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IV Notation and Conventions

[] The time variation of electrical fields is taken proportional to \exp(-{\mathrm{i}}\omega t). [] The complex dieelectric constant \varepsilon is written as

\varepsilon=\varepsilon^{\prime}+{\mathrm{i}}\varepsilon^{{\prime\prime}}=\varepsilon^{\prime}+{\mathrm{i}}\frac{\sigma^{\prime}}{\omega}, (4.1)

where \varepsilon^{\prime}(\varepsilon^{{\prime\prime}}) are the real (imaginary) parts of \varepsilon, and \sigma^{\prime} is the real part of the complex conductivity \sigma. [] SI units will be used. [] Values for \varepsilon are given in multiples of \varepsilon _{0}=8,8542\times 10^{{-12}}\,\mathrm{f}/\mathrm{m}, the permittivity of free space, and the units for \sigma^{\prime} are then \text{S}/\text{m}. S/m. [] The conductivity is written as

\sigma=\sigma^{\prime}+{\mathrm{i}}\sigma^{{\prime\prime}}=\sigma^{\prime}+{\mathrm{i}}\omega(1-\varepsilon^{\prime}). (4.2)

[] The relationship between \sigma and \varepsilon is also written as

\varepsilon(u)=1-\frac{\sigma}{u}, (4.3)

where the notation u={\mathrm{i}}\omega has been used.

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

\varepsilon _{W}=\varepsilon^{\prime}_{W}+{\mathrm{i}}\frac{\sigma^{\prime}_{W}}{\omega}, (4.4)

for water, and

\varepsilon _{R}=\varepsilon^{\prime}_{R}, (4.5)

for rock. [] In calculations the values \varepsilon^{\prime}_{W}=79\varepsilon _{0} and \varepsilon^{\prime}_{R}=7\varepsilon _{0} will be used. [] Dimensionless frequencies are introduced by setting \omega _{0}=\sigma^{\prime}_{W}/\varepsilon^{\prime}_{W}=1 for the relaxation frequency of water, and this also fixes \varepsilon^{\prime}_{W}=\sigma^{\prime}_{W}.

[] The average local dielectric constants \varepsilon _{B}(\omega;\phi) and \varepsilon _{C}(\omega;\phi) at porosity \phi are in general unknown. [] Nevertheless, some general statements can be made. [] The relation

\sigma^{\prime}_{B}(\omega=0;\phi)=0 (4.6)

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

\displaystyle\sigma^{\prime}_{C}(\omega;\phi=0)=\sigma^{\prime}_{R}=0, (4.7a)
\displaystyle\sigma^{\prime}_{C}(\omega;\phi=1)=\sigma^{\prime}_{W}, (4.7b)
\displaystyle\varepsilon^{\prime}_{C}(\omega;\phi=0)=\varepsilon^{\prime}_{B}(\omega;\phi=0)=\varepsilon^{\prime}_{R}, (4.7c)
\displaystyle\varepsilon^{\prime}_{C}(\omega;\phi=1)=\varepsilon^{\prime}_{B}(\omega;\phi=1)=\varepsilon^{\prime}_{W}, (4.7d)

and they are valid for all frequencies \omega.

[] Finally, the local simplicity hypothesis will be cast into mathematical form by requiring that \sigma^{\prime}_{C}(\omega=0;\phi) can be expanded for samll \phi as

\sigma^{\prime}_{C}(0;\phi)=\phi(c_{1}+C_{2}\phi+\dots). (4.8)

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

[\varepsilon^{\prime}_{B}(0;\phi)]^{{-1}}=(1-\phi)[B_{1}+B_{2}(1-\phi)+\dots] (4.9)

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

[] For the local porosity distribution, it will be assumed that \mu _{0}=\mu _{1}=0 in equation (2.8) and thus \mu(\phi)=\widetilde{\mu}(\phi). [] The local percolation probability has to assume the limiting values

\displaystyle\lambda(\phi=0)=0, (4.10a)
\displaystyle\lambda(\phi=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.