# Assessment of AC Corrosion Probability in Buried Pipelines with a FEM-Assisted Stochastic Approach
## 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 ofinterference 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 A/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 [PERSON] 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].