# Fragility Analysis for Water-Retaining Structures Resting on Spatially Random Soil

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Probabilistic Seepage Analysis

#### 2.2. Fragility Curve

#### 2.2.1. Method A

_{f}/n

_{d}) that would be observed in a carefully drawn flow net [28,34], $\overline{Q}$ is determined regardless of $\u2206H$.

#### 2.2.2. Method B

## 3. Fragility Curve for Seepage through the Soil Foundation of the Water-Retaining Structure

#### 3.1. Soil Properties and Analysis Conditions

#### 3.2. Stochastic Simulations

#### 3.2.1. Fragility for Piping

#### 3.2.2. Fragility for Seepage Rate

^{−6}m

^{3}/s/m, the fragility curve had a steep slope, resulting in a significant increase in the probability of failure, even with a slight increase in water level (see Figure 10a,b). This suggests that the structure may experience reduced reliability, even with a small increase in the flood head. As the critical seepage rate increased, the fragility curve became less steep. For example, with a critical seepage rate of 2 × 10

^{−5}m

^{3}/s/m, the probability of failure remained low, even when the water level reached 14 m. In addition, the degree of reduction in reliability due to the water level increase was small, even at higher water levels. Figure 10c compares the fragility curves obtained from Method A and Method B, showing that the two results are in good agreement.

#### 3.2.3. Fragility for Uplift Force

#### 3.3. Effect of ${COV}_{k}$ on the Fragility Curve

#### 3.4. Effect of Variation in the Cutoff Length

## 4. Discussion

## 5. Summary and Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- USACE. Risk-Based Analysis for Flood Damage Reduction Studies; Engineer Manual 1110-2-1619; US Army Corps of Engineers: Washington, DC, USA, 1996; Available online: https://apps.dtic.mil/sti/pdfs/ADA401763.pdf (accessed on 27 November 2023).
- Simm, J.; Tarrant, O. Development of fragility curves to describe the performance of UK levee systems. In Proceedings of the Twenty-Sixth International Congress on Large Dams, Vienna, Austria, 1–7 July 2018. [Google Scholar]
- Rossi, N.; Bačić, M.; Kovačević, M.S.; Librić, L. Development of fragility curves for piping and slope stability of river levees. Water
**2021**, 13, 738. [Google Scholar] [CrossRef] - Kennedy, R.P.; Cornell, C.A.; Campbell, R.D.; Kaplan, S.; Perla, H.F. Probabilistic seismic safety study of an existing nuclear power plant. Nucl. Eng. Des.
**1980**, 59, 315–338. [Google Scholar] [CrossRef] - Tsompanakis, Y.; Lagaros, N.D.; Psarropoulos, P.N.; Georgopoulos, E.C. Probabilistic seismic slope stability assessment of geostructures. Struct. Infrastruct. Eng.
**2010**, 6, 179–191. [Google Scholar] [CrossRef] - Porter, K. A Beginner’s Guide to Fragility, Vulnerability, and Risk; University of Colorado Boulder: Boulder, CO, USA, 2018; Available online: https://www.sparisk.com/pubs/Porter-beginners-guide.pdf (accessed on 27 November 2023).
- Cho, S.E. Failure distribution analysis of shallow landslides under rainfall infiltration based on fragility curves. Landslides
**2020**, 17, 79–91. [Google Scholar] [CrossRef] - Vorogushyn, S.; Merz, B.; Apel, H. Development of dike fragility curves for piping and micro-instability breach mechanisms. Nat. Hazards Earth Syst. Sci.
**2009**, 19, 1383–1401. [Google Scholar] [CrossRef] - Wu, X.Z. Development of fragility functions for slope instability analysis. Landslides
**2015**, 12, 165–175. [Google Scholar] [CrossRef] - Martinović, K.; Reale, C.; Gavin, K. Fragility curves for rainfall-induced shallow landslides on transport networks. Can. Geotech. J.
**2018**, 55, 852–861. [Google Scholar] [CrossRef] - McKenna, G.; Argyroudis, S.A.; Winter, M.G.; Mitoulis, S.A. Multiple hazard fragility analysis for granular highway embankments: Moisture ingress and scour. Transp. Geotech.
**2021**, 26, 100431. [Google Scholar] [CrossRef] - Hall, J.W.; Dawson, R.J.; Sayers, P.B.; Rosu, C.; Chatterton, J.B.; Deakin, R. A methodology for national-scale flood risk assessment. Proc. Inst. Civ. Eng.-Water Marit. Eng.
**2003**, 156, 235–247. [Google Scholar] [CrossRef] - Dawson, R.J.; Hall, J.; Sayers, P.; Bates, P.; Rosu, C. Sampling-based flood risk analysis for fluvial dike systems. Stoch. Environ. Res. Risk Assess.
**2005**, 19, 388–402. [Google Scholar] [CrossRef] - Wojciechowska, K.; Pleijter, G.; Zethof, M.; Havinga, F.; Haaren, D.V.; Ter Horst, W. Application of fragility curves in operational flood risk assessment. In Proceedings of the 5th International Symposium on Geotechnical Safety and Risk, Rotterdam, The Netherlands, 13–16 October 2015; IOS Press: Amsterdam, The Netherlands, 2015; pp. 528–534. [Google Scholar] [CrossRef]
- Mainguenaud, F.; Peyras, L.; Khan, U.T.; Carvajal, C.; Sharma, J.; Beullac, B. A probabilistic approach to levee reliability based on sliding, backward erosion and overflowing mechanisms: Application to an inspired Canadian case study. J. Flood Risk Manag.
**2023**, 16, e12921. [Google Scholar] [CrossRef] - Fenton, G.A.; Griffiths, D.V. Extreme hydraulic gradient statistics in stochastic earth dam. J. Geotech. Geoenviron. Eng.
**1997**, 123, 995–1000. [Google Scholar] [CrossRef] - Griffiths, D.V.; Fenton, G.A. Probabilistic analysis of exit gradients due to steady seepage. J. Geotech. Geoenviron. Eng.
**1998**, 124, 789–797. [Google Scholar] [CrossRef] - Srivastava, A.; Sivakumar Babu, G.L.; Haldar, S. Influence of spatial variability of permeability property on steady state seepage flow and slope stability analysis. Eng. Geol.
**2010**, 110, 93–101. [Google Scholar] [CrossRef] - Ahmed, A.A. Stochastic analysis of seepage under hydraulic structures resting on anisotropic heterogeneous soils. J. Geotech. Geoenviron. Eng.
**2012**, 139, 1001–1004. [Google Scholar] [CrossRef] - Cho, S.E. Probabilistic analysis of seepage that considers the spatial variability of permeability for an embankment on soil foundation. Eng. Geol.
**2012**, 133–134, 30–39. [Google Scholar] [CrossRef] - Liu, L.L.; Cheng, Y.M.; Jiang, S.H.; Zhang, S.H.; Wang, X.M.; Wu, Z.H. Effects of spatial autocorrelation structure of permeability on seepage through an embankment on a soil foundation. Comput. Geotech.
**2017**, 87, 62–75. [Google Scholar] [CrossRef] - Tan, X.; Wang, X.; Khoshnevisan, S.; Hou, X.; Zha, F. Seepage analysis of earth dams considering spatial variability of hydraulic parameters. Eng. Geol.
**2017**, 228, 260–269. [Google Scholar] [CrossRef] - Hekmatzadeh, A.A.; Zarei, F.; Johari, A.; Haghighi, A.T. Reliability analysis of stability against piping and sliding in diversion dams, considering four cutoff wall configurations. Comput. Geotech.
**2018**, 98, 217–231. [Google Scholar] [CrossRef] - Johari, A.; Heydari, A. Reliability analysis of seepage using an applicable procedure based on stochastic scaled boundary finite element method. Eng. Anal. Bound. Elem.
**2018**, 94, 44–59. [Google Scholar] [CrossRef] - Liu, K.; Vardon, P.J.; Hicks, M.A. Probabilistic analysis of seepage for internal stability of earth embankments. Environ. Geotech.
**2019**, 6, 294–306. [Google Scholar] [CrossRef] - Chi, F.; Breul, P.; Carvajal, C.; Peyras, L. Stochastic seepage analysis in embankment dams using different types of random fields. Comput. Geotech.
**2023**, 162, 105689. [Google Scholar] [CrossRef] - Griffiths, D.V.; Fenton, G.A. Seepage beneath water retaining structures founded on spatially random soil. Géotechnique
**1993**, 43, 577–587. [Google Scholar] [CrossRef] - Griffiths, D.V.; Fenton, G.A. Three-dimensional seepage through spatially random soil. J. Geotech. Geoenviron. Eng.
**1997**, 123, 153–160. [Google Scholar] [CrossRef] - Robbins, B.A.; Griffiths, D.V.; Fenton, G.A. Random finite element analysis of backward erosion piping. Comput. Geotech.
**2021**, 138, 104322. [Google Scholar] [CrossRef] - Cho, S.E. Probabilistic assessment of seepage stability of soil foundation under water retaining structures by fragility curves. J. Korean Geotech. Soc.
**2021**, 37, 41–54. [Google Scholar] [CrossRef] - Cho, S.E. Probabilistic seepage analysis by the finite element method considering spatial variability of soil permeability. J. Korean Geotech. Soc.
**2011**, 27, 93–104. [Google Scholar] [CrossRef] - Lacasse, S.; Nadim, F. Uncertainties in characterizing soil properties. In Proceedings of the Uncertainty 96, Madison, WI, USA, 31 July–3 August 1996; ASCE: Reston, VA, USA, 1996; pp. 49–75. [Google Scholar]
- Vanmarcke, E.H. Random Fields: Analysis and Synthesis; World Scientific: Singapore, 1983. [Google Scholar]
- Fenton, G.A.; Griffiths, D.V. Risk Assessment in Geotechnical Engineering; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Kennedy, R.P.; Ravindra, M.K. Seismic fragilities for nuclear power plant risk studies. Nucl. Eng. Des.
**1980**, 79, 47–68. [Google Scholar] [CrossRef] - Baker, J.W. Efficient analytical fragility function fitting using dynamic structural analysis. Earthq. Spectra
**2015**, 31, 579–599. [Google Scholar] [CrossRef] - Jeon, O.H.; Jung, W.Y. Fragility analysis of weir structure subjected to flooding water damage. Int. J. Civ. Environ. Eng.
**2018**, 12, 75–80. Available online: https://publications.waset.org/10008936/pdf (accessed on 27 November 2023). - Bodda, S.S.; Gupta, A.; Ju, B.S.; Jung, W.Y. Fragility of a weir structure due to scouring. Comput. Eng. Phys. Model.
**2020**, 3, 1–15. [Google Scholar] [CrossRef] - Freeze, R.A. A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour. Res.
**1975**, 11, 725–741. [Google Scholar] [CrossRef] - Hoeksema, R.J.; Kitanidis, P.K. Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res.
**1985**, 21, 563–572. [Google Scholar] [CrossRef] - Sudicky, E.A. A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process. Water Resour. Res.
**1986**, 22, 2069–2083. [Google Scholar] [CrossRef] - Duncan, J.M. Factors of safety and reliability in geotechnical engineering. J. Geotech. Geoenviron. Eng.
**2000**, 126, 307–316. [Google Scholar] [CrossRef] - El-Ramly, H.; Morgenstern, N.R.; Cruden, D.M. Probabilistic stability analysis of a tailings dyke on presheared clay-shale. Can. Geotech. J.
**2003**, 40, 192–208. [Google Scholar] [CrossRef]

**Figure 2.**Finite element discretization of the soil beneath the water-retaining structure (modified from [31]).

**Figure 3.**Results of the MCS (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$, $H=17\text{}\mathrm{m}$), PDF, and CDF: (

**a**) exit gradient; (

**b**) seepage rate; (

**c**) uplift force.

**Figure 4.**Results of the MCS (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$): PDF and CDF of the exit gradient at different water levels: (

**a**) PDF; (

**b**) CDF.

**Figure 5.**Variation in the estimated mean, standard deviation, and COV of the exit gradient with the water level (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$): (

**a**) Variation in the mean of the exit gradient (RMSE = 3.0496 × 10

^{−4}); (

**b**) Variation in the standard deviation of the exit gradient (RMSE = 3.354 × 10

^{−3}); (

**c**) Variation in the COV of the exit gradient (RMSE = 7.2841 × 10

^{−3}).

**Figure 6.**Fragility curves for piping with respect to varying critical hydraulic gradients (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$; dots are the MCS results): (

**a**) Method A; (

**b**) Method B; (

**c**) Comparison of fragility curves obtained from Method A and Method B.

**Figure 7.**Function fitting results by Method B according to the number of data (${i}_{crit}$ = 0.3).

**Figure 8.**Results of the MCS (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$): the PDF and CDF of the seepage rate at different water levels: (

**a**) PDF; (

**b**) CDF.

**Figure 9.**Variation in the estimated mean, standard deviation, and COV of the seepage rate with the water level (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$): (

**a**) variation in the mean of the seepage rate (RMSE = 0.0); (

**b**) variation in the standard deviation of the seepage rate (RMSE = 1.1836 × 10

^{−9}); (

**c**) variation in the COV of the seepage rate (RMSE = 7.9706 × 10

^{−5}).

**Figure 10.**Fragility curves for the seepage rate with respect to varying critical seepage rates (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$; dots are the MCS results): (

**a**) Method A; (

**b**) Method B; (

**c**) Comparison of fragility curves obtained from Method A and Method B.

**Figure 11.**Results of the MCS (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$): the PDF and CDF of the uplift force at different water levels: (

**a**) PDF; (

**b**) CDF.

**Figure 12.**Variation in the estimated mean, standard deviation, and COV of the uplift force with the water level (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$): (

**a**) Variation in the mean of uplift force (RMSE = 1.4511 × 10

^{−2}); (

**b**) Variation in the standard deviation of uplift force (RMSE = 3.9815 × 10

^{−1}); (

**c**) Variation in the COV of uplift force (RMSE = 7.4303 × 10

^{−4}).

**Figure 13.**Fragility curves for uplift with respect to varying critical uplift forces (${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$, ${l}_{v}=2\text{}\mathrm{m}$; dots are the MCS results): (

**a**) Method A; (

**b**) Method B; (

**c**) Comparison of fragility curves obtained from Method A and Method B.

**Figure 14.**Influence of the ${COV}_{k}$ on the estimated mean, standard deviation, and COV of exit gradient, seepage rate, and uplift force obtained from simulations (fixed ${l}_{h}=20\text{}\mathrm{m}$ and ${l}_{v}=2\text{}\mathrm{m}$); (

**a**) mean of exit gradient; (

**b**) mean of seepage rate; (

**c**) mean of uplift force; (

**d**) standard deviation of exit gradient; (

**e**) standard deviation of seepage rate; (

**f**) standard deviation of uplift force; (

**g**) COV of exit gradient; (

**h**) COV of seepage rate; (

**i**) COV of uplift force.

**Figure 15.**Fragility curves for the seepage with respect to varying ${COV}_{k}$ (fixed ${l}_{h}=20\text{}\mathrm{m}$ and ${l}_{v}=2\text{}\mathrm{m}$): (

**a**) fragility curves for the piping (${s}_{crit}={i}_{e\left(\mathrm{d}\mathrm{e}\mathrm{t}\right)}=0.48$); (

**b**) fragility curves for the uplift (${s}_{crit}={F}_{\left(\mathrm{d}\mathrm{e}\mathrm{t}\right)}=475\text{}\mathrm{k}\mathrm{N}/\mathrm{m}$); (

**c**) fragility curves for the seepage rate (${s}_{crit}={Q}_{\left(\mathrm{d}\mathrm{e}\mathrm{t}\right)}=1.949\times {10}^{-5}\text{}{\mathrm{m}}^{3}/\mathrm{s}/\mathrm{m}$).

**Figure 16.**Typical realizations of the random field and the corresponding analysis results when $H$ is 20 m (fixed ${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$ and ${l}_{v}=2\text{}\mathrm{m}$): (

**a**) cutoff length = 0 m; (

**b**) cutoff length = 1.5 m; (

**c**) cutoff length = 3.0 m; (

**d**) cutoff length = 4.5 m.

**Figure 17.**Fragility curves for the seepage with respect to varying cutoff length (fixed ${COV}_{k}=0.5$, ${l}_{h}=20\text{}\mathrm{m}$ and ${l}_{v}=2\text{}\mathrm{m}$): (

**a**) fragility curves for the piping (${i}_{cr}=0.48$); (

**b**) fragility curves for the uplift (${F}_{cr}=475\text{}\mathrm{k}\mathrm{N}/\mathrm{m}$); (

**c**) fragility curves for the seepage rate (${Q}_{cr}=1.949\times {10}^{-5}\text{}{\mathrm{m}}^{3}/\mathrm{s}/\mathrm{m}$).

**Table 1.**Statistical properties of soil parameters used for seepage analysis (lognormal distribution).

Parameter | ${\mathit{\mu}}_{\mathit{k}}$(m/s) | ${\mathit{C}\mathit{O}\mathit{V}}_{\mathit{k}}$ | Autocorrelation Distance (m) |
---|---|---|---|

K | $1\times {10}^{-5}$ | 0.3, 0.5, 0.7, 1.0 | ${l}_{h}=20$ ${l}_{v}=2$ |

**Table 2.**Estimated $\theta $ and $\beta $ from the MCS results to calibrate Equation (20) for the piping failure mode of the water-retaining structure (Method B).

${i}_{crit}$ | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.0 |

$\theta $ | 2.981 | 4.470 | 5.960 | 7.450 | 8.940 | 10.427 | 11.913 | 13.392 | 14.862 |

$\beta $ | 0.346 | 0.346 | 0.346 | 0.346 | 0.346 | 0.346 | 0.346 | 0.345 | 0.345 |

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**MDPI and ACS Style**

Cho, S.-E.
Fragility Analysis for Water-Retaining Structures Resting on Spatially Random Soil. *Water* **2023**, *15*, 4165.
https://doi.org/10.3390/w15234165

**AMA Style**

Cho S-E.
Fragility Analysis for Water-Retaining Structures Resting on Spatially Random Soil. *Water*. 2023; 15(23):4165.
https://doi.org/10.3390/w15234165

**Chicago/Turabian Style**

Cho, Sung-Eun.
2023. "Fragility Analysis for Water-Retaining Structures Resting on Spatially Random Soil" *Water* 15, no. 23: 4165.
https://doi.org/10.3390/w15234165