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

(4.1) |

where are the real (imaginary) parts of , and is the real part of the complex conductivity . [64.1.4.3] SI units will be used. [64.1.4.4] Values for are given in multiples of , the permittivity of free space, and the units for are then . S/m. [64.1.4.5] The conductivity is written as

(4.2) |

[64.1.4.6] The relationship between and is also written as

(4.3) |

where the notation 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

(4.4) |

for water, and

(4.5) |

for rock. [64.1.5.3] In calculations the values and will be used. [64.1.5.4] Dimensionless frequencies are introduced by setting for the relaxation frequency of water, and this also fixes .

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

(4.6) |

must hold for all . [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:

(4.7a) | |||

(4.7b) | |||

(4.7c) | |||

(4.7d) |

and they are valid for all frequencies .

[64.2.2.1] Finally, the local simplicity hypothesis will be cast into mathematical form by requiring that can be expanded for samll as

(4.8) |

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

(4.9) |

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

[64.2.3.1] For the local porosity distribution, it will be assumed that in equation (2.8) and thus . [64.2.3.2] The local percolation probability has to assume the limiting values

(4.10a) | |||

(4.10b) |

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