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# IV Notation and Conventions

[64.1.4.1] The time variation of electrical fields is taken proportional to exp-iωt. [64.1.4.2] The complex dieelectric constant ε is written as

 ε=ε′+i⁢ε′′=ε′+i⁢σ′ω, (4.1)

where εε′′ are the real (imaginary) parts of ε, and σ is the real part of the complex conductivity σ. [64.1.4.3] SI units will be used. [64.1.4.4] Values for ε are given in multiples of ε0=8,8542×10-12f/m, the permittivity of free space, and the units for σ are then S/m. S/m. [64.1.4.5] The conductivity is written as

 σ=σ′+i⁢σ′′=σ′+i⁢ω⁢1-ε′. (4.2)

[64.1.4.6] The relationship between σ and ε is also written as

 ε⁢u=1-σu, (4.3)

where the notation u=iω has been used.

[64.1.5.1] The constituent materials are assumed to be rock forming the matrix and water filling the pore space. [64.1.5.2] Their dielectric constants are

 εW=εW′+i⁢σW′ω, (4.4)

for water, and

 εR=εR′, (4.5)

for rock. [64.1.5.3] In calculations the values εW=79ε0 and εR=7ε0 will be used. [64.1.5.4] Dimensionless frequencies are introduced by setting ω0=σW/εW=1 for the relaxation frequency of water, and this also fixes εW=σW.

[64.2.1.1] The average local dielectric constants εBω;ϕ and εCω;ϕ at porosity ϕ are in general unknown. [64.2.1.2] Nevertheless, some general statements can be made. [64.2.1.3] The relation

 σB′(ω=0;ϕ)=0 (4.6)

must hold for all ϕ. [64.2.1.4] It expresses the fact that the blocking geometry is nonconducting. [64.2.1.5] The following relations for the low- and high-porosity limits are also obvious:

 σC′(ω;ϕ=0)=σR′=0, (4.7a) σC′(ω;ϕ=1)=σW′, (4.7b) εC′(ω;ϕ=0)=εB′(ω;ϕ=0)=εR′, (4.7c) εC′(ω;ϕ=1)=εB′(ω;ϕ=1)=εW′, (4.7d)

and they are valid for all frequencies ω.

[64.2.2.1] Finally, the local simplicity hypothesis will be cast into mathematical form by requiring that σC(ω=0;ϕ) can be expanded for samll ϕ as

 σC′⁢0;ϕ=ϕ⁢c1+C2⁢ϕ+…. (4.8)

[64.2.2.2] Correspondingly, for the blocking geometries the real dielectric constant diverges as ϕ1, which is a thin-plate effect. [64.2.2.3] Local simplictiy is assumed to imply that the expension

 εB′⁢0;ϕ-1=1-ϕ⁢B1+B2⁢1-ϕ+… (4.9)

is vaild for ϕ<1. [64.2.2.4] Note that equation (4.7d) implies a discontinuity at ϕ=1.

[64.2.3.1] For the local porosity distribution, it will be assumed that μ0=μ1=0 in equation (2.8) and thus μϕ=μ~ϕ. [64.2.3.2] The local percolation probability has to assume the limiting values

 λ(ϕ=0)=0, (4.10a) λ(ϕ=1)=1. (4.10b)

The first equation states that there are no conducting geometries with porosity 0, and the second says that there exist no blocking geometries at porosity 1.