# Global Sensitivity Analysis of Groundwater Related Dike Stability under Extreme Loading Conditions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{10}[1]. As a result, many flood prone areas have an extensive network of artificially elevated levees or dikes, which, along Europe’s major rivers, add up to a length of approximately 60,000 km [2]. To ensure the safety of people living behind dikes, continuous maintenance and reinforcements are needed to warrant the stability of dikes and their proper functioning during high water events. Climate change, e.g., earlier snow melt or an increase in extreme precipitation events in the upstream drainage area [3], poses a new threat that may increase the risk of a society to flooding [4]. To maintain safety levels under changing climatic conditions, major investments are needed for dike maintenance and reinforcement, of which the costs for the latter are in the order of EUR 1–20 million per kilometer [1]. Improved knowledge of the processes during and following a high-water event that can result in dike failure is crucial for more cost-effective dike reinforcements, which may reduce the total expenditures on dike reinforcements substantially and can support more societally acceptable flood defense measures [5].

## 2. Materials and Methods

#### 2.1. Case-Study Schematization

#### 2.2. Coupled Hydrology-Stability Model

#### 2.2.1. Hydrological Model Setup

#### 2.2.2. Dike Slope Stability Model Setup

^{2}is imposed.

#### 2.3. Workflow and Parameters for Global Sensitivity Analysis

#### 2.4. Parameter Prioritization

#### 2.5. Factor Fixing Procedure

#### 2.6. Factor Sampling

#### 2.7. Trend Identification and Interaction Qualification

## 3. Results

#### 3.1. Exemplary Results of the Hydro-Stability Model

#### 3.2. Parameter Priorization

#### 3.2.1. Factor Fixing on Globally Minimized Input Vector

#### 3.2.2. Delta Moment Independent Measure

#### 3.3. Trend Identification

#### 3.4. Subsurface Interaction Qualification

## 4. Discussion and Practical Application

#### 4.1. Global Sensitivity Indices for Groundwater Induced Dike Failure

#### 4.2. Toward Flooding

^{2}, while the mean phreatic level in the upper subsurface layer decreases from 2.36 to 1.93 m when the substrate material changes from clay to sand. Accordingly, the average safety factor increases from 1.06 to 1.29. This indicates not only that dike failure as a result of macro-instability is less likely due to occur with a sand subsurface, but also that potential instabilities are less threatening for a dike breach, as their volume is much smaller.

#### 4.3. Limitations Regarding Sampling, Hydrology and Subsurface Uncertainty

#### 4.4. Suggestions for Further Research

#### 4.5. Case Study: Application of the Database for Fast High-Resolution Dike Safety Assessment

#### 4.5.1. Case Study Methodology

#### 4.5.2. Case Study Results

#### 4.5.3. Case Study Discussion and Conclusions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Workflow of Global Sensitivity Analysis, focusing on the pre-processing by factor fixing and performing the model runs. The specific measures used to qualify and quantify sensitivity are discussed in Figure 4.

**Figure 3.**Subsurface types used in the analysis as seen in the soil textural triangle, modified from [39].

**Figure 4.**Visual explanation of both SA methods used. The elementary effect $\mathrm{EE}$ (

**A**) uses a fixed step (${\mathsf{\delta}}_{\mathrm{i}}$, equation 3) of input factor ${\mathrm{X}}_{\mathrm{i}}$ from a random starting points and measures the change in result $\mathrm{Y}$. Note that all arrows are of the same size in the X direction, representing the fixed step size. The DMI method (

**B**) is based on the area difference (highlighted in red) between the continuous unconditional probability density function ${\mathrm{f}}_{\mathrm{Y}}\left(\mathrm{y}\right)$ and a conditional unconditional probability density function ${\mathrm{f}}_{\mathrm{Y}|{\mathrm{X}}_{\mathrm{i}}}\left(\mathrm{y}\right)$, which is based on a sample of the unconditional input vector.

**Figure 5.**Example results of the coupled hydro-stability model. Left two columns: steady state with maximum loading equal to dike crest. Right two columns: falling water levels from dike crest to dike toe. Continuous colored lines indicate hydraulic heads at the depth of the dotted lines with a corresponding color. The black curved lines indicate the sliding planes on the inner and outer slope of the dike associated with the minimal safety factor. ${t}_{0}$ indicates the steady-state results at maximum pressure, and ${t}_{e}$ shows the results with water levels returned to the dike toe elevation. For each situation the safety factors of basal sliding (${F}_{lat}$ ), inner slope stability (${F}_{inner}$ ) and outer slope stability (${F}_{outer}$) are presented.

**Figure 6.**Factor Elementary Effects, leading to the factor rank per failure mechanism (

**A**). Fixing the insensitive factors results in the final scatter plot between the results of the unconditional (unfixed) input, and conditional (partly fixed) input (

**B**), leading to a combined ${r}^{2}$ of 0.985. Eight parameters (selection x-axis) are selected for the Monte-Carlo analysis, namely dike slope (${D}_{s}$ ), dike height (${D}_{s}$ ), dike material type (${D}_{typ}$ ), upper layer material type (${U}_{typ}$ ), lower layer material type (${L}_{typ}$ ), upper layer thickness (${U}_{thck}$ ), drainage spacing ($D{r}_{s}$ ) and foreland width (${F}_{w}$ ).

**Figure 7.**Results of global sensitivity analysis (DMI-method). The bars indicate the Delta Moment-Independent measure (${\delta}_{i})\text{}$ per failure mechanism where higher values indicate a greater sensitivity of the factor of safety to that variable (Table 1). Error bars show the values $\pm $ 1 standard deviation (1 $\sigma $ ). Wider error bounds indicate greater variance in the sensitivity for that parameter, possibly caused by parameter interaction.

**Figure 8.**Probability of failure $p\left(F\le 1\right)$ given that the selected input factor is fixed at the given value on the x-axis, and all other input factors are not fixed. Where $p\left(F\le 1\right)$ equals 0 all calculated dikes are stable, and with a $p\left(F\le 1\right)$ of 1, all calculated dikes fail. A large difference in the probability of failure between neighboring bars indicates a strong local importance of that factor.

**Figure 9.**Response surfaces of dike stability $F$ on different combinations of dike and subsurface properties. Low factors of safety clearly coincide with high (darker blue) hydraulic heads for each failure mechanism. Note the different scales for both the y-axis ($F$ ) as the colors (heads).

**Figure 10.**Probability of slip surface location and phreatic surface for different types of upper layer (${U}_{typ}$) material. Brighter colors indicate a larger probability of, respectively, the slip surfaces or phreatic surfaces to occur at that location.

**Figure 11.**Comparison between official preliminary dike safety assessment and safety factors derived from our database. The histograms (

**A**) of the safety factors corresponding to the sufficiently safe (green) or insufficiently safe (red) dike segments clearly show a clear distinction in our database between these segments. Spatially (

**B**), both inner and outer slope stability show a much larger variation in safety factors than the official dike assessment suggests.

Parameter | Symbol | Range | Unit |
---|---|---|---|

Dike Height | ${D}_{h}$ | 3–10 | m |

Dike crest width | ${D}_{w}$ | 2–5 | m |

Dike slope | ${D}_{s}$ | 0.2–1 | m/m |

Dike type | ${D}_{typ}$ | C-CL-L-SL-S | - |

Upper layer thickness | ${U}_{thck}$ | 0.3–1.9 | m |

Upper layer type | ${U}_{typ}$ | C-CL-L-SL-S | - |

Lower layer thickness | ${L}_{thck}$ | 5–10 | m |

Lower layer type | ${L}_{typ}$ | C-CL-L-SL-S | - |

Foreland width | ${F}_{w}$ | 0–100 | m |

Drainage depth | $D{r}_{d}$ | 0.1–2 | m |

Drainage spacing | $D{r}_{s}$ | 1–20 | m |

Riverbed slope | ${R}_{d}$ | 0.33 | m/m |

River depth | ${R}_{d}$ | $0.9*({U}_{thck}$$+{L}_{thck}$) | m |

Flood height | $H$ | $Dh$ | m |

Drawdown time | ${T}_{d}$ | 1 | days |

**Table 2.**Subsurface types used in the model, related to the ${D}_{typ}$, ${U}_{typ}$ and ${L}_{typ}$ parameters. The subsurface type is linked to the hydraulic conductivity (${K}_{sat}$ ), drained cohesion ($c\prime $ ), effective friction angle ($\varphi $ ), the bulk unit weight ($\gamma $ ) and the saturated bulk unit weight (${\gamma}_{sat}$ ).

Subsurface Type | Abbreviation | ${\mathit{K}}_{\mathit{s}\mathit{a}\mathit{t}}(m{d}^{-1})$ | $\mathit{c}\prime \left(kPa\right)$ | $\mathit{\varphi}{(}^{o})$ | $\mathit{\gamma}\left(kN{m}^{-3}\right)$ | ${\mathit{\gamma}}_{\mathit{s}\mathit{a}\mathit{t}}\left(kN{m}^{-3}\right)$ |
---|---|---|---|---|---|---|

Clay | C | 0.13 | 5.0 | 17.5 | 17 | 17 |

Clay-Loam | CL | 0.18 | 4.0 | 22.5 | 18 | 18 |

Loam | L | 0.19 | 1.0 | 30.0 | 20 | 20 |

Sandy Loam | SL | 1.54 | 0.5 | 31.25 | 19.5 | 19.5 |

Sand | S | 8.94 | 0.0 | 32.5 | 18 | 20 |

**Table 3.**Minimum $F\left(X\right)$ per failure mechanism and parameter values of input vector X resulting in that minimum $F\left(X\right)$. See Table 1 for the abbreviations and units of the parameters.

${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{D}}_{\mathit{h}}$ | ${\mathit{D}}_{\mathit{w}}$ | ${\mathit{D}}_{\mathit{s}}$ | ${\mathit{D}}_{\mathit{t}\mathit{y}\mathit{p}}$ | ${\mathit{U}}_{\mathit{t}\mathit{h}\mathit{c}\mathit{k}}$ | ${\mathit{U}}_{\mathit{t}\mathit{y}\mathit{p}}$ | ${\mathit{L}}_{\mathit{t}\mathit{h}\mathit{c}\mathit{k}}$ | ${\mathit{L}}_{\mathit{t}\mathit{y}\mathit{p}}$ | ${\mathit{F}}_{\mathit{w}}$ | $\mathit{D}{\mathit{r}}_{\mathit{d}}$ | $\mathit{D}{\mathit{r}}_{\mathit{s}}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Basal sliding | 0.69 | 9.97 | 2.14 | 1:1 | Sand | 0.31 | Clay | 5.07 | Clay | 50 | −1.05 | 20.0 |

Inner slope stability | 0.00 | 9.74 | 3.50 | 1:1 | Sand | 1.10 | Clay-Loam | 7.50 | Clay-Loam | 50 | −1.05 | 10.5 |

Outer slope stability | 0.26 | 9.34 | 4.61 | 1:1 | Sandy Loam | 1.82 | Loam | 5.25 | Clay-Loam | 85 | −0.62 | 20.0 |

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**MDPI and ACS Style**

van Woerkom, T.; van Beek, R.; Middelkoop, H.; Bierkens, M.F.P.
Global Sensitivity Analysis of Groundwater Related Dike Stability under Extreme Loading Conditions. *Water* **2021**, *13*, 3041.
https://doi.org/10.3390/w13213041

**AMA Style**

van Woerkom T, van Beek R, Middelkoop H, Bierkens MFP.
Global Sensitivity Analysis of Groundwater Related Dike Stability under Extreme Loading Conditions. *Water*. 2021; 13(21):3041.
https://doi.org/10.3390/w13213041

**Chicago/Turabian Style**

van Woerkom, Teun, Rens van Beek, Hans Middelkoop, and Marc F. P. Bierkens.
2021. "Global Sensitivity Analysis of Groundwater Related Dike Stability under Extreme Loading Conditions" *Water* 13, no. 21: 3041.
https://doi.org/10.3390/w13213041