Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Numerical and stochastic models of flowin 3D Discrete Fracture Networks
J-R. de Dreuzy 1, J. Erhel 2, G. Pichot2, B. Poirriez 3
1CNRS, Geosciences Rennes, 2INRIA, Rennes, 3INSA, IRISA, Rennes
Special Semester on Multiscale Simulation and Analysis in Energy andthe Environment
Linz, October 2011
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Context of the SAGE team
common team to INRIA and IRISA in Rennes, Brittanyhttp ://www.irisa.fr/sage
MICAS project (funded by ANR)
collaboration with Geosciences Rennes, University of Lyon, University ofPoitiers, University of Rennes.
study of flow in fractured media and fractured porous media
study of flow and transport in heterogeneous porous media
uncertainty quantification : random transmissivty fields and random networksof fractures
software platform H2OLab, with libraries MPFRAC and PARADIS
Some other related projects
GEOFRAC, with Estime team at Inria Rocquencourt, on fractured porousmedia
GRT3D, global reactive transport model, with Andra and Momas group(nuclear waste disposal)
H2OGuilde, a user interface for H2OLab
Hydromed, with Morocco and Tunisia, on uncertainty quantification andinverse problems
Arphymat, with Archeosciences Rennes and IPR, on prehistoric fires
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Outline
1 Problem descriptionGeometryFlow equations
2 Derivation of the linear systemMixed Hybrid Finite Element MethodMortar method
3 Linear solvers
4 Domain Decomposition methodDomain decomposition of a DFNPCG applied to the Schur complementResults with Schur
5 Conclusion
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Geometry
Flow equations
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Stochastic Generation of DFN
Following a Discrete Fracture Network approach,fractures are planes with the following random properties :
Parameter Random distributionlength power lawshape disks / ellipsesposition uniformorientation uniformConductivity homogeneous
/ correlated log-normal
Example of DFN with 217 fractures
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Geometry
Flow equations
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Stochastic Generation of DFN
The broad natural fracture length distribution is modeled by a power lawdistribution (Bour et al, 2002) :
p(l)dl =1
a− 1
l−a
l−a+1min
dl ,
where p(l)dl is the probability of observing a fracture with a length in the interval[l , l + dl ], lmin is the smallest fracture length, and a is a characteristic exponent.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Geometry
Flow equations
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Flow model
Current assumptions :
The rock matrix is impervious : flow is only simulated in the fractures,
Study of steady state one phase flow,
There is no longitudinal flux in the intersections of fractures.
Flow equations within each fracture Ωf :
∇ · u(x) = f(x), for x ∈ Ωf ,
u(x) = −T (x)∇p(x), for x ∈ Ωf ,
p(x) = pD(x), on ΓD ∩ Γf ,
u(x).ν = qN(x), on ΓN ∩ Γf ,
u(x).µ = 0, on Γf \(Γf ∩ ΓD) ∪ (Γf ∩ ΓN),
ν (resp. µ ) outward normal unit vectors
T (x) a given transmissivity field (unit [m2.s−1]), f(x) ∈ L2(Ωf ) sources/sinks.
Continuity conditions in each intersection :
pk,f = pk , on Σk , ∀f ∈ Fk ,∑f∈Fk
uk,f .nk,f = 0, on Σk ,
with Fk the set of fractures with Σk (the k-th intersections) on the boundary,
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Conforming mesh at the intersections
Mixed-Hybrid Finite Element Method (MHFEM) for DFNs
Ref. J. Erhel et al., SISC, Vol. 31, No. 4, pp. 2688-2705, 2009
Makes it easy to deal with complex geometry ;
Conforming mesh at the fracture intersections ;
A linear system with only trace of pressure unknowns :
AΛ = b,
with A a symmetric positive definite matrix, the flux at the edges and themean pressure are then easily derived locally on each triangle.
Specific mesh generation :
1 A first discretization of boundaries andintersections is done in 3D by using elementarycubes.
2 The discretization of the boundaries andintersections within the fracture f is obtained bya projection of the previous voxel discretizationwithin the fracture plane.
3 Some local corrections are made to ensuretopological properties.
4 Once the borders and intersections arediscretized, a 2D mesh of each fracture isgenerated, using triangular elements.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Channelling in large fractures
In DFN, flow is highly channelled = an opportunity to reduce the number ofunknowns and the computational cost, by using a non conforming mesh atintersections.
How to apply a mortar method [Arbogast et al., SINUM 37(4) (2000)] to DFN ?
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Mixed-Hybrid Mortar Finite Element Method applied to DFN
Mixed-Hybrid Mortar Finite Element Method (MHMFEM) for DFNs
Ref. G. Pichot et al., Applicable Analysis, Volume 89, 1629-1643, 2010Ref. G. Pichot et al., SISC, In revision
A method to mesh the fractures independently and to refine the chosen fracturesusing a posteriori estimators.
Same advantages as MHFEM
A simple mesh generation
A reduced number of unknowns while keeping a solution of good quality
a complex numerical method with Mortar conditions
A new specific mesh generation :For each fracture f, choose a mesh step, then
1 A first discretization of boundaries andintersections is done in 2D, by using elementarysquares, it leads to a stair-case likediscretizations.
2 Some local corrections are made to ensure sometopological properties.
3 Once the borders and intersections arediscretized, a 2D mesh of each fracture isgenerated, using triangular elements.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Meshing procedure : Example
The discretization of intersections is non matching⇒ Mortar conditions are required to ensure the continuity of heads and fluxes.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Mortar principle
Mortar method principle :
It consists in choosing arbitrarily for each intersection a master fracture (m) and aslave fracture (s).
Classical Mortar approach : each edge is either master or slave
Specific Mortar method : some edges have several master or slave properties
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Master and slave unknowns
Hydraulic head NotationCell PInterior edge Λin
Intersection edge ΛΣ
Master edge Λm
Slave edge Λs
Jump of flux NotationIntersection edge QΣ
Master edge Qm
Slave edge Qs
Conforming mesh : Λm = Λs = ΛΣ
Classical Mortar conditions : relations between master and slave unknowns
Specific Mortar conditions : new relations with intersection unknowns
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Mortar global conditions
Trace of hydraulic head Jump of flux
Λs = CΛm Qm + CT Qs = 0
ΛΣ = AmΛm + AsΛs AmT QΣ = Qm
AsT QΣ = Qs
with C a block matrix of dimension NsxNm, with blocks (Ck) of dimensionNk,sxNk,m for which each block represents the L2-projection from the master sideto the slave side with coefficients Cln, l ∈ 1, ...,Nk,s, n ∈ 1, ...,Nk,m :
Cln =
(|Em
n ∩ E sl |
|E sl |
),
where the notation |E | stands for the length of the edge E .
As and Am are ponderation matrices that gives ΛΣ as the mean of Λm and Λs.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Equations
D P −(
Rin RΣ(Am + AsC))( Λin
Λm
)= f,
(Min MΣ(Am + AsC)(MΣ(Am + AsC))T (Am + AsC)T BΣ(Am + AsC)
)(ΛinΛm
)−(
Rin RΣ(Am + AsC))T
P− v = 0.
obtained by inverting locally Poiseuille’s law on each triangle and by expressing thefluxes in terms of traces of hydraulic head and mean heads in the followingequations :
First set of equations : mass conservation
Second set of equations : continuity of flux through inner edges
Third set of equations : continuity of flux through intersection edges
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Linear system
For all cases, with conforming or non conforming mesh, we get a linear system ofthe form D −R
−RT M
( PΛ
)=
(fv
)Then the system reduces to :
AΛ = c,
with A = M − RT D−1 R, Λ =
(ΛinΛm
)and c = v + RT D−1f.
Assuming the transmissivity is locally symmetric positive definite, and the networkis connected, the matrix
J =
D −R
−RT M
is symmetric and, with the presence of Dirichlet boundary conditions within at leastone fracture, it is positive definite.
Then A is also symmetric positive definite.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Consistency of the results
Criteria checked for all simulations :
Null sum of the fluxes over all the system
Null sum of the fluxes over all intersections between fractures
Boundary conditions satisfied
Continuity of the flux on inner edges (that is egdes that are not intersection).
50 fractures, 315 intersections, 41967 edges, Mean head computation
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Convergence criterium
Numerical convergence is estimated via a discrete relative L2 error :
1 A computation is performed on a fine mesh Tη that gives a reference meanpressure pη
2 Simulations are performed on coarsened grids Th of mesh step h > η.
The mean head obtained on coarse meshes, ph, are then compared with pη[Martin et al. 2005] :
||ph − pη ||2L2(Ω)=
∑Tη∈Tη (Πηph − pη)2|Tη |∑
Tη∈Tη (pη)2|Tη |,
|Tη | the area of the triangle Tη ∈ TηΠηph the projection of ph onto the fine mesh Tη .
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Mixed HybridFinite ElementMethod
Mortar method
Linear solvers
DomainDecompositionmethod
Conclusion
Convergence analysis
On 25 Monte-Carlo simulations :
Parameter Random distributionlength power lawshape disksposition uniformorientation uniform
Parameter Valuea 3.5L/lmin 2NMC 25Mesh step from 0.05 to 0.09Density 2
Log10 of the Convergence criterium vs mesh scale
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Solving the linear system
Linear system to solve :
In the previous section, we have seen that using MHFEM or MHFEM with Mortarleads to a linear system in terms of trace of hydraulic head unknowns :
AΛ = c
with A a symmetric positive definite matrix.
Solvers tested
Direct solver :Cholesky factorization : it is robust but memory and CPUexpensive
Iterative solver : multigrid method : it is efficient in most cases but it fails insome cases
Iterative solver : Preconditioned Conjugate Gradient (PCG) method : it isrobust with a multigrid preconditioner
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Experiments with stochastic DFN
Ref. B. Poirriez, Ph-D thesis, University of Rennes, December 2011
Generation of 417 random connected Discrete Fracture Networks
Experiments using MPFRAC software and scientific platform H2OLab
Sequential computations done with a multiprocessor with 16 GB memory,windows server
10 random simulations for each set of parameters
Increasing the density of fracturesParameter valuedimension L/lmin 3power law exponent 2.5 ;3.5 ;4.5density 2 ;3 ;4 ;5 ;6 ;7mesh step 0.04
Reducing the mesh stepParameter valuedimension L/lmin 3power law exponent 2.5 ;3.5 ;4.5density 4mesh step 0.01 ;0.02 ;0.03 ;0.04 ;0.05 ;0.06 ;0.07 ;0.08 ;0.09
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Results with a direct solver (UMFPACK)
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 5
1 0 0
1 0 1
1 0 2
CPU T
ime
s y s t e m s i z e n
U M F P A C K
Increasing the density of fractures
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6
1 0 0
1 0 1
1 0 2
1 0 3
U M F P A C K
CPU T
ime
m e s h n b
Reducing the mesh step
387 systems can be solved with this computer
CPU and memory requirements increase rapidly with the size
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Results with Algebraic MultiGrid (Boomer-AMG from Hypre)
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 5
1 0 0
1 0 1
1 0 2
A M G
CPU T
ime
s y s t e m s i z e n
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 50
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
Nb of
cycle
s
s y s t e m s i z e n
A M G
Increasing the density
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 6
1 0 0
1 0 1
1 0 2
1 0 3
A M G
CPU T
ime
s y s t e m s i z e n
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 60
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0 A M G
Numb
er of
cycles
s y s t e m s i z e n
Reducing the mesh step
375 systems can be solved with less than 1000 cycles
convergence rate highly depends on the network
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Results with PCG + AMG (from Hypre)
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 5
1 0 0
1 0 1
1 0 2
P C G
CPU T
ime
s y s t e m s i z e n
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 50
1
2
3
4
5
6
Nb of
iterat
ions
s y s t e m s i z e n
P C G
Increasing the density
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 61 0 0
1 0 1
1 0 2
1 0 3
P C G
CPU T
ime
s y s t e m s i z e n
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 61
2
3
4
5
6
7
8
9
1 0P C G
Numb
er of
iterat
ions
s y s t e m s i z e n
Reducing the mesh step
all systems can be solved
convergence rate depends on the network
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
A geometric partition of the network
The network : Ω = ∪iΩi , i = 1,..., Nf , Nf total number of fractures.A specific matrix shape : block angular form
A1,1 . . . . . . . . . A1,Nf +1
.... . . 0
...... Ai,i Ai,Nf +1
... 0. . .
...AT
1,Nf +1 . . . ATi,Nf +1 . . . ANf +1,Nf +1
One block for each fracture
Same memory complexity as a 2D problem
Many fractures : they can be grouped into fracture sets
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Domain decomposition of a Fracture Network
Partition into fracture sets
The network is decomposed into Nk fracture sets Fk , k = 1, ...Nk ,Fk = ∪iΩi , i ∈ Ik
Each subdomain is a connected fracture set to ensure that the submatrix isirreducible
Some subdomains are floating, with no Dirichlet boundary condition
block angular form of the matrix
The matrix is permuted and partitioned into blocks B = PT AP =
Nk∑k=1
Bk ,
with P a permutation matrix.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Schur complement with subdomains
Local block matrix : For a block k, k = 1, ...Nk :
Bk =
0 . . . . . . . . .... Bk,k Bk,Nk+1
.... . .
...... B
Tk,Nk+1 . . . B
(k)Nk+1,Nk+1
,B =∑k
Bk
The block Bk,k is symmetric positive definite (SPD).
Schur Complement and local Schur complement :
S =
Nk∑k=1
RTk SkRk , with Sk = B
(k)Nk+1,Nk+1 − BT
k,Nk+1B−1k,kBk,Nk+1
The Schur complement matrix S is SPD but Sk can be singular.
Equivalent system : Sx = b,
with b = bNk+1 −∑k
BTk,Nk+1B−1
k,kbk
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Preconditioned Conjugate Gradient
M spd matrixInitialisation :
Choose x0
r0 = b− S x0
z0 = Mr0
p0 = z0
Iterations : For j = 0, 1, ... until convergence
qj = Spj ⇐ Computation of matrix-vector product
αj =rTj rj
pTj qj
xj+1 = xj + αjpj
rj+1 = rj − αjqj
zj+1 = Mrj+1 ⇐ Preconditioning
βj =rTj+1zj+1
rTj zj
pj+1 = zj+1 + βjpj
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Neumann Neumann preconditioning
Mn = ∆
Nk∑k=1
RTk S−1
k Rk∆
If the subdomain is not floating, the matrix Sk is non singular and Sk=Sk
If the subdomain is floating, Sk singular ⇒ Non singular approximation Sk
∆ is a diagonal matrix which accounts for the number of subdomainsinvolved
Mn is not given explicitly
Solving z = Mnr
z = Mnr = ∆
Nk∑k=1
RTk zk , with zk = S−1
k rk , with rk = Rk∆r
Solving Skzk = rk is equivalent to solve Bk
(xkzk
)=
(0rk
)with
Bk =
(Bk,k Bk,Nk+1
BTk,Nk+1 B
(k)Nk+1,Nk+1
)The matrix Bk is symmetric positive definite.
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Computing the local products
Matrix-vector computation
q = Sp =
Nk∑k=1
qk , with qk = RTk SkRkp
qk = RTk
(B
(k)Nk+1,Nk+1Rkp− BT
k,Nk+1B−1k,kBk,Nk+1Rkp
)Cholesky factorization : before iterations
Bk = LkLTk with Lk =
(Lk,k 0
Lk,Nk+1 L(k)Nk+1,Nk+1
)
This gives the Cholesky factorization Bk,k = Lk,kLTk,k
and the Cholesky factorization Sk = L(k)Nk+1,Nk+1L
(k)TNk+1,Nk+1
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Deflation preconditioning
Ref. J. Erhel et al., SIMAX, Volume 21, no. 4, pp. 1279-1299, 2000
M is any preconditioner, for example M = Mn
Projection matrixZ a basis of a coarse subspacerestriction onto the subspace : Sd = ZT SZProjection P = I − SZS−1
d ZT
Balancing preconditioning
Mb = PT MP + ZS−1d ZT
Deflation preconditioning
Md = PT MP
with the initial choicex0 = ZS−1
d ZTb
Deflation and balancing are equivalent for this initial choice[R. Nabben and C. Vuik, SISC, 27(5), 2006]
Sketch of the proofZT r0 = 0 and ZT rk = 0Thus Md rk = PT MrkAnd, during PCG iterations, Md = Mb = PT M
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
SCHUR-Solve implementation(in MPFRAC)
as many subdomains as fractures Nf
mutual factorization
deflation with the signature of subdomains : Z of size Nf
implemented as a library and interfaced with MPFRAC in H2OLab
experiments with a conforming mesh
more details in Baptiste’s Poirriez Ph-D manuscript
to be defended in December 2011
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Results with SCHUR-Solve
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 5
1 0 0
1 0 1
1 0 2
S c h u r
CPU T
ime
s y s t e m s i z e n
1 0 5 2 x 1 0 5 3 x 1 0 5 4 x 1 0 5 5 x 1 0 5 6 x 1 0 5 7 x 1 0 50
2 0
4 0
6 0
8 0
1 0 0
S c h u r
Nb of
iterat
ions
s y s t e m s i z e n
Increasing the density
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 6
1 0 0
1 0 1
1 0 2
1 0 3 S c h u r
CPU T
ime
s y s t e m s i z e n
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 60
2 0
4 0
6 0
8 0
1 0 0
S c h u r
Nb of
iterat
ions
s y s t e m s i z e n
Reducing the mesh step
all systems can be solved
convergence not too sensitive to the network
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Domaindecompositionof a DFN
PCG applied tothe Schurcomplement
Results withSchur
Conclusion
Comparison of solvers
6 x 1 0 4 8 x 1 0 4 1 0 5 2 x 1 0 5 4 x 1 0 5 6 x 1 0 5 8 x 1 0 5 1 0 6 2 x 1 0 6 4 x 1 0 6 6 x 1 0 6 8 x 1 0 60 , 1
1
P C G A M G U m f S c h u r
relati
ve CP
U tim
e
s y s t e m s i z e nt < 1 E 5 t < 1 . 5 5 E 5 t < 2 . 3 E 5 t < 3 . 6 E 5 t < 5 . 5 E 5 t < 8 E 5 t < 6 E 5
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
8 0
9 0
1 0 0
Faste
st(%)
s y s t e m s i z e n
u m f a m g p c g s c h u r
Schur is reliable and fast
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
On-going and future work
Build a posteriori estimators to optimize mesh generation
Develop a parallel software
Run large scale DFNs simulations to study scalability
Run Monte Carlo simulations to derive upscaling rules
Numerical andstochastic
models of flowin 3D Discrete
FractureNetworks
J-R. de Dreuzy, J. Erhel , G.
Pichot, B.Poirriez
Outline
Introduction
Problemdescription
Derivation ofthe linearsystem
Linear solvers
DomainDecompositionmethod
Conclusion
Important announcement
DD21 will be in Rennes, in June 25-29, 2012.International conference on Domain Decomposition methodsco-organized by Inria and University of Caenwith an unforgettable excursion to Mont Saint-Michel in Normandy
Save the date !
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