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Porous struct.ures frequently 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 Howand 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. For the modeJling of porous media in this thesis we avail ourselves 01' the methods of stochastics. Hence the goal we pursue cannot be the exact artificial rcconstruction of a natural porous structure but rather t.he analysis 01' its stochastic propertics. This will allow HS to quantitatively cornparc different realizations 01' the same stochastic model with respcct 1,0 their geometrical properties and analyse their variability, Our basic proccdure is as follows: We first rneasure the properties 01' interest (mainly the Minkowski-functionals) and thcn fit a stochastic model such that in average the roalizations of the model reprodnce the measured values for those characteristies. In this thesis, we basically eoncentrate on three simple stochastie models, which are the germ-grain type Boolean Model, the Tnmcated Gaussian Fields and the Gibbsian Model. Our fOCHS regarding the selection of stoehastic models to eonsider was more on analytical tractahility 01' a specific model than on its power 1,0 generate realistie artificial reproductions 01' porous structures. This is the main reaSOll why we did not cOIJsider in this thesis the more complicated realistic: models ofthe hard-eore type (such as the cherry-pit model).

Modelling of random Porous Media using Minkowski-Functionals