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

1.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

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:

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

ADVANTAGES: Non-iterative solution fast Layers thickness is allowed to be greater than in classical models Robust Flux discretisation: Flux q i 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 1. Ross (2003) numerical solution (3)

1.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

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)

Basha (1999) analytical solution 8 soils with Gardner parameters (Mualem 1976 et Bresler 1978) Constant surface flux=15mm/h during 10h Initially dry profile 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

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

I(t), I(q) 3 characteristics soils (sand, clay, loam) θ(z=0)=θ s Initially dry profile, h surf =  x=20cm  x=40cm  x=10cm Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990

I(t), I(q) 3 characteristics soils (sand, clay, loam) θ(z=0)=θ s Initially dry profile, h surf =  x=20cm  x=40cm  x=10cm Results on infiltration are sensitive to the discretization, especially on clayey soils: A finer discretization is needed close to the soil surface Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990

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

E=0.28 Profile 10 layers

E=0.44 Profile 10 layers

E=0.60 Profile 10 layers

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

E=0.96 Profile 100 layers

E=0.96 Profile 100 layers

E=0.97 Profile 100 layers

E=0.97 Profile 100 layers

E=0.98 Profile 100 layers

E=0.98 Profile 100 layers

E=0.97 Profile 100 layers

E=0.97 Profile 100 layers

E=0.98 Profile 100 layers

E=0.98 Profile 100 layers

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%) 2.2. Another reference numerical solution

1.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

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: 3. Account for vegetation processes (1)

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. 3. Account for vegetation processes (2)

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