Hydro-Mechanical Analysis of Breach Processes Due to Levee Failure

UNIVERSITÉ D’AIX-MARSEILLE

ÉCOLE DOCTORALE SCIENCE POUR L’INGÉNIEUR: MÉCANIQUE, PHYSIQUE, MICRO ET NANOÉLECTRONIQUE

THÈSE

HYDRO-MECHANICAL ANALYSIS OF BREACH PROCESSES DUE TO LEVEE FAILURE

PRÉPARÉE À IRSTEA D’AIX-EN-PROVENCE

POUR OBTENIR LE GRADE DE DOCTEUR DE L’UNIVERSITÉ D’AIX-MARSEILLE

SPÉCIALITÉ MÉCANIQUE DES SOLIDES

PAR LIU ZHENZHEN

SOUTENUE PUBLIQUEMENT LE 03 JUILLET 2015 DEVANT LE JURY COMPOSÉ DE

Frédéric Golay Rapporteur MCF, Université du Sud Toulon Var Didier Marot Rapporteur PR, IUT de Saint-Nazaire Fabien Anselmet Examinateur PR, Ecole Centrale de Marseille Said EI Youssoufi Examinateur PR, Université de Montpellier Stéphane Bonelli Directeur de thèse DR, IRSTEA, Aix en Provence

Introduction

Floods can have serious consequence in floodplains. Worldwide, many areas can become flooded or are at risk, and numerous people risk being displaced. Determining flood risks is an effective method of ensuring safety against inundation. The accurate analysis of the failure of flood protection structures is the first step in mapping the resulting inundation in a floodplain. Thus, hydro-mechanical analyses of breach processes due to the failure of embankment dams and levees are conducted. As one of the main cause of the serious failure of embankment dams and levees, piping (or the internal erosion of soil) failure process is simulated in Chapter 2. The pipe flow with erosion mechanism is employed into the pipe enlargement model for embankment dams and levees. The hydraulic head variation in the upstream, the trail water conditions in the downstream, the collapse of the pipe top and the transition to a breach are taken into account in the pipe enlargement model. Once a breach is formed, it begins to enlarge downward and laterally. The hydraulic erosion in the breach and episodic retreat of the headcut are considered as controlling mechanisms during breach widening processes. A simply headcut migration model based on the soil tensile strength is presented in Chapter 3 to simulate the critical length of the headcut. Input parameters of this model are: the breach geometry, the soil properties and the flow situations in the breach. Heights of embankment dams ranging from 2 to 12 meters are used to calculate the critical lengths of the headcut under different soil tensile strengths ranging from 0 to 18 kPa. This analytical model is verified by the traditional side slope stability analysis method with three typical embankment dam scales (2, 4 and 6 meters high) in Chapter 4. Due to quick variations of flow streamlines in the breach, distributions of shear stress on the boundary of the breach are complex with the effects of the secondary flow. To accurate estimate erosion processes, a simple numerical model is proposed to calculate averaged lateral and bottom shear stresses and the depth-averaged velocity in Chapter 5. It accounts

for turbulent effects via the dimensionless eddy viscosity coefficient  , and secondary flow effects via the dimensionless secondary flow parameters k1 and k2 . A large-scale test is used to validate our numerical model in Chapter 6. The comparisons with measured data are with good agreements.

Contents

Chapter 1. Bibliography 1.1. Floods and flood risk ...... 3 1.1.1. Floods in China ...... 3 1.1.2. Floods in France ...... 12 1.2. Breach models of embankment dam and levee failures ...... 15 1.2.1. Introduce...... 15 1.2.2. Experimental breach models ...... 16 1.2.3. Physical breach models ...... 20 1.3. Laboratory and field dam breach tests ...... 35 1.3.1. IMPACT project in the EU (from 2001 to 2004) ...... 35 1.3.2. Overtopping research in the US ...... 37 1.3.3. Headcut migration tests ...... 42 1.4. Conclusions ...... 46

Chapter 2. Pipe enlargement processes 2.1. Literature review of the failure of embankment dams and levees due to internal erosion and piping ...... 49 2.1.1. Introduction ...... 49 2.1.2. Initiation and continuation of erosion ...... 49 2.1.3. Pipe enlargement ...... 50 2.1.4. Formation of breach ...... 54 2.1.5. Breach widening ...... 55 2.2. Piping flow with erosion ...... 55 2.2.1. General introduction of pipe flow ...... 55 2.2.2. Soil surface erosion by tangential flow ...... 56 2.2.3. Pipe flow with erosion ...... 57

2.3. Soil erodibility ...... 58 2.3.1. Soil erodibility tests ...... 58 2.3.2. Erosion rate coefficient ...... 60 2.3.3. Critical shear stress ...... 62 2.4. Model for pipe enlargement ...... 63 2.4.1. Model description ...... 63 2.4.2. Model calculation ...... 66 2.4.3. Computational algorithm ...... 69

2.5. Conclusions...... 69

Chapter 3. Breach widening processes 3.1. Literature review of the breach widening processes ...... 73 3.1.1. The formation of a breach ...... 73 3.1.2. Breach enlargement ...... 73 3.1.3. Final breach ...... 80 3.2. The breach width expression ...... 82 3.2.1. Simplifications of breach geometry ...... 82 3.2.2. Calculation of the breach width ...... 84 3.2.3. General classification of breach widening ...... 84 3.2.4. The knowledge of the two variables ...... 85 3.3. Headcut migration ...... 87 3.3.1. Failure modes ...... 87 3.3.2. Sketch of a headcut ...... 89 3.3.3. Failure criterions ...... 90 3.3.4. Soil shear strength ...... 90 3.3.5. Bending failure calculations ...... 94 3.3.6. Shear failure calculations ...... 103 3.4. Analytical results and discussion...... 109 3.4.1. The specific soil weight ...... 110 3.4.2. The water head in the breach ...... 110 3.4.3. The ratio of the depth of erosion to the water head ...... 110 3.4.4. Infiltration coefficient ...... 111 3.4.5. Analytical results ...... 115 3.5. Conclusions...... 120

Chapter 4. Verification of headcut migration by the limit equilibrium method 4.1. Limit equilibrium method ...... 123 4.1.1. Introduce...... 123 4.1.2. The theory of LEM ...... 123 4.1.3. Failure criterion ...... 125 4.1.4. Spencer method ...... 128 4.2. Model created in Slope/W ...... 130 4.2.1. Definition of geometry of levee and eroded notch ...... 130 4.2.2. Soil properties input into SLOPE/W ...... 131 4.2.3. Determination of the critical length of the headcut ...... 132 4.2.4. Mathematical model verification ...... 134 4.3. Results and discussions ...... 136 4.4. Conclusions ...... 136

Chapter 5. Shear stress distributions in the breach 5.1. Literature review of flow conditions and boundary shear stress distributions in the breach ...... 141 5.1.1. Experiments in rectangular open channels ...... 141 5.1.2. Flow conditions in the breach ...... 144 5.2. Shear stress calculations ...... 145 5.2.1. Depth-averaged Navier-Stokes equation ...... 145 5.2.2. The bed shear stress...... 146 5.2.3. The vertical depth-averaged Reynolds stress ...... 146 5.2.4. The secondary flow ...... 147 5.2.5. The streamwise velocity variation...... 148 5.2.6. Re-formulation of the depth-integrated equation ...... 148 5.2.7. Analytical solutions ...... 149 5.3. Analytical results and discussion ...... 151 5.3.1. Bottom shear stresses ...... 151 5.3.2. Lateral shear stresses ...... 151 5.3.3. The streamwise depth-averaged velocity ...... 152 5.3.4. Curves...... 152

III

5.4. Conclusions...... 163

Chapter 6. Validation by tests 6.1. Presentation of the test ...... 167 6.1.1. Test introduce ...... 167 6.1.2. Test setup ...... 167 6.1.3. Test procedure ...... 171 6.1.4. Results ...... 173 6.2. Test simulation ...... 178 6.2.1. The pipe enlargement process ...... 178 6.2.2. The breach widening processes ...... 178 6.3. Results and discussion ...... 180 6.3.1. Results ...... 180 6.3.2. Discussion ...... 181 6.4. Conclusions...... 181

Conclusions 183

References 185

Appendix A A.1. The security and protection levels of levees ...... 193 A.2. The classification of security level in China ...... 193 A.3. Flood frequency ...... 193

Appendix B B.1 Simulation setup ...... 195 B.2 Boundary conditions...... 197 B.3 Results and discussions ...... 198 B.4 Conclusions...... 198

CHAPTER 1

Bibliography

Floods can have serious consequence in floodplains. Worldwide, many areas can become flooded or are at risk, and numerous people risk being displaced. In developing countries, more people are affected by floods. Determining flood risks is an effective method of ensuring safety against inundation. The accurate analysis of the failure of flood protection structures is the first step in mapping the resulting inundation in a floodplain. This bibliography first introduces floods and flood risks in China and in France. The statuses of embankment dams and levees are also briefly mentioned. Then, breach models of embankment dam and levee failure developed in recent years are summarized, and simple comparisons of these models are performed. To better understand the failure processes and failure mechanisms, the last section presents some recently conducted laboratory and field failure tests of embankment dams and levees.

1.1. Floods and flood risk

1.1.1. Floods in China Floods are a substantial challenge for many countries worldwide. In China, approximately 10% of the land is below river flood levels, covering 40% of the population, 35% of farmland and 70% of industrial and agricultural centers. Floods have occurred frequently throughout history, causing substantial losses of human life and damage. More than 2,700 large floods have been recorded in China over the past 2,000 years. Faced with social and economic development, China is facing greater pressures in terms of flooding risks. The status of levees and embankment dams constructed to prevent flooding along rivers, lakes and coasts will be introduced. The large flood disasters over the past 100 years as well as their corresponding damage will be summarized.

1.1.1.1. Status of embankment dams and levees

1.1.1.1.1. River system and coastline China has approximately 430,000 km of rivers, and over 50,000 rivers have drainage basins that are larger than 100 km2. Most of these rivers flow from east to west and drain into the Pacific Ocean. Two-thirds of the area of China is influenced by the monsoon climate. Rainstorms and floods are concentrated in the monsoon areas of China through 7 main river systems (the Yangtze River, Yellow River, Huai River, Haihe River, Songhua River, Liao River and Pearl River). The total drainage areas of these 7 river systems constitute more than 4.3 million km2 of land (70% of the total drainage area in China), with approximately 1.2 billion residents being affected. Therefore, these are the main areas of concern in terms of flood protection. The Yangtze River (Changjiang) is the largest river in China and the third longest in the world. It flows for 6,300 km across southwest, central and eastern China. The Yangtze River drainage area encompasses nearly 20% of China’s total land. This river also plays an important role in the history, culture, and economics of China. The Yellow River (Huanghe), called the mother river of China, is 5,464 km long and is the second largest river in the country. It flows across mountains upstream, plateaus midstream and low plains downstream. Flows with high sediment concentrations make it the most sediment-laden river in the word. In recorded history, it has flooded more than 1,500 times and changed its main course 18 times.

Chapter 1. Bibliography

Less than 15m Total 120000 97246 100000 85160 80000 74603 63160 60000

40000 36226 17406 20000 Number of dams in China in dams of Number 0 22 0 1950 1986 2005 2012 Years

Figure 1-1. Dams constructed in China from 1950 to 2012.

The coastline of China stretches for 18,400 km and includes approximately 14,000 km of island coastline (Wang, 1987). Cities with high population densities and high economic development have developed near the coast. Coastal floods occur along the edges of oceans and are driven predominantly by storm surges and wave damage. Such floods are usually consequences of typhoons or tropical storms and threaten human lives and property.

1.1.1.1.2. Embankment dams A detailed survey of all large dams (more than 15 meters) was performed by ICOLD in 1958. A total of 9315 dams were catalogued in the first edition published in 1964 using data from 48 member countries (Flogl2010). The present edition of ICOLD considers 39,188 large dams constructed worldwide in 130 member countries, and 24,395 (62%) of them are embankment dams. Many embankment dams have heights of less than 15 meters and commonly have heights in the range of 5-10 meters. They play an important role in flood prevention and irrigation, but they have received less attention. At the end of 2012, there were 98,002 dams in China, and 97,246 of them have been constructed. Although the history of dams in China originates in 200 BC (the Dujiangyan project), most the dams were constructed after 1950 (shown in Figure 1-1). A total of 22 dams had been constructed in China before 1950 (large dams more than 15 meters high registered by ICOLD). Dams less than 15 meters account for approximately 80% of all

1.1. Floods and flood risk

dams, and most of them are embankment dams. As a result of their simple construction technique and easily obtained materials, a large number of embankment dams were constructed between the 60s and 80s of the last century. The constructed embankment dams played an important role in preventing floods and generating electricity at the beginning of the new country. They have been in use for approximately 30-50 years. As a result of the limited knowledge of these embankment dams’ construction processes and the lack of maintenance, they are facing an aging problem. Floods in China always occur at embankment dams.

1.1.1.1.3. Levees By 2012, approximately 294,000 km of levees had been constructed in China, in which 6,690,000 km are up to standards differentiated by security levels (shown Table 1-1). They protect approximately 33,000,000 hm2 of farmlands and 0.4 billion residents. As aspects of flood control projects, they play an important role in preventing flooding from rivers and lakes, especially in the mid-downstream of the Yangtze River (the levee of Jingjiang), in the downstream of the Yellow River (the levee of Huanghe), at the levee of Huaibei at the Huai River and at the levee of Beijiang at the Pearl River. The levees (security levels of 1, 2 and 3) of the Yangtze River have a total length of approximately 33,100 km, and those of the Yellow River have a total length of approximately 10,794 km. They represent the first and second longest levees in the main drainage areas. Table 1-1 clearly shows the different roles of the main drainage areas in flood control projects in China. Security level: In China, five different security levels for levees are differentiated by “flood control standards” to draft the design and construction standards for levees. The different security levels mean that the constructed levee can remain intact and protect floodplains subject to the corresponding flood frequency. Level 1: flood frequency ≥ 100 years; Level 2: flood frequency between 50 and 100 years; Level 3: flood frequency between 30 and 50 years; Level 4: flood frequency between 20 and 30 years; Level 3: flood frequency between 10 and 20 years (see Appendix A).

The levee of Jingjiang The levee of Jingjiang, located at the north bank of the Jing River (a part of the downstream of the Yangtze River), is 182.35 km long and spans from Zaolingang of the Jingzhou District to the south of Jianli District in Jingzhou City. It protects approximately 5,000,000 inhabitants and 5,336 km2 of farmland in the north of Jingjiang’s plains. As a part of the downstream of the Yangtze River, discharges increase at the inflow of the river’s branches, and the sediment follows the riverbed. Thus, the river in this area bends,

Chapter 1. Bibliography

and the river bed becomes higher than the protected floodplain. Floods have occurred frequently throughout this area’s history. In 1954, the levee of Jingjiang had suffered a total of 2440 floods, including 1791 occurrences of piping and 268 occurrences of overtopping. In 1998, it had suffered 90 floods, and 49 of them included occurrences of overtopping, and 30 of them included occurrences of piping. Because it is located in an alluvial plain, most of the foundation of the levee of Jingjiang is composed of sand and gravel, with a small amount of cohesive soil on the surface. The levee is also constructed with the same material as the foundation, with a thin impermeable layer. As a result of reinforcements over time, the width of the levee has increased from 8 to 12 m, with a levee slope of 1:3 inside and 1:3-1:5 outside. The platforms inside and outside are approximately 30-50 m.

The levee of the Yellow River The levee of the Yellow River spans 1,371.227 km and originates at the Meng District at the north bank and from Zhengzhou City at the south bank to the end of the river. The Yellow River flows in braided streams downstream, i.e., a network of smaller channels that weave in and out of each other. Because of the presence of a large amount of silt, the riverbed is raised and higher than the surrounding landscape. If a breach of the levee occurs, the resulting large disaster would threaten human lives and property. The construction of the levee of the Yellow River dates back to 500 BC. During its long history, reinforcement and maintenance continued. As a result of technological limitations faced over time, the quality of the levee remains below the requirements for preventing large floods. The three most recent reinforcements were performed after 1949 in three steps. The first step was to raise the levee height and reduce the levee slope in the upstream and downstream. Because the river bed continually rises due to sediment deposition, the second step was to widen the levee crest and reduce the deposition onto the river bed. Various important dams were constructed in the midstream of the river to control the discharge and flow velocity. Since the end of the last century, the river bed elevation has been well controlled. Thus, the third step initiated at the beginning of this century focused on reinforcing the levee foundation, repairing the cracks in the levee body, covering the downstream slope using cohesive soil, planting forests along the outside of the levee body, preventing further sediment deposition, and reducing the river bed in the downstream.

1.1. Floods and flood risk

Table 1-1. Summary of the lengths of the levees of the 7 main drainage areas and Tai Lake drainage.

Drainage areas Security levels Total and coastline 1 2 3

Yangtze River 1335 5280 26485 33100

Yellow River 1371 1657 7766 10794

Huai River 1716 2142 3858

Haihe River 599 2947 2562 6108

Songhua River 80 2502 232 2814

Liao River 289 2588 1368 4245

Pearl River 667 1001 1017 2685

Tai Lake 131 358 318 807

Others 118 315 2061 2494

Total 6306 18790 41809 66905

The levee of the Yellow River has been naturally divided into two parts by Aikou, where the river bed becomes narrow. Before Aikou, the levee height is approximately 11-12 m, and the levee crest is approximately 9-10 m, with a levee slope of 1:3 inside and outside. After Aikou, The levee is approximately 7 to 10 m in height. The levee crest is approximately 7-8 m, with a levee slope of 1:3 outside and 1:2.5-1:3 inside.

The coastal levees The construction of coastal levees has been concentrated since the foundation of the new China in 1949. By 2001, 13,476 km of coastal levees had been built, and 5,965 km of them were up to standards. The construction of coastal levees in China remains well below standards, especially when considering climate change, global warming and extreme weather. Coastal levees are normally constructed using soil or mixed soil and gravel material, and some of them are constructed using rock or concrete because of large waves and fast flows.

Chapter 1. Bibliography

Table 1-2. Flood frequency and distribution in the 7 main drainage areas over the last 100 years.

Flood frequency* Drainage areas Total >5% 5%-10% 10%-20%

Yangtze River 6 19 33 58

Yellow River 4 4 15 23

Huai River 4 9 14 27

Haihe River 3 5 10 18

Songhua River 3 4 16 23

Liao River 3 6 17 26

Pearl River 5 5 16 26

Total 28 52 121 201 *Flood frequency: see Appendix A.

1.1.1.2. Flood disasters over the past 100 years Over the past 100 years, floods in China have periodically occurred and have often caused considerable destruction of property and loss of life.

1.1.1.2.1. Floods in the 7 main drainage areas During the last century, floods with a frequency of less than 20% occurred approximately 201 times in the 7 main drainage areas. Floods with a frequency of 5% occurred 28 times (approximately 14% of all floods). Floods with a frequency of between 5 and 10% occurred 52 times (approximately 26% of all floods). Floods with a frequency of between 10 and 20 % occurred 121 times (approximately 60% of all floods). Table 1-2 shows that floods have occurred in the Yangtze River approximately 58 times, and the other 6 main drainage areas suffered almost the same number of floods. Flood frequency: the flood frequency describes the probable frequency of occurrence of a given flood.

1.1.1.2.2. Deadliest flood disasters

1.1. Floods and flood risk

Among the most recent major flood events over the past 100 years are those of 1931, 1954, 1991, and 1998.

Floods in the year 1954 This flood event is considered to be the worst flood disaster over the last century in China. Before this series of disastrous events, China experienced a long drought from 1928 to 1930. Following this period, in the winter of 1930, heavy snowstorms hit China. When spring began, the winter snowstorms melted into the river. In the summer of 1931, particular heavy rainfalls caused by extreme cyclone activity (7 cyclones in July, 1931) continued throughout the summer. This period of unusual weather caused the major rivers in the center of China (the Yangtze River, Yellow River and Huai River) to rise and overflow. According to the Chinese government, 150,000 people lost their lives, and many people starved following these floods.

Floods in 1954 In 1954, a series of floods hit the downstream of the Yangtze River. At the beginning of May in 1954, a rainy season caused by the so-called plum rain came early throughout the Yangtze River basin and stopped over. In the middle of June, three heavy rains hit the midstream and downstream of the Yangtze River over the course of 9 days. After several weeks of heavy rainfall along the Yangtze River, the flood levels continued to rise. More than 1800 km of flood levels broke historic records. This disaster killed 30,000 people, with many perishing through starvation and disease after the flooding.

Floods in 1991 Floods devastated the central and eastern parts of China in 1991. Torrential rain due to plum rain continually hit the eastern part of China over the course of 56 days (an above- average occurrence) in the Huai River basin, especially in the Anhui and Jiangsu provinces. It resulted in the overflowing of rivers and caused heavy flooding throughout the provinces. This flooding also resulted in approximately 5112 fatalities and huge economic losses.

Floods in 1998 A series of floods occurred in various major river systems of the Songhua River basin, the Pearl River basin and, especially, the Yangtze River basin throughout China. In the summer of 1998, the midstream and downstream of the Yangtze River basin were subject to continuous, heavy rainstorms that moved away for a short period of time. Because the

Chapter 1. Bibliography

water levels of the large lakes remained at a high level, flooding of the entire Yangtze River basin first affected the basin of two main lakes in this area (the Poyang Lake basin and the Dongting Lake basin). These lakes play an important role in storing flood water from the upstream and midstream of the Yangtze River. Because the floods first affected the downstream of the Yangtze River, additional flood water could not be stored when the strong floods occurred at the midstream. Therefore, after flooding at the midstream, water in the midstream remained at a high level for several days. Therefore, floods affected the entire basin of the Yangtze River. Floods also affected the Songhua River basin and the Pearl River basin in the northeast and south of China. In total, 29 provinces suffered flooding. A total of 222,900 million m2 of farmland had been submerged, and 137,800 million m2 had been destroyed. The flooding resulted in 4,150 dead and 37.47 billion dollars in economic losses. Although heavy rainfalls were the main causes of the floods, various manmade factors contributed: (1) A lack of flood storage area in the drainage basin. Lakes in the midstream and downstream of the Yangtze River have experienced significant depositions of sediment and have been reclaimed for farmland, resulting in nearly half of the lake’s surface disappearing. Flood capacity has reduced to a low level. (2) Vegetable and woodland reduce flood drainage. Trees and plants around lakes and the upper edge of the river bed reduced the surface area of the river and reduced the width of the river. Flood water ran slowly, which made the water in the downstream remain at a high level. (3) Most of the levees are located on the alluvial plains and have fragile foundations. Platforms inside and outside of the levees have not been well protected.

1.1.1.2.3. Dam failure Based on the statistics provided by the Ministry of Water Resources, from 1954 to the end of 2012, 3484 dam failure incidents were recorded in China (He et al., 2005, Jie and Sun, 2009). The most catastrophe dam failures were concentrated in the 70s of the last century during the Cultural Revolution. In 1973, 1974 and 1975, there were 556, 396 and 291 dam failures, respectively. At the beginning of this century, from 2004 to 2012, 33 dams failed, which was much fewer than in the last century.

1.1.1.2.4. Damage caused by floods As one of the most severe natural disasters in the world, floods in China resulted in approximately 260,960 human lives being lost and substantial amounts of starvation in the 20th century. Countless houses and roads were destroyed during or after floods. The average inundations areas are listed in Table 1-3. The losses of human lives in the 21th

1.1. Floods and flood risk

century have decreased compared to previous years, except in 2010. In 2010, a series of floods occurred in the south of China, resulting in many deaths. The statistics of losses of human lives at the beginning of this century are shown in Figure 1-2.

1.1.1.3. Conclusions Because it is threatened by floods, China has a long history of flood control. After the establishment of the new country, increasingly more attention was paid to flooding. Advanced systems of flood control infrastructure exist for many large rivers. They are generally supported by effective flood emergency response systems, including flood forecasting.

Table 1-3. Statistics of average damage caused by floods in the second half of the last century.

1950- 1960- 1970- 1980- 1990- Damages 1959 1969 1979 1989 1999

Inundation (1E8 m2) 736 737 536 1042 1609

Losses of human life (persons) 8571 4091 5181 4349 3904

3500 3222 3000

2500 1942 2000 1819 1605 1551 1522 1500 1282 1203 1024 1000 673 774 633 538 519 500

Losses of human lives (Person)lives human ofLosses 0

Years

Figure 1-2. Losses of human lives caused by floods between 2000 and 2013.

Chapter 1. Bibliography

1.1.2. Floods in France

1.1.2.1. The main river systems in France

1.1.2.1.1. Loire river (Maurin et al., 2013) The Loire is the longest river in France. It flows for 1006 km from Ardèche to the Atlantic Ocean. The drainage areas constitute approximately 117,000 km2, which covers more than one fifth of French territory. Along it, there are approximately 600 km of levees, which protect approximately 1,000 km2 from flooding. The main levees are concentrated along the Royal stream. Their construction dates back to the middle ages and the Kings of France. There are three important levee systems that should be mentioned.

The Orléans’s levee The length of the Orléans’s levee is approximately 43.5 km, and this levee protects approximately 63,000 people. A flood probability of 1/200 is considered as the protection level. This corresponds to a flood of 6,100 m3/s with a water level of 5.77 m, which is equal to the historic flooding level of the George V Bridge. The security level of this levee system corresponds to a flood of 4,600 m3/s with a water level of 4.60 m. The definitions of the security level and protection level are given in Appendix A.

The Tours levees The Tours Val is situated on an island, with a length of 60,160 m, and protects 135,000 people. It consists of three levees:

 The levee of Loire (28,800 m long and 8.15 m high): A flood probability of 1/500 is considered as the protection level. This corresponds to a flood of 6,600 m3/s with a water level of 8.64 m. The security level represents a flood of 4,700 m3/s and corresponds to a flood probability of 1/70 with a water level of 7.09 m.

 The levee of Cher (26,900 m long and 8.00 m high): A flood flow of 1,500 m3/s is considered as the protection level. This corresponds to a flood of 1,500 m3/s with a water level of 6.83 m. The security level represents a flood of 1,000 m3/s and corresponds to a flood probability of 1/50 with a water level of 6.83 m.

 The levee of Canal (1,960 m long and 7.00 m high)

The Authion levee The length of the Authion levee is approximately 43.5 km. This levee protects approximately 63,000 people. A flood probability of 1/100 is considered as the protection

1.1. Floods and flood risk

level. This represents a flood of 6,620 m3/s with a water level of 6.31 m. The security level of this levee system corresponds to a flood of 5,800 m3/s with a water level of 5.64 m. The total length of these three levee systems is approximately 170 km, and they protect approximately 230,000 people. The levee system of the Loire River had not been subject to major flood peaks until three great floods occurred in the middle of the 19th century, which caused considerable losses.

1.1.2.1.2. The course of the Garonne River The Garonne River was extensively diked following flooding in 1875, when 500 people lost their lives, including 200 in Toulous. The dikes did not, however, prevent the loss of another 200 lives in the 1930 flood. Although over 90% of the land subject to flooding and protected by dikes alongside the Garonne is agricultural, the populations of a number of large towns, including 40,000 people in Toulouse and 25,000 people in Agen, remain directly at risk. The statuses of dikes along the Garonne exhibit wide variations.

1.1.2.1.3. The Rhône The Rhône is the most powerful river in France. It flows from Suisse, passes through the southeast of France and flows into the Mediterranean Sea. The Rhône has a total length of approximately 813 km. The river divides into two breaches, known as the Great Rhône and the Little Rhône. The resulting delta constitutes the Camargue region, which is protected from flooding by 200 km of levees. The Rhône has experienced numerous floods, which lead to the construction of the levees.

1.1.2.2. Status of levees and embankment dams France has approximately 8600 km of levees, including 510 km of marine levees. Most of these levees are class B (30%) and class C (40%) levees and are situated in 4 main departments: les Bouches-du-Rhône, l’Isère, le Vauclues and la Gironde (Garry et al., 2002). In France, most levees were constructed between the 19th century and the middle ages. This means that they were built using backfill through several simple steps (connecting different sections and enhancement) without prior geotechnical study and the use of heavy mechanical devices (for soil compaction). If a levee was not well compacted or impermeable, it is very likely to contain breaches or damage to its foundation (Mériaux et al., 2012).

1.1.2.3. Recent flood disasters

Chapter 1. Bibliography

1.1.2.3.1. Floods of Rhône in 2003 From the first to the fifth of December 2003, most of the Rhône’s midstream and downstream tributaries had experienced very large floods. There were 7 victims, 27,000 evacuations and significant economic losses. It was the third highest severity flood after the historic floods in 1840 and 1856. The historical record flow of 11,500 m3/s at Beaucaire has not been exceeded.

1.1.2.3.2. Floods of Rhône in 1993 and 1994 In the autumn of 1993, from 22 September to 10 October, floods swept down the Rhône. The maximum discharge flow reached 9,800 m3/s at Beaucaire. Approximately 244 km2 of land were inundated. These floods caused approximately a dozen deaths and economic losses of approximately 124 million euros. In January of 1994, the Rhône and its tributaries experienced another series of floods. Fortunately, there were no casualties. However, this flooding still caused huge economic losses of 55 million euros and a large inundated area of 180 km2.

1.1.2.3.3. Floods of Var in 2010 In 2010, the floods of Var, des Bouches-du-Rhône du Vaucluse began with particularly heavy rain; soon after, an orange alert was declared in 11 departments. In the center and east of the region, the rainfall ranged from approximately 150 to 397 mm. The floods caused 26 deaths, 50 million euros worth of agricultural losses and over 1 billion euros worth of damage.

1.1.2.3.4. Other major floods Other major floods occurred at the beginning of this century and are listed in Table 1-4.

1.2. Breach models of embankment dam levee failure

Table 1-4. Other major floods at the beginning of this century.

Number of Victims Economic losses Locations Period communes (persons) (euro) PACA, Languedoc 2011 400 6 330 million Roussillon et Corse Poitou Charente et Pays 2010 1560 53 1.2 billion de la Loire Quart Sud-Est de la 2008 590 - 100 million France Rhône-Alpes 2008 336 - 60 million Meurthe et Moselle 2006 300 - -

Gard 2005 241 2 -

Floods of Rhône 2003 7 -

Gard 2002 419 24 1.2 billion

Marseille 2000 - 3 -

Laïta à Quimperlé 2000 - - - “-” means the formation is not clear.

1.2. Breach models of embankment dam and levee failures

1.2.1. Introduce To satisfy flood forecasting requirements and support flood emergency response systems, research on flooding models for embankment dams and levees has been performed by many researchers. These models focus their computational efforts on the routing of floods (breach discharge hydrographs, flow velocities, and flow depths) and their potential damage to protected plains. However, the simulation of discharge flow hydrograph through the breach is also recognized as a crucial process for accurately predicting the routing of floods. In recent decades, much progress has been achieved; such progress can be subdivided into two main categories: 1) Experimental breach models

Chapter 1. Bibliography

For an experimental breach model, data must be collected from real failure cases of embankment dams or through laboratory or field tests. Clearly, the failure of an embankment depends on numerical factors, including the height of water behind it, the inflow conditions, volume of water contained, embankment geometry, slope cover, materials and construction method. Therefore, the prediction of breach parameters (the peak discharge flow, the breach size and the time to failure) has been performed using regression equations based on reservoir characteristics. These predicted breach parameters can be directly used in flood forecasting systems or as input in breach models of the failure of embankment dams as parameters to achieve increasingly accurate estimations. 2) Physical breach models In contrast to experimental breach models based on data collection and regression analyses, a physical breach model focuses on the failure mechanism during the entire process. Based on observations of tests and real failure cases, the enlargement of piping in embankments, the flow conditions, soil erosion and soil block stability during breach processes have been simulated. This is performed with respect to the physical failure processes occurring in each embankment dam. Before introducing these two categories of breach models, it is necessary to understand the purposes of the simulations of these breach models. Table 1-5 provides illustrations of a typical discharge flow hydrograph and its corresponding breach processes, which simply represent the relationship between the discharge flow and breach processes. The details will be given in the following chapters.

1.2.2. Experimental breach models Estimations of certain breach parameters are always required as inputs for certain flooding model to assess the potential damage (Pierce et al., 2010). Since the 1970s, the empirical regression analysis of case study data has been developed to estimate breach parameters. Such analysis focuses on predicting the peak discharge flow and the breach geometry using formulas obtained through regression analyses of historical breach data. Certain models define the breach by its shape, final size and time to failure and assume the development of the breach as a parametric process. Figure 1-3 shows a typical shape of a breach assumed by these models.

1.2. Breach models of embankment dam levee failure

Table 1-5. Illustration of a typical discharge flow hydrograph through a breach and its corresponding breach processes.

Phases Discharge flow hydrograph Descriptions

t t 0 4 A typical discharge flow The main illustration hydrograph from the initiation of of the discharge flow erosion to the end of the failure hydrography

t0 t 1 No initiation of failure, and the embankment dam is safe Safe

The initiation of surface erosion/internal erosion due to  t1 t 2 overtopping/piping results in the The initiation of formation of a breach. erosion The discharge flow through it is visible but remains stable.

As the erosion moves, a breach  t2 t 3 develops. The formation of A large discharge flow quickly breach forms

t3 t 4 After a breach is fully formed, it grows laterally and deeply until The enlargement of no more flow occurs through the breach until the end breach. of the failure

Chapter 1. Bibliography

Figure 1-3. A typical shape of the breach assumed.

1.2.2.1. Prediction of peak discharge flows The prediction of peak discharge flows is a direct approach used to study the consequences of embankment dam breaches. Since the 1980s, multiple methods have been developed to quickly estimate peak discharge flows (given in Table 1-6).The volume of water contained in the reservoir and the height of the water in the reservoir at the start of the formation of the breach are considered as the main influences on the peak discharge flow. Thus, the predicted peak discharge flow is always related to the height of the water ( Hw ), the volume of water contained in the reservoir (Vw ), the dam factor ( HVw w ) and multiple regression relationships, including the average width ( B ) and length ( L ) of the embankment dam. The Soil Conservation Service (SCS) (Brevard et al., 1981) proposed the existence of a relationship between the height of the water behind an embankment dam ( Hw ) and the peak discharge ( Qp ). They used 13 cases studies to relate Hw to Qp . The USBR

(Reclamation, 1982) expression proposed a relation similar to that of the SCS for Hw and

Qp , and it was developed using case study data from 22 failed dams, including several concrete arch and gravity dams. Several studies have related the peak discharge flow to the volume of water contained in the reservoir at failure (Evans, 1986, Singh and Snorrason, 1984). Various investigators (MacDonald and Langridge Monopolis, 1984, Froehlich, 1995, Costa, 1985) have referred to a dam factor ( HV ) as an index of the energy w w‐ expenditure at dam failure. Based on the measured peak discharge flows at dam failure, various equations have been developed to predict Qp using the dam factor.

1.2. Breach models of embankment dam levee failure

Table 1-6. Predictions of peak discharge flows using experimental breach models.

Number Relationship Investigator Equations of case  1.85 SCS 13 QHP16.6 w Height of water

( Hw ) 1.85 USBR 21 QHP19.1 w

Volume of Singh and 8 QV1.176 0.47 water contained Snorrason PW in the reservoir 0.53 Evans 29 QVP 0.72 w (Vw )

0.48 Hagen 6 QVHP 1.205 w w  Dam factor MacDonald 0.41 23 QVHP 1.175 w w  ( HVw w ) and

0.42 Costa 31 QVHP 0.763 w w 

0.295 1.24 Froehlich 22 QVHP 0.607 w w 

Multiple QVHW 0.863 0.335 1.833 0.663 relationships P w w  Pierce 25/14 0.493 1.205 0.226 QVHLP 0.012 w w 

1.2.2.2. Prediction of breach parameters To predict breach discharge flows, certain methods first estimate breach parameters and then calculate the peak discharge flow using an existing breach model. Breaches in embankment dams and levees are usually assumed to be trapezoidal or rectangular in shape; therefore, the shape and size of the breach are always defined by a base width and side slope angle or a breach width (Wahl, 2004, Mohamed et al., 2002). The time to failure is a critical parameter affecting the outflow hydrograph and the consequences of dam failure, especially when populations at risk are located close to a dam and available warning and evacuation times strongly affect losses of life. The time to failure always means an active development phase with rapid increases in flow rates for piping failures and obvious erosion cuts through the upstream face for overtopping failures.

Chapter 1. Bibliography

Experimental formulae based on regression analyses are summarized in Table 1-7 and are used to estimate breach parameters, including average breach width ( B ), eroded volume (

Ver ), and failure time (t). Equations for predicting the side slope angle are rare, and most researchers use an assumed angle. Froehlich (Froehlich, 1995) suggested simply assuming side slopes of 0.9:1 (H:V) for piping failures. Visser (Visser, 1998) assumed the side slope angle to be equal to the sand internal friction angle in his sand-dike breach model. Zhu (Zhu, 2006) suggested a critical side slope that is a function of the soil properties of the embankment. The function depended on the soil cohesion and internal friction angle.

Table 1-7. Predictions of breach parameters.

Number Breach width ( B ) / Model Failure time (t) of cases Eroded volume (Ver )

0.769 VHer 0.0261(V w w ) MacDonald (Cohesive) 0.364 and Langridge- 42 t 0.0179 Ver  0.852 Monopolis VHer0.00348(V w w ) (Non-cohesive)

0.32 0.19 0.53 0.9 Froehlich 43 B 0.1803 K0 Vw h b t 0.00254 Vw h b

Bureau of BH 3 t 0.011 B Reclamation w

1.2.3. Physical breach models Before the 90s in the last century, many physical breaching models had been developed in Europe and in the USA in an attempt to simulate the processes of dam breaches and corresponding floods for the management and forecasting of flood risks. All such models have been successful, but none of them have been widely used. We continue to face many flood risks, especially at the beginning of this century. Therefore, the study of physical breaching models has recently seen great progress. Some detailed descriptions of these physical breach models are reviewed below.

1.2.3.1. BEED (Singh et al., 1988) The BEED model describes dynamic water-sediment transport during dam failures. Both surface erosion and side slope collapse are incorporated. The model assumes a

1.2. Breach models of embankment dam levee failure

homogeneous dam and requires a specific size, shape, and location of the initial breach. The selection of these initial data is mostly based on engineering justification from past experience rather than on any solid theoretical basis. The breach shape can be rectangular, trapezoidal or triangular and is mainly specified by historical data. The simulation begins after a breach is developed on the crest of the dam.

1.2.3.2. BRES (Zhu, 2006, Visser, 1998) BRES was developed by Visser (Visser, 1998) for simulating the breach of a homogeneous non-cohesive embankment dam due to overtopping failure. Based on observations of tests, the breach processes are divided into 5 continuous stages. A small trapezoidal breach at the top of the embankment dam is assumed as the initial situation. Stage 1 presents the steeping of the embankment downstream slope from an initial value a0 up to a critical value a1 . When the critical value is attained, stage 2 starts, and the discharge flow continues to erode the downstream slope, which remains a constant value a1 . Then, the crest moves toward the upstream until the end of this stage, when the crest is entirely moved by downstream erosion and disappears. In the meantime, no deep erosion occurs on the crest (see Figure 1-4 and Figure 1-5). In stage 3, because the crest has already vanished, erosion starts to deepen the breach bottom toward the base of the embankment at a constant slope angle a1 . As the breach deepens, it widens at a constant slope angle of c1 . As the breach reaches the base of the embankment bottom, stage 3 ends. During this stage, both the bottom erosion and side slope erosion are controlled by the erosion rate of the bottom. Stage 4 describes the critical flow stage. The breach continues to grow in both the vertical and horizontal directions and is controlled by the bottom erosion rate and side slope erosion rate. The side slope angles remain at the critical angle (see Figure 1-6). As the discharge flow changes from critical to subcritical, stage 4 ends and stage 5 starts. In this stage, lateral erosion is the main erosion process. The breach continues to enlarge until the end of the breach. Based on the theory of erosion processes and headcut migration, the simulation of homogeneous cohesive embankment dams was extended by Zhu (Zhu, 2006). Their method retains the same five-stage division method of breach processes as in the preview model but emphasizes headcut migration as the main theory of the formation of a breach (see the next section). Figure 1-7 shows the development of a headcut of an embankment dam with non-erodible foundation, and Figure 1-8 shows the instability of the headcut and forces acting on it. The increased breach width in stage 4 and stage 5 is shown in Figure 1-9. Although the migration of the headcut results in the increased breach width, the flow shear stress at the breach bottom determines the rate of increase of the breach width.

Chapter 1. Bibliography

Figure 1-4. Illustrations of the breach processes of a homogeneous non-cohesive embankment dam. The cross section of the embankment dam after the initiation of a breach and the following breach processes in stage 1, stage 2 and stage 3 (Visser, 1998).

Figure 1-5. Illustration of the increase in the breach width along with the deep erosion on the downstream slope in stage 3 (Visser, 1998).

Figure 1-6. Illustration of the increase in the breach width after a breach is fully formed in stage 4 and stage 5 (Visser, 1998).

1.2. Breach models of embankment dam levee failure

Figure 1-7. Illustrations of the cross section of the headcut development of an embankment dam with a non-erodible foundation (Zhu, 2006).

Figure 1-8. Illustration of the forces acting on a headcut (Zhu, 2006).

Figure 1-9. Illustrations of the breach enlargement (Zhu, 2006).

Chapter 1. Bibliography

1.2.3.3. Rupro (Hanson and Cook, 2004, Paquier and Recking, 2004) Rupro was developed by IRSTEA to study the erosion of a homogeneous embankment dam caused by overtopping and piping failure. It is integrated in CastorDigue (flood forecasting software) as a module for breaches. The Bernoulli equation is used to calculate the flow characteristics of overtopping and piping. The erosion process is estimated by the sediment transport equation of Meyer-Peter and Muller. Three stages are considered for the breach evolution. The first stage includes the initiation of a breach, which is neglected in Rupro because the initiation of the breach is usually invisible. The second stage begins with visible flow and erosion through a pipe (piping, see Figure 1-10) or a rectangular breach (overtopping, see Figure 1-11) until the embankment dam is fully eroded through. The last stage continues until the final breach is attained. In this stage, the breach, controlled by soil sediment transport along the wetted perimeter, widens and deepens. During the IMPACT project (see below), Rupro was tested and demonstrated to properly reflect testing results. The wall shear stress decreased by a factor of 0.6 because of the difference between the bottom shear stress and the side wall shear stress. The wetted perimeter was calculated as twice the water depth, which means that no further erosion occurs on the bottom of the breach.

1.2.3.4. SIMBA (Temple et al., 2005, Hanson et al., 2005b) Simplified Breaching Analysis (SIMBA) was developed at the USDA-ARS Hydraulic Engineering Research Unit, Stillwater, OK (USDA - U.S. Department of agriculture) (Temple et al., 2005). It is a research-focused model used to analyze data from large-scale laboratory tests for developing and refining algorithms needed for the creation of an application-focused model. The erosion technology developed in SIMBA has now been incorporated into WINDAM B (Morris et al., 2012). SIMBA focuses on headcut erosion in homogeneous cohesive embankments during overtopping failure. SIMBA makes the following assumptions: the headcut begins at the top of the downstream slope; meanwhile, the crest remains at its original level until the headcut reaches the upstream slope. The illustration of the formation and movement of the headcut is described in Figure 1-12 and Figure 1-13. Breach channel side slope stability is analyzed by the model to determine an average side slope angle, which is kept constant during the entire analysis.

1.2. Breach models of embankment dam levee failure

Figure 1-10. The evolution of the pipe in stage 2 and the enlargement of the breach in stage 3 of piping failure (Paquier, 2007).

Figure 1-11. The evolution of a rectangular notch in stage 2 and the enlargement of the breach in stage 3 due to overtopping (Paquier, 2007).

Chapter 1. Bibliography

Figure 1-12. Illustration of headcut formation and movement (Hanson et al., 2005b).

Figure 1-13. Sketch of forces on a headcut failure element (Robinson and Hanson, 1994).

1.2. Breach models of embankment dam levee failure

1.2.3.5. HR Breach Model (Temple et al., 2005, Morris and Hassan, 2009) The HR Breach model was originally developed by Mohamed Hassan as part of an HR Wallingford research program for simulating both the overtopping and piping failure of embankment dams (Morris et al., 2012). For overtopping failure, either surface erosion or headcut can be simulated, although the latter reproduces processes as defined by Temple (Temple et al., 2005) for the SIMBA model. The user can choose different erosion models for different types of embankment dams. The model also considers the influence of the grass and rock cover of the outside slope of embankment dams in the breach formation process. For the performance of the grass cover, parameters for grass performance without any added safety factors can be used. The advantage of the HR BREACH model is the ability to simulate zoned embankment dams (Figure 1-14) (Morris, 2011). Different layers of material can be distinguished by different soil erodibility, and thus, soil erodibility is an important parameter. This model determines the erosion rate in each zone and then the overall breach processes.

Figure 1-14. Illustrations of the zoned approach in the HR BREACH model (Morris, 2011).

1.2.3.6. WinDam B (Visser et al., 2010) WinDAM B was developed by the Agricultural Research Service (ARS) and the NRCS for estimating earthen embankment overtopping. The model integrates headcut theory as the main failure process. It consists of three parts: (1) the failure of vegetal cover and the development of concentrated flows due to erosion on the surface; (2) the downward erosion of concentrated flows resulting in headcut formation; and (3) the movement of headcuts downward and upstream results in the breach of earthen embankments and auxiliary spillways of dams. Each phase is described by a set of threshold-rate relationships based on the process mechanics. A headcut erodibility index ( K h ) describes the resistance of the exposed geologic materials to erosive attack during the last part of the process.

Chapter 1. Bibliography

Currently, WinDAM can only estimate the flood routing due to the overtopping failure of homogenous earthen embankments. A user of WinDAM must provide input hydrography using other software because the hydrology component is not included. This model begins with an evaluation of the surface failure of the downstream slope with vegetal (grass) protection, riprap protection, and no protection (bare soil). Vegetation is described with a vegetal cover factor, i.e., a maintenance code. The riprap is described using the diameter

D50 , uniformity, porosity, and thickness of the riprap layer. Manning’s value is used to describe the embankment without protection. The main output items of WinDAM are the crest elevation and peak outflow, overtopping flow duration and dam fill advance rate coefficient.

1.2.3.7. BREACH (Fread, 1988) BREACH was developed by Fread (Fread, 1988) for predicting embankment dam failure due to overtopping and piping. The model is capable of simulating homogenous and zoned dams composed of two different materials. Two different downstream slope configurations are considered: grass cover and a cover composed of a material with a larger grain size. It is a sediment-transport-capacity-based erosion model with assumptions for breach size formation using a board-crested weir equation (for overtopping) and an orifice flow equation (for piping). BREACH computes the outflow hydrograph and the breach parameters used in DAMBRK and SMPDBK (in below). For overtopping failure, the following three processes are assumed. The first stage begins at the downstream slope, and a small rectangle rivulet is assumed to exist. Erosion occurs only along the downstream slope, and the bottom of the flow channel remains equal to the dam crest. As the overtopping flow erodes into the downstream slope, the dam crest moves upstream. When the bottom of the erosion channel reaches the upstream side, the second stage starts, and the breach bottom begins to quickly erode. This stage continues until the bottom of the channel reaches the dam’s foot. The development of the breach depth and cross section in both stage 1 and stage 2 are shown in Figure 1-15. Figure 1-16 shows the corresponding development of the breach width. Two main mechanisms determine the width of the breach. The initial rectangular shape is controlled by the water depth. Then, as the breach depth increases to a critical value, the side slope becomes unstable and collapses, and the breach becomes trapezoidal in shape. cis the side slope angle formed after collapse and is controlled by the soil properties (internal friction angle, cohesive and unit weight) and breach geometry. The overtopping flow erosion rate in the vertical direction is assumed to be the same as that of the side slope. When the entire body of the

1.2. Breach models of embankment dam levee failure

embankment dam is completely removed, the third stage begins to substantially widen the breach. No additional deep erosion is assumed, and the same side slope collapse mechanism is used in this stage. For piping flow, an initial rectangular piping channel is assumed. Then, the pipe starts to increase vertically upward and downward. The collapse of the pipe is assumed once the flow transitions from orifice to weir, at which time all of the collapsed soil is transported along the breach channel. Then, the breach is formed and begins to greatly widen following the same process as in stage 3 for the overtopping failure.

1.2.3.8. DAMBRK (Feliciano Cestero et al., 2014) DAMBRK is a flood forecasting model developed by the U.S. National Weather Service. A simple embankment dam breach model is integrated in this model to compute the discharge outflow and simulate the flood routing downstream. The geometry of the breach is illustrated in Figure 1-17. Three parameters can be used to define the breach as exhibiting a triangular, rectangular or trapezoidal shape: the bottom width of the breach Bb

, the height of the breach hb and the slope of the breach side wall 1: z (vertical : horizontal) ( z ranges from 0 to 2). At breach initiation, a triangular notch is formed on the downstream surface. When it reaches to a point, the breach begins to widen and deepen until the bottom of the dam is reached. The slope of the breach side wall remains constant during the processes. The broad-crested weir formula (for overtopping) and orifice formula (for piping) are used for the hydraulic calculations, and no sediment transport is considered.

1.2.3.9. DamBreach (Wang and Bowles, 2006) DamBreach was developed by Wang for simulating the overtopping failure of non- cohesive embankment dams. The model defines overtopping failure as three erosion processes based on the BREACH model (see above). The first stage begins at the initiation of the breach and continues until the embankment dam crest is vertically eroded by 0.61 m. Then, the second stage begins and continues until the entire embankment dam is eroded. The third stage concerns the enlargement of the breach. During the breach processes, DamBreach calculates the discharge outflow by solving the two-dimensional shallow water equations. In stage 1, a comparison of the erosion rate and the sediment transport rate is performed to choose the smaller of the two for the calculation of the eroded soil quantity. In stage 2, the same erosion mechanism as in stage 1 is used. The breach enlarges due to the undercut of soil mass by erosion. This means that in each time step, the eroded soil as well as the collapsed soil are removed. Soil mass collapse occurs once the bottom of the

Chapter 1. Bibliography

mass is undercut. Meanwhile, it deepens via the removal of unstable soil mass using a 3D slope stability method. This erosion model, soil mass collapse model and 3D slope stability model continue to be used in stage 3.

1.2.3.10. Conclusions Table 1-8 provides a simple description of these physical models, including (1) the failure modes integrated in the models, (2) the materials considered, and (3) the type of embankment dams modeled. Table 1-9 summaries the literature review of available physical models for simulating the breach of embankment dams and levees; some of them remain under development. Among these models, the overtopping failure mode was developed more widely and earlier than was the piping failure mode. The initiation, formation and growth of an overtopping failure have been well recognized. The exploitation of the piping failure mode has normally been based on the overtopping failure mode. The enlargement of the pipe is always assumed as a continuous process based on the theory of sediment transport.

Table 1-8. Failure modes and types of embankment dams integrated in the physical breach models

Types of embankment Failure modes Material dams No. Models Non- Overtopping Piping Cohesive Homogeneous Zoned cohesive 1 BEED √ √ √

2 BREACH √ √ √ √ √ √

3 BRES √ √ √ √

4 DamBreach √ √ √

5 DAMBRK √ √

HR √ √ √ √ √ √ 6 BREACH 7 Rupro √ √ √ √ √

8 SIMBA √ √ √

9 WinDam B √ √ √ √

1.2. Breach models of embankment dam levee failure

Figure 1-15. Illustration of the development of the embankment dam cross section due to overtopping in stage 1 and stage 2 (Fread, 1988).

Figure 1-16. Illustration of the development of the breach width due to overtopping in stage 1 and stage 2 (Fread, 1988).

Figure 1-17. Illustration of embankment dam breach formation integrated in the flood forecasting model DAMBRK (Feliciano Cestero et al., 2014).

Table 1-9. Physical breach models of embankment dam failure.

Models Flow Sediment Breach Crest degradation and Lateral Validation Comments Failure characteristics transport/Erosion Morphology recession/ breach widening mode* BEED Broad-crested Sediment Trapezoidal Surface erosion; Teton dam and No further slope O weir formula transport Huaccoto sliding occurs after the Breach enlarges via removal of (Einstein-Brown) breach bottom is unstable soil and soil erosion reached BREACH Overtopping: Modified Smart Overtopping: Using sediment transport equation Teton dam and a (1) Grass cover on O/P broad-crested formula to estimate the rate of erosion; dam in Lawn downstream face; (2) rectangular/ weir formula; Lake; Landslide Three materials: an Breach enlargement is governed Piping: orifice trapezoidal blockage of the inner core, an outer by erosion rate of breach bottom, flow/weir Mantaro River in portion, and a thin Piping: as performed for lateral erosion. formula rectangular Peru layer; (3) Short computational time

BRES Broad-crested Excess shear Trapezoidal Surface erosion: downstream A series of tests (1) Headcut and O weir formula stress equation slope surface retreats/ Headcut (EC IMPACT estimation of slope migration; project) stability are included; (2) Simple slope The rate of widening is only stability mechanism governed by the rate of flow shear erosion. Dam 2D St. Venant Sediment Undetermined Surface erosion until the crest is Laboratory tests (1) Multiple breach O equations transport formula eroded through/Headcut and IMPACT locations under wind Breach (Smart)/Sediment movement checked by 3D slope project and wave actions; (2) erosion rate stability prediction; Zoned dam; (3)

Models Flow Sediment Breach Crest degradation and Lateral Validation Comments Failure characteristics transport/Erosion Morphology recession/ breach widening mode* Headcut movement Dam Breach widens via removal of checked by 3D model eroded soil and collapsed soil Breach under the stability check and erosion calculation DAMBRK Overtopping: Linear Concerns the Linear rate governed by inputted Teton dam, (1) It is capable of O/P broad-crested predetermined breach bottom parameters of breach time and Buffalo Creek flood wave movement; weir formula; erosion width dam shape; Coal-Waste Dam, (2) The accuracy of Piping: orifice Johntown Dam, the model depends on Breach increases at a linear rate flow/weir Toccoa Dam and the input data based on the inputted parameters formula Laurel Run Dam of the final breach width and breach time HR Breach Overtopping: Excess shear Overtopping: Surface erosion (non-cohesive), Two tests from (1) Both homogeneous O/P Model variable weir stress equation vertical side headcut processes (cohesive) and USDA-ARS and non-cohesive formula/1D slope with bending failure (cohesive core); Stillwater soils/composite or

steady non- undercut laboratory zoned structures; (2) Side slopes remain vertical during uniform flow Grass/rock surface Piping: breach process; soil transport, Three tests from equation; protection; (3) circular, erosion and soil mass collapse are the IMPACT Including headcut and Piping: orifice vertical side included. project discrete block failure flow formula slope width processes undercut Rupro Bernoulli Meyer-Peter and Overtopping: Surface erosion; Two tests by Lnh (1) Computational O/P equation Muller rectangular and Irstea; Tests times are short; The rate of widening is equal to of CADAM and Piping: the rate of deepening (2) Few parameters IMPACT projects circular

Models Flow Sediment Breach Crest degradation and Lateral Validation Comments Failure characteristics transport/Erosion Morphology recession/ breach widening mode* SIMBA Broad-crested Excess shear Trapezoidal Headcut migration; (1) Only for O weir formula stress equation homogeneous Breach widening rate is calculated condition; (2) Headcut using excess shear stress equation movement through with critical flow conditions; crest WinDam Input Excess shear Surface erosion/formation and (1) Capable of O hydrograph stress equation movement of headcut different slope through protections: Vegetal, embankment riprap, and no protection; (2) Integrates the formation and movement of headcut * O means the overtopping failure mode, and P means the piping failure mode.

1.3. Laboratory and field dam breach tests

Based on laboratory and field dam breach tests on failure by overtopping and piping conducted in the last 10 years, the understanding of the breach processes of dam/flood defense structures has been improved. Various laboratory experiments and field tests of the failure of embankment dams in the EU (IMPACT project), China, and the US (ARS) will be introduced.

1.3.1. IMPACT project in the EU (from 2001 to 2004) A long-term study (CADAM, IMPACT and FLOODsite) in Europe was conducted to reduce uncertainty in predicting extreme flood conditions and improve the predictions of risks considering flood defense structures. Each project addressed different aspects of flood risk management. The IMPACT (from 2001 to 2004) project undertook a program of field and lab work to collect reliable data on breaches to better understand failure processes and assess model performance. Based on various conclusions drawn by the CADAM (from 1998 to 2000) project, such as the limitations concerning the identification of breach formation processes (structure failure mechanisms, breach formation mechanisms and breach location) by existing models, the mechanisms of debris movement and soil sediment transport, and flood routing in protected area, a series of tests was performed under IMPACT to improve the reliability and accuracy of dam breach analysis and to address the assessment of risks from flooding. Under the IMPACT project, 5 field tests were undertaken during 2002 and 2003 using embankments 4-6 m high in Norway (Vaskinn et al., 2004). Figure 1-18 shows the breach processes at one of the 5 field tests. This test included a rockfill dam with a moraine core. The height of the dam was 6 m, with a 3-m-large crest and 1:1.5 ratio for both the downstream and upstream slope. The dam crest was overtopped, and the combined discharge in the depression and over the crest was observed (Figure 1-18 (a) and (b)). After the initiation of the breach (Figure 1-18 (c)), discharge flow continued for a few minutes (Figure 1-18 (d)).

Chapter 1. Bibliography

(b) The combined discharge flow in the depression (a) The dam crest was overtopped and over the crest

(c) The breach is fully formed (d) The breach developed and enlarged Figure 1-18. Pictures from a test of a rockfill dam with a moraine core (Vaskinn et al., 2004).

A series of 22 laboratory tests were undertaken during the same period, the majority of which at a scale of 1:10 to the field tests. During the five field tests, a cohesive, non- cohesive and rockfill with moraine core composite embankment was constructed for the overtopping tests. A homogenous moraine and rockfill with moraine core composite embankment were used for the piping tests. The 22 laboratory tests were divided into 3 series: series 1 included 9 small-scale tests with non-cohesive material undertaken using different embankment geometries and breach locations. Series 2 included 8 tests with two different cohesive materials and different embankment geometries, compaction efforts and moisture contents. Series 3 included 5 tests of pipe initiation. Figure 1-19 shows the breach formation during the series 2 tests. More than one headcut was observed on the downstream face (see Figure 1-19 (a)). Then, the headcuts combined into one deep headcut and migrated upstream (see Figure 1-19 (b)). Once the breach was fully developed, material below water was eroded, which lead to the collapse of soil mass (see Figure 1-19 (c)). Upstream slope erosion was also observed on the upstream face (see Figure 1-19 (d)).

1.3. Laboratory and field dam breach tests

(a) Several headcuts were formed (b) Headcut combined into one deep headcut

(c) Soil mass failure of side slope (d) Upstream erosion Figure 1-19. Pictures of the breach processes of the small, cohesive embankment dam from the series 2 test (Hassan et al., 2004).

The objectives of these tests were to understand the formation of breaches in homogeneous cohesive and non-cohesive embankments failed by overtopping and to assess the initiation of the piping mechanism and dimensions for homogeneous embankments. Based on the observations and data collection, the failure processes of different embankments have been analyzed. (1) Creaking, piping formation and headcut migration were all observed. (2) The first phase in the external erosion of the downstream slope due to overtopping erosion was slow and gradual until the upstream edge was reached; subsequently, the breaching was rapid and dramatic. (3) The opening of the breach first progressed down to the base of the dam before expanding laterally, and the sides of the breach were vertical in all materials. (4) The initiation of the piping took a significant amount of time, and the dam breach did not occur until the erosion had proceeded up to the dam crest.

1.3.2. Overtopping research in the US The US Dam Safety community has similar needs as the EU and performs similar activities. Based on the results obtained in spillway erosion research, the ARS

Chapter 1. Bibliography

(Agricultural Research Service of the US Department of Agriculture) has expanded spillway erosion research into embankment overtopping erosion research. The following four research projects have been conducted to better understand how an embankment will perform during overtopping failure: 1) steep vegetated and bare channel tests; 2) large- scale flume studies of headcut migration; 3) large-scale tests of breach initiation and formation; and 4) large-scale tests of breach widening. All of these projects have been focused on erosion mechanics during failure.

1.3.2.1. Steep vegetated and bare channel tests A 3-m-high embankment was constructed in an outdoor laboratory under green Bermuda grass, dormant Bermuda grass, green fescue, and bare conditions. The movement of erodible boundary-supporting vegetation resulting in the initiation of the failure of the grass channel was observed and compared with the bare earth channel.

1.3.2.2. Large-scale flume studies of headcut migration Large-scale flume studies of headcut migration: A total of 46 tests were conducted in a 1.8- m-wide, 29-m-long flume with 2.4-m-high sidewalls. Two soils, red sandy clay soil and silty sand soil, were used in various configurations and under various conditions. Observations of the tests showed that the initial overtopping flow results in sheet and rill erosion, with one or more master rills developing into a series of over-falls, therein resulting in a large headcut. Then, the headcut migrates from the downstream to the upstream edge of the embankment crest until the headcut reaches the upstream crest. Then, the headcut migrates into the reservoir, lowering the crest until the crest’s downward erosion reaches the breach bottom.

1.3.2.3. Large-scale tests of breach initiation and formation Large-scale tests of breach initiation and formation: Seven large-scale overtopping tests were conducted to obtain information relevant to the erosion processes of cohesive embankment breach failures. These overtopping failure tests included 2.3-m- and 1.5-m- high cohesive embankments with 3 horizontal to 1 vertical upstream and downstream side slopes. Three different soils, from silty sand to lean clay, were used. The reservoir water level, embankment erosion, and discharge were measured throughout the tests. Breach initiation times were observed to range from 0.07 to 11.6 times the breach formation time. This means that the breach initiation time can be quite lengthy and often greater than the breach formation time. One of these tests is shown in Figure 1-20.

1.3. Laboratory and field dam breach tests

(a) (b) (c)

(d) (e) (f) Figure 1-20. Erosion processes during overtopping test; (a) Rills and cascade of small overfalls during breach initiation at t = 7 min; (b) Consolidation of small overfalls during headcut migration at t = 13 min; (c) Headcut at downstream crest, transition from breach initiation to headcut migration at t = t1 ≈ 16 min; (d) Headcut at upstream crest, transition from headcut migration into crest lowering at t=t2 ≈ 16 min; (e) Flow through breach during crest lowering at t = 40 min; (f) Transition from crest lowering to breach widening at t = t3 ≈ 51 min (Hanson et al., 2005a).

1.3.2.4. Large-scale tests of breach widening Large-scale tests of breach widening: Three large-scale earthen embankments were constructed to evaluate the widening of the breach. The tested embankments were 1.3 m in height, with a 1.8 m crest and 3:1 (H:V) downstream and upstream slope. A 0.3-m-wide notch in the center of the embankment was created. All of the embankments were made of cohesive homogeneous materials, ranging from silty sand to lean clay (Hunt et al., 2005). The width of the breach during widening was monitored during the test. The breach widened in all three tests to a large extent as the breach sides were undermined, causing mass failures to occur. The undermining was observed to occur as a detachment process, resulting in the development of an overhang. A crack along the surface was observed at the initiation of the mass failure of the bank. The crack slowly opened, and the material eventually toppled into the breach opening.

Chapter 1. Bibliography

Table 1-10. Summary of laboratory and field tests conducted in the UK, Norway and in the US.

Projects Test Materials Geometry/Descriptions Observations series HR Series 1 Homogeneous Height: 0.5 m Headcut was not observed; Once the Walling non-cohesive Slope: 1:2/1:1.7 breach was fully developed, the ford in material below the water level Crest width: 0.3/0.2 m the UK eroded. This undermines the slopes Different embankment and leads to block failure; geometries, breach locations Upstream slope erosion was also and times before failure observed; Series 2 Homogeneous Height: 0.6 m More than one headcut was observed cohesive Slope: 1:1/1:2 on the downstream face; The headcuts combine into one deep Different embankment headcut, which migrates upstream geometries, compaction and then erodes downward; efforts and moisture contents Once the breach is fully developed,

the material below the water level is eroded. This undermines the slopes and leads to block failure; Upstream slope erosion was also observed; Series 3 Homogeneous moraine Norway 1-02 Homogeneous Height: 6 m Headcut development was observed. clay fill US Slope: 1:2 The width of the headcut remained equal to the width of the initial DS Slope: 1:2 notch. When the headcut had moved A 0.5-m-deep and 3-m-wide back to the upstream side of the notch at the top of the dam dam, the breach rapidly developed; was made to initiate the breach 2-02 Homogeneous Height: 5 m During the breach initiation phase, gravel US Slope: 1:1.7 headcut development proceeded similar to that in the clay dam (Test DS Slope: 1:1.7 1-02). Following the breach This embankment was initiation, vertical erosion finished composed of a moraine core after 5 minutes, and the horizontal that was vibratory roller erosion finished after 5-10 minutes; compacted in 0.5 m layers 1-03 Rock fill with Height: 6 m Further concerns regarding the central moraine US Slope: 1:1.5 geometry of the layer construction were recorded on video; core DS Slope: 1:1.5 Core slope: 5:1 The moraine core was vibratory roller compacted in 0.5 m layers

1.3. Laboratory and field dam breach tests

Projects Test Materials Geometry/Descriptions Observations series Norway 2-03 Rock fill with Height: 6 m After opening the trigger device of central moraine US Slope: 1:1.5 the preformed pipe, a sinkhole rapidly formed on top of the dam. core DS Slope: 1:1.5 The sinkhole formed a notch through Core slope: 4:1 the dam, and the dam failed in the A pipe was perforated along same manner as by overtopping; certain lengths, filled with and surrounded by sand from the bottom of the dam to the top

3-03 Homogeneous Height: 4.5 m The failure was very rapid; moraine US Slope: 1:1.3 An obvious peak discharge was DS Slope: 1:1.3 formed; Crest width: 3 m Pipe collapsed suddenly when the pipe diameter reached nearly the A pipe was perforated along dam height; certain lengths, filled with and surrounded by sand from the bottom of the dam to the top ARS in Channel Channels were Height:3 m US tests tested under 4 Six 0.9-m-wide channels (Hassan conditions were cut into the et al., embankment to allow tests to 2004) be conducted on individual sections Headcut Red sandy clay Channel width: 1.8 m height: The global rate of movement for a migration soil and silty 2.4 m set of flow conditions and soil sand soil Soils were used in various material properties appeared to be configurations and uniform; conditions Headcut migration rates for each test were determined based on linear regression of the observed headcut position versus time; Breach Three different Height: 2.3/1.5 m The reservoir water level, initiation/ soils, from silty Slope: 1:3 embankment erosion, and discharge formation sand to lean hydrograph were measured clay throughout the tests; Lead to a four-stage description of the embankment breach processes; Breach Homogeneous Height: 1.3 m The breach width during widening widening Slope: 1:3 was monitored during the test; A 0.3-m-wide notch in the The width of the breach opening was center of the embankment monitored over time to a maximum was created width of 5.5 m; The rate of widening was strongly influenced by the compaction water content.

Chapter 1. Bibliography

1.3.3. Headcut migration tests Through observations of the various tests, headcut migration has been observed as the controlling failure process. Various other tests have been conducted recently to better understand these processes (in China, the UE, and the US). The small-scale laboratory tests involve different flume sizes from 60 to 120 cm and include various embankment materials and profiles. The large-scale tests in the field involve scales varying from 2 and 6 m with different foundations and materials. Some of these tests were briefly introduced in the last section, and the experiments summarized here focus on the study of headcut migration.

1.3.3.1. Tests in the Netherlands A total of 5 laboratory experiments of embankment dam breaches due to overtopping failure were performed at the Delft University of Technology in the Netherlands (Zhu, 2006, Zhu et al., 2011). The tests were conducted in a straight flume that was 35.5 m long, 0.8 m wide and 0.85 m deep. Another flume was placed next to the main flume for water storage and sediment trapping. One of the tests was conducted with pure sand (test s), and the other four tests were conducted with different mixtures of sand, silt and clay (from test 1 to test 4). All five tests had the same profiles: 75 cm height, 40 cm crest length, 60 crest width, and 1:2 inner and outer embankment slopes. The developments of the embankment profiles during test 1, test 2, test 3 and test 4 are shown in Figure 1-21. During the test, the headcut erosion process was observed during the growth of the breach for embankments constructed both with cohesive materials and pure sand. Erosion is usually initiated at locations close to the toe of the embankment when overtopped.

1.3.3.2. Tests in China A total of 5 tests of embankment overtopping failure were performed at the Changjiang river scientific research institute, Changjiang River Water Resources Commission in China (Zhao et al., 2014). The tests were conducted in a 60-m-long flume, with a height of 3 m and a width of 3 m. A total of 4 tests had equal heights of 1.2 m, and a small test had a height of 0.6 m. All of the tests had a same upstream slope of 1:1 and the same crest width of 0.6 m. Two different downstream slopes of 1:3 and 1:2 were used for these 5 tests. A small notch with a width of 0.5 m was excavated in the side of the embankment crest for the 4 large tests and in the middle of the embankment crest (see Figure 1-22 (a)). All of these tests were conducted with a mixture of clay and silt.

1.3. Laboratory and field dam breach tests

(a) (b)

(c) (d) Figure 1-21. Embankment profile development of (a) test 1, (b) test 2, (c) test 3 and (d) test 4 (Zhu et al., 2011).

After initiation of the test, the headcut was first observed at the toe of the downstream surface. Then, several headcuts were formed over the entire downstream surface (see Figure 1-22 (b)). After erosion and headcut migration on the downstream surface, a breach was fully formed and began to widen. A hole is observed at the bottom of the side slope, and the eroded material was washed away to the downstream. The secondary flow in the breach channel was observed to play an important role. As the erosion of the soil at the toe of embankment side slope continued, slope stability was destroyed, and a soil material block collapsed into the breach channel (see Figure 1-22 (c)). This is the main reason for the increase in the breach width. As shown in Figure 1-22 (d), at the end of the test, a negative slope was observed. Erosion was found at the toe of the side slope, and on the upper side, a nearly vertical wall was generated. The headcut migration processes on the downstream surface and during the enlargement of the breach were both observed in these tests.

Chapter 1. Bibliography

(a) (b)

(c) (d) Figure 1-22. Images of embankment dam breach process observed in the tests; (a) Initial situation with a small preset notch on the side of the embankment crest; (b) Several headcuts on the downstream surface; (c) Collapse of soil mass of side slope; (d) Side slope view at the end of the test (Zhao et al., 2014).

1.3.3.3. Tests at the University of South Carolina, US A total of 8 experiments were conducted at the University of South Carolina, US (Feliciano Cestero et al., 2014). All of them had the same geometry: a height of 0.25 m, bottom width of 1.6 m, and crest width of 0.1 m. All of the tests were conducted in a 17-m- long, 0.5-m-wide and 0.7-m-deep horizontal flume. The storage reservoir was 1.39 m3 up to the level of the crest. Each of the experimental setups was constructed with different soil compositions. Figure 1-21 (a) and (b) show 3D views of the breach evolution over time. The recorded breach width during the experiments is shown in Figure 1-23 (c), and the recorded headcut migration is shown in Figure 1-23 (d). wis the measured breach width, and xh is the headcut migration distance.

1.3. Laboratory and field dam breach tests

(a) (b)

(c) (d) Figure 1-23. (a) 3D view of breach evolution over time: initial situation; (b) 3D view of breach evolution over time: during the breach; (c) Evolution of the breach width during the experiments; (d) Headcut migration during the experiments (Feliciano Cestero et al., 2014).

The observations of these tests indicate that headcut erosion plays an important role in the process of breach formation in embankment dam and levee failure. Headcut erosion has been widely recognized as the key erosion feature of surface erosion due to overtopping failure. The mechanism of the retreat of the headcut has been developed by many researchers (Hanson et al., 2005a, Zhu, 2006, Feliciano Cestero et al., 2014, Hanson et al., 2005b, Vaskinn et al., 2004). Recently, increasingly more experiments have indicated that it is also the control mechanism that determines the retreat of the side slope during the enlargement of a breach (Zhao et al., 2014, Hunt et al., 2005). Although the headcut is the key driving force behind both of these breach processes, the forces acting on it and the

Chapter 1. Bibliography

erosion processes that generate it are different. Currently, no mathematical model can describe the retreat of the headcut during the breach-widening process. Physical embankment dam breach models focus on the breach formation processes and not on the breach-widening process. Normally, after a breach is fully formed, the peak discharge flow and primary water surface begin to reduce based on observations of embankment dam failures (Hanson and Cook, 2004). However, for the failure of a levee, the water surface level is more stable compared to that of a reservoir. Thus, the breach- widening process remains significant, and the discharge flow can also increase. Various physical breach models consider the widening of the breach as a continuous erosion process of the side slope controlled by the bottom erosion rate (BREACH, BRES_Zhu), side slope erosion rate (BRES_Visser) or by the bottom erosion rate reduced by a proportionality factor (Rupro). Other physical breach models consider the widening of the breach as discontinuous processes resulting from soil erosion and episodic soil mass failure of the side slope. DamBreach checks the soil mass failure of side slopes using a 3D slope stability method. In addition, it uses both the sediment transport method and the bottom soil erosion method to calculate the erosion of soil masses. Therefore, the erosion rate on the side slope of a breach is assumed to be equal to the bottom erosion rate or to be proportional to the bottom erosion rate. The study of discharge flow in a breach and the shear stress distribution on the side slope and on the bottom is not sufficient.

1.4. Conclusions

CHAPTER 2

Pipe enlargement processes

The objective of this chapter is to setup a model to simulate the pipe development process in embankment dams and levees. Section 2.1 reviews the literature on the internal erosion and piping failure of embankment dams and levees, including initiation of the internal erosion and the development of pipes. Section 2.2 provides a literature review of the characteristics of pipe flows with erosion. Then, it provides a study of the soil erodibility, including introducing the tests conducted, the definition of the erosion rate coefficient and the critical shear stress. In section 2.4, a model is presented for simulating pipe development processes in embankment dams and levees. It accounts to the soil erosion law and turbulent flow effects on the large scale embankment dams and levees. The validation of this model by a large-scale test will be given in Chapter 6.

2.2. Pipe flow with erosion

2.1. Literature review of the failure of embankment dams and levees due to internal erosion and piping

Piping, or the internal erosion of soil, is the main cause of the serious failure of embankment dams and levees in terms of the risk of downstream areas being flooded (Bonelli and Brivois, 2008). Internal erosion occurs when water flows through a cavity, crack, and/or other continuous voids within embankment dams and levees. These openings may be the result of inadequate compaction during construction, differential settlement, desiccation, earthquakes, burrowing animals, and/or decay of woody vegetation roots (Hanson et al., 2010). Once openings are formed, they will quickly be enlarged by erosion of the surrounding wall. Embankment dams and levees sometimes fail in a short period of time, resulting in serious consequences to the downstream valley and the floodplain. Thus, research on internal erosion and piping is meaningful for flood risk management. Here, it is necessary to distinguish between the term piping and internal erosion failure. Because many internal erosion failures or accidents result in a tunnel or pipe-shaped erosion feature through the embankment, they are often referred to as piping failures. However, the term piping failure indicates the form of the internal erosion failure. However, other researchers have defined piping as another mechanism that causes seepage incidents to make a distinction with the internal erosion mechanism (McCook, 2004). Mainly, the key elements of our research are the progression of piping in the embankment; therefore, piping and internal erosion failures have not been distinguished in this research.

2.1.1. Introduction Through case studies of failures, the failure processes of embankment dams and levees due to internal erosion and piping have been divided into the following phases (Bonelli, 2013, Fell et al., 2003): the initiation and continuation of erosion, the enlargement of the pipe, the formation of a breach, and the widening of a breach. Illustrations of these phases are given in Figure 2-1 and Figure 2-2.

2.1.2. Initiation and continuation of erosion The internal erosion and piping of embankment dams and levees can be initiated by four main mechanisms: concentrated leak erosion, backward erosion, contact erosion and suffusion (Bonelli, 2013). Illustrations and descriptions of these four main mechanisms as well as their continuations are given in Table 2-1. The initiation of erosion may not always lead to a failure, but the existence of filters or transition zones can always lead to the continuation of erosion. According to the descriptions in Table 2-1, the continuations of

Chapter 2. Pipe enlargement processes

concentrated leak erosion and backward erosion can generate a pipe in the embankment, but the continuation of contact erosion and suffusion erosion does not always produce this result.

2.1.3. Pipe enlargement The initiations and continuations of internal erosion and piping in embankment dams and levees have been described in the previous section. Clearly, not every instance of the initiation of internal erosion and piping can generate a pipe. For other consequences, such as the instability of the downstream slope and the clogging of the permeable layer, are not considered in this section. Therefore, the initiation of pipe enlargement results from the lack of effective means to control the internal erosion that leads to a pipe. Subsequently, pipe enlargement progresses. Figure 2-4 shows the pipe enlargement during two large-scale internal erosion tests conducted at the HERU (Fell et al., 2003). Both of the tests had the same geometry: height of 1.3 m, crest with of 1.8 m and upstream and downstream slopes of 3:1. A 40-mm- diameter continuous steel pipe was placed in the embankment to initiate internal erosion. They were constructed with different soil materials. Figure 2-5 shows images obtained from the video of an internal erosion failure test at Rossaage (details of this test are given in Chapter 6).

2.2. Pipe flow with erosion

(a) The initiation of erosion (b) The continuation of erosion

(b) The enlargement of pipe (d) The formation of a breach

Figure 2-1. Pipe initiation, development and formation in the embankment initiated by the initiation of concentrated leak erosion (Bonelli, 2013).

(a) The initiation of erosion (b) The continuation of erosion

(b) The enlargement of a pipe (d) The formation of a breach

Figure 2-2. Pipe initiation, development and formation in the embankment initiated by the initiation of backward erosion (Bonelli, 2013).

Chapter 2. Pipe enlargement processes

Table 2-1. Illustrations and descriptions of four main mechanisms of initiation of internal erosion and piping (Courivaud, 2004).

Mechanisms Illustrations Descriptions Continuations of initiation

Erosion occurs when water flows through a cavity, crack, Concentrated and/or other continuous void A pipe is formed leak erosion within the embankment dam and levee

Soil particles are removed Backward under a cohesive surface or at A pipe is formed erosion the flow exit, and then, erosion progresses upward

This can cause a Fine soil particles are removed cavity within the along an interface between a unsaturated fill, Contact coarse material layer and a fine an erosion pipe, a erosion material layer under the action weak area in of the flow crossing the coarse embankment dam, material layer or clogging

Increasing more Fine soil particles are fine soil particles transported by the flow in the Suffusion begin to move and voids between the matrix of are transported by coarse materials flow

Based on observations of these tests and literature review, we note the following:

 Outflow through the pipe with soil concentration;  The opening of the pipe is uniform;  The pipe maintains its circular shape during opening;  Soil around the pipe can support its circular form against collapse;

2.2. Pipe flow with erosion

(a) Pipe enlargement at 5 minutes in test P1 (b) Pipe enlargement at 8 minutes in test P1

Figure 2-3. Pipe enlargement observed in test P1 at the HERU (Hanson et al., 2010).

Figure 2-4. Pipe enlargement observed in test P4 at the HERU (Hanson et al., 2010).

Figure 2-5. Pipe enlargement observed in test 3-2003 at Rossaaga.

Chapter 2. Pipe enlargement processes

(a) Before pipe collapse at 13 minutes (b) After pipe collapse at 13 minutes

Figure 2-6. Pipe collapse observed in the test P1 at the HERU (Hanson et al., 2010).

(a) Before pipe collapse at 13:55:46 (b) After pipe collapse at 13:56:40 Figure 2-7. Pipe collapse observed in test 3-2003 at Rossaaga.

2.1.4. Formation of breach As the pipe becomes large, a breach will eventually be formed. The transition from a pipe to a breach is a certain result of pipe failure. Once the transition occurs, the pipe will disappear and be replaced by a breach, which is a different failure process. As the pipe enlarges in the embankment dam, the positive arching formed by soil above it becomes weak. There exists a moment when the positive arching cannot support itself and subsequently collapses. This type of sudden collapse has been observed in tests (see Figure 2-6 and Figure 2-7). Various researchers have also defined this procedure in their piping failure models:

 The flow changes from orifice control to weir control when the head on the pipe is less than the pipe diameter. Upon reaching the instant of flow transition from

2.2. Pipe flow with erosion

orifice to weir, the remaining material above the top of the pipe and below the top of the dam is assumed to collapse (Fread, 1988).

 A stability analysis is conducted to evaluate the soil mass above the pipe. A safety factor of the soil above is defined as the ratio of the cohesion of the soil and the soil side areas to the soil weight (Morris et al., 2012).

 The vault of the pipe collapses after the diameter of the pipe has reached 2/3 of the dam height (Paquier and Recking, 2004). Experiment-based assumptions and theory-based analysis of soil stability have been developed. In this process, there are several issues affecting the enlargement of pipes: (1) the rate of pipe enlargement and the erodibility of the soil, (2) how the pipe expands and collapses, and (3) the flow characteristics in the pipe (see next sections).

2.1.5. Breach widening Once the pipe collapses, a breach is formed. The discharge flow will continues to enlarge the breach. The details of this process will be presented in the next chapter.

2.2. Piping flow with erosion

Once a pipe is formed in an embankment dam or levee, flow through the pipe is driven by a pressure difference between upstream and downstream. This pipe may be completely or partially filled with the flow. Therefore, the flow is fully or partially bounded by soil surfaces. When there is a flow through the pipe, it erodes soil and carries away the eroded particles. First, it is necessary to examine the soil surface erosion by tangential flow. The extension to piping flow with erosion is presented in the next part of this section.

2.2.1. General introduction of pipe flow Let us examine a horizontal pipe with a radius of R and a length of L . The pipe is assumed to be filled with flow. Figure 2-8 shows a typical average velocity profile and relative magnitude of the shear stress for a turbulent flow in the pipe. The flow profile of the turbulent flow is uniform, with a sharp decrease near the pipe wall. Based on the theory of no-slip conditions, the flow speed at the wall of the pipe is zero. The larger the Reynolds number is, the more uniform the velocity profile is. In turbulent flows, the average flow velocity v is considered as steady; therefore, it is always used as the flow velocity in the pipe.

Chapter 2. Pipe enlargement processes

Figure 2-8. A sketch of a horizontal pipe flow with the velocity profile of a turbulent pipe flow and the tensile strength distributions.

Figure 2-9. Tangential flow over erodible soil surface. On the upper water side, soil particles transport to this side, and on the lower solid side, water enters. As erosion progresses, more soil material is eroded toward the flow phase, and the singular interface (  ) degenerates toward the soil phase (Bonelli, 2012).

2.2.2. Soil surface erosion by tangential flow When the flow travels over an erodible soil surface, a transition between soil and water can occur. This transition produces an interface between soil and water: on one side of this interface, the water contains soil particles, and on the other side, the soil contains water. The mechanics of soil surface erosion by tangential flow is detailed by Bonelli and Brivois (Bonelli, 2012, Bonelli and Brivois, 2008). As shown in Figure 2, an area (  , a singular interface) between the soil and the flow is defined as an intermediary to exchange momentum with a two-phase model. On the upper water side, soil particles transport to this

2.2. Pipe flow with erosion

side, and on the lower solid side, water enters. As erosion progresses, more soil material is eroded toward the flow phase, and the singular interface (  ) degenerates toward the soil phase.

The tangential flow with erosion and the boundary condition of the interface can be defined by the following equations (Bonelli, 2012):

 Conservation equation in  u 0 in  (2.1) u   ( u )  u   T in  (2.2) t 

 Conditions on 

w  soil uu n, u  qD  (  1) c , on  (2.3) flow  flow

C e       (c ) if c c   d (2.4)  0 otherwise where  is the tangential stress on the interface  . The expressions of these equations are complex. To use this mechanism of soil surface erosion by tangential flow into pipe flow with erosion, certain simplifications are necessary.

2.2.3. Pipe flow with erosion High shear stress at the interface of the soil and flow can generate hydraulic erosion and induce the enlargement of the pipe. Using the Integrated Reduced Navier-Stokes/Prandtl equations with erosion (Bonelli, 2012, Lachouette et al., 2008), the characteristics of pipe flow erosion have been described using a momentum equation (2.5), turbulent flow equation (2.6) and a mass jump equation (2.7) on the interface. The quantity dv/ dt on the left-hand side of equation (2.5) accounts for some transitory effect, and the quantity vm is the contribution of the moving wall to the momentum balance. The enlargement of the pipe resulting from erosion has been described as equation (2.8). This equation shows that the erosion starts once the applied flow shear stress ( b ) exceeds the critical shear stress ( c ) of the soil (Temple, 1985, Stein and Nett, 1997, Kandiah, 1974, Partheniades, 1965, Bonelli, 2012).

Chapter 2. Pipe enlargement processes

R dv R w vm  p  b (2.5) 2 dt R0

2 b  w f b v (2.6)

dR m   (2.7) soil dt

C e      dR  (b c ) if b c   d (2.8) dt  0 otherwise

3 where w is the density of water (kg/m ), R is the pipe radius (m), R0 is the initial pipe radius (m), v is the flow velocity (m/s), m is the mass flux crossing the interface 2 (kg/m /s), P is the driving pressure (Pa),  b is the applied shear stress (Pa),  c is the critical shear stress of the soil (Pa), fb is the turbulent friction factor in the pipe, soil is the 3 density of the soil (kg/m ), c is the interface celerity (m/s), Ce is the erosion rate 3 coefficient (s/m), and d is the dry density of the soil (kg/m ).

2.3. Soil erodibility

The erosion rate resulting in the enlargement of the pipe is calculated using equation (2.8). The erosion rate is considered to be zero when the hydraulic shear stress is less than the critical shear stress. In this equation, there are two parameters, Ce and  c , that determine the rate of enlargement. Ce is a property of the material and is obtained from soil tests. In addition, it determines the soil’s resistance to erosion (Wan and Fell, 2004). These two parameters are extremely complex to determine. To measure them, various soil erodibility testing methods in the laboratory and in the field have been developed.

2.3.1. Soil erodibility tests

2.3. Soil erodibility

Soil erodibility tests have been developed by many investigators over many years to quantify the erodibility of soil materials. Following with understanding the failure mode of embankment dams and levees, soil erodibility tests have been improved to model various situations, including the overtopping flow on the crest and downstream surface of embankment dams at the initiation of breach failure processes, piping erosion and concentrated leak erosion in embankment dams, and discharge flows eroding soil mass at the base of side slopes during breach widening processes. These tests can be grouped into the following categories:

 Rotating cylinder tests Rotating cylinder tests (Moore and Masch, 1962) use a soil block suspended and submerged inside a rotating cylindrical chamber, which induces a flow around the specimen to cause erosion. The critical shear stress and erosion rate are determined to estimate the erodibility parameters. This was an early test method has and has been used as a research tool for studying the relationship between erosion characteristics and various fundamental soil properties.

 Flume tests Flume tests are an open-channel erosion test conducted in an open channel (Gibbs, 1962; Kandiah and Arulanandan, 1974). Several small-scale and large-scale flume tests have been conducted to measure the erosion of cohesive soil.

 Jet erosion tests Jet erosion tests were developed to measure erodibility parameters in the laboratory as well as in situ (Hanson, 1991, Hanson and Cook 2004). The procedures of the test and the development of analytical methods were presented by many researchers (Hanson and Cook, 1997, Al-Madhhachi et al., 2013). This test uses a submerged hydraulic jet to produce scour erosion, similar to processes that occur at a headcut or a free overfall. Therefore, it is always used for the measurement of soil erosion parameters in spillways or on the downstream face of embankment dams due to overtopping failure.

 Hole erosion tests Hole erosion tests were developed by Wan and Fell (Wan and Fell (2002, 2004a, b) to measure the erosion properties of soils. They have been refined by Benahmed and Bonelli (Benahmed et al., 2012). This test utilizes an internal flow through a hole pre-drilled in a test specimen, similar to a piping erosion configuration. Thus, it is always used to assess the erosion characteristics of the soil in embankment dams due to piping failure.

Chapter 2. Pipe enlargement processes

Considering the erodibility of the soil materials of embankment dams and levees, jet erosion tests and hole erosion tests have been more widely applied. Based on the study of the different range of soil materials and different erosion and flow situations, the soil erosion rates and critical shear stresses obtained using these two tests may be different. Wahl performed two series of hole erosion tests and jet erosion tests with individual specimens prepared with the same compaction effort and moisture content. Comparisons of the erosion rate coefficients and critical shear stress values have been conducted. The HET was found to indicate higher critical shear stress and higher erosion rate index values for all of the soils. The erosion rate coefficients obtained by jet erosion tests were approximately 0.75 to 1 orders of magnitude greater than those obtained from the hole erosion test, and the critical shear stresses were approximately 2 to 3 orders of magnitude lower in the jet erosion tests compared to in the hole erosion tests.

2.3.2. Erosion rate coefficient According to the descriptions of soil erodibility tests, different testing methods can yield different erodibility parameter estimations. The erosion rate coefficient and the critical shear stress have been observed to vary over several orders of magnitude. To properly choose erodibility parameters for use in embankment dam breach models, their measurement and classification are reviewed.

k d is often used as the erosion rate coefficient in jet erosion tests and Ce in hole erosion tests. Based on in situ jet erosion tests in the loess areas of the Midwestern USA, Hanson and Simon (Hanson and Simon, 2001) classified the values of k d into 4 classes; the lower the class, the greater the resistance to erosion. The Bureau of Reclamation extended the classifications of k d values into 5.5 orders of magnitude, from 0.001 to 300. The classifications of k d values are shown in Figure 2-10. As obtained by hole erosion tests (Wan and Fell, 2004), the proportionality constants for erosion rates vary over several orders of magnitude in soils of engineering interest. For convenience, the erosion rate index ICHET  log( e ) was presented by Wan to describe the progress of soil erosion of embankment dams due to piping failure. The soil erodibility decreases with increasing erosion rate index. Six groups are classified corresponding to the erosion rate index from 1 to 6. Large values indicate decreasing erosion rate or increasing erosion resistance (see Table 2-2).

2.3. Soil erodibility

Figure 2-10. Hole erosion test and jet erosion test data collected by the Bureau of Reclamation in field and laboratory tests since 2007. Erodibility classifications are from Hanson and Simon (Hanson and Simon, 2001).

Table 2-2. Qualitative description of rates of progression of internal erosion or piping for soils with specific erosion rate indices (Wan and Fell, 2004).

Group Erosion rate index Description

number ICHET  log( e ) 1 <2 Extremely rapid 2 2-3 Very rapid 3 3-4 Moderately rapid 4 4-5 Moderately slow 5 5-6 Very slow 6 >6 Extremely slow

Using numerical analyses of pipe enlargement based on experimental results, the erosion rate index was extended by Bonelli (Bonelli, 2013) to predict the time to failure (from detection to maximum pipe diameter) due to piping erosion failure of embankment dams and levees. Under the assumption of a constant pressure drop (this gives more conservative

Chapter 2. Pipe enlargement processes

results and is reasonable for analyses of levee failure with a constant water level upstream), the time to failure has been observed to be highly dependent on the soil erosion index (see Figure 2-11). For example, if the failure indicator is of the order of magnitude of 2, the failure time is less than one hour, which means that the failure will take place very quickly after visual detection. If I HET  4 , the failure time is more than one day, which means that failure will not occur for several days. Two different dam heights of 5 and 10 m were considered, and it is obvious that the failure times for these two classic dams are approximate.

Figure 2-11. The failure time ( tu ) as a function of the erosion rate indicator ( I HET ) for

two dam heights (Hdam=5 m and Hdam=100) (Bonelli, 2013).

2.3.3. Critical shear stress As described by equation (2.8), the erosion rate is considered to be zero when the hydraulic shear stress is less than the critical shear stress. This is defined as the stress at which soil detachment begins or the condition that initiates soil detachment (Clark and Wynn, 2006). Therefore, it classifies the soil’s resistance to erosion. There are two main approaches used to determine the critical shear stress:

 Laboratory and field tests (cohesive soil)  Relationship between other soil parameters (non-cohesive soil)

2.3. Soil erodibility

Laboratory and field tests (see the last section) are a common method of measuring the critical shear stress of cohesive soils. By considering the impacts of biological, physical and chemical factors, field tests can better address the disadvantages of laboratory tests. For non-cohesive soils, the estimation of the critical shear stress is based on the grain size and the soil density according to Shield (Shields, 1936) or the relationship of the erosion index according to Wan (Wan and Fell, 2002). Table 2-3 shows the approximate estimates and likely range of initial shear stress versus hole erosion index.

Table 2-3. Approximate estimates and likely range of initial shear stress ( c ) versus hole

erosion index ( I HET ) (Bonelli, 2013).

Initial shear stress ( c ) (Pa) Hole erosion Non-cohesive soil Cohesive soil behavior behavior index ( I HET ) Best Likely Best Likely estimate range estimate range <2 2 1-5 1 0.5-2 2-3 2 1-5 1 0.5-2 3.5 5 2-20 2 1-5 4 25 10-50 5 2-10 5 60 25-100 5 2-10 6 100 60-140 5 2-10

2.4. Model for pipe enlargement

To simulate pipe enlargement in embankment dams and levees (dams), a model based on the extension of the model of pipe flow with erosion is presented. This section introduces the simulation of pipe enlargement, the change from pipe flow to open-channel flow, the judgment of the sudden collapse of the pipe top and the calculation of the pressure drop between the up and down streams.

2.4.1. Model description

2.4.1.1. General description

Chapter 2. Pipe enlargement processes

This model simulates the enlargement of a pipe, as shown in Figure 2-11. This dam is homogeneous and consists of only one material (soil density, soil ; soil dry density, d ; the turbulent friction factor in the pipe, fb ; and the erosion rate coefficient Ce ). A straight, horizontal and cylindrical pipe with an initial radius of the pipe is assumed to already be initiated. The pipe has already appeared but only in the dam. The geometries of the dam are described in Table 2-2. Therefore, the length of pipe’s center line when the pipe radius equals R can be expressed as

LR L c ()()() H  R s up  s down  L b  R s up  s down (2.9)

The characteristics of the upstream flow are described based on the water depth near the dam, yup . Therefore, a relationship between the storage reservoir and the water depth or the variation in water depth upstream is needed.

Table 2-4. The geometries of the dam.

Dam Geometries

Height Hdam

Crest length Lc

Bottom length Lb

Upstream slope sup  Horizontal/ Vertical

Downstream slope sdown  Horizontal/ Vertical

Pipe initial radius R0 Pipe radius R

2.4.1.2. Simplification of pipe enlargement processes

2.4.1.2.1. Definition of initial pipe radius The erosion of the pipe occurs when the driving force is larger than the resistance force of the soil. Based on the study of Bonelli (Bonelli, 2013), the critical radius of the pipe depends not on the flow regime and embankment dam height but on the critical shear stress of the soil on the wall of pipe. The critical radius is considered as a proportion of the critical shear stress, with a range between 6 and 6 cm ( Rc 0.06 c in cm). For example, silty sand with low plasticity and dispersive clays have lower critical shear stresses, as low as 1 Pa; therefore, the critical radius of the pipe can be as narrow as 1 mm.

2.4. Model for pipe enlargement

Figure 2-12. The simplified longitudinal profile of the embankment and foundation simplified oflongitudinal and The in the model. 2-12. Figure profile the embankment

Chapter 2. Pipe enlargement processes

2.4.1.2.2. Pipe enlargement Once the pipe radius is larger than the critical radius, the enlargement of the pipe begins. The pipe is assumed to be uniformly enlarged by hydraulic erosion and maintain its circular cross section shape (see Figure 2-13 (a) and (b)).

2.4.1.2.3. Pipe collapse The collapse of the pipe top is the transition of the embankment dam failure from the pipe enlargement process to the breach widening process. In this model, the pipe is assumed to collapse when the top of pipe reaches the crest of the embankment dam (see Figure 2-13 (c) and (d)).

Figure 2-13. Pipe enlarges after initiation until the collapse of the pipe top. (a) The critical situation of pipe erosion; (b) The enlargement of the pipe (b) and (c); (d) The sudden collapse of the pipe top, upon which the breach widening process begins.

2.4.2. Model calculation

2.4.2.1. Pressure drop in the pipe

2.4.2.1.1. Pressure drop described by pressure difference The pipe flow is driven by a difference in pressure between the pipe entrance (due to the water head in the reservoir near the dam) and the pipe exit (due to the water head in the downstream). The total pressure drop through the embankment may be described as the difference between the pressure in the up- and downstream as

pT  w g() y in  y out (2.10)

2.4. Model for pipe enlargement

where pT is the total pressure drop (Pa); yin (m) is the water head at the pipe entrance, considered to be equal to the water head at the reservoir or river near the dam and always given by the discharge hydrograph in the river or the characteristics of the storage reservoir; and yout is the water head at the pipe exit, and the determination of this value will be discussed below.

2.4.2.1.2. Pressure drop described by head loss

2 In addition, the total pressure drop can also be defined as a singular head loss (1 / (2k w v ) ) and pressure drop in the pipe ( p ) by equation (2.11).

1 p   p () k  k v2 (2.11) T2 in out w where v is the average flow velocity in the pipe (m/s); kin is the singular head loss coefficients at the entrance of the pipe, and kout is that at the exit of the pipe.

2.4.2.2. Moment equation To use the theory of pipe flow with erosion in this model, the transitory effect of the pipe flow is assumed to be neglected. Therefore, the momentum equation (2.5) can be simplified as equation (2.12):

R vm  P b (2.12) R0

The driving pressure P in the pipe may be expressed as equation (2.13). This equation expresses the relationship between the pressure drop and the driving pressure in the pipe.

R p P  0 (2.13) 2LR

2.4.2.3. Enlargement of the pipe Under the uniform assumptions for the pipe enlargement process, the pipe enlargement evolution equation is formed as equation (2.14):

Ce dR  (b  c ) if  b   c  d (2.14) dt  0 otherwise

2.4.2.4. Calculation of flow velocity in the pipe

Chapter 2. Pipe enlargement processes

By combining equations (2.10)-(2.14) and equations (2.6)-(2.7), equation (2.12) becomes

2LR soil C e 2  kin k out2 L R f b  2 ytytin()()(())()() out  w fvtvt b   c    vt (2.15) R( t )w g  d  2 g gR 

1) Evolution of upstream water level ( yin () t ) In the upstream, the variation in water level in the river, inflow and outflow of the reservoir, and the relationship between the storage reservoir and water head are usually given as the upstream boundary condition.

2) Downstream water level ( yout () t ) In the downstream, the discharge flow through a breach will propagate in the floodplain/ watershed, where two different flow regimes can form: supercritical flow and subcritical flow. Under subcritical flow conditions, the flow is controlled by the flow condition downstream, and information will be transmitted upstream. This condition leads to backwater/tail-water effects. Supercritical flow is controlled by the flow condition upstream, and the disturbance will be transmitted downstream. Considering these conditions, a comparison of water levels downstream, ydown , and at the exit of the pipe, yexit , is necessary. In this model, the higher level is chosen as the controlled water head.

yout max( y down , y exit ) (2.16)

2.4.2.5. Calculation of discharge flow in the pipe By obtaining the mean flow velocity, the discharge flow (Q) through pipe may be deduced using the continuity equation (2.17):

Q()()() t v t A t (2.17) where A() t is the cross-section area (m2). If the upstream water head is higher than the top of the pipe, Aequals the cross-section of the pipe; see equation (2.18). If the upstream water head is lower than the top of pipe, A may be expressed using equation (2.19): A()() t  R t 2 (2.18)

A() t R () t2  () t  sin2()/2  t   (2.19) where  ()t may be expressed as:

 (t ) acr cos 1  yout ( t ) / R ( t )  (2.20)

2.4. Model for pipe enlargement

2.4.2.6. Calculation of pipe radius By applying the equation (2.6) in the pipe enlargement equation (2.14), the increase of the pipe radius, R , can be expressed by equation (2.21). Therefore, the pipe radius may be expressed as equation (2.22). t C e 2   2    wfb v()() tc  dt if w f b v t c R() t   0 d (2.21)  0 otherwise

R()()() t R t   R t   t (2.22)

2.4.3. Computational algorithm A simple iterative algorithm is used to computer for the flow, erosion, and pipe properties. An estimated incremental erosion width is used at each time step to start the iterative computation. The flow chart of numerical simulation is shown in Figure 2-14. 1) The time step is specified as 2 seconds;

2) The input data: the geometries of the dam ( Hdam , Lc , Lb , R0 sup and sdown ); the

soil properties ( Ce , soil , d , fb ); the variation in water head upstream ( yin () t ); 3) Compare the water head at the entrance of the pipe with the top of the pipe;

4) Compute the water head at the exit of the pipe ( yout ); 5) Compare the water head at the exit of the pipe with the downstream water head; 6) Compute the mean flow velocity and the discharge flow in the pipe ( v and Q ); 7) Compute the erosion and the increase in pipe radius ( R); 8) Compute the new pipe radius ( R );

9) Check if the top of the pipe remains smaller than the dam height ( 2RH dam ?); 10) The end of one loop.

2.5. Conclusions

A pipe enlargement model is proposed to simulate the evolution of the pipe in embankment dams and levees. In this model, the turbulent pipe flow with erosion mechanism is employed as well as the soil erosion law. Meanwhile, the hydraulic head variation in the upstream, the trail water conditions in the downstream, the collapse of the pipe top and the transition to a breach are taken into account. The validation of this model by a large-scale test will be given in Chapter 6.

Chapter 2. Pipe enlargement processes

Figure 2-14. The flow chart of the numerical simulation.

CHAPTER 3

Breach widening processes

Many researchers have developed mathematical models of the enlargement of pipes in embankment dams, the formation of breaches due to overtopping and the corresponding discharge flow. Little attention has been given to the simulation of widening processes during a breach. In fact, after a breach is formed, the water level in the river and the amount of water stored in the reservoir remain relative high. Thus, a large amount of flow should be discharged through the breach until the end of failure. Consequently, it is important to study breach widening processes to predict integral discharge flow during a breach. In this chapter, section 3.1 includes a literature review of breach widening in embankment dams and levee failure. Breach widening processes, including the formation, enlargement and final geometry of the breach, are described based on tests presented in the literature. In section 3.2, definitions of the breach width, the geometry of the breach, and headcut migration are provided. Based on these definitions, a simply headcut migration model based on the soil tensile strength is presented in section 3.3 to simulate the critical length of the headcut based on the breach geometry, the soil properties and the flow situations in the breach. In the fourth section, the results are obtained by extending the analytical model to the general situation. The conclusions are given in the last section.

3.1. Literature reviews

3.1. Literature review of the breach widening processes

Knowledge of breach widening mechanisms is important for defining the geometry, breach width and discharge flow of a breach. Recent laboratory and field experiments have improved our knowledge of breach widening mechanisms.

3.1.1. The formation of a breach

3.1.1.1. The breach initiation The formation of a breach, also called embankment dam and levee failure, can be caused by different situations. In the last chapter, we describe how internal erosion and piping cause failure. In addition, the overtopping flow, sliding of the downstream slope, earthquakes, and other causes can generate failure. Figure 3-1 illustrations of the formations of a breach. Regardless of the cause of embankment dam failure, the breach begins to enlarge after it is formed. Figure 3-2 shows a typical longitudinal profile and breach cross-section.

3.1.1.2. Initial breach geometry Figure 3-2 provides a general definition of the initial geometry of a breach. Breaches are difficult to observe during actual failure. Some breach pictures taken during the failure tests are shown in Figure 3-3. In these pictures, the upper parts of the breach can be observed above the water surface, where the side walls are nearly vertical. Because of the high soil concentration flow through it, the foundations and lower portions of the breach in the upstream regions cannot be observed directly.

3.1.2. Breach enlargement Regardless of how a breach is formed, the embankment dam or levee is separated into two banks, the ends of which are the two side walls of the breach. Before this process, the failure is not complete because the discharge flowing through the breach could still remove soil blocks and particles and enlarge the breach. Relatively large flow volumes result from breaches into a floodplain or downstream valley. Thus, how a breach becomes enlarged is important when studying its discharge flow hydrograph.

Chapter 3. Breach widening processes

(a) The internal erosion and piping processes

(b) The overtopping processes

(d) Erosion of the downstream foot Figure 3-1. Illustrations of the formations of a breach due to internal erosion and piping, overtopping, and other causes.

3.1. Literature reviews

(a) The longitudinal profile of a breach.

(b) The cross-section of a breach.

Figure 3-2. A typical sketch of a breach.

(a) Homogenous gravel dam

(b) Homogenous cohesive dam (c) Homogenous moraine dam Figure 3-3. Pictures of different breach geometries in tests.

Chapter 3. Breach widening processes

3.1.2.1. Breach widening tests Several photos of breaches taken during tests are presented from the literature. Figure 3-4 shows a field failure test of a homogeneous moraine dam with a height of 4.3 m, and Figure 3-5 shows another field failure test of a cohesive homogeneous dam with a height of 1.3 m. The breach widening processes of a river levee with a height of 4 m are shown in Figure 3-6. The main difference between the failure tests conducted on the embankment dam and levee is the direction of incoming flow. For the two cases shown in Figure 3-4 and Figure 3-5, the direction of incoming flow is perpendicular to the axis of the embankment dam. By contrast, the direction of incoming flow is parallel to the axis of the levee for the case shown in Figure 3-6, which is consistent with actual cases. As shown in Figure 3-4, flow came from the upstream reservoir and embankment dam and widened the left and right banks. Because of the erosion at the foot of the dam (Figure 3-4 (a)), an overhanging soil block was formed that eventually collapsed (Figure 3-4 (b)). As shown in Figure 3-4 (a) and Figure 3-4 (b), tensile cracks appeared from the crest to the side wall. The breach width was measured using a video taken during the test. As shown in Figure 3-4 (c), the breach width evolved in steps. The widths of the collapsed soil blocks were between 0.6 and 1.4 m. The final widths of the breach on the left and right banks were not very different. Similar to that described in Figure 3 4, the breach became wider as the side walls began to overhang, which resulted in block failures. A crack along the surface was observed at the initiation of soil block failure from the side banks. A measurement of the breach width (Figure 3-5 (b)) showed that widening occurred in an episodic fashion on the left and right banks. The widths of failing soil blocks varied from 0.2 to 1 m, with an average of 0.6 m. The average width of 0.6 m was approximately half the height of the embankment (Hunt et al., 2005). The final widths of the breaches on the left and right banks were approximately equal. In contrast with the last two field tests of embankment dam failure, Figure 3-6 show the breach widening processes of a levee failure test. The flow direction in this test was parallel to the axis of the levee. Once the initial notch proceeded to a breach (Figure 3-6 (a)), it progressed in the upstream and downstream directions (compared with the flow direction in the flume) (Figure 3-6 (b)). Once most of the cross-section of the levee was eroded where the notch was located, the breach mainly progressed in the downstream direction (Figure 3-6 (c)). Figure 3-6 (d) show the continuous progression of the breach width in the downstream direction.

3.1. Literature reviews

(a) Overhanging side walls and a tensile (b) Collapse of a soil block from the side wall crack on the right bank

8 6 4 Left 2 Right 0 -2 -4 Breach width (m) -6 -8 -10 13:40:00 13:47:12 13:54:24 14:01:36 14:08:48 14:16:00 Time (hh:mm:ss)

(c) Episodic breach widening measurements from the left and right banks (from 13:56:14 to 14:16:00) Figure 3-4. Photos of breach widening obtained from the literature.

Chapter 3. Breach widening processes

(b) Episodic breach widening measurements (a) Formation of a crack prior to failure from the left and right banks

Figure 3-5 A photo taken during breach widening tests and a figure showing the measured breach widths (Hunt et al., 2005).

(a) Initial notch and flow direction (b) Beginning of widening

(c) Acceleration of widening especially along the (d) Deceleration of widening flow direction Figure 3-6. Photos of breach widening process taken by tests (Kakinuma and Shimizu, 2014).

3.1. Literature reviews

3.1.2.2. Observations obtained from the literature Some common phenomenon is observed by tests:

1) Several common phenomena observed by researchers: 2) Discharge flow through the breach erodes soil particles under the water surface; 3) Episodic collapses of soil blocks from the side walls enlarge the breach width on the upper side;

4) Side walls above the water surface remain vertical and constant until an episodic collapse is encountered;

5) Overhanging soil blocks have been observed on the side walls; 6) The breach enlargement process is not continuous; 7) Before collapse, tensile cracks have been observed on the crest that quickly extend throughout the side walls and reduce soil block failure;

8) The breaches in the two banks evolve approximately the same during embankment dam failure, but are different for levee failure.

3.1.2.3. Existing models for simulating breach enlargement According to the literature review, several models were developed to simulate the breach widening process. The evolution of existing models for simulating breach widening processes can be simplified as follows:

 Continuous process with sediment transport Early models have considered breach widening as a continuous process by using the sediment transport capacity to calculate soil erosion and to calculate the discharge flow using the broad-crested weir formula and the 1D and 2D St Venant equations (Wang and Bowles, 2007, Paquier and Recking, 2004, Mohamed et al., 2002).

 Continuous process with soil erosion Next, the theory of river sediment transport was recognized as distinct from hydraulic erosion, which reflects the erodibility of soil during failure. In addition, the rate of erosion has widely been used to estimate losses in soil volume or the breach length with time (Hunt et al., 2005, Temple et al., 2005). It has been concluded that hydraulic stress should play an important role in calculating the rate of erosion (Bonelli, 2013, Visser, 1998, Zhu et al., 2005). Therefore, a relationship between the hydraulic stress and the rate of erosion has been presented as a governing equation for simulating the breach width.

Chapter 3. Breach widening processes

 Discontinuous process and instability of soil block in side walls Some models have considered that the breach is enlarged due to the instability of the soil blocks on the two side walls of the breach. Wang (Wang and Bowles, 2006) developed a 3D model to analyze the stabilities of sidewalls due to soil erosion and sediment transport. The enlargement of the breach resulted from the movement of the unstable soil block.

3.1.3. Final breach The final breach is reached when the soil erosion process is finished and the side walls of the breach remain constant. The soil erosion process ends when weak flow velocity cannot erode more soil from the breach. In the case of levee failure, the same process that occurs during embankment dam failure may occur. Meanwhile, the erosion process may end due to an increase in the water level in the floodplain, which can decrease the flow velocity and the pressure difference between the upstream and downstream.

(a) Breach geometry viewed from downstream at (b) Breach geometry viewed from upstream at the the end of test end of test (Zhao et al., 2014)

(c) The failure of a dam in Chouteau County in 2011 in US Figure 3-7. Finial breach geometry at the end of the embankment dam test and the actual failure case.

3.2. The breach width expression

After a breach is fully formed, it begins to widen and deepen through hydraulic discharge flow. From the observations, experiments and literature review presented in the last section, the discharge flow through a breach erodes the soil material from the bottom and toe of the side slope, resulting in an eroded notch. Above the water surface, nearby vertical sidewalls are observed. As the erosion and evolution of the eroded notch continues, the slope stability is broken and the soil material block collapses in the breach channel. Because this phenomenon is similar to the headcut advance that governs the processes of overtopping flow (Hanson et al., 2005, Zhu, 2006, Zhao et al., 2013), it is referred to as headcut migration. An illustration of the headcut migration of the side slopes during breach widening processes is shown in Figure 3-8. In detail, the headcut is formed because of hydraulic erosion under the water surface, which migrates along the axis of the embankment dam/levee through the episodic collapse of the soil mass. The migration of a headcut on the side slopes of a breach is illustrated in Figure 3-8. An illustration of the headcut migration of a side wall during a breach widening processes is shown in Figure 3-8. In detail, an eroded notch is formed due to hydraulic erosion under the water surface. The side wall migrates along the axis of the embankment dam/levee due to the episodic collapse of the soil mass.

3.2.1. Simplifications of breach geometry To simulate the breach widening processes, the geometry of the breach must be simplified. Regardless, of the different embankment dam failure modes, the breach begins to widen once it is completely formed. An initial rectangle cross-section is assumed before the breach begins to widen. Based on the experimental observations, the breach in this process is assumed with vertical walls above the water surface and overhanging walls under the water surface (see Figure 3-9). The width of the breach is determined by the distance between the two side walls. Generally, a large final breach width will be achieved relative to the height of the breach. Thus, the width of the breach is measured as the distance between the two side walls at the crest. Along with the discharge flow through the breach, the two side walls retreat due to the eroded notch generated by hydraulic erosion under the water surface. On the left side of the breach, the width of the soil block is Ll , and the erosion time resulting in the failure of the side slope is tel . On the right side of the breach, the width of the soil block is Lr and the erosion time is ter (see Figure 3-9).

Chapter 3. Breach widening processes

Figure 3-8. Illustration of the headcut migration of the side slopes during breach widening processes of embankment dam/levee failure.

Figure 3-9. 2D sketch of the breach width.

3.2. The breach width expression

3.2.2. Calculation of the breach width

Based on the above descriptions and simplifications of breach geometry, the stepwise enlargement of the breach width be presented using equation(3.1).

B t  t   B() t  Ll  L r (3.1)

t  min( tel , t er ) (3.2)

Where, B is the width of the breach, t is the time needed for each episodic collapse of headcut, tel is the erosion time that results in an eroded notch and the failure of the headcut on the left side of the breach and ter is the erosion time on the right side of the breach. Influenced by the coming flow directions in the reservoir and protected by the embankment dam or in the river protected by the levee, the flow conditions in the breach are different. These differences determine the erosion rates of the two side walls and the breach width. Previously, no physical model of breach widening has addressed the different influences of erosion on the two side walls due to flow conditions. In these studies, breach widths on both of the side walls were always assumed equal. This assumption was consistent with most field and laboratory test results. For example, in a flumes test, the axis of the embankment dam is along the cross-section of the flume, and the arriving flow is perpendicular to the embankment dam. This process is common during the failure of an embankment dam that protects a reservoir. However, this process is not always consistent with river levee failure cases. In the normal section of the river, flow is generally parallel to the levee and flow conditions in the breach are different from those in previous cases.

3.2.3. General classification of breach widening According to the directions of coming flow, breach widening processes are divided into the two following types:

 Symmetrical breach widening It is commonly assumed that the breach widening is symmetrical in many physical models used for breach widening processes. Before initiation of failure, water in the reservoir is relatively stable, and the incoming water flows perpendicular to the embankment dam. As shown in Figure 3-10, the conditions of the flow coming into the breach upstream of the embankment dam can be considered as symmetrical. Thus, the hydraulic erosion generated by discharge flow can be considered the same on the two side walls. In addition, breach

Chapter 3. Breach widening processes

widths are the same on the two side walls of the breach. Based on this assumption, equation (3.1) for the breach width can be rewritten to form equation (3.3). B t  t   B( t )  2 L (3.3)

t  te (3.4) Where, the number 2 on the right side of the equation indicates that the two side walls have the same contributions to the breach width; L is the width of the headcut LLLL  r ; and

te is the erosion time te t el  t er .  Independent breach widening In contrast with the symmetrical breach widening, the two side walls always have a different degree of erosion and, consequently, different degrees of breach widening. As shown in Figure 3-11, the flow in the river is parallel to the embankment dam, and the hydraulic erosion on the two side walls of the breach generated by discharge flow through the breach is obviously different. The widening of the breach is contributed by the headcut migration of the two side walls. Equation (3.1), which is used for the breach widening, remains the same in this case.

3.2.4. The knowledge of the two variables The processes of breach widening due to side wall recession result from soil erosion and headcut migration. Soil erosion can provide the erosion time, te , and headcut migration can provide the headcut width L.

3.2.4.1. Soil erosion The excess shear stress erosion equation is used to determine the amount of soil eroded from the notch. It is assumed that the rate of soil erosion is proportional to the effective shear stress in excess of the critical shear stress (Hanson and Cook, 2004). This rate provides the erosion time and predicts the possibility of side wall migration.

3.2.4.2. The headcut migration process The headcut migration equation describes the physical process of side wall retreat that governs the widening of the breach. It is possible to link the headcut migration process to soil erosion. The episodic failure of the soil block results from soil erosion on the toe of the side slope, where the hydraulic shear stress is assumed to be a driving force in eroding the soil mass. The unstable soil block falls into the breach and results in the enlargement of the breach. The entire process can be regarded as the headcut migration process.

3.2. The breach width expression

Figure 3-10. Tope view of the breach with symmetrical incoming flow.

Figure 3-11. Tope view of the breach with parallel incoming flow.

Chapter 3. Breach widening processes

3.3. Headcut migration

As observed in the laboratory, soil mass failures more likely result from headcuts falling into the breach due to the evolution of the eroded notch through soil erosion (Hassan et al., 2004, Hunt et al., 2005, Castedo et al., 2013, Hayakawa and Matsukura, 2010). Removal of soil from the toe of the side walls results in an eroded notch. As this eroded notch evolves, the soil block above it becomes a cantilever-like beam that is fixed at the end of the side wall and is free at the other end of the breach. Removal of soil from the toe of the side slope effectively removes physical support for the upper part and results in mass failure. These processes are shown in Figure 3-12, in which the dashed lines show the evolution of an eroded notch.

Figure 3-12. Illustration of the headcut migration process of a side wall due to hydraulic soil erosion during the breach widening processes.

3.3.1. Failure modes As described above, two possible failure modes likely occur when considering the cantilever-like beam (the generated headcut from the evolution of the eroded notch due to hydraulic soil erosion at the toe of the side wall).

1) Bending failure As the eroded notch evolves, a crack is formed at the crest. Next, the crack expands along the entire side wall until the headcut fails. This process is a typical mode of a cantilever- like beam failure (see Figure 3-13 (a)).

2) Shear failure Another possible failure mode is the shear failure, which results from sliding of the entire soil mass along the fixed end of the cantilever-like beam (see Figure 3-13 (b)).

3.3. Headcut migration

(a) Bending failure (b) Shear failure

Figure 3-13. Illustrations of the failure modes of a headcut (a cantilever-like beam).

 1 H

 h 2 H w

hw  2 h e s h ze e e s z

0 x l e (b) Coordinate system of an eroded (a) Sketch of a headcut notch Figure 3-14. Headcut notation.

Chapter 3. Breach widening processes

3.3.2. Sketch of a headcut A 2D sketch of a simplified headcut cross-section is shown in Figure 3-14 (a). H is the embankment height, hw is the water head in breach, he is the erosion height, and le is the width of the erosion width (equals to the width of the headcut). The soil mass of the headcut is divided into three different zones according to its specific weight (influenced by water conditions) and shapes as follows:

1) Zone 1 : The first zone is rectangular, above the water surface, and has a specific

weight of 1 ; 2) Zone 2 : The second zone is the rectangular part of the soil mass under the water

surface, with a specific weight of  2 ; 3) Zone 3 : The third zone is the triangle portion of the soil mass under the water

surface with a specific weight of  2 . Three surfaces of this headcut are visible under the water surface, the right side, left side and under-side. The right side surface of the headcut is assumed as a vertical plane (a vertical line in this cross-section), and the under-side surface is assumed as a plain (a straight line in this cross-section). According to these assumptions, the two border lines of the headcut may be expressed as shown in equation (3.5) in the coordinate system of Figure 3-14 (b).

x l h  z  h (right side)  e e w  h (3.5) e   z( x ) x 0 x le (under-side)  le

When simplifying the headcut geometry and dividing the soil zones, the following assumptions have been made:

1) The crest level: The crest level near the generated headcut is assumed as horizontal with a uniform height, and the irregularities of the crest near the generated breach are not considered.

2) The height of the headcut: The height of the headcut is calculated using the height of the embankment dam;

3) Side walls of the headcut: Above the water surface, the side walls of a headcut are assumed vertical towards the breach. According to experimental observations, a vertical side wall assumption above the water surface is reasonable.

3.3. Headcut migration

4) The water head near the headcut: The water level near the headcut is assumed to equal the average water level in the breach.

3.3.3. Failure criterions To determine the migration of the headcut due to mass soil failure under a given loading (dead weight of soil mass, hydrostatic pressure of the water head in the breach and water pressure in the soil due to soil permeability), the failure criterion should be checked at all critical points.

3.3.3.1. Bending failure criterion The moments of the forces applied on the headcut result in bending moments and bending stress. The top of the headcut is under tension and the lower part is under compression.

Failure may occur when the maximum tensile stress,  t _ max , equals the tensile strength, Rt , of the soil as follows:

 t_ max  R t (3.6)

3.3.3.2. Shear failure criterion According to the increase of the eroded notch, the soil mass of the headcut tends to slide over the cross-section through shearing failure. Thus, the failure criterion may be expressed as follows:

max   (3.7) where, max is the maximum shear stress of the headcut applied along the failure surface by the driving force and  is the shear strength of the soil. Calculations of the maximum shear stress and the determination of the soil tensile stress are described in the next section.

3.3.4. Soil shear strength Soil materials used in embankment dams are commonly obtained from one local area. Thus, these materials are constructed of all types of geologic materials. Generally, the tensile strength of soil has been given less attention by engineers. However, based on observations of field and laboratory tests, tensile cracks have been found at the crest of the embankment dam. The extension of these cracks results in the failure of the soil block. Thus, the soil shear strength plays an important role when analyzing breach widening processes and performing additional studies.

3.3.4.1. Tensile strength tests

Chapter 3. Breach widening processes

Different tensile tests have been conducted to recognize the probability of tensile crack development and to determine the soil tensile strength. According to the principle of loading, tensile tests were divided into the followings types by Vanicek (Vanicek, 2013):

a) Axial tensile test (direct tension test) Specimens are commonly tested in the horizontal position to eliminate the influences of their own weight. The tested samples are uniform cross sections with uniformly distributed tensile stresses (Vanicek, 2013).

b) Triaxial tensile test The triaxial tensile test is one of the most popular tests because it is versatile and offers one the opportunity to easily study a wide range of parameters (Saada and Townsend, 1981).

c) Bending test Bending tests were mainly conducted to investigate the state of the compacted horizontal layer in the clay core of earth and rockfill dams because tensile cracks can develop during their deflection. The principle of the bending test consists of loading the tested soil beam with a pair of forces in the middle portion of the sample. Two advantages of such loading are that the shearing force is zero in the central part of the tested beam and that the bending moment is constant, which represents a typical case of pure bending. The outermost fibers are either under tension or compression (Vanicek, 2013).

d) Hollow cylinder test The hollow cylinder tensile test can be used to test soils under tensile stress. In this test, the specimen is placed between two smooth annular platens and covered from the inside and outside surfaces by two thin rubber membranes. The ability to measure the radial stresses and deformations allows one to measure the tensile stress (Al-Hussaini, 1981).

e) Indirect (Brazilian) tensile test The Brazilian tensile test is an indirect tensile test method that is often used in rock mechanics. The sample is easy to prepare from the obtained core drill and the load transfer is not difficult. Determination of the tensile strength by using this method is based on a series of theoretical assumptions (Vanicek, 2013). The direct test method is generally considered as the most reliable approach. Generally, the obtained tensile strength from a direct tensile test is highly accurate. However, direct tensile tests are difficult to perform due to problems related to specimen preparation and

3.3. Headcut migration

test procedures. The convenience of the indirect tensile test makes it useable for tensile tests (Li et al., 2014).

Figure 3-15. Division of tensile tests according to the principle of loading. a) Axial tensile tests (direct tension test). b) Triaxial tensile test. c) Bending test. d) Hollow cylinder test. e) Indirect (Brazilian) tensile test. (Vanicek, 2013).

3.3.4.2. Development of tensile failure criteria One problem for shear failure criteria is their inability to correctly predict the tensile strength of the material, essentially because the failure criterion predicts the tensile strength of a material by assuming shear failure. However, brittle materials are known to fail on the planes normal to the direction of the tensile stress, which would suggest a tensile mode of failure rather than a shear mode. Therefore, the failure plane is

Chapter 3. Breach widening processes

perpendicular to III R t , where Rc is the tensile strength of the material (Meyer and Labuz, 2013). Because the tensile strength value of the shear failure criterion has no physical meaning, a modified Mohr-Coulomb failure criterion was proposed to account for the transition from shear to tensile failure modes (Paul, 1961). Paul assumed that a brittle material will fail according to the values predicted by equation (3.8) when  III R t and by equation (3.9) when   R (see Figure 3-16). III t

I  IIIKR p  c (3.8)

 III R t (3.9)

(1 sin ) K  (3.10) p (1 sin )

2c cos R  (3.11) c (1 sin )

Where,  is the internal friction angle, c is the cohesion of the soil, and Rc is the uniaxial compressive strength of the material. As shown in Figure 3-16, the failure envelope is composed of two straight lines. When III R t , the failure envelope is the same as the failure envelope of the MC failure criterion. When III R t , the perpendicular envelope presents the tensile failure of the material. Only one parameter, Rt , is used in this equation and is important in the headcut migration model. A study of the order of magnitude is presented below.

Figure 3-16. Failure envelope in the principal stress plane (Meyer and Labuz, 2013).

3.3. Headcut migration

3.3.4.3. The order of magnitude According to equation (3.11), the uniaxial compressive strength can be estimated using soil properties. Predicting the tensile strengths of some soils is possible because some relationships between the tensile strengths and uniaxial compressive strengths of materials are available. Generally, embankment dams are constructed of all types of geologic materials. For information regarding the characteristics and soil properties of embankment dams, please refer to the procedures established by different departments in different countries. Different classification systems exist around the world. Normally, four groups of embankment dam soil materials are considered: clay, silt, sand and gravel. Although the particle size distributions for each classification are not the same, common orders of magnitude of the soil cohesion and internal friction angle are given in Table 3-1.The orders of magnitude of soil properties are only based on engineering experience and can be used in primary evaluations.

Table 3-1. Orders of magnitude of soil properties classified by the particle size of embankment dams (Xiao, 2006). Classifications Cohesion (kPa) Internal friction angle (◦) Clay/Silt 10-200 0-30 Sand 10 28-36 Gravel 10 32-40

The uniaxial compressive strength can be roughly estimated using equation (3.11) and the order of magnitude suggested in Table 3-1.For example, in soil materials with a cohesion of 100 kPa and an internal friction angle of 10°, the calculated compressive strength is approximately 275 kPa. Based on the relationship RRc/ t = 5 to 25, the corresponding tensile strengths range from 11 to 55.

3.3.5. Bending failure calculations As shown in Figure 3-17, the normal stress on the fixed end section of the headcut consists of two portions: (1) the normal stress resulted from the bending moments  M , and (2) the normal stress resulted from the normal forces  P on the section. The normal stress is the sum of  M and  P , and can be expressed using equation (3.12). If distributions of the normal stress due to bending moments are assumed as linear, the distributions of the normal stress can be illustrated as shown in Figure 3-17.

 MP   (3.12)

Chapter 3. Breach widening processes

Figure 3-17. Shear stress distributions on the fixed end section of the headcut.

Because the headcut is under tension in the upper part and under compression in the lower part, the maximum tensile stress,  t _ max , potentially occurs on the top of the headcut when assuming a linear stress distribution. This relationship can be written as follows:

MHP  t _ max   (3.13) 2IH0

Where, “-” means that the tensile stress of soil on the section is positive and the compressive stress is negative. M is the bending moment on the fixed end of the headcut per unit width and may be expressed as the sum of the moments of forces in equation (3.14). P is the normal forces on section. In addition, H / 2 is the distance from the neutral axis to the top of the headcut, and I0 is the moment of inertia of the lengthwise cross-section, which is assumed rectangular with a width of unit. Thus, the moment of inertia may be expressed using equation (3.15).

MMM  (3.14) GP dead weight water pressure

H 3 I  (3.15) 0 12

3.3. Headcut migration

Figure 3-18. The bending moment generated by dead weight.

MP MP

MP2 MP2b MP2

hw hw MP1 MP1

(a) Low permeability (b) High permeability Figure 3-19. Moments of force due to water pressure depending of the soil permeability.

Chapter 3. Breach widening processes

Where, MG is the moment of dead weight. As shown in Figure 3-18, the dead weight of the headcut is divided into three parts, and the calculations of these dead weights are presented in the next section. In addition, M P is the moment of water pressure forces. As shown in Figure 3-19, two situations should be considered. (1) Soil materials with low permeabilities (Figure 3-19 (a)): When a soil material has low permeability, the water pressure inside the embankment dam body can be neglected. (2) Soil material with high permeability (Figure 3-19 (b)): When a soil material has a high permeability, the water pressure inside the embankment dam body should be considered. The calculation details are given in the next section.

3.3.5.1. Moment of dead weight per unit width

The moment of dead weigth, MG , per unit width is simply the dead weight multiplied by the distance from the horizontal centers of gravity to the origin (the fix end of the cantilever-like bream). Considering the three zones of soil mass, it may be expressed as:

The moment of dead weight, MG , per unit width is the dead weight multiplied by the distance from the horizontal centers of gravity to the origin (the fixed end of the cantilever- like beam). Considering the three zones of soil mass, MG may be expressed as follows:

MAXAXAXG 1 1 1   2() 2 2  3 3 (3.16)

Where, A1 , A2 and A3 are the areas of the three zones respectively, and may be expressed from equation (3.17) to equation (3.19):

A1  le() H  h w (3.17)

A2  le() h w  h e (3.18)

1 A l  h (3.19) 3 2 e e

X1 , X 2 and X3 are the distances from the horizontal centers of gravity to the origin (the fixed end of the headcut), and may be expressed from equation (3.20) to equation (3.22):

2 le() H h w X1  (3.20) 2A1

2 le() h w h e X 2  (3.21) 2A2

3.3. Headcut migration

2 le h e X 3  (3.22) 6A3

Thus, the dead weight bending moment equation (3.16) can be transformed as follows:

2 2 2 le()() H h w l e h w  h e l e h e  MG 1   2    (3.23) 2 2 6 

If the parameter e is defined as the ratio of the depth of erosion to the water head in the breach, the bending moment of dead weight can be expressed by the length of erosion, le , the parameter, e , and the properties and geometries of the embankment dam.

he  e  (3.24) hw

2 1hw   2 h w  MG l e H 1    (3  2e )  (3.25) 2HH  6 

3.3.5.2. Moments of water pressure and normal forces per unit width The moments of water pressure per unit width is consist of three parts: (1) the moment of the water pressure force applied on the under-side of the headcut (also called under pressure); (2) the moment of water pressure force applied on the right side of the headcut (in the breach); and (3) the moment of water pressure force applied on the left side of the headcut (in the soil):

MMMM   (3.26) PPPP1 2 2b under pressure right side left side  in the soil in the breach

The water pressure stress applied on the surface of the headcut may be expressed as the specific weight of the water multiplied by the water head. The two borders of the headcut have been described in equation (3.5). To express the influences of water pressure force in soils on the right side of the headcut, another right side border must be described. The water pressure stress applied on each of the borders can be expressed in each equation (see the coordinate system in Figure 3-14 (b)).

1) The under water pressure stress

u( x ) w h w  z ( x )  , 0

Chapter 3. Breach widening processes

2) The right side water pressure stress

u() z w h w  z  , h e  z  h w (3.28) 3) The left side water pressure stress

u( z ) w h w  z  , 0  z  h w (3.29)

3.3.5.2.1. Moment of the under water pressure force The moment of the under water pressure, M , per unit may be expressed as the integral of P1 the differential under water pressure force multiplied by the distance from the central axis of the cross-section along the under surface of the headcut. When considering the unit width of the headcut, the central axis of the cross-section (rectangular) is the center of the left side of the headcut (0, H / 2 ) (see Figure 3-14 (b)). The differential under water pressure force may be expressed as the under water pressure stress multiplied by the differential length of the under surface of the headcut, ds :

dP1  u() x ds (3.30)

Where, u() x is expressed by equation(3.27). To simply this calculation, the differential under water pressure force, dP1 , is divided into horizontal, dP1x , and vertical, dP1z , components (see equation (3.31)). Thus, the moments of M and M generated by these P1x P1z two components can be written separately as shown in equation (3.32) and equation (3.33).

dP1x dP 1cos , d P 1 z  dP 1 sin  (3.31)

h e H M dP()  z (3.32) P1x  1 x 0 2

le M  dP x (3.33) P1z  1 z 0

Where, “-” means that the direction of moments is anticlockwise,  is the angle between the under water pressure force and the horizontal direction, and the differential length of the under surface of the headcut can be transformed as follows:

dx dz ds   (3.34) sin cos 

3.3. Headcut migration

Thus, the moment of under water pressure force, M , is the summation of M and M . P1 P1x P1z By integrating equations (3.27), (3.30), (3.31) and (3.34) and M may be expressed as P1 follows:

h l e H e h M M  M ()()()  z h  z dz   h  e x xdx P1 P 1x P 1 z 2 w w  w w l 0 0 e (3.35) 3 he Hh w h eH h w  2 h w h e 2   w     he ()  l e  3 2 4 2  2 3 

3.3.5.2.2. The moment of the right side water pressure force The moment of water pressure per unit width, M , at the right side of the headcut may be P2 expressed as the integral of the water pressure force, dP2 , multiplied by the distance from the origin ( H / 2 ,0) along the right side surface of the headcut :

h w H M dP()  z (3.36) P2  2 2 he

dP2  u() x dz (3.37)

Where, u() z is expressed by equation (3.28) with a range of he z  h w . By integrating this equation, M can be obtained as follows: P2

hw 3 2 3 H hw Hh w h w Hh e h e h w H  2  MP u( z )(  z ) dz  w (   )   w     he  (3.38) 2  2 6 4 2 3 2 4  he  

3.3.5.2.3. The moment of the left side water pressure (1) Soil with low permeability For soils with low permeability, the pore pressure in the soil can be neglected (see Figure 3-19 (a)). Thus, the moment of the left side water pressure force per unit width may be expressed as follows: M  0 (3.39) P2b

(2) Soil with high permeability For soils with high permeability, pore pressure is present in the soil and should be considered as a contribution to the moment of forces on the fixed end of the headcut (see

Chapter 3. Breach widening processes

Figure 3-19 (b)). Thus, the moment of the left side water pressure force per unit may be expressed as follows:

h w H M dP() z  (3.40) P2b  2 b 0 2

dP2b  u() x dz (3.41)

Where, u() z is expressed by equation (3.29) with a range of 0 z  hw . By integrating this equation, M can be obtained as follows: P2b h w H h3 Hh 2 M u( z )( z  ) dz  (w  w ) (3.42) P2  w 0 2 6 4

In nature, no soil materials have zero or full permeability. Real soils have permeabilities between these extreme cases. Thus, a definition of parameter  i , which is the infiltration coefficient, is necessary when studying real cases.

0 if low permability h3 Hh 2  M  (w  w ) ,   1 if high permability (3.43) P2b i w i  6 4  (0,1) otherwise

By integrating equations (3.24), (3.35), (3.38) and (3.43) into equation (3.26), the equation of moments of water pressure per unit width may be transformed as follows:

 1 H h  e 2     w 2 MP w( ) h w l e (1 i )( ) h w  (3.44) 3 2 4 6 

The bending moment of the headcut per unit width generated by the moments of dead weight and the moments of water pressure forces can be expressed as the sumof these moments (see equation (3.45)). In this equation,  is a simplification parameter.

2 1hw   2 h w  2  w 2 H hw M le H1    (32) e   l e h w (23)(1)(  e     i  w h w  ) 2HH  6  6 4 6 (3.45) l2 HH h e B (1   )  h2 (  w ) 2i w w 4 6

B hw  h w 2   11    (  2  w ) 1   e  (3.46) HH  3 

100

3.3. Headcut migration

3.3.5.3. Normal forces

The normal forces on section P may be expressed using equation (3.47).

PPPP 1x  2  2 b (3.47)

Where, P1x is the normal portion of the water pressure on the under surface of the headcut

(the headcut notation is shown in Figure 3-14), P2 is the normal water pressure on the right side surface of the headcut, and P2b is the water pressure on the left side surface of the headcut. “-” in equation (3.47) means that the compressive force is positive and the tensile force is negative on the section.

3.3.5.3.1. Normal force due to under water pressure By integrating equations (3.30) and (3.31), the normal portion of the water pressure on the under surface of the headcut, P1x may be expressed as follows: h e 1          2 P1x dP 1 x  dP 1 cos  w ( hzdz w ) w hh w e w h e (3.48) 0 2

3.3.5.3.2. Normal force due to the right side water pressure By integrating equation (3.37) the normal water pressure on the right side surface of the headcut, P2 may be expressed as follows: h w 1 P dP ()() h  z dz   h  h 2 (3.49) 2 2  w w2 w w e he

3.3.5.3.3. Normal force due to the left side water pressure By integrating equation (3.41)the normal water pressure on the left side surface of the headcut, P2b may be expressed as follows:  h i w 1        2 2 P2b dP 2 b  w() i h w z dz w i h w (3.50) 0 2

By integrating equations (3.48), (3.49) and (3.50) into equation (3.47), the normal forces may be expressed using equation (3.51). 1 P h2(1   2 ) (3.51) 2 w w i

3.3.5.4. Critical length of headcut

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A critical situation occurs when the maximum tensile stress, t _ max , reaches the tensile strength of the soil, Rt , (see the failure criterion used in equation(3.6)), and the headcut falls down to the breach and the side wall retreats. The critical length of the headcut can be calculated by integrating equations (3.45) and (3.51) into equation (3.13) and replacing the maximum tensile stress by the soil tensile strength.

H  1 3 h   2   2     2  w leB  R t w h w(1 i ) (1 i ) w h w (2 ) (3.52) 3  2HHH 2 

3.3.6. Shear failure calculations The shear failure of the headcut may occur when the maximum shear stress applied by the driving forces exceeds the shear strength of the soil.

3.3.6.1. The shear stress on the potential failure surface

The maximum shear stress per unit width,  max , applied along the failure surface is generated by the dead weight of the headcut and water pressure and can be expressed as follows:

WUs V  max   (3.53) HH dead weight under pressure in the breach

Where,  is the maximum tensile strength along the potential failure surface (the fixed end of the headcut); Ws is the dead weight of the headcut; and UV is the vertical water pressure force on the headcut.

3.3.6.1.1. Dead weight The calculationed headcut dead weights are divided into three parts due to their shapes and specific weights (see Figure 3-20 (b)):

1 W l()() H  h   l h  h   l h (3.54) s1 e w 2 e w e2 2 e e

By integrating equation (3.24), equation (3.54) may be transferred as follows:

hw   2 h w  Ws l e H 1 1    (2  e )  (3.55) HH  2 

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3.3. Headcut migration

1 Shear Shear stress stress   2 

2

(a) Dead weight (b) Water pressure

Figure 3-20. Shear stress of the headcut due to dead weight and water pressure.

3.3.6.1.2. Water pressure Because only the vertical component of the water pressure can contribute to the shear force along the failure section, the calculated water pressure force only considers the vertical component (see Figure 3-20 (b)).

se  UVV u() s ds (3.56) 0

Where, uV () s is the vertical component of the water pressure stress, and may be expressed by equation(3.57), and ds is the differential element of the length on which water pressure stresses apply, and can be expressed as shown in equation(3.58) :

l u()() s e u x (3.57) V 2 2 he l e

2 h  ds1  e  dx (3.58) le 

After integration along the lower surface of the headcut, the vertical water pressure force on the headcut can be obtained as follows:

l U e (2 h  h ) (3.59) V w2 w e

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By integrating equation (3.24), equation (3.59) may be transformed as:

 Uw l h (2  ) (3.60) V2 e w e

3.3.6.1.3. Shear stress per unit width Finally, by integrating equation (3.55) and equation (3.60) into equation(3.53), the maximum shear stress can be obtained as follows:

S max   le (3.61) Where,  S obtained as follows:

S hw  h w 1   11    (  2  w ) 1   e  (3.62) HH  2 

3.3.6.2. The normal stress on the potential failure surface

The normal stress on the potential failure surface per unit width,  n , is generated by the normal force applied and can be written as follows:

U   N (3.63) n H

Where, H is the length of the potential failure surface and equals the height of the levee, and U N is the normal force per unit width applied on the potential failure surface. In this forcing system (see Figure 3-21) three portions of water pressure contribute to the normal force.

UUUU   (3.64) NNNN1 2 2b under pressure pressure at the right side pressure at the left side  in the soil in the breach

Where,U is the under pressure force per unit width generated by the water pressure under N1 the headcut, U is the water pressure force per unit width at the right side of the N2 headcut generated by the water head in the breach, and U is the water pressure force per N2b unit width on the left side of the headcut generated by the water head in the soil.

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3.3. Headcut migration

Normal Normal stress stress

n n

hw hw

(a) Low permeability (b) High permeability

Figure 3-21. Normal stress due to water pressure.

3.3.6.2.1. Under pressure force The magnitude of the under pressure force, U , is determined by the integrating the N1 differential of the under pressure force along the under surface of the headcut as follows:

se U u() s ds (3.65) NN1  0

Where, uN () s is the normal component of the under water pressure stress,u() x , and can be written as follows:

h u()() x e u x (3.66) N 2 2 he l e

By integrating the equations (3.27), (3.58) and (3.66) into equation (3.65), the under pressure force per unit width is obtained as follows:

2 h h  le h U e1  e  ( h  e x ) dx (3.67) N1 w2 2 l w l he l e e  0 e

To simply equation (3.67), the definition of parameter  e in equation (3.24) is integrated. Thus, equation (3.67) can be transformed into equation (3.68).

 Uw h2 (2  )  (3.68) N1 2 w e e

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3.3.6.2.2. Water pressure force on the right side Because the right side of the headcut is vertical and the water pressure stress is normal to the surface (see Figure 3-21), the normal force on the right side per unit width can be written as follows:

hw U u() z dz (3.69) N2  he

By integrating equation (3.28) and equation (3.24), equation (3.69) can be transformed as follows:

 Uw h2(1  ) 2 (3.70) N2 2 w e

3.3.6.2.3. Water pressure force on the left side Similar to the calculation of water pressure force on the right side of the headcut, the water pressure force on the left side can be written as follows:

hw U  u() z dz (3.71) N2b i  0

Where, “-” in this equation indicates the direction of the water pressure force on the left side is opposite of the other two forces. In addition, parameter i is the infiltration coefficient discussed in the last section, and the range of its value is from 0 to 1. By integrating equation (3.29), the water pressure force on the left side per unit width can be expressed as follows: 1 U   h2 (3.72) N2b i2 w w

3.3.6.2.4. The normal stress per unit width Finally, by integrating equations (3.68), (3.70) and (3.72) into equation (3.64), the normal forces per unit width applied on the potential failure surface U N may be expressed as shown in equation (3.73) and the corresponding normal stress may be expressed as follows:

 Uw (1   ) h2 (3.73) N2 i w

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3.3. Headcut migration

(1  )  i  h2 (3.74) n2H w w

3.3.6.3. Soil shear strength The soil shear strength, , of the potential failure surface is mainly governed by effective stresses and soil strength parameters. Based on the failure criterion of Mohr-Coulomb,  consists of a cohesive component and a frictional component.

c '  n tan  (3.75)

Where, c'is the effective cohesion and  is the effective soil internal friction angle.

3.3.6.4. Calculating the length of erosion Due to the shear failure criterion (see equation (3.7)), the soil mass slides along the failure surface when the shear stress reaches the soil shear strength. The critical situation occurs when the maximum shear stress equals the soil shear strength (max   ). At this point, the critical length of erosion, le , may be expressed as shown in equation (3.76) by integrating equation (3.75) and equation (3.61).

2 1 (1  ) h     i w  leS  c' w H   tan  (3.76)  2 H  

From this equation, the critical length of erosion was obtained as a function of the side wall geometry (the height of headcut, H ), soil properties (the effective cohesion, c', and the effective internal friction angle,  ' ), water head in the breach ( hw ) and the soil impermeability condition (  i ). To find a similar expression for the critical length of erosion as a function of the soil tensile strength, Rt , and other parameters, an analysis of the soil properties is needed.

A relationship between the effective cohesion, c' and the tensile strength, Rt , may be expressed as follows (Fang and Hirst, 1973):

R c '  t (3.77) 

Where,  is the ration of tensile strength to effective cohesion.

Based on the Mohr-Coulomb criterion, the compressive strength, Rc , can be written as follows:

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cos R 2 c (3.78) c 1 sin 

If r is defined as the ration of compressive to tensile strength r Rc/ R t , equation (3.78) may be transferred as follows:

cos rR 2 c (3.79) t 1 sin 

Thus, if the compressive to tensile strength ratio, r , is known, can be written as equation (3.80) by comparing with equation (3.77) and equation (3.79):

2 cos '   (3.80) r(1 sin ')

Where,  can also be expressed as described by Fang (Fang and Hirst, 1973):

I  0.34  p (3.81) 100

The solution of  in equation (3.81) requires the value of the soil plasticity index, I p .

If the order of magnitude of the ratio of compressive to tensile strength, r , is given as 10, then the order of magnitude of the effective internal friction angle is given as 30°, and the effective cohesion can be obtained as c' 2.88 Rt . The equation of (3.76) can be rewritten as follows:

2 1 h        w leS 2.88 R t 0.29(1 i ) w H    (3.82)  H  

3.4. Analytical results and discussion

When analyzing the two failure modes of the headcut and soil tensile strength, the B S expressed headcut critic length due to bending failure, le , and the shear failure, le , were obtained (see equation (3.83) and equation (3.84)). The potential failure is a combination of these two situations, which means that the smaller critical length of headcut calculated by these two failure mode equations controls the critical headcut failure (equation (3.85)).

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3.4. Analytical results and discussion

1) Bending failure mode:

H  1 3 h  B  2   2     2  w leB  R t w h w(1 i ) (1 i ) w h w (2 ) (3.83) 3  2HHH 2 

2) Shear failure mode:

2 1 h   S      w leS 2.88 R t 0.29(1 i ) w H    (3.84)  H  

BS le min( l e , l e ) (3.85) where the constants are the height of the headcut, H , and the specific weights of the soil under and above the water surface, 1 and 2 . The variables include the water head in the breach, hw , the ratio of the depth of erosion to the water head in the breach, e , the infiltration coefficient, i , and the soil tensile strength, Rt .

3.4.1. The specific soil weight

The soil specific weight was defined separately as 1 for upper part and  2 for lower part as follows: G Se   (3.86) w 1 e

Where G is the specific gravity, S is the degree of saturation, and e is the void ratio. From this equation, the difference between 1 and  2 is due to saturation. For illustration purposes, a soil with the specific gravity of 2.6 and a void ratio of 0.8 is considered. If the soil saturation is assumed as 1 for the soil under the water surface and as 0.7 for the soil above the water surface, the specific weights are 18.8 kN/m3 and 17.5 kN/m3, respectively.

3.4.2. The water head in the breach The water head in the breach always varies with the discharge flow. Thus, the parameter

 w is defined to express the relative water head in the breach. The range of  w is normally between 0 and 1 and can be larger than 1 for overtopping failure cases. h   w (3.87) w H

3.4.3. The ratio of the depth of erosion to the water head

The ratio of the depth of erosion and the water head in the breach, e h e/ h w , is defined by equation (3.24). This ratio indicates the degree of soil erosion at the foot of the headcut.

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by equation (3.24). This ratio indicates the degree of soil erosion at the foot of the headcut. Figure 3-22 shows a sketch of the evolution of the eroded notch at the foot of the headcut.

Figure 3-22. Sketch of the evolution of the eroded notch at the foot of the headcut.

According to the literature review, the influence of the side wall on the distribution of the boundary shear stress in the rectangular open-channel is concentrated in the lower region where the eroded depth is less than 0.65 times the channel height (Nezu, 1993, Zheng and Jin, 1998, Hunt et al., 2005). Because the erosion at the foot of the headcut is controlled by the shear stress distribution, the value of e is assumed to equal 0.65. The simulations of distributions of lateral (side wall) and bottom shear stress due to the turbulent flow in closed ducts are given in Appendix B. Ducts have been defined as trapezoid to simulate the side wall due to the overhang of the headcut. Parameter Slope is the cosine of the angle on the corner of the erosion, and it reflects the degree of the overhang. Ducts with different Slope have been simulated under different aspect ratios ranging from 1 to 5 (ratio of the width to depth of the duct). Results are interesting but not complete.

3.4.4. Infiltration coefficient In real cases, the soil materials used in embankment dams and levees vary widely, along with their permeabilities. Figure 3-23 shows the following illustrations of some typical water pressure distributions in soils during breach widening processes: (a) a case with low soil permeability and no soil water pressure; (b) a case of high soil permeability with a water table that is equal to that in the breach and for which the distribution of water pressure in the soil is controlled by the water head in breach; and (c) a case between the former two extremal situations in which the distribution of soil water pressure is controlled by the water table in the soil. As the erosion process evolves, the soil in the headcut gradually infiltrates and the water table in the soil varies. The last situation is the most similar to the actual failure situations of the breach widening process.

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3.4. Analytical results and discussion

(a) Soil with low permeability

(b) Soil with high permeability

(c ) Infiltration during the erosion process

Figure 3-23. Illustrations of the typical water pressure stress distributions on the headcut.

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Chapter 3. Breach widening processes

The distributions of water pressure stress on the soil based on the influences of the infiltration coefficient, i , are shown in Figure 3-24. The water pressure stress along the right side of the headcut follows a linear distribution with a distance of  h from the foot i w of the headcut. When  is equal to 1, it corresponds to a soil with a high permeability, and i the water pressure is distributed on the right side of the headcut over a distance of hw . When  equals 0, it corresponds to a soil with low permeability, and no water is i distributed on the right side of the headcut (see equation (3.88)).

0 if low permeability  i   1 if high permeability (3.88)  (0,1) otherwise

If we consider the cross-section of the headcut, or the direction of discharge flow, we can explain why soil saturation is influenced by the infiltration coefficient. Figure 3-25 shows illustrations of water tables in an embankment dam under normal conditions. In the upper part of embankment body (separated by the core wall or by the central line) that is influenced by the water head in the upstream reservoir or river, the water table in the soil remains high. Next, the water table decreases through the dam body into the lower portion of the dam. It is obvious that the drainage and core wall play important roles in reducing the water table in the embankment dam body. However, during the breaching processes, the upstream water head remains relatively high, and the soil of the upper part of the embankment body is normally considered as 100% saturated. Figure 3-26 shows the potential soil saturation in the breach during the breaching processes of an embankment dam with a drainage and core wall. In the lower part of the dam, the soil saturation conditions may be different due to the permeability of the dam. The calculations should consider the following different situations.

a) If i  0, the soil in the lower portion should be considered as dry (see Figure 3-26 (a));

b) If i 1, the soil in the lower portion should be considered as totally saturated (Figure 3-26 (c));

c) If 0i  1, the soil in the lower portion should be considered as a situation that is between these two extreme situations (Figure 3-26 (b)).

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3.4. Analytical results and discussion

i hw

Figure 3-24. Illustrations of the distributions of water pressure stress on the headcut

considering the influence of the infiltration coefficient, i .

(a) (b) (c) Figure 3-25. Illustrations of typical water tables in different types of embankments: (a) Homogeneous, (b) Homogeneous with a drainage, and (c) Homogeneous with drainage and a core wall.

(a) (b) (c) Figure 3-26. Illustrations of potential distributions of the soil saturation in the breach due to different soil permeabilities in the downstream portion of an embankment dam: (a)

i  0 ; (b) 0i  1; (c) i  1.

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3.4.5. Analytical results To extend the analysis to general situations, the embankment geometries and soil specific weights are given in Table 4. The range of the levee height is from 2 to 12 meters, which is a common range for natural and man-made flood protected structures. A relative water depth of 0.8 was chosen, which corresponds to the relative high water head in the breach. The ratio of the depth of erosion to the water head was determined and is discussed in the last section. An experience value of 0.65 was chosen (see section 3.4.3). As discussed in the last section, the specific weight of the saturated soil depends on the degree of saturation and the void ratio of a given soil. These parameters are not always available. Thus, the specific soil weight and the specific saturated soil weight are set as 18 k N/m3. The range of tensile soil strength is considered as 0 to 18 kPa, which corresponds to a soil cohesion range of 0 to 51.84 kPa.

Table 3-2. Definition of parameters in this analytical model.

Parameters Vaules

The levee height (m) 2 - 12

The relative water head  w 0.8

The ratio of erosione 0.65

3 Specific weight of the soil 1 (kN/m ) 18

3 Specific weight of the saturated soil  2 (kN/m ) 18

The tensile strength of the soil Rt (kPa) 0 -18

By applying the above parameters into the analytical equations (1.75), (1.76) and (1.77), interesting results were obtained. Two extreme situations were calculated: (1) i 1soil of the embankment body with high permeability; and (2) i  0.5 soil of the embankment body with moderate permeability. For each situation, the critical lengths of the headcut under different soil tensile strengths (and the corresponding soil cohesion) and for a range of embankment dam heights (from 2 to 12 meters) were obtained, as shown in Figure 3-27 and Figure 3-28, respectively.

1) Embankment body soil with high permeability ( i 1)

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3.4. Analytical results and discussion

In Figure 3-27, the critical lengths of the headcut are between 0 and 3.09 m. When the tensile strength of the soil is 0, the critical lengths of the headcut for the embankment dams of different heights are equal to 0. The maximum critical lengths of erosion are observed when the tensile strength of the soil reaches the maximum value of 18 kPa. For all embankment dams of different heights, the critical lengths of the headcut increase as the soil tensile strength increases. The critical headcut length range for a 2 meter tall embankment dam is 0 to 1.26 m. Thus, the ranges for 4, 6, 8, 10, and 12 meter tall embankment dams are from 0 to 1.78, 0 to 2.18, 0 to 2.52, 0 to 2.8, and 0 to 3.09, respectively. All of the curves pass through the same line, which passes through the origin, has a slope of 0.35 and is called the shear failure line. The failure modes of the headcut correspond to the shear failure on this line and the bending failure on the right side of this line.

2) Embankment body soil with moderate permeability ( i  0.5) In Figure 3-28, the critical lengths of the headcut are between 0.16 and 3.25 m, which is a little larger than results obtained under the situation of high soil permeability. The failure modes of the headcut for all the embankment dams with different soil tensile strengths are the bending failure. No the shear failure occurs. When soil tensile strengths equal to 0, the critical lengths of the headcut range from 0.16 to 1m corresponding to a range of embankment dam height from 2 to 12m.

3) Comparisons of the critical lengths of headcuts under the above two situations for each embankment dam with a different height. To understand the distributions of the critical lengths of the headcut for a given embankment dam and how they are influenced by the soil infiltration coefficient, i , comparisons of the critical lengths of the headcuts under two soil permeability conditions for each embankment dam with a different were made (see Figure 3-29). For a 2 meter tall embankment dam, the critical lengths of the headcuts obtained under two soil permeability

( i 1 and i  0.5 ) have the same magnitude. The differences between calculated critical lengths of headcuts under the two soil permeability situations for embankment dams with heights of 4, 6, 8, 10, and 12 meters are not obvious when the bending failure mode works as the failure mode of the headcut. But the differences are obvious when the shear failure mode works as the failure mode of the headcut.

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Figure 3-27. The critical lengths of the headcuts under different soil tensile strengths and

effective cohesion for a range of embankment dam heights (from 2 to 12 meters ; i  1, saturated situation).

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3.4. Analytical results and discussion

Figure 3-28. The critical lengths of the headcuts under different soil tensile strengths and effective cohesion for a range of embankment dam heights (from 2 to 12 meters; i  0.5).

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Chapter 3. Breach widening processes

Figure 3-29. Comparisons of the critical lengths of the headcuts under two extreme soil

infiltration situations of i  1 (saturated situation) and i  0.5 for a range of embankment dams from 2 to 12 meters talls.

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3.5. Conclusions

A simply headcut migration model based on the soil tensile strength is presented to simulate the critical length of the headcut. The failure of the headcut is determined by the bending and shear failure criterions. Calculations of the critical length of the headcut are given in equations (3.81), (3.82), and (3.83). Input parameters of this model are: the breach geometry (the levee height), the soil properties (the soil tensile strength/the soil compressive strength, saturated and normal specific weights of the soil), the flow situations in the breach (the relative water head and the ratio of erosion) and the soil permeability parameter. A common range of embankment dam heights from 2 to 12 meters are used to calculate the critical lengths of the headcut under different soil tensile strengths (ranging from 0 to 18 kPa). It is concluded that critical lengths of headcut increase as soil tensil strengths increase. Both of the bending failure and shear failure modes play a role on the failure of the headcut when the soil permeability equals to 1. For the case of soil with low permeability ( i  0.5 ), only bending failure mode determine the migration of the headcut. The differences between calculated critical lengths of headcuts under two soil permeability

( i 1 and i  0.5 ) are not obvious. This analytical model is verified by the traditional side slope stability analysis method in the next Chapter.

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CHAPTER 4

Verification of headcut migration by the limit equilibrium method

Side wall recession during the breach widening process was defined in the previous chapter. A headcut migration model based on the soil tensile strength was presented. The objective of this chapter is to verify this mathematical model by using the limit equilibrium method (LEM). First, this chapter briefly introduces the LEM, the choice of failure criterion for the material and the choice of an analysis method. In the second section, the side wall stability analyses are modeled by Slope/W using LEM for three typical levee scales (2, 4 and 6 meters high). The critical lengths of the headcuts for each case are proposed. Finally, the comparisons with the previous numerical model are given as well as the conclusions.

4.1. Limit equilibrium method

4.1.1. Introduce In the case of side wall recession during breach widening, the location of the soil block is more important in terms of the consequences of failure. This order of importance occurs because the generated amount of failure mass is generally removed over a short period of discharge flow and because the length of the headcut determines the width of the breach and the corresponding discharge flow. Thus, as described in the last chapter, the headcut migration process can cause episodic failure in the side walls of the breach. The LEM presents the basic principles for safely designing natural earth and rock slopes. The LEM is the most common approach for analyzing slope stability and can identify potential failure mechanisms and derive safety factors for particular geotechnical situations. The LEM is an appropriate choice for assessing the stability of side walls during embankment dam breach and considers the geometry, soil properties and groundwater conditions of the side walls. Thus, the LEM was selected to verify the tensile strength based headcut migration model.

4.1.2. The theory of LEM The LEM requires trial failure surfaces and optimization analyses to locate the critical failure surface. In addition, LEM discretizes the trial failure surface into smaller slices and treats each individual slice as a unique sliding block (see Figure 4-1). Each slice is affected by a general system of forces (see Figure 4-2). Many different solution techniques for the methods of slices have been developed over several decades. The differences between these methods depend on the solution of statics equations (moments, forces or moments and forces) and on the assumed relationship between the inter-slice shear and normal forces. Figure 4-1 illustrates a typical sliding mass discretized into slices and the possible forces on the slice. As shown in Figure 4-2, one slice is removed from the slope and is acted on by normal and shear forces (Cheng et al., 2007).

The Fs parameter is defined as the ratio of the ultimate shear strength on the base of slices to the mobilized shear stress at incipient failure, Fs  f /  . The failure state means that the shear strength of the slip surface is mobilized by the same amount to bring the sliding body into a limit state.

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Chapter 4. Verification of headcut migration

Figure 4-1. Slice discretization and slice forces in a sliding mass.

Figure 4-2. Forces on a slice within the trail sliding mass.

Where, X is the vertical inter-slice shear force, E is the horizontal inter-slice normal force, W is the weight of the slice mass, P is the normal force on the base of the slice, and S is the shear force mobilized on the base of the slice.

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4.1. Limit equilibrium method

4.1.3. Failure criterion

4.1.3.1. Mohr-Coulomb failure criterion The shear strength along sliding surface is determined by the soil failure criterion. The Mohr-Coulomb failure criterion is widely used to describe the strength of the soil mass. The MC theory can be thought of as a set of linear equations in principal stress space that represent a shear failure surface for an isotropic material and are not affected by the intermediate principal stress  2 . Thus, the MC theory can be written in terms of normal stress  and shear stress  as follows:

c  tan  (4.1)

Where,  is cohesion, and  the angle of internal friction.

Figure 4-3. Mohr-Coulomb failure envelope on a Mohr diagram.

A representation of the MC failure envelope on a Mohr diagram is shown in Figure 4-3

(Meyer and Labuz, 2013). The dashed circle with a diameter of Rc represents the state of stress for uniaxial compression testing, with R = uniaxial compressive strength. The c dashed circle with a diameter of Rt represents the tensile strength predicted from the MC failure criterion. However, this value is generally much larger than that observed from actual testing. For example, with  = 30° and RR/ = 3, the tensile strength, R , is c t t approximately 1/10 of the R based on experiments. This discrepancy results from the c specimen failing in a tensile mode rather than in the shear failure mode predicted by the

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MC failure criterion. A non-linear failure envelope is necessary to describe the behavior of material while considering the tensile strength. It is often convenient to represent the MC failure criterion in terms of principal stresses (the major principal stress  1 and the minor principal stress  3 ) as follows:

1kp  3  R c (4.2)

1 sin k  (4.3) p 1 sin

cos R 2 c (4.4) c 1 sin

cos R 2 c (4.5) t 1 sin Embankment dams are constructed of materials that usually consist of soil, but may also include moraine, rock, or crushed paving materials. It is important to note that a non-linear failure envelope is needed to describe the behavior exhibited by embankment materials over the entire stress state range.

4.1.3.2. Hoek-Brown failure criterion The Hoek-Brown failure criterion promotes the nonlinear envelop of the Mohr-Coulomb failure criterion. This model evolved in stages over a few decades and has been updated several times in response to its uses (Hoek, 1988, Hoek et al., 2002, Eberhardt, 2012). A generalized form of the criterion is shown in equation (4.6).

 3 a 1  3 Rc() m b  s (4.6) Rc

Where,  1 and  3 are major and minor principal stresses at failure, and Rc and Rt are the compressive and tractive strength of the intact rock, and mb , s and a are three parameters estimated by observations of in situ experiments. All of the parameters were obtained according to the Geological Strength Index (GSI) as follows:

GSI 100 m m exp( ) (4.7) b i 28 GSI 100 s  exp( ) (4.8) 9

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4.1. Limit equilibrium method

1 1 a () eGSI /15  e  20/3 (4.9) 2 6 Considering the large variations in the materials used in embankment dams and levees, it is acceptable to assume the presence of intact rock (GSI=100). Therefore, s is equal to 1, a is equal to 0.5, and mb is equal to mi . mi , which is the ratio of the compressive strength to the tractive strength of the intact rock, mi R c/ R t . For embankment dams and levee materials, it is possible to choose with an order of 10 as the ratio of compressive strength to tractive strength. Equation (4.6) can be rewritten as equation (4.10). In addition, we can convert the above equation to a relationship between the shear strength,  , and normal strength,  .

2 1  3 mi R t  3  R t (4.10)

1 m R()     R2 (4.11) 2 i t t 1 ()    (4.12) 2 1 3 1 ()    (4.13) 2 1 3

4.1.3.3. Comparison of MC and Hoke-Brown failure criterion The soil tensile strength predicted by the Mohr-Coulomb failure criterion is larger than that obtained experimentally. Thus, the Mohr-Coulomb failure criterion does not consider the soil tensile strength. However, the soil tensile strength plays an important role in the numerical toppling failure model. Thus, the Hoke-Brown failure criterion is introduced here under the material conditions imposed by assuming an intact rock with an internal friction angle of 35 degree and mi  10 . Thus, the Mohr-Coulomb (equation (4.2)) and Hoke-Brown (equation (4.10)) failure criteria can be converted using equations (4.14) and (4.15) as follows:

  1 3.73  1 (4.14) RRc c

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Chapter 4. Verification of headcut migration

   1 3 10 3  1 (4.15) RRRc c c

Figure 4-4 shows failure envelopments of a special material of embankment dam (intact rock, an internal friction angle of 35 degree and mi R c/ R t  10 ) over a range of

RRt  3  c . By considering a range of 0  3  Rc , a nonlinear failure curve (Hoke- Brown) fits well the stress states than a straight line (Mohr-Coulomb). By considering soil tensile stress state ( Rt  3  0 ), the Hoek-Brown failure criterion provides a reasonable description of material response. Figure 4-4 shows how the failure of an embankment dam made of a special material (intact rock, an internal friction angle of 35 degree and mi R c/ R t  10 ) evolves over a range of

RRt  3  c . By considering a range of 0  3  Rc , the stress states are fit by a nonlinear failure curve (Hoke-Brown) and a straight line (Mohr-Coulomb). By considering the soil tensile stress state ( Rt  3  0 ), the Hoek-Brown failure criterion provides a reasonable description of the material response.

σ1/Rc 4

3.5    1 3 10 3  1 RRR 3 c c c

2.5   1 3.73  1 RR 2 c c

1.5

 3  Rt 1 Mohr-Coulomb Hoek-Brown 0.5

0 -0.4 -0.2 0 0.2 0.4 0.6 0.8

σ3/Rc

Figure 4-4. Comparison of the Mohr-Coulomb and Hoek-Brown failure criterion ( mi  10 and   35 ).

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4.1. Limit equilibrium method

4.1.4. Spencer method As previously mentioned, based on the integration of moment and force statics equations and on assumptions regarding the relationships between inter-slice shear and normal stress, more than ten limit equilibrium methods have been developed for slope stability analysis.

In this study, the Spencer method is considered to compute the safety factor, Fs , which is integrated in SLOPE/W (a LEM based software). The Spencer method satisfies the force equilibrium and moment equilibrium and considers the inter-slice forces as inter slice shear and normal forces by defining a constant relationship between them, as shown in equation (4.16). Thus, the Spencer method can be used for both circular and non-circular failure surfaces. Two factors of safety are computed from the force equilibrium equation ( F f ) and the moment equilibrium equation ( Fm ). When summing the moments around the center of rotation, the value of F can be derived using the moment equilibrium equation (4.17). m

When summing the forces in the horizontal direction for the overall slope, the F f value can be derived using the force equilibrium equation (4.18). The normal force, P , for these two equations can be evaluated by summing the vertical forces on each slice as shown by equation (4.19). The factor of safety, Fs , is determined at the point where the two curves cross.

XE/ tan (4.16)

c l ( p  ul ) tan   R F  (4.17) m Wx c l ( p  ul ) tan   cos  F  (4.18) f  Psin c l ul tan   sin  WXX()RL   F  F (4.19) f sin tan  cos  F Where, c is the effective cohesion,  is the effective angle of friction, p is the normal stress on the surface of each slide, l is the length of the slide, and u is the pore-water pressure. At each pint of the Hoek-Brown failure criterion in the shear-normal stress expression, the tangent line of this point corresponds to a couple of c and  values. Thus, the expression of the factor of safety uses the form of c and  .

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Chapter 4. Verification of headcut migration by LEM

4.2. Model created in Slope/W

A model using limit equilibrium method is created in SLOPE/W. In this study, the failure of a soil mass (resulting in the headcut migration) is considered as having a safety factor,

Fs of 1. The main procedures required during the simulation are listed below.

 Definition of the geometry of the levee side slope and eroded notch;  Soil properties input in SLOPE/W;  Choice of failure criterion.

4.2.1. Definition of geometry of levee and eroded notch

Table 4-1. Geometry of levee and parameters used in Slope/W.

Levee height H (m) 2, 4, and 6

Levee crest Flat

Slope surface above notch Vertical

The relative water head  w 0.8

The ratio of erosion  e 0.65

Foundation depth (m) 4

The infiltration coefficient  (see i 1 Chapter 3)

Figure 4-5. Definitions and simplification of the geometry of a levee, an eroded notch and a foundation with SLOPE/W.

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4.4. Results and discussions

The definitions and simplifications of the geometry of a levee, an eroded notch and a foundation are shown in Figure 4-5. The levee crest is assumed to remain at the same level near the breach. The levee side slope surface above the water level is treated as vertical. A sharp triangle is defined as the eroded notch, which can be modified manually to simulate the continual lateral erosion process. Meanwhile, the depth of erosion remains the same as the water level in the breach. During flooding, a breach is normally widened and deepened simultaneously by discharge flow. Different forms of the foundation near the levee side slope that result from discharge flow are possible, including no eroded and deepened, less eroded and deepened and well eroded and deepened. Therefore, complex hydrodynamic water pressure and hydraulic shear stress can be generated. However, in this levee side wall recession model, only the hydrostatic water pressure is considered. Thus, the formation of a foundation near the levee side slope has less influence on the headcut migration process. As shown in Table 2, levees with a small range of height from 2 to 6 meters are simulated. The depth of the foundation is 4 meters for each different levee height. The water head in the breach is set as 0.8 relative to the levee height.

4.2.2. Soil properties input into SLOPE/W To implement this model in SLOPE/W, three different material properties are given for the levee side wall, the eroded notch and the levee foundation, respectively. To model the eroded notch at the foot of the headcut, the region of the eroded notch is treated specially with very low soil strength to transform the water pressure to the edge of the headcut under the surface. Table 4-2 shows the parameters of the Hoek-Brown failure criterion that were used to represent the soil strength of the levee (see equation (4.6)). The Hoek and Brown model is a nonlinear shear strength model that accounts for the soil tensile strength. Considering the material of the embankment dams and levees used here, it is important to use this failure criterion. The soil strength properties of the levee foundation and eroded notch are given in Table 4-3.

Table 4-2. The parameters of the Hoek-Brown failure criterion used for the soil strength of the levee material in the SLOPE/W.

Parameters mb a s Values 10 0.5 1

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Table 4-3. Soil strength properties of the eroded notch and foundation used in the SLOPE/W.

Unit weight  The internal friction Region Strength model (kN/m3) angle  (°) Eroded notch 0.1 0 Mohr-Coulomb Foundation 18 35 Mohr-Coulomb

4.2.3. Determination of the critical length of the headcut The model using the limited equilibrium method built by SLOPE/W was introduced in the last section. To simulate the lateral erosion process, the lengths of the eroded notch were manually changed step by step (see Figure 4-6 from (a) to (c)). The increase in the length of the eroded notch during each step was determined by the safety factor ( Fs ). Thus, at the beginning of erosion, when the safety factor was still large, the increase in each step was set to a relatively larger value. When the safety factor decreased to less than 1, the increase was refined. The minimum increase in each step was set to 0.1 m. For each of the eroded notch geometries, the safety factors were calculated sequentially under the same hydraulic and geotechnical situations. Thus, a series of safety factors was obtained and recorded. The safety factor for a 2-meter tall levee was calculated and recorded. The increase in the length of the eroded notch was stopped when the safety factor was less than 1. To find a case that corresponded better to a safety factor of 1, the increase in the length of the eroded notch was refined to calculate the corresponding safety factors that approached Fs  1 . Next, we traced the safety factors for each soil tensile strength. Finally, we obtained a trend for the safety factors using the headcut length in Figure 4-7. When the soil tensile strength was 2 kPa, the safety factor changed decreased from 1.25 to 0.63 as the erosion of the notch geometry progressed from 0 to 0.5 m. When the soil tensile strength was 4 kPa, the safety factor decreased from 1.98 to 0.26 as the length of eroded notch increased from 0 to 0.6 meters. The same variation trends regarding the safety factors of the different soil materials (over a tensile strength range of 2 to 10 kPa) were observed due to the progress of the eroded notch geometry. If a line is drawn where the factor of safety equals 1, we can easily identify the point of intersection by using the variation of the safety factor (see Table 4-4). According to the definition of the critical headcut length, this value corresponds to the critical length of the side wall headcut. Thus, when the length of the headcut reaches this value, the side wall is in a critical situation and can collapse.

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4.4. Results and discussions

(a) (b)

Table 4-4. The factors of safety calculated in each step corresponding to the evolution of the eroded notch for a 2-meter tall levee, and the obtained critical lengths of the headcut.

Tensile strength  t_ c  2  t_ c  4  t_ c  6  t_ c  8  t_ c  10 (kPa) Length of headcut le Fs le Fs le Fs le Fs le Fs (m)/ factor of safety 0 1.09 0 1.75 0 2.24 0 2.93 0 3.57

0.2 1.25 0.3 1.98 0.3 2.58 0.6 3.01 0.6 3.75

0.4 1.02 0.5 1.86 0.6 2.43 0.8 2.47 1.0 1.15

0.5 0.60 0.6 0.26 0.7 0.76 0.9 0.34 1.1 0.31

Critical length of 0.40 0.55 0.69 0.87 1.04 headcut (m) le_ c

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Chapter 4. Verification of headcut migration by LEM

4.5 4 3.5 3 2.5 2 1.5 1 The factor of safety (Fs)safetyof factorThe 0.5 Fs=1 0 0 0.2 0.4 0.6 0.8 1 1.2 Length of recession (m)

Ts=2kPa Ts=4kPa Ts=6kPa Ts=8kPa Ts=10kPa

Figure 4-7. The factors of safety for a levee with a height of 2 meters under different tensile strengths recorded as increases in the headcut length.

4.2.4. Mathematical model verification To verify the mathematical toppling failure model, a series of levees over a range from 2 to 6 meters in height was simulated using the LEM. The method introduced in the previous section was used to determine the critical length of the headcut for each levee dimension. The factors of safety calculated in each step corresponding to the evolution of the eroded notch with heights of 4 and 6 meters, and the obtained critical lengths of the headcut are listed in Table 4-5 and Table 4-6. All of the critical lengths of headcut for different levee dimensions (2, 4 and 6 meters in height) are listed in Table 4-7. As shown in Figure 4-7, it is obvious that the simulated critical length of the headcut of each embankment dam and levee varied as the soil tensile strength changed. Thus, the soil tensile strength is a significant index. Considering a 2 meter tall levee, the side slopes of the levee retreated from a width of 0.5 to 1.1 meters as the soil mass collapsed. Considering a tall levee with a height of 6 meters, the critical soil mass width was between approximately 1.6 and 2.4 meters.

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4.4. Results and discussions

Table 4-5. The factors of safety calculated in each step corresponding to the evolution of an eroded notch in a 4-meter tall levee, and the obtained critical lengths of the headcut.

Tensile strength  t_ c  4  t_ c  6  t_ c  8  t_ c  10  t_ c  12 (kPa) Length of headcut le Fs le Fs le Fs le Fs le Fs (m)/ factor of safety 0 1.07 0 1.44 0 1.62 0 2.04 0 2.17

0.4 1.25 0.4 1.65 0.5 1.95 0.5 2.27 0.5 2.58

0.6 1.16 0.6 1.60 1.0 1.81 1.0 2.12 1.0 2.43

0.8 1.08 0.9 1.17 1.2 1.44 1.3 2.02 1.4 2.30

0.9 0.98 1.0 0.98 1.3 0.86 1.4 0.45 1.5 0.30

Critical length of 0.88 0.99 1.27 1.37 1.46 headcut (m) le_ c

Table 4-6. The factors of safety calculated in each step corresponding to the evolution of an eroded notch in a 6-meter tall levee, and the obtained critical lengths of the headcut.

Tensile strength  t_ c  6  t_ c  8  t_ c  10  t_ c  12  t_ c  14 (kPa) Length of headcut le Fs le Fs le Fs le Fs le Fs (m)/ factor of safety 0 1.19 0 1.34 0 1.60 0 1.81 0 2.02

0.3 1.24 0.3 1.48 0.5 1.73 0.5 1.94 0.5 2.14

0.8 1.22 0.8 1.51 1.0 1.71 1.0 1.92 1.2 2.14

1.2 1.15 1.2 1.48 1.5 1.66 1.5 1.80 1.5 2.08

1.5 1.04 1.6 1.15 1.7 1.63 1.8 1.53 1.9 1.98

1.6 0.97 1.7 0.91 1.8 1.28 1.9 0.83 2.0 0.97

Critical length of 1.55 1.66 1.74 1.83 1.98 headcut (m) le_ c

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Chapter 4. Verification of headcut migration by LEM

Table 4-7. The critical lengths of the headcuts of 2, 4 and 6-meter tall levees under different tensile strengths.

Tensile Levee height Levee height Levee height strength (kPa) H =2m H =4m H =6m

2 0.40 - -

4 0.55 0.88 -

6 0.69 0.99 1.55

8 0.87 1.27 1.66

10 1.04 1.37 1.74

12 - 1.46 1.83

14 - - 1.98

4.3. Results and discussions Comparisons of the critical headcut lengths using the soil tensile strength based mathematical model and the simulated results obtained by LEM are shown in Figure 4-8. For a 2 meters high levee, simulated calculations were concentrated over a range of soil tensile strengths of 2 to 10 kPa. For the 4 and 6 meter tall levees, the simulated calculations were concentrated over soil tensile strength range of 4 to 12 and 6 to 14 kPa, respectively. As shown in Figure 4-8, the headcut lengths simulated by LEM increased as the soil tensile strengths increased. The headcut lengths under different soil tensile strengths obtained by LEM fit the mathematical failure models well. The simulated results obtained by LEM agree reasonably well with the failure model based on the mathematical tensile strength.

4.4. Conclusions The numerical headcut migration model based on the soil tensile strength is verified by using limit equilibrium method for three typical levee scales (2, 4 and 6 meter high). The Hoek-brown failure criterion is considered to be reasonable to use as the soil strength of the headcut (levee body). The calculations of critical lengths of the headcuts for each case have been conducted and good agreements are obtained by comparing with the numerical model.

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4.4. Results and discussions

3 H=2m_LEM H=4m_LEM 2.5 H=6m_LEM H=2m H=4m 2 H=6m

1.5

1 Critical length of headcut (m) headcut oflength Critical

0.5

0 0 2 4 6 8 10 12 14 16 18 Tensile strength (kPa)

Figure 4-8. Comparison of the critical lengths of headcuts using the failure model based on tensile strength and the simulated results calculated by LEM.

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CHAPTER 5

Shear stress distributions in the breach

In the last chapter, the soil erosion of side walls under water surface is accessed by the excess shear stress equation. In this equation, the only variable is the hydraulic shear stress. Estimating the hydraulic shear stress in the breach is to accurately calculate soil erosion. The main object of this chapter is to estimate the lateral and bottom distributions of the shear stress in the breach resulted from discharge flow.

A literature review of the flow conditions and shear stress distributions in the breach is included in section 5.1. Section 5.2 introduces the calculation of the shear stress distributions on the boundary of the breach and the streamwise depth-averaged velocity in the breach. It accounts for secondary flow effects via the dimensionless eddy viscosity coefficient  , and the dimensionless secondary flow parameters k1 and k2 . The calculated shear stress distributions and the averaged streamwise depth-averaged velocity are given in section 5.3 as well as the discussion.

5.1. Literature review

5.1. Literature review of flow conditions and boundary shear stress distributions in the breach When the flow enters into the breach, the cross-section of flow suddenly becomes several times smaller than the cross-section of the reservoir or river. Due to the convergence of flow in the breach, streamlines quickly change directions. In this case, it exist streams perpendicular to the main streamlines of the flow (called secondary flow). Thus, the flow in the breach is with complex conditions characterized by the existence of secondary flow. The distributions of boundary shear stress around the wetted perimeter of the breach depend on the shape of the breach and the structure of secondary flow. The importance of understanding boundary shear stress distributions in the breach is demonstrated by the use which is made of local or mean boundary shear stress in hydraulic equations concerning erosion problem (Knight et al., 1984). Until now, no accurate measurement of velocity and shear stress in the breach has made its success in literature. However, some numerical models have been developed to measure velocity and shear stress distributions in rectangular open channel based on field measurements, assumptions of secondary flow structure and the lateral momentum transfer. According to the literature review, some experiments conducted have been reported concerning the boundary shear stress distributions in a smooth rectangular channel.

5.1.1. Experiments in rectangular open channels Knight (Knight et al., 1984) presented some experimental results for distributions of shear stress in smooth rectangular open channels. The shape of a rectangular open channel was defined by the ratio of the breadth ( B ) to depth ( H ). The aspect ratios were modified for adaptions of different channel shapes. In order to study distributions of boundary shear stress, the local side walls  w and bed shear stress b were divided by their respective mean values  w and b , and tabulated against z/ H and 2y / B , in which z is vertical distance above bed, and y is lateral distance from one corner (see figure Figure 5-1). Figure 5-2 (a) showed a series of distributions of shear stress in side walls with the aspect ratio ranging from 0.58 to 2.0. It could be seen that at the bottom of side walls the minimum shear stress was appeared and the maximum was not appeared on the surface of flow but in the lower part of side walls. Figure 5-2(b) showed a series of distributions of shear stress in bed with aspect ratio ranging from 3.91 to 7.73. The minimum shear stress was always appeared at the corner and it increased along lateral distance from the corner. If we compare the two series of distributions of shear stress in both Figure 5-2 (a) and (b), it

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Chapter 5. Shear stress distributions in the breach

is obvious that along with the incremental change in BH/ , the distributions of shear stress in side walls (Figure 5-2 (a)) and bed (Figure 5-2 (b)) were difference. However, the influence of aspect ratio differ was slight.

Figure 5-1.The sketch of an open channel within experiments.

Figure 5-2. Experimental results of shear stress distributions on the bed and side walls of rectangular open channels with aspect ratios (B/H) ranging from 0.58 to 7.73 (asterisk lines) (Knight et al., 1984).

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5.1. Literature review

An experimental equation has been developed by Jiang ((Jiang, 1999)) for the prediction of lateral distributions of shear stress in open channels. Jiang compared the model equations with experimental results of Knight, Myers, and Cokljat. Figure 2 (a), (b) and (c) showed comparisons of prediction results and the experimental date with aspect ratios ranging from 2 to 7.73. On each figure, the minimum shear stress appeared in the side wall of bed and the maximum appeared in the middle of the bed. According to comparisons of these three pictures, it can be seen that along with the increase of aspect ratios, distributions of shear stress on the bottom became more stable. Figure 2 (b) showed comparisons of prediction result and the different experimental data with aspect ratios ranging from 0.5 to 2.28. It can be seen that from the bottom to top of the side wall, shear stresses quickly increased and then slowly increased until near the water surface where it decreased a little. The minimum shear stress occurred on the bottom of the side wall, and the maximum occurred near the water surface. It also indicated that the aspect ratio on prediction results had little effect on the distributions of shear stress on side walls.

(a) Shear stress distributions on the bed with the (b) Shear stress distributions on the bed with the aspect ration of 2 aspect ratio of 3.94

(c) Shear stress distributions on the bed with the aspect ratio of 7.73 (d) Shear stress distributions on side walls Figure 5-3.Comparisons of prediction results and experimental data of boundary shear stress distributions on the bottom and side walls of rectangular open channels with different aspect ratios (Jiang, 1999).

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Chapter 5. Shear stress distributions in the breach

5.1.2. Flow conditions in the breach As mentioned in Chapter 3 section 3.2.4, the flow in the breach has been widely classified into two categories by the directions of the incoming flow: (2) symmetrical incoming flow and (2) parallel incoming flow. The streamlines of flow velocities are complex and shown in Figure 5-4 and Figure 5-5. It is the principle flow that is along with flow direction. In the cross-section, it is characterized by existence of secondary flow (Blanckaert and De Vriend, 2004, Jiang, 1999). Secondary flow is a helical like motion existing on the cross-section of the flow. It has been usually observed in river bends (Hey and Thorne, 1975). In addition, it has been recognized as an important cause of soil sediment and deposition (Kitanidis and Kennedy, 1984). In river bends, the existence of secondary flow removes soil particle from outer bank to inner. That makes the outer bank more stepper and the inner one gradual. By observations of breach failure tests (Zhao et al., 2014), the secondary flow also existed in the breach. Due to advecting flow momentum, the secondary flow has an important influence on distributions of velocity and the boundary shear stress in the breach, and thereby erosion of the breach (Blanckaert and De Vriend, 2004).

Figure 5-4. Illustration of the distributions of flow velocities in the breach with parallel incoming flow.

144

5.2. Shear stress calculations

Figure 5-5. Illustration of the distributions of flow velocities in the breach with symmetrical incoming flow.

5.2. Shear stress calculations

As the development of the headcut erosion, observed by Vaskinn, Morris, Visser and Zhao (Vaskinn et al., 2004, Zhao et al., 2014, Visser, 1998, Morris et al., 2007), the lateral erosion started to play an important role in the breach widening processes. The discharge flow in the breach can generate vertical lateral side walls. This can be ascribed to the soil cohesion, and to the breach water pressure on a non-saturated and low permeable soil. Due to the secondary flow (helicoidal flow) in the breach channel, the under mining process triggered the erosion at the side toe of the embankment, which makes the side wall of the breach into negative. The soil blocks of side wall then collapse due to the unbalanced situation. Therefore, it is the secondary flow that stimulates the lateral development of the breach channel and makes the breach width increase directly.

5.2.1. Depth-averaged Navier-Stokes equation For an incompressible fluid, the governing steady Navier-Stokes equation for the streamwise motion in an open channel with a plane bed inclined in the streamwise direction is as follows:

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Chapter 5. Shear stress distributions in the breach

2  u ()() uv  uw  xxxy   xz  p w         g sin  (5.1) x  y  z   x  y  z  x

Where, x,, y z are the streamwise, the lateral and the normal directions respectively; u,, v w  are the velocity components in the (,,)x y z directions; g is the gravitational constant;  xx is the Reynolds stress on the section; xy is the Reynolds stress on the lateral side;  xz is the Reynolds stress on the bed; p is the water static pressure;  is the bottom slope; w is the water density. The schematic of the channel geometry and coordinate system is shown in Figure 5-6.

Figure 5-6. Schematic of the channel geometry and coordinate system.

Considering a longitudinal flow yields, at the first order p  h  g cos  w (5.2) xw  x  xx  0 (5.3) x

Where, hw is the water depth. Therefore, equation (5.1) becomes as follows:

2  u ()() uv  uw  xy  xz w  w     w gS0   (5.4) x  y  z    y  z  Weight component    Bed Reynolds stress Streamwise velocity variation Secondary flow Lateral Reynolds stress

146

5.2. Shear stress calculations

Where, S0 is the channel energy slope, and may be expressed as follows:

h S sin  cos  w (5.5) 0 x Therefore, the depth-integrated equation may be expressed as follows:

 h u2       w  hw() uv   h w xy     gh S    (5.6) wx w  y w w0  y b

Where,  b is the value of  xz on the bottom.

h 1 w u2  u 2 () z dz (5.7) hw 0

h 1 w uv  u()() z v z dz (5.8) hw 0

h 1 w uv  u()() z v z dz (5.9) hw 0

5.2.2. The bed shear stress

The bed shear stressb can be related to the stream wise velocity by introducing the dimensionless friction factor f as follows:

f   U 2 (5.10) b w 8 Where, U is the depth-averaged axial velocity, and may be expressed as follows:

h 1 w U(,)(,,) x y  u x y z dz (5.11) hw 0

5.2.3. The vertical depth-averaged Reynolds stress Shiono and Knight (Shiono and Knight, 1991) proposed to write the vertical depth- averaged Reynolds stress as a function of the stream wise velocity gradient by introducing the dimensionless eddy viscosity coefficient  as follows:

U    (5.12) xy w xy y

147

Chapter 5. Shear stress distributions in the breach

Where, xy is the depth-averaged eddy viscosity. It is often related to the local friction velocity U* and the depth hw by the dimensionless eddy viscosity coefficient  and defined by equation (5.13).

xy U* h w (5.13)

 b U*  (5.14) w

5.2.4. The secondary flow The second term in the left hand side of the stream wise momentum equation (5.6) can be denoted as the secondary flow,     hw () uv     (5.15) w y

For the sake of simplicity, it is assumed that the variation of the water level is negligible in the lateral direction hw(,)() x y h w x . This assumption must be removed when considering a breach side with a slope.

1) The assumption of Tang and Knight The research by Tang and Knight (Tang and Knight, 2008) showed that uv(x,y) can be considered as linear with y close to the wall. Then, uv  A(x)y . Since A has the same dimension as g , the expression of uv(x,y) may be rewritten by introducing the dimensionless secondary flow parameter k1 as follows: ()uv (,)()()x y k x gS x (5.16) y 1 0

Finally, the secondary flow,  , can be rewritten as

(,)()()()x y  k1 xw gh w x S 0 x (5.17)

2) The assumption of Ervine The assumption of Ervine (Ervine et al., 2000) is as follows:

2 uv(,)()(,) x y k2 x U x y (5.18)

Therefore, the secondary flow equation (5.15) becomes as follows:

148

5.2. Shear stress calculations

U 2 (,)()()(,)x y  k x h x x y (5.19) 2 w w y

3) Mixture assumption Considering the above two cases, for the sake of generality, it is assumed that the secondary flow equation is as follows:

U 2  (,)()()()()(,)x y w h w x k1 x gS 0 x  k 2 x x y  (5.20) y 

5.2.5. The streamwise velocity variation Shiono and Knight (Shiono and Knight, 1991) ignored the streamwise velocity variation in a straight channel. However, the streamwise variation should be considered due to the 3D flow behavior in the breach. Therefore the streamwise velocity variation is included in the analysis. The assumption of Liu (Liu et al., 2014) is as follows:

u 2  A (5.21) x Since A has the same dimension as g, it may be assumed to re-write the equation (5.21) by introducing the dimensionless parameter c1 as follows:

u 2 (,)()x y c x g (5.22) x 1

Considering the turbulent flow and for the sake of simplicity, it is assumed that u2 U2 . Therefore, the variation of the velocity can be transferred as follows:

2   hw u   2 h  Uw   h c() x g (5.23) wx w  x w w 1

5.2.6. Re-formulation of the depth-integrated equation Finally, by integrated the bed shear stress equation (5.10), the vertical depth-average Reynolds stress equation (5.12), the secondary flow equation (5.20)and the streamwise variation of the velocity equation (5.23), the depth-integrated equation (5.6) can be rewritten as follows:

149

Chapter 5. Shear stress distributions in the breach

1/2  2 2 f   U   U2  f hw  hw  U   h w k2   U     ghw  S0(1  k 1 )  c 1   0 (5.24) y8   y   y 8  x 

The more concise expression of the equation (5.24) may be written as follows:

2UU 2  2 a a  a U2  a  0 (5.25) 1y2 2  y 3 4

1/2 1 2 f  a1   hw   (5.26) 2 8 

a2 hw k 2 (5.27)

f h a   w (5.28) 3 8 x

a4 ghw  S 0(1  k 1 )  c 1  (5.29)

The expression of the depth-integrated form of the streamwise Reynolds averaged Navier– Stokes (RANS) equation is more general than that given by Shiono and Knight (Shiono and Knight, 1991).

5.2.7. Analytical solutions

For the sake of simplicity, a symmetrical flow is assumed as U (0)  0 (y=0 is located at the breach center). Taking the Laplace transform, it yields

a a U 2 (0) V() s  4 3 (5.30) s( a3 s ( a 2  a 1 s ))

2 2 Where, V(y)  U (y)  U (0) . The inverse Laplace transform yields

a a U 2 (0) V()() y 4 3 A y (5.31) a3

     a2 y   a 2 y  y  A( y ) 1  exp   sinh   cosh    (5.32) 2a1    2 a 1   2 a 1  

2 Where,   (a2 )  4a1a3 .

150

5.3. Analytical results and discussion

U( y ) V ( y )  U 2 (0) (5.33)

The unknown value U 2 (0) can be obtained by assuming that the velocity vanishes at the

2 vertical wall (U (b)  0 ). Therefore, it is obtained that U (0)  V(b) . The equation (5.33) can be rewritten as follows:

U()()() y V y  V b (5.34)

By integrating the expression of U2 (0)  V(b) into equation (5.31), it is obtained that a a V() b V()() b 4 3 A b (5.35) a3

The solution is as follows:

a A() b V() b  4 (5.36) a3 (1 A ( b )) The results yield a a a V() b  4 (5.37) 4 3 1 A ( b )

a A() y V() y  4 (5.38) a3 1 A ( b )

a A()() y A b V()() y V b  4 (5.39) a3 1 A ( b )

V a( a  )( a   )  a y  y  (y ) 4 2 2 exp 2  sinh  (5.40)    y2a1 a 3  1  A ( b )  2 a1   2 a 1 

5.3. Analytical results and discussion

To calculate the averaged lateral and bottom shear stresses on the boundary of rectandular open channel, the analytical results have been obtained by considering the secondary flow effects.

5.3.1. Bottom shear stresses

The bottom bed shear stress at the center of the channel,b(0) , can be expressed using equation (5.41).

151

Chapter 5. Shear stress distributions in the breach

f (0)  U 2 (0) (5.41) b w 8

Where, U (0) is the depth-averaged velocity at the center of the channel U(0) V (0)  V ( b ) .

5.3.2. Lateral shear stresses From above analyses, the vertical depth-averaged lateral shear stresses may be expressed using equation (5.42).

1 f V ()()b   h b (5.42) xy2 w w 8 y

Where, V() s is the Laplace operator, and may be expressed using equation (5.43). The parameters in this equation can be determined using equation (5.44).

a A() y V() y  4 (5.43) a3 1 A ( b )

V a( a  )( a   )  a y   y  (y ) 4 2 2 exp 2  sinh      y2a1 a 3  1  A ( b )  2 a1   2 a 1  1/2 1 2 f  f hw a1 hw  , a 2  h w k 2 , a 3   , a 4  ghw  S 0 (1  k 1 )  c 1  2 8  8 x

2  (a2 )  4 a 1 a 3 (5.44)

     a2 y  a 2  y  y  A( y ) 1  exp  sinh   cosh    2a1   2 a 1   2 a 1   h S sin  cos  w 0 x

5.3.3. The streamwise depth-averaged velocity The transverse variation streamwise depth-averaged velocity may be expressed using the equation (5.45).

U()()() y V y  V b (5.45)

5.3.4. Curves

152

5.3. Analytical results and discussion

In the expression of the transverse variation streamwise deth-averaged velocity, there are two variables:

1) The water depth in the channel, hw ; 2) The half-width of the channel bottom, b . Meanwhile, the expression is determined by five coefficients:

1) The dimensionless eddy viscosity coefficient,  ;

2) The dimensionless friction factor, f ;

3) The dimensionless secondary flow parameter of Tang and Knight, k1 ;

4) The dimensionless secondary flow parameter of Ervine, k2 ;

5) The dimensionless streamwise velocity variation parameter of Liu et al, c1 ;

Parameters used to plot the curves of the averaged lateral and bottom shear stresses,  xy ()b and b(0) , the depth-averaged velocity at the center of the channel and the channel energy slope, S0 , are given in Table 5-1.

Table 5-1. Set of parameters used to plot the curves.

Notations Parameters Values g Gravitational constant 9,81 m/s2 The water density 3 w  1000 kg/m The water depth in the channel hw 4 m b The half-width of the breach 4 m   The breach bottom slope 0 f The dimensionless bed friction factor 0,1   The dimensionless eddy viscosity coefficient 0,1 The dimensionless secondary flow parameter k 0 1 of Tang and Knight The dimensionless secondary flow parameter k 0 2 of Ervine The dimensionless streamwise velocity c1 variation parameter of Liu 0,001

153

Chapter 5. Shear stress distributions in the breach

5.3.4.1. The water depth in the channel hw

Ranges of the water depth from 0 to 4 meters have been chosen to plot the curves. The other parameters are given in Table 5-1. Figure 5-7 shows evolutions of the averaged lateral and bottom shear stresses for different water depths ranging from 0 to 4 meters. It can be observed that the averaged lateral and bottom shear stresses have the same magnitude along with the increase of the water depth in the channel. The differences between values of them are smaller. Figure 5-8 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different water depths ranging from 0 to 4 meters. Taking into account the value of the breach bottom slope (considered as zero), the channel energy slope can be expressed as S0  hw /  x . In this figure, both of the streamwise depth-averaged velocity and the channel energy slope rise as the water depth increases.

5.3.4.2. The half-width of the breach b

Ranges of the channel bottom form 0 to 20 meters have been chosen to plot the curves. It corresponds to a breach width ranging from 0 to 40 meters. The other parameters are given in Table 5-1. Figure 5-9 shows evolutions of the averaged lateral and bottom shear stresses for different half-widths of the channel bottom ranging from 0 to 20 meters. It can be seen that averaged lateral shear stresses decrease as the half-width of the channel bottom increases. The opposite trend is observed for the averaged bottom shear stresses. The channel bottom width plays a role only when it is less than 10 meters. After that, the increase of the channel bottom width has no influence on the averaged lateral and bottom shear stresses. Figure 5-10 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different half-widths of the channel bottom ranging from 0 to 20 meters. In this figure, the streamwise depth-averaged velocity rises as the half width of the channel bottom increases. On the contrary, the channel energy slope decreases as the half width of the channel bottom increases. The influence of the channel bottom width plays a role only when it is less than 10 meters.

5.3.4.3. The dimensionless bed friction factor f

The dimensionless bed friction factor is Darcy’s coefficient of pressure loss. A range of it from 0.02 to 0.49 has been chosen. The other parameters are given in Table 5-1. Figure

154

5.3. Analytical results and discussion

5-11 shows evolutions of averaged lateral and bottom shear stresses for different friction factors ranging from 0.02 to 0.49. We can see that both of the averaged lateral and bottom shear stresses increase as the dimensionless bed friction factor increases. The increase of the dimensionless bed friction has more influence on the lateral shear stresses than that on the bottom shear stresses. Figure 5-12 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different friction factors ranging from 0.02 to 0.49. The stremwise depth-averaged velocity decreases as the dimensionless bed friction factor increases and the trend of the channel energy slope is contrary.

5.3.4.4. The dimensionless eddy viscosity coefficient  

The dimensionless eddy viscosity coefficient has been introduced by Shiono and Knight (Shiono and Knight, 1991) to express the moment transfer. A range of it from 0.01 to 0.09 has been chosen. The other parameters are given in Table 5-1. Figure 5-13 shows evolutions of the averaged lateral and bottom shear stresses for different dimensionless eddy viscosities coefficient ranging from 0.01 to 0.9. From this figure, the averaged lateral shear stresses increase as the dimensionless eddy viscosity coefficient increases. On the contrary, the dimensionless eddy viscosity coefficient has less influence on the averaged bottom shear stresses. Figure 5-14 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different dimensionless eddy viscosities coefficient ranging from 0.01 to 0.9. From the figure, we can see that the dimensionless edd viscosity coefficient plays the same role as the dimensionless bed friction factor.

5.3.4.5. The dimensionless secondary flow parameter of Tang and Knight k1

The dimensionless secondary flow parameter k1 has been introduced by Tang and Knight (Tang and Knight, 2008). It expresses the velocity variation and can be determined by experiments. A range of it from 0 to 0.9 has been chosen. The other parameters are given in Table 5-1. Figure 5-15 shows evolutions of the averaged lateral and bottom shear stresses for different secondary flow parameters k1 ranging from 0 to 0.9. We can see that The averaged lateral and bottom shear stresses have the same magnitude and trends as the dimensionless secondary flow parameter k1 increases. Figure 5-16 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different secondary flow parameters k1

155

Chapter 5. Shear stress distributions in the breach

ranging from 0 to 0.9. In this figure, it is observed that the channel energy slope increases as the increase of the dimensionless secondary flow parameters k1 increases and the trend of the streamwise depth-averaged velocity is contrary.

5.3.4.6. The dimensionless secondary flow parameter of Ervine k2

The dimensionless secondary flow parameter k2 has been introduced by Ervine (Ervine et al., 2000). A range of it from 0 to 0.035 has been chosen and the other parameters are given in Table 5-1. Figure 5-17 shows evolutions of the averaged lateral and bottom shear stresses for different secondary flow parameters k2 ranging from 0 to 0.035. From this figure, we can see that the averaged lateral shear stresses increase as the dimensionless secondary flow parameter k2 increases. To the contrary, the dimensionless secondary flow parameter k2 has little influence on the averaged bottom shear stresses. Figure 5-18 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different secondary flow parameters k2 ranging from 0 to 0.035. In this figure, it is observed that as the increase of the dimensionless secondary flow parameter k2 the streamwise depth-averaged velocity increases, and on the contrary the channel energy slope decreases.

5.3.4.7. The dimensionless streamwise velocity variation parameter of Liu c1

The dimensionless streamwise velocity variation parameter c1 has been introduced by Liu (Liu et al., 2014). It expresses the streamwise velocity variation that is significant for breaching discharge flow along the streamwise direction. A range of it from 0 to 0.0055 has been chosen. Figure 5-19 shows evolutions of the averaged lateral and bottom shear stresses for different streamwise velocity variation parameters ranging from 0 to 0.0055. In this figure, both of the averaged lateral and bottom shear stresses increase as the dimensionless streamwise velocity variation parameter c1 increases. Figure 5-20 shows evolutions of the streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different streamwise velocity variation parameters ranging from 0 to 0.0055. We can see that both of the streamwise depth- averaged velocity and the channel energy slope increase as the dimensionless streamwise velocity variation parameter c1 increases.

156

5.3. Analytical results and discussion

120

100

Stresses (Pa) Stresses 40

20 tauxy (Pa) lateral stress taub (Pa) bed stress 0 0 1 2 3 4

hw (m)

Figure 5-7. The averaged lateral and bottom shear stresses for different water depths ranging from 0 to 4 meters.

3 0.005 0.0045 2.5 0.004 0.0035 2 0.003 S0 1.5 0.0025

0.002 Slope

Velocity (m/s) Velocity 1 0.0015 0.001 0.5 U(0) 0.0005 S0 0 0 0 1 2 3 4

hw (m)

Figure 5-8. The streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different water depths ranging from 0 to 4 meters.

157

Chapter 5. Shear stress distributions in the breach

160

140

120

100

60 Stresses (Pa) Stresses 40

20 tauxy (Pa) lateral stress taub (Pa) bed stress 0 0 4 8 12 16 20 b (m)

Figure 5-9. The averaged lateral and bottom shear stresses for different half-widths of the channel bottom ranging from 0 to 20 meters.

3.5 0.007

3 0.006

2.5 0.005

2 0.004 S0

1.5 0.003 Slope

Velocity (m/s) Velocity 1 0.002

0.5 U(0) 0.001 S0 0 0 0 4 8 12 16 20 b (m)

Figure 5-10. The streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different half-widths of the channel bottom ranging from 0 to 20 meters.

158

5.3. Analytical results and discussion

320

280

240

200

160

120 Stresses (Pa) Stresses 80

40 tauxy (Pa) lateral stress taub (Pa) bed stress 0 0 0.1 0.2 0.3 0.4 0.5 Dimensionless friction factor f

Figure 5-11. The averaged lateral and bottom shear stresses for different friction factors ranging from 0.02 to 0.49.

3.5 0.01 0.009 3 0.008 2.5 0.007 0.006 2 S0 0.005 1.5

0.004 Slope

Velocity (m/s) Velocity 1 0.003 0.002 0.5 U(0) 0.001 S0 0 0 0 0.1 0.2 0.3 0.4 0.5 Dimensionless friction factor f

Figure 5-12. The streamwise depth-averaged velocity at the center of the channel for different friction factors ranging from 0.02 to 0.49.

159

Chapter 5. Shear stress distributions in the breach

400 tauxy (Pa) lateral stress 350 taub (Pa) bed stress

300

250

200

150 Stresses (Pa) Stresses 100

0 0 0.2 0.4 0.6 0.8 1 Dimensionless eddy viscosity λ

Figure 5-13. The averaged lateral and bottom shear stresses for different dimensionless eddy viscosities coefficient ranging from 0.01 to 0.9.

3.5 0.012

3 0.01

2.5 0.008

2 S0 0.006 1.5 Slope

Velocity (m) Velocity 0.004 1

0.002 0.5 U(0) S0 0 0 0 0.2 0.4 0.6 0.8 1 Dimensionless eddy viscosity λ

Figure 5-14. The streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different dimensionless eddy viscosities coefficient ranging from 0.01 to 0.9.

160

5.3. Analytical results and discussion

120 tauxy (Pa) lateral stress taub (Pa) bed stress 100

Stresses (Pa) Stresses 40

0 0 0.2 0.4 0.6 0.8 1

Secondary flow parameter k1

Figure 5-15. The averaged lateral and bottom shear stresses for different secondary flow

parameters, k1 , ranging from 0 to 0.9.

3 0.014

2.5 0.012

0.01 2

0.008 S0 1.5 0.006 Slope

Velocity (m/s) Velocity 1 0.004

0.5 U(0) 0.002 S0 0 0 0 0.2 0.4 0.6 0.8 1

Secondary flow parameter k1

Figure 5-16. The streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different secondary flow parameters, k1 , ranging from 0 to 0.9.

161

Chapter 5. Shear stress distributions in the breach

480

400

320

240

Stresses (Pa) Stresses 160

80 tauxy (Pa) lateral stress taub (Pa) bed stress 0 0 0.01 0.02 0.03 0.04

Secondary flow parameter k2

Figure 5-17. The averaged lateral and bottom shear stresses for different secondary flow

parameters, k2 , ranging from 0 to 0.035.

3.6 0.005

3.4 0.004

3.2 0.003 S0 3

0.002 Slope

Velocity (m/s) Velocity 2.8

0.001 2.6 U(0) S0 2.4 0 0 0.01 0.02 0.03 0.04

Secondary flow parameter k2

Figure 5-18. The streamwise depth-averaged velocity at the center of the channel and the

channel energy slope S0 for different secondary flow parameters, k2 , ranging from 0 to 0.035.

162

5.3. Analytical results and discussion

250

200

150

100 Stresses (Pa) Stresses

50 tauxy (Pa) lateral stress taub (Pa) bed stress 0 0 0.002 0.004 0.006

Streamwise velocity variation c1

Figure 5-19. The averaged lateral and bottom shear stresses for different streamwise velocity variation parameters ranging from 0 to 0.0055.

5 0.012

0.01 4

0.008 3 S0 0.006

2 Slope

Velocity (m/s) Velocity 0.004

1 0.002 U(0) S0 0 0 0 0.001 0.002 0.003 0.004 0.005 0.006

Streamwise velocity variation c1

Figure 5-20. The streamwise depth-averaged velocity at the center of the channel and the channel energy slope S0 for different streamwise velocity variation parameters ranging from 0 to 0.0055.

163

Chapter 5. Shear stress distributions in the breach

5.4. Conclusions

Considering the effects of the secondary flow, a model for estimating the streamwise depth-averaged velocity and averaged lateral and bottom shear stresses is proposed, as shown in equations (5.41), (5.42), and (5.45). Equations are derived from the depth integrated of Navier-Stokes equation for the streamwise direction. It accounts for turbulent effects via the dimensionless eddy viscosity coefficient  , and secondary flow effects via the dimensionless secondary flow parameters k1 and k2 .  is introduced to express the moment transfer of the turbulent flow. k1 is introduced by Tang and Knight for simplifying secondary flow effects using the product of the gravitational constant g and the channel energy slope S0 . k2 is introduced by Ervine for simplifying secondary flow effects using the product of the water density w , the water depth hw , and the streamwise depth-averaged flow velocity U . A breach with 8m wide and 4m of water depth is used within this numerical model to study the sensitivity of parameter. It is concluded that the dimensionless friction factor f , the dimensionless eddy viscosity coefficient  , and the dimensionless secondary flow parameters k2 have obvious influences on the averaged lateral shear stress. The water depth hw , the dimensionless secondary flow parameters k2 , and the dimensionless streamwise velocity variation parameter c1 have few influence on the averaged lateral shear stress. The half-width of the bottom b effects both of the averaged lateral and bottom shear stresses when it is smaller than 10m. The validation of this simple numerical model is still required.

164

CHAPTER 6

Validation by tests

A large-scale test is used here to validate the breach failure model of an embankment dam and levee. The test setup and test procedure are described in section 6.1. The measured discharge flow results and evolution of the breach width are listed. Simulated results are obtained by inputting the parameters needed for the breach failure model as described in the second, third and fifth chapters. The simulation of the test dam is conducted in section 6.2. Comparisons with the measured data are provided in section 6.3 as well as discussions. Conclusions are provided in section 6.4.

6.1. Presentation of the test

6.1.1. Test introduce To improve the accuracy of estimating the breach formation of the embankment dam subject to internal erosion, a large-scale test was performed during the FP5 IMPACT (investigation of extreme flood processes and uncertainty) project.

6.1.2. Test setup A homogenous moraine embankment dam was built in the autumn of 2003 near the town of Moi Rana in the middle of Norway. The test site is show in Figure 6-1 and is located approximately 600 m downstream of Rössvatn Dam. Rossvatn Dam has three identical bottom outlet gates with a total capacity of approximately 450 m3/s at the full supply level. These gates were used to regulate the flow during the tests. The water levels upstream (in the reservoir between the test dam and Rössvatn Dam) and downstream of the test dam were measured using water level gauges. The positions of the water level gauges are shown in Table 6-1. VM1 and VM2 measured the water level upstream the test dam. VM1 is located just upstream of the rapid varying flow and VM2 is located just downstream of the rapid varying flow to obtain the upstream water level near the test dam. VM3 and VM5 were positioned downstream of the test dam to measure discharge flow from the test site. VM3 is a V-notch weir that was designed to measure discharge flow of less than 100 l/s during initial phase. VM5 is a tail-water level gauge that was used to determine discharges greater than 10 m3/s and was located approximately 200 m downstream of the test dam. Regarding unsteady flow during dam failure, the rating curve of VM5 should not be applied for a discharge flow of less than 5 m3/s. The test site and embankment dam were instrumented and monitored to collect inflow, outflow, and pore water pressure data from the dam body and detailed information on breach initiation, formation, and progression. This homogeneous moraine dam was built using 10% sand and 90% gravel (d50 = 7 mm). The properties of the dam materials are listed in Table 1-2 and are obtained by testing (Vaskinn et al., 2004). Photos of the test dam are shown in Figure 1-3. A sketch of the test dam is shown, and the geometry of the test dam is shown in Table 1-3. A 0.2 m pipe was built in the bottom of the dam as a trigger mechanism to initiate internal erosion (see Figure 1-4). The pipe was filled with homogenous sand and a sand layer. At the downstream end, the pipe was closed before testing.

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Chapter 6. Validation by tests

Table 6-1. The positions and uses of water level gauges.

Water level Positions relative to the Usages Capability gauges test dam

Upstream; Measure water level in VM1 Before rapid varying the reservoir flow Upstream; Measure water level VM2 After rapid varying upstream of the test flow dam Measure discharge It is capable for flow VM3 Downstream flow less than 100 l/s Measure discharge It is incapable for VM5 200m downstream flow flow less than 5 m3/s

Figure 6-1. The test site at Rössvatn.

168

6.1. Presentation of the test

Rossvatn Dam

Figure 6-2. Locations of the water level gauges used. Map contour lines for every 1m. The flow direction is indicated using red arrows.

(a) View from the downstrem surface of the test dam divided by (b) The crest of the test dam. Before the test, the reservoir upstrem of the test dam 2*1m grids; the trigger pipe is closed at the downstream end; was filled with water.

Figure 6-3. Photos of the test dam.

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Chapter 6. Validation by tests

(a) The cross-section of the test dam.

(b) The longitunal section of the test dam.

Figure 6-4. Sketch of the test dam.

Table 6-2. Properties of the test dam materials.

Moraine Values

Soil dry density (kg/m3) 2160

Soil density (kg/m3) 2341

Porosity 0.244

Angle of friction (°) 42

D50(mm) 7

Moisture content 0.06

Cohesion (kN/m2) 20

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6.1. Presentation of the test

Table 6-3. The geometries of the test dam.

Geometries Values

Bottom elevation (m) 364.8

Dam height (m) 4.3

Crest width (m) 3

Bottom width (m) 15.04

Upstream slope 1:1.4

Downstream slope 1:1.4

6.1.3. Test procedure In this test, embankment dam failure due to internal erosion was studied. Between opening of the valve of the pipe and the end of the erosion process, embankment failure lasted approximately 38 minutes. Before starting the test, the reservoir between the test dam and the Rossvatn Dam was filled with water. Six minutes after the opening the pipe, the discharge flow began to increase quickly. Over the next 10 minutes, the water level in the reservoir quickly decreased until the incoming flow from the flow gates of the Rossvatn Dam arrived. The water head upstream of the test dam and recorded by VM2 is shown in Figure 6-5. The incoming flow to the reservoir from the Rossvatn Dam is also given. According to our observations, the entire test area can be divided into the following five sub-processes: pipe initiation, pipe enlargement, pipe collapse, breach enlargement and final breach width. The pictures obtained by recorded video during the test are shown in Figure 6-6.

6.1.3.1. Pipe initiation By opening of the valve at the downstream end of the pipe (13:34:00), the sand was flushed out and the internal erosion started. Soon after, at 13:40:00 the leakage occurred around the pipe due to the lake of any filter, and the flow increased rapidly resulting in a tunnel through the test dam. From that on, it is considered that the initial process of internal erosion was end and the pipe began to progressive quickly and obviously.

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Chapter 6. Validation by tests

20.0 4.5 Inflow 18.0 Outflow 4 Water level 16.0 3.5 14.0 3 12.0 /s) 3 2.5 10.0 2

Flow(m 8.0 1.5 Water (m) level 6.0

4.0 1

2.0 0.5

0.0 0 13:40:00 13:50:05 14:00:10 14:10:15 14:20:20 14:30:24 Time (hh:mm:ss)

Figure 6-5. Water depth upstream of the test dam measured by VM2; The inflow hydrograph from the flood gates of Rossvatn Dam; The outflow hydrograph measured by VM5 located 200m downstream of the test dam.

6.1.3.2. Pipe enlargement The main process of pipe enlargement was from 13:40:00 to 13:56:14. The maximum pipe diameter reached about 4.2 m. It means that the pipe enlarged from 0.2m to 4.2 m within 974 seconds. From Figure 6-6 (a), we can see the discharge flow through the pipe with soil concentration and a small round pipe in the bottom. In Figure 6-6(b), we can see that the top of the pipe was still stable and the diameter of the pipe almost reached the crest. Meanwhile, the discharge flow through the pipe increased and still with soil concentration. By comparing the Figure 6-6(a) and the Figure 6-6(b), the top of the pipe kept stable, and the pipe diameter increased as well as the discharge flow. But the form of the pipe was still kept circle and the discharge flow was with high soil concentration.

6.1.3.3. Pipe collapse At 13:56:14, the top of the pipe collapsed into the breach. Soon after, the collapsed soil block was quickly washed away quickly by discharge flow. Figure 6-6 (c) was taken just after the collapsed of the top. From this picture, we can see that a breach with two vertical side walls separates the test dam into two banks. The collapse of the top of the pipe occurred suddenly (within few seconds).

172

6.1. Presentation of the test

6.1.3.4. Breach enlargement At 13 :56 :14, a breach was formed resulted from the collapse of the top of pipe. Two vertical walls were observed on the two banks of the test dam. From Figure 6-6 (c), we can see that the water level in the upstream remained high and the discharge flow through the breach was relative large. One minute after the pipe collapsed, the soil was eroded and washed away by flow at the foot of two side walls of the breach on the two banks. The soil on the upper side of the side walls remained stable. Next, a crack appeared on the crest of the right bank of the test dam and extended downward (see Figure 6-6 (d)), and the soil block on the right bank fell into the breach along the extended crack (headcut migration). The discharge flow was quickly washed away from the fallen soil block and the width of the breach was enlarged. Similarly, the same episodic collapse of the soil block occurred on the two side walls of the breach. Accordingly, the breach continued enlarging until 14:16:00, at which no additional erosion occurred and the final width of the breach was reached.

6.1.3.5. Final breach The end of the main erosion process occurred at 14:16:00 when the final breach width was reached. From Figure 6-6 (f), it is clear that the upstream water was low and that the discharge flow through the breach was slow and clear with less suspended sediment. The inflow into the test embankment was controlled by the flood gates of Rossvatn Dam located 600 m upstream. The water level upstream of the test dam was measured by VM1 and VM2, and the data used in our research results were recorded by VM2. The water level quickly decreased from 13:40:00 to 13:49:14 before remaining constant at approximately 1.527 m and increasing again at 13:52:38. From the test observations, a quick decrease in the water level resulted from the quick enlargement of the pipe. Meanwhile, inflow from the flood gates occurred later. The data from the outflow hydrograph were measured by VM5, which was located approximately 200 m downstream in the valley. However, VM5 could not measure discharge flow of less than 5 m3/s. Thus, the outflow weekly increases from 13:48:39 to 13:53:39 before strongly increasing to 14:08:39, at which time the peak discharge (171.01 m3/s) is reached.

6.1.4. Results

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Chapter 6. Validation by tests

The water level measured by the VM2 gauge was used as the water level in the reservoir, and the discharge flow (see Figure 6-7 ) was measured by the VM5 gauge downstream of the test site. As supplementary information, Figure 6-8 shows the breach width evaluation. If we separate the test dam into two banks based on the axis of the initial pipe, the width of the pipe and the breach on the left and right banks can be recorded. Positive values indicate the evolution of the breach on the left bank of the test dam and negative values indicate evolution of the breach on the right. After the pipe collapsed, the two banks evolved in significant steps (see Figure 6-9). From the video, it is clear that headcut migrations occurred every several minutes or seconds. The length of the headcut (at the crest) and the relative water depth (water depth/embankment height) upstream of the test dam were also recorded and varied from 0.6 m to 1.4 m. As shown in Figure 6-10, the reservoir water level decreases to below the top of the pipe and then increases from 13:48:00 to 13:56:14. If we consider the video, the outflow at the test site increases to a maximum value before decreasing. This change in outflow can easily be seen in the video. Obviously, VM5 did not measure the change of outflow during pipe development because the outflow is measured at VM5 and is located 200 m downstream of the test site. Thus, the outflow cannot be used directly as the discharge flow of the test dam.

174

6.1. Presentation of the test

(a) Pipe initiation (b) Pipe enlargement

(e) Headcut migration on left bank (f) Final breach width Figure 6-6. Pictures of failure processes of the test dam taken by video. The enlargement of the breach width is shown in pictures (g) through picture (f) and resulted from headcut migration in the two banks.

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Chapter 6. Validation by tests

180 4.5 Discharge flow 160 4 Water head in upstream 140 3.5 /s) 3 120 3

100 2.5

80 2

60 1.5 Water (m) level Didcharge flow(m 40 1

20 0.5

0 0 13:40:00 13:52:58 14:05:56 14:18:53 14:31:51 14:44:48 Time (hh:mm:ss)

Figure 6-7. The discharge flow measured by VM5 200m downstream of the test dam.

16 4.5

14 4

3.5 12 3 10 2.5 8 2 6 Water (m) level

Breachwidth (m) 1.5 4 1 Breach width 2 0.5 Water head in upstream 0 0 13:40:00 13:48:39 13:57:17 14:05:56 14:14:34 14:23:12 Time (hh:mm:ss)

Figure 6-8. The breach width measured by video from initiation of the pipe to the final breach.

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6.1. Presentation of the test

0 Left -2 Right

Breachwidth (m) -4

-6

-8

-10 13:40:00 13:47:12 13:54:24 14:01:36 14:08:48 14:16:00 Time (hh:mm:ss)

Figure 6-9. The breach widths measured separately at the two embankment banks by video from the initiation of the pipe to the final breach. Positive values indicate the evolution of the breach on the left bank of the dam and negative values indicate on the right bank.

1.6 1.2

1.4 1

1.2 ) w 0.8 1

0.8 0.6

0.6 0.4 Recession Recession length (m)

0.4 Relativewater (α depth 0.2 0.2 Recession length Relative water depth 0 0 13:54:37 13:59:48 14:05:00 14:10:11 14:15:22 14:20:33 Time (hh:mm:ss)

Figure 6-10. The length of the toppling failure mass measured by video on the right and left banks and the relative water depth in the upstream area near the breach.

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Chapter 6. Validation by tests

6.2. Test simulation As the description of test procedures, the failure of the test dam consists of two main processes: (1) the enlargement of the pipe in the test dam, and (2) the episodic breach widening processes after the collapse of the pipe top. To applying the numerical models developed into this test dam, two processes are divided as followings.

6.2.1. The pipe enlargement process Table 6-4 shows parameters input in the analytical model proposed in Chapter 2. Soil properties and dam geometries are given in Table 6-2 and Table 6-3. The water depth upstream of the test dam measured by VM2 is used as upstream boundary condition. The initial water depth downstream of the test dam is set as zero.

Table 6-4. Parameters input in the analytical model for the pipe enlargement process.

Parameters Values Initial pipe diameter (m) 0.02 Final pipe diameter (m) 4.3 Turbulent friction factor 0.005

Singular head loss coefficient ( kin k out ) 0.4

Erosion coefficient (Ce , s/m ) 0.0455

Critical shear stress (c , Pa) 0

6.2.2. The breach widening processes Considering the direction of the incoming flow (Perpendicular to the test dam) and the position of the predrilled pipe (in the center of the test dam bottom), the analysis of symmetrical widening process is chosen. To apply the simple headcut migration model into the calculation of the stepwise enlargement of the breach width (see equation ), the determination of the erosion time is needed (see section 3.2.3 in Chapter 3). B t  t   B( t )  2 L ( t ) (6.1)

t  te (6.2) where, B is the width of the breach, L is the length of the headcut, 2 means that lengths of the headcut on each side of the breach are same due the symmetrical widening process assumption. t is the time spent to generate the failure of the headcut, and te is the erosion time spent on the eroded notch to reach the critical length.

178

6.2. Test simulation

If the excess shear equation is used to calculate the soil erosion rate on the corner of the headcut (see equation (6.3)), the erosion time spent to reaches to a critical length of the eroded notch that generates the failure of the headcut can be obtained. This critical length of the eroded notch equals to the critical length of the headcut due to the assumption of failure modes in Chapter 3. Thus, the erosion time is obtained. dl e k ()   (6.3) dt d v cv where, le is the length of the eroded notch on the corner of the headcut, kd is the erosion coefficient (see Table 6-5), v is the lateral shear stress in the breach, and v is the critical lateral shear stress in the breach (see Table 6-5).

Table 6-5. Parameters input in analytical model for the breach widening process.

Parameters Values Initial breach width (m) 4.3 Final breach width (m) - Dimensionless friction factor ( f ) 0.001

Erosion coefficient ( kd , m/s/Pa ) 0.002

Critical shear stress ( cv , Pa) 10

The Bernoulli equation is used to calculate the flow characteristics in the breach.

2 2  y y4 y  yup  y 1    2 (6.4) 2 y2   up y yup  

Q gBy3 (6.5) Q V  (6.6) By

where, yup is the upstream water head, y is the critical water depth in the breach,  is the coefficient of singular head loss and set as 0.5,  is the coefficient of the linear head loss and ignored, Q is the discharge flow, and V is the flow velocity. The lateral shear stress is assumed as the same as the bottom shear stress in the breach, and can by calculated by using equation (6.7). f     V 2 (6.7) c b w 8

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Chapter 6. Validation by tests

where, f is the dimensionless friction factor (see Table 6-5). Computational algorithm is shown as follows:

1) The time step is specified as t  2 seconds; 2) The input data: all the parameters in Table 6-2, Table 6-3, Table 6-4, and Table 6-5, as well as upstream water head;

3) Compute the critical water depth in the breach y ;

4) Compute the discharge flow Q , flow velocity V , and lateral shear stress c in the breach;

5) Compute the increase in the erosion length le   t  k d() c   cv ;

6) Compute the length of erosion le l e   l e 7) Compute the critical length of headcut L (see equations (3.81), (3.82) and (3.83)); 8) Compare the length of the erosion and the critical length of headcut. If the length of

erosion is smaller than the critical length of the headcut (le  L ), no failure occurs and the calculation continues to the next loop; if the length of erosion is larger than the critical length of the headcut, failure occurs and the width of the breach is

enlarged (le  L ) as BBL  2 . Meanwhile the length of erosion is set as zero (

le  0 ) to prepare for the next loop. 9) The end of one loop.

6.3. Results and discussion

6.3.1. Results The simulated discharge flow in the breach is shown in Figure 6-11. The discharge flow weakly increased after initiation of the pipe at 13:40:40. It reached the first peak value of 20m3/s at 13:45:06, and then it decreased. The secondary peak discharge flow was about 150m3/s at 14:00:00. Some small discharge flows appeared around the secondary peak value. Simulated evolutions of the breach width (called pipe before the collapse of the pipe top) are shown in Figure 6-12. The first quick increase of the pipe diameter appeared before that the pipe diameter was about 3m at 13:47:26. Then it slowly increased until the collapse of the pipe top at 13:56:12. After that, the breach stepwise enlarged. In each step,

180

6.3. Results and discussion

the breach width ranged from 1 to 1.4m. The simulation ended at 14:16:10 when the breach width was approximately 22m.

6.3.2. Discussion It should be note that the discharge flow hydrograph was measured 200m downstream of the test dam by the water level of VM5. VM5 is incapable for measuring the flow less than 5m3/s. From the video taken at the site of the test dam, it was clearly observed that a peak discharge flow appeared before the collapse of the pipe top. Then it decreased due to the drop of water level in the reservoir upstream of the test dam. Until the arriving of the inflow from upstream water gates, the water level increased as well as the discharge flow. Thus, the measured discharge flow does not show this process due to technical and geographical reasons. But the simulated results can well express it. The simulated secondary peak discharge flow is about 10% smaller than measured one. If we compare the volume of discharge flow before the peak discharge flow, we get a good agreement between the measured and simulated results. The simulation of the stepwise enlargement of the breach width has good agreements at the beginning of the breach widening process. At the end of the breach widening process, the measured failure soil mass width decreased significantly in each step. However, the simulated results cannot well simulate this phenomenon.

6.4. Conclusions

It can be concluded that the simulation of the pipe enlargement process has good agreements with the measured data. The variation of water conditions in upstream can be well considered. Both of the pipe diameter and the discharge flow are well simulated. The stepwise enlargement of the breach with is well simulated at the beginning of the breach widening process, but not well simulated at the end of this process. The accurate estimation of the erosion process at the corner of the headcut is still needed.

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Chapter 6. Validation by tests

180 4.5 Discharge flow_measured 160 Discharge flow_simulated 4 Water head in upstream 140 3.5 /s) 3 120 3

100 2.5

80 2

60 1.5 Water (m) level Didcharge flow(m 40 1

20 0.5

0 0 13:40:00 13:52:58 14:05:56 14:18:53 14:31:51 14:44:48 Time (hh:mm:ss)

Figure 6-11. Comparisons of the measured and simulated discharge flow in the breach of the test dam.

24 4.5

21 4

3.5 18 3 15 2.5 12 2 9 Water (m) level

Breachwidth (m) 1.5 6 1 Breach width_measured 3 Breach width_simulated 0.5 Water head in upstream 0 0 13:40:00 13:48:39 13:57:17 14:05:56 14:14:34 14:23:12 Time (hh:mm:ss)

Figure 6-12. Comparisons of the measured and simulated evolutions of the breach width of the test dam.

182

Conclusions

According to the literature review of floods in the history and statuses of embankment dam and levee, it is recognized that challenges for flood risks are still great. The breach processes have been simulated by many researchers using physical based models and experimental regression equations from the end of the last century. However, none of them has been wildly used until now. At the beginning of this century, benefiting from the development of experiments and the soil erosion theory, knowledge of the breach processes has been progressed. A pipe enlargement model is proposed to simulate the evolution of the pipe in embankment dams and levees. In this model, the turbulent pipe flow with erosion mechanism is employed as well as the soil erosion law. Meanwhile, the hydraulic head variation in the upstream, the trail water conditions in the downstream, the collapse of the pipe top and the transition to a breach are taken into account. This model is applied to a large-scale test analysis (4.3m high) and good results are obtained. A simply headcut migration model based on the soil tensile strength is presented to simulate the critical length of the headcut. The failure of the headcut is determined by the bending and shear failure criterions. Input parameters of this model are: the breach geometry (the levee height), the soil properties (the soil tensile strength/the soil compressive strength, saturated and normal specific weights of the soil), the flow situations in the breach (the relative water head and the ratio of erosion) and the soil permeability parameter. A common range of embankment dam heights from 2 to 12 meters are used to calculate the critical lengths of the headcut under different soil tensile strengths (ranging from 0 to 18 kPa). Both of the bending failure and shear failure modes play a role on the failure of the headcut when the soil is saturated. When the soil is unsaturated, only the bending failure mode determines the migration of the headcut. The numerical headcut migration model based on the soil tensile strength is verified by using limit equilibrium method for three typical levee scales (2, 4 and 6 meter high). The Hoek-brown failure criterion is considered to be reasonable to use as the soil strength of

183

Conclusions

the headcut (levee body). The calculations of critical lengths of the headcuts for each case have been conducted and good agreements are obtained by comparing with the numerical model. Considering the effects of the secondary flow, a model for estimating the streamwise depth-averaged velocity and averaged lateral and bottom shear stresses in the breach is proposed. Equations are derived from the depth integrated of Navier-Stokes equation for the streamwise direction. It accounts for turbulent effects, secondary flow effects, and the streamwise depth-averaged flow velocity gradient. A breach with 8m wide and 4m of water depth used within this numerical model to study the sensitivity of parameter. It is established that the lateral shear stress on the breach side can be greater than the bottom bed shear stress due to the secondary flow influence. A large-scale test of dam failure is simulated by using the pipe enlargement and breach widening models proposed. The simulation of the pipe enlargement process has good agreement with the measured data. The variation of water conditions in upstream can be well considered. Both of the pipe diameter and the discharge flow are well simulated. The stepwise enlargement of the breach with is well simulated at the beginning of the breach widening process.

184

References

AL-HUSSAINI, M. 1981. Tensile properties of compacted soils. Laboratory shear strength of soil. Edited by RN Young and FC Townsend. American Society for Testing and Materials. ASTM STP, 740, 207-225. AL-MADHHACHI, A., HANSON, G., FOX, G., TYAGI, A. & BULUT, R. 2013. Measuring soil erodibility using a laboratory “mini” JET. T. ASABE, 56, 901-910. BENAHMED, N., BONELLI, S. Internal erosion of cohesive soils: laboratory parametric study. 6th International Conference on Scour and Erosion, 2012. BLANCKAERT, K. & DE VRIEND, H. 2004. Secondary flow in sharp open-channel bends. Journal of Fluid Mechanics, 498, 353-380. BONELLI, S. 2012. Erosion of Geomaterials, John Wiley & Sons. BONELLI, S. 2013. Erosion in geomechanics applied to dams and levees, John Wiley & Sons. BONELLI, S. & BRIVOIS, O. 2008. The scaling law in the hole erosion test with a constant pressure drop. International Journal for Numerical and Analytical Methods in Geomechanics, 32, 1573-1595. BREVARD, J. A., THEURER, F. D., COMER, G. H. & SERVICE, U. S. S. C. 1981. Simplified Dam-breach Routing Procedure, Soil Conservation Service. CASTEDO, R., FERNÁNDEZ, M., TRENHAILE, A. S. & PAREDES, C. 2013. Modeling cyclic recession of cohesive clay coasts: Effects of wave erosion and bluff stability. Marine Geology, 335, 162-176. CHENG, Y. M., LANSIVAARA, T. & WEI, W. B. 2007. Two-dimensional slope stability analysis by limit equilibrium and strength reduction methods. Computers and Geotechnics, 34, 137-150.

185

References

CLARK, L. & WYNN, T. 2006. Methods for Determining Streambank Critical Shear Stress and Erodibility: Implications for Erosion Rate Predictions. COSTA, J. E. 1985. Floods from dam failures, Denver, Colo, U.S. Dept. of the Interior, Geological Survey : Open-File Services Section, Western Distribution Branch. EBERHARDT, E. 2012. The Hoek–Brown Failure Criterion. Rock mechanics and rock engineering, 1-8. ERVINE, D. A., BABAEYAN-KOOPAEI, K. & SELLIN, R. H. 2000. Two-dimensional solution for straight and meandering overbank flows. Journal of Hydraulic Engineering, 126, 653-669. EVANS, S. G. 1986. The maximum discharge of outburst floods caused by the breaching of man-made and natural dams. Canadian Geotechnical Journal, 23, 385-387. FANG, H. & HIRST, T. 1973. A method for determining the strength parameters of soils. Highway Research Record, 463, 45-50. FELICIANO CESTERO, J. A., IMRAN, J. & CHAUDHRY, M. H. 2014. Experimental Investigation of the Effects of Soil Properties on Levee Breach by Overtopping. Journal of Hydraulic Engineering. FELL, R., WAN, C. F., CYGANIEWICZ, J. & FOSTER, M. 2003. Time for development of internal erosion and piping in embankment dams. Journal of Geotechnical and Geoenvironmental Engineering, 129, 307-314. FREAD, D. 1988. BREACH, an erosion model for earthen dam failures, Hydrologic Research Laboratory, National Weather Service, NOAA. FROEHLICH, D. 1995. Peak Outflow from Breached Embankment Dam. Journal of Water Resources Planning and Management, 121, 90-97. GARRY, G., BALLAIS, J.-L. & MASSON, M. 2002. La place de l'hydrogéomorphologie dans les études d'inondation en France méditerranéenne/The contribution of hydrogeomorphology in flood hazard assessment: a review of the situation in southern France. Géomorphologie: relief, processus, environnement, 8, 5-15. HANSON, G. & COOK, K. 1997. Development of excess shear stress parameters for circular jet testing. ASAE Paper, 972227. HANSON, G. & COOK, K. 2004a. Apparatus, test procedures, and analytical methods to measure soil erodibility in situ. Applied engineering in agriculture, 20, 455-462.

186

References

HANSON, G. & COOK, K. Determination of material rate parameters for headcut migration of compacted earthen materials. Proc. ASDSO Dam Safety Conf., Phoenix, AZ [CD-ROM]. Assoc. State Dam Safety Officials, Lexington, KY, 2004b. HANSON, G., COOK, K. & HUNT, S. 2005a. Physical modeling of overtopping erosion and breach formation of cohesive embankments. Transactions of the ASAE, 48, 1783-1794. HANSON, G. & SIMON, A. 2001. Erodibility of cohesive streambeds in the loess area of the midwestern USA. Hydrological processes, 15, 23-38. HANSON, G., TEJRAL, R., HUNT, S. & TEMPLE, D. Internal erosion and impact of erosion resistance. Proceedings of the 30th US Society on dams annual meeting and conference, 2010. 773-784. HANSON, G. J., TEMPLE, D., MORRIS, M., HASSAN, M. & COOK, K. Simplified breach analysis model for homogeneous embankments: Part II, Parameter inputs and variable scale model comparisons. Proceedings of 2005 the US Society on Dams Annual Meeting and Conference, Salt Lake City, Utah, 2005b. 163-174. HASSAN, M., MORRIS, M., HANSON, G. & LAKHAL, K. 2004. Breach formation: Laboratory and numerical modeling of breach formation. Association of State Dam Safety Officials: Dam Safety. HAYAKAWA, Y. S. & MATSUKURA, Y. 2010. Stability analysis of waterfall cliff face at Niagara Falls: An implication to erosional mechanism of waterfall. Engineering geology, 116, 178-183. HE, X., WANG, Z., HUANG, J. & DING, L. 2005. Accident statistics and analyses of embankment dams and reservoirs in China. Chinese Hydraulic Engineering Society 2005 Annual Conference Proceedings. HEY, R. D. & THORNE, C. R. 1975. Secondary flows in river channels. Area, 191-195. HOEK, E. The Hoek-Brown failure criterion-a 1988 update. Proc. 15th Canadian Rock Mech. Symp, 1988. Toronto, Dept. Civil Engineering, University of Toronto, 31- 38. HOEK, E., CARRANZA-TORRES, C. & CORKUM, B. 2002. Hoek-Brown failure criterion-2002 edition. Proceedings of NARMS-Tac, 267-273.

187

References

HUNT, S., HANSON, G., COOK, K. & KADAVY, K. 2005. Breach widening observations from earthen embankment tests. Transactions of the ASAE, 48, 1115- 1120. JIANG, S. 1999. Lateral distribution of boudary shear in open-channels. Master, University of Regina. JIE, J. & SUN, D. 2009. National statistics and embankment dam break analysis. Water resources and hydropower engineering 40, 124-128. KAKINUMA, T. & SHIMIZU, Y. 2014. Large-Scale Experiment and Numerical Modeling of a Riverine Levee Breach. Journal of Hydraulic Engineering, 140. KANDIAH, A. 1974. Fundamental aspects of surface erosion of cohesive soils. University of California, Davis. KITANIDIS, P. K. & KENNEDY, J. F. 1984. Secondary current and river-meander formation. Journal of Fluid Mechanics, 144, 217-229. KNIGHT, D. W., DEMETRIOU, J. D. & HAMED, M. E. 1984. Boundary shear in smooth rectangular channels. Journal of hydraulic engineering, 110, 405-422. LACHOUETTE, D., GOLAY, F. & BONELLI, S. 2008. One-dimensional modeling of piping flow erosion. Comptes Rendus Mécanique, 336, 731-736. LI, J., TANG, C., WANG, D., PEI, X. & SHI, B. 2014. Effect of discrete fibre reinforcement on soil tensile strength. Journal of Rock Mechanics and Geotechnical Engineering, 6, 133-137. LIU, C., WRIGHT, N., LIU, X. & YANG, K. 2014. An analytical model for lateral depth- averaged velocity distributions along a meander in curved compound channels. Advances in Water Resources, 74, 26-43. MACDONALD, T. & LANGRIDGE MONOPOLIS, J. 1984. Breaching Charateristics of Dam Failures. Journal of Hydraulic Engineering, 110, 567-586. ‐ MAURIN, J., BOULAY, A., FERREIRA, P., TOURMENT, R. & BEULLAC, B. Études de dangers des digues de classe A de la Loire et de ses affluents-retour d'expérience. Digues maritimes et fluviales de protection contre les submersions- 2ème colloque national-Digues2013, 2013. 209-216. MCCOOK, D. A comprehensive discussion of piping and internal erosion failure mechanisms. Proceedings, annual dam safety conference, ASDSO. Phoenix, Arizona, 2004.

188

References

MÉRIAUX, P., MONIER, T., TOURMENT, R., MALLET, T., PALMA LOPES, S., MAURIN, J. & PINHAS, M. L’auscultation des digues de protection contre les inondations: un concept encore à inventer. 2012. Colloque CFBR:«Auscultation des barrages et des digues-Pratiques et perspectives. MEYER, J. P. & LABUZ, J. F. 2013. Linear failure criteria with three principal stresses. International Journal of Rock Mechanics and Mining Sciences, 60, 180-187. MOHAMED, M., SAMUELS, P., MORRIS, M. & GHATAORA, G. Improving the accuracy of prediction of breach formation through embankment dams and flood embankments. International Conference on Fluvial Hydraulics, 2002a. C:\Users\Zhenzhen.Liu\Documents\Mission\JabRef\Slope Stability\Mohamed2002_Improving the accuracy of prediction of breach formation through embankment dams and flood embankments.pdf. MOHAMED, M., SAMUELS, P., MORRIS, M. & GHATAORA, G. 2002b. Improving the accuracy of prediction of breach formation through embankment dams and flood embankments. International Conference on Fluvial Hydraulics. MORRIS, M. & HASSAN, M. 2009. Breach initiation and growth: Physical processes. HR Wallingford, UK, www. f loodsite. net (May 26, 2013).

MORRIS, M., HASSAN,〈 M. & VASKINN,〉 K. 2007. Breach formation: Field test and laboratory experiments. Journal of Hydraulic Research, 45, 9-17. MORRIS, M., HASSAN, M., WAHL, T., TEJRAL, R., HANSON, G. & TEMPLE, D. 2012. Evaluation and development of physically-based embankment breach models. Comprehensive Flood Risk Management: Research for Policy and Practice, Boca Raton, Florida: CRC Press. MORRIS, M. W. 2011. Breaching of earth embankments and dams. Open University. NEZU, I. 1993. Turbulence in open-channel flows. PAQUIER, A. Testing a simplified breach model on Impact project test cases. PROCEEDINGS OF THE CONGRESS-INTERNATIONAL ASSOCIATION FOR HYDRAULIC RESEARCH, 2007. 342. PAQUIER, A. & RECKING, A. 2004. Advances on breach models by Cemagref during Impact Project. 4th Project Workshop, Zaragoza, Spain, 12. PARTHENIADES, E. 1965. Erosion and deposition of cohesive soils. Journal of the Hydraulics Division, ASCE, 91, 105-139.

189

References

PAUL, B. 1961. A modification of the Coulomb-Mohr theory of fracture. Journal of Applied Mechanics, 28, 259-268. PIERCE, M., THORNTON, C. & ABT, S. 2010. Predicting Peak Outflow from Breached Embankment Dams. Journal of Hydrologic Engineering, 15, 338-349. RECLAMATION, U. S. B. O. 1982. Guidelines for Defining Inundated Areas Downstream from Bureau of Reclamation Dams, U.S. Department of the Interior, Bureau of Reclamation. ROBINSON, K. & HANSON, G. 1994. A deterministic headcut advance model. Transactions of the ASAE, 37, 1437-1443. SAADA, A. & TOWNSEND, F. 1981. State of the art: laboratory strength testing of soils. Laboratory shear strength of soil, ASTM STP, 740, 7-77. SHIELDS, A. 1936. Application of similarity principles and turbulence research to bed- load movement. Soil Conservation Service. SHIONO, K. & KNIGHT, D. W. 1991. Turbulent open-channel flows with variable depth across the channel. Journal of Fluid Mechanics, 222, 617-646. SINGH, K. P. & SNORRASON, A. 1984. Sensitivity of outflow peaks and flood stages to the selection of dam breach parameters and simulation models. Journal of Hydrology, 68, 295-310. SINGH, V., SCARLATOS, P., COLLINS, J. & JOURDAN, M. 1988. Breach erosion of earthfill dams (BEED) model. Natural Hazards, 1, 161-180. STEIN, O. & NETT, D. 1997. Impinging jet calibration of excess shear sediment detachment parameters. Transactions of the ASAE, 40, 1573-1580. TANG, X. & KNIGHT, D. W. 2008. A general model of lateral depth-averaged velocity distributions for open channel flows. Advances in Water Resources, 31, 846-857. TEMPLE, D. 1985. Stability of grass-lined channels following mowing. Transactions of the ASAE American Society of Agricultural Engineers. TEMPLE, D., HANSON, G., NEILSEN, M. & COOK, K. 2005. Simplified breach analysis model for homogeneous embankments: Part I, Background and model components. Proceedings of the 2005 US Society on Dams Annual Meeting and Conference, Salt Lake City, Utah, 151-161.

190

References

VANICEK, I. 2013. The importance of tensile strength in geotechnical engineering. Acta Geotechnica Slovenica, 1, 5-17. VASKINN, K. A., LØVOLL, A., HÖEG, K., MORRIS, M., HANSON, G. & HASSAN, M. 2004. Physical modeling of breach formation–large scale field tests. Dam safety 2004. VISSER, K., HANSON, G., TEMPLE, D., LOBRECHT, M., NEILSEN, M., FUNDERBURK, T. & MOODY, H. WINDAM B Earthen Embankment Overtopping Analysis Software. Joint Federal Interagency Sedimentation and Hydrologic Modeling, 2010. VISSER, P. J. 1998. Breach erosion in sand-dikes. Coastal Engineering Proceedings, 1. WAHL, T. 2004. Uncertainty of Predictions of Embankment Dam Breach Parameters. Journal of Hydraulic Engineering, 130, 389-397. WAN, C. F. & FELL, R. 2002. Investigation of Internal Erosion and Piping of Soils in Embankment Dams by the Soil Slot Erosion Test and the Hole Erosion Test, University of New South Wales, School of Civil and Environmental Engineering. WAN, C. F. & FELL, R. 2004. Investigation of rate of erosion of soils in embankment dams. Journal of Geotechnical and Geoenvironmental Engineering, 130, 373-380. WANG, Z. & BOWLES, D. S. 2006. Three-dimensional non-cohesive earthen dam breach model. Part 1: Theory and methodology. Advances in water resources, 29, 1528- 1545. WANG, Z. & BOWLES, D. S. 2007. A numerical method for simulating one-dimensional headcut migration and overtopping breaching in cohesive and zoned embankments. Water resources research, 43. XIAO, R. 2006. Soil Mechanics, Peking University Press. ZHAO, G., VISSER, P. & PEETERS, P. 2014. Large Scale Embankment Breach Experiments in Flume. ZHAO, G., VISSER, P. J., PEETERS, P. & VRIJLING, J. K. 2013. Headcut migration prediction of the cohesive embankment breach. Engineering Geology, 164, 18-25. ZHENG, Y. & JIN, Y.-C. 1998. Boundary shear in rectangular ducts and channels. Journal of Hydraulic Engineering, 124, 86-89.

191

References

ZHU, D., LEE, C., QIAN, Q. & CHEN, G. 2005. A concise algorithm for computing the factor of safety using the Morgenstern Price method. Canadian geotechnical journal, 42, 272-278. ZHU, Y. 2006. Breach growth in clay-dikes. Hohai University ZHU, Y., VISSER, P., VRIJLING, J. & WANG, G. 2011. Experimental investigation on breaching of embankments. Science China Technological Sciences, 54, 148-155.

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Appendix A

A.1. The security and protection levels of levees

1) The security level of levees The security level is defined as the water level in the river above which the probability of failure of the levee is no longer considered negligible.

2) The protection level of levees The protection level is the level of water in the river above which the protected area begins to be flooded without breaking the dam: the overflow over the top of the dam or the weir (if present). The concept of the security level refers to the risk of rupture, and the concept of the protection level refers to the overflow. In a perfectly reliable containment system, the safety level is as the same as or above the protection level. This means that the break before the overflow is impossible. The fracture risk appears only when the flowing water reaches a significant height or the maximum capacity of the weir dam.

A.2. The classification of security level in China

In China, five different security levels for levees are differentiated by “flood control standards” to draft the design and construction standards for levees. The different security levels mean that the constructed levee can remain intact and protect floodplains subject to the corresponding flood frequency. Table A-1 shows the classification of the security level in China.

A.3. Flood frequency

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Appendix A

The term of 100 year flood is a flood with recurrence interval of 100 years: one that has a 1% chance of occurring in any given year. The recurrence interval is the time period over which it is likely that a particular magnitude flood will occur.

Table A-1. The security level of levees in China.

Security levels Flood frequency

Level 1 ≥ 100 years

Level 2 ≥ 50 years, and ≤ 100years

Level 3 ≥ 30 years, and ≤ 50years

Level 4 ≥ 20 years, and ≤ 30years

Level 5 ≥ 10 years, and ≤ 20years

194

Appendix B. Boundary shear stress distributions

Appendix B

Simulations of boundary shear stress distributions in ducts

According to the erosion law, calculations of boundary shear stresses in the breach are significant to accurate estimate hydraulic erosion during breach widening processes. Thus, the object is to simulate the distributions of boundary shear stress. There are two series of simulations conducted:

1) The flow through trapezoid smooth closed ducts; 2) The flow through rectangular smooth open ducts. To simulate the enlargement process of the breach, different ratios of the width to depth (B:H) of ducts are conducted for both of two series. To study the effects of the overhang degree on the distributions of the shear stress, closed smooth ducts are set with different Slope ( Slope  cot is the cosine of the bottom angle of the duct).

B.1 Simulation setup

1) Series 1: Trapezoid smooth closed ducts The flow in trapezoid smooth closed ducts was simulated by using Fluent. In Figure B-1, the aspect ratio of B:H is set as 1, 3 and 5. In order to express the overhang degree of the side wall, Slope is defined as the cosine of the bottom angle of the duct (see Figure B-1). It was set as 0, 0.5 and 1. When the Slope =0, it corresponds to a vertical side wall; when the slope =1, it corresponds to the overhang with a degree of 45 at the bottom corner. L is the length of the smooth closed duct set as 10 times to the width of the duct. In total, nine different numerical experiments were performed, details for which are found in Table B-1.

195

Appendix B

Hexahedral elements were used for the whole fluid zone with the element size of 6mm. At the bottom and side wall boundaries, layer meshes were refined as 2mm. Figure B-1 shows the fluid zone geometry of the smooth closed ducts as well as the elements setup. The Renault numbers ranged from 9960 to 16932.

2) Series 2: Rectangular smooth open ducts The flow in rectangular smooth open ducts was also simulated by using Fluent, which is more similar with the breach widening discharge flow. The aspect ratios were set as 0.5, 1 and 2. The other setups of this series were as the same as in the series 1. The Renault numbers for this series ranged from 16932 to 32869.

Table B-1. Physical parameters within simulations of series 1 and series 2.

Series 1 Series 2

B:H 1 3 5 0.5 1 2 Slope 0/0.5/1 0/0.5/1 0/0.5/1 0 0 0

H Y θ Cross section B Z

Figure B-1. Part of the employed elements for the cases in series 1.

H Y Cross section B Z Figure B-2. Part of the employed elements for the cases in series 2.

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Appendix B. Boundary shear stress distributions

B.2 Boundary conditions

1) Series 1 : Trapezoid smooth closed ducts At the bottom and side walls of the ducts the no-slip boundary conditions were applied.

2) Series 2 : Rectangular smooth open ducts Different form boundary conditions applied on the top of the smooth closed ducts, specified shear condition were used. The disributions of the dimensionless wall distance y+ on the boundary are shown in Figure B-3.

Table B-2. Boundary conditions in simulations.

Boundary conditions Series 1 Series 2

Inflow Velocity-inlet Velocity-inlet Outflow Outflow Outflow Bottom No slip No slip Side walls No slip No slip Top No slip Specified shear condition

Figure B-3. The distributions of wall Y+ at the bottom and side wall for series 1 and series 2.

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Appendix B

B.3 Results and discussions

1) Series 1: Trapezoid smooth closed ducts Simulated distributions of bottom and side wall shear stress generated by turbulent flow in ducts are obtained as shown in Figure B-4 and Figure B-5. Comparisons are mainly concentrated in ducts with the Slope of 0, 0.5 and 1 when the aspect ratios equal to 1, 3, and 5. It can be seen that shear stress distributions on the bottom increase along the Y direction (from the corner to the middle of the duct) until approximately 20% of the bottom width. Then, it remains constant. As the increase of the Slope , trends of shear stress distributions vary a little within a range of the former 20% of the bottom width. The same trends can be seen for the distributions of shear stress on the side wall. Along the Z direction (from the bottom to top of the duct), shear stress distributions increase until approximately 20% of the depth. Then it remains constant until near the top of the duct where it reachs approximately 80% of the depth. As the increase of the Slope , trends of shear stress distributions vary a little within a range of the former 20% of the depth and the later 20% of the depth.

2) Series 2 : Rectangular smooth open ducts Figure B-6 shows the simulated distributions of bottom shear stress in rectangular smooth open ducts. It can be seen that, along Y direction (from the corner to the middle of the bottom) the shear stress distributions increase until approximately 0.3 times bottom width. Then, it remains constant. As the increase of aspect ratios (B:H), the influence of side wall to the bottom shear stress distribution decreases. The simulated distributions of side wall shear stress are compared with some experiment data (see Figure B-7). Along Z direction, shear stress distributions increase. Then, it remains constant from approximately 0.2 times the depth.

B.4 Conclusions

To better estimate shear stresses on the boundary of the breach, simulations of distributions of bottom and side wall shear stress due to the turbulent flow in smooth closed/open ducts are conducted. The definition of the trapezoid duct with the Slope varying from 0 to 1 is to simulate the shape of the eroded notch on the side wall of the breach generated by erosion. The enlargement process of the breach is simulated by changing aspect ratios of the ducts.

198

Appendix B. Boundary shear stress distributions

It is concluded that, the influence of the side wall on the bottom and side wall shear stress distributions only plays a role near the corner of the duct. The influenced region along Z direction is about 0.2 times the depth from the corner, and 0.4 times the bottom width of the duct along Y direction. As the increase of aspect ratios (B:H), the influence of side wall on the bottom and side wall shear stress distributions are not significant. In additions, the aspect ratio in simulation is less than 5 (2 for series 2). However, the ratio of the breach with to depth can reach to 10 or more in reality. Due to the limitation of the computing power and deadline of the thesis, there are still many works.

199

Appendix B

Figure B-4. Comparisons of the simulated distributions of bottom (y direction) and side wall (z direction) shear stress with different slope conditions ( Slope =0, 0.5, and 1) when the aspect ratios equal to BH: =1, 3, and 5.

200

Appendix B. Boundary shear stress distributions

Figure B-5. Comparisons of the simulated distributions of bottom (y direction)and side wall (z direction) shear stress with different aspect ratios ( BH: =1, 3, and 5) when side slopes equal to Slope =0, 0.5, and 1.

201

Appendix B

Figure B-6. Comparisons of the simulated distributions of bottom shear stress in rectangular open ducts under different aspect ratios ( BH: =0.5, 1, and 2).

Figure B-7. Comparisons of the simulated distributions of side wall shear stress and the experimental results in rectangular open ducts under different aspect ratios ( BH: =0.5, 1, and 2) (Knight et al., 1984, Myers and Elsawy, 1975).

202