This section develops a basis for calculating conditional weld failure probabilities. In the model of piping reliability, the conditional weld failure probability, pL/F, represents the likelihood of a weld flaw propagating to a significant structural failure. However, for Code Class 1 piping there is no service experience data available to support a direct estimation of pL/F. Therefore, the options for calculating the conditional failure probability when the service experience data consists of zero events include applications of (1) probabilistic fracture mechanics modeling and (2) Bayesian modeling. Since the objective of this Base Case was to directly utilize insights from service experience data reviews, the latter approach was selected. Before defining the input to the Bayesian modeling, it is useful to organize the available service experience according to piping classification and severity of observed failures. Figure D.23 shows the conditional pipe failure probability as a function of observed through-wall flow rate for reactor coolant pressure boundary (RCPB) piping (Code Class 1), Code Class 2 and 3 piping, and ASME B31.1 (non-Code) piping.

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Figure D.23 Likelihood of Structural Failure According to Service Experience with

Light Water Reactor Piping System

The empirical data that is used to construct the chart in Figure D.23 represents approximately 7,200 recorded pipe failure events representing almost 9,000 reactor-years of commercial nuclear power plant operation. It is important to note that the chart covers a wide variety of piping systems, from Reactor Coolant Pressure Boundary (RCPB) piping, Safety Injection and Recirculation piping, and Auxiliary Cooling piping to ASME B31.1 piping (or non-Code piping). Figure D.23 represents our current state-of-knowledge with respect to the probability of pipe failure and it provides an upper bound for estimates of conditional pipe failure probability. Noteworthy is the fact that only for ASME B31.1 (non-Code) piping systems have significant structural failures been observed (that is, failures equivalent to double-ended guillotine breaks). For the other classes of piping it is necessary to do data extrapolations beyond the Category 1 through-wall flow rates. In this Base Case study, a Bayesian approach is used for this extrapolation and by acknowledging service data insights. For the Code Class 1 piping, it is also important to recognize that according to the available service experience data, through-wall flow rates greater than 38 lpm (10 gpm) have been primarily attributed to failure of small bore piping. It is unlikely that the structural integrity of the Class 1 piping is equal to or less that that of Code Class 2 and 3 piping for which Category 1 LOCA events have been observed.

The Beta Distribution has some convenient and useful properties for use in Bayes’ updating. The analysis starts by defining a prior distribution that represents the analyst’s understanding of piping performance given the presence of some sort of degraded condition. The prior distribution is defined by selecting an appropriate set of initial values for parameters A(a) and B(P), denoted as APrior and BPrior. Then, when looking at the relevant service experience data, if there are “N” structural failures of a certain magnitude and “M” successes

(or degraded conditions that were repaired before progressing to a structural failure), the Bayes’ updated, or posterior distribution is also a Beta Distribution with the following parameters:

APosterior APrior + N BPosterior BPrior M

The above explains how the Beta Distribution can be used to estimate conditional weld failure probabilities. The challenge is to justify the selected parameters when the evidence is zero structural failures. Certainly, it can be argued the ASME B31.1 service experience data represents a very conservative upper bound for the conditional weld failure probability.

Selecting a well justified set of “A” and “B” parameters is not a trivial task. One basic ground rule should be for the “weight” of the field experience data to determine the shape of the posterior Beta Distribution. However, many different parameter combinations will produce the same predicted mean value. Where very little evidence is available about the parameters, constrained non-informative priors may be selected. For such a case, one can say that the “A” parameter has to be a small number.

In this Base Case study, the prior “A” and “B” parameters are defined by first deriving a constraint for the prior mean value of the conditional failure probability and then fixing the “A” parameter at 1.0 for stress corrosion cracking and 2.0 for thermal fatigue to account for the fact that according to available service experience data, thermal fatigue cracks propagate in the through-wall direction considerably faster than flaws caused by stress corrosion cracking. The process for developing conditional failure probabilities starts by deriving a point-estimate of pLF for small-diameter piping given susceptibility to stress corrosion cracking (SCC). This point estimate is based on Jeffrey’s non-informative prior and service experience data. There have been 42 through-wall flaws and zero large leaks in small-bore BWR piping. This gives a point estimate of 1.2×10-2, which is used as a “fix point” for determining conditional weld failure probabilities for other pipe sizes. The relationship between pipe size (diameter and wall thickness) and the conditional failure probability is assumed to follow a power law of the form:

Plif = a x DNb (D.8)

Where, “DN” is the nominal pipe size in [mm]. Decreasing trends correspond to negative values of b. Parameters a and b are determined for pLF = 1.2×10-2 and DN = 25. Point estimates of pLF for other pipe sizes are derived using the power law for conditional failure probabilities and assuming that the general shape of the curve is similar to that of piping susceptible to vibratory fatigue for which pLF = 2.5/DN. Next the predicted conditional weld failure probability using the power law approach is used to determine the Beta Distribution parameter “B.” For Class 1 piping, engineering judgment, as portrayed by Figure D.23 is used to assign values to the prior Beta Distribution parameters. The proposed Beta Distribution posterior parameters for this Base Case study are summarized in Table D. 14.

Table D.14 Proposed Beta Parameters for Code Class 1 Piping

Degradation Mechanism	Pipe Size	Parameter B in Beta Posterior (“Large Leak”)
DN	NPS
SCC (APrior = 1)	300	12	1,262 (APost = 1; M = 0)
550	22	1,496 (APoSt = 1; M = 0)
700	28	1,700 (APoSt = 1; M = 0)
Thermal Fatigue (APrior = 2)	90	3-3/4	227 (APost = 2; M = 0)
350	14	592 (APost = 2; M = 0)

D.5.3 Conditional Failure Probability and Flow Rate

The conditional failure probabilities derived in the previous section are assumed applicable to Cat0 LOCA. It is furthermore assumed that for a significant primary piping breach to occur there has to be a through-wall flaw coinciding with a plant operational mode change or an unusual or severe loading condition such that the leakage exceeds a Cat0 LOCA. The service data collection (e. g., PIPExp) includes numerous examples where pressure pulses or spikes caused by changing flow conditions following a plant operational mode change have resulted in non-active leaks11 becoming active leaks. The physics of such transitions from non-active to active leaks are complex and location-dependent (e. g., function of flaw size and pipe stresses). Some published work exists on the correlation between crack propagation and plant transient history [D.25]. Using available empirical data, the uncertainties in such crack growth assessments are considerable, however.

In this analysis a simple parametric approach is applied to the estimation of weighted conditional failure probabilities (CL) of a pressure boundary breach that exceeds a Cat0 flow rate threshold value. This approach is described through the event tree in Figure D.24. An undetected, or detected but monitored through-wall is exposed to a pressure pulse or unusual loading condition before a decision to perform manual, controlled reactor shutdown. The pressure pulse or unusual loading condition is characterized as a subjectively defined probability distribution.

Through-wall Defect (Cat0)	Unusual Load	Unstable Crack Growth	LOCA Category
Moderate і—————————————————	Cat1
		Cat0
	Moderate-to-high		Cat2
			Cat0,Cat1
	High |——————————————————	Cat3
	1—————	Cat0,Cat1
	Very High |—————————————————	Cat4
	1————————————	Cat0,Cat1
	Severe і—————————————————-	Cat5
	1————————————	Cat0,Cat1
	Extreme і—————————————————	Cat6

Cat0,Cat1 —1

Figure D.24 Event Tree for Definition of LOCA Categories

In the cases of “moderate-to-high” to “extreme”, the term “unusual” implies a loading condition beyond that resulting from anticipated transients including manual and automatic reactor/turbine trips. The conditional probability of an unusual or severe loading condition is described by five sets of subjective 3-bin discrete and overlapping probability distributions as summarized in Table D.15. These DPDs are combined with the weld failure rate distributions and conditional weld failure probability distributions by using a Monte Carlo merge technique. [11]

Category	Flow Rate (v) Intervals [gpm]	DPD for Severe Loading
CL-Hk1i	CL-Med	CL-Low	PHl2h	pMed	pLow
0	10 < v < 100	N/A	N/A	N/A	N/A	N/A	N/A
1	100 < v < 1500	.80	.50	.20	.2	.6	.2
2	1500 < v < 5000	.32	.20	.08	.2	.6	.2
3	5000 < v < 25,000	.13	.08	.03	.2	.6	.2
4	25,000 < v < 100,000	.05	.03	.01	.2	.6	.2
5	100,000 < v < 500,000	.02	.01	.005	.2	.6	.2
6	v > 500,000	.01	.005	.002	.2	.6	.2

Service data on water hammer events provides a justification for the chosen DPDs. From PIPExp, a point estimate for CL-WH-Cat6 is approximately 4.9E-03, which is based on two events involving severe overloading (including plastic deformation) of a pipe section in 411 recorded water hammer events. This is taken as a best estimate CL-value for calculating a Cat6 LOCA. Figure D.24 includes the rules for how the DPDs are applied to the LOCA frequency calculation. The Cat0 and Catl LOCAs include contributions from each loading condition associated with Cat2 or larger pressure boundary breach. In other words, the calculation accounts for the possibility that an ‘unusual’ loading condition may not result in a global or catastrophic pressure boundary breach. Given a through-wall flaw and severe overload, Figure D.25 shows the conditional failure probability as a function of pipe size.

Figure D.25 Conditional Probability of Weld Failure Given Through-Wall Flaw and Severe

Overloading