# Prediction of Corroded Pipeline Failure Pressure Based on Empirical Knowledge and Machine Learning
## 2 Methodology  
### Empirical Formula  
To evaluate the integrity of pipelines with corrosion defects, several widely accepted criteria have been developed to predict the remaining strength and failure pressure. These include the ASME B31G, Modified B31G, PCORRC criteria, and DNV RP-F101. Each method employs different approaches and assumptions, which are outlined below.  
(1) ASME B31G  
The ASME B31G method is a foundational approach for assessing the remaining strength of corroded pipelines. It was formulated based on experimental burst data from real-world corroded pipelines and assumes that the corroded area has a parabolic shape for short defects (length \(L\leq 20 Dt\)) and a rectangular shape for long defects (length \(L>20 Dt\)). The failure pressure is calculated using the specified minimum yield strength (SMYS) and a geometry correction factor known as the Folias factor, which accounts for stress concentration effects [15]. The failure pressure is calculated as Equation (1).  
\[P_{F}=\left\{\begin{array}{cc}\frac{2t\sigma}{D}\left[\frac{1-(2/3)(d/t)}{1 -(2/3)(d/t)/M}\right]&L\leq\sqrt{20 Dt}\\ \frac{2t\sigma}{D}\left(1-\frac{d}{t}\right)&L>\sqrt{20 Dt}\end{array}\right. \tag{1}\]  
where \(\sigma\) is the flow stress of the pipeline steel, defined as \(\sigma\) = 1.1 _SMYS_ (specified minimum yield strength); \(t\) is the thickness of the pipeline; \(D\) is the outer diameter; \(d\) is the corrosion depth; and \(M\) is the Folias factor, which accounts for the stress concentration caused by radial deflection of the pipe. \(M\) can be calculated by Equation (2).  
\[M=\sqrt{1+0.8\left(\frac{L}{\sqrt{Dt}}\right)^{2}} \tag{2}\]  
Addressing the over-conservatism of the original ASME B31G, the Modified ASME B31G method adjusts the defect profile to better match actual corrosion shapes. This method replaces the parabolic factor with a more accurate 0.85 coefficient, resulting in a more realistic representation of the corroded area. The failure pressure, which can be calculated as Equation (3), is recalculated using an adjusted flow stress and a revised Folias factor that considers the defect length [16].  
\[P_{F}=\frac{2t\sigma}{D}\left[\frac{1-0.85(d/t)}{1-0.85(d/t)/M}\right] \tag{3}\]  
where the flow stress \(\sigma=SMYS+68.95\) (MPa) and the Folias factor \(M\) is adjusted based on the defect length \(L\), which can be calculated by Equation (4).  
\[M=\begin{cases}\sqrt{1+0.6275\Big{(}\frac{L}{\sqrt{Dt}}\Big{)}^{2}-0.003375\Big{(} \frac{L}{\sqrt{Dt}}\Big{)}^{4}}&L\leq\sqrt{50 Dt}\\ 3.3+0.032\Big{(}\frac{L}{\sqrt{Dt}}\Big{)}^{2}&L>\sqrt{50 Dt}\end{cases} \tag{4}\]  
The PCORRC criteria, developed by [PERSON], focus on predicting the remaining strength of pipelines with moderate to high toughness. This method uses finite element analysis to model elliptical defects and incorporates a plastic collapse failure mode. The failure pressure is derived from the ultimate tensile strength (UTS) and accounts for the defect length through an exponential function, providing a more nuanced prediction of residual strength [17]. The PCORRC criteria can be expressed as Equation (5).  
\[P_{F}=\frac{2t\sigma_{UTS}}{D}\Bigg{[}1-\frac{d}{t}\Bigg{(}1-\exp\Bigg{(}-0.15 7\frac{L}{\sqrt{D(t-d)/2}}\Bigg{)}\Bigg{)}\Bigg{]} \tag{5}\]  
where \(\sigma_{UTS}\) is the ultimate tensile strength of the pipeline steel.  
(4) DNV RP-F101  
The DNV Recommended Practice (DNV RP-F101) is a comprehensive guideline for assessing the integrity of corroded pipelines. This method assumes a rectangular defect shape and calculates the failure pressure as Equation (6).  
\[P_{F}=\frac{2t\sigma_{UTS}}{D-t}\Bigg{[}\frac{1-d/t}{1-d/tQ}\Bigg{]} \tag{6}\]  
where \(Q\) is a geometry correction factor that accounts for the defect length \(L\) and other geometrical parameters. \(Q\) can be calculated as Equation (7).  
\[Q=\sqrt{1+0.31\Big{(}\frac{L}{\sqrt{Dt}}\Big{)}^{2}} \tag{7}\]