# Numerical Modeling of the Effects of Toe Configuration on Throughflow in Rockfill Dams

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## Abstract

**:**

## 1. Introduction

## 2. Background

**K**represents the global element matrix, defining each element’s geometry and material properties;

**h**is the primary unknown vector consisting of the total head at each node. Lastly,

**q**is the resultant vector also called the nodal flow vector, defined by the boundary conditions. This system of equations is iteratively solved so that each element in addition to the whole domain satisfies the governing equation.

## 3. Materials and Methods

#### 3.1. Physical Model

**Figure 5.**Sketch of the model displaying the rockfill shell, along with the base and core, with two regions for (

**a**) internal and (

**b**) external toe configuration. The coordinate system, drawn in blue, is placed at the origin. The locations of the installed pressure sensors are listed as P1–P10 (see Table 1).

#### 3.2. Numerical Model

#### 3.2.1. Material Properties

#### 3.2.2. Mesh

#### 3.2.3. Boundary Conditions

#### 3.2.4. Calibration and Evaluation

## 4. Results

#### 4.1. General Results from Numerical Modeling

#### 4.1.1. No Toe

#### 4.1.2. External Toe

#### 4.1.3. Internal Toe

#### 4.1.4. Combined Toe

#### 4.2. Performance Evaluation

#### 4.3. Laminar vs. Turbulent Flow in Numerical Models

## 5. Discussion

#### 5.1. Boundaries

#### 5.2. Pressure Development

#### 5.3. Calibration

#### 5.4. Application and Future Recommendations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- International Commission on Large Dams (ICOLD). World Register on Dams. General Synthesis. 2020. Available online: http://www.icold-cigb.org/GB/World_register/general_synthesis.asp (accessed on 25 April 2021).
- Sigtryggsdóttir, F.G.; Snæbjörnsson, J.T.; Grande, L.; Sigbjörnsson, R. Interrelations in multi-source geohazard monitoring for safety management of infrastructure systems. Struct. Infrastruct. Eng.
**2016**, 12, 327–355. [Google Scholar] [CrossRef] - International Commission on Large Dams (ICOLD). Dam Failures Statistical Analysis; Bulletin 99-3; ICOLD: Paris, France, 1995. [Google Scholar]
- Solvik, O. Throughflow and stability problems in rockfill dams exposed to exceptional loads. In Proceedings of the 17th International Congress on Large Dams, Vienna, Austria, 17–21 June 1991. [Google Scholar]
- Marulanda, A.; Pinto, N.L.S. Recent Experience on Design, Construction and Performance of CFRD Dams, J. Barry Cooke volume. In Proceedings of the International Symposium on Concrete Face Rockfill Dams and 20th International Commission on Large Dams Congress, Beijing, China, 18 September 2000; pp. 279–299. [Google Scholar]
- Cruz, P.T.; Materón, B.; Freitas, M. Concrete Face Rockfill Dams; CRC Press: Leiden, The Netherlands, 2009. [Google Scholar]
- Javadi, N.; Mahdi, T.F. Experimental investigation into rockfill dam failure initiation by overtopping. Nat. Hazards
**2014**, 74, 623–637. [Google Scholar] [CrossRef] - Siddiqua, S.; Blatz, J.; Privat, N. Evaluating turbulent flow in large rockfill. J. Hydraul. Eng.
**2011**, 137. [Google Scholar] [CrossRef] - Toledo, M.; Morera, L. Design of Overtopping-Resistant Rockfill Dams; CRC Press: Leiden, The Netherlands, 2015; pp. 133–142. [Google Scholar]
- Morán, R.; Toledo, M. Research into protection of rockfill dams from overtopping using rockfill downstream toes. Can. J. Civ. Eng.
**2011**, 38, 1314–1326. [Google Scholar] [CrossRef] - Morán, R.; Toledo, M.A.; Larese, A.; Monteiro-Alves, R. A procedure to design toe protections for rockfill dams against extreme through-flows. Eng. Struct.
**2019**, 195, 400–412. [Google Scholar] [CrossRef] - Ravindra, G.H.R. Hydraulic and Structural Evaluation of Rockfill Dam Behavior When Exposed to Throughflow and Overtopping Scenarios. Ph.D. Thesis, Norwegian University of Science and Technology, Trondheim, Norway, 2021. [Google Scholar]
- Kiplesund, G.H.; Ravindra, G.H.R.; Rokstad, M.M.; Sigtryggsdóttir, F.G. Effects of toe configuration on throughflow properties of rockfill dams. J. Appl. Water Eng. Res.
**2021**. [Google Scholar] [CrossRef] - Hiller, P.H. Survey of Placed Riprap on the Downstream Slopes of Rockfill Dams; Report B1-2016-1; Norwegian University of Science and Technology: Trondheim, Norway, 2016. [Google Scholar]
- Ravindra, G.H.R.; Sigtryggsdóttir, F.G.; Asbølmo, M.; Lia, L. Toe support conditions for placed ripraps on rockfill dams—A field survey. VANN
**2019**, 3, 185–199. [Google Scholar] - Ravindra, G.H.R.; Sigtryggsdóttir, F.G.; Høydal, O. Non-linear flow through rockfill embankments. J. Appl. Water Eng. Res.
**2019**, 7, 247–262. [Google Scholar] [CrossRef] - Hiller, P.H.; Aberle, J.; Lia, L. Displacements as failure origin of placed riprap on steep slopes. J. Hydraul. Res.
**2018**, 56, 141–155. [Google Scholar] [CrossRef][Green Version] - Ravindra, G.H.R.; Sigtryggsdóttir, F.G.; Lia, L. Buckling analogy for 2D deformation of placed ripraps exposed to overtopping. J. Hydraul. Res.
**2021**, 59, 109–119. [Google Scholar] [CrossRef] - Ravindra, G.H.; Gronz, O.; Dost, J.B.; Sigtryggsdóttir, F.G. Description of failure mechanism in placed riprap on steep slope with unsupported toe using smartstone probes. Eng. Struct.
**2020**, 221, 111038. [Google Scholar] [CrossRef] - GeoSlope. Heat and Mass Transfer Modeling with GeoStudio, 2nd ed.; Geoslope International Ltd.: Calgary, AB, Canada, 2017. [Google Scholar]
- Kheiri, G.; Javdanian, H.; Shams, G. A numerical modeling study on the seepage under embankment dams. Model. Earth Syst. Environ.
**2020**, 6, 1075–1087. [Google Scholar] [CrossRef] - Papagiannakis, A.; Fredlund, D. A steady state model for flow in saturated–unsaturated soils. Can. Geotech. J.
**1984**, 21, 419–430. [Google Scholar] [CrossRef] - GeoSlope. Add-In: Non-Darcy Flow, Theoretical Background; Geoslope International Ltd.: Calgary, AB, Canada, 2018. [Google Scholar]
- Engelund, F. On the laminar and turbulent flows of groundwater through homogeneous sand. Akad. Tek. Vidensk.
**1953**, 3, 1–105. [Google Scholar] - Wilkins, J.K. Flow of water through rockfill and its application to the design of dams. N. Z. Eng.
**1955**, 10, 382–387. [Google Scholar] - Dudgeon, C.R. An experimental study of the flow of water through coarse granular media. Houille Blanche
**1966**, 7, 785–801. [Google Scholar] [CrossRef][Green Version] - Ferdos, F.; Yang, J. Characterization of Hydraulic Behaviours of Coarse Rock Materials in a Large Permeameter. J. Geosci. Environ. Prot.
**2013**, 1, 1–6. [Google Scholar] [CrossRef] - Ferdos, F.; Ekström, I. Hydraulic Conductivity of Coarse Rockfill used in Hydraulic Structures. Transp. Porous Media
**2015**, 108, 367–391. [Google Scholar] [CrossRef] - Parkin, A.K.; Trollope, D.H.; Lawson, J.D. Rockfill Structures Subject to Water Flow. J. Soil Mech. Found. Div.
**1966**, 92, 135–151. [Google Scholar] [CrossRef] - López-Acosta, N.P. Numerical and Analytical Methods for the Analysis of Flow of Water Through Soils and Earth Structures. In Groundwater; Javaid, M.S., Ed.; IntechOpen: Rijeka, Croatia, 2016; Chapter 5. [Google Scholar] [CrossRef]
- Van Genuchten, M. A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils1. Soil Sci. Soc. Am. J.
**1980**, 44, 892–898. [Google Scholar] [CrossRef][Green Version] - Van Genuchten, M.; Pachepsky, Y. Hydraulic Properties of Unsaturated Soils. In Encyclopedia of Earth Sciences Series; Springer: New York, NY, USA, 2011; pp. 368–376. [Google Scholar] [CrossRef]
- Mace, A.; Rudolph, D.L.; Kachanoski, R.G. Suitability of Parametric Models to Describe the Hydraulic Properties of an Unsaturated Coarse Sand and Gravel. Ground Water
**1998**, 36, 465–475. [Google Scholar] [CrossRef] - Yang, X.; You, X. Estimating Parameters of Van Genuchten Model for Soil Water Retention Curve by Intelligent Algorithms. Appl. Math. Inf. Sci.
**2013**, 7, 1977–1983. [Google Scholar] [CrossRef][Green Version] - Hoffmans, G.J.C.M. The Influence of Turbulence on Soil Erosion; Eburon Academic Publishers: Delft, The Netherlands, 2012; pp. 55–56. [Google Scholar]

**Figure 1.**Sketch of throughflow situations for an embankment dam with a central core. (

**a**) Normal conditions with seepage through the core. (

**b**) Accidental load situation with overtopping of the core leading to large throughflow.

**Figure 2.**Displaying the investigated toe configurations, where (

**a**) shows no toe configuration, (

**b**) external toe, (

**c**) internal toe and (

**d**) combined toe configuration.

**Figure 4.**Representation of different flow regimes through porous media. Where (

**a**) shows laminar flow through uniform small-grained material and (

**b**) demonstrates turbulent flow condition through material with coarser grains and larger voids.

**Figure 8.**Input boundary flux shown for (

**a**) $q=1\times {10}^{-3}$ m${}^{3}$/s, (

**b**) $q=2\times {10}^{-3}$ m${}^{3}$/s and (

**c**) $q=4\times {10}^{-3}$ m${}^{3}$/s.

**Figure 9.**Numerical modeling results for the four toe configurations; column (

**a**) shows results for $q=1.0\times {10}^{-3}$ m${}^{3}$/s, column (

**b**) $q=2.0\times {10}^{-3}$ m${}^{3}$/s and column (

**c**) $q=4.0\times {10}^{-3}$ m${}^{3}$/s. The colored contours display the water pressure shown in the legend, while the phreatic surface is displayed as the dotted blue line.

**Figure 10.**Numerical modeling results juxtaposed to physical modeling results. The stippled line displays the outline of the dam and toe configuration.

**Figure 11.**Detail showing flux vectors at the crest of the no toe configuration for $q=4.0\times {10}^{-3}$ m${}^{3}$/s. The maximum flux shown in the top left corner is 0.0068 m${}^{3}$/s/m${}^{2}$; the vectors are magnified by a factor of two to increase visibility.

Sensor | P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P9 | P10 |
---|---|---|---|---|---|---|---|---|---|---|

Position (m) | 0.29 | 0.54 | 0.93 | 1.13 | 1.33 | 1.53 | 1.73 | 1.93 | 2.14 | 2.44 |

$\mathit{\rho}$ | ${\mathit{d}}_{10}$ | ${\mathit{d}}_{50}$ | ${\mathit{d}}_{60}$ | ${\mathit{c}}_{\mathit{u}}$ | |
---|---|---|---|---|---|

Shell | 2720 | 1.2 | 6.5 | 9.0 | 7.50 |

Toe | 2860 | 37 | 52 | 55 | 1.42 |

(kg/m${}^{3}$) | (mm) | (mm) | (mm) | ( - ) |

Vol. Water Content | Fitting Param. | Hydr. Cond. | Tortuoisity & Connectivity | Form Drag Constant | Fluid Temp. | |||
---|---|---|---|---|---|---|---|---|

Sat. | Res. | $\mathit{\alpha}$ | n | k (m/s) | l | FDC | T (°C) | |

Shell | 0.1500 | 0.015 | 8 | 2 | 0.003 | −0.5 | 1.50 | 20 |

Toe | 0.40 | 0.040 | 15 | 4 | 0.100 | −1.0 | 0.75 | 20 |

Limit | [0–1] | [0–1] | - | >1 | - | - | - | - |

**Table 4.**Comparison of pressure reduction for different toe configurations relative to no toe configuration for physical and numerical models.

Physical Model Rel. Pressure Reduction (%) | Numerical Model Rel. Pressure Reduction (%) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Toe Config. | (L/s) | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P2 | P3 | P4 | P5 | P6 | P7 | P8 |

External | 1.0 | −1 | 3 | 4 | 7 | 8 | 9 | 12 | 0 | −1 | −1 | −1 | −1 | −2 | −3 |

1.5 | −2 | 2 | 2 | 3 | 4 | 5 | 7 | 0 | −1 | −1 | −1 | −1 | −2 | −3 | |

2.0 | −4 | −1 | −1 | 1 | 1 | 3 | 6 | −1 | −1 | −2 | −2 | −2 | −2 | −3 | |

2.5 | −3 | 0 | 1 | 2 | 3 | 5 | 8 | 0 | 0 | 0 | −1 | −1 | −1 | −1 | |

3.0 | −2 | 1 | 2 | 3 | 4 | 6 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

3.5 | −1 | 5 | 6 | 6 | 5 | 8 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | |

4.0 | 0 | 5 | 5 | 6 | 5 | 8 | 4 | −1 | 0 | 0 | 0 | 0 | 0 | 0 | |

Internal | 1.0 | −1 | −24 | −34 | −53 | −80 | −77 | −73 | 0 | −29 | −36 | −46 | −68 | −75 | −75 |

1.5 | −1 | −17 | −26 | −46 | −78 | −75 | −69 | −1 | −27 | −33 | −43 | −66 | −73 | −72 | |

2.0 | −3 | −18 | −26 | −44 | −77 | −74 | −67 | −1 | −25 | −31 | −40 | −64 | −72 | −69 | |

2.5 | −1 | −16 | −24 | −42 | −76 | −72 | −64 | 0 | −20 | −26 | −35 | −59 | −66 | −59 | |

3.0 | 0 | −14 | −21 | −39 | −74 | −70 | −61 | −1 | −22 | −27 | −37 | −62 | −70 | −65 | |

3.5 | 0 | −11 | −18 | −36 | −73 | −68 | −61 | −1 | −22 | −27 | −36 | −61 | −69 | −64 | |

4.0 | 0 | −11 | −18 | −36 | −72 | −67 | −59 | −1 | −21 | −26 | −35 | −60 | −68 | −62 | |

Combined | 1.0 | 2 | −41 | −48 | −59 | −77 | −74 | −67 | 0 | −29 | −35 | −46 | −71 | −77 | −74 |

1.5 | 2 | −35 | −41 | −53 | −75 | −72 | −64 | 0 | −26 | −32 | −43 | −67 | −74 | −68 | |

2.0 | 2 | −33 | −39 | −50 | −73 | −70 | −60 | 0 | −24 | −30 | −40 | −64 | −71 | −63 | |

2.5 | 2 | −32 | −36 | −47 | −71 | −68 | −56 | 0 | −20 | −25 | −35 | −60 | −65 | −54 | |

3.0 | 2 | −26 | −31 | −43 | −69 | −65 | −53 | 0 | −22 | −27 | −37 | −62 | −68 | −58 | |

3.5 | 3 | −15 | −20 | −39 | −71 | −64 | −54 | 0 | −21 | −26 | −36 | −61 | −67 | −56 | |

4.0 | 3 | −15 | −19 | −36 | −70 | −63 | −53 | 0 | −21 | −26 | −35 | −60 | −66 | −55 |

Legend | 0–20% | 21–40% | 41–60% | 61–80% |

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**MDPI and ACS Style**

Smith, N.S.; Ravindra, G.H.R.; Sigtryggsdóttir, F.G. Numerical Modeling of the Effects of Toe Configuration on Throughflow in Rockfill Dams. *Water* **2021**, *13*, 1726.
https://doi.org/10.3390/w13131726

**AMA Style**

Smith NS, Ravindra GHR, Sigtryggsdóttir FG. Numerical Modeling of the Effects of Toe Configuration on Throughflow in Rockfill Dams. *Water*. 2021; 13(13):1726.
https://doi.org/10.3390/w13131726

**Chicago/Turabian Style**

Smith, Nils Solheim, Ganesh H. R. Ravindra, and Fjóla Guðrún Sigtryggsdóttir. 2021. "Numerical Modeling of the Effects of Toe Configuration on Throughflow in Rockfill Dams" *Water* 13, no. 13: 1726.
https://doi.org/10.3390/w13131726