Establish Effective Lower Bounds of Watershed Slope for Traditional Hydrologic Methods

Manoj KC and Xing Fang

May 11-13, 2009For TxDOT project 0-6382

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DIFFUSION HYDRODYNAMIC MODEL(DHM)

It couples two-dimensional overland flow and one-dimensional open channel flow.

It is based on the diffusion form of the St. Venant equation dominated by pressure, gravity, and friction forces.

It is a numerical model based on an explicit, integrated finite-difference scheme.

It accommodates several important hydraulic effects like backwater effects, channel overflow, combined overland flow and storage effects, and ponding which are neglected by kinematic methods.

The catchment is represented by topographic elevation and geometric data.

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DIFFUSION HYDRODYNAMIC MODEL(DHM)

Some previous applications of the model include:- rainfall-runoff modeling,- development of synthetic S-graphs for unit hydrograph

studies,- modeling of the flooding of the watershed due to open

channel deficiencies,- large scale flood plain dam-break analysis,- small scale flood plain dam-break analysis,- temporary flood control debris-basin failure onto a broad

plain.3

GOVERNING EQUATION OF DHM

The 1-D governing equation of DHM:

- where Qx is the flow rate; x and t are spatial and temporal coordinates; Ax is the flow area; H is water surface elevation; n is manning’s roughness coefficient, R is hydraulic radius; and mx is the momentum quantity defined as follows.

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GOVERNING EQUATION OF DHM

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For convective acceleration model

For local acceleration model

For coupled model

For DHM

COMPUTER PROGRAM FOR DHM

DHM computer code consists of 2-D flood plain model, 1-D channel model, and an interface sub-model.

The interface model updates the water surface elevations of grid (flood plain) and channel elements at specified time intervals.

The kinematic routing technique can also be revoked by setting KMODEL to 1 for the comparison between two models.

Variable time step algorithm used in the model dramatically reduces the computational time.

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INPUT FILE FOR DHMSeparate program has been developed for the creation of input

data file for DHM by researcher at Auburn University - the input file contains the following information:

- minimum and maximum allowable time step - decrement and increment time step - total simulation time and output time- tolerance for depth change and surface detention- number and dimensions of grids in floodplain and channel- north, east, south and west cells, manning’s roughness

coefficient, elevation and depth of the grid surrounding - similar data for channel elements- rainfall data, critical output node and output node for discharge

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TESTING OF DHM DHM was tested for channel-floodplain model of a

hypothetical watershed. It was also tested for 5 other catchments for -sensitivity of DHM S-graph to effective rainfall intensity-sensitivity of DHM S-graph to watershed area and slope-sensitivity of DHM S-graph to manning’s friction factor-sensitivity of DHM S-graph to catchment shape DHM was also used to verify results for 2-D overland

simulations by Su and Fang (2003).

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0

500

1000

1500

2000

2500

3000

3500

4000

4500

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Disc

harg

e (c

fs)

Time (hr)

Catchment 5

3 in/hr

2 in/hr

1 in/hr

0.5 in/hr

TESTING OF DHMDischarge variation for different excess rainfall intensity

The outflow node is indicated by the arrow.9

TESTING OF DHMSensitivity of DHM S-Graph to effective rainfall intensityS-Graph varied dependingupon the effective rainfallintensity.When it is normalized withrespect to maximumdischarge and lag, the S-graphs became nearlycoincident indicating the dominance of lag value in this technique (lag from 0to 50% Qp)

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TESTING OF DHMSensitivity of DHM S-graph to watershed areaFor this case the nodal elevations are held constant,as the grid size is increasedcatchment size is increased but the slope is decreased.Watershed lag differed depending upon area and slope but when normalized produced similar graphs.

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0

10

20

30

40

50

60

70

80

90

100

0 100 200 300

(Q/Q

max

)*10

0%

(time/lag)*100%

Synthetic S-graph for catchment 5 for various grid size for 1 in/hr effective rainfall

GRID 500'

GRID 750'

GRID 1500'

GRID 3000'

TESTING OF DHMSensitivity of DHM S-graph to watershed slopeFor this case the nodal elevations are changed accordingly as the cross-fall slope while the grid size is held constant.Watershed lag differed depending upon slope but when normalized produced similar graphs.

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0

50

100

0 100 200 300 400 500

(Q/Q

max

)*10

0%

(time/lag)*100%

Synthetic S-graph for catchment 5 for various cross slope ,1 in/hr effective rainfall (750-ft grids)

0.10%

0.40%

0.80%

1.50%

CROSSFALL SLOPE

TESTING OF DHMSensitivity of DHM S-graph to Manning’s friction factorManning’s friction factorwas varied to get theidentical S-graph forconstant effective rainfall.Watershed lag differed depending upon manning’s roughness factor, lag was directly proportional to theroughness factor.

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0

50

100

0 100 200 300 400 500

(Q/Q

max

)*10

0%

(time/lag)*100%

Variation in Synthetic S-graph for catchment 5 with variation in Manning's Friction Factor ,1 in/hr effective rainfall (750-ft grid)

0.0150.0300.0400.0500.100

MANNING'S FRICTION FACTOR

Sensitivity of DHM S-graph to the combination of catchment area and effective rainfall intensity

Area = 5 miles 2 Area =10 miles 2

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0

50

100

0 100 200 300 400 500

(Q/Q

max

)*10

0%

(time/lag)*100%

Variation in Synthetic S-graph for catchment 5 with variation in effective rainfall for catchment area of 5 sq. miles

1 in/hr

0.5 in/hr

2 in/hr

3 in/hr

EFFECTIVE RAINFALL INTENSITY

0

50

100

0 100 200 300 400 500

(Q/Q

max

)*10

0%

(time/lag)*100%

Variation in Synthetic S-graph for catchment 5 with variation in effective rainfall for catchment area of 10 sq. miles

1 in/hr

0.5 in/hr

2 in/hr

3 in/hr

EFFECTIVE RAINFALL INTENSITY

TESTING OF DHMFrom these tests it can be inferred that DHM produced S-graphs are relatively insensitive to

constant effective rainfall intensity, watershed area and slope, and the Manning’s roughness factor.

The S-graphs so produced changed very slightly to the combination of changing rainfall intensity and catchment area.

The watershed lag decreases with increasing constant effective rainfall intensity.

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The watershed lag and the constant effective rainfall intensity, i are related byLag(i)=k1*ik2

k1, k2 are constants for a watershed.From regression analysis,k2 = -0.401 for each watershed.The watershed lag varied highly for low effective rainfall.

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y1 = -0.401x + 0.148

y2 = -0.402x + 0.187

y3 = -0.401x + 0.081

y4 = -0.401x + 0.105

y 5= -0.400x + 0.172

-1

-0.5

0

0.5

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

y=lo

g(la

g)

hr

x=log(intensity) in/hr

Lag versus Effective Rainfall Intensity(i) Correlation

catchment 1

catchment 2

catchment 3

catchment 4

catchment 5

For low-effective rainfall intensity the catchment behaved as non-linear system in terms of watershed lag values.For high intensity of effective rainfall, the variation in lag values is small and the watershed is expected to behave as a linear system.

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y = 1.483x-0.38

R² = 0.997

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

y =

lag

(hr)

x = intensity (in/hr)

Lag versus Effective Rainfall Intensity(i) Correlation for high rainfall intensities

Data y = -0.179*x + 1.503 Power (Data)

y = -0.399x + 0.172R² = 1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

y=lo

g(la

g) h

r

x=log(intensity) in/hr

Lag versus Effective Rainfall Intensity(i) Correlation for high and low rainfall intensities

high intensity rainfall data

low intensity rainfall data

TESTING OF DHMWhen the backwater effects are neglected (i.e.,when flows are all freedraining) the kinematic and diffusion hydro-dynamic models produce similar results as shown inthe figure aside.

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0

500

1000

1500

2000

2500

3000

0 1 2 3 4 5 6 7

Disc

harg

e (c

fs)

time (hr)

Comparision of DHM vs Kinematic model

dhm 0.5 in/hr

kinematic 0.5 in/hr

dhm 1 in/hr

kinematic 1 in/hr

dhm 2 in/hr

kinematic 2 in/hr

TESTING OF DHMThe S-Graph from kinematic model when it is normalizedproduces similar results as that of DHM.

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0

50

100

0 100 200 300 400 500

(Q/Q

max

)*10

0%

(time/lag)*100%

Synthetic S-Graph by Kinematic Wave Model,catchment-5

0.5 in/hr

1 in/hr

2 in/hr

3 in/hr

EFFECTIVE RAINFALL INTENSITY

The figures show the resulting hydrographs for the three storm patterns of variable rainfall input

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0.00

2.00 1.60 1.20 0.80 0.40 0.00

0 1 2 3 4 5 6 7 8 9 10

0

10

20

0

500

1000

1500

0 1 2 3 4 5 6 7

Rain

fall in

tens

ity (in/

hr)

Disc

harg

e (cfs)

Storm time (hr)

Runoff Hydrographs (Catchment-1) for Storm A

DHM

Kinematic

Linear UH method

rainfall pattern

0.000.67

1.332.00

1.330.67

0.00

0 1 2 3 4 5 6 7 8 9 10

0

10

20

0

500

1000

1500

0 1 2 3 4 5 6 7

Rain

fall in

tens

ity (in/

hr)

Disc

harg

e (cfs)

storm time (hr)

Runoff Hydrographs (Catchment-1) for storm-B

DHM

Kinematic

Linear UH method

rainfall pattern

The hydrographs from the diffusion and kinematic wave models are similar.The hydrograph from the linear unit hydrograph method gives different resultmostly in peak discharge depending on the type of storm pattern.

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0.00 0.40 0.80 1.20 1.60 2.00

0.00

0 1 2 3 4 5 6 7 8 9 10

0

10

20

0

500

1000

1500

0 1 2 3 4 5 6 7

Rain

fal i

nten

sity

(in/

hr)

Disc

harg

e (c

fs)

Storm time (hr)

Runoff Hydrographs (Catchment-1) for rainfall pattern-C

DHM

Kinematic

Linear UH method

rainfall pattern

The figure showsthe hydrographs produced by the diffusion and thekinematic wave models for the different values ofManning’s friction factors.The peak dischargedecreases as the Manning’s friction factor is increased significantly.

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0

500

1000

1500

2000

0 1 2 3 4 5 6 7

Disc

harg

e (c

fs)

Storm time (hr)

Comparasion of Kinematic and Diffusion Routing Techniques at different Manning's Roughness Factor for Rainfall pattern-C

Kinematic (n=0.1)

Diffusion (n=0.1)

Diffusion (n=0.05)

Kinematic (n=0.05)

Kinematic (n=0.01)

Diffusion (n=0.01)

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Rain

fall

(in/h

r)

Time (hr)

Storm Pattern C

TESTING OF DHMThe DHM model was also tested for Su-Fang 2-D numerical

model of the overland flow: The model was tested in a rectangular watershed (basin) of

length 35 m and width 10 m with grid size of 5 m with constant effective rainfall over 12 hours.

The traveling time was calculated from the relationship of discharge versus time at the point of 95.5% of the maximum discharge.

The maximum discharge calculated from DHM model for various tests as given in the following table is in good agreement with the value by the Rational Method.

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Recreation of the Table 1 from Dehui Su and Xing Fang-2003showing the simulated Travel Time from DHM.

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The figure shows the variation of discharge with time for the test T1-n1 as given in the table for each mesh.The travel time is directly proportional to the travel length of the flow in the water-shed.The travel time is also directly proportional to the Manning’s roughness coefficient.

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0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0 2 4 6

Disc

harg

e(cm

s)

time (min)

n=0.01, S=0.005, I=88.9 mm/hr

L=35

L=30

L=25

L=20

L=15

L=10

L=5

The figure shows the variation of discharge with time for the test for smaller and even zero slope.The travel time is inversely proportional to the basin slope of the watershed. The tests in table t2-i shows that travel time decreases withincrease in rainfall intensity.

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0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

0 5 10 15 20 25 30

Disc

harg

e (cm

/sec

)

Time (min)

L=35.0 m,i=88.9mm/hr, n=0.02

T3-S1 T3-S2 T3-S3 T4-S4 T5-S5 T5-S6

s=0.0001

s=0.0

s=0.001

s=0.01

s=0.05

s=0.1

Testing DHM for Small Scale Testing Plots

Rainfall Test:The figure shows one of the testing plots (6 ft wide by 30 ft long) for the Rainfall Test performed byDr. Ming-Han Li and hisstudents through theTxDOT project 4404-2.

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Series-1 : gross rainfall intensity for total storm period Series-2: gross rainfall intensity considering initial

abstractions for the first six minutes. Series-3: Effective rainfall intensity considering initial

abstractions (TR = 0.6545”, initial loss = 0.187”, excess = 0.076”) Series-4: Field data from Ming-Han Li’s experiment.

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0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30 35

Runo

ff (g

pm)

Time (min)

Bare clay (Run 1,surface slope=0.43%, rainfall intensity=1.87 in/hr,n=0.012)

DHM(I=1.87 "/hr, 0-21 min) (1) DHM(I=1.87 "/hr, 6-21 min) (2)

DHM (i = 0.305"/hr, 6-21 min) (3) Run-1 (Ming-Han's Test) (4)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 5 10 15 20 25 30 35 40 45 50

Runo

ff (g

pm)

Time (min)

Bare clay (Run 2,surface slope=0.43%, rainfall intensity=1.98 in/hr,n=0.012)

DHM(I=1.98 "/hr, 0-31 min) (1) DHM(I=1.98 "/hr, 14-31 min) (2)DHM(I=0.376 "/hr, 14-31 min) (3) Run-2 (Ming-Han's Test) (4)

Series Peak Q (gpm)1 & 2 3.50

3 0.574 0.68

Series Peak Q (gpm)1 & 2 3.70

3 0.704 0.84

Discussion“Infiltration rate = 0.0358 – 0.0598 in/hr”Antecedent soil moisture (on bare clay, lawn and pasture):

8 – 54%

Resulting infiltration loss will be 0.0125 – 0.0209” over 21 rainfall input <<< estimated from runoff hydrograph.

The figures show the comparisons of runoff variation between the field tests performed by Dr. Ming-Han Li (TxDot-4404-2) and the result from DHM.

The rainfall loss affected the peak discharge to a greater extent than the travel time.

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Impulse Runoff TestThe figure shows one of the testing plots (6 ft wide by 30 ft long) forthe impulse runoff testperformed by Dr. Ming-Han Li and his students through the TxDOT project 4404-2.The hydroseeder was used as the water sourcefor the reservoir andthe overflow from a weir as inflow hydrograph.

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“Measure travel time after water overtopped the weir until the waterfront reaches the outlet” –(Cahill and Li, 2005)The overflow from the reservoir through the weir at t = 0 is not equal to the flow from the hydroseeder, the inflow hydrograph as input for the DHM model was delayed.The calculation shows the overflow from the weir equals the inflow from the hydroseeder takes 10 seconds (approximately) after the water overtopped the weir for the flow rate of 41.2 gpm.

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Time (min) Discharge (gpm)0 01 Q9 Q

10 0

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10

Inflo

w fr

om th

e w

eir (

gpm

)

Inflow Time (min)

Non-delayed Discharge

Delayed Discharge

The series-1 in the above graph is the recreation of the figure F.1 from Ming-Han Li’s Report TX-04/0-4404-2.

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0

5

10

15

20

25

30

35

40

45

50

0 0.5 1 1.5 2 2.5 3 3.5

Inflo

w R

ate

from

ups

trea

m e

nd (

gpm

)

Travel Time (min)

Impulse Rainoff Test (Surface Type=Bareclay,Surface Slope=0.43%)

Ming-Han's Test (1) Nonzero flow at the outlet (2) Q = 80% Qp (3) Q = 100% Qp (4)

Q starts at 1 minutes

The series-2, 3, 4 in the above graph are the results produced from DHM model which show the variation of travel time with ending flow rates. The input discharge is delayed by 1 min.The travel time for series-2 is the time when flow just starts to appear at the outlet of the plot; for series-3 and -4, they are the travel times corresponding to 80% and 100% of peak discharge at the outlet respectively.

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Discharge 100%PD 80%PDFirst

flow at outlet

100%PD 80%PDFirst

flow at outlet

100%PD 80%PDFirst

flow at outlet

Ming-Han's Test

(gpm) (min) (min) (min) (min) (min) (min) (min) (min) (min) (min)17.44 2.00 0.87 0.57 2.22 1.12 0.82 2.53 1.41 1.00 1.2820.61 1.92 0.83 0.53 2.35 1.08 0.77 2.48 1.38 0.95 1.0225.36 2.02 0.79 0.48 2.46 1.04 0.72 3.21 1.35 0.90 1.2026.95 2.16 0.78 0.48 2.43 1.03 0.72 2.79 1.34 0.90 1.3128.53 2.76 0.77 0.48 2.73 1.02 0.72 3.12 1.33 0.87 1.1736.46 2.76 0.73 0.42 2.82 0.98 0.66 2.88 1.29 0.81 0.7041.21 2.43 0.71 0.39 3.03 0.96 0.63 3.18 1.28 0.78 0.73

Input Discharge starts @ 0 min Input Discharge starts @ 0.5 min Input Discharge starts @ 1 min

(hydroseeder)

The flow line in green shows the shortestpath for travel time in the test plot (Ming-Han Li’s experiment). The travel time is the time when the flow starts to occur atthe outlet.The flow line in black shows the longerpath for travel time in the test plot since the water can only flow out from the middle node. The travel time therefore is longer than that of green flow line.

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FUTURE TAKS Integration of rainfall loss process in the model.

Verification Ming-Han Li other test results-Rainfall tests-Impulse runoff tests

Testing of the model with real watershed’s data.

Testing of the model from the data collected for the current project by Ming-Han Li.

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