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Use of Invasion Percolation Models to Study the Secondary Migration of Oil and Related Problems

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Use of Invasion Percolation Models to Study the Secondary Migration of Oil and Related Problems Use of Invasion Percolation Models To Study the Secondary Migration of Oil and Related Problems G. Wagner Department of Physics Faculty of Mathematics and Natural Sciences University of Oslo 1997 AJEX- rV0--$ 3? Use of Invasion Percolation Models To Study the Secondary Migration of Oil and Related Problems A Thesis in Partial Fulfillment of the Requirements for the Degree of Doctor Scientiarum Gerhard Wagner Department of Physics, University of Oslo P.O.BOX 1048 Blindern N-0316 Oslo, Norway 1997 OismmmoN of rm document @ unumited Summary The thesis treats simulations of the slow displacement of a wetting fluid by a non-wetting fluid in porous media and in a single fracture. The simulations were based on the invasion percolation model. New modified versions of the invasion percolation model are presented that simulate migration, fragmenta ­ tion and coalescence processes of the clusters of non-wetting fluid. The result­ ing displacement patterns were characterized by scaling laws. Section 1 presents a general overview over the content of the thesis and connects the problem of slow two-phase flow to percolation theory. The rel­ evance of slow immiscible displacement processes in industry and technology is briefly discussed. In section 2 the mechanisms controlling immiscible dis­ placements are discussed, and in section 3 the invasion percolation model is introduced. In section 4 new modified versions of the invasion percolation model are pre­ sented and applied to simulate an imbibition process in a porous medium and the migration of a cluster of non-wetting fluid through a porous medium satu­ rated with a wetting fluid. In particular, simulations of the secondary migration of oil through porous homogeneous rock are discussed. Qualitative simulations of fluid migration through heterogeneous porous media are discussed in section 5. Fractured rocks represent extreme cases of inhomogeneous porous media. Simulations of the slow displacement of a wet­ ting fluid by a non-wetting fluid in a single fracture using the standard invasion percolation model are presented in the second part of section 5. Section 6 summarizes the findings presented in the papers PI - P8. In Pa­ per PI experiments and simulations are reported in which the fragmentation of invason percolation-like structures of non-wetting fluid in a porous medium saturated with a wetting fluid was studied. A more detailed description of the work is given in paper P2. This paper is also concerned with a scenario in which a cluster of non-wetting fluid migrated through a porous medium that was sat­ urated with a wetting fluid. The migration was driven by continously increas­ ing buoyancy forces. These experiments and simulations are presented in brief form also in paper P3. In paper P4, the same scenario was studied theoretically and by simulations using a simplified percolation model of fluid migration in one dimension. Pa­ per P5 contains a study of the migration model in two dimensions, simulat­ ing constant buoyancy forces. Simulations of fluid migration, in particular the secondary migration of oil, in both two- and three-dimensional media are dis­ cussed in paper P6. The porous media considered were not homogeneous but V had multi-affine properties. Other two-dimensional random media considered in P6 represented a single fracture. Slow immiscible displacement processes in single fractures were studied quantitatively in papers P7 and P8. Single fractures were modeled using frac­ tal geometries. Simple models for a single fracture included a self-affine rough surface and a plane surface, or two self-affine rough surfaces. The aperture fields obtained from these models were spatially correlated. In P7, the prop ­ erties of invasion percolation clusters growing on a self-affine topography are discussed. In P8 the cross-over from correlated cluster growth to uncorrelated growth is described and related to the properties of the aperture fields. VI DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document Deutsche Zusammenfassung Die Dissertation behandelt Simulierungen von langsamen Verdrangungspro- zessen in denen eine netzende Flussigkeit dutch eine nicht-netzende Flussig- keit in einem porosen Medium verdrangt wird. Die Simulationen waren auf dem Invasionsperkolations-Modell 1 (IP Modell) basiert. Es werden neue, mo- difizierte Varianten des IP Modells dargestellt die die Migration, das Fragmen- tieren und das Zusammenschmelzen von Blasen der nicht-netzenden Fliissig- keit simulieren. Die daraus resultierenden Verdrangungsmuster werden dutch Skalierungsgesetzecharakterisiert. Kapitel 1 beinhaltet eine generelle Ubersicht uber den Inhalt der Disserta­ tion und verbindet das Problem der langsamen Zweiphasen-Stromung mit der Problematik der Perkolationstheorie. Die Relevanz langsamer Verdrangungs- prozesse ohne Durchmischung in Bezug auf Industrie und Technik wird kurz diskutiert. In Kapitel 2 werden die Mechanismen von Verdrangungspro- zessen ohne Durchmischung behandelt, und in Kapitel 3 wird das IP Modell eingefuhrt. In Kapitel 4 werden neue, modifizierte Varianten des IP Modells prasentiert und zur Simulation eines Imbibitionsprozesses 2 in einem porosen Medium ver- wendet. Im gleichen Kapitel wird die Anwendung der Modelle auf die Simu­ lation der Migration einer Blase nicht-netzender Flussigkeit dutch ein poroses, mit einer netzenden Flussigkeit gesattigtes Medium diskutiert. Simulationen der sekundaren Migration von Erdol dutch porosen, homogenen Pels werden im besonderen diskutiert. Qualitative Simulationen von Flussigkeitsmigration durch heterogene po- rose Medien werden in Kapitel 5 diskutiert. Fragmentierter Pels ist ein extre­ mes Beispiel eines inhomogenen porosen Mediums. Simulationen der langsa ­ men Verdrangung einer netzenden Flussigkeit durch eine nicht-netzende Flus­ sigkeit in einer einzelnen Felsspalte, basierend auf dem gewdhnlichen IP Mo­ dell, werden im zweiten Teil von Kapitel 5 besprochen. In Kapitel 6 werden die Ergebnisse, die in den Artikeln PI - P8 dargestellt werden, zusammengefasst. In Artikel PI wird von Experimenten und Simu­ lationen berichtet, in welchen die Fragmentierung von IP-artigen Strukturen nicht-netzender Flussigkeit in einem porosen, mit einer netzenden Flussigkeit gesattigten Medium untersucht wird. Eine ausfuhrlichere Beschreibung der Arbeit wird in Artikel P2 gegeben. Artikel P2 behandelt ebenfalls ein Szenario, in dem eine nicht-netzende Flussigkeitsblase durch ein poroses, mit einer netzenden Flussigkeit gesattig- 1 Invasion percolation model ^Imbibition Vll ten Medium migrierte. Die Migration war durch kontinuierlich zunehmende Auftriebskrafte angetrieben. Diese Experimente und Simulierungen werden in zusammengefasster Form auch in Artikel P3 dargestellt. In Artikel P4 wird das gleiche Szenario theoretisch und durch Simulationen in einer Dimension studiert, basierend auf einem vereinfachten Perkolations- modell fur Fliissigkeitsmigration. Artikel P5 enthalt eine Studie des Migra- tionsmodells in zwei Dimensionen unter dem Einfluss konstanter Auftriebs­ krafte. Simulationen von Fliissigkeitsmigration, insbesondere der sekunda- ren Migration von Erdol, in sowohl zwei- als auch drei-dimensionalen Medien werden in Artikel P6 behandelt. Die dort betrachteten porosen Medien waren nicht homogen, sondern hatten multi-affine Eigenschaften. Andere dort be- handelte zwei- dimensionale, zufallig geordnete 3 Medien reprasentierten eine einzelne Felsspalte. Langsame Verdrangungsprozesse ohne Durchmischung in einzelnen Fels- spalten wurden in den Artikeln P7 und P8 quantitativ untersucht. Die Spalten wurden auf der Basis fraktaler Geometric modelliert. Einfache Modelle fiir eine einzelne Spalte beinhalteten eine selbst-affine4 rauhe Flache und eine ebene Flache, oder zwei selbst-affine rauhe Flachen. Diese Modelle lieferten raumlich korrelierte Spaltoffnungsfelder. In P7 wurden die Eigenschaften von auf selbst- affinen Topologien gewachsenen Invasionsperkolations-Clustern 5 untersucht. In P8 wird der Ubergang 6 von korreliertem Wachstum der Cluster zu unkorre- liertem Wachstum beschrieben und mit den Eigenschaften der Spaltoffnungs ­ felder in Verbindung gebracht. 3 random 4 self-affine 5Invasion percolation clusters 6 cross-over vin Preface The work presented in this thesis was carried out in the years 1992-1996 at the Cooperative Phenomena Group at the Physics Department, University of Oslo. Much of the work in this group is concerned with the growth of patterns - the structures that arise if oil is displacing water in a porous rock, or if a brittle plate is fracturing under the influence of external stress. Most patterns in nature are formed under non-equilibrium conditions, and the resulting pat­ terns are very complex and often have fractal properties. The complexity is connected to the large number of discrete units or "particles" that "cooperatively" interact with each other in such systems. Although the group is mainly doing experimental basic research, computer simulations have always been an important part of the work. The simulations attempt to complete the the­ oretical understanding emerging from the experiments, and occasionally even serve as a moti ­ vation to conduct additional experiments. In this climate of fruitful interactions between lab­ oratory work and computer modeling, I had the great pleasure to be admitted and to become part of the group. I am immensely grateful to my former supervisor Harry Thomas from the University
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