1. 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

2. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Overview Rationale and Background Methodology Available Data Results and Discussion Conclusions

3. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Rationale and Background Tile Drains  Controlling 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

4. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School 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

5. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Rationale and Background Objectives  To 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

6. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Methodology Tile Drain Problem and Parameter Definition Schematic of the Tile Drain Problem a Recharge Soil Surface h(x,t) Impervious barrier Water table at time t Initial Water table Tile Drain L no-flow boundary ( x=0 ) x z

7. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Methodology Mathematical Development  Saturated horizontal flow Water flux in x -direction throughout the saturated thickness Continuity equation Boussinesq equation x z a h(x,t) L Q(x,t) K: hydraulic conductivity S: drainable porosity

8. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School 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 z

9. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Methodology  Unsaturated vertical flow Time dependant recharge (from sharp-front analogy)

10. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Methodology  After simplification, tile drain response is expressed as:

11. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Methodology  Parameter estimation based on method of moments.

12. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Available Data Description of Experimental Site  Purdue 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 lysimeter  Slurry walls to create a hydrologically isolated “box”  Dimensions L = 5m, a = 0.53m, and z = 1m.

13. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Available Data Events used in this study Calibration Event: Event 2, single burst  Using first moment Event12345 Date2/20/19973/13/19974/8/20025/11/20025/9/2002 Total Precipitation (cm)3.052.972.393.161.14 Number of Bursts31221 Hours of Observed Flow6454326217

14. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion Comparison of observed hydrographs and the transfer function model for Event 2, calibration event.

15. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion  maintained at 0.265 hr -1 for all events  C from zeroth moment, and t 0 from rainfall hyetograph Event12345 Date2/20/19973/13/19974/8/20025/11/20025/9/2002 Total Precipitation (cm)3.052.972.393.161.14 Number of Bursts31221 Hours of Observed Flow6454326217 t 0 (hr): First Burst47862 t 0 (hr): Second Burst 11-1922- t 0 (hr): Third Burst31---- C (cm 2 /hr): First Burst2647000768700036100001042000561600 C (cm 2 /hr): Second Burst 90760-402500935100- C (cm 2 /hr): Third Burst241700----

16. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion Comparison of observed hydrographs and model results for Event 1.

17. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion Comparison of observed hydrographs and model results for Event 3.

18. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion Comparison of observed hydrographs and model results for Event 4.

19. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion Comparison of observed hydrographs and model results for Event 5.

20. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Results and Discussion Evaluation of Model Performance based on Error- Statistics Event12345 R2R2 0.950.980.920.980.88 E N-S 0.850.970.770.960.81

21. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School of Civil engineering Purdue University Conclusions A solution for the response of a single tile drain  Semi-analytical  Three-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

22. EWRI World Water & Environmental Resources Congress May 17, 2005 Rao S. Govindaraju School 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