Dam Break Modeling in a Cascade of Small EarthenDams: Case Study of the Cizina River in the Czech Republic

water Article Dam Break Modeling in a Cascade of Small Earthen Dams: Case Study of the ˇ Cižina River in the Czech Republic Jaromírˇ Ríha , Stanislav Kotaška * and Lubomír Petrula * Faculty of Civil Engineering, Institute of Water Structures, Brno University of Technology, Veveˇr í 331/95, 602 00 Brno, Czech Republic; [email protected].cz *Correspondence: [email protected].cz (S.K.); [email protected].cz (L.P.); Tel.: +420-541-147-761 (S.K.); +420-541-147-756 (L.P.) Received: 26 June 2020; Accepted: 14 August 2020; Published: 17 August 2020   Abstract: Failures of small dams can pose a serious threat to people and property even if the size of the schemes is relatively low. In many cases, small dams are situated in a cascade along streams, meaning that the failure of the uppermost dam may cause the dams downstream to fail. In this paper, a cascade of three small reservoirs, Lichnov II (14.6 m high), Lichnov III (10 m high), and Pocheˇn (8.5 m high), is the subject of the dam break analyses carried out via various methods such as empirical formulae, analogy, and hydraulic modeling. The dam-break flood routing was simulated using a shallow water flow hydraulic model. The simulations confirm that the attenuation effect of the peak discharge is governed by the flood volume, slope, and morphology of the floodplain and increases with the distance from the breached dam following an approximately exponential trend. When estimating peak discharge, empirical formulae derived for a single dam break should be applied carefully as they may underestimate the peak outflow by up to 10% in the case of a dam cascade. The attenuation volume of small reservoirs is small when compared to the flood volume, meaning that the attenuation of the peak discharge usually varies between 5–10%. Keywords: dam break; dam overtopping; internal erosion; cascade of dams; erodibility 1. Introduction Every dam represents a certain threat due to the possibility of its failure, which may lead to a dam-break flood in the area downstream. Statistics show that the most frequent causes of earth-fill dam failures are overtopping and internal erosion [ 1 – 3 ]. The safety of small dams, ponds, and dry reservoirs is still topical. In the Czech Republic, there are about 20,000 small dams. During extreme floods in 1997, 2002, and 2010, dozens of collapsed small dams were reported [ 4 ]. Therefore, dam break studies, including dam-break flood propagation modeling are also carried out for systems of small dams. An example of the application of dam break modeling to a cascade of small earthen dams is presented in this case study of the ˇ Cižina River in the Czech Republic. For Emergency Action Plans, it is necessary to specify the characteristics of the dam-break flood and the extent of the endangered area. The knowledge concerning such flood-prone areas aids in the development of warning systems and evacuation plans. Various hydraulic and sediment transport models can be used for this purpose [5–8]. Numerous authors have dealt with the issue of dam breach modeling. In the 1970s, Cristofano [ 1 ] collected data from historical failures of earth-fill dams occurring up to 1965, based on which he performed a simulation of the onset of erosion. In the early eighties, the BRDAM model [ 9 ] was used to simulate the erosion of an earth dam in the case of overtopping or internal erosion. Water 2020,12, 2309; doi:10.3390/w12082309 www.mdpi.com/journal/water Water 2020,12, 2309 2 of 21 Subsequent authors [ 10 ] improved the model by specifying the place of origin, simulating the erosion at thelowestpointofthedamcrest. Thesurfaceerosionproceedingonthedownstreamslopewasmodeled. The model used continuity equations, sediment transport equations, and equations for the formation of the dam breach opening via surface erosion. Nogueria [ 11 ] subsequently presented a 1D model using Saint–Venant’s equations to describe unsteady flow and Exner’s, Meyer–Peter, and Muller’s formula for sediment transport. Almost at the same time, Fread [ 12 ] published the numerical models BREACH and Dam Break Forecasting model (DAMBRK), which allowed the simulation of a dam breach via both overtopping and internal erosion, including the effect of lateral slope due to a landslide. That same year, Fread [ 13 , 14 ] introduced the National Weather Service dam-break flood forecasting model (NWS DAMBRK) numerical model. It was based on complex hydrodynamics and erosion processes, and marked the first inclusion of soil strength characteristics within such a model. An improved model, BEED [ 15 ], also simulated erosion processes during a dam break. The existing empirical and numerical procedures were summarized by Wahl [ 3 ], who proposed an inventory for the development and further improvement of numerical models. These works were followed by ˇ R í ha and Danˇeˇcek [ 16 ], Tingsanchali, and Chinnarasri [ 17 ], and others using different approaches to derive a breach outflow hydrograph using different erosion formulae. Later, a group of authors at HR Wallingford [ 18 – 20 ] developed the numerical model named HR BREACH now Embankment Breach (EMBREA), which allows the input of parameter intervals characterizing the material properties of soils, and then performs Monte–Carlo simulations to find the most unfavorable result of the simulation. In 2005, Hanson developed the Simplified Breach Analysis (SIMBA) now Windows Dam Analysis Modules (WINDAM) model for the Agricultural Research Service and Natural Resources Conservation Service in the US [ 21 ]. Currently, the newest version of the program (WINDAM C) is able to model the overtopping and internal erosion of embankment dams, as well as head-cut erosion and the failure of protective layers. In 2012, the AREBA software was developed by [ 22 ]. The software is capable of simulating a dam break via overtopping or internal erosion in homogeneous and composite embankments. The numerical module Breach_Macchione [ 23 ] was adopted to the code Basement in 2012 by Voltz et al. [ 24 ]. The model is based on the application of two-dimensional shallow-water equations to breaching processes occurring in non-cohesive embankments due to overtopping. Wu [ 25 ] developed A Simplified Physically-Based Dam/Levee Breach (DL Breach), which simulates dam and levee breaching via overtopping and internal erosion in homogeneous and composite materials. It can simulate the failure of cover surfaces and can perform Monte–Carlo simulations. Recently, researchers have incorporated simplified modules into River Analysis System (HEC-RAS), A modelling system for Rivers and Channels (MIKE11 DB), WOLF, Rubar 20TS, and other software for modeling water flow in inundation areas [ 5 , 26 – 28 ]. Recent work by King and Simonovic [ 29 ] presents a deterministic Monte–Carlo simulation concept used for the assessment of dam safety management control. In the case of the breaching of dams in a cascade, the resulting dam-break flood depends on numerous factors like the rate of breaching of the upper dam, the ratio of the volumes of the reservoirs in the cascade, and the ability of the lower reservoirs to attenuate the flood wave. The distance between individual reservoirs and the characteristics of the floodplain are also important factors. Therefore, the dam-break flood routing along a cascade of reservoirs may vary according to the failure mode of the uppermost dam and to the current water level and free volume of the reservoirs in the cascade. Usually, the most unfavorable scenario is used for Emergency Action Plans. Only a few studies deal with dam breaks in a cascade of reservoirs [ 30 – 33 ]. Liu et al. [ 31 ] describe a procedure for the numerical simulation of a dam breach due to overtopping in a cascade together with risk analysis. This includes the 1D flow open channel modeling of flood propagation. Stoyanova and Coombs [ 32 ] use the methodology applied in the AREBA software. Both simple two-reservoir and more complex four-reservoir cascade scenarios are dealt with. Cai et al. [ 30 ] applied Bayesian networks for the quantification of the flood risk due to dam breaks in a cascade. Studies [ 30 – 33 ] focused on large dams with the possibility of outflow discharge regulation. Water 2020,12, 2309 3 of 21 The lessons learned during past extreme floods [ 2 , 4 , 34 ] show that on average, about ten small dams fail in the Czech Republic during extreme flood events and that such dams are more vulnerable to breaching than large dams, which is primarily due to their insufficient spillway capacity. As a rule, such schemes have a flood attenuation volume, which is relatively small when compared to the volume of the arriving extreme flood. Their fixed spillways usually have a limited capacity due to the low importance of the dams. This study aims to fill a gap in the research concerning the breaching of small embankment dams in a cascade. It deals with a cascade of three relatively new small embankment dams in the ˇ Cižina River catchment in the Moravian–Silesian Region of the Czech Republic. One of the aims was to demonstrate the effect of the distance between dams on dam-break flood attenuation. In our case study, the two upper reservoirs are quite close to one another, while the lower one is located more remotely downstream along the ˇ Cižina River. A complete set of technical data about the dams was available. The lowest dam, Pocheˇn, has already experienced overtopping twice but it resisted in both cases and did not fail. Based on the results of the analysis, typical dam-break flood routing characteristics are summarized for a cascade of small dams. The method proposed in the study may be used for the development of Emergency Action Plans containing dam break and flood routing analyses for the affected area below a dam. The results of such analyses contain flood arrival times and maps with flood zones where water velocities and depths are included. 2. Case Study 2.1. Study Area The case study covers an area of the ˇ Cižina River catchment in the Moravian–Silesian Region of the Czech Republic (Figure 1) that extends for about 103 km2. Water 2020, 12, x FOR PEER REVIEW 3 of 21 such schemes have a flood attenuation volume, which is relatively small when compared to the volume of the arriving extreme flood. Their fixed spillways usually have a limited capacity due to the low importance of the dams. This study aims to fill a gap in the research concerning the breaching of small embankment dams in a cascade. It deals with a cascade of three relatively new small embankment dams in the Čižina River catchment in the Moravian–Silesian Region of the Czech Republic. One of the aims was to demonstrate the effect of the distance between dams on dam-break flood attenuation. In our case study, the two upper reservoirs are quite close to one another, while the lower one is located more remotely downstream along the Čižina River. A complete set of technical data about the dams was available. The lowest dam, Pocheň, has already experienced overtopping twice but it resisted in both cases and did not fail. Based on the results of the analysis, typical dam-break flood routing characteristics are summarized for a cascade of small dams. The method proposed in the study may be used for the development of Emergency Action Plans containing dam break and flood routing analyses for the affected area below a dam. The results of such analyses contain flood arrival times and maps with flood zones where water velocities and depths are included. 2. Case Study 2.1. Study Area The case study covers an area of the Čižina River catchment in the Moravian–Silesian Region of the Czech Republic (Figure 1) that extends for about 103 km 2 . Figure 1. Location of the area. In the catchment area, three small reservoirs named Lichnov II, Lichnov III, and Pocheň are arranged in a cascade (Figure 2). The cascade was selected because the distance between the dams differs significantly, allowing the demonstration of the impact of distance on dam-break flood attenuation. The two upper reservoirs are quite close to one another, while the lower one (Pocheň) is Figure 1. Location of the area. Water 2020,12, 2309 4 of 21 In the catchment area, three small reservoirs named Lichnov II, Lichnov III, and Pocheˇn are arrangedin a cascade(Figure2). Thecascadewas selected because the distance between the dams differs significantly, allowing the demonstration of the impact of distance on dam-break flood attenuation. The two upper reservoirs are quite close to one another, while the lower one (Pocheˇn) is about 15 km downstream from the Lichnov III dam. A complete set of technical data about the dams was available. The lowest dam, Pocheˇn, has already experienced overtopping twice, but it resisted in both cases and did not fail. Water 2020, 12, x FOR PEER REVIEW 4 of 21 about 15 km downstream from the Lichnov III dam. A complete set of technical data about the dams was available. The lowest dam, Pocheň, has already experienced overtopping twice, but it resisted in both cases and did not fail. Figure 2. Map of the area. 2.2. Description of the Small Dams The main characteristics of the studied dams are described in Table 1. Table 1. Description of the small dams. Basic Characteristic of Dam Lichnov II Lichnov III Pocheň Units Dam height 14.6 10 8.5 (m) Crest width 5 4 4 (m) Upstream slope 2 3 3 (1/X) Downstream slope 2.8 2.6 2.3 (1/X) Material classification Gravel clay Loam gravel Gravel clay (–) The Lichnov II dam is located about 1 km above the village of Lichnov on Tetřeví Stream, which is a tributary of the Čižina River. The dam was built in 2015. The main purpose of the reservoir is to provide flood protection to the villages downstream. The dam is homogenous and is made of sandy to gravelly clay. The maximum height of the dam above the local terrain is 16 m, and the dam crest is 5 m wide. A longitudinal section along the dam crest is shown in Figure 3, while a typical cross-section is displayed in Figure 4. The dam crest elevation at the left abutment is 0.85 m lower than the dam crest due to the presence of an emergency spillway. This place where overtopping may start, and water flows rapidly along the downstream slope was considered to be the potential point of origin of a breach due to overtopping. At the toe of the right abutment, extensive leakage through disintegrated colluvial soil below the base of the dam was identified via technical surveillance during the first trial filling of the reservoir. It is considered that there is a threat of backward erosion piping and a possible breach at this location. Figure 3. Longitudinal section of the Lichnov II dam with indicated breach locations. Figure 2. Map of the area. 2.2. Description of the Small Dams The main characteristics of the studied dams are described in Table 1. Table 1. Description of the small dams. Basic Characteristic of Dam Lichnov II Lichnov III Pocheˇn Units Dam height 14.6 10 8.5 (m) Crest width 5 4 4 (m) Upstream slope 2 3 3 (1/X) Downstream slope 2.8 2.6 2.3 (1/X) Material classification Gravel clay Loam gravel Gravel clay (–) The Lichnov II dam is located about 1 km above the village of Lichnov on Tetˇrev í Stream, which is a tributary of the ˇ Cižina River. The dam was built in 2015. The main purpose of the reservoir is to provide flood protection to the villages downstream. The dam is homogenous and is made of sandy to gravelly clay. The maximum height of the dam above the local terrain is 16 m, and the dam crest is 5 m wide. A longitudinal section along the dam crest is shown in Figure 3, while a typical cross-section is displayed in Figure 4. The dam crest elevation at the left abutment is 0.85 m lower than the dam crest due to the presence of an emergency spillway. This place where overtopping may start, and water flows rapidly along the downstream slope was considered to be the potential point of origin of a breach due to overtopping. At the toe of the right abutment, extensive leakage through disintegrated colluvial soil below the base of the dam was identified via technical surveillance during the first trial filling of the reservoir. It is considered that there is a threat of backward erosion piping and a possible breach at this location. Water 2020,12, 2309 5 of 21 Water 2020, 12, x FOR PEER REVIEW 4 of 21 about 15 km downstream from the Lichnov III dam. A complete set of technical data about the dams was available. The lowest dam, Pocheň, has already experienced overtopping twice, but it resisted in both cases and did not fail. Figure 2. Map of the area. 2.2. Description of the Small Dams The main characteristics of the studied dams are described in Table 1. Table 1. Description of the small dams. Basic Characteristic of Dam Lichnov II Lichnov III Pocheň Units Dam height 14.6 10 8.5 (m) Crest width 5 4 4 (m) Upstream slope 2 3 3 (1/X) Downstream slope 2.8 2.6 2.3 (1/X) Material classification Gravel clay Loam gravel Gravel clay (–) The Lichnov II dam is located about 1 km above the village of Lichnov on Tetřeví Stream, which is a tributary of the Čižina River. The dam was built in 2015. The main purpose of the reservoir is to provide flood protection to the villages downstream. The dam is homogenous and is made of sandy to gravelly clay. The maximum height of the dam above the local terrain is 16 m, and the dam crest is 5 m wide. A longitudinal section along the dam crest is shown in Figure 3, while a typical cross-section is displayed in Figure 4. The dam crest elevation at the left abutment is 0.85 m lower than the dam crest due to the presence of an emergency spillway. This place where overtopping may start, and water flows rapidly along the downstream slope was considered to be the potential point of origin of a breach due to overtopping. At the toe of the right abutment, extensive leakage through disintegrated colluvial soil below the base of the dam was identified via technical surveillance during the first trial filling of the reservoir. It is considered that there is a threat of backward erosion piping and a possible breach at this location. Figure 3. Longitudinal section of the Lichnov II dam with indicated breach locations. Figure 3. Longitudinal section of the Lichnov II dam with indicated breach locations. Water 2020, 12, x FOR PEER REVIEW 5 of 21 Figure 4. Typical cross-section of the Lichnov II dam. The length of Tetřeví Stream between the Lichnov II and Lichnov III reservoirs is relatively small, about 500 m. The valley is relatively steep and narrow and is covered by forest and meadows. The small detention dam Lichnov III is located approximately 400 m upstream of the village of Lichnov. It was built in 2018. Its purpose is to protect the village and to transform incoming flood waves from the Lichnov II dam and from the area adjacent to the reservoir. The dam body is homogeneous. It is made of sandy to gravelly clay and is very similar to that of Lichnov II. The dam height is about 12 m; see Table 1 for its main characteristics and parameters. As a dam-break flood arriving from the Lichnov II dam would be sudden, it is not probable that a gradually evolving piping process would develop, and so only overtopping was considered as a potential failure mode. The place of overtopping was similar to that of Lichnov II, i.e., at the auxiliary crest spillway at the left abutment (Figure 5). Figure 5. Longitudinal section of the Lichnov III dam with indicated possible failure locations. About 600 m downstream of the Lichnov III dam, Tetřeví Stream flows into the Čižina River, which passes through the village of Lichnov. Numerous civil structures such as bridges and culverts are located there, and the area along the 3.5 km long river reach is urbanized with numerous buildings. Further, on the downstream of the village, the Čižina River continues into a region of fields and meadows covered with trees. This section of the Čižina River stretches for about 6 km, after which the river enters the Pocheň reservoir. The small dam named Pocheň is located on the Čižina River about 5 km upstream of its junction with the larger Opava River. The Pocheň dam was built between 1973 and 1975 as a source of water for agricultural activities in nearby areas. During the extreme flood event in the year 1996, the dam was seriously damaged by overtopping but not breached. Due to this damage, reconstruction of the dam body took place, and the spillway capacity was improved over the following years. The dam height is about 9.5 m, and the dam is constructed from local gravelly clay. The geodetic survey indicated the lowest crest elevation is approximately in the middle of the crest length (see Figure 6). Due to the short duration of the extreme load, a progressive piping failure would not develop, and so only the overtopping failure mode was taken into account. Figure 4. Typical cross-section of the Lichnov II dam. The length of Tetˇrev í Stream between the Lichnov II and Lichnov III reservoirs is relatively small, about 500 m. The valley is relatively steep and narrow and is covered by forest and meadows. The small detention dam Lichnov III is located approximately 400 m upstream of the village of Lichnov. It was built in 2018. Its purpose is to protect the village and to transform incoming flood waves from the Lichnov II dam and from the area adjacent to the reservoir. The dam body is homogeneous. It is made of sandy to gravelly clay and is very similar to that of Lichnov II. The dam height is about 12 m; see Table 1for its main characteristics and parameters. As a dam-break flood arriving from the Lichnov II dam would be sudden, it is not probable that a gradually evolving piping process would develop, and so only overtopping was considered as a potential failure mode. The place of overtopping was similar to that of Lichnov II, i.e., at the auxiliary crest spillway at the left abutment (Figure 5). Water 2020, 12, x FOR PEER REVIEW 5 of 21 Figure 4. Typical cross-section of the Lichnov II dam. The length of Tetřeví Stream between the Lichnov II and Lichnov III reservoirs is relatively small, about 500 m. The valley is relatively steep and narrow and is covered by forest and meadows. The small detention dam Lichnov III is located approximately 400 m upstream of the village of Lichnov. It was built in 2018. Its purpose is to protect the village and to transform incoming flood waves from the Lichnov II dam and from the area adjacent to the reservoir. The dam body is homogeneous. It is made of sandy to gravelly clay and is very similar to that of Lichnov II. The dam height is about 12 m; see Table 1 for its main characteristics and parameters. As a dam-break flood arriving from the Lichnov II dam would be sudden, it is not probable that a gradually evolving piping process would develop, and so only overtopping was considered as a potential failure mode. The place of overtopping was similar to that of Lichnov II, i.e., at the auxiliary crest spillway at the left abutment (Figure 5). Figure 5. Longitudinal section of the Lichnov III dam with indicated possible failure locations. About 600 m downstream of the Lichnov III dam, Tetřeví Stream flows into the Čižina River, which passes through the village of Lichnov. Numerous civil structures such as bridges and culverts are located there, and the area along the 3.5 km long river reach is urbanized with numerous buildings. Further, on the downstream of the village, the Čižina River continues into a region of fields and meadows covered with trees. This section of the Čižina River stretches for about 6 km, after which the river enters the Pocheň reservoir. The small dam named Pocheň is located on the Čižina River about 5 km upstream of its junction with the larger Opava River. The Pocheň dam was built between 1973 and 1975 as a source of water for agricultural activities in nearby areas. During the extreme flood event in the year 1996, the dam was seriously damaged by overtopping but not breached. Due to this damage, reconstruction of the dam body took place, and the spillway capacity was improved over the following years. The dam height is about 9.5 m, and the dam is constructed from local gravelly clay. The geodetic survey indicated the lowest crest elevation is approximately in the middle of the crest length (see Figure 6). Due to the short duration of the extreme load, a progressive piping failure would not develop, and so only the overtopping failure mode was taken into account. Figure 5. Longitudinal section of the Lichnov III dam with indicated possible failure locations. About 600 m downstream of the Lichnov III dam, Tetˇrev í Stream flows into the ˇ Cižina River, which passes through the village of Lichnov. Numerous civil structures such as bridges and culverts are located there, and the area along the 3.5 km long river reach is urbanized with numerous buildings. Further, on the downstream of the village, the ˇ Cižina River continues into a region of fields and meadows covered with trees. This section of the ˇ Cižina River stretches for about 6 km, after which the river enters the Pocheˇn reservoir. The small dam named Pocheˇn is located on the ˇ Cižina River about 5 km upstream of its junction with the larger Opava River. The Pocheˇn dam was built between 1973 and 1975 as a source of water for agricultural activities in nearby areas. During the extreme flood event in the year 1996, the dam was seriously damaged by overtopping but not breached. Due to this damage, reconstruction of the dam Water 2020,12, 2309 6 of 21 body took place, and the spillway capacity was improved over the following years. The dam height is about 9.5 m, and the dam is constructed from local gravelly clay. The geodetic survey indicated the lowest crest elevation is approximately in the middle of the crest length (see Figure 6). Due to the short duration of the extreme load, a progressive piping failure would not develop, and so only the overtopping failure mode was taken into account. Water 2020, 12, x FOR PEER REVIEW 6 of 21 Figure 6. Longitudinal section of the Pocheň dam with an indicated place of overtopping. Downstream of the Pocheň reservoir, the Čižina River passes through an area of forest, flows below several road and railway bridges, and then enters the Opava River close to the village of Skrochovice. 3. Methods Both empirical formulae and comparison with real incidents were applied for the first preliminary estimates of the dam break peak discharges. The obtained values provide an idea about the range of the peak outflows and provide a general framework for numerical modeling. Due to the lack of relevant incidents and reliable data from real breached dams, these equations suffer from a great deal of inaccuracy and are mainly useful for the preliminary estimation of parameters, which can then be determined more precisely by numerical models. The dam break simulation was carried out using the numerical model of breaching due to overtopping and internal erosion (the latter phenomenon at the uppermost dam only), which is normally applied to individual dams. The peak discharges were compared with the results of empirical formulae and with real cases involving the failure of small dams. Two-dimensional shallow-water flow hydraulic models were applied for the simulation of the dam-break flood propagation into the areas between the dams. The flow and erosion parameters were searched via Monte–Carlo random sampling to achieve the highest peak outflow discharge below the dam. 3.1. Empirical Formulae In order to obtain an initial idea about the peak discharges, published empirical formulae were applied. This approach is the simplest but least accurate way of predicting dam breach parameters. In dam-break simulations, direct calibration of models is usually impossible. One of the methods of verifying numerical models is to compare the modeling results with the outputs of retrospective studies of real incidents and failures. Unlike, e.g., hydrological floods, there is a lack of dam-break calibration data worldwide due to a lack of relevant incidents and reliable data from real breached dams. The empirical formulae discussed in this study (Table 2) are based on such real cases. It is true that statistical analysis is difficult due to the heterogeneity, inaccuracy, and scatter of the dam break data when subjected to regression analysis. However, the published empirical formulae and the analogy method are frequently used in dam break analysis for the comparison of real incidents with the results of numerical analysis. One of the aims of this study was to demonstrate the difference between a single dam break and a cascade when applying empirical formulae. Table 2. Summary of formulae used for estimating the peak breach outflow Qb. Author, Reference Formulae for Qb *) (m3/s) Figure 6. Longitudinal section of the Pocheˇn dam with an indicated place of overtopping. Downstream of the Pocheˇn reservoir, the ˇ Cižina River passes through an area of forest, flows below several road and railway bridges, and then enters the Opava River close to the village of Skrochovice. 3. Methods Both empirical formulae and comparison with real incidents were applied for the first preliminary estimates of the dam break peak discharges. The obtained values provide an idea about the range of the peak outflows and provide a general framework for numerical modeling. Due to the lack of relevant incidents and reliable data from real breached dams, these equations suffer from a great deal of inaccuracy and are mainly useful for the preliminary estimation of parameters, which can then be determined more precisely by numerical models. The dam break simulation was carried out using the numerical model of breaching due to overtopping and internal erosion (the latter phenomenon at the uppermost dam only), which is normally applied to individual dams. The peak discharges were compared with the results of empirical formulae and with real cases involving the failure of small dams. Two-dimensional shallow-water flow hydraulic models were applied for the simulation of the dam-break flood propagation into the areas between the dams. The flow and erosion parameters were searched via Monte–Carlo random sampling to achieve the highest peak outflow discharge below the dam. 3.1. Empirical Formulae In order to obtain an initial idea about the peak discharges, published empirical formulae were applied. This approach is the simplest but least accurate way of predicting dam breach parameters. In dam-break simulations, direct calibration of models is usually impossible. One of the methods of verifying numerical models is to compare the modeling results with the outputs of retrospective studies of real incidents and failures. Unlike, e.g., hydrological floods, there is a lack of dam-break calibration data worldwide due to a lack of relevant incidents and reliable data from real breached dams. The empirical formulae discussed in this study (Table 2) are based on such real cases. It is true that statistical analysis is difficult due to the heterogeneity, inaccuracy, and scatter of the dam break data when subjected to regression analysis. However, the published empirical formulae and the analogy method are frequently used in dam break analysis for the comparison of real incidents with the results of numerical analysis. Water 2020,12, 2309 7 of 21 Table 2. Summary of formulae used for estimating the peak breach outflow Qb. Author, Reference Formulae for Qb*) (m3/s) Holomek, ˇ Ríha (2000) [35]Qb=mp2ghbHk1.5 +0.4B−b hdH1.5 ki The following breach dimensions can be used: USACE (1980) [36]b=3hd&B=5hd ICOLD (1974) [37]b=1.6hd&B=3.6hd Šimek (1988) [38]b=1.2hd&B=3.2hd Froehlich (1995a, 1995b) [39,40]Qb=0.607Vw0.295hw1.24 Webby (1996) [41]Qb=0.0443 ·g0.5V0.367 Ah1.40 w Bulletin 111 (1998) [42]Qb=0.07(ghd)0.5h2 dVb h3 d0,5 Vischer, Hager (1998) [43]Qb=1.25·10−2Ha hdqgB2 ahd3 *) An explanation of the variables can be seen in the section “List of symbols”. One of the aims of this study was to demonstrate the difference between a single dam break and a cascade when applying empirical formulae. 3.2. The Analogy Method A more reliable peak breach outflow may be derived via the comparison of data from real incidents at dams with similar characteristics (dam type, height, reservoir volume) to the studied dam. The data may be obtained from publications and extensive dam failure databases (both national and international) [ 2 , 3 , 34 , 42 , 44 , 45 ]. In Table 3, dams similar to those studied are mentioned together with basic relevant characteristics (see also Chapter 2). Table 3. Summary of the characteristics of the real breached dams and the studied dams. Cases Dam Name Dam Height (m) Reservoir Volume (ths. m3)Crest Width (m) Breach Width (m) Peak Outflow (m3/s) Real cases Metly 8.50 1930 4.0–6.9 35.0 +15.0 *) 550 Velký Bˇelˇcický6.70 1063 7.4 42.0 600 Buffalo Creek 14.00 483.5 107.0 148.0 1416 Kelly Barnes 11.50 505 6.1 6.7 680 Studied dams Lichnov II 14.60 400.6 5.0 - - Lichnov III 10.00 117.9 4.0 - - Pocheˇn **) 8.50 780 4.5 - 163 *) Two breach openings. **) The dam was not completely breached. 3.3. Embankment Dam Breach Modeling A simple parametric model based on hydraulic and transport modules was used for the simulation. The tailor-made code was developed in MATLAB software to allow the dam breaching process to be modified easily according to local conditions and the failure mode. The main output of the dam break analysis is the outflow hydrograph Q(t), which results from the failure. The following steps were performed before the modeling took place. A careful investigation was performed with regard to the dam type and structure (material, emergency spillway position, dam surface, type of pavement on the dam crest, etc.), as well as the local geological conditions (subbase material, material discontinuities). The probable place of failure initiation was located; for an overtopping failure, the place with the minimum dam crest or auxiliary spillway height was chosen. For a piping failure, the location with the weakest subbase providing the highest seepage was chosen. 1. The shape of the dam break opening was rectangular for overtopping and circular for piping. 2. The flood hydrograph Q in (t) entering the reservoir was provided by Czech Hydrometeorological Institute. Water 2020,12, 2309 8 of 21 3. The bathymetry of the reservoirs was taken from a geodetic survey of the reservoir area. 4. The initial reservoir water level H 0 (t) was taken from the dam operation manual. It corresponds to the live storage water level. The following relation between the reservoir inflow, outflow, and a change in water level holds: dV dt=Qin(t)−Qb(H)−Qf(H)(1) where Vis the reservoir water volume (m 3 ), tis time (s), Q in is the reservoir inflow (m 3 /s), Q b is the breach outflow (m 3 /s) (in the following text, Q bo denotes overtopping discharge and Q bp piping discharge), Q f is the outflow through appurtenant works (outlets, spillways, etc.) in (m 3 /s), and His the elevation of the water level (m a. s. l.). The outflow through the appurtenant works governed by the reservoir water level was calculated using equations taken from standard dam hydraulics [ 46 ]. The equations used and related coefficients were determined according to the arrangement of the hydraulic structures. 3.3.1. Overtopping In the case of dam overtopping, the breach was approximated using a 1D model. The breach started at the lowest point of the dam crest, where the first overtopping occurred. The overtopping width along the dam crest was suggested as an initial breach width, b 0 . The shape of the breach opening was approximated by a rectangle, and the erosion progress was applied in both the downward and the lateral direction. The water moving along the downstream slope was assumed to exhibit quasi-steady uniform flow [ 3 , 13 , 45 ] with no submergence from the downstream side. The resistance of the dam body against the surface erosion was evaluated with respect to the water flow velocity at the downstream slope, with non-scouring velocity being used for the evaluation. Parallel uniform gradual backward erosion of the downstream slope was assumed [ 13 ]. The angle of the downstream slope and the width of the dam crest were set as input parameters, along with hydraulic parameters such as the discharge coefficient m, Manning roughness n, and the non-scouring velocity v ns and erosion parameters of the soil α1and α2, together with initial conditions (see below). The breach outflow during overtopping Q bO was calculated using an overflow formula for a rectangular broad crested weir [47]: QbO =mb p2g(H−Z)1.5 (2) where mis the weir coefficient, bis the width of the breach opening (m), gis the acceleration due to gravity (m/s2), and Zis the breach bottom elevation (m a. s. l.). Using the overtopping discharge obtained from Equation (2), the water depth h f and velocity v f along the slope was derived from the Chezy equation [47]: hf=      Qb0n bpsinβ      0.6 (3) vf=psinβ nh0.67 f(4) where nis the roughness of the downstream slope, and βis the downstream slope angle. At the moment, when the flow velocity along the slope exceeds the non-scouring velocity v ns (the value is an input parameter), erosion occurs and the breach starts to increase in width and depth [34]: dZ dt=−α1vf, for vf>vns (5) Water 2020,12, 2309 9 of 21 db dt=α2vf, for vf>vns (6) where α1and α2are erosion parameters. Initial conditions specify the reservoir water level H 0 , the lowest dam crest elevation Z 0 , and initial overtopping width b 0 at the beginning of the solution (t =t 0 ). The set of Equations (1) to (6) was solved numerically using the Newton method with a small-time step ∆t=1 s. 3.3.2. Piping For the piping model, a circular opening with a constant leakage pipe diameter and constant shear stress and hydraulic gradient along the pipe were assumed [ 19 – 22 , 28 , 48 ]. The input parameters were the length of the pipe L, the initial reservoir water level H 0 , the elevation of the pipe outflow H outflow , the critical shear stress along the walls of the pipe τc , the soil erodibility C e , the initial pipe diameter D0, and absolute roughness ∆. The outflow through the pipe Q bP and average water velocity v pip in the pipe were determined using pipe hydraulics [47]: QbP =1 qα+λL D Sq2gH−Houtflow(7) vpip =QbP S(8) where α is the kinetic energy (Coriolis) coefficient, Lis the length of the pipe (m), Dis the pipe diameter (m), λ is the friction loss coefficient, Sis the pipe cross-sectional area (m 2 ), and H outflow is the pipe outflow elevation (m a. s. l.). Using the water velocity in the pipe, the average hydraulic gradient along the pipe I E and shear stress τacting on the pipe walls were obtained: IE=λ1 D v2 pip 2g(9) τ=ρgD 4IE(10) where ρ is the water density. When the critical shear stress τc on the pipe wall was exceeded, erosion and an increase in the pipe diameter initiated. The rate of erosion . ε and the change in pipe radius drwere determined according to equations suggested by Wan and Fell [48]: . ε=Ce(τ−τc)(11) dr= . ε ρb dt(12) where . ε is the rate of erosion (kg/s/m 2 ), C e is the coefficient of soil erosion (s/m), ρb is the bulk density of the eroded soil (kg/m3), and r=D/2 is the pipe radius. The initial reservoir water level H 0 and initial pipe diameter D 0 are the initial conditions. As in the case of overtopping, the solution of Equations (1) and (7) to (12) was carried out using the Newton method with a very small-time step of ∆t=1 s. During the solution, the relation between the pipe diameter and the thickness of the roof of the conduit (overburden) was checked. After the pipe diameter exceeded the overburden thickness, it was assumed that the soil layer above the opening collapsed and the process continued similarly as in the case of overtopping. The width of the breach opening corresponded to the pipe diameter at the instant of the overburden collapse, while the breach bottom was assumed to be at the top of the seepage pipe at the instant of the collapse. Water 2020,12, 2309 16 of 21 the dam-break flood simulation was terminated at the profile where the peak discharge induced by the dam break dropped below the 100-year hydrological flood discharge. A summary of the simulations is shown in Figure 13, where the flood hydrographs are plotted starting with the initial hydrological flood that entered the Lichnov II reservoir (on the left) and induced the dam-break flood progression through the system of three reservoirs along the Tetˇrev í Stream and the ˇ Cižina River. In Table 6, the dam breach peak discharges obtained from simulations are compared with the results of the comparisons with real incidents and with data from empirical equations. The comparison indicates that the use of comparisons and empirical equations for a dam break in a cascade may be confusing, especially if the breached dams are very close to one another. These methods may be used for the comparison of the breach peak discharge failures at single dams, but with some caution due to the significant uncertainty involved. Water 2020, 12, x FOR PEER REVIEW 16 of 21 Figure 13. Progression of the dam-break flood in selected critical profiles in the area. Figure 14 shows a map depicting the zone inundated by a dam-break flood corresponding to the worst-case scenario identified by Monte–Carlo optimization. The map displays the maximum water depth reached during the event. Figure 14. Map of the maximum water depths on the map base, depth in meters. 5. Discussion The simulation of a dam break in a cascade consists of two basic procedures, namely single dambreak analysis (Sections 3.1 to 3.3), and the analysis of the dam-break-induced flood between the dams (Section 3.4). Basically, two failure modes—overtopping and piping—should be simulated at the uppermost dam. These modes are completely independent; they occur for different reasons at different places in the dam body, and at different times (Figures 3 and 8). The initiation of piping is governed by the threshold hydraulic gradient when critical shear stress is exceeded. Therefore, piping usually starts at the rising limb of the flood hydrograph and progresses during the arrival of the flood peak at the maximum water level in the reservoir. This may happen before dam crest overtopping occurs. At the moment when the “roof” of the pipe becomes thin (here this is assumed to be when it is equal to the “pipe’s” diameter), the overburden collapses, and the simulation switches to the overtopping mode at the location of the original piping. This may result in a temporary drop in outflow during piping. Figure 13. Progression of the dam-break flood in selected critical profiles in the area. Figure 14 shows a map depicting the zone inundated by a dam-break flood corresponding to the worst-case scenario identified by Monte–Carlo optimization. The map displays the maximum water depth reached during the event. Water 2020, 12, x FOR PEER REVIEW 16 of 21 Figure 13. Progression of the dam-break flood in selected critical profiles in the area. Figure 14 shows a map depicting the zone inundated by a dam-break flood corresponding to the worst-case scenario identified by Monte–Carlo optimization. The map displays the maximum water depth reached during the event. Figure 14. Map of the maximum water depths on the map base, depth in meters. 5. Discussion The simulation of a dam break in a cascade consists of two basic procedures, namely single dambreak analysis (Sections 3.1 to 3.3), and the analysis of the dam-break-induced flood between the dams (Section 3.4). Basically, two failure modes—overtopping and piping—should be simulated at the uppermost dam. These modes are completely independent; they occur for different reasons at different places in the dam body, and at different times (Figures 3 and 8). The initiation of piping is governed by the threshold hydraulic gradient when critical shear stress is exceeded. Therefore, piping usually starts at the rising limb of the flood hydrograph and progresses during the arrival of the flood peak at the maximum water level in the reservoir. This may happen before dam crest overtopping occurs. At the moment when the “roof” of the pipe becomes thin (here this is assumed to be when it is equal to the “pipe’s” diameter), the overburden collapses, and the simulation switches to the overtopping mode at the location of the original piping. This may result in a temporary drop in outflow during piping. Figure 14. Map of the maximum water depths on the map base, depth in meters. Water 2020,12, 2309 17 of 21 5. Discussion The simulation of a dam break in a cascade consists of two basic procedures, namely single dam-break analysis (Sections 3.1–3.3), and the analysis of the dam-break-induced flood between the dams (Section 3.4). Basically, two failure modes—overtopping and piping—should be simulated at the uppermost dam. These modes are completely independent; they occur for different reasons at different places in the dam body, and at different times (Figures 3and 8). The initiation of piping is governed by the threshold hydraulic gradient when critical shear stress is exceeded. Therefore, piping usually starts at the rising limb of the flood hydrograph and progresses during the arrival of the flood peak at the maximum water level in the reservoir. This may happen before dam crest overtopping occurs. At the moment when the “roof” of the pipe becomes thin (here this is assumed to be when it is equal to the “pipe’s” diameter), the overburden collapses, and the simulation switches to the overtopping mode at the location of the original piping. This may result in a temporary drop in outflow during piping. The duration of the piping failure until the maximum breach opening developed was more than 1 h (Figure 8). The worst-case scenario of an overtopping failure occurred at the receding limb of the inflow hydrograph, at the moment when the reservoir was completely full, and critical velocity along the downstream slope was reached. The optimization procedure showed that the outflow from a reservoir is predominantly governed by the size of the breach opening, and the reservoir water level in relation to the base of the breach opening. Thus, the peak outflow is limited by the location of the first overtopping, the morphology of the dam profile, and the erodibility of the dam material. With large dams, the impact of the breaching of an upper dam on the failure risk of a downstream one is usually governed by the ratio between the released volume from the upper reservoir and the flood attenuation volume of the lower one. In the case of small dams, the flood attenuation volume is relatively small when compared to the volume of the arriving extreme flood, with the ratio varying between 1:10 and 1:25. The attenuation of peak discharge usually varies between 5–10%. Therefore, the attenuation effect of small reservoirs is not so important for dam break analysis in the case of a cascade of small dams. The capacity of the bottom outlets at small dams usually has a negligible effect on the flood passing through the reservoir and may be neglected. At lower-lying small dams in a cascade where the reservoir only provides a limited attenuation effect, the rising limb is steep (Figure 9), and the increase in the reservoir water level is quite rapid (5–10 min). In the case of the scheme investigated here, the overtopping and breaching started shortly after the rapid increase in the reservoir water level. About 1 h was needed for the initiation of the piping, and, therefore, the expected failure mode was primarily overtopping. Therefore, the piping failure mode was not taken into account for the downstream dams. The results of the presented study prove that flood attenuation between reservoirs located in close proximity to one another is practically negligible and that simplifications can be applied (Figure 9) [ 52 ]. Flood attenuation increases with the distance from the breached dam and follows an approximately exponential trend. Our results (Figure 13) correspond well with the results of parametric simulations published in [ 53 ], where the longitudinal slope, the dam-break flood volume, the width, and the “roughness” of the channel between the reservoirs are taken into account. The empirical formulae tested in this study were derived using real collapses of single dams. When used for peak discharge estimation, empirical formulae derived for a single dam break should be applied carefully, as they may underestimate the peak outflow by up to 10% in the case of a dam cascade. 6. Conclusions The paper describes the analysis of a dam break process in a cascade of small dams, including the simulation of flood routing along the adjacent streams. The aim is to fill a gap in the existing research concerning the breaching of small embankment dams in a cascade. This is a specific type of event Water 2020,12, 2309 18 of 21 which may differ from a dam break in a cascade of large dams, where the reservoir attenuation volume may play a significant role. The case study concerns an area along Tetˇrev í Stream and the ˇ Cižina River in the Moravian-Silesian Region of the Czech Republic. The area contains three small dams: Lichnov II, Lichnov III, and Pocheˇn. A simplified parametric model for piping and overtopping erosion with parameter optimization was used for the dam break simulations, while a standard 2D unsteady shallow-water flow model was applied for the flood in the areas downstream of each reservoir. As no calibration data were available, the comparison of the simulation results was carried out with the use of empirical formulae and by means of comparison with real small dam failures. It is recommended that the solution for a dam cascade be divided into separate sequences corresponding to the dam breaching process and flood routing along the streams downstream of the dams. The location of the first overtopping and piping event should be identified based on the dam geometry, the arrangement of appurtenant works (namely the auxiliary spillway), observed leaks, etc. It is desirable to carry out at least a simplified optimization procedure, which guarantees that a combination of parameters providing the worst possible realistic scenario for the simulated event is obtained. When applying empirical formulae or comparison with real dam failures to a dam cascade, one should be aware that these methods may underestimate the peak discharge by up to 10%. All flood hydrographs (Figure 13) showing arrival times and flood zones with water depth (Figure 14) provide important information for warning and emergency plans, and for evacuation procedures. Further research should focus on the comprehensive probabilistic analysis of dam failures and flood routing between reservoirs in order to create a basis for the development of risk maps for endangered areas. Author Contributions: Conceptualization, S.K. and L.P.; methodology, J.ˇ R., L.P.; software, S.K., L.P. and J.ˇ R.; validation, S.K.; formal analysis, S.K.; investigation, S.K., L.P. and J.ˇ R.; resources, J.ˇ R.; data curation, J.ˇ R.; writing—original draft preparation, L.P. and S.K.; writing—review and editing, S.K. and J.ˇ R.; visualization, S.K.; supervision, J.ˇ R.; project administration, J. ˇ R.; funding acquisition, S.K., L.P. and J.ˇ R. All authors have read and agreed to the published version of the manuscript. Funding: “This research was funded by Determination of uncertainties of selected hydraulic problems, grant number FAST-J-20-6315” and Probability assessment of internal instability in earth structures and in hydraulic structures’ foundations, grant number FAST-S-19-5714. Conflicts of Interest: The authors declare no conflict of interest. List of Symbols Bwidth of the breach opening in the crest (m) bwidth of the breach opening (m) b0initial overtopping width (m) bspill overflow width at the auxiliary spillway/lowest point of the dam crest (m) Baaverage breach width (m) BLdam bottom elevation (m a. s. l.) Cecoefficient of soil erosion (s/m) CLdam crest elevation Dpipe diameter (m) D0initial pipe diameter (m) gacceleration due to gravity (m/s2) Helevation of water level (m a. s. l.) H0initial reservoir water level (m a. s. l.) Hacharacteristic dimension of the reservoir (-) Hkwater depth in the reservoir at the time of peak inflow (m) Haux auxiliary spillway crest elevation (m a. s. l.) hddam height (m) hfwater depth along the slope (m) Water 2020,12, 2309 19 of 21 Houtflow elevation of the pipe outflow (m a. s. l.) HSV bottom outlet axis elevation (m a. s. l.) hwhydraulic depth in the breach (m) IEaverage energy gradient along the pipe (-) Kwelevation of water level (m a. s. l.) Llength of the pipe (m) mweir coefficient (-) nManning roughness coefficient of the downstream slope (s/m1/3) Qbbreach outflow (m3/s) QbO breach outflow during overtopping (m3/s) QbP outflow via the pipe (m3/s) Qfoutflow through appurtenant works (outlets, spillways, etc.) (m3/s) Qin reservoir inflow (m3/s) rpipe radius (m) Spipe cross-sectional area (m2) ttime (s) Vreservoir water volume (m3) VA volume in the reservoir under the lowest point of the breach at the time failure starts (m3) Vbreservoir water volume (m3) vfvelocity along the slope (m/s) vns non-scouring velocity (m/s) vpip average water velocity in the pipe (m/s) Vwvolume in the reservoir for the elevation of water level Kw(m3) Zbreach bottom elevation (m a. s. l.) 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