# A Quantitative Framework for Propagation Paths of Natech Domino Effects in Chemical Industrial Parks: Part I—Failure Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Propagation Rules of Domino Effects Triggered by Natech Events

#### 2.1. Interaction between Equipment Units

- (i)
- Bidirectional-acting relationship. The intensity of the escalation vector generated by an equipment unit is larger than the escalation threshold of the target equipment unit, as is shown by T1 and T3 in Figure 1a. When T1 first occurs in an accident, it may damage T3, and when T3 first occurs in an accident, it may cause T1 to be damaged.
- (ii)
- Single-acting relationship. The intensity of the escalation vector generated by one equipment unit is larger than the escalation threshold of the target equipment unit, while the intensity of the escalation vector generated by the other equipment unit is less than the escalation threshold of the target equipment unit, as is shown by T1 and T4 in Figure 1a. When T1 occurs first in an accident, it may damage T4, while when T4 occurs first in an accident, it will not lead to the destruction of T1.
- (iii)
- No interaction relationship. The intensity of the escalation vector generated by the two equipment units is less than the escalation threshold of the target equipment unit. For example, T1 and T2 in Figure 1a cannot damage each other in the case of an accident.

#### 2.2. Synergistic Effects

## 3. Methodology

#### 3.1. Failure State Combination of Primary Equipment Units Caused by Natural Disasters

#### 3.2. Propagation Path Probabilities of Domino Effects Triggered by Natural Disasters

#### 3.2.1. Equipment Escalation Matrix and Escalation Probability Matrix

_{ij}= 1, otherwise EQ

_{ij}= 0.

#### 3.2.2. Failure State Transition Probability Matrix

- (i)
- The primary failure state combination caused by natural disasters is uncertain, as shown in Figure 2, and all seven failure state combinations have the possibility to be the primary failure state.
- (ii)
- Only undamaged equipment units can be escalated to damaged equipment units, but damaged equipment units cannot be escalated to undamaged equipment units; that is, they are non-maintainable equipment units in the propagation process. For example, as shown in Figure 2, T2 is not damaged in S1 failure state, and can be damaged when escalating to S4, but S4 cannot be converted to S1.
- (iii)
- The failure state with a large number of damaged equipment units cannot be converted into a failure state with a smaller number of damaged equipment units. For example, S7 cannot be converted to S5, and S5 cannot be converted to S2.
- (iv)
- The failure states of the same number of damaged equipment units cannot be mutually converted, such as S4, S5, and S6.
- (v)
- The state combination with a small number of damaged equipment units will escalate to the state combination with a larger number of damaged equipment units, and the state combination with a larger number of damaged equipment units may escalate directly from the primary state combination (such as S1 → S7), or from the intermediate state combination with a smaller number of damaged equipment units (such as S1 → S4 → S7). The specific possible evolution path is shown in Figure 2 (a total of 18 possible paths, as listed in Supplementary Material S2).

_{ij}(k) depends on the comparison result of the failure state of the k-th equipment unit in the two failure state combinations $S(i)$ and $S(j)$, as shown in Table 1.

#### 3.2.3. Propagation Path Probabilities of Domino Effects Based on Directed Graph

- The failure state combination S(i), i = 1, is set as the primary scenario, and the path sets are stored in the path cell array RA, which is initialized to a null set.
- The main path matrix MP is used to store the ID of primary failure state combination i, and its adjacent node set matrix is obtained by the function successors (ST, MP(end)), which is stored in the cell array AP.
- If the main path matrix MP is not a null set, the last item of MP will be pushed onto the top of matrix AT, which indicates the un-accessed adjacency failure states, while the primary failure state will otherwise be reset to S(i + 1).
- If AT is a null set, whereby namely there is no adjacent failure states or all the adjacent failure states have been visited, such that the path cell array RA will be used to store the new path in main path MP, and its corresponding path probability will be obtained and put in PR, then the last items of MP and AP will be deleted, and the algorithm program will continue.
- If AT is a null set, the last item of AT will be the top item of MP, and other items of AT will be put in AP, then the adjacent failure states of the new items in MP will be put in the top item of AP.
- If all the primary failure states have been visited, the algorithm program will be terminated, and all the possible paths and their probabilities can be obtained from the cell arrays RA and PR.

## 4. Case Study

#### 4.1. Overview of the Case Study

#### 4.2. Accident Consequence Analysis and Accident Escalation Probability

#### 4.3. Primary Failure State Assessment

^{−10}is considered almost impossible and its impact on risk can be neglected [38], this paper only considers the failure state combinations with primary failure probabilities larger than 10

^{−10}. There are thirty-five primary accident scenarios with probabilities larger than 10

^{−10}, and the top ten most likely primary failure state combinations are shown in Table 3. It can be seen from Table 3 that the most likely primary failure state combination is S24. In this combination, T3 and T6 have the highest probability of being damaged by flood, so T3 and T6 being simultaneously damaged has the highest probability, which is $3.03\times {10}^{-3}$. Among the top ten most likely primary accident scenarios, the probability ranges from 10

^{−3}to 10

^{−7}, and most of the combinations have T3 or T6 tanks, indicating that these two tanks have a large impact on the primary failure scenario, which can be considered as the most likely primary equipment unit.

#### 4.4. Propagation Path Analysis

#### 4.4.1. Most Likely Propagation Path

#### 4.4.2. Propagation Paths at All Levels

^{−10}.

#### 4.4.3. Failure Probability Distribution of Tanks at All Levels

#### 4.5. Accident Scenario Analysis

- Impact of Natech events

- 2.
- Impact of domino effects

- 3.
- Impact of multi-level domino effects

## 5. Conclusions

^{−3}to 10

^{−7}, and most of the top ten primary failure state combinations contain T3 or T6 tanks, indicating that these two tanks can be considered as the most likely primary equipment unit. The probability analysis of the propagation path of the domino effects induced by natural disasters in the case study shows that the primary failure state has the greatest impact on the propagation probability of domino effects, and the cumulative failure probability of each tank is greatly affected by the primary failure state. The number of propagation paths of the single-level domino effect is the least, but the cumulative failure probability is the largest, and the influence of the multi-level domino effect cannot be ignored. The failure probability analysis of each tank in different scenarios shows that Natech events and multi-level domino effects have a significant impact on the failure probability of tanks, even resulting in the failure probability of some tanks increasing by several orders of magnitude. Ignoring the impacts both of natural disasters and the multi-level domino effect will lead the failure probability of tanks to be underestimated.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**An example of the evolution process of the accidental chains of domino effects triggered by natural disasters.

**Figure 2.**Schematic diagram of propagation paths of domino effects caused by natural disasters for three key equipment units.

**Table 1.**The possible results of the comparison of the k-th units of the Boolean vectors $S(i)$ and $S(j)$.

$\mathit{S}{(\mathit{i})}_{\mathit{k}}$ | $\mathit{S}{(\mathit{j})}_{\mathit{k}}$ | CR_{ij}(k) | State Description |
---|---|---|---|

1 | 0 | 0 | Impossible to escalate |

1 | 1 | 1 | Damaged, no impact |

0 | 1 | ${P}_{k}^{i}$ | Damaged probability |

0 | 0 | 1 − ${P}_{k}^{i}$ | Undamaged probability |

Tank ID | Volume (m^{3}) | Diameter (m) | Height (m) | Inventory | Tank Type | $\mathbf{Density}\text{}({\mathbf{kg}/\mathbf{m}}^{3})$ | Filling Level | Failure Probability |
---|---|---|---|---|---|---|---|---|

T1 | 10,173 | 30 | 14.4 | Gasoline | Internal floating roof | 750 | 0.58 | $5.21\times {10}^{-6}$ |

T2 | 10,173 | 30 | 14.4 | Gasoline | Internal floating roof | 750 | 0.36 | 0.0343 |

T3 | 10,173 | 30 | 14.4 | Gasoline | Internal floating roof | 750 | 0.28 | 0.6564 |

T4 | 11,434 | 34 | 12.6 | Naphtha | Internal floating roof | 750 | 0.56 | $1.69\times {10}^{-4}$ |

T5 | 11,434 | 34 | 12.6 | Naphtha | Internal floating roof | 750 | 0.73 | $4.51\times {10}^{-7}$ |

T6 | 18,086 | 40 | 14.4 | Crude oil | External floating roof | 810 | 0.23 | 0.9576 |

T7 | 18,086 | 40 | 14.4 | Crude oil | External floating roof | 810 | 0.45 | $4.59\times {10}^{-4}$ |

T8 | 18,086 | 40 | 14.4 | Crude oil | External floating roof | 810 | 0.64 | $2.39\times {10}^{-7}$ |

NO. | Primary Failure State | Primary Equipment | Probability of Primary Failure State | NO. | Primary Failure State | Primary Equipment | Probability of Primary Failure State |
---|---|---|---|---|---|---|---|

1 | S24 | T3, T6 | $3.03\times {10}^{-3}$ | 6 | S16 | T2, T3 | $4.77\times {10}^{-6}$ |

2 | S6 | T6 | $1.59\times {10}^{-3}$ | 7 | S2 | T2 | $2.50\times {10}^{-6}$ |

3 | S3 | T3 | $1.34\times {10}^{-3}$ | 8 | S80 | T3, T6, T7 | $1.39\times {10}^{-6}$ |

4 | S60 | T2, T3, T6 | $1.08\times {10}^{-3}$ | 9 | S34 | T6, T7 | $7.29\times {10}^{-7}$ |

5 | S19 | T2, T6 | $5.64\times {10}^{-5}$ | 10 | S74 | T3, T4, T6 | $5.13\times {10}^{-7}$ |

NO. | Propagation Path | Path Probability | NO. | Propagation Path | Path Probability |
---|---|---|---|---|---|

1 | S24 → S80 | $5.41\times {10}^{-4}$ | 6 | S24 → S154 | $2.10\times {10}^{-4}$ |

2 | S6 → S34 | $5.16\times {10}^{-4}$ | 7 | S24 → S115 | $1.80\times {10}^{-4}$ |

3 | S24 → S135 | $3.92\times {10}^{-4}$ | 8 | S24 → S204 | $1.52\times {10}^{-4}$ |

4 | S6 → S80 | $3.89\times {10}^{-4}$ | 9 | S24 → S60 | $1.51\times {10}^{-4}$ |

5 | S6 → S24 | $2.94\times {10}^{-4}$ | 10 | S24 → S151 | $1.44\times {10}^{-4}$ |

Level | Number of Paths | Path Cumulative Probability | Maximum Path Probability | Maximum Path | Minimum Path Probability |
---|---|---|---|---|---|

L1 | 873 | $4.30\times {10}^{-3}$ | $5.41\times {10}^{-4}$ | S24 → S80 | $4.75\times {10}^{-13}$ |

L2 | 8222 | $4.10\times {10}^{-3}$ | $1.29\times {10}^{-4}$ | S6 → S34 → S157 | $7.44\times {10}^{-15}$ |

L3 | 33,553 | $3.30\times {10}^{-3}$ | $3.54\times {10}^{-5}$ | S24 → S135 → S251 → S255 | $1.50\times {10}^{-16}$ |

L4 | 69,644 | $1.50\times {10}^{-3}$ | $1.10\times {10}^{-5}$ | S6 → S34 → S157 → S244 → S255 | $2.49\times {10}^{-18}$ |

L5 | 77,536 | $3.95\times {10}^{-4}$ | $4.35\times {10}^{-6}$ | S6 → S34 → S157 → S217 → S254 → S255 | $2.45\times {10}^{-19}$ |

L6 | 44,238 | $5.12\times {10}^{-5}$ | $8.48\times {10}^{-7}$ | S6 → S34 → S157 → S217 → S246 → S254 → S255 | $6.57\times {10}^{-20}$ |

L7 | 10,176 | $2.45\times {10}^{-6}$ | $1.16\times {10}^{-7}$ | S6 → S34 → S92 → S157 → S217 → S246 → S254 → S255 | $6.06\times {10}^{-20}$ |

Accident Scenarios | Natech Events | Multi-Level Domino Effects | Single-Level Domino Effects | Conventional Accident Scenarios |
---|---|---|---|---|

a | √ | × | × | √ |

b | √ | √ | × | √ |

c | √ | × | √ | √ |

d | × | × | × | √ |

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**MDPI and ACS Style**

Yang, Y.; Chen, G.; Zhao, Y.
A Quantitative Framework for Propagation Paths of Natech Domino Effects in Chemical Industrial Parks: Part I—Failure Analysis. *Sustainability* **2023**, *15*, 8362.
https://doi.org/10.3390/su15108362

**AMA Style**

Yang Y, Chen G, Zhao Y.
A Quantitative Framework for Propagation Paths of Natech Domino Effects in Chemical Industrial Parks: Part I—Failure Analysis. *Sustainability*. 2023; 15(10):8362.
https://doi.org/10.3390/su15108362

**Chicago/Turabian Style**

Yang, Yunfeng, Guohua Chen, and Yuanfei Zhao.
2023. "A Quantitative Framework for Propagation Paths of Natech Domino Effects in Chemical Industrial Parks: Part I—Failure Analysis" *Sustainability* 15, no. 10: 8362.
https://doi.org/10.3390/su15108362