analytical study of a preconditioner for fractured porous ... · porous medium : model fracture as...

$: Analytical Study of a Preconditioner for Fractured Porous ... · porous medium : model fracture as dimension d −1 interface between subdomains. 1 W 2 S W R p=0 p=p u.n=0 u.n=0 L$
Analytical Study of a Preconditionerfor Fractured Porous Media

Laila Amir Michel Kern Jean Roberts

INRIA Rocquencourt, France

SIAM GS07 ConferenceSanta Fe

March 19 – 22 2007

Amir, Kern & Roberts DD for fractured media GS07, March 2007 1 / 21

Plan

1 Introduction

2 Continuous problem

3 Discrete problem

4 Eigenvalues and condition numbers


Outline

1 Introduction


3 Discrete problem



Problem setting

Fractured porous medium : one large fracture, modeledindividually.Fracture width is small compared to transverse dimension ofporous medium : model fracture as dimension d − 1 interfacebetween subdomains.

1Ω

2

Σ

Ω

R

p=

0

p=

p

u.n=0

u.n=0

L2

L1

H

Pressure continuous through the fractureFlow in fracture due to velocity jump between two subdomainsAmir, Kern & Roberts DD for fractured media GS07, March 2007 4 / 21

Outline

1 Introduction


3 Discrete problem



Mathematical model

Flow in subdomainsdiv ui = 0ui = −Ki∇pi

in Ωi

Pressure continuity

pi = pf , i = 1, 2

Flow in fracturedivf uf = u1.n − u2.nuf = −dKf∇pf

in Σ

d width of fracture, Ki (i = 1, 2), Kf permeabilitiesCf. Alboin, Jaffre, Roberts (2002)


Domain decomposition formulation

Introduce Steklov-Poincaré (DtN) operator on Ωi :

Si p = −Ki(ui .n)|Σ, where :

div ui = 0ui = −Ki∇pi

in Ωi

ui .n = 0 on ∂ΩNi

pi = 0 on ∂ΩDi

pi = p on Σ

DD formulationdivf uf = S1pf + S2pf

uf = −dKf∇pfin Σ

uf = 0 on ∂Σ.

Solve by conjugate gradient, solve one flow problem per subdomain ateach iteration.


Numerical example

A possible preconditionerAs Laplacian is 2nd order, and DtN operator is 1st order,preconditioning by Laplacian in the fracture should be effective (cfAmir, Jaffre, Roberts, 2005)

Influence of fracture permeabilityNumber of iterations when Kf varies :

dKf 103 102 101 100 10−1 10−2

No prec. 68 50 50 47 36 34Laplace prec. 7 10 16 29 38 38

Preconditioner more effective for large values of Kf . Why ?


Influence of the fracture permeability

dKf = 1 dKf = 102

dKf = 103 dKf = 105


Fourier analysis of the continuous problem

Theorem (Eigenstructrure of the DtNoperator)

Let wn(y) = cos(nπyH

), then

Swn =KH

nπ

tanh nπL/Hwn

H

L

Ω

u.n=0

p=

0

p=

pR

Eigenvalues of the global operator

λexn =

K1

Hnπ

tanh nπL1/H+

K2

Hnπ

tanh nπL2/H+ dKf

n2π2

H2

Last part (fracture Laplacian) always dominant


Outline

1 Introduction


3 Discrete problem



Numerical discretization

Mixed finite elementsLowest order RT elementUniform mesh→ difference equations

i−1/2,j

pij

ui+1/2, j

vi,j−1/2

vi,j+1/2

h

u

Analysis of the discrete Steklov-Poincaréoperator

S operator maps pj = pNj to uN+1/2j

Determine eigenstructure of S, analysissimilar to Chan (87)


Discrete equations (1)

16

hk

(ui−1/2,j + 4ui+1/2,j + ui+3/2,j

)=pij − pi+1,j

16

hk

(2u1/2,j + u3/2,j

)=− p1j

16

hk

(uN−1/2,j + 2uN+1/2,j

)=pNj − pj

16

hk

(uij−1/2 + 4uij+1/2 + uij+3/2

)=pij − pij+1

ui1/2 = uiM+1/2 =0

ui+1/2j − ui−1/2j + uij+1/2 − uij−1/2 = 0


Discrete equations (2)

Special form for solutionpi j =pi cos(nπjh/H)

ui+1/2 j =ui+1/2 cos(nπjh/H)

ui j+1/2 =v i sin(nπ(j + 1/2)h/H)

Eliminate pi , v i , difference equation for ui+1/2, withσn = sin2(nπh/(2H)) :

(3−2σn)(ui+3/2−2ui+1/2 +ui−1/2)−2σn(ui+3/2 +4ui+1/2 +ui−1/2) = 0

with boundary conditions

(3− 4σn)u3/2 − (3 + 2σn)u1/2 =0

(3− 4σn)uN−1/2 − (3 + 2σn)uN+1/2 =12Kσnpn


The discrete Steklov Poincaré operator

Characteristic equation

(3− 4σn)r2 − 2(3 + 2σn)r + (3− 4σn) = 0

Roots 0 < r− < 1 < r+, γn =r−

r+.

Theorem (Eigenvalues ofSteklov-Poincaré operator)

λn =Kh

√12σn

3− σn

1 + γN+1n

1− γN+1n

,

RemarkFor fixed n, limh→0 λn = λex

n . 0 5 10 15 200

10

20

30

40

50

60

n

Eig

enva

lues

ApproxExact


Outline

1 Introduction


3 Discrete problem



Eigenvalues of the unpreconditioned operator

Eigenvalues of the “fracture operator”

λn =K1

h

√12σn

3− σn

1 + γN1+1n

1− γN1+1n

+K2

h

√12σn

3− σn

1 + γN2+1n

1− γN2+1n

+ 4dKfσn

h2 .

0 5 10 15 200

500

1000

1500

2000

n

Eig

enva

lues

0 5 10 15 200

500

1000

1500

2000

2500

3000

3500

4000

ApproximateExact


Two preconditioners

A = S1 + S2 + L, Si is Steklov-Poincaré for subdomain i , L is flow in

the fracture Lp = −dKfd2pdx2 .

Condition number, no preconditioning

κNP =

Kh√

12σN

3− σN

1 + γN+1N

1− γN+1N

+ 2dKf σN

Kh√

12σ1

3− σ1

1 + γN+11

1− γN+11

+ 2dKf σ1

limKf→∞ κNP =σN

σ1, like for Laplace eqn.

limKf→0 κNP = O(N), like DD

Laplacian preconditioning Use L as a preconditionerDD preconditioning Use S1 + S2 as a preconditioner.


Eigenvalues for 3 cases

Top : Noprec.,

Middle :Lap. prec.,

Bottom :DD prec.

0 10 200

50

100

150

n

Eig

enva

lues

Kf=1e−2

0 10 200

50

100

n

Eig

enva

lues

Kf=1e−2

0 10 201

1.5

2

n

Eig

enva

lues

Kf=1e−2

0 10 200

2000

4000

n

Eig

enva

lues

Kf=1e−2

0 10 201

1.5

2

nE

igen

valu

es

Kf=1e−2

0 10 200

50

100

n

Eig

enva

lues

Kf=1e−2

0 10 200

2

4x 10

5

n

Eig

enva

lues

Kf=1e−2

0 10 201

1.005

1.01

1.015

n

Eig

enva

lues

Kf=1e−2

0 10 200

2000

4000

6000

nE

igen

valu

es

Kf=1e−2


Condition numbers when Kf varies

.1e2

1.

0.1

Kf

0.0750.050.025

.1e3

25

15

20

10

5

.1e31. .1e2.1e−2

Kf

.1.1e−1

Cond. number vs Kf , no prec., Laplacian prec., DD prec.


Conclusions – perspectives

Analysis of discrete Steklov – Poincaré operator for mixed finiteelements on a regular gridCondition number for discrete fracture problem, asymptoticbehavior

Better understanding of low Kf valuesUse for designing a better preconditioner, for all values of Kf .


analytical study of a preconditioner for fractured porous ... · porous medium : model fracture as...

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