# Comparative Study on the Efficiency of Simulation and Meta-Model-Based Monte Carlo Techniques for Accurate Reliability Analysis of Corroded Pipelines

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## Abstract

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## 1. Introduction

## 2. Limit State Functions of Corroded Pipelines Based on the Steel Grade

**X**denotes a vector of input random variables, and ${P}_{Burst,i}$ and ${P}_{O}$ are the burst and operating pressures at the ith defect, respectively. Several models have been developed to model the burst pressure of corroded pipelines over the years. Keshtegar and Mohamed [46] reviewed 35 empirical models for the burst pressure in terms of the development basis, equation forms, advantages, and limitations. Furthermore, the authors compare the effectiveness of the assessed burst pressure models to a large-scale experimental test database, in which conclusions revealed that all the empirical models have a tendency to miss-estimate the real values for outside ranges of their development. Rafael Amaya-Gómez et al. [47] conducted a similar study in which several burst pressure models were examined using various criteria and the same conclusions were reached. Recently, Mohamed El Amine et al. [36] have overcome this drawback by developing new probabilistic models based on the pipelines grade as low (e.g., X46 and X52), mid (e.g., X60 and X65), and high (e.g., X80 and X100) strength steel pipelines, whereas Equations (2)–(4) give their formulas, in the same respect.

## 3. Approaches for Structural Reliability Analysis

#### 3.1. Simulation Techniques Based on Monte Carlo Method

#### 3.1.1. Monte Carlo Simulation (MCS)

**X**). The failure zone is defined as points with g(

**X**) < 0, while the safe zone is defined as points with g(

**X**)> 0; the limit state function is defined as the border between these two zones, g(

**X**) = 0. The ratio of the number of samples in the failure zone to the total samples is used to determine the failure probability as follows:

#### 3.1.2. Importance Sampling (IS)

#### 3.1.3. Subset Simulation (SS)

#### 3.1.4. Directional Simulation (DS)

**A**is the unit direction vector. Thus, the failure probability can be written as follows [59]:

**A**variables. ${E}_{A}[]$ is a directional simulation operator of the vector

**A**. $r\left(a\right)$ presents the radius length in the simulation direction of A = a (Figure 2). More details regarding this approach are available in Ref [57].

#### 3.1.5. Line Sampling (LS)

#### 3.2. Meta-Models Based on Monte Carlo Simulation

- Using the PDFs of the variables to generate a sequence of random numbers.
- Recalling the LSF based on the generated series of random numbers.
- Generating MCS samples and predicting new values of the LSF.

#### 3.2.1. Kriging for Modeling the Performance Function Response

#### 3.2.2. Artificial Neural Network for Modeling the Performance Function Response

**X**) is the output of ith neuron in the hidden layer, which is expressed as:

^{th}input variable in the input layer to ith neuron in the hidden layer, b

_{i}is a constant.

## 4. Illustrative Examples

_{0}), Lognormal (i.e., ${\sigma}_{u}$ and ${\xi}_{High}$), Gumbel (i.e., ${\xi}_{Low}$), and Freschet (i.e., ${\xi}_{Med}$). The candidate pipelines have different ranges of design (i.e., D and t) and material (i.e., ${\sigma}_{y}$ and ${\sigma}_{u}$) parameters, although the material characteristic distributions may alter as the pipeline ages (i.e., elapsing time) [76]. The mean of these parameters was taken as the nominal values of the real-cases pipelines (X52, X65, and X100), while a low coefficient of variation (COV) value was given to consider the uncertainties of the manufacturing or human measuring error, whereas the attributed distributions are based on the works of [36,48]. In this work, the corrosion defect depth is suggested to be varied in a range of 15% t to 75% t to account for the degree of severity of the defects at different growth stages. The mean value of corrosion defect length is set to be 200 mm with a normal distribution as suggested by [36,77]. The operating pressure varies between 5 to 25 MPa, where its fluctuations are modelled by a normal distribution and a COV of 0.1 as referred in [44]. The model error distributions and values were adopted based on the works of Mohamed El Amine et al. [36].

## 5. Results and Discussion

_{f}), and the number of call-functions (g-call). It is worth noting that the reliability index (ꞵ) and failure probability (P

_{f}) have an inverse relationship based on the standardized normal distribution, as shown in Equation (21).

#### 5.1. Monte Carlo Simulation Accuracy

_{f}) when the number of simulations supplied is adequate to cover all conceivable areas based on the LSFs. However, this will incur significant computational expenses. As a result, determining the optimal number of simulations for completing the reliability analysis is critical. In general, the coefficient of variation (CoV) is utilized as an indicator for illustrating the stability of the MCS performance. For simple LSFs, a value of less than 5% is necessary for stable and reliable results using the MCS; consequently, in our study, because the LSFs are highly nonlinear, the required value of CoV is set to be less than 0.0005, and Equation (22) represents the procedure for estimating the CoV for the MCS as follows:

_{0}= 10 MPa. As it can be observed, all of the reliability analysis results in terms of failure probability are more stable and produce the same outcomes when no fewer than 10

^{6}simulations are used. According to Figure 6b, the values of CoV naturally decrease as the number of simulations increases. Using the 0.0005 threshold condition, the least number of simulations necessary to obtain correct MCS results using the aforementioned basic random variables (i.e., d/t = 0.45; L = 200 mm; and P

_{0}= 10 MPa) is 10

^{6}. The achieved failure probabilities using 10

^{6}simulations are ${P}_{f}^{X52}=0.003;{P}_{f}^{X65}=0.006$ and ${P}_{f}^{X100}=0.001$, with corresponding CoV values of $Co{V}^{X52}=0.00033;Co{V}^{X65}=0.00015$ and $Co{V}^{X100}=0.00078$, respectively. Another observation is that the X65 pipeline provides a higher probability of failure than the X52 pipeline. This is due to the selected basic random variables in Table 1, where the X52 presents a pipeline with a relatively large wall-thickness and diameter (D = 914.4 mm, t = 20.6 mm) compared to the X65 pipeline (D = 762 mm, t = 17.5 mm). Overall, 10

^{6}simulations were chosen to perform the reliability analysis findings, utilizing MCS.

#### 5.2. Performance Evaluation of the Simulation Method

_{f}) decreases as the operating pressure (P

_{0}) increases, which is acceptable given the effect of this parameter on the strength condition of corroded pipelines. However, based on the table and figure findings, this is not the case for P

_{f}values produced using the DS and LS techniques for the X52 and X65 pipelines, where results are chaotic and inaccurate. The DS technique was unable to solve the problem for the X100, which may be attributed to the approach’s failure to identify the direction vector since the given LSF of the high strength pipeline is complicated and extremely nonlinear. IS only needed 15,000 simulations to achieve P

_{f}values, however the results are less accurate. SS technique appears to produce the most reliable and accurate results among the other simulation approaches by attaining good P

_{f}values with minimal g-call, which varies between 57,000 and 6000 for low to high P

_{f}values. According to the results of SS, when it comes to low failure probability instances, such as P

_{0}= 5 MPa for the three examples, the needed number of simulations utilizing MCS must greatly rise to at least 10

^{12}, resulting in a high computational burden.

_{f}values decrease as the corrosion depth increases. This comment has previously been shown via various studies, in which findings indicated that the corrosion depth is the key factor reducing pipeline strength as it directly reduces the thickness of the pipe-walls. Unlike findings obtained by varying the operating pressure, simulation-based techniques produced more accurate P

_{f}values for varying the corrosion defect depths. This can be attributed to the comparatively high achieved P

_{f}values by varying d/t to varying the operating pressure. As a result, all methods use fewer g-call functions. As an example, for d/t = 0.15, the P

_{f}values for the X52, X65, and X100 pipelines are 1.83 × 10

^{−4}, 9.96 × 10

^{−4}, and 1.77 × 10

^{−4}, respectively, but for P

_{0}= 5 MPa, the MCS was unable to reliably establish the P

_{f}values due to a lack of simulations. Regardless of the efficiency of the simulation approaches, DS techniques fail to produce any findings for the X100 pipeline for the same reasons stated previously. On the other hand, the LS approach produced the fewest g-calls, followed by the SS strategy, which produced correct results.

#### 5.3. Performance Evaluation of the Meta-Models

_{0}over 10 MPa for the three pipelines, indicating that these techniques are highly efficient when compared to the needed g-call using simulation-based methods. It has been observed that the meta-models cannot reach correct solutions when the failure probability is low, as in the case of P

_{0}= 5 MPa. This disadvantage can be mitigated by further adjusting the meta-models’ governing parameters. Overall, the findings of the reliability analysis obtained using meta-models and presented in Figure 9 are similar to those obtained using MCS. The same statements are made in the case of corrosion depth variation, where findings show that the performance of both meta-models is adequate, with greater accuracy utilizing the ANN-MCS technique.

#### 5.4. Relative Error in Comparison to Monte Carlo Simulation

## 6. Conclusions

- Although the MCS is an accurate technique for assessing the failure probability of corroded pipelines, the simulation offered is quite huge when compared to all other simulation and meta-model approaches.
- The risk of collapse failure of the three pipelines is shown to be more sensitive to changes in operating pressure. Obtaining the failure probability for specific operating pressures was discovered to be more challenging than adjusting the corrosion defect depths. This is because the operating pressure is the most critical factor in the limit-state functions that describes the load, and therefore any alterations will result in noticeable observations in the probability of failure findings.
- When compared to MCS, subset simulation was shown to be the most accurate simulation-based technique.
- The results showed that meta-models, particularly the ANN-MCS technique, produce very accurate results that match the MCS solutions almost completely. For the three corroded pipes, ANN-MCS has the lowest g-call (600) and the lowest relative error (percentage). Based on the overall results, ANN-MCS is regarded as the best performing technique for accurate reliability analysis of corroded pipes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**The effect of simulation number on the reliability outcomes, (

**a**) failure probability versus simulation number (N); (

**b**) CoV versus simulation number (N).

**Figure 7.**Variation of the reliability index values corresponding to various operating pressures using simulation-based approaches.

**Figure 8.**Variation of the reliability index corresponding to various corrosion depth to wall-thickness ratios using simulation-based approaches.

**Figure 9.**Variation of the reliability index values corresponding to various operating pressures using meta-models-based approaches.

**Figure 10.**Variation of the reliability index values corresponding to various corrosion depth to wall thickness ratios using meta-models-based approaches.

**Figure 11.**Estimated relative error percentage using simulation and meta-models (

**a**) versus operating pressures, (

**b**) versus corrosion to wall-thickness ratios.

Category | High | Mid | Low | |||
---|---|---|---|---|---|---|

Grade | X100 | X65 | X52 | |||

Random Variables | Description | Mean | Mean | Mean | CoV | Distribution |

$D$ | Outer diameter of the pipe, mm | 1320 | 762 | 914.4 | 0.03 | Normal |

$t$ | Wall thickness of the pipe, mm | 22.9 | 17.5 | 20.6 | 0.06 | Normal |

${\sigma}_{y}$ | Yield stress, MPa | 740 | 467 | 358 | 0.07 | Normal |

${\sigma}_{u}$ | Ultimate tensile strength, MPa | 813 | 576 | 455 | 0.08 | Lognormal |

$d$ | Depth of corroded defect, mm | 3.44–17.18 | 2.6–13.13 | 3.1–15.45 | 0.1 | Normal |

$L$ | Length of corroded defect, mm | 200 | 200 | 200 | 0.05 | Normal |

${P}_{0}$ | Operating pressure, MPa | 5–25 | 5–25 | 5–25 | 0.1 | Normal |

${\xi}_{High}$ | 0.079 | Lognormal | ||||

${\xi}_{Med}$ | Model error | 1.025 | 1.026 | 0.993 | 0.088 | Frechet |

${\xi}_{Low}$ | 0.021 | Gumbel |

**Table 2.**Comparative reliability analysis results using simulation methods at various operating pressures and a mean of corrosion defect geometries of d/t = 0.45 and L = 200 mm.

Reliability Analysis Results | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

MCS | IS | SS | DS | LS | |||||||

Pipeline Grade | P_{0} | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call |

X52 | 5 | 0 | 10-6 | 0 | 15,000 | 1.20 × 10^{−12} | 57,000 | 1.50 × 10^{−10} | 55,700 | 7.80 × 10^{−8} | 33,000 |

10 | 0.003045 | 10-6 | 0.003533 | 15,000 | 0.003258 | 15,000 | 0.00342 | 29,929 | 0.003025 | 6000 | |

15 | 0.2571 | 10-6 | 0.257 | 15,000 | 0.2463 | 6000 | 0.8515 | 17,143 | 0.2585 | 4800 | |

20 | 0.7947 | 10-6 | 0.7927 | 15,000 | 0.78467 | 6000 | 0.013855 | 19,176 | 0.203 | 4800 | |

25 | 0.97 | 10-6 | 0.97167 | 15,000 | 0.972 | 6000 | 0.13089 | 27,311 | 0.025 | 4800 | |

X65 | 5 | 0 | 10-6 | 0 | 15,000 | 6.34 × 10^{−7} | 30,000 | 4.18 × 10^{−7} | 38,039 | 1.20 × 10^{−6} | 7200 |

10 | 0.0065 | 10-6 | 0.0064 | 15,000 | 0.005954 | 15,000 | 0.0048 | 30,935 | 0.0065 | 6000 | |

15 | 0.1173 | 10-6 | 0.1161 | 15,000 | 0.121 | 9000 | - | - | 0.1179 | 6091 | |

20 | 0.3953 | 10-6 | 0.3944 | 15,000 | 0.41 | 6000 | 0.0634 | 12,816 | 0.394 | 4800 | |

25 | 0.671 | 10-6 | 0.669 | 15,000 | 0.673 | 6000 | 0.0028 | 11,219 | 0.32788 | 4800 | |

X100 | 5 | 0 | 10-6 | 0 | 15,000 | 2.90 × 10^{−8} | 36,000 | - | - | 4.90 × 10^{−8} | 12,006 |

10 | 0.00122 | 10-6 | 0.001333 | 15,000 | 0.001241 | 18,000 | - | - | 0.001347 | 10,914 | |

15 | 0.0489 | 10-6 | 0.048 | 15,000 | 0.05 | 9000 | - | - | 0.0495 | 10,550 | |

20 | 0.245 | 10-6 | 0.252 | 15,000 | 0.241 | 6000 | - | - | 0.24645 | 9078 | |

25 | 0.521 | 10-6 | 0.52 | 15,000 | 0.511 | 7618 | - | - | 0.478 | 7618 |

**Table 3.**Comparative reliability analysis results using simulation methods at various corrosion depth to wall-thickness ratios and a mean of operating pressure P

_{0}= 10 and corrosion length L = 200 mm.

Reliability Analysis Results | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

MCS | IS | SS | DS | LS | |||||||

Pipeline Grade | d/t | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call |

X52 | 0.15 | 1.83 × 10^{−4} | 106 | 0.0002 | 15,000 | 1.33 × 10^{−4} | 21000 | 1.40 × 10^{−4} | 44,400 | 1.90 × 10^{−4} | 6000 |

0.3 | 0.00076 | 106 | 0.00067 | 15,000 | 0.00078 | 18,000 | 0.00057 | 41,848 | 0.00079 | 6000 | |

0.45 | 0.003045 | 106 | 0.003533 | 15,000 | 0.003258 | 15,000 | 0.00342 | 29,929 | 0.003025 | 6000 | |

0.6 | 0.0105 | 106 | 0.0113 | 15,000 | 0.0105 | 12,000 | 0.0082 | 23,160 | 0.0107 | 5950 | |

0.75 | 0.033 | 106 | 0.032 | 15,000 | 0.0325 | 12,000 | 0.0251 | 20,430 | 0.03348 | 4800 | |

X65 | 0.15 | 9.96 × 10^{−4} | 106 | 0.001 | 15,000 | 1.14 × 10^{−3} | 18,000 | 1.30 × 10^{−3} | 36,100 | 9.80 × 10^{−4} | 7190 |

0.3 | 0.00216 | 106 | 0.0022 | 15,000 | 0.0021 | 15,000 | 0.0024 | 31,525 | 0.00224 | 6002 | |

0.45 | 0.0065 | 106 | 0.0064 | 15,000 | 0.005954 | 15,000 | 0.0048 | 30,935 | 0.0065 | 6000 | |

0.6 | 0.021 | 106 | 0.021 | 15,000 | 0.0215 | 12,000 | 0.0143 | 26,696 | 0.021 | 6000 | |

0.75 | 0.073 | 106 | 0.074 | 15,000 | 0.074 | 9000 | 0.046 | 23,207 | 0.073 | 6000 | |

X100 | 0.15 | 1.77 × 10^{−4} | 106 | 0.000133 | 15,000 | 1.36 × 10^{−4} | 21000 | - | - | 1.91 × 10^{−4} | 11,900 |

0.3 | 0.000418 | 106 | 0.00013 | 15,000 | 0.00046 | 18,000 | - | - | 0.00044 | 12,006 | |

0.45 | 0.00122 | 106 | 0.001333 | 15,000 | 0.001241 | 18,000 | - | - | 0.001347 | 10,914 | |

0.6 | 0.0049 | 106 | 0.0047 | 15,000 | 0.0045 | 15,000 | - | - | 0.005 | 10,914 | |

0.75 | 0.02 | 106 | 0.02 | 15,000 | 0.0207 | 12,000 | - | - | 0.0208 | 9458 |

**Table 4.**Comparative reliability analysis results using meta-models at various operating pressures and a mean of corrosion defect geometries of d/t = 0.45 and L = 200 mm.

Reliability Analysis Results | |||||||
---|---|---|---|---|---|---|---|

MCS | Kriging-MCS | ANN-MCS | |||||

Pipeline Grade | P_{0} | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call |

X52 | 5 | 0 | 106 | 0 | 600 | 0 | 600 |

10 | 0.003045 | 106 | 0.0029 | 600 | 0.0029 | 600 | |

15 | 0.2571 | 106 | 0.2565 | 600 | 0.2624 | 600 | |

20 | 0.7947 | 106 | 0.7942 | 600 | 0.8038 | 600 | |

25 | 0.97 | 106 | 0.9753 | 600 | 0.9735 | 600 | |

X65 | 5 | 0 | 106 | 0 | 600 | 0 | 600 |

10 | 0.0065 | 106 | 0.0054 | 600 | 0.0062 | 600 | |

15 | 0.1173 | 106 | 0.12 | 600 | 0.12 | 600 | |

20 | 0.3953 | 106 | 0.39 | 600 | 0.386 | 600 | |

25 | 0.671 | 106 | 0.6613 | 300 | 0.6632 | 600 | |

X100 | 5 | 0 | 106 | 0 | 600 | 0 | 600 |

10 | 0.00122 | 106 | 0.0011 | 600 | 0.0012 | 600 | |

15 | 0.0489 | 106 | 0.0535 | 600 | 0.0514 | 600 | |

20 | 0.245 | 106 | 0.246 | 600 | 0.244 | 600 | |

25 | 0.521 | 106 | 0.5195 | 300 | 0.535 | 600 |

**Table 5.**Comparative reliability analysis results using meta-models at various corrosion depth to wall-thickness ratios and a mean of operating pressure P

_{0}= 10 and corrosion length L = 200 mm.

Reliability Analysis Results | |||||||
---|---|---|---|---|---|---|---|

MCS | Kriging-MCS | ANN-MCS | |||||

Pipeline Grade | d/t | P_{f} | g-Call | P_{f} | g-Call | P_{f} | g-Call |

X52 | 0.15 | 1.83 × 10^{−4} | 106 | 1.00 × 10^{−4} | 600 | 1.00 × 10^{−4} | 600 |

0.3 | 0.00076 | 106 | 0.0009 | 600 | 0.0008 | 600 | |

0.45 | 0.003045 | 106 | 0.0029 | 600 | 0.0029 | 600 | |

0.6 | 0.0105 | 106 | 0.0099 | 600 | 0.0098 | 600 | |

0.75 | 0.033 | 106 | 0.0316 | 600 | 0.034 | 600 | |

X65 | 0.15 | 9.96 × 10^{−4} | 106 | 0 | 600 | 6.00 × 10^{−4} | 600 |

0.3 | 0.00216 | 106 | 0.0002 | 600 | 0.0023 | 600 | |

0.45 | 0.0065 | 106 | 0.0054 | 600 | 0.0062 | 600 | |

0.6 | 0.021 | 106 | 0.0152 | 600 | 0.0184 | 600 | |

0.75 | 0.073 | 106 | 0.0755 | 600 | 0.071 | 600 | |

X100 | 0.15 | 1.77 × 10^{−4} | 106 | 0 | 600 | 2.00 × 10^{−4} | 600 |

0.3 | 0.000418 | 106 | 0 | 600 | 0.0006 | 600 | |

0.45 | 0.00122 | 106 | 0.0011 | 600 | 0.0012 | 600 | |

0.6 | 0.0049 | 106 | 0.0016 | 600 | 0.005 | 600 | |

0.75 | 0.02 | 106 | 0.0202 | 600 | 0.0215 | 600 |

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## Share and Cite

**MDPI and ACS Style**

Seghier, M.E.A.B.; Spyridis, P.; Jafari-Asl, J.; Ohadi, S.; Li, X.
Comparative Study on the Efficiency of Simulation and Meta-Model-Based Monte Carlo Techniques for Accurate Reliability Analysis of Corroded Pipelines. *Sustainability* **2022**, *14*, 5830.
https://doi.org/10.3390/su14105830

**AMA Style**

Seghier MEAB, Spyridis P, Jafari-Asl J, Ohadi S, Li X.
Comparative Study on the Efficiency of Simulation and Meta-Model-Based Monte Carlo Techniques for Accurate Reliability Analysis of Corroded Pipelines. *Sustainability*. 2022; 14(10):5830.
https://doi.org/10.3390/su14105830

**Chicago/Turabian Style**

Seghier, Mohamed El Amine Ben, Panagiotis Spyridis, Jafar Jafari-Asl, Sima Ohadi, and Xinhong Li.
2022. "Comparative Study on the Efficiency of Simulation and Meta-Model-Based Monte Carlo Techniques for Accurate Reliability Analysis of Corroded Pipelines" *Sustainability* 14, no. 10: 5830.
https://doi.org/10.3390/su14105830