# Comparative Assessment of Different Modelling Schemes and Their Applicability to Inland Small Reservoirs: A Central Europe Case Study

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{13%}) within the range 0.32–0.56 m, wave period 1.32–2.11 s and run-up (R

_{2%}) 0.84–1.68 m under conditions of design wind speed 25 m·s

^{−1}. Results obtained by CEM, SPM models predict lower values than STN and ASABE models. Since the height difference between the dam crest and still water level in the reservoir is only 0.90 m, we can expect overtopping of the crest by waves after the critical wind speed is exceeded.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Case Study

^{2}and the reservoir volume 97 665 m

^{3}). The upstream slope is protected by concrete blocks up to the level of 180.15 m a.s.l. The downstream slope, the crest, and the rest of the upstream slope has vegetative grass protection (Figure 2). The reservoir meets the criteria of a small water reservoir—the water volume up to 2 million m

^{3}and a dam height of up to 9.0 m, and the maximum harmless flood flow rate up to 60 m

^{3}·s

^{−1}compliant to Slovak standards [23].

#### 2.2. Wind

_{10}and applying an appropriate value for drag coefficient C

_{d}. The drag coefficient depending on the elevation and stability (the difference between air and water temperature) of the atmosphere above the water level.

_{A}which is calculated directly from wind speed in 10 m u

_{10}. The wave prediction model of STN does not consider abovementioned physical effects and the equations for determination of wave parameters are expressed in terms of u

_{10}directly (Table 1).

_{ef}[19]. On this premise, the American scientist Saville developed a method to determine the effective fetch F

_{ef}, which has been validated on the Fort Peck and Denison reservoirs in the 1950s [19]. The technique assumes that the wind blows effectively in the main direction on the area in a sector of 2 × 45° on each side of the main direction. The effectiveness of each sector of fetch is determined by the ratio of the length to the length of a segment that would result from fetch of unlimited width. The effective fetch length F

_{ef}is determined from 15 rays F

_{i}starting from the investigated point, so that the middle ray is identical with the main wind direction and 7 rays on each side are plotted at intervals of φ = 6° (i.e., up to an angle of 42°) to the opposite bank of the reservoir (Figure 3). The method was modified and implemented into STN 75 0255; however, the point of longest fetch length in the reservoir must be found. The effective length of fetch in accordance with the standards can be calculated using the following formula:

_{i}, plotted by φ = 3° from the middle ray (Figure 3). The method is also used in calculations by the ASABE investigating the wind conditions on reservoirs:

#### 2.3. Wave Parameters

_{13%}) which represents the wave of height with a 13% probability of exceedance. Second one parameter is the period of waves (T) described the time of two consequent wave crests pass through the given point. The characteristic period could be designated as mean period T

_{m}, or average zero-crossing period [16,17,21]. The specific value of period is widely applied in subsequent calculations. The peak period T

_{p}is identical with the peak spectral energy density, hence the waves with T

_{p}carry the largest part of energy in wave spectra. The relationship between mean and peak period is expressed by the equation T

_{p}= 1.2T

_{m}.

_{0}(H

_{13%}) and peak period T

_{p}in deep water conditions (Table 2).

#### 2.4. Wave Run-Up

_{1%}. The wave run-up of 1% exceedance probability R

_{1%}is calculated by means of the following formula, where k

_{d}is the slope roughness coefficient and k

_{p}is the coefficient based on the angle of slope α, wavelength L

_{0}and wave height H

_{1%}according to the STN [16,17]:

_{i}to calculate other values of wave run-up based on the exceedance probability.

## 3. Results and Discussion

_{0}has a length of 471 m. To determine the effective fetch length, eight more rays with 3° angle increments were plotted in compliance with the SPM and ASABE method. Their lengths ranged between 280 m and 451 m, the calculated fetch length has a final value of 405 m. According to the CEM, the fetch length was considered as a linear distance between the point of analysis and the opposite bank in the main wind direction, i.e., 471 m. According to the STN, 14 other rays were plotted with 6° increments and lengths of 157–440 m.

_{ef}and the radial line φ

_{0}to decrease the overtopping of dam crest. Therefore, the length of 471 m instead of F

_{ef}= 273 m, was used for the following calculations. The determination of the fetch length in compliance with the above methods is affected by the two islands in the western part of the reservoir, which represent a wave shadow and thus reduce the wave-forming surface (Figure 3).

_{13%}and the wave period T

_{p}) were calculated by the equations in Table 1 and Table 2 for the appropriate fetch length above the water level. The maximum design wind speed at the reference height of 10 m above the water surface u

_{10}was considered 25 m·s

^{−1}according to the STN recommendation.

#### 3.1. Wave Height H_{13%}

_{13%}was performed by four models (CEM, SPM, ASABE, STN) based on the selected wind speed at the point of analysis. The lowest values of wave height were calculated by the CEM model—0.32 m. In contrast, the highest values of the wave height with the same wind speed are gained using the ASABE model—0.56 m. The estimates of the STN and SPM reach 0.42 m and 0.38 m, respectively (Figure 4).

^{−1}, afterward the values are lower than by ASABE but higher than CEM or SPM, respectively. The models developed for the conditions of the seas and oceans provide lower estimates of the wave heights compared to the models adapted to the conditions of water reservoirs, which was also confirmed by experimental measurements on irrigation reservoirs in the USA by Ozeren and Wren [9]. Underestimated results of fetch-unlimited model discussed Pelikán and Koutný [29], comparing the measured data with models.

#### 3.2. Wave Periods Tp and Wavelength L

^{−1}. The resulting maximum periods of around 2 s correspond to the results of a former study of the gravitational oscillating waves on dams and reservoirs, based on which Lukáč and Abaffy [18] stated that a wave period of 5 s at maximum can be expected in Central European dams of usual dimensions.

_{p}and wavelength L increase for a given fetch length and with the increasing wind speed [10]. Thus, the relative depth d/L drops below the critical value, and the wave begins to be affected by the reservoir bottom.

_{2%}was selected to verify the overtopping of the dam crest.

#### 3.3. Wave Run-Up

_{1%}= 1.4H

_{13%}. The value of the coefficient dependent on the roughness of the slope reinforcement was selected k

_{d}= 1.0, whereas the slope is protected by concrete blocks with the smooth surface up to 180.15 m a.s.l. The rest of the slope above the blocks up to the dam crest (180.40 m a.s.l.) is only covered by grass, which can also be considered as smooth surface for the purposes of wave run-up stress. Coefficient k

_{p}dependent on the L

_{0}/H

_{1%}ratio achieved values 2.35–2.65 for the considered range of wind speed.

_{2%}= 1.68 m, followed by the STN model with the value of 1.35 m, SPM 1.05 m and CEM 0.84 m, respectively (Figure 6). Since the SWL in the reservoir is only 0.90 m below the dam crest, the dam overtopping by wind-driven waves can be expected from the critical wind speed, which will cause wave run-up R

_{2%}> 0.90 m. This critical wind speed is 27 m·s

^{−1}in the case of the CEM model, 22 m·s

^{−1}in the case of the SPM model, and only 15 m·s

^{−1}in the case of the ASABE model.

^{−1}, at which the wave parameters are H

_{0}≈ H

_{13%}= 0.28 m, T

_{0}= 1.73 s, and the wavelength L

_{0}= 4.7 m.

#### 3.4. Possible Results Application

_{2%}in relation to the dam crest (180.40 m a.s.l.) or in relation to the level of the upstream reinforcement (180.15 m a.s.l.). In case of the design of the maximum wave run-up reach at the wind speed of 25 m·s

^{−1}up to the dam crest, the SWL of the water in the reservoir should be reduced to 179.05 m a.s.l. according to the STN model. If the considered maximum reach of the wave run-up is the top of the upstream armor, the SWL should be 178.80 m a.s.l., which is about 0.70 m below the current general state. Such a quick regulation of the water level in the reservoir without permanent operation staff is not possible. So far, the dam crest of Kolíňany water reservoir has not been overflowed and overtopped, even during the extreme flood situation in 2010 and 2020.

## 4. Conclusions

_{2%}at the wind speed of 25 m·s

^{−1}is 0.37 m, which indicates relatively high level of uncertainty and the selection of suitable calculation model for specific conditions must be considered. STN and ASABE models provide higher estimations of the set boundary conditions. Therefore, they can be used to calculate appropriate dam parameters and design stabilization measures with higher requirements regarding safety. Critical wind speeds have been established by iteration corresponding to each model. Dam crest overtopping by wind-driven waves at the still water level 179.50 m a.s.l. can be expected in case of excess of critical wind speeds.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Jurík, Ľ.; Pierzgalski, E.; Hubačíková, V. Vodné Stavby v Krajine, Malé Vodné Nádrže. [Water Structures in the Landscape. Small Water Reservoirs] 1. vyd; Slovenská poľnohospodárska univerzita v Nitre: Nitra, Slovakia, 2011. (In Slovak) [Google Scholar]
- Zhong, X.; Jiang, X.; Li, L.; Xu, J.; Xu, H. The impact of socio-economic factors on sediment load: A case study of the Yanhe River watershed. Sustainability
**2020**, 12, 2457. [Google Scholar] [CrossRef] [Green Version] - Mailhot, A.; Talbot, G.; Ricard, S.; Turcotte, R.; Guinard, K. Assessing the potential impacts of dam operation on daily flow at ungauged river reaches. J. Hydrol. Reg. Stud.
**2018**, 18, 156–167. [Google Scholar] [CrossRef] - Chongxun, M.; Fanggui, L.; Mei, Y.; Rangyong, M.; Guikai, S. Risk analysis for earth dam overtopping. Water Sci. Eng.
**2008**, 1, 76–87. [Google Scholar] - Meer, J.W.V.D. Erosion Strength of Inner Slopes of Dikes Against Wave Overtopping. Preliminary Conclusions after Two Years of Testing with the Wave Overtopping Simulator, v 1.1. ComCoast report to Rijkswaterstaat. Summary Report. Available online: http://vandermeerconsulting.nl/downloads/summary_report_overtopping_tests_v11.pdf (accessed on 25 August 2008).
- Kubinský, D.; Weis, K. Zmeny retenčného objemu v priestore Belianskej vodnej nádrže. [Retention volume changes of Beliansky water reservoir]. Acta Hydrol. Slovak.
**2013**, 13, 108–113. (In Slovak) [Google Scholar] - Pelikán, P.; Marková, J. Wind effect on water surface of water reservoirs. Acta Univ. Agric. Silvic. Mendel. Brun.
**2013**, 61, 1823–1828. [Google Scholar] [CrossRef] [Green Version] - Pullen, T.; Allsop, N.W.H.; Bruce, T.; Kortenhaus, A.; Schüttrumpf, H.; Meer, J.W.V.D. EUROTOP, Wave Overtopping of Sea Defences and Related Structures–Assessment Manual. Available online: http://overtopping-manual.com/eurotop/downloads/ (accessed on 28 August 2007).
- Ozeren, Y.; Wren, D.G. Predicting wind-driven waves in small reservoirs. Am. Soc. Agric. Biol. Eng.
**2009**, 52, 1213–1221. [Google Scholar] - U.S. ARMY Corps of Engineers. Coastal Engineering Manual. Engineer Manual 1110-2-1100; USACE: Washington, DC, USA, 2011; Volume I–VI.
- Thomas, T.J.; Dwarakish, G.S. Numerical wave modelling–A review. Aquat. Procedia
**2015**, 4, 443–448. [Google Scholar] [CrossRef] - Fitzgerald, C.J.; Taylor, P.H.; Orszaghova, J.; Borthwick, A.G.L.; Whittaker, C.; Raby, A.C. Irregular wave runup statistics on plane beaches: Application of a Boussinesq-type model incorporating a generating-absorbing sponge layer and second-order wave generation. Coast. Eng.
**2016**, 114, 309–324. [Google Scholar] [CrossRef] [Green Version] - Xiang, Y.; Fu, Z.; Meng, Y.; Zhang, K.; Cheng, Z. Analysis of wave clipping effects of plain reservoir artificial islands based on MIKE21 SW model. Water Sci. Eng.
**2019**, 12, 179–187. [Google Scholar] [CrossRef] - Farrok, O.; Ahmed, K.; Tahlil, A.D.; Farah, M.M.; Kiran, M.R.; Islam, M.R. Electrical power generation from the oceanic wave for sustainable advancement in renewable energy technologies. Sustainability
**2020**, 12, 2178. [Google Scholar] [CrossRef] [Green Version] - Coastal Engineering Research Center. Shore Protection Manual, 4th ed.; U.S. Government Printing Office: Washington, DC, USA, 1984.
- UNMS. STN 75 0255. Calculation of Wave Effects on Waterworks and Weir Basins, Czechoslovakia, 1988. Available online: https://sutn.sk/eshop/public/standard_detail.aspx?id=69467 (accessed on 24 November 2020).
- UNMZ. CSN 75 0255 Calculation of Wave Effects on Hydrotechnic Structures on Water Reservoirs, Czechoslovakia, 1988. Available online: http://technicke-normy-csn.cz/750255-csn-75-0255_4_31249.html (accessed on 24 November 2020).
- Lukáč, M.; Abaffy, D. Vlnenie na Nádržiach, Jeho Účinky a Protiabrázne Opatrenia. [Waves on Reservoirs, Its Effects and Anti-Abrasion Measures]; Ministerstvo Lesného a Vodného Hospodárstva SSR: Bratislava, Czechoslovakia, 1980. (In Slovak) [Google Scholar]
- Kratochvil, S. Determination of parameters of wind-driven gravitational waves on deep dam reservoirs and lakes. Vodohospodársky Časopis
**1970**, 18, 239–255. [Google Scholar] - Linhart, J. Abrasion Processes on the Kníničky Dam Reservoir; Sborník Československé společnosti zeměpisné: Praha, Czechoslovakia, 1954. [Google Scholar]
- Pelikán, P.; Šlezingr, M. Parameters of wind driven waves on Nove Mlyny water reservoir. In Water Management and Hydraulic Engineering 2015; Institute of Water Structures, FCE, BUT: Brno, Czech Republic, 2015; pp. 55–64. [Google Scholar]
- Říha, J.; Špano, M. The influence of current on the height of wind wave run-up, a comparison of experimental results with the Czech National Standard. J. Hydrol. Hydromech.
**2012**, 60, 174–184. [Google Scholar] [CrossRef] [Green Version] - UNMS. STN 73 6824/b. Small Water Reservoirs, Czechoslovakia, 1988. Available online: https://sutn.sk/eshop/public/standard_detail.aspx?id=69372 (accessed on 24 November 2020).
- Melby, J.A. Wave Runup Prediction for Flood Hazard Assessment. ERDC/CHL TR-12-24 Technical Report; Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development Center: Vicksburg, MS, USA, 2012. [Google Scholar]
- Diwedar, A.I. Investigating the effect of wave parameters on wave runup. Alex. Eng. J.
**2015**, 55, 627–633. [Google Scholar] [CrossRef] [Green Version] - Stockdon, H.F.; Thompson, D.M.; Plant, N.G.; Long, J.W. Evaluation of wave runup predictions from numerical and parametric models. Coast. Eng.
**2014**, 92, 1–11. [Google Scholar] [CrossRef] - Park, H.; Cox, D.T. Empirical wave run-up formula for wave, storm surge and berm width. Coast. Eng.
**2016**, 115, 67–78. [Google Scholar] [CrossRef] [Green Version] - Mase, H. Random wave run-up height on gentle slope. J. Waterw. Port. Coast. Ocean. Eng.
**1989**, 115, 649–661. [Google Scholar] [CrossRef] - Pelikán, P.; Koutný, L. Hindcast of wind driven wave heights in water reservoirs. Soil Water Res.
**2016**, 11, 205–211. [Google Scholar] - World Meteorological Organization. Guide to Wave Analysis and Forecasting, 2nd ed.; WMO No. 702: Geneva, Switzerland, 1998. [Google Scholar]
- Lichner, M.; Kašiarová, H.; Novák, J. Banskoštiavnické Tajchy. [Historic Water Reservoirs in Vicinity of Banská Štiavnica]TELEM; TLM spol. s.r.o.: Liptovský Mikuláš, Slovakia, 1997; p. 109s. ISBN 80-967757-0-7. (In Slovak) [Google Scholar]
- Gacko, I.; Muchová, Z.; Jurík, Ľ.; Šinka, K.; Fabian, L.; Petrovič, F. Decision making methods to optimize new dam site selections on the Nitra river. Water
**2020**, 12, 2042. [Google Scholar] [CrossRef]

**Figure 3.**Methods of effective fetch calculation to the investigated point on water reservoir Kolíňany.

Method | Magnitude | Unit |
---|---|---|

STN | ${u}_{10}$ | m·s^{−1} |

CEM | ${C}_{d}=0.001\left(1.1+0,035{u}_{10}\right)$ | – |

${u}_{*}=\sqrt{{C}_{d}}{u}_{10}$ | m·s^{−1} | |

SPM, ASABE | ${u}_{A}=0.71{u}_{10}{}^{1,23}$ | – |

Method | H_{0} [m] | T_{p} [s] |
---|---|---|

STN | $0.0026\frac{{u}_{10}{}^{1.06}{F}^{0.47}}{{g}^{0,53}}$ | $0.46\frac{{u}_{10}{}^{0.44}{F}^{0.28}}{{g}^{0.72}}$ |

CEM | $\frac{{u}_{*}{}^{2}}{g}0.0413{\left(\frac{gF}{{u}_{*}{}^{2}}\right)}^{\frac{1}{2}}$ | $\frac{{u}_{*}}{g}0.751{\left(\frac{gF}{{u}_{*}{}^{2}}\right)}^{\frac{1}{3}}$ |

SPM | $\frac{{u}_{A}{}^{2}}{g}0.0016{\left(\frac{gF}{{u}_{A}{}^{2}}\right)}^{\frac{1}{2}}$ | $\frac{{u}_{A}}{g}0.286{\left(\frac{gF}{{u}_{A}{}^{2}}\right)}^{\frac{1}{3}}$ |

ASABE | $\frac{{u}_{A}{}^{2}}{g}0.0025{\left(\frac{gF}{{u}_{A}{}^{2}}\right)}^{0.44}$ | $\frac{{u}_{A}}{g}0.4147{\left(\frac{gF}{{u}_{A}{}^{2}}\right)}^{0.28}$ |

R_{%} | R_{0.1%} | R_{1%} | R_{2%} | R_{5%} | R_{10%} | R_{13%} | R_{30%} | R_{50%} |
---|---|---|---|---|---|---|---|---|

k_{i} | 1.10 | 1.00 | 0.96 | 0.91 | 0.86 | 0.85 | 0.76 | 0.68 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pelikán, P.; Hubačíková, V.; Kaletová, T.; Fuska, J.
Comparative Assessment of Different Modelling Schemes and Their Applicability to Inland Small Reservoirs: A Central Europe Case Study. *Sustainability* **2020**, *12*, 10692.
https://doi.org/10.3390/su122410692

**AMA Style**

Pelikán P, Hubačíková V, Kaletová T, Fuska J.
Comparative Assessment of Different Modelling Schemes and Their Applicability to Inland Small Reservoirs: A Central Europe Case Study. *Sustainability*. 2020; 12(24):10692.
https://doi.org/10.3390/su122410692

**Chicago/Turabian Style**

Pelikán, Petr, Věra Hubačíková, Tatiana Kaletová, and Jakub Fuska.
2020. "Comparative Assessment of Different Modelling Schemes and Their Applicability to Inland Small Reservoirs: A Central Europe Case Study" *Sustainability* 12, no. 24: 10692.
https://doi.org/10.3390/su122410692