*Article* **Calibrating the Impressed Anodic Current Density for Accelerated Galvanostatic Testing to Simulate the Long-Term Corrosion Behavior of Buried Pipeline**

**Yoon-Sik So 1, Min-Sung Hong 1, Jeong-Min Lim 1, Woo-Cheol Kim <sup>2</sup> and Jung-Gu Kim 1,\***


**Abstract:** Various studies have been conducted to better understand the long-term corrosion mechanism for steels in a soil environment. Here, electrochemical acceleration methods present the most efficient way to simulate long-term corrosion. Among the various methods, galvanostatic testing allows for accelerating the surface corrosion reactions through controlling the impressed anodic current density. However, a large deviation from the equilibrium state can induce different corrosion mechanisms to those in actual service. Therefore, applying a suitable anodic current density is important for shortening the test times and maintaining the stable dissolution of steel. In this paper, to calibrate the anodic current density, galvanostatic tests were performed at four different levels of anodic current density and time to accelerate a one-year corrosion reaction of pipeline steel. To validate the appropriate anodic current density, analysis of the potential vs. time curves, thermodynamic analysis, and analysis of the specimen's cross-sections and products were conducted using a validation algorithm. The results indicated that 0.96 mA/cm<sup>2</sup> was the optimal impressed anodic current density in terms of a suitable polarized potential, uniform corrosion, and a valid corrosion product among the evaluated conditions.

**Keywords:** galvanostatic test method; underground infrastructure; long-term corrosion; carbon steel

#### **1. Introduction**

With the recent development of general industry, the demand for various types of pipeline has increased. Numerous infrastructures have been built in downtown underground areas. Most of these underground infrastructures are aimed at achieving long-term use, since they are generally difficult to maintain or replace. Therefore, it is essential to verify the long-term corrosion behavior of buried metallic structures.

In fact, despite the various developments, underground structural failures continue to occur [1–3]. Here, the corrosion of metallic infrastructures in underground soil is a major issue that presents numerous safety and economic concerns [4–8]. In short, pipelines can be damaged by corrosion, which can lead to the failure of the structures within a soil environment. However, it is difficult to detect the failure of a large underground system, which means understanding the long-term corrosion behavior is crucial to mitigating unpredictable failures. As such, various studies have been conducted on the long-term corrosion mechanism for metals in a soil environment. The majority of these studies involved the use of immersion tests to analyze the corrosive characteristics of the metals [9–12]. However, obtaining the results of this type of test requires a long period of time (at least several months). It is also difficult to maintain the same environmental conditions during the entire test period, making it difficult to yield reproducible results. Since an immersion test is not suitable for evaluating the long-term corrosion properties of materials, an appropriate acceleration test must be considered. Here, electrochemical acceleration methods

**Citation:** So, Y.-S.; Hong, M.-S.; Lim, J.-M.; Kim, W.-C.; Kim, J.-G. Calibrating the Impressed Anodic Current Density for Accelerated Galvanostatic Testing to Simulate the Long-Term Corrosion Behavior of Buried Pipeline. *Materials* **2021**, *14*, 2100. https://doi.org/10.3390/ ma14092100

Academic Editor: Vít Kˇrivý

Received: 30 March 2021 Accepted: 20 April 2021 Published: 21 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

present the most effective approach for simulating long-term corrosion. Among the various methods, galvanostatic tests allow for accelerating the surface chemical reactions through controlling the impressed anodic current density. It is thus a suitable method for long-term corrosion studies.

The impressed anodic current density and the time can be calculated using Faraday's law [13]. Applying an appropriate anodic current density is the key factor here, since the expected corrosion reaction cannot be achieved if the impressed density is too high [14]. Meanwhile, accelerated corrosion testing methods must allow for shortening the test time and inducing the same mechanism of degradation as that in actual service. However, there exists no international standard for these tests. With this in mind, this study was aimed at providing an academic standard for anodic current density that can be applied to accelerate and simulate the long-term corrosion of metals in a soil environment. Thus, a potentiodynamic (PD) polarization test was conducted to determine the corrosion current density for carbon steel in synthetic groundwater. To determine the most appropriate anodic current density, galvanostatic (GS) tests were performed at four different anodic current densities using an acceleration period that represented the one-year corrosion reaction of a specific pipeline. Each anodic current density value was determined according to the corrosion current density, which was obtained from the PD polarization measurements and through the use of Faraday's law. Meanwhile, an optical microscope (OM) was used to observe the surface and cross-section morphologies of the specimens, while X-ray diffraction (XRD) analysis was used to confirm the species of each oxide following the galvanostatic experiments.

#### **2. Materials and Methods**

#### *2.1. Specimen and Solution*

The specimen was cut into cuboid-shaped pieces with dimensions of 10 × <sup>10</sup> × 5 mm3. The chemical composition of the tested pipeline steel is presented in Table 1 (ASTM A 139), while the composition of the synthetic soil solution is presented in Table 2. The results were obtained from three soil environment sites close to an operating pipeline.

**Table 1.** Composition of the tested steel specimen.


**Table 2.** Composition of the tested solution.


#### *2.2. Electrochemical Analysis to Optimize the Impressed Anodic Current Density for GS Testing*

To evaluate the corrosion resistance of pipeline steel, a PD polarization test was conducted using a multi-potentiostat/galvanostat instrument (VMP-2, Bio-Logic Science Instruments, Seyssinet-Pariset, France). Meanwhile, a three-electrode cell was constructed using pipeline steel as the working electrode (WE), two pure graphite rods as the counter electrode (CE, Qrins, Seoul, Korea), and a saturated calomel electrode as the reference electrode (RE, Qrins). Prior to conducting the electrochemical tests, the specimens were abraded with 600-grit silicon carbide paper. These prepared steel surfaces were then covered with silicone rubber, leaving an area of 0.25 cm2 unmasked before they were exposed to a synthetic soil solution at 60 ◦C under aerated conditions, and then rinsed with ethanol and finally dried using nitrogen gas. Prior to all the electrochemical tests, the specimens were immersed in a test solution for 3 h to attain a stable surface. The PD polarization measurements were performed at a potential sweep of 0.166 mV/s from an

initial potential of −250 mV vs. an open circuit potential (OCP) up to a final potential of 0 VSCE. A GS test was then performed to accelerate corrosion reaction of steel after 3 h of OCP measurements at four different anodic current densities. Each of the impressed anodic current density values was determined according to the corrosion current density, which was obtained from the PD polarization measurements.

#### *2.3. Surface Analysis*

An OM (SZ61TRC, Olympus Korea Co., Seocho-gu, Seoul, Korea) was used to observe the surface morphology and the cross-section of each specimen, while each oxide product was analyzed via XRD (Rigaku Ultima III X-ray diffractometer, Tokyo, Japan) analysis with Cu Kα<sup>1</sup> radiation (λ = 1.54056 Å) over a 2θ range of 20◦–70◦, using a step-size of 0.017◦ and a step-time of 1 s, to confirm the species of each oxide following the galvanostatic experiments.

#### **3. Results and Discussion**

#### *3.1. Corrosion Behavior of Pipeline Steel in a Synthetic Soil Solution*

The PD polarization curve related to a synthetic soil solution at a pH of 6.4 and a temperature of 60 ◦C is shown in Figure 1, while the PD results are summarized in Table 3.

**Figure 1.** The PD polarization curve of steel in a synthetic soil solution at pH 6.4 and a temperature of 60 ◦C.


To calculate the corrosion current density, the Tafel extrapolation method (as described in the equation below) was applied. Here, Equation (1) describes the linear relationship between the over-potential and the log scale current density [14,15]:

$$\mathbf{u}\mathbf{\dot{u}} = \mathbf{a} \pm \beta\_{\mathbf{a},\mathbf{c}} \log|\mathbf{i}| \tag{1}$$

where a = −βalog(i0) or βclog(i0), β<sup>a</sup> ∼= (RT/(1 − α)nF) is the Tafel slope of the anodic polarization curve, β<sup>c</sup> ∼= (RT/αnF) is the Tafel slope of the cathodic polarization curve, i0 is the exchange current density, α is the charge transfer coefficient, n is the charge number, R is the gas constant (8.314 J/[mol·K]), and T is the absolute temperature [K]. From Equation (1), a linear relationship was derived, and the corrosion current density was measured as approximately 27.96 μA/cm<sup>2</sup> according to the Tafel extrapolation in the PD polarization curve (Figure 1). No passivation behavior of the anodic polarization

curve was observed, which means that in a synthetic soil solution, pipeline steel will be homogeneously corroded.

#### *3.2. Corrosion Acceleration Using the GS Method*

The mass loss of pipeline steel can be calculated for each current and experiment time by applying Faraday's law. The mass loss by PD test for the one-year corrosion of pipeline steel is given in the following Equation (2):

$$\text{cm} = \text{ita/nF} = \text{(27.96 } \mu\text{A/cm}^2 \times 1 \text{ year} \times 55.84) / \text{(2} \times 96500 \text{ C)} = 0.255 \text{ g/cm}^2 \tag{2}$$

where m is mass loss, i is corrosion current density, F is Faraday's constant (96,500 C/ equivalent), n is the number of equivalents exchanged, a is the atomic weight, and t is time [16]. To accelerate the one year of corrosion, the impressed anodic current density and test time were set accordingly. The impressed anodic current densities were selected to investigate a wide range as possible, starting from the maximum output (24 mA, 3435.29 times faster than corrosion rate) of the potentiostat instrument (VMP-2) before being reduced to 0.024 mA (3.43 times faster than corrosion rate) in 1/10 stages. Meanwhile, the exposure times were also determined to maintain the same theoretical metal loss for each test. With the reduction in exposure time, the acceleration coefficient, which is the ratio between impressed anodic current density and corrosion current density, increased sharply. All of these variables are detailed in Table 4.

**Table 4.** Experimental conditions and calculated variables.


The logarithmic shape of the curves in Figure 2 follows Equation (3), known as the Sand equation [17]. This equation defines the quantitative relationship between the impressed anodic current density and time:

$$(\text{i}\pi^{1/2})/(\text{C}\_0\text{\*}) = (\text{nFD}\_0^{1/2}\pi^{1/2})/2\tag{3}$$

where i is the impressed current density, τ is the transition time, C0\* is the bulk concentration of the reactant, n is the coefficient number, and D0 is the diffusion coefficient of the reactant. When the impressed current can no longer be supported by the intended metal dissolution reaction, the potential changes to an alternative electron-transfer reaction [17]. However, in most of the accelerated GS corrosion tests, the reactants were the metal itself. Therefore, the probability that the concentration of the reactants reaches zero is extremely low. In Figure 2, since the transition point could not be observed, i.e., the potential vs. time curve exhibited a logarithmic curve shape, the GS tests were appropriately performed under all conditions. Nevertheless, there were differences in the stability of the curve shape for each condition. As shown in Figure 2a, the wavering potential curve was recorded only at the slowest reaction rate of 0.096 mA/cm2. When the reaction rate is slower, oxides will have more opportunity to adsorb onto the electrode surface. Hence, it can be expected that there will be an obstruction of the reaction area comprising the laminated oxide layer. Consequently, in this experiment, the 0.096 mA/cm2 condition appeared to be invalid for accelerating a homogeneous corrosion using the GS method.

**Figure 2.** Potential (VSCE) vs. time (h) curves during the GS test at 60 ◦C. The shape of the graph follows the Sand equation. (**a**) Entire curves with breaks; (**b**) graph from 0 to 250 h; (**c**) graph from 0 to 2.5 h.

Meanwhile, the measured potential of each condition was different depending on the impressed anodic current density. As the impressed anodic current density increased, the WE potential increased to more positive values and was generated at a higher current during a relatively short period of time, with the initial and final values of the measured potentials shown in Figure 3a. Here, it was clear that the measured WE potential depended on the impressed anodic current density. Therefore, thermodynamic analysis was then conducted to verify the electrochemically accelerated reaction. The reactions and the Nernst equations that primarily occur in the Fe-H2O system at 60 ◦C are described in terms of

Equations (4)–(6) [13], while these are also presented in terms of a pH-potential diagram in Figure 3b.

$$\text{Fe}^{2+} + \text{e}^- = \text{Fe}, \varepsilon\_{\text{[Fe2+/Fe]}} = -0.199 + 0.033 \log[\text{Fe}^{2+}], \text{[V}\_{\text{SCE}}] \tag{4}$$

$$\text{Fe(OH)}\_{3} + 3\text{H}^{+} + \text{e}^{-} = \text{Fe}^{2+} + 3\text{H}\_{2}\text{O}, \text{pH} = 6.65 - 0.5 \log[\text{Fe}^{2+}] \tag{5}$$

Fe(OH)2 + 2H+ = Fe2+ + 2H2O, <sup>ε</sup>[Fe2+/Fe(OH)3)] = 1.298 <sup>−</sup> 0.177 pH <sup>−</sup> 0.066 log[Fe2+], [VSCE] (6)

In Figure 3b, the metal ions appeared to be stable in the range of –0.397 to 0.561 VSCE when the soluble ion activity was 10−<sup>6</sup> at pH = 6.4 and 60 ◦C, as shown in the Pourbaix diagram. Meanwhile, as Figure 3a shows, the 0.096 and 0.96 mA/cm2 current densities had initial and final potentials of −0.651 and −0.244 VSCE, −0.597 and −0.206 VSCE, respectively, while the 9.6 mA/cm2 current density had an initial potential of −0.295 and a final potential of −0.153 VSCE. These three conditions indicated the region where the metal ions were both stable and sensitive to corrosion (see Figure 3b). As such, the corrosion of carbon steel will be accelerated accordingly. However, the 96 mA/cm<sup>2</sup> current density had extreme potentials, with an initial potential of 0.648 VSCE and a final potential of 0.960 VSCE. In this potential range, a metal-dissolution reaction is no longer stable, meaning the thermodynamic stability of the pipeline steel will change from iron to iron oxide/hydroxide ions, based on the Pourbaix diagram [18]. In addition, an oxygen-evolution reaction may occur close to 1 VSCE. As such, the 96 mA/cm2 condition, as a current value for accelerating the corrosion process, cannot be reasonably accepted.

**Figure 3.** (**a**) Initial and final potentials during the GS test, and (**b**) Pourbaix diagram of an Fe-H2O system (activity of Fe2+: 10−6, 60 ◦C).

Meanwhile, Table 5 shows the parameters following the accelerated tests for corrosion over a one-year period. During the GS test, the electrode was exposed to a water-based solution, with the primary reaction being the production of electrons via metal dissolution. However, a self-corrosion reaction also occurred [19], which consumed the electrons generated by the dissolution of iron from the WE surface [20]. Due to this self-corrosion reaction, the generated electrons were not completely transported to the CE; rather, they reacted with the water and oxygen at the WE's surface. As a result, the mass loss in the GS acceleration test became larger than the theoretical mass loss calculated using Faraday's law. As such, in electrochemically accelerated tests, the self-corrosion ratio in relation to the corrosion rate could be around 20–30%, as shown in Table 5. Nevertheless, the electrochemical acceleration test had the advantage of a significantly reduced testing time. For example, while with the 0.96 mA/cm<sup>2</sup> condition, a 28.9% self-corrosion ratio was indicated, the testing time was generally reduced from one year to around 10 days. Thus, the electrochemical acceleration method still has merits despite the self-corrosion aspect.


**Table 5.** Measured parameters of the one-year corrosion-accelerated specimen as a function of impressed anodic current density.

<sup>1</sup> Corrosion rate: calculated values from mass loss by GS test. <sup>2</sup> Self-corrosion rate: difference between corrosion rate by PD test and GS test. <sup>3</sup> Self-corrosion ratio: (self-corrosion rate/corrosion rate) × 100.

#### *3.3. Analysis of Surfaces*

Cross-sectional images of the tested specimens obtained using the OM are shown in Figure 4. Most of the images clearly indicated a uniform corrosion. However, at the lowest anodic current density of 0.096 mA/cm2, different behavior, which indicated localized corrosion, was observed. As shown in Figure 2, an unstable potential was recorded at the anodic current density of 0.096 mA/cm2. As noted above, when the reaction rate is slower, oxides will have more opportunity to adsorb onto the electrode's surface. Hence, most of the iron oxides produced via the self-corrosion reaction covered the reaction area at the anodic current density of 0.096 mA/cm2. Consequently, an accelerated corrosion reaction occurred at the localized site of the narrow reaction area, due to the layer of iron oxide formed on the surface.

**Figure 4.** Cross-sectional images following the galvanostatic test at different applied current densities: (**a**) 0.096 mA/cm2; (**b**) 0.96 mA/cm2; (**c**) 9.6 mA/cm2; (**d**) 96 mA/cm2.

Meanwhile, phase analysis of the oxides under each anodic current density condition was conducted using the XRD instrument, with the major phase of the oxides shown in Figure 5. Here, the corrosion product of the lower anodic current density group was mainly comprised of goethite. However, the higher anodic current density condition (96 mA/cm2) showed different XRD peaks, which represented ferrihydrite and magnetite. According to the existing literature, the main iron oxide phase in general soil environments is goethite [10,20–24]. Furthermore, the XRD results at the 0.096 and 0.96 mA/cm2 conditions were similar to those obtained for actual environment iron oxides in previous studies. Therefore, these conditions accurately reflected the long-term corrosion behavior of buried pipeline.

**Figure 5.** The XRD results following the galvanostatic tests based on current density: G: goethite (α-FeO[OH]); F: ferrihydrite (Fe2O3·0.5H2O); M: magnetite (Fe3O4).

#### *3.4. Validation of the Galvanostatically Accelerated Testing*

To establish an appropriate impressed anodic current density for the accelerated test, a validation process was conducted for all tests, based on the process shown in Figure 6. This process was initially implemented to assess the inflection points in the potential vs. time curves observation, and thus to confirm the effect of the environment during the test. In the galvanostatic test, all the curves exhibited a stable logarithmic shape, indicating that the pipeline steel reacted well. Meanwhile, the validation process involved comparing the measured potentials with the Pourbaix diagram. Here, the majority of the conditions are within a thermodynamically stable Fe ion range. However, the 96 mA/cm2 condition departed from this range and was thus excluded from the valid current density range. Next, the validation process was continued in terms of the cross-sectional images of the steel pipeline specimen to confirm the presence of uniform corrosion, which was indeed clearly confirmed by the majority of the images. However, the lowest anodic current density (0.096 mA/cm2) indicated different corrosion behavior (localized corrosion) and was thus also excluded from our determination of a valid anodic current density. Finally, the validation process involved analyzing the corrosion product using XRD analysis. Here, 0.96 mA/cm<sup>2</sup> was found to be the optimal anodic current density for reproducing long-term corrosion behavior.

**Figure 6.** Flowchart showing the validation process for the GS accelerated testing.

#### **4. Conclusions**

In this study, the impressed anodic current density for a GS test aimed at determining long-term corrosion behavior was evaluated using an electrochemical test, OM observation, and XRD analyses. To verify the accelerated test, analysis of the potential vs. time curves, thermodynamic analysis, and analysis of the cross-sections and products of the specimen were performed. During the GS test, based on the laminated oxide layer, the most unstable potential form was recorded at the slowest reaction rate (0.096 mA/cm2). Meanwhile, the highest anodic current density (96 mA/cm2) demonstrated extreme potentials that were out of the Fe ion's stable range. The majority of the OM images clearly indicated uniform corrosion. However, the slowest condition at the anodic current density of 0.096 mA/cm2 indicated localized corrosion. The XRD peaks at the 0.096 and 0.96 mA/cm<sup>2</sup> conditions corresponded to a corroded buried pipeline product (goethite), while the 9.6 and 96 mA/cm<sup>2</sup> conditions indicated the presence of different oxides. In conclusion, the anodic current density of 0.96 mA/cm<sup>2</sup> was found to be the most suitable for conducting the GS acceleration testing of carbon steel in a soil environment. Based on this finding, an appropriate

validation process was established for an accelerated corrosion test aimed at predicting long-term corrosion lifetimes. Furthermore, it is expected to help determine the reliable impressed anodic current density by applying the validation process to design accelerating metal corrosion.

**Author Contributions:** Conceptualization, Y.-S.S. and W.-C.K.; methodology, Y.-S.S. and J.-M.L.; validation, M.-S.H.; investigation, J.-M.L. and W.-C.K.; writing—original draft preparation, Y.-S.S.; writing—review and editing, M.-S.H. and J.-G.K.; supervision, J.-G.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the program for fostering next-generation researchers in engineering of National Research Foundation of Korea (NRF), funded by the Ministry of Science and ICT (2017H1D8A2031628). This work also was supported by an NRF grant funded by the Korean Government (NRF-2020-Research Staff Program) (NRF-2020R1I1A1A01074866).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article material.

**Acknowledgments:** This research was supported by the Korea District Heating Corporation (No. 0000000014524).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Structural Integrity of Steel Pipeline with Clusters of Corrosion Defects**

**Maciej Witek**

Gas Engineering Group, Warsaw University of Technology, 20 Nowowiejska St., 00-653 Warsaw, Poland; maciej.witek@pw.edu.pl

**Abstract:** The main goal of this paper is to evaluate the burst pressure and structural integrity of a steel pipeline based on in-line inspection results, in respect to the grouping criteria of closely spaced volumetric surface features. In the study, special attention is paid to evaluation of data provided from the diagnostics using an axial excitation magnetic flux leakage technology in respect to multiple defects grouping. Standardized clustering rules were applied to the corrosion pits taken from an in-line inspection of the gas transmission pipeline. Basic rules of interaction of pipe wall metal losses are expressed in terms of longitudinal and circumferential spacing of the features in the colony. The effect of interactions of the detected anomalies on the tube residual strength evaluated according to the Det Norske Veritas Recommended practice was investigated in the current study. In the presented case, groups of closely-spaced defects behaved similarly as individual flaws with regard to their influence on burst pressure and pipeline failure probability.

**Keywords:** steel pipeline; in-line inspection results; interacting corrosion defects; structural integrity

#### **1. Introduction**

Degradation of underground steel structures during their service lives leads to occurrence of volumetric surface defects and reduction of the tube wall thickness, as it is shown in Figure 1. The steel pipelines are buried almost on whole their length, and the properties of the soil are the most important factor of the corrosion; however, there are many other parameters such as an influence of the straight currents. A corrosion rate of high pressure steel pipelines needs to be controlled by their operators during the maintenance. Periodic in-line inspections (ILI), using an axial excitation magnetic flux leakage technology (Figure 2), are usually performed by gas transmission grid operators to detect and size the tube wall metal losses during the certain time intervals. If more than one diagnostic survey is performed at on a steel pipeline, so-called defect matching can be performed in order to evaluate the growth of the corrosion in the specific maintenance conditions [1]. Direct application of theoretical fracture mechanics methods to the assessment of the volumetric features provided from the in-line inspection is not appropriate due to uncertainty of the in-line inspection tool results highlighted by the author in [2]. The steel pipe wall burst was analysed by the author in [3]. However, a lot of studies deal with investigation of strength and structural integrity of steel pipelines with wall metal losses and longitudinally-oriented grooves similar to cracks using different methodologies, for instance, the latest publications applying a finite element method [4] and a linear elastic fracture failure mode [5]. A rupture pressure prediction model for steel tubes affected by the stray current corrosion based on artificial neural network was applied in [6].

**Citation:** Witek, M. Structural Integrity of Steel Pipeline with Clusters of Corrosion Defects. *Materials* **2021**, *14*, 852. https://doi.org/10.3390/ ma14040852

Academic Editor: Vít Kˇrivý Received: 10 January 2021 Accepted: 4 February 2021 Published: 10 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** A photograph of corrosion colonies on the steel pipe surface. Author's source.

**Figure 2.** Axial excitation magnetic flux leakage inspection technology: a diagnostics tool (at the top) and a measurement principle (at the bottom). Author's source.

Many research projects, in which the failure behaviour and assessment of cylindrical shells containing adjacent corrosion indications were investigated, have been carried out worldwide over the past 50 years. From those studies the conclusions are as follows. The interaction between each pair of metal losses within a group is governed by several parameters, namely, spacing, tube outside diameter, wall thickness, depth of anomalies, and shapes of defects. Among these factors, the distance between each pair of volumetric flaws is most significant. Rules of interaction of pipe wall metal losses are generally expressed in terms of longitudinal and circumferential spacing of the features in the cluster and are studied in many publications, for example [7]. The authors applied a finite element method to find new interaction rules of corrosion flaws for longitudinal and circumferential aligned metal losses as well as to compare these rules to the available grouping standards. However, the failure pressure of a cluster of closely spaced corrosion pits is generally smaller than the rupture pressures in the case when the defects are considered as isolated. This reduction in the corroded pipe strength occurs due to the possible interaction between the adjacent tube wall metal losses.

The influence of the results of analysis of the anomaly colonies on the rupture of the pipe subjected to the internal pressure was evaluated in the present study. There are two main subjects of interest in the investigation of closely spaced interacting corrosion defects. The first issue is the rules of grouping the metal losses, whereas the other one is prediction of the failure pressure of the pipe with adjacent defects. This research is focused on the assessment of tube wall volumetric surface flaws in respect to the criteria of interaction. In the present paper, DNV-RP-F101:2010 Det Norske Veritas Recommended practice [8] standard is applied to find defects which can be considered as clusters within the population of the external surface metal losses taken from the diagnostics data.

The mechanical reliability of the underground infrastructure within long-term operation, counted in decades, can be analysed as a stochastic process of random degradation of the structure elements. The algorithms calculating the probability of failure can be divided into methods based on functions of random parameters [9] and a theory of random variables which is applied in the current paper similarly as in [10,11]. There are plenty of publications which analyse the time dependent structural integrity relying on the stochastic models and on the results of the in-service diagnostics as well. Many publications, for example, [12], as well as the author's work [13], calculate the failure probability considering defects as single isolated corrosion pits. However, a few publications focus on the interacting corrosion areas and, for this reason, this problem is the subject of the current research. The aim of the present study is to calculate the failure probability of the steel pipeline failure based on the in-line inspection data containing groups of indications. A limited number of publications focus on the multiple colonies of corrosion features and, for this reason, this issue is the subject of the current research. The following words: group, colony, cluster are synonyms and were used exchangingly in the text.

#### **2. Clustering Rules for Tube Wall Metal Losses**

There are two main subjects of interest in the investigation of the interacting pipe wall metal losses. The first issue are coincidence rules, whereas the other one is prediction of the failure pressure of a tube with adjacent colonies of defects. Section 2.1 presents the grouping criteria for the pipe wall volumetric anomalies detected by the axial excitation magnetic flux leakage in-line inspection tools. Section 2.2 contains the standardized assessment level 2 methodology for burst pressure calculations of a steel pipe with the interacting metal losses.

#### *2.1. Grouping Criteria for Volumetric Defect Colonies*

According to the Benjamin Adilson's classification presented in [14], there are three types of interactions of volumetric features caused by corrosion. Type 1 is a group of metal losses separated circumferentially; however, their individual profiles overlap when projected into the longitudinal plane through the wall thickness. Type 2 is found in colonies in which the flaws are or are not longitudinally aligned and their individual profiles do not overlap when projected onto the longitudinal plane through the wall thickness, i.e., their projected individual profiles are separated by the length of the full pipe wall-thickness. Type 3 is a combination of 2 above mentioned types. The described types are graphically shown in Det Norske Veritas Recommended practice [8].

The rules of interactions establish the limit value of the distance between two individual defects in the colony, beyond which the interaction is negligible. For the purpose of analysis of interactions, the longitudinal spacing *sl* and the circumferential distance *sc* of each metal loss in the group are usually verified. A majority of the currently available rules of clustering the corrosion flaws adopt expressions containing two following conditions to be met [8]:

(a) longitudinal spacing along the pipe axis is less than *sLi* ≤ (*sl*)lim

$$(s\_l)\_{\rm lim} = 2.0 \sqrt{Dt} \tag{1}$$

and

(b) circumferential spacing *sci* ≤ (*sc*)lim

$$(s\_\varepsilon)\_{\text{lim}} = \pi \sqrt{Dt} \tag{2}$$

where: *sli*—longitudinal spacing of each defect in the colony, [mm]; *sci*—circumferential spacing of each anomaly in the group, [mm]; *D*—tube outside diameter, [mm]; *t*—pipe wall nominal thickness, [mm].

If the cluster is composed of more than two metal losses, the rules of interaction are applied to all possible pairs of adjacent defects within the group and the above mentioned criteria are verified. In the present study, the evaluation rules applied to the repeated in-line inspection data of gas transmission pipeline are presented.

#### *2.2. Analytical Assessment of Interacting Defects*

Det Norske Veritas Recommended practice Corroded pipelines level 2 methodology was developed for the calculation of the burst pressure of thin-walled cylindrical shells with interacting volumetric surface colonies and is applied in the current paper. The combined length of the interacting metal losses is calculated as follows [8]:

$$l\_{nm} = l\_m + \sum\_{i=n}^{i=m-1} (l\_i + s\_i) \tag{3}$$

where:

*li*—axial length of each interacting metal loss from *n* to *m*, [mm];

*si*—circumferential distance of each interacting metal loss from *n* to *m*, [mm].

*lnm*—combined length of defects in the longitudinal direction [mm].

The effective depth of the combined flaw formed from all *n* to *m* of the interacting metal losses is calculated as follows [8]:

$$d\_{nm} = \frac{\sum\_{i=n}^{i=m} d\_i l\_i}{l\_{nm}} \tag{4}$$

where:

*dnm*—effective depth of the metal loss combined from *n* to *m*, [mm];

*d*i—circumferential depth of each interacting metal loss from *n* to *m*, [mm];

The rupture pressure of the steel pipe with the combined corrosion colony formed from all *n* to *m* of the interacting metal losses is calculated as follows:

$$P\_{burst} = \frac{2tf\_{\text{fl}}\left(1 - \frac{d\_{\text{nm}}}{t}\right)}{(D - t)(1 - \frac{d\_{\text{nm}}}{tM\_{\text{nm}}})} \tag{5}$$

where:

*Pburst*—burst pressure of the steel pipe with the combined defect formed from all *n* to *m*, [MPa];

*fu*—ultimate tensile strength of the steel, [MPa];

*Mnm*—Folias factor, [–], corresponding to the combined length of defects in the longitudinal direction is expressed as:

$$M\_{nm} = \sqrt{1 + 0.31 \left(\frac{l\_{nm}}{\sqrt{Dt}}\right)^2} \tag{6}$$

#### **3. In-Line Inspection Indications Clustering**

In order to illustrate the methodology for the interactions of defects described in Section 2, the clustering criteria were implemented to evaluate the possible material loss colonies along the studied gas transmission grid. The case study considers a 711 mm outer diameter cathodically protected pipeline of the total length of 147.267 km and the tube wall thickness of 10.5 mm. The maximum operating pressure value of all the pipeline sections is *MOP* = 5.5 MPa. The material used is equivalent to L360NE steel grade, according to EN-ISO 3183, with the following average parameters: ultimate tensile strength *σ<sup>U</sup>* = 554.7 MPa, average yield stress *σ<sup>Y</sup>* = 370 MPa and average elasticity modulus of steel *E* = 202 GPa. The pipe as well as the mechanical properties of the girth weld were confirmed by the destructive tests conducted on the steel coupons taken from the pipeline after 13 years of operation, directly after the first diagnostic [15]. Two in-line inspections were performed with the use of axial excitation magnetic flux leakage diagnostic tools. A period of time between the first and the second survey was 12 years. In this paper, all analyses are based on the real data for the external metal losses of the tube wall. Relying on the report of the first ILI, the number of 1347 external corrosion pits were found on the outside surface of the wall. Based on the data of the second ILI, the amount of 2838 external surface flaws were found. The number of 72 external closely spaced metal losses taken from the second inspection data were classified as groups for detailed considerations. For the research purpose, three of groups were described below in details and shown in Figures 3–5. The limit of longitudinal spacing along the pipe axis in the studied case needs to be less than *sL* ≤ (*sL*)lim = 176.9 mm and circumferential spacing *sc* ≤ (*sc*)lim = 277.8 mm. Dimensions of the defects from group 1 are summarized in Table 1 and the graphical presentation of features spacing is shown in Figure 3. Description of defects in colonies are as follows: C1D1—cluster 1, defect 1.

**Figure 3.** Graphical presentation of volumetric defects in colony 1. Source: Author's analysis.

**Figure 4.** Graphical presentation of metal losses spacing in group 2. Source: Author's analysis.

**Figure 5.** Graphical presentation of defects spacing in group 3. Source: Author's analysis.


**Table 1.** Anomaly dimensions in cluster 1. Source: Author's analysis.

Group 1 contains four flaws separated circumferentially; however, their individual profiles overlap when projected onto the longitudinal plane through the wall thickness (interaction type 1), as it is illustrated in Figure 3. Due to the limit of circumferential spacing for each pair of metal losses of the studied pipeline (*sc*)lim = 271.5 mm, only two indications, C1D2 and C1D4, can be considered as interacting with the combined length of *lnm* = 27.5 mm and the maximum depth of 22% of the pipe wall thickness which corresponds to 2.42 mm. The effective depth of the external metal loss combined from C1D2 and C1D4 calculated according to Equation (4) is equal to *dnm* = 3.43 mm.

Dimensions of the defects from group 2 are summarized in Table 2. Cluster 2 contains six indications divided into two subgroups whose individual profiles overlap in the circumferential as well as in the longitudinal plane through the wall thickness (interaction type 1 and type 2), as it is shown in Figure 4. Due to the limit of circumferential spacing for the studied pipeline (*sc*)lim = 271.5 mm and a value of longitudinal spacing along the pipe axis less than *sL* ≤ (*sL*)lim = 172.8 mm of each pair, only two anomalies, C2D3 and C2D4, can be considered as interacting with the combined length of *lnm* = 29.0 mm and the maximum depth of 31% of wall thickness which corresponds to the value of 3.41 mm. The effective depth of the metal loss combined from C2D3 and C2D4 calculated according to Equation (4) is equal to *dnm* = 5.10 mm.

**Table 2.** Metal loss sizing in group 2. Source: Author's analysis.


Dimensions of the metal losses from cluster 3 are summarized in Table 3. Colony 3 is the group of five anomalies separated circumferentially; however, four individual profiles overlap when projected onto the longitudinal plane through the wall thickness (interaction type 1), as it is illustrated in Figure 5. Due to the limit of circumferential spacing for the considered pipeline (*sc*)lim = 271.5 mm and a value of longitudinal spacing along the pipe axis less than *sL*≤ (*sL*)lim = 172.8 mm of each pair, only the two flaws C3D3 and C3D4 can be considered as interacting with the combined length of *lnm* = 33.5 mm and maximum depth of 22% of pipe wall thickness which corresponds to the value of 2.42 mm. The effective depth of the metal loss combined from C3D3 and C3D4 calculated according to Equation (4) is equal to *dnm* = 4.0 mm.


**Table 3.** Flaws dimensions in the colony 3. Source: Author's analysis.

The results of the plastic burst pressure calculations according to Equations (5) and (6), in the three analysed cases of colonies of the volumetric defects, are summarized in Table 4.

**Table 4.** The results of the pipe wall plastic collapse calculated according to DNV-RP-F101 standard. Source: Author's analysis.


From Table 4, it can be concluded that rupture pressure calculated according to Det Norske Veritas Recommended practice for every analysed group of flaws is in the difference range of 2 bar compared to the assessment of the individual features. The results of analytical calculation of burst pressure of the defected pipes, taking into consideration interactions of metal losses obtained from the in-line inspection, show a little influence of closely spaced corrosion flaws on the burst pressure of the studied case.

#### **4. Probability of Pipeline Rupture**

In order to estimate a failure probability in a long operation time, the limit state based on a pressure difference between the plastic collapse of the remaining pipe wall thickness and an expected value of the gas working pressure as a random variable were employed.

#### *4.1. Calculation Methodology*

The limit state function of the plastic collapse of the *j*-th pipeline section affected by a part-wall reduction caused by corrosion is expressed as:

$$\text{g}(\stackrel{\rightarrow}{X}) = \begin{array}{c} \text{O}\_{f\text{j}} \ - \text{ O}\text{P}\_{\text{max}} \end{array} \tag{7}$$

where:

*g*( → *X*)—limit state function of the tube wall plastic collapse;

→ *X*—vector of random variables related to the pipeline segment;

*Pf j*—failure pressure of the j-th pipeline section affected by corrosion, [MPa];

*OPmax*—maximum operating pressure of the segment, [MPa].

The time-dependent theoretical failure pressure for a straight pipe with a part-wall volumetric surface defect is a function of the following variables:

$$P\_{f\bar{j}}(T) = f(t, \text{ d}, \text{l}, \text{D}, \text{c}\_{\text{d}}, \text{a}\_{\text{\textquotedblleft}c}, \text{c}\_{\text{L}\text{\textquotedblright}}f\_{\text{u}\text{\textquotedblright}}T) \tag{8}$$

where:

*T*—time period, [year];

*cL*—axial corrosion rate for the defect length, [mm/year];

*cd*—proportionality coefficient of a power law function for the defect depth, [mm/year*α*];

*α*—exponential coefficient of a power law function for the flaw depth;

*fu*—ultimate tensile strength of the material used in design. [MPa].

Due to active corrosion of the pipe wall, the pipeline reliability decreases with time of the system operation. Increments of dimensions of metal losses during operation are described for the feature depth in radial direction with the following functions:

$$d(T) = d\_{mcm}(T\_0) + c\_d \cdot T^\alpha \tag{9}$$

and for the length of the longitudinally-oriented tube external surface features in the axial direction:

$$L(T) = L(T\_0) + c\_L \cdot T \tag{10}$$

Failure probability for a pipeline with corrosion grooving with time (*T*) can be calculated as follows:

$$\operatorname{Prof}\_{\vec{\beta}}\left(\stackrel{\rightarrow}{X},T\right) = P[\![\mathcal{G}(\stackrel{\rightarrow}{X},T)\!] \le 0] \tag{11}$$

$$P[\mathcal{g}(\stackrel{\rightarrow}{X},T) \le 0] \;= \int\limits\_{\mathcal{g}(\stackrel{\rightarrow}{X},T) \le 0} f(\mathbf{x}\_{i\prime},T)d\mathbf{x}\_{i} \tag{12}$$

where:

*Po fj*—probability of failure of the j-th pipeline segment with an active corrosion defect, [1/year].

The failure occurs when *g* → *X*, *T* ≤ 0. For a specific time period, Monte Carlo numerical simulation is conducted by random generating of numbers for variables *Pf j*, with respect to statistical distribution of the input parameters specific for a segment. For each evaluation of limit state function (7), occurrence of *g* → *X*, *T* 0 is counted.

Probability of rupture of the j-th section of the pipeline, with the assumption of independence of *n* individual failures *Pofjt* of tubes connected in series, is calculated as

$$\operatorname{Prof}\_{\vec{\jmath}}^{\left(\rightarrow} \vec{X}, T \right) \;= \; 1 - \prod\_{t=1}^{n} (1 - \operatorname{Prof}\_{\vec{\jmath}t}^{\rightarrow}(\vec{X}, T)) \tag{13}$$

where:

*n*—number of pipe wall metal losses, [–].

For each external corrosion flaw, the total number of failure events *Nf* is determined after Monte Carlo samples were generated and rupture probability of a single defect as a function of time can be obtained using the following Equation.

$$\operatorname{Pof}\_{\mathfrak{H}}\left(\stackrel{\rightarrow}{\mathcal{X}},T\right) \;= \frac{N\_f}{\mathcal{MC}}\tag{14}$$

where:

*Nf*—total number of failure events when *g* → *X*, *T* ≤ 0;

*MC*—total number of Monte Carlo trials at the specific time step for calculations of the *j*-th pipeline segment failure probability.

#### *4.2. Input Data for Structural Integrity Evaluation*

For the input parameters specified below, the pipe diameter and the wall thickness are modelled as random variables relying on the tube manufacturer's certificates. The random variables listed below have been obtained from the two repeated diagnostics on the same pipeline. For ILI of oil and gas pipelines, it is a common practice to track the same anomaly in different inspections (i.e., so-called defect matching) based on the longitudinal and circumferential positions of the indication reported by ILI tools. Taken into consideration the clustering criteria, only six pairs of flaws from the total amount of 138 fully matched external corrosion pits were classified as interacting. Eventually, a number of 132 external metal losses are selected for structural reliability estimation taken into consideration the above mentioned interacting criteria.

The external pipe surface corrosion coefficients for the cathodically protected underground structures are derived from the literature [16] and from the author's other papers [13]. The reliability estimations of the pipeline were conducted with coefficients of corrosion growth rates according to Equations (9) and (10): *cd* = 0.164 mm, *α* = 0.78, *cl* = 1.4 mm/year. The linear length increment of the metal loss of the initial length *L* = 174 mm with a function of service time is presented in Figure 6.

**Figure 6.** The linear length increment of the detected metal loss of the initial length l = 174 mm and a growth rate of 1.4 mm/year within the predicted pipeline service time. Source: Author's analysis.

The list of statistical distribution of input parameters for a structural integrity evaluation is presented in Table 5. Computations of failure probability were conducted with the use of an academic license of GoldSim software.


**Table 5.** Statistical distribution of input parameters for reliability evaluation. Source: Author's analysis.


**Table 5.** *Cont.*

#### **5. Pipeline Structural Reliability Considering Flaws Grouping**

Integrity computations, taking into consideration the plastic collapse of the remaining wall thickness of the tube, were carried out relying on the in-line inspections data for the considered case. Due to a corrosion increment of the steel, the structural reliability decreases with time of the underground pipeline operation. The plot of the burst pressure (green line, right axis) and a probability of rupture logarithmic graph (red line, left axis) for the longest single metal loss of initial length *L* = 322 mm and maximum depth of 12% of the wall thickness, detected during the second ILI, as a function of service time for the considered case, is presented in Figure 7. The failure probability calculated for service time within 60 years starting from the second in-line inspection, even for non-repaired longest defect, is low and remain lower than a related code-based target value set for a pipeline safety class high as not higher than 10−<sup>3</sup> per annum [8]. In the later years of maintenance, e.g., when the operation life of the studied pipeline is more than 40 years, a rate of the failure probability increase is strong, which means the rapid aging process of the steel underground structure.

**Figure 7.** Change of rupture pressure and failure probability with time for X52 DN 700 *MOP* = 5.5 MPa pipeline with the longest detected defect and the initial depth of 12%. Source: Author's analysis.

The plot of the burst pressure and a probability of rupture logarithmic graph for cluster 1 with the combined length of *lnm* = 27.5 mm and the maximum depth of 22% of the wall thickness as a function of pipeline service time for the considered case is presented in Figure 8. The failure probability calculated for service time within 60 years starting from the second diagnostics, even for non-repaired defect colony 1, is low and remain lower than a related code-based target value set as not higher than 10−<sup>3</sup> per annum. In the later years

of pipeline service, exceeding 40 years, the failure probability increase is strong, showing the rapid aging effect of the steel buried structure.

**Figure 8.** Change of rupture pressure and failure probability with time for X52 DN 700 *MOP* = 5.5 MPa pipeline with defect group1. Source: Author's analysis.

The graph of the burst pressure and a probability of rupture for flaw group 2 with the combined length of *lnm* = 29.0 mm and the maximum depth of 3.41 mm of the pipe wall thickness as a function of the operating time for the considered case is presented in Figure 9. The failure probability calculated for pipeline service in time, after 53 maintenance years, remains higher than a related code-based target value set for a safety class high as not exceeding 10−<sup>3</sup> per annum.

**Figure 9.** Change of burst pressure and failure probability with time for L360NE DN 700 *MOP* = 5.5 MPa pipeline with metal loss group 2. Source: Author's analysis.

The plot of the burst pressure and a probability of rupture logarithmic graph for cluster 3 with the total length of *lnm* = 33.5 mm and the maximum depth of 22% of the wall thickness which corresponds to the value of 2.42 mm is shown in Figure 10. The probability of burst calculated for a pipeline operating period not more than 60 years, is low and remains lower than a related code-based target value set for a safety class high as not exceeding than 10−<sup>3</sup> per annum. If the operation period of the studied buried pipeline is more than 40 years, a rate of the failure probability increase is strong, which means the rapid aging effect of the steel.

**Figure 10.** Change of burst pressure and failure probability with time X52 DN 700 *MOP* = 5.5 MPa pipeline with defect group 3. Source: Author's analysis.

#### **6. Conclusions**

The calculation results of the burst pressure of the defected pipes relying on Det Norske Veritas Recommended practice Corroded pipelines, taking into consideration interactions of metal losses obtained from in-line inspections, show a little influence of closely spaced indications on the rupture pressure of the considered steel pipeline.

The probability of pipeline burst in the studied corrosion colonies cases do not differ significantly from the corresponding case when features are assessed as isolated. In the case considered in the present paper, grouping of closely spaced defects is almost consistent compared to the assessment of the individual flaws in respect of the burst pressure calculations and the probability estimations. The burst probability computations of the studied pipeline are independent of the results of corrosion grouping indications due to both small areas and the depth of the defects, which are the most important impact factors. The failure probability calculated for the pipeline service time within 50 years, starting from the second in-line inspection, even for non-repaired corrosion clusters, is low and remains lower than a related code-based target value set for a safety class high as not exceeding 10−<sup>3</sup> per annum. In the later years of the studied pipeline operation, beyond 40 years, the structural integrity decrease is strong, which means the rapid aging degradation of the steel underground structure.

**Funding:** The author declare that the research has not received external funding. This paper was cofinanced under the research grant of the Warsaw University of Technology supporting the scientific activity in the discipline of Environmental Engineering, Mining and Power Engineering.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare that there are no conflict of interest that are directly or indirectly related to the research.

#### **References**

