# Prediction of Earth Dam Seepage Using a Transient Thermal Finite Element Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analytical Method

^{2}), $\mu $ is the fluid viscosity (Pa∙s), and $\nabla P$ is the pressure gradient (Pa/m).

^{2}), which can be described as heat flowing per time per unit area, $k$ is the thermal conductivity (W/m∙K), and $\nabla u$ is the temperature gradient (K/m). Assuming that the flow velocity is equivalent to the heat flux,

^{3}) of the material assuming the heat capacity and density remain constant with respect to time. Applying the law of conservation of energy results in Equation (14) in which the rate of heat accumulation is equal to the negative of the derivative of the heat flow at point $x$,

#### 2.2. Finite Element Method

^{5}°C while the other end was kept at 0 °C. The remaining edges were perfectly insulated to represent a 1D flow model. A schematic of the flow model is shown in Figure 4. The model was simulated from a timespan of 1 year to 10,000 years at order of magnitude intervals to show the time dependence of the system from initial conditions, transient flow, and steady state, i.e., when the flux at the inlet is equal to the flux at the outlet, ${q}_{a}={q}_{b}$. The input parameters used in the analytical and FEA model are listed in Table 1.

^{10}s ($\approx $803 years) for the system to reach steady-state conditions.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Mohr–Coulomb failure criteria of a cylindrical sample. (

**B**) Mohr circle depicting the failure criteria.

**Figure 2.**Illustration of tensile and shear failure of a 2D cylindrical sample. The arrows indicate the direction of the force being applied to the sample.

**Figure 3.**Mohr–Coulomb failure criteria for dry soil and rock (green and black, respectively) and a “seepage” soil/rock assuming complete seepage of the hydrostatic head of water.

**Figure 4.**(

**Top**): Schematic of the 1D flow model used in the analytical and FEA model comparison. (

**Bottom**): Snapshot of a portion of the gridded FEA model and the corresponding scale.

**Figure 5.**Pressure distribution along the sample with time intervals ranging from 1 year to 10,000 years. The analytical solution is represented by solid and dashed lines while the FEA model results are depicted by data points. Note that the FEA results are in agreement with the analytical solution.

**Figure 6.**Flow rate along the length of the sample from time intervals of 1 year to 10,000 years. B) Zoomed-in flow rates from 100 years to 10,000 years to show the transition from transient to steady-state flow.

**Figure 7.**Zoomed-in flow rates from Figure 6 for time intervals of 100 years to 10,000 years to show the transition from transient to steady-state flow. The FEA model is represented by the data points, and the analytical model is shown by the lines/dashes.

Darcy Flow | Heat Equation | ||||
---|---|---|---|---|---|

Variable | Value | Variable | Value | ||

${u}_{a}$ | $\left(\mathrm{P}\mathrm{a}\right)$ | 6.89 × 10^{5} | ${u}_{a}$ | $(\xb0\mathrm{C})$ | 6.89 × 10^{5} |

${u}_{a}$ | $\left(\mathrm{P}\mathrm{a}\right)$ | 0.00 | ${u}_{a}$ | $(\xb0\mathrm{C})$ | 0.00 |

${u}_{o}$ | $\left(\mathrm{P}\mathrm{a}\right)$ | 0.00 | ${u}_{o}$ | $(\xb0\mathrm{C})$ | 0.00 |

$L$ | $\left(\mathrm{m}\right)$ | 1.00 | $L$ | $\left(\mathrm{m}\right)$ | 1.00 |

$\rho $ | $(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3})$ | 1.72 × 10^{3} | $\rho $ | $(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3})$ | 1.72 × 10^{3} |

${k}_{D}$ | $\left({\mathrm{m}}^{2}\right)$ | 1.43 × 10^{−15} | $k$ | $(\mathrm{W}/\mathrm{m}\xb7\mathrm{K})$ | 1.43 × 10^{−12} |

$\mu $ | $(\mathrm{P}\mathrm{a}\xb7\mathrm{s})$ | 1.00 × 10^{−3} | $\alpha $ | (${\mathrm{m}}^{2}/\mathrm{s}$) | 4.00 × 10^{−12} |

$\varphi $ | 35.7% | ${c}_{p}$ | ($\mathrm{J}/\mathrm{k}\mathrm{g}\xb7\mathrm{K}$) | 2.07 × 10^{−4} |

**Table 2.**Minimum, maximum, and average flow across the sample at the specified time intervals compared to the theoretical Darcy flow (Equation (4)) from the parameters in Table 1. Note that if the sample has a minimum flow of zero, the average flow was not considered.

Min. Flow | Max. Flow | Avg. Flow | Theoretical Darcy Flow | |
---|---|---|---|---|

(W/m^{2}) | (W/m^{2}) | (W/m^{2}) | (m/s) | |

1 year | 0.00 × 10^{0} | 5.15 × 10^{−5} | ||

10 years | 0.00 × 10^{0} | 1.61 × 10^{−5} | 9.85 × 10^{−7} | |

100 years | 0.00 × 10^{0} | 5.09 × 10^{−6} | ||

Approx. time | 2.96 × 10^{−7} | 2.96 × 10^{−7} | 9.93 × 10^{−7} | |

1000 years | 4.32 × 10^{−7} | 1.57 × 10^{−6} | 9.90 × 10^{−7} | |

10,000 years | 9.86 × 10^{−7} | 9.86 × 10^{−7} | 9.86 × 10^{−7} |

Variable | Min. | Base | Max | |
---|---|---|---|---|

${u}_{a}$ | $\left(\mathrm{P}\mathrm{a}\right)$ | 1.72 × 10^{5} | 6.89 × 10^{5} | 1.21 × 10^{6} |

$L$ | $\left(\mathrm{m}\right)$ | 0.25 | 1.00 | 1.75 |

$\rho $ | $(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3})$ | 4.31 × 10^{2} | 1.72 × 10^{3} | 3.02 × 10^{3} |

${k}_{D}$ | $\left({\mathrm{m}}^{2}\right)$ | 3.58 × 10^{−16} | 1.43 × 10^{−15} | 2.50 × 10^{−15} |

$\mu $ | $(\mathrm{P}\mathrm{a}\xb7\mathrm{s})$ | 2.50 × 10^{−4} | 1.00 × 10^{−3} | 1.75 × 10^{−3} |

$\varphi $ | 8.9% | 35.7% | 62.4% |

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**MDPI and ACS Style**

Wise, J.; Hunt, S.; Al Dushaishi, M.
Prediction of Earth Dam Seepage Using a Transient Thermal Finite Element Model. *Water* **2023**, *15*, 1423.
https://doi.org/10.3390/w15071423

**AMA Style**

Wise J, Hunt S, Al Dushaishi M.
Prediction of Earth Dam Seepage Using a Transient Thermal Finite Element Model. *Water*. 2023; 15(7):1423.
https://doi.org/10.3390/w15071423

**Chicago/Turabian Style**

Wise, Jarrett, Sherry Hunt, and Mohammed Al Dushaishi.
2023. "Prediction of Earth Dam Seepage Using a Transient Thermal Finite Element Model" *Water* 15, no. 7: 1423.
https://doi.org/10.3390/w15071423