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

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

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

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

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

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

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

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

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

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

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 =

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 = Transport CPU time increases with variance Transport CPU time slightly sensitive to correlation length

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 = Transport CPU time increases for small Peclet numbers Transport CPU time slightly sensitive to correlation length

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

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

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

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

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

Particle tracker Parallel CPU time

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