2 Geometrical Problems in Porous Media

[374.3.1] A two-component porous sample S=P∪M is defined as the union of two closed subsets P⊂R3 and M⊂R3 where P denotes the pore space (or component 1 in a heterogeneous medium) and 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 χ⁢P⁢r of a set P. [374.3.4] It is defined for arbitrary sets P as

[page 375, §1]    [375.1.1] The geometrical problems in porous media arise because in practice the pore space configuration χ⁢P⁢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 χ⁢P⁢r.

[375.2.1] While it is becoming feasible to digitize samples of several m⁢m3with a resolution of a few μ⁢mthis is not possible for larger samples. [375.2.2] For this reason the true pore space P is often replaced by a geometric model P~. [375.2.3] One then solves the problem for the model geometry and hopes that its solution u~ obeys u~≈u in some sense. [375.2.4] Such an approach requires quantitative methods for the comparison of P and the model 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 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.