Some Difficulties in Modeling Water and Solute Transport in Soils Ph. ACKERER IMFS STRASBOURG

1. Some Difficulties in Modeling Water and Solute Transport in Soils Ph. ACKERER IMFS STRASBOURG ackerer@imfs.u-strasbg.fr With the help of B. Belfort, H. Beydoun, F. Lehmann and A. Younès.

2. Hillslope hydrology 0.36 km2, 1000-750 m Contact: Bruno AMBROISE (IMFS) The Ringelbach catchment

3. Saturated area Discharge (from B. Ambroise, IMFS)

4. Mathematical models • Darcy – Richards eq. • Soil hydraulic properties • Parameter measurements • Direct methods • Indirect methods • Numerical methods • Highly non linear PDEs • Very strong parameters contrasts • Long term simulation • ‘Flat’ geometry Hillslope hydrology

5. (from UMR LISAH, Montpellier) Usual concepts and mathematical models __________________________________________________________________________________ Model concept

6. -7 10 Continuum Mec. (Stokes, Hagen-Poiseuille, …) -5 10 -3 10 Scale (m) REV Darcy, Richards, Water retention curves , …. -1 KT 10 1 10 KL Q1 Usual concepts and mathematical models __________________________________________________________________________________ Model scale

7. Mass conservation Generalized Darcy’s law Richards’ equation Usual mathematical models – conservation laws __________________________________________________________________________________

8. Mualem, 1976 Van Genuchten, 1981 Usual mathematical models – Soil hydraulic properties __________________________________________________________________________________ Pore-size distribution models

9. Water content W: fraction of particle distribution Pore radius Ri: average particle radius for fraction i rb : soil density rp : particle density n : number of particle a : 1.35 – 1.40 Water pressure g : surface tension q : contact angle Usual mathematical models – Soil hydraulic properties __________________________________________________________________________________ Particle-size distribution(Arya & Paris, 1981)

10. Robbez-Masson, UMR LISAH, Montpellier

11. Macropores in un-colonised and colonised soil (from Pierret et al., 2002)

12. Hierarchy of flow/transport models for variably-saturated structured media (after Altman et al., 1996)

13. From Tuller & Or, 2001

14. Some recent concepts __________________________________________________________________________________ New mathematical models Richard’s equation with alternative h(q) and K(q) Network models Alternative models

15. Modified Van Genuchten, Vogel et al. (1998, 2001) Soil Hydraulic Properties, h(q) and K(q) __________________________________________________________________________________ Pore-size distribution models

16. Soil Hydraulic Properties, h(q) and K(q) __________________________________________________________________________________ Kosugi, 1996

17. Soil Hydraulic Properties, h(q) and K(q) __________________________________________________________________________________ Pore-scale models (Tuller & Or, 2002)

18. Soil Hydraulic Properties, h(q) and K(q) __________________________________________________________________________________ Pore-scale models (Tuller & Or, 2002) (a) Fitted liquid saturation for silt loam soil with biological macropores. (b) Predicted relative hydraulic conductivity. (Note that 1 J kg-1 =  10-2 bar.) (from Tuller & Or, 2002)

19. Soil Hydraulic Properties, h(q) and K(q) __________________________________________________________________________________ Pedotransfer functions(Wösten, 2001)

20. Mualem, 1976 Prunty & Casey, 2002 Soil Hydraulic Properties, h(q) and K(q) __________________________________________________________________________________ Smooth functions

21. Network models __________________________________________________________________________________ Kinematic–dispersive wave model (Di Pietro et al., 2003)

22. Alternative models __________________________________________________________________________________ Two-phase flow using Lattice Boltzmann approach From Pan et al., 2004

23. Water retention curve from Pan et al., 2004. Alternative models __________________________________________________________________________________

24. Spatial variability and scales __________________________________________________________________________________ Parameter estimation Direct measurements and interpolation Indirect estimation by inverse approach

25. Spatial variability and scales __________________________________________________________________________________ (Ptak, Teutsch, 1994)

26. Spatial variability and scales __________________________________________________________________________________ Interpolation Conditioning Measurement locations Interpolation Conditioning Probability distribution of indicator 1 Probability distribution of indicator 2 . . . .

27. Pk = Pk / (S Pi) . . . Spatial variability and scales __________________________________________________________________________________ Probability normalization Integrated density function

28. init (30 cm) Ksat Spatial variability and scales __________________________________________________________________________________ Experimental site in Alsace

29. Water Nitrate Nitrate Fluxes after 8 weeks Fluxes after 20 weeks Nitrate Water Water Fluxes after 16 weeks Spatial variability and scales __________________________________________________________________________________

30. Inverse methods __________________________________________________________________________________ Parameter identification by inverse approaches Generalized least-square approach

31. Inverse methods __________________________________________________________________________________ Experimental set-up

32. Inverse methods __________________________________________________________________________________ Computed and measured variables

33. Inverse methods __________________________________________________________________________________ Parameter estimation and validation

34. Covariance matrix Parameter uncertainty Sensitivity matrix Inverse methods __________________________________________________________________________________ First order confidence interval

35. Inverse methods __________________________________________________________________________________ Correlation matrix

36. Measurements: Ym,1 = y(p) + e1 Measurements: Ym,i = y(p) + ei Measurements: Ym,n = y(p) + en Min(J(p)) Min(J(p)) Min(J(p)) Parameters and computed variable pc,1 Yc,1 Parameters and computed variable pc,i Yc,i Parameters and computed variable pc,n Yc,n Exp. Covariance matrix Inverse methods __________________________________________________________________________________ Virtual data set P, y(p) First Monte Carlo approach

37. Observations Yo = y(p) + ek Measurements: Ym,n = Yo + en Measurements: Ym,1 = Yo + e1 Measurements: Ym = Yo + ei Min(J(p)) Min(J(p)) Min(J(p)) Parameters and computed variable pc,n Yc,n Parameters and computed variable pc,1 Yc,1 Parameters and computed variable pc,i Yc,i Exp. Covariance matrix Inverse methods __________________________________________________________________________________ Virtual data set P, y(p) Second Monte Carlo approach

38. Inverse methods __________________________________________________________________________________ Comparison between 1er order and Monte Carlo Approaches

39. Conclusions __________________________________________________________________________________ Many challenges remain: Understanding of processes and their mathematical modelling Parameter scaling: from measurements to element size Soil heterogeneity description Accurate of numerical codes will be of great help

40. References Frontis Workshop on Unsaturated-Zone Modeling: Progress, Challenges and Applications, Wageningen, The Netherlands 3-5 October 2004. http://library.wur.nl/frontis/unsaturated/ Arya & Paris, Soil Sci. Soc. Am. J.,1981 Binayak P. Mohanty, Water Res. Res, 1999 Di Pietro et al., J. of Hydrology ,2003 Pan et al., Water Res. Res., 2004 Pierret et al., Géoderma, 2002 Prunty & Casey, Vadose Zone J, 2002 Tulle & Or, Vadose Zone J, 2002 Vogel et al., Adv. Water Res., 2001 Vogel & Roth, J of Hydrology, 2003 Wösten, J. of Hydrology., 2001