Homogenization and porous media

Heike Gramberg, CASA Seminar Wednesday 23 February 2005

by Ulrich Hornung

Chapter 1: Introduction

• Diffusion in periodic media– Special case: layered media

• Diffusion in media with obstacles

• Stokes problem: derivation of Darcy’s law

Overview

We start with the following Problem

• Let with bounded and smooth

• Diffusion equation

Diffusion equation (Review)

NR

0

D

a x u x f x x

u x u x x

• is rapidly oscillating i.e.

• a=a(y) is Y-periodic in with periodicity cell

Assumptions

a

xa x a x for all

NR

1 0 1 1N iY y y y y i N , , , , ,

• has an asymptotic expansion of the form

• and are treated as independent variables

Ansatz

u

20 1 2u x u x y u x y u x y , , ,

x xy

1x y

• Comparing terms of different powers yields

where

Substitution of expansion

20

11 0

02 1

0

0

0

0

,:

: , ,

: , ,

,

yy

yy xy yx

yy xy yx

xx

u x y

u x y u x y

u x y u x y

u x y f x

A

A A A

A A A

A

ab a ba y A :

• Terms of order :

since is Y-periodic we find

Solution2

0 0u x y u x,

0 ,u x y

0 0y ya y u x y ,

• Terms of order :

separation of variables

where is Y-periodic solution of

y y j y ja y w y a y e

1

1 01

j j

N

y y y xj

a y u x y a y u x

,

1 01

, j

N

j xj

u x y w y u x

jw y

• Terms of order integration over Y

using for all Y-periodic g(y):

0 :

2 1

0 0

, ,

yy xy yx

xx

u x y u x y

u x f x

A A A

A

Y

dy

0iyYg y dy

01

0,

i i j

N

y j ij x xYi j

a y w y dy u f x

PropositionsProposition 1: The homogenization of the

diffusion problem is given by

where is given by

0

D

A u x f x x

u x u x x

ijA a

iij y j ijY

a a y w y dy

Proposition 2:

a. The tensor A is symmetric

b. If a satisfies a(y)>>0 for all y then

A is positive definite

Remarks

• are uniquely defined up to a constant• are uniquely defined• Problem can be generalized by considering

• Eigenvalues of A satisfy Voigt-Reiss inequality:

where

jw

, a a x y A A x

ija

11a a

:Y

f f dy

Example: Layered Media

• Assumption: • Then

and is Y-periodic solution of

1, , N Na y y a y

1 0

0

for

for

N N

N N

d ddy dyN j N

d ddy dyN j N

a y w y j N

a y w y j N

1, ,j N j Nw y y w y

jw

Proposition 3:

a) If , then

b) The coefficients are given by

01 1

0

, ,

Ny da

N N Nd

a

w y y y

0 for andjw j N

ija11 for

otherwisea

ij

ij

i j Na

a

1, , N Na y y a y

Remarks

• Effective Diffusivity in direction parallel to layers is given by arithmetic mean of a(y)

• Effective Diffusivity in direction normal to layers is given by geometric mean of a(y)

• Extreme example of Voigt-Reiss inequality

Media with obstacles

• Medium has periodic arrangement of obstacles

B

G

• Standard periodicity cell

• Geometric structure within

• Assumption:

Formal description of geometry

B

G

\ B = B G = B

Diffusion problem

• Diffusion only in

• Assumptions: and

B

0

0

D

a x u x f x x

a x u x x

u x u x

B

xa x a

20 1 2u x u x y u x y u x y , , ,

• Comparing terms of different powers yieldsSubstitution of expansion

20

11 0

02 1

0

0

0

0

yy

yy xy yx

yy xy yx

xx

u x y

u x y u x y

u x y u x y

u x y f x

A

A A A

A A A

A

,:

: , ,

: , ,

,

10

00 1

11 2

0

0

0

,:

: , ,

: , ,

y

x y

x y

a y u x y

a y u x y u x y

a y u x y u x y

with boundary conditions on

Lemmas

• Lemma 1: for and

• Lemma 2 (Divergence Theorem):

for Y-periodic

,g g x y

y y yf g f g f g

,g g x y

, ,y g x y dy g x y d y

B

,f f x y

• Terms of order : for

using Lemmas 1 and 2 we find

therefore

Solution2

0 0u x y u x,

0 0y ya y u x y ,

yB

0 0

20

0 0

0 ,

| , |

, ,

y y

y

y

u a y u x y dy

a y u x y dy

u x y a y u x y d y

B

B

• Terms of order : for

• with boundary condition for

1

1 01

j j

N

y y y xj

a y u x y a y u x

,

yB

1u on

1 01

j

N

y j xj

u x y u x

,

• separation of variables

where is Y-periodic solution of

1 01

, j

N

j xj

u x y w y u x

jw y

y y j y j

y j j

a y w y a y e y

w x y e y

,

B

• Terms of order

using Lemma 2 and boundary conditions:

hence is solution of

0 :

2 1

0 0

, ,

yy xy yx

xx

u x y u x y

u x f x

A A A

A

Bdy

01

0,

i i j

N

y j ij x xi j

a y w y dy u f x

B

2 1 2 1

0

yy yx y xu u dy a y u u d y

BA A

0 0u u x

PropositionProposition 4: The homogenization of the

diffusion problem on geometry with obstacles is given by

where is given by

0

D

A u x f x x

u x u x x

ijA a

iij y j ija a y w y dy B

Remarks• Due to the homogeneous Neumann conditions

on integrals over boundary disappear• Weak formulation of the cell problem

where is characteristic function of

0,j j Ya w e

1

0

yy

y

B

G

y B

Stokes problem

2

0

0

v x p x x

v x x

v x x

B

B

• For media with obstacles

• Assumptions

20 1 2

20 1 2

, , ,

, , ,

v x v x y v x y v x y

p x p x y p x y p x y

Solution

• Comparing coefficients of the same order– Stokes equation:

– Conservation of mass:

– Boundary conditions:

10 0 0

00 1 0

0

: ,

: , , ,

y

y y x

p x y p p x

v x y p x y p x y

10 0: ,y v x y

0 1 0, , for v x y v x y y

for yB

for yB

• With we get for

• Separation of variables for both

where are solution of

0 0jx j xjp x e p x

0 1 0

0 0

, ,

,jy y j xj

v x y p x y e p x

v x y

yB

10 0

1 0

,

,

j

j

j xj

j xj

v x y w y p x

p x y y p x

0

0

y j y j j

y j

j

w y y e y

w y y

w y y

B

B

and j jw

0 1 and v p

Darcy’s law

• Averaging velocity over

where is given by

B

10 0: , u v x y dy u K p x B

,ij i j

K k

ij jik w y dyB

Conservation of mass

• Term of order in conservation of mass

• Integration over yields

0

1 0 0, ,y xv x y v x y

1

1 1

0

,

, ,

x y

Y

u x v x y dy

v x y d y v x y d y

B

B

Proposition

Proposition 5: The homogenization of the Stokes problem is given by

Proposition 6: The tensor K is symmetric and positive definite

1 0 , u K p x u

Conclusions

• We have looked at homogenization of the Diffusion problem and the Stokes problem on media with obstacles

• Solutions of the homogenized problems can be expressed in terms of solutions of cell problems

• The homogenization of the Stokes problem leads to the derivation of Darcy’s law