblem is the change in water quality as water travels through corroding ies in the distribution system. This imact is caused by corrosion roducts entering the water, causing for examle, red water. Major lifetime costs of water distribution are the costs of the loss of water and of the ie reairs needed to revent leaks. The most effective methods of reventing corrosion in water mains include the alication of surface coatings and the addition of inhibitors, e.g., chromate, nitrate, and hoshate, to control the internal corrosion (Snoeyink and Wagner 1996), and the alication of surface coatings and the use of cathodic rotection to control the external corrosion (Baeckmann et al.1997). Kiene and Wable (1996) conduct an extensive series of laboratory studies at Lyonnaise des Eaux (LDE) in France directed toward understanding chlorine decay in a cast iron ie. Based on these studies, they develo a general zeroth-order function that relates the ratio of chlorine concentration, i.e., the ratio of current chlorine concentration and the initial chlorine concentration, with the rate of internal corrosion in ie segments. In a subsequent study, Kiene et al. (1998) and Zhang et al. (2008) identify the relative imact of a number of factors, including 30

46 Total Organic Carbon (TOC), initial free chlorine concentration, internal corrosion of iron ies, and the amount of fixed biomass, on the degree of chlorine decay in water distribution ies. They conclude that the chlorine consumtion in uncoated cast iron ies due to internal surface corrosion is high comared with the other water quality factors and therefore the corrosion of cast iron ie is a major factor in chlorine decay in such ies. Snoeyink and Wagner (1996) reort on a laboratory study that show that while the rate of corrosion may be high when a cast iron ie is installed, it decreases to a stable value within 18 months. This chater determines the robability of mechanical failure due to the thinning of the ie wall caused by internal corrosion in cast iron ies. This robabilistic analysis identifies and incororates the relationshis between water quality factors, e.g., chlorine concentration ratio, hydraulic erformance, e.g., velocity, and the rate of internal corrosion. The robability of mechanical failure for a given exosure time may then be considered as one measure of ie condition and incororated into the lanning of rehabilitation and relacement of ies in water distribution systems. MCS and FORM- and SORM-based aroaches for estimating the robability of mechanical failure due to thinning of the ie wall caused by internal corrosion are used herein. The most basic and accurate method is the MCS, while erhas the most efficient are aroximate methods such as those based on FORM and SORM. One of the requirements for using the FORM method is the linearity or near linearity of the erformance function. The accuracy of FORM is influenced by the nonlinearity of the erformance function but SORM does not suffer the same roblem. While the rimary contribution of this study is the develoment of the erformance function that describes the robability of mechanical failure as a function of ie wall thickness due to internal corrosion, a secondary contribution is the evaluation of the efficiency of these aroximation aroaches. In the following two sections, a zeroth-order chlorine decay model develoed by Kiene and Wable (1996) that describes the relationshi between chlorine concentration ratio and the rate of internal corrosion, and the assumed linear relationshi between the rate of internal corrosion and the corrosion deth (based on observations made by Snoeyink and Wagner 1996), are reviewed. The corrosion deth is estimated based on the assumtion that the rate of corrosion is constant throughout the exosure time, at least when the ie is more than 18 months old. Next, a 31

47 erformance function is develoed that defines mechanical ie failure as a case in which the ie wall corrosion deth, which is a function of the chlorine concentration ratio and the rate of internal corrosion, is less than or equal to the allowable reduction of the ie wall thickness. A robability of failure analysis is then conducted using this erformance function for two different allowable ie reduction criteria, and two ie diameters. The imortance of the variable distributions is also investigated. 2.2 CHLORINE DECAY -CORROSION RATE MODEL Chlorine decay modeling in distribution systems is based on the consumtion of chlorine due to the oxidation reactions within the bulk water, a first-order reaction, and to the oxidation reactions occurring at the ie wall, a first or zeroth-order reaction (Vesconselous et al. 1996). Reduction of the chlorine residual is the most common water-quality concern and is a function of time of travel from the chlorine source, initial chlorine concentration, chlorine decay constant, and the rate of internal corrosion. Since the amount of exosed surface area of the ies in a distribution system is large comared with the volume of water traveling through the ies, the effect of the ie wall reactions on chlorine decay is normally significant. Corrosion is the oxidation of metallic iron (Fe 0 ) to ferrous iron (Fe 2+ ) as shown in Equation Fe Fe 2e (2.1) The intensity of the internal corrosion is controlled by the hysical-chemical conditions in the ie network such as the water temerature, H, carbonate balance, and chlorine concentration and the resence of biofilm (Barbier at al. 1990). At temeratures and H values common in water suly distribution networks the ferrous iron may reduce the chlorine concentration. When free chlorine, Cl 2, is dissolved in water, it reacts raidly to form hyochlorous, HOCl, and hydrochloric acid, HCl. Because hyochlorous acid is a stronger oxidant than hydrochloric acid under temeratures and H values common in water distribution networks, it is considered as the main oxidant of ferrous iron as indicated in Equation Fe HOCl H 2Fe Cl H 2O (2. 2) 32

48 Kiene and Wable (1996) roose a theoretical zeroth-order rate model for chlorine decay that is a function of the corrosion rate, assuming that the corrosion reaction is limited by the roduction rate of ferrous iron, Fe 2+, and that the free chlorine is the only oxidant of ferrous iron, i.e., HOCl is the main oxidant. This model is: d[ Cl ] 2 dt Cr K D (2.3) Where d [ Cl2 ] is the rate of chlorine decay. dt Kiene, et al. (1998) undertake a ilot-scale study to determine the relationshi between chlorine concentration ratio and the corrosion rate under tyical field conditions to verify the theoretical model as indicated in Equation 2.3. According to these ilot-scale exeriments, the chlorine concentration due to the corrosion rocess could be modeled based on the integration of Equation 2.3: C C t 0 K Cr 1 t (2.4) C D 0 All the exeriments conducted by Kiene et al. (1998) confirm that chlorine concentration, C t, can affect the internal corrosion rate considerably, and verify that a zeroth-order relationshi, e.g., Equations 2.3 and 2.4 is reresentative of the reactions that occur in gray cast iron. The best-fit value of the exerimental chlorine decay constant, K, from these exeriments is 1.36 * 10-6 min -1, about twice the theoretical value (Kiene et al. 1998). However, estimates of K and the ratio of chlorine concentration, C t /C0, vary among different studies. Several factors may cause such differences and the most imortant of theses are: 1) inaccurate measurement of the corrosion rate in exerimental studies, 2) variation in water quality and hydraulic conditions, and 3) the hysical condition of the ie examined (Kiene and Wable 1996). In this chater, the chlorine decay constant, the ratio of chlorine concentration, and water velocity are considered to be uncertain values in the estimation of ie wall thickness and thus in the robability of mechanical failure. 33

49 2.3 CORROSION MODELS The loss of ie wall due to the corrosion rocess can be either uniform around the ie or localized. The corrosion henomenon is comlex, and the rate of wall loss due to both internal and external corrosion, as well as whether the rate is constant or variable, has been the subject of debate (Ahammed and Melchers 1994; Melchers 1996; and Snoeyink and Wagner 1996). In all cases, however, corrosion rate is generally considered to be high during early ie ages and then stabilizes at a certain value thereafter. In the first hase a raid growth of the corrosion its occurs because any corrosion roducts formed on the ie surface are orous and have oor rotective roerties. In the second hase a slow growth of the corrosion its leading to a constant corrosion deth occurs due to the strong rotective roerties of the ie. This comlexity of the corrosion henomena leads to the different corrosion models and different suggested definitions of corrosion based on the different assumtions made by researchers. Corrosion has been defined in terms of weight loss of the ie material, maximum corrosion deth, or average corrosion deth (Ahammed and Melchers 1994). In this chater, a uniform corrosion rocess with a constant rate is assumed since over a long eriod, i.e., in old ies, the localized corrosion its grow to the oint where they are in contact with one another, making individual sites of tuberculation indistinguishable and causing the ie to aear to have a uniform layer covering the surface (Benjamin et al. 1996). Therefore, a linear corrosion model is emloyed for determining the corrosion deth, d*. Considering a time of exosure, T, which reresents the age of the ie, the relationshi between the internal corrosion rate and the corrosion deth, is assumed to be: Cr d * T (2.5) In ractice, layers of corrosion roducts may accumulate on the ie surface, and the variation in hydraulic conditions may cause sloughing of the corrosion roducts. Thus the corrosion deth may vary commensurately. Sloughing effects are not modeled herein. 2.4 PROBABILISTIC ANALYSIS In this study, the mechanical failure due to internal corrosion is defined to occur when the corrosion deth is more than the maximum accetable decrease in ie wall thickness set by the 34

50 utility manager. The erformance function of a ie segment is defined as the difference between the resistance, R, and load, L. The resistance is the strength or caacity of the system, e.g., a maximum accetable decrease in the thickness of the ie; whereas the load is the result of external conditions or influences uon a system, e.g., the corrosion deth. The corrosion deth may be estimated as a function of known factors, such as ie diameter and length, and random variables, such as the chlorine decay constant and water velocity. The erformance function, G(X), may be exressed in terms of these random variables, x, as: G ( X ) R L (2.6) The maximum accetable decrease in the ie thickness is selected by utility managers to determine the failure oint in the ie segment. In this study, two criteria for the failure oint are examined to evaluate the robability of failure of the ie throughout the exosure time. If the ie thickness is reresented as t, these criteria are exressed as d =t /2 and d =t /4, where d is the maximum accetable decrease in ie thickness defined by the difference between the outer, D 0, and inner, D, ie diameters, (D 0 -D)/2. Figure 2.1 shows a section of a ie segment with its outer and inner diameters, and its ie thickness. t D D 0 Figure 2.1. Section of a ie segment with outer and inner diameters The load used in the erformance function is the corrosion deth determined by Equation 2.5 rearranged as: 35

51 Equation 2.4 can be written in terms of C r as follows: d* C T (2.7) r C r Ct D 1 C0 (1 ) (2.8) C K t 0 The ratio of the length of the ie and the velocity of water may be substituted for travel time in Equation 2.8 to exlicitly account for the hydraulic conditions of the system. Thus, Equation 2.8 may be exressed as: C Ct D V C0 (1 (2.9) C K L r ) For the failure criterion in which the maximum accetable decrease in ie wall thickness is t /2, the erformance function may be exressed as: D0 D G ( X ) d * (2.10) 4 Substituting Equations 2.7 and 2.9 in Equation 2.10 yields: 0 G( X ) D 0 4 D [ C (1 0 C C t 0 D V ) ] T K L (2.11) For a given age of ie or time of exosure, if G(X) is less than zero, then the ie is considered to have failed and the robability of failure occurring is: P (G) 0 (2.12) In this analysis, random variables are considered to be chlorine decay constant, K, the water velocity, V, and the ratio of chlorine concentration, C t /C 0, which reresents the concentration of chlorine as water travels through the ie. MCS and FORM- and SORM-based aroaches are alied to estimate the robability of failure. While MCS is the most accurate method, the FORM- and SORM-based aroaches are also examined to investigate the effect of nonlinearity of the erformance function on estimates of the robability of mechanical failure. Here, the latter method is exected to erform better than the former method in highly nonlinear cases. 36

52 The robability of failure is estimated for a 6.1 m length of 203 and 548 mm diameter ie throughout the exosure time. The ie wall thickness rior to corrosion is considered to be 12 and 20 mm, for the 203 and 548 mm diameter ies, resectively. The initial chlorine concentration is assumed to be 1 mg/l. The robability of mechanical failure may be useful in identifying the remaining service life and in addressing long-term lanning for the rehabilitation and relacement of water mains in distribution systems. Therefore, the time of exosure examined in this study is years, i.e., T= in Equation 2.5. The starting oint of 10 years is well above the 18 month eriod reorted by Snoeyink and Wagner (1996) in which raid corrosion occurs, allowing for the rate of corrosion to be stabilized at a constant value. Often data required to define robability distribution functions (PDFs) are not available. Therefore, re-defined distributions based on limited information are considered. Table 2.1 summarizes the subjectively derived robability distributions for the inut variables used in this study (obtained from Keine et al. 1998; and Mays 2000). Statistical distributions such as the Normal, Lognormal, and Gumbel distributions are examined to show how these distributions may affect the robability of failure at a given exosure time. All robability estimates are determined using the Relan software, Version 7 (Foschi and Li 2005). Table2.1. Characteristics of inut variables for the robabilistic model of ie failure Pie diameter Parameter Source Minimum Mean StdDev. Maximum Chlorine concentration rate, Ct/C0 Mays, mm chlorine decay constant (mg/l), K Kiene et al., * * * *10-6 Water velocity (m/s), V Mays, Chlorine concentration rate, Ct/C0 Mays, mm chlorine decay constant (mg/l), K Kiene et al., * * * *10-6 Water velocity (m/s), V Mays, The distributions of all inut variables are assumed to be indeendent. However, in reality cases in which the variables are correlated are ossible. For examle, the ratio of chlorine 37

53 concentration is related to the hydraulic conditions, e.g., water velocity, and the water velocity is also related to the ie diameter (Menaia et al. 2003). In ie networks, for smaller ie diameter, the velocity of water is higher and the higher velocity of water may also increase the chlorine consumtion as water travels through the ie. Therefore, cases in which variables have a range of degree of correlation are examined to estimate the imact of this assumtion. 2.5 RESULTS The data in Figure 2.2 rovide the results of the MCS, and FORM- and SORM-based * aroaches for a 203 mm diameter ie, a failure criteria of d / 2, and normal distributions for all random variables. The difference in robability of failure is less than five ercent in the first 50 years of exosure using the MCS, and FORM- and SORM-based aroaches. The nonlinearity of the erformance function leads to a substantial difference between the FORM and SORM-based results after the 50 years of exosure time. For this case, the SORM-based aroach erforms well for exosure times greater than 50 years. The likelihood of failure increases to nearly 50% in the 80 th year of ie age. This indicates that the effect of internal corrosion on ie wall thickness and thus mechanical failure is lower than the effect of external corrosion on such failures. Sadiq et al. (2004) estimate a 50% likelihood of failure in the 70 th year of ie age in the case of external corrosion for the same tye of ie. t P Figure 2.2. Probability of failure throughout the exosure time, for a normal distribution using MCS and FORM- and SORM-based aroaches (203 mm ie diameter and failure criteria of * d / 2) t P 38

54 Cumulative robability failure (%) The data in Figure 2.3 rovide the results of the SORM-based analysis using Normal, Lognormal, and Gumbel distributions for all random variables of the analysis for a 203 mm ie * diameter and the failure criteria of d / 2. The results show that the maximum difference t P among these distributions is less than 15%. The robability of failure is less than 5% in the first 80 years of exosure under all distribution tyes Normal distribution 10 Log-normal distribution Gumbel distribution Exosure time (year) Figure 2.3. Probability of failure throughout the exosure time, for all statistical distributions * using a SORM-based aroach (203 mm ie diameter and failure criteria of d / 2) t P The robability of failure is also estimated for a 548 mm ie diameter for the failure criteria of * d / 2 and comared with the robability of failure for a 203 mm ie diameter (see Figure t P 2.4). The results show that for the 548 mm ie the robability of failure is 50% in the 90 th year of exosure. The robability of failure is increased in smaller ie diameters and this is likely due to the higher average velocities in such ies. 39

55 Figure 2.4. Probability of failure throughout the exosure time, for a normal distribution and * various ie diameters using a SORM-based aroach (failure criteria of d / 2 ) t P The data in Figure 2.5 show how the robability of failure varies throughout the exosure time for different failure criteria for a 203 mm ie diameter. The results show that with maximum accetable decreases of d =t /2 and d =t /4, the robability of failure is 50%, when the ie is 80 years old and 38 years old, resectively. Figure 2.5.Probability of failure throughout the exosure time, for a normal distribution and two failure criteria using a SORM-based aroach (203 mm ie diameter) 40

56 Cumulative robability of failure (%) The data in Figure 2.6 show the robability of failure throughout the exosure time assuming that all variables are uncorrelated, exactly correlated (correlation coefficient of 1.0) and somewhat correlated (correlation coefficient of 0.5). The results indicate that the maximum difference between robabilities based on these assumtions is less than 10%. When the variables are considered to be correlated, the random variables that are artially or highly correlated may counter balance each other and cause only a moderate variation in robability of failure at a given exosure time No correlation 10 Partial correlation High correlation Exosure time (year) Figure 2.6. Probability of failure throughout the exosure time, for a normal distribution with different correlation assumtions using a SORM-based aroach (203 mm ie diameter and * failure criteria of d / 2 ) t P 2.6 SENSITIVITY ANALYSIS A sensitivity analysis is conducted to identify the variables that have the most significant effect on the reliability index, β, and subsequently on the robability of failure. The sensitivity of the reliability index to each of the three variables (C t /C 0, K, V) is evaluated by the rate of change of the β with resect to each of the variables, called the sensitivity factor. These rates are given as the artial derivatives of β with resect to each of the variables. The sensitivity factors are determined using the Relan software, Version 7 (Foschi and Li 2005). The sensitivity factors for each of the three variables (C t /C 0, K, V) are shown in Figure 2.7 for the range of exosure time. The results indicate that the sensitivity factor for the chlorine 41

57 Sensitivity factor concentration ratio, C t /C 0, is higher than that for any other variable in terms of their contribution to the robability of mechanical failure after 30 years of ie age. This imlies that the robability of failure is most sensitive to the ratio of chlorine concentration, C t /C 0, which reresents the current concentration of chlorine as water travels through the ie and is deendent on the initial chlorine concentration and the rate of corrosion. Thus, the analysis of boosting chlorine throughout a system, its influence on the rate of corrosion and on the chlorine concentration ratio is recommended as a means of further extending ie life. Data in Figure 2.7 show that while the variable K has the most influence on the robability of failure before 30 years of ie age, its imortance is greatly reduced thereafter. The value of K reresents the intensity of internal corrosion rate and should be examined carefully to show the real value of internal corrosion rate in the ie segment Value for Ct/C0 Value for K 0.2 Value for V Exosure time (year) Figure 2.7. Sensitivity factors for inut arameters throughout the exosure time, for a normal distribution using a SORM-based aroach (203 mm ie diameter and failure criteria of * d / 2 ) t P The results also show that the velocity of water has a minimal effect on the robability of failure among these variables before 80 years of ie age. 42

58 2.7 SUMMARY AND CONCLUSIONS Chlorine consumtion in cast iron ie due to internal corrosion is high comared with other water quality factors including Total Organic Carbon, TOC, initial free chlorine concentration, and the amount of fixed biomass (Kiene et al. 1998). The internal corrosion of cast iron ie may cause the loss of ie wall and this is one of the major factors contributing to ie mechanical failure. Considering the effect of internal corrosion on the ie wall and thus mechanical failure, robabilistic analyses are used herein to estimate the robability of mechanical failure throughout the exosure time based on a zeroth-order model that reresents the relationshi between the chlorine decay and the internal corrosion rate. Using MCS and the SORM-based aroach leads to more accurate estimates of the robability of failure comared with the FORM-based aroach. Probability of failure throughout the exosure time is estimated for two different ie diameters. The robability of failure is greater for smaller diameter ies because the mean velocity is greater in such ies. The sensitivity of the robability of failure to different variables is estimated to determine which variables have the most influence throughout the exosure time. Comared with the hydraulic erformance, e.g., water velocity, the water quality factors, e.g., ratio of chlorine concentration and the chlorine decay constant, are the most imortant variables influencing the robability of failure. Probability of failure for a given exosure time may be thought of as a surrogate for the service life of a ie. This information may be useful in addressing the lanning rocess for the rehabilitation and relacement of the system infrastructure. The imortance of the chlorine decay constant in modeling water quality has been extensively studied. The correct measurement of K should be determined to reflect the aroriate internal corrosion rate in the ie network. Therefore, advances in infrastructure asset management may benefit from further analysis of the chlorine decay constant to determine the accurate internal corrosion rate in the ie network. The robabilistic analysis conducted in this work is based on the assumtions that the chlorine is the only oxidant of ferrous iron, Fe 2+, the internal corrosion rate is uniform and has a constant value throughout the exosure time, i.e., T=10-200, and water quality failure and sloughing effects are not modeled. The risk of failure may be further heightened if the water quality and sloughing effects are considered. Also, variation of the corrosion rate may cause 43

59 significant changes in ie failure results. These considerations should be investigated in future studies. Further studies that may be beneficial include: exanding the erformance function to include other modes of failure, most imortantly water quality failure due to the release of corrosion roducts, e.g., contributing to red water; and undertaking robability analyses that include both internal and external corrosion. External corrosion has been shown to affect ie wall and thus mechanical failure significantly and the risk of failure may be further heightened by internal corrosion. A sensitivity analysis can be conducted to determine the contribution of uncertain inut variables of both external and internal corrosion to the variability of the time to failure. These analyses may be useful to determine a comrehensive method to assist in the develoment of future asset management strategies. ACKNOWLEDGEMENTS The authors gratefully acknowledge the insights and advice of Professor R. O. Foschi at the University of British Columbia. Financial suort for this research was rovided by a Natural Science and Engineering Research Council of Canada (NSERC) Discovery Grant to the second author, Grant Award Number

60 References Ahammed, M., and Melchers, R.E. (1994). "Reliability of underground ielines subject to corrosion." ASCE Journal of Transortation Engineering, 120 (6). Andreou, S. A., D. H. Marks, and Clark R.M. (1987 a). "A new methodology for modelling break failure atterns in deteriorating water distribution systems: Theory."Advance Water Resources, 10, Andreou, S. A., D. H. Marks, and Clark R.M (1987b). "A new methodology for modeling break failure atterns in deteriorating water distribution systems: Alications." Advance Water Resources, 10, Baeckmann, W., Schwenk, W., Prinz., W. (1997). Handbook of Cathodic Corrosion Protection: Theory and Practice of Electrochemical Protection Processes, Gulf Professional Publishing, 3 RD Edition, ISBN-10, Benjamin, M.M., Sontheimer, H., and Leroy, P. (1996). "Internal Corrosion of Water Distribution Systems." Chater 2 (Cooerative Research Reort, Denver, Colo.) American Water Works Association (ISBN: ). Barbier, J. P., Cordonnier, J. and Gabriel, J. M. (1990). "Corrosion metalique et henomenes d eaux reux rouges-utilisation d inhibiteurs. " T.S.M.-L EAU, Benjamin, M. M., Sontheimer, H. and Leroy, P. (1996). "Internal Corrosion of Water Distribution Systems. " Chater 2, (Cooerative Research Reort, Denver, Colo.) American Water Works Association (ISBN: ). Foschi, R. O., and Li, H. (2005). Reliability Analysis Software, Version 7. University of British Columbia, Vancouver, B.C. Kiene, L., Lu, W. and Levi, Y. (1998). "Relative imortance of the henomena resonsible for chlorine decay in drinking water distribution systems." Water Science and Technology, 38 (6), Kiene, L., and Wable, M. (1996). "Characterization and Modeling of Chlorine Decay in Distribution System." Chater 2, Denver, Colo. Water Works Association and Research Foundation and American Water Works Association (ISBN: ). Kirmeyer, G.J., Richards, W. and Smith, CD. (1994). "An Assessment of Water Distribution Systems and Associated Research Needs." American Water Works Association Research Foundation, (90658), Denver, Colorado. 45

61 Menaia, J., Coelho, S.T., Loes, A., Fonte, E., and Palma, J. (2003). "Deendency of bulk chlorine decay rates on flow velocity in water distribution networks." Journal of IWA, Vol 3 No Mays, L. (2000). Water Distribution Handbook, Deartment of Civil and Environmental Engineering, Arizona State University, McGraw-Hill. Melchers, A.M. (1996). "Reliability of underground ielines subject to corrosion." ASCE Journal of Transortation Engineering, 120 (6), Rajani, B., and Mc Donald, S. (1995). "Water Mains Break Data on Different Pie Materials for 1992 and 1993." NRCC, Reort No. A , Ottawa, ON. Sadiq, R., Rajani, B. and Kleiner, K. (2004). "Probabilistic risk analysis of corrosion associated failures in cast iron water mains." Reliability Engineering and System Safety, 86, Sadiq, R., Syed, A.I., Kleiner, Y., (2007). "Examining the Imact of Water Quality on the Integrity of Distribution Infrastructure." NRCC, and AwwaRF, Denver, Colo. Snoeyink, V.L. and Wagner, I. (1996). "Internal Corrosion of Water Distribution Systems." Chater 1, (Cooerative Research Reort, Denver, Colo.) American Water Works Associations (ISBN: ). Vasconcelos, J.J., Grayman, W.M., Kiene, L., Wable, O., Biswas, P., Bhari, A., Rossman, L.A. and Clark, RM. (1996). "Characterization and Modeling of Chlorine Decay in Distribution System." Chater 5, Denver, Colo. AwwaRF and American Water Works Association (ISBN: ). Zhang, Z., Stout, J., Yu, V., Vidic, R., (2008). "Effect of ie corrosion scales on chlorine dioxide consumtion in drinking water distribution systems" Journal of IWA, Water Research, 42,

62 CHAPTER 3 PROBABILITY OF FAILURE ANALYSIS DUE TO INTERNAL AND EXTERNAL CORROSION IN CAST IRON PIPES A version of this chater has been submitted for ublication H. Yamini and B. J. Lence. Probability of Failure Analysis due to Internal and External Corrosion in Cast Iron Pies. 47

63 PREFACE In the revious chater the robability of failure due to internal corrosion is estimated based on the criteria that a failure occurs when the corrosion deth is more than the maximum accetable decrease in ie wall thickness set by the utility manager. Internal corrosion may lead to water leakage and loss of hydraulic caacity caused by the loss of ie wall thickness and build u of corrosion roducts which also reduce the resistance caacity of the ie segment External corrosion can also lead to the develoment of corrosion its that may reduce the resistance caacity of the ie segment to resist stresses uon a ie segment. It has been shown to affect the ie mechanical failure significantly. The risk of mechanical failure may be further heightened if internal corrosion is considered. This chater develos a methodology for estimating the robability of mechanical failure of cast iron ies due to both internal and external corrosion. Two failure oints are defined to estimate the robability of failure due to the internal corrosion and to both the internal and external corrosion. The first failure oint is defined as the time at which the stress due to water ressure exceeds the ie strength caacity considering only internal corrosion. The second failure oint is defined as the time at which total stresses exceed the ie strength caacity considering the effect of both internal and external corrosion. Using the failure definitions allows for a comarison to be made with the failure definition for the robability of failure due to the external corrosion estimated by Sadiq et al. (2004). Each definition is based on the difference between the stresses acting on a ie segment and the ie strength caacity. A robabilistic analysis is used that incororates a linear internal corrosion model as well as the Two-hase nonlinear external corrosion model (Rajani et al., 2000) to calculate the robability of ie failure. The Monte Carlo Simulation (MCS), and First Order Reliability Method (FORM)- and Second Order Reliability Method (SORM)-based aroaches are used to erform the robabilistic analysis. The results indicate that the likelihood of failure of a cast iron ie due to internal corrosion is nearly 50% by the 110 th year of exosure and the likelihood of failure of a cast iron ie due to both internal and external corrosion is nearly 50% by the 45 th year of exosure. A sensitivity analysis is conducted to determine the sensitivity of the estimates of the robability of failure to each of the uncertain inuts of internal and external corrosion. 48

64 This analysis reveals that the contribution of external and internal corrosion arameters to the variability of time to failure is more significant than the contributions of other arameters. In the next chater, a methodology is develoed to integrate the robability of failure analysis with an aroriate robability-based redictive ie breakage model insired by survival analysis to estimate the condition of assets in a redefined lanning eriod. 49

65 3.1 INTRODUCTION Cast iron ies have been used to transfer otable water for more than 500 years, and iron ie corrosion has been a roblem for as long. Old cast iron ies, which reresent a large ortion of the total ie length in water distribution systems, as well as their financial cost to water utilities, have been the subject of extensive research over the ast 20 years in Canada and the USA (Kirmeyer et al. 1994; Kleiner 1998; Selvakumar et al. 2002; Rajani and Tesfamariam 2005). These ies have a tendency to corrode externally and internally under aggressive environmental conditions. Corrosion of ies is a major factor in the weakening of ie integrity and leads to several roblems, including ie breaks, loss of hydraulic caacity, and oor water quality (Mays 2000). Water utilities in Canada and in the USA invest caital and oerating funds to maintain the water mains of distribution systems. Much of this exenditure is to reair or relace ageing cast iron mains that have suffered from corrosion throughout the ie exosure time. It is estimated that the financial costs for ie reair and relacement exceeds $1 billion annually in Canada alone (Rajani and Tesfamariam 2005). Also, Selvakumar et al. (2002) reort that the total costs of ie reair and relacement exceeds $3.7 billion annually in the USA. The most effective methods of reventing corrosion in water mains include the alication of surface lining and addition of inhibitors, e.g., chromate, nitrate, hoshate, to control the internal corrosion (Snoeyink and Wagner 1996), and the alication of surface coating and the use of cathodic rotection to control the external corrosion (Baeckmann et al. 1997). Corrosion of cast iron ies in distribution systems can lead to the develoment of corrosion its that may reduce the resistance caacity of the ie segment, resulting in mechanical failure. The mechanical failure of ies is mostly the result of this structural weakening due to externally, environmental, and internally, oerational, imosed stresses and the vulnerability of ies due to their structural roerties, tye of material, ie-soil interaction, and inadequate ie installation. Pie vulnerability due to these latter rocesses has not been modeled because of the varied, uncertain, and comlex factors involved. This chater develos a methodology to determine the robability of failure due to the thinness of the ie wall caused by internal and external corrosion in cast iron ies. This robabilistic analysis identifies and incororates variables related to water quality, e.g., the chlorine decay rate, hydraulic erformance, e.g., velocity and internal water ressure, and 50

66 external stresses, e.g., stress due to soil ressure, and the rate of internal and external corrosion to estimate the robability of failure of a ie segment. The robability of failure for a given exosure time may then be considered as one measure of condition and incororated into the lanning of rehabilitation and relacement of ies in water distribution systems. The high degree of uncertainty associated with factors that affect ie failures warrants a robabilistic-based analytical aroach. A detailed uncertainty analysis is required to quantify the robability of ie failures at a given time. MCS and FORM- and SORM-based aroaches for estimating the robability of mechanical failure due to the thinness of the ie wall caused by internal and external corrosion are used herein. While the rimary contribution of this study is the develoment of the erformance function that describes the robability of mechanical failure as a function of ie wall thickness due to internal and external corrosion, a secondary contribution is the evaluation of the efficiency of these aroximation aroaches. In the following two sections, the external stresses acting on a ie segment and corrosion models are reviewed. Next, functions that describe the imact on ie integrity of internal corrosion and both internal and external corrosion are defined. Then erformance functions that define mechanical ie failure due to the thinness of the ie wall caused by 1) internal corrosion as a case in which the stress due to water ressure exceeds the ie strength caacity and 2) both internal and external corrosion as a case in which the total stresses, e.g., stresses due to internal water chemistry and ressure, soil, traffic, and frost ressures, exceed the ie strength caacity, are develoed. A sensitivity analysis which describes the imortance of the variables related to internal and external corrosion, hydraulic erformance, and all stresses acting on the ie, is undertaken thereafter. 3.2 EXTERNAL AND INTERNAL STRESSES AND CORROSION MODELS Internal, e.g., internal fluid ressure, and external, e.g., soil ressure, loads may cause stress in both the circumferential and longitudinal directions. Ahammed and Melchers (1994) show that if a ie is uniformly loaded and suorted along its length, then circumferential stresses may dominate. A large number of factors influence the loads acting on ie walls including the width of the trench, the unit weight of the soil, and the surface load. 51

67 Rajani et al. (2000) develo a model of external and internal stresses including all circumferential and longitudinal stresses. They define the hoo or circumferential stress,, as a combination of stresses due to internal fluid ressure, F, soil ressure, S, frost ressure, L, and traffic loads, V. Similarly the axial, or longitudinal, stress is defined as a combination of these stresses and ressure gradients, T, due to temerature differences. They show that the effect of internal fluid ressure is minor comared with environmental ressures for determining longitudinal stress, while the effect of internal ressure is significant comared with environmental ressures for determining circumferential stress. Rajani et al. (2000) develo the following formulation for total circumferential stress: F S L V (3.1) Where F = i 2 D t (3.2) S = 3 K E m t 3 B 2 d 3 C K d d E i t D 3 D (3.3) L = f frost S = f 3 K B C 2 m d d frost (3.4) 3 3 E t 3 K d i D E t D V = 3 A K m [ E I c 3 t C t 3 F K E d i t D D 3 ] (3.5) Rajani et al. (2000) recommend the consideration of frost ressure, which is aroximately a multile of 0 to1 (i.e., 0 to 100%) of the soil ressure. Seasonal variations in temerature may be the cause of the variation in frost ressure. Chater 1, Figure 1.3 shows the circumferential forces acting on a ie and the orientation of failures that may occur due to the environmental and oerational loads. A review of existing stress models (Rajani and Kleiner 1996; Rajani et al. 2000) for cast iron ies in water distribution systems shows that the total circumferential stress model develoed by Rajani et al. (2000) (Equation 3.1) is the most comrehensive models of ie 52

68 Maximum corrosion deth (mm) failure. This model incororates most of the identified stresses that may act on a ie segment. In this chater, the effect of the loads in the circumferential direction is considered under the assumtion that the ie is uniformly loaded and suorted along its length to estimate the overall likelihood of ie failure. Rajani et al. (2000) develo the Two-hase emirical corrosion model which is based on the assumtion that the corrosion rate is generally high during early ie ages and then stabilizes at a certain value thereafter. Figure 3.1 shows the form of the Two-hase external corrosion model as well as the linear internal corrosion model develoed in Chater 2. The model redictions are the relationshi between external and internal corrosion deth, for the Two-hase and linear model, resectively, and the ie exosure time. During the first hase of the Two-hase external corrosion model, a raid exonential it growth occurs and in the second hase a reduced linear it growth occurs due to the corrosion self-inhibiting rocess. This it deth is defined as: ct d * at b(1 e ) (3.6) ext This model is emloyed in this chater to evaluate the robability of failure due to external corrosion throughout the exosure time. 8.0 Two-hase external corrosion model (Rajani et al., 2000) Linear internal corrosion model (Yamini and Lence, 2009) Exosure time (year) Figure 3.1. Predictive external and internal corrosion models for corrosion deths 53

69 In Chater 2, an internal corrosion model is develoed that relates the rate of internal corrosion to the ratio of chlorine concentration, i.e., the ratio of current chlorine concentration and the initial chlorine concentration as water travels through the ie, chlorine decay constant, and the velocity of the water in a ie segment with a 6.1 m length and 203 mm diameter. Figure 3.1 also shows a linear relationshi between the corrosion deth caused by internal corrosion and the ie exosure time. The internal corrosion model is defined as: Ct D V d * int Cr T [ C (1 ) ] T (int) 0 (3.7) C K L In this work, corrosion models are used to determine the external, d* ext, and internal corrosion deth, d* int, which are in turn used to define the resistance caacity of the ies. For convenience, in all equations related to stresses, the original ie thickness, t, is relaced with residual ie thickness (t res ) to calculate the robability of failure. The residual ie thickness is defined as: tres t d * (3.8) 0 Where d* d * ext d (3.9) * int 3.3 PIPE STRENGTH CAPACITY Rajani et al. (2000) state that the structural strength caacity, Y, of cast iron is related to the corrosion it deth. They show that increasing the corrosion it deth causes decreases in the ie residual strength over the ie exosure time. This model is described by the following relationshi: Y d * a1( ) t res b1 ( d K * q / t res a n ) S (3.10) 54

70 3.4 PROBABILITY OF FAILURE DUE TO INTERNAL CORROSION The mechanical failure due to the thinness of the ie wall caused by internal corrosion is defined as the case in which the stress due to water ressure exceeds the ie strength caacity under the assumtion that no environmental ressures affect the ie failure throughout the exosure time. The erformance function of a ie segment is defined as the difference between the resistance, R, and load, L. The resistance is the strength or caacity of the system, e.g., a ie strength caacity, whereas the load is the result of environmental and oerational conditions on the system, e.g., stress due to water ressure. The erformance function, G(X), may be exressed in terms of random variables, x, as: G ( X ) R L (3.11) The maximum accetable stress may be selected by utility managers to determine the failure oint in the ie segment. In this chater, two criteria for the failure oint are examined to evaluate the robability of failure of the ie throughout the exosure time. These are the oints at which resulting resistance caacity due to 1) internal corrosion and 2) the combination of internal and external corrosion, are less than the loads on the system. For the first failure criterion in which the stress due to water ressure exceeds the ie strength caacity, the erformance function may be exressed as: G ( X ) (3.12) Substituting Equations 3.2 and 3.10 in Equation 3.12 and relacing t with t res, yields: Y F G( X ) d * a1( t res int ) b 1 K ( d * q int / t res n ) S id 2t res (3.13) If the linear internal corrosion model is used (Equation 3.7), resulting in an internal corrosion deth, d * in t, Equation 3.8 can be rewritten as: Substituting Equation 3.14 in Equation 3.13 yields: Ct D V tres t ( d * in ) t {[ C0 (1 ) ] T} t (3.14) C K L 0 55

71 G( X ) { [ a 1 Ct D V [ C0 (1 ) ] T C0 K L ( ) Ct D V [ t C0 (1 ) T] C K L 0 b1 ( n [ C 0 (1 K q Ct ) C 0 D K V L T ]/[ t C 0 (1 Ct ) C 0 D K V L S T]) ] 2 [ t C 0 (3.15) i D (1 Ct ) C 0 D K V L } T] For a given age of ie or time of exosure, if G(X) is less than zero, then the ie is considered to have failed and the robability of failure occurring is: P (G) 0 (3.16) In this analysis, the random variables are considered to be the water quality variables including the chlorine decay constant, K, and the ratio of chlorine concentration (Ct/C 0 ), the hydraulic characteristics including the water velocity, V, and water ressure, i, and the ie hysical characteristics including the ie diameter, D, and ie original thickness, t. MCS, FORM- and SORM-based aroaches are alied to estimate the robability of failure using Equation While MCS is the most accurate method, the FORM- and SORM-based aroaches are alied in order to investigate the erformance of these techniques for this roblem i.e., to investigate the effect that the nonlinearity of the erformance function has on estimates the robability of failure. 3.5 PROBABILITY OF FAILURE DUE T0 INTERNAL AND EXTERNALCORROSION Sadiq et al. (2004) use robability analysis to estimate the reliability of the ie in the face of external corrosion associated failure in cast iron water mains. They define the failure time as the time at which the total stresses exceed the ie strength caacity. The erformance function relates all the variables of the stresses acting on the ie, the ie residual strength, and the external corrosion, and is defined as: G ( X ) (3.17) Y Substituting Equations 3.2 through 3.5 in Equation 3.1 for considering circumferential stresses, and Equations 3.1 and 3.10 in Equation 3.17 for estimating the erformance function, yields: 56

72 G( X ) a 1 d * ( t int ext res ) K 3 K I C F 2 q i D m c t res m d d res [1 f ] 3 3 frost 3 3 b 2 1 S tres ( d * [ 3 ] 3 int / t ) A E tres K d i D E tres K d i D ext res n E t D 3 K (3.18) B C E t D If the Two-hase external corrosion (Equation 3.6) and linear internal corrosion model (Equation 3.7) are used, resulting in a corrosion it deth, rewritten as: d * ext d * int. Equation 3.8 may be ct Ct D V tres t ( d * ext d * in ) { t ([ at b(1 e )] [ C0 (1 ) T ])} t (3.19) C K L Substituting Equations 3.19 in Equation 3.18 yields the general erformance function G(X) which may be used to calculate the robability of failure due to the thinness of the ie wall caused by both external and internal corrosion. Often data required to define robability distribution functions (PDFs) are not available. Therefore, re-defined distributions based on limited information are considered. Sadiq et al. (2004) identify some robability distributions for inut variables for erforming a risk analysis of external corrosion associated failures in water mains. These distributions, along with those used in a similar study of the robability of failure due to the thinness of the ie wall caused by internal corrosion in Chater 2, are used in this analysis. The robability distribution functions for inut variables related to all environmental and oerational stresses, resistance caacity, and corrosion rates are used for evaluating robability of failure throughout the exosure time. The tye of distributions, minimum and maximum value, and standard deviation for each inut variable are rovided in Table

73 Table3.1 Probability distributions of inut variables for calculating robability of failure Variable Symbol Units Tye of Min. Mean Stdev. Max. distributio ns Water ressure i MPa Normal Pie diameter D mm Normal Wall thickness t mm Normal Toughness corrosion coefficient Constants used for determining strength caacity a 1 b 1 Uniform Uniform Normal Multile of it width L Uniform Toughness exonent S Normal chlorine decay constant K mg/l Normal 6.32* * * *10-6 Water velocity V m/s Lognormal Pie length A mm Normal Ratio of chlorine concentration C t /C 0 Normal Bending moment coefficient Km Lognormal Unit weight of soil N/mm 2 Normal 18.85* *10-7 Width of ditch B d mm Normal Calculation coefficient C d Lognormal Modulus elasticity of the E MPa Normal ie Deflection coefficient K d Lognormal Final itting rate constant a mm/yr. Normal Pitting deth scaling b mm Normal constant Corrosion rate inhibition factor c yr. -1 Normal Frost load multilier f frost Uniform 0 1 Imact factor Ic Normal Surface load coefficient Cs Lognormal Wheel load traffic F N Normal RESULTS AND DISCUSSION The data in Figure 3.2 rovide the results of the MCS, and the FORM- and SORM-based aroaches for a 6.1 m, 203 mm diameter ie, under the failure criteria for which the stress is 58

74 due to water ressure exceeding the ie strength caacity, and is only affected by internal corrosion. This ie length and diameter were chosen by Sadiq et al. (2004) in their analysis and the same ie tye is selected herein in order to comare this work with their analysis. The nonlinearity of the erformance function leads to a difference between the FORM and SORM results after aroximately 40 years of exosure time. The likelihood of failure increases to nearly 50% in the 110 th year of ie age using MCS and the SORM-based aroach. This indicates that the effect of internal corrosion on mechanical failure is lower, but on the same order of magnitude, as the effect of external corrosion on such failures. Sadiq et al. (2004) estimate a 50% likelihood of failure in the 70 th year of ie age in the case of external corrosion for the same tye of ie. P 50 =110 yr. Figure3.2. Probability of failure due to the thinness of the ie wall caused by internal corrosion throughout the exosure time using MCS, FORM- and SORM- based aroaches (failure criteria of F Y ) The data in Figure 3.3 rovide the results of the MCS, and FORM- and SORM-based aroaches for which the failure criteria is defined as the case in which the total stresses exceed the ie strength caacity, affected by both external and internal corrosion. The likelihood of failure increases to nearly 50% in the 45 th year of ie age. While external corrosion has been shown to affect the ie mechanical failure significantly and the effect of internal corrosion on 59

75 mechanical failure is lower than that of external corrosion, the results show that the risk of mechanical failure may be further heightened if internal corrosion is considered. ` 70 ) (% re 60 ilu fa 50 f o 40 ility b a b30 ro e20 tiv la u10 m u C 0 P 50 =45 yr. FORM-based Aroach SORM-based Aroach MCS Exosure time (year) Figure 3.3. Probability of failure due to the thinness of the ie wall caused by external and internal corrosion throughout the exosure time using MCS, and FORM- and SORM-based aroaches (failure criteria of Y ) 3.7 SENSITIVITY ANALYSIS A sensitivity analysis is conducted to identify the variables that have the most significant effect on the reliability index, β, and subsequently on the estimate of the robability of failure. The sensitivity of the reliability index to each of the water quality factors (C t /C 0, K), hydraulic erformance variables (V, i ), and ie structural characteristics related to its strength caacity is estimated and described in terms of sensitivity factors. The sensitivity factors rovide the rate of change of the reliability index with resect to each of the random variables. These rates are given as the artial derivatives of β with resect to each of the variables. The sensitivity factors are determined using the Relan software, Version 7 (Foschi and Li 2005). The results in Figure 3.4 indicate that the robability of failure due to only internal corrosion is most sensitive to the chlorine decay constant, K, the velocity of water, V, and the ratio of chlorine concentration, C t /C 0, which reresents the current concentration of chlorine as 60

76 Sensitivity factor water travels through the ie. These variables have the most influence on the robability of failure over the ie life. While the variable K has the most influence on the robability of failure rior to 105 years of ie age, after this age the velocity becomes the most imortant variable throughout the life of the ie. The correct measurement of K should be determined to reflect the aroriate internal corrosion rate in the ie network. Therefore, advances in infrastructure asset management may benefit from further analysis of the chlorine decay constant to determine the accurate internal corrosion rate in the ie network K V Ct/C Exosure time (year) Water ressure() Pie thickness(t) Constant(a1) The multilier(l) Chlorine decay constant(k) Pie length(a) Pie diameter(d) Toughness correction(a) Tensile strength equation(b1) Toughness exonent(s) Water velocity(v) Ratio of chlorine consumtion(ct/c0) Figure 3.4. Sensitivity factors for inut variables throughout the exosure time using a SORMbased aroach (internal corrosion, failure criteria of F Y ) The cost effective allocation of investments in these activities is time deendent, and will benefit from further analysis of how the chlorine decay coefficient changes as ies age. The ratio of chlorine concentration, Ct/C 0, is significant throughout the exosure time and is deendent on the initial chlorine concentration and the rate of corrosion. Thus, the analysis of chlorine boosting throughout a system, its influence on the rate of internal corrosion and on the chlorine concentration ratio is recommended as a means of further extending ie life. 61

77 Sensitivity factor The results also show that the other variables in the erformance function have a minimal effect throughout the ie exosure time. The data in Figure 3.5 rovide the results of the sensitivity analysis of variables for redicting robability of failure due to the ie wall thickness considering both external and internal corrosion. These results show that two of the variables of the internal corrosion model (K, Ct/C 0 ), and one of the variables of external corrosion model (the scaling constant for corrosion deth, b) are the most significant variables. Variable b has the maximum effect on the external corrosion and this imlies that the future research should invest on the validation of the external corrosion model K V Ct/C0 b Exosure time (year) Internal ressure Pie diameter Wall thickness Toughness corrosion coeff. Constant used for beta Tensile strength equation Multilier coeff. Toughness exonent Chlorine decay rate Water velocity Pie length Ratio of chlorine consumtion Bending moment coeff. Unit w eight of soil Width of ditch Calculation coeff. Modulus of elasticity of ie Deflection coeff. Final itting rate constant Pitting deth scaling constant Corrosion rate inhibition factor Frost load multilier Imact factor Surface load coeff. Wheel load traffic Figure3.5. Sensitivity factors for inut variables throughout the exosure time using a SORMbased aroach (external and internal corrosion, failure criteria of Y ) 62

78 Another imortant variable is the water velocity which reresents the hydraulic condition of the ie segment. While the corrosion rate constant, K, has the most influence on the robability of failure rior to 40 years of ie age, after that age the velocity becomes the most imortant variable throughout the life of the ie. The value of K in early ages is higher because of the higher chlorine consumtion and corrosion rate at these ages. In early ages, a raid growth of the corrosion occurs because any corrosion roducts formed on the ie surface are orous and have oor rotective roerties. After some years, a slow growth of the corrosion leading to a constant corrosion deth occurs due to the strong rotective roerties of the ie. 3.8 SUMMARY AND CONCLUSIONS The water quality variables, e.g., the ratio of chlorine concentration, can affect the ie wall thickness caused by internal corrosion and thus contribute to the ie mechanical failure. Considering the effect of internal corrosion on the ie failure, a robabilistic analysis is used to estimate the robability of mechanical failure over exosure time. The Two-hase external corrosion model is used in this chater and is combined with the internal corrosion model develoed in Chater 2 to estimate the robability of overall mechanical failure of a ie segment throughout the exosure time. The robability of failure throughout the exosure time is estimated for two failure criteria using MCS, and FORM- and SORM-based aroaches. The first criteria reresents the ie failure due to the thinness of the ie wall caused by internal corrosion when the stress due to water ressure exceeds the ie strength caacity and the second criteria reresents ie failure due to the thinness of the ie wall caused by both internal and external corrosion when the total stresses exceed the ie strength caacity. The sensitivity factors are estimated to determine the variables that have the most influence on the reliability index and subsequently on the robability of failure throughout the exosure time. The contribution of chlorine residual, i.e., causing internal corrosion, soil roerties, i.e., causing external corrosion, and hydraulic conditions to the estimates of the robability of failure are investigated. Comared with the all other variables in the environmental and other oerational imosed stresses, the water quality factors in the internal corrosion model, e.g., the ratio of chlorine concentration, the scaling constant in external corrosion model, and the water velocity are the most imortant variables influencing on the robability of failure. This result 63

79 oints to a conclusion that the major effort should be invested in refining the internal and external corrosion models and their related variables. The imortance of the chlorine decay constant in modeling water quality has been extensively studied. The correct measurement of K should be determined to reflect the aroriate internal corrosion rate in the ie network. Therefore, advances in infrastructure asset management may benefit from further analysis of the chlorine decay constant to determine the accurate internal corrosion rate in the ie network. The robability of mechanical failure due to the thinness of the ie wall caused by external and internal corrosion may be used to obtain the likelihood of failure for a given residual ie wall thickness and the number of ie breaks for a given exosure time as surrogates for condition. This information may be useful in identifying the remaining service life and addressing the lanning rocess for the rehabilitation and relacement of water mains in distribution systems. ACKNOWLEDGEMENTS The authors gratefully acknowledge the insights and advice of Professor R. O. Foschi at the University of British Columbia. Financial suort for this research was rovided by a Natural Science and Engineering Research Council of Canada (NSERC) Discovery Grant to the second author, Grant Award Number

80 References Ahammed, M., and Melchers, R.E. (1994). "Reliability of underground ielines subject to corrosion." ASCE Journal of Transortation Engineering, 120 (6). Baeckmann, W., Schwenk, W., Prinz., W. (1997). Handbook of Cathodic Corrosion Protection: Theory and Practice of Electrochemical Protection Processes, Gulf Professional Publishing, 3 RD Edition, ISBN-10, Burmaster, D.E., and Hull, D.A. (1996). "Using lognormal distributions and lognormal robability lots in robabilistic risk assessments." Human and Ecological Risk Assessment, 3(2), Foschi, O., R. and Li, H. (2005). Reliability Analysis Software, Version 7. University of British Columbia, Vancouver, B.C. Kleiner,Y. (1998). "Long-term Planning Methodology for Water Distribution System Rehabilitation." Water Resources Research, Vol.34, No.8, Kirmeyer, G.J., Richards, W. and Smith, CD. (1994). "An Assessment of Water Distribution Systems and Associated Research Needs." American Water Works Association Research Foundation, (90658), Denver, Colorado. Mays, L. (2000). Water Distribution Handbook, Deartment of Civil and Environmental Engineering, Arizona State University, McGraw-Hill. Rajani, B. and Kleiner, Y. (1996). "Comrehensive review of structural deterioration of water mains: hysically based models." Institute for Research in Construction, National Research Council Canada, Ottawa, Ont., Canada K1A 0R6. Rajani, B. and Tesfamariam, S. (2005). "Estimating time to failure of aging cast iron water mains under uncertainties." Institute for Research in Construction, National Research Council Canada (NRC), Ottawa, Ontario, Canada. Rajani, B., and Mc Donald, S. (1995). "Water Mains Break Data on Different Pie Materials for 1992 and 1993." National Research Council of Canada, Reort No. A , Ottawa,ON. Rajani, B., Makar, J., McDonald, S. Zhan, C., and Kuraoka, S., Jen C-K., and Veins, M. (2000). "Investigation of grey cast iron water mains to develo a methodology for estimating service life." American Water Works Association Research Foundation, Denver, CO. Sadiq, R., Rajani, B. and Kleiner, K. (2004). "Probabilistic risk analysis of corrosion associated failures in cast iron water mains." Reliability Engineering and System Safety, 86,

81 Selvakumar, A., Clark, R., Sivaganejan, M. (2002) "Costs for Water Suly Distribution System Rehabilitation", EPA/600/JA-02/404. Snoeyink, V.L. and Wagner, I. (1996). "Internal Corrosion of Water Distribution Systems." Chater 1, (Cooerative Research Reort, Denver, Colo.) American Water Works Associations (ISBN: ). Yamini, H. and Lence, B.J. (2009). "Probability of Failure Analysis due to Internal Corrosion in Cast Iron Pies." Acceted for ublication in ASCE Journal of Infrastructure Systems, IS/2007/ In rint for December Issue. 66

82 CHAPTER 4 Condition Assessment of Cast Iron Pies in Water Distribution Systems A version of this chater has been submitted for ublication. H. Yamini, B. J. Lence, and R. O. Foschi. Condition Assessment of Cast Iron Pies in Water Distribution System. 67

83 PREFACE Most water distribution systems in Canada and the USA are in oor condition and are deteriorating over time (Rajani and Mc Donald, 1995). Estimating the service life of the ies and undertaking long term lanning for their relacement deends on rediction of the future conditions of the water mains in the system. Estimates of the ie condition rovide utilities with a means of rioritizing ie reair, and lanning for rehabilitation, and relacement. In cast iron ies, the mechanical failure of a ie due to internal and external corrosion is a function of the ie wall thickness and thus robability of failure for a given residual ie wall thickness may be considered as an indicator of ie condition. In this chater, the likelihood of failure for a given residual ie wall thickness at a given ie age is obtained and a modeling strategy insired by survival analysis is used to estimate the number of ie failures at that age. The annual number of ie breaks and the likelihood of ie failure for a given residual wall thickness throughout the exosure time are reasonable indicators of the structural state of the water mains in water distribution systems. This analysis relies on the robabilistic analysis of ie failure due to internal and external corrosion conducted in Chater 3 of the thesis. Pelletier et al. (2003) estimate the hazard and survival functions for cast iron ies with diameters of between 100 and 200 mm using ie break data for the municialities of Chicoutimi, Gatineau and Saint-Georges, Quebec. Pelletier (2000) and Mailhot et al. (2000) show that the best-fit function for the hazard function from installation to the first break is a Weibull distribution, and for all subsequent breaks is an Exonential distribution. Pelletier et al. (2003) also determine the survival functions for first breaks, based on their data. In this work, the robability of failure estimates determined in Chater 3 are used to estimate discrete robability density and hazard functions for secific ie diameter (203 mm) and these discrete estimates are fit to a Weibull and Exonential distributions to estimate continuous forms of these functions. The survival function is then estimated based on these continuous functions. The estimated survival function comares well with those determined by Pelletier et al. (2003) based on data for cities having similar ie sizes. An aroach develoed by Pelletier (2000) is then alied, along with the estimated continuous robability density, hazard, and survival functions, to estimate the number of ie breaks that may be exected throughout the exosure time. 68

84 4.1. INTRODUCTION It is well understood that cast iron water mains in North America are in oor condition and are deteriorating raidly (Rajani and Mc Donald, 1995). These ies are also the oldest ies, and have a tendency to corrode externally and internally under aggressive environmental conditions. Corrosion of ies is a major factor in the weakening of ie integrity and leads to several roblems, including ie breaks and loss of hydraulic caacity (Mays 2000). The resulting ie breaks are also otential sources of contamination of the water distribution system. Water utilities are therefore facing the roblem of managing deteriorating networks in the most efficient way ossible to maintain existing and future levels of service. In addition, further exansion of water distribution systems and the need for renewal of ies increase financial stress on resent budgets. Water utilities in Canada and in the USA invest caital and oerating funds to maintain the water mains of distribution systems. Much of this exenditure is to reair or relace aging cast iron mains that have suffered from external and internal corrosion throughout the exosure time. It is estimated that the financial costs for ie reair and relacement exceeds $1 billion annually in Canada alone (Rajani and Tesfamariam, 2005). Also, Selvakumar et al. (2002) reort that the total costs of ie reair and relacement exceeds $3.7 billion annually in the USA. To meet redetermined levels of service for a deteriorating water distribution system, longterm lanning is needed to revent system failures and to reair or relace water mains just before a failure has occurred. The major objective of long-term lanning is to maintain or imrove the system reliability and oeration regularity given a limited budget. The long-term lanning of the renewal of water distribution systems requires the ability to redict system reliability and assess condition. The robability of failure for different ie segments discussed in Chaters 2 and 3 estimates the otential of these ies to break at different exosure times. In order to assess condition, general assessment of the structural behavior of ies is needed, as are tools to estimate the ie failure behavior. To make an assessment, data regarding the characteristics of ies and their breakage histories must be collected. Unfortunately, many municialities only have historical records for a decade, while their ie ages are much greater. So, it may be difficult to trace the deterioration of the ies over time. Also, field-based condition assessment is costly and only rovides information for a ie segment at a articular 69

85 oint in time. There is a need to imrove the level of accuracy in ie condition assessment that does not require a long history of data (De Silva et al. 2002). This chater develos an aroach for assessing condition of water mains throughout the exosure time using the robability of failure due to internal and external corrosion and a modeling strategy insired by survival analysis resented by Pelletier (2000) and Mailhot et al. (2000). The survival analysis is used to redict the failure behavior of water mains throughout the exosure time. The condition assessment aroach may then be incororated into lans of rehabilitation and relacement of ies in water distribution systems. In the following sections, the discrete robability density and hazard functions of ie failures for a secific ie diameter are estimated based on the robability analysis conducted in Chater 3. Then the robability density and hazard functions are fit to Weibull and Exonential distributions to estimate continuous forms of these functions and the continuous survival function is determined based on these functions. The estimated survival functions are shown to comare well with those determined by Pelletier et al. (2003) for communities with ies of a similar diameter. Next, robability density, hazard, and survival functions are used to determine the number of ie failures throughout the exosure time. The likelihood of failure for a given residual ie wall thickness is also estimated as a surrogate for ie condition using a robabilistic analysis. The number of ie failures and likelihood of failure for a given residual ie wall thickness may then be considered as indicators of the structural condition of water mains ESTIMATING THE NUMBER OF PIPE FAILURES DUE TO CORROSION Estimates of the cumulative robability of failure due to corrosion The cumulative robability of failure as a function of exosure time is evaluated based on the results of the analysis conducted in Chater 3. When total environmental, e.g., due to soil, traffic, and frost, and oerational, e.g., due to water ressure, stresses aroach the ie strength caacity, the chance of ie breakage is increased and the overall reliability of a ie segment is reduced. Modeling stresses and internal and external corrosion acting on a ie segment involves uncertainties in the model and its inut arameters. The high degree of uncertainty associated with the arameters that affect ie failures warrants a robabilistic-based analytical aroach. 70

86 The concentration of chlorine residual, as one of the water quality arameters, can influence the internal corrosion rate and contribute to ie mechanical failure. Considering the effect of internal corrosion on the ie failure, a robabilistic analysis is used to estimate the robability of mechanical failure throughout the exosure time in Chater 2. The Two-hase external corrosion model (Rajani et al. 2000) is then combined with the internal corrosion model to estimate the robability of overall mechanical failure of a ie segment throughout the exosure time in Chater 3. The failure criterion is defined as the time at which the stress due to environmental and oerational stresses exceeds the ie strength caacity. The results indicate that the likelihood of failure of a cast iron ie due to internal and external corrosion is nearly 50% by the 45 th year of exosure. Figure 4.1 shows the cumulative robability distribution throughout the exosure time using a SORM-based aroach. All robability estimates are determined using the Relan software, Version 7 (Foschi and Li 2005). P 50 =45 yr. Figure 4.1. Probability of failure due to external and internal corrosion throughout the exosure time, for various statistical distributions using SORM-based aroach (203 mm ie diameter) The data in Figure 4.1 and a modeling aroach insired by survival analysis conducted by Pelletier (2000) and Mailhot et al. (2000) are used in this work to determine the number of ie failures throughout the exosure time. Pelletier et al. (2003) estimate the hazard and survival functions for cast iron ies with diameters of between 100 and 200 mm using a limited history of ie break data for the Quebec communities of Chicoutimi, Gatineau and Saint-Georges. Pelletier (2000) and Mailhot et al. (2000) show that the best-fit function for the hazard function 71

87 from installation to the first break is a Weibull distribution, and for all subsequent breaks, is an Exonential distribution. Pelletier et al. (2003) also determine the survival function for first breaks, based on their data. The cumulative robability of failure data in Figure 4.1 are used herein to estimate robability density and hazard functions and thus the survival functions. Pelletier (2000) develos an aroach for estimating the number of ie breaks throughout the exosure time that integrate the robability density, hazard, and survival functions and this aroach is used herein for estimating this indicator of the condition of a ie segment Estimates of the number of ie failures The robability distribution function, f (T), may be estimated by evaluating the continuous or discrete derivative of the cumulative robability function of ie failure with resect to time. In this chater, the discrete values of the robability distribution function are estimated by evaluating the derivative of the cumulative robability function shown in Figure 4.1, where the derivative is the ratio of the difference between the discrete values of the cumulative robability of failure at times T 1 and T 2 and the difference between discrete values of T 1 and T 2. The discreet value of robability distribution function is defined as Equation 4.1. F( T ) F( T ) 2 1 f ( T2 ) (4.1) T2 T1 The hazard function, or instantaneous failure rate, λ (T), is the robability that a ie exeriences a failure between time T 1 and T 2 given that the ie segment is oerating at time zero and has survived to time T 1 (Kaur and Lamberson 1977). The discrete hazard function may be determined by evaluating Equation 4.2 which relates the hazard function, the robability distribution function and the cumulative robability function, F (T), of ie failure at discrete oints. f ( T ) ( T ) (4.2) 1 F( T ) Pelletier (2000) and Mailhot et al. (2000) determine that the Weibull distribution best describes the eriod of time to failure from ie installation to the first ie break, called the first break order, and that the Exonential distribution best describes the eriod of time to subsequent 72

88 Probability dencity function, f ( T) breaks, i.e., time to failure from the first to second ie break, the second to the third, etc. The Weibull and Exonential distributions are rovided in Equations 4.3 and 4.4, resectively: ( 1 T ) k( kt) (4.3) ( ) k T (4.4) 2 The Weibull distribution is defined by two arameters, k and and the Exonential distribution is a secial case of the Weibull distribution where =1, with only one arameter, k 2. Data in Figure 4.2 show the discrete robability distribution function of ie failure based on the cumulative robability function of ie failure determined in Chater 3 and the best-fit Weibull distribution for these data. Data in Figure 4.3 also show the discrete hazard function for the first order breaks, based on Equation 4.2 and the discrete robability distribution function, as well as the best-fit Weibull distribution for these data. The arameters of the Weibull distribution, k and, are and 0.017, resectively. The estimate of, being greater than one, corresonds to the case where the risk of ie failure is low at early ie ages and increases over the exosure time. The Exonential distribution that describes the hazard function for all subsequent breaks is also fit to these data, resulting in an estimate of k 2 of SORM Analysis Fit t ed W eibull Dist ribut ion Ex osure t im e ( y e a rs) Figure 4.2. Discrete robability density function fit to the Two-arameter Weibull distribution function (=1.552, k=0.017) 73

89 Ha za rd f unct ion Ex osure t im e ( y e a rs) SORM Analysis Fit t ed W eibull Dist ribut ion Figure 4.3. Discrete hazard function fit to the Two-arameter Weibull distribution function (=1.552, k=0.017) Pelletier (2000) estimates the number of ie breaks at a given time, based on continuous estimates of the robability density, hazard and survival functions for the data ublished in Pelletier et al. (2003). Assuming that the reference time T=0 corresonds to the year of ie installation, the number of ie breaks for a given ie segment between time T 1, and T 2, n (T 1, T 2 ), are estimated as: n( T1, T2 ) [ S( T1 ) S( T2 )] k2{ T2[1 S( T2 )] T1[1 S( T1 )] dt t f1( T)} (4.5) The survival function is the comlement of the cumulative robability of failure, and is an estimate of the likelihood of a ie surviving at a given exosure times T 1 and T 2. Kalbfleisch and Prentice (1980) define the survival function, and show that it may be reresented by a Weibull distribution at time T, as: S ( T ) ex[ ( kt) ] (4.6) For the estimate of the continuous hazard function for the first order breaks shown in Figure 4.3, the resulting survival function is shown in Figure 4.4. The estimated number of breaks, for an assumed length of ie of 1000 m, can then be determined based on the estimated robability density and survival functions and these are rovided in Figure 4.5. T T

90 Cumulative number of ie failures Survival function Exosure time (years) Figure 4.4 Survival function based on the hazard function for first order breaks In this analysis, the inut arameters associated with environmental and oerational loads and the structural integrity of the ie due to external and internal corrosion are taken into account in estimating the average number of ie failures. The factors contributing to the number of ie breaks over exosure time are comrised of water quality considerations, e.g., the ratio of chlorine concentration, hydraulic erformance, e.g., water velocity and internal water ressure, external stresses, e.g., stress due to soil ressure, traffic and frost loads, and the rate of internal and external corrosion Exosure time (years) Figure 4.5. Cumulative number of ie failures based on the results of SORM-based aroach fitted to the Weibull and Exonential distributions 75

91 4.3 ESTIMATES THE LIKELIHOOD OF FAILURE FOR A GIVEN RESIDUAL PIPE WALL THICKNESS The robability of mechanical failure of a ie segment at a given ie age is related to its relative residual ie wall thickness, t res, based on the following load- resistance equation, which is based on the modification of work conducted by Rajani et al. (2000), and develoed in Chater 3. The erformance function, G(X), is determined based on the assumtion that failure occurs when the total environmental, e.g., due to soil, traffic, and frost, and oerational, e.g., due to water ressure, stresses surass the ie strength caacity. It is defined as: G( X ) Where ( d * K 3 K I C F 2 q id m c s res m d [1 f ] S 3 3 frost 3 2 int / ) t ext tres n res A [ E tres 3 K d i D ] E tres 3 E t D 3 K B (4.7) ct Ct D V tres t ( d * ext d * int ) t { at b(1 e ) [(1 ) ] T} (4.8) C K L For a given age of ie or time of exosure, if G(X) is less than zero, then the ie is considered to have failed and the robability of failure occurring is: 0 C d K d E i t res D 3 D P (G) 0 (4.9) The cumulative robability of failure, for a 6.1 m length of 203 mm diameter with an initial ie wall thickness of 12 mm, is estimated for a given residual ie wall thickness and a given ie age, and these data are shown in Figure 4.6. This analysis may be alied for different ie diameters to determine how the likelihood of failure for a given residual ie wall thickness varies with ie age and ie diameter. 76

92 Cum ula t ive roba bilit y of f a ilure ( % ) Re sidua l ie w a ll t hickne ss ( m m ) T= 5 y ear T= 10 y ear T= 15 y ear T= 20 y ear T= 25 y ear T= 30 y ear T= 35 y ear T= 40 y ear T= 45 y ear T= 50 y ear T= 55 y ear T= 60 y ear T= 65 y ear T= 70 y ear Figures 4.6. Cumulative robability of failure vs. residual ie wall thickness, for a SORMbased aroach, for different exosure times (ie ages) 4.4 RESULTS AND DISCUSSION The data in Figure 4.3 show the estimated discrete hazard function given the results of the SORM-based analysis for a secific ie diameter. These results indicate that the discrete hazard function declines after the first 40 years of ie age. Given that the frequency of breakage of aging ies is increasing (Sadiq et al. 2004), one might exect the failure rate, or hazard function, to increase due to the ie deterioration throughout the ie exosure. Two imortant assumtions made in this study may have contributed to the decreasing failure rate observed. The first assumtion is that the internal and external corrosion is considered to be a selfinhibiting rocess whereby the rotective roerties of corrosion roducts reduce the corrosion rate over time (Ahammed and Melchers 1994; Snoeyink and Wagner 1996), and then stabilize at a constant rate thereafter. If this were not the case, the corrosion rate may have continued to increase with time, thereby increasing the failure rate. The second assumtion is that the focus of the external corrosion analysis is on a single corrosion it. If on the other hand, the robability of failure was determined based on an assumtion that many external corrosion its were being generated over the length of the ie segment as it ages, Sadiq et al. (2004) observes that the robability of ie failure would increase, causing an increase of the failure. In this 77

93 study, the robability of failure analysis deals solely with the contribution of a single external corrosion it. More research is needed to analyze the formation of corrosion its over the length of the ie segment and thus to define the estimate of failure frequency observed in aging water mains. The data in Figure 4.4 show the continuous survival function at different ie ages estimated based on the Wiebull distribution, i.e., based on the hazard function that reflects the time to failure from installation to first break. The survival function reresents the likelihood of a ie surviving to a given exosure time T. The higher the curve, the longer it takes for the first failure to occur. These data show that at about 10 years after installation, the ie segment has not failed. Data in Figure 4.5 show the estimated cumulative number of ie failures er 1000 m of cast iron ie. The results show that the number of failures increases throughout the exosure time. For examle, after 45 years of ie installation, the cumulative number of ie breaks is about 100. This may be considered as a diagnostic tool for condition assessment because asset managers are facing roblems associated with determining how and what to evaluate, defining what constitutes condition and indices, and determining how to use data related to the condition of the assets. The number of ie failures may also be estimated for ies of different diameters. Data in Figures 4.6 show the cumulative robability of failure for a given residual ie wall thickness for different exosure times, i.e., ie ages. This may be considered as a second indicator of ie condition. For examle, for a ie segment with a 12mm original wall thickness that is 45 years old, the likelihood of failure is about 50% when the residual ie wall thickness is 7.8mm. 4.5 TOWARD ASSESSING PIPE CONDITION AND REPLACEMENT STRATEGIES The goal of infrastructure management is to minimize life-cycle rehabilitation costs for an infrastructure system while meeting an accetable erformance level. Therefore, an economic analysis is essential for determining an otimum relacement time for water mains that are failing mostly due to corrosion, and the budgetary needs for future reairs under various relacement and rehabilitation strategies. To undertake an economic analysis, the following inut data are required: 1) the rojected number of ie failures throughout the exosure time; 2) the cost of reairing one break or 78

94 failure; and 3) the discount rate used in converting future exenditures to resent value (Shamir and Howard 1972). If these inut data are known, utility managers may evaluate: 1) the resent value (dollars) of all future reairs for the existing ie as a function of the ie age; 2) the resent value (dollars) of relacing the existing ie with a new ie as a function of the ie age; and 3) the total life-cycle cost as a function of ie age, which is the sum of the resent values of all future ie reairs and relacements. The otimal year for ie relacement is equal to the year at which the total life-cycle cost is a minimum. The robability of failure for a given residual ie wall thickness and age may be used as a comlementary condition indicator to assess the service life of a ie segment. Deending on discount rates, and costs of reairing a ie, it may be more or less conservative than the results of an economic analysis of the otimal year of ie relacement. For examle, assume that a utility manager would only accet a likelihood of failure of 50% for any age of ie. For a 6.1 m long ie with a 203 mm diameter, and an original ie wall thickness of 12 mm, at the 45 th year of ie age, and a residual ie wall thickness of 7.8 mm, the likelihood of failure is 50% (see Figure 4.6). Therefore if a given ie has a wall thickness of less than or equal to 7.8 mm at this ie age, the utility may choose to reair the ie. If, on the other hand, based on an economic analysis which includes an estimate of the number of ie failures (see Section 4.2), the otimal year of ie relacement is year 50, the utility manager may have waited to relace the ie at its 50 th year. 4.6 MODEL VALIDATION A survey of over 200 municialities in the Province of Quebec reveals that few municialities have maintained comuterized records of ie breaks for longer than a decade (Fougeres et al. 1998). Pelletier et al. (2003) fit these data to Weibull, i.e., time to failure from installation to first break, and Exonential distributions, i.e., time to failure from the first to second ie break, the second to the third, etc, for determining the survival function and redicted behavior of ie breaks throughout the exosure time. The survival function based on the Weibull distribution is resented for the three municialities, where A corresonds to the City of Chicoutimi, B to Gatineau, and C to Saint-Georges, is shown in Figure 4.7. Preliminary analyses conducted by Pelletier et al., (2003) with data from Chicoutimi and Gatineau show that ie segments installed after 1960 have a different ie failure behavior than those installed 79

95 before Therefore, two sets of arameters are estimated for the two municialities. A1 and B1 reresent all ies installed before 1960, and A2 and B2 reresent those installed after 1960, for Chicoutimi and Gatineau, resectively. The reason for this behavior is not well understood, but it may be related to the ie installation technique, variation in soil characteristics, and water quality in the ie over time. Figure 4.7. Comaring the survival functions for three municialities with survival function associated with the Weibull Distribution using the SORM-based aroach Data in Figure 4.7 indicate that ie segments for B2 and A2 grous are at the highest risk for subsequent failures while ie segments in grou C face a moderate, and those in grou A1 face the lowest risk. The results show that the continuous survival function associated with the SORM-based analysis (model) develoed in Chater 3 and estimated based on the Weibull distribution lies within the survival functions for three municialities with different historical records of ie breaks. Figure 4.7 also shows that the survival function derived from this work has nearly the same behavior of the data related to the City of Saint-Georges (C), which has the moderate risk of subsequent failures. 80