1. High performance flow simulation in discrete fracture networks and heterogeneous porous media Jocelyne Erhel INRIA Rennes Jean-Raynald de Dreuzy Geosciences Rennes Anthony Beaudoin LMPG, Le Havre Damien Tromeur-Dervout CDCSP, Lyon Géosciences Rennes SIAM Conference on Computational Geosciences Santa Fe March 2007

2. Physical context: groundwater flow Spatial heterogeneity Stochastic models of flow and solute transport -random velocity field -random solute transfer time and dispersivity Lack of observations Porous geological media fractured geological media Flow in highly heterogeneous porous medium 3D Discrete Fracture Network

3. Head Numerical modelling strategy Numerical Stochastic models Simulation results Physical model natural system Simulation of flow and solute transport Characterization of heterogeneity Model validation

4. Natural Fractured Media Fractures exist at any scale with no correlation Fracture length is a parameter of heterogeneity n(l)~l -2.7 probability density n ( l ) Fracture length l Site of Hornelen, Norwegen

5. Discrete Fracture Networks with impervious matrix Stochastic computational domain length distribution has a great impact : power law n(l) = l - a 3 types of networks based on the moments of length distribution  mean  variation 2 < a < 3  mean  variation  third moment 3 < a < 4  mean  variation  third moment a > 4

6. Permeability field in porous media Simple 2D or 3D geometry stochastic permeability field finitely or infinitely correlated Multifractal D 2 =1.7 finitely correlated medium Multifractal D 2 =1.4

7. Output of simulations in 2D fracture networks : upscaling

8.  Darcy law and mass conservation v = - K*grad (h) in Ω div (v) = f in Ω  Boundary conditions Given head Nul flux 3D fracture network Given Head Nul flux 2D porous medium Flow equations

9. Uncertainty Quantification methods Probabilistic framework Given statistics of the input data, compute statistics of the random solution stochastic permeability field Kstochastic network Ω stochastic flow equations stochastic velocity field

10. Monte-Carlo simulations For j=1,…M sample network Ωj compute vj sample permeability field Kj

11. Spatial discretization 2D heterogeneous porous medium Finite volume and regular grid 3D Discrete Fracture Network Mixed Finite Elements and non structured grid

12. Meshing a 3D fracture network Direct mesh : poor quality or unfeasible Projection of the fracture network: feasible and good quality

13. Mesh and flux computation in 3D fracture networks

14. Discrete flow numerical model Linear system Ax=b b: boundary conditions and source term A is a sparse matrix : NZ coefficients Matrix-Vector product : O(NZ) opérations Direct linear solvers: fill-in in Cholesy factor Regular 2D mesh : N=n 2 and NZ=5N Regular 3D mesh : N= n 3 and NZ=7N Fracture Network : N and NZ depend on the geometry N = 8181 Intersections and 7 fractures

15. 2D heterogeneous porous medium memory size and CPU time with PSPASES Theory : NZ(L) = O(N logN)Theory : Time = O(N 1.5 ) variance = 1, number of processors = 2 Sparse direct linear solvers

16. 3D fracture network memory size and CPU time with PSPASES NZ(L) = O(N) ?Time = O(N) ? Theory to be done Sparse direct linear solvers

17. 2D heterogeneous porous medium CPU time with HYPRE/AMG variance = 1, number of processors = 4 residual=10 -8 Linear complexity of BoomerAMG Sparse iterative linear solvers

18. Flow computation in 2D porous medium Finitely correlated permeability field Impact of permeability variance matrix order N = 10 6 PSPASES and BoomerAMG independent of variance BoomerAMG faster than PSPASES with 4 processors matrix order N = 16 10 6

19. Parallel algorithms Domain decomposition

20. parallel sparse linear solvers 2D heterogeneous porous medium Direct and multigrid solvers Parallel CPU time variance = 9 matrix order N = 10 6 matrix order N = 4 10 6

21. Current work and perspectives Current work Iterative linear solvers for 3D fracture networks 3D heterogeneous porous media Subdomain method with Aitken-Schwarz acceleration Transient flow in 2D and 3D porous media Solute transport in 2D porous media Grid computing and parametric simulations Future work Porous fractured media with rock Well test interpretation Site modeling UQ methods