# Probabilistic Analysis of Floods from Tailings Dam Failures: A Method to Analyze the Impact of Rheological Parameters on the HEC-RAS Bingham and Herschel-Bulkley Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Mathematical Rheological Models and Parameterization

_{y}is the yield stress or limit stress; µ is the dynamic viscosity of the mixture or dynamic viscosity; and dv

_{x}/dz is the strain rate or velocity gradient.

_{y}[45]. In this model, the shear stress and strain rate exhibit a non-linear relationship, serving as a generalized proposition of the Bingham plastic fluid. This relationship is described by Equations (3) and (4), indicating the sum of yield and viscous/turbulent stresses [46].

_{v}); maximum volumetric concentration of solids (Max C

_{v}); and the coefficients a and b of the exponential yield stress curve. These parameters were determined in view of the predefined HEC-RAS approaches. For the Herschel-Bulkley model, the following parameters were defined: the yield stress (τ

_{y}), the consistency index (K), and the behavior index (n). In this case, the predefined approach of the exponential curve, i.e., the calibration parameters a and b, was not used since these parameters are usually adjusted based on the yield stress (τ

_{y}) of the Bingham model. Thus, Herschel-Bulkley yield stress (τ

_{y}) values obtained from previous studies were used.

#### 2.2. Parameter Intervals and Sampling

_{v}, Max C

_{v}, a, and b were subject to variation. The decision to employ independent random sampling for these parameters is justified by the inherent limitation of HEC-RAS, which utilizes a fixed C

_{v}value. Consequently, the resulting values of yield stress (τ

_{y}) and dynamic viscosity (μ) remain constant when C

_{v}is employed in their calculation. Additionally, to ensure physically meaningful parameter combinations, the value ranges and relationships between the parameters were adjusted to yield τ

_{y}and μ values within the ranges documented in the literature (Table 1), following a similar approach [24].

_{v}) in the context of hyperconcentrated flow was defined as 20% to 60%. This interval was chosen to cover a range of flow conditions, including floods, mudflows, and hyperconcentrated flows. Based on the literature, the study [54] suggested a C

_{v}range of 20% to 50% for floods and mudflows, while [54] proposed a C

_{v}range of 30% to 60% specifically for hyperconcentrated flows.

_{v}), the chosen interval considered a maximum value of 61.5%, which corresponds to the value reported in [53]. This value is pre-defined in HEC-RAS [46], and since it exceeds the upper limit of the adopted interval for solid volume concentration (C

_{v}of 60%), it was established as the maximum value for the Max C

_{v}interval.

_{v}and Max C

_{v}were independently sampled, along with the parameters a and b in the Bingham model. Subsequently, the sampled values of Max C

_{v}were reorganized to ensure that within each parameter set, the value of Max C

_{v}was always greater than C

_{v}. Another criterion in combining these parameters was to adhere to the limits of dynamic viscosity (μ) reported in the literature, with a minimum μ of 0.002 Pa∙s [9] and a maximum μ of 100.0 Pa∙s [38]. In brief, the Max C

_{v}samples were rearranged, based on the C

_{v}samples, in such a way that the relationship between the parameters satisfied two criteria: Max C

_{v}> C

_{v}, and the relationship between Max C

_{v}and C

_{v}in the calculation of μ using the Maron and Pierce method [46] yielded values within the range of 0.002–100.0 Pa∙s.

_{v}of 65% in the slump test. In accordance with the C

_{v}range of 20% to 60% adopted in this study, the value of 1726.33 Pa was considered the maximum value for τ

_{y}, also obtained in the slump test for a C

_{v}of 60%.

_{v}, τ

_{y}, K, and n. For this test, the ranges defined in the literature review (Table 2) were maintained, as the parameters K and n are calibration parameters of the H-B model and do not have a direct relationship that allows for evaluating or limiting their ranges. The range of solid volume concentration (C

_{v}) was kept from 20% to 60% [14,54].

_{y}values in the H-B model remained the same as observed in previous studies. Although they share the same name, the yield stresses of the Bingham and H-B models have different values for the same fluid. Thus, the adopted yield stress range considers only the values adjusted according to the H-B model.

_{v}) in tailings was compared to the normal distribution using measured data [57]. The authors in [27] fitted normal, lognormal, gamma, and beta statistical distributions for rheological parameter sets. Due to the scarcity of studies and data used in fitting probability distributions in the literature for these parameters, the approach in [24] was adopted in this paper, which used a uniform distribution to account for the variation of all rheological parameters tested.

#### 2.3. Automation of Sensitivity Analysis in HEC-RAS

#### 2.4. Case Study: The Hydrodynamic Model

^{6}m

^{3}. It is located in a virtual mountainous terrain, with a confined V-shaped downstream valley along the first 3.6 km and a flat-floored urban area valley ending in a lake along approximately 17.6 km. It should be noted that for this study, it was assumed that the dam is an earth dam storing undefined mining tailings.

Parameters | Data | Froehlich [64] |
---|---|---|

Failure Mode | Overflow | |

Total volume (V_{w}) (1.000 m^{3}) | 38,276.34 | |

Average breach width (B_{m}) (m) | 116.27 | |

Minimum breach width (m) | 55.27 | |

Elevation of the dam crest (m) | 272.00 | |

Elevation of the base of the dam (m) | 211.00 | |

Elevation of the bottom of the breach (m) | 211.00 | |

Dam height (m) | 61.0 | |

Height of the breach (H_{b}) (m) | 61.0 | |

Time of breach formation (h) | 0.54 | |

Left Lateral Slope (H:1V) | 1.0 | |

Right Side Slope (H:1V) | 1.0 | |

Mode of progression | Sine Curve |

^{2}. The computational mesh consists of cells with dimensions of 50 × 50 m, with a refinement of 40 × 40 m in the downstream thalweg of the confined valley, and cells of 100 × 100 m in the lake area. The downstream boundary conditions are defined as normal depth in two sections, with a slope of 0.005 m/m for the thalweg and 0.0001 m/m for the downstream lake. The computational time step used in the simulations was one second, satisfying the Courant condition with a maximum value of unity. The shallow water equations (SWE) were employed to perform the simulations.

^{3}/s.

## 3. Results

#### 3.1. Probabilistic Maps Related to Flooded Areas

#### 3.2. Arrival Times and Maximum Depth Variation along the Valley

#### 3.3. Rheological Parameters vs. Simulated Areas

^{2}and 50.8 km

^{2}(Figure 7), respectively, and the area results showed a COV of 17.9%. It is observed that most simulations returned values close to the maximum one: about 68% of the simulations resulted in flooded areas within a 5% variation from the maximum simulated area (from 48.2 km

^{2}to 50.8 km

^{2}). It revealed that most sets of rheological parameters resulted in higher flooded area values. The relationships of the four parameters (C

_{v}, Max C

_{v}, a and b) with the flooded area indicated a tendency toward reduction in the flooded area with the increase of these input parameters, as indicated by the Spearman correlation coefficient (negative). This influence was most noticeable between the flooded area and the values of b, the exponent of the yield stress curve, which indicated the strongest correlation (ρ = −0.7301) (Figure 7). The parameters C

_{v}and a showed moderate correlation with the flooded area, with Spearman’s correlation coefficients equal to −0.4116 and 0.4606, respectively. The parameter Max C

_{v}showed a weak correlation with the flooded area (ρ = −0.0529). Thus, the simulations that returned smaller flooded areas are mainly related to the increase in the value of the parameter b.

^{2}to 50.4 km

^{2}, with a coefficient of variation of 25.7%. A very strong correlation (ρ = −0.9738) is observed between the increase in yield stress (τ

_{y}) and the reduction in flooded area (Figure 8). The variation of the parameter C

_{v}showed a weak correlation with the flood-simulated areas (ρ = 0.2371), and the parameters K and n showed no correlation (respectively, ρ = 0.0113 and ρ = 0.0241).

^{2}. It is observed that this maximum flooded area threshold refers to physical limitations related to the terrain, roughness, runoff volume, or different sets of rheological parameters, which were defined by the modeler and considered for the different scenarios. The results showed that, in these conditions, both models were able to represent the worst-case scenario with different sets of rheologically assigned variables.

#### 3.4. Rheological Parameters vs. H_{mean}

_{max}) in the computational mesh cells for the areas of interest (A1 and A2) (Figure 9) were evaluated according to Equation (6). This equation calculates the maximum average runoff height (H

_{mean}) for each area by summing up the maximum flow depth (H

_{max}) in each flooded cell (H > 0) within each area and dividing the result by the total number of flooded cells (N

_{WC}) for each area [24].

_{WC}is the total number of flooded cells (h > 0); and H

_{max}is the maximum flow depth.

_{mean}and each parameter were obtained for each area of interest for the Bingham model (Figure 10) and for the H-B model (Figure 11).

_{v}in regions A1 and A2 (0.0520 and 0.0640, respectively). Between H

_{mean}and b, it is verified that in both regions, A1 and A2, the larger values of b led to higher values of H

_{mean}, with strong correlations in region A1 (0.6902) and in region A2 (0.6896).

_{y}led to the increase of H

_{mean}, with Spearman correlation strong in region A1 (0.8255) and very strong in region A2 (0.9317). Spearman’s correlation between parameter n and depths shows a weak correlation for both regions, equal to 0.1937 and 0.0881, respectively. By the formulation of the H-B model, a more evident effect of the parameters K and n on the results was expected; therefore, the parameter ranges may have evidenced the impact of the yield stress (τ

_{y}) on the results. Therefore, in this scenario, the parameter τ

_{y}was the one that caused the most noticeable effect on the flooded area and the maximum average runoff height results.

## 4. Discussion

_{mean}frequency distribution showed a normal distribution of this result, and in region A2, the results were uniformly distributed.

_{v}was the one that showed less correlation. In the H-B model, the yield stress (τ

_{y}) was the parameter that showed greater correlation with the variation of results, and the behavior index (n) expressed less correlation. In the results obtained by [24], it was observed that the FLO-2D model (Quadratic rheological model) is mainly sensitive to the parameter β

_{1}that composes the rheological curve of the dynamic viscosity, affecting mainly the results of the simulated flow depth. The study presented in [66] indicates that the results of maximum debris flow depths were more affected by the variation of the turbulent coefficient and the friction coefficient of the Voellmy rheological model in the 2D dynamic modeling with the RAMMS v.1.3.16 (Rapid Mass Movements) software. It is noticed that different hydrodynamic models have different sensitivities, which could justify future comparative tests using different approaches from this perspective.

## 5. Conclusions and Recommendations

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Parameter [Unit] | Parameter Range | Reference | Tailings | Test/Method | Rheological Model/Fitted Equation |
---|---|---|---|---|---|

Volumetric concentration of solids (C_{v})[%] | 21.21–35.78 | [47] | Coal | n.a. | n.a. |

27.98–46.65 | [47] | Copper | |||

36.50–38.62 | [47] | Gold | |||

36.34–44.62 | [47] | Lithium | |||

11.09–19.66 | [47] | Nickel | |||

25.48–31.25 | [47] | Uranium | |||

11.82–50.11 | [47] | Zinc | |||

18.0–52.5 | [7] | Base metal | |||

21.8–55.3 | [48] | Iron | |||

32.5–65.0 | [48] | Iron | |||

9.3–48.4 | [9] | Gold | |||

34.75–57.14 | [49] | Iron | |||

58 | [2] | Iron | |||

47 | [37] | Iron | |||

23 | [52] | Iron | |||

Maximum volumetric concentration of solids (Max C_{v})[%] | 61.5 | [53] | - | - | n.a. |

53 | [7] | Base metal | - | ||

49.0–62.0 * | [50] | Copper | [67] ** | ||

59.9–76.5 * | [51] | Iron | [67] ** | ||

57.3–68.0 * | [51] | Gold | [67] ** | ||

Dynamic viscosity (μ) [Pa∙s] | 0.03–0.49 | [7] | Base metal | Viscometer | Bingham |

0.15–2.69 | [48] | Iron | Viscometer | Bingham | |

0.002–0.311 | [9] | Gold | Viscometer | Bingham | |

0.0071–0.4457 | [49] | Iron | Rheometer | Quadrática | |

1.09–1.46 | [8] | Pyrophyllite | Rheometer | Bingham | |

50 | [37] | Iron | Calibration | Bingham | |

30.0–100.0 | [38] | Iron | Calibration | Full Bingham | |

Yield stress (τ_{y})[Pa] | 2.122–48.535 | [47] | Coal | Viscometer | Herschel-Bulkley |

0.641–93.50 | [47] | Copper | Viscometer | Herschel-Bulkley | |

0.6–1.5 | [47] | Gold | Viscometer | Herschel-Bulkley | |

1.048–11.165 | [47] | Lithium | Viscometer | Herschel-Bulkley | |

1.564–37.110 | [47] | Nickel | Viscometer | Herschel-Bulkley | |

2.769–9.411 | [47] | Uranium | Viscometer | Herschel-Bulkley | |

0.652–100.260 | [47] | Zinc | Viscometer | Herschel-Bulkley | |

26.0–638.0 | [7] | Base metal | Viscometer | Bingham | |

19.36–602.82 | [48] | Iron | Viscometer | Bingham | |

59.59–2396.53 *** | [48] | Iron | Slump test | Bingham | |

0.5–181.0 | [9] | Gold | Viscometer | Bingham | |

0.085–118.0 | [49] | Iron | Rheometer | Quadrática | |

12.0–23.0 | [8] | Pyrophyllite | Rheometer | Herschel-Bulkley | |

9.7–251.9 | [27] | Synthetic | Rheometer | Herschel-Bulkley | |

100.0–1000.0 | [37] | Iron | Calibration | Bingham | |

750.0–1000.0 | [38] | Iron | Calibration | Full Bingham | |

a [Pa] | 1 | [7] | Base metal | Viscometer | Exponential |

21.381 | [48] | Iron | Viscometer | ||

0.0065 | [48] | Iron | Slump test | ||

0.08 | [52] | Iron | Calibration | ||

1.00 × 10^{−7} | [49] | Iron | Rheometer | ||

~0.04–3.40 | [10] | Copper | Rheometer | ||

~0.40–3.45 | [10] | Iron | Rheometer | ||

b [-] | 12.2 | [7] | Base metal | Viscometer | Exponential |

90.874 | [48] | Iron | Viscometer | ||

20.47 | [48] | Iron | Slump test | ||

40 | [52] | Iron | Calibration | ||

39.278 | [49] | Iron | Rheometer | ||

~1.2–5.0 | [10] | Copper | Rheometer | ||

~1.2–5.5 | [10] | Iron | Rheometer | ||

Consistency index (K) [Pa∙s ^{n}] | 0.034–6.409 | [47] | Coal | Viscometer | Herschel-Bulkley |

0.008–130.0 | [47] | Copper | Viscometer | ||

0.108–0.221 | [47] | Gold | Viscometer | ||

0.222–1.515 | [47] | Lithium | Viscometer | ||

0.154–2.001 | [47] | Nickel | Viscometer | ||

0.065–0.097 | [47] | Uranium | Viscometer | ||

0.428–14.720 | [47] | Zinc | Viscometer | ||

0.69–1.96 | [8] | Pyrophyllite | Rheometer | ||

Flow behavior index (n) [-] | 0.4–1.0 | [47] | Coal | Viscometer | Herschel-Bulkley |

0.192–1.347 | [47] | Copper | Viscometer | ||

0.705–0.744 | [47] | Gold | Viscometer | ||

0.766–1.020 | [47] | Lithium | Viscometer | ||

0.450–0.602 | [47] | Nickel | Viscometer | ||

0.742–0.913 | [47] | Uranium | Viscometer | ||

0.306–0.577 | [47] | Zinc | Viscometer | ||

0.84–1.14 | [8] | Pyrophyllite | Rheometer | ||

0.50–1.50 | [37] | Teórico | Calibration |

_{v}= 1-pm’, to estimate the maximum concentration based on bed porosity (pm’); *** C

_{v}of 65% in Slump Test; n.a.: not applicable.

## Appendix B

## Appendix C

Specification | Bingham | Herschel-Bulkley H-B |
---|---|---|

Model area | 69.8 km^{2} | |

2D mesh resolution | 40 m on the slope centerline in the embedded valley 100 m in the downstream lake 50 m for the rest of the model | |

Number of template cells | 21,148 | |

Equation | Shallow Water Equations | |

Simulation time frame | 20 h | |

Maximum number of computational iterations | 20 | |

Computational Interval | Adjustable based on Courant: maximum = 1.0; minimum = 0.45 1.0–16.0 s | |

Machine used | Processor AMD Ryzen 7 3700X. 8-Core. 3.6 GHz processing speed (4.4 GHz Turbo). 16 GB DDR4 RAM and 4 GB/s M,2 SSD | AMD Ryzen 3 3200G. 4-Core. 3.6 GHz processing speed. 8 GB DDR4 RAM and 6 GB/s Sata SSD |

Average time per simulation | 3.233 min/simulation | 5.284 min/simulation |

## Appendix D

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**Figure 5.**Variation of arrival time along the downstream talweg for simulations using the (

**a**) Bingham model and (

**b**) Herschel-Bulkley (H-B) model.

**Figure 6.**Variation of maximum depths along the downstream talweg for the (

**a**) Bingham and (

**b**) H-B models.

**Figure 7.**Relationship between flooded areas and C

_{v}, Max C

_{v}, a, and b in 1000 simulations with the Bingham model.

**Figure 8.**Relationship between inundated areas and C

_{v}, τ

_{y}, K, and n in 1000 simulations with the H-B model.

**Figure 10.**Relationships between maximum average runoff height (H

_{mean}) and the parameters C

_{v}, Max C

_{v}, a, and b in the areas of interest, A1 and A2—Bingham Model results.

**Figure 11.**Relationships between the maximum average runoff height (H

_{mean}) and the parameters C

_{v}, τ

_{y}, K, and n in the areas of interest, A1 and A2—H-B model results.

**Table 1.**Value ranges of rheological parameters defined for the sensitivity assessment of HEC-RAS using Bingham and Herschel-Bulkley models.

Rheological Model | Varied Parameters | Literature Value Ranges | Adopted Interval |
---|---|---|---|

Bingham | C_{v} (%) | 9.3–65.0 | 20.0–60.0 |

Max C_{v} (%) | 49.9–76.5 | 49.0–61.5 | |

a (Pa) | 0.0000001–3.45 | 0.067–3.450 | |

b | 1.2–40.0 | 1.2–10.359 | |

Herschel-Bulkley (H-B) | C_{v} (%) | 20.0–60.0 | |

τ_{y} (Pa) | 0.6–251.9 | ||

K (Pa∙s^{n}) | 0.008–130.0 | ||

n | 0.192–1.5 |

Results/Parameters | Bingham | Herschel H-B | |||||||
---|---|---|---|---|---|---|---|---|---|

C_{v} | Max C_{v} | a | b | C_{v} | τ_{y} | K | n | ||

H_{mean} | Region A1 | 0.4777 | 0.052 | 0.4357 | 0.6902 | −0.2728 | 0.8255 | 0.4149 | 0.1937 |

Region A2 | 0.4737 | 0.064 | 0.4323 | 0.6896 | −0.2667 | 0.9317 | 0.2096 | 0.0881 |

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**MDPI and ACS Style**

Melo, M.; Eleutério, J.
Probabilistic Analysis of Floods from Tailings Dam Failures: A Method to Analyze the Impact of Rheological Parameters on the HEC-RAS Bingham and Herschel-Bulkley Models. *Water* **2023**, *15*, 2866.
https://doi.org/10.3390/w15162866

**AMA Style**

Melo M, Eleutério J.
Probabilistic Analysis of Floods from Tailings Dam Failures: A Method to Analyze the Impact of Rheological Parameters on the HEC-RAS Bingham and Herschel-Bulkley Models. *Water*. 2023; 15(16):2866.
https://doi.org/10.3390/w15162866

**Chicago/Turabian Style**

Melo, Malena, and Julian Eleutério.
2023. "Probabilistic Analysis of Floods from Tailings Dam Failures: A Method to Analyze the Impact of Rheological Parameters on the HEC-RAS Bingham and Herschel-Bulkley Models" *Water* 15, no. 16: 2866.
https://doi.org/10.3390/w15162866