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

ee

e

b

ese

b

es

hhhh

hhhh

KKhh

hh

si 1 si 1

si si /32/1

eeses

e

h

hhhhKnhK

hhn

KhdhhK

si 1

si 1

• Brooks and Corey (1964) model to describe soil hydraulic properties:

• Kirchhoff potential or degree of saturation used as calculation variable:

1

zh)h(K

zt

• Spatial discretisation :mass budget on layer n°i

iii qqQdtd

1

• Time discretisation:

0,1σ 1

ii qq

tQi

iiiiiii dScSbSa 11

• Tri-diagonal matrix:

• Taylor development at first order :

11

0i

i

ii

i

iii S

SqS

Sqqq

i-1

i

q i-1

q i

h i-1

h i

h i+1

xi

i+1

1. Ross (2003) numerical solution (2)

• ADVANTAGES:

Non-iterative solution fastLayers thickness is allowed to be greater than in classical

modelsRobust

• 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.

i

iiii

i

iiii Z

KKZ

Kq

11

12/1 1

• calculation : at each time step and for each node Hypothesis: if the pressure is hydrostatic, flux will be null

zKq

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

hexpKhK s

hexprsr

Sols (m-1) Ks (m.sec-1) s Chino clay 0.0685 2.29E-07 0.532 Lamberg clay 32.7 3.34E-04 0.537 Peat 0.104 6.13E-07 0.47 Touched silt loam 1.56 4.86E-06 0.469 Oso Flasco fine

sand

7.2 2.00E-04 0.266 Crab Creek sand 46.6 1.27E-04 0.375 Rehovot sand 15.74 7.64E-05 0.44 Ida silt clay loam 6.7 4.17E-06 0.53

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

layer 1

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.10

0.20

layer 2

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.10

0.20

layer 3

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.10

layer 4

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.10

layer 5

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.06

0.12

layer 6

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.04

0.08

layer 7

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.02

0.05

layer 8

time (h)

wat

er c

onte

nt (

m3.

m-3

)

0 2 4 6 8 10

0.0

0.02

layer 9

time (h)

wat

er c

onte

nt (

m3.

m-3

)0 2 4 6 8 10

0.0

0.01

0

Ross (2003)

Basha (1999)

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)=θs

• Initially dry profile, hsurf=0

123

4

5

6

7

8

9

10

x=20cm

x=40cm

x=10cm exp * 11* * ln

1I

t I

Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990

clay

time (h)

cum

ulat

ive

infil

tratio

n (m

m)

0 2 4 6 8 10

020

4060

8010

012

014

0

Ross (2003)analytical solution

• I(t), I(q)

• 3 characteristics soils (sand, clay, loam)• θ(z=0)=θs

• Initially dry profile, hsurf=0

123

4

5

6

7

8

9

10

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

exp * 11* * ln1

It I

Cumulative infiltration: Parlange et al. 1985, Haverkamp et al. 1990

clay 15 layers

time (h)

cum

ulat

ive

infil

tratio

n (m

m)

0 5 10 15 20

050

100

150

200

Ross (2003)analytical solution

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)

42 ** * *

* * *

1ln 111 1 1

1 2 4

z

z z

cc cz z

z

qc czq q c q

1 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.28

Profile 10 layers

HaverkampRoss (2003)

2 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.44

Profile 10 layers

HaverkampRoss (2003)

3 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.60

Profile 10 layers

HaverkampRoss (2003)

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.

42 ** * *

* * *

1ln 111 1 1

1 2 4

z

z z

cc cz z

z

qc czq q c q

1 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.96

HaverkampRoss (2003)

Profile 100 layers

2 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.96

Profile 100 layers

HaverkampRoss (2003)

3 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.97

Profile 100 layers

HaverkampRoss (2003)

4 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.97

Profile 100 layers

HaverkampRoss (2003)

5 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.98

Profile 100 layers

HaverkampRoss (2003)

6 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.98

Profile 100 layers

HaverkampRoss (2003)

7 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.97

Profile 100 layers

HaverkampRoss (2003)

8 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.97

Profile 100 layers

HaverkampRoss (2003)

9 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.98

Profile 100 layers

HaverkampRoss (2003)

10 h

teneur en eau (m^3/m^3)

prof

onde

ur (m

)

0.0 0.1 0.2 0.3 0.4 0.5

-2.0

-1.5

-1.0

-0.5

0.0

E=0.98

Profile 100 layers

HaverkampRoss (2003)

• 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:

( ) 1 ,hK h Ex z tt z z

3. Account for vegetation processes (1)

1 2, , ,Ex z t z z g z TP

(1 exp( ))exp( )

bl

bl

TP ETP a LAIEP ETP a LAI

iiiiiii dScSbSa 11

• 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