A Process-Based Transfer Function Approach

to Model Tile Drain

HydrographsMazdak Arabi, Jennifer Schmidt

and Rao S. Govindaraju

World Water & Environmental Resources Congress 2005

May 17, 2005

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Overview

Rationale and BackgroundMethodologyAvailable DataResults and DiscussionConclusions

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Rationale and Background

Tile DrainsControlling the height of the water table Earlier planting More developed root system

Expedite the transport of nutrients and pesticides to surface waters

Water quality problems

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Rationale and Background

Previous Work (Reviewed by Youngs, 1999 )Numerical solutions of Richards’ equation

Solutions using the concept of specific yield in Boussinesq’s equation

Method of continuous succession of steady states

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Rationale and Background

ObjectivesTo develop a mathematical model for tile

drain response to rainfall events Transfer function from physical principles

Unsaturated vertical flow Saturated horizontal flow Parameter estimation in the context of method of

moments

To evaluate model performance utilizing data from a field study

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Methodology

Tile Drain Problem and Parameter Definition

Schematic of the Tile Drain Problem

a

RechargeSoil Surface

h(x,t)

Impervious barrier

Water table at time t

Initial Water table

Tile Drain

L

no-flow boundary (x=0)

x

z

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Methodology

Mathematical DevelopmentSaturated horizontal flow

Water flux in x-direction throughout the saturated thickness

Continuity equation

Boussinesq equation

x

hahKtxQ

)(),(

),()(

txix

Q

t

ahS

)(])([ tix

hahK

xt

hS

x

z

a

h(x,t)

L

Q(x,t)

K: hydraulic conductivityS: drainable porosity

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Methodology

Initial Conditions

Boundary Conditions a

Soil Surface

h(x,t)Water table at time t

Initial water table

Tile Drain

L

no-flow boundary

(x=0)

x

zatxh )0,(

0),0(

x

tQ

atLh ),(

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Methodology

Unsaturated vertical flow Time dependant recharge (from sharp-front analogy)

;tt,

t

Btt,0

)t(i0

0

;1

10

eSK

Wt

/1/1

WK

z B

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Methodology

After simplification, tile drain response is expressed as:

000

0

tt,)tt(A~

)t

tln()tA

~exp(C

tt,0

)t(Q

2

2'

SL4

KaA~

;A~

LBC

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Methodology

Parameter estimation based on method of moments.

A~

)tA~

(Ei)tA

~exp(

A~1

Cm 000

00

20

021 tA~

)tA~

(Ei

A~

)tA~

(Ei)tA

~exp(

A~3

Cm

)tA~

(EiA~2

)tA~

(EiA~

t)tA

~(Ei

A~t2

)tA~

exp(A~t

)tA~

exp(A~9

Cm 030

20

020

020

032

x

dtt

)texp()x(Ei;

00 t 0

0

n0t

n0n dt)tt(A

~)

t

tln()tA

~exp(C)tt(dt)t(Q)tt(m

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Available Data

Description of Experimental SitePurdue Water Quality Field Station (WQFS) Silty clay loam Glacial till at approximately 2 m below the surface The field contains cracks and other features A group of 48 plots each with a 10m by 24m clay

lysimeterSlurry walls to create a hydrologically isolated

“box” Dimensions L = 5m, a = 0.53m, and z = 1m.

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Available Data

Events used in this study

Calibration Event: Event 2, single burstUsing first moment

Event 1 2 3 4 5

Date 2/20/1997 3/13/1997 4/8/2002 5/11/2002 5/9/2002

Total Precipitation (cm) 3.05 2.97 2.39 3.16 1.14

Number of Bursts 3 1 2 2 1

Hours of Observed Flow 64 54 32 62 17

-1hr 0.265~ A

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

Comparison of observed hydrographs and the transfer function model for Event 2, calibration event.

Time (hr)

Q(c

m3 /h

r)

Rai

nfal

l(cm

)

10 20 30 40 50 60

10 20 30 40 50 60 70 80

0

100000

200000

300000

400000

500000

600000

700000

800000

900000 0

0.5

1

1.5

2

2.5

3

Observed: AverageObserved: Plot 13Observed: Plot 18HyetographSimulated

Event 2

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

maintained at 0.265 hr-1 for all events C from zeroth moment, and t0 from rainfall hyetograph

Event 1 2 3 4 5

Date 2/20/1997 3/13/1997 4/8/2002 5/11/2002 5/9/2002

Total Precipitation (cm) 3.05 2.97 2.39 3.16 1.14

Number of Bursts 3 1 2 2 1

Hours of Observed Flow 64 54 32 62 17

t0 (hr): First Burst 4 7 8 6 2

t0 (hr): Second Burst 11 - 19 22 -

t0 (hr): Third Burst 31 - - - -

C (cm2/hr): First Burst 2647000 7687000 3610000 1042000 561600

C (cm2/hr): Second Burst 90760 - 402500 935100 -

C (cm2/hr): Third Burst 241700 - - - -

A~

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

Comparison of observed hydrographs and model results for Event 1.

Time (hr)

Q(c

m3 /h

r)

Rai

nfal

l(cm

)

10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0

200000

400000

600000

800000

1E+06

1.2E+06 0

1

2

3

4

5

Observed: AverageObserved: Plot 13Observed: Plot 14Observed: Plot 15Observed: Plot 18Observed: Plot 33HyetographSimulated

(a) Event 1

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

Comparison of observed hydrographs and model results for

Event 3.

Time (hr)

Q(c

m3 /h

r)

Rai

nfal

l(cm

)

10 20 30 40 50 60 70

10 20 30 40 50 60 70

0

100000

200000

300000

400000

500000

600000

700000 0

0.5

1

1.5

2

2.5

3

3.5

4

Observed (Plot 12)HyetographSimulated

(b) Event 3

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

Comparison of observed hydrographs and model results for

Event 4.

Time (hr)

Q(c

m3 /h

r)

Rai

nfal

l(cm

)

10 20 30 40 50 60 70 80

10 20 30 40 50 60 70 80

0

100000

200000

300000

400000

500000

600000

700000

800000

900000 0

0.5

1

1.5

2

2.5

Observed: AverageObserved: Plot 10Observed: Plot 12HyetographSimulated

(c) Event 4

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

Comparison of observed hydrographs and model results for

Event 5.

Time (hr)

Q(c

m3 /h

r)

Rai

nfal

l(cm

)

5 10 15 20 25 30

5 10 15 20 25 30

0

100000

200000

300000

400000

500000

600000

700000 0

0.5

1

1.5

2

2.5

3

3.5

4

Observed (Plot 11)HyetographSimulated

(d) Event 5

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Results and Discussion

Evaluation of Model Performance based on Error-Statistics

;

)()(

))((

1

2

1

2

2

12

N

ii

N

ii

N

iii

PPOO

PPOO

R

N

ii

N

iii

SN

OO

PO

E

1

2

2

1

)(

)(

0.1

Event 1 2 3 4 5

R2 0.95 0.98 0.92 0.98 0.88

EN-S 0.85 0.97 0.77 0.96 0.81

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Conclusions

A solution for the response of a single tile drainSemi-analyticalThree-parameter transfer function (note:

all parameters have physical interpretation) : time lag for infiltrated water to reach the

groundwater table C : scaling parameter that ensures mass balance : a function of soil properties and geometry of

the plot

0t

A~

EWRIWorld Water & Environmental Resources

CongressMay 17, 2005

Rao S. GovindarajuSchool of Civil engineering

Purdue University

Conclusions

Tile-drain response scales linearly with the infiltrated depth raised to a power; i.e. as .

Satisfactory model performance, especially for events with data from multiple experimental plots

The solution is amenable to moment analysis thereby allowing for parameter estimation

/1W