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A parallel scientific software for heterogeneous hydrogeoloy
Conference on Parallel CFD
Antalya, Turkey May 2007
Jocelyne Erhel INRIA Rennes
Jean-Raynald de Dreuzy Geosciences Rennes
Anthony Beaudoin
LMPG, Le Havre
Etienne Bresciani
INRIA Rennes
Damien Tromeur-Dervout
CDCSP, Lyon
Partly funded by Grid’5000
french project
From Barlebo et al. (2004)
Dispersion
Flow
Injection of tracer
Tracer evolution during one year (Made, Mississippi)
Heterogeneous permeability
Physical context: groundwater flow
Flow governed by the heterogeneous permeability
Solute transport by advection and dispersion
Physical context: groundwater flow
Spatial heterogeneity
Stochastic models of flow and solute transport
-random velocity field-random solute transfer time and dispersivity
Lack of observationsPorous geological mediafractured geological media
Flow in highly heterogeneous porous medium
3D Discrete Fracture Network
Head
Numerical modelling strategy
NumericalStochasticmodels
Simulationresults
Physical model
natural system
Simulation of flowand solute transport
Characteriz
ation of
heterogeneity
Model validation
Hydrolab scientific software
Object-oriented and modular with C++Parallel algorithms with MPIEfficient numerical librariesfree software
PHYSICAL MODELS
PorousMedia
FractureNetworks
FracturedPorous
Hydrolab platform: physical models
Physical equations
Permeability field in porous media
Simple 2D or 3D geometrySimple 2D or 3D geometrystochastic permeability fieldstochastic permeability field
finitely or infinitely correlatedfinitely or infinitely correlated
MultifractalD2=1.7
finitely correlated medium
MultifractalD2=1.4
D=1
dw=2
10 h
100 h
D=2
dw=2
Well test Interpretation
Generalized flux equation
r
hr
rr
T
t
hS wdD
D21
1
Cristalline aquifer ofPloemeur (Brittany, France)
D=1.5dw=2.8
0 250 500 750 100010-1
100
101
2=9
2=6,25
2=4
2=2.25
2=1
2=0.25
DL(t
)
tN
0 500 1000-2
-1
0
1
2 2=9
2=6.25
2=4
2=2.25
2=1
2=0.25
DT(t
N)
tN
Longitudinal dispersion Transversal dispersion
Macro-dispersion analysis
Natural Fractured Media
Fractures exist at any scale with no correlation
Fracture length is a parameter of heterogeneity
0.1 1 10 10010
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
n(l)~l-2.7
prob
abili
ty d
ensi
ty
n(l)
Fracture length l
Site of Hornelen, Norway
Discrete Fracture Networks with impervious matrix
Stochastic computational domainlength distribution has a great impact : power law n(l)=l-a
3 types of networks based on the moments of length distribution
Existing meanNo variation2 < a < 3
Existing meanExisting variationNo third moment3 < a < 4
Existing meanExisting variationExisting third momenta > 4
Output of simulations in 2D fracture networks : upscaling
Pas de réseaux
p , param ètre de percolation (échelle, densité de fractures)
Théorie de la percolation >3a
: longueur de corré lation
M odèle à deux échelles propres 2< <3a
M odèle de superposition de fractures
infinies <2a
Saturated medium: one water phaseSaturated medium: one water phase Constant density: no saltwaterConstant density: no saltwater Constant porosity and constant viscosityConstant porosity and constant viscosity Linear equationsLinear equations Steady-state flow or transient flowSteady-state flow or transient flow Inert transport: no coupling with chemistryInert transport: no coupling with chemistry No coupling between flow and transportNo coupling between flow and transport No coupling with heat equationsNo coupling with heat equations No coupling with mechanical equationsNo coupling with mechanical equations Classical boundary conditionsClassical boundary conditions Classical initial conditionsClassical initial conditions
Physical equations
Flow equations: Darcy law and mass conservationFlow equations: Darcy law and mass conservationTransport equations: advection and dispersionTransport equations: advection and dispersion
NUMERICAL METHODS
PDE solversEDO/DAE solversLinear solversParticle tracker Multilevel methods
Monte-Carlo methodUQ methodsParametric simulations
Hydrolab platform: numerical methods
Darcy law and mass conservationDarcy law and mass conservation
BoundaryBoundary conditions conditions
Given head
Nul flux
3D fracture network3D fracture network
Giv
en
Head
Giv
en
H
ead
Nul flux
Nul flux
2D porous medium2D porous medium
Steady-state flow equations
Uncertainty Quantification methods
Probabilistic framework
Given statistics of the input data,
compute statistics of the random solution
stochastic permeability field K stochastic network Ω
stochastic flow equations
stochastic velocity field
n(l)=l-a
Monte-Carlo simulations
For j=1,…,M
sample network Ωj
compute vj
sample permeability field Kj
Spatial discretization
2D heterogeneous porous mediumFinite volume and regular grid
3D Discrete Fracture NetworkMixed Finite Elements and non structured grid
Meshing a 3D fracture network
• Direct mesh : poor quality or unfeasible
• Projection of the fracture network: feasible and good quality
Mesh and flux computation in 3D fracture networks
Parallel algorithms
Domain decomposition
Parallel computing facilities
16,8 millions of unknowns in 100 seconds
with 16 processors
©INRIA/Photo Jim Wallace
Numerical model
Clusters at Irisa
Grid’5000
project
Funded by
French
Government
and Brittany
council
Discrete flow numerical model
Linear system Ax=b
b: boundary conditions and source termA is a sparse matrix : NZ coefficientsMatrix-Vector product : O(NZ) opérationsDirect linear solvers: fill-in in Cholesy factor
Regular 2D mesh : N=n2 and NZ=5NRegular 3D mesh : N= n3 and NZ=7N
Fracture Network : N and NZ depend on the geometry
N = 8181
Intersections and 7 fractures Parallel sparse linear solvers
2D heterogeneous porous medium2D heterogeneous porous medium
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
Sparse direct linear solvers
2D heterogeneous porous medium2D heterogeneous porous medium
CPU time with HYPRE/AMGCPU time with HYPRE/AMG
Linear complexity of BoomerAMG
Sparse iterative linear solvers
variance = 1, number of processors = 4residual=10-8
Flow computation in 2D porous medium
Finitely correlated permeability fieldFinitely correlated permeability field
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
parallel sparse linear solvers
2D heterogeneous porous medium2D heterogeneous porous medium
Direct and multigrid solversDirect and multigrid solvers
Parallel CPU timeParallel CPU time
variance = 9
matrix order N = 106 matrix order N = 4 106
Solute transport
Fix
ed
head
an
d C
=0
Fix
ed
head
an
d
C/
n=
0
Nul flux and C/ n = 0
Nul flux and C/ n=0
inje
ctio
n
Advection-dispersion equationBoundary conditionsInitial condition
injection
Solute transport : particle tracker
Many independent particlesBilinear interpolation for V
Homogeneous molecular diffusion Dm
Stochastic differential equationFirst-order explicit scheme
Parallel particle trackerBunch of independent particles
Subdomain decomposition
1 and Pe=1 3 and Pe=100
Solute transport in 2D heterogeneous porous media
Permeability Particles
Horizontal velocity Vertical velocity
particle tracker : convergence analysis
Np = 1000 is a good trade-off between efficiency and convergence
Longitudinal dispersion
Transversal dispersion
Pure advection Advection-diffusion with Pe=100
Grid size 8192*8192 with 32 processors
More efficient in pure advection caseSlightly more efficient with moderately heterogeneous media
particle tracker : impact of diffusion and heterogeneity
particle tracker : parallel performances
Grid size 4096*4096
Good speed-up in any configuration
Current work and perspectives
Current work
• Iterative linear solvers for 3D fracture networks
• 3D heterogeneous porous media
• Subdomain method with Aitken-Schwarz acceleration
• Transient flow in 2D and 3D porous media
• Grid computing and parametric simulations
Future work
• Porous fractured media with rock
• Site modeling
• UQ methods