A mathematical scaling approach to macroscopic heterogeneity of composite and porous media is introduced. It is based on weak limits of uniformly bounded measurable functions. The limiting local porosity distributions that were introduced in the work [Adv. Chem. Phys. XCII, 299–424 (1996)] are found to be related to Young measures of a weakly convergent sequence of local volume fractions. The Young measures determine frequency dependent complex dielectric functions of multiscale media within a generalized self-consistent effective medium approximation. The approach separates scales by scale factor functions of regular variation. It renders upscaled results independent of the shape of averaging windows upon reaching the scaling limit.
REFERENCES
1.
2.
M.
Hubbert
, Int. Assoc. Sci. Hydrol. Bull.
2
, 23
(1957
).3.
S.
Whitaker
, Transp. Porous Media
1
, 3
(1986
).4.
C.
Marle
, Int. J. Eng. Sci.
20
, 643
(1982
).5.
M.
Hassanizadeh
and W. G.
Gray
, Adv. Water Res.
2
, 131
(1979
).6.
A.
Lemmer
and R.
Hilfer
, J. Comput. Phys.
281
, 970
(2015
).7.
K.
Mattila
, T.
Puurtinen
, J.
Hyväluoma
, R.
Surmas
, M.
Myllys
, T.
Turpeinen
, F.
Robertsen
, J.
Westerholm
, and J.
Timonen
, J. Comput. Sci.
12
, 62
(2016
).8.
B.
Biswal
, P.
Øren
, R.
Held
, S.
Bakke
, and R.
Hilfer
, Image Anal. Stereol.
28
, 23
(2009
).9.
S.
Roth
, B.
Biswal
, G.
Afshar
, R.
Held
, P.
Øren
, L.
Berge
, and R.
Hilfer
, AAPG Bull.
95
, 925
(2011
).10.
J.
Hu
and P.
Stroeven
, Cem. Concr. Res.
35
, 233
(2005
).11.
S.
Whitaker
, Ind. Eng. Chem.
61
(12
), 14
(1969
).12.
13.
14.
R.
Bayford
, Annu. Rev. Biomed. Eng.
8
, 63
(2006
).15.
A.
Haas
and A.
Revil
, Water Resour. Res.
45
, W10202
, https://doi.org/10.1029/2009wr008160 (2009
).16.
A.
Barbetta
, C.
Cametti
, G.
Rizzitelli
, and M.
Dentini
, Soft Matter
8
, 1120
(2012
).17.
R.
Hilfer
, Phys. Rev. B
44
, 60
(1991
).18.
E.
Haslund
and B.
Nøst
, Geophysics
63
, 149
–153
(1998
).19.
R.
Hilfer
, Adv. Chem. Phys.
XCII
, 299
(1996
).20.
J.
van Vleck
, The Theory of Electric and Magnetic Susceptibilities
(Clarendon
, Oxford
, 1932
).21.
E.
Sanchez-Palencia
, Non-Homogeneous Media and Vibration Theory
, Lecture Notes in Physics (Springer Verlag
, Berlin
, 1980
), Vol. 127.22.
V.
Jikov
, S.
Kozlov
, and O.
Oleinik
, Homogenization of Differential Operators and Integral Functionals
(Springer
, Berlin
, 1994
).23.
J.
Ball
, in Partial Differential Equations and Continuum Models for Phase Transitions
, Lecture Notes in Physics, edited by M.
Rascle
, D.
Serre
, and M.
Slemrod
(Springer
, Berlin
, 1989
), Vol. 344, pp. 207
–215
.24.
C.
Castaing
, P.
Raynaud de Fitte
, and M.
Valadier
, Young Measures on Topological Spaces
(Kluwer
, Dordrecht
, 2004
).25.
F.
Boger
, J.
Feder
, R.
Hilfer
, and T.
Jøssang
, Physica A
187
, 55
(1992
).26.
R.
Hilfer
, Physica A
194
, 406
(1993
).27.
R.
Hilfer
, in Räumliche Statistik und Statistische Physik
, Lecture Notes in Physics, edited by D.
Stoyan
and K.
Mecke
(Springer
, Berlin
, 2000
), Vol. 554, p. 203
.28.
B.
Biswal
, C.
Manwart
, R.
Hilfer
, S.
Bakke
, and P.
Øren
, Physica A
273
, 452
(1999
).29.
R.
Hilfer
, Transp. Porous Media
46
, 373
(2002
).30.
R.
Hilfer
, J.
Widjajakusuma
, and B.
Biswal
, Granular Matter
2
, 137
(1999
).31.
R.
Hilfer
, J.
Widjajakusuma
, and B.
Biswal
, Physica A
318
, 319
(2003
).32.
C.
Andraud
, A.
Beghdadi
, E.
Haslund
, R.
Hilfer
, J.
Lafait
, and B.
Virgin
, Physica A
235
, 307
(1997
).33.
B.
Biswal
, C.
Manwart
, and R.
Hilfer
, Physica A
255
, 221
(1998
).34.
B.
Biswal
and R.
Hilfer
, Physica A
266
, 307
(1999
).35.
J.
Widjajakusuma
, B.
Biswal
, and R.
Hilfer
, Comput. Mater. Sci.
16
, 70
(1999
).36.
J.
Widjajakusuma
, C.
Manwart
, B.
Biswal
, and R.
Hilfer
, Physica A
270
, 325
(1999
).37.
P.
Yin
and G.
Zhao
, Int. J. Rock Mech. Min. Sci.
70
, 82
(2014
).38.
J.
Thovert
, F.
Yousefian
, P.
Spanne
, C.
Jacquin
, and P.
Adler
, Phys. Rev. E
63
, 061307
(2001
).39.
J.
Liu
, G.
Pereira
, and K.
Regenauer-Lieb
, J. Geochem. Explor.
144
, 84
(2014
).40.
M.
Talukdar
, O.
Torsaeter
, M.
Ioannidis
, and J.
Howard
, Transp. Porous Media
48
, 101
(2002
).41.
J.
Petrasch
, F.
Meier
, H.
Friess
, and A.
Steinfeld
, Int. J. Heat Fluid Flow
29
, 315
(2008
).42.
K.
Wan
and Q.
Xu
, Sci. China
57
, 953
(2014
).43.
R.
Hilfer
, B.
Nøst
, E.
Haslund
, Th.
Kautzsch
, B.
Virgin
, and B. D.
Hansen
, Physica A
207
, 19
(1994
).44.
Z.
Fishman
and A.
Bazylak
, J. Electrochem. Soc.
158
, B841
(2011
).45.
L.
Keller
, L.
Holzer
, P.
Schuetz
, and P.
Gasser
, J. Geophys. Res.: Solid Earth
118
, 2799
, https://doi.org/10.1002/jgrb.50228 (2013
).46.
P.
Cosenza
, D.
Pret
, A.
Giraud
, and S.
Hedana
, Mech. Mater.
84
, 55
(2015
).47.
N.
Bingham
, C.
Goldie
, and J.
Teugels
, Regular Variation
(Cambridge University Press
, Cambridge
, 1987
).48.
H.
Bauer
, Maß-und Integrationstheorie
(Walter de Gruyter
, Berlin
, 1992
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
