I Introduction

[page 60, §1]   
[60.1.1.1] A long-standing problem of considerable scientific and technological importance is to improve the understanding of geometric-dielectric correIations in porous materials. [60.1.1.2] The scientific problem is to find out which properties of the complicated random geometry of the pore space have a significant influence on the electrical and dielectric properties. [60.1.1.3] The engineer, on the other hand, is interested in the inverse problem of how to infer geometrical features of the porous medium from its dielectric response. [60.1.1.4] The most prominent examp1e for the technoIogica1 importance of this question arises in the context of well-log interpretation for petroIeum or water exploration.[1]

[60.1.2.1] My objective in this paper is to present a simple theoretical framework which allows a systematic study of how the pore-space geometry of an insulating porous material influences the low-frequency dielectric response when the pore space is filled with a conductor. [60.1.2.2] The approach is based on a novel geometric characterization of porous media via 1ocaI porosity distributions and local perco1ation probabilities which will be defined in Section II. [60.1.2.3] These geometric quantities are conceptually well defined and experimentally observable. [60.1.2.4] The dielectric properties will be calculated directly from the geometric characteristics using mean-field theory, scaling theory, and numerical techniques. [60.1.2.5] The results are intended to be applicable to the physics of water- or brine-saturated clay-free rocks. [60.1.2.6] Such rocks can have very large re1ative dielectric permittivities, sometimes exceeding those of the constituent materials by a factor of or more.[1, 2, 3, 4, 5] [60.1.2.7] As a second hallmark for the dielectric response of water-saturated rocks, the dielectric response of water-saturated rocks, many investigators report a power-law behavior for the dc conductivity with porosity known as Archie’s law.[6] [60.1.2.8] Holwech and Nøst have recently measured the frequency-dependent response of water-filled sintered glass beads and showed that strong dielectric dispersion and Archie’s law can arise as a purely geometrical effect.[8] [60.2.0.1] It will be assumed throughout this paper that electrochemical effects are absent.

[60.2.1.1] Despite numerous efforts,[9] no theoretical framework has been found to date which encompasses both aspects, strong dielectric enhancement and Archie’s law, simultaneous1y. [60.2.1.2] Recent theoretical approaches can be divided into two categories: [60.2.1.3] The first class[10, 11, 12, 13, 14, 15, 16, 17] attempts to calculate the dielectric properties starting from highly simplified or indirect geometric models for the pore space. [60.2.1.4] The difficulty is that most models are too idealized to be compared even to the simplest experimental model system. [60.2.1.5] The second category[18, 19, 20, 21, 22, 23] does not attempt to incorporate the pore-space geometry, but concentrates instead on a sometimes sophisticated phenomenological approach.

[60.2.2.1] Geometrical theories fall again into two families: percolation theories[10, 11, 12] and grain mixture models.[13, 14, 15, 16] [60.2.2.2] Percolation theories predict a finite porosity below which the conductivity vanishes. [60.2.2.3] Such models are attractive because they capture to some extent the geometry of a random network of pores, and they give rise to a strong dielectric enhancement near the percolation threshold. [60.2.2.4] However, they have been ruled out by the argument[13] that the pore spaces of realistic systems appear to remain connected down to zero porosity (Archie’s law). [60.2.2.5] In addition, strong dielectric enhancement does not appear to be correlated with a particular porosity threshold. [60.2.2.6] Grain mixture models, on the other hand, concentrate on the connectedness of the pore space. [60.2.2.7] They are attractive because they incorporate to some extent the consolidation processes though which porous rocks are formed. [60.2.2.8] Similar to abstract resistor network models,[10] they are usually not very explicit about the microgeometry underlying a specific model. [60.2.2.9] While predicting grain-shape-dependent exponents for Archie’s law the grain mixture models need to assume the presence of strongly platelike grains to obtain dielectric enhancement. [page 61, §0]    [61.1.0.1] It was pointed out by Wong, Koplik and Tomanic[17] however, that the cementation index in Archie’s law is not determined by grain shape. [61.1.0.2] On the experimental side, it was concluded[5] for the case of White stone that the spheroidal grain model cannot account simultaneously for the observed frequency-dependent conductivity and dielectric constant.

[61.1.1.1] Let me conclude the brief discussion of recent theories with phenomenological approaches which form the second large category. [61.1.1.2] Most approaches[18, 19, 20, 21] are based on the spectral representation of the complex dielectric functions developed by Fuchs[22] and Bergman.[23] [61.1.1.3] These theories start from an abstract pole spectrum which can be adjusted to reproduce the features of experimental data.[6] [61.1.1.4] Unfortunately, the pole spectrum or its properties cannot be related to the pore-space geometry without prior geometric modeling. [61.1.1.5] Therefore, the theory by itself, while separating material and geometric aspects, does not lead to a better understanding of geometric-dieletric correlations in porous media.

[61.1.2.1] To develop a better understanding of geometric-dielectric correlations in porous media, it is first necessary to develop a suitable geometric characterization as a starting point. [61.1.2.2] Instead of considering the different “sizes” of pores as the fundamental source of randomness in porous media, the present approach suggests to consider the porosity itself as the fundamental source of randomness in porous media, the present approach suggest to consider the porosity itself as the fundamental random variable. [61.1.2.3] This leads immediately to suggest local porosity distributions and local percolation probabilities as partial geometric characterizations of the complex pore space.

[61.1.3.1] Based on these new geometric characterizations, the following questions will be discussed in the paper: validity and theoretical origin of static and dynamic scaling laws, including Archie’s law, and the universality of scaling indices (Sec. V), geometric mechanisms of dielectric enhancement (Sec. VI), model calculations for the interplay between geometric characteristics and frequency-dependent dielectric response (Sec. VI), and theoretical aspects of how to obtain geometric information from the dielectric response (Sec. VII). [61.1.3.2] It must be emphasized again that dielectric dispersion in this paper results only from the randomness in the pore-space geometry. [61.1.3.3] In particular, electrochemical effects, which are important in real rocks, are not considered. [61.1.3.4] The dielectric response is found to be surprisingly sensitive to the geometrical features encapsuled in local porosity distributions, and it is hoped that the present investigation might be a step towards developing ultimately a “dielectric spectroscopy” of porous media.