A Process-Based Transfer Function Approach to Model Tile Drain Hydrographs Mazdak Arabi, Jennifer Schmidt and Rao S. Govindaraju World Water & Environmental Resources Congress 2005 May 17, 2005
Dec 22, 2015
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