Risk evaluation of tailings dam based on game theory and matter-extension theory

: The safety and stability of tailings dam are directly related to the safety of life and property of downstream residents and the ecological and environmental safety of downstream rivers and farmland. During the operation stage of the tailings pond, safety assessment shall be carried out from seepage damage of the tailings dam, instability of the dam foundation, structural damage of the dam body, flooded roof, and safety management, etc. Aiming at a tailings pond in the running stage of Guangxi Province in China, by establishing the evaluation index system and its grading standard of the tailings dam-break, the subjectivity and objectivity of the evaluation index are taken into consideration by the combination of matter-element extension theory and game theory. Meanwhile, in order to solve the problem of the defect of the correlation function when the eigenvalue exceeds the controlled field and of the incommensurability for different units, a linear dimensionless method is adopted. The results show that the grade of tailings dam-break is I, and the eigenvalue of grade variable is 1.3, that is, the tailings pond is a “normal tailing reservoir”, but there is a risk of evolution to the “sickness tailing reservoir”. Thus, corresponding measures must be taken to prevent. In addition, the result of the two methods is consistent by comparing with the unascertained measure method, which not only validates the accuracy of the evaluation method, but also embodies the unique advantage of possessing the eigenvalue of grade variable to alert its evolution direction.


Introduction
Tailing pond is an important facility of metal or nonmetal mining mineral concentrator used to store "waste" and is the significant major hazard in mine production.The tailing ponds in China have the characteristics of large quantity, high dam body, and large potential energy.And the number of "Trilateral Tailing Pond" and "Head of Tailing Pond" is a lot.Once an accident occurs, it will directly threaten the lives and property of the surrounding residents and cause social instability.According to statistics [1], by the end of 2012 China had a total of 12,273 tailing ponds, of which 1223 were dangerous and sick, and 2213 were inactivated, accounting for about 28%.So, the security situation is quite severe for tailing pond.Document [2] counted accidents of tailing ponds for nearly 12 years, which the dam-break accident accounted for about 56%, and other types of accidents can induce a dam-break accident.Compared with other types of accident, damage caused by the dam-break accident is the largest.For example, September 8, 2008, Xinta mine in Xiangfen County, Linfen City, Shanxi Province occurred the dam-break accident, resulting in that 271 people was killed, and 35 others were injured, and the direct economic loss was as high as several hundred million [3].Therefore, it is necessary to study the stability of tailings dam.Safety and stability of tailings dam are always the research focus of scholars around the world.LI Xibing et al. [4] by using fault tree analysis of tailing dam-break during flood season, found the way caused by dam-break accident and taken effective measures to prevent the flooding accident; Chen Jianhong et al. [5] through use of SLIDE software, combining with Fellenius method and Bishop method, checked and verified SLIDE software which can efficiently, accurately, intuitively reflect the slope stability of tailings dam.Wang YingBo et al. [6] through the effective identification of the safety risk of the tailings dam and the necessary analysis, the risk assessment index system of the tailings dam is established, and then the fuzzy analytic hierarchy process is used to evaluate the risk.Cai Sijing et al. [7] studied the risk of a dam break under the action of the earthquake, and used the fuzzy evaluation method.Zheng Xin et al. [8] combined with the set pair analysis theory and analytic hierarchy process to evaluate the tailings dam, which is evaluated from both qualitative and quantitative aspects, and get its security level.On the basis of the theory of vulnerability, Mei Guodong et al. [9] established the evaluation system of dam break vulnerability and the evaluation method of disaster risk loss assessment.The safety stability and risk evaluation of tailings dam are discussed in the previous paragraph.The fault tree evaluation method can only give the risk factor which has a great impact on the dam, but does not draw substantial conclusions that the tailings dam is safe.Fuzzy comprehensive evaluation method can quantify fuzzy information, but it is more subjective.The set-pair analysis theory, although evaluated from the qualitative and quantitative aspects, was over-reliance on subjective experience.The unascertained measure evaluation method not only objectively reflects the influence degree of each evaluation index on the dam break, but can get the safety grade of the tailings dam.However, it completely ignores the influence of subjective factors such as the characteristics of each tailings dam on the evaluation according to objective data.At present, these methods have a certain effect on the prevention of the dam break, and also has some guiding significance to the actual project.However, in the process of evaluation, they exist a variety of ambiguity, complexity, and uncertainty, especially in the determination of the weight, either by subjective experience or completely objective, and risk assessment should consider both objective facts and subjective factors.Therefore, the evaluation results obtained by using these methods will inevitably be lack fidelity to some extent.Due to its ambiguity and complexity, that is, the contradictory problem, the problem not only from the qualitative point of view that the tailings dam break caused by the impact of indicators, but also from the quantitative point of view given the value of the indicators, and ultimately the safety grade was determined.The matter-element extension method is a kind of method to solve the contradiction between the qualitative and quantitative point of view.It cannot only reflect the influence degree of each evaluation index on the dam break, but also can more accurately determine the safety grade of risk assessment.The method has been widely used in mining engineering, geotechnical engineering and other fields [10][11] [12].In addition, in order to solve the problem of complete objectivity or complete subjectivity in the evaluation, the game theory is introduced, which in the process of determining each evaluation index, not only considers the objective facts of the evaluation indexes, but also takes the subjective factors of the evaluation indexes into account.

Methods
The theory of matter-element extension [12,13] was put forward by Cai Wen in 1983 for how to use formal tools to solve contradictory issues and was from two aspects of qualitative and quantitative to solve contradictory problems.It mainly consists of three pillars: the basicelement theory, extension set theory and extension logic, and matter-element extension model solved contradictory problems is based on basic-element as logic cells to represent the process of contradictions.Matter-element extension method was that, firstly by determining the evaluation index system of tailings dam, a risk evaluation criteria of tailings pond dam-break was established; and classical domain, section domain, and matter-element to be evaluated were ensured; secondly, the elementary correlation function value and the weight of each evaluation index was determined by matter-element to be evaluated; finally, the risk grade of tailings pond dambreak was confirmed.

Classical domain, section domain, and
matter-element to be evaluated 1) Classical domain The theory of matter element extension is based on names of things, its attributes and values as three basic orderly elements to describe things, that is, matter element R= (names of things, characteristic of things, values) = (N, C, V).If the risk grade of tailings dam-break is m, the classical domains can be expressed as: Where P is the whole evaluation levels of risk evaluation of tailings pond dam-break; the v pk is the value range of evaluation indicator C k .
3) Matter-element to be evaluated For risk grade P of certain tailings to be evaluated, the collection and analysis of the evaluation index data can be expressed as matter-element R w .
Where P w is risk grade of certain tailings to be evaluated; v wk is the specific value of evaluation index C k .

Index specifications based on membership functions
From the past research, application of matter-element extension method often encountered two problems [11]: Firstly, the eigenvalue of the matter-element to be evaluated may exceed the range of section domain, resulting in correlation functions failure; Secondly, the evaluation index of different units caused incommensurability.In order to solve these two problems, this paper adopted linear dimensionless method [14], the membership functions was expressed as: 1) Economic benefits index (the bigger, the better) 2) Cost index (the litter, the better) Where v′ wk is the k-th evaluation index value after dimensionless; v wk is the k-th evaluation index value before dimensionless; max (v pk ) and min (v pk ) are respectively the maximum value and minimum value of section domain of the k-th evaluation index.

Elementary correlation functions
Elementary correlation functions values of risk evaluation of tailings pond dam-break index and the corresponding weight value were used to determine extension correlation degree, elementary correlation functions at the j-th level of the matter-element to be evaluated (v wk ) can be expressed as: Where d=j or p, and ρ(v wk , v dk ) is used to represent the distance of a point to the interval in the extension theory, that is, "extension distance".

2.4
The comprehensive weighting method based on game theory Matter-element extension evaluation method to determine the weight values commonly used subjective weighting method (AHP, expert surveys, etc.) or objective weighting method (simple correlation function, entropy, etc.).Whether the subject weighting method or objective weighting method, only consider effects of unilateral factors.However, in the actual evaluation, the weight of evaluation index is not only affected by objective factors, but is influenced by subjective factors.In order to integrate the two influencing factors, the paper drew on the comprehensive weighting method based upon the game theory from the literature [15].It was to optimize and integrate the objective and subjective weighting method, and finally got the comprehensive weight of matter-element extension method.In order to improve science and rationality of evaluation index weight, and avoid subjective and objective onesidedness, y different methods can respectively be used empowerment of evaluation index weights.So y ( , , , ),( 1,2, , ) were obtained.The weight vector of arbitrary linear combination can be expressed as: In order to obtain a satisfactory index weight vector, formula (8) can be optimized so that deviation of u and each u i is minimization.Thus, the game model can be expressed as: And then based on the differential nature of the matrix, formula ( 9) can be converted into linear equations of first derivative.
Using Matlab software calculated formula (10), the solution was 1 2 ( , , , ) k     .u can be expressed as: Then formula (11) was normalized.So, u  can be expressed as: 2.5 Extension correlation of the j-th level for tailings dam-break risk assessment By formula ( 6) and ( 12), extension correlation with the jth level for tailings dam-break risk assessment can be calculated by: Where u k * is the weight value of the k-th risk evaluation index of a tailings dam in the whole evaluation index system; K j (v wk ) is elementary correlation functions value of the k-th risk evaluation index for matter-element (v wk ) in the j-th level.

Risk grade of tailings dam-break
According to the principle of maximum correlation degree, matter-element to be evaluated depending on the maximum value of extension correlation degree.So, the risk grade of tailings dam-break is: By formula (14), the risk assessment grade of tailings dam to be evaluated is j 0 .To more accurately determine the risk grade of the tailings dam, it can be measured by the level variable eigenvalue j * .

Geological setting
A tailing pond was built in 1998 and is a valley-type and upstream, and its annual tailings output of the concentrator is 50,000 t.Bulk density of the tailings is 2.0t/m 3 , and catchment area of the tailing pond is 1.26km 2 , and overall design height of the dam is 33.5m, and total storage capacity is 1.5 million m 3 , and freeboard is 1.02 m, and dry sands length is 51 m, and the current tailings pile approximately is 320,000 m 3 .The initial dam is the impervious masonry dam, and crest elevation is 458.5m, and height of the dam is 22m, and the downstream slope ratio is 1:2.Late the tailings adopted tailings dam constructed with the upstream method, and the average accumulation slope is 1: 3. The final tailings' accumulation elevation is 470m and the total height of dam about 33.5 m.At present, the tailings' accumulation elevation is 460.5m, and the total height of the dam is 24m.

Evaluation index system and its grading standard
The selection of risk evaluation indicators for the tailings dam-break is based on the previous research results and combined with the dam-break accident of tailings in the past.It could be discovered that the cause of tailings dambreak resulted from the interaction of multiple factors.However, risk evaluation indicator composed of various factors often tended to have some relevance.Thus, the relevance of risk evaluation indicators for tailings dambreak was studied, and excluded greater relevance of evaluation indicators.Besides, due to the emphasis of the actual situation and influence factors of each tailing pond was different, and combined with the characteristics of tailings, the tailings eventually formed a relatively independent risk assessment indexes system.The risk evaluation indicators of tailings dam-break were divided into 5 categories and 14 items [16,17,18]. 1) Seepage failure Seepage field has a great impact on tailings dam, and can directly induce dam-break caused by instability of the dam.Seepage failure is usually caused by seepage field occurring damage, and height of saturation line and bulk density of the tailings are the main reason for seepage failure.Therefore, the height of saturation line (denoted as C1, similarly hereinafter) and bulk density (C2) were used as the evaluation indicators to measure the seepage failure.

2) Instability of dam foundation
Instability of the dam foundation is usually a very important reason for tailings dam-break.The size of antisliding ability for tailings dam can directly affect the stability of the dam, while the size of anti-sliding ability is generally determined by the anti-sliding ability of dam and current situation of tailings' accumulation.However, the anti-sliding ability depended on average particle size (C3), and the existing situation of tailings' accumulation was determined by downstream slope ratio (C4), current dam height (C5) and seismic intensity (C6).

3) Damage to dam structure
The extent of damage to dam structure is an important factor influencing tailings dam-break and is generally used to measure the size of cracks for the dam.Therefore, the transverse crack coefficient (C7), the longitudinal crack coefficient (C8) and horizontal crack coefficient (C9) were used as the evaluation indicators to measure damage dam structure.

4) Flood overtopping
Due to poor permeability of tailings dam, and if the problems of flood-control and drainage emerged, when floods, it was likely to cause flood overtopping, resulting in tailings dam-break.Thus, the flood control capacity coefficient (C10), the height of beach top and the water level (C11), and drainage facility's intact coefficient (C12) were as the measuring indexes.

5) Safety management
Safety management of tailings was through the establishment of a sound safety management system, and early investigation to find various hazards of the tailings dam, then take effective preventive measures to prevent dam -break accident.It can be divided into daily safety management and emergency safety management, respectively corresponding to routine management coefficient (C13) and emergency management coefficient (C14).Based on relevant regulations and parameters of China's "Technical Specification for Tailings pond", "Design Guidelines for Tailings Dams", "Safety Supervision Regulations for Tailings Dams", and "Technical Specifications for Safety Monitoring of Tailings Dams", and so on , and reference to the results of research on dam break risk classification of tailings reservoirs [5] [20], and according to the stability, flood-control capacity and safety management of the tailings and other safety factors, the corresponding risk levels of tailings dam can be divided into four levels: Ⅰ (normal tailing reservoir), Ⅱ(sickness tailing reservoir), Ⅲ (risk tailing reservoir), Ⅳ(danger tailing reservoir).Besides, according to the membership functions, the height of saturation line, bulk density, average particle size and downstream slope ratio, etc. were economic benefits index, while the cost index contained only current dam height.So, risk evaluation criteria and membership range of the dam-break for tailings pond were shown in table 1. Note: the membership range of each evaluation indicator value in the brackets

2) Section domain
According to the membership range in table 1, its section domain is Matter-element to be evaluated According to the current actual situation with the tailings pond, and basing on its investigation and study, its parameter values and membership degree of 14 evaluation indexes for the tailings pond were shown in table 2. Daily management coefficient 0.87 (0.87) Emergency management coefficient 0.7(0.7)The membership degree of risk evaluation index for tailings dam-break in table 2, the matter-element to be evaluated for the membership degree is

4) Calculation of elementary correlation function values
According to formula (6) and formula (7), elementary correlation function values of risk evaluation index for tailings dam-break at all levels can be calculated.For example, if

5) Determination of weight
In the engineering application, AHP was used to determine the subjective weights of evaluation indexes, and the objective weights of evaluation indexes were determined by the elementary correlation function.The specific steps respectively referenced document [19] and [10].According to the formula (10), .Thus, the comprehensive weight value of each evaluation index after normalization was as shown in Table 4.

Results
The extension correlation of risk level for tailings dambreak can be obtained by inputting the elementary correlation function values and the comprehensive weight values in table 3 and table 4 to the formula (13).Besides, according to the principle of maximum correlation degree(formula ( 14)), the risk grade of tailings dam-break can be calculated.Finally, by inputting calculated values of formula ( 13) and ( 14) to formula ( 15) and ( 16), the level variable eigenvalue can be acquired (See Table 5).So, the five aspects should be strengthened to manage, preventing its evolution to the low grade.

Comparison and Analyses
In the literature [5], the concrete theory and detailed step of using the unascertained measure evaluation method to carry out the tailings dam risk assessment are introduced in detail.First, the method provides a quantitative analysis of the tailings dam by establishing an unascertained measure function of each evaluation factor; Secondly, the weight of each evaluation index is calculated by using the information entropy, and the grade is determined according to the confidence criterion.And its evaluation indicators, evaluation criteria, and weight values were shown in Table 1, Table 2 and Table 4. So, the single index evaluation matrix of the tailings is  k    , the risk grade of the tailings dam-break was Ⅰ.The two discriminant results of the unascertained degree evaluation method are consistent, and its the risk grade is I. Thus, the results of matter-element extension method and the unascertained measure method are consistent, which verifies the correctness of matter-element extension method applied to risk assessment of tailings dam-break, and is in accordance with the actual situation of the project.The difference of matter-element extension method and other methods is that it has the eigenvalue of grade variable, and can be called "the evolution of value".It not only can accurately determine the risk grade of tailings dam-break, but can determine the evolution direction of risk grade for tailings dam-break, so as to take effective measures to prevent the evolution of the risk grade from high to low.

Conclusions
In this paper, on the basis of China's safety standards, specifications, and regulations for tailings ponds, on-site investigation and research were conducted on a tailings pond in Guangxi Province, China.The risk of tailing dambreak is mainly embodied in five aspects, such as seepage failure, dam foundation failure, dam structure destruction, flood overtopping and safety management.By establishing evaluation index system and evaluation criteria, the theory of matter element extension and game theory are combined to evaluate its safety a. are applied to evaluate its safety level.The results show that the grade of tailings dam-break is grade I, which is normal tailings reservoir, and its the eigenvalue of grade variable is 1.3.Therefore, there is a risk of evolution to dam-break level II (ie, sickness tailing reservoir).From the evaluation results, we need to strengthen safety management, flood control facilities and flood control capacity, and take necessary preventive measures to prevent their evolution to the sickness tailing reservoir.The method based on game theory and matter-element extension theory has advantages over other evaluation methods, and its superiority is embodied in two aspects.On one hand, the method based on game theory and matter-element extension theory takes into account the influence of both subjective factors and objective facts on evaluation; On the other hand, it has the level variable eigenvalue, namely "the evolution of value",which cannot only accurately determine the tailings dam risk grade, but also determine the evolution direction of risk level for tailings dam-break, so as to take effective measures to improve the management of technology and risk level of tailings dam, and prevent evolution of the risk grade from high to low.The safety and stability evaluation of tailings dam is a complicated problem.It is necessary to combine monitoring data and experimental analysis data of the tailings pond, and choose an excellent evaluation method to accurately assess it.The purpose is to find out in advance the problems, so as to take necessary measures to prevent dam-break accidents.

Table 1
Risk evaluation criteria of dam-break for tailings reservoir

Table 2
Parameter values and its membership degree of risk evaluation index

Table 5
Assessment results of dam-break for tailings reservoir As Table5shown, the risk grade of the tailings dam-break was Ⅰ (normal tailing reservoir), but the level variable eigenvalue was 1.3.That is to say, the actual risk grade of the tailings dam-break was 1.3 of between I and II.By the level variable eigenvalue, the tailings pond may evolve into sickness tailing pond if decision makers did not take preventive measures.According to the comprehensive weight values of Table4, daily management coefficient, emergency management coefficient, drainage facility's intact coefficient, the height of saturation line and flood control capacity coefficient of proportion was quite large.