SCIENCE

Statistical Modeling of Pitting Corrosion and Pipeline ReliabilityA.K. Sheikh,* J.K. Boah,** and D.A. Hanen***

ABSTRACTPitting corrosion is treated as a time-dependent stochastic damage process characterized by an exponential or logarithmic pit growth. Data from water injection pipeline systems and from the published literature are used to simulate the sample functions of pit growth on metal surfaces. Perforation occurs when the deepest pit extends through the thickness of the pipe. Because the growth of the deepest pit is of stochastic nature, the time-to-perforation is modeled as a random variable that can be characterized by a suitable reliability model. It is further demonstrated that this time-to-first-leak is often distributed according to the Weibul) extreme value reliability model. KEY WORDS: extreme value and Weibult models, leak growth, pipeline reliability, pitting corrosion, probability models, statistical models. be used to determine the probability that a specific depth wilt occur or the number of observations that must be made to find a particular pit depth. Aziz 3 used extreme probability paper in the study of the pitting of aluminum alloys. Aziz's data when plotted on extreme value probability paper produced straight lines, indicating an extreme value distribution. By extrapolating from the plotted lines, one can make certain predictions. For example, with Aziz's two-week data it can be shown that the probability of obtaining a pit depth of 760 .m or less is 0.999 and that the probability of obtaining a pit greater than 760 m is only 1 in 1000, even though the observed deepest pit depth was 580 m. In a recent paper, Ishikawa et al. 7 discussed the validity of the extreme value analysis of plain carbon steel pipe carrying river water for cooling water turbines. This is analogous to water injection piping systems, where water used in the injection system may be more corrosive than the river water referred to in the analysis of Ishikawa et al. The extrapolation of pitting data obtained from the laboratory or from a small portion of the structure (such as a small section of a pipeline) to large-scale field installations has also been successfully carried out with the aid of extreme value statistics by Eldredge, Hawn, 5 and Aziz. 8 In the literature on pitting corrosion, there is substantial experimental evidence indicating that the dimensions of pits (of any type) at a given instant of time and in any of a variety of environments may be characterized by the log normal distribution. For example, the results of extensive studies by Flaks" on the size distribution of the pits on aluminum alloys under various atmospheric conditions suggest the validity of the log normal distribution. Summerson and others 12 used a normal distribution to analyze pit depth measurements on aluminum. By plotting the square root of the pit depth, \/f, against frequency, a reasonably normal distribution was obtained. From these curves, the mean square root of the pit depth, the standard deviation, and the probability error could be determined. This approach requires the measurement of many pit depths. Once it is established that a normal frequency distribution is a function of the square root of the pit depth, a sampling procedure can be developed that will allow the measurements of a smaller number of randomly selected pits to give a predetermined confidence level. The variability of corrosion rate in a given pipeline system can also be characterized by the normal distribution model.

INTRODUCTIONIn pipeline pitting corrosion data analysis, the quantities involved are corrosion rate, R; pit depth, h; maximum pit depth, D,,,; and time-to-leak, T. These quantities are random variables and show scatter around their average values: R, h, D,,, a,,, and T. Each of these random variables can be modeled by a suitable probability model. A number of statistical correlations have been proposed to quantify the distributive parameters involved in pitting.' -10 Gumbel' -2 is given credit for the original development of the extreme value statistical approach to pit-depth distribution. The procedure is to measure maximum pit depths on several replicate specimens that have pitted, then arrange the pit depth values in order of increasing rank. A plotting position F(D R ) for each order of ranking is necessary. This is obtained by substitution in the relationship F(D R ) = R/(n + 1) where R is the order of ranking, and n is the total number of specimens or values. For example, the plotting position for the second value of 10 would be 2/(10 + 1) = 0.1818. These values are plotted on the ordinate of extreme value probability paper vs their respective maximum pit depths. If a straight line is obtained, it shows that extreme value statistica apply. Extrapolation of the straight line canSubmitted for publication May 1988; revised December 1988. ' Mechanical Engineering Department, King Fand University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia. " Carrier Corporation, Carrier Parkway, P.O. Box 4808, Syracuse, NY 13221. M.W. Kellogg Company, Houston, TX.

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0010-9312/90/000059/$3.00/0 1990, National Association of Corrosion Engineers

CORROSION March 1990

S CIEN C E

STATISTICAL MODELSPlausible statistical models for random quantities, such as the normal distribution, log normal distribution, exponential distribution,

extreme value Gumbel distribution, and Weibuil distribution,'' 2,13will be used in modeling D m ., T, R, and N. These models are described using a general notation of random variable X, which can

obtained with hazard rate X = rj 1 . Furthermore, as 13 increases, the Weibuil distribution tends to approximate the normal distribution. At (3 = 3.57, this approximation is the best. In the case of X = 0, Equation (5) reduces to a two-parameter Weibuil model, where the scale parameter, i, also represents the "characteristic value" of the random variable X.

be replaced by R, h,

Dmax,

N, or T according to the context of the

MODELING OF PIT-DEPTH DISTRIBUTIONThe typical pitting of metals develops in the period when a moisture film remains on the metal surface, It can therefore be

discussion. The statistical distributions are described below: 1. The normal (Gaussian) distribution is a two-parameter probability density function represented by f(x) = (1/ y \/) exp[-(X - ) 2/2v 2 ] where and o are parameters equal to the mean standard deviation a x , respectively. 2.x ( 1)

considered that the depth of the pit h, recorded at time

T,

has suc-

cessive values h, < h 2 < ... < h at various stages of growth according to the total time that the metal surface is wet.

Let the increase in pitting depth at theand

ith

wetting be randomly

proportional to the strength of the independently distributed exter-

nal influences Ki and to the depth of the pitting h._, at the moment of wetting the metal surface, then h, - h ; _, = K i f(h,_,), where K, isthe intensity of the external influences depending on the depth of the electrolyte film, its evaporation rate, and the composition of

The log normal probability density function is represented by f(x) = 1/(Va X) exp{-[ n(X) - ] 2/2a for X > 02 ]}

dissolved gases; f(h (2)

1)

is an intensity function that determines the

amount of corrosion and depends on the condition of the metallic phase, the presence of anodic and cathodic inclusions, and protective layers of corrosion producers. Thus

where

Q

is the shape parameter and is the scale pan nKi

rameter or "characteristic" life. 3. The exponential probability density function is represented by the expression

n

i=,

1

= 1 (hi - hi-,)/f(hi-,) = 1 i=,

Ah i-,

i=, f(h;_,)

(8)

f(X) = 1xe -Txfor X za 0= 0 otherwise

(3)

Assuming that f(hEquation (8) as

1

) = h, and that Ah, - 0, we can express

where parameter k is a real, positive constant. E K,i=1

= f ho" dh/h = n hh

n

- n h o = n(h n /h o )

(9)

4. The extreme value (Gumbel) distribution F(Xmax) f' f(X ma )dX ma , is represented by F(Xmax) = DL )( max - Xmax] = P[Y Y] = F(Y) F(Y) = exp[-e ']-

(4)

The distribution of the sum

Ki

when n is sufficiently large (i.e., with

protracted exposure of the specimens) can be considered, with the most common assumptions concerning the distribution law of the

for -co < Y < m, whereY = a(X max - X 0 ) is the reduced variate,

independent random quantity K ; , to be normal. (This is the centra/ limit theorem.) Consequently, the quantity log h is distributed asymptotically normal or the pitting depth h is distributed logarithmically normal. Figure 1 illustrates the histogram of the pitting depth, h,

X max = the largest extreme values being analyzed, X. = the most probable value of the distribution or mode, 1/a = measure of variance (v 2 ) given as (V6 /ir) of the standard deviation of the distribution, i.e.,a = (-rr/)/a.

which was observed by Flaks, 11 when 760 samples of an aluminum alloy were exposed for seven years in the atmosphere of an open-hearth shop main building. This experimentally obtained histogram was found to be represented by a log normal model, by

using the method of moment and X 2 test. An important feature ofthe log normal model is its marked skewness. The log normal distribution approaches an exponential distribution when the scatter in data increases, i.e., when the coefficient of variation of data reaches 1, (K -+ 1, which means v - 0). This corresponds to the situation where the mode is very close to the origin. For example, Figure 2(b), where the distribution shows a long upper tail, illustrates such a situation, where both log normal and exponential models can be used to characterize the pit distribution. For clear distinction to establish the superiority of one of the model, it is necessary to use statistical tests for goodness of fit. However, Figure 2(a) is clearly represented by an exponential distribution. It is essential to point out that the histograms (Figures 1 and 2) and their fitted models represent the initial (or parent) distributions of the pit depths. The extreme value selected from each sample represent extreme value data, and method of modeling such a data is discussed below.

5. The Weibull distribution is a three-parameter probability model represented by the expression F(X) = 1 - exp{-[(X - X )/n]a} wheren >0,(3>0,0