dambrk modeling methodology
TRANSCRIPT
DAMBRKModeling Methodology
Dam Failures: Worldwide Statistics
Piping, Seepage, Slides, Earthquake--------------------------------------------- 40 - 50%
Overtopping --------------------------------------------------------------------------- 30 - 35%
Misc. ------------------------------------------------------------------------------------- 15 - 30%
Overtopping
Only overtopping can be predicted by a few of the current Dam - BreachFlood Models.
Overtopping prediction subject to errors in the Rainfall - Runoff HydrologicModel used to predict the inflow to the reservoir.
Extent of overtopping to cause dam - failure is not well understood. It ishighly dependent on the type of dam (concrete or earthfill).
Earthfill — Duration and Magnitude of overtopping flow
Concrete — Magnitude of overtopping flow
Maximum Possible WaveBefore Failure
Figure 1: Dam just before failure.
Just after complete and sudden failure
Figure 2: Dam just after failure.
Flood Wave Attenuation
Figure 3: Location of various H values.
Teton: H1 ≈ 57 ft at DamH2 ≈ 14 ft at 9 miles
H3 ≈ 9 ft at 60 miles
Travel Time of Flood Wave Peak
Mi. Hr.
Teton: 1 to 4 Mi/Hr Rexburg 16 5
Idaho Falls 50 34
Buffalo Creek: 4 to 7 Mi/Hr Stowe 7 1
Man 16 3
Toccoa: 4 Mi/Hr College 0.7 0.2
Dam - Breach Flood Routing Models
Some Improvements During the Last Decade:
1. 1-Dimensional EquationsDAMPENS of Unsteady Flow
2. Effects of Downstream Dams and/or Bridge - Embankments on FloodPropagation
3. Breach Dynamics
a) Time of Formation
b) Size
c) Shape
Not with standing the above improvements, errors of 2 feet or more can beexpected in the flow peak profile.
This is evidence by comparison of various Dam - Breach Flood Medels withthe observed flooding produced by the Teton, Toccoa, Laurel Run, andBuffalo Creek dam - failure floods.
Outflow from Dam
Affected by:
1. Size and shape of beach as a function of time
2. Height of dam
3. Storage volume of reservoir
4. Inflow to reservoir
5. Downstream channel conditions (channel size, roughness) which maycause submerged weir flow through the breach
Dam - Break Wave Transformation
Affected by:
1. Initial shape
2. Storage volume of downstream valley
3. Acceleration components of wave
4. Resistance to flow through downstream valley
Figure 4: Small Storage Volume in Reservoir
Figure 5: Large Storage Volume in Reservoir
7. Affect of uncertainty in parameters of breach dampens outdownstream
• Raising n will
• Locally raise WSEL
• Increase W.S. slope downstream
• Decrease travel time
• Changing n typically changes Q by about the same percent
• Stage depends on XS shape
Figure 6: Pre-forecast flash flood envelope.
DAMBRK (Dam Break)
1. Develops outflow hydrograph due to dam failure.
2. Failure may be partial and time-dependent.
3. Simulates spillway flows, overtopping flows, time-dependent gated flows.
4. Routes outflow hydrograph hydraulically through downstream valleyusing expanded form of 1-D Saint - Venant Equations.
5. Considers affects of: downstream dams, bridges, levees, tributaries, off-channel storage areas, river sinuosity, backwater from tides.
6. Flow may be Newtonian (water) or non-Newtonian (mud/debris)
7. Produces output of: high water profile along valley, flood arrival times,flow/stage hydrographs.
Expanded Saint - Venant Equations - 1
( )( )
[ ]( )
∂∂
∂∂
Q
x
s A A
tq
o++
− =(
0
( )( )
( )( )
( )( )
∂∂
∂∂
∂∂
sQ
t
Q A
xgA
h
xS S S Lf e i+ + + + +
+ =2
0/
Where:
Q = Flow
x = Distance along river
t = Time
A = Active cross-sectional area of flow
Ao = Inactive (Dead) cross-sectional area
q = Lateral inflow or outflow
s = Sinuosity factor
g = Acceleration due to gravity
h = Water surface elevation
Sf = Friction slope due to boundary friction
Si = Internal friction slope due to non-Newtonian fluid properties
L = Momentum effect of lateral inflow/outflow
Expanded Saint-Venant Equations - 2
SQ Q
Kf = 2
( )S
k Q A
g xe =∆
∆/
2
2
( ) ( )( )S
b Q
AD
b
Di b
o
b
b
b
=+
++
+
κγ
τ κ2 2
21
1
/
Where:
K = Channel conveyance factor
k = Expansion/contraction loss factor
κ = Apparent viscosity of fluid
γ = Unit weight of fluid
τo = Shear strength of fluid
b = 1/m where: m = Power of fluid’s stress/strain relation
m = 1 if Newtonian fluid or Binghan plastic fluid
D = A/B = Hydraulic depth where: A = Wetted area
B = Top width of activecross-section
Internal Boundaries - 1
Q Q Qs b= + where: Qs = Flow through structure
Qb = Flow through breach
Dams
( ) ( ) ( )Q c L h h c A h h c L h h Qs s s s g g g d d d t= − + − + − +1 5 0 5 1 5. . .
( ) ( )[ ]Q c k b h h z h hb v s b b= − + −31 2 451 5 2 5
. .. .
kh h
h hst b
b
= −−−
−
10 27 8 0 67
3
. . .
Where:
cs,Ls,hs = Spillway coefficient, length, crested elevation
cg, Ag, hg = Gate coefficient, area, sill elevation
cd, Ld, Hd = Dam crest coefficient, length, crested elevation
Qt = Head independent flow (turbines, etc.)
cv = Velocity of approach correction
ks = Submergence correction
b, z, hb = Breach bottom width, side slope, bottom elevation
b b t to f= / where: bo = Final bottom width
t = Timetf = Time of failure for breach
h = Water surface elevation upstream of structure
ht = Water surface elevation downstream of structure
Internal Boundaries - 2
Q Q Qs b= + where: Qs = Flow through structure
Qb = Flow through breach
Bridges( ) ( )Q c g A h h C k h hs t d s c= − + −2
0 5 1 5. .
Where:C = Bridge flow coefficient
A = Bridge flow area
hc, ce, Le = Bridge embankment crest elevation, flow losscoefficient, length
ceLe = Cd = Coefficient of discharge
Special Features of DAMBRK
Floodplain Compartments
Figure 7: Location of floodplain compartments.
Tributary Inflows
Figure 8: Location of tributary inflows
Landslide Waves
Figure 9: Location of landslide waves
DAMBRK
1. Simulation of flows which change with time and location betweensubcritical and supercritical
2. Improvement of numerical robustness of 4 pt. implicit solution
3. Nonlinear behavior of breach formation
4. Interactive, user-friendly data input
Breach
1. Predicts breach size, formation time, and shape for earthen dams(embankments) and naturally-formed landslide blockages.
2. Predicts outflow hydrograph due to breach initiated by overtopping orpiping of the earthen dam.
3. Considers:
a) Effect of downstream face cover
b) Non-homogeneity of dam materials
c) Slope stability
d) Reservoir inflows, reservoir outflow, reservoir storage.
Breach Erosion Uses
Modified Meyer - Peter & Muller Equation
( )QD
D
D
nS DS Ds c=
−3 64 0 005490
30
0 2
1 150
23
. .
.
. τ
Where:
D90 = Grain size (mm) for which 90% is finer
D50 = Grain size (mm) for which 50% is finer
D30 = Grain size (mm) for which 30% is finer
n D= 0 013 500 167. .
D = Hydraulic depth of flow
S = Slope of breach
τc = Shield’s critical slope which is a function of the D50 grain size
Figure 10: Teton outflow hydrograph produced by BREACH model.
DAM-BREAK Flood Forecasting ModelSome General Requirements:
1. Wide applicability
2. Reasonably small computational requirements
3. Data input which varies from a minimum level to a maximum level
4. Data input must be obtainable with available sources
5. Computational scheme must be robust (stable)
DAM- BREAK Model Components1. Breach description (shape vs. time)
2. Downstream Flood Routing
3. Reservoir routing to produce outflow hydrograph
a) Storage routing
b) Dynamic routing
Features of DAM-BREAK Flood Forecasting ModelOutflow from reservoir:
1. Reservoir inflow
2. Reservoir storage characteristics
3. Spillway (uncontrolled and gated) and turbine flows
4. Crest overflow
5. Breach outflow (broad crested weir flow with submergence correction)
Breach Characteristics:
1. Time dependent geometry
2. Triangular, rectangular or trapezoidal shape
3. Erosion formed breach with time dependent width starting at top ofdam
4. Piping formed breach with time dependent width starting at anyprescribed elevation
5. Collapse failure, constant width, approaching instantaneous
Reservoir routing:1. Storage type, assumes level pool condition
2. dynamic type, considers negative wave and/or inflow flood waveeffects
Downstream routing (Dynamic):1. One-dimensional unsteady flow equations
2. Conservation form of equations
3. Weighted 4-pt implicit non-linear finite difference solution
4. Variable time step
5. Variable reach lengths between cross sections
6. Option to create cross-sections via linear interpolation
7. Off channel storage effects
8. Subcritical or supercritical flows
9. Lateral inflows from tributaries
10. Lateral outflow (loss function)
11. Manning roughness coefficient function of distance and stage
12. Expansion - contraction losses
13. Downstream boundary - generated loop rating or flow controlstructure
14. Initial conditions automatically computed via gradually varied steadyflow equations
15. Internal computational checks to provide robust computationalprocedure
Additional capabilities of model:1. Multiple dam capabilities
2. Supercritical reach - subcritical reach sequence capability
3. Bridge embankment effects
4. Model easily used for only downstream routing
5. Tributary dam failure can be analyzed by two applications of model
6. Feasible computational requirements (Weton analysis required lessthan 20 sec. cpu time)
7. Minimal data acceptance for generating approximate results
Breach Characteristics
Type of Dam Ave. Breach Width (b) Time of Failure (τ)
Earth (Well constructed) 2Hd < b < 5Hd 0.1 ≤ τ ≤ 0.5
Earth 2Hd < b < 5Hd 0.1 ≤ τ ≤ 0.5
Flag Pile b ≥ 0.8 w τ ≤ .2
Concrete (gravity) b ≤ 0.5 w τ ≤ .2
Concrete (arch) b ≥ 0.8 w τ ≤ .1
Figure 11: Location of equation variables.
Triangular Breach:
Figure 12: Location of equation variables in a triangular breach.
Rectangular Breach:
Figure 13: Location of equation variables in a rectangular reach.
Trapezoidal Breach:
Figure 14: Location of equation variables in a trapezoidal reach.
Reservoir Hydraulics
Note: Breach starts forming when h ≥ HF
Figure 15: Reservoir profile view.
Dam Breach Outflow
Figure 16: Location of equation variables for breach outflow.
Q Q Qb s= +
Where:
Qb = Breach flow
Qs = Spillway and other outflow
( ) ( )Q K c BBt
h h z h hb s vb
b b= − + −
31 2 451 5 2 5
. .. .
τ
( ) ( ) ( )Q K c h h cg h h c h h Qs ss s s b d d t= − + − + − +1 5 0 5 1 5. . .
Where:Ks = 1.0 (If r ≤ 0.67)
Ks = 1.0 - 27.8(rs - 0.67)3 (If r > 0.67)
r = (h-ht)/(h-hb)
Kss = 1.0 (If rs ≤ 0.67)
Kss = 1.0 - 27.8(rs - 0.67)3 (If rs > 0.67)
rs = (h-ht)/(h-hs)
cy = 1.0 + 0.023(V2/(h-hb)
tb = Time since breach began forming
Outflow HydrographReservoir (level pool) routing
I Qs
t− =
∆∆
( )∆sA A
h hs s=+
−
’’
2
( )( )A A
h h
tQ Q I Is s+
−+ + − − =’ ’
’ ’∆
0
Where:As = Surface area
As’ = Surface area at t-∆t
h = Water surface elevation
h’ = Water surface elevation at t-∆t
∆t = Time step
Q = Total instantaneous outflow
Q’ = Outflow at t-∆t
I = Inflow
I’ = Inflow at t-∆t
Dynamic wave routing
Downstream boundary: Q Q Qb s= +
A h Ah h
tQ h Q I Is s( )
’( ) ’ ’’+
−+ + − − =
∆0
Where:
( ) ( ) ( ) ( ) ( )Q h c h h c h h c h h c h h c h hb b s g d( ) = − + − + − + − + − +1 2 3 4 5
32
52
32
12
12
Solve A h Ah h
tQ h Q I Is s( )
’( ) ’ ’’+
−+ + − − =
∆0 by Newton-Raphson Iterations
for h. Then use h to compute Q(h). Then advance to the next time andrepeat.
Flood Routing
Saint-Venant equations of unsteady flow:
Conservation of mass:
∂∂
∂∂
( )
( )
(
( ))Q
x
A A
tq+
+− =0
0
Conservation of momentum:
∂∂
∂∂
∂∂
∂∂
( )
( )
( / )
( )
( )
( )
./
( / )
( )
Q
t
Q A
xgA
h
xS S L
SQ Q
A RR A B
Sk
g
Q A
x
L fv
f e
f
e
+ + + +
− =
=
=
=
=
2
2
2
2
0
2 2
2
23
Where:
Q = Flow, cfs
A = Cross-section area, ft
B = Top width, ft
x = Distance along river, ft
t = Time, sec
q = Lateral inflow (+) or outflow (-)
h = Water surface elevation, ft
g = 37.2 ft/sec2
Figure 17: Location of top width.
Types of Hydraulic Flood Routing Methods
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
( )
( )
( / )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( / )
( )
( )
( )
( )
( )
( / )
Q
t
Q A
xgA
h
xS
h y z
h
x
y
x
z
x
y
xS
Q
t
Q A
xgA
y
xS S
SgA
Q
t gA
Q A
f
f
f
+ + +
=
= +
= + = −
+ + − +
=
= − −
2
0
2
0
2
0
0
1 1
( )
( )
( )
:
( )
( ):
( )
( )
( )
( )
( / )
( ):
x
y
xS
S S
Sy
x
Sy
x gA
Q
t gA
Q A
s
f
o
− +
≈
≈ −
≈ − − −
∂∂
∂∂∂∂
∂∂
∂∂
0
0
0
21 1
Kinematic Routing
S Diffusion Routing
S Dynamic Routing
f
f
Routing Model Selection
Kinematic: T S
n q Ep
p
01 6
1 2 0 2
0 2.
. .
.
φ>
Diffusion: T S n
q Ep o
p
0 7 0 6
0 4
0 003. .
.’
.
φ>
q aQ
ka
aS
n m
m
mm
m
p
pm
=
=+
=++
=++
+5
2 5
12
53
159
1
1
3 53
3 5
0
2
.
( )
( )
’
φ
φ
m = Cross section shape parameter
Tp = Time of rise of hydrograph, hr
S0 = Bottom slope ft/ft
n = Manning n
qp = Unit width peak discharge, (ft3/sec)/ft
E = Allowable routing error, percent
Boundary ConditionsUpstream: Flow Q Q tj
11+ = ( )
Water elevation h h tj1
1+ = ( ) (not in DAMBRK)
Downstream: Flow Q Q tNj+ =1 ’( )
Water elevation h h tNj+ =1 ’( )
Critical flow Q gA
BNj
N
j
+
+
=
1
132
12
Rating curve Q QNj+ =1 ’
Q Q Q Qh h
h hk k kNj
N
k k
’ ( )( )
= + −−−+
+
+1
1
1
Synthetic loop rating Qn
AR SNj
Nj+ +=1 1149 2
312
.( )
Location of Downstream Boundary
Figure 18: Example 1 of correct downstream boundaries.
Figure 19: Example 2 of correct downstream boundaries.
Figure 20: Example 3 of correct downstream boundaries.
Flood-Plain Model Plan View
Figure 21: Plan view of the flood-plain model.
Flood-Plain Model Cross-Sections
Figure 22: Cross-section locations for flood-plain model.
Volume Losses During Downstream Routing
qV P
LmL
f
= −0 00458.
TT T
=+1 2
2
Figure 23: Location of equation variables T1 and T2.
P: 0 < P < .3
Teton: P = .25 (Wide valley (10 mi.), Irrigation levels, Canals)
Buffalo Creek: P = .25 (Sludge)
Narrow Valley: 0 < P < 0.5 (Silt, Loam)
Wide Valley: 0.5 < P < .10 (Sand, Loam)
Time Distribution: qQ Q
Q Qqi
j ij
i
im
i
=−−
max
Qmaxi obtained from linear interpolation between max flow at dam andmax flow at downstream extremity of valley (QmaxD)
Usually, simulate DAM-BREAK first with qm=QLL=0. QmaxD=0
Then simulate with QmaxD obtained from first simulation.
Landslides
Figure 24: Landslide cross-sections.
Comparison of “NWS DAMBRK” and “HEC-1”
Outflow of Hydrograph:
1. Initial breach width is the same as the final breach width in HEC-1 butin DAMBRK initial width is zero.
2. Tailwater submergence effects are considered in DAMBRK butneglected in HEC-1.
3. DAMBRK uses either level pool (storage routing) or the St. Venantunsteady flow equations for the reservoir routing while HEC-1 usesonly level pool routing.
Downstream Routing:
1. DAMBRK uses St. Venant equations while HEC-1 DB uses ModifiedPools (storage routing)
2. HEC-1 does not consider the effects of:
a) multiple dams
b) bridges/embankments
c) flow losses
d) dead storage areas
e) special treatment of floodplains
f) land slides
Typical Difficulties using DAMBRK
1. Data errors
2. Data not entered correctly
3. Model parameters not understood
4. Starting with too complex of a problem
5. Subcritical/Supercritical flow
6. Expanding/Contracting cross-sections
7. Wide, float overbank (floodplain)
8. ∆x, ∆t too large
Interpolation of Cross-section
Example of incorrect data (BS top widths)
Cross-section “1” 0 100. 1000. 1200.
Cross-section “2” 0 60. 95. 400. 450. 950. 1075.
Incorrect because of: 1) Unequal number of top widths
2) Lack of geometric similarity
Figure 25: Examples of incorrect cross-section top widths.
Example of incorrect data (BS top widths)
Figure 26: Cross-section 1
Cross-section “1” 0 100. 600. 700. 850.
Cross-section “2” 0 60. 105. 750. 830.
Incorrect because of: 1) Lack of geometric similarity
Figure 27: Cross-section 2.
Simplify the problem to start.
Then, add complexities one or two at a time.
Example: Variable geometry and roughness
2 bridges
Levees along both sides
1. Prismatic geometry, constant roughness
2. Variable geometry, variable roughness
3. Add 1 bridge, then the other
4. Add levees
Figure 28: Example of mixed flow.
∆x reduction at severe changes in water surface slope
Figure 29: Cross-section refinement.
Subcritical/Supercritical Flow
Figure 30: Example of supercritical and subcritical flow.
1. Adjust Manning n so that flow is subcritical or supercritical througout thereach
Sn
Dc = 77000
2
13
Where:
Sc = Critical slope, ft/mi
n = Manning n
D = A/B = Hydraulic depth, ft
2. Use internal boundary condition for short reach through rapids
a) rating curve
b) critical flow/depth relation
3. Split the total reach into separate reaches of only subcritical orsupercritical
4. New algorithm may be the approach to use
Wide Overbanks
1. Friction Slope
( )S
n Q
A R
n Q
Af
AB
= =2 2
2
2 2
22 2 2 243
43. .
Figure 31: Observed change in hydraulic radius with elevation.
SQ
Kf =2
2
Figure 32: Change in hydraulic radius.
Where:
Kn
A Rn
A Rn
A Rc
c cL
L LR
R R= + +149 149 1494
343
43
. . .
Figure 33: Conveyance ratio.
2. One solution: Round overbanks
Figure 34: Rounding of overbank edges.
New model will also allow conveyance/height descriptions ofcross-sections.
Numerical Problems with Very Wide, Flat Flood Plains
Figure 35: Modeling dead storage areas.
Time Step Selection
∆tT
mp≤
Figure 36: Time step selection is important to catch the peak value.
∆
∆
t czT
D
ze
D e
p≤
=−− −
011
1
2 2
0
2
2 2 2
12
.
( )θ
Where:
Do = Initial Hydraulic Depth
e = 1 - error
θ = Weight factor; 0.5 ≤ θ ≤ 1
C = KV
K =5
3
2
3 1−
+m
m; 1.2 ≤ K ≤ 1.67
V =Q
A
Distance Step Selection∆ ∆x c t≤
c = 0.68C
Parameter Selection
1. Area (A)
2. Manning n
Cross-section Area of Channel-Valley
Errors due to:
1. Field measurements
2. Sampling interval
3. Linear variation of contour interval of topographic maps
Figure 37: Approximating cross-section top widths.
A
Am
h
h
h
h
A
A
e
p
m
e em
∝ +
=
+
+
( )
( )
1
1
5
3 5
∆
Figure 38: Error in flood depth caused by error in cross-sectional area for various shaped sectionshaving steady flow.
Estimation of Manning n: Steep Gradient Streams
Sm > 10 ft/mi S > .002 ft/ft
nS
R≈ 0 4
0 4
0 16..
.
Composite n:
Figure 39: Diagram of composite n value.
nn P n P n P n P
P P P P=
+ + ++ + +
1
21 2
22 3
23 4
24
1 2 3 4
12
Off-Channel Storage at a Point
Figure 40: Dead storage areas.
Use triangular distribution between 1 and 3.
Figure 41: Triangular distribution for off-channel storage.
SABSS
L
BSSSA
L
=
∴ =
22
Say l = 1/2 mi = 2640 ft
If SA = 456,000 ft2 @ elevation 52, then BSS = ( )2 456000
2640 = 345 ft.
If SA = 102,000 ft2 @ elevation 45, then BSS = ( )2 102000
2640= 77 ft.
Complicated Off-Channel Storage
Figure 42: River confluence can cause dead storage areas.
Figure 43: Triangular distribution for complicated off-channel storage.
OK as long as ∆t gD LT≥ , that is, disturbance propagates upstream in ≤ 1time step.