Thermo hydro mechanical coupling for underground waste storage - - PowerPoint PPT Presentation

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Thermo hydro mechanical coupling for underground waste storage - - PowerPoint PPT Presentation

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Thermo hydro mechanical coupling for underground waste storage simulations Clment Chavant - Sylvie Granet Romo Frenandes EDF R&D 1 december 20 , 2006 Journe Momas Outline Underground waste storage concepts Main

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december 20 , 2006 Journée Momas

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Clément Chavant - Sylvie Granet – Roméo Frenandes EDF R&D

Thermo hydro mechanical coupling for underground waste storage simulations

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Outline

  • Underground waste storage concepts
  • Main phenomena and modelisation
  • Coupling
  • Numerical difficulties
  • Spatial discretisation for flows and stresses
  • Simulation of the excavation of a gallery
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Underground waste storage concepts(1/3)

Alvéole C Alvéole CU

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Underground waste storage concepts

(2/3)

  • C waste Cell
  • Alvéole de déchet B
  • B waste cell
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Underground waste storage concepts

(3/3)

  • Complex geometry
  • Heterogeneous materials

Rock at initial state or damaged rock Concrete Engineered barriers (sealing, cells closure) Fill materials Gaps Steel : container liners

  • Different physical behaviour

Mechanical, thermal , chemical, hydraulic

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Main phenomena and modelisation (1/2)

  • Flows

 2 components (air or H2 and water ) in 2 phases (liquid and gaz)  Transport equations :

  • Pressures

Velocities

  • Darcy

+ diffusion + phase change for each phase within each phase (dissolution /vaporization )

  • Sorption curve

pgz=paspvp plq=pwpad

Mgz ρgz =1−C vp Mas ρas Cvp Mvp ρvp Cvp=

pvp pgz

M lq ρlq = K int. klq

rel Slq

μlq −∇ plqρ lqg Mgz ρgz = K int.k gz

rel Slq

μgz −∇ pgzρgz g ¿

{¿ ¿¿

¿

Mvp ρvp − Mas ρas =−Fvp∇ Cvp Mad−Mw=−F ad∇ ρad

ρad Mad

  • l =

pad K H

dpvp ρvp = dpw ρw hvp

m −hw m dT

T

P c=f  Slq=P gz−Plq

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Main phenomena and modelisation (2/2)

  • Mechanical behaviour

Plastic and brittle behaviour or the rock Dilatance effect at rupture stage Swelling of Engineered materials .

10 20 30 40 50 60

  • 0,02
  • 0,015
  • 0,01
  • 0,005

0,005 0,01 0,015 0,02 0,025 0,03

Axial strain Lateral strain Deviatoric stress

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Numerical difficulties (1/3)

  • For flows

 Non linear terms induce hyperbolic behaviour

  • Kind of equation :
  • Stiff fronts can appear
  • Big capillary effect
  • > No « mean » pressure

∂ u ∂ t − ∂2um ∂ x2 =0

m2 m=2 m2

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Numerical difficulties (2/3)

  • Example of a desaturation problem

Initial state :

Near desaturation transition zone gas pressure tends to zero

Pression de gaz (VF_RE)

0,00E+00 2,00E+04 4,00E+04 6,00E+04 8,00E+04 1,00E+05 1,20E+05 1,40E+05

  • 0,5
  • 0,3
  • 0,1

0,1 0,3 0,5 X Pgz(Pa) 1s 10s 100s 1000s 5000s

Sat=0,7 Porosity=0,3 Sat=1 Porosity=0,05

Saturation (VF_RE)

0,7 0,75 0,8 0,85 0,9 0,95 1

  • 0,2
  • 0,1

0,1 0,2 X S 1s 10s 100s 1000s 5000s

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Numerical difficulties (3/3)

  • Instabilities due to brittle behaviour

B Pre peak damage Fractured rock Post peak damage

γe≤γ p≤γult

0γ p≤γe

σ1−σ3≤0,7 σ1−σ3 peak

~2,2 m ~1 m

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Coupling (1/2)

  • Incidence of flow on mechanical behaviour

Standard notion of pore pressure Equivalent pore pressure definition for partially saturated media

  • Taking into account of interfaces in thermodynamic formulation
  • Incidence material deformation on flow

Porosity change Straight increase of permeability with damage π=Sα pα−2 3∫Sl

1

pcSdS

homogenization

DF DM ?

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Lower pressure values Coupling (2/2)

  • Thermal evolution -> Mechanic

 Thermal expansion

  • Thermal evolution -> flow

 Changes in Viscosity, diffusivity coefficients

  • Flow, mechanic -> thermal evolution

 No effect

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Numerical methods for flow

  • Choice of principal variables

Capillary pressure/gas pressure

  • Air mass balance ill conditioned for S=1

Saturation/water pressure

  • Air mass balance becomes :

ϕ ∂ ρa 1−S  ∂t −∇ρa ka  S ∇ pa=0 P cS≈ A1−S

  • 0. 6

k aS≈1−S

3

g S≈ pc h S≈A1−S

2,6

At S=1 We have

∂ S ∂t =0

ϕ 1−S  ∂ pe ∂ t ϕg S −pe ∂ S ∂t −∇ [pahS∇ S ]−∇ [paka S∇ pe]=0

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Numerical methods for flow and mechanic : spatial discretisation

  • Goals

A stable, monotone method for flow Easy to implement in a finite element code

  • Method 1 :pressure and displacement EF P2/P1 lumped formulation
  • OK for consolidation modelling
  • OK for desaturation test (Liakopoulos)

Standard EF Lumped EF Saturation Air Pressure

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Numerical methods for flow and mechanic : spatial discretisation

  • Limitations of previous formulation

Poor quadrature rule induces lack of accuracy in stresses evaluations Instabilities appear when simulating gas injection problem

  • CFV/DM (control finite volume/dual mesh)
  • Goal

formulation VF compatible with architecture of EF software

  • Principle
  • To use primal mesh for EF formulation of mechanical equations
  • To construct a finite volume cell surrounding each node of the primal mesh
  • To write mass balance on that polygonal cell

K

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CFM/DM (control finite volume/dual mesh)

  • Model equation
  • Mass balance:
  • Up winding
  • Loop over elements of primal mesh

AK mK

n1−mK n1

t Δ ∑

L

T KLku

KL

n1u

L

n1−u

K

n1=0

∑

e∈ΤK

A

K

e mK ,e n1−mK , p,e n

t Δ ∑

e ∑ L∈e≠K

T

KL

e k u

KL

n1u

L

n1−u

K

n1=0

T

KL

e =

dI ,H dK, L =−∫e ∇ λK.∇ λ K

si u

L

n1¿u

K

n1u

KL

n1=u

L

n1

∂ mu  ∂t ∇ .F u ; F=k u∇ u

K L I J H

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CFM/DM (control finite volume/dual mesh)

  • Theoretical predictions :

stable, convergent and monotone for Delaunay meshes

  • Example
  • For stretching H/L > 4/3 no convergence is achieved

L H

Gas injection Pg 10 years H/L=1 S 10 years H/L=4/3

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CFM/DM (control finite volume/dual mesh)

  • Remark about interfaces

Differences between material properties can induce discontinuities. It is better to ensure constant properties over the control cell

OK No OK

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Numerical modelisation of brittle rocks

  • Equilibrium equation

Mechanical law of behaviour

  • Resulting weak formulation

Div σf =0

σ=F  , ε α

∫OMEGA

: σ εu¿∫OMEGA f . u¿=0 ∀u¿

  • Lak of ellipticity
  • Possible bifurcations
  • Instabilities

is not positive

∂ σ ∂ ε

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Regularisation method

  • Main idea :

 Introduce some term bounding gradients of strain

  • Second gradient
  • Simplified second gradient : micro gradient dilation model

 For a dilatant material we can regularise only the volumic strain

∫OMEGA :

σ εu

¿∫OMEGA D.∇ ε :∇ ε ¿∫OMEGA f .u ¿=0∀u ¿

∫OMEGA D. u

Δ . u Δ

¿

∫OMEGA :

σ εu

¿∫OMEGA D.∇Tr ε.∇Tr ε ¿∫OMEGAf .u ¿=0∀u ¿

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Mixed formulation of micro gradient dilation model

  • Weak formulation
  • Possible free energy displacements modes
  • Two ways

 Enhancing degree of discretisation for  Using penalisation

∫OMEGA :

σ εu

¿∫OMEGA D.∇ θ.∇ θ ¿−∫OMEGA λ∇ .u ¿−θ ¿∫OMEGA λ ¿∇.u−θ∫

OMEGA f .u

¿=0∀ u ¿ ,λ ¿

P0 P1 P2 Triangles P0 Q1 Q2 Quadrangles Approximation spaces

u

θ

λ

∫OMEGA σ w:ε w=0

∫e ∇.w=0 ∀e

λ

∫OMEGA :

σ εu

¿∫OMEGA D.∇ θ.∇ θ ¿−∫OMEGA λ∇ .u ¿−θ ¿∫OMEGA λ ¿∇.u−θr∫OMEGA ∇.u ¿−θ ¿. ∇ .u−θ ∫OMEGA f .u ¿=0

∀ u

¿,λ ¿

w

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Benchmark Momas : Simulation of an Excavation under Brittle Hydro-mechanical behaviour

  • Cylindrical cavity
  • Excavation simulation
  • Initial conditions : anisotropic state of stress (11.0MPa, -15.4MP) ;

water pressure (4.7 Mpa)

X Y

θ

Radius of cavity : 3 meters Horizontal length for calculation domain : 60 meters Vertical length for calculation domain : 60 meters Permeability : Time of simulation for excavation : 17 days Time of simulation for consolidation : 10 years

10

−12m. s−1

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Benchmark Momas : Simulation of an Excavation under Brittle Hydro-mechanical behaviour

  • Mechanical elasto-plastic formulation
  • Drucker-Prager Yield Criterion
  • Plastic rule : associated formulation
  • Softening :

decrease of cohesion / shear plastic deformation

σ1

'

σ2

'

σ3

'

Space of effectives stresses

Triaxial stress state of Drücker-Prager law for a confinement of 2 MPa

1 2 3 4 5 6 7 8 0.005 0.01 0.015 0.02 DY SXX-SYY (Mpa)

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Benchmark Momas : Localisation Phenomenon

  • Coupled modelling – After 10 years – Shear bandings

Shear bands are related to softening model Localisation bands are influenced by the mesh Hydro-mechanical coupling provides no regularisation Coupled modelling Coarse mesh Coupled modelling Refined mesh

0.15m 0.015m

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Benchmark Momas : simulation with Micro Gradient Dilation Model

Band width is always greater than 2 elements

  • > We can hope this result is independent of the spatial discretisation

Displacement / deconfinement

  • > Regularisation gives results for higher deconfinement ratio
  • > With a coarser mesh, simulation stops earlier

0.05m 0.015m Spatial discretisation with triangle elements Visualisation of Shear bandings on Gauss Points during the excavation phasis

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Conclusions

  • Simulation of nuclear waste storage requires to solve non

linear coupled equations in heterogeneous media

  • Some simulations need to solve jointly difficulties relative to

two phase flow and brittle mechanical behaviours

  • Control finite volume/dual mesh is a reliable method for

coupling darcean flows and mechanic

  • Regularised brittle models seem to be useful in coupled

(saturated) simulations

  • Simplified second gradient (Micro Gradient Dilation) model

gives reasonable results