# Delving into Earth Dam Dynamics: Exploring the Impact of Inner Impervious Core and Toe Drain Arrangement on Seepage and Factor of Safety during Rapid Drawdown Scenarios

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Soil Material Characterization

#### 2.2. Study Design

#### 2.2.1. Lab-Scale Models

#### 2.2.2. Numerical Models

#### 2.3. General Cases Investigated in the Numerical Modeling

- The first equation computes the factor of safety with respect to the moment equilibrium (Fm).
- The second equation calculates the factor of safety considering the horizontal force equilibrium (Ff).

_{m}) [29], as summarized in Equation (1).

_{u}” all contribute to the comprehensive analysis. Additionally, the bulk density “ɣ,” the height of the embankment “H”, the angle of shearing resistance with respect to effective stress “ø,′” and the cohesion with respect to effective stress “c′” play essential roles in this geotechnical assessment.

#### 2.4. Validation of Seepage Discharge Calculations

#### 2.5. Statistical Methods

#### 2.5.1. Investigation of the Relationship between Seepage and Stability Parameters

#### 2.5.2. Variance Analysis

## 3. Results

#### 3.1. Seepage Analysis from 1:1 (V:H) Slope

#### 3.1.1. Without a Drain

^{−5}m

^{3}/s at the upstream side and −1.60 × 10

^{−5}m

^{3}/s at the downstream side. However, when the embankment experienced a drawdown rate of 1 m per day, a water flow rate of −1.1 × 10

^{−8}m

^{3}/s was obtained at the upstream face and 9.73 × 10

^{−7}m

^{3}/s was retrieved at the downstream base of the embankment. From the water mass flux data, it becomes evident that under steady-state conditions, a water mass flux of 0.022 kg/s/m

^{2}was discerned at the upstream face of the embankment. Notably, a larger water mass flux of 0.045 kg/s/m

^{2}was observed at the downstream toe of the embankment. Interestingly, when the embankment was exposed to a drawdown rate of 1 m per day, the water mass flux escalated. Specifically, a heightened value of 2.05 × 10

^{−5}kg/s/m

^{2}was extracted from the upstream face, while the downstream toe exhibited a water mass flux of 0.012 kg/s/m

^{2}.

#### 3.1.2. With a 30 m Drain

^{−7}m

^{3}/s at the upstream side, along with a minor discharge of −5.10 × 10

^{−11}m

^{3}/s at the downstream side. In contrast, under the condition of a drawdown rate of 1 m per day, the embankment exhibited a notably diminished water flow rate of 7.78 × 10

^{−23}m

^{3}/s at the upstream face, while similarly minimal outflow of −1.70 × 10

^{−11}m

^{3}/s was observed at the downstream base. Under steady-state conditions, an observable water mass flux of 0.03384 kg/s/m

^{2}was noted at the embankment’s upstream face. Meanwhile, a water mass flux of 1.13 × 10

^{−6}kg/s/m

^{2}was identified at the downstream toe, indicating intensified water movement in this specific area. During the embankment’s exposure to a drawdown rate of 1 m per day, a water mass flux of 1.79 × 10

^{−5}kg/s/m

^{2}was extracted from the upstream face, whereas the downstream toe showcased a water mass flux of 3.65 × 10

^{−7}kg/s/m

^{2}.

#### 3.2. Seepage Analysis from 1:2 (H:V) Slope

#### 3.2.1. Without a Drain

^{−5}m

^{3}/s at the upper part of the upstream side and 6.56 × 10

^{−7}m

^{3}/s at the midpoint of the upstream face, along with a minor outflow of −5.10 × 10

^{−6}m

^{3}/s at the downstream side. Conversely, in the context of a 1 m per day drawdown rate, the embankment displayed a notably reduced water flow rate of 0 m

^{3}/s at the upper part of the upstream face, −5.40 × 10

^{−9}m

^{3}/s at the midpoint of the upstream face, and a similarly modest outflow of −1.70 × 10

^{−11}m

^{3}/s at the downstream base. Also, under conditions of equilibrium, a noticeable water mass flux of 0.02 kg/s/m

^{2}was observed at the upper part of the embankment’s upstream face and 0.0012 kg/s/m

^{2}at the midpoint of the upstream face. Impressively, a relatively greater water mass flux of 0.023 kg/s/m

^{2}was identified at the downstream toe, indicating heightened water movement in that specific region. Significantly, the scenario shifted when the embankment experienced a drawdown rate of 1 m per day. The water mass flux encountered a substantial surge, with an increased value of 1.28 × 10

^{−5}kg/s/m

^{2}observed at the upper part of the upstream face and 1.06 × 10

^{−5}kg/s/m

^{2}at the midpoint of the upstream face. Concurrently, the downstream toe showcased a water mass flux of 0.02 kg/s/m

^{2}. These findings underscored the embankment’s sensitivity to variations in flow conditions, illustrating how such alterations can lead to noteworthy adjustments in the flux of water masses across distinct sections of the embankment. In a nutshell, Table 5 succinctly summarizes the outcomes derived from the seepage investigation carried out employing a 1:2 (horizontal to vertical) ratio and without the presence of a drain.

#### 3.2.2. With a 30 m Drain

^{−5}m

^{3}/s at the upper segment of the upstream side and 9.64 × 10

^{−7}m

^{3}/s at the midpoint of the upstream face, coupled with a negligible outflow of 2.07 × 10

^{−14}m

^{3}/s at the downstream side. In a contrasting scenario, under the circumstances of a drawdown rate of 1 m per day, the embankment showcased a considerably diminished water flow rate of 1.57 × 10

^{−23}m

^{3}/s at the upper section of the upstream face, −4.70 × 10

^{−9}m

^{3}/s at the midpoint of the upstream face, and a comparably modest outflow of 2.19 × 10

^{−15}m

^{3}/s at the downstream base.

#### 3.3. Seepage Analysis from 1:4 (H:V) Slope

#### 3.3.1. Without a Drain

^{2}) was noted at the upper region of the embankment’s upstream face and 8.66 × 10

^{−5}kg/s/m

^{2}at the midpoint of the upstream face. Remarkably, a comparatively higher water mass flux of 0.0115 kg/s/m

^{2}was detected at the downstream toe, indicating elevated water movement within that specific area. Importantly, the scenario altered when the embankment encountered a drawdown rate of 1 m per day. The water mass flux underwent a substantial increase, with an elevated measurement of 6.90 × 10

^{−6}kg/s/m

^{2}observed at the upper portion of the upstream face and 9.74 × 10

^{−6}kg/s/m

^{2}at the midpoint of the upstream face. Concurrently, the downstream toe showcased a water mass flux of 0.012 kg/s/m

^{2}.

#### 3.3.2. With a 30 m Drain

^{2}) was derived from the upper portion of the embankment’s upstream face, under conditions of equilibrium. Additionally, a measure of 0.0001 kg/s/m

^{2}was observed at the midpoint of the upstream face. Furthermore, a markedly lower water mass flux of 2.63 × 10

^{−16}kg/s/m

^{2}was identified at the downstream toe. When the embankment encountered a drawdown rate of 1 m per day, the water mass flux experienced a significant upsurge. Specifically, a reading of 6.89 × 10

^{−6}kg/s/m

^{2}was noted at the upper region of the upstream face, while a value of 8.20 × 10

^{−6}kg/s/m

^{2}was registered at the midpoint of the upstream face. Simultaneously, the downstream toe exhibited a notably constrained water mass flux of 1.83 × 10

^{−16}kg/s/m

^{2}.

#### 3.4. Seepage Analysis from 1:2 (H:V) Slope and Central Core

#### 3.4.1. Without a Drain

^{−7}m

^{3}/s at the upper segment of the upstream side, 61.59 × 10

^{−8}m

^{3}/s at the midpoint of the upstream face, and an outflow of 2.14 × 10

^{−6}m

^{3}/s at the downstream side. Conversely, with a drawdown rate of 1 m per day, the water flow rate reduced significantly to −1.20 × 10

^{−22}m

^{3}/s at the upper portion of the upstream face, accompanied by 2.20 × 10

^{−8}m

^{3}/s at the midpoint of the upstream face and an outflow of 2.06 × 10

^{−6}m

^{3}/s at the downstream base. Furthermore, under equilibrium conditions, the water mass flux was 0.001 kg per second per square meter (kg/s/m

^{2}) at the upper region of the embankment’s upstream face and 2.98 × 10

^{−5}kg/s/m

^{2}at the midpoint of the upstream face, while the downstream toe exhibited a heightened water mass flux of 0.0079 kg/s/m

^{2}, indicating increased water movement in that specific zone. Notably, these dynamics shifted when the embankment encountered a drawdown rate of 1 m per day, with water mass flux values of 5.29 × 10

^{−5}kg/s/m

^{2}at the upper part of the upstream face, 3.95 × 10

^{−5}kg/s/m

^{2}at the midpoint of the upstream face, and 0.0065 kg/s/m

^{2}at the downstream toe.

#### 3.4.2. With a 30 m Drain

^{2}) at the upper region of the embankment’s upstream face. This measure stood in contrast to the reading of 2.98 × 10

^{−5}kg/s/m

^{2}observed at the midpoint of the upstream face, denoting a lower yet still noticeable value. Interestingly, the downstream toe displayed a significantly reduced water mass flux, quantified at 1.70 × 10

^{−11}kg/s/m

^{2}. A critical transformation occurred as the embankment underwent a drawdown rate of 1 m per day. The water mass flux experienced a marked escalation, with a heightened reading of 5.29 × 10

^{−5}kg/s/m

^{2}observed at the upper portion of the upstream face. This reading was in contrast to the measurement of 3.95 × 10

^{−5}kg/s/m

^{2}taken at the midpoint of the upstream face, subtly altered due to the imposed drawdown condition. Concurrently, the downstream toe maintained a water mass flux of 1.66 × 10

^{−11}kg/s/m

^{2}, indicative of its consistent albeit minor influence in the revised seepage dynamics. In essence, the nuanced variations in water mass flux underscored the intricate interplay between structural integration, positional influence, and hydraulic behavior. The data highlighted how manipulating these factors can orchestrate significant shifts in the complex tapestry of seepage patterns along the embankment.

#### 3.5. Slope stability Analysis

#### 3.6. Analysis of Variance

#### 3.6.1. Matric Suction

#### 3.6.2. Water Conductivity

#### 3.6.3. Factor of Safety

#### 3.6.4. Factor of Safety for Models with a Central Core

#### 3.7. Seepage Discharge Analysis

#### 3.8. Relationships among Different Parameters

## 4. Discussion

^{−5}m

^{3}/s at the upstream side. Conversely, an intriguing observation was made at the downstream side, where a negative water flow rate of −1.60 × 10

^{−5}m

^{3}/s was recorded. The presence of this negative value could indicate a reversal of water flow direction, suggesting that seepage might have occurred into the embankment at that particular location. However, the scenario transformed when the embankment was subjected to a drawdown rate of 1 m per day. During this dynamic condition, the water flow behavior shifted significantly. The upstream face of the embankment displayed a notably reduced water flow rate of −1.1 × 10

^{−8}m

^{3}/s, indicating an inward flow, which might be attributed to rapidly decreasing water levels. On the other hand, the downstream base of the embankment exhibited contrasting behavior, with a relatively higher water flow rate of 9.73 × 10

^{−7}m

^{3}/s, suggesting potentially increased seepage outflow as the water levels decreased. The existing literature also highlights that as the water level in front of the slope steadily decreases, there is a corresponding gradual augmentation in the seepage gradient within the slope’s seepage field. This alteration becomes more pronounced as the rate of descent increases [32].

^{2}was observed at the embankment’s upstream face, indicating the movement of water through the embankment. Strikingly, a more substantial water mass flux of 0.044967 kg/s/m

^{2}was recorded at the downstream toe, emphasizing the heightened water movement in that specific area. This could be attributed to the differences in hydraulic gradients and local flow patterns within the embankment. Remarkably, when the embankment was subjected to a drawdown rate of 1 m per day, the water mass flux experienced a notable escalation. Specifically, an elevated value of 2.05 × 10

^{−5}kg/s/m

^{2}was extracted from the upstream face, signifying a heightened rate of water movement through the embankment. Simultaneously, the downstream toe exhibited a water mass flux of 0.01 kg/s/m

^{2}, reinforcing the significance of this location in terms of water seepage. The results reveal the intricate interplay between various factors, such as slope configuration and drawdown rate, showcasing the dynamic behavior of seepage within embankments. The observed patterns of water flow and water mass flux underscore the complexity of seepage processes and necessity of considering these factors when designing and managing embankment systems.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The general outlook of the investigated embankments (

**a**) 1:1 (V:H) slope, (

**b**) 1:2 (V:H) slope, (

**c**) 1:4 (V:H) slope, and (

**d**) 1:2 (V:H) slope with a core.

**Figure 3.**Water conductivity and matric suction without a drain under (

**a**) steady-state, (

**b**) instantaneous, (

**c**) 10-day, and (

**d**) 1 m per day drawdown.

**Figure 4.**Water conductivity and matric suction with a 30 m drain under (

**a**) steady-state, (

**b**) instantaneous, (

**c**) 10-day, and (

**d**) 1 m per day drawdown.

**Figure 5.**Trends in the factor of safety values for (

**a**) 1:1 (V:H) without toe drain and instantaneous drawdown (

**b**) 1:1 (V:H) without toe drain and 10-day drawdown (

**c**) 1:1 (V:H) without toe drain and 1 m per day drawdown (

**d**) 1:4 (V:H) with 30 m toe drain and instantaneous drawdown (

**e**) 1:4 (V:H) with 30 m toe drain and 10-day drawdown (

**f**) 1:4 (V:H) with 30 m toe drain and 1 m per day drawdown.

Slope (V:H) | Drain Size | Top Width (m) | Bottom Width (m) | Initial Water Level (m) |
---|---|---|---|---|

1:1 | No drain | 20 | 70 | 20 |

5 m | 20 | 70 | 20 | |

15 m | 20 | 70 | 20 | |

30 m | 20 | 70 | 20 | |

1:2 | No drain | 20 | 120 | 20 |

5 m | 20 | 120 | 20 | |

15 m | 20 | 120 | 20 | |

30 m | 20 | 120 | 20 | |

1:4 | No drain | 20 | 220 | 20 |

5 m | 20 | 220 | 20 | |

15 m | 20 | 220 | 20 | |

30 m | 20 | 220 | 20 | |

Central core: 1:2 (V:H) Core slope: 1:1 upstream and 1:1.5 downstream | No drain | 20 | 120 | 20 |

5 m | 20 | 120 | 20 | |

15 m | 20 | 120 | 20 | |

30 m | 20 | 120 | 20 |

Conducted Test | Description | Unit | Value | |
---|---|---|---|---|

Shell | Core | |||

Atterberg’s limits | Liquid Limit | % | 25.86 | 45.2 |

Plastic Limit | % | 11.85 | 36.4 | |

Compaction characteristics | Checking for dry unit weight | kN/m^{3} | 20 | 18.05 |

Direct shear | Cohesion (c) | kPa | - | 19 |

Internal angle of friction | 33 | 18 | ||

Permeability of the shell | Ability to allow the flow of water | m/s | 6.45 × 10^{−6} | |

Permeability of the core | Ability to allow the flow of water | m/s | - | 5.2 × 10^{−8} |

Coefficient of consolidation of the core | The rate of undergoing consolidation | m^{2}/s | - | 1.48 × 10^{−8} |

Coefficient of compressibility of the core | The compressibility or volume change characteristics | m^{2}/Kg | - | 3.26 × 10^{−3} |

Natural moisture content | The equilibrium moisture condition | % | 1.54 | 36.8 |

Grain size distribution | Soil classification | Silty sand | Compacted clay | |

Diameter at passing 60% | mm | 1.525 | 0.05 | |

Diameter at passing 10% | mm | 0.089 | 0.002 |

Parameter | Steady-State | 1 m per Day | ||
---|---|---|---|---|

Upstream | Downstream | Upstream | Downstream | |

Water Pressure (kPa) | - | - | −134.721 | 2.093 |

Water Total Head (m) | - | - | 5.548 | 0.2131 |

Water Pressure Head (m) | - | - | −13.7373 | 0.213 |

Water Rate (m^{3}/s) | 1.35 × 10^{−5} | −1.60 × 10^{−5} | −1.1 × 10^{−8} | 9.73 × 10^{−7} |

Cumulative Water Volume (m^{3}) | - | - | 0.459122 | 14.354 |

Water Mass Rate (kg/s) | 0.013 | −0.01611 | −1.1 × 10^{−5} | 0.001 |

Cumulative Water Mass (kg) | - | - | 459.139 | 14354.78 |

Water X Flux (m^{3}/s/m^{2}) | 2.13 × 10^{−5} | 4.50 × 10^{−5} | −1.3 × 10^{−8} | 9.82 × 10^{−6} |

Water Y Flux (m^{3}/s/m^{2}) | −5.70 × 10^{−6} | −1.00 × 10^{−6} | −1.6 × 10^{−8} | 2.77 × 10^{−6} |

Water Flux (m^{3}/s/m^{2}) | 2.20 × 10^{−5} | 4.50 × 10^{−5} | 2.05 × 10^{−8} | 1.02 × 10^{−5} |

Water Mass X Flux (kg/s/m^{2}) | 0.021 | 0.045 | −1.3 × 10^{−5} | 0.01 |

Water Mass Y Flux (kg/s/m^{2}) | −0.006 | −0.001 | −1.6 × 10^{−5} | 0.003 |

Water Mass Flux (kg/s/m^{2}) | 0.022 | 0.045 | 2.05 × 10^{−5} | 0.010 |

Water X-Gradient | −0.462 | −0.977 | 0.271 | −0.213 |

Water Y-Gradient | 0.125 | 0.023 | 0.334 | −0.060 |

Water Gradient | 0.479 | 0.978 | 0.431 | 0.222 |

Parameter | Steady-State | 1 m per Day | ||
---|---|---|---|---|

Upstream | Downstream | Upstream | Downstream | |

Water Pressure (kPa) | −141.364 | 0 | ||

Water Total Head (m) | 5.585 | 0 | ||

Water Pressure Head (m) | −14.415 | 0 | ||

Water Rate (m^{3}/s) | 1.73 × 10^{−7} | −5.10 × 10^{−11} | 7.78 × 10^{−23} | −1.70 × 10^{−11} |

Cumulative Water Volume (m^{3}) | 1.25 × 10^{−16} | −8.30 × 10^{−5} | ||

Water Mass Rate (kg/s) | 0.017296 | −5.10 × 10^{−8} | 7.78 × 10^{−20} | −1.70 × 10^{−8} |

Cumulative Water Mass (kg) | 9.02 × 10^{−14} | −0.083 | ||

Water X Flux (m^{3}/s/m^{2}) | 3.23 × 10^{−5} | −8.0124114 × 10^{−316} | −1.10 × 10^{−8} | −2.596 × 10^{−316} |

Water Y Flux (m^{3}/s/m^{2}) | −1.00 × 10^{−5} | −1.10 × 10^{−9} | −1.40 × 10^{−8} | −3.60 × 10^{−10} |

Water Flux (m^{3}/s/m^{2}) | 3.38 × 10^{−5} | 1.13 × 10^{−9} | 1.79 × 10^{−8} | 3.65 × 10^{−10} |

Water Mass X Flux (kg/s/m^{2}) | 0.032 | −8.013 × 10^{−313} | −1.10 × 10^{−5} | −2.596 × 10^{−313} |

Water Mass Y Flux (kg/s/m^{2}) | −0.010 | −1.10 × 10^{−6} | −1.40 × 10^{−5} | −3.60 × 10^{−7} |

Water Mass Flux (kg/s/m^{2}) | 0.034 | 1.13 × 10^{−6} | 1.79 × 10^{−5} | 3.65 × 10^{−7} |

Water X-Gradient | −0.701 | 1.742 × 10^{−311} | 0.261 | 5.644 × 10^{−312} |

Water Y-Gradient | 0.222 | 2.45 × 10^{−5} | 0.327 | 7.93 × 10^{−6} |

Water Gradient | 0.736 | 2.45 × 10^{−5} | 0.418 | 7.93 × 10^{−6} |

Parameter | Steady-State | 1 m per Day | ||||
---|---|---|---|---|---|---|

Upstream | Midpoint | Downstream | Upstream | Midpoint | Downstream | |

Water Pressure (kPa) | −141.364 | 100.249 | 0 | −138.903 | −79.772 | 0 |

Water Total Head (m) | 5.585 | 20 | 0 | 5.836 | 1.644 | 0 |

Water Pressure Head (m) | −14.415 | 10.222 | 0 | −14.164 | −8.134 | 0 |

Water Rate (m^{3}/s) | 1.24 × 10^{−5} | 6.56 × 10^{−7} | −5.10 × 10^{−6} | 0 | −5.40 × 10^{−9} | −2.80 × 10^{−6} |

Cumulative Water Volume (m^{3}) | - | - | - | −1.50 × 10^{−16} | 0.309 | −12.18 |

Water Mass Rate (kg/s) | 0.012 | 0.0007 | −0.005 | 0 | −5.40 × 10^{−6} | −0.003 |

Cumulative Water Mass (kg) | - | - | - | −1.50 × 10^{−13} | 309.077 | −12178.5 |

Water X Flux (m^{3}/s/m^{2}) | 1.62 × 10^{−5} | 5.31 × 10^{−7} | 2.30 × 10^{−5} | −1.00 × 10^{−8} | −8.90 × 10^{−9} | 1.53 × 10^{−5} |

Water Y Flux (m^{3}/s/m^{2}) | −1.00 × 10^{−5} | −1.10 × 10^{−6} | 3.87 × 10^{−10} | −7.80 × 10^{−9} | −5.60 × 10^{−9} | −2.50 × 10^{−6} |

Water Flux (m^{3}/s/m^{2}) | 1.93 × 10^{−5} | 1.19 × 10^{−6} | 2.30 × 10^{−5} | 1.28 × 10^{−8} | 1.06 × 10^{−8} | 1.55 × 10^{−5} |

Water Mass X Flux (kg/s/m^{2}) | 0.016 | 0.0005 | 0.023 | −1.00 × 10^{−5} | −8.90 × 10^{−6} | 0.015 |

Water Mass Y Flux (kg/s/m^{2}) | −0.01 | −0.001 | 3.87 × 10^{−7} | −7.80 × 10^{−6} | −5.60 × 10^{−6} | −0.003 |

Water Mass Flux (kg/s/m^{2}) | 0.019 | 0.001 | 0.023 | 1.28 × 10^{−5} | 1.06 × 10^{−5} | 0.015 |

Water X-Gradient | −0.35 | −0.012 | −0.5 | 0.227 | 0.062 | −0.332 |

Water Y-Gradient | 0.227 | 0.023 | −8.40 × 10^{−6} | 0.176 | 0.039 | 0.055 |

Water Gradient | 0.42 | 0.026 | 0.5 | 0.287 | 0.073 | 0.337 |

Parameter | Steady-State | 1 m per Day | ||||
---|---|---|---|---|---|---|

Upstream | Midpoint | Downstream | Upstream | Midpoint | Downstream | |

Water Pressure (kPa) | 0 | 100.25 | 0 | −141.592 | −83.357 | 0 |

Water Total Head (m) | 20 | 20 | 0 | 5.562 | 1.278 | 0 |

Water Pressure Head (m) | 0 | 10.2 | 0 | −14.44 | −8.49977 | 0 |

Water Rate (m^{3}/s) | 1.63 × 10^{−5} | 9.64 × 10^{−7} | 2.07 × 10^{−14} | 1.57 × 10^{−23} | −4.70 × 10^{−9} | 2.19 × 10^{−15} |

Cumulative Water Volume (m^{3}) | 2.78 × 10^{−16} | 0.849 | 2.32 × 10^{−8} | |||

Water Mass Rate (kg/s) | 0.016 | 0.00096 | 2.07 × 10^{−11} | 1.57 × 10^{−20} | −4.70 × 10^{−6} | 2.19 × 10^{−12} |

Cumulative Water Mass (kg) | 3.05 × 10^{−13} | 849.3 | 2.32 × 10^{−5} | |||

Water X Flux (m^{3}/s/m^{2}) | 2.21 × 10^{−5} | 7.81 × 10^{−7} | 0 | −9.70 × 10^{−9} | −7.70 × 10^{−9} | 0 |

Water Y Flux (m^{3}/s/m^{2}) | −1.40 × 10^{−5} | −1.60 × 10^{−6} | 1.51 × 10^{−12} | −7.60 × 10^{−9} | −5.10 × 10^{−9} | 1.60 × 10^{−13} |

Water Flux (m^{3}/s/m^{2}) | 2.64 × 10^{−5} | 1.75 × 10^{−6} | 1.51 × 10^{−12} | 1.23 × 10^{−8} | 9.27 × 10^{−9} | 1.60 × 10^{−13} |

Water Mass X Flux (kg/s/m^{2}) | 0.022 | 0.0008 | 0 | −9.70 × 10^{−6} | −7.70 × 10^{−6} | 0 |

Water Mass Y Flux (kg/s/m^{2}) | −0.014 | −0.002 | 1.51 × 10^{−9} | −7.60 × 10^{−6} | −5.10 × 10^{−6} | 1.60 × 10^{−10} |

Water Mass Flux (kg/s/m^{2}) | 0.026 | 0.002 | 1.51 × 10^{−9} | 1.23 × 10^{−5} | 9.27 × 10^{−6} | 1.60 × 10^{−10} |

Water X-Gradient | −0.480 | −0.017 | 0 | 0.228 | 0.059 | 0 |

Water Y-Gradient | 0.313 | 0.034 | −3.30 × 10^{−8} | 0.178 | 0.039 | −3.50 × 10^{−9} |

Water Gradient | 0.573 | 0.038 | 3.29 × 10^{−8} | 0.29 | 0.07 | 3.47 × 10^{−9} |

Parameter | Steady-State | 1 m per Day | ||||
---|---|---|---|---|---|---|

Upstream | Midpoint | Downstream | Upstream | Midpoint | Downstream | |

Water Pressure (kPa) | 0 | 98.07 | 0 | −131.92 | −66.66 | 0 |

Water Total Head (m) | 20 | 20 | 0.243 | 6.548435 | 3.20 | 0.243 |

Water Pressure Head (m) | 0 | 10 | 0 | −13.4516 | −6.78 | 0 |

Water Rate (m^{3}/s) | 1.12 × 10^{−5} | 8.75 × 10^{−8} | 8.46 × 10^{−15} | 3.39 × 10^{−9} | −7.90 × 10^{−23} | −1.20 × 10^{−8} |

Cumulative Water Volume (m^{3}) | - | - | - | −0.018 | −0.57 | −0.0003 |

Water Mass Rate (kg/s) | 0.011 | 8.75 × 10^{−5} | 8.46 × 10^{−12} | 3.39 × 10^{−6} | −7.90 × 10^{−20} | −1.20 × 10^{−5} |

Cumulative Water Mass (kg) | - | - | −17.843 | −568.4 | −0.318 | |

Water X Flux (m^{3}/s/m^{2}) | 8.86 × 10^{−6} | 2.10 × 10^{−8} | 1.15 × 10^{−5} | −5.70 × 10^{−9} | −9.10 × 10^{−9} | 1.15 × 10^{−5} |

Water Y Flux (m^{3}/s/m^{2}) | −8.50 × 10^{−6} | −8.40 × 10^{−8} | −1.70 × 10^{−14} | −3.90 × 10^{−9} | −3.60 × 10^{−9} | 2.48 × 10^{−8} |

Water Flux (m^{3}/s/m^{2}) | 1.23 × 10^{−5} | 8.66 × 10^{−8} | 1.15 × 10^{−5} | 6.90 × 10^{−9} | 9.74 × 10^{−9} | 1.15 × 10^{−5} |

Water Mass X Flux (kg/s/m^{2}) | 0.009 | 2.10 × 10^{−5} | 0.0115 | −5.70 × 10^{−6} | −9.10 × 10^{−6} | 0.012 |

Water Mass Y Flux (kg/s/m^{2}) | −0.009 | −8.40 × 10^{−5} | −1.70 × 10^{−11} | −3.90 × 10^{−6} | −3.60 × 10^{−6} | 2.48 × 10^{−5} |

Water Mass Flux (kg/s/m^{2}) | 0.012 | 8.66 × 10^{−5} | 0.012 | 6.90 × 10^{−6} | 9.74 × 10^{−6} | 0.012 |

Water X-Gradient | −0.193 | −0.00046 | −0.25 | 0.113 | 0.044 | −0.25 |

Water Y-Gradient | 0.185 | 0.0018 | 3.68 × 10^{−10} | 0.079 | 0.017 | −0.0005 |

Water Gradient | 0.267 | 0.0019 | 0.25 | 0.138 | 0.047 | 0.25 |

Parameter | Steady-State | 1 m per Day | ||||
---|---|---|---|---|---|---|

Upstream | Midpoint | Downstream | Upstream | Midpoint | Downstream | |

Water Pressure (kPa) | 0 | 98.07 | −2.38 | −134.7 | −70.03 | 0 |

Water Total Head (m) | 20 | 20 | −7.50 × 10^{−15} | 6.27 | 2.86 | 0 |

Water Pressure Head (m) | 0 | 10 | −0.243 | −13.73 | −7.14 | 0 |

Water Rate (m^{3}/s) | 1.28 × 10^{−5} | 1.02 × 10^{−7} | −2.20 × 10^{−19} | 9.47 × 10^{−23} | 4.71 × 10^{−23} | 8.53 × 10^{−22} |

Cumulative Water Volume (m^{3}) | −1.10 × 10^{−16} | −0.436 | 7.19 × 10^{−15} | |||

Water Mass Rate (kg/s) | 0.012832 | 0.0001 | −2.20 × 10^{−16} | 9.47 × 10^{−20} | 4.71 × 10^{−20} | 8.53 × 10^{−19} |

Cumulative Water Mass (kg) | −1.10 × 10^{−13} | −436.482 | 7.19 × 10^{−12} | |||

Water X Flux (m^{3}/s/m^{2}) | 1.02 × 10^{−5} | 2.45 × 10^{−8} | 0 | −5.60 × 10^{−9} | −7.60 × 10^{−9} | 0 |

Water Y Flux (m^{3}/s/m^{2}) | −9.90 × 10^{−6} | −9.80 × 10^{−8} | 2.63 × 10^{−19} | −4.00 × 10^{−9} | −3.20 × 10^{−9} | 1.83 × 10^{−19} |

Water Flux (m^{3}/s/m^{2}) | 1.42 × 10^{−5} | 1.01 × 10^{−7} | 2.63 × 10^{−19} | 6.89 × 10^{−9} | 8.20 × 10^{−9} | 1.83 × 10^{−19} |

Water Mass X Flux (kg/s/m^{2}) | 0.01 | 2.45 × 10^{−5} | 0 | −5.60 × 10^{−6} | −7.60 × 10^{−6} | 0 |

Water Mass Y Flux (kg/s/m^{2}) | −0.01 | −9.80 × 10^{−5} | 2.63 × 10^{−16} | −4.00 × 10^{−6} | −3.20 × 10^{−6} | 1.83 × 10^{−16} |

Water Mass Flux (kg/s/m^{2}) | 0.01 | 0.0001 | 2.63 × 10^{−16} | 6.89 × 10^{−6} | 8.20 × 10^{−6} | 1.83 × 10^{−16} |

Water X-Gradient | −0.22 | −0.0005 | 0 | 0.12 | 0.041 | 0 |

Water Y-Gradient | 0.214 | 0.002 | −3.10 × 10^{−14} | 0.083 | 0.017 | −4.00 × 10^{−15} |

Water Gradient | 0.308 | 0.002 | 3.09 × 10^{−14} | 0.145 | 0.044 | 4.00 × 10^{−15} |

**Table 9.**Summary of the results from the seepage analysis with 1:2 (H:V), without a drain, but with a central core.

Parameter | Steady-State | 1 m per Day | ||||
---|---|---|---|---|---|---|

Upstream | Midpoint | Downstream | Upstream | Midpoint | Downstream | |

Water Pressure (kPa) | 0 | 100.25 | −0.32 | −94.827 | −51.36 | −0.47 |

Water Total Head (m) | 20 | 20 | 0.41 | 10.331 | 4.541 | 0.399 |

Water Pressure Head (m) | 0 | 10.2 | −0.033 | −9.671 | −5.24 | −0.05 |

Water Rate (m^{3}/s) | 9.41 × 10^{−7} | 1.59 × 10^{−8} | 2.14 × 10^{−6} | −1.20 × 10^{−22} | −2.20 × 10^{−8} | 2.06 × 10^{−6} |

Cumulative Water Volume (m^{3}) | 3.89 × 10^{−16} | −2.4 | 5.49 | |||

Water Mass Rate (kg/s) | 0.001 | 1.59 × 10^{−5} | 0.002 | −1.20 × 10^{−19} | −2.20 × 10^{−5} | 0.002 |

Cumulative Water Mass (kg) | - | - | - | 5.00 × 10^{−13} | −2402.54 | 5492.2 |

Water X Flux (m^{3}/s/m^{2}) | 8.79 × 10^{−7} | 1.33 × 10^{−8} | 7.23 × 10^{−6} | −4.60 × 10^{−8} | −3.60 × 10^{−8} | 6.34 × 10^{−6} |

Water Y Flux (m^{3}/s/m^{2}) | −5.30 × 10^{−7} | −2.70 × 10^{−8} | −1.60 × 10^{−6} | −2.50 × 10^{−8} | −1.70 × 10^{−8} | −1.40 × 10^{−6} |

Water Flux (m^{3}/s/m^{2}) | 1.03 × 10^{−6} | 2.98 × 10^{−8} | 7.41 × 10^{−6} | 5.29 × 10^{−8} | 3.95 × 10^{−8} | 6.49 × 10^{−6} |

Water Mass X Flux (kg/s/m^{2}) | 0.0009 | 1.33 × 10^{−5} | 0.007 | −4.60 × 10^{−5} | −3.60 × 10^{−5} | 0.006 |

Water Mass Y Flux (kg/s/m^{2}) | −0.0005 | −2.70 × 10^{−5} | −0.002 | −2.50 × 10^{−5} | −1.70 × 10^{−5} | −0.001 |

Water Mass Flux (kg/s/m^{2}) | 0.001 | 2.98 × 10^{−5} | 0.007 | 5.29 × 10^{−5} | 3.95 × 10^{−5} | 0.006 |

Water X-Gradient | −0.019 | −0.0003 | −0.42 | 0.459 | 0.107 | −0.4 |

Water Y-Gradient | 0.01 | 0.0006 | 0.093 | 0.252 | 0.05 | 0.09 |

Water Gradient | 0.02 | 0.0006 | 0.428 | 0.524 | 0.118 | 0.4 |

**Table 10.**Summary of the results from the seepage analysis with 1:2 (H:V), 30 m drain, and central core.

Parameter | Steady-State | 1 m per Day | ||||
---|---|---|---|---|---|---|

Upstream | Midpoint | Downstream | Upstream | Midpoint | Downstream | |

Water Pressure (kPa) | 0 | 100.2493 | −4.378 | −94.83 | −51.36 | −4.378 |

Water Total Head (m) | 20 | 20 | −1.30 × 10^{−9} | 10.33 | 4.54 | −1.20 × 10^{−9} |

Water Pressure Head (m) | 0 | 10.22 | −0.45 | −9.67 | −5.24 | −0.446 |

Water Rate (m^{3}/s) | 9.41 × 10^{−7} | 1.60 × 10^{−8} | −1.70 × 10^{−14} | 2.32 × 10^{−23} | −2.20 × 10^{−8} | −1.60 × 10^{−14} |

Cumulative Water Volume (m^{3}) | −3.90 × 10^{−16} | −2.4 | −4.30 × 10^{−8} | |||

Water Mass Rate (kg/s) | 0.0009 | 1.60 × 10^{−5} | −1.70 × 10^{−11} | 2.32 × 10^{−20} | −2.20 × 10^{−5} | −1.60 × 10^{−11} |

Cumulative Water Mass (kg) | −3.90 × 10^{−13} | −2402.3 | −4.30 × 10^{−5} | |||

Water X Flux (m^{3}/s/m^{2}) | 8.79 × 10^{−7} | 1.33 × 10^{−8} | 0 | −4.60 × 10^{−8} | −3.60 × 10^{−8} | 0 |

Water Y Flux (m^{3}/s/m^{2}) | −5.30 × 10^{−7} | −2.70 × 10^{−8} | 1.70 × 10^{−14} | −2.50 × 10^{−8} | −1.70 × 10^{−8} | 1.66 × 10^{−14} |

Water Flux (m^{3}/s/m^{2}) | 1.03 × 10^{−6} | 2.98 × 10^{−8} | 1.70 × 10^{−14} | 5.29 × 10^{−8} | 3.95 × 10^{−8} | 1.66 × 10^{−14} |

Water Mass X Flux (kg/s/m^{2}) | 0.0009 | 1.33 × 10^{−5} | 0 | −4.60 × 10^{−5} | −3.60 × 10^{−5} | 0 |

Water Mass Y Flux (kg/s/m^{2}) | −0.001 | −2.70 × 10^{−5} | 1.70 × 10^{−11} | −2.50 × 10^{−5} | −1.70 × 10^{−5} | 1.66 × 10^{−11} |

Water Mass Flux (kg/s/m^{2}) | 0.001 | 2.98 × 10^{−5} | 1.70 × 10^{−11} | 5.29 × 10^{−5} | 3.95 × 10^{−5} | 1.66 × 10^{−11} |

Water X-Gradient | −0.019 | −0.0003 | 0 | 0.459 | 0.11 | 0 |

Water Y-Gradient | 0.012 | 0.00066 | −2.80 × 10^{−9} | 0.252 | 0.05 | −2.80 × 10^{−9} |

Water Gradient | 0.022 | 0.0006 | 2.82 × 10^{−9} | 0.52 | 0.118 | 2.76 × 10^{−9} |

Slope | Drain Condition | Drawdown Rate | Min | Max | Median | STD |
---|---|---|---|---|---|---|

1 to 1 | Without drain | Insta | 0.622 | 1.026 | 0.827 | 0.053 |

10-day | 0.808 | 1.019 | 0.827 | 0.061 | ||

1 m per day | 0.815 | 1.029 | 0.868 | 0.072 | ||

5 m drain | Insta | 0.626 | 1.030 | 0.827 | 0.055 | |

10-day | 0.809 | 1.023 | 0.827 | 0.062 | ||

1 m per day | 0.816 | 1.033 | 0.871 | 0.074 | ||

15 m drain | Insta | 0.643 | 1.052 | 0.827 | 0.054 | |

10-day | 0.812 | 1.044 | 0.827 | 0.070 | ||

1 m per day | 0.817 | 1.055 | 0.880 | 0.081 | ||

30 m drain | Insta | 0.682 | 1.109 | 0.828 | 0.057 | |

10-day | 0.816 | 1.101 | 0.827 | 0.089 | ||

1 m per day | 0.820 | 1.110 | 0.900 | 0.101 | ||

1 to 2 | Without drain | Insta | 1.221 | 1.568 | 1.355 | 0.075 |

10-day | 1.310 | 1.567 | 1.384 | 0.064 | ||

1 m per day | 1.327 | 1.568 | 1.387 | 0.074 | ||

5 m drain | Insta | 1.221 | 1.568 | 1.358 | 0.075 | |

10-day | 1.312 | 1.567 | 1.385 | 0.064 | ||

1 m per day | 1.328 | 1.568 | 1.387 | 0.073 | ||

15 m drain | Insta | 1.224 | 1.578 | 1.367 | 0.074 | |

10-day | 1.317 | 1.578 | 1.387 | 0.066 | ||

1 m per day | 1.332 | 1.578 | 1.390 | 0.076 | ||

30 m drain | Insta | 1.234 | 1.596 | 1.380 | 0.071 | |

10-day | 1.326 | 1.596 | 1.390 | 0.070 | ||

1 m per day | 1.337 | 1.596 | 1.392 | 0.081 | ||

1 to 4 | Without drain | Insta | 2.040 | 2.659 | 2.126 | 0.130 |

10-day | 2.044 | 2.655 | 2.265 | 0.168 | ||

1 m per day | 2.115 | 2.654 | 2.305 | 0.168 | ||

5 m drain | Insta | 2.040 | 2.659 | 2.127 | 0.130 | |

10-day | 2.044 | 2.655 | 2.265 | 0.168 | ||

1 m per day | 2.115 | 2.654 | 2.307 | 0.168 | ||

15 m drain | Insta | 2.041 | 2.660 | 2.133 | 0.132 | |

10-day | 2.049 | 2.656 | 2.267 | 0.166 | ||

1 m per day | 2.121 | 2.656 | 2.317 | 0.166 | ||

30 m drain | Insta | 2.046 | 2.668 | 2.148 | 0.136 | |

10-day | 2.066 | 2.664 | 2.285 | 0.163 | ||

1 m per day | 2.133 | 2.664 | 2.335 | 0.164 | ||

1 to 2 with a central core | Without drain | Insta | 1.171 | 1.564 | 1.176 | 0.069 |

10-day | 1.114 | 1.562 | 1.253 | 0.122 | ||

1 m per day | 1.186 | 1.563 | 1.296 | 0.119 | ||

5 m drain | Insta | 2.152 | 2.185 | 2.185 | 0.006 | |

10-day | 2.172 | 2.213 | 2.213 | 0.007 | ||

1 m per day | 2.172 | 2.213 | 2.213 | 0.008 | ||

15 m drain | Insta | 2.152 | 2.185 | 2.185 | 0.006 | |

10-day | 2.172 | 2.213 | 2.213 | 0.007 | ||

1 m per day | 2.172 | 2.213 | 2.213 | 0.008 | ||

30 m drain | Insta | 2.152 | 2.185 | 2.185 | 0.006 | |

10-day | 2.172 | 2.213 | 2.213 | 0.007 | ||

1 m per day | 2.172 | 2.213 | 2.213 | 0.008 |

Slope | Drain | p-Value | STATUS (Is p-Value < Alpha Value (0.05)?) |
---|---|---|---|

1 to 1 | No drain | 1.7 × 10^{−175} | TRUE |

5 m | 4.3 × 10^{−197} | TRUE | |

15 m | 4.7 × 10^{−233} | TRUE | |

30 m | 8.8 × 10^{−142} | TRUE | |

1 to 2 | No drain | 5 × 10^{−220} | TRUE |

5 m | 5 × 10^{−238} | TRUE | |

15 m | 0 | TRUE | |

30 m | 1.6 × 10^{−252} | TRUE | |

1 to 4 | No drain | 3.6 × 10^{−286} | TRUE |

5 m | 1.5 × 10^{−291} | TRUE | |

15 m | 0 | TRUE | |

30 m | 0 | TRUE |

Slope | Drain | p-Value | STATUS (Is p-Value < Alpha Value (0.05)?) |
---|---|---|---|

1 to 1 | No drain | 5.21 × 10^{−40} | TRUE |

5 m | 5.17 × 10^{−51} | TRUE | |

15 m | 1.88 × 10^{−67} | TRUE | |

30 m | 7.08 × 10^{−42} | TRUE | |

1 to 2 | No drain | 2.38 × 10^{−50} | TRUE |

5 m | 6.41 × 10^{−94} | TRUE | |

15 m | 1.389 × 10^{−104} | TRUE | |

30 m | 9.11 × 10^{−41} | TRUE | |

1 to 4 | No drain | 6.17 × 10^{−94} | TRUE |

5 m | 1.18 × 10^{−102} | TRUE | |

15 m | 4.516 × 10^{−155} | TRUE | |

30 m | 7.147 × 10^{−137} | TRUE |

Slope | Drain | p-Value | STATUS (Is p-Value < Alpha Value (0.05)?) |
---|---|---|---|

1 to 1 | No drain | 2.31 × 10^{−4} | TRUE |

5 m | 1.01 × 10^{−4} | TRUE | |

15 m | 9.79 × 10^{−5} | TRUE | |

30 m | 9.39 × 10^{−5} | TRUE | |

1 to 2 | No drain | 5.91 × 10^{−5} | TRUE |

5 m | 6.22 × 10^{−5} | TRUE | |

15 m | 9.30 × 10^{−5} | TRUE | |

30 m | 1.69 × 10^{−4} | TRUE | |

1 to 4 | No drain | 1.92 × 10^{−4} | TRUE |

5 m | 1.96 × 10^{−4} | TRUE | |

15 m | 2.05 × 10^{−4} | TRUE | |

30 m | 2.42 × 10^{−4} | TRUE |

Drawdown Rate | p-Value | Status (Is p-Value < Alpha Value (0.05)?) |
---|---|---|

Instantaneous | 2.6 × 10^{−130} | TRUE |

10-day | 3.7 × 10^{−99} | TRUE |

1 m per day | 7.82 × 10^{−97} | TRUE |

Matric Suction (kPa) | Water X-Conductivity (m/s) | Time (d) | Factor of Safety | |
---|---|---|---|---|

Matric Suction (kPa) | 1 | |||

Water X-Conductivity (m/s) | −0.729 | 1 | ||

Time (d) | 0.950 | −0.587 | 1 | |

Factor of Safety | −0.864 | 0.750 | −0.699 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Utepov, Y.B.; Mkilima, T.; Aldungarova, A.K.; Shakhmov, Z.A.; Akhazhanov, S.B.; Saktaganova, N.A.; Abdikerova, U.B.; Budikova, A.M.
Delving into Earth Dam Dynamics: Exploring the Impact of Inner Impervious Core and Toe Drain Arrangement on Seepage and Factor of Safety during Rapid Drawdown Scenarios. *Infrastructures* **2023**, *8*, 148.
https://doi.org/10.3390/infrastructures8100148

**AMA Style**

Utepov YB, Mkilima T, Aldungarova AK, Shakhmov ZA, Akhazhanov SB, Saktaganova NA, Abdikerova UB, Budikova AM.
Delving into Earth Dam Dynamics: Exploring the Impact of Inner Impervious Core and Toe Drain Arrangement on Seepage and Factor of Safety during Rapid Drawdown Scenarios. *Infrastructures*. 2023; 8(10):148.
https://doi.org/10.3390/infrastructures8100148

**Chicago/Turabian Style**

Utepov, Yelbek Bakhitovich, Timoth Mkilima, Aliya Kairatovna Aldungarova, Zhanbolat Anuarbekovich Shakhmov, Sungat Berkinovich Akhazhanov, Nargul Amanovna Saktaganova, Uliya Baktybaevna Abdikerova, and Aigul Moldashevna Budikova.
2023. "Delving into Earth Dam Dynamics: Exploring the Impact of Inner Impervious Core and Toe Drain Arrangement on Seepage and Factor of Safety during Rapid Drawdown Scenarios" *Infrastructures* 8, no. 10: 148.
https://doi.org/10.3390/infrastructures8100148