# A Simplified Method for Leakage Estimation of Clay Core Dams with Different Groundwater Levels

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Basic Assumptions

#### 2.2. Simplified Calculation Method

#### 2.2.1. Groundwater Level below the Reservoir Bottom

_{1}and 1:m

_{2}. The seepage flux can be calculated as follows [43]:

^{2}/s; ${k}_{0}$ represents the hydraulic conductivity of the rock-soil body close to the reservoir, m/s; $B$ represents the width of the pool level in the cross-section of the reservoir, m; $H$ is height of the pool level, m; and $A$ is the coefficient depending on the value of $B/H$, ${m}_{1}$, and ${m}_{2}$. The value of $A$ can be determined empirically by Figure 3 [43].

#### 2.2.2. Groundwater Level above the Reservoir Bottom

#### 2.3. Case Verification

^{−9}m/s. Figure 15b is the cross-section of the reservoir. The width of the bottom of the reservoir ${B}_{0}$ is 59 m, the slope on the left side of the reservoir is 1:1.3, and the slope on the right side of the reservoir is 1:1.2. The rock-soil body can be divided into four layers under the reservoir, and the thicknesses and hydraulic conductivities are listed in Table 1.

## 3. Engineering Application

^{2}, accounting for about 27.1% of the catchment area of the Jinsha River (above Yibin). The average annual discharge of the estuary is 1930 m

^{3}/s, and the annual runoff is 60.9 billion m

^{3}. The Lianghekou hydropower station is located on the main stream of the Yalong River in Yajiang Country, Sichuan Province, China (shown in Figure 18).

^{3}. The dam crest elevation is 2875.00 m a.s.l., and the excavation elevation of the core bottom is 2580.00 m a.s.l. The top width of the core is 6 m, the top elevation is 2874.00 m a.s.l., and the upstream and downstream slope ratio of the core are both 1:0.2. The bottom width of the core is 153 m and the bottom elevation is 2583.00 m a.s.l. The cross-sections of the dam body and the reservoir are shown in Figure 19 and Figure 20, and the region in magenta is the range of the core. In Figure 19 and Figure 20, different areas of the dam body have been marked. Circled Roman numerals indicate different rock grades and “f+ numerals” indicate different faults.

#### 3.1. Analytical Solution

#### 3.2. Numerical Solution

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

## Abbreviations

Variables | Meanings |

$H$ | Height of the pool level |

${H}_{1}$ | Distance between the groundwater level and the earth surface line |

${H}_{2}$ | Downstream water level |

${B}_{0}$ | Length of the reservoir bottom in the cross-section |

${B}_{1}$ | Length of the top of the longitudinal section of the core |

${B}_{2}$ | Length of the bottom of the longitudinal section of the core |

${\delta}_{1}$ | Geometric parameter of core dimension, shown in Figure 5 |

${\delta}_{2}$ | Geometric parameter of core dimension, shown in Figure 5 |

${\delta}_{3}$ | Geometric parameter of core dimension, shown in Figure 5 |

${\delta}_{0}$ | Geometric parameter of core dimension, shown in Figure 5 |

$l$ | Length upstream of the dam body |

${\beta}_{1}$ | Angle of the upstream slope of the core |

${\beta}_{2}$ | Angle of the downstream slope of the core |

${\alpha}_{1}$ | Angle parameter, shown in Figure 6 |

${\alpha}_{2}$ | Angle parameter, shown in Figure 6 |

${k}_{c}$ | Hydraulic conductivity of the core |

${k}_{i}$ | Hydraulic conductivities of different layers of the rock-soil body |

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**Figure 2.**Groundwater level far below the reservoir bottom. (

**a**) Unsteady free leakage stage. (

**b**) Steady free leakage stage.

**Figure 15.**Simple model of the clay core dam. (

**a**) Cross-section of the core. (

**b**) Cross-section of the reservoir.

**Figure 16.**Streamline directions and piezometric lines of the core. (

**a**) Downstream water level above the reservoir bottom. (

**b**) Downstream water level below the reservoir bottom.

**Figure 17.**Streamline directions and piezometric lines of the reservoir bottom. (

**a**) Groundwater level above the reservoir bottom. (

**b**) Groundwater level below the reservoir bottom (L = 5 m). (

**c**) Groundwater level below the reservoir bottom (L = 40 m).

Layers | Thicknesses (m) | Hydraulic Conductivities (m/s) |
---|---|---|

Layer 1 | 20 | 1 × 10^{−6} |

Layer 2 | 25 | 5 × 10^{−7} |

Layer 3 | 27 | 1 × 10^{−7} |

Layer 4 | 29 | 1 × 10^{−8} |

Position of Downstream Water Level Condition | Above the Reservoir Bottom | Below the Reservoir Bottom |
---|---|---|

H_{2} = 5 m | ||

Analytical solution (m^{3}/s) | 6.54 × 10^{−5} | 9.11 × 10^{−5} |

Numerical solution (m^{3}/s) | 6.02 × 10^{−5} | 8.47 × 10^{−5} |

Absolute error (m^{3}/s) | 0.52 × 10^{−5} | 0.64 × 10^{−5} |

Relative error (%) | 7.95% | 7.03% |

Location of Groundwater Level Condition | Above the Reservoir Bottom | Below the Reservoir Bottom | |
---|---|---|---|

H_{1} = 5 m | L = 5 m | L = 40 m | |

Analytical solution (m^{2}/s) | 2.74 × 10^{−4} | 3.65 × 10^{−4} | 1.55 × 10^{−4} |

Numerical solution (m^{2}/s) | 2.51 × 10^{−4} | 3.33 × 10^{−4} | 1.41 × 10^{−4} |

Absolute error (m^{2}/s) | 0.23 × 10^{−4} | 0.32 × 10^{−4} | 0.14 × 10^{−4} |

Relative error (%) | 8.39% | 8.77% | 9.03% |

Different Zones | Hydraulic Conductivities (m/s) |
---|---|

Core | 3.0 × 10^{−9} |

Filter layer 1 | 2.47 × 10^{−5} |

Filter layer 2 | 2.47 × 10^{−5} |

Transition layer | 3.24 × 10^{−3} |

Rockfill area 1 | 1.1 × 10^{−2} |

Rockfill area 2 | 1.1 × 10^{−2} |

Concrete floor | 5 × 10^{−10} |

Grouting curtain | 3 × 10^{−7} |

Different Layers | Hydraulic Conductivities (m/s) |
---|---|

q > 100 Lu | 1.3 × 10^{−4} |

10 Lu < q ≤ 100 Lu | 2.7 × 10^{−5} |

3 Lu < q ≤ 10 Lu | 3.2 × 10^{−6} |

1 Lu < q ≤ 3 Lu | 9.0 × 10^{−7} |

q < 1 Lu | 1.3 × 10^{−7} |

**Table 6.**Values of the corresponding parameters when the normal pool level is 2865.00 m a.s.l. and the downstream water level is 2610 m a.s.l.

Parameters | Values |
---|---|

$H$ | 282 m |

${H}_{1}$ | 27 m |

${H}_{2}$ | 27 m |

${B}_{0}$ | 29.5 m |

${B}_{1}$ | 621.7 m |

${B}_{2}$ | 29.5 m |

${\delta}_{1}$ | 153 m |

${\delta}_{2}$ | 56.4 m |

${\delta}_{3}$ | 96.6 m |

${\delta}_{0}$ | 40.2 m |

$l$ | 20 m |

${\beta}_{1}$ | 78.69° |

${\beta}_{2}$ | 78.69° |

${\alpha}_{1}$* | 42.27° |

${\alpha}_{2}$* | 45° |

Leakage Area | Core | Reservoir Bottom | Total Leakage |
---|---|---|---|

Analytical solution (m^{3}/d) | 196.54 | 6129.93 | 6326.47 |

Numerical solution (m^{3}/d) | 182.22 | 5593.14 | 5775.36 |

Absolute error (m^{3}/d) | 14.32 | 536.79 | 551.11 |

Relative error (%) | 7.29% | 8.76% | 8.71% |

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**MDPI and ACS Style**

Yang, C.; Shen, Z.; Xu, L.; Shen, H.
A Simplified Method for Leakage Estimation of Clay Core Dams with Different Groundwater Levels. *Water* **2022**, *14*, 1961.
https://doi.org/10.3390/w14121961

**AMA Style**

Yang C, Shen Z, Xu L, Shen H.
A Simplified Method for Leakage Estimation of Clay Core Dams with Different Groundwater Levels. *Water*. 2022; 14(12):1961.
https://doi.org/10.3390/w14121961

**Chicago/Turabian Style**

Yang, Chao, Zhenzhong Shen, Liqun Xu, and Hongjie Shen.
2022. "A Simplified Method for Leakage Estimation of Clay Core Dams with Different Groundwater Levels" *Water* 14, no. 12: 1961.
https://doi.org/10.3390/w14121961