1. 1 Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha 1,2, A. Beaudoin 1, J. Erhel 1 and J.R. De Dreuzy 2 1 - IRISA – INRIA de Rennes 2 - Géosciences de Rennes

2. 2 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver - PSPASES  Iterative solver – HYPRE Results and performances Results and performances  Tests and architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

3. 3 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver - PSPASES  Iterative solver – HYPRE Results and performances Results and performances  Tests and architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

4. 4 Geometrical model Discrete fracture network 3D fracture network 2D porous media Heterogeneous hydraulic conductivity field H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

5. 5  Equations Q = - K*grad (h) div (Q) = 0  Boundary conditions Flow fluid model Fixed head Nul flux 3D fracture network Fixed head Nul flux 2D porous media H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

6. 6 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  direct solver – PSPASES  iterative solver – HYPRE Result and performances Result and performances  tests and architecture  complexity and scalability analysis with PSPASES  comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

7. 7 Numerical method and parallel matrix generation Conforming triangular mesh Regular grid Assembled by all corresponding sub-domains linear systems Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005  Mixed Hybrid Finite Element Method  Type of mesh  Global linear system

8. 8 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver – PSPASES  Iterative solver – HYPRE Results and performances Results and performances  Tests and architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005

9. 9 Parallel Linear solver  Direct solver – PSPASES  sparse matrices  symmetric positive definite matrices  distributed-memory paradigm and MPI library  P (number of processors) = 2^N  Cholesky factorization  Iterative solver – HYPRE  sparse matrices  V-cycle multi-grid algorithm (SMG)  distributed-memory paradigm and MPI library H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

10. 10 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver – PSPASES  Iterative solver - HYPRE Results and performances Results and performances  Parallel architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005

11. 11 Parallel architecture distributed memory 2 nodes of 32 bi – processors (Proc AMD Opteron 2Ghz with 2Go of RAM) Parallel architecture H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

12. 12 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver – PSPASES  Iterative solver - HYPRE Results and performances Results and performances  Parallel architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005

13. 13 Test 1 : CPU time of matrix generation, linear solving and flow computation obtained with two processors Test 1 : CPU time of matrix generation, linear solving and flow computation obtained with two processors H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media Complexity analysis with PSPASES

14. 14 Test 2 : Memory requirements for both applications Test 2 : Memory requirements for both applications H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media Complexity analysis with PSPASES

15. 15 Scalability analysis with PSPASES Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005 PNTpR 22,6 10 5 5.601.20 10 6 81,05 10 6 11.331.18 10 6 324,19 10 6 25.701,04 10 6 42,6 10 5 2.921.15 10 6 161,05 10 6 6.061.11 10 6 644,19 10 6 13.081,05 10 6 PNTpR 22,6 10 5 13.1010006 81,05 10 6 22.065942 324,19 10 6 38.41341 42,6 10 5 7.9416508 161,05 10 6 16.054083 644,19 10 6 No value 2D medium 3D fracture network Test 1 : Efficiency Test 1 : Efficiency

16. 16 Scalability analysis with PSPASES Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005 Test 2 : Speed-up Test 2 : Speed-up

17. 17 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver - PSPASES  Iterative solver - HYPRE Results and performances Results and performances  Parallel architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005

18. 18 Comparison between PSPASES and HYPRE  Test 1 : Linear solving time H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media PSPASESHYPRE

19. 19 Comparison between PSPASES and HYPRE  Test 2 : Speed-up HYPRE PSPASES H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

20. 20 Outline Problem presentation Problem presentation  Geometrical and flow fluid model  Numerical method and parallel matrix generation Parallel linear solver Parallel linear solver  Direct solver – PSPASES  Iterative solver – HYPRE Results and performances Results and performances  Parallel architecture  Complexity and scalability analysis with PSPASES  Comparison between PSPASES and HYPRE Conclusions and future work Conclusions and future work H. Mustapha INRIA, Parco sept 2005 Parallel Simulations of Underground Flow in Porous and Fractured Media

21. 21 Conclusions  Conclusions Future work  Future work Parallel Simulations of Underground Flow in Porous and Fractured Media H. Mustapha INRIA, Parco sept 2005 Direct solver PSPASES Iterative solver HYPRE Scalable in the case of 2D medium Not scalable in the case of 3D fracture network A 3D extension of parallel software in the case of 2D porous media Parallel software for simulating solute migration Efficient for large linear systems