Pore-scale modelling of transport phenomena in homogeneous porous media

Diedericks, Gerhardus Petrus Jacobus (1999-12)

Dissertation (Ph.D) -- University of Stellenbosch, 1999.

Thesis

ENGLISH ABSTRACT: The main purpose of this study is to develop deterministic, process-based models of incompressible Newtonian flow and electrical c01iduction in homogeneous, anisotropic porous media. The foundation of the models is provided by the volume averaging theory which is used to obtain the macroscopic balance equations for momentum transport and electrical conduction. These volume averaged equations contain, amongst others, integral terms over the fluid-solid surface area where the integrands are related to the microscopic fluxes of the transport quantities. The closure modelling is conducted by employing a pore-scale model which requires explicit assumptions regarding the mean geometric properties of the porous medium microstructure and accounts for the configuration of the fluid-solid surface area. The pore-scale model also provides an estimate of the microscopic flow paths. The average geometry of different anisotropic materials, namely two types of foamlike materials, granular porous media and fibre beds, is captured in representative unit cells which form the core of the physical pore-scale model. This particular type of closure modelling further requires a direct transformation of microscopic fluxes to the macroscopic level. It is indicated, in context of the volume averaging theory, that microscopic fluxes may be estimated by the respective macroscopic channel average fluxes. The transformation of the microscopic flux to the channel average flux is accomplished through the flux related tortuosity tensor. New definitions for the tortuosity and lineality as second-order tensors are proposed for porous media in general. Novel names, semantically in line with the respective physical meanings, are proposed for these quantities. It is shown that the definitions produce results which conform with several other published results and are applicable to anisotropic media. Application of the modelling technique to Newtonian flow results in momentum transport equations valid for both the Darcy and Forchheimer flow regimes. The coefficients appearing in these equations are expressed in terms of fluid properties and measurable geometric features of the porous medium. The predictions of the anisotropic foamlike materials are validated against experimental pressure gradient measurements for flow through a high porosity, anisotropic knitted wire mesh rolled up to form a cylindrical plug. The predictions compare reasonably well with the experimental results. The modelling approach is also applied to electrical conduction in anisotropic porous media saturated with an electrically conductive fluid. A macroscopic form of Ohm's law is derived as well as deterministic expressions for the formation factor. The formation factor predictions for isotropic porous media are compared to several experimental measurements as well as to semi-empirical expressions. The predictions compare favourably to the measurements.

AFRIKAANSE OPSOMMING: Die hoofdoel van hierdie studie is om deterministiese, proses-gebaseerde modelle van onsaamdrukbare Newton-vloeiers en elektriese geleiding in homogene, anisotrope poreuse media te ontwikkel. Die basis van die modelle is die volumetriese homogeniseringsteorie wat gebruik word om die makroskopiese balansvergelykings vir momentum transport en elektriese geleiding te verkry. Die makroskopiese vergelykings bevat, onder andere, integrasieterme oor die benatte oppervlakte waar die integrand afhanklik is van die mikroskopiese vloede van die transport groothede. Die sluitingsmodellering word uitgevoer deur gebruik te maak van 'n porie-skaal model wat spesifieke aannames in verband met die gemiddelde geometriese eienskappe van die mikrostruktuur vereis en inkorporeer die konfigurasie van die benatte oppervlakte. Die model verskaf ook 'n afskatting van die mikroskopiese vloeipatrone. Die gemiddelde geometrie van verskillende anisotrope materiale, naamlik twee tipes sponse, korrelagtige materiale en vesels, word in verteenwoordigende eenheidselle inkorporeer en vorm die kern van die fisiese porie-skaal model. Hierdie spesifieke sluitingsmodellering vereis verder 'n direkte transformasie van mikroskopiese vloede na die makroskopiese vlak. Daar word aangetoon, in konteks van die volumetriese homogeniseringsteorie, dat die mikroskopiese vloede afgeskat kan word deur die ooreenkomstige makroskopiese kanaal-gemiddelde vloede. Die transformasie van die mikroskopiese vloed na die kanaal-gemiddelde vloed word bewerkstellig deur die vloed-gekoppelde tortuositeistensor. Nuwe definisies vir die tortuositeit en linealiteit as tweede or de tensors word voorgestel vir poreuse media in die algemeen. N uwe name wat semanties ooreenstem met die onderskeie fisiese betekenisse word ook voorgestel vir hierdie groothede. Daar word aangetoon dat die definisies ooreenstem met gepubliseerde result ate en dat dit van toepassing is op anisotrope materiale. Toepassing van die modelleringstegniek op Newton vloei lei tot momentum transportvergelykings wat geldig is oor beide die Darcy en Forchheimer vloeigebiede. Die koeffisiente wat in hierdie vergelykings verskyn word uitgedruk in terme van vloeier eienskappe en meetbare geometriese eienskappe van die poreuse medium. Die voorspellings vir die anisotrope sponse word vergelyk met eksperimentele drukgradiente vir vloei deur 'n hoe porositeit, anisotrope gevlegde metaaldraad wat opgerol is om 'n silindriese prop te vorm. Die voorspellings vergelyk goed met die eksperimentele metings. Die modelleringsaanslag word ook toegepas op elektriese geleiding in anisotrope poreuse materiale wat versadig is met 'n elektries geleidende vloeistof. 'n Makroskopiese vorm van Ohm se wet word afgelei asook deterministiese uitdrukkings vir die formasiefaktor. Die formasiefaktor voorspellings wat van toepassing is op isotrope materiale word vergelyk met verskeie eksperimentele metings asook met semi-empiriese uitdrukkings. Die voorspellings vergelyk gunstig met die eksperimentele metings.

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