[64.2.4.1] Consider now the direct problem with

[64.2.5.1] For strongly peaked

(5.1) |

for

(5.2) |

for

(5.2) |

for

[65.2.1.1] In the high-frequency limit (

(5.3) |

where

(5.3) |

for the real dielectric constant, and

(5.4) |

for real part of the conductivity.
[65.0.1.2] The sign in equation (5.3a)
has to be chosen such that

[65.0.2.1] In this subsection it will be shown that equation (3.2)
does indeed imply the existence of a percolation threshold,
also for general

[65.0.3.1] To identify the percolation transition,
consider the low-frequency limit.
[65.0.3.2] Expanding eq. (3.2) around

(5.5) |

for the effective dc conductivity

(5.6) |

is found.
[65.0.3.4] This equation for

(5.7) |

and the effective-medium value

[65.1.1.1] For the effective real dielectric constant

(5.8) |

[65.1.1.3] For

(5.9) |

where

[65.1.2.1] Equation (5.2) can be treated analogously
to the case of ordinary percolation.[30, 31][65.1.2.2] The effective conductivity

(5.10a) | |||

(5.10b) | |||

(5.10c) |

[65.1.2.5] [page 66, §0] For case (a) the solution of eq. (5.6) is obtained as

(5.11) |

where

(5.12) |

[66.1.0.1] The integral exists because of eq. (4.8)
and condition (5.10a).
[66.1.0.2] Note that eq. (5.11) is valid for all

[66.1.1.1] The situation is very different for case (c).
[66.1.1.2] If

(5.13) |

where the conductivity exponent

(5.13) |

is now no longer universal.
[66.1.1.3] It depends on the behavior of

(5.14) |

[66.1.2.1] For the real dielectric constant, eq. (5.8)
is valid below

(5.15) |

is obtained, where

(5.16) |

analogous to eqs. (5.11) and (5.12).
[66.1.2.5] Again, the expected value

in analogy with eq. (5.13b) for the conductivity exponent

[66.1.3.1] The central result of this section
is the identification of a percolation transition
underlying the random geometry
of porous media.
[66.2.0.1] The control parameter for the transition
is neither the bulk porosity

[66.2.1.1] In this section the scaling theory for the percolation transition[32, 33] is applied in the present context.[34] [66.2.1.2] Therefore, the present section goes beyond the effective-medium equation (3.2). [66.2.1.3] Scaling theory starts from the assumption that the complex dielectric constant can be written as

(5.17) |

with the scaling function

(5.18) |

for

(5.18) |

for

(5.18) |

for

(5.19) |

for

(5.20) |

for

(5.20) |

for

[66.2.2.1] For one-dimensional systems,
the effective-medium approximation is known
to be asymptotically exact for class (a) distributions.[31][66.2.2.2] If this remains true in higher dimensions,
then the scaling theory can be applied to porous media by identifying
the prefactors

[66.2.3.1] Consider now the case of finite frequencies

(5.21) |

for the conductivity, and

(5.22) |

for the dielectric constant, where

(5.23) |

and

(5.24) |

where

(5.25) |

and

(5.26) |

[67.1.0.6] These results predict a divergence of

[67.1.1.1] Most publications on the electrical properties of porous media discuss the phenomenological relationship[8] between dc conductivity and bulk porosity:

(5.27) |

called “Archie’s law”, which is usually written in terms
of the formation factor

[67.2.1.1] Sedimentary and related rocks arise from sedimentation
and subsequent compactification, cementation,
and other physicochemical processes.
[67.2.1.2] The bulk porosity

(5.28) | |||

(5.29) |

then eq. (5.19) implies that

(5.30) |

[67.2.1.12] This already resembles eq. (5.27).
[67.2.1.13] In particular, if it happens that

(5.31) |

as long as condition (5.10a) remains satisfied during the cementation process. [67.2.1.14] If the system falls under condition (5.10c), however, the cementation index becomes

(5.31) |

[page 68, §0]
[68.1.0.1] Note that

(5.31) |

[68.1.0.4] The surprising result is that the simplest form for

[68.1.1.1] A second interesting consequence of this section
is that it predicts similar scaling laws for the dielectric
constant in the high-porosity limit

(5.32) |

[68.1.1.4] Here the “dilution exponent”

(5.33) |

where

(5.33) |

[68.1.1.5] The exponents