dambrk modeling methodology

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



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.



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.

dambrk modeling methodology

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