**5. Practical application: coupled water flow-slope stability analyses of a tailings dam**

#### **5.1. Introduction**

Mining waste (tailings) is disposed of in structures called tailings dams. The most important difference between a tailings dam and a typical water storage dam in the conventional waterretaining sense is that tailings dams do not contain an engineered water barrier and they are built and used simultaneously. For this reason, it is common for the originally conceived project to undergo constant modifications. Current technological advances allow for the numerical modelling of steady and transient state flow analyses for these structures, and along with adequate monitoring, their stability can be verified with coupled water flow-slope stability analyses before the occurrence of any potential errors. In the following sections, a cross section of a tailings dam is analysed (**Figure 5**), and the importance and benefits that result from the numerical model are highlighted; in addition, further recommendations are given for this type of analysis.

**Figure 5.** Maximum tailings dam cross section for two-dimensional model.

#### **5.2. Material properties**

The properties of tailings dams are variable because they depend largely on the origin of the materials used for their construction. Several authors have performed tests with different materials and have specified typical values for these [37–40]. Based on these references, in **Table 3**, the properties of the materials from the case study considered in the numerical model of this section are provided.


**Table 3.** Tailings properties for the tailings dam considered.

Assuming that the materials of the tailings dam are found in unsaturated state, for this analysis, the SWCC and hydraulic conductivity functions must be determined. If laboratory data are unavailable, it is possible to use estimates with the aid of material index properties, such as grain-size distribution curves, which are considered in the calculations performed here (**Figure 6**).

**Figure 6.** Grain-size distribution curves for tailings considered in the numerical model.

#### **5.3. Unsaturated soil properties functions**

**5. Practical application: coupled water flow-slope stability analyses of a**

Mining waste (tailings) is disposed of in structures called tailings dams. The most important difference between a tailings dam and a typical water storage dam in the conventional waterretaining sense is that tailings dams do not contain an engineered water barrier and they are built and used simultaneously. For this reason, it is common for the originally conceived project to undergo constant modifications. Current technological advances allow for the numerical modelling of steady and transient state flow analyses for these structures, and along with adequate monitoring, their stability can be verified with coupled water flow-slope stability analyses before the occurrence of any potential errors. In the following sections, a cross section of a tailings dam is analysed (**Figure 5**), and the importance and benefits that result from the numerical model are highlighted; in addition, further recommendations are

The properties of tailings dams are variable because they depend largely on the origin of the materials used for their construction. Several authors have performed tests with different materials and have specified typical values for these [37–40]. Based on these references, in **Table 3**, the properties of the materials from the case study considered in the numerical model

**Material USCS classification** *k* **(m/s)** *c* **' (kPa)** *φ* **' (°)**

Fine material Silt with sand ML 1.582 × 10−8 0 25 Coarse material Silty sand SM 1.000 × 10−6 0 35

**Name Symbol**

**tailings dam**

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**5.1. Introduction**

given for this type of analysis.

**5.2. Material properties**

of this section are provided.

**Table 3.** Tailings properties for the tailings dam considered.

**Figure 5.** Maximum tailings dam cross section for two-dimensional model.

For the dam materials analysed in the present case study, the SWCC (that represent the relationship between water content and soil suction) of **Figure 7** were determined by the following method:


**Figure 7.** Soil-water characteristic curves of tailings dam materials.

In **Figure 8**, the hydraulic conductivity functions considered in the present dam case study are presented. A great similarity between the laboratory data and the estimated data can be observed. Thus, the use of estimation methods when laboratory data are unavailable repre‐ sents a feasible solution and potential advantage; however, such models should be used with caution and rationality.

**Figure 8.** Hydraulic conductivity functions of the tailings dam materials.

#### **5.4. Two-dimensional modelling for steady-state seepage conditions**

A steady-state seepage analysis was performed with Eq. (5), which can be solved numerically by the finite element method using the Seep/W algorithm [41]. For two-dimensional models, boundary conditions should be adequately defined in addition to discretizing the flow regions the best possible. Greater emphasis must be placed on areas that require greater detail in the expression of results, where numerical difficulties may be presented, or for areas with high contrast in the permeability of materials by orders of magnitude. In **Figure 9**, the boundary conditions and the discretization of the flow regions are detailed.

**Figure 9.** Boundary conditions and discretization model for the tailings dam with Seep/W [41].

**Figure 7.** Soil-water characteristic curves of tailings dam materials.

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**Figure 8.** Hydraulic conductivity functions of the tailings dam materials.

**5.4. Two-dimensional modelling for steady-state seepage conditions**

A steady-state seepage analysis was performed with Eq. (5), which can be solved numerically by the finite element method using the Seep/W algorithm [41]. For two-dimensional models, boundary conditions should be adequately defined in addition to discretizing the flow regions the best possible. Greater emphasis must be placed on areas that require greater detail in the expression of results, where numerical difficulties may be presented, or for areas with high

caution and rationality.

In **Figure 8**, the hydraulic conductivity functions considered in the present dam case study are presented. A great similarity between the laboratory data and the estimated data can be observed. Thus, the use of estimation methods when laboratory data are unavailable repre‐ sents a feasible solution and potential advantage; however, such models should be used with Based on the steady-state seepage analysis, it is possible to estimate the position of the phreatic surface of the tailings dam and for the different pond levels (**Figure 10**).

**Figure 10.** Variation in the phreatic surface due to reduction of the freeboard with Seep/W [41].

Based on the previous results, the stability of the structures of the tailings dam can be evaluated by a GLE method, such as the Morgenstern-Price method, to define the slip surface for each level of the pond. In **Figure 11**, the slip surfaces are indicated, which were obtained from the Slope/W algorithm [42] and from considering the results of the steady-state groundwater flow analyses.

**Figure 11.** Slip surfaces due to phreatic surfaces for different pond levels with Slope/W [42].

#### **5.5. Two-dimensional modelling for transient conditions**

A transient model (variable over time) was performed to evaluate the influence of rainfall on the tailings dam defined in **Figure 5** by the numerical solution of Eq. 13. In addition, the behaviour of the pond levels is studied when the level varies 3.00 m with respect to the crest. In this case, the boundary conditions are modified to be a time-dependent function while the discretization of the model was maintained without modifications with respect to the mesh used for the steady-state analysis.

The aforementioned model allows for the distinct stages of the analysis to be defined. The first set of calculations corresponds to the period of 0–48 h, after which, a storm would cause an immediate increase in water level of the pond (**Figure 12**). An evaluation of the infiltration due to rainfall during this period is also shown in **Figure 13**.

**Figure 12.** Variation in the water surface within the tailings structure due to rainfall and loss of freeboard for 48-h peri‐ od with Seep/W [41].

**Figure 13.** Behaviour of the pore-water pressure near the ground surface due to rainfall and loss of freeboard.

Short analysis periods for materials of low permeability are often inadequate because they might not be able to clearly determine the influence of water. For example, in the previous analysis during the water filling of pond, the water surfaces variation was unable to be clearly distinguished. Similarly, rainfall was shown to have low significance because it was unable to infiltrate to a considerable enough depth to affect the slope stability.

**5.5. Two-dimensional modelling for transient conditions**

to rainfall during this period is also shown in **Figure 13**.

used for the steady-state analysis.

180 Groundwater - Contaminant and Resource Management

od with Seep/W [41].

A transient model (variable over time) was performed to evaluate the influence of rainfall on the tailings dam defined in **Figure 5** by the numerical solution of Eq. 13. In addition, the behaviour of the pond levels is studied when the level varies 3.00 m with respect to the crest. In this case, the boundary conditions are modified to be a time-dependent function while the discretization of the model was maintained without modifications with respect to the mesh

The aforementioned model allows for the distinct stages of the analysis to be defined. The first set of calculations corresponds to the period of 0–48 h, after which, a storm would cause an immediate increase in water level of the pond (**Figure 12**). An evaluation of the infiltration due

**Figure 12.** Variation in the water surface within the tailings structure due to rainfall and loss of freeboard for 48-h peri‐

**Figure 13.** Behaviour of the pore-water pressure near the ground surface due to rainfall and loss of freeboard.

Short analysis periods for materials of low permeability are often inadequate because they might not be able to clearly determine the influence of water. For example, in the previous Therefore, long-term analyses for these types of materials are more representative because they allow remark the water surfaces variation over a longer period of time. Thus, the results of these analyses are shown in **Figure 14**.

**Figure 14.** Water surfaces variation within the tailings structure for a transient analysis of 20 years with Seep/W algo‐ rithm [41].

Following the previous procedure and considering the pond levels indicated in **Figure 12**, the slip surfaces were defined by the Morgenstern-Price method with the Slope/W algorithm [42]. It is worth mentioning that a stability analysis for a period of 48 h did not provide significant results because during this time, the structure was not considerably influenced by the water. However, for long-term conditions, such stability analyses could play an important role in the study.

**Figure 15.** Variation in the factor of safety in the tailings dam over a period of 20 years.

In **Figure 15**, the variation of factor of safety over time is presented, wherein the variation in the position of water surface over time is considered (**Figure 14**). A decrease in stability over time is also observed. For this type of evaluations, numerical difficulties may be presented that are subsequently reflected in the result, which leads to erroneous behaviour. Therefore, it is recommended to manipulate the variables that may affect the model, such as the calculated time intervals (smaller time increments may lead to a more detailed response), the finite elements mesh, and the convergence parameters. An adequate manipulation of these variables can significantly improve the results of the numerical model [43].

#### **5.6. Three-dimensional numerical modelling**

### *5.6.1. Model geometry*

In current numerical modelling, it is common for 3D domains to be extruded, or in other words, two-dimensional sections are assigned a certain thickness. Then, the model is extended along a horizontal axis, resulting in a continuous, three-dimensional model.

Extrusions are recommended depending on the type of problem to be solved. For example, extrusions are suitable for analyses of protection levees or embankments with regular geome‐ tries that extend over long distances. In the case of structures with irregular geometries, such as dams, it is more suitable that calculations consider the specific topography of the site for these to be more representative. This latter method requires a greater amount and detail of information for the numerical model to be successfully resolved. However, it is convenient in this case to standardize certain surface areas of the model to avoid an excessive discretization of each region. In **Figure 16**, the geometries assumed for both cases performed here are shown.

**Figure 16.** Geometries: (a) 3D model extrusion and (b) 3D realistic model.

#### *5.6.2. Boundary conditions*

In assigning boundary conditions, sufficient regions must be created to allow the site-specific conditions to be adequately represented and well defined, which thus leads to optimum modelling results. In **Figure 17**, the conditions assigned in both scenarios of the case study analysis are shown.

**Figure 17.** Boundary conditions of the 3D models considered in the calculations with SVFlux algorithm [44].

#### *5.6.3. Discretization of the model*

elements mesh, and the convergence parameters. An adequate manipulation of these variables

In current numerical modelling, it is common for 3D domains to be extruded, or in other words, two-dimensional sections are assigned a certain thickness. Then, the model is extended along

Extrusions are recommended depending on the type of problem to be solved. For example, extrusions are suitable for analyses of protection levees or embankments with regular geome‐ tries that extend over long distances. In the case of structures with irregular geometries, such as dams, it is more suitable that calculations consider the specific topography of the site for these to be more representative. This latter method requires a greater amount and detail of information for the numerical model to be successfully resolved. However, it is convenient in this case to standardize certain surface areas of the model to avoid an excessive discretization of each region. In **Figure 16**, the geometries assumed for both cases performed here are shown.

In assigning boundary conditions, sufficient regions must be created to allow the site-specific conditions to be adequately represented and well defined, which thus leads to optimum modelling results. In **Figure 17**, the conditions assigned in both scenarios of the case study

**Figure 17.** Boundary conditions of the 3D models considered in the calculations with SVFlux algorithm [44].

can significantly improve the results of the numerical model [43].

a horizontal axis, resulting in a continuous, three-dimensional model.

**Figure 16.** Geometries: (a) 3D model extrusion and (b) 3D realistic model.

**5.6. Three-dimensional numerical modelling**

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*5.6.1. Model geometry*

*5.6.2. Boundary conditions*

analysis are shown.

The mesh generation for 3D models is perhaps the most complicated part of this process and even more so when considering site-specific topographic conditions. During these processes, the benefits of the extruded 2D models are evident because the generation of the mesh, as well as its distribution, is more regular. Moreover, 3D models of realistic topographies make mesh generation more difficult; in addition, 3D models require a greater number of elements to adapt to the model. In **Figure 18**, the distribution of the meshes obtained with the SVFlux algorithm is shown [44], and in **Table 4**, a comparison is made of the number of elements required for each model.

**Figure 18.** Finite element mesh used for calculation of the 3D models with SVFlux algorithm [44].


**Table 4.** Comparison of the number of finite elements for different analysis conditions.

#### *5.6.4. Water flow analyses*

Important differences are found when 3D model extrusions are compared to 3D models that consider site-specific topography. In **Figure 19**, it can be observed that the distribution of the hydraulic heads tends to vary in both cases. The 3D model extrusion shows a constant dissipation in the hydraulic head; however, this is not very representative of this type of structure. On the other hand, the 3D realistic model that considers site-specific topography shows a more variable behaviour in the distribution of the hydraulic head, which can be considered to be more representative of the analysed case study.

**Figure 19.** Distribution of hydraulic heads (m) with SVFlux algorithm [44].

The previous results can be verified by comparing the distributions of the hydraulic heads at the maximum cross section of the structure. Theoretically, the distributions of the 2D model, 3D model extrusion, and 3D realistic model with site-specific topography should be nearly identical or extremely similar considering the similar conditions and inputs for the analysis. In **Figure 20**, this comparison is shown.

**Figure 20.** Variation in the hydraulic heads (m) for different analysis criteria.

#### *5.6.5. Three-dimensional slope stability analysis*

Several studies have demonstrated the importance of defining the relationship between the two-dimensional and three-dimensional models. The majority of studies agree in that the values for the factors of safety obtained from the 2D slope stability analyses are conservative, and these values tend to increase upon considering 3D models, which occurs because these calculations consider steady-state water conditions and disregard the filtration forces gener‐ ated within the structure [45, 46]. This phenomenon is observed in **Figure 21**, where the results of a slope stability analysis during dry conditions are shown. In this case, the 3D model extrusion and 3D realistic model (considering site-specific topography) did not present significant differences.

On the contrary, in **Figure 22**, the safety factors obtained for each scenario from a slope stability analysis with the linear *phi-b* model are shown, which consider water flow in the tailings structure. For these cases, the differences between each of the analyses and their criteria are distinguishable. According to the previous results, it is important to consider the water flow through the specific medium under study because it can significantly affect the stability of the earth structures. In this case, the 2D and 3D model extrusions show a high value for the factor of safety in comparison with the 3D realistic model. These differences can be attributed to the irregular topography, which causes the distribution of the hydraulic head to exhibit non-linear behaviour.

**Figure 19.** Distribution of hydraulic heads (m) with SVFlux algorithm [44].

**Figure 20.** Variation in the hydraulic heads (m) for different analysis criteria.

*5.6.5. Three-dimensional slope stability analysis*

significant differences.

In **Figure 20**, this comparison is shown.

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The previous results can be verified by comparing the distributions of the hydraulic heads at the maximum cross section of the structure. Theoretically, the distributions of the 2D model, 3D model extrusion, and 3D realistic model with site-specific topography should be nearly identical or extremely similar considering the similar conditions and inputs for the analysis.

Several studies have demonstrated the importance of defining the relationship between the two-dimensional and three-dimensional models. The majority of studies agree in that the values for the factors of safety obtained from the 2D slope stability analyses are conservative, and these values tend to increase upon considering 3D models, which occurs because these calculations consider steady-state water conditions and disregard the filtration forces gener‐ ated within the structure [45, 46]. This phenomenon is observed in **Figure 21**, where the results of a slope stability analysis during dry conditions are shown. In this case, the 3D model extrusion and 3D realistic model (considering site-specific topography) did not present

**Figure 21.** Factor of safety for distinct slope stability analyses in dry conditions (without water flow) with SVSlope al‐ gorithm [47].

**Figure 22.** Factor of safety for distinct slope stability analyses considering water flow through the tailings structure with SVSlope algorithm [47].

It is important to highlight that this type of analysis requires more exhaustive evaluations. In this case, only the importance of these calculations is reinforced, and methodological sugges‐ tions are proposed. However, additional numerical analyses and their respective validation in the field are imperative to adequately define the behaviour of these types of numerical models.
