numerical and stochastic models of flow in 3d discrete ... · numerical and stochastic models of ow...

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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription

Geometry

Flow equations


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription

Geometry

Flow equations


Linear solvers


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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription

Geometry

Flow equations


Linear solvers


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,



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Mixed HybridFinite ElementMethod

Mortar method

Linear solvers


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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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 ?



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


Conclusion

Meshing procedure : Example

The discretization of intersections is non matching⇒ Mortar conditions are required to ensure the continuity of heads and fluxes.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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 .



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription



Mortar method

Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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


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


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


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



375 systems can be solved with less than 1000 cycles

convergence rate highly depends on the network



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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


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


P C G


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


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



all systems can be solved

convergence rate depends on the network



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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.



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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


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



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


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



all systems can be solved

convergence not too sensitive to the network



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers




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(%)


u m f a m g p c g s c h u r

Schur is reliable and fast



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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



FractureNetworks


Pichot, B.Poirriez

Outline

Introduction

Problemdescription


Linear solvers


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 !

Sponsors are welcome...

numerical and stochastic models of flow in 3d discrete ... · numerical and stochastic models of ow...

Documents