Numerical simulation of solute transport in heterogeneous porous media
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Numerical simulation of solute transport
in heterogeneous porous media
A. Beaudoin, J.-R. de Dreuzy, J. ErhelA. Beaudoin, J.-R. de Dreuzy, J. Erhel Workshop High Performance Computing at LAMSINENIT-LAMSIN, Tunisia, November 27 - December 1st, 2006
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2D Heterogeneous 2D Heterogeneous permeability fieldpermeability fieldStochastic model Y = ln(K)Stochastic model Y = ln(K)with correlation functionwith correlation function
2( ) expY YY
C
rr
31 Y
Physical modelPhysical model
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Flow modelFlow model
v = - K*grad (h)
div (v) = 0
Fix
ed
head
Fix
ed
head
Nul flux
Nul flux
Steady-state caseDarcy equationMass conservation equationBoundary conditions
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Transport modelTransport modelF
ixed
head
an
d C
=0
Fix
ed
head
an
d d
C/d
n=
0
Nul flux and C=0
Nul flux and C=0
Advection-dispersion equationBoundary conditionsInitial condition
dC/dt + div(v C - d gradC) = f
injection
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Numerical flow Numerical flow simultionssimultions
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 simulationNumerical transport simulation
Particle tracker
rZdtddttMvtMdttM 2
Many independent particlesBilinear interpolation for v
injection
d
l
vvvv
ldt m
yyxx
m
2,
,,,maxmin
2
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Examples of simulations with Examples of simulations with σσ=2=2
Pe=∞ Pe=10
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Sparse direct solverSparse direct solver
memory size and CPU time with memory size and CPU time with PSPASESPSPASES
Theory : NZ(L) = O(N logN) Theory : Time = O(N1.5)
variance = 1, number of processors = 2
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Multigrid sparse solverMultigrid sparse solver
CPU time with HYPRE/AMGCPU time with HYPRE/AMG
variance = 1, number of processors = 4residual=10-8
Linear complexity of BoomerAMG
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Transport with particle trackerTransport with particle tracker
CPU timeCPU time
Linear complexity of particle tracker
variance = 1, number of processors = 4
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Sparse linear solversSparse linear solvers
Impact of permeability varianceImpact of permeability variance
matrix order N = 106
PSPASES and BoomerAMG independent of varianceBoomerAMG faster than PSPASES with 4 processors
matrix order N = 16 106
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Particle trackerParticle tracker
Impact of permeability variance and Impact of permeability variance and correlation lengthcorrelation length
number of particles injected = 1000, Peclet number = number of processors P = 64 and matrix order N = 134.22 106
Transport CPU time increases with varianceTransport CPU time slightly sensitive to correlation length
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Particle trackerParticle tracker
Impact of Peclet number and correlation Impact of Peclet number and correlation lengthlength
number of particles injected = 2000, variance = 9.0,number of processors P = 64 and matrix order N = 134.22 106
Transport CPU time increases for small Peclet numbersTransport CPU time slightly sensitive to correlation length
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Parallel architectureParallel architecturedistributed memorydistributed memory
2 nodes of 32 bi – processors 2 nodes of 32 bi – processors (Proc AMD Opteron 2Ghz with 2Go (Proc AMD Opteron 2Ghz with 2Go
of RAM)of RAM)
Parallel architectureParallel architecture
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Parallel algorithms and Data Parallel algorithms and Data distributiondistribution
Domain decomposition into slicesGhost cells at the boundaries
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Parallel matrix generation using FFTWParallel sparse solverParallel particle tracker
Parallel algorithms and Data Parallel algorithms and Data distributiondistribution
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Direct and multigrid solversDirect and multigrid solvers
Parallel CPU timeParallel CPU time
variance = 9
matrix order N = 106 matrix order N = 4 106
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Direct and multigrid solversDirect and multigrid solvers
Speed-upSpeed-up
matrix order N = 106 matrix order N = 4 106
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Particle trackerParticle tracker
Parallel CPU timeParallel CPU time
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Flow and transport computationsFlow and transport computations
SummarySummary
• 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