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Numerical simulation of solute transport in heterogeneous porous media A. Beaudoin, J.-R. de Dreuzy, J. Erhel Workshop High Performance Computing at LAMSIN ENIT-LAMSIN, Tunisia, November 27 - December 1st, 2006

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2D Heterogeneous permeability field Stochastic model Y = ln(K) with correlation function Physical model

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Flow model v = - K*grad (h) div (v) = 0 Fixed head Nul flux Steady-state case Darcy equation Mass conservation equation Boundary conditions

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Transport model Fixed head and C=0 Fixed head and dC/dn=0 Nul flux and C=0 Advection-dispersion equation Boundary conditions Initial condition dC/dt + div(v C - d gradC) = f injection

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Numerical flow simultions Finite Volume Method with a regular mesh ; N =Nx Ny cells Large sparse structured matrix A of order N with 5 entries per row Linear system Ax=b

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Numerical transport simulation Particle tracker Many independent particles Bilinear interpolation for v injection

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Examples of simulations with σ=2 Pe= Pe=10

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Sparse direct solver memory size and CPU time with PSPASES Theory : NZ(L) = O(N logN)Theory : Time = O(N 1.5 ) variance = 1, number of processors = 2

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Multigrid sparse solver CPU time with HYPRE/AMG variance = 1, number of processors = 4 residual=10 -8 Linear complexity of BoomerAMG

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Transport with particle tracker CPU time Linear complexity of particle tracker variance = 1, number of processors = 4

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Sparse linear solvers Impact of permeability variance matrix order N = 10 6 PSPASES and BoomerAMG independent of variance BoomerAMG faster than PSPASES with 4 processors matrix order N = 16 10 6

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Particle tracker Impact of permeability variance and correlation length number of particles injected = 1000, Peclet number = number of processors P = 64 and matrix order N = 134.22 10 6 Transport CPU time increases with variance Transport CPU time slightly sensitive to correlation length

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Particle tracker Impact of Peclet number and correlation length number of particles injected = 2000, variance = 9.0, number of processors P = 64 and matrix order N = 134.22 10 6 Transport CPU time increases for small Peclet numbers Transport CPU time slightly sensitive to correlation length

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Parallel architecture distributed memory 2 nodes of 32 bi – processors (Proc AMD Opteron 2Ghz with 2Go of RAM) Parallel architecture

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Parallel algorithms and Data distribution Domain decomposition into slices Ghost cells at the boundaries

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Parallel matrix generation using FFTW Parallel sparse solver Parallel particle tracker Parallel algorithms and Data distribution

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Direct and multigrid solvers Parallel CPU time variance = 9 matrix order N = 10 6 matrix order N = 4 10 6

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Direct and multigrid solvers Speed-up matrix order N = 10 6 matrix order N = 4 10 6

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Particle tracker Parallel CPU time

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Flow and transport computations Summary PSPASES is efficient for small matrices HYPRE-AMG and PSPASES are not sensitive to the variance HYPRE-AMG is efficient for large matrices HYPRE-AMG and PSPASES are scalable Particle tracker is sensitive to Peclet number Particle tracker is efficient transport requires less CPU time than flow for large matrices

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