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2 Geometrical Problems in Porous Media

[374.3.1] A two-component porous sample \mathbb{S}=\mathbb{P}\cup\mathbb{M} is defined as the union of two closed subsets \mathbb{P}\subset\mathbb{R}^{3} and \mathbb{M}\subset\mathbb{R}^{3} where \mathbb{P} denotes the pore space (or component 1 in a heterogeneous medium) and \mathbb{M} denotes the matrix space (or component 2). [374.3.2] For simplicity only two-component media will be considered throughout this paper, but most concepts can be generalized to media with an arbitrary finite number of components. [374.3.3] A particular pore space configuration may be described using the characteristic (or indicator) function \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}}) of a set \mathbb{P}. [374.3.4] It is defined for arbitrary sets \mathbb{P} as

\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}})=\begin{cases}1&\text{for~}{\boldsymbol{r}}\in\mathbb{P}\\ 0&\text{for~}{\boldsymbol{r}}\notin\mathbb{P}.\end{cases} (1)

[page 375, §1]    [375.1.1] The geometrical problems in porous media arise because in practice the pore space configuration \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}}) is usually not known in detail. [375.1.2] On the other hand the solution of a physical boundary value problem would require detailed knowledge of the internal boundary, and hence of \chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\boldsymbol{r}}).

[375.2.1] While it is becoming feasible to digitize samples of several mm^{3}with a resolution of a few \mu mthis is not possible for larger samples. [375.2.2] For this reason the true pore space \mathbb{P} is often replaced by a geometric model \widetilde{\mathbb{P}}. [375.2.3] One then solves the problem for the model geometry and hopes that its solution \widetilde{u} obeys \widetilde{u}\approx u in some sense. [375.2.4] Such an approach requires quantitative methods for the comparison of \mathbb{P} and the model \widetilde{\mathbb{P}}. [375.2.5] This in turn raises the problem of finding generally applicable quantitative geometric characterization methods that allow to evaluate the accuracy of geometric models for porous microstructues. [375.2.6] The problem of quantitative geometric characterization arises also when one asks which geometrical characteristics of the microsctructure \mathbb{P} have the greatest influence on the properties of the solution u of a given boundary value problem.

[375.3.1] Some authors introduce more than one geometrical model for one and the same microstructure when calculating different physical properties (e.g. diffusion and conduction). [375.3.2] It should be clear that such models make it difficult to extract reliable physical or geometrical information.