Comprehensive Diagnosis Method of the Health of Tailings Dams Based on Dynamic Weight and Quantitative Index

: As a dangerous source of man-made debris ﬂow with high potential energy, tailings dams can cause huge losses to people’s lives and property downstream once they break, and their safety control problem is particularly prominent. The health diagnosis of tailings dams is a complex and nonlinear problem full of uncertainty. At present, the health diagnosis of tailings dams is mostly qualitative evaluation or quantitative analysis aiming at a single index, so this study puts forward a comprehensive quantitative diagnosis method of tailings dam health based on dynamic weight. Slope stability, deformation stability and seepage stability are taken as project layers, and the diagnosis index system of the tailings dam is constructed. The quantitative methods of diagnosis indexes of project layers are proposed. For the dam slope stability project, the safety factor and the reliability index of tailings dams are determined based on the Monte Carlo method, which can consider the uncertainty of tailings material parameters. For the deformation stability project, the normal operation values of deformation rate and deformation amount are determined by analyzing the in situ observation data and combining them with the numerical simulation results. For the seepage stability project, through the analysis of seepage and stability, the relationship curve between the depth of saturation line and the safety factor of anti-sliding stability is established. The norms method is used to determine the quantitative standards for the diagnosis indexes of the basic layer. Based on the analytical hierarchy process method and the penalty variable weight method, the method of dynamic weight of the project layer index is proposed. The proposed methods are applied to a practical engineering project. The results show that the methods can accurately reﬂect the health status of tailings dams. This study provides a new method for evaluating the safety of tailings dams.


Introduction
The tailings pond is a place for storing tailings, and the tailings dam is a dam structure around the tailings pond, which is a key project to ensure the normal operation of the tailings pond. At present, there are about 8869 non-coal mine tailings ponds in China, among which there are about 1112 "overhead tailings ponds", accounting for 14.3%. Tailings pond accidents rank 18th in the ranking of hidden dangers in the world, and their hazards are second only to nuclear radiation and nuclear explosions. [1,2]. On 8 September 2008, the tailings dam of Xinta Mining in Xiangfen, Shanxi Province, China collapsed, resulting in the deaths of at least 277 people and having an extremely bad social impact [3]. On 4 August 2014, the Mount Polley tailings dam in Canada broke due to design reasons, flooded large forests and lakes, and seriously damaged the ecological environment [4]. On 5 November 2015, the Samarco tailings dam in Brazil was liquefied and collapsed due to a As the basis of comprehensive diagnosis, whether the diagnosis index is appropriate or not will directly affect whether the diagnosis result is reasonable and accurate. In this paper, the diagnosis index system is divided into a project layer, an effect-quantity layer and a basic layer. With reference to related literatures [27,28] and norms [29,30], combined with the tailings dam failure modes and hazard factors, the health diagnosis index system of tailings dam was established. As shown in Figure 1, the project layer of the index system includes three aspects: slope stability, deformation stability and seepage stability, and the effect quantity layer is obtained by further concretization of the project layer. The slope stability project includes the deterministic safety factor and the reliability index, and the reliability index is added to consider the randomness of physical and mechanical characteristics of tailings. As a non-linear material, tailings will deform significantly during the life cycle of the tailings dam. Normal consolidation deformation is beneficial to the stability of the tailings dam, but for tailings with strong cementation characteristics and structural properties, excessive deformation will cause its strength to be lost. The deformation stability project is divided into two indexes: the deformation rate and the total deformation. The tailings dam is a permeable structure, and the saturation line is the lifeline. Therefore, the depth of the saturation line is taken as an index of the seepage project. The basic layer is the bottom layer diagnosis index in the index system, including the in situ monitoring data of the tailings dam and the corresponding calculation results obtained by numerical analysis based on the structure of the tailings dam and the physical and mechanical properties of the tailings. The main contents of this paper are as follows: (a) The diagnosis index system is constructed and the health grade is determined. (b) Based on the analytic hierarchy process and the penalty variable weight method, the dynamic weight method of the diagnosis index is proposed. (c) The quantified methods and standards for the basic-layer diagnosis index such as the slope stability, the deformation stability and the seepage stability are proposed. (d) The method was applied to tailings dam I in Brazil, and its applicability and accuracy were verified.

Diagnosis Index
As the basis of comprehensive diagnosis, whether the diagnosis index is appropriate or not will directly affect whether the diagnosis result is reasonable and accurate. In this paper, the diagnosis index system is divided into a project layer, an effect-quantity layer and a basic layer. With reference to related literatures [27,28] and norms [29,30], combined with the tailings dam failure modes and hazard factors, the health diagnosis index system of tailings dam was established. As shown in Figure 1, the project layer of the index system includes three aspects: slope stability, deformation stability and seepage stability, and the effect quantity layer is obtained by further concretization of the project layer. The slope stability project includes the deterministic safety factor and the reliability index, and the reliability index is added to consider the randomness of physical and mechanical characteristics of tailings. As a non-linear material, tailings will deform significantly during the life cycle of the tailings dam. Normal consolidation deformation is beneficial to the stability of the tailings dam, but for tailings with strong cementation characteristics and structural properties, excessive deformation will cause its strength to be lost. The deformation stability project is divided into two indexes: the deformation rate and the total deformation. The tailings dam is a permeable structure, and the saturation line is the lifeline. Therefore, the depth of the saturation line is taken as an index of the seepage project. The basic layer is the bottom layer diagnosis index in the index system, including the in situ monitoring data of the tailings dam and the corresponding calculation results obtained by numerical analysis based on the structure of the tailings dam and the physical and mechanical properties of the tailings.

Classification of Health Levels
Comprehensive health diagnosis needs to classify the diagnosis results and refer to relevant literatures and norms [29], and the classification is mainly based on the fourth and third levels. For example, the Chinese standard Tailing facilities design code [30] is

Classification of Health Levels
Comprehensive health diagnosis needs to classify the diagnosis results and refer to relevant literatures and norms [29], and the classification is mainly based on the fourth and third levels. For example, the Chinese standard Tailing facilities design code [30] is divided into four levels according to the degree of safety: normal pond, disease pond, risk pond, and dangerous pond. Guidelines on the safe design and operating standards for tailings storage [31] issued by Australia is divided into three levels: high, significant and low. Because the definition of safety degree in the four levels is mainly qualitative description, the boundary between risk pond and dangerous pond is relatively fuzzy. Therefore, the Sustainability 2022, 14, 3068 4 of 13 study divides the health levels of tailings dams into three levels. Referring to medical comments on human health, the levels are determined as healthy, diseased, and dangerous.

Standardization Method of Health Value
The effect quantity of each diagnosis index is different, and it is not commensurable. Therefore, the index diagnosis result is processed to make it dimensionless, that is, it is converted into the form of health value, and the influence of index unit and numerical magnitude is excluded. The health value interval is set to [0, 1], in which the health value of 1 is the most ideal health state. The specific classification criteria are shown in Table 1. The health value of the basic index is calculated by standardized Equation (1): where X is the index health value; U is the diagnosis parameter of the basic index; u 1 is the threshold value of the healthy and diseased level; u 2 is the threshold value of the diseased and dangerous level;.u * 1 . and u * 2 . are the upper and lower limits of the diagnosis parameters, respectively, and the health values exceeding the limits are 1 and 0.

The Analytical Hierarchy Process
The analytic hierarchy process is a comprehensive evaluation method that combines qualitative and quantitative analysis by objectively describing subjective judgments [32]. It has a wide range of applications in comprehensive evaluation [33]. The steps of the analytic hierarchy process to determine the weight of index are as follows: (1) Hierarchical structure reflects the relationship between indexes, and the proportion of each index in the same target layer is quantitatively analyzed by the judgment matrix A = (a ij ) n×n . The judgment matrix is positive and the reciprocal matrix is constructed by comparing the factors in pairs, and is generally represented by the scale of 1-9. (2) The constructed judgment matrix has a certain degree of inconsistency, so its rationality is checked for consistency. When the consistency index CR < 0.1, the consistency of the judgment matrix is considered acceptable, and the weight coefficients are allocated reasonably. (3) The maximum eigenvalue λ max and the corresponding eigenvector x are obtained by solving the judgment matrix A = (a ij ) n×n , and the weights of indexes are obtained by normalizing eigenvector x.

The Penalty Variable Weight Method
The deterioration of each index in the project layer will have a significant impact on the health of the tailings dam. When a certain index deteriorates while other indexes are still in healthy state, the traditional fixed-weight diagnosis ignores the impact of the deterioration index. The penalty variable weight method can adjust the weight of the index according to the change of health value, thereby highlighting the diagnosis index that has Sustainability 2022, 14, 3068 5 of 13 deteriorated. The method increases the influence of the index with the lower health value in the comprehensive diagnosis, so as to diagnose the overall health status of the tailings dam more reasonably and accurately.
According to the definition of the penalty variable weight function [34], the three axioms of normalization, continuity and monotonicity need to be satisfied, and, combined with the characteristics of the tailings dam, the variable weight function is constructed as follows: where S(x) is the value of variable weight; x is the health value of the index. The variable weight function is composed of three sections: non-penalty function, penalty function and heavy penalty function, which respectively correspond to three health states: healthy, diseased, and dangerous. The variable weight value of each index is calculated by Equation (3): i is the fixed weight of the ith index.

Diagnosis Method of the Index of Effect Quantity Layer
The data dimensions of each index of the basic layer are different, and the reflections on the health status of the tailings dam are also different. Therefore, according to the characteristics of diagnosis indexes, the reasonable quantitative methods are selected to analyze the indexes, such as numerical simulation, statistical analysis and so on. Then, the norms method and the confidence interval method are used to determine the quantitative standard of the index, to complete the quantitative diagnosis of the indexes.

Slope Stability Project
The typical profile of the tailings dam is selected, and the stability of the tailings dam is calculated by the limit equilibrium method to obtain the slope safety factor corresponding to deterministic parameters. Considering the variability of tailings material and taking the mean and standard deviation of its physical and mechanical parameters as random variables, the reliability analysis is carried out by the Monte Carlo method. A large number of tailings parameter combinations are extracted and their anti-sliding stability safety factors are calculated respectively to determine the probability of the tailings dam break and the reliability index. Quantitative diagnosis is performed by combining the calculation results of the deterministic safety factor and reliability index. The quantitative standard adopts the norms method, and the specific quantitative standard is: (1) Regarding the deterministic safety factor index, the minimum safety factor stipulated in the code is regarded as u 1, which is the threshold value of the healthy and diseased level. u 1 *u 1 is taken as u * 1 that is the upper limit value. The norm [35] describes that the tailings dams with a minimum safety factor of less than 0.95 times the stipulated value belong to the dangerous reservoir, so this value is taken as u 2 , which is the threshold value of the diseased and dangerous level, and the safety factor of 1 is taken as u * 2 , which is the lower limit value. (2) Regarding the reliability index, referring to the value in the norm [35], the specific values are shown in Table 2. The second category of destruction is suitable for sudden brittle destruction, and is mostly applied to concrete structures with higher requirements. Therefore, the reliability index of the tailings dam refers to the first category of destruction. According to the corresponding relationship between reliability index and failure probability, the hierarchical corresponding relationship between the structural safety level and the probability of tailings dam failure is shown in Table 3. In the norms [29], the magnitude of the probability of dam failure is taken as the basis for the classification of risk significance, and the significance difference of different health levels is reflected by enlarging or reducing the failure probability by one magnitude. According to the reference manual, the value in Table 2 is taken as u 1 , and the value is reduced by two orders of magnitude as u * 1 . The value in Table 2 is enlarged by one order of magnitude as u 2 , and the u 2 is enlarged by one order of magnitude as u * 2 .

Deformation Stability Project Deformation Rate
The normal operating value of deformation rate is determined by statistical analysis of historical monitoring data, such as curve fitting and statistical regression. The quantitative standard adopts the norms method, which stipulates that the yellow warning value is 1.3 times the normal operating value, the orange warning value is 2 times the normal operating value, and the red warning value is 3 times the normal operating value. Therefore, u * 1 ,u 1 ,u 2 , and u * 2 are 1, 1.3, 2 and 3 times of the normal operating value respectively.

Total Deformation
After the tailings dam stops filling sub-dam, there is no new load on the upper part, and the deformation of the dam body is mainly the secondary consolidation deformation of tailings. For tailings with cementing properties, creep deformation may lead to loss of strength and eventually instability failure. The quantification of the total deformation index adopts the numerical analysis. Through forward and inverse analysis of the creep deformation of the tailings dam, the total deformation of the dam body can be calculated more accurately. Deformation failure is defined as a penetrating failure area formed by excessive deformation.
Taking the safety factor of deformation as the quantitative index, the greater the safety factor, the greater the safety margin of the total deformation index. When the safety factor is 1, the monitored deformation of the dam body reaches the destruction deformation, that is, the tailings dam is on the verge of instability. Therefore, u * 2 and u 2 take 1 and F s , which is the slope safety factor specified in the norm. Referring to relevant norms and study, u 1 and u * 1 take 2 and 2 * F s .

Seepage Stability Project
The depth of saturation line of the tailings dam has an important influence on the stability of the slope, so the quantitative basis of the saturation line index is the coupling relationship between the depth of saturation line and the stability of the slope.
The distribution of the saturation line under different dry beach lengths is obtained by seepage calculation. Then, the stability safety factors of the dam slope corresponding to different saturation lines are calculated, so as to establish the coupling relationship between the depth of saturation line and the stability safety factor. According to the relationship curve, u 1 takes the depth of saturation line corresponding to F s . u * 1 is twice of u 1 , and u 2 is 0.95 times of F s . u * 2 is the minimum depth of saturation line specified in the norm. Finally, the flow chart of this paper can be shown in Figure 2. deformation, that is, the tailings dam is on the verge of instability. Therefore, 2 u and 2 u take 1 and s F , which is the slope safety factor specified in the norm. Referring to relevant norms and study, 1 u and * 1 u take 2 and 2 * s F .

Seepage Stability Project
The depth of saturation line of the tailings dam has an important influence on the stability of the slope, so the quantitative basis of the saturation line index is the coupling relationship between the depth of saturation line and the stability of the slope.
The distribution of the saturation line under different dry beach lengths is obtained by seepage calculation. Then, the stability safety factors of the dam slope corresponding to different saturation lines are calculated, so as to establish the coupling relationship between the depth of saturation line and the stability safety factor. According to the relationship curve, 1 u takes the depth of saturation line corresponding to s F . * 1 u is twice of 1 u , and 2 u is 0.95 times of s F . * 2 u is the minimum depth of saturation line specified in the norm.
Finally, the flow chart of this paper can be shown in Figure 2.

Case Study
The tailings dam B-I at Córrego do Feijão Iron Ore Mine (dam I) located in Brumadinho, Brazil, was constructed in 1976. The tailings discharge was stopped in July 2016. Before the dam break, the height of the tailings dam was 86 m and the storage capacity was 12.7 Mm 3 . On 25 January 2019, the tailings dam broke, and about 9.7 Mm 3 tailings flowed out of the pond, killing 235 people and flooding an area of about 40 km 2 downstream [36]. Figure 3 is the satellite image taken before the dam break, in which ◆ is the monitoring point of the saturation line and • is the monitoring point of the deformation.

Case Study
The tailings dam B-I at Córrego do Feijão Iron Ore Mine (dam I) located in Brumadinho, Brazil, was constructed in 1976. The tailings discharge was stopped in July 2016. Before the dam break, the height of the tailings dam was 86 m and the storage capacity was 12.7 Mm 3 . On 25 January 2019, the tailings dam broke, and about 9.7 Mm 3 tailings flowed out of the pond, killing 235 people and flooding an area of about 40 km 2 downstream [36]. Figure 3 is the satellite image taken before the dam break, in which is the monitoring point of the saturation line and • is the monitoring point of the deformation.

Diagnosis of the Slope Stability Project
Tailings dam I is a Class 3 dam, and the minimum safety factor specified in the norm [30] is 1.3. According to the quantitative method of the effect quantity index, the quantitative standards of deterministic safety factor index and reliability index are determined. Specific diagnostic criteria are shown in Table 4. Table 4. Diagnosis criteria of the effect quantity indexes (tailings dam I).

Diagnosis of the Slope Stability Project
Tailings dam I is a Class 3 dam, and the minimum safety factor specified in the norm [30] is 1.3. According to the quantitative method of the effect quantity index, the  Table 4. The typical section of the tailings dam is selected to establish a two-dimensional model, as shown in Figure 4. The iron content of the tailings is more than 50%, which makes the bulk density of the tailings very high, about 26 kN/m 3 . The material parameters required in the stability calculation refer to the test data [37], as shown in Table 5.
determined. Specific diagnostic criteria are shown in Table 4. The typical section of the tailings dam is selected to establish a two-dimensional model, as shown in Figure 4. The iron content of the tailings is more than 50%, which makes the bulk density of the tailings very high, about 26 kN/m 3 . The material parameters required in the stability calculation refer to the test data [37], as shown in Table 5.  The calculation results of stability and reliability are shown in Figure 5. The most dangerous sliding surface of the tailings dam is located between the downstream toe and the fourth sub-dam, and its deterministic safety factor is 1.307, which is basically consistent with the results of the accident investigation report. The failure probability is 1.45 × 10 −3 , and the reliability index is 2.704. The diagnostic results determined by the quantified standard and the Equation (1) show that the health value of the safety factor index is 0.6759, and the health value of the reliability index is 0.4376.
The calculation results of stability and reliability are shown in Figure 5. The most dangerous sliding surface of the tailings dam is located between the downstream toe and the fourth sub-dam, and its deterministic safety factor is 1.307, which is basically consistent with the results of the accident investigation report. The failure probability is 1.45 × 10 −3 , and the reliability index is 2.704. The diagnostic results determined by the quantified standard and the Equation (1) show that the health value of the safety factor index is 0.6759, and the health value of the reliability index is 0.4376.

Deformation Rate Index
The deformation monitoring data is derived from the radar monitoring data, including the vertical component and a small amount of the horizontal component. Because the horizontal component is close to the noise level, the vertical deformation is used for diagnosis of the deformation rate index. Figure 6 shows the vertical deformation curve of the points, with No.1, No.

Deformation Rate Index
The deformation monitoring data is derived from the radar monitoring data, including the vertical component and a small amount of the horizontal component. Because the horizontal component is close to the noise level, the vertical deformation is used for diagnosis of the deformation rate index. Figure 6  Based on the monitoring data from 6 January 2018 to 20 December 2018, it can seen that the deformation in the historical period has no obvious acceleration stage, and the overall deformation is uniform. Therefore, the average deformation rate in the period is taken as the normal operation value. According to the determination method of quantification standard, the quantification standard of each point is obtained, which is shown in Table 4. The standard is used to diagnose the average deformation rate of each point within one month before the dam break, and the healthy value of the deformation rate index before the dam break is obtained. The specific results are shown in Table 6.

Total Deformation Index
Because of the high iron content in the tailings, the oxidation of iron will lead to cementation between particles. The test results [37] also show that under the constant load, the continuous creep of the tailings may lead to obvious and rapid strength loss, so the strain-softening model is adopted in the simulation calculation.
The initial stress state is obtained by simulating the accumulation process of the tailings dam. Then the creep deformation is added to the calculation, and the Based on the monitoring data from 6 January 2018 to 20 December 2018, it can seen that the deformation in the historical period has no obvious acceleration stage, and the overall deformation is uniform. Therefore, the average deformation rate in the period is taken as the normal operation value. According to the determination method of quantification standard, the quantification standard of each point is obtained, which is shown in Table 4. The standard is used to diagnose the average deformation rate of each point within one month before the dam break, and the healthy value of the deformation rate index before the dam break is obtained. The specific results are shown in Table 6.

Total Deformation Index
Because of the high iron content in the tailings, the oxidation of iron will lead to cementation between particles. The test results [37] also show that under the constant load, the continuous creep of the tailings may lead to obvious and rapid strength loss, so the strain-softening model is adopted in the simulation calculation.
The initial stress state is obtained by simulating the accumulation process of the tailings dam. Then the creep deformation is added to the calculation, and the deformation is increased step by step, and the stability of the tailings dam after creep is calculated step by step until the tailings dam is destroyed. The investigation report shows that when the added creep is consistent with the deformation indicated by the monitoring data, the strength loss of tailings will be caused and dam failure will occur. Combined with the diagnostic standard shown in Table 4, the safety factor of the total deformation is 1, and the total deformation index of the tailings dam is determined to be in a dangerous state and the health value is 0.

Diagnosis of the Seepage Stability Project
The seepage field of the dry beach with lengths of 50 m, 100 m, 125 m, 150 m and 200 m is calculated. The results show that the saturation lines all overflow from the drainage body between the fifth and sixth sub-dams, and then flow into the downstream channel through the drainage channel on the dam surface, which is consistent with the actual situation. By calculating the stability safety factor of the dam slope corresponding to the seepage field under various working conditions, the relationship curves between the depth of saturation line at PZ-4C, PZ-5C, PZ-24C and PZ-23C points and the stability safety factor of the dam slope are established, as shown in Figure 7. According to the curve, the quantification standard of each point is determined, as shown in Table 4. The diagnosis data adopts the average value of the depth of the saturation line within one month before the dam break, and the diagnosis results are shown in Table 7.

Comprehensive Health Diagnosis
The health value of the project-layer index is obtained by summarizing the health value of the effect quantity index, and the health values of the slope stability project, the deformation stability project and the seepage stability project are 0.5568, 0.2080 and 0.7377, respectively, as shown in Table 8. The slope stability project is in a diseased state, and the deformation stability project is in a dangerous state. Before the dam break, the seepage field of the tailings dam is in good condition, and the depth of saturation line has a downward trend year by year, and the seepage stability project is in healthy state.  According to the curve, the quantification standard of each point is determined, as shown in Table 4. The diagnosis data adopts the average value of the depth of the saturation line within one month before the dam break, and the diagnosis results are shown in Table 7.

Comprehensive Health Diagnosis
The health value of the project-layer index is obtained by summarizing the health value of the effect quantity index, and the health values of the slope stability project, the deformation stability project and the seepage stability project are 0.5568, 0.2080 and 0.7377, respectively, as shown in Table 8. The slope stability project is in a diseased state, and the deformation stability project is in a dangerous state. Before the dam break, the seepage field of the tailings dam is in good condition, and the depth of saturation line has a downward trend year by year, and the seepage stability project is in healthy state. The analytical hierarchy process method was used to weight the project-layer indexes, and eight experts were invited to give judgment matrix considering the importance of each index to the overall health status of the tailings dam and the operation characteristics of the tailings dam.
The consistency indexes of the judgment matrices are all less than 0.1, and the consistency test meets the requirements. The weight of the project-layer index is obtained by solving the eigenvalue of the judgment matrices. As can be seen from Table 9, the opinions of experts are relatively unified, and they all think that the slope stability project is more important. Combining the opinions of eight experts by the arithmetic average method, the fixed weights of the project-layer index are w = (0.4483, 0.2612, 0.2905). The health value of the slope stability project falls within the diseased state. Therefore, the penalty function should be adopted in Equation (2), and its value of variable weights should be determined to be 2.5813. Similarly, the values of variable weights of the project layer indexes obtained by Equation (2) are S = (2.8513, 16.3090, 1.000), and the final variable weights calculated by Equation (3) are w 0 = (0.2193, 0.7309, 0.0498). The health value of the tailings dam based on dynamic weight is 0.3109 calculated by the weighted average method, that is, the tailings dam before the dam break was in a dangerous state, which is consistent with the actual situation of the tailings dam. The result proves the accuracy and applicability of this method. The health value of the tailings dam based on fixed weight is 0.5283. The traditional diagnosis method based on fixed weight will lead to the distortion of diagnosis results, while the dynamic weight can effectively solve the problem of state imbalance and improve the reliability of diagnosis results.

Conclusions
(1) The index system of comprehensive diagnosis of tailings dam health is established, and the dynamic change of index weight is realized based on the analytical hierarchy process and the penalty variable weight method, which increases the importance of the deterioration index in comprehensive diagnosis and makes the diagnosis result more accurate and reasonable. (2) Based on numerical simulation and the statistical analysis method, the diagnosis method of the indexes of effect quantity layer is put forward, and the quantitative standard of each index is determined. The comprehensive diagnosis method of tailings dam health based on monitoring data is put forward, and the quantitative diagnosis of tailings dam health status is realized. (3) This method was applied to tailing dam I, and the health value of 0.3109 indicates that the tailings dam is in a dangerous state before the dam failure, which is consistent with the actual situation and verifies the accuracy and applicability of the method.
The comprehensive diagnosis based on the dynamic weight of tailings dam is very valuable. This method overcomes the influence of artificial subjective judgment and provides a new method for evaluating the safety of tailings dams.