Understand physical phenomena Manage water resources Prevent risks of pollution

1. J. Erhel, INRIA Rennes

2. J. Erhel, INRIA Rennes Objectives of groundwater numerical models • Understand physical phenomena • Manage water resources • Prevent risks of pollution • Help in remediation

3. J. Erhel, INRIA Rennes MICAS project E R O H + High Performance computing Software engineering SAGE RENNES MICAS CDCSP LYON TRANSF RENNES Applied mathematics Geophysics LMPG LE HAVRE 7 great challenges in hydrogeology and a scientific software platform http://www.irisa.fr/sage/micas

4. J. Erhel, INRIA Rennes Porous Media Fracture Networks H2OLab scientific software platform Fractured Porous media • Physical equations • Steady-state or transient flow • Advection-diffusion

5. J. Erhel, INRIA Rennes Stochastic models Porous geological media fractured geological media Lack of observations Spatial heterogeneity Stochastic models of flow and solute transport -random velocity field -random solute transfer time and dispersivity 3D Discrete Fracture Network Flow in highlyheterogeneousporous medium

6. J. Erhel, INRIA Rennes Scientific challenge Dispersion of inert solute in 2D porous media and in 3D porous media

7. Macro dispersion in heterogeneous porous media • Random dataK log-normal and exponential correlation • Flow equations Nul flux injection Fixed head Fixed head • Advection-dispersionequations Nul flux • Asymptotic behavior of dispersion coefficients ? • Impact of heterogeneity factor , correlation length λand Peclet number Pe ?

8. J. Erhel, INRIA Rennes Macro dispersion Pure convection Articles WRR 2007 and WRR 2008 18/20

9. J. Erhel, INRIA Rennes Effect of molecular diffusion Articles WRR 2007 and WRR 2008 19/20

10. J. Erhel, INRIA Rennes Effect of hydrodynamical local dispersion Article WRR 2010 20/20

11. J. Erhel, INRIA Rennes 3D Simulations work in progress 2/20

12. J. Erhel, INRIA Rennes Scientific challenge Flow numerical model in 3D fracture networks

13. J. Erhel, INRIA Rennes Natural fractured media Fractures existatanyscalewith no correlation Fracture lengthis a parameter of heterogeneity Site of Hornelen, Norwegen

14. J. Erhel, INRIA Rennes Generated Discrete Fracture Networks a=2.5 a=3.5 a=4.5

15. J. Erhel, INRIA Rennes Numerical model with conforming mesh Article SISC 2009 Physicalequations imperviousmatrix Poiseuille’slaw and mass continuity in each fracture Continuity of hydraulichead h and flux V.nateach intersection Spatial discretization conformingmesh mixed hybridfiniteelementmethod easy to apply interface conditions

16. J. Erhel, INRIA Rennes Numerical model with non conforming mesh Article Applicable Analysis 2010 and preprint submitted Interface conditions writtenusingmortarspaces Geometricallyconforming intersections Slave side and master side Hydraulichead on slave sideis L2 projection of hydraulichead on master side Mass continuitythrough an intersection edge Geometrically non conforming intersections Intersections partlycommon to more than 2 fractures (because of projections) Severalunknowns for each intersection edge: intersection, master, slave Relations betweentheseunknowns

17. J. Erhel, INRIA Rennes Hydraulicheadwith conforming and non conformingmesh

18. J. Erhel, INRIA Rennes Scientific challenge High Performance Computing

19. J. Erhel, INRIA Rennes Large scale simulations with 2D porous media Transversal dispersion Longitudinal dispersion Each curve represents 100 simulations on domains with 67.1 millions of unknowns high performance computing is required

20. J. Erhel, INRIA Rennes Parallel performances Articles PARCO 2006, EUROPAR 2007 solute transport: particletracker flow: sparselinearsolver

21. J. Erhel, INRIA Rennes Sparse linear solver for 3D domains Work in progress 3D porous media: AMG 3D fracture networks : PCG+AMG

22. J. Erhel, INRIA Rennes Scientific challenge Stochastic simulations

23. J. Erhel, INRIA Rennes Monte-Carlo simulations For j=1,…,Ns For j=1,…,Ns Compute D(j,t)using a random walker method generatepermeabilityfield K(j,x) using a regularmesh Compute V(j,x)using a finite volume method End For Spread of mass: E[S(ω,t)] ≈1/Nsj S(j,t) End For

24. J. Erhel, INRIA Rennes Theoretical convergence analysis Article submitted to SINUM Severalassumptions of regularity Error E = E[S(ω,t)] - 1/Ns 1/Npjk (Xk-X) (Xk-X)T || E || ≤ C (1/ √Ns + 1/ √Np + ∆t + ∆x |ln(∆x)|) Work by J. Charrier and A. Debussche

25. J. Erhel, INRIA Rennes Numerical convergence analysis Article in preparation Pure advection Pe=1000 Fast convergence of Monte Carlo in the ergodic case

26. J. Erhel, INRIA Rennes Conclusion and perspectives

27. J. Erhel, INRIA Rennes Achievements so far • Porous media: numericalstochasticmethod for flow and solute transport in large 3D heterogeneousdomains • Fractured media: numericalstochasticmethod for flow in large 3D Discrete Fracture Networks • Large linearsystems: use of algebraicmultigridmethod • Uncertainty Quantification: convergence analysis of coupled flow and transport Monte-Carlo simulations • High-performance computing: two-levelparallelism of random simulations • Software development: collaborative platform H2OLAB using software engineering tools

28. J. Erhel, INRIA Rennes Perspectives • Macro-dispersion analysis in 3D porous media • Upscalingrules in 3D Discrete Fracture Networks • Domaindecompositionmethods • Uncertainty Quantification methods for non ergodicfields • Towardspetascale and exascalecomputing • Integration of new modules into H2OLab