High performance flow simulation in discrete fracture networks and heterogeneous porous media...
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Transcript of High performance flow simulation in discrete fracture networks and heterogeneous porous media...
High performance flow simulation in discrete fracture networks and heterogeneous porous media
Jocelyne Erhel INRIA Rennes
Jean-Raynald de Dreuzy Geosciences Rennes
Anthony Beaudoin
LMPG, Le Havre
Damien Tromeur-Dervout
CDCSP, Lyon
GéosciencesRennes
SIAM Conference onComputational Geosciences
Santa Fe March 2007
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
Natural Fractured Media
Fractures exist at any scale with no correlationFracture 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, Norwegen
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
mean variation2 < a < 3
mean variation third moment3 < a < 4
mean variation third momenta > 4
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
2( ) expY YY
C
rr
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
Darcy law and mass conservationDarcy law and mass conservation
v = - K*v = - K*grad (hgrad (h) in ) in ΩΩ
div (v) = f in div (v) = f in ΩΩ
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
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
Monte-Carlo simulations
For j=1,…M
sample network Ωj
compute vj
M
j
vjM
vE1
1)(
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
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
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
3D fracture network 3D fracture network
memory size and CPU time with PSPASESmemory size and CPU time with PSPASES
NZ(L) = O(N) ? Time = O(N) ?Theory to be done
Sparse direct linear solvers
2D heterogeneous porous medium2D heterogeneous porous medium
CPU time with HYPRE/AMGCPU time with HYPRE/AMG
variance = 1, number of processors = 4residual=10-8
Linear complexity of BoomerAMG
Sparse iterative linear solvers
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
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 • Solute transport in 2D porous media • Grid computing and parametric simulations
Future work• Porous fractured media with rock • Well test interpretation• Site modeling• UQ methods