Modelling of random Porous Media using Minkowski-Functionals
Maik A. Berchtold · Repository for Publications and Research Data (ETH Zurich) · 2007
Porous struct.uresfrequently anse in nature and belong 1,0 the preferred materiale studied in materials science.They are encountered in a myriad of shapes and structural variations, Among the exarnples of natural porous media we find rocks, soils, sporiges 01' biological tissue and the most prominent porous basic materiale being concrctc, ceramies or foams.TL is therefore not surprising that since the seventies of t.he last century porous media have belonged to the core focus of interdisciplinary rescarch.They are of interest not only to the mathematician, goo-physicist, biologist and ehernist but -not least in considcration of the sheer mass of data to be handled -also for the computer science and imagc-processing communitics.Due to the fast-paced devclopment of synchrotron technology, high-resolution irnages of three-dimensional porous specimens have finally become availablo in recent years.The Institute 1'01' Terrestrial Ecology at the Swiss Federal Institute of Technology (ETH) and the Paul-Scherrer-Institut (PSI) in Villigen are involved in the making 01' such high-resolution images for sand soils 01' various granularity which by courtesy they rnade available to us Ior use in the prescnt thesis.The geometrical strueture of the pore space is known to have a major impact on the How-and transport-properties in porous media such as pcrrneability.Unfortunately neither thc specific nature of t.his impact is prcsently known nor which exactly arc the decisive charactoristics of the porc space responsible 1'01' this impact.In this thesis wc mainly concentrate on a certain simple dass of geometrioal characteristics, t.he so-called Minkowski-functionals, They cornprise well-known olementary geometrical quantities such as volume (of the pore and also the solid phase}, surface (01' the boundary between pore and solid phase), the integral of mean curvaturc and the Euler-charactcrist.ic, the latter being an important connectivity-mcasure well-known in differential topology.The use of Minkowski-functionals to summarize the inforrnation content of a porous specimen can be theoretically justified by the famous Hadwiger-Theorem.cantains furt her the proofs of same fundamental theorems used ta uutline the basic thcory which were omitted in thc main text and also contains alternative proofs to some of our own reslllts.