Fritz R. FiedlerUniversity of IdahoDepartment of Civil Engineering

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

2 0 m in u te s 4 0 m in u te s

0.00

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20.00

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35.00

0 5 10 15 20

time (min)

dis

char

ge

(mm

/hr)

1993

1994

Veg. Dist. 1

Veg. Dist. 2

h

t+

p

x+

q

y- q l

0

p

t+

x

p

h+

g h2

+y

p q

h- g h ( S - S ) +

p

hq = 0

2 2

o x f x l

q

t+

y

q

h+

g h2

+x

p q

h- g h ( S - S ) +

q

hq = 0

2 2

o y f y l

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16 18 20 22 24 26

time(min)

disch

arge

(mm

/hr)

Cv=0Cv=0.2Cv=0.4Cv=0.6Cv=0.8Cv=1.0

Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain

What is shallow discontinuous flow?

Shallow: depth << wavelength vertical acceleration negligible depth-averaged NS equations

Discontinuous: both dry and wet areas shocks topographic control infiltration variability

What is complex terrain?

Topography with characteristic length scales (amplitude and wavelength) similar to flow depth two-dimensional flow

Examples

Flooding inundation mapping dam breaks

Overland Flow hydraulics hydrologic response

Wetlands and Estuaries, and Tidal Flats

Physical Objectives

Determine how Dynamic Surface Interactions affect Hydrologic ResponseEvaluate the Effects of Grazing

– degenerates plant community • changes infiltration• changes microtopography

Study Area Description

Central Plains Experimental RangelandLight- and heavy-grazed enclosures 1/2-hour, 100-year rain: ~100 mm/hr 1-hour, 100-year rain: ~75 mm/hrPatchy vegetation

CPER

Outline

Field MeasurementsMathematical ModelResults

Infiltration Measurements

Disc infiltrometersLight- and heavy-grazed areasBare and vegetated

Infiltration Variability

High K vegetated (locally high elevation)Low K bare (locally low elevation)

Microtopography

The ground surface topography with approximately the same order amplitude and frequency as the overland flow depth in a given situation:

–related to rainfall intensity–related to infiltration characteristics–caused by vegetation growth

Ground Microtopography

Shaded Relief Map

0 1 0 0 2 0 0

x (c m )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

Mathematical Modeling

Infiltration spatial variability (G-A model)Microtopography (2-D dynamic equations)Uniform rainfallSimplified flow resistance

Surface Water Equations

h

t+

p

x+

q

y- q l

0

p

t+

x

p

h+

g h2

+y

p q

h- g h ( S - S ) +

p

hq = 0

2 2

o x f x l

q

t+

y

q

h+

g h2

+x

p q

h- g h ( S - S ) +

q

hq = 0

2 2

o y f y l

Numerical Challenges

Non-linear hyperbolic systemStrong source terms (sometimes “stiff”)Small depths / dry areas (discontinuous)Large gradients in dependent variables

Vector Form

)()()(

USUHUGU

= y

+ x

+ t

Vector Form

U = h , p , qT[ ]

G U( ) = p ,p

h+

g h

2,

p q

h

2 2T[ ]

H U( ) = q ,q

h+

g h

2,

q p

h

2 2T[ ]

S U( ) 1 = q , - g hz

x-

K p

8 h-

p

hq (

p

x+

p

y) ,

- g hz

y-

K q

8 h-

q

hq (

q

x+

q

y)

l

o

2 l

2

2

2

2

o

2 l 1

2

2

2

2

T

[

]

Basic MacCormack Scheme

j , kn + 1

x y y x j , kn = L ( t / 2 ) L ( t / 2 ) L ( t / 2 ) L ( t / 2 )U U2 2 1 1

j , k*

j , kn

j , kn

j-1 , kn

x ; j , kn = -

t

2 x ( - ) +

t

2 U U G G S

j , k**

j, kn

j, k*

j+ 1 , k*

j, k*

x ; j , k* = 0 .5 - -

t

2 x ( - ) +

t

2 U U U G G S

Lx1 Operator:

Friction Slope: Point-Implicit Treatment

)p(Opp

SSS 2

n

fxnfx

1nfx

pSx

fxt - 1

1 = D

SGGUU nkj, x;x

nk1,-j

nkj,x

nkj,

*kj,

2

tD + ) - (

x2

tD - =

Convective Acceleration Upwinding

h

p -

h

pn

kj,

2n kj,

nk1,j+

2n k1,j+

SGGUU nkj, x;

nk1,-j

nkj,

nkj,

*kj,

2

tD + ) - (

x2

tD - =

Smoothing Function

h + h2 + h

|h + h2 - h|

t

x =

**k1,-j

**kj,

**k1,j+

**k1,-j

**kj,

**k1,j+

kj,

) , (max = kj,k1,j+2k1/2,j+

)h-h( - )h-h( + h = h **k1,-j

**kj,k1/2,-j

**kj,

**k1,j+k1/2,j+

**kj,

***kj,

) , (max = kj,k1,j2k1/2,j

Lateral Inflow

l j,k j,kaveq = r - f

ponded:

y

q +

x

p +

th

+r = inon

non

nkj,

kj,a

tF - F

= fn

kj,1+n

kj,avekj,

tK = + F

+ Fln - F - F kj,

kj,kj,n

kj,

kj,kj,1+n

kj,kj,kj,

nkj,

1+nkj,

not ponded:

x

q +

x

p +r = i

non

non

kj,a

i = f kj,aave

kj,

High-performance computing

Fortran Loop optimizations

most dependencies eliminated unrolling, fusion single-stride memory access

Shared-memory parallel processing PC environment

Comparative Numerical Examples

Steady state kinematic wave solution (analytical)Dam break problem (analytical)Published results

Iwagaki, 1955 (experimental)Woolhiser et al., 1996 (characteristics- based)

Dam Break Problem

500

600

700

800

900

1000

wate

r su

rface e

lev

ati

on

(cm

)

400 600 800 1000 1200 1400 1600 x (m)

model results analytical solution

dam

Microtopographic Surface

Overland Flow Depths

Flow Depths and Velocity

Spatial Distribution ofInfiltration Parameters

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (cm )

Flow Channels

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

2 0 m in u te s 4 0 m in u te s

Overland Flow Depths

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

0 .0 c m

0 .2 c m

0 .6 c m

1 .0 c m

1 .4 c m

1 .8 c m

2 .2 c m

2 .6 c m

3 .0 c m

2 0 m in u te s 4 0 m in u te s

Cumulative Infiltration

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

1 .4 c m

2 .0 c m

2 .6 c m

3 .2 c m

3 .8 c m

4 .4 c m

5 .0 c m

5 .6 c m

6 .2 c m

2 0 m in u te s 4 0 m in u te s

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

0 5 10 15 20

time (min)

dis

char

ge

(mm

/hr)

1993

1994

Veg. Dist. 1

Veg. Dist. 2

Simulated vs. Measured

Simulated Grazing Effects

0

5

10

15

20

25

30

0 5 10 15 20 25

time (min)

dis

char

ge

(mm

/hr)

Heavy-grazedLight-grazed

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

2 0

1 6 0

3 0 0

4 4 0

5 8 0

7 2 0

8 6 0

1 0 0 0

0 1 0 0 2 0 0

x (cm )

0

1 0 0

2 0 0

3 0 0

4 0 0

5 0 0

6 0 0

7 0 0

8 0 0

y (c m )

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

Spatial Distribution of ReynoldsNumber and log(f )

Re

0.1 1 10 100

f

1e+1

1e+2

1e+3

1e+4

1e+5

1e+6

1e+7

1e+8

20 minutes

f = 11713 Re-1.51

R2 = 0.67

Cross-Sectional Mean ReynoldsNumber vs. Friction Factor

Distribution of log(KS)

0.00 100.00 200.000.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

-11.00

-10.50

-10.00

-9.50

-9.00

-8.50

-8.00

-7.50

-7.00

-6.50

-6.00

-5.50

-5.00

-4.50

-4.00

-3.50

x (cm )

y (c

m)

0.00 100.00 200.000.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

x (cm )

y (c

m)

Plane Slope, Variable Ks

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16 18 20 22 24 26

time(min)

disch

arge

(mm

/hr)

Cv=0Cv=0.2Cv=0.4Cv=0.6Cv=0.8Cv=1.0

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02 1.E-01 1.E+00

Mean Depth (cm)

Un

it D

isc

ha

rge

(c

m/s

)

CV=1.0

CV=0.8

CV=0.6

CV=0.4

CV=0.2

CV=0.0

Mean Depth vs DischargeVariable KS

Effect of Microtopographic Amplitude

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02 1.E-01 1.E+00

Mean Depth (cm)

Uni

t D

isch

arge

(cm

/s)

20% reduced40% reduced60% reducedactual20% increasedplane surface

Mean Depth vs DischargeVariable Microtopography

Conclusions

Plane approximation gross distortionVegetation controls responseAverage/effective K not applicableInteractive infiltration importantReynolds No. - Friction FactorK-W assumption

Watch Your Step!