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Studies of Tracer Dispersion and Fluid Flow in Porous Media

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Studies of Tracer Dispersion and Fluid Flow in Porous Media /S/£f'J'> NO- - Studies of Tracer Dispersion and Fluid Flow in Porous Media A Thesis in Partial Fulfillment of the Requirements for the Degree of Doctor Scientiarum MASTER Thomas Rage Department of Physics, University of Oslo P.O.BOX 1048 Blindern N-0316 Oslo, Norway 1996 WSTHBWIOM OF THIS DOCUMENT IS tMJMTH) DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original Preface This thesis presents main parts of the project work I performed in the period 1993 to 1996 as a doctor student in the Condensed Matter Group at the Department of Physics, University of Oslo. The project was part of the Cooperative Phenomena Program, initiated by Torstein Jpssang and Jens Feder. My work during that period included studies of tracer dispersion and fluid flow in porous media as well as numerical studies of diffusion limited aggregation (DLA) and related growth models. In order to create a consistent and comprehensive thesis, only work related to tracer dispersion and fluid flow is presented here. Financial support was provided by the Royal Norwegian Research Council (NFR) through a DEMINEX-scholarship, Grant No. 101874/410, and is gratefully acknowl ­ edged. I am indebted to Jens Feder, Paul Meakin, and Torstein Jossang who super ­ vised my work and taught me the physics of condensed matter on the scientific, administrative, and social level. When I joined the group as a student with a ger­ man Master’s degree on the subject of chaos (and relatively little knowledge about modern condensed matter physics), they took a chance in hiring me. I have hope to believe that they didn ’t regret. I would like to thank Unni Oxaal for her endless will to prepare experimental results such that they could be compared to the results of my computational studies. To work with Eirik Flekkpy was a challenging pleasure. His numerical approach towards dispersion being different from mine, the friendly competition between us had inspiring influence on my work. Finn Roger, Thomas Walmann and Anders Malthe-Sprenssen were most helpful in my efforts to make computers do the kind of things that I wanted them to do. I acknowledge valuable suggestions of Geri Wagner, Ragnhild Halvorsrud, and Thomas Walmann who read an early version of the thesis. Finally, I mention that this thesis could not have been written without the support from my family and friends, especially Beate, Johannes, and my parents. Oslo, November 1996 (Thomas Rage) ii Abstract This thesis is based on computational studies of tracer dispersion and fluid flow in porous media. Its objective is to explore the connection between the topology of a porous medium and its macroscopic transport properties. In porous media, both diffusion and convection contribute to the dispersion of a tracer. The investigation of the combined action of diffusion and convection is therefore an important ingredient in the work presented here. The differential equations that govern fluid flow in porous media are solved numerically by finite-difference schemes. To model tracer transport, a finite-difference scheme and a Monte Carlo method were employed. The thesis is divided into two parts. The first part (Section 1-7) provides an introduction to the subject of tracer dispersion and fluid flow in porous media, dis­ cusses its scientific and industrial relevance, establishes the required mathematical framework, and presents the numerical methods that were used. Some specific stud­ ies serve as illustrations in the otherwise relatively brief presentation. The reader will be referred to standard textbooks and reviews for deeper discussions. The second part of the thesis consists of reproductions of five scientific papers, listed on page 39. All five publications are based on original research that was carried out as part of the doctorate work presented in this thesis. In the following, these five papers are referred to as PI - P5. Paper 1 (PI) discusses a so-called echo-experiment on tracer dispersion. In such experiments a tracer is first convected into a porous media, and then - by flow reversal - echoed back towards its initial position. Echo-experiments have become an important tool to probe a number of aspects of flow and dispersion in porous media. The paper investigates the influence of finite Reynolds number on the outcome of echo-experiments. By comparing results of experiments and simulations, it is shown that at surprisingly low values of the Reynolds number effects of non­ linear inertial forces lead to a visible deformation of the returned tracer. Apart from its scientific relevance, this observation may be of significant importance in the interpretation of results from echo-experiments in natural porous media. Paper 2 (P2) is a study on tracer dispersion and fluid flow in periodic arrays of discs. The accuracy of the numerical methods is demonstrated by comparing results from simulations with some well-established analytical predictions. The connection between microscopic and macroscopic dispersion mechanisms is investigated by cal­ culating macroscopic dispersion coefficients for samples of different porosity and a large range of Peclet number. Since relatively little is known about macroscopic dispersion coefficients, the detailed study of the dispersive properties of such simple geometries adds valuable information to the subject of tracer dispersion in porous media. In detail, it is demonstrated that the mechanisms of mechanical disper ­ sion in periodic media and in natural (non-periodic) porous media are substantially different. The results of P2 illustrate the close connection between the topology of a porous medium and its macroscopic transport properties. To quantify geometric charac ­ teristics of a porous medium (such as homogeneity, connectedness, and tortuosity) and to rigorously relate them to macroscopic transport coefficients is, however, still an unfinished enterprise. In this context, Paper 3 (P3) presents first measurements on the percolation probability distribution of a sandstone sample. Local porosity theory predicts that this simple geometric function of a porous medium is of domi­ nant importance for its macroscopic transport properties. It is demonstrated in P3 that the measurement of percolation probability distributions on digitized samples presents only little difficulty. Results are consistent with theoretical expectations m on the shape of the obtained curves. The subject of Paper 4 (P4) is essentially iden ­ tical to the one of P3, but discusses our results from a somewhat different point of view. Paper 5 (P5) presents a number of qualitative results on tracer dispersion and two-phase flow in rough fractures. Faults and fractures play a dominant role for geo­ logical phenomena (such as metamorphism and mineral deposition) , waste disposal by deep-well injection, and the transport of hydrocarbon fluids from the source rock to a reservoir. Due to the small length scales and large time scales that the trans ­ port in a natural fracture involves, experiments are difficult to perform. Moreover - since the void space is usually not accessible - they yield only limited information. Consequently, relatively few quantitative results on flow and transport in fractures exist. It is demonstrated in P5 that, using simple but realistic models and readily available computer resources, many aspects of transport through fractures can be studied numerically on scales large enough to allow quantitative questions to be addressed. IV Zusammenfassung Diese Abhandlung basiert auf computergestiitzten Studien fiber die Ausbreitung (Dispersion) eines Indikators und Fliissigkeitsstromungen in porosen Median. Ziel ist, die Verbindung zwischen der Topologie eines porosen Mediums und seinen makroskopischen Transporteigenschaften zu erforschen. In porosen Median tra- gen sowohl Diffusion als auch Konvektion zur Dispersion eines Indikators bei. Die Untersuchung der kombinierten Wirkung von Diffusion und Konvektion ist da- her wesentlicher Bestandteil der Abhandlung. Die Differentialgleichungen , denen Stromungsprozesse in porosen Medien unterliegen, warden numerisch durch Diffe- renz-Verfahren gelost. Simulationen des Transports eines Indikators beruhen auf einem Differenz-Verfahren , sowie einer Monte Carlo Methode. Die Abhandlung gliedert sich in zwei Teile. Der erste Teil (Abschnitt 1-7) gibt eine Einfiihrung in die Physik der Dispersion und der Stromung in porosen Medien, diskutiert deren wissenschaftliche und wirtschaftliche Relevanz, prasentiert den notwendigen mathematischen Apparat, und erlautert die benutzten numerischen Methoden. Einige einfache Studien dienen als Illustrationen in dieser ansonsten recht knapp gehaltenen Einfiihrung. Der Leser wird auf Lehrbiicher und Ubersichts- artikel zu den jeweiligen Themen verwiesen. Der zweite Teil dieser Abhandlung umfafit Reproduktionen von fiinf wissenschaft- lichen Arbeiten, die auf Seite 39 aufgelistet sind. Alia fiinf Publikationen basieren auf originaren Ergebnissen der Doktorarbeit, die diese Abhandlung reprasentiert. Die Publikationen sind im folgenden mit P1-P5 bezeichnet. Die erste Publikation (PI) diskutiert ein sogenanntes Echo-Experiment. In solchen Experimenten wird ein Indikator mittels einer stromenden Fliissigkeit zuerst in ein porosen Medium transportiert und anschliebend - durch Umkehr der Stro- mung - zuriick zu seiner Ausgangsposition getrieben. Echo-Experimente haben sich zu einem bedeutenden Werkzeug bei der Untersuchung zahlreicher Aspekte der Stromung und Dispersion
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