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Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.

EVALUATION OF A FAST NUMERICAL SOLUTION OF THE 1D RICHARD’S EQUATION AND INCLUSION OF VEGETATION PROCESSES. Varado N., Ross P.J., Braud I., Haverkamp R., Kao C. Workshop DYNAS, December 6-8, 2004. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003)

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Varado N., Ross P.J., Braud I., Haverkamp R., Kao C.

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  1. EVALUATION OF A FAST NUMERICAL SOLUTION OF THE 1D RICHARD’S EQUATION AND INCLUSION OF VEGETATION PROCESSES Varado N., Ross P.J., Braud I., Haverkamp R., Kao C. Workshop DYNAS, December 6-8, 2004

  2. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003) 2. How to evaluate the numerical solution ? • Use of analytical solutions: • Moisture profile • Cumulative infiltration • Use of a numerical h-iterative solution 3. A sink term to account for the water extraction by roots • Inclusion within the numerical solution • Test the accuracy of the vadose zone module

  3. 1. Ross (2003) numerical solution (1) • 1D Richards equation • Brooks and Corey (1964) model to describe soil hydraulic properties: • Kirchhoff potential or degree of saturation used as calculation variable:

  4. i-1 h i-1 q i-1 i xi h i q i h i+1 i+1 1. Ross (2003) numerical solution (2) • Spatial discretisation : mass budget on layer n°i • Time discretisation: • Taylor development at first order : • Tri-diagonal matrix:

  5. 1. Ross (2003) numerical solution (3) • Flux discretisation: Flux qi between layers i and i+1 is expressed from Darcy low written with Kirchhoff potential and hydraulic conductivity of each layer. •  calculation :at each time step and for each node • Hypothesis:if the pressure is hydrostatic, flux will be null • ADVANTAGES: • Non-iterative solutionfast • Layersthickness is allowed to be greater than in classical models • Robust

  6. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003) 2. How to evaluate the numerical solution ? • Use of analytical solutions: • Moisture profile • Cumulative infiltration • Use of a numerical h-iterative solution 3. A sink term to account for the water extraction by roots • Inclusion within the numerical solution • Test the accuracy of the vadose zone module

  7. 2.1. Analytical solutions • With the Brooks and Corey model, no analytical solution describes the moisture profile. • Moisture profile with simplified soil properties description: Basha (1999) : linear solution • Cumulative infiltration with BC models: Parlange et al. (1985) Haverkamp et al. (1990)

  8. Basha (1999) analytical solution • Gardner (1958) model: allows the analytical formulation of the Kirchhoff potential. • Modification of the Ross (2003) numerical solution to deal with the same soils characteristics description • Huge simplification • 8 soils with Gardner parameters (Mualem 1976 et Bresler 1978) • Constant surface flux=15mm/h during 10h • Initially dry profile

  9. α=1.56x10-2 cm-1 Ks=4.86x10-4 cm.s-1 Touched Silt Loam

  10. I(t), I(q) 3 characteristics soils (sand, clay, loam) θ(z=0)=θs Initially dry profile, hsurf=0 Dx=10cm 1 2 Dx=20cm 3 4 5 Dx=40cm 6 7 8 9 10 Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990

  11. I(t), I(q) 3 characteristics soils (sand, clay, loam) θ(z=0)=θs Initially dry profile, hsurf=0 Dx=10cm 1 2 Dx=20cm 3 4 5 Dx=40cm 6 7 8 9 10 Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990 Results on infiltration are sensitive to the discretization, especially on clayey soils: A finer discretization is needed close to the soil surface

  12. Haverkamp (personal communication): moisture profile with the Brooks and Corey model. z(q, θ ) Initially dry profile, θ(z=0)=θs, hsurf=0 3 characteristics soils (sand, clay, loam)

  13. Profile 10 layers E=0.28

  14. Profile 10 layers E=0.44

  15. Profile 10 layers E=0.60

  16. Haverkamp (personal communication): moisture profile with the Brooks and Corey model. z(q, θ ) Initially dry profile, θ(z=0)=θs, hsurf=0 3 characteristics soils (sand, clay, loam) The soil column needs to be homogeneously discretized from the surface to the bottom.

  17. Profile 100 layers E=0.96

  18. Profile 100 layers E=0.96

  19. Profile 100 layers E=0.97

  20. Profile 100 layers E=0.97

  21. Profile 100 layers E=0.98

  22. Profile 100 layers E=0.98

  23. Profile 100 layers E=0.97

  24. Profile 100 layers E=0.97

  25. Profile 100 layers E=0.98

  26. Profile 100 layers E=0.98

  27. 2.2. Another reference numerical solution • Comparison with a SVAT model: SiSPAT (Braud et al., 1995), which provides a reference h-iterative solution (Celia et al. 1990) • Coupled resolution of heat and water transfers • Fine discretization (around 1 cm) • Numerous validations under distinct pedo-climatic conditions. • Raining and evaporation periods • Systematic tests on 3 characteristic soil types, various climate forcing and initial conditions • Systematic underestimation of the evaporation flux (-2%) and overestimation of water content in the first layer (8%)

  28. A fast non iterative solution of the 1D Richards’ equation (Ross, 2003) 2. How to evaluate the numerical solution ? • Use of analytical solutions: • Moisture profile • Cumulative infiltration • Use of a numerical h-iterative solution 3. A sink term to account for the water extraction by roots • Inclusion within the numerical solution • Test the accuracy of the vadose zone module

  29. 3. Account for vegetation processes (1) • Inclusion of a sink term within the Richards’ equation (Feddes et al. 1978). • Does not affect the resolution of the tridiagonal matrix • Ex(z,t) from literature: Li et al. (2001) account for water stress and provides a compensation by the deeper layers still humid. • Linear function of a PET • Interception like a reservoir • No resolution of the energy budget; use of a partition law:

  30. 3. Account for vegetation processes (2) • Test of the accuracy of the vadose zone module with the SiSPAT model • Test on a soybean dataset • Underestimation of soil evaporation greater than on bare soil • Overestimation of water content in the first layer • Low relative error on transpiration • Different partition of the energy between the use of a PET or the resolution of the energy budget.

  31. Conclusion • Fast, accurate and robust numerical solution • Validation against analytical solutions and a numerical solution. • Inclusion of a sink term to account for vegetation processes • Another formulation of the evaporation flux? • Problem of partition of the energy • Vadose zone module. • Inclusion within a large scale hydrological model

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