IV Notation and Conventions

[64.1.4.1] The time variation of electrical fields is taken proportional to exp⁡-i⁢ω⁢t. [64.1.4.2] The complex dieelectric constant ε is written as

where ε′⁢ε′′ 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=8,8542×10-12⁢f/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

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

where the notation u=i⁢ω 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

for water, and

for rock. [64.1.5.3] In calculations the values εW′=79⁢ε0 and εR′=7⁢ε0 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 εB⁢ω;ϕ and εC⁢ω;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

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:

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

[64.2.2.2] Correspondingly, for the blocking geometries the real dielectric constant diverges as ϕ→1, which is a thin-plate effect. [64.2.2.3] Local simplictiy is assumed to imply that the expension

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 μ⁢ϕ=μ~⁢ϕ. [64.2.3.2] The local percolation probability has to assume the limiting values

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.