# Wave Run-Up on Mortar-Grouted Riprap Revetments

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

_{5/40}or LMB

_{10/60}(light mass with stone mass of 5–40 kg and 10–60 kg, respectively, according to DIN EN 13383-1 [10]). Either the entire pore volume of the riprap is grouted with mortar, creating an impermeable top layer with a rough revetment surface, or it is partially grouted, creating a rough, porous, and permeable top layer. Beneath the top layer, mostly geotextiles are used as a separation and filter layer. Figure 1 gives an impression of MGRR top layers with different amounts of mortar, either in experimental (Figure 1a,c) or in situ settings (Figure 1b,d) (all additional information can be found in Appendix A).

#### 1.2. Wave Run-Up

_{u}

_{2%}, which is exceeded by 2% of the incoming waves. The main influences on wave run-up are the wave parameters, slope angle, and overall geometry of the embankment, shallow foreshores, and the hydraulic properties of the embankment or revetment on which the incoming waves run up, such as roughness, porosity, and permeability [11,12,13,14].

- they are the most widely used models for design purposes
- they are easy to use and comprehensible
- full-scale hydraulic model tests on MGRRs have been conducted, thus, the number of experiments is not suited for machine learning techniques
- we test the ability of different parameters for describing wave run-up for different roughnesses, porosities, and permeabilities, as these parameters vary considerably for MGRRs.

_{u}

_{2%}on rubble mound structures is calculated using Equation (2):

_{m}

_{0}is the mean spectral wave height, γ

_{f}the influence factor for roughness, γ

_{b}the influence factor for a berm, γ

_{β}the influence factor for oblique wave attack, and ξ

_{m−}

_{1,0}the surf similarity parameter. The surf similarity parameter ξ

_{m}

_{−1,0}is a fictitious parameter, because it combines the wave height at the toe of the structure with the deep water wavelength, which is calculated using the wave period T

_{m}

_{−1,0}at the toe of the structure. For rubble mound revetments, the influence factor γ

_{f,surging}is equal to γ

_{f}for ξ

_{m}

_{−1,0}≤ 1.8, and linearly increases ξ

_{m}

_{−1,0}> 1.8, thereby accounting for the decreasing influence of roughness on wave run-up for surging waves:

_{i}= 1 for a smooth impermeable revetment without a berm and for perpendicular wave attack. The influence factor γ

_{f’}is called the “influence factor due to roughness”, but is used in the EurOtop Manual [8] and by many researchers, e.g., [22,23], as a factor that summarizes the influence of roughness, porosity, and permeability. Because these influences are summarized in a single influence factor, this factor is determined for each revetment type as a constant or depending on wave parameters and structural parameters of the revetment type being investigated. The influence factors γ

_{f}for different revetment types have been summarized by various authors (e.g., [8,24,25]); readers are referred to these and citations therein for a further read. As a non-exhaustive summary, Table 1 provides influence factors for some rough, porous, and permeable revetments.

_{p}is based on the peak period of the spectrum T

_{p}. As no further insight into the wave run-up on MGRRs can be gained from the existing equations of the second type because they are specific for only one revetment type, they are not discussed further here. For a more comprehensive overview of existing run-up equations for revetments see for example [13,24], or more recently [14].

_{F,dim}is in the case of irregular waves calculated using the water depth h, wave height H

_{m}

_{0}, peak period T

_{p}, and empirical constants A

_{0}and A

_{1}:

_{m}

_{0}/L

_{p}< 0.0225, where L

_{p}is the wave length corresponding to the peak period T

_{p}) and non-breaking waves (criterion according to [31]: H

_{m}

_{0}/L

_{p}> 0.0225). For non-breaking waves, Hughes [31] gives:

_{f,M}= 0.505, which in case of a smooth impermeable revetment is γ

_{f,M}= 1 [27]. In general, using the wave momentum flux parameter to describe wave run-up gives better results for non-breaking than for breaking waves, as the run-up tongue is, in this case, well-characterized by a wedge, which is assumed in the derivation of the parameter. For breaking waves, however, this assumption can be violated, and therefore, it is expected that the accuracy of the wave momentum flux parameter for describing wave run-up is reduced [14,31,34]. Nevertheless, in addition to Hughes [27], other researchers used the wave momentum flux parameter to describe wave run-up for breaking and non-breaking waves alike, making use of only one equation (Equation (10)) for all of the wave conditions, e.g., [34] for irregular waves on rubble mound breakwaters and [14] for regular waves on rough, porous, and permeable slopes.

_{r}= H

_{r}/H

_{i}, they gave the empirical equation:

^{TM}with cot(α) = 1.5 and a permeable core, Muttray et al. [32] give a = 1.65 for TMA spectra with no wave breaking on the foreshore. While it is useful to sketch the physical process of wave run-up in this way, the usefulness of the model for design purposes depends not only on the quality of the description of wave run-up as a function of the wave reflection coefficient, but also on the quality of the model that describes the wave reflection coefficient as a function of wave parameters and structural parameters of the revetment or breakwater.

- To analyze the wave run-up process in the presence of partially or fully mortar-grouted riprap
- To determine the appropriate influence factors in the framework of the EurOtop equations

## 2. Materials and Methods

_{s}= 1.4 m and peak periods of up to T

_{p}= 12 s [36].

#### 2.1. Revetment Configurations

_{5/40}, while most MGRRs have a thickness of 0.6 m [38]. Before the first test phase, both revetments were partially grouted with 80 l/m

^{2}mortar. Assuming a porosity of 0.45 before grouting (riprap dumped in dry conditions and with medium density according to [39]), this resulted in permeable revetments after grouting, with a porosity of 0.25 (t = 0.4 m) and 0.32 (t = 0.6 m), respectively.

^{2}, confirming the assumption of a porosity of 0.45 before grouting, which results in an available pore volume of 180 l/m

^{2}(0.45 × 0.4 m × 1000 l/m

^{3}) before grouting.

^{2}, confirming that the lower part of the cross-section of the top layer still exhibits a free pore volume and is only partially grouted. The potential free pore volume before any grouting of the top layer took place (before test phase 1) was 270 l/m

^{2}(0.45 × 0.6 m × 1000 l/m

^{3}), thus, a pore volume of 90 l/m

^{2}was still available in the lower part of the top layer in test phase two. The overall volumetric porosity of the revetment equaled 0.16, while it was at the same time impermeable for any flow perpendicular to the embankment. The revetments were grouted by hand by the staff of experienced contractors that have been grouting MGRRs for decades.

_{10,H50mm}= 2.86 × 10

^{−3}m/s, thickness 8 mm, characteristic opening size 0.1 mm), which was directly placed on the sand embankment. The toe structure of the revetment below the lowest wave run-down was made of concrete blocks (see Figure 2a and Figure 3) in order to support the revetment. By this means, an efficient construction of the revetment was ensured, covering only the relevant section of the slope with grouted riprap material. The crest of the embankment was constructed of concrete blocks behind which any overtopping water was collected in a basin and discharged back into the flume during the tests. The cross-section of the revetment on the north side of the flume is shown in Figure 3.

_{5/40}, with a median weight of G

_{50}= 23.5 kg and a narrow grading, with d

_{85}/d

_{15}= 1.5.

#### 2.2. Instrumentation

_{m}

_{−1,0}, the fictitious wave length in deep water according to the EurOtop Manual [8] was calculated:

#### 2.3. Experimental Program

_{m}

_{0}= 0.38–0.94 m and wave periods from T

_{m}

_{−1,0}= 2.7–9.0 s were generated, which resulted in surf similarity parameters ξ

_{m}

_{−1,0}= 1.55–4.64. In this way, a wide range of wave-loading conditions and breaker types were covered. The still water level in all experiments was set to 4 m at the toe of the structure. A list of the experimental conditions used for the experiments described in this work is provided with all results (see Section 3) in the appendix.

#### 2.4. Detection of the Wave Run-Up Height

^{®}MATLAB R2018b, Natick, MA, USA) routines, including various filter functions. The LIDAR data was used to determine the water surface elevation on the revetment using the difference between the measured surface of the revetment and the measured surface (water and revetment) at each time step. These elevation data were filtered for outliers, which are detected by criteria describing non-physical values, such as measured points above the canal height or wave run-up velocities exceeding a physically meaningful range. Considering the systematic error of the 2D-LIDAR scanner, to identify the top of the leading edge of the water surface on the revetment, the minimum layer thickness was defined with a limit of 25 mm. From this data, time series of the wave run-up were established.

_{u}

_{2%}that is exceeded by 2% of the incoming waves was determined.

#### 2.5. Error Evaluation of Different Equations and Parameters

_{u}

_{2%,p}denotes the predicted run-up height, R

_{u}

_{2%,m}is the measured run-up height, and N is the number of experiments conducted. Note, however, that in Equations (15)–(17), the run-up height R

_{u}

_{2%}in meters was used instead of its dimensionless form in order to directly compare the predicted values of the empirical Equation (2) and Equations (10)–(12) used for modeling wave run-up on MGRRs, because these equations make use of either R

_{u}

_{2%}/H

_{m}

_{0}or R

_{u}

_{2%}/h. Consequently, bias and RMSE are in unit meters.

## 3. Results

#### 3.1. Relative Wave Run-Up Height as a Function of Surf Similarity Parameter ξ_{m−1,0}

_{u}

_{2%}/H

_{m}

_{0}dependent on the surf similarity parameter ξ

_{m}

_{−1,0}for the partially grouted and fully grouted MGRRs. Only results with less than 2% wave overtopping events and without wave breaking in the flume are presented. For reference, Equation (2) for smooth impermeable revetments (γ

_{f}= 1) and a riprap revetment with two layers (γ

_{f}= 0.55) are plotted in Figure 5 as well.

_{m}

_{−1,0}< 3, and these are well-described by Equation (2), scatter increases for ξ

_{m}

_{−1,0}> 3. Regarding the fully grouted MGRR, there is less scatter, with the exception of two particularly high wave run-up heights for ξ

_{m}

_{−1,0}= 2.36 and ξ

_{m}

_{−1,0}= 4.62. The influence factor γ

_{f}was determined by least square fitting of Equation (2) to the measurements, yielding influence factors of 0.72 and 0.86 for the partially grouted and re-grouted MGRR, respectively. Because there was no berm and all waves approached the revetment perpendicularly, the influence factors for a berm γ

_{b}and oblique wave attack γ

_{β}were set to unity. Statistical parameters for all models are given in the discussion section to compare them more easily.

_{f}= 0.88 for the fully grouted and γ

_{f}= 0.79 for the partially grouted MGRR.

_{f}= 0.72) were lower than for t = 0.4 m (γ

_{f}= 0.79). Since the same amount of mortar (80 l/m

^{2}) was used for both revetments, the 0.6 m thick revetment had a higher porosity and thus a higher permeability than the 0.4 m thick revetment. During wave run-up, water could therefore penetrate the existing pore space more easily. The pore volume after partially grouting for the 0.6 m thick revetment was 270 l/m

^{2}− 80 l/m

^{2}= 190 l/m

^{2}, and for the 0.4 m thick revetment was 180 l/m

^{2}− 80 l/m

^{2}= 100 l/m

^{2}(see also Table 2). The roughness was also higher in the case of the thicker revetment, as the riprap was less embedded in the mortar.

_{f}= 0.86 (t = 0.6 m) and γ

_{f}= 0.88 (t = 0.4 m). Since both cover layers were impermeable, the different wave run-up heights could only be caused by different roughness of the cover layers. This is also supported by the fact that the differences in wave run-up heights are largest for small surf similarity parameters and become smaller for larger surf similarity parameters. A decreasing influence of roughness on wave run-up heights with increasing surf similarity parameters is widely described in the literature, see for example [8,26,28]. This is modeled in Equation (2) by the steady increase of the influence factor γ

_{f,surging}with increasing surf similarity parameter ξ

_{m}

_{−1,0}> 1.8, which becomes γ

_{f}= 1 when ξ

_{m}

_{−1,0}≥ 10. However, as described in the EurOtop manual, the maximum relative wave run-up height does not become the same as for smooth impermeable revetments for all types of revetments. For example, in the case of a two-layer riprap revetment on an impermeable embankment, the maximum relative wave run-up height is R

_{u}

_{2%}/H

_{m}

_{0}= 3 [8]. However, this maximum is not reached until ξ

_{m}

_{−1,0}≈ 7.6, which is out of the range of the investigated surf similarity parameters of this work. Further research towards the maximum relative wave run-up height will be required to cover that range of surf similarity parameters.

#### 3.2. Relative Wave Run-Up Height as a Function of the Momentum Flux Parameter M_{F}

_{u}

_{2%}/h dependent on the dimensionless momentum flux parameter M

_{F}/(ρgh

^{2}) for MGRRs. Equation (10) with corresponding influence factors γ

_{f,M}(see caption of Figure 6) as well as Equation (10) for smooth impermeable revetments (γ

_{f,M}= 1) and a riprap revetment (two layers, γ

_{f,M}= 0.51) are also plotted in Figure 6 for comparison.

_{m}

_{0}/L

_{p}> 0.0225 as breaking waves, 7 of 17 tests on the fully grouted MGRRs and 3 of 16 tests for the partially grouted MGRRs were conducted with breaking waves in the GWK tests. However, Equation (10) for breaking waves was used to describe all results, as Hughes [27] also does for riprap revetments. Hughes [27] justifies using a single equation, as there was no clear difference in run-up behavior for breaking and non-breaking waves according to the breaker criterion for the data of [30,33], and as the wave run-up heights for riprap are well-described by Equation (10). This is the case for MGRRs as well.

#### 3.3. Relative Wave Run-Up Height as a Function of the Modified Surf Similarity Parameter ϕ

_{u}

_{2%}/H

_{m}

_{0}as a function of the modified surf similarity parameter ϕ for MGRRs. According to the breaker criterion defined by Hammeken Arana [14], which classifies waves with tanα/(H

_{s}/h) < 1.4 as breaking waves, all tests on the MGRRs were tests with non-breaking waves. This suggests the applicability of Equation (11), which according to [14] is valid only for non-breaking waves. The modified surf similarity parameter ϕ has, to the authors’ knowledge, up until now not been used to describe wave run-up on rough, porous, and permeable revetments under irregular waves, so no comparison for a lower limit of relative wave run-up can be provided. As a comparison for an upper limit, Equation (11) for smooth impermeable revetments with a = 2.11 and b = −0.17 according to [14] is plotted in Figure 7. To make the predictions with Equation (11) comparable to the results for MGRRs, it is assumed that H

_{s}= H

_{m}

_{0}.

#### 3.4. Relative Wave Run-Up Height as a Function of the Reflection Coefficient C_{r}

_{u}

_{2%}/H

_{m}

_{0}dependent on the reflection coefficient C

_{r}for MGRRs. Equation (12) with corresponding empirical factors a (see caption of Figure 8) as well as, for reference, Equation (12) for a rubble mound breakwater with cot(α) = 1.5 (a = 1.65 [32]) is plotted in Figure 8.

_{r}≈ 0.4 and C

_{r}≈ 0.8. No distinction was made between breaking and non-breaking waves when wave run-up was described as a function of the reflection coefficient [32].

_{m}

_{−1,0}, as well as Equation (18) [48], with c

_{1}= 1.0 and c

_{2}= 9.4 for partially grouted and c

_{1}= 1.0 and c

_{2}= 6.7 for fully grouted MGRRs.

_{r}denotes the reflected wave height and H

_{i}the incident wave height.

## 4. Discussion

#### 4.1. 2D-LIDAR Measurements

_{m}

_{−1,0}(see Figure 5). While there was only small scatter in the results of the partially grouted MGRRs for surf similarity parameters ξ

_{m}

_{−1,0}< 3, scatter increased for ξ

_{m}

_{−1,0}> 3 and is higher for the top layer with t = 0.6 m compared to the top layer with t = 0.4 m. This could be because the LIDAR scanner gives a line measurement of the wave run-up only. During the tests in the GWK, differences in wave run-up perpendicular to the embankment slope were observed visually. Using the video data, this difference is estimated to be at maximum roughly ΔR

_{u}

_{2%}/H

_{m}

_{0}≈ 0.25 for partially grouted MGRRs and smaller for fully grouted MGRRs. Thus, using a line measurement of wave run-up like an LIDAR scanner for revetments with bigger surface irregularities may introduce arbitrary scatter depending on the position of the line scan on the revetment. This resulting scatter seems to occur in the same magnitude for all wave conditions and tests presented in this paper. It therefore generally accounts for some scatter, but does not explain the increase in scatter for ξ

_{m}

_{−1,0}> 3.

#### 4.2. Comparison of Models and Corresponding Parameters

_{u}

_{2%}/H

_{m}

_{0}= f(C

_{r})) performs best, as it gives the lowest RMSE and SI, but the differences between the quality of the model predictions are generally small. All models give an absolute bias < 0.02 m, a RMSE < 0.12 m, and a SI < 8.17 %. Equation (11) (R

_{u}

_{2%}/H

_{m}

_{0}= f(ϕ)) performs worst judged by the statistical parameters, but also judging by the qualitative comparison between predicted and measured values (cf. Section 3.3).

_{u}

_{2%}/H

_{m}

_{0}= f(ξ

_{m}

_{−1,0})) performs best (zero bias, RMSE < 0.11 m and SI < 6 %), although there are two exceptionally high wave run-up heights for ξ

_{m}

_{−1,0}= 2.36 and ξ

_{m}

_{−1,0}= 4.62 (cf. Section 3.1). The prediction of the reflection coefficient as a function of the surf similarity parameter is very good for fully grouted MGRRs (see Figure 9) and better for partially grouted MGRRs because of the aforementioned reasons, yielding better input parameters to Equation (12) (R

_{u}

_{2%}/H

_{m}

_{0}= f(C

_{r})). However, wave run-up on fully grouted MGRRs is not as well described as by Equation (2) (see Table 3).

_{u}

_{2%}/H

_{m}

_{0}= f(ϕ)) performs worst of all models for fully grouted MGRRs. For both fully grouted and partially grouted MGRRs, there is a sharp increase in wave run-up height for ϕ < 0.2 that cannot be modeled by Equation (11) (cf. Section 3.3).

_{p}= 1–2.86 s [14], so the number of incident waves may roughly have been 400 at maximum. Thus, the above stated absolute values of these parameters (a = 2.11 and b = −0.17 [14]) for smooth impermeable embankments, due to the way they were determined, should not be regarded as reliable or valid as long as a comparison to validated physical model tests with irregular waves is missing. This could explain why, using these values, in some instances lower wave run-up heights are predicted for smooth impermeable revetments than were measured for MGRRs (cf. Figure 7).

_{u}

_{2%}/h, it needs to be kept in mind that the water depth was kept at a constant level of 4 m during the tests. Therefore, no empirical data is available to determine how well Equation (10) performs and whether γ

_{f,M}would remain constant in case the water depth changes.

#### 4.3. Further Discussion

## 5. Conclusions

- The wave run-up heights on MGRRs are generally lower than for smooth impermeable revetments and higher than for non-grouted riprap revetments. Partially grouted MGRRs, due to their roughness, porosity, and permeability, reduce wave run-up heights from 21% to 28%, and fully grouted MGRRs, due to their roughness, reduce wave run-up heights from 12% to 14% compared to smooth impermeable revetments.
- Influence factors for the state-of-the-art design guideline EurOtop have been determined for four widely used revetment configurations, which can now be used for design purposes.
- For the results acquired in the model tests, wave run-up for all MGRR configurations is best described by the state-of-the-art EurOtop equation, which describes wave run-up as a function of the surf similarity parameter ξ
_{m}_{−1,0}. The model of Hughes [27], which describes wave run-up as a function of the momentum flux parameter M_{F}, also gives very good results. Models describing wave run-up as a function of the modified surf similarity parameter ϕ or the reflection coefficient C_{r}give poorer results for MGRRs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Configuration Nr. | H_{m}_{0} [m] | T_{m}_{−1,0} [s] | T_{p} [s] | L_{m}_{−1,0} [m] | R_{u}_{2%} [m] | R_{u}_{2%}/H_{m}_{0} [-] | R_{u}_{2%}/h [-] | ξ_{m}_{−1,0} [-] | M_{F}/(ρgh^{2}) [-] | Φ [-] | C_{r} [-] | H_{m}_{0}/L_{p} [-] | tan(α)/(H_{m}_{0}/h) [-] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.375 | 2.73 | 3.01 | 11.60 | 0.80 | 2.14 | 0.201 | 1.85 | 0.021 | 0.639 | 0.24 | 0.027 | 3.56 |

1 | 0.389 | 6.14 | 7.08 | 58.92 | 0.92 | 2.36 | 0.230 | 4.10 | 0.048 | 0.279 | 0.58 | 0.005 | 3.43 |

1 | 0.383 | 3.53 | 4.01 | 19.48 | 0.90 | 2.35 | 0.225 | 2.38 | 0.028 | 0.488 | 0.32 | 0.015 | 3.48 |

1 | 0.577 | 6.14 | 6.58 | 58.78 | 1.32 | 2.29 | 0.330 | 3.36 | 0.075 | 0.229 | 0.55 | 0.009 | 2.31 |

1 | 0.759 | 6.13 | 6.85 | 58.57 | 1.72 | 2.27 | 0.431 | 2.93 | 0.112 | 0.200 | 0.51 | 0.010 | 1.76 |

1 | 0.765 | 7.84 | 9.24 | 95.87 | 2.10 | 2.74 | 0.525 | 3.73 | 0.139 | 0.156 | 0.63 | 0.006 | 1.74 |

1 | 0.389 | 4.39 | 4.89 | 30.06 | 0.94 | 2.42 | 0.236 | 2.93 | 0.035 | 0.390 | 0.43 | 0.010 | 3.43 |

1 | 0.579 | 8.36 | 10.10 | 109.07 | 1.47 | 2.54 | 0.367 | 4.57 | 0.105 | 0.168 | 0.70 | 0.004 | 2.30 |

1 | 0.659 | 9.04 | 12.18 | 127.54 | 2.03 | 3.08 | 0.508 | 4.64 | 0.141 | 0.145 | 0.71 | 0.003 | 2.02 |

1 | 0.667 | 3.97 | 4.27 | 24.61 | 1.46 | 2.19 | 0.364 | 2.02 | 0.067 | 0.329 | 0.31 | 0.023 | 2.00 |

1 | 0.764 | 4.41 | 5.08 | 30.41 | 1.65 | 2.15 | 0.411 | 2.10 | 0.092 | 0.277 | 0.34 | 0.019 | 1.75 |

1 | 0.853 | 4.87 | 5.30 | 36.97 | 1.90 | 2.22 | 0.474 | 2.19 | 0.112 | 0.237 | 0.38 | 0.019 | 1.56 |

1 | 0.765 | 4.11 | 4.44 | 26.32 | 1.65 | 2.16 | 0.413 | 1.96 | 0.084 | 0.297 | 0.31 | 0.025 | 1.74 |

1 | 0.580 | 4.85 | 5.36 | 36.68 | 1.35 | 2.33 | 0.338 | 2.65 | 0.065 | 0.289 | 0.43 | 0.013 | 2.30 |

1 | 0.572 | 4.05 | 4.47 | 25.63 | 1.30 | 2.28 | 0.326 | 2.23 | 0.055 | 0.348 | 0.34 | 0.018 | 2.33 |

1 | 0.580 | 4.85 | 5.36 | 36.76 | 1.35 | 2.33 | 0.338 | 2.65 | 0.065 | 0.289 | 0.43 | 0.013 | 2.30 |

2 | 0.375 | 2.73 | 3.01 | 11.60 | 0.83 | 2.21 | 0.207 | 1.85 | 0.021 | 0.639 | 0.24 | 0.027 | 3.56 |

2 | 0.389 | 6.14 | 7.08 | 58.92 | 0.99 | 2.54 | 0.247 | 4.10 | 0.048 | 0.279 | 0.58 | 0.005 | 3.43 |

2 | 0.383 | 3.53 | 4.01 | 19.48 | 0.91 | 2.37 | 0.227 | 2.38 | 0.028 | 0.488 | 0.32 | 0.015 | 3.48 |

2 | 0.577 | 6.14 | 6.58 | 58.78 | 1.50 | 2.61 | 0.376 | 3.36 | 0.075 | 0.229 | 0.55 | 0.009 | 2.31 |

2 | 0.759 | 6.13 | 6.85 | 58.57 | 1.90 | 2.51 | 0.476 | 2.93 | 0.112 | 0.200 | 0.51 | 0.010 | 1.76 |

2 | 0.765 | 7.84 | 9.24 | 95.87 | 2.21 | 2.89 | 0.552 | 3.73 | 0.139 | 0.156 | 0.63 | 0.006 | 1.74 |

2 | 0.389 | 4.39 | 4.89 | 30.06 | 0.95 | 2.44 | 0.237 | 2.93 | 0.035 | 0.390 | 0.43 | 0.010 | 3.43 |

2 | 0.579 | 8.36 | 10.10 | 109.07 | 1.58 | 2.74 | 0.396 | 4.57 | 0.105 | 0.168 | 0.70 | 0.004 | 2.30 |

2 | 0.659 | 9.04 | 12.18 | 127.54 | 2.15 | 3.26 | 0.537 | 4.64 | 0.141 | 0.145 | 0.71 | 0.003 | 2.02 |

2 | 0.667 | 3.97 | 4.27 | 24.61 | 1.56 | 2.34 | 0.390 | 2.02 | 0.067 | 0.329 | 0.31 | 0.023 | 2.00 |

2 | 0.764 | 4.41 | 5.08 | 30.41 | 1.82 | 2.38 | 0.454 | 2.10 | 0.092 | 0.277 | 0.34 | 0.019 | 1.75 |

2 | 0.853 | 4.87 | 5.30 | 36.97 | 1.99 | 2.33 | 0.497 | 2.19 | 0.112 | 0.237 | 0.38 | 0.019 | 1.56 |

2 | 0.765 | 4.11 | 4.44 | 26.32 | 1.78 | 2.32 | 0.445 | 1.96 | 0.084 | 0.297 | 0.31 | 0.025 | 1.74 |

2 | 0.580 | 4.85 | 5.36 | 36.68 | 1.48 | 2.55 | 0.370 | 2.65 | 0.065 | 0.289 | 0.43 | 0.013 | 2.30 |

2 | 0.572 | 4.05 | 4.47 | 25.63 | 1.35 | 2.36 | 0.337 | 2.23 | 0.055 | 0.348 | 0.34 | 0.018 | 2.33 |

2 | 0.580 | 4.85 | 5.36 | 36.76 | 1.45 | 2.50 | 0.362 | 2.65 | 0.065 | 0.289 | 0.43 | 0.013 | 2.30 |

Configuration Nr. | H_{m}_{0} [m] | T_{m}_{−1,0} [s] | T_{p} [s] | L_{m}_{−1,0} [m] | R_{u}_{2%} [m] | R_{u}_{2%}/H_{m}_{0} [-] | R_{u}_{2%}/h [-] | ξ_{m}_{−1,0} [-] | M_{F}/(ρgh^{2}) [-] | Φ [-] | C_{r} [-] | H_{m}_{0}/L_{p} [-] | tan(α)/(H_{m}_{0}/h) [-] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

3 | 0.768 | 4.11 | 4.44 | 26.36 | 1.76 | 2.29 | 0.439 | 1.95 | 0.085 | 0.296 | 0.36 | 0.025 | 1.74 |

3 | 0.856 | 4.87 | 5.43 | 37.08 | 2.07 | 2.42 | 0.518 | 2.19 | 0.114 | 0.237 | 0.43 | 0.019 | 1.56 |

3 | 0.760 | 6.12 | 6.85 | 58.42 | 1.98 | 2.61 | 0.496 | 2.92 | 0.112 | 0.200 | 0.59 | 0.010 | 1.75 |

3 | 0.386 | 3.52 | 3.68 | 19.38 | 1.17 | 3.04 | 0.293 | 2.36 | 0.027 | 0.488 | 0.43 | 0.018 | 3.45 |

3 | 0.390 | 4.39 | 4.53 | 30.09 | 0.99 | 2.54 | 0.248 | 2.93 | 0.033 | 0.389 | 0.56 | 0.012 | 3.42 |

3 | 0.582 | 6.13 | 7.06 | 58.59 | 1.57 | 2.69 | 0.392 | 3.34 | 0.080 | 0.228 | 0.65 | 0.007 | 2.29 |

3 | 0.763 | 7.83 | 8.47 | 95.67 | 2.21 | 2.90 | 0.553 | 3.73 | 0.131 | 0.156 | 0.71 | 0.007 | 1.75 |

3 | 0.664 | 9.04 | 12.18 | 127.65 | 2.25 | 3.40 | 0.564 | 4.62 | 0.142 | 0.145 | 0.79 | 0.003 | 2.01 |

3 | 0.393 | 6.14 | 6.57 | 58.90 | 1.11 | 2.82 | 0.277 | 4.08 | 0.046 | 0.277 | 0.71 | 0.006 | 3.39 |

3 | 0.576 | 4.05 | 4.55 | 25.63 | 1.52 | 2.64 | 0.380 | 2.22 | 0.056 | 0.347 | 0.41 | 0.018 | 2.31 |

3 | 0.584 | 8.37 | 9.36 | 109.46 | 1.74 | 2.98 | 0.435 | 4.56 | 0.100 | 0.167 | 0.79 | 0.004 | 2.28 |

3 | 0.752 | 3.59 | 3.90 | 20.10 | 1.83 | 2.43 | 0.457 | 1.72 | 0.075 | 0.343 | 0.29 | 0.032 | 1.77 |

3 | 0.727 | 3.34 | 3.79 | 17.45 | 1.76 | 2.43 | 0.441 | 1.63 | 0.070 | 0.374 | 0.28 | 0.033 | 1.83 |

3 | 0.941 | 4.26 | 4.77 | 28.31 | 2.32 | 2.47 | 0.580 | 1.83 | 0.121 | 0.258 | 0.32 | 0.027 | 1.42 |

3 | 0.929 | 3.87 | 4.42 | 23.32 | 2.18 | 2.35 | 0.545 | 1.67 | 0.113 | 0.286 | 0.29 | 0.030 | 1.44 |

3 | 0.908 | 3.66 | 4.10 | 20.87 | 2.05 | 2.26 | 0.513 | 1.60 | 0.104 | 0.306 | 0.27 | 0.035 | 1.47 |

3 | 0.722 | 3.16 | 3.43 | 15.61 | 1.70 | 2.36 | 0.426 | 1.55 | 0.064 | 0.397 | 0.26 | 0.039 | 1.85 |

4 | 0.768 | 4.11 | 4.44 | 26.36 | 1.89 | 2.46 | 0.472 | 1.95 | 0.085 | 0.296 | 0.36 | 0.025 | 1.74 |

4 | 0.856 | 4.87 | 5.43 | 37.08 | 2.21 | 2.58 | 0.552 | 2.19 | 0.114 | 0.237 | 0.43 | 0.019 | 1.56 |

4 | 0.760 | 6.12 | 6.85 | 58.42 | 2.08 | 2.74 | 0.520 | 2.92 | 0.112 | 0.200 | 0.59 | 0.010 | 1.75 |

4 | 0.386 | 3.52 | 3.68 | 19.38 | 1.17 | 3.02 | 0.291 | 2.36 | 0.027 | 0.488 | 0.43 | 0.018 | 3.45 |

4 | 0.390 | 4.39 | 4.53 | 30.09 | 1.03 | 2.63 | 0.257 | 2.93 | 0.033 | 0.389 | 0.56 | 0.012 | 3.42 |

4 | 0.582 | 6.13 | 7.06 | 58.59 | 1.65 | 2.83 | 0.412 | 3.34 | 0.080 | 0.228 | 0.65 | 0.007 | 2.29 |

4 | 0.763 | 7.83 | 8.47 | 95.67 | 2.19 | 2.88 | 0.548 | 3.73 | 0.131 | 0.156 | 0.71 | 0.007 | 1.75 |

4 | 0.664 | 9.04 | 12.18 | 127.65 | 2.25 | 3.38 | 0.562 | 4.62 | 0.142 | 0.145 | 0.79 | 0.003 | 2.01 |

4 | 0.393 | 6.14 | 6.57 | 58.90 | 1.12 | 2.84 | 0.279 | 4.08 | 0.046 | 0.277 | 0.71 | 0.006 | 3.39 |

4 | 0.576 | 4.05 | 4.55 | 25.63 | 1.62 | 2.81 | 0.405 | 2.22 | 0.056 | 0.347 | 0.41 | 0.018 | 2.31 |

4 | 0.584 | 8.37 | 9.36 | 109.46 | 1.82 | 3.11 | 0.454 | 4.56 | 0.100 | 0.167 | 0.79 | 0.004 | 2.28 |

4 | 0.752 | 3.59 | 3.90 | 20.10 | 1.91 | 2.54 | 0.477 | 1.72 | 0.075 | 0.343 | 0.29 | 0.032 | 1.77 |

4 | 0.727 | 3.34 | 3.79 | 17.45 | 1.80 | 2.47 | 0.450 | 1.63 | 0.070 | 0.374 | 0.28 | 0.033 | 1.83 |

4 | 0.941 | 4.26 | 4.77 | 28.31 | 2.23 | 2.37 | 0.557 | 1.83 | 0.121 | 0.258 | 0.32 | 0.027 | 1.42 |

4 | 0.929 | 3.87 | 4.42 | 23.32 | 2.10 | 2.26 | 0.525 | 1.67 | 0.113 | 0.286 | 0.29 | 0.030 | 1.44 |

4 | 0.908 | 3.66 | 4.10 | 20.87 | 2.04 | 2.25 | 0.510 | 1.60 | 0.104 | 0.306 | 0.27 | 0.035 | 1.47 |

4 | 0.722 | 3.16 | 3.43 | 15.61 | 1.73 | 2.39 | 0.431 | 1.55 | 0.064 | 0.397 | 0.26 | 0.039 | 1.85 |

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**Figure 1.**MGRRs made of LMB

_{5/40}, (

**a**) partially grouted (top layer thickness t = 0.6 m, grouted with an amount of mortar of v

_{g}= 80 l/m

^{2}, Coastal Research Center’s Large Wave Flume (GWK) hydraulic model test configuration 1), (

**b**) partially grouted in the upper part, fully grouted in the lower part of the top layer (t = 0.4 m, upper part v

_{g}≈ 60 l/m

^{2}, revetment in the field), (

**c**) fully grouted (t = 0.4 m, grouted with v

_{g}= 180 l/m

^{2}, GWK hydraulic model test configuration 4), and (

**d**) fully grouted (t = 0.6 m, revetment in the field).

**Figure 2.**MGRRs in the GWK: (

**a**) view from the empty flume to the revetment with the toe structure in the lower part of the picture. (

**b**) Wave run-up event on a fully grouted MGRR in the GWK with high aeration of the up-rushing wave after the wave plunged on the revetment; view is from the revetment to the flume.

**Figure 4.**Longitudinal cross-section (

**a**) and top view (

**b**) of the GWK showing the wave paddle at the left, the instrumentation along the flume, and the revetment on the right, with the 2D-LIDAR above.

**Figure 5.**Relative wave run-up height R

_{u}

_{2%}/H

_{m}

_{0}as a function of the surf similarity parameter ξ

_{m}

_{−1,0}(

**a**) for the partially grouted (▲) and re-grouted impermeable (■) MGRR with a thickness of t = 0.6 m, as well as the best fit EurOtop equation with γ

_{f}= 0.72 and γ

_{f}= 0.86, respectively. (

**b**) R

_{u}

_{2%}/H

_{m}

_{0}for the partially grouted (●) and fully grouted (▼) MGRR with a thickness of t = 0.4 m, as well as the best fit EurOtop equation with γ

_{f}= 0.79 and γ

_{f}= 0.88, respectively. For reference, Equation (2) for smooth impermeable revetments (γ

_{f}= 1 [8]) and riprap revetments (two layers, γ

_{f}= 0.55 [8]) is given.

**Figure 6.**Relative wave run-up height R

_{u}

_{2%}/h as a function of the dimensionless momentum flux parameter M

_{F}/(ρgh

^{2}) for (

**a**) the partially grouted (▲) and re-grouted impermeable (■) MGRR with a thickness of t = 0.6 m, as well as the best fit Equation (10) with γ

_{f,M}= 0.65 and γ

_{f,M}= 0.76, respectively. (

**b**) R

_{u}

_{2%}/h for the partially grouted (●) and fully grouted (▼) MGRR with a thickness of t = 0.4 m, as well as the best fit Equation (10) with γ

_{f,M}= 0.70 and γ

_{f,M}= 0.77, respectively. For reference, Equation (10) for smooth impermeable revetments (γ

_{f,M}= 1 [27]) and riprap revetments (two layers, γ

_{f,M}= 0.51 [27]) is given.

**Figure 7.**Relative wave run-up height R

_{u}

_{2%}/H

_{m}

_{0}as a function of the modified surf similarity parameter ϕ for (

**a**) the partially grouted (▲) and re-grouted impermeable (■) MGRR with a thickness of t = 0.6 m, as well as the best fit Equation (11) with a = 1.87, b = −0.18 and a = 2.07, b = −0.18, respectively. (

**b**) R

_{u}

_{2%}/H

_{m}

_{0}for the partially grouted (●) and fully grouted (▼) MGRR with a thickness of t = 0.4 m, as well as the best fit Equation (11) with a = 1.90, b = −0.22 and a = 2.13, b = −0.18, respectively. For reference, Equation (11) for smooth impermeable revetments (a = 2.11; b = −0.17 [14]) is given.

**Figure 8.**Relative wave run-up height R

_{u}

_{2%}/H

_{m}

_{0}as a function of the reflection coefficient C

_{r}for (

**a**) the partially grouted (▲) and re-grouted impermeable (■) MGRR with a thickness of t = 0.6 m, as well as the best fit Equation (12) with a = 1.63 and a = 1.77, respectively. (

**b**) R

_{u}

_{2%}/H

_{m}

_{0}for the partially grouted (●) and fully grouted (▼) MGRR with a thickness of t = 0.4 m, as well as the best fit Equation (12) with a = 1.73 and a = 1.81, respectively. For reference, Equation (12) for a rubble mound breakwater (a = 1.65 [32]) is given.

**Figure 9.**Reflection coefficient C

_{r}as a function of the surf similarity parameter ξ

_{m}

_{−1,0}for the partially grouted (▲,●; SI = 6.88%) as well as fully grouted and re-grouted impermeable MGRR (■,▼; SI = 3.97%).

**Table 1.**Overview of some influence factors γ

_{f}as well as characteristics of the revetment type for which they are valid and the equation in this work for which the influence factor was determined.

Type of Revetment | Influence Factor γ_{f} | Equation Nr. in This Work | Revetment Characteristics | Reference | ||
---|---|---|---|---|---|---|

Rough | Porous | Permeable | ||||

Grass | 0.9–1.0 | (2) | (x) | [8] | ||

Polyurethane bonded gravel | ≈0.75 | mod. (2) | (x) | x | x | [22,23] |

Basalt | 0.9 | (2) | (x) | (x) | (x) | [8] |

Stepped revetments | 0.4–0.9 | (4) | x | [26] | ||

Two layers of rock | 0.55 | (2) | x | x | x | [8] |

Two layers of rock | 0.51 | (10) | x | x | x | [27] |

Pattern placed revetments with enhanced roughness | 0.65–0.85 | (4) | x | (x) | (x) | [28] |

**Table 2.**Characteristics of the MGRRs in the GWK (Data from [38]).

Partially Grouted | Fully Grouted | |||
---|---|---|---|---|

Section | North | South | North | South |

Configuration Nr. | 1 | 2 | 3 | 4 |

Top layer thickness t [m] | 0.6 | 0.4 | 0.6 | 0.4 |

Amount of mortar v_{g} [l/m^{2}] | 80 | 80 | 80+100 | 180 |

Estimated porosity n before grouting [-] | 0.45 | 0.45 | 0.45 | 0.45 |

Porosity n after grouting [-] | 0.32 | 0.25 | 0.16 | 0 |

Pore volume V_{p} after grouting [l/m^{2}] | 190 | 100 | 90 | 0 |

**Table 3.**Influence factors and empirical parameters for the MGRRs in the GWK, as well as corresponding statistical parameters.

Partially Grouted | Fully Grouted | ||||||
---|---|---|---|---|---|---|---|

Section | North | South | North | South | |||

Equation | Nr. | 1 (▲) | 2 (●) | 3 (■) | 4 (▼) | ||

(2) | $\frac{{R}_{u2\%}}{{H}_{m0}}=min\left(\right)open="\{">\begin{array}{c}1.65\times {\gamma}_{f}\times {\xi}_{m-1,0}\\ 1.00\times {\gamma}_{f,surging}\times \left(4-\frac{1.5}{\sqrt{{\xi}_{m-1,0}}}\right)\end{array}$ | γ_{f} | 0.72 | 0.79 | 0.86 | 0.88 | |

bias | −0.02 | 0 | 0 | 0 | |||

RMSE | 0.09 | 0.08 | 0.11 | 0.10 | |||

SI | 6.27 | 5.51 | 6.00 | 5.56 | |||

(10) | $\frac{{R}_{u2\%}}{h}=4.4{\left(\mathit{tan}\alpha \right)}^{0.7}{\left(\frac{{M}_{F}}{\rho gh}\right)}^{0.5}\times {\gamma}_{f,M}$ | γ_{f,M} | 0.65 | 0.70 | 0.76 | 0.77 | |

bias | 0.01 | 0.02 | 0.01 | 0 | |||

RMSE | 0.11 | 0.11 | 0.13 | 0.12 | |||

SI | 7.86 | 7.38 | 7.22 | 6.68 | |||

(11) | $\frac{{R}_{u2\%}}{{H}_{s}}=a{\varphi}^{b}=a{\left({\xi}_{m-1,0}\times \frac{h}{{L}_{m-1,0}}\right)}^{b}$ | a | 1.87 | 1.90 | 2.07 | 2.13 | |

b | −0.18 | −0.22 | −0.18 | −0.18 | |||

bias | 0.01 | 0.02 | 0.03 | 0.05 | |||

RMSE | 0.12 | 0.10 | 0.16 | 0.19 | |||

SI | 8.17 | 6.84 | 9.26 | 10.29 | |||

(12) | $\frac{{R}_{u2\%}}{{H}_{m0}}=a\left(1+{C}_{r}\right)$ | a [-] | 1.63 | 1.73 | 1.77 | 1.81 | |

bias | 0 | −0.01 | −0.01 | 0 | |||

RMSE | 0.09 | 0.07 | 0.11 | 0.10 | |||

SI | 6.11 | 4.72 | 6.37 | 5.68 |

_{u}

_{2%}[m] (see Section 2.5).

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**MDPI and ACS Style**

Kreyenschulte, M.; Schürenkamp, D.; Bratz, B.; Schüttrumpf, H.; Goseberg, N.
Wave Run-Up on Mortar-Grouted Riprap Revetments. *Water* **2020**, *12*, 3396.
https://doi.org/10.3390/w12123396

**AMA Style**

Kreyenschulte M, Schürenkamp D, Bratz B, Schüttrumpf H, Goseberg N.
Wave Run-Up on Mortar-Grouted Riprap Revetments. *Water*. 2020; 12(12):3396.
https://doi.org/10.3390/w12123396

**Chicago/Turabian Style**

Kreyenschulte, Moritz, David Schürenkamp, Benedikt Bratz, Holger Schüttrumpf, and Nils Goseberg.
2020. "Wave Run-Up on Mortar-Grouted Riprap Revetments" *Water* 12, no. 12: 3396.
https://doi.org/10.3390/w12123396