1 / 32

1 rst Couplex Workshop, July 26-27 2001

1 rst Couplex Workshop, July 26-27 2001. 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

brady-day
Download Presentation

1 rst Couplex Workshop, July 26-27 2001

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 1 rst Couplex Workshop, July 26-27 2001 Simulation of the Couplex 1 test case and preliminary results of Couplex 2 H. HOTEIT 1,2, Ph. ACKERER1, R. MOSE1 1IMFS STRASBOURG 2IRISA RENNES 1. Mathematical and Numerical Models 2. Couplex 1: Final results 3. Couplex 2: Preliminary results 4. Ongoing works

  2. 1 rst Couplex Workshop, July 26-27 2001MATHEMATICAL MODELS q : specific discharge (Darcy velocity [L/T]) K : permeability tensor of the porous medium [L/T] D : dispersion tensor defined by : Dm : molecular diffusion coefficient [L2/T], I : the unit tensor [-], aL,aT : longitudinal and transversal dispersivity [L].

  3. RT0 basis functions defined by : j=1,...,nf Aj : face (3D) or edge (2D) of E nAj : outward normal vector to Aj 1 rst Couplex Workshop, July 26-27 2001NUMERICAL METHODS : Mixed Finite Element • 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 The flux is given by :

  4. Darcy’s law discretization with with hE : average head over E, ThAi : average head over the face or edge i. Continuity discretization Fluxes preservation 1 rst Couplex Workshop, July 26-27 2001NUMERICAL METHODS : Mixed Finite Element

  5. 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 1 rst Couplex Workshop, July 26-27 2001NUMERICAL 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, mEi : linear basis functions, nn : number of nodes per element. • The scheme is fully explicit and second order in time (Runge-Kutta scheme).

  6. Step 1 : • : the flux through A, positive if pointed outside • : norm of A (length, surface). • Step 2 : • : depending on the sign of 1 rst Couplex Workshop, July 26-27 2001NUMERICAL METHODS : Discontinuous Finite Element

  7. min(E)/max (E) : min/max value of over each element which has a common node with E. min(i)/max (i) : min/max of over each element containing i Constraints : Extrema : Optimization : then E 1 rst Couplex Workshop, July 26-27 2001NUMERICAL METHODS : Discontinuous Finite Element Step 3 : stabilization with a slope limiting procedure

  8. 1 rst Couplex Workshop, July 26-27 2001COUPLEX I • Main difficulties : • - flatness of the domain (Lx=25000 m et Ly=695 m) ; • - high hydraulic conductivity contrasts (about 106 ) ; • - high dispersivity coef. contrasts (higher than 103) ; • - high computational accuracy (C < 10-12 with max (CSOURCE ) = 0.0015) ; • - boundary conditions which generate a flow pattern • which is not // to the geological structures • - very long time predicition.

  9. 1 rst Couplex Workshop, July 26-27 2001COUPLEX 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 Mesh N. elements N. nodes N. unknowns DX DY M1 2241 2352 4592 540 m 3 m M2 13608 13843 27450 100 m 3 m M3 54432 54901 109332 50 m 2 m

  10. 1 rst Couplex Workshop, July 26-27 2001COUPLEX I : Discretization Mesh M2 : NE= 1360 NN = 13843 NU= 27450

  11. 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 1 rst Couplex Workshop, July 26-27 2001COUPLEX I - FLOW SIMULATION Double precision : Tol. PCG : 10-14. Average MBE : 1.6 10-11 m3/j Max. MBE : 1.0 10-9 m3/j. Average RQmin : 64 . Quadruple precision : Tol. PCG : 10-30. Average MBE : 10-29 m3/j Max. MBE : 3.10-27 m3/j. Average RQmin : 10-24 and Max RQmin 10-20

  12. 1 rst Couplex Workshop, July 26-27 2001COUPLEX I - FLOW SIMULATION- Head distribution and pathlines

  13. 1 rst Couplex Workshop, July 26-27 2001COUPLEX I Vertical velocity profiles

  14. 0.01 0.001 1.e-6 1.e-7 1.e-12 1 rst Couplex Workshop, July 26-27 2001COUPLEX I - FLOW Darcy ’ s velocity norm min=3 10-9 m/y

  15. 1000 10 5 4 3 2.5 2 1.5 1 0.5 Grid Peclet number distribution : 1 rst Couplex Workshop, July 26-27 2001COUPLEX I - IODE

  16. 1 rst Couplex Workshop, July 26-27 2001COUPLEX I

  17. 1 rst Couplex Workshop, July 26-27 2001COUPLEX I

  18. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : Domain Boundary conditions : - Periodic for vertical faces - Dirichlet for horizontal faces - Fourier at alveoli (red) Simulated elements - silica - Cesium

  19. Fourrier type boundary conditions on the glass-bentonite interface rm : precipitate concentration (M/L3) nm : precipitation speed (L/T) Sm : solubility (M/L3) 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : MATHEMATICAL MODELS Silica Transport Model rp : precipitate concentration (M/L3) np : precipitation speed (L/T) lp : inverse of the specific surface (L) Sp : solubility (M/L3) fp : porosity (-) Kds : partition coefficient (L3/M) rsol : solid density (M/L3)

  20. Fourrier type boundary conditions on the glass-bentonite interface 1 rst Couplex Workshop, July 26-27 2001COULEX II : MATHEMATICAL MODELS Cesium Transport Model Rm : precipitate concentration (M/L3) nm : precipitation speed (L/T) Sm : solubility (M/L3) l0 : degradation coef. (T-1) f : porosity (-) : initial number of moles of silica

  21. Fourier type boundary conditions on the glass-bentonite interface t: 10-2 mol/m2/year Fourier type boundary conditions on the glass-bentonite interface rm : precipitate concentration (M/L3) nm : precipitation speed (L/T) Sm : solubility (M/L3) 1 rst Couplex Workshop, July 26-27 2001COUPLEX II - Preliminary calculation Assumption : A1 : Cs = 0.54 mol/m3 (saturation, instantaneous precip.) A2 : Cs = 0.099 mol/m3 (initial concentration) A1. is reasonnable, A2. gives an underestimate of the dissolution time For one alveole : A1 : Input flux = 2.15 mol/m2/year Silica dissolved after : 42 500 years A2 : Input flux = 5.54 mol/m2/year Silica dissolved after : 16 500 years

  22. Estimation of the non linearity: Inflow Initial number of moles of Silica : f(t) ? 1 rst Couplex Workshop, July 26-27 2001COUPLEX II - Preliminary calculation Cesium Transport Model (In the buffer only) Assumptions: A1 : N0(t)=N0 A2 :Silica flux : 5.54 10-2 mole/m2/year A3 : No out-fluxes Maximum concentration of Cesium : C0 = 0.2 10-3 mol/m3 k = 0.15 m3/mol WEAK NON LINEARITY

  23. 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: Numerical strategy : Iterate until : Check if : The residual due to the solver is defined by: Replacing xk+1 by xk+Dx : 1 rst Couplex Workshop, July 26-27 2001COUPLEX II - Numerical model

  24. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II - Fluxes at Fourier boundary

  25. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : CESIUM Concentration distribution at T = 100 years

  26. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : CESIUM Concentration distribution at T = 1000 years

  27. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : CESIUM Concentration distribution at T = 5000 years

  28. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : CESIUM Concentration distribution at T = 10000 years

  29. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : CESIUM Concentration distribution at T = 100000 years

  30. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : CESIUM Concentration distribution at T = 106 years 3D Z=cst

  31. Total mass in the domain Flux at lower and upper boundaries Cumulative mass balance error 1 rst Couplex Workshop, July 26-27 2001COUPLEX II - Cesium distribution

  32. 1 rst Couplex Workshop, July 26-27 2001COUPLEX II : Ongoing works Ongoing works : - Extend to other nuclides - Mesh convergence study with a new elementary cell

More Related