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

Thermo hydro mechanical coupling for underground waste storage simulations Clément Chavant - Sylvie Granet – Roméo Frenandes EDF R&D 1 december 20 , 2006 Journée Momas

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 2 december 20 , 2006 Journée Momas

Underground waste storage concepts (1/3) Alvéole CU Alvéole C 3 december 20 , 2006 Journée Momas

Underground waste storage concepts (2/3)  C waste Cell  Alvéole de déchet B  B waste cell 4 december 20 , 2006 Journée Momas

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 5 december 20 , 2006 Journée Momas

Main phenomena and modelisation (1/2)  Flows  2 components (air or H 2 and water ) in 2 phases (liquid and gaz)  Transport equations : • Pressures Velocities p gz = p as  p vp M gz M as M vp  C vp = p gz  p vp =  1 − C vp   C vp ρ gz ρ as ρ vp p lq = p w  p ad • Darcy + diffusion + phase change for each phase within each phase ( dissolution /vaporization ) dp vp dp w rel  S lq  m  dT M vp M as m − h w = K int . k lq M lq  −∇ p lq  ρ lq g   h vp = − =− F vp ∇ C vp ρ lq μ lq ρ vp ρ w T ρ vp ρ as = K int . k gz rel  S lq  M gz  −∇ p gz  ρ gz g  ρ gz μ gz ρ ad p ad M ad − M w =− F ad ∇ ρ ad ¿ ol = { ¿ ¿¿ K H M ad ¿ • Sorption curve P c = f  S lq = P gz − P lq 6 december 20 , 2006 Journée Momas

Main phenomena and modelisation (2/2)  Mechanical behaviour  Plastic and brittle behaviour or the rock  Dilatance effect at rupture stage Deviatoric stress 60 50 40 30 20 10 Lateral strain Axial strain 0 -0,02 -0,015 -0,01 -0,005 0 0,005 0,01 0,015 0,02 0,025 0,03  Swelling of Engineered materials . 7 december 20 , 2006 Journée Momas

Numerical difficulties (1/3)  For flows  Non linear terms induce hyperbolic behaviour • Kind of equation : ∂ t − ∂ 2 u m ∂ u ∂ x 2 = 0 • Stiff fronts can appear m = 2 m  2 m  2 • Big capillary effect -> No « mean » pressure 8 december 20 , 2006 Journée Momas

Numerical difficulties (2/3)  Example of a desaturation problem  Initial state : Sat=0,7 Sat=1 Porosity=0,05 Porosity=0,3 Near desaturation transition zone gas pressure tends to zero Pression de gaz (VF_RE) Saturation (VF_RE) 1,40E+05 1 1,20E+05 0,95 1s 1s 1,00E+05 0,9 10s 10s Pgz(Pa) 8,00E+04 100s 0,85 100s S 6,00E+04 1000s 1000s 0,8 4,00E+04 5000s 5000s 0,75 2,00E+04 0,7 0,00E+00 -0,2 -0,1 0 0,1 0,2 -0,5 -0,3 -0,1 0,1 0,3 0,5 X X 9 december 20 , 2006 Journée Momas

Numerical difficulties (3/3)  Instabilities due to brittle behaviour ~2,2 m ~1 m B Post peak damage γ e ≤ γ p ≤ γ ult Fractured rock 0  γ p ≤ γ e Pre peak damage  σ 1 − σ 3  ≤ 0,7  σ 1 − σ 3  peak 10 december 20 , 2006 Journée Momas

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 π = S α p α − 2 1 3 ∫ S l p c  S  dS  Incidence material deformation on flow  Porosity change  Straight increase of permeability with damage ? D F D M homogenization 11 december 20 , 2006 Journée Momas

Coupling (2/2)  Thermal evolution -> Mechanic  Thermal expansion Lower pressure values  Thermal evolution -> flow  Changes in Viscosity, diffusivity coefficients  Flow, mechanic -> thermal evolution  No effect 12 december 20 , 2006 Journée Momas

Numerical methods for flow  Choice of principal variables  Capillary pressure/gas pressure • Air mass balance ill conditioned for S=1 ∂  ρ a  1 − S   −∇  ρ a k a  S  ∇ p a  = 0 ϕ ∂ t  Saturation/water pressure • Air mass balance becomes : ∂ p e ∂ t  ϕ  g  S  − p e  ∂ S ∂ t −∇ [ p a h  S  ∇ S ] −∇ [ p a k a  S  ∇ p e ] = 0 ϕ  1 − S  g  S  ≈ p c 0. 6 P c  S  ≈ A  1 − S  At S=1 2,6 3 h  S ≈ A  1 − S  k a  S  ≈  1 − S  ∂ S We have = 0 ∂ t 13 december 20 , 2006 Journée Momas

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 Standard EF Lumped EF • OK for desaturation test (Liakopoulos) Saturation Air Pressure 14 december 20 , 2006 Journée Momas

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 15 december 20 , 2006 Journée Momas

CFM/DM (control finite volume/dual mesh) ∂ m  u   Model equation ∇ . F  u  ; F = k  u  ∇ u ∂ t  Mass balance: n  1 − m K n  1 m K T KL k  u n  1   u n  1  = 0  ∑ n  1 − u A K Δ t KL L K L K  Up winding J H n  1 ¿ u n  1 u n  1 = u n  1 si u I L L K KL L  Loop over elements of primal mesh n  1 − m K , p,e n e m K ,e e k  u n  1  u n  1  = 0 ∑  ∑ e ∑ n  1 − u A T Δ t K KL KL L K e ∈ Τ K L ∈ e ≠ K d I ,H e = =− ∫ e ∇ λ K . ∇ λ K T d K, L KL 16 december 20 , 2006 Journée Momas

CFM/DM (control finite volume/dual mesh)  Theoretical predictions :  stable, convergent and monotone for Delaunay meshes  Example H L Gas injection Pg 10 years H/L=1 S 10 years H/L=4/3  For stretching H/L > 4/3 no convergence is achieved 17 december 20 , 2006 Journée Momas

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 18 december 20 , 2006 Journée Momas

Numerical modelisation of brittle rocks  Equilibrium equation Mechanical law of behaviour σ = F  , ε α  Div σ  f = 0 ∂ σ is not positive ∂ ε  Resulting weak formulation σ ε  u ¿   ∫ OMEGA f . u ¿ = 0 ∀ u ¿ ∫ OMEGA : •Lak of ellipticity •Possible bifurcations •Instabilities 19 december 20 , 2006 Journée Momas

Regularisation method  Main idea :  Introduce some term bounding gradients of strain  Second gradient σ ε  u ¿   ∫ OMEGA D . ∇ ε : ∇ ε ∫ OMEGA : ¿  ∫ OMEGA f . u ¿ = 0 ∀ u ¿  Simplified second gradient : micro gradient dilation model  For a dilatant material we can regularise only the volumic strain σ ε  u ¿   ∫ OMEGA D . ∇ Tr  ε  . ∇ Tr  ε ¿   ∫ OMEGA f . u ∫ OMEGA : ¿ = 0 ∀ u ¿ ∫ OMEGA D . u ¿ Δ . u Δ 20 december 20 , 2006 Journée Momas

Mixed formulation of micro gradient dilation model  Weak formulation ¿ , λ σ ε  u ¿   ∫ OMEGA D . ∇ θ . ∇ θ ¿ − ∫ OMEGA λ  ∇ . u ¿   ∫ OMEGA λ ¿ = 0 ∀  u ¿  ∫ OMEGA : ¿  ∇ . u − θ   ∫ ¿ − θ OMEGA f . u λ θ u Approximation spaces Quadrangles Q2 Q1 P0 Triangles P2 P1 P0  Possible free energy displacements modes w ∫ OMEGA σ  w  : ε  w  = 0 ∫ e ∇ . w = 0 ∀ e  Two ways λ  Enhancing degree of discretisation for  Using penalisation σ ε  u ¿   ∫ OMEGA D . ∇ θ . ∇ θ ¿ − ∫ OMEGA λ  ∇ . u ¿   ∫ OMEGA λ ¿  ∇ . u − θ   r ∫ OMEGA  ∇ . u ¿  .  ∇ . u − θ   ∫ OMEGA f . u ∫ OMEGA : ¿ − θ ¿ − θ ¿ = 0 ∀  u ¿  ¿ , λ 21 december 20 , 2006 Journée Momas

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) Y Radius of cavity : 3 meters Horizontal length for calculation domain : 60 meters Vertical length for calculation domain : 60 meters X θ Permeability : − 12 m . s − 1 10 Time of simulation for excavation : 17 days Time of simulation for consolidation : 10 years 22 december 20 , 2006 Journée Momas