Risk Assessment of Dike Based on Risk Chain Model and Fuzzy Influence Diagram

Abstract

For the risk assessment of flood defense, a comprehensive understanding of risk factors affecting dike failure is essential. Traditional risk assessment methods are mostly based on experts’ experience and focus on just one type of failure mode of flood defensive structures. The risk resources, including the analytical factors and non-analytical factors, were summarized firstly according to the general experience of dikes. The uncertainty of the resources that affect dike safety can be quantified by membership degree. Hence, a fuzzy influence diagram based on fuzzy mathematics was proposed to assess the safety of the dikes. We evaluated the multi-failure modes at the same time by a fuzzy influence diagram. Taking a dike as an example, the expected value of the dike failure was 6.25%. Furthermore, the chance of damage to this dike was “very unlikely” according to the descriptive term of the Intergovernmental Panel on Climate Change (IPCC). The evaluation result was obtained as a probabilistic value, which enabled an intuitive perception of the safety of the dikes. Therefore, we provided some reasonable suggestions for project management and regular maintenance. Since the proposed method can account for uncertainties, it is well suited for the risk assessment of dikes with obvious uncertainties.

1. Background

Dikes play an important role in the flood defense system of China. It was reported there were around 320,000 km of dikes to protect more than 640 million people and numerous cities [1]. Therefore, how to manage dikes using a reasonable method is an important task for reducing flood disaster. Because of some historical reasons, dikes in China were constructed in ancient times, and many dikes were made with local materials. As a result, the design standard and strength of the dikes located in remote areas of China are relatively low. Therefore, conducting the risk assessment of the dikes and discovering the hidden dangers in time are significant to protect the life and property of every citizen in the dike-protected areas.

In recent years, the traditional management of the dikes gradually changed to risk management. Developed countries, such as the UK, the Netherlands, and Japan, have taken the lead in introducing risk management methods into the risk assessment of dikes. For example, Mouri et al. [2] adopted the probability method to conduct the risk assessment of flooding considering natural and societal factors. Marijnissen et al. [3] used an extensive probabilistic method to re-evaluate the multifunctional dikes in the Netherlands considering the society and economy. Balistrocchi et al. [4] discussed the advantages of the two methods, i.e., numerical simulations and fragility curves, to study the failure probability analysis of levees. He proposed a model to calculate the failure probability of a mammal bioerosion levee considering the peak flow discharge and flood duration. Agbaje et al. [5] studied the influence of the effective friction angle on slope stability based on the Grag deposits in the East of England, UK. It pointed out that the total factor of safety was usually higher, leading to overestimation of the stability of a slope.

There are three main failure modes in China: overtopping, scouring, and seepage [6]. If the crest of the dike is below the run-up level, the waves will cause a flow of water over the top of the dike, across the crest, and onto the landside slope. This phenomenon is called overtopping. The pore pressure affects the stability of the levee body and/or foundation through infiltration and affects the back erosion piping or slope stability. This phenomenon is called seepage failure. Scouring failure is the damage caused by the water flow and waves, which usually leads to dangerous situations such as the inner slope collapse of the dike. Overtopping failure is considered a kind of hydrology failure, and the scouring and seepage failures are considered structural failures. However, most of the risk analyses focused on one of the above three failure mechanisms by only considering the analytical factors. For example, the risk analysis of overtopping was used to measure the overtopping failure [7,8,9,10,11,12,13,14]; the scouring analysis and slope stability analysis were conducted only for the dike body [15,16,17,18,19]; and most seepage analyses were also performed separately [20,21,22]. There were a few pieces of literature that assessed dike damage by considering failure modes with both analytical and non-analytical factors [23].

In another way, the traditional models, including the analytic hierarchy process, event tree method, and fault tree method, are mainly used to perform the system risk assessment. Song et al. [24] studied the risk assessment of tunnel construction based on the fuzzy analytic hierarchy process. Wang et al. [25] used a fuzzy analytic hierarchy process to identify the safety mode of the section of Jingnan dike that is located in Jingzhou City, China. Gu et al. [26] proposed an experts’ weighting model for the safety evaluation of a dike using fuzzy mathematics and the dynamic clustering method. The analytic hierarchy process mostly relies on the subjective experts’ experience and evaluates only using the appearance, which often leads to deviation in the evaluation results [27]. Fault tree and event tree analysis methods are widely used; however, the accurate probability value of basic events needs to be known in the calculation process [28].

The influence diagram method is a risk assessment method proposed by Howard [29]. The influence diagram is composed of two parts, the risk node and the risk relationship. It can represent the directed graph of the mutual influence between the risk elements and can provide relatively accurate calculation results for risk assessment. Moreover, it is also suitable for the risk assessment of system engineering. A fuzzy influence diagram is a combination of the fuzzy set, fuzzy transformation method, and influence diagram method to carry out probabilistic inferences on uncertain events. During the inference process, the occurrence probability of an event and the occurrence probability of other dependent events can be fully evaluated. Due to the advantages of intuition and easy understanding, the application scope of the fuzzy influence diagram is gradually expanding. Based on the 3D Hall structure model of system engineering, the risk identification method of overseas nuclear power projects was optimized by considering the idea of risk chain and risk map [30]. Xu et al. [31] used the fuzzy influence diagram to evaluate the construction safety risk of power transmission and transformation engineering construction projects. Quan et al. [32,33] conducted a qualitative risk analysis on the general contracting of overseas EPC power engineering projects using the fuzzy influence diagram method. Later, Zhang et al. [34] introduced the fuzzy influence diagram method into the risk assessment process of information security and introduced fuzzy theory to describe risk factors that are difficult to quantify and to obtain good results. Lin et al. [35] applied the fuzzy influence diagram method to the safety management of underground pipelines and focused on the analysis of the risks caused by external force damage. To the best knowledge of the authors, this is the first time using the risk chain model and fuzzy influence diagram method to evaluate dike safety. Theoretically, the fuzzy influence diagram method is based on fuzzy mathematics. The introduction of state fuzzy sets, frequency fuzzy sets, and membership degrees can fully consider the uncertainty of the risk factors and at the same time reduce the subjectivity of experts’ judgment. It is suitable to perform the risk assessment for the dikes with uncertainty and complexity.

This present study aimed to (1) summarize the risk resources that affect dike safety based on the general experience of the dikes; (2) analyze the relationships between the risk resources and risk events; (3) construct a fuzzy influence diagram based on (1) and (2); and (4) take a fictitious dike as an example to illustrate the proposed method and apply the description term of IPCC to the result.

2. The Risk Chain Model and the Fuzzy Influence Diagram

2.1. The Risk Chain Model

A fuzzy influence diagram is a graph composed of multiple risk chains, and each risk chain can be regarded as an independent logical relationship. A risk chain is established by adding nodes such as risk sources, risk factors, and risk events in layers. Within an assessment system, different risk chains should ultimately point to the same risk consequence. The risk chain structure is shown in Figure 1.

2.2. Fuzzy Set and Membership Function

2.2.1. The Definition of Fuzzy Set and Membership

The introduction of fuzzy sets in this paper is limited to the case where the universe of discourse is a finite set, denoted by $A=\left\{a_1, a_2,\cdots ,a_n\right\}$. Mapping on the domain $A$, ${\mu }_{\mathit{X}}(\mathbf{·}):A\to \left[0, 1\right]$, then $a\to {\mu }_{\mathit{X}}\left(a\right)$, the fuzzy set $X$ was determined. ${\mu }_X(\mathbf{·})$ is the membership function of $X$, and ${\mu }_X\left(a\right)$ is the membership degree of a to the fuzzy set $X$, denoted as:

$$X=\left\{\left(a, {\mu }_X\left(a\right)\right)|a\in A\right\}$$

(1)

The algorithm of the fuzzy influence graph is based on fuzzy set operation and uses a directed graph to describe the state, frequency, and relationship between nodes, and then it carries out a risk assessment for the system.

2.2.2. The Operations on Fuzzy Sets

We briefly describe the three operation notations of the fuzzy set used in this paper. The symbol “×” is a Cartesian product. If $X$ and $Y$ are fuzzy sets of two universes $A$ and $B$, respectively, then the fuzzy relation $R$ from $X$ to $Y$ is:

$${\mu }_R={\mu }_{X\times Y}\left(a_i, b_j\right)=\min \left[{\mu }_X\left(a_i\right), {\mu }_Y\left(b_j\right)\right]$$

(2)

A fuzzy relation is a special kind of fuzzy set, and the result is a fuzzy matrix.

The symbol “∘” is called fuzzy synthesis, that is, dot product. If $R$ is a fuzzy relation from $X$ to $Y$, similarly, if $S$ is a fuzzy relation from $Y$ to $Z$, then the fuzzy composition of $R$ and $S$ is:

$${\mu }_{R\circ S}\left(a_i, b_j\right)=\max \left\{\min \left[{\mu }_R\left(a_i,b_j\right), {\mu }_S\left(a_i,b_j\right)\right]\right\}$$

(3)

The result of fuzzy synthesis is still a fuzzy matrix, representing the fuzzy relationship from $X$ to $Z$.

The symbol “$\cup$” is called the “union” of fuzzy sets and has the following operations:

$${\mu }_{R\cup S}\left(a_i, b_j\right)=\max \left[{\mu }_X\left(a_i,b_j\right), {\mu }_Y\left(a_i,b_j\right)\right]$$

(4)

It should be pointed out here that “Cartesian product” and “dot product” should operate on fuzzy sets of different universes, and “union” should operate on fuzzy sets in the same universe.

2.3. The Process of the Fuzzy Influence Diagram

The fuzzy influence diagram introduces the state fuzzy set and the frequency fuzzy set in the numerical layer to describe the data structure of the node and uses the fuzzy relationship to describe the relationship between the node variables in the function layer. The calculation process of the fuzzy influence diagram is explained below [36].

2.3.1. The Calculation of Independent Node Frequency Matrix

If $X$ has no immediate predecessor, it is called an independent node, and it is assumed that the possible state vector of the independent node $X$ is:

$$P_X={\left\{P_{X_1},{ P}_{X_2},\cdots ,P_{X_n}\right\}}^{\mathrm{T}}$$

(5)

where $P_{X_1},P_{X_2},\cdots ,P_{X_n}$ is the state fuzzy set defined by the risk assessor according to the actual situation. The frequency vector of an independent node $X$ is:

$$f_X={\left\{f_{X_1},{ f}_{X_2},\cdots ,f_{X_n}\right\}}^{\mathrm{T}}$$

(6)

where $f_{X_1},f_{X_2},\cdots ,f_{X_n}$ is the state fuzzy set defined by the risk assessor according to the actual situation. The frequency vector of an independent node $X$ is:

$$F_X=\left(f_{X_1}\times P_{X1}\right)\cup \left(f_{X_2}\times P_{X_2}\right)\cup \cdots \cup \left(f_{X_n}\times P_{X_n}\right)$$

(7)

2.3.2. The Calculation of the Frequency Matrix of Dependent Nodes

If $X$ is made up of m random nodes $Y_1,Y_2,\cdots ,Y_m$ as its immediate predecessors, then $X$ is called a dependent node.

$$F_{XP}=F_{Y_1}\cup F_{Y_2}\cup \cdots \cup F_{Y_m}$$

(8)

Define $R_{X{Y}_1}$ as the fuzzy relationship from node $Y_1$ to dependent node $X$:

$$R_{X{Y}_1}=\left(P_{Y_{11}}\times P_{X_i}\right)\cup \left(P_{Y_{12}}\times P_{X_i}\right)\cup \cdots \cup \left(P_{Y_{1n}}\times P_{X_i}\right)$$

(9)

where $P_{Y_{11}},P_{Y_{12}},\cdots ,P_{Y_{1n}}\in P_{Y_1}$, $P_{X_i}\in \left\{P_{X_1}{,P}_{X_2},\cdots ,P_{X_n}\right\}=P_X$. The fuzzy relationship from node $Y_m$ to node $X$ is:

$$R_{X{Y}_m}=\left(P_{Y_{m1}}\times P_{X_i}\right)\cup \left(P_{Y_{m2}}\times P_{X_i}\right)\cup \cdots \cup \left(P_{Y_{mn}}\times P_{X_i}\right)$$

(10)

Then, the joint $R_{XP}$ of the fuzzy relationship of all immediate predecessors of the dependent node $X$ is:

$$R_{XP}=R_{X{Y}_1}\cup R_{X{Y}_2}\cup \cdots \cup R_{X{Y}_m}$$

(11)

The frequency matrix of the dependent node $X$ can be obtained by the dot product relationship:

$$F_X=F_{XP}\circ R_{XP}$$

(12)

2.3.3. Result Analysis

The membership degree of the random result is selected from the frequency matrix $F_X$ of the risk consequence node by Equation (13), and the probability value of the result is calculated by Equation (14):

$$\max \left(f_{X_i}\mathbf{·}{\displaystyle \sum _{\Omega X}{\mu }_{X_i}}\right)$$

(13)

$$P\left(X_i\right)=\frac{{\mu }_{X_i}}{{\displaystyle \sum _{\Omega X}{\mu }_{X_i}}}$$

(14)

According to the probability description proposed by the Intergovernmental Panel on Climate Change (IPCC), this paper used seven levels for the result to give a corresponding language description of the possibility of the risk result [27,37], as shown in

Table 1. The probabilistic result of IPCC.

Probability Range	Descriptive Term
<1%	Extremely unlikely
1~10%	Very unlikely
10~33%	Unlikely
33~66%	Medium likelihood
66~90%	Likely
90~99%	Very likely
>99%	Virtually certain

.

3. The Evaluation Process of the Fuzzy Influence Diagram

3.1. The Construction of the Frequency Fuzzy Set

For the convenience of calculation, combined with the experts’ experience on the dike, the frequency vector of the risk node of the dike was uniformly simplified to $f={\left\{\mathit{High}, Medium, Low\right\}}^{\mathrm{T}}$, as shown in

Table 2. The state and frequency of risk nodes.

Nodes	The Potential Risk State	The Frequency
Flooding	big	low
	middle	medium
	small	high
Illegal operation	big	low
	middle	medium
	small	high
Low skills	big	low
	middle	medium
	small	high
Reduced cohesion	big	low
	middle	high
	small	medium
Increase of water content	big	medium
	middle	low
	small	high
Void ratio change	big	low
	middle	medium
	small	high
Friction angle change	big	low
	middle	high
	small	medium
……	……	……

. The frequency was processed as follows, and 0 to 1 were decomposed into (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0), indicating the occurrence frequency of risk nodes. From the definition of fuzzy sets and membership degrees, the three fuzzy sets and their membership degrees of the frequency vector f of the dike risk node can be expressed as follows:

$$High=\left\{\frac{0.7}{0.5},\frac{0.8}{0.7},\frac{0.9}{0.9},\frac{1.0}{1.0}\right\}Medium=\left\{\frac{0.3}{0.2},\frac{0.4}{0.8},\frac{0.5}{1.0},\frac{0.6}{0.8},\frac{0.7}{0.2}\right\}Low=\left\{\frac{0.0}{1.0},\frac{0.1}{0.9},\frac{0.2}{0.7},\frac{0.3}{0.5}\right\}$$

where the number above the symbol “—” is the frequency of node occurrence, and the number below represents the membership degree of the frequency. The number 0.7/0.5 means that the risk node is in the High Set, and the membership degree is 0.5 when the frequency of occurrence is 0.7. The number 0.7/0.2 means that the risk node is in the Medium Set, and the membership degree is 0.2 when the frequency is 0.7.

3.2. The Construction of the State Fuzzy Set

The state vector of the risk node in the fuzzy influence diagram is uniformly simplified as $P={\left\{\mathit{Increase} \mathit{max}\left(\mathit{IMAX}\right), \mathit{Increase} \mathit{medium}\left(\mathit{IMID}\right), \mathit{Increase} \mathit{min}\left(\mathit{IMIN}\right)\right\}}^{\mathrm{T}}$. The language description of the fuzzy set in the state vector has the same meaning as the description of the relationship between the nodes in

Table 3. The fuzzy relationship of different risk nodes.

Relationship between Different Nodes	Relationship Description
	The Degree of Change in Disadvantage for the Risk Nodes	The Results Corresponding to the Degree of Change of the Risk Nodes
Flooding → higher water level	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Illegal operation → reduced drainage capacity	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Low skills → reduced drainage capacity	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Reduced cohesion → dike body damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Water content → dike body damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Reduced cohesion → dike foundation damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
water content → dike foundation damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Void ratio → dike body damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Friction angel → dike foundation damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Void ratio → dike foundation damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Friction angle → dike foundation damage	big, middle, and small, respectively	Increase max, increase medium, increase min, respectively
Higher water level → overtopping	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Reduced drainage → overtopping	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Damage of dike body → seepage	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Damage of dike foundation → seepage	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Damage of dike body → scouring	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Damage of dike foundation → scouring	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Overtopping → dike failure	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Seepage → dike failure	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively
Scouring → dike failure	Increase max, medium, min, respectively	Increase max, increase medium, increase min, respectively

Take the “Flooding → higher water level” and “Higher water level → overtopping” as examples to illustrate Table 3. When the flooding is big, middle, and small, it can cause an increase in water level to maximum, medium, and minimum, respectively. When the increase of higher water level is maximum, medium, and minimum, it can cause a degree of overtopping to maximum, medium, and minimum, respectively.

. The state fuzzify sets and their membership functions are expressed as follows:

$$\mathit{IMAX}=\left\{\frac{8\%}{0.5},\frac{10\%}{0.8},\frac{12\%}{1.0}\right\} \mathit{IMID}=\left\{\frac{4\%}{0.5},\frac{6\%}{1.0},\frac{8\%}{0.5}\right\} \mathit{IMIN}=\left\{\frac{0\%}{1.0},\frac{2\%}{0.5},\frac{4\%}{0.2}\right\}$$

It is more appropriate to describe the states of some parts of nodes in the fuzzy influence diagram with certainty. Hence, it is necessary to artificially fuzzify them. In this paper, the states of all risk source nodes were artificially fuzzified. The membership functions are as follows:

$$\mathit{Big}=\left\{\frac{\mathit{big}}{1.0},\frac{\mathit{middle}}{0.0},\frac{\mathit{small}}{0.0}\right\} \mathit{Middle}=\left\{\frac{\mathit{big}}{0.0},\frac{\mathit{middle}}{1.0},\frac{\mathit{small}}{0.0}\right\} \mathit{Small}=\left\{\frac{\mathit{big}}{0.0},\frac{\mathit{middle}}{0.0},\frac{\mathit{small}}{1.0}\right\}$$

3.3. Risk Assessment of One Dike Case

This paper sorted out the analytical factors and non-analytical factors that lead to dike failure and selected some factors that have an impact on the safety status of the dike as the risk source for analysis. The emergence of risk sources causes the occurrence of risk factors, which in turn leads to the occurrence of three common failure modes of dikes. The three failure modes comprehensively lead to dike failure (risk consequences). The constructed fuzzy influence diagram of dike failure is shown in Figure 2. According to the frequency fuzzy set, state fuzzy set, and fuzzy relationship constructed above, the calculation process of the risk chain “illegal operation-reduced drainage—overtopping—dike failure” was taken as an example, and the remaining evaluation process is similar to it.

Step 1, The frequency matrix $F_{\mathit{illegal}\_operation}$ of illegal operation risk source nodes is calculated by Equations (5)–(7):

$$F_{\mathit{illegal}\_operation}=\left[\begin{array}{cccc}& \mathit{big}& \mathit{middle}& \mathit{small}\\ 0.0& 1.0& 0.0& 0.0\\ 0.1& 0.9& 0.0& 0.0\\ 0.2& 0.7& 0.0& 0.0\\ 0.3& 0.5& 0.2& 0.0\\ 0.4& 0.0& 0.8& 0.0\\ 0.5& 0.0& 1.0& 0.0\\ 0.6& 0.0& 0.8& 0.0\\ 0.7& 0.0& 0.2& 0.5\\ 0.8& 0.0& 0.0& 0.7\\ 0.9& 0.0& 0.0& 0.9\\ 1.0& 0.0& 0.0& 1.0\end{array}\right]$$

(15)

In the same way, the frequency matrix of low-risk source nodes of security skills can be obtained.

Step 2, The joint frequency matrix $F_{\mathit{union}}$ of all immediate predecessor nodes of the node with reduced drainage capacity is calculated by Equation (8):

$$F_{union}=\left[\begin{array}{cccc}& \mathit{big}& \mathit{middle}& \mathit{small}\\ 0.0& 1.0& 0.0& 0.0\\ 0.1& 0.9& 0.0& 0.0\\ 0.2& 0.7& 0.0& 0.0\\ 0.3& 0.5& 0.2& 0.0\\ 0.4& 0.0& 0.8& 0.0\\ 0.5& 0.0& 1.0& 0.0\\ 0.6& 0.0& 0.8& 0.0\\ 0.7& 0.0& 0.2& 0.5\\ 0.8& 0.0& 0.0& 0.7\\ 0.9& 0.0& 0.0& 0.9\\ 1.0& 0.0& 0.0& 1.0\end{array}\right]$$

(16)

Step 3, The fuzzy relationship matrix from the node of illegal operation risk source to the node of drainage capacity reduction risk factor is calculated by Equation (10):

$$R_{\mathit{illegal}\_\mathit{operation}\to \mathit{reduced}\_\mathit{drainage}}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ big& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\\ middle& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ small& 1.0& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\end{array}\right]$$

(17)

In the same way, the fuzzy relation matrix of the lower drainage capacity node caused by the low skills node can be obtained.

Step 4, The joint matrix of the fuzzy relationship of all immediately preceding nodes of the node that leads to the reduction of drainage capacity is calculated by Equation (11):

$$\begin{array}{ll}{R}_{\mathit{union}}& {=R}_{\mathit{illegal}\_\mathit{operation}\to \mathit{reduced}\_\mathit{drainage}}\cup R_{\mathit{low}\_skills\to reduced\_drainage}\\ & =\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ big& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\\ middle& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ small& 1.0& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\end{array}\right]\end{array}$$

(18)

Similarly, the joint matrix of other fuzzy relations can be obtained.

Step 5, The frequency matrix of the nodes with reduced drained capacity is calculated by Equation (12):

$$F_{\mathit{reduced}\_drainage}=F_{union}\circ R_{union}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0.0& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\\ 0.1& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 0.9\\ 0.2& 0.0& 0.0& 0.0& 0.0& 0.5& 0.7& 0.7\\ 0.3& 0.0& 0.0& 0.2& 0.2& 0.5& 0.5& 0.0\\ 0.4& 0.0& 0.0& 0.5& 0.8& 0.5& 0.0& 0.0\\ 0.5& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ 0.6& 0.0& 0.0& 0.5& 0.8& 0.5& 0.0& 0.0\\ 0.7& 0.5& 0.5& 0.2& 0.8& 0.2& 0.0& 0.0\\ 0.8& 0.7& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\\ 0.9& 0.9& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\\ 1.0& 1.0& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\end{array}\right]$$

(19)

In the same way, the frequency matrix of the node of rising water level in front of the dikes can be obtained. Thus far, the frequency matrix, the frequency joint matrix, the fuzzy relationship matrix between the nodes, and the fuzzy relationship joint matrix of any node can be obtained by analogy with the calculation process above.

Corresponding to the fuzzy relationship between the nodes in Table 3, the fuzzy influence diagram is completely calculated by analogy to the above calculation process. Only the calculation results of the key nodes in the subsequent calculation process of the fuzzy influence diagram are given here.

The relationship matrix from the drainage capacity reduction node to the overtopping risk event node is as follows:

$$R_{\mathit{reduced}\_\mathit{drainage}\to \mathit{overtopping}}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0\%& 1.0& 0.5& 0.0& 0.0& 0.0& 0.0& 0.0\\ 2\%& 0.5& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\\ 4\%& 0.2& 0.2& 0.5& 0.5& 0.5& 0.0& 0.0\\ 6\%& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ 8\%& 0.0& 0.0& 0.5& 0.5& 0.5& 0.5& 0.5\\ 10\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 0.8\\ 12\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\end{array}\right]$$

(20)

The union of the fuzzy relationship matrices of all the immediate nodes of the overtopping risk event node is:

$$\begin{array}{ll}{R}_{\mathit{union}}& {=R}_{\mathit{higher}\_\mathit{water}\to \mathit{overtopping}}\cup R_{\mathit{reduced}\_\mathit{drainage}\to \mathit{overtopping}}\\ & =\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0\%& 1.0& 0.5& 0.0& 0.0& 0.0& 0.0& 0.0\\ 2\%& 0.5& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\\ 4\%& 0.2& 0.2& 0.5& 0.5& 0.5& 0.0& 0.0\\ 6\%& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ 8\%& 0.0& 0.0& 0.5& 0.5& 0.5& 0.5& 0.5\\ 10\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 0.8\\ 12\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\end{array}\right]\end{array}$$

(21)

The frequency matrix of overtopping risk event nodes can be obtained as:

$$F_{\mathit{over}topping}=F_{union}\circ R_{union}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0.0& 0.0& 0.0& 0.5& 0.5& 0.5& 0.8& 1.0\\ 0.1& 0.0& 0.0& 0.5& 0.5& 0.5& 0.8& 0.9\\ 0.2& 0.0& 0.0& 0.5& 0.5& 0.5& 0.7& 0.7\\ 0.3& 0.2& 0.2& 0.5& 0.5& 0.5& 0.5& 0.5\\ 0.4& 0.2& 0.2& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.5& 0.2& 0.2& 0.5& 1.0& 0.5& 0.5& 0.5\\ 0.6& 0.2& 0.2& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.7& 0.5& 0.5& 0.5& 0.8& 0.5& 0.2& 0.2\\ 0.8& 0.7& 0.5& 0.2& 0.2& 0.2& 0.0& 0.0\\ 0.9& 0.9& 0.5& 0.2& 0.2& 0.2& 0.0& 0.0\\ 1.0& 1.0& 0.5& 0.2& 0.2& 0.5& 0.0& 0.0\end{array}\right]$$

(22)

The joint frequency matrix of the three risk event nodes of overtopping, scouring, and seepage is:

$$F_{union}{=F}_{\mathit{over}topping}\cup F_{seepage}\cup F_{scouring}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0.0& 1.0& 0.5& 0.5& 0.5& 0.5& 0.8& 1.0\\ 0.1& 0.9& 0.9& 0.5& 0.5& 0.5& 0.8& 0.9\\ 0.2& 0.7& 0.7& 0.5& 0.5& 0.5& 0.8& 0.9\\ 0.3& 0.5& 0.5& 0.5& 0.5& 0.5& 0.5& 0.5\\ 0.4& 0.2& 0.2& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.5& 0.2& 0.2& 0.5& 1.0& 0.5& 0.5& 0.5\\ 0.6& 0.2& 0.2& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.7& 0.5& 0.5& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.8& 0.7& 0.7& 0.5& 0.5& 0.5& 0.7& 0.7\\ 0.9& 0.9& 0.9& 0.5& 0.5& 0.5& 0.8& 0.9\\ 1.0& 1.0& 1.0& 0.5& 0.5& 0.5& 0.8& 1.0\end{array}\right]$$

(23)

The fuzzy relationship matrix of the overtopping failure mode to the dike failure node is:

$$R_{\mathit{overtopping}\to \mathit{dike}\_failure}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0\%& 1.0& 0.5& 0.0& 0.0& 0.0& 0.0& 0.0\\ 2\%& 0.5& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\\ 4\%& 0.2& 0.2& 0.5& 0.5& 0.5& 0.0& 0.0\\ 6\%& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ 8\%& 0.0& 0.0& 0.5& 0.5& 0.5& 0.5& 0.5\\ 10\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 0.8\\ 12\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\end{array}\right]$$

(24)

$$R_{\mathit{union}}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0\%& 1.0& 0.5& 0.0& 0.0& 0.0& 0.0& 0.0\\ 2\%& 0.5& 0.5& 0.2& 0.0& 0.0& 0.0& 0.0\\ 4\%& 0.2& 0.2& 0.5& 0.5& 0.5& 0.0& 0.0\\ 6\%& 0.0& 0.0& 0.5& 1.0& 0.5& 0.0& 0.0\\ 8\%& 0.0& 0.0& 0.5& 0.5& 0.5& 0.5& 0.5\\ 10\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 0.8\\ 12\%& 0.0& 0.0& 0.0& 0.0& 0.5& 0.8& 1.0\end{array}\right]$$

(25)

The frequency matrix of risk consequence of dike failure nodes is:

$$F_{\mathit{dike}\_failure}=F_{union}\circ R_{union}=\left[\begin{array}{cccccccc}& 0\%& 2\%& 4\%& 6\%& 8\%& 10\%& 12\%\\ 0.0& 1.0& 0.5& 0.5& 0.5& 0.5& 0.8& 1.0\\ 0.1& 0.9& 0.5& 0.5& 0.5& 0.5& 0.8& 0.9\\ 0.2& 0.7& 0.5& 0.5& 0.5& 0.5& 0.8& 0.9\\ 0.3& 0.5& 0.5& 0.5& 0.5& 0.5& 0.5& 0.5\\ 0.4& 0.2& 0.2& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.5& 0.2& 0.2& 0.5& 1.0& 0.5& 0.5& 0.5\\ 0.6& 0.2& 0.2& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.7& 0.5& 0.5& 0.5& 0.8& 0.5& 0.5& 0.5\\ 0.8& 0.7& 0.5& 0.5& 0.5& 0.5& 0.7& 0.7\\ 0.9& 0.9& 0.5& 0.5& 0.5& 0.5& 0.8& 0.9\\ 1.0& 1.0& 0.5& 0.5& 0.5& 0.5& 0.8& 1.0\end{array}\right]$$

(26)

By Equation (13), the row with the largest product value of the product of the frequency is selected, and the sum of the membership degrees of the row is calculated. In this example, it is the row corresponding to the frequency equal to 1.

$$\left\{\frac{0\%}{1.0}, \frac{2\%}{0.5}, \frac{4\%}{0.5}, \frac{6\%}{0.5}, \frac{8\%}{0.5}, \frac{10\%}{0.8}, \frac{12\%}{1.0}\right\}$$

(27)

The change probability of the dike failure node calculated by Equation (14) is:

$$\begin{array}{ll}P\left(0\%,2\%,4\%,6\%,8\%,10\%,12\%\right)& =\left(1.0,0.5,0.5,0.5,0.5,0.8,1.0\right)/4.8\\ & =\left(0.2083,0.1042,0.1042,0.1042,0.1042,0.1667,0.2083\right)\end{array}$$

(28)

The expected value of the increased risk can be calculated:

$$\begin{array}{ll}{E}_{\mathit{failure}}& =\frac{0\%\times 1+\left(2\%+4\%+6\%+8\%\right)\times 0.5+10\%\times 0.8+12\%\times 1}{4.8}\\ & =6.25\%\end{array}$$

(29)

Referring to the descriptive term of the IPCC probability results (Table 1), it was concluded that the chance of this example dike was “very likely”. Figure 3 and Figure 4 show the probability distribution and cumulative distribution of the increased risk, respectively. The top three of the increased risk were 0%, 12%, and 10%, and the cumulative probability from 10% to 12% was 37.5%. The occurrence probability was relatively high, which was different from the result “very likely”. Actually, the expected value of 6.25% of structure failure cannot be accepted for flood defense. The two conclusions from cumulative cure and the expected value had the same meanings, indicating that a risk exists. Hence, the probabilistic result of IPCC was not suitable for the dikes perfectly. Developing a similarly descriptive term for dikes is an urgent task for hydraulic experts.

4. Summary and Conclusions

Based on the influence diagram, this paper proposed a risk assessment method for dikes with a risk chain model. We summarized the common risk resources that affect the safety of dikes based on general experiences. The relationships between the different risk nodes were also deduced. The whole process of risk assessment was displayed by taking an example of a hypothesis dike. The probability of dike damage was 6.25%, illustrating that the chance of dike damage is “very likely” according to the descriptive term proposed by the Intergovernmental Panel on Climate Change (IPCC). The advantages of the proposed method are summarized as follows.

(1) All the analytical risk factors and non-analytical risk factors can theoretically be integrated into the influence diagram. The risk resources in Table 2 are only the common ones. The other risk resources can be further added as a supplement when this method is used to evaluate different dikes.

(2) This method is based on fuzzy mathematics and uses the membership degree function to describe the uncertainty of the analytical risk factors and non-analytical factors, which is more prone to reality.

(3) The different failure modes can be considered during the process of risk assessment at the same time.

(4) The risk assessment is obtained as a probabilistic result, which enables an intuitive perception of dike safety. We can provide reasonable suggestions for the engineering management and regular maintenance of dikes.

Expected for the risk assessment of dikes, the risk chain model and fuzzy influence diagram method can be applied to other hydraulic structures and geotechnical engineering. To obtain more accurate assessment results in engineering applications, more comprehensive risk sources should be selected, and field surveys or/and tests should be conducted to obtain real engineering data for calculation.