1 Simulation of the Couplex 1 test case and preliminary results of Couplex 2 H. HOTEIT 1,2, Ph. ACKERER 1, R. MOSE 1 1 IMFS STRASBOURG 2 IRISA RENNES 1. Mathematical and Numerical Models 2. Couplex 1: Final results 3. Couplex 2: Preliminary results 4. Ongoing works 1 rst Couplex Workshop, July

2 q : specific discharge (Darcy velocity [L/T]) K : permeability tensor of the porous medium [L/T] D : dispersion tensor defined by : D m : molecular diffusion coefficient [L 2 /T], I : the unit tensor [-],  L,  T : longitudinal and transversal dispersivity [L]. 1 rst Couplex Workshop, July MATHEMATICAL MODELS

3 -The variable and its gradient are approximated simultaneously with the same order of convergence -The mass is conserved locally (over each element E) -They can easily handle full tensors especially for dispersion -They enforce the continuity of the fluxes across the interelement boundaries 1 rst Couplex Workshop, July NUMERICAL METHODS : Mixed Finite Element The flux is given by : RT0 basis functions defined by : j=1,...,nf A j : face (3D) or edge (2D) of E n Aj : outward normal vector to A j

4 1 rst Couplex Workshop, July NUMERICAL METHODS : Mixed Finite Element h E : average head over E, Th Ai : average head over the face or edge i. Darcy’s law discretization with Continuity discretization Fluxes preservation

5 1 rst Couplex Workshop, July NUMERICAL METHODS : Discontinuous Finite Element The hyperbolic part of the transport equation is solved by DFE. C is approximated by : : nodal value of C in element E, m Ei : linear basis functions, nn : number of nodes per element. Defining: the linear variation of C on edge/face A inside of E the linear variation of C on edge/face A outside of E The scheme is fully explicit and second order in time (Runge-Kutta scheme).

6 1 rst Couplex Workshop, July NUMERICAL METHODS : Discontinuous Finite Element Step 2 : : depending on the sign of Step 1 : : the flux through A, positive if pointed outside : norm of A (length, surface).

7 1 rst Couplex Workshop, July NUMERICAL METHODS : Discontinuous Finite Element Step 3 : stabilization with a slope limiting procedure min(i)/max (i) : min/max of over each element containing i min(E)/max (E) : min/max value of over each element which has a common node with E. E Optimization : Constraints : then Extrema :

8 1 rst Couplex Workshop, July COUPLEX I Main difficulties : - flatness of the domain (Lx=25000 m et Ly=695 m) ; - high hydraulic conductivity contrasts (about 10 6 ) ; - high dispersivity coef. contrasts (higher than 10 3 ) ; - high computational accuracy (C < with max (C SOURCE ) = ) ; - boundary conditions which generate a flow pattern which is not // to the geological structures - very long time predicition.

9 1 rst Couplex Workshop, July COUPLEX I : Discretization Domain discretization Quadrangular elements which respect the geological structure The element should not be too flat The mesh is refined in the neighborhood of the source The refinement is located inside a structure and not at the interface MeshN. elements N. nodes N. unknowns  X  Y M m3 m M m3 m M m2 m

10 1 rst Couplex Workshop, July COUPLEX I : Discretization Mesh M2 : NE= 1360 NN = 13843NU= 27450

11 1 rst Couplex Workshop, July COUPLEX I - FLOW SIMULATION Following criteria are used to study grid convergence: - head distribution - water fluxes at boundaries - pathlines and travel time with starting points at source corners - water balance for each element - dimensionless flux error defined by Double precision : Tol. PCG : Average MBE : m 3 /j Max. MBE : m 3 /j. Average RQmin : 64. Quadruple precision : Tol. PCG : Average MBE : m 3 /j Max. MBE : m 3 /j. Average RQmin : and Max RQmin

12 1 rst Couplex Workshop, July COUPLEX I - FLOW SIMULATION - Head distribution and pathlines

13 1 rst Couplex Workshop, July COUPLEX I Vertical velocity profiles

14 1 rst Couplex Workshop, July COUPLEX I - FLOW Darcy ’ s velocity norm min= m/y e-7 1.e-6 1.e-12

15 1 rst Couplex Workshop, July COUPLEX I - IODE Grid Peclet number distribution :

16 1 rst Couplex Workshop, July COUPLEX I

17 1 rst Couplex Workshop, July COUPLEX I

18 1 rst Couplex Workshop, July COUPLEX II : Domain Boundary conditions : - Periodic for vertical faces - Dirichlet for horizontal faces - Fourier at alveoli (red) Simulated elements - silica - Cesium

19 Silica Transport Model  p : precipitate concentration (M/L 3 ) p : precipitation speed (L/T) p : inverse of the specific surface (L) S p : solubility (M/L 3 )  p : porosity (-) K ds : partition coefficient (L 3 /M)  sol : solid density (M/L 3 ) Fourrier type boundary conditions on the glass-bentonite interface  m : precipitate concentration (M/L 3 ) m : precipitation speed (L/T) S m : solubility (M/L 3 ) 1 rst Couplex Workshop, July COUPLEX II : MATHEMATICAL MODELS

20 Cesium Transport Model  m : precipitate concentration (M/L 3 ) m : precipitation speed (L/T) S m : solubility (M/L 3 ) 0 : degradation coef. (T -1 )  : porosity (-) : initial number of moles of silica Fourrier type boundary conditions on the glass-bentonite interface 1 rst Couplex Workshop, July COULEX II : MATHEMATICAL MODELS

21 1 rst Couplex Workshop, July COUPLEX II - Preliminary calculation Fourier type boundary conditions on the glass-bentonite interface  m : precipitate concentration (M/L 3 ) m : precipitation speed (L/T) S m : solubility (M/L 3 ) Fourier type boundary conditions on the glass-bentonite interface  : mol/m 2 /year Assumption : A1 : Cs = 0.54 mol/m 3 (saturation, instantaneous precip.) A2 : Cs = mol/m 3 (initial concentration) A1. is reasonnable, A2. gives an underestimate of the dissolution time For one alveole : A1 : Input flux = 2.15 mol/m 2 /yearSilica dissolved after : years A2 : Input flux = 5.54 mol/m 2 /yearSilica dissolved after : years

22 Cesium Transport Model 1 rst Couplex Workshop, July COUPLEX II - Preliminary calculation Maximum concentration of Cesium : C 0 = mol/m 3  = 0.15 m 3 /mol Estimation of the non linearity: Inflow Assumptions: A1 : N 0 (t)=N 0 A2 : Silica flux : mole/m 2 /year A3 : No out-fluxes WEAK NON LINEARITY (In the buffer only) Initial number of moles of Silica : f(t) ?

23 1 rst Couplex Workshop, July COUPLEX II - Numerical model The retardation coefficent R is calculated from the concentration of the previous iteration step (fixed point method). Stopping criterion is based on the maximum value of the residual. It is linearly dependent on the change in the primary variables. For iteration k, the residual is defined by: Replacing x k+1 by x k +  x : The residual due to the solver is defined by: Numerical strategy : Iterate until : Check if :

24 1 rst Couplex Workshop, July COUPLEX II - Fluxes at Fourier boundary

25 1 rst Couplex Workshop, July COUPLEX II : CESIUM Concentration distribution at T = 100 years

26 1 rst Couplex Workshop, July COUPLEX II : CESIUM Concentration distribution at T = 1000 years

27 1 rst Couplex Workshop, July COUPLEX II : CESIUM Concentration distribution at T = 5000 years

28 1 rst Couplex Workshop, July COUPLEX II : CESIUM Concentration distribution at T = years

29 1 rst Couplex Workshop, July COUPLEX II : CESIUM Concentration distribution at T = years

30 1 rst Couplex Workshop, July COUPLEX II : CESIUM Concentration distribution at T = 10 6 years 3D Z=cst

31 1 rst Couplex Workshop, July COUPLEX II - Cesium distribution Flux at lower and upper boundaries Total mass in the domain Cumulative mass balance error

32 1 rst Couplex Workshop, July COUPLEX II : Ongoing works Ongoing works : - Extend to other nuclides - Mesh convergence study with a new elementary cell