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1 Introduction

[page 203, §1]   
[203.1.1] An important subclass of heterogeneous and disordered systems are porous materials which can be loosely defined as mixtures of solids and fluids [20, 1, 55, 30]. [203.1.2] Despite a long history of scientific study the theory of porous media or, more generally, heterogeneous mixturesa (This is a footnote:) a including solid-solid and fluid-fluid mixtures continues to be of central interest for many areas of fundamental and applied research ranging from geophysics [26], hydrology [43, 7], petrophysics [36] and civil engineering [21, 19] to the materials science of composites [17].

[203.2.1] My primary objective in this article is to review briefly the application of local porosity theory, introduced in [27, 28, 30], to the geometric characterization of porous or heterogeneous media. [203.2.2] A functional theorem of Hadwiger [23, p. 39] emphasizes the importance of four set-theoretic functionals for the geometric characterization porous media (see also the paper by Mecke in this volume). [203.2.3] In contrast herewith local porosity theory has emphasized geometric observables, that are not covered by Hadwigers theorem [29, 31, 25]. [203.2.4] Other theories have stressed the importance of correlation functions [63, 60] or contact distributions [38, 46, 61] for characterization purposes. [203.2.5] Recently advances in computer and imaging technology have made threedimensional microtomographic images more readily available. [203.2.6] Exact microscopic solutions are thereby becoming possible and have recently been calculated [66, 68, 11]. [203.2.7] Moreover, the availability of threedimensional microstructures allows to test approximate theories and geometric models and to distinguish them quantitatively.

[203.3.1] Distinguishing porous microstructures in a quantitative fashion is important for reliable predictions and it requires apt geometric observables. [203.3.2] Examples of important geometric observables are porosity and specific internal surface area [page 204, §0]    [6, 20]. [204.0.1] It is clear however, that porosity and specific internal surface area alone are not sufficient to distinguish the infinite variety of porous microstructures.

[204.1.1] Geometrical models for porous media may be roughly subdivided into the classical capillary tube and slit models [20], grain models [61], network models [22, 15], percolation models [16, 54], fractal models [34, 53], stochastic reconstruction models [49, 1] and diagenetic models [51, 4]. [204.1.2] Little attention is usually paid to match the geometric characteristics of a model geometry to those of the experimental sample, as witnessed by the undiminished popularity of capillary tube models. [204.1.3] Usually the matching of geometric observables is limited to the porosity alone. [204.1.4] Recently the idea of stochastic reconstruction models has found renewed interest [1, 50, 70]. [204.1.5] In stochastic reconstruction models one tries to match not only the porosity but also other geometric quantities such as specific internal surface, correlation functions, or linear and spherical contact distributions. [204.1.6] Similar ideas have been proposed in spatial statistics [61]. [204.1.7] As the number of matched quantities increases one expects that also the model approximates better the given sample. [204.1.8] My secondary objective in this review will be to compare simple stochastic reconstruction models and physically inspired diagenesis models with the experimental microstructure obtained from computer tomography [11].