# 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′+iε′′=ε′+iσ′ω, | | (4.1) |

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=8,8542×10-12f/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′+iσ′′=σ′+iω1-ε′. | | (4.2) |

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

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

for water, and

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

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=ϕc1+C2ϕ+…. | | (4.8) |

[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

εB′0;ϕ-1=1-ϕB1+B21-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.