Thermo hydro mechanical coupling for underground waste storage simulations Clément Chavant - Sylvie Granet Roméo Frenandes EDF R&D 1
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
Underground waste storage concepts(1/3) Alvéole CU Alvéole C 3
Underground waste storage concepts (2/3) C waste Cell Alvéole de déchet B B waste cell 4
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
Main phenomena and modelisation (1/2) Flows 2 components (air or H2 and water ) in 2 phases (liquid and gaz) Transport equations : Pressures Velocities Mgz pgz =pas pvp plq=pw pad ρ gz = 1 C vp M as ρ as C vp Darcy + diffusion + for each phase within each phase M lq ρ lq M gz ρ gz = = K int. k rel lq S lq μ lq K int. k rel gz S lq μ gz Mvp M as = F vp Cvp ρvp ρas M ad Mw = F ad ρad plq ρ lq g pgz ρ gz g { Sorption curve 6 P c=f Slq =P gz P lq Mvp ρ vp Cvp = pvp pgz phase change (dissolution /vaporization ) dpvp dpw m dt = hm h vp w ρ vp ρw T ρ ad Mol ad = pad KH
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 0-0,02-0,015-0,01-0,005 0 0,005 0,01 Swelling of Engineered materials. 7 0,015 0,02 0,025 0,03 Axial strain
Numerical difficulties (1/3) For flows Non linear terms induce hyperbolic behaviour Kind of equation : u 2 u m =0 t x2 Stiff fronts can appear m=2 m 2 m 2 Big capillary effect 8 -> No «mean» pressure
Numerical difficulties (2/3) Example of a desaturation problem Initial state : Sat=1 Porosity=0,05 Sat=0,7 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 10s 0,85 100s 0,8 1000s -0,1 1000s 4,00E+04 5000s 2,00E+04 0 0,1 0,2 X 9 100s 6,00E+04 0,7-0,2 10s 8,00E+04 5000s 0,75 1s 1,00E+05 Pgz(Pa) S 0,9-0,5-0,3 0,00E+00-0,1 0,1 X 0,3 0,5
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 σ1 σ3 peak 10
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 2 1 π=s α p S pc S ds 3 l α Incidence material deformation on flow Porosity change Straight increase of permeability with damage homogenization 11 DF? DM
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
Numerical methods for flow Choice of principal variables Capillary pressure/gas pressure Air mass balance ill conditioned for S=1 ϕ ρa 1 S t ρ a k a S p a =0 Saturation/water pressure Air mass balance becomes : ϕ 1 S pe t P c S A 1 S k a S 1 S 13 3 ϕ g S pe 0. 6 S [ pa h S S ] [ pa k a S pe ]= 0 t g S pc 2,6 h S A 1 S At S=1 We have S =0 t
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
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
CFM/DM (control finite volume/dual mesh) m u.f u ; F =k u u t Model equation Mass balance: n 1 m n 1 m K K AK Δt T KL k un 1 un 1 un 1 =0 KL L L K K Up winding H si un 1 un 1 un 1 =un 1 L K KL I L Loop over elements of primal mesh A e Τ n 1 n m m e K,e K, p,e K K Δt e L e K d I,H T = = e λ K. λ K KL dk, L e 16 T e k un 1 un 1 un 1 =0 KL KL L K J L
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
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 18 No OK
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 OMEGA σ:ε u OMEGA f. u =0 u Lak of ellipticity Possible bifurcations Instabilities 19
Regularisation method Main idea : Introduce some term bounding gradients of strain Second gradient OMEGA σ:ε u OMEGA D. ε : ε OMEGA f.u =0 u Simplified second gradient : micro gradient dilation model For a dilatant material we can regularise only the volumic strain OMEGA σ:ε u OMEGA D. Tr ε. Tr ε OMEGA f.u =0 u OMEGA D.Δu.Δu 20
Mixed formulation of micro gradient dilation model Weak formulation OMEGA σ:ε u OMEGA D. θ. θ OMEGA λ.u θ OMEGA λ.u θ OMEGA f.u =0 u,λ λ Approximation spaces u θ Quadrangles Q2 Q1 P0 Triangles P2 P1 P0 Possible free energy displacements modes OMEGA σ w :ε w =0 w e. w=0 e Two ways Enhancing degree of discretisation for Using penalisation 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 21
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 θ X Permeability : 12 10 m. s 1 Time of simulation for excavation : 17 days Time of simulation for consolidation : 10 years 22
Benchmark Momas : Simulation of an Excavation under Brittle Hydro-mechanical behaviour Mechanical elasto-plastic formulation ' σ1 Drucker-Prager Yield Criterion Space of effectives stresses ' Plastic rule : associated formulation σ2 Triaxial stress state of Drücker-Prager law for a confinement of 2 MPa 8 7 SXX-SYY (Mpa) Softening : decrease of cohesion / shear plastic deformation σ3 ' 6 5 4 3 2 1 0 0 0.005 0.01 DY 23 0.015 0.02
Benchmark Momas : Localisation Phenomenon Coupled modelling After 10 years Shear bandings 0.015m 0.15m Coupled modelling Coarse mesh Coupled modelling Refined mesh Shear bands are related to softening model Localisation bands are influenced by the mesh Hydro-mechanical coupling provides no regularisation 24
Benchmark Momas : simulation with Micro Gradient Dilation Model Spatial discretisation with triangle elements Visualisation of Shear bandings on Gauss Points during the excavation phasis 0.05m 0.015m 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 25
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 26