CORROSION—1: Study helps model buried pipeline pitting corrosion

New deterministic and stochastic models can perdict the evolution of pitting corrosion depth and rate distributions from observed soil properties. Researchers performed a total of 259 excavations in 3 years gathering data that serve as the basis for these models.

Background

Recent reports place pipeline corrosion costs in Canada and the US at $7 billion/year.¹ Corrosion is the second and third leading cause, respectively, of pipeline failures in North America and Europe.^2-4 Pitting corrosion causes a higher percentage of failures than other corrosion mechanisms.

Corrosion pitting caused roughly 70% of external corrosion-related hazardous liquid spills and gas leaks in cross-country European oil and gas pipelines from the early 1970s through the mid-2000s, according to incidents reported to the European Gas Pipeline Incident Data Group (EGIG) and Conservation of Clean Air and Water in Europe (CONCAWE).^{3 4} In the US, pitting corrosion caused 79% of total incidents reported to the Office of Pipeline Safety (OPS) for onshore transmission oil and gas pipelines between 2002 and 2008.²

In Mexico, a recent study identified about 99% of external corrosion-related incidents reported between 1994 and 2004 as pinhole-type leaks produced by corrosion pitting (OGJ, Nov. 27, 2006, p. 60).

Its high incidence has caused multiple modeling studies to focus on corrosion pitting of underground oil and gas pipelines.^5-11 Given that localized corrosion is extremely difficult to model on the basis of electrochemical reactions, statistical approaches have been used instead of electrochemical principles in developing predictive models for pitting corrosion in underground pipelines.

Available statistical approaches for modeling pitting corrosion use pit depth as the dependent corrosion variable and various environmental factors as the independent (predictor) variables.^5-9 Three key elements underlie these models: the law governing the time dependence of pit depth, the data available on environmental factors, and the model used to correlate the measured pit depth to these factors.

Despite the scientific and technical progress in modeling pitting corrosion in underground pipelines, several issues relevant to this damage mechanism remain and should be properly addressed. The need for more accurate predictive models for pit growth fully based on pipeline and soil characteristics, the need for a better understanding of the statistical properties of pitting depth and rate distributions, and the need for holistic approaches exploiting the symbiosis between the deterministic and stochastic models describing pipeline corrosion-pitting in soils all continue to guide pipeline engineers.

This article summarizes the results of recent study aimed at addressing these needs. Part 1, presented here, describes new deterministic and stochastic predictive models for external pitting corrosion in underground pipelines. The deterministic model considers local chemical and physical properties of soil as well as pipeline coating to predict the time dependence of pitting depth and rate in a range of soils.

Researchers used this model, based on results from a field study, to conduct Monte Carlo simulations establishing the probability distribution of pitting depth and growth rate in the studied soils and their evolution over the pipeline lifetime.

In the last stage of the study, researchers developed an empirical Markov chain-based stochastic model for predicting the evolution of pitting corrosion depth and rate distributions from the observed properties of the soil. Part 2 (next week) will illustrate these models using real-life case studies involving the Monte Carlo simulated and experimental pit depth distributions.

Field study

The values of the soil and pipeline variables listed in Table 1 show a 3-year period at 259 excavation sites at underground pipelines operating in southern Mexico for up to 50 years.¹² External corrosion-caused metal losses with a diameter equal to or less than two times WT qualified as pits. In each exposed pipeline segment in situ measurements (redox potential, pH, pipe-to-soil potential, and soil resistivity) determined maximum pit depth and the value of selected local soil properties.

Soil samples taken from the excavation site determined the rest of the properties considered in the model (soil texture, water content, bulk density, and dissolved chloride, bicarbonate, and sulfate ion concentrations) in a laboratory setting. Pipeline age and coating type also made up part of the collected data sets. Experimental details regarding this field study are published elsewhere.¹²

Data analysis

The weight ratio of the three soil separates (sand, silt, and clay) determined soil texture.¹³ Textural classes used by the US Department of Agriculture categorized the soil samples. Identified soil categories included:

Clay (110 samples).
Sandy clay loam (79 samples).
Clay loam (61 samples).
Silty clay loam (6 samples).
Silty clay (2 samples).
Silt loam (1 sample).

Only the first three categories contained a large enough number of samples to ensure statistical significance. Standard exploratory statistical tools allowed analysis of the distribution of measured variables, examined the correlation between them, and determined the presence of outliers. Table 2 shows the results of the statistical fitting of measured data for all observed samples (all) and for clay (c), sandy clay loam (scl), and clay loam (cl) samples.

A scoring model previously reported by the authors accounted for the influence of pipe coating type on pit growth.12 Practical criteria quantifying the general body of practical experience reported in the literature on the susceptibility of pipeline coatings to failure provide the basis for this model (Table 3).^{9 13 14}Table 3 also gives the probability of occurrence of each score used in the Monte Carlo simulations.

Depth model

Maximum pit depth was modeled using Equation 1 as a power function of the pipeline service time (t).¹²

Researchers performed a multivariate regression analysis with dmax as the dependent variable and exposure time and soil and pipe characteristics as the independent (predictor) variables. Equations 2 and 3 provided the best correlation between the proportionality and exponent parameters and the predictor variables.

Table 4 gives the adjusted regression coefficients, ki and ni, for each soil category. It also reports the value fitted for the pit initiation time (t₀) in each soil class. The pit depth model described by Equations 1-3 is more sensitive to pH value, pipe-to-soil potential, pipe coating type, bulk density, water content, and the dissolved chloride content, in that order.

A Monte Carlo simulation¹² yielded unbiased estimates of the model parameters (kt and nt) in each soil class for typical conditions. Table 4 also shows the values of k_t and n_t.

Fig. 1 shows the time evolution of the average maximum pit depth under average (typical) conditions in each soil class. Equation 1 together with the information presented in Table 4 allowed creation of these curves. Table 4 and Fig. 1 show the order of corrosivity predicted by the proposed model as clay > clay loam > sandy clay loam. These findings agree with those from previous works on pitting corrosion of low carbon steel in soils.^{5 7 8 15}

Rate model

The time derivative of Equation 1 for the maximum pit depth yielded the functional form of the corrosion pitting rate. Accordingly, Equation 4 models time dependence of the average pit growth rate of deeper pits (v_m(t)).

This approach has an advantage in that the value of the proportionality and exponent parameters of Equation 4 can be determined from the adjusted regression coefficients in Table 4. The resulting values for of k' and n' are shown in Table 4. Together with the value of t₀ in this table they allow derivation of an empirical formula for the time evolution of the average pitting rate for the four soil categories.

Fig. 2 shows the evolution of the average pit growth rate with pipeline service time for a 40-year span of each soil class. For long exposure periods (30 years and longer), the corrosion rate in all soil classes attains a slightly decreasing value, which is unique for each soil category, and shows a relative variation of less than 10% over a 10-year period, beginning at 30 years. In pipeline integrity management practice pitting rate values within such limits are considered relatively constant.

Probabilistic distributions

Study of the probability distributions of the maximum pit depth and growth rate used Monte Carlo simulations.16 Equations 1-4, used together with information from Tables 2-4, simulated these distributions in each soil class for different service times. Researchers drew random values of the predictor variables from the probability distributions fitted to the observed corrosion data (Table 2). Random predictors evaluated pitting depth and rate models 5,000 times.

For each soil class, equally spaced, 2.5-year apart, simulation times simulated Year 5 to Year 40. In most cases, generalized extreme value (GEV) distribution proved the best model for fitting the distributions obtained by means of the Monte Carlo framework.¹⁷ The value of the shape parameter (z) of the fitted GEV distribution determined whether the Weibull (z < 0), Gumbel (z = 0), or Fréchet (z > 0) distribution best fit the simulated data.

Fig. 3 shows the time evolution of nonparametric estimates of the variance of the Monte Carlo-simulated maximum pit depth distribution in each soil class. This figure illustrates the stochastic nature of pitting corrosion in all soil categories. The larger the exposure time is, the higher the uncertainty in the estimates as measured by the variance of the maximum pit depth distribution. Fig. 3 also reveals the variability in the predicted maximum pit depth increasing with soil corrosivity.

Fig. 4 shows the time evolution of the nonparametric estimates of the variance of the Monte Carlo-simulated pitting rate distribution for each soil category. For long-term exposure, the variance of the pitting corrosion rate shows a slightly decreasing trend with time, attaining a relatively constant value differing considerably from one soil to another. Pipeline corrosion literature has already reported this result.⁹

For sufficiently long exposure, the deepest pits are theoretically the most likely to grow, such that the growth rate of the deepest pits largely determines the range of pitting rates. These grow with a steadily decreasing rate because of the stability attained during the long-term diffusion-controlled pit growth. Other factors, such as the buildup of corrosion products, also reduce and stabilize the variability of the pitting corrosion rate as exposure time increases.

Monte Carlo simulations are especially suitable for investigating the shape of the pitting depth and rate probability distributions. The temporal variation of the shape parameter of the GEV distribution fitted to the simulated distributions ascertained which of the three extreme value distributions (Weibull, Gumbel, or Frechet) best fits the maximum pit depth distribution during the lifetime of the pipelines in each soil class.

For relatively short exposure periods, the Weibull distribution is the most appropriate for description of the simulated extreme pit depth data. For long-term exposure, the Gumbel distribution and then the Fréchet distribution become the best fitting distributions. This is the case for all studied soil categories and may explain why different distribution types have been reported for maximum pit depth in previous studies.^{7 10 11}

Fig. 5 illustrates how the shape parameter of the fitted GEV distribution varies with exposure times for the all soil class.

The increase in the shape parameter of the GEV distributions fitted to the Monte-Carlo-simulated maximum pit-depth distributions can be associated with the fact that, for sufficiently long exposures, the deeper pits are the most apt to continue growing. This causes the maximum pit depth distribution to skew toward large pit depths and the tail at large depths to become more extensive, as Fig. 5 shows. This behavior also occurs for the micropitting regime of aluminum alloys in tap water¹⁸ and for line pipe steel in laboratory-simulated soil corrosion conditions.¹⁹

The simulations also revealed the shape parameter of the maximum pit depth distribution as different for different soil categories and that the higher the soil corrosivity and the longer the exposure time, the larger the value of the shape parameter. Other authors previously reported this result in a practical extreme value analysis of steel storage tank pitting.²⁰ Fig. 6 shows how the type of extreme distribution varies from one soil class to another for a 20-year service lifetime.

The increase in the variance of the distribution of maximum pit depths with exposure time observed in Fig. 3 seems to be related to the sustained growth of the deepest pits. The difference in the growth behavior of the deepest pits with respect to the rest of the pit population increases with the corrosivity of the soil. This explains why the Frechet distribution becomes the best fit for the distribution of maximum pit depths earlier in clay soils than in the rest of the soil categories.

In the case of the corrosion pitting rate (as in the case for the maximum pit depth), the Weibull, Gumbel, and Frechet models become the best fitting distributions as the exposure time increases. The Frechet distribution best fits the pitting rate data for exposures longer than 20 years in all soil environments considered in this study. The previously explained long-term stability of pit growth explains these results. A further detailed discussion of this subject can be found elsewhere.¹⁶

Fig. 7, for example, shows how the GEV distribution fit to the Monte Carlo-simulated pitting rate distributions evolves with time for the sandy clay loam soil class. After long-term service times, the deepest pits in the pipeline are the most likely to grow, but they do so with a slightly decreasing rate (Fig. 2). The lower rate limit is zero, concentrating the mass of the pitting rate distribution at the lower rate values (left) with increased exposure time.

Concurrently, the higher growth rates of new and short-life shallow pits, though less numerous, contribute to the increased importance of the upper tail of the distribution. Therefore, the variance of the pitting rate distribution becomes more dependent on the less frequent extreme growth rate of the short-life pits. The Fréchet distribution provides the best model to fit the right-skewed, leptokurtic Monte Carlo-simulated distributions of long-term pitting rate data.

Based on these results, the long-term (≥25 years) the Frechet distribution reasonably approximates pit growth rate in the soils considered in this work, with parameters and cumulative probability functions (CDFs) shown in Table 5 and Fig. 8, respectively.

The 80% probability bound (x₈₀) for the pit growth rate obtained by the present study is lower than those published elsewhere.^{9 21 22} Taking the all soil class as an example, Table 5 shows the value of x₈₀ as 0.087 mm/year (3.5 mils/year). This value is four times less than the NACE recommended 80% upper bound of 0.4 mm/year (~16 mils/year)²¹ and three times less than the maximum rate of 0.31 mm/year (~12 mils/year) proposed in ASME B31.8S code.²²

The differences between the value of x₈₀ for the all soil category in Table 5 and the NACE and ASME recommended bounds for pit growth rate are easily explainable. The results of up-to-17-year corrosion tests of cathodically unprotected, bare steel coupons in a range of soils provide the basis for the NACE and ASME recommendations.²¹ These conditions contrast with those of the pipelines considered in this study, which are under the influence of more realistic operating conditions with moderated potential for corrosivity.

Table 2 presents information corroborating studied pipelines as having been exposed to weakly-acidic soil environments with a relatively high resistivity (~5,000 W–cm), a redox potential above 100 mv, and a pipe-to-soil potential (natural or imposed) close to the optimum level of cathodic protection.

Markov chain model

A continuous-time, non-homogenous linear growth (pure birth) Markov process²³ modeled pipeline corrosion pitting.²⁴ This model assumes the pipe wall is divided into N discrete states and pit depth at any point in time t can be represented by a discrete random variable D with probability P {D(t) = i} = p_i(t), i = 1, 2….N. It also assumes the probability that if damage at the i-th state jumps one state during a short interval of time ∆t can be written as λi(t)∆t + o(δt).

For a linear growth Markov process with intensities λi(t) = iλ(t), Equation 5 governs the probability pitting damage presently in state i will be in state j (j ≥ i) at some later time.²³

The solution to this equation provides the conditional probability of transition of the pitting damage from the m-th to the ℓ-th state (ℓ ≥ m) in the time interval (t_i,t); that is, p_m,ℓ(t_i,t) = P {D(t) = ℓ |D(t_i) = m}. Equation 6 depicts this probability in closed form.²³

Equation 6 describes the probability of increasing pit depth from m to ℓ in an interval of length t – t_i as following a negative binomial distribution NegBin(r,p) with parameters r = m and p = p_s. The authors recently have shown that when the power function given by Equation 1 accurately describes the time evolution of maximum pit depth, the probability ps in Equation 6 can be obtained using Equation 7.²⁴

Knowing the pitting corrosion initiation time (t₀) and the pitting exponent factor (n) allows prediction of the evolution of corrosion pitting damage in a given pipeline from the measured (or assumed) initial pit depth distribution (p_m(t_i)) and the observed local soil and pipe characteristics.

For example the results of in-line inspection of the pipeline can predict p_m(t_i) with ti as the time of the inspection. In addition, soil and pipe properties can estimate t₀ and n, using Equation 3 together with information shown in Tables 2 and 4. Equation 6 together with Equation 8 can determine the probability of pitting damage depth at any future time, p_l (t), once p_m(t_i) has been obtained and ps has been determined from t₀ and n.²⁴

Furthermore, p_{m, l} (t_i,t), allows an estimate of the probability distribution of the damage rate f (v; m, t_i, t) associated with the pitting corrosion process over the interval of length ∆t = t – t_i, when the damage depth is at the m-th state in the moment of time t_i. Equation 9 provides this estimation.

Using Equation 10 from f (v; m, t_i, t) provides a straightforward estimation of the probability distribution of the damage rate associated with the entire pipeline pit population.

Acknowledgments

The authors acknowledge the work done by the members of field inspection crews, who gathered the pitting corrosion data on which this study is based.

References

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The authors

Julio C. Velazquez is an associate professor at the Instituto Politecnico Nacional (IPN) of Mexico. He holds a BS in chemistry and an MS in physical metallurgy from IPN and currently is pursuing a PhD at the IPN in topics related to pipeline corrosion.

Francisco Caleyo ([email protected]) is a full professor at the IPN. He is a senior researcher at the Convenio de Investigacion y Desarrollo de Integridad Mecanica (CIDIM) of the IPN. He holds a BS in physics and an MS in materials science from Universidad de La Habana, and a PhD in materials science from Universidad Autonoma del Estado de Mexico.

Alma Valor ([email protected]) is an assistant professor at University of Havana. She is currently an invited researcher at the CIDIM. She holds a BS and MS in physics and mathematics from Lomonosov Moscow State University and a PhD in advanced technology from IPN.

Victoria Venegas ([email protected]) is a postdoctoral fellow at the CIDIM of the IPN. She holds a BS from Universidad de La Habana and an MS and PhD in materials science from IPN.

Jose Hiram Espina Hernandez is a professor of IPN. He is a researcher at the CIDIM of the IPN. He holds a BS in electrical engineering, an MS in automation and a PhD in technical sciences from the Instituto Superior Politecnico Jose Antonio Echeverria of Havana.

Jose Manuel Hallen ([email protected]) is a full professor of the IPN and heads CIDIM, which has service contracts with Pemex to conduct mechanical integrity and risk analyses of onshore and offshore pipelines. He holds a BS and MS in physical metallurgy from IPN and a PhD in physical metallurgy from the University of Montreal.

Maximino Ricardez Lopez ([email protected]) is a senior pipeline integrity specialist with Petroleos Mexicanos (Pemex). He currently works as pipeline integrity vice-manager at Pemex Exploracion y Produccion, Region Sur. He holds a BS in electrical engineering from Universidad Veracruzana and an MS in pipeline management engineering from Universidad de las Americas.

Correction

In the articles “US construction plans slide; pipeline companies experience flat 2003, continue mergers,” by Warren R. True and Jeannie Stell, and “US gas carriers see 2004 net jump; construction plans rebound,” by Christopher E. Smith, Warren R. True, and Jeannie Stell, Table 7 (OGJ, Aug. 23, 2004, p. 64 and OGJ, Sept. 12, 2005, p. 64, respectively) contained errors. The tabulation below and the one on the previous page are corrected versions of the affected portions of each table.