EVALUATION OF A FAST NUMERICALSOLUTION OF THE 1D RICHARD’SEQUATION 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 1DRichards’ 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 waterextraction 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

hhh

h

K

Khh

h

h

≥=≥=

<

=<

=

+−

si 1 si 1

si si /32/1

θθ

( )

( ) eeses

e

h

hhhhKn

hK

hhn

KhdhhK

≥−+−

=

<−

== ∫ ∞−

si 1

si 1

φφ

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

• Kirchhoff potential or degree of saturation used as calculationvariable:

+∂

∂

∂

∂=

∂

∂1

z

h)h(K

zt

θ

• Spatial discretisation :

mass budget on layer n°i

iii qqQdt

d−= −1

• Time discretisation:

[ ]0,1ó 1 ∈−=Δ

Δ−

σσii qq

t

Qi

iiiiiii dScSbSa =Δ+Δ+Δ +− 11

• Tri-diagonal matrix:

• Taylor development at first order :

Δ

∂

∂+Δ

∂

∂+= +

+1

1

0i

i

ii

i

iii S

S

qS

S

qqq σσ

i-1

i

q i-1

q i

h i-1

h i

h i+1

Δxi

i+1

1. Ross (2003) numerical solution (2)

• ADVANTAGES:

Non-iterative solution fast

Layers 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 writtenwith Kirchhoff potential and hydraulic conductivity of each layer.

( )i

iiii

i

iiii Z

KKZ

KqΔ

−−−+=

Δ

−−= +

++

+

φφωω

φφ 11

12/1 1

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

zKq

∂∂

+=φ

1. Ross (2003) numerical solution (3)

1. A fast non iterative solution of the 1DRichards’ 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 waterextraction 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, noanalytical solution describes the moistureprofile.

– Moisture profile with simplified soil propertiesdescription: Basha (1999) : linear solution

– Cumulative infiltration with BC models: Parlange etal. (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 α=

( ) ( )hexprsr αθθθθ −+=

Sols α (m-1) Ks (m.sec-1) θs 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 theanalytical formulation of theKirchhoff potential.

•Modification of the Ross (2003)numerical solution to deal with thesame soils characteristicsdescription•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 8 100.

00.

060.

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)w

ater

con

tent

(m

3.m

-3)

0 2 4 6 8 10

0.0

0.01

0

Ross (2003)

Basha (1999)

Touched Silt Loam _=1.56x10-2 cm-1

Ks=4.86x10-4 cm.s-1

• I(t), I(q)

• 3 characteristics soils (sand, clay, loam)• _(z=0)=_s

• Initially dry profile, hsurf=0

123

4

5

6

7

8

9

10

Δx=20cm

Δx=40cm

Δx=10cm

()exp*11**ln1ItIββββ−+=−−

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

clay

time (h)

cum

ulat

ive

infil

trat

ion

(mm

)

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)• _(z=0)=_s

• Initially dry profile, hsurf=0

123

4

5

6

7

8

9

10

Δx=20cm

Δx=40cm

Δx=10cm

Results on infiltration are sensitiveto the discretization, especially on clayeysoils:

A finer discretization is neededclose to the soil surface

()exp*11**ln1ItIββββ−+=−−

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

clay 15 layers

time (h)

cum

ulat

ive

infil

trat

ion

(mm

)

0 5 10 15 20

050

100

150

200

Ross (2003)analytical solution

Haverkamp (personal communication): moistureprofile with the Brooks and Corey model.

• z(q, _ )

• Initially dry profile, _(z=0)=_s, hsurf=0• 3 characteristics soils (sand, clay, loam)

42*******1ln11111124zzzccczzzqcczqqcqλλθθ++−=+−+−−+

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): moistureprofile with the Brooks and Corey model.

• z(q, _ )

• Initially dry profile, _(z=0)=_s, hsurf=0• 3 characteristics soils (sand, clay, loam)

• The soil column needs to be homogeneously discretizedfrom the surface to the bottom.

42*******1ln11111124zzzccczzzqcczqqcqλλθθ++−=+−+−−+

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 thefirst layer (8%)

2.2. Another reference numerical solution

1. A fast non iterative solution of the 1DRichards’ 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 waterextraction 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 etal. 1978).

• Does not affect the resolution of the tridiagonal matrix

• Ex(z,t) from literature: Li et al. (2001) account for water stressand 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,hKhExzttzzθ∂∂∂=+−∂∂∂

3. Account for vegetation processes (1)

()()()()12,,,ExztzzgzTPαθαθ=

(1exp())exp()blblTPETPaLAIEPETPaLAI=−−=−

iiiiiii dScSbSa =Δ+Δ+Δ +− 11

• Test of the accuracy of the vadose zone module withthe 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