A. Beaudoin 1 , J.R. De Dreuzy 2 , J. Erhel 1 and H. Mustapha 1
Numerical simulation of solute transport in heterogeneous porous media A. Beaudoin, J.-R. de Dreuzy,...
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Transcript of Numerical simulation of solute transport in heterogeneous porous media A. Beaudoin, J.-R. de Dreuzy,...
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
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
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
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
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
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
Examples of simulations with Examples of simulations with σσ=2=2
Pe=∞ Pe=10
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
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
Transport with particle trackerTransport with particle tracker
CPU timeCPU time
Linear complexity of particle tracker
variance = 1, number of processors = 4
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
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
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
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
Parallel algorithms and Data Parallel algorithms and Data distributiondistribution
Domain decomposition into slicesGhost cells at the boundaries
Parallel matrix generation using FFTWParallel sparse solverParallel particle tracker
Parallel algorithms and Data Parallel algorithms and Data distributiondistribution
Direct and multigrid solversDirect and multigrid solvers
Parallel CPU timeParallel CPU time
variance = 9
matrix order N = 106 matrix order N = 4 106
Direct and multigrid solversDirect and multigrid solvers
Speed-upSpeed-up
matrix order N = 106 matrix order N = 4 106
Particle trackerParticle tracker
Parallel CPU timeParallel CPU time
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