# Assessment of AC Corrosion Probability in Buried Pipelines with a FEM-Assisted Stochastic Approach
## 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]:  
\[R_{s}=R_{p}+R_{l}+R_{pol} \tag{1}\]  
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:  
\[\
abla\cdot(\sigma\
abla\Phi)=0 \tag{2}\]  
where \(\Phi\) is the electric scalar potential and \(\sigma\) the medium electrical conductivity. The symbol \(\
abla\) denotes the divergence operator. The solved quantity is the electric scalar potential \(\Phi\), 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 cm\({}^{2}\); the thickness of the pore is \(d=3\) mm, and the pore resistivity is \(\rho_{p}=\rho=100\)\(\Omega\)m; 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 dimeric view of the pore model \begin{table}
\begin{tabular}{c c c c} \hline
\begin{tabular}{c} Equivalent Transmission \\ Line Circuit \\ \end{tabular} & Current density & Probabilistic \\ \hline \end{tabular}  
* See [32; 33]. b Partial element equivalent circuit method, see [34].  
\end{table} 
Table 1: Summary of calculation methodologies and evaluated quantities. 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 \(\alpha=90^{\circ}\) 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 \(\alpha=5^{\circ}\). 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 \(\alpha\) decreases and that, apart from \(\alpha=90^{\circ}\) (which represents a cylindrical pore), the spread resistance calculated analytically with (1) is larger than that calculated with finite element analysis.