# Slope Stability Evaluation Due to Reservoir Draw-Down Using LEM and Stress-Based FEM along with Mohr–Coulomb Criteria

^{*}

## Abstract

**:**

## 1. Introduction

- -
- To evaluate the approach of LEM and stress-based FEM for defining safe allowable draw-down rates.
- -
- To study the influence of change in permeability of upstream dam shell in the FOS.
- -
- To examine the effect of horizontal upstream filter material in the FOS.
- -
- To evaluate the plastic behaviors on the upstream slope during RDD.

## 2. Materials and Methods

#### 2.1. Case Study

#### 2.2. Material Properties

#### 2.2.1. Soil Water Characteristic Curve (SWCC)

#### 2.2.2. Hydraulic Conductivity/Permeability (${k}_{w}$)

#### 2.2.3. Angle of Friction (${\varphi}^{\prime}$) and Cohesion (${c}^{\prime}$)

#### 2.2.4. Modules of Elasticity (E) and Poisson’s Ratio (v)

#### 2.3. Governing Equations

#### 2.3.1. Darcy’s Law

#### 2.3.2. 2D Partial Differential Flow Water Equation

#### 2.3.3. Hydraulic Conductivity/Permeability (${k}_{w}$)

#### 2.3.4. Coefficient of Volume Compressibility (${m}_{v}$)

#### 2.3.5. Mohr–Coulomb Theory

#### 2.3.6. Methods for Factor of Safety (FOS)

#### 2.3.7. Limit Equilibrium Method

#### 2.3.8. Stress-Based FEM Method

#### 2.4. Model Parameters

#### 2.5. Correlation Matrix

## 3. Results

#### 3.1. Steady-State

#### 3.1.1. Stress Distribution

#### 3.1.2. Factor of Safety (FOS)

#### 3.2. Transient State

#### 3.2.1. Nonlinear Behavior Analysis

#### 3.2.2. Pressure and Seepage Discharge Variation

#### 3.2.3. Time-Dependent FOS and Pore Water Pressure Using LEM

## 4. Discussion

#### 4.1. Steady-State Conditions

#### 4.2. Transient State

## 5. Conclusions

- The long-term steady-state analysis has resulted in a similar FOS of 1.92 with LEM and 1.89 with stress-based FEM. The identical slip circles resulted in similar base stress distribution, using both models. As a result, similar FOS values were obtained.
- Considering the as-built design dam with 8 h of RDD, a critical FOS of less than 1.3 was obtained. It was classified as unsafe according to the guidelines. A safe allowable draw-down rate of 0.76 m/h for 20 h was identified to reach the minimum critical FOS criteria.
- The sensitivity analyses’ test results have shown that the FOS values are significantly dominated by upstream dam shell permeability and draw-down rates. The slow draw-down rates and quick release of excess pore water pressure are important to ensure safe sliding stability. The additional test case with a horizontal filter provided adequate FOS for all draw-down rates.
- The coupled stress-based FEM for the nonlinear behavior analysis was carried out in the case of 8 h of RDD. The local plastic deformation in the upstream slope and at the toe of the dam has been observed, which was not seen in the equilibrium method.
- To ensure dam safety during RDD, the allowable draw-down rate should be evaluated. The appropriate permeability of the dam shell and horizontal filters must be considered during the design and construction phase.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The evolution of the river bed from 2006 to 2010 is represented by (

**a**), and (

**b**) shows sediment deposition in the MMHPP reservoir at the right bank of headworks [21].

**Figure 2.**Instability of the slope after the draw-down is represented by (

**a**,

**b**) after the draw-down. * Represents detail image.

**Figure 4.**Soil water characteristic curve, Rahardjo [24].

**Figure 5.**SWCC for different materials defined by Khire and Bosscher [25].

**Figure 6.**Hydraulic conductivity versus Matric suction, Khire and Bosscher [25].

**Figure 7.**Stress–strain relationship [28].

**Figure 9.**Free body diagram of a slice showing the shear and normal forces (SLOPE/W) [32].

**Figure 19.**At 8 h with a draw-down rate of 1.91 m/h—upstream dam shell (${k}_{w}=1\times {10}^{-4}$ m/s).

**Figure 20.**At 8 h with a draw-down rate of 1.91 m/h—upstream dam shell (${k}_{w}=8\times {10}^{-4}$ m/s).

**Figure 21.**At 8 h with a draw-down rate of 1.91 m/h—three-layer horizontal filters (cross-section length = 33 m, 15 m, 8.5 m and height = 1 m with ${k}_{w}=1\times {10}^{-2}$ m/s).

**Figure 23.**Left represents FOS and right illustrates pore water pressure at location A for different draw-down rates using LEM (Upstream dam shell (${k}_{w}$ = ${10}^{-4}$ m/s)).

**Figure 24.**Left represents FOS and right illustrates pore water pressure at location A for different draw-down rates using LEM (Upstream dam shell (${k}_{w}$ = $8\times {10}^{-4}$ m/s)).

**Figure 25.**Left represents FOS and right illustrates pore water pressure at location A for different draw-down rates using LEM (Upstream three horizontal filter layers (${k}_{w}$ = $1\times {10}^{-2}$ m/s)).

**Table 1.**Cohesion and internal angle of friction, Ameratunga [27].

Material | ${\mathit{\varphi}}^{\prime}$ (${}^{\circ}$) | |
---|---|---|

Soils | Soft and firm clay of medium to high plasticity, silty clays, loose variable clayey fills, loose sandy silts (use ${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\u20135\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$) | 17–25 |

Stiff sandy clays, gravelly clays, compacted clayey sands and sandy silts, compacted clay fill (use ${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\u201310\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$) | 26–32 | |

Gravelly sands, compacted sands, controlled crushed sandstone, and gravel fill, dense well-graded sands (use ${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\u20135\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$) | 32–37 | |

Weak weathered rock, controlled fills of roadbase, gravelly and recycled concrete (use ${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\u201325\phantom{\rule{3.33333pt}{0ex}}\mathrm{kPa}$) | 36–43 | |

Rocks | Chalk (${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\mathrm{kPa}$) | 35 |

Weathered granite (${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\mathrm{kPa}$) | 33 | |

Fresh basalt (${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\mathrm{kPa}$) | 37 | |

Weak sandstone (${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\mathrm{kPa}$) | 32 | |

Weak siltstone (${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\mathrm{kPa}$) | 28 | |

Weak mudstone (${c}^{\prime}=\phantom{\rule{4.pt}{0ex}}0\mathrm{kPa}$) | 32 |

**Table 2.**Elastic constants of various soil materials, AASHTO [29].

Soil Type | Typical Range of Young’s Modulus (MN/m^{2}) | Poisson’s Ratio |
---|---|---|

Clay: Soft sensitive | 2.4–15 | 0.4–0.5 undrained |

Medium stiff to stiff | 15–50 | |

Very stiff | 50–100 | |

Loess Silt | 15–60 | 0.1–0.3 |

2–20 | 0.3–0.35 | |

Fine Sand: Loose | 7.5–10 | 0.25 |

Medium Dense | 10–20 | |

Dense | 20–25 | |

Sand: Loose | 10–25 | 0.25–0.35 |

Medium Dense | 25–50 | 0.3–0.4 |

Dense | 50–75 | |

Gravel: Loose | 25–75 | 0.2–0.35 |

Medium Dense | 75–100 | 0.3–0.4 |

Dense | 100–200 |

**Table 3.**Inter-slice force characteristics and relationships (SLOPE/W) [32].

Method | Inter-Slice Normal (E) | Inter-Slice Shear (X) | Inclination of X/E Resultant, and X-E Relationship |
---|---|---|---|

Ordinary or Fellenius | No | No | No inter-slice forces |

Bishop’s Simplified | Yes | No | Horizontal |

Janbu’s Simplified | Yes | No | Horizontal |

Spencer | Yes | Yes | Constant |

Morgenstern–Price | Yes | Yes | Variable; user function |

Corps of Engineers-1 | Yes | Yes | Inclination of a line from crest to |

Corps of Engineers-2 | Yes | Yes | Inclination of ground surface at top of slice |

Lowe-Karafiath | Yes | Yes | Average of ground surface and slice base inclination |

Janbu Generalized | Yes | Yes | Applied line of thrust and moment equilibrium of slice |

Sarma-vertical slices | Yes | Yes | $X=C+Etan\varphi $ |

**Table 4.**Equations of statics satisfied (SLOPE/W) [32].

Method | Moment Equilibrium | Force Equilibrium |
---|---|---|

Ordinary or Fellenius | Yes | No |

Bishop’s Simplified | Yes | No |

Janbu’s Simplified | No | Yes |

Spencer | Yes | Yes |

Morgenstern–Price | Yes | Yes |

Corps of Engineers-1 | No | Yes |

Corps of Engineers-2 | No | Yes |

Lowe-Karafiath | No | Yes |

Janbu Generalized | Yes (by slice) | Yes |

Sarma-vertical slices | Yes | Yes |

Materials | ${\mathit{k}}_{\mathit{w}}$ (m/s) | Vol. WC | Residual = 10% WC | ${\mathit{m}}_{\mathit{v}}$ (kPa${}^{-1}$) | Unit Wt (kN/m${}^{3}$) | Friction Angle ($\mathit{\varphi}$) | c (kPa) | Modulus of Elasticity (MPa) | Poisson’s Ratio |
---|---|---|---|---|---|---|---|---|---|

Clay core | $1\times {10}^{-8}$ | 0.50 | 0.05 | $3.3\times {10}^{-5}$ | 16 | 25 | 10 | 30 | 0.45 |

Dam shell | $1\times {10}^{-4}$ | 0.40 | 0.040 | $2\times {10}^{-5}$ | 20 | 30 | 8 | 50 | 0.3 |

Fine filter | $1\times {10}^{-3}$ | 0.30 | 0.030 | $4\times {10}^{-5}$ | 18 | 32 | 2 | 25 | 0.25 |

Coarse filter | $1\times {10}^{-2}$ | 0.25 | 0.025 | $1.6\times {10}^{-5}$ | 22 | 35 | 1 | 60 | 0.3 |

Rip-rap | $1\times {10}^{-1}$ | 0.20 | 0.020 | $1\times {10}^{-5}$ | 27 | 37 | 0.5 | 95 | 0.35 |

Upstream side of the dam at location A with a draw-down rate $1.91\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathrm{/}\mathrm{h}$ after $8\phantom{\rule{3.33333pt}{0ex}}\mathrm{h}$ | |

Hydraulic conductivity
$\left(\right)$ m/s | Seepage discharge $\left(\right)$ |

$1\times {10}^{-4}$ | $2.81\times {10}^{-5}$ |

$8\times {10}^{-4}$ | $8.86\times {10}^{-5}$ |

Three-layers horizontal filter $\phantom{\rule{3.33333pt}{0ex}}=1\times {10}^{-2}$ conductivity at location A is $1\times {10}^{-4}$ | $3.96\times {10}^{-5}$ |

**Table 7.**Minimum required factors of safety: new earth and rock-fill dams, US Army Corps of Engineers [33].

Analysis Condition ${}^{1}$ | Required Minimum Factor of Safety | Slope |
---|---|---|

End-of-Construction (including staged construction) ${}^{2}$ | 1.3 | Upstream and Downstream |

Long-term (Steady seepage, maximum storage pool, spillway crest or top of gates) | 1.5 | Upstream and Downstream |

Maximum surcharge pool ${}^{3}$ | 1.4 | Downstream |

Rapid Draw Down | 1.1–1.3 ${}^{4,5}$ | Upstream |

Overall Minimum FOS | |||||||
---|---|---|---|---|---|---|---|

Draw-Down | Draw-Down Rate | Dam Shell $\left(\right)$ | Dam Shell $\left(\right)$ | Upstream Horizontal Filter | |||

Using LEM | Using FEM | Using LEM | Using FEM | Using LEM | Using FEM | ||

8 h | $1.91\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{h}$ | 1.28 | 1.27 | 1.43 | 1.41 | 1.56 | 1.54 |

20 h | $0.76\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{h}$ | 1.35 | 1.33 | 1.49 | 1.47 | 1.59 | 1.57 |

40 h | $0.38\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{h}$ | 1.42 | 1.38 | 1.53 | 1.51 | 1.60 | 1.58 |

60 h | $0.25\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{h}$ | 1.46 | 1.42 | 1.54 | 1.52 | 1.61 | 1.59 |

80 h | $0.19\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{h}$ | 1.49 | 1.44 | 1.55 | 1.53 | 1.62 | 1.60 |

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**MDPI and ACS Style**

Pandey, B.R.; Knoblauch, H.; Zenz, G.
Slope Stability Evaluation Due to Reservoir Draw-Down Using LEM and Stress-Based FEM along with Mohr–Coulomb Criteria. *Water* **2023**, *15*, 4022.
https://doi.org/10.3390/w15224022

**AMA Style**

Pandey BR, Knoblauch H, Zenz G.
Slope Stability Evaluation Due to Reservoir Draw-Down Using LEM and Stress-Based FEM along with Mohr–Coulomb Criteria. *Water*. 2023; 15(22):4022.
https://doi.org/10.3390/w15224022

**Chicago/Turabian Style**

Pandey, Binaya Raj, Helmut Knoblauch, and Gerald Zenz.
2023. "Slope Stability Evaluation Due to Reservoir Draw-Down Using LEM and Stress-Based FEM along with Mohr–Coulomb Criteria" *Water* 15, no. 22: 4022.
https://doi.org/10.3390/w15224022