Homogeni z ation and porous media

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# Homogeni z ation and porous media - PowerPoint PPT Presentation

Homogeni z ation and porous media. by Ulrich Hornung Chapter 1: I ntroduction. Heike Gramberg , CASA Seminar Wednesday 23 February 2005. Overview. Diffusion in periodic media Special case: layered media Diffusion in media with obstacles Stokes problem: derivation of Darcy’s law.

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## Homogeni z ation and porous media

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### Homogenization and porous media

by Ulrich Hornung

Chapter 1: Introduction

Heike Gramberg, CASA Seminar Wednesday 23 February 2005

Overview
• Diffusion in periodic media
• Special case: layered media
• Diffusion in media with obstacles
• Stokes problem: derivation of Darcy’s law
Diffusion equation (Review)
• Let with bounded and smooth
• Diffusion equation
Assumptions
• is rapidly oscillating i.e.
• a=a(y) is Y-periodic in with periodicity cell
Ansatz
• has an asymptotic expansion of the form
• and are treated as independent variables
Substitution of expansion
• Comparing terms of different powers yields
• where
Solution
• Terms of order :
• since is Y-periodic we find
Terms of order :

separation of variables

where is Y-periodic solution of

Terms of order integration over Y

• using for all Y-periodic g(y):
Propositions

Proposition 1: The homogenizationof the

diffusion problem is given by

where is given by

Proposition 2:
• The tensor A is symmetric
• If a satisfies a(y)>a>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 l of A satisfy Voigt-Reiss inequality:

where

Example: Layered Media
• Assumption:
• Then

and is Y-periodic solution of

Proposition 3:
• If , then
• The coefficients are given by
Remarks
• Effective Diffusivity in direction parallel to layers is given by arithmeticmean of a(y)
• Effective Diffusivity in direction normal to layers is given by geometricmean of a(y)
• Extreme example of Voigt-Reiss inequality

e

Media with obstacles
• Medium has periodic arrangement of obstacles
Formal description of geometry
• Standard periodicity cell
• Geometric structure within
• Assumption:
Diffusion problem
• Diffusion only in
• Assumptions: and
Substitution of expansion
• Comparing terms of different powers yields
Lemmas
• Lemma 1: for and
• Lemma 2 (Divergence Theorem):

for Y-periodic

Solution
• Terms of order : for
• using Lemmas 1 and 2 we find
• therefore
Terms of order : for
• with boundary condition for
separation of variables

where is Y-periodic solution of

Terms of order

• using Lemma 2 and boundary conditions:
• hence is solution of
Proposition

Proposition 4: The homogenizationof the

diffusion problem on geometry with obstacles is given by

where is given by

Remarks
• Due to the homogeneous Neumann conditions on integrals over boundary disappear
• Weak formulation of the cell problem

where is characteristic function of

Stokes problem
• For media with obstacles
• Assumptions
Solution
• Comparing coefficients of the sameorder
• Stokes equation:
• Conservation of mass:
• Boundary conditions:
• Separation of variables for both

where are solution of

Darcy’s law
• Averaging velocity over

where is given by

Conservation of mass
• Term of order in conservation of mass
• Integration over yields
Proposition

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

Proposition 6: The tensorKis symmetric and positive definite

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