Parallel Simulations of Underground Flow in Porous and

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 1

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

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

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

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

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

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

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

Parallel Simulations of Underground Flow in Porous and Fractured Media 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 9

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

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

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

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

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

Parallel Simulations of Underground Flow in Porous and Fractured Media Scalability analysis with PSPASES Test 1 : Efficiency n 2 D medium 3 D fracture network P N Tp R 2 2, 6 105 5. 60 1. 20 106 2 2, 6 105 13. 10 10006 8 1, 05 106 11. 33 1. 18 106 8 1, 05 106 22. 06 5942 32 4, 19 106 25. 70 1, 04 106 32 4, 19 106 38. 41 341 4 2, 6 105 2. 92 1. 15 106 4 2, 6 105 7. 94 16508 16 1, 05 106 6. 06 1. 11 106 16 1, 05 106 16. 05 4083 64 4, 19 106 13. 08 1, 05 106 64 4, 19 106 H. Mustapha INRIA, Parco sept 2005 No value 15

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

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

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

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

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

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