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Large dam-reservoir systems: guidelines and tools to estimate loads resulting from natural hazards

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Abstract

This paper examines hydrodynamic pressure diagrams due to earthquakes acting on distinct configurations of the upstream face of a dam, considering the reservoir length and different bathymetries of the reservoir bottom. Two suitable mathematical models are used to obtain the dynamic pressure on the dam, and the reservoir free surface oscillation. Conceptual mathematical models are proposed to study the impact of a landslide triggered by an earthquake, and its progress into the reservoir. The waves formed when a landslide advances into the reservoir are tested with the help of laboratory results. These waves and their propagation in the reservoir are studied using experimental data and numerical results, including wave-type analyses based on dimensionless parameters. Two distinct 1DH and 2DV numerical models based on different mathematical formulations are tested. A discussion of physical and numerical results is detailed in a general risk context and uncertainty associated with the input data in a deterministic model. Numerical simulations are performed for the upper and lower limits of the sliding mass velocity diagram which is obtained as a result of the intrinsic uncertainty of the stochastic nature of the friction angle. Finally, the findings are discussed and some conclusions drawn.

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Abbreviations

A :

Slide mass section

a c :

Wave crest amplitude

a M :

Maximum wave crest amplitude

\( \ddot{a}_{g} \) :

Horizontal acceleration of the ground motion

A p , A s :

Coefficients for progressive and standing waves

b :

Slide width

c :

Velocity of sound in water (c ≈ 1,438 ms−1)

c c :

Wave velocity

C i :

Constant (0.0, 0.5 or 1.0)

C d :

Drag coefficient for the slide

C D  = 0.085:

(constant)

\( C_{p} \left( {x,z} \right) = {{\bar{p}\left( {x,z} \right)} \mathord{\left/ {\vphantom {{\bar{p}\left( {x,z} \right)} {p_{h} \left( H \right)}}} \right. \kern-\nulldelimiterspace} {p_{h} \left( H \right)}} \) :

Normalized pressure coefficient

Cε1=1.42, Cε2 = 1.68:

(constants)

F = F(x,z,t):

VOF-function

\( F = {{v_{s} } \mathord{\left/ {\vphantom {{v_{s} } {\sqrt {gh_{0} } }}} \right. \kern-\nulldelimiterspace} {\sqrt {gh_{0} } }} \) :

Froude number

g :

Acceleration due to gravity

\( \vec{g} = g\left( {g_{x} ,g_{z} } \right) \) :

Body force

h :

Total water depth

H :

Initial water level; reservoir water depth close to the dam; height

H p :

Height with a different slope

h 0 :

Initial reservoir water level; still water depth

i, j:

Indexes; point and element

k :

Wave number; turbulent kinetic energy

k p , k s :

Wave numbers for progressive and standing waves

L :

Length; weave length

m s :

Sliding mass

M :

Mass of the sliding block

M w :

Submerged mass

n :

Normal to an element

p :

Progressive wave index

p = p(x, z, t):

Mean pressure

\( p = \bar{p}\left( {x,z,w} \right)e^{iwt} \) :

Hydrodynamic pressure

P :

Production of turbulent kinetic energy

\( \bar{p} = \bar{p}\left( {x,z,w} \right) \) :

Frequency response function for hydrodynamic pressure

\( p_{h} \left( z \right) = \rho g z \) :

Hydrostatic pressure

p t (x, z):

Total pressure on the upstream face of a dam

r :

Radius vector

R :

Rate-of-strain

s :

Slide thickness; standing wave index

S=s/h0:

Dimensionless slide thickness, boundary

S j :

Length of element j

t :

Time

T :

Thickness

t s :

Sliding time

\( \bar{u} \) :

Horizontal mean velocity

\( \vec{u} = \left[ {u \, \left( {x,z,t} \right), \, w \, \left( {x,z,t} \right)} \right] \) :

Mean velocity vector

\( V = {{V_{s} } \mathord{\left/ {\vphantom {{V_{s} } {\left( {bh_{0}^{2} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {bh_{0}^{2} } \right)}} \) :

Dimensionless slide volume

v s :

Sliding mass velocity

v sm :

Slide velocity when the mass centre reaches the water surface

V s :

Slide volume

w :

Frequency of the horizontal ground motion; angular frequency

W :

Width

x :

Space

x, z:

Rectangular coordinates

α:

Bank slope; slide impact angle

\( \ddot{a}_{g} \) :

Acceleration of the movement

β:

Angle between an element situated at the reservoir bottom and the x-axis

∆z :

Height of the mass centre positions

\( \varepsilon = \varepsilon \left( {x,z,t} \right) \) :

Dissipation rate

\( \phi \) :

Friction angle; velocity potential

\( \varphi = \left( {{6 \mathord{\left/ {\vphantom {6 7}} \right. \kern-\nulldelimiterspace} 7}} \right) \, \alpha \),:

With α being the slide impact angle

\( \eta = h - H + \xi \) :

Surface elevation

\( \eta_{por} \approx 0.40 \) :

Porosity

v :

Kinematic viscosity

v T :

Eddy viscosity

θ:

Angle between the x-axis and the normal to an element situated at the upstream face of a dam; porosity function; parameter

\( \rho = \rho_{w} \) :

Density of water

\( \rho_{s} \) :

Density of sliding mass

\( \sigma_{k} = 1.0, \sigma_{\varepsilon} = 1.3 \) :

(constants)

\( \varsigma_{0} \) :

Amplitude of the harmonic motion

\( \tau \left( \xi \right) \) :

Bottom shear stress

\( \xi \) :

Bed elevation

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Acknowledgments

The authors wish to acknowledge the Portuguese Foundation for Science and Technology for support under the project POCTI/ECM/2688/2003.

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Authors and Affiliations

  1. Department of Civil Engineering, IMAR/FCTUC, Polo II of the University, 3030-788, Coimbra, Portugal

    José Simão Antunes do Carmo & Rita Fernandes de Carvalho

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  1. José Simão Antunes do Carmo
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  2. Rita Fernandes de Carvalho
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Correspondence to José Simão Antunes do Carmo.

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Antunes do Carmo, J.S., de Carvalho, R.F. Large dam-reservoir systems: guidelines and tools to estimate loads resulting from natural hazards. Nat Hazards 59, 75–106 (2011). https://doi.org/10.1007/s11069-011-9740-9

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