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This article discusses the often catastrophic flows that result from the sudden collapse of high dams that protect low-lying land. The mechanisms causing dam breaks are not dealt with; for this the reader is referred to the extensive literature existing on this subject, see for example  Zhang et al. (2016) <ref>Zhang, L., Peng, M., Chang, D. and Xu, Y. 2016. Dam failure and Risk Assessment. John Wiley and Sons, Singapore</ref> and Almog et al. (2011) <ref>Almog, E., Kelham, P and King, R. 2011. Modes of dam failure and monitoring and measuring techniques. Environmental Agency,UK https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/290819/scho0811buaw-e-e.pdf</ref> and to the article [[Stability of rubble mound breakwaters and shore revetments]].



==Introduction==

[[image:Dijkdoorbraak1953.jpg|thumb|left|300px|Fig. 1. Sea dikes protecting low-lying polders in the Netherlands were breached during the extreme storm surge of 31 January 1953.]] 



Dam failure can lead to disastrous situations. Most dam-break tragedies are related to the collapse of reservoir dams in mountain rivers. The failure of sea dikes can also cause major disasters,  although the level difference in this case is not as large. A dramatic example is the failure of more than hundred sea dikes in the Netherlands during the extreme storm surge of 1953, see Fig. 1. Many of these dikes protected polders lying a few meters below the average sea level, while the storm surge level in the tidal inlets reached up to five meters. Rapid flooding killed almost two thousand people who were unable to flee in time to safe places <ref>Gerritsen, H. 2005. What happened in 1953? The Big Flood in the Netherlands in retrospect. Phil. Trans. R. Soc. A 363: 1271–1291</ref>. The most common failure mechanisms of earth dams are related to overtopping and seepage (also called piping or internal erosion)<ref>Richards, K.S. and Reddy, K.R. 2007. Critical appraisal of piping phenomena in earth dams.
Bull. Eng. Geol. Environ. 66: 381–402</ref><ref>Jiang, X., Woerman, A., Chen, P., Huang, Q. and Chen, H. 2020. Mechanism of the progressive failure of non-cohesive natural dam slopes. Geomorphology 363, 107198</ref>. In the case of the 1953 storm surge, overtopping and subsequent scour of the interior dike slope was the most important dike failure mechanism (see [[Overtopping resistant dikes]]). 







[[image:DamBreakFlowPrinciple.jpg|thumb|center|600px|Fig. 2. Left panel: Schematic representation of water retention behind a dam. Right panel: Positive downstream and negative upstream surges following instantaneous dam removal.]]


The consequences of dam failure have been studied for more than a century. It nevertheless remains a challenging topic due to the high non-linearity of the flood wave propagation. The problem can be tackled with numerical models, but rough estimates and insight into the tidal wave dynamics can be obtained with analytical solution methods.


==Frictionless dam-break flow: analytical solution==

Analytical solutions relate to idealized situations as depicted in Fig. 2. The initial situation consists of an infinite reservoir with a water level that is <math> h_0 </math> higher than the horizontal ground level downstream of the dam. Dam break is simulated by instantaneous removal of the dam. This causes a positive surge in the positive <math> x </math>-direction and a negative surge in the negative <math> x </math>-direction. The floor is dry in front of the positive surge (water level <math>h = 0</math>) and the water body behind the negative surge is undisturbed (water level <math>h = h_0 </math> and current speed <math> u = 0 </math>).

[[image:SouthForkDamFailure1889.jpg|thumb|left|300px|Fig. 3. The earth-filled South Fork Dam on Lake Conemaugh (Pennsylvania, US) collapsed in 1889, killing 2,209 people in downstream villages.]]





The study of dam break-flow was triggered by several dam failures in the 19th century, in particular the breach of the South Fork Dam in Pennsylvania (USA) in 1889, Fig. 3. Three years later Ritter <ref> Ritter, A. 1892. Die Fortpflanzung der Wasserwellen. Zeitschrift des Vereines Deutscher Ingenieure 36: 947-954 </ref> published an exact analytical solution for dam-break flow by assuming that frictional effects can be ignored. The derivation is given in box 1. According to this solution, the tip of the positive surge advances at high speed (supercritical flow) given by <math> u_f = 2 c_0</math>, where <math>c_0 = \sqrt {gh_0}</math> and <math> g </math> is the gravitational acceleration. The shape of the positive and negative surge is a concave-up parabola,


<math>
h(x,t) =
\begin{cases}
h_0 , \qquad x < -c_0 t  \\
\Large\frac{h_0}{9}\normalsize \; (2 -  \Large\frac{x}{c_0 t}\normalsize)^2 , \quad -c_0 t<x<2c_0 t     \qquad (1) \\
0 , \qquad x > 2 c_0 t 
\end{cases}
</math>

<math>u(x,t) = \Large\frac{2 c_0}{3}\normalsize \, (1 + \Large\frac{x}{c_0 t}\normalsize) , \quad -c_0 t < x < 2 c_0 t  \; .  \qquad (2)</math>



This solution is shown in Fig. 4.  According to Eqs. (1, 2), the discharge through the breach per unit width is given by

<math>q = \frac{8}{27} h_0 \sqrt{g h_0}. \qquad  (3) </math> 


Figure 4 also shows the shape of the dam break wave that was observed in laboratory experiments. For small values of <math> x /(c_0 t) </math>, the frictionless solution corresponds fairly well with the observations. However, the front zone is different: the shape is a bull nose, rather than a sharp edge. The front also advances more slowly; the speed of the front decreases with time and is closer to <math> u_f = c_0 </math> than to <math> u_f = 2 c_0 </math>. The reason for this difference is the neglect of frictional effects that are important in the thin fluid layer near the front.


[[image:DamBreakFlowDerivation.jpg|thumb|center|900px|Box 1. Derivation of the solution for frictionless dam break flow.]]


==Dam-break flow with friction==

[[image:DamBreakWaveProfiles.jpg|thumb|left|400px|Fig. 4. Wave profile after dam break. The red curve corresponds to the frictionless solution Eq. 1. The blue band represents data from different laboratory experiments performed at times <math>\small t \sim (40-80) \, \sqrt {h_0 / g}\normalsize </math> after removal of the dam <ref>Dressler, R. 1954. Comparison of theories and experiments for the hydraulic dam-break wave. Proc. Int. Assoc. Scientific Hydrology Assemblée Générale, Rome, Italy 3 (38)M 319–328</ref> <ref name=S>Schoklitsch, A. 1917. Über Dambruchwellen. Kaiserliche Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse, Sitzungberichte IIa, 126: 1489–1514</ref><ref>Cavaillé,Y. 1965. Contribution à l’étude de l’écoulement variable accompagnant la vidange brusque d’une retenue. Publ. Scient. et Techn. du Ministère de l’Air, 410, Paris, France, 165</ref>. The frictionless solution only depends on <math>x/(c_0 t)</math>. Observed wave profiles also depend on time <math>t</math> because of the decreasing wave tip speed.]]


When friction terms are included in the flow equations, there is no exact analytical solution. It is generally assumed that the frictionless flow equations are approximately valid in a short period after dam break and for small values of <math>x / (c_0 t) </math>. In the front zone, where the water depth is small, the momentum balance is dominated by frictional momentum dissipation. If the inertia terms <math>\partial u / \partial t + u \partial u / \partial x </math> in the momentum equation are ignored, then the current in the front region is mainly determined by the balance of gravitational acceleration <math> g \partial h / \partial x </math> and shear stress <math>\tau</math>, the latter term being proportional to the square of the flow velocity. With such models the shape of the bore in the friction-dominated zone is given by<ref> Baldock T.E. and Torres-Freyermuth, A. 2020. Numerical study of the flow structure at a swash tip propagating over a rough bed. Coastal Engineering 161, 103729
</ref> 

<math>h(s) \propto s^a  , \quad  a = 1/3 – 1/2  \;  ,</math> 

where <math>s</math> is the distance measured from the wave front, and where it has been assumed that the flow velocity in the tip region does not (strongly) depend on <math>s</math><ref>Dressler, R.F. 1952. Hydraulic resistance effect upon the dambreak functions. J. Res. Natl. Bureau of Standards, 49(3): 217–225</ref><ref name=W>Whitham, G.B. 1955. The effects of hydraulic resistance in the dam-break problem. Proc. Roy. Soc. of London, Serie A, 227: 399–407</ref><ref>Hogg, A.J. and Pritchard, D. 2004. The effects of hydraulic resistance on dam-break and other shallow inertial flows. J. Fluid Mech. 501: 179–212</ref><ref name=C09>Chanson, H. 2009. Application of the method of characteristics to the dam break wave problem. Journal of Hydraulic Research 47: 41–49</ref><ref name=DLL>Deng, H., Liu, H. and Lu, S. 2018. Analytical Study of Dam-Break Wave Tip Region. J. Hydraul. Eng. 144: 4018015</ref>. This shape is similar to what is observed in flume experiments (Fig. 4)<ref name=S/>.  At the bore front, the flow has a strong downward component and it is highly turbulent. Field observations of very high suspended sediment  concentrations just behind the bore front demonstrate the strong erosion capacity of dam break flows<ref>Khezri, N. and Chanson, H. 2012. Inception of bed load motion beneath a bore. Geomorphology 153: 39–47</ref> (see also the article [[Tidal bore dynamics]]). 

The bore front speed <math>u_f</math> decreases with time. At small times  <math>t<<t_{\infty}</math>, the bore front speed varies approximately  as <ref name=W></ref> 

<math>u_f \approx c_0 \, (2 – 3.45 \sqrt{t / t_{\infty}}) \; , \qquad (4) </math> 

where <math> t_{\infty} = \sqrt{(h_0/g)}/ c_D </math> and <math>c_D</math> is the friction coefficient (usual values in the range <math>\approx (2-10)\, 10^{-3}</math>). 
 
At large times <math>t \ge t_{\infty}</math> the bore front speed varies approximately  as <ref name=C09></ref><ref name=DLL></ref>

<math>u_f \approx c_0 \sqrt{ t_{\infty} / 3 t}  \; .  \qquad (5)</math> 

It should be emphasized that these approximate formulas have been derived for very large (infinite) reservoirs in which the backward wave is not reflected.

==Field observations==

Data of dam break discharges can be derived from the rate at which the reservoir level is falling. Complementary field data can be retrieved from analyzing the situations before and after the dam break. Detailed experimental data of dam break events only exist for laboratory settings.

Data from numerous reservoir dam breaks in the past have provided empirical formulas for the maximum discharge <math>Q_{max}</math> through the breach <ref>Froehlich, D.C. 2016. Predicting Peak Discharge from Gradually Breached Embankment Dam. Journal of Hydraulic Engineering, ASCE 04016041</ref><ref name=W> Wang, B., Chen, Y., Wu, C., Peng, Y., Song, J., Liu, W. and Liu, X. 2018. Empirical and semi-analytical models for predicting peak outflows caused by embankment dam failures. Journal of Hydrology 562: 692–702</ref>. From this data it was deduced that the width of the breach is often a function of the water volume <math>V</math> in the reservoir (volume above breach level). A rough estimate of the width is given by<ref name=F8>Froehlich, D.C. 2008. Embankment Dam Breach Parameters and Their Uncertainties. Journal of Hydraulic Engineering, ASCE 134: 1708-1721</ref> <math> B \approx 0.3 V ^ {1/3} </math>. Empirical formulas for the maximum dam break discharge then yield <ref>Webby, M. G. 1996. Discussion of ‘Peak outflow from breached embankment dam.’ by D. C. Froehlich. J. Water Resour. Plann. Management 122:4(316), 316–317</ref> <ref name=W></ref>

<math>Q_{max} \approx 0.04 \sqrt{g} \, V^{0.37} h_0^{1.4} \approx 0.15 h_0 \, B \, \sqrt{g h_0} \, V^{0.04} h_0^{-0.1}  .  \qquad  (6)</math> 

From this follows an empirical estimate for the order of magnitude of the maximum discharge per unit width,

<math> q_{max} \approx  0.22 h_0 \, \sqrt{g h_0} ,  \qquad (7)</math>

where we have considered a large reservoir volume (<math>V \approx 10^7 m^3</math>) and a water depth <math>h_0 \approx 15 m</math>. The empirical estimate (7) is about 75% of the estimate (3) given by the frictionless flow solution.


==Numerical models==
The analytical methods for describing dam-break flow provide a qualitative picture and some first-order estimates for the bore profile and surge speed. Major assumptions underlying the simplifications are, (1) hydrostatic pressure throughout the dam break flow, (2) uniform flow velocity throughout the bore tip region, and (3) the reservoir and outflow region can be represented by an unbounded prismatic channel. For actual field situations, the dam-break flow must be modeled numerically. A correct simulation of the backward surge requires models of the Boussinesq type that take into account vertical fluid accelerations<ref>Castro-Orgaz, O. and Chanson, H. 2017. Ritter’s dry-bed dam-break flows: positive and negative wave dynamics. Environ Fluid Mech (2017) 17:665–694</ref>. Practical situations are generally characterized by the complex geometry of a highly irregular river bed that evolves during the dam-break event. Current numerical models cannot easily meet the requirements for performing accurate simulations of such real events. 


==Related articles==
:[[Tidal bore dynamics]]
:[[Tsunami]]


==References==
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{{author
|AuthorID=120
|AuthorFullName=Job Dronkers
|AuthorName=Dronkers J}}

[[Category:Physical coastal and marine processes]]
[[Category:Hydrodynamics]]
[[Category:Coastal protection]]