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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

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

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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

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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

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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

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

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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

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

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

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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

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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

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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

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

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

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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

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

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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

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

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

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

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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

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