1 Introduction

[page 373, §1]   
[373.1.1] A crucial prerequisite for the prediction of transport parameters of porous media is a suitable characterization of the microstructure [13, 1, 38, 22]. [373.1.2] Despite a long history of scientific study the microstructure of porous media continues to be investigated in many areas of fundamental and applied research ranging from geophysics [18], hydrology [30, 5], petrophysics [26] and civil engineering [14, 12] to the materials science of composites [10].

[373.2.1] My primary objective in this article is to review briefly the application of local porosity theory, introduced in [19, 20, 22], as a method that provides a scale dependent geometric characterization of porous or heterogeneous media. [373.2.2] A functional theorem of Hadwiger [16, p. 39] emphasizes the importance of four set-theoretic functionals for the geometric characterization of porous media. [373.2.3] In contrast herewith local porosity theory has emphasized geometric observables, that are not covered by Hadwigers theorem [21, 23, 17]. [373.2.4] Other theories have stressed the importance of correlation functions [44, 42] or contact distributions [27, 32, 43] for [page 374, §0]    characterization purposes. [374.0.1] Recently advances in computer and imaging technology have made threedimensional microtomographic images more readily available. [374.0.2] Exact microscopic solutions are thereby becoming possible and have recently been calculated [46, 47, 7]. [374.0.3] Moreover, the availability of threedimensional microstructures allows to test approximate theories and geometric models and to distinguish them quantitatively.

[374.1.1] Distinguishing porous microstructures in a quantitative fashion is important for reliable predictions and it requires apt geometric observables. [374.1.2] Examples of important geometric observables are porosity and specific internal surface area [4, 13]. [374.1.3] It is clear however, that porosity and specific internal surface area alone are not sufficient to distinguish the infinite variety of porous microstructures.

[374.2.1] Geometrical models for porous media may be roughly subdivided into the classical capillary tube and slit models [13], grain models [43], network models [15, 8], percolation models [9, 37], fractal models [25, 36], stochastic reconstruction models [33, 1] and diagenetic models [35, 2]. [374.2.2] 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. [374.2.3] Usually the matching of geometric observables is limited to the porosity alone. [374.2.4] Recently the idea of stochastic reconstruction models has found renewed interest [1, 34, 48]. [374.2.5] 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. [374.2.6] As the number of matched quantities increases one expects that also the model approximates better the given sample. [374.2.7] Matched models for sedimentary rocks have recently been subjected to a quantitative comparison with the experimentally obtained microstructures [7].