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 Matrix Computations and Scientific Computing Seminar Berkeley, 26 October 2005

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

3. 3D fracture network with impervious matrix Parallel Simulations of Underground Flow in Porous and Fractured Media 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  third moment 3 < a < 4  mean  variation 2 < a < 3  mean  variation  third moment a > 4

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

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

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

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

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

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

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(N 1.5 ) Slope about 1Slope about 1.5

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

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

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

17. Scalability analysis with PSPASES : isoefficiency Parallel Simulations of Underground Flow in Porous and Fractured Media PNTpR 20.26 10 6 5.601.20 10 6 81.05 10 6 11.331.18 10 6 324.19 10 6 25.701,04 10 6 40.26 10 6 2.921.15 10 6 161.05 10 6 6.061.11 10 6 644.19 10 6 13.081,05 10 6 PNTpR 20.26 10 6 13.10 81.05 10 6 22.06 324.19 10 6 38.41 40.26 10 6 7.94 161.05 10 6 16.05 644.19 10 6 No value 2D medium3D fracture network

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

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

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

21. Perspectives porous medium : large sigma, up to 9 and large N, up to 10 8 porous medium : 3D problems, N up to 10 12 porous medium : scaling, iterative refinement, multigrid adapted to heterogeneous permeability field 3D fracture networks : large N, up to 10 9 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