1 Modélisation et simulation appliquées au suivi de pollution des nappes phréatiques Jocelyne Erhel Équipe Sage, INRIA Rennes Mesures, Modélisation et Simulation INSA, Rennes, Juin 2010

J. Erhel, INRIA Rennes2 L’eau potable en Bretagne:  70% eaux de surface  Quelques captages profonds © ©Yves Chaux

J. Erhel, INRIA Rennes3

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5 Understand physical phenomena Manage water resources Prevent risks of pollution Help in remediation Objectives of groundwater numerical models

6J. Erhel, INRIA Rennes SAGE RENNES Software engineering CDCSP LYON Applied mathematics TRANSF RENNES Geophysics LMPG LE HAVRE EROH+EROH+ High Performance computing MICAS 7 great challenges in hydrogeology and a scientific software platform MICAS project

7J. Erhel, INRIA Rennes Porous Media Fracture Networks Physical equations Steady-state or transient flow Advection-diffusion Fractured Porous media H2OLab scientific software platform

8J. Erhel, INRIA Rennes Lack of observations Spatial heterogeneity Stochastic models of flow and solute transport -random velocity field -random solute transfer time and dispersivity Porous geological media fractured geological media Flow in highly heterogeneous porous medium3D Discrete Fracture Network Stochastic models

9J. Erhel, INRIA Rennes Scientific challenge Dispersion of inert solute in 2D porous media and in 3D porous media Scientific challenge Dispersion of inert solute in 2D porous media and in 3D porous media

Fixed head Nul flux injection Advection-dispersion equations Random data K log-normal and exponential correlation Flow equations Asymptotic behavior of dispersion coefficients ? Impact of heterogeneity factor , correlation length λ and Peclet number Pe ? 10 Macro dispersion in heterogeneous porous media

11J. Erhel, INRIA Rennes Macro dispersion Pure convection Articles WRR 2007 and WRR 2008 Macro dispersion Pure convection Articles WRR 2007 and WRR /20

12J. Erhel, INRIA Rennes Effect of molecular diffusion Articles WRR 2007 and WRR 2008 Effect of molecular diffusion Articles WRR 2007 and WRR /20

13J. Erhel, INRIA Rennes Effect of hydrodynamical local dispersion Article WRR 2010 Effect of hydrodynamical local dispersion Article WRR /20

3D Simulations work in progress 3D Simulations work in progress 2/20 14J. Erhel, INRIA Rennes

15J. Erhel, INRIA Rennes Scientific challenge Flow numerical model in 3D fracture networks Scientific challenge Flow numerical model in 3D fracture networks

16J. Erhel, INRIA Rennes Site of Hornelen, Norwegen Fractures exist at any scale with no correlation Fracture length is a parameter of heterogeneity Natural fractured media

a=2.5 a=3.5 a=4.5 17J. Erhel, INRIA Rennes Generated Discrete Fracture Networks

18J. Erhel, INRIA Rennes Physical equations impervious matrix Poiseuille’s law and mass continuity in each fracture Continuity of hydraulic head h and flux V.n at each intersection Spatial discretization conforming mesh mixed hybrid finite element method easy to apply interface conditions Numerical model with conforming mesh Article SISC 2009 Numerical model with conforming mesh Article SISC 2009

19J. Erhel, INRIA Rennes Interface conditions written using mortar spaces Geometrically conforming intersections Slave side and master side Hydraulic head on slave side is L2 projection of hydraulic head on master side Mass continuity through an intersection edge Geometrically non conforming intersections Intersections partly common to more than 2 fractures (because of projections) Several unknowns for each intersection edge: intersection, master, slave Relations between these unknowns Numerical model with non conforming mesh Article Applicable Analysis 2010 and preprint submitted and preprint submitted Numerical model with non conforming mesh Article Applicable Analysis 2010 and preprint submitted and preprint submitted

20J. Erhel, INRIA Rennes Hydraulic head with conforming and non conforming mesh Hydraulic head with conforming and non conforming mesh

21J. Erhel, INRIA Rennes Scientific challenge High Performance Computing Scientific challenge High Performance Computing

22J. Erhel, INRIA Rennes Longitudinal dispersion Transversal dispersion Each curve represents 100 simulations on domains with 67.1 millions of unknowns high performance computing is required Large scale simulations with 2D porous media

23J. Erhel, INRIA Rennes solute transport: particle tracker flow: sparse linear solver Parallel performances Articles PARCO 2006, EUROPAR 2007 Parallel performances Articles PARCO 2006, EUROPAR 2007

24J. Erhel, INRIA Rennes Sparse linear solver for 3D domains Work in progress Sparse linear solver for 3D domains Work in progress 3D porous media: AMG3D fracture networks : PCG+AMG

25J. Erhel, INRIA Rennes Scientific challenge Stochastic simulations Scientific challenge Stochastic simulations

26J. Erhel, INRIA Rennes For j=1,…,N s End For Monte-Carlo simulations For j=1,…,N s Compute V(  j,x) using a finite volume method generate permeability field K(  j,x) using a regular mesh Compute D(  j,t) using a random walker method End For Spread of mass: E[S(ω,t)] ≈1/N s  j S(  j,t)

27J. Erhel, INRIA Rennes Several assumptions of regularity Error E = E[S(ω,t)] - 1/N s 1/N p  j  k (X k -X) (X k -X) T || E || ≤ C (1/ √Ns + 1/ √Np + ∆t + ∆x |ln(∆x)|) Work by J. Charrier and A. Debussche Theoretical convergence analysis Article submitted to SINUM Theoretical convergence analysis Article submitted to SINUM

28J. Erhel, INRIA Rennes Pure advectionPe=1000 Numerical convergence analysis Article in preparation Numerical convergence analysis Article in preparation Fast convergence of Monte Carlo in the ergodic case

29J. Erhel, INRIA Rennes Conclusion and perspectives

30J. Erhel, INRIA Rennes Porous media: numerical stochastic method for flow and solute transport in large 3D heterogeneous domains Fractured media: numerical stochastic method for flow in large 3D Discrete Fracture Networks Large linear systems: use of algebraic multigrid method Uncertainty Quantification: convergence analysis of coupled flow and transport Monte-Carlo simulations High-performance computing: two-level parallelism of random simulations Software development: collaborative platform H2OLAB using software engineering tools Achievements so far

31J. Erhel, INRIA Rennes Macro-dispersion analysis in 3D porous media Upscaling rules in 3D Discrete Fracture Networks Domain decomposition methods Uncertainty Quantification methods for non ergodic fields Towards petascale and exascale computing Integration of new modules into H2OLab PerspectivesPerspectives