Vulnerability Assessment of Process Vessels in the Event of Hurricanes

Abstract

Hurricanes are multi-hazard natural hazards that can cause severe damage to chemical and process plants via individual or combined impact of strong winds, torrential rainfall, floods, and hitting waves especially in coastal areas. To assess and manage the vulnerability of process plants, failure modes and respective failure probabilities both before and after implementing safety measures should be assessed. However, due to the uncertainties arising from interdependent failure modes and lack of accurate and sufficient historical data, most conventional quantitative risk assessment techniques deliver inaccurate results, which in turn lead to inaccurate risk assessment and thus ineffective or non-cost-effective risk management strategies. Bayesian network (BN) is a probabilistic technique for reasoning under uncertainty with a variety of applications is system safety, reliability engineering, and risk assessment. In this chapter, applications of BN to vulnerability assessment and management of process vessels in the event of hurricanes are demonstrated and discussed.

1. Introduction

The impact of natural disasters such as earthquakes, hurricanes, and floods on chemical and process plants can be quite catastrophic due to the possibility of damage to process units and subsequent release of large quantity of hazardous chemicals which may lead to disastrous environmental pollution [1] or fires and explosions [2, 3, 4, 5, 6, 7, 8]. Such technological accidents, which involve release of chemical substances into the environment, due to the impact of natural hazards are also known as Na-techs [9]. Katrina and Harvey are the hurricanes, which hit the U.S. in 2005 and 2017, respectively, are the first and second costliest events in the U.S. history and considered as noteworthy examples for Na-techs and their substantial damage to the environment and the industry. Katrina (2005) caused extensive damage to chemical and process plants and resulted in release of ∼ 8 million gallons of oil, making it the second largest environmental pollution in the U.S after the Deep Horizon disaster in the Gulf of Mexico in 2010. Similarly, Harvey (2017) caused substantial damage to refineries and petrochemical plants in the Huston Greater Area, leading to release of ∼ 2000 tons of chemicals into the environment as of September 2017 [10, 11, 12]. Table 1 includes a list of industrial plants that suffered direct damages during Hurricane Harvey [10].

Company name	Type of industry	Type of toxic chemicals
Invista	Plastic Manufacturer	Polytetramethylene ether glycol, Tetrahydrofuran, and 1,4 Butane Diol
LyondellBasell	Plastic Manufacturer	Polymers
LyondellBasell	Refinery	Gasoline, Diesel, Benzene, Paraxylene, Orthoxylene
Celanese Chemical	Chemicals Manufacturer	Methanol
Dow	Plastic Manufacturer	Polyurethanes
Eastman	Plastic Manufacturer	Non-phthalate plasticizers
Indorama Ventures	Chemicals Manufacturer	Ethylene oxide
Indorama Ventures	Plastics Manufacturer	Glycols
Arkema	Chemicals Manufacturer	Sulfuric substances
ExxonMobil	Refinery	Acrylic acid
Enterprise	Refinery	Hydrocarbon fuels
Chevron Phillips	Petrochemical	Hydrocarbon fuels
Shell	Refinery	NGL fractionators

Table 1.

Type of industrial plants damaged during the Hurricane Harvey [10].

Among the process units, storage tanks – and atmospheric storage tanks in particular – have reportedly been the most vulnerable type of industrial vessels during hurricanes both due to their thin shell, which makes them very susceptible to lateral forces exerted by wind and flood, and due their high volume-weight ratio, which makes them susceptible to buoyancy force exerted by flood and heavy rainfall as components of hurricanes [11, 12, 13, 14]. During Harvey in 2017, a total of 47 incidents of tank damage was reported with a total spillage 2 million liters of gasoline, crude oil, and other chemicals [15].

Damage to storage tanks can cause considerable environmental pollutions. During Katrina, ruptured storage tanks released several millions of gallons of oil while during Harvey, more than two dozen storage tanks ruptured, spilling ∼ 550 cubic meters of crude oil and gasoline into surrounding areas [10]. Figure 1 shows the release of crude oil from to storage tanks, which have been removed from their foundations due to flood forces during Hurricane Katrina (2005).

Figure 1.

Release of crude oil due to displacement of storage tanks from their bases due to storm surges during Hurricane Katrina, 2005 [10].

Environmental pollution, however important, is not the only concern with regard to Na-techs. Release of petrochemicals – which in most cases are highly flammable and explosive – from damaged storage tanks can readily result in fire or explosion particularly in the presence of a variety of ignition sources and hot surfaces in the chemical and process plants. Such fires and explosions can easily go out of control due to large quantities of released chemicals and vicinity of storage tanks, capable of resulting in a sequence of fires and explosions – known as domino effects [16, 17].

Compared to conventional technological accidents that are caused by random failures or human error, risk assessment and management of Na-techs are more challenging [9, 18] due to the presence of more uncertain parameters and interdependent and complex failure mechanisms. The uncertainty consists of aleatory uncertainty arising from the randomness of natural disasters, resistance and vulnerability of process vessels to combined forces – wind, flood, rainfall, lightning – during hurricanes, and epistemic uncertainty arising from lack of sufficiently accurate and large historical databases.

Probability theory has effectively been used to account for uncertainties arising from environmental and operational random variable involved in risk assessment of Na-techs [19, 20, 21, 22, 23, 24, 25]. Even given the probability of random variables (depth of flood inundation, speed of wind, etc.), conventional quantitative risk assessment (QRA) techniques such as fault tree analysis fall short in modeling interdependent failure modes and common root causes arising from concurrent impact of hurricane forces on process vessels.

Bayesian network (BN) is a probabilistic tool for handling uncertainty and modeling dependencies [26, 27]. Previous studies demonstrated that BN can effectively accommodate and handle a variety of probabilities in the form of point probabilities [28], probability distributions [29, 30], fuzzy probabilities [31], and imprecise probabilities [32, 33].

In the present chapter, we demonstrate how BN can be employed to assessing the vulnerability of process vessels to the forces of hurricanes. Section 2 reviews the constituent natural hazards in the event of a hurricane and the resulting failure modes with emphasis on atmospheric storage tanks as the most vulnerable type of process vessels. Section 3 reviews the fundamentals of BN. Section 4 demonstrates the application of BN to modeling, combining, and calculating the failure probabilities of an illustrative storage tank. The main outcomes of the study are summarized in Section 5.

2. Failure mechanisms during hurricanes

According to the Hurricane Hazard Mitigation Handbook for Public Facilities [34], winds, rains, and flood (or storm surge in coastal areas) are the three natural hazards or forces of hurricanes responsible for damages, including Na-techs. In line with [34], in the present study, hurricanes are considered as multi-hazard natural disasters consisting of strong winds, floods, and heavy rainfalls.

2.1 Flood

Compared to seismic or wind related Na-techs, the one caused by floods have received relatively less attention; this has mainly been due to the scarcity of experimental or high-resolution field observations. Only for a limited number of floods were the flood inundation depths registered and even for fewer floods were the flood speeds recorded.

According to [14], 272 flood related Na-techs were reported in Europe and the U.S. from 1960 to 2007, with the aboveground storage tanks as the most frequently damaged equipment (74% of cases), including atmospheric storage tanks, floating roof tanks, and pressurized tanks. Displacement of storage tanks (due to rigid sliding or floatation) and subsequent disconnection or rupture of attached pipelines, shell rupture due to lateral forces or debris impact, and the collapse of equipment were reportedly the main failure modes during floods. Godoy [13] reported similar failure modes while investigating the process plants affected by the Hurricanes Katrina in Louisiana and Texas, U.S. These failure modes are depicted in Figure 2.

Figure 2.

Flood related failure modes for storage tanks.

Due to a lack of sufficiently accurate and reliable historical data, a majority of previous studies has employed analytical or numerical techniques to model the abovementioned failure modes and estimate the respective failure probabilities [12, 20, 22, 23, 24, 25].

In these studies, limit state equations (LSEs) that were developed based on “physics of failure” models were used to calculate the failure probabilities. Such physics-of-failure models consider forces (loads) exerted by floods on target vessels (e.g., buoyancy force, which tends to float the tank) and the resistance of the vessel (e.g., total weight of the tank, which tend to resist the floatation). The developed LSEs can then be coupled with Monte Carlo simulation to generate artificial databases that in turn can be used to develop fragility curves [23] or be combined with other QRA techniques, such as BN [12, 24] for probabilistic reasoning.

2.2 Wind

Shell buckling is the main failure mode caused by wind [35] especially if the storage tank is empty or less than half-filled. According to Godoy [13], several storage tanks were reported to suffer shell buckling during Hurricane Katrina, mostly due to strong winds rather than flood. High winds can enforce critical pressures on tank shells. The magnitude of this pressure depends on several parameters, the most important of which being the wind speed and the shape and size of the storage tank.

Considering cylindrical storage tanks, however, the height of the tank does not play a key role [36, 37] as does the tank’s circumference. As such, the wind pressure is usually considered as constant along the tank’s height [38]. Similar to the flood related failure modes, it has been a common practice to develop LSEs for wind related failure mode – i.e., shell buckling and shell damage due to windborne debris – based on the load-resistance relationships. Considering the shell buckling, for instance, external pressure exerted by wind (i.e., load) and the internal pressure of the tank (i.e., resistance) were taken into account to develop the related LSE [12].

2.3 Rainfall

Heavy rainfalls before, during and after passage of hurricanes were rarely accounted for as a separate natural hazard directly capable of causing damage to process vessels. In that sense, contribution of heavy rainfalls to Na-techs has been indirectly taken into account via the runover of water bodies and rivers due to heavy rainfall and consequent flooding of process plants.

Hurricane Harvey, however, demonstrated the potential of heavy rainfall as a natural hazard, which can cause significant damage to process vessels standalone or in combination with other forces of the hurricane. Large amount of rainfall that fell during Hurricane Harvey caused damage to 400 “floating roof storage tanks”, including sinking of 14 roofs. A floating-roof storage tank is a tank the roof of which is not fixed and rather floats on the surface of chemicals stored inside the tank.

3. Bayesian network

Bayesian network (BN) [26] is a directed acyclic graph consisting of random variable nodes and directed edges connecting the nodes, with the edges directed from parents to children. A BN can mathematically be defined as BN = (G, θ), where G is the network structure – nodes and edges – and θ is the network parameters – conditional probabilities of child nodes given their parents. Figure 3 shows an illustrative BN consisting of 5 nodes. X1 and X2 are the root nodes (equivalent to the basic events of a fault tree), X3 is an intermediate event (equivalent to an intermediate event of a fault tree), and X4 and X5 are the leaf nodes (equivalent to the top event of a fault tree). The root nodes are assigned marginal probability distributions while the other nodes are assigned conditional probability distributions. Such conditional probabilities reflect the type (usually causal) and strength of the impact parent nodes have on their child nodes.

Figure 3.

BN consisting of 5 random variables (nodes). X1 and X2 are the root nodes and parents of X3. X3 is an intermediate node, the child of X1 and X2 and the parent of X4 and X5. X4 and X5 are the leaf nodes and the children of X3.

Considering the local dependencies and chain rule, the joint probability distribution of the random variables in a BN can be presented as the product of marginal and conditional probabilities considering only the immediate parents of a child node:

$$P\left(X_1,X_2,\dots ,X_n\right)=\prod _{i=1}^{n}P\left(X_i|\mathit{pa}\left(X_i\right)\right)$$E1

For example, for the BN in Figure 3, P (X1, X2, X3, X4, X5) = P (X1) P(X2) P(X3|X1, X2) P(X4|X3) P(X5|X3). BN can be used for either forward or backward reasoning. In forward reasoning, for example, the knowledge about X1 can be used to deduce about X5, whereas in backward reasoning, the knowledge about X5 can be used to infer about X1. The main advantages of BN over conventional techniques such as fault tree analysis are its capability of considering conditional dependencies, handling multistate variables, and applying Bayes’ rule for belief updating (backward reasoning) [28]. Using the Bayes’ theorem, BN can update the probabilities assigned a priori to the random variables in the presence of new information about the states of its nodes:

$$P\left(X|E\right)=\frac{P\left(X,E\right)}{P\left(E\right)}$$E2

The parameters of the BN—the marginal and conditional probabilities needed to quantify the model—can be assigned by subject matter experts or be estimated from a dataset by applying machine learning techniques such as the maximum likelihood estimation. Table 2 summarizes some of the advantages of BN against fault tree.

Fault tree	Bayesian network
Has only one top event	Can have more than one leaf node
Handles only binary variables	Handles a variety of multi-state variable
Assumes variables are independent	Can consider dependencies
N/A	Conducts probability updating
Can be used only for forward reasoning	Can be used for both forward and backward analyses

Table 2.

Modeling advantages of BN as opposed to fault tree.

4. Vulnerability assessment of storage tanks

As previously discussed, one way to analyze the failure mechanism and to generate failure data required for probabilistic vulnerability assessment of process vessels is application of “physics of failure” to develop LSEs. The benefit of LSEs is twofold: From one side, they help better understand random variables that contribute to the failure mechanisms and should thus be considered in the BN, and from the other side, they can be coupled with, for instance, Monte Carlo simulation to generate datasets required for estimating the parameters—especially, the conditional probabilities—of the BN [12, 23, 24].

To demonstrate the abovementioned methodology, we consider floatation as one of the frequently reported failure modes (failure mechanism) for above-ground atmospheric storage tanks. Since such storage tanks are usually unanchored (not bolted or cemented to their foundations), the buoyancy force of flood in Eq. (3) can make the storage tank float should it overcome the total weight of the tank—i.e., the weight of the tank in Eq. (4) plus the weight of chemical in the tank in Eq. (5)—as the only resisting force. Considering the bouncy as the only moving force and the total weight of the tank as the only resistance, the LSE of the tank due to the floatation can be modeled as in Eq. (6):

$$F_B={\rho }_w.g.\pi \frac{D^2}{4}S$$E3

$$W_T={\rho }_T.g.\left(\mathit{\pi DH}+2\pi \frac{D^2}{4}\right).t$$E4

$$W_L={\rho }_L.g.\pi \frac{D^2}{4}h$$E5

$$\mathit{LSE}=F_B-W_L-W_T$$E6

where $F_B$ is the buoyancy force; $W_T$ and $W_L$ are the weight of the tank and the contained chemical, respectively; ${\rho }_w$, ${\rho }_T$, and ${\rho }_L$ are, respectively, the density of flood water, of the tank structure’s material, and of the chemical; g is the gravitational acceleration; D and H are, respectively, the diameter and height of the tank; S is the depth of flood; h is the depth of chemical in the tank; t is the shell thickness of the tank. Given the LSE in Eq. (6), if LSE > 0, or simply $F_B>W_L+W_T$, the tank floats. The parameters in Eqs. (3)–(6) are depicted in Figure 4. Considering the depth of flood (S) and the depth of chemical (h) as the only random variables (the other parameters can reasonably be considered as constants), $F_B>W_L+W_T$ can be further simplified as:

Figure 4.

Parameters used to develop the LSE required for floatation failure mechanism of an unanchored atmospheric storage tank [23].

$$S>\mathit{ah}+b$$E7

where a and b are constants:

$$a=\frac{{\rho }_L}{{\rho }_w}$$E8

$$b=\frac{W_T}{{\rho }_w.g.\pi .\frac{D^2}{4}}$$E9

According to Eq. (7), the probability of floatation $P_{\mathit{\text{float}}}$ can be calculated as:

$$P_{\mathit{\text{float}}}=P\left(S>\mathit{ah}+b\right)=1-P\left(S<\mathit{ah}+b\right)$$E10

Given the probability distribution functions of S and h as $f_S\left(S\right)$ and $f_h\left(h\right)$, and considering that the level of chemical in the tank cannot exceed H (in practice, 15-20% of the tank’s height is usually left for safety purposes), Eq. (10) can be expanded as:

$$P_{\mathit{\text{float}}}=1-P\left(S<\mathit{ah}+b\right)=1-\underset{0}{\overset{H}{\int }}\left({\int }_0^{\mathit{ah}+b}{f}_S\left(S\right).\mathit{dS}\right)f_h\left(h\right)\mathit{dh}$$E11

4.1 Analytical approach

For some types of probability distributions, P(S < ah +b) can easily be calculated with no need for calculating the double integral in Eq. (11). One type of such probability distributions is Normal distribution. Given two normally distributed random variables as X ∼ Normal (${\mu }_x,{\sigma }_x$) and Y ∼ Normal (${\mu }_y,{\sigma }_y$), any linear function of X and Y is a normally distributed random variable as Z ∼ Normal (${\mu }_z,{\sigma }_z$). For instance, if Z = mX + nY, the mean value ${\mu }_z$ and standard deviation ${\sigma }_z$ of Z can be calculated as:

$${\mu }_z=m{\mu }_x+n{\mu }_y$$E12

$${\sigma }_z=\sqrt{m^2.{\sigma }_x^2+n^2.{\sigma }_y^2}$$E13

Therefore, if S ∼ Normal (${\mu }_s,{\sigma }_s$) and h ∼ Normal (${\mu }_h,{\sigma }_h$), then Q = S – ah is a linear function of S and h and thus normally distributed as:

$${\mu }_Q={\mu }_s-a{\mu }_h$$E14

$${\sigma }_Q=\sqrt{{\sigma }_s^2+a^2.{\sigma }_h^2}$$E15

As a result:

$$\mathrm{P}\left(\mathrm{S}<\mathrm{ah}+\mathrm{b}\right)=\mathrm{P}\left(\mathrm{S}-\mathrm{ah}<\mathrm{b}\right)=\mathrm{P}\left(\mathrm{Q}<\mathrm{b}\right)={\mathrm{\Phi }}_Q\left(b\right)$$E16

where ${\mathrm{\Phi }}_Q\left(.\right)$ is the cumulative distribution function for normal distribution. Consequently:

$$P_{\mathit{\text{float}}}=P\left(S>\mathit{ah}+b\right)=1-P\left(S<\mathit{ah}+b\right)=1-{\mathrm{\Phi }}_Q\left(b\right)$$E17

To demonstrate the application of Eq. (17), consider an unanchored crude oil storage tank with diameter of D = 91m, height of H = 6m, and shell thickness of t = 15mm. The characteristics of the tank, crude, and flood are presented in Table 3.

Parameter	Symbol (unit)	Value or probability distribution
Tank diameter	D (m)	91
Tan height	H (m)	6
Shell thickness	t (mm)	15
Density of shell material	${\rho }_T$ (kg/m^3)	7900
Density of flood water	${\rho }_w$ (kg/m^3)	1024
Density of crude oil	${\rho }_L$ (kg/m^3)	900
Depth of crude in the tank	h (m)	fs(S) = Normal (1.0, 1.0)
Depth of flood water	S (m)	fh(h) = Normal (1.5, 0.1)

Table 3.

Parameters used for the illustrative storage tank and hitting flood.

Given the values in Table 3, the values of a and b are calculated via Eqs. (8) and (9) as a = 0.88 and b = 0.26. Subsequently, using Eqs. (14) and (15):

$${\mu }_Q={\mu }_s-a{\mu }_h=1-0.88\times 1=0.62$$

$${\sigma }_Q=\sqrt{{0.1}^2+{0.88}^2\times 1^2}=0.89$$

Knowing that Q is a normal random variable Q ∼ Normal (0.62, 0.89), using Eq. (17), $P_{\mathit{\text{float}}}=1-{\mathrm{\Phi }}_Q\left(b\right)=1-{\mathrm{\Phi }}_Q\left(0.26\right)=0.66$.

4.2 Numerical approach

Given the probability distribution of S and h, the probability of flotation in Eq. (10) can be approximated numerically using, among others, Monte Carlo simulation. Table 4 show part of the dataset generated for S and h given their probability distributions. Obviously, for each pair of S-h dataset, if Q > 0.26 the tank floats. The last column of Table 4 assign 1 to each pair if Q > 0.26 (the tank floats) and 0 if Q < 0.26 (the tank does not float). Having a sufficiently large dataset, the mean value of the last column denotes the probability of floatation. Given 1000 samples, $P_{\mathit{\text{float}}}$ = 0.676, which is close enough to the probability calculated using the analytical approach.

Sample No	h ∼ N(1.0, 1.0)	S ∼ N(1.0, 0.1)	Q = S – 0.88h	1 if Q > 0.26; 0 otherwise
120	0.089	1.608	1.529	1
203	0.724	1.592	0.955	1
311	1.410	1.599	0.359	1
457	1.657	1.508	0.049	0
590	1.768	1.745	0.189	0
602	1.481	1.454	0.150	0
740	1.129	1.417	0.423	1

Table 4.

Sample of dataset generated by Monte Carlo simulation given the probability distributions of S and h.

4.3 Bayesian network approach

Instead of solving Eq. (10) or Eq. (11) analytically or numerically, the BN in Figure 5 can be used to model the tank flotation and estimate its probability. Knowing that S and h are the only random variables that contribute to the flotation, the BN should consist of two root nodes “S” and “h”, and a leaf node, “Floatation”.

Figure 5.

BN for modeling and assessing the probability of tank floatation.

S and h in Eq. (11) are continuous variables, but their corresponding nodes in the BN are discrete nodes. As such, S and h need to be discretized into a finite set of states. The probabilities of the states can then be calculated using the probability distributions of S and h, that is, $f_S\left(S\right)$ and $f_h\left(h\right)$. Subsequently, the conditional probability table for node “Floatation” can readily be populated with 1s and 0s given Eq. (7), that is, 1 if S > ah + b, and 0 otherwise. Given the BN in Figure 5, the probability of floatation is estimated as 0.62, sufficiently close to the results obtained by solving the integrals analytically (Section 4.1) or numerically (Section 4.2).

While by increasing the resolution of nodes h and S (increasing their states) the calculated probability for the node Floatation can further be improved, such refinement comes at a substantially increased modeling time and effort. It is because the size of the conditional probability table for the node Floatation increases exponentially with the size of the probability tables (number of states) for nodes h and S.

Following the same procedure for the other failure modes, the final BN for assessing the combined failure probability can be developed as in Figure 6. In Figure 6, the nodes “V_wind” and “V_flood” denote the speed of wind and flood, respectively; “H_ac” is the height of accumulated rainfall on the roof of storage tank; “Wind_buckle” and “Flood_buckle” represent shell buckling failure mode due to lateral forces of wind and flood, respectively; “Slide” denotes rigid sliding of the tank due to lateral force of flood, and “Roof-sinking” denotes the failure mode due to accumulation of rainfall on the top.

Figure 6.

Final BN for assessing the total failure probability considering all the failure modes [12].

5. Conclusions

In this chapter, we discussed failure modes and vulnerability of process vessels – with emphasis on atmospheric storage tanks – in the event of hurricanes. It was demonstrated how physics-of-failure models and Bayesian network can be used to assess the failure probabilities of storage tanks due to the impact of flood, wind, and rainfall.

Although the physics-of-failure models can directly be used to develop mathematical relationships for estimating the failure probabilities (see Eq. (11) as an example), such mathematical relationships cannot seem to efficiently account for dependencies when modeling and combining interdependent failure modes. These interdependencies may arise when one root node contributes to several failure modes or when one failure mode excludes or contributes to another failure mode. An example for such interdependencies includes the accumulation of rainfall on the rooftop of a floating- roof tank (“H_ac” in Figure 6), which causes the roof to sink (“Roof_sinking”) but reduces the likelihood of tank’s floatation (“Floatation”) and sliding (“Slide”) due to its added weight.

The application of BN, on the other hand, enables the analyst to consider the dependencies among the failure modes, which would otherwise result in inaccurate failure probabilities. Table 5 summarizes the modeling features of the methodologies discussed in this chapter.

Methodology	Feasibility	Accuracy
Analytical (Section 4.1)	Low	Exact solution
Numerical (Section 4.2)	High	Depends on sample size
BN (Section 4.3)	High	Depends on discretization resolution

Table 5.

A comparison between the discussed methodologies.

Acknowledgments

The financial supports by the Natural Sciences and Engineering Research Council of Canada (NSERC) via Discovery Grant, and by the Faculty of Community Services, Ryerson University, Canada, via Publication Grant is much appreciated.