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A. Beaudoin 1 , J.R. De Dreuzy 2 , J. Erhel 1 and H. Mustapha 1

Sparse linear solvers applied to parallel simulations of underground flow in porous and fractured media. A. Beaudoin 1 , J.R. De Dreuzy 2 , J. Erhel 1 and H. Mustapha 1. 1 - IRISA / INRIA, Rennes, France 2 - Department of Geosciences, University of Rennes , France.

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A. Beaudoin 1 , J.R. De Dreuzy 2 , J. Erhel 1 and H. Mustapha 1

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  1. Sparse linear solvers applied to parallel simulations of underground flow in porous and fractured media A. Beaudoin1, J.R. De Dreuzy2, J. Erhel1 and H. Mustapha1 1 - IRISA / INRIA, Rennes, France 2 - Department of Geosciences, University of Rennes, France Matrix Computations and Scientific Computing Seminar Berkeley, 26 October 2005

  2. Parallel Simulations of Underground Flow in Porous and Fractured Media 2D heterogeneous porous medium Heterogeneous permeability field Y = ln(K) with correlation function

  3. Parallel Simulations of Underground Flow in Porous and Fractured Media 3D fracture network with impervious matrix 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

  4. Equations Q = - K*grad (h) div (Q) = 0 Fixed head Nul flux Parallel Simulations of Underground Flow in Porous and Fractured Media Flow model • Boundary conditions 2D porous medium 3D fracture network Nul flux Fixed head Fixed head Nul flux

  5. Parallel Simulations of Underground Flow in Porous and Fractured Media Numerical method for 2D heterogeneous porous medium Finite Volume Method with a regular mesh Large sparse structured matrix with 5 entries per row

  6. Parallel Simulations of Underground Flow in Porous and Fractured Media Sparse matrix for 2D heterogeneous porous medium zoom n=32

  7. Parallel Simulations of Underground Flow in Porous and Fractured Media Numerical method for 3D fracture network Mixed Hybrid Finite Element Method with unstructured mesh Conforming triangular mesh Large sparse unstructured matrix with about 5 entries per row

  8. Parallel Simulations of Underground Flow in Porous and Fractured Media Sparse matrix for 3D fracture network zoom N = 8181 Intersections and 7 fractures

  9. Parallel Simulations of Underground Flow in Porous and Fractured Media Complexity analysis with PSPASES Memory requirements for matrices A and L

  10. Parallel Simulations of Underground Flow in Porous and Fractured Media Complexity analysis with PSPASES CPU time of matrix generation, linear solving and flow computation obtained with two processors

  11. Parallel Simulations of Underground Flow in Porous and Fractured Media 2D porous medium : memory size and CPU time with PSPASES Theory : NZ(L) = O(N logN) Theory : Time = O(N1.5) Slope about 1 Slope about 1.5

  12. Parallel Simulations of Underground Flow in Porous and Fractured Media 3D fracture network : memory size and CPU time with PSPASES Theory to be done NZ(L) = O(N) ? Time = O(N) ?

  13. Parallel Simulations of Underground Flow in Porous and Fractured Media 2D porous medium : condition number estimated by MUMPS To be ckecked : scaling or not

  14. Parallel Simulations of Underground Flow in Porous and Fractured Media 2D porous medium : residuals with PSPASES

  15. Parallel Simulations of Underground Flow in Porous and Fractured Media Parallel architecture Parallel architecture distributed memory 2 nodes of 32 bi – processors (Proc AMD Opteron 2Ghz with 2Go of RAM)

  16. Parallel Simulations of Underground Flow in Porous and Fractured Media Scalability analysis with PSPASES : speed-up

  17. Parallel Simulations of Underground Flow in Porous and Fractured Media Scalability analysis with PSPASES : isoefficiency 2D medium 3D fracture network

  18. Parallel Simulations of Underground Flow in Porous and Fractured Media 2D porous medium : number of V cycles with HYPRE/SMG

  19. Parallel Simulations of Underground Flow in Porous and Fractured Media Comparison between PSPASES and HYPRE/SMG : CPU time HYPRE PSPASES

  20. Parallel Simulations of Underground Flow in Porous and Fractured Media Comparison between PSPASES and HYPRE/SMG : speed-up PSPASES HYPRE

  21. Parallel Simulations of Underground Flow in Porous and Fractured Media Perspectives • porous medium : large sigma, up to 9 and large N, up to 108 • porous medium : 3D problems, N up to 1012 • porous medium : scaling, iterative refinement, • multigrid adapted to heterogeneous permeability field • 3D fracture networks : large N, up to 109 • model for complexity and scalability issues • 2-level nested dissection • subdomain method • parallel architectures : up to 128 processors • Monte-Carlo simulations • grid computing with clusters for each random simulation • parallel advection-diffusion numerical models

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