Assessment of AC Corrosion Probability in Buried Pipelines with a FEM-Assisted Stochastic Approach

Abstract

In this paper, a stochastic approach is combined with field theory and circuit methods to study how the geometrical and electrical properties of holidays (defects or pores in the insulating coating) in a metallic pipeline influence the probability of exceeding the current density limit for corrosion. Three-dimensional FEM simulations are conducted to assess the influence of the shape and electrical resistivity of the pore on the computed spread resistance value. The obtained results are then used to evaluate the probability of exceeding a given current density value for different sizes of pore and soil resistivities. Finally, a case of 50 Hz interference along a pipeline-transmission line routing is examined. The probabilistic approach presented in this paper allows the pipeline sections more subjected to the induced AC corrosion risk to be identified to be used as an auxiliary tool for adopting preventive protection countermeasures. Lastly, unlike most papers devoted to assessing electromagnetic interference on pipelines, the present work uses a probabilistic rather than a deterministic approach, representing its main novelty aspect.

1. Introduction

Long metallic structures installed nearby high-voltage alternate current (HVAC) power lines or electrified AC railway lines are often subjected to electrical stresses due to the time-varying current flowing through the line conductors [1]. A typical example is steel pipelines for transporting liquids or gases, which—for economic reasons—often share the same corridors with electrical transmission lines [2] or AC railways [3,4]. Unless effective protective measures are put in place, buried steel pipelines may be subjected to electrochemical corrosion over time because of the current density flowing from the pipeline metal to earth through imperfections of the pipeline coating [5] (holidays). The corrosion phenomenon may eventually lead to perforations under long-term exposure, thus endangering the transportation and operation safety of the pipeline [6]. While this work focuses on interferences at typical power frequencies (50 or 60 Hz), it is worth adding that pipelines may also be influenced by lightning strikes hitting nearby power lines or failures of nearby electrical apparatuses. In these cases, pipes may be subjected to transients characterized by larger magnitudes and a wider frequency spectrum [7].

Interference on pipelines is usually classified using three distinct mechanisms, namely inductive, conductive, and capacitive coupling. Capacitive coupling is essential only when the victim is located above ground, and conductive coupling occurs only when the earth is actively used as a return path by the interference source, such as in the case of phase-to-ground fault conditions of a three-phase line. Out of these, inductive interference is the only mechanism to which pipelines are exposed under all operational conditions of nearby transmission lines and substations [8]. In general, the three coupling mechanisms of interference may also occur simultaneously [9]. Many efforts have been devoted over the last decades to the development of computational techniques for predicting interference levels. Typical physical quantities of interest include the pipe-to-soil voltage and the pipeline longitudinal and radial current densities. The radial current component (normal to the pipeline geometrical axis) eventually reaches the soil through the pipe insulation layer. It is considered responsible for corrosion when the magnitude exceeds $30 \mathrm{A}/{\mathrm{m}}^2$ [10].

Computational approaches for calculating these quantities are generally grouped into two main groups, i.e., methods based either on circuit theory/transmission-line theory or field theory. Techniques belonging to the first category require a discretization of the system into an ensemble of smaller sections that share some kind of electromagnetic coupling. Each section is described by means of lumped parameters for circuit approaches and distributed parameters for transmission line methods [11]. The numerical values of the parameters are usually found utilizing exact or approximated analytical formulas. A classic example is constituted by several approximate expressions developed over time to evaluate the well-known Carson’s series for calculating self- and mutual impedances of earth-return conductors [12].

Instead, field theory methods are based on obtaining an approximated solution of Maxwell’s partial differential equations in space (and time for unsteady problems). These can be further subdivided into integral and differential methods. Integral methods are generally very efficient for open-boundary problems and thus have also been employed in this field. An example is the shifting complex images method by Andolfato et al. [13] which combines the moments and the images method. In contrast to integral methods, where only the surfaces of the field sources are discretized, the whole physical domain is discretized when a differential approach is adopted, such as the finite-difference time-domain (FDTD) or the finite element method (FEM) [14]. Since AC interference modelling problems are essentially three-dimensional and given the considerable size of corridors, field-theory-based models have mostly been used with circuital approaches. Hence, FEM is used to infer physical information on selected cross sections of the corridors, subsequently used to build an equivalent circuit representation of the whole geometry. The models in [15,16,17,18,19,20,21] are examples of efforts in this direction. These have been applied to study complex non-parallel configurations in conjunction with complex soil models and the presence of multiple conductors. Some work, such as [22] and later on [23], has also shown the advantages of combining FEM (used only for the pipeline region) with the boundary element method (BEM). Notably, several commercial programs feature comprehensive modelling capabilities regarding electrical interference on pipelines (and grounding systems), often allowing one to perform studies using circuit and electromagnetic field theory models. Examples of such computer programs include CDEGS [24], which originated from the seminal modelling works of Dawalibi and colleagues [25], Elsyca [26,27] and XGSLAB [28,29].

Table 1. Summary of calculation methodologies and evaluated quantities.

Paper	Applied Methodology	Physical Quantities Evaluated	Type of Calculation
Taflove and Dabkowski, 1979 [1] Dawalibi and Southey, 1989 [25] Djogo and Salama, 1997 [31]	Equivalent Transmission Line Circuit a	Induced voltage	Deterministic
Christoforidis et al., 2005 [16,17]	2D FEM + Circuit theory	Induced voltage Induced current	Deterministic
Micu et al., 2013 [15]	2D FEM + Circuit theory	Induced voltage	Deterministic
Wu et al., 2017 [7]	Equivalent Transmission Line Circuit (time domain)	Induced voltage Induced current	Deterministic
Cristofolini et al., 2018 [18] Popoli et al., 2019 [19]	2D FEM	Induced voltage Induced current	Deterministic
Lucca, 2019 [30]	Equivalent Transmission Line Circuit	Current density	Probabilistic
Popoli et al., 2019 [20] Popoli et al., 2020 [21]	2D FEM + Circuit theory (Quasi-3D method)	Induced voltage Induced current	Deterministic
Muresan et al., 2021 [8]	Equivalent Transmission Line Circuit (EMTP-RV), PEEC b (XGSLab)	Induced voltage	Deterministic
Moraes et al., 2023 [9]	Equivalent Transmission Line Circuit (EMTP)	Induced voltage Induced current	Deterministic
This work	Equivalent Transmission Line Circuit	Current density	Probabilistic

^a See [32,33]. ^b Partial element equivalent circuit method, see [34].

summarizes the calculation methodologies employed in the above-discussed works. For each work, the evaluated physical quantities are also reported. A feature shared by most of the listed approaches and corresponding computer codes is that deterministic approaches are applied to estimate the physical quantities of interest. In contrast, this work uses field theory and circuit methods combined with a stochastic approach to study how holidays’ geometrical and electrical properties influence the probability of exceeding the current density limit for corrosion. A similar approach has been used in [30] without considering the presence of coating defects, which are the main focus of the present manuscript.

In this work, three-dimensional FEM simulations are performed to study the electrical properties of a holiday (pore) in the insulation layer of a metallic pipeline. The focus is set on the influence of the shape and electrical resistivity of the pore on the obtained value of spread resistance. Once the physical characteristics of a typical holiday have been defined, an evaluation of the leakage current density is performed using a probabilistic method. The computed quantity is the probability of exceeding a given current density value. In this way, the influence of both pore and external soil resistivity on exceeding the current density limit is assessed. Finally, the developed probabilistic approach is applied to a more complex case by computing the probability of exceeding the limit along a pipeline transmission line routing.

The parametric FEM simulations on the holiday spread resistance are described in Section 2. The probabilistic evaluation of the current density is discussed in Section 3.

2. Finite Element Analysis of the Spread Resistance of a Coating Defect

Corrosion effects in pipelines due to induced AC voltages are related to the spread resistance of a coating fault, which results from three contributions, the resistance of the medium located in the coating fault (pore or holiday resistance, $R_p$), the leakage resistance in the soil $R_l$ and the polarisation resistance of the bare metal at the bottom of the coating fault [35]:

$$R_s=R_p+R_l+R_{pol}$$

(1)

Usually, only the first two contributions are considered in the literature as the polarization resistance $R_{pol}$ It is much smaller than the other two terms [35] and can thus be neglected [36]. In this paper, to evaluate the spread resistance of a coating fault with an angle between the coating and the coating fault gutter, a finite element analysis is carried out.

Specifically, the DC Conduction solver available in Ansys Maxwell 3D is used [37]. It solves the equation:

$$\nabla \cdot \left(\mathsf{\sigma }\nabla \mathsf{\Phi }\right)=0$$

(2)

where $\mathsf{\Phi }$ is the electric scalar potential and $\sigma$ the medium electrical conductivity. The symbol $\nabla$ denotes the divergence operator. The solved quantity is the electric scalar potential $\mathsf{\Phi }$, from which the current density $\overline{J}$ and electric field $\overline{E}$ are derived.

In Figure 1a the vertical cross-section of the finite element model for a defect with finite coating thickness d is shown. The quantities $r_a$ and $r_b$ represent the radii of the top and bottom bases of the pore, respectively. A constant 1 A current is applied to the top base of the pore, which has an area of 1 cm²; the thickness of the pore is $d=3$ mm, and the pore resistivity is ${\rho }_p=\rho =100$ Ωm; the bottom base of the pore is set to 0 V potential and the lateral surface of the pore is set to be insulating. The 3D dimetric view of the pore model is shown in Figure 1b, with an indication of the boundary and excitation conditions for each face of the object. Some calculations performed when the medium inside the pore has a constant resistivity equal to that of the soil show that the current density is not uniform along the pore depth (i.e., the thickness of the coating). This can be seen in Figure 2 and Figure 3, where the current density is plotted in a vertical and a horizontal cross-section of the pore, respectively. This result accounts for calculating the pore resistance with finite element analysis, as the analytical calculation assumes a uniform current density along the pore thickness:

$$R_p={\rho }_p\frac{d}{\pi r_a r_b}={\rho }_p\frac{d}{\pi r_a\left[d\cot \left(\alpha \right)+r_a\right]}.$$

(3)

The comparison between the pore resistance $R_p$ calculated with the finite element method (FEM) and with (3) is shown in Figure 4, where $R_p$ is plotted versus the angle between the coating and the coating fault gutter; the angle $\alpha =90°$ represents a cylindrical pore. The figure shows that the difference between the FEM calculation and (3) increases for a decreasing $\alpha$, with the percent error reaching about 80% for $\alpha =5°$.

The FEM model is then modified by including a block of resistivity $\rho$ representing the soil; the top face of this block is in contact with the bottom face of the pore, thus allowing the spread resistance to be calculated by applying a constant 1 A current on the top face of the pore and setting a 0 V potential to the faces of the soil block. The FEM model is depicted in Figure 5. The spread resistance is obtained by dividing the potential of the top face of the pore by the 1 A current applied. Analytically, the leakage resistance in (1) can be evaluated as the grounding resistance of a circular plate of radius $r_a$ as $R_l=\frac{\rho }{2r_a}$, whereas the pore resistance is estimated with (3). The comparison of the spread resistance versus $\alpha$ obtained with the FEM model and analytically with (1) is plotted in Figure 6. This figure shows that the spread resistance decreases as α decreases and that, apart from $\alpha =90°$ (which represents a cylindrical pore), the spread resistance calculated analytically with (1) is larger than that calculated with finite element analysis. Considering a cylindrical pore of radius $r_a$ and a resistivity ${\rho }_p=\frac{\rho }{10}=10$ Ω.m (value supported by field measurements, see [38]), the spread resistance evaluated with (1) is 4731 Ω, very close to the value obtained in the case of a pore with $\alpha =5°$ and ${\rho }_p=100$ Ω.m, which is 4627 Ω and represents the minimum value of spread resistance obtained for a pore with an angle. Therefore, in the following analysis for simplicity a cylindrical pore with an averaged resistivity value of ${\rho }_p=\frac{\rho }{10}$ is considered. This configuration is conservative as the spread resistance equals the minimum value of the spread resistance obtained for a pore with an angle between the coating and the coating fault gutter.

3. Probabilistic Evaluation of the Leakage Current Density through a Holiday in the Coating

Almost always, power frequency electromagnetic interference problems among power/railway lines and pipelines/telecommunication cables are based on a purely deterministic approach that assumes that all the parameters characterizing the calculation model are known and expressed by specific values. Nevertheless, this is not always the case; in certain situations, some of the model parameters have a random behavior that suggests a different approach to the problem that necessarily involves concepts of probabilistic-statistical nature.

Just the issue which is the object of this paper has this kind of characteristic because the area size of the holidays present in the insulating coating of the pipeline can be considered, without any doubt, a random parameter.

Therefore, under the assumption that area A of the holiday is represented by a random variable, this section proposes a probabilistic method to assess the value of the current density leaking from a holiday, present in the pipe insulating coating, to the soil. More explicitly, one may pose the problem as follows by starting from these two assumptions:

• If V is the value of the induced voltage in a specific point along the pipeline due to the electromagnetic influence produced by nearby High Voltage power lines or electrified railway lines.
• If, at the same point, a holiday in the pipeline insulating coating is present.

Then, evaluate the probability that the induced AC current density, due to the electromagnetic interference from nearby power/railway lines, exceeds the limit value J_lim established by the standards. See, for example, [10] or [35].

One can summarize the basic steps of the whole calculation procedure in the flow chart in Figure 7.

Before presenting the formulas to assess the above-mentioned probability, it is necessary to summarize some basic relationships, given in [39], that are here reported for convenience.

3.1. Relationships among Induced Voltage, Current Density and Holiday Area

In the previous paragraph devoted to the FEM simulations, it has been shown that the simple model of the holiday inside the pipeline insulating coating given by a cylindrical pore having area A, height d (equal to the thickness of the coating) and constant resistivity ρ_p is a fair approximation. (See Figure 8 for a schematic drawing).

According to it, one has that the formula for the spread resistance R_s presented in many papers and textbooks, see for example [5,10], is given by:

$$R_s=\frac{\rho }{4}\sqrt{\frac{\pi }{A}}+\frac{{\rho }_{p}d}{A}$$

(4)

the first addendum represents the earth (or leakage) contribution, while the second represents the pore contribution. If $V$ is the AC-induced pipe potential in the point where the holiday coating is located, one has that $V/R_s$ represents the AC leakage current to soil; therefore, it is possible to define an average current density $J$ through the holiday (see Appendix A) by means of:

$$J=\frac{V}{{\rho }_{p}d+\frac{\rho }{4}\sqrt{\pi A}}$$

(5)

This quantity has to be compared with the limit value $J_{lim}$ suggested by the standards to assess the AC corrosion risk for the pipeline, i.e., it has to be $J<J_{lim}$. Thus, from Formula (5), one must have:

$$\sqrt{\pi A}\le \frac{4}{\rho }\left(\frac{V}{J_{lim}}-{\rho }_{p}d\right)$$

(6)

that yields:

$$A\le A_{max}\left(V\right)=\frac{16}{\pi {\rho }^2}\left(\frac{V}{J_{lim}}-{\rho }_{p}d\right)$$

(7)

$A_{max}$ represents, for a given value of the induced potential V, the maximum value of the holiday area such that one can have a current density exceeding $J_{lim}$.

At the same time, being the left-hand-side of Formula (6) a positive quantity, it must be $V\ge J_{lim} {\rho }_{p}d$. Thus, one obtains a threshold value $V_{min}$ for the induced potential; such a quantity represents the minimum value for the induced potential V able to generate a current density greater than $J_{lim}$ leaking to the soil through the holiday. Therefore, one has:

$$V\ge V_{min}=J_{lim}{\rho }_{p}d$$

(8)

Notice that $V_{min}$ does not depend on soil resistivity but only on the resistivity of the material inside the pore.

Finally, it is useful to emphasize that the above formulas involve only the area size of the holiday and are not based on a particular shape. See Appendix A for further details.

3.2. Probability Distribution of the Holidays Area

Based on the experimental data and their processing presented in [39], one has that the empirical probability density distribution $w\left(A\right)$ of the holidays area is sufficiently well fitted by a log-normal distribution given by:

$$w\left(A\right)=\frac{1}{A\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}\frac{{\left(ln A-\mu \right)}^2}{\sigma }}$$

(9)

In (9), the parameters $\mu$ and $\sigma$ characterizing the distribution have the following values: $\mu$ = −8.207 and $\sigma$ = 1.419.

3.3. Probability of Exceeding the Limit Value J_lim

By considering Formulas (7)–(9), it is possible to write the following relationship expressing the probability that a leakage current $J$, flowing through a holiday located in a point at potential $V$, exceeds the limit value $J_{lim}$. One has:

$$P(J(V)\ge J_{\lim })=P(A\le A_{\max }(V))=\left\{\begin{array}{cc}\underset{0}{\overset{A_{\max }(V)}{{\displaystyle \int }}}\frac{1}{A\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}\frac{{(\ln A-\mu )}^2}{\sigma }}dA& if V>V_{\min }\\ 0& otherwise\end{array}\right.$$

(10)

Therefore, if the induced voltage profile is known along all the pipeline routes, the above formula calculates the probability that the leakage current $J$ exceeds $J_{lim}$ at any point of the pipeline provided that a holiday is present at the same point. In such a way, it is possible to identify along the pipeline route the sections that are more exposed to the induced AC corrosion risk.

3.4. Influence of the Main Parameters

In this paragraph, some plots are presented that describe the probability of exceeding the value $J_{lim}$ = 30 A/m² (suggested by [10,35]) versus the induced AC potential $V$ by varying the main parameters involved in the model that is: the soil resistivity $\rho$, the resistivity of the material inside the pore ρ_p and the insulating coating thickness $d$. As far as the parameter ${\rho }_p$ is concerned, two cases are considered:

(1)

The resistivity of the material inside the pore is equal to the soil resistivity, i.e.,: ${\rho }_p=\rho$;

(2)

The resistivity ${\rho }_p$ is one order of magnitude smaller than $\rho$ i.e., ${\rho }_p=\rho /10$. This hypothesis is suggested by some results based on field measurements and described in [38].

In Figure 9, the probability, of the induced AC current density exceeding the value of 30 A/m² versus the induced potential is shown; in particular, the curves have been plotted in correspondence with different values of soil resistivity by considering both cases (1) and (2).

Figure 9 clearly confirms what is also known from experience, i.e., the probability of exceeding the limit value of 30 A/m² is much higher for low and medium-low soil resistivities than for high and medium-high resistivities. Moreover, as one could expect, the case with ${\rho }_p=\rho /10$ is associated with higher probabilities because the spread resistance R_s is smaller than the cases with ${\rho }_p=\rho$.

Figure 10 has been plotted by considering different values for soil resistivity and very different values of the coating thickness i.e., $d$ = 1 mm and $d$ = 10 mm, representing the extreme values inside the typical range for d. Moreover, it has been assumed that ${\rho }_p=\rho /10$.

By looking at Figure 10, since ${\rho }_p=\rho /10$, one has that the second addendum in (4) contributes only in a minor way to the spread resistance $R_s$; for such a reason, the probability curves corresponding to the same value $\rho$ differ between them only slightly. Therefore, one can conclude that if the resistivity inside the pore is much smaller than the soil resistivity, the coating thickness has a scarce influence on the probability of exceeding the 30 A/m² limits.

3.5. Example of Application to a Real Case of 50 Hz Interference

In this paragraph, one considers a 50 Hz interference case between a 380 kV power line carrying, under normal operating conditions, a balanced current of 630 A and a nearby pipeline 28.5 km long having a coating thickness of 3 mm; the soil resistivity is 100 Ωm; the layouts of the two plants are shown in Figure 11 where one can notice a close approaching between them for almost the whole pipeline route.

By means of a two-step calculation method based on treating both power line and pipeline by means of equivalent transmission line models one can obtain the 50 Hz induced voltage along the pipeline versus the pipeline abscissa $s$ (see Figure 12).

By applying (10), one can determine the probability that the limit of $30 \mathrm{A}/{\mathrm{m}}^2$ for the induced current density is exceeded in a generic point of abscissa s along the pipeline provided that a holiday exists in the insulating coating in the same point. As far as the pore resistivity is concerned, the two hypotheses previously presented have been considered i.e., ${\rho }_p=\rho$ and ${\rho }_p=\rho /10$. The results are shown in Figure 13.

As one can expect, the probability profiles of Figure 13 are like the induced voltage profile of Figure 12 just because, according to Formula (4), the current density J is directly proportional to the voltage V. According to the expectations, the plot in Figure 13 corresponding to the case ${\rho }_p=\rho$ is always lower than the one corresponding to the case ${\rho }_p=\rho /10$ and in certain intervals, along the pipeline route it is zero.

4. Conclusions

This paper presents an approach consisting of field theory and circuit methods combined with a stochastic approach for analysing the probability of exceeding the current density limit for corrosion in metallic pipelines. The FEM analysis shows that the pore shape influences the values of spread resistance; in particular, pores with an angle between the coating and the coating fault gutter have a resistance that decreases when the angle decreases. It is shown that conservative results from the point of view of the probability of exceeding the current density limits for pipeline corrosion are found when the pore has a cylindrical shape, with a resistivity for the medium inside the pore equal to one-tenth of the soil resistivity. The method allows plots showing the induced voltage along the pipeline to be found; based on these plots and knowledge of the probabilistic distribution of holidays area, it is possible to individuate the sections along the pipeline route, more exposed to potential AC corrosion risk. Therefore, the proposed calculation method can be considered an auxiliary tool, especially at the design stage of new power lines or pipelines, to adopt suitable protective measures against AC corrosion.