Mechanics of Unsaturated Soils for Engineering

numerical
GU modelling of Nuclear
Waste
DU Disposal

ENPC
BehrouzUNITN
Gatmiri

Ecole Nationale
UPC des Ponts et Chaussées
UNINA

Second MUSE School in Paris, May 17 2006

“Marie Curie” Research Training Network

Mechanics of Unsaturated Soils for Engineering

Outline
GU
•Formulation for THM
DU problems in Multiphase media
in CERMES, integrated in θ-STOCK Code
•Review of THM formulation in Barcelona
•An alternative ENPC
formulation
•Some applications to Nuclear
UNITN Waste Disposal
problem
UPC
UNINA

“Marie Curie” Research Training Network

 Objective
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Geologic Formation or host medium

= Soil (Clay) or rock (Granite)

Compacted Clay

container

Access Tunnels

BG (enresa, 1994) Mechanics of Unsaturated Soils for Engineering (MUSE)

 Soil
T Terminology
Particles Air
“Marie Curie” Research Training Network

GU Constituents (Unsaturated
DU and Thermal Condition):

• Solid skeleton

ENPC
• Water :
Water - Gas Form
UNITN (water vapour)
Air
(dry) Air - Liquid Form
Dissolved
Air UPC
Water Vapour Water Humidity •UNINA
Air
Soil In Gas phase
Particles
Dissolved in water

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Soil
T
Particles Air
“Marie Curie” Research Training Network

GU
DU
Mass densities in medium

ENPC
Water
(
ρma =nρa 1 − Srw + HSrw )
UNITN
Air
(dry) Air ρmw = θ w ρ w + ( n − θ w ) ρ v
Dissolved
Air
Water Vapour Water
UPC
Humidity ( )
ρmw = nSrw ρ w + n 1 − Srw ρ v
UNINA

ρms = (1-n ) ρs
Soil
Particles

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

Mechanics of Unsaturated Soils for Engineering

•Mass balance equations for different phases

Three equations
GU for three Phases
•Momentum and energy
DUbalance equation for medium

Two equations for medium

•Constitutive laws for the different constituents

ENPC
•Generalized Darcy law for advective liquid water and dry gas
•Philip and de Vries law forUNITN
water vapour
•Mechanical constitutive laws
UPC
•Conductive, advective and latent heat transfer
UNINA
•Water retention curve (n and Sr changes)
•State equations of liquid and gas density (ρl and ρg )
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 Solid skeleton equations
¾ Equilibrium equation :
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- SaturatedGU (σ ' ij − δ ijbp w ), j + b i = 0

- Unsaturated DUij − δ ijp g ), j + p g,i + b i = 0
(σ
¾ Constitutive relations :
Non linear thermo-elastic (hyperbolic)
ª Dry ENPC
ª Saturated (Thermal state surface notion of ‘ e ’)
UNITNnotion of e and Sr)
ª Unsaturated (Thermal state surface

Thermo-elastic-plastic
ª Dry UPC
UNINA
ª Saturated (Thermal loading surface)
ª Unsaturated ( modified Barcelona model)

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Thermal non linear elastic model of unsaturated soil

¾ Constitutive d(σ ij − δ ij p g ) = Ddε − Fd(p g − p w ) − CdT

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laws 1 ∂e
F=GU
DDs−1 , Ds−1 = β sm , β s =
1 + e ∂ (p g − p w )
DU
1 ∂e
C = DD t−1 , D t−1 = β t m , β t =
1 + e ∂T
1 ª Thermal state surface concept :

ENPC- Void ratio (Gatmiri, 1997):

T0 0.8

0.6

UNITN1 + e0
T>T0 e

e= −1
0.4
σ − pg σ − p g p g − p w 1− m
[a e ( ) + b e (1 − )( )]
p atm σc p atm
0.2
exp[ ]exp[c e (T − T0 )]
UPC
0

2.5 K b (1 − m)
UNINA
0
Succion H kPaL
5
0
2.5
7.5

10
7.5
5

Contrainte HkPaL
Ö Compatibility with the other equations of
non linear elastic model
10

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Thermal non linear elastic model of unsaturated soil

Degree of saturation state surface :

Stress dependent
Suction dependent
1

T0 9 Different propositions
0.5
Sr isothermes :
- Brooks et Corey (1964),
- van Genuchten (1980),
T>T0 0
0 - Lloret et Alonso (1985),
10
2.5
- Gatmiri (1994)
Contrainte HkPaL 5
5
Succion HkPaL
- Thomas et He (1997)
7.5

0 10
Thermal surface (Gatmiri 1997)
Sr = 1 − [a s + bs (σ − pg )][1− exp(cs (pg − p w ))]exp(ds (T − T0 ))
a s , bs , cs , d s = constants

En modifiant les paramètres ÖModéliser différents types de sol

 Thermo-elastic-plastic model for
unsaturated soils, theoretical aspects
Gatmiri and Jenab 2000

 4 components of model :

-Elastic properties
- Loading surface
- Hardening rule
- Plastic potential

BG
 Thermo-elastic-plastic model for
unsaturated soils, theoretical aspects

9 1/4 : Elastic properties

⎧ * ⎛ ki 2 ⎞
⎡p p ⎤ ⎫ ks ⎛ s + Patm ⎞
Ω = ⎨(α 0 + α 2 ∆T )∆T ( p − pg ) + ⎜⎜
'
+ α1∆T + α 3 ∆T ⎟⎟ pg ⎢ ( Ln − 1) + 1⎥ ⎬ − Ln⎜⎜ ⎟
⎟
⎩ ⎝ 1 + e0 ⎠ ⎣ pg pg ⎦ ⎭ 1 + e0 ⎝ sg + Patm ⎠

∂ Ω '

and ε ve = ' ⇒
∂p

⎡ ki ⎤ dp ⎛ ks 1 ⎞ ⎡ * p⎤
dεv = ⎢
e
+ (α1 +α3∆)∆T ⎥ + ⎜⎜ ⎟⎟ds + ⎢(α0 + 2α2∆T ) + (α1 + 2α3∆T ) ⋅ Ln ⎥dT
⎣1+ e0 ⎦ p ⎝1+ e0 s + Patm ⎠ ⎣ pg ⎦

kT
2 e
ε qe = (ε 3 − ε 3e ) = 1 (q − qg )
3 3G
 Thermo-elastic-plastic model for
unsaturated soils, theoretical aspects
9 2/4 : Loading surface

f = q 2 − M 2 (p + p s )(p 0 − p ) = 0,

ps = ps (T,s) = k⋅ s⋅ f(∆T) = k⋅ s⋅ (1−αT)

λ ( 0 )− k T
⎛ pc ⎞ λ (s )− k T
p 0 = p 0 ( ε , T , s ) = p ⎜⎜ c ⎟⎟
p
v
c

⎝p ⎠
p c = p c ( ε pv , T ) = p *0 + 2( α 1∆ T + α 2 ∆ T ∆ T )

⎛
p *0 = p *0 ( ε pv , T ) = p c 0 ⋅ exp ⎜⎜
1
[e 1 ] ⎞
− e 2 + (1 + e 0 ) ε pv ⎟⎟
⎝ λ (0 ) − k T ⎠
 Thermo-elastic-plastic model for
unsaturated soils, theoretical aspects
9 2/4 : Loading Surface

Opposite effects of increase of s and T

 Thermo-elastic-plastic model for
unsaturated soils, theoretical aspects
9 3/4 : Hardening law
dp0* v 1 + e0
= dε p
= dε p

p0* λ (0) − kT λ ( 0) − k T
v v

9 4/4 : Plastic potential : associated law (g=f)

ª Knowing
- dε = dε + dε
e p

- dε = ( D e ) dσ + ( D s ) ds + ( DT ) dT
e −1 −1 −1

∂f
- dε = dλ
p

∂σ
- Condition of consistency :
∂f ∂f ∂f ∂f
dσ + ds + dT + p dε vp = 0
∂σ ∂s ∂T ∂ε v
 Thermo-elastic-plastic model for
unsaturated soils, theoretical aspects
Incremental constitutive law

dσ = (De − Dp)dε − (DeT + DeTp − DeTe

p
)dT − (Des + Desp − Dese
p
)ds

1 ⎛ ⎛ ∂f T ⎞ e ⎞ ⎛ e ∂f ⎞
D = p
⎜ ⎜ ∂σ ⎟ D ⎟ ⎜ D ⎟
H − H cr ⎝ ⎝ ⎠ ⎠⎝ ∂σ ⎠

D eT = D e DT−1
D =D D
e −1

( )
es s
p
= β =
m α α
[ + 2 ∆ + (α α+ 2 ∆ ) ( )]
p m Ds=β sm = ks 1 m
T T
D T T
T0 2 T Ln1 3 T T
g
1+e0 s+ Patm
= 1 ⎛ ∂f ⎛ ∂f ⎞ ⎞ ⎛ ∂f ⎛ e ∂f ⎞ ⎞
D H − H ⎜ ∂T ⎜ D ∂σ ⎟ ⎟ 1
p e
eT
⎝ ⎝ ⎠⎠ =
Des H −Hcr ⎜ ∂s ⎜ D ∂σ ⎟⎟
p

⎝ ⎝ T ⎠⎠
cr

⎛ ⎛ ∂f ⎞ ⎞ ⎛ ∂ f ⎞
T

= 1
D H − H ⎜ ⎜ ∂ σ ⎟ D ⎟ ⎜ D ∂σ ⎟
p e
⎛ ⎛ ∂f ⎞ ⎛ e ∂f ⎞ ⎞
= 1
Dese H −Hcr ⎜ ⎜ ∂σ ⎟ Des⎜ D ∂σ ⎟ ⎟
⎝ ⎝ ⎠ ⎠⎝ ⎠
eTe eT p
cr

⎝ ⎠ ⎝ ⎠
⎝ ⎠
 Thermo-elastic-plastic model of unsaturated soil

 Four components of the model

Loading surface
“Marie Curie” Research Training Network

-Elastic properties
GU
- Loading surface DU

- Hardening rule
- Plastic potential
ENPC
Opposite effects of increase of s and T
Incremental constitutive law UNITN

dσ = (De − Dp)dε − (DeTUPC

+ DeTp − DeTe
p
)dT − (Des + Desp − Dese
p
)ds
UNINA
1 ⎛ ⎛ ∂f ⎞ e ⎞ ⎛ e ∂f ⎞
T

D =
p
⎜ ⎜ ∂σ ⎟ D ⎟ ⎜ D ⎟
H − H cr ⎝ ⎝ ⎠ ⎠⎝ ∂σ ⎠

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Water constituent - Transfers
Vapour transfer : (Philip et de Vries, 1957)
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GU q vap = − D0 .ν .α .θa .∇ρ v

⎧ρ =ρ h
⎪ v 0 , ρ 0 = ρ 0 ( T) DU(Wan Vijk 1963, Geraminzad et Saxena, 1986)
⎨ ψg , ψ = ψ(θ, T )
⎪ h = exp(
⎩ RT
) , h = h( θ ) (Philip et de Vries, 1957)

∇ρv = h∇ρ0 + ρ0∇h

ENPC
d ρ0 dh
∇ρv = h ∇T + ρ 0
dT dθ UNITN
d ρ0 ρv g ∂ψ
qvap = − D0 .ν .α UPC
.θ a h∇T − D0 .ν .α .θ a . ∇θ
dT RT ∂θ
UNINA
q vap
V= = − DTv ∇T − Dθv ∇θ
ρw
BG Mechanics of Unsaturated Soils for Engineering (MUSE)
Mechanics of Unsaturated Soils for Engineering

Liquid phase transfer :

Generalized Darcy law
GU
qw K = K ( e, θ , T )
U= = − K ∇ (ψ + z ) DU
ρw ψ = ψ (θ ,T )
U = − K (θ , T ) ∇ψ − K (θ , T ) ∇Z σ (T )
ψ= ψ r (θ )
σr
∂ψ ∂ψ
ENPC
∇ψ (θ , T ) = ∇T + ∇θ
∂T ∂θ UNITN
ψ r (θ ) dσ (T ) σ (T ) dψ r
U = − K (θ , T ) ∇T − K (θ , T ) ∇θ − K (θ , T ) ∇Z
σr dTUPC σ r dθ
UNINA
qw
U= = − DTw ∇T − Dθw ∇θ − Dw ∇Z
ρw

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 Water constituent - Transfers
Permeability
General form :
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GU K w = k rw (S r ) ⋅ K (n ) / µ w
DU
⎛ µ w (Tr ) ⎞
d
⎛ S − S ru ⎞ 1
K w = K w 0 ⋅ 10αe ⋅ ⎜⎜ r ⎟⎟ ⎜⎜ ⎟⎟
⎝ 1 − S ru ⎠ ⎝ µ w (T ) ⎠
0.75

ENPC
with
krw
a = 1.2 ´ 10-9 ; 0.5

UNITN
α = 5.0 ;
Sru = 0.05 ; 0.25

d = 3; UPC
UNINA
0
25
THCL
0
µ = 0.6612 * (T − 229) −1.562 (kPa s) 1
0.75 75
50

0.5
(Kaye & Laybe, 1973) Sr
0.25
0
100

Kw en m/s

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Water Constituent – Mass Conservation
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→ Humidity mass conservation :

GU
∂ρ m DU
= −div (ρ w (U + V ))
∂t
ρ m = θρ w
+ (nENPC
− θ )ρ v = nS rρ w + n (1 − S r )ρ v
UNITN
θ = nS r
UPC
U : Liquid phase velocity UNINA
V : Water vapour velocity

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Air Constituent - Transfer
Air transfer :
“Marie Curie” Research Training Network

qg GU ⎛ ⎛ Pg ⎞ ⎞
Vg = = − K g ⎜⎜ ∇⎜⎜ DU ⎟ + ∇Z ⎟ 0.75

ρg γ ⎟ ⎟
⎝ ⎝ ⎠ g ⎠ krg 0.5

γg
K g = c [e(1 − Sr )]
d 0.25

µg ENPC
0
0 0.25
0
0.25 0.5
0.5
0.75 0.75
Sr 1 1 e

Air mass conservation : UNITN

∂
UPC )] = −div (ρ V ) − div (ρ HU )
[nρ a (1 − S r + HS
∂t
r a g a
UNINA

H : Henry Coefficient

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Heat - Transfer and Conservation of energy

→ Heat transfer
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[CPwρ w U + Cpvρ w V + Cpgρ g Vg ](T − T0 ) + h fg (ρ w V + ρ v Vg )

Q = −λgradT +GU
DU
λ = (1 − n )λ s + θ λ w + (n − θ )λ v
• Could be assumed under different forms :
- Sr (Thomas et al., 1994)
- γd (Kersten,
ENPC1949)

→ Energy conservation UNITN

∂ϕ
UPC
+ divQ = 0
∂t
ϕ = C (T − T ) + (n −UNINA
T 0 θ )ρ h v fg

C T = (1 − n )ρ s C PS + θρ w C Pw + (n − θ )ρ v C PV + (n − θ )ρ g C Pg

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Initial and boundary conditions of thermo-hydro-
mechanic problem

{
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Initial conditions (t = 0) u(x,t) = u0(x,t) , Pw(x,t) = Pw0(x,t)

GU
Pg(x,t) = Pg0(x,t) , T(x,t) = T0(x,t).
DU
For the domain Ω and the limits Γ

ENPC

UNITN

¾ Mechanical UPC
boundary conditions : Γ= Γσ∪Γu et Γσ∩Γu =«
¾ Boundary conditions - Liquid :
UNINA
Γ= Γh∪ΓQh et Γh∩ΓQh=«
¾ Boundary conditions - Air : Γ= Γg∪ΓQg et Γg∩ΓQg=«
¾ Thermal boundary conditions : Γ= ΓT∪ΓQT et ΓT∩ΓQT=«

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Numerical treatment of the problem
Finite Element Method
“Marie Curie” Research Training Network

 Writing the variational formulation

On the set of fieldsGUof allowable virtual displacement (V), of water pore
pressure (Qw), of the gas pressure
DU (Qg) and of temperature (T)

ÂWriting the approximate variational formulation

on the set of sub-space of V, Qw, Qg and T defined on the mesh
nodes, ENPC

 Semi-discretization in the space : UNITN

{u(x,t)} = N {u (t)} , {Pw(x,t)} =N {Pw(t)} , {Pg(x,t)} = N {Pg(x)} , {T(x,t)} = N {T(x)}.
UPC
 Time discretization by Θ-method. UNINA

 Final matrix form

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Final Matrix Form
“Marie Curie” Research Training Network

GU
⎡ [R ] [Cw ] [DU
C ]
g [CT ] ⎤ ⎧ ∆u ⎫
⎢[C ]
⎢ wu [Cww ] + θ∆t[K ww ] [Cwg ]+ θ∆t[K wg ] [CwT ] + θ∆t[K wT ]⎥⎥ ⎪⎪∆Pw ⎪⎪
⎬=
[ ] [C ]+ θ∆t[K ] [C ]+ θ∆t[K ] [ ]
⎢ Cgu gw gw gg gg [ ] ⎨
CgT + θ∆t K gT ⎥ ⎪ ∆Pg ⎪
⎢
[C ] + θ∆t[K ] [C ]+ θ∆t[K ]
⎢⎣ [CTu ] Tw Tw Tg Tg [CTT ] + θ∆t[K TT ]⎥⎥⎦ ⎪⎩ ∆T ⎪⎭
ENPC
⎧ ∆Fσ ⎫ ⎧ 0 ⎫
⎪θ∆t∆F ⎪
⎪ w⎪
⎪F − [K ]P − K P − [K ]T ⎪
⎪ w0 ww w 0 UNITN
wg [ ]
g0 wT 0⎪
⎬ + ∆t ⎨
⎨
⎪ θ ∆ t∆ Fg⎪
F
⎪ g0 − [
K gw Pw0 ]− K gg P[ ]
g0 − K [ ]
gT T0 ⎪
⎬
⎪⎩ θ∆t∆FT ⎪⎭ ⎪⎩ FT 0 −UPC [ ]
[K Tw ]Pw 0 − K Tg Pg 0 − [K TT ]T0 ⎪⎭
UNINA
θ-STOCK Code
(Gatmiri 1997)
BG Mechanics of Unsaturated Soils for Engineering (MUSE)
 Validation and applications of θ-stock code

• Heat transfer in dry soil - Carter et Booker (1989)

“Marie Curie” Research Training Network

•Thermomechanical behaviour
GU of a dry medium with a heat source
•Strip foundation ultimate load - Chen (1975)
DU

• Consolidation of saturated soil - Terzaghi et Frölich (1936)

• Thermoelastic consolidation - Aboustit et al. (1985)
•Thermo-hydro-mechanical response of saturated porous media -Giraud
(1993) ENPC
•Thermoelastoplastic consolidation of saturated soil- Lewis et Schrefler
(1998) UNITN
• Recharge of the aquifers with free surface with consideration of
unsaturated zone - Vauclin et al. (1979)
UPC
• Multiphase flow in the porous media - Liakopoulos (1965)
UNINA
• Mass and heat transfer in an unsaturated soil - Pollock (1986)
• Heating of an unsaturated clay sample - Villar et al. (1993)
B. Jenab Ph.D. (2000)

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Thermohydromecanical modeling of Multiphase media in θ-STOCK
(1994-2000)
“Marie Curie” Research Training Network

•First step
GU of θ-STOCK, A FINITE ELEMENT SOFTWARE
Preparation
DU
8 DRY SOIL
–NON-LINEAR ELASTIC MODEL
–THERMAL VON-MISES ELASTOPLASTIC MODEL

8SATURATED SOIL
ENPC
NON-LINEAR ELASTIC MODEL

→THERMAL VOID
UNITN
RATIO STATE SURFACE

aFINAL REPORT ( GATMIRI, 1995)

UPC
aINT. J. NUMER.UNINA
ANAL. METHODS GEOMECH (MARCH, 1997

ELASTOPLASTIC MODEL

→TEMPERATURE-DEPENDENT YEILD SURFACE

a FINAL REPORT ( GATMIRI, 1996)
aNAFEMS (1997)
 8 UNSATURATED SOIL
“Marie Curie” Research Training Network

NON-LINEAR ELASTIC MODEL

GU
→NET STRESS, SUCTION DU
AND TEMPERATURE AS STATE VARIABLES

→TWO TEMPERATURE-DEPENDENT STATE SURFACES

• Void ratio state surface

• DegreeENPC
of saturation state surface

→PHASE CHANGES
UNITN
a FINAL REPORT ( GATMIRI, MARS 1997)
aNAFEMS'99 (April 1999) , IJOE ( 2002)
a BULLETIN
UPCDE LIAISON LPC (2004)
ELASTOPLASTIC MODEL UNINA

→TEMPERATURE AND SUCTION-DEPENDENT YEILD SURFACE

a Ph.D. of B. JENAB (October 2000)

 •SECOND STEP
MODEL DEVELOPMENT IN Code_Aster ( EDF)
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1998:
GU
DEVELOPMENT OF SATURATED MODELS
NON LINEAR ELASTIC MODEL
DU
ELASTOPLASTIC MODEL

1999:
DEVELOPMENT OF UNSATURATED MODEL
NONLINEAR ELASTIC MODEL

2000-2003: ENPC
VALIDATION AND JUSTIFICATION
COMPARISON WITH: UNITN
ANALYTICAL SOLUTIONS
OTHER NUMERICAL RESULTS
UPC
EXPERIMENTAL RESULTS
ENGINEERING APPLICATIONS ( 2D and 3D)
UNINA
FICHE N° 38, In the frame of Project of
“Aval de cycle” EDF
6th KIWIR, Paris 2001
NUMGE, Paris 2002; RECIFE 2002; Tehran 2002
and …
 Thermo-hydro-chemo-mechanic behaviour of
unsaturated media (2001-2005)
“Marie Curie” Research Training Network

Master of Hosseini
GU (2001)
Ph.D. Thesis of Ghasemzadeh
DU (2002-2005)

Θ-STOCK with a new module in THCM

ENPC
5.50 5.50

5.00 5.00

4.50 34.00 4.50 34.00

UNITN
32.00 32.00
30.00 30.00
4.00 4.00
28.00 28.00
26.00 26.00
3.50 24.00 3.50 24.00
22.00 22.00
20.00 20.00
3.00 3.00

UPC
18.00 18.00
16.00 16.00
2.50 14.00 2.50 14.00
12.00 12.00

UNINA
10.00 10.00
2.00 2.00
8.00 8.00
6.00 6.00
1.50 4.00 1.50 4.00
2.00 2.00
0.00 0.00
1.00 -2.00 1.00 -2.00

0.50 0.50

0.00 0.00
0.00 0.50 1.00 1.50 2.00 2.50 3.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00

NAPL distribution contours in massif

Mechanics of Unsaturated Soils for Engineering

ACTIVITIES PERFORMED IN THE UPC

¾ Fully coupled T-H-M formulation in porous media (Olivella et al., 1994)

¾ Implementation in a finite element code CODE_BRIGHT (Olivella et al., 1996).

¾ The T-H-M formulation is performed in multiphase and multispecies approach. The

code is able of solver non-saturated multiphase flow in each medium under non-
isothermal conditions.
Vapour flux (diffusion) GAS
Heat in Heat out
Liquid phase: Water Water
water + dissolved air evaporation Heat flux condensation
(conduction and advection)

Solid phase
Liquid flux (advection) LIQUID

Heat flux (conduction and advection)

Gas phase:
dry air + water vapour
Heat flux (conduction) SOLID
Water in

Thermal and hydraulic interaction in an unsaturated porous media

subjected to hydration and to heating (Gens and Olivella, 2001).

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

GOVERNING EQUATIONS
¾ Mass balance of each specie

∂
Mass balance of solid (θ s (1 − φ ) ) + ∇ ⋅ ( js ) = 0
∂t
∂ w
Mass balance of water
∂t
(θ l Slφ + θ gw S gφ ) + ∇ ⋅ ( jlw + jwg ) = f w

∂ a
Mass balance of air
∂t
(θ l Slφ + θ ga S gφ ) + ∇ ⋅ ( jla + jag ) = f a

¾ Momentum balance for the medium ∇⋅σ + b = 0

¾ Internal energy balance for the medium

∂
∂t
( Es ρ s (1 − φ ) + El ρl Slφ + Eg ρ g S gφ ) + ∇ ⋅ (i c + jE + jE + jE ) = f E
s l g

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

GOVERNING EQUATIONS (continued)

¾ Constitutive equations
Thermal constitutive law
Heat conduction is governed by Fourier’s law: i c = − λ ∇T
Hydraulic constitutive law
The advective fluxes of fluid phase are computed by generalized Darcy’s law:
krα
qα = −K α ( ∇Pα − ρα g ) α = l, g Kα = k krα = f ( S rα )
µα
The non advective fluxes of species inside the fluid phase are computed by Fick’s law:
iαi = − Dαi ∇ωαi i = w, a α = l, g
The water retention curve is the link between the degree of saturation of the medium
and the water potential (suction):
−λ
⎛ 1
⎞
Sl − S rl ⎜ ⎛ Pg − Pl ⎞1−λ ⎟
Se = = 1+
Sls − S rl ⎜ ⎜⎝ P ⎟⎠
Van Genuchten model
⎟
⎝ ⎠
mercredi 28 juin 2006
School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

GOVERNING EQUATIONS (continued)

Mechanical constitutive law
A T-H-M constitutive equation have the contributions of strains, temperature and fluid
pressure:
• • • •
σ = Dε + f s + t T

Different constitutive models have been implemented in CODE_BRIGHT:

¾ Elasticity (with thermal and pore pressure term)
¾ Nonlinear elasticity
¾ Viscoplasticity for saline materials
¾ Viscoplasticity for granular materials
¾ Viscoplasticity for unsaturated soils based on BBM
¾ Thermoelastoplastic model for soils
¾ Damage-elastoplastic model for argillaceous rocks

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

APPLICATIONS
¾ FEBEX PROJECT

Large scale heating test mock-up (Sánchez M., 2004)

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

¾ MONT TERRI PROJECT

HEATER EXPERIMENT (HE)

T-H-M numerical simulation of a large scale heating test (Muñoz and Alonso, 2005)

BHE-0

CONCRETE FLOOR

2.00 m
RESIN PLUG

SAND BACKFILL

2.00 m
HEATER TUBE

2.00 m

HEATER
1.00 m

BUFFER
Vertical section of HE Experiment
Dismantling of the HE Experiment

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

Niche floor (z = 0.00m) Niche floor (z = 0.00m) Niche floor (z = 0.00m)

Desaturated
zone

Borehole BHE-0

Degree of saturation distribution Temperature distribution Liquid pressure distribution

Hydration Phase Heating Phase Heating Phase

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

ENGINEERED BARRIER EMPLACEMENT EXPERIMENT (EB)

H-M numerical simulation of hydration phase in EB Experiment (Hoffmann, 2005)

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

H-M numerical simulations of EB niche excavation (Muñoz et al., 2005)

Calculated damaged zone

0.6 m / [0.5 – 0.75 m]

(Schuster, 2002)

RH = 93%
(s = 10 MPa)
t = 95 days
0.3 m / [0.1 – 0.15 m]
(Schuster, 2002)

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

Damage-elastoplastic constitutive model for argillaceous rocks

(Vaunat, J. & Gens, A., 2003)
V de
Void → e = V ; dε v = −
VS (1 + e )
V de
Bond → e = b ; dε =− b
b V
S
vb (1 + e )

Matrix → eM =
(V + VV )
b
; dε = −
deM
VS vM (1 + e )

d ε ijM = d ε ij + d ε
ijb
↓ ↓ ↓
Matrix Composite Bond
behaviour behaviour behaviour
dσ ijM d σ ij dσ
ijb
⎛
⎜ −L ⎞
⎟
dσ ijM = (1 + χ ) dσ ij + χ dσ ; χ = χ0 ⋅e ⎝ 2 ⎠
ijb
mercredi 28 juin 2006
School "Applied Unsaturated Soils Mechanics"
Mechanics of Unsaturated Soils for Engineering

¾ FULL COUPLING T-H-C-M in Code_Bright (Guimaraes, 2002)

mercredi 28 juin 2006

School "Applied Unsaturated Soils Mechanics"
 VALIDATION TEST
MONTMORILLONITE CLAY
STEEL
“Marie Curie” Research Training Network

T0= 20 °C T = 100°C
Sr0 = 0.5 GU
GU
e0 = 0.72 DU
DU

14.6 cm
T °C
ENPC
ENPC
UNITN
UNITN

UPC
UPC UNINA
T = 28°C UNINA
15. cm

ALL BOUNDARIES ⇒ NO MOISTURE AND AIR TRANSFER

EXCEPT AIR THROUGH THE UPPER BOUNDARY
 NUMERICAL RESULTS
TEMPERATURE CONTOURS (°C)
“Marie Curie” Research Training Network

0.14
GU
0.12
DU 59 34 30

100
95
90
0.10 85
80
75

0.08 ENPC 70
65
60
UNITN
32 29 30
55
0.06 50
45
UPC 40
35
UNINA
0.04
30
28
25
0.02

29 28 28
0.00
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
 NUMERICAL RESULTS
DEGREE OF SATURATION CONTOURS
“Marie Curie” Research Training Network

0.14
GU
DU .47 .47 .47
0.12

0.525
0.520
0.10
0.515
.48 .48 .48
0.510
ENPC 0.505
0.500
0.08
0.495
UNITN.48 .50
0.490
.50
0.485
0.06 0.480

UPC 0.475
0.470
UNINA 0.465
0.04 .52 .52 .51
0.460
0.455
0.450
0.02 0.445

.51 .53 .52

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15

 NUMERICAL RESULTS
VOID RATIO CONTOURS
“Marie Curie” Research Training Network

0.14
GU
DU
.76 .76 .77

0.12
0.785

0.780
0.10 .74 .74 .74 0.775

ENPC
0.770

0.765
0.08 0.760
.72
UNITN
.76 .72
0.755

0.750
0.06
0.745

UPC .71 .72

0.740

UNINA
.71 0.735
0.04
0.730

0.725

0.02
.73 .70 .71

0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15

 fiche n° 38, project of “AVAL DE CYCLE” EDF

Saturated engineering and geologic barriers Modelling

“Marie Curie” Research Training Network

GU non linear elastic and elastoplastic

Linear elastic,
DU
1: Axisymetric modelling of a two-materiel case
• 200 m of length 1000 m of length
2: Coupled THM analysis, two dimensional in plane strain
ENPC
Excavation steps,
Lining modelling, UNITN

Stability analysis of galleries and the disposal wells

UPC
during:
UNINA
construction period,
thermal loading of massif

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 fiche n° 38, project of “AVAL DE CYCLE” EDF
“Marie Curie” Research Training Network

Unsaturated engineering and geologic barriers Modelling

GU
DU
Linear elastic and non linear elastic

1:Axisymetric modelling of a bi-materiel case

ENPC

2:Axisymetric modelling of geologic

UNITNbarrier with
excavation and thermal loading
3:Axisymetric modellingUPC
of geologic barrier with
excavation, placing the engineeredUNINA
barrier and thermal
loading

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Coupled THM analysis, Plane strain case
“Marie Curie” Research Training Network

GU
DU

ENPC

UNITN

UPC
UNINA

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Horizontal displacement contours
“Marie Curie” Research Training Network

GU
DU

initial One month

ENPC

UNITN

UPC
UNINA

One year 10 years

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Horizontal displacement contours
“Marie Curie” Research Training Network

GU
DU

50 years ENPC 300 years

UNITN

UPC
UNINA

100 years 1000 years

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Pore pressure contours, non linear case
“Marie Curie” Research Training Network

GU
DU

One year ENPC 10 years

UNITN

UPC
UNINA

100 years 300 years

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 Unsaturated model, axisymetric analysis
“Marie Curie” Research Training Network

GU
DU

ENPC

UNITN

UPC
UNINA
Geometry and boundary conditions

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 6,0E-04
t= 0
t= 3 mois
5,0E-04

d é p la c e m e n t ( m )
2,5E-04
t= 6 mois
“Marie Curie” Research Training Network

1,5E-04 t= 9 mois
4,0E-04
5,0E-05 t= 1 an

GU

d é p la c e m e n t (m )
t= 5 ans
-5,0E-05
3,0E-04 0 1 2 t= 10 ans
x (m)

DU
t= 50 ans

2,0E-04 t= 100 ans

t= 300 ans

t= 500 ans
1,0E-04
t= 1000 ans

t= 3000 ans
0,0E+00

-1,0E-04
0 20 40 60 80 100 120 140 160 180 200

ENPC
x (m)

1,4E+02

140,0
1,2E+02 120,0

UNITN
t= 0
100,0

température
t= 3 mois
80,0
1,0E+02 t= 6 mois
60,0
t= 9 mois
40,0
t= 1 an
température

8,0E+01 20,0
t= 5 ans
0,0

UPC
t= 10 ans
0 1 2 3 4 5
6,0E+01 x (m ) t= 50 ans
t= 100 ans

UNINA
t= 300 ans
4,0E+01 t= 500 ans
t= 1000 ans
t= 3000 ans
2,0E+01

0,0E+00
0 20 40 60 80 100 120 140 160 180 200
x (m)

BG Mechanics of Unsaturated Soils for Engineering (MUSE)

 8,0E+07
8,0E+07
7,0E+07
7,0E+07 6,0E+07 t= 0

Succion (Pa)
5,0E+07
t= 3 mois
4,0E+07
6,0E+07 t= 6 mois
3,0E+07
“Marie Curie” Research Training Network

2,0E+07 t= 9 mois
1,0E+07
5,0E+07 t= 1 an
Succion (Pa) 0,0E+00

GU
0 0,5 1 1,5 2 t= 5 ans
4,0E+07 x (m ) t= 10 ans
t= 50 ans

3,0E+07
DU t= 100 ans
t= 300 ans
t= 500 ans
2,0E+07
t= 1000 ans
t= 3000 ans
1,0E+07

0,0E+00
0 20 40 60 80 100 120 140 160 180 200
x (m)

ENPC
1,10E+07
x=0,71 m
x=6,21 m
1,00E+07 x=12,5 m

UNITN
x=105 m
9,00E+06

8,00E+06
Succion (Pa)

7,00E+06 UPC
6,00E+06 UNINA
5,00E+06

4,00E+06

3,00E+06
0,01 0,1 1 10 100 1000 10000
tem ps (ans)
 0,00045

x=0,71 m
0,0004 x=6,21 m
x=12,5 m
0,00035 x=105 m
“Marie Curie” Research Training Network

0,0003

Déplacement (m)
0,00025
GU
DU
0,0002

0,00015

0,0001

0,00005

0
0,01 0,1 1 10 100 1000 10000

120
ENPC tem ps (ans)

x=0,71 m
x=6,21 m

100
UNITN x=12,5 m
x=105 m

80

UPC
température

UNINA
60

40

20

0
0,01 0,1 1 10 100 1000 10000
tem ps (ans)
Mechanics of Unsaturated Soils for Engineering

GU
Merci pour votre patience
DU

ENPC

UNITN

UPC
UNINA

“Marie Curie” Research Training Network