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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

Homogenization and porous media

by Ulrich Hornung

Chapter 1: Introduction

Heike Gramberg, CASA Seminar Wednesday 23 February 2005

overview
Overview
  • Diffusion in periodic media
    • Special case: layered media
  • Diffusion in media with obstacles
  • Stokes problem: derivation of Darcy’s law
diffusion equation review
Diffusion equation (Review)
  • We start with the following Problem
  • Let with bounded and smooth
  • Diffusion equation
assumptions
Assumptions
  • is rapidly oscillating i.e.
  • a=a(y) is Y-periodic in with periodicity cell
ansatz
Ansatz
  • has an asymptotic expansion of the form
  • and are treated as independent variables
substitution of expansion
Substitution of expansion
  • Comparing terms of different powers yields
  • where
solution
Solution
  • Terms of order :
  • since is Y-periodic we find
slide8
Terms of order :

separation of variables

where is Y-periodic solution of

slide9

Terms of order integration over Y

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

Proposition 1: The homogenizationof the

diffusion problem is given by

where is given by

slide11
Proposition 2:
  • The tensor A is symmetric
  • If a satisfies a(y)>a>0 for all y then

A is positive definite

remarks
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
Example: Layered Media
  • Assumption:
  • Then

and is Y-periodic solution of

slide14
Proposition 3:
  • If , then
  • The coefficients are given by
remarks15
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
media with obstacles

e

Media with obstacles
  • Medium has periodic arrangement of obstacles
formal description of geometry
Formal description of geometry
  • Standard periodicity cell
  • Geometric structure within
  • Assumption:
diffusion problem
Diffusion problem
  • Diffusion only in
  • Assumptions: and
substitution of expansion19
Substitution of expansion
  • Comparing terms of different powers yields
lemmas
Lemmas
  • Lemma 1: for and
  • Lemma 2 (Divergence Theorem):

for Y-periodic

solution21
Solution
  • Terms of order : for
  • using Lemmas 1 and 2 we find
  • therefore
slide22
Terms of order : for
  • with boundary condition for
slide23
separation of variables

where is Y-periodic solution of

slide24

Terms of order

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

Proposition 4: The homogenizationof the

diffusion problem on geometry with obstacles is given by

where is given by

remarks26
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
Stokes problem
  • For media with obstacles
  • Assumptions
solution28
Solution
  • Comparing coefficients of the sameorder
    • Stokes equation:
    • Conservation of mass:
    • Boundary conditions:
slide29
With we get for
  • Separation of variables for both

where are solution of

darcy s law
Darcy’s law
  • Averaging velocity over

where is given by

conservation of mass
Conservation of mass
  • Term of order in conservation of mass
  • Integration over yields
proposition32
Proposition

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

Proposition 6: The tensorKis symmetric and positive definite

conclusions
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